Dynamic Behavior of Materials, Volume 1

Conference Proceedings of the Society for Experimental Mechanics Series

For other titles published in this series, go to www.springer.com/series/8922

Tom Proulx Editor

Dynamic Behavior of Materials, Volume 1 Proceedings of the 2011 Annual Conference on Experimental and Applied Mechanics

Editor Tom Proulx Society for Experimental Mechanics, Inc. 7 School Street Bethel, CT 06801-1405 USA [email protected]

ISSN 2191-5644 e-ISSN 2191-5652 ISBN 978-1-4614-0215-2 e-ISBN 978-1-4614-0216-9 DOI 10.1007/978-1-4614-0216-9 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011929862 © The Society for Experimental Mechanics, Inc. 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

Dynamic Behavior of Materials represents one of eight volumes of technical papers presented at the Society for Experimental Mechanics Annual Conference & Exposition on Experimental and Applied Mechanics, held at Uncasville, Connecticut, June 13-16, 2011. The full set of proceedings also includes volumes on Mechanics of Biological Systems and Materials, Mechanics of Time-Dependent Materials and Processes in Conventional and Multifunctional Materials, MEMS and Nanotechnology; Optical Measurements, Modeling and, Metrology; Experimental and Applied Mechanics, Thermomechanics and Infra-Red Imaging, and Engineering Applications of Residual Stress. Each collection presents early findings from experimental and computational investigations on an important area within Experimental Mechanics. The Dynamic Behavior of Materials conference track was organized by: Vijay Chalivendra, University of Massachusetts Dartmouth; Bo Song, Sandia National Laboratories; Daniel Casem, U.S. Army Research Laboratory This Volume represents an ever growing area of broad interest to the SEM community, as evidenced by the increased number of papers and attendance in recent years. This track was initiated in 2005 and reflects our efforts to bring together researchers interested in the dynamic behavior of materials and structures, and provide a forum to facilitate technical interaction and exchange. The Sessions within this track are organized to cover the wide range of experimental research being conducted in this area by scientists around the world. The following general technical research areas are included:     

Composite Materials Dynamic Failure and Fracture Dynamic Materials Response Novel Testing Techniques Low Impedance Materials

vi

     

Metallic Materials Response of Brittle Materials Shock and Blast Loading Optical Techniques for Imaging High Strain Rate Material Response Simulation & Modeling of Dynamic Response & Failure Dynamic Response of Transparent Materials

The contributed papers span numerous technical divisions within SEM. It is our hope that these topics will be of interest to the dynamic behavior of materials community as well as the traditional mechanics of materials community. The track organizers thank the authors, presenters, organizers and session chairs for their participation and contribution to this track. We are grateful to the SEM TD chairs who cosponsored and organized sessions in this track (e.g., Composite Materials, Optical Techniques for Imaging High Strain Rate Events). The SEM support staff is also acknowledged for their devoted efforts in accommodating the large number of submissions this year. The opinions expressed herein are those of the individual authors and not necessarily those of the Society for Experimental Mechanics, Inc. Bethel, Connecticut

Dr. Thomas Proulx Society for Experimental Mechanics, Inc

Contents

1

Punch Response of Gels at Different Loading Rates M. Foster, P. Moy, R. Mrozek, J. Lenhart, T. Weerasooriya, U.S. Army Research Laboratory

2

A Kolsky Torsion Bar Technique for Characterization of Dynamic Shear Response of Soft Materials X. Nie, W. Chen, R. Prabhu, J.M. Caruthers, Purdue University; T. Weerasooriya, U.S. Army Research Laboratory

3

Loading Rate Effect on the Tensile Failure of Concrete and Its Constituents Using Diametrical Compression and Direct Tension S. Weckert, Defence Science and Technology Organisation; T. Weerasooriya, C.A. Gunnarsson, U.S. Army Research Laboratory

4

Influence of Strain-rate and Confining Pressure on the Shear Strength of Concrete P. Forquin, Université Paul Verlaine-Metz

29

5

Dynamic Tensile Properties of Steel Fiber Reinforced Concrete R. Chen, National University of Defense Technology; Y. Liu, X. Guo, Beijing Institute of Technology; K. Xia, University of Toronto; F. Lu, National University of Defense Technology

37

6

Effect of Liquid Environment on Dynamic Constitutive Response of Reinforced Gels S. Padamati, V.B. Chalivendra, A. Agrawal, P.D. Calvert, University of Massachusetts Dartmouth

43

7

Ballistic Gelatin Characterization and Constitutive Modeling D.S. Cronin, University of Waterloo

51

8

Strain Rate Response of Cross-linked Polymer Epoxies Under Uni-axial Compression S. Whittie, P. Moy, A. Schoch, J. Lenhart, T. Weerasooriya, U.S. Army Research Laboratory

57

9

Strength and Failure Energy for Adhesive Interfaces as a Function of Loading Rate T. Weerasooriya, C.A. Gunnarsson, R. Jensen, U.S. Army Research Laboratory; W. Chen, Purdue University

67

10

Fracture in Layered Plates Having Property Mismatch Across the Crack Front U.H. Bankar, A. Rajesh, P. Venkitanarayanan, Indian Institute of Technology Kanpur

77

11

Stress Variations and Particle Movements During Penetration Into Granular Materials H. Park, W.W. Chen, Purdue University

85

12

Sand Particle Breakage Under High-pressure and High-rate Loading Md.E. Kabir, W. Chen, Purdue University

93

1

11

13

viii 13

Experimental and Numerical Study of Wave Propagation in Granular Media T. On, K.J. Smith, P.H. Geubelle, J. Lambros, University of Illinois at Urbana-Champaign; A. Spadoni, C. Daraio, California Institute of Technology

95

14

Communication of Stresses by Chains of Grains in High-Speed Particulate Media Impacts W.L. Cooper, Air Force Research Laboratory

99

15

Effects of Thermal Treated on the Dynamic Facture Properties Using a Semi-circular Bend Technique T.B. Yin, University of Toronto/Central South University; X.B. Li, Central South University; K.W. Xia, S. Huang, University of Toronto

16

Development and Characterization of a PU-PMMA Transparent Interpenetrating Polymer Networks (t-IPNs) K.C. Jajam, S.A. Bird, M.L. Auad, H.V. Tippur, Auburn University

17

Dynamic Ring-on-Ring Equibiaxial Flexural Strength of Borosilicate Glass X. Nie, W. Chen, Purdue University

123

18

Stress-strain Response of PMMA as a Function of Strain-rate and Temperature P. Moy, C.A. Gunnarsson, T. Weerasooriya, U.S. Army Research Laboratory; W. Chen, Purdue University

125

19

Dynamic Behavior of Three PBXs with Different Temperatures J.L. Li, National University of Defense and Technology/Chinese Academy of Engineering and Physics; F.Y. Lu, R. Chen, J.G. Qin, P.D. Zhao, L.G. Lan, S.M. Jing, National University of Defense and Technology

135

20

Dynamic Compressive Properties of A PBX Analog as a Function of Temperature and Strain Rate J. Qin, Y. Lin, F. Lu, National University of Defense Technology; Zh. Zhou, Beijing Institute of Technology; R. Chen, J. Li, National University of Defense Technology

21

Dynamic Response of Shock Loaded Architectural Glass Panels P. Kumar, A. Shukla, University of Rhode Island

147

22

A Dynamic Punch Method to Quantify the Dynamic Shear Strength of Brittle Solids S. Huang, K. Xia, F. Dai, University of Toronto

157

23

A Sensored Projectile Impact on a Composite Sandwich Panel M. Mordasky, W. Chen, Purdue University

165

24

Cut Resistance and Fracture Toughness of High Performance Fibers J.B. Mayo, Jr., Tuskegee University/U.S. Army Research Laboratory; E.D. Wetzel, U.S. Army Research Laboratory

167

25

Kolsky Tension Bar Techniques for Dynamic Characterization of Alloys B. Song, H. Jin, B.R. Antoun, Sandia National Laboratories

175

26

Prediction of Dynamic Forces in Fire Service Escape Scenarios M. Obstalecki, J. Chaussidon, P. Kurath, G.P. Horn, University of Illinois at Urbana-Champaign

179

27

Tensile Behavior of Kevlar 49 Woven Fabrics over a Wide Range of Strain Rates J.D. Seidt, T.A. Matrka, A. Gilat, G.B. McDonald, The Ohio State University

187

109

117

141

ix 28

The Effect of Loading Rate on the Tensile Behavior of Single Zylon Fiber C.A. Gunnarsson, T. Weerasooriya, P. Moy, Army Research Laboratory

195

29

Statistical Analysis of Fiber Gripping Effects on Kolsky bar Test J.H. Kim, N.A. Heckert, S.D. Leigh, H. Kobayashi, W.G. McDonough, R.L. Rhorer, K.D. Rice, G.A. Holmes. National Institute of Standards and Technology

205

30

Perpendicular Yarn Pull-out Behavior Under Dynamic Loading J. Hong, Purdue University; J. Lim, Hyundai Motor Company; W.W. Chen, Purdue University

211

31

Dynamic Response of Homogeneous and Functionally Graded Foams When Subjected to Transient Loading by a Square Punch C. Periasamy, H. Tippur, Auburn University

32

Dynamic Strain Measurement of Welded Tensile Specimens Using Digital Image Correlation K.A. Dannemann, R.P. Bigger, S. Chocron, Southwest Research Institute; K. Nahshon, Naval Surface Warfare Center Carderock Division

217

33

Ultra High Speed Full-field Strain Measurements on Spalling Tests on Concrete Materials F. Pierron, Arts et Métiers ParisTech; P. Forquin, Paul Verlaine University

221

34

Contact Mechanics of Impacting Slender Rods: Measurement and Analysis A. Sanders, I. Tibbitts, D. Kakarla, University of Utah; S. Siskey, J. Ochoa, K. Ong, Exponent, Inc.; R. Brannon, University of Utah

229

35

Solenoid Actuated, Rail Mounted, Aircraft Payload Release Mechanisms C.L. Reynolds, Dynetics, Inc.; J.A. Gilbert, University of Alabama in Huntsville

237

36

Finite Element Modeling of Ballistic Impact on Kevlar 49 Fabrics D. Zhu, McGill University; B. Mobasher, S.D. Rajan, Arizona State University

249

37

Optimal Pulse Shapes for SHPB Tests on Soft Materials M. Scheidler, J. Fitzpatrick, R. Kraft, U.S. Army Research Laboratory

259

38

Dynamic Tensile Characterization of Foam Materials B. Song, H. Jin, W.-Y. Lu, Sandia National Laboratories

269

39

On Measuring the High Frequency Response of Soft Viscoelastic Materials at Finite Strains S. Teller, R. Clifton, T. Jiao, Brown University

273

40

The Blast Response of Sandwich Composites With a Graded Core: Equivalent Core Layer Mass vs. Equivalent Core Layer Thickness N. Gardner, A. Shukla, University of Rhode Island

281

41

Effects of High and Low Temperature on the Dynamic Performance of the Core Material, Face-sheets and the Sandwich Composite S. Gupta, A. Shukla, University of Rhode Island

289

42

Influence of Texture and Temperature on the Dynamic-tensile-extrusion Response of High-purity Zirconium D.T. Martinez, C.P. Trujillo, E.K. Cerreta, J.D. Montalvo, J.P. Escobedo-Diaz, Los Alamos National Laboratory; V. Webster, Case Western Reserve University; G.T. Gray, III., Los Alamos National Laboratory

213

297

x 43

Modeling and DIC Measurements of Dynamic Compression Tests of a Soft Tissue Simulant S.P. Mates, R. Rhorer, A. Forster, National Institute of Standards and Technology; R.K. Everett, K.E. Simmonds, A. Bagchi, Naval Research Laboratory

307

44

Measurement of R-values at Intermediate Strain Rates Using a Digital Speckle Extensometry J. Huh, Y.J. Kim, H. Huh, Korea Advanced Institute of Science Technology

317

45

Study of Strain Energy in Deformed Insect Wings H. Wan, H. Dong, Y. Ren, Wright State University

323

46

Experimental Study of Cable Vibration Damping A. Maji, Y. Qiu, University of New Mexico

329

47

Dynamic Thermo-mechanical Response of Austenite Containing Steels V.-T. Kuokkala, Tampere University of Technology; S. Curtze, Tampere University of Technology/Oxford Instruments Nano Analysis; M. Isakov, M. Hokka, Tampere University of Technology

337

48

Investigation into the Spall Strength of Cast Iron G. Plume, C.-E. Rousseau, University of Rhode Island

343

49

Development of Brick and Mortar Material Parameters for Numerical Simulations C.S. Meyer, U.S. Army Research Laboratory

351

50

Electrical Behavior of Carbon Nanotube Reinforced Epoxy Under Compression N. Heeder, A. Shukla, University of Rhode Island; V. Chalivendra, University of Massachusetts Dartmouth; S. Yang, K. Park, University of Rhode Island

361

51

Effect of Curvature on Shock Loading Response of Aluminum Panels P. Kumar, University of Rhode Island; J. LeBlanc, Naval Undersea Warfare Center; A. Shukla, University of Rhode Island

369

52

Deformation Measurements and Simulations of Blast Loaded Plates K. Spranghers, Vrije Universiteit Brussel; D. Lecompte, Royal Military Academy; H. Sol, Vrije Universiteit Brussel; J. Vantomme, Royal Military Academy

375

53

The Blast Response of Sandwich Composites With Bi-axial In-plane Compressive Loading E. Wang, University of Illinois Urbana-Champaign; A. Shukla, University of Rhode Island

383

54

Dynamic Response of Porcine Articular Cartilage and Meniscus under Shock Loading Y.-C. Juang, L. Tsai, National Kaohsiung University of Applied Sciences; H.R. Lin, Southern Taiwan University

393

55

Dynamic Response of Beams Under Transverse Impact Loadings D. Goldar, Sharda University

399

56

Constitutive Model Parameter Study for Armor Steel and Tungsten Alloys S.J. Schraml, U.S. Army Research Laboratory

409

57

A Scaled Model Describing the Rate-dependent Compressive Failure of Brittle Materials J. Kimberley,G. Hu, K.T. Ramesh, Johns Hopkins University

419

58

Experimental Verification of Negative Phase Velocity in Layered Media A.V. Amirkhizi, S. Nemat-Nasser, University of California, San Diego

423

xi 59

Gas Gun Impact Analysis on Adhesives in Sandwich Composite Panels M. Mordasky, W. Chen, Purdue University

425

60

Damage Analysis of Projectile Impacted Laminar Composites B.S. Nashed, J.M. Rice, Y.K. Kim, V.B. Chalivendra, University of Massachusetts Dartmouth

427

61

Rate Sensitivity in Pure Ni Under Dynamic Compression K.N. Jonnalagadda, Indian Institute of Technology Bombay

439

62

Temperature Effect on Drop-weight Impact of Woven Composites Y. Budhoo, Vaughn College of Aeronautics and Technology; B. Liaw, F. Delale, The City College of New York

443

63

Dynamic Mode-II Characterization of a Woven Glass Composite W.-Y. Lu, B. Song, H. Jin, Sandia National Laboratories

455

64

Rate Dependent Material Properties of an OFHC Copper Film J.S. Kim, Korea Railroad Research Institute; H. Huh, Korea Advanced Institute of Science and Technology

459

65

Zirconium: Probing the Role of Texture Using Dynamic-tensile-extrusion C.P. Trujillo, J.P. Escobedo-Diaz, G.T. Gray, III., E.K. Cerreta, D.T. Martinez, Los Alamos National Laboratory

467

Punch Response of Gels at Different Loading Rates Mark Foster [email protected] Paul Moy [email protected] Randy Mrozek [email protected] Joe Lenhart [email protected] Tusit Weerasooriya [email protected] Army Research Laboratory Weapons and Materials Research Directorate Bldg 4600 Deer Creek Loop Aberdeen Proving Ground, MD 21005-5069 ABSTRACT Synthetic soft polymer gels have many advantages over protein-based gels that are derived from animal collagen and bones such as stability at room temperature and prolonged shelf life. In addition, the ability to tailor the formulations and processes of synthetic gel to control mechanical properties both isotropically or anisotropically is another essential feature in order for gels to mimic the spectrum of biological tissues. However, it is impractical to physically characterize all aspects of every gel available. To do so would require production of a significant amount of material to accommodate all the varying tests needed for a comprehensive study. A novel punch test was developed as a simple solution to obtain mechanical responses at different loading rates without the production of a large amount of sample material. The gels used in this effort are 10% and 20% ballistic gelatin, the commercially marketed PermaGel™, and triblock copolymer gels. The experimental setup is discussed, and the results are presented and compared to a previous study that discussed the tensile behavior of these soft materials. INTRODUCTION Traditionally, ballistic gelatin is used extensively to examine the penetration depth of firearms. While useful as a base material for qualitative comparison, the applications of this biologically derived gelatin become very limited when used as a tissue simulant for quantitative testing. Next generation armor and protection systems require an understanding of injury mechanisms that has not yet been realized [1-3]. However, ballistic gelatins inconsistent viscoelastic properties, high sensitivity to temperature, and aging effects all prove detrimental to producing data to validate proposed material models [4]. Synthetic gelatins are therefore a promising solution because networks of cross-links and polymer chains can be tuned to resemble the mechanics of biological tissues. Unfortunately, such a wide range of controllable properties necessitates laborious and repetitive testing for full characterization.

T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_1, © The Society for Experimental Mechanics, Inc. 2011

1

2 Much prior work has been performed on soft tissue stimulants. Methods have been established in constrained and unconstrained compression at a variety of strain rates [2,5,6]. Often this work is to measure a shear modulus for use in various constitutive models. While existing models can be useful at determining material response at lower rates of strain, a fully nonlinear model is required at higher rates [7]. Wu et al. [6] also demonstrated that friction can cause a non-uniform deformation state in unconfined compression tests which contradicts assumptions in traditional models. In a study on the effects of temperature, aging time, and strain rate to the penetration depth of ballistic gel, Cronin and Falzon [4] found that tensile strain dominates over shear strain during failure. This also contradicts many current models of tissues that are based on shear failure criteria. Microscale indentation and rheology tests have been performed, proving that viscoelastic and hyperelastic models are applicable at small (physiological) deformations, but large strain behavior of nonlinear tissues warrants further analysis [8-11]. Other methods of modeling human tissue behavior involve porcine, bovine, rat or mouse tissue. Snedeker et al. [8] performed impact experiments on porcine and human kidneys, and found large differences in stress to failure. Fracture tear tests have also found basic energy dissipation and tear resistance values [12]. Moy et al. [13,14] have used digital image correlation techniques to measure strain field qualities in notched gelatin tensile specimens. This technique provided a measurement of the maximum tensile strain and energy required for a crack to propagate through gelatin. This interest led to the development of a simple punch penetration test to screen the multitude of possible gelatin formulations into a smaller group of gels that resemble the mechanical behaviors of ordnance gelatins. Various gelatin materials were subjected to a constant-rate displacement by a 6.35 mm hemispherical penetrator tip. These materials included synthetic polymer gels, ballistic gels, and Permagel™, a commercially available ballistic gel replacement. Then puncture data was compared to gelatin fracture tear data from Moy et al. [13] to ensure that the basic puncture test can be used as a screening process to find suitable tissue surrogates without an indepth investigation into each material. Materials Three different synthetic polymer gels were made from different concentrations of Poly(styrene-b-ethylene-cobutylene-b-styrene) (SEBS) G1652 as-received from Kraton Polymers (Houston, TX, USA) and mineral oil asreceived from Aldrich Chemical (Milwaukee, WI, USA). These three gels were mixed in sheets with concentrations of 70, 80, and 90 percent mineral oil to SEBS polymer. Ballistic gelatin was made from Bloom 250 Type A ordnance gelatin mix as-received from GELITA USA (Sioux City, IA). Each batch was mixed in water according to manufacturer directions, poured into a sheet, refrigerated at 3.89°C, and tested the following day. The two types of ballistic gel typically used for bullet penetration testing are either 10 or 20 percent by mass gelatin, and both are accounted for in this work [4]. Permagel™ is a transparent material designed to have similar properties to 10% ballistic gelatin, but without its inherent disadvantages. The material was used as-delivered from USALCO (Browns Mills, NJ) and was cut from a large base block, molded into sheets and allowed to cool before testing. The group of gelatins tested is included in Table 1.

3 Table 1: Collection of Materials Tested Material Concentration Source Mineral Oil/SEBS polymer

Permagel Bloom 250A Ordnance Gel

70/30 80/20 90/10 N/A (by weight to water) 10% 20%

Aldrich Chemical/ Kraton Polymers

USALCO GELITA, USA

Test Methods A 4mm thick specimen was bonded to one side of a standard 7/16 inch washer, excess material was trimmed and the specimen was tested to full failure with a 6.35 mm hemispherical indenter. All specimens were inspected for bubbles or debris prior to gluing. An elevated acrylic table fixture was used to provide clearance for full specimen failure and give ample space for observation. A custom 45° triangle mirror fixture was then designed to fit underneath the elevated fixture to observe the specimen during the indentation. Figure 1 shows the test fixture. After the specimens fully adhered to the washer, the entire sample was placed on the top of the fixture centered over the middle 12.7 mm diameter hole in the acrylic plate. While the washer did not provide clamping pressure by any means, it did give an open space on the top gelatin surface to fill with lubricant. Friction between the acrylic and the gelatin provided ample resistance to prevent any slippage during the experiment.

Figure 1: Punch Test Fixture with 45° Mirror An Instron 8871 servo-hydraulic load frame was used to control and directly measure the load and displacement required to indent and to puncture the gel specimens. Three different displacement rates were addressed: 12.7, 127 and 1270 mm/min. These gave a wide range of material characteristics while remaining within the capabilities of the load frame. The overall machine setup is in Figure 2. Lubrication was used in order to minimize friction between the gel and indenter. For ballistics gel an olive oil was used, but silicone oil of similar viscosity was used for the Permagel™ and synthetic polymer gels, to ensure the lubricant did not affect the gel material being tested. Prior to each experiment, the indenter was lowered until it just contacted the gelatin surface.

4

Figure 2: Machine Setup with Accompanied Gel Punch Fixture

RESULTS AND DISCUSSION Without lubrication, friction between the indenter and the gelatin was quite evident in the preliminary testing. It was observed that when oil was omitted specimens would exhibit a “cork” type failure characteristic of a shearing failure instead of the desired Mode I tearing failure. Load and displacement data obtained for all experiments at each extension rate can be seen in Figures 3a through c. Figures 4a to 4c provides a magnified view of the gels at the lower load range. This data includes all testing, and it is clear from the amount of overlap that the test is inherently repeatable. The gelatins displayed a close agreement between certain pairs of material, such as the close agreement between 80/20 and Permagel™ for all three rates. However, ballistic gels do not exhibit the relaxation portion of the load-displacement curve that the other 4 materials expressed around 8 mm displacement. This can clearly be seen in the 1270 mm/min data where the 20% ballistic gel deviates from 70/30 data curve at around 7.5 mm extension. Note that prior to this point there is a close agreement between the 20% ballistic gel and 70/30 curves. The ballistic gels also show lower overall extensions than the other materials. When the 10% ballistic and Permagel™ gelatins are compared across the three rates studied, the load response converges as the displacement rate increases. This compels the idea that Permagel™ could be used as a replacement for the 10% ballistic gelatin, but only at higher rates.

5

Figure 3: Load vs. Extension for all Gel Materials Tested at (a) 12.7 mm/min, (b) 127 mm/min, and (c) 1270 mm/min Displacement Rate

6 6

4 80/20 90/10 Permagel 10% Ballistic

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(4c) Figure 4: Expanded Load vs. Extension for Punch Experiments at the Lower Load Region at (a) 12.7 mm/min, (b) 127 mm/min, and (c) 1270 mm/min Displacement Rate The maximum displacement to failure was compared in Figure 5 across the three displacement rates of 12.7, 127, and 1270 mm/min. Two distinct trends are clear. Most interesting is the distinct difference in trend between the mineral oil and the Permagel™/ballistic gel. The synthetic mineral oil gels exhibit a much larger stretch ratio than the ballistic gelatins or Permagel™ at a higher rate of displacement, which suggests a different strengthening mechanism may occur at these higher displacement velocities. This could easily be attributed to the structure of polymer chains and amount of crosslinking in the mineral oil gelatins which do not occur in the others. As the manufacturer claims Permagel™ is similar to the ballistic gels but requires a larger overall extension to fully penetrate, which is shown from the data at all rates of displacement.

7 60

Maximum Extension (mm)

50 40 30 20 10 0

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100 Displacement Rate (mm/min) 70/30 80/20

90/10 Permagel

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Figure 5: Maximum Extension as a Function of Displacement Rate for all Gel Punch Experiments A similar analysis was performed with the maximum load to failure in Figure 6. The 80/20 formulation ratio of mineral oil to SEBS polymer closely agrees with the Permagel™ at all displacement rates studied. Unlike the 70/30 gelatin which gave a much larger penetration resistance. Despite a similarity in extension to failure between the 70/30 and Permagel™ at lower displacement rates, the punch test showed a vast difference in stiffness of the 70/30 gelatin. While miniscule, the loads of the 90/10 polymer gel resemble the 10% ballistic gel across all displacement rates. 30

Maximum Load (N)

25 20 15 10 5 0

10

100 Displacement Rate (mm/min) 70/30 80/20

90/10 Permagel

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Figure 6: Maximum Load as a Function of Displacement Rate for all Gel Punch Experiments

8 Each load-extension curve was integrated to determine the overall energy required to fully puncture the gelatin. These values were then correlated to a prior work by Moy et al. which tested the fracture characteristics of both ballistic gel concentrations, and Permagel™ [13]. A Mode I test was used with a dogbone shaped specimen that had a small pre-crack. The crack propagation was then inspected using high speed cameras with DIC to acquire surface strains near and around the crack tip. The load-extension curve was also integrated to obtain fracture energy values. This comparison is shown in Figure 7 a, b, and c. It is important to note the difference in scaling between the two data sets, for they are not in perfect alignment in any case. However, considering the differences between the two test methods, the results are quite similar in trend. It is very easy to see the agreement in the 20% ballistic gel for example, where the fracture energy is close to a factor of 5x higher than the punch energy. This comparison validates the punch method presented here as a simple screening process as it is a much less demanding test.

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Figure 7a: Comparison between Punch Energies and Fracture Energies for Permagel™ [13]

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Figure 7b: Comparison between Punch Energies and Fracture Energies for 10% Ballistic Gel [13]

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Figure 7c: Comparison between Punch Energies and Fracture Energies for 20% Ballistic Gel [13] CONCLUSIONS A significant number of synthetic gelatins offers multiple solutions for potential replacement of ballistic gelatin. More importantly, polymer gels can be tailored to have characteristics to actual biological tissues. To adequately examine the load response of every gel available would be very demanding. In an effort to find a suitable tissue surrogate to validate theoretical models for next-generation armor and protection systems, several gelatin materials were compared in a simplified puncture test. Ballistic gels, mineral oil gels, and the commercially available Permagel™ were punctured to full failure using a hemispherical indenter across extension rates of 12.7, 127 and 1270 mm/min. Similar initial load responses were seen in the 10% ballistic, 80/20 mineral oil/SEBS, and

10 Permagel™ at each rate despite the ballistic gelatins lower extension and maximum load to failure. Through a comparison with previous mode I gel fracture data, the punch technique provides an easy screening method to classify a gelatin as a possible tissue simulant. REFERENCES 1. Song, B., Ge, Y., Chen, W., and Weerasooriya, T. Radial Inertia Effects in Kolsky Bar Testing of ExtraSoft Specimens. Experimental Mechanics, 47, pp. 659-670. 2007. 2. Saraf, H., Ramesh, K.T., Lennon, A.M., Merkle, A.C., and Roberts, J.C. Mechanical Properties of Soft Human Tissues Under Dynamic Loading. Journal of Biomechanics, 40, pp. 1960-1967. 2007. 3. Van Sligtenhorst, C., Cronin, D. S., and Brodland, G. W. High Strain Rate Compressive Properties of Bovine Muscle Tissue determined using a split Hopkinson bar apparatus. Journal of Biomechanics, 39, pp 1852-1858. 2006. 4. Cronin, D.S., and Falzon, C. Characterization of 10% Ballistic Gelatin to Evaluate Temperature, Aging and Strain Rate Effects. Proceedings of the 2010 SEM Annual Conference, Indianapolis, IN. 2010. 5. Kwon, J., and Subhash, G., Compressive Strain Rate Sensitivity of Ballistic Gelatin. Journal of Biomechanics, 43. pp 420-425. 2010. 6. Wu, J.Z., Dong, R.G., and Schopper, A.W. Analysis of Effects of Friction on the Deformation Behavior of Soft Tissues in Unconfined Compression Tests. Journal of Biomechanics, 37, pp 147-155. 2004. 7. Cronin, D.S., and Falzon, C. Dynamic Characterization and Simulation of Ballistic Gelatin. Proceedings of the 2009 SEM Annual Conference. Albuquerque, N.M. 2009. 8. Snedeker, J.G., Barbezat, M., Niederer, P., Schmidlin, F.R., and Farshad, M. Strain Energy Density as a Rupture Criterion for the Kidney: Impact Tests on Porcine Organs, Finite Element Simulation, and a Baseline Comparison Between Human and Porcine Tissues. Journal of Biomechanics, 38, pp 993-1001. 2005. 9. Lin, D.C., Shreiber, D. I., Dimitriadis, E. K., and Horkay, F. Spherical Identation of Soft Matter Beyond the Hertzian Regime: Numerical and Experimental Validation of Hyperelastic Models. Biomechanics and Modeling in Mechanobiology, 8 (5), pp 345-358, 2009. 10. Wu, J.Z., Dong, R.G., Smutz, W.P., and Schopper, A.W. Nonlinear and Viscoelastic Characteristics of Skin under Compression; Experiment and Analysis. Biomedical Materials and Engineering. 13 (4), pp 373-385, 2003. 11. Clark, A.H., Richardson, R.K., Ross-Murphy, S.B., and Stubbs, J.M. Structural and Mechanical Properties of Agar/Gelatin Co-gels. Small Deformation Studies. Macromolecules. 16, pp 1367-1374, 1983. 12. Furukawa, H., Kuwabara, R., Tanaka, Y., Kurokawa, T., Na, Y., Osada, Y., and Gong, J.P. Tear Velocity Dependence of High-Strength Double Network Gels in Comparison with Fast and Slow Relaxation Modes Observed by Scanning Microscopic Light Scattering. Macromolecules. 41, pp 7173-7178, 2008. 13. Moy, P. Gunnarsson, C. A. and Weerasooriya, T. Tensile Deformation and Fracture of Ballistic Gelatin as a Function of Loading Rate. Proceedings of the 2009 SEM Annual Conference. Albuquerque, NM. 2009. 14. Moy, P. Foster, M. Gunnarsson, C. A. and Weerasooriya, T. Loading Rate Effect on Tensile Failure Behavior of Gelatins under Mode I. Proceedings of the 2010 SEM Annual Conference. Indianapolis, IN. 2010.

A Kolsky Torsion Bar Technique for Characterization of Dynamic Shear Response of Soft Materials Xu Nie1*, Weinong Chen1, Rasika Prabhu2, James M. Caruthers2, Tusit Weerasooriya3 1

AAE&MSE schools, Purdue University. 2ChE school, Purdue University. 3Army Research Laboratory * Corresponding author: Xu Nie, 701 W. Stadium Ave. West Lafayette, IN 47907-2045 Email: [email protected]

ABSTRACT A novel Kolsky torsion bar technique is developed and successfully utilized to characterize the high strain rate shear response of a rate-independent end-linked polydimethylsiloxane (PDMS) gel rubber with a shear modulus of ~10 KPa. The results show that the specimen deforms uniformly under constant strain rate and the measured dynamic shear modulus well follows the trend determined by dynamic mechanical analysis (DMA) at lower strain rates. Contrastive Kolsky compression bar experiments are also performed on the same gel material with annular specimens. The dynamic moduli obtained from compression experiments, however, are an order of magnitude higher than those predicted by the torsional technique, due to the pressure caused by the radial inertia and end constraints. INTRODUCTION Characterization of dynamic response of soft biological tissues has seen a tremendous rise in the past decade. Among all the published non-oscillatory high rate results, dynamic uniaxial compression/tension has generated the most popular group of data [1], and its experimental conditions have also been extensively investigated [2]. There are two major issues associated with the axial loading conditions when the strain rate is high: 1. Dynamic stress (or force) equilibrium across the specimen length, and 2. Radial inertia induced pressure by strain rate and strain acceleration. A preliminary solution to minimize the inertia effect is to punch a hole in the center of the specimen, for which the pressure was greatly reduced by creating a stress-free inner surface. However, for materials as soft as human brain tissues whose elastic moduli are typically in the range of 0.1-10 KPa, even the reduced pressure in an annular sample can be sufficiently high to overshadow the intrinsic material response. To separate the pressure from the intrinsic mechanics response of soft materials, a pure shear loading condition is desired. In this paper, we present a newly developed desktop Kolsky torsion bar technique for the characterization of high rate shear mechanical properties of soft materials. The effectiveness of this torsion bar technique was demonstrated by our calibration experiments on the end-linked polydimethylsiloxane (PDMS) gel rubber. EXPERIMENTS AND RESULTS A typical oscilloscope record of the modified Kolsky torsion bar experiment is shown in Fig. 1. The trace noted as “incident bar signal” is measured by the strain gages mounted on the incident bar, while the other trace is taken from the torque sensor which connects to the external ring adapter. Since the gel material under investigation has extremely low wave impedance compared to that of the incident bar, most of the incident wave is reflected back. Consequently, the reflected pulse would not see any noticeable difference, both in shape and amplitude, from the incident pulse. We used the incident wave to calculate the shear strains in the T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_2, © The Society for Experimental Mechanics, Inc. 2011

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specimen so that a better-quality signal can be used directly from the oscilloscope reading. The stress-strain curves of five different samples loaded at shear strain rate of ~1000s-1 are displayed in Fig. 2. Although some discrepancies were found on the five measured stress-strain curves, all of them exhibited linear elasticity when the strain is beyond 8%. In order to compare the shear modulus with those obtained from DMA tests, and thus evaluate the validity of our Kolsky torsion bar experiment, the tangential of these stress-strain curves in Fig. 2 were measured in the strain range from 8% to the maximum strain on each curve. As mentioned before, the purpose of conducting current dynamic torsional experiments on soft materials is to directly acquire their shear constitutive properties, which in the past were mostly inferred from the dynamic compression results. To compare the measured modulus value from uniaxial compression experiments with those obtained from DMA and Kolsky torsion bar experiments, dynamic compressive experiments on this same PDMS gel were also conducted at comparable strain rates. The results are plotted in Fig. 3. The dynamic shear elastic modulus of gel measured with torsion bar technique follows the trend of DMA test results, while the same material exhibited much higher modulus value (about an order of magnitude) when it was under dynamic compression. Such a large discrepancy between the two dynamic testing techniques and the analysis of the discrepancy reveal that the Kolsky torsion bar experiment is necessary to characterize the shear behavior of extra soft materials under high strain rate loading conditions.

Fig. 1 The original signals of torsional experiments

Fig. 2 Shear stress-strain curve of PDMS at 1000/s by Kolsky torsion bar technique

Fig. 3 Comparison of shear modulus obtained by different testing techniques at different shear rates REFERENCE: [1] Bo Song, Weinong Chen, Yun Ge and Tusit Weerasooriya, “Dynamic and quasi-static compressive response of porcine muscle”, Journal of Biomechanics, 40, 2999-3005, 2007 [2] Bo Song and Weinong Chen, “Dynamic stress equilibration in split Hopkinson pressure bar tests on soft materials”, Experimental Mechanics, 44, 300-312, 2004

Loading Rate Effect on the Tensile Failure of Concrete and Its Constituents using Diametrical Compression and Direct Tension

Samuel Weckert1 [email protected] Tusit Weerasooriya2 [email protected] C. Allan Gunnarson2 [email protected] 1

Defence Science and Technology Organisation Edinburgh, South Australia, 5111 2

Army Research Laboratory Weapons and Materials Research Directorate Bldg 4600 Deer Creek Loop Aberdeen Proving Ground, MD 21005-5069 ABSTRACT The loading rate effect on the tensile failure strength of concrete and its constituent materials has been investigated. Concrete is inherently weaker in tension than compression so tensile failure represents the dominant failure mode. Understanding the failure characteristics of concrete, particularly at high loading rate, is important for developing modeling capabilities, in particular for predicting spallation damage and fragmentation. Several concretes, and their constituents, have been investigated at different loading rates to understand the tensile failure behavior as a function of loading rate. In this paper, experimental procedures that were used are discussed, and results from two different tensile testing methods, direct tension and diametric compression (Brazilian/split-tension), are presented for several of these materials. INTRODUCTION Tensile failure is a vulnerable failure mode for concrete as it is much weaker in tension compared with other modes of failure such as compression. Typically the tensile strength is an order of magnitude less than the compressive strength. High strain rate tensile testing of concrete is important for weapons effects problems, such as penetration and explosive loading, where the loading rates are very high and tensile failure can occur as spallation damage in a target. Materials can behave differently at high strain rates, so material characterization in this regime is important for developing accurate material models for simulations. Direct tension experiments produce a nominally uniaxial tensile stress state, however, it can be difficult to implement because of issues associated with gripping the sample. This is particularly the case for brittle 1

This work was undertaken as a collaborative effort between Australia and the U.S. while Sam Weckert was on a 6 month attachment at ARL in the High Rate Mechanics and Failure branch under the Scientists and Engineers Exchange Program in 2010.

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14 materials, such as concrete, where it is not possible to use conventional grips or threaded joints. High strength adhesives can be used to grip normal concrete; however they are not strong enough for testing the new generation of high strength concretes. Notched specimens can be used to reduce the cross-sectional area [1], however, this creates a stress concentration which leads to an under prediction of the tensile strength. Dog-bone specimen geometries are also possible, however these are difficult implement in brittle materials. The diametric compression test, also known as the Brazilian or Split-tension test, offers an alternate test method to indirectly obtain the material tensile response. This test induces tensile stresses within the specimen by point diametric compression of a disc shaped sample. This permits the use of simpler compression testing apparatus to obtain tensile material response data. Other tensile test techniques also include three or four point bend tests and high rate specific spallation experiments [2]. This report presents the tensile strength for various concretes and their constituent materials (mortar and aggregate) using the diametric compression technique. Tests are conducted at high loading rate using a Split Hopkinson Pressure Bar (SHPB) apparatus and intermediate and low loading rates using an Instron hydraulic test machine for comparison. This allows an investigation of the loading rate effects for each of these materials. Direct tension tests at high loading rate have also been conducted for several of the concrete materials to allow a comparison of tensile strengths from the direct tension and diametric compression test methodologies. MATERIALS The tensile strengths of five different materials were investigated in this study: 1. SAM35 concrete: a 3500psi (~24MPa) minimum quasi-static unconfined compressive strength concrete produced by the US Engineering and Research Development Center (ERDC) [3]. It contains small limestone aggregate components up to approximately 8mm in size. 2. Mortar: prepared from a commercially available mix - Drypack basic mortar sand and cement, Adelaide Brighton Cement Limited, Australia. 3. Granite: charcoal black granite, Starrett, True Stone Tech Division, MN, US. 4. Ultra High Performance Concrete (UHPC): a reactive powder concrete reinforced with steel fibers of length 12.7mm and diameter 0.2mm, randomly distributed through the concrete at 6.2% by weight [4], samples obtained from Australia. 5. Alcatraz concrete: obtained from a tourist commercial vendor at the Alcatraz prison in San Francisco, California. The SAM-35 represents a common concrete mix and the mortar and granite are representative of typical concrete constituents (however, the SAM-35 contains limestone aggregate not granite). The UHPC is a new generation high strength concrete which was tested to evaluate its enhanced characteristics. The diametric compression tests used disc shaped specimens with a nominal thickness of ¼” (6.35mm). The mortar, UHPC and Alcatraz concrete specimens had a nominal diameter of 20mm, whereas the SAM-35 and granite specimens had a nominal diameter of 1” (25.4mm). Direct tension tests at high loading rate were also performed on the mortar and SAM35 concrete using cylindrical samples with ¾” diameter and ¾” length. The size of the specimen geometry was dictated by the available experimental equipment and was fairly small relative to the size of the aggregrate components in the concrete and the steel fiber distribution for the UHPC. In addition, the material heterogeneity is further amplified in the diametric compression test where only a portion of the sample is in tension. Consequently it is expected that there will be considerable scatter in the results and so a minimum of five tests were conducted for each material at each loading rate. EXPERIMENTAL METHODOLOGY Diametric compression test technique The diametric compression (Brazilian/split tension) test uses a circular disc sample, which is point loaded at diametrically opposite points in compression. This test methodology produces a biaxial stress state where a tensile stress is induced perpendicular to the compressive stress along the loading axis and the material fails in tension. The stress state produced with this loading condition is discussed in detail in [5]. The diametric

15 compression test methodology gives the tensile failure strength of the material, however, no pre or post-peak stress-strain response can be obtained. A problem with the point loading used in this test methodology, particularly for brittle materials, is that the sample is subject to high stress concentrations at the external load contact points. Thus failure may initiate at these contact points rather than in the induced tensile region in the bulk of the specimen, which invalidates the test. To reduce the stress concentrations at the contact points, wooden bearing strips are recommended for distributing the load in quasi-static diametric compression tests [6,7], however, these are not suitable for high rate tests because of the reflections of stress waves and material impedance effects. For high rate diametric compression tests, other researchers have suggested several techniques to overcome this problem, both with the objective of spreading the load over a small area at the sample sides to reduce the stress concentrations. The first method is to cut flat areas onto the sides of the sample at loading points [8]. The second method is to maintain a circular disc shaped sample, but use concave curved input/output bars for loading the sample [9], as shown in figure 1. It is this second method, which was adopted at all loading rates for the tests presented here.

Figure 1: Curved input/output bars for loading the disc specimen

Figure 2 shows the normal equation used to obtain the tensile strength, σ, for a diametric compression test.

Figure 2: Compressive loading and tensile stress diagram/equation for diametric compression setup

where P is the axial compressive load, D is the specimen diameter and t is the specimen thickness. A modification to this equation is used in [9] to account for the spreading of the load at the contact points. However, the modified equation introduces a contact width parameter between the sample and the curved loading platens, which is difficult to measure. Thus the modified equation is not used for the work presented here. For a contact width of 2.5mm for a 25mm diameter sample, the stress is only reduced by 4% by the modified equation, so the implication of ignoring this correction factor is relatively small. Low and intermediate loading rate experiments The low and intermediate rate diametric compression tests were performed using a 5000lb Instron hydraulic test machine. The tests were conducted with a constant compressive displacement rate of 0.001mm/s for the low rate and 1mm/s for the intermediate rate experiments. Instrumentation for these tests included the load cell and displacement transducer in the test machine and high-speed video to record the material loading and failure process. High loading rate experiments

16 The high rate diametric compression tests were performed on a compression Split-Hopkinson Pressure Bar (SHPB) with 1¼” diameter aluminum input and output bars. Background on the SHPB and test methodology is provided in [10]. The SHPB input and output bars were instrumented with semiconductor strain gauges. The semiconductor gauges have a much higher sensitivity compared with traditional metal foil strain gauges and are essential for measuring the small strains associated with testing concrete in tension. A comparison of the signals from the two gauge types is shown in figure 3, which illustrates the noise reduction using the higher sensitivity semiconductor gauges.

Figure 3: Comparison of metal foil and semiconductor strain gauges - voltage signal (left); and strain signal (right)

A 24” long striker bar, accelerated by compressed nitrogen, was used to impact the input bar to produce the compressive incident pulse. This was used to load the sample at a compressive displacement rate of approximately 1000mm/s. The striker bar had a flat impact end, however a small amount of silicon grease on the impact face was used for shaping the incident pulse. This has the effect of damping high frequency components (ringing) associated with the impact and increases the pulse rise time to load the specimen more gradually. Ramping the load in this way is critical for allowing time for the stress to equilibrate, through multiple wave reverberations, in the sample. This is particularly important for brittle materials, which only undergo minimal strain before failure and equilibrium needs to be achieved before this time for a valid fracture strength to be reported. Pulse-shaping of the incident pulse for SHPB experiments is discussed further in [11]. Figure 4 shows a detail of the high loading rate experimental setup.

Figure 4: High rate loading experimental setup

17 Figure 5 shows experimental results performed here with and without pulse shaping. The shape of the incident pulse is changed significantly and the comparison between the transmitted and incident minus reflected pulses, which is proportional to the force on either side of the sample (ie. the stress equilibrium condition), is greatly improved.

Figure 5: Test with no pulse-shaping (left); and with pulse-shaping using silicon grease (right)

The specimen loading and failure process for the high rate tests was imaged using a Shimadzu Hypervision HPV2 high-speed camera. This camera produces 102 frames at 312x260 resolution at a frame rate of up to 1 million frames per second. The camera was synchronized to the strain gauge on the output bar, so the images could be related to the stress history in the sample from the transmitted pulse. The high-speed images provided a visual assessment that the crack was initiating in the centre of the specimen rather than at the loading contact points. Tensile strain and strain rate Using the diametric compression technique it is easy to measure the compressive displacement rate associated with a test. This can be calculated from the reflected pulse for the SHPB high rate tests and is available from the machine displacement transducer for the intermediate and low rate tests. However, in measuring the tensile strength of the material, it is the tensile strain and strain rate, which is of interest. For diametric compression tests, other researchers have instrumented the sample with a strain gauge [8,12] to measure the tensile strain. However, by gluing a strain gauge to the sample where it is expected to fail, this has the potential to reinforce the material and influence the tensile failure strength and crack initiation defect point. Other non-invasive techniques for measuring the tensile strain include full field optical techniques such as moire interferometry [13, 14] and Digital Image Correlation (DIC) [9,13,15,16]. The DIC technique involves high speed imaging of a random speckle pattern (natural or painted) on the specimen surface. The deforming images are correlated spatially at each time step to calculate the sample deformations and strain fields. The DIC technique was attempted here, however, it was found that the strain in concrete before failure was too small to reliably measure displacement fields using this technique. The camera resolution and speckle size were the limiting factors for the strain sensitivity and the background noise. Consequently, it was not possible to measure the tensile strain and strain rate for the diametric compression tests reported here and the rate sensitivity is reported using the stress loading rate instead. The stress loading-rate divided by the quasi-static material elastic modulus has been used by other researchers [12,17] to estimate the tensile strain rate from diametric compression tests. However, this is avoided here due to apprehension in using quasi-static values for the modulus, which may potentially be different under dynamic loading conditions. High loading rate direct tension tests

18 The high rate direct tension tests were conducted using a tension SHPB with ¾” steel input and output bars instrumented with semiconductor strain gauges. The tension SHPB is shown schematically in figure 6. It uses a hollow striker bar which impacts a flange at the end of the input bar to produce the tensile pulse. The samples were glued to detachable platens, which screwed into the input and output bars. The glue used was Sikadur crack fix structural epoxy, which was found to have a high rate bond strength of 20-25MPa when used in this application. This was high enough to successfully test the mortar and SAM35 concrete, however not strong enough for the UHPC and granite. Several tests were attempted for the UHPC, however these resulted in failure at the glue line rather than in the material itself. Figure 7 shows some of the direct tension specimens for mortar, SAM35 and UHPC. Note the very large aggregate size for SAM35 concrete.

Figure 6: Split Hopkinson Pressure Bar schematic for high rate direct tension testing

Figure 7: High rate direct tension specimens

19 RESULTS SAM35 concrete The results for the SAM35 concrete are shown in figure 8. It was tested using the diametric compression technique at low, intermediate and high loading rates and also at high rate in direct tension. The results show significant scatter and this can be attributed to the material in-homogeneity (due to the aggregate components) for the small specimen size used here. The mean tensile strength at low rate (quasi-static loading conditions) was 2.5MPa. This is approximately 10% of the indicated quasi-static compressive strength of 24MPa (3500psi).

Figure 8: SAM35 concrete tensile strength versus loading stress rate

The SAM35 concrete exhibited a strong loading rate effect, with the tensile strength increasing with loading stress rate. This increase was fairly linear over the loading rate range considered here and the mean tensile strength of 5.5MPa at high rate was more than double the low rate (quasi-static) value of 2.5MPa. The data points for direct tension at high rate are also shown in Figure 8. The comparison between the diametric compression and direct tension test techniques at high rate was very good for this material.

20 Mortar The mortar was also tested at low, intermediate and high loading rates using the diametric compression technique and at high rate in direct tension. These results are shown figure 9. The mortar material is more homogeneous compared with the SAM35 concrete. It still contains sand particles within a cement matrix, however it has no large aggregate components. This resulted in less scatter of the data points for the diametric compression tests. The mean tensile strength at low and intermediate loading rates was approximately the same, 3MPa, and increased to 4MPa at the high loading rate. Thus, the loading rate effect on tensile strength for this material is only evident at the high rate and differs from that seen with the SAM35 concrete. The comparison between the two test techniques at high rate was poor for the mortar material. The tensile strength measured by the diametric compression technique was significantly lower than that measured by the direct tension tests (although there was significant scatter in the direct tension results). Thus, compared with the SAM35 concrete results, it seems that the agreement between the two test techniques may be material dependent. The direct tension test produces a nominally uniaxial tensile stress state, whereas the diametric compression test produces a more complex biaxial tension-compression stress state as discussed in [5]. Thus it is possible that the SAM35 concrete response is similar under both stress states, whereas the mortar response is different due to different macro and microstructural mechanisms.

Figure 9: Mortar tensile strength versus loading stress rate

The effect on specimen disc thickness for the diametric compression tests was investigated for the mortar. The diametric compression results shown in figure 9 were obtained using disc samples with nominally 20mm diameter (D) and ¼” (6.35mm) thickness (L), resulting in a cylinder with L/D of approximately 0.3. In comparison, the direct tension tests were performed with cylindrical specimens with an L/D=1 (20mm diameter, 20mm thickness/length). Consequently, the diametric compression tests at high rate were repeated with specimens with an L/D=1 (although in a different orientation to the direct tension tests) and these results are presented in table 1 and figure 10. The tensile strength determined by the diametric compression technique should not depend on the specimen thickness, however, the mean tensile strength increased by approximately 10% using the longer specimens (L/D=1). This is possibly due to inertial material confinement effects. However, despite the small increase, there was still significant disparity with the direct tension results.

21 Table 1: Mortar high rate tensile strength results – effect of specimen geometry

Mortar High Rate Tensile Strength (MPa) Direct Tension Diametric Compression L/D~0.3 L/D=1 L/D=1 4.1 4.7 8.2 3.6 4 8.2 4.1 4.5 5.7 3.7 4 7.3 4.1 4.7 5.7 Mean: 3.9 Mean: 4.4 Mean: 7.0

Figure 10: Tensile strength of mortar as a function of specimen L/D and experiment type

22 Granite The tensile strength of the granite samples was evaluated at the low, intermediate and high loading rates using the diametric compression technique. These results are presented in figure 11. At low rate the mean tensile strength was approximately 15MPa. This increased to approximately 21MPa for the intermediate loading rate and to 22.5MPa at high rate. These values are substantially higher than those for the SAM35 concrete and mortar samples and is due to the strong crystalline microstructure of the granite. The increase in tensile strength with loading rate was more pronounced over the low to intermediate rate range, which is in contrast to the mortar response where the increase was only seen over the intermediate to high rate range. Combining these two responses, we would expect a fairly linear increase over low to intermediate to high rate. This is exhibited in the behaviour of the SAM35 concrete which is a composite of these two representative components.

Figure 11: Granite tensile strength versus loading stress rate

The high strength glue used for the SAM35 concrete and mortar direct tension tests was not strong enough for the granite so no high rate direct tension tests could be performed for this material. Thus there is no comparison between the two test techniques for this material.

23 Alcatraz concrete The concrete sample obtained from the Alcatraz prison in San Francisco was tested at the three loading rates in diametric compression and its tensile strength results are presented in figure 12. The material response was similar to that of the SAM35 concrete. It showed an increase in tensile strength with loading rate, which was fairly linear over the range tested. The mean tensile strength at low rate was approximately 5.5MPa, increased to 6.6MPa at the intermediate loading rate and to approximately 8MPa at the high loading rate. The Alcatraz concrete was stronger than the SAM35 concrete, however, the SAM35 concrete is a relatively low strength concrete with only 3500psi or 24MPa quasi-static compressive strength.

Figure 12: Alcatraz concrete tensile strength versus loading stress rate

24 Ultra-high performance concrete The tensile strength results for the Ultra-High Performance (UHP) concrete, tested at the three loading rates in diametric compression, are shown in figure 13. The UHP concrete was substantially stronger than the other concretes tested, with tensile fracture strengths similar to that of the granite. The mean tensile strength at low rate was approximately 15.5MPa, this increased 21MPa at the intermediate loading rate and to 24.5MPa at the high rate. Thus, similar to the granite, the increase in tensile strength with loading rate was more pronounced for the low to intermediate loading rate.

Figure 13: Ultra high performance concrete tensile strength versus loading stress rate

As discussed earlier, the UHP concrete contains short steel fibers of length 12.7mm and diameter 0.2mm, which are randomly distributed throughout the concrete at 6.2% by weight. In work by Williams, et al [18], a similar steel fibre reinforced high strength concrete material, Cor-Tuf, was tested at low strain rate in direct tension both with and without the fibers. They observed a slightly lower peak stress in the material containing the fibers, which suggests that the steel fibers are not contributing to increase the peak tensile strength (rather the opposite). The lower strength is presumably because of the failure initiation at the weak interfaces introduced between the steel fibers and the concrete. However, the benefit of the steel fibres is assumably in their ability to increase the tensile post peak load bearing capacity (the ‘ductility’) of the material. This is evident by the condition of the samples after the tests reported here. Whereas the other materials investigated fractured into several pieces, typical of a brittle material response, the UHP concrete fragments stayed held together by the steel fibres, maintaining some residual strength in the sample. Isaacs, et al [1] also found a sustained load bearing post peak response for the UHP concrete, which they attributed to the steel reinforcing fibers. They conducted preliminary tests on the UHP concrete at high strain rate -1 (approximately 100s ) using a direct tension SHPB. They conducted two tests on 76.2mm diameter specimens which were notched to 57.2mm diameter. The samples were glued to the input/output bars using epoxy and the notch was required to reduce the concrete cross section area to prevent failure at the glue line (as the UHPC tensile strength is higher than available epoxies). They reported peak stresses of 16.6Mpa and 15.4MPa for the two tests. This is lower than the mean peak tensile strength of 24.7MPa at high rate reported here. This discrpeancy is due to the difference in test methodologies. The notched specimen used by Isaac, et al creates a stress concentration at the notch and so the peak stress is less than true tensile strength of the material. Whereas for the tests results reported here, the diametric compression test produces a biaxial stress state and so the induced tensile region is not pure uniaxial tension. The difference in specimen size, which affects the lateral

25 interia and confinement in the sample, may also play a role. Finally, the tensile strain rate for the tests reported here was probably lower, although it has been shown here that the strength decreases with decreasing strain rate. DISCUSSION A summary of the diametric compression results (mean values) for the five materials is shown in figure 14. This shows an increase in tensile strength with loading rate for all of the materials tested. For the mortar the increase only occurred from the intermediate to high loading rates, whereas for the granite the increase was most significant at the low to intermediate loading rates. For the SAM35 and Alcatraz concretes, which are both combinations of these two representative components, the response was a combination of these individual material responses, where the tensile strength increased fairly linearly over the total loading rate range investigated. The Alcatraz concrete was stronger than the SAM35 concrete, however, the rate of increase in tensile strength with loading rate was similar for both concretes, as shown in figure 14, where the two lines are parallel. The UHP concrete was substantially stronger than the other concretes tested, with a similar tensile strength to the granite. However, the failure response of the UHPC was much less brittle compared to the granite. The UHP concrete exhibited post fracture residual load bearing capacity due to the steel reinforcing fibres, which has also been reported by other researchers [1,18].

Figure 14: Summary of average tensile strength versus loading stress rate for the five materials tested

The material rate sensitivity for the five materials was reported using the loading stress rate. It is recognized that most rate dependent numerical model models require the material strength as a function of strain rate, not stress rate. However, it is difficult to determine the tensile strain and corresponding tensile strain rate using the diametric compression technique. The DIC technique was attempted, however, was not successful here due to the small strains in concrete before failure. The high-speed camera was setup to image the entire sample surface. It is possible that the DIC technique could be successful by zooming in on the sample center and using a smaller speckle size to improve resolution and provide greater strain measurement sensitivity. This was not pursued here. The DIC technique has been successfully used for measuring the tensile strain in diametric compression tests by [9], however, the material tested (polymer bonded sugar explosive stimulant) most likely produced larger strains than those seen in concrete.

26 It is also to possible to convert from stress rate to strain rate through the elastic modulus of the material; however, the modulus is usually obtained from quasi-static experiments and may also be rate dependent. Consequently, this conversion has not been performed here and the material tensile strength is maintained as a function of stress rate. The experimental results presented in figures 8, 9, 11-13 show significant scatter and this is a consequence of the heterogeneous nature of many of the materials tested. This results from the aggregate components in the concrete and the steel fibers in the UHP concrete. The specimen sizes used here (20mm & 25.4mm diameter) were relatively small; this was influenced by the available experimental apparatus, specifically, the diameter of the split Hopkinson pressure bars used for high rate testing. The Australian standard for quasi-static concrete testing [7] specifies specimen size requirements relative to aggregate size and recommends 150mm diameter samples where the aggregate is less than 40mm and 100mm diameter samples where the aggregate is less than 20mm.The Defense Science and Technology Organization (DSTO) in Australia are currently building a 100mm diameter split Hopkinson pressure bar system, capable of testing concrete in both direct tension and compression. Thus, high rate material characterization of larger sized concrete specimens will be the subject of continuing future work. The limited comparison between the diametric compression and direct tension test techniques performed here at high rate suggests that the agreement between the two techniques is material dependent. However, more materials need to be tested to confirm this. The different test techniques produce different stress states and so it is feasible that while one material might behave similarly under the two stress states, another material might have a different response for each stress state. The comparison for the SAM35 concrete was very good, whereas the results differed for the mortar. However, the diametric compression technique enables an estimation of the tensile response of high strength brittle materials, such as granite and the ultra-high strength concrete, which would be very difficult to test otherwise in direct tension. It also permits the use of more common compression test apparatus, particularly in the case of high rate testing using a split Hopkinson pressure bar. CONCLUSIONS The tensile strength for various concretes and their representative constituent components was determined at three loading rates using the diametric compression (Brazilian/ split-tension) technique. All five materials tested showed an increase in tensile strength with increasing loading rate. Direct tension tests at high loading rate were also performed for several of the materials. The diametric compression and direct tension test techniques produce different stress states and agreement between the two methods appears to be material dependent. However, this finding is preliminary and more tests are required to confirm this. The diametric compression test remains a useful technique. It enables tensile testing of high strength brittle materials, such as granite and ultra-high performance concrete, which are very difficult to test otherwise in direct tension. However the diametric compression technique does not provide any information on the post fracture response of the material, which is important for materials such as the ultra-high performance concrete. This requires direct tension testing. REFERENCES 1.

J. Isaacs, J. Magallanes, M. Rebentrost & G. Wight, Exploratory dynamic material characterization tests th on ultra-high performance fibre reinforced concrete, Proceedings of 8 International Conference on Shock and Impact Loads on Structures, Adelaide, Australia, 2009.

2.

H. Schuler, C. Mayrhofer & K. Thoma, Spall experiments for the measurement of the tensile strength and fracture energy of concrete at high strain rates, International Journal of Impact Engineering, Vol 32, Issue 10, pp1635-1650, 2006.

3.

E. M. Williams, S. A. Akers & P. A. Reed, Laboratory characterization of SAM-35 concrete, US Army Corps of Engineers, Engineer Research and Development Center, ERDC/GSL TR-06-15, 2006.

27 4.

B. A. Graybeal, Material Property Characterisation of Ultra-High Performance Concrete, US Department of Transportation, Federal Highway Administration, FHWA-HRT-06-103, 2006.

5.

H. J. Petroski & R. P. Ojdrovic, The concrete cylinder: stress analysis and failure modes, International Journal of Fracture, Vol 34, pp263-279, 1987.

6.

ASTM C 496/C 496M, Standard test method for splitting tensile strength of cylindrical concrete specimens, ASTM International, 2004.

7.

AS 1012.10 Methods of testing concrete, Method 10: Determination of indirect tensile strength of concrete cylinders (‘Brazil’ or splitting test), Australian Standards, 2000.

8.

Q.Z. Wang, W. Li & H. P. Xie, Dynamic split tensile test of flattened Brazilian disc of rock with SHPB setup, Journal of Mechanics of Materials, Vol 41, pp252-260, 2009.

9.

S. G. Grantham, C. R. Siviour, W. G. Proud & J. E. Field, High-strain rate Brazilian testing of an explosive stimulant using speckle metrology, Journal of Measurement Science and Technology, Vol 15, pp1867-1870, 2004.

10. G. T. Gray III, Classic Split-Hopkinson Pressure Bar Testing, ASM Handbook Volume 8 Mechanical Testing and Evaluation, 2000. 11. D. J. Frew, M. J. Forrestal & W. Chen, Pulse shaping techniques for testing brittle materials with a split Hopkinson pressure bar, Experimental Mechanics, Vol 42, No. 1, March 2002, pp93-106 12. M. L. Hughes, J. W. Tedesco & C. A. Ross, Numerical analysis of high strain rate splitting-tensile tests, Computers & Structures, Vol 47, No 4/5, pp653-671, 1993. 13. J. E. Field, S. M. Wally, W. G. Proud, H. T. Goldrein, C. R. Siviour, Review of experimental techniques for high rate deformation and shock studies, International Journal of Impact Engineering, Vol 30, pp725775, 2004. 14. T. C. Chen, W. Q. Yin, P. G. Ifju, Shrinkage measurement in concrete materials using cure reference method, Society for Experimental Mechanics, 2009. 15. C. R. Siviour, S. G. Grantham, D. M. Williamson, W. G. Proud, J. E. Field, Novel measurements of material properties at high rates of strain using speckle metrology, The Imaging Science Journal, Vol 57, pp326-332, 2009. 16. C. R. Siviour, A measurement of wave propagation in the split Hopkinson pressure bar, Measurement Science and Technology, IOP Publishing, Vol 20, 2009. 17. D. E. Lambert & C. A. Ross, Strain rate effects on dynamic fracture and strength, International Journal of Impact Engineering, Vol 24, pp985-998, 2000. 18. E. M. Williams, S. S. Graham, P. A. Reed, T. S. Rushing, Laboratory characterization of Cor-Tuf concrete with and without steel fibres, US Army Corps of Engineers, Engineer Research and Development Center, ERDC/GSL TR-09-22, 200

Influence of strain-rate and confining pressure on the shear strength of concrete Pascal FORQUIN1 1

Laboratoire d'Etude des Microstructures et de Mécanique des Matériaux (LEM3), Université Paul Verlaine - Metz, Ile du Saulcy, 57045 Metz Cedex 1, France [email protected]

ABSTRACT. The paper presents an experimental method used to investigate the shear behaviour of concrete and rock-like materials in quasi-static and dynamic loading. This method is based on the use of PunchThrough Shear (PTS) specimen and a passive confining cell. PTS sample is a short cylinder in which two cylindrical notches are performed. The displacement of the central zone beside the peripheral zone produces a shear fracture in the ligament. Metallic (steel or aluminium) confining ring allows inducing a confining pressure in the fractured zone due to the dilating behaviour of concrete under shear deformation. The experimental configuration has been designed through a series of numerical simulations in which the Drucker-Prager plasticity model is used for modelling the concrete behaviour. Computations showed the necessity to practice radial notches in the peripheral zone of the sample for deducing the radial stress in the ligament from data of strain gages glued on the outer surface of the confining ring. Finally this experimental method was employed to analyse the strain-rate and pressure sensitivity of dry and wet concrete under shear loading. 1. Introduction Concrete structures as bridges, nuclear power stations or bunkers can be exposed to intensive dynamic loadings such as earthquakes, industrial accidents or projectile-impacts. During such loading, tensile and shear damage modes as spalling, scabbing, cratering, shear fracturing can be observed [1, 2, 3, 4]. In a case of penetration of a rigid projectile in the core of a target triaxial compression and shear stresses are generated, and the inertia of the surrounding material creates a passive confinement in front of the projectile. To improve the understanding and modelling of concrete behaviour under such confined loading, quasi-static and dynamic shear tests have been developed in LEM3 laboratory. In the last decades, several experimental techniques have been used to characterise the confined or unconfined shear strength and mode II fracture toughness of concretes and rock like materials. Compression-Shear Cube tests were pioneered by Rao [5] to investigate the influence of confining pressure on the mode II fracture toughness of marble and granite. The author observed a linear increase of the stress intensity factor in mode II (KIIC) with the confining level. The Short Beam Compression test relies on the use of samples with 2 notches orientated perpendicular to the loading direction. This technique allowed investigating the quasi-static and dynamic unconfined shear strength of concretes [6, 7]. The authors noted low strain-rate sensitivity in comparison to results obtained in dynamic tension. The Punch Through Shear Test (Fig. 1a) was introduced by Watkins [8] and used more recently by Backers [9] to determine the stress intensity factor in mode II of six types of rock as function of confining pressure. The cylindrical specimen includes two cylindrical notches on top and bottom faces. In a first step, the specimen is subjected to a pure hydrostatic pressure. In a second step an axial load is added to create the shearing state. This technique was also used by Montenegro et al [10] to investigate the shear behaviour of concrete and fracture energy of T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_4, © The Society for Experimental Mechanics, Inc. 2011

29

30

concrete as function of the hydrostatic pressure (Fig. 1b). The authors noted a significant increase of strength and fracture energy with confining pressure. However, supposedly due to strong difficulties in performing dynamic tests with a hydraulic confinement, very few authors have investigated the confined shear strength of concrete and rock-like materials in a wide range of strain-rate. In the present work, in a similar way than in quasi-oedometric compression tests [11, 12] a passive confining cell has been applied to Punch-Through-Shear specimens to characterise the confined shear strength of concrete over a wide range of strain-rates (Fig. 1c). Inner diameter of the lower notch and outer diameter of upper notch coincide so a straight cylindrical fracture surface is obtained (Fig. 1c). Numerical simulations of shear testing, experimental method, data processing and some experimental results are detailed in the present paper. Compression crown Confining cell Shear surface Compression (c) plug (b) Fig. 1. Punch-Through-Shear tests (a) conducted by Backers [9], (b) by Montenegro et al. [10] (c) and in the present work.

(a)

2. Experimental method 2.1. Numerical simulation of PTS tests 2D-axisymetric (Fig. 2a) and 3D (Fig. 2b) numerical simulations of shear tests have been performed to set the dimensions of the confining cell and concrete sample. The Drucker-Prager model was used for the specimen to describe the pressure sensitivity and dilation behaviour of concrete. The length of the ligament (10 mm) allows ensuring an almost homogeneous shear stress field in the ligament. Moreover the confining ring is not centred toward the symmetry plane of the specimen to get a uniform pressure between the confining ring and the concrete sample (Fig. 2a). Finally, aluminium and steel confining rings 10 mm thick and 15 mm in height have been considered for quasi-static and dynamic experiments.

Specimen

Crown Crown

Confining ring

Confining ring

(a)

Radial notch

Fig. 2. (a) Mesh of 2D axisymetric computation, (b) Mesh of 3D computation.

(b)

31

Furthermore, 3D computations (Fig. 2b) have been performed to investigate the influence of radial notches on the stress fields in the sample. It was demonstrated that 4 radial notches at 90° through the outer part of the specimen allow avoiding a self confinement of the sample and any radial cracks triggered during the test. 2.2. Experimental procedure The experimental devices used in quasi-static and dynamic shear testing are presented on Fig. 3a and 3b. Prior to each test a bi-component resin (Chrysor®) has been used to fill the gap between the sample and the confining ring. Moreover, the mean radial stress in the ligament (sheared zone) is deduced from the contact force between the sample and the confining vessel knowing the contact surface between the specimen and the ring (Scontact) and the area of the fracture surface (Sligament):

σ radial =

S contact Pcontact , Sligament

(1)

Radial notches

Strain gauges placed on the confining cell

(a)

(b)

LVDT Concrete sample Damping system

Laser interferometer

(c)

(d)

Fig. 3. Shear tests performed on R30A7 standard concrete. (a) Concrete specimen with radial notches after shear testing, (b) Concrete sample without radial notches after shear testing, (c) Device used in quasi-static shear testing, (d) Device used in dynamic shear testing and high-speed hydraulic press (LEM3). In a similar way than in quasi-oedometric test [11, 12] the contact pressure between the sample and the confining ring is deduced from data of strain gages glued on the metallic cell (Fig. 3a and 3b). A high-speed hydraulic press has been used to perform shear tests for jack speed above 0.01 m/s. Above this loading rate, a damping system is included (Fig. 3d) to assure a correct balance of the specimen (equal input and output forces) and to avoid any shock waves. In dynamic tests laser extensometers have been used to deduce the axial displacement of the central part of the

32

specimen. The mean shear stress in the ligament is obtained from the axial load according to Equation (2):

σ shear =

Faxial Sligament

(2)

2.3. Tested concrete Basic mechanical properties and composition of the tested concrete are gathered in Table 1. R30A7 concrete is a standard concrete containing hard siliceous aggregates with a grain size from 2 to 8 mm, sand, cement and water. All the specimens have been cored out from large blocks and stored in water saturated by lime in order to avoid the dissolution of portlandite in water. Two sets of specimens have been considered for shear testing: saturated specimens have been kept in water until few minutes before testing. “Dry” specimens were oven-dried at 60ºC during several weeks until a constant weight has been achieved. Table 1. Composition and quasi-static compression strength of dry and wet R30A7 concretes Composition

R30A7 [13]

Aggregates [kg/m3]

1008

Sand [kg/m3]

838

Cement [kg/m3]

263

Water [kg/m3]

169

Water/Cement

0.64

Mechanical properties

Wet R30A7

Dry R30A7

2380

2290

Young modulus (GPa) [14]

40

30

Quasi-static compression strength (MPa) [13]

32

42

Density

3. Measurement results 3.1. Influence of lateral pressure A series of confined shear experiments has been conducted with both confining rings (aluminium or steel rings) and varying the concrete moisture and loading rate. A comparison of shear tests performed with dry samples is proposed on Fig. 4. After the stress peak, a strong softening behaviour is noted followed by a plateau. Similar trends have been reported in hydraulic confinement PTS experiments [10]. The plateau results supposedly from the friction on the fractured surface. Moreover the lateral pressure between the confining cell and the specimen deduced from data of strain gages is also plotted. The resulting radial stress in the shear zone is computed from Eq. 1. According to Fig. 4 the maximum radial stress reaches about 52 MPa (in absolute value) with the steel confining ring whereas it does not exceed 38 MPa with the aluminium cell. This difference of confining pressure may explain the difference in shear strength observed between both tests. A confirmation that pressure sensitivity of concrete under shear loading may be explored through PTS experiments with passive confining cells varying the stiffness of the ring.

33

(b) (a) Fig. 4. Results of quasi-static shear tests performed on dry specimens with (a) aluminium and (b) steel confining cells. Strain-rate: 2e-5/s, concrete samples with radial notches. 3.2. Influence of moisture content Experimental results obtained with dry and wet specimens in quasi-static loading with an aluminium alloy confining vessel are reported on Fig. 5. In wet specimen the maximum shear stress reached 41 MPa whereas a maximum value of 54 MPa was obtained in dry sample. Again a strong decrease of shear stress is observed after the stress peak. Again the change of lateral pressure between the confining cell and the specimen is reported on the same plot as well as the resulting radial stress in the shear zone. Oppositely to Fig. 4 a similar radial stress about 30 MPa (in absolute value) is observed in both tests (dry and wet specimens) in the shear zone when the peak shear stress is reached.

(a) (b) Fig. 5. Results of quasi-static shear tests performed on dry (a) and wet (b) specimens. Strain-rate: 2e-5/s, aluminium confining cell, concrete samples with radial notches. 3.3. Influence of strain-rate A series of shear tests has been performed with dry and wet samples and the aluminium confining ring varying the strain-rate from 2e-5/s to 4/s. Experimental results are gathered on Fig. 6. The shear strength of wet samples is markedly lower than that of dry specimens whatever the loading rate.

34

Furthermore, a very limited increase of strength is pointed out for both sets of specimens (dry and wet sets). This result significantly differs to the conclusion obtained for concrete under tensile loading. Indeed, as shown by several authors [4, 14, 15] the tensile strength of wet concrete is double in a similar strain-rate range (1e-6/s - 1/s) whereas a quasi-nil rate-effect is noted with dry specimens. Thus, the well-known Stephan effect supposed to explicate the strain-rate sensitivity of wet concrete in tension is supposedly inoperative under shear loading. 70 y = 64,36x0,017

Maximum shear stress (MPa)

60

y = 46,99x0,017

50 40 30 20 10

Dry R30A7 concrete Wet R30A7 concrete

0 1,0E-05

1,0E-04

1,0E-03

1,0E-02

1,0E-01

1,0E+00

1,0E+01

Strain-rate (1/s)

Fig. 6. Quasi-static and dynamic shear tests performed on dry and wet specimens (aluminium confining cell, concrete samples with radial notches). Summary An experimental method is proposed to investigate the shear behaviour of concrete under quasistatic and dynamic loading. The set-up based on PST (Punch Through Shear) testing technique relies on the use of a passive confining ring. Radial notches are also performed on each sample so the radial stress in the ligament may be deduced from the measurement of strain-gauges glued on the confining ring skirting around the effect of self confinement due to the peripheral part of the specimen. Geometries of the specimen and of the ring have been defined through a series of numerical simulations. Experiments have been conducted on dry and water saturated concrete samples over a large range of strain-rate with two confining cells (steel and aluminum rings). The obtained results show a higher strength with dry samples than in wet ones and with the steel ring than with the aluminum ring. Furthermore, oppositely to previous results obtained in the literature in tension, both sets of concrete (dry and wet sets) show very small strain-rate sensitivity in the considered range of strainrate. The so-called Stephan effect is though to be inoperative under shear loading. References [1] Li Q.M., Reid S.R., Wen H.M. (2005), Telford A.R., Local impact effects of hard missiles on concrete targets, Int J. Impact Eng. 32, 224-284. [2] Forquin P., Arias A., Zaera R. (2008), Role of porosity in controlling the mechanical and impact behaviours of cement-based materials, Int. J. Impact Eng., 35 (3), 133-146. [3] Forquin P., Hild F. (2008), Dynamic Fragmentation of an Ultra-High Strength Concrete during Edge-On Impact Tests, ASCE J Eng Mech, 134 (4), 302–315.

35

[4] Forquin P., Erzar B. (2010), Dynamic fragmentation process in concrete under impact and spalling tests, Int. J. Fracture, 163 : 193 – 215. [5] Rao Q. (1999), Pure shear fracture of brittle rock. Doctoral thesis, Division of rock Mechanics, Lulea University, Sweden. [6] Watkins J., Liu K.L.W. (1985), A finite element study of the short beam test specimen under mode II loading. Int. J. Cement Composites and Lightweight. 7, 39-47. [7] Ross C. A., Jerome D.M., Tedesco .J.W, et Hughes M. (1996), Moisture and Strain Rate on Concrete Strenght. ACI Material Journal 93. 293-299 [8] Watkins J. (1983), Fracture toughness test for soil-cement samples in mode II. Int. J. Fract. 23, 135-138. [9] Backers T. (2004) Fracture Toughness Determination and Micromechanics of Rock Under Mode I and Mode II Loading. Ph.D. thesis, University of Potsdam, Germany. [10] Montenegro O., Sfer, Carol I. (2007), Characterization of concrete in mixed mode fracture under confined conditions. ICEM13 conference, Alexandroupolis, Greece [11] Forquin P., A. Arias, R. Zaera. (2007), An experimental method of measuring the confined compression strength of geomaterials, Int. J. Solids Struct., 44 (13), pp. 4291-4317. [12] Forquin P., G. Gary, F. (2008b), Gatuingt. A testing technique for concrete under confinement at high rates of strain, Int. J. Impact Eng., 35 (6), 425-446. [13] Vu X.H., Malecot Y., Daudeville L., Buzaud E. (2009), Experimental analysis of concrete behavior under high confinement: effect of the saturation ratio, Int. J. Solids Struct., 46 1105-1120. [14] Erzar B., Forquin P. (2010), Experiments and mesoscopic modelling of dynamic testing of concrete, Mechanics of Materials (submitted for publication). [15] Cadoni E., Labibes K., Albertini C., Berr M., Giangrasso, M. (2001), Strain-rate effect on the tensile behaviour of concrete at different relative humidity levels, Materials Structures, 34, 21-26.

Dynamic Tensile Properties of Steel Fiber Reinforced Concrete

R. Chen1*, Y Liu2, X. Guo2, K. Xia3, and F. Lu1 1. College of Science, National University of Defense Technology, 410073 Changsha, P.R. China 2. National Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, 100081 Beijing, P.R. China 3. Department of Civil Engineering, University of Toronto, Toronto, Ontario, Canada M5S 1A4 ABSTRACT: This paper presents experimental results on three kinds of concretes, plain concrete (PC), 1.5% and 3% steel fiber reinforced concrete (SFRC), subjected to dynamic tensile loading. The cylinder splitting (Brazilian disc) specimens are loaded by a modified Split Hopkinson Pressure Bar (SHPB) with various loading rates (100~500 GPa/s). From the experiments it is found that there is a significant enhancement in tensile strength with increasing loading rates. Crack gauges mounted on the specimen showed that the average fracture velocity of 3% SFRC during the test is 730 m/s whereas that of PC is 790 m/s. Both the tensile stress history and the recovered specimen have demonstrated that SFRC has superior resistance to crack initiation and crack propagation as compared with PC. Keywords: Steel fiber reinforced concrete (SFRC); Brazilian Disc; SHPB; Fracture velocity INTRODUCTION Fiber reinforcement is one of the most important modification methods to alter the brittle nature of plain concrete (PC). The use of steel fiber reinforced concrete (SFRC) has been continuously increasing during the past decades because of its enhancement of material performance in toughness and crack control [1]. With the addition of steel fibers, SFRC shows an enormous increase in strength, toughness, and ductility from static mechanical tests [2]. As a result, SFRC has been widely used in many civil engineering structures. However due to its limitations in the flexibility and resistance to shrinkage cracking, it has been rarely used in bridge pavement. The success of SFRC in structural engineering has encouraged the development of newer high performance materials for critical infrastructures subjected to extreme loadings. Among these loading cases, intense dynamic loading is a unique one because of the worldwide increase of terrorist attacks against civilian targets. It is thus critical to better understand the impact resistance of SFRC and methods to enhance its performance under such loadings [3]. Over the past few years, significant progress has been made in the characterization of dynamic properties of SFRC [4-7]. However, the tensile strength has rarely been measured. It is thus the objective of this work to quantify the dynamics tensile behaviour of PC and SFRC. We use the Brazilian disc sample and apply the dynamic load with a modified split Hopkinson pressure bar (SHPB) system. The influence of the loading rate and fiber volume fraction on dynamic tensile strength is studied. To illustrate the tensile failure process, crack velocities are monitored. EXPERIMENT Sample preparation and geometry The concrete matrixes were designed for accommodating the volume fraction (Vf) of 0, 1.5% and 3% of steel fibers. The following materials were used in the fabrication of SFRC specimens: tap water, cement, steel fiber, standard sand, Silica fume, fly ash, water reducer and steel fiber. Tables 1 and 2 present the properties of raw material and the compositions of each component.

*

Corresponding author. Tel: +8673184573276; fax: +8673184573297. E-mail address:[email protected]

T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_5, © The Society for Experimental Mechanics, Inc. 2011

37

38 Table1. The properties and manufactures of raw material Properties Cement Standard Sand Silica Fume Fly Ash Water Reducer Steel Fiber

Manufacture

Description

Density g/cm3

Grade 42.5 Dia.: 0.25-0.65mm 1250 Head Type I by JGJ28-86 Poly Ling salt water reducer Dia.: 0.15-0.2mm Average length: 15mm

2.59 2.15 2.2 1.06

Huxin Cement Co., Ltd. Xiamen ISO Standard Sand Co., Ltd. GaofengMiners Powder Co., Ltd. Chonghui Fly Ash Co., Ltd. JinSheng Tci. and Tech. Co., Ltd

7.8

Xintu Engineering Fiber Co., Ltd.

Table2 The compositions of SFRC/PC Water (W/B) 1# 2# 3#

0.2 0.2 0.2

Cement (C) 1 1 1

Binder (B) Silica Fume (SF/C) 0.3 0.3 0.3

Fly Ash (FA/C) 0.25 0.25 0.25

Sand (S/B)

Water Reducer (Solid Content)

Steel Fiber (Vf)

1.18 1.18 1.18

1% 1% 1%

0 1.5% 3.0%

The mixture was poured into a steel cubic mold (300 mm) for 2 hours. The block obtained was then cured in a standard condition of 20 ℃ and >95% relative humidity for 28 days. After curing, concrete cores with a nominal diameter of 50 mm were drilled from the block and then sliced to obtain discs with an average thickness of 24 mm. All the disc samples were polished afterwards resulting in a surface roughness variation of less than 0.5% of the sample thickness. The modified Split Hopkinson Pressure Bar system A 40 mm diameter SHPB system was employed as the loading apparatus in this study, as shown in Fig. 1. When the striker hit the incident bar, it generates an incident pulse in the incident bar. The incident wave travels through the flange without resistance because there is a gap between the flange and the rigid mass in the beginning. The incident pulse propagates along the incident bar to hit the sample, leading to reflected stress wave and transmitted stress wave. Denote the incident wave, reflected wave and transmitted wave by εi, εr, and εt. Based on the one-dimensional stress theory, and assuming the stress equilibrium prevails during dynamic loading (i.e., εi +εr =εt), we can determine the histories of the tensile stress σ(t) within the sample as [8]：

σ (t ) =

A0 E ε (t ) πRB 0 t

where E0 is the Young’s modulus of the bar, and A0 is the area of the bar; R is the radius of the sample and B is the thickness of the sample. Rigid mass Striker

Incident bar

Specimen

Transimitted bar

Absorbtion bar

Crack Gauge

Pulse shaper Flange

Fig. 1. Schematics of the preloaded spit Hopkinson pressure bar (SHPB) system with the Brazilian disc specimen. In order to ensure single pulse loading, the momentum trap technique is adopted in our Hopkinson bar setup. Detailed

39 explanation of the SHPB procedure and momentum trap can be found in the literature [9, 10]. EXPERIMENT RESULT Fig. 2 shows the original and recovered SFRC specimens. The recovered crack gauge showed that the crack travels along the crack gauge. It also can be observed as the steel fibers are pulled-out from the fragments.

Fig. 2 The virgin and recovered SFRC specimen. Dynamic force balance Fig. 3 shows the forces on both ends of the sample. The dynamic force on one side of the sample is the sum of the incident and reflected force waves, and the dynamic force on the other side of the sample is the transmitted force wave. It can be seen from Fig. 2 that the dynamic forces on both sides of the samples are almost identical during the whole dynamic loading period. We thus can use the equation mentioned above to determine the tensile strength. 300

Force (kN)

200 100 0 -100

In. Tr. Re. In.+Re.

-200 -300

0

40

80

120

160

Time (μs) Fig. 3. Dynamic force balance during a typical preloaded SHPB with the Brazilian disc specimen.

Effects of loading rate The tensile strength is the maximum value of the tensile stress history. There is a approximately linear region in σ(t), and its slope is taken as the fracture loading rate, with which σ varies. Fig. 4 shows some examples of stress versus time curves for different loading rates and two materials. One can observe how the maximum strength is reached for the three loading rates in different time due to the imposed loading rate. In the case of PC, the duration of the fracture process is about 120, 90 and 60 μs for 142, 253 and 409 GPa/s, respectively; while for the 1.5% FRC, it has been obtained 120, 100, 80 and 68 μs for 192, 233, 340 and 0.449 GPa/s, respectively. Fig.5 shows stress histories of different types of specimens with similar loading rate of 330 GPa/s. The tensile strength of the plain concrete is 14.7 MPa, while the one of 1.5% SFRC and 3% SFRC are 16.9 MPa and 19.3 MPa respectively. The postpeak ductility in the stress-time curve for the test of SFRC specimen shows the effect of reinforced fibers, as the steel fibers are being pulled-out from the materials. In addition, the post-peak ductility increases with increasing fiber volume fractions.

40 Fig.6 shows the conclusion of our tests. The tensile strength increase with increasing loading rates and fiber volume fractions. The slops of different materials are similar to each other.

20

25

Vf=0

142

253

409

12 8

Loading Rate (GPa/s)

192

232

340

449

15 10 5

4 0

Vf=1.5%

20

Stress (Mpa)

16

Stress (Mpa)

Loading Rate (GPa/s)

0

0

40

80

120

160

200

0

50

100

150

200

250

Time (μs) Time (μs) (a) (b) Fig. 4. Stress versus time curves for different loading rates: (a) PC; (b) 1.5% SFRC

25 Loading Rate: ~330GPa/s Vf : 0 1.5%

Stress (Mpa)

20

3%

15 10 5 0

0

50

100

150

200

250

Time (μs)

Fig. 5. Dynamic tensile histories of SFRC/PC specimens with various volume fractions. 22.5 0 1.5% 3.0%

Strength (Mpa)

20.0 17.5 15.0 12.5 10.0

0

100

200

300

400

500

600

Loading Rate (GPa/s)

Fig. 6. Tensile strength increase with increasing loading rates and fiber volume fractions. Fracture velocity A crack gauge was mounted on the specimen during the test. The typical signal of the crack gauge and corresponding loading history are shown in Fig. 7. There is a significant post-peak ductility after the crack travels through the specimen, which indicates the pull-out of the fiber from the concrete matrix. The details of the crack gauge can be seen from the insert of Fig.

41 7. The distance between each wile in the crack gauge is 1.1 mm, and the time step can be measured by the jump of the crack gauge signal. From which, the crack velocity can be measured for each test, as shown in Fig. 8(a). The average crack velocities of three types of specimens are shown in Fig. 8(b). The average crack velocity of PC is 790 m/s where as the ones of 3% SFRC is 730 m/s. The phenomenon demonstrates that one of the important properties of SFRC is its superior resistance to cracking and crack propagation, which is also proved by both the tensile stress history and the recovered specimen. 5

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(a) (b) Fig. 8. The crack velocities of different conditions: (a) the crack velocity of each test, and (b) the effects of fibers’ volume fraction. CONCLUSION We present the dynamic tensile properties of PC, 1.5% SFRC and 3% SFRC with the loading rates from 100 to 500 GPa/s. The experiments were conducted on the modified SHPB system with the Brazilian disc specimens. The results show that there is a significant enhancement in tensile strength with increasing loading rates and the volume of steel fiber. Crack gauges mounted on the specimen show that the average fracture velocity of SFRC during the test is 730 m/s whereas that of PC is 790 m/s. The result demonstrates the fibers do substantially increase the post-cracking ductility, or energy absorption of the material. ACKNOWLEDGMENTS This work was supported by the open fund of National Key Laboratory of Explosion Science and Technology through Grant No. KFJJ08-1, and the Natural Science Foundation of China (NSFC) through Grant No. 10872215 & 10902100. K.X. acknowledges the support by Natural Sciences and Engineering Research Council of Canada (NSERC) through Discovery Grant No. 72031326.

42 REFERENCE [1] Daniel J.I., Gopalaratnam V.S., Galinat M.A., et al., "Report on Fiber Reinforced Concrete", (2002). [2] Cadoni E., Meda A., and Plizzari G.A., "Tensile behaviour of FRC under high strain-rate", Materials and Structures. 42(9): 1283-1294 (2009). [3] Bindiganavile V. and Banthia N., "Generating dynamic crack growth resistance curves for fiber reinforced concrete", Experimental Mechanics. 45(2): 112-122 (2005). [4] Wang Z.L., Wu L.P., and Wang J.G., "A study of constitutive relation and dynamic failure for SFRC in compression", Construction and Building Materials. 24(8): 1358-1363 (2010). [5] Wang Z.L., Liu Y.S., and Shen R.F., "Stress-strain relationship of steel fiber-reinforced concrete under dynamic compression", Construction and Building Materials. 22(5): 811-819 (2008). [6] Lok T.S. and Zhao P.J., "Impact response of steel fiber-reinforced concrete using a split Hopkinson pressure bar", Journal of Materials in Civil Engineering. 16(1): 54-59 (2004). [7] Liu Y.S. and Chen M.C., "Study on mechanical properties of ultra-short steel fiber reinforced concrete under dynamic compression". in: Vol. Innovation & Sustainability of Modern Railway Proceedings of Ismr' 2008. 240-247 (2008). [8] Berenbaum R. and Brodie I., "Measurement of the tensile strength of brittle materials", British Journal of Applied Physics. 10(6): 281-286 (1959). [9] Xia K., Nasseri M.H.B., Mohanty B., et al., "Effects of microstructures on dynamic compression of barre granite", International Journal of Rock Mechanics and Mining Sciences. 45(6): 879-887 (2008). [10] Chen W.W. and Song B., "Split Hopkinson (Kolsky) Bar Design, Testing and Applications", Mechanical Engineering Series. 388 (2010).

Effect of Liquid Environment on Dynamic Constitutive Response of Reinforced Gels Sashank Padamati, Vijaya B. Chalivendra*, Animesh Agarwal, Paul D. Calvert Dynamic Material Testing Laboratory University of Massachusetts, Dartmouth, MA, 02747 Corresponding author: [email protected] ABSTRACT Quasi-static and dynamic compressive behavior of three different types of hydrogels used for soft tissue applications are tested using a modified split Hopkinson pressure bar. Three kinds of hydrogels: (a) Epoxy hydrogels, (b) Epoxy hydrogels reinforced with definite orientation of three-dimensional polyurethane fibers and (c) fumed silica nano particles reinforced hydrogels with different weight fractions are considered in this study. Swellability of the all the hydrogels considered are studied and controlled by mixing different ratios of jeffamines and epoxides. The three dimensional pattern of the fibers are generated by a rapid robo-casting technique. Split Hopkinson pressure bar (SHPB) was used for dynamic loading and a pulse shaping technique was used to increase the rising time of the incident pulse to obtain dynamic stress equilibrium. A novel liquid environment technique was implemented to observe the dynamic behavior of hydrogels when immersed in water. Experiments were carried out at dynamic loading conditions for different strain rates with and without water environment. Results show that the hydrogels are rate sensitive. Also the yield strength of hydrogels decreased and elongation percent increased when they were immersed in water. INTRODUCTION Mimicking human tissues with artificial materials which are bio-compatible and bio-degradable are of vast interest. Hydrogels are extensively used in biomedical applications [1] especially in tissue engineering. They are used as scaffolds to guide the growth of new tissues. Hydrogels are highly corrosion-resistant and their swelling nature provides an aqueous environment comparable to soft tissues. Hydrogels are water swollen polymer networks, which have a tendency of absorbing water when placed in aqueous environment. As the hydrogels tend to swell when fully saturated, these are suitable for biological conditions and are ideal in use of drug delivery. These are hydrophilic polymers that can absorb up to 1000 times their dry weight in water. This high water content makes it a resemblance to living tissues. When hydrogels are completely saturated with water they tend to have poor mechanical properties. Usage of hydrogels as biological material in different parts of human body is increasing at a rapid pace. With a huge advantage of varying the properties, hydrogels are being used as a replacement for many damaged parts of the body. One example of such a part is human cartilage, which is subjected to high amounts of loads. Recent studies show many researchers have tested and published the mechanical properties of the hydrogels in Quasi-static compression state. Testing soft materials for the dynamic behavior has been a challenging task. Many researchers have worked on soft materials and have achieved success in testing them. Mechanical properties of soft materials like lungs, stomach, heart and liver have been found out at dynamic loading conditions [2]. Dynamic compression tests using polymer split Hopkinson pressure bar (SHPB) apparatus were performed on bovine tissues [3]. PAG gels, an alternative for tissue stimulant has been tested for the dynamic response [4]. Subhash et al., has found the compressive strain rate behavior of ballistic gelatin. All the above testing was performed under open to air environment. Most of the tissues in human body are surrounded with some type of aqueous environment. This environment change in the body will possess the tissue to behave with different properties. Hence testing the hydrogels in open air environment might vary their properties as compared to aqueous environment. In this paper a novel technique for testing the hydrogels under aqueous environment in both static and dynamic loading conditions were designed, developed and implemented. Attempts were made to increase the mechanical properties of hydrogels by adding glass fibers and were partially successful [6]. In this study three-dimensional polyurethane fibers and fumed silica nano particles are reinforced into different hydrogels to improve low mechanical properties. A novel liquid environment chamber built with acrylic glass is developed for testing hydrogels in water. The need of testing materials at high strain rates developed a method which produced mechanical properties of materials at elevated strain rates which in recent time very well known as split Hopkinson pressure bar. This method was widely used for

T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_6, © The Society for Experimental Mechanics, Inc. 2011

43

44 testing materials such as metals as they have high impedance values. However, soft materials like foams and hydrogels have very low impedance values which resist the use of SHPB. The main disadvantage for dynamic testing of soft materials using SHPB arises in capturing the transmitted pulse. Soft materials such as hydrogels have extremely low impedance values, which transmits a very low magnitude pulse into the transmission bar. This makes it very hard to capture the transmitted pulse. In literature, few approaches were adopted to magnify the transmitted pulse, (i) Using a hollow transmission bar Song and Chen, 2004, this technique magnifies the magnitude of the transmitted pulse up to 10 times. However, the magnitude of the pulse in case of hydrogels is so low when compared to incident pulse increasing it 10 times still makes it barely visible, (ii) Using polymeric bars Liu and Subhash, 2006, these bars can provide decent transmitted pulse but it requires lots of assumptions. The wave attenuation and dispersion in the bars requires idealized assumptions about the bar material properties and extensive mathematical treatment of pulses should be obtained in the experiments. To overcome this issue, foil gages were replaced with semiconductor gages on input and output bars. Semiconductor gages have a very high gage factor and these gages have 50-75 times more sensitivity than normal foil gages [4]. This technique is vastly used for characterizing ultra soft and very low impedance materials. Biological materials such as tissues, soft bones and hydrogels are some materials which have been using this technique in recent times. In this paper semiconductor gages are used for data acquisition. It can be noticed from the above studies that characterization of hydrogels was successfully studied in open to air environment. It was identified from the above studies that there was no detailed study conducted to understand the effect of liquid environment on hydrogels under static and dynamic loading conditions. Hence this paper mainly focused on studying the quasi-static and dynamic constitutive behavior of reinforced hydrogels inside liquid environment. EXPERIMENTAL DETAILS Materials and preparation Three types of hydrogels were considered in this study and are epoxy-amine based. Three kinds of hydrogels are (a) Epoxy hydrogels, (b) Three dimensional Polyurethane fiber reinforced hydrogels and (c) Nano particle reinforced hydrogels. Polyethyleneglycol diglycidylether(PEGDGE) was purchased from Sigma Aldrich. Jeffamine® ED 600 is donated from Huntsman Chemical Company. 15 Minutes Degassing

PEGDGE

ED 600

Mould Water

3 hours

24 hours

48 hours

Oven at 700C

Curing at room temperature

Curing in Deionized water

Fully swelled Hydrogel

Figure 1 Material preparation procedure Amines were cross linked to epoxides in aqueous solution to form hydrogels. An equal molar ratio of PEGDGE and ED 600 are mixed together with 20-30% weight fraction of H2O. The entire mixture is stirred at a regular pace for 15 min. Thus formed mixture is poured in to acrylic molds and placed in oven at 65-700C for 3 hours. These molds are then cured at room temperature for 24 hours. These samples are placed in de-ionized water for 48 hours to completely swell to get fully saturated epoxy amine hydrogels. Figure 1 shows the pictorial view of the material preparation

Figure 2 Asymtek dispensing system

Figure 3 Three dimensional optical image of PU fiber construct

45 Figure 2 shows a dispensing system machine were 3D fibers are extruded. This machine is produced from Asymteck. As Figure 2 shows the machine consists of a disposable syringe, a needle and a gas pressure apparatus to pump the solution on to the surface. Required patterns of fibers can be extruded from the machine through a computer program. The machine moves in all the three directions which make it more flexible to generate complex design patterns.10-20% polyurethane solution were prepared in dimethylformamide(DMF). The fibers are extruded through 100µ EFD needle and deposited into 3D structure by pressure driven Asymteck Automove 402 dispensing system as shown in Figure 3. The fiber formation process is based on wet spinning process and deposition technique is similar to robocasting technique. These 3D structures are placed in to hydrogel matrix and the solution is ultrasonicated to form fiber reinforced hydrogel. Fumed silica nano particles were purchased from Sigma Aldrich. Nano particles are measured for weight and are mixed with epoxy amine hydrogel. This mixture is ultrasonicated in a water bath up to 10 min allowing the particles to disperse. In this paper 2%, 4% and 6% are three different weight fractions of particles used.

Figure 4 Split Hopkinson pressure bar The split Hopkinson pressure bar was used for dynamic testing of hydrogels. Traditionally SHPB consists of an incident bar and a transmission bar and are all made of Aluminum 7075-T651. The striker bar used in these experiments has a diameter of 12.7mm and length 304.79mm. Incident and transmission bars have the diameter of 12.7 mm. Incident bar is 1828.80mm long and transmission bar is up to a length of 1220mm. A Copper C11000 pulse shaper was used in the testing to obtain dynamic stress equilibrium and is placed using KY jelly at the impact end of the incident bar as shown in Figure 4. The specimen is sandwiched between incident bar and transmission bar. Specimen has the thickness of 2.36mm and diameter of 6.35mm. Molybdenum disulfide lubricant is applied between specimen and the contacting surfaces of bars to minimize the friction. When the striker bar impacts the incident bar, an elastic compressive pulse is generated. This pulse deforms the pulse shaper mounted at the incident bar and creates a ramp in the pulse. This pulse propagates through the incident bar and reaches the specimen bar interface. When it reaches the interface due to impedance mismatch between the bar and specimen some part of the pulse (reflected pulse) is reflected back in to the incident bar and some transmits through the specimen (transmission pulse) to the transmission bar. Strain gages are mounted on both the bars which provide time-resolved measures of the elastic strain in the pulses. For softer materials strain gages were replaced with semi conductor gages as the gage factor of semiconductor gage are very high when compared to foil gages which gives a very high sensitivity to semiconductor gages. Typical pulses obtained from SHPB using semiconductor gages are shown in Figure 5.

Figure 5 Typical pulses obtained from SHPB

Figure 6 Liquid environment specimen fixture

46 A novel environmental chamber was designed for SHPB at the interface of incident and transmission bar as shown in Figure 6. This fixture ensures that specimen is placed in between bars and an aqueous environment is surrounded around the specimen during the testing time. The fixture is made of acrylic glass has rubber washers fixed in both the bars ensuring no leak of aqueous solution. Sample tests were ran with the liquid environment chamber and ensured that the setup doesn’t restrict the movement of bars. The specimen undergoes uniform homogenous deformation and the analysis based on one-dimensional wave theory. In order to make the experiment valid when testing soft materials the specimen should be in dynamic equilibrium and should have a near constant strain rate. In this paper, using SHPB we obtained a dynamic stress-strain behavior of hydrogels at a constant strain rate range of 3600-4000/s.

(a)

(b)

Figure 7 Specimen holders in Quasi-static compression (a) without water (b) with water conditions The quasi-static compression tests are performed on Instron Materials Testing Machine 5585. Experiments were performed at slow loading rates of 1mm/min. The tests were continued until the specimen is completely crushed at specified loading rate. A setup was also built for testing the hydrogels under water in quasi-static loading conditions. Figure 7 shows the liquid environmental setup built for quasi-static compression experiments RESULTS AND DISCUSSION

5

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As discussed above a novel environmental chamber was implemented in this study, consistency experiments were ran and the results are showed in Figure 8. Results show that the methods implemented and the material behavior in all tested cases remained same.

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47 As the hydrogels are soft materials dynamic testing of these materials is a challenging task. The data that we obtain from the tests should be trustable. So, experiments for consistency were performed on all the hydrogels. Currently 2% nano particle hydrogels were randomly considered and presented in this paper. 1.2

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Figure 9 Comparison of True Stress vs. True Strain curves under quasi-static compression (a) epoxy and nano composite hydrogel (b) fiber reinforced hydrogel Figure 9(a) shows the comparison of the data obtained from epoxy gels and nano particle gels. Figure 9 clearly indicates the increase of strength of material of pure epoxy gels as the nano particles are added to it. We can also observe that the rate of increase of stress of the material has a significant change when tested in with and without water conditions. When tested in water the material has a low rate of increase and higher elongation percentages. These results conforms that the materials such as hydrogels with a water content of 95% as their volume subjects to a different behavior when tested in different environmental conditions. Figure 9(b) shows the comparison of fiber reinforced hydrogels in with and without water environment condition. The curves shows that the material behavior is non linear. The compressive stress of the material increased with the increase in strain percent. But the flow stress in this case has a highly nonlinear effect when compared to epoxy and nano particle gels. The non linearity in the material behavior is due to the fibrous structure reinforcement in the hydrogels matrix. The fibers reinforced in the hydrogel when subjected to compression show more stability towards the material allowing it to take higher loads. The material when tested in normal atmospheric conditions has yield strength of 4MPa and has percentage elongation of 70%. Some resistance was observed around the specimen as a result yield strength of the material decreased increasing the percentage elongation of the specimen. 1.8

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Figure 10 Comparison of True Stress vs. True Strain curves of hydrogels considered under fatigue loading

48 From the quasi-static experimental data we observed that the data obtained for epoxy gels and nano particle reinforced gels has a linear increase in the stress in both the experimental conditions. And the fiber reinforced hydrogels had a highly non linear increase. So, to confirm the results and to see if there is any hysteresis effect in these specimens fatigue loading experiments was conducted. Fatigue loading was done at a rate of 1 mm/min and up to 10 cycles. The specimens were subjected to an elongation of 20%. Figure 10 shows the results of fatigue loading on all the hydrogels considered in this study. From the figure we can see that there is no significant hysteresis observed in epoxy and nano hydrogels. But the fiber reinforced gels have been subjected to hysteresis confirming the non linear behavior observed in the quasi-static experiments. 6

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Figure 11 Comparison of True Stress vs. True Strain curves under dynamic compression on epoxy and nano composite hydrogel (a) without water (b) with water conditions Figure 11(a) shows the dynamic true stress- strain curves for a constant strain rate plotted against the quasi-static true stressstrain curves for each hydrogel specimen without water conditions. It can be seen from figures that all the hydrogels are rate sensitive and show significant rate sensitivity. The constitutive response in all the above cases is non linear. The rate of increase of yield stress in case of without water experiments is very high. The yield strength of the material is reached with lesser percentage elongation. It can be observed from the graphs that the flow stress has a significant change when compared to the quasi-static conditions. Figure 11(b) shows the dynamic true stress- strain curves for a constant strain rate plotted against the quasi-static true stressstrain curves for each hydrogel specimen with water conditions. It can be seen from figure that the materials even when immersed in water are rate sensitive and show significant rate sensitivity. The constitutive response of the materials when immersed in water shows a high non linearity. The pulses when compared to that of regular without water experiments showed a significant change in the strength, elongation and signature of the pulses. The strength of the materials decreased, percentage elongation of the materials increased when tested in water. The reason for this effect in the material as mentioned in the above sections, a plasticizing effect is observed by the material when they are immersed in water and tested. The rate of increase of yield stress in case of with water experiments is less when compared to without water. As mentioned the water experiments produce plasticizing effect, as a result the material experiences high percent elongations. Even in the case of water experiments it can be observed from the graphs that the flow stress has a significant change when compared to the quasi-static conditions. Figure 12 shows the true stress vs. true strain curve for fiber reinforced hydrogel for both the experimental conditions. Fiber reinforced hydrogels also experiences same behavior as that of the other gels. They are rate sensitive and have the highest yield strength of all the hydrogels tested. Fiber reinforced hydrogel have the highest dynamic yield strength of 12MPa followed with 6% nano particle hydrogels with 6MPa and epoxy gel with 4MPa in without water conditions.

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Figure 12 Comparison of True Stress vs. True Strain curves of fiber hydrogel under dynamic compression CONCLUSIONS In this paper, a detailed experimental study was conducted to investigate the effect of water environment on three different hydrogel. The change in their material properties was showed significantly. Following are the major outcomes of this study: Quasi-static characterization 

A linear true stress vs. true strain response was observed with a large deformation in both epoxy hydrogels and nano particle reinforced hydrogels.



A non linear true stress vs. true strain response was observed for fiber reinforced hydrogels. These hydrogels also possess large percentage elongations.



Fatigue loading experiments showed no hysteresis in both epoxy hydrogel and nano particle reinforced hydrogel. This also confirms the hydrogel under low loading rates possess linearity.



In case of PU fiber reinforced hydrogels, fatigue loading on the specimens showed a significant change in the hysteresis. This confirms the fiber hydrogels even under low loading rates show non linearity.



In water the material is experiencing softening effects which increase the percentage elongation at break.

Dynamic characterization 

Epoxy hydrogel showed significant rate sensitivity at different strain rates. Considerable difference in yield stress from quasi-static conditions to dynamic conditions was observed and the difference ranges between 0.5MPa and 3.5MPa.



In water epoxy hydrogel showed almost similar characteristics as that of the regular experiment. The yield stress of the material is decreased by increasing the percentage elongation. Considerable difference in yield stress from quasistatic conditions to dynamic conditions was observed and the difference ranges between 0.5MPa and 2.0MPa.



Nano particle reinforced hydrogels also showed a significant change in their properties when tested under water by reducing the yield stress. Considerable difference in yield stress from quasi-static conditions to dynamic conditions was observed and the difference ranges between 0.5MPa – 0.7MPa and 4.0MPa – 5.5MPa. Under water the yield stress values show 0.5MPa – 0.7MPa and 1.5MPa – 3.0MPa.



Fiber reinforced hydrogels also show a decrease in yield stress and increase in elongation when tested under water environment.

50 

In conclusion the rate of increase of flow stress is less in water environment and constitutive response is non linear in all the hydrogels considered.

REFERENCES 1.

Metters AT, and Lin CC, Biodegradable Hydro gels: Tailoring Properties and Function through Chemistry and Structure, Biomaterials, Wong JY, and Bronzino JD, Eds. NY: CRC Press, 5.1-5.44, 2007

2.

Saraf.H Ramesh, K.T., Lennon, A.M., Merkle, A.C., Roberts, J.C., Measurement of the dynamic bulk and shear response of soft human tissues, Experimental Mechanics 47, 439-449, 2007.

3.

Sligtenhorst, C.V., Cronin, D.S., Broadland, G.W., High strain rate compressive properties of bovine muscle tissue determined using a split Hopkinson bar apparatus, Journal of Biomechanics 39, 1852-1858, 2007.

4.

Paul Moy, Tusit Weerasooriya, Thomas F. Juliano, Mark R. VanLandingham, and Wayne Chen, Dynamic Response of an Alternative Tissue Simulant, Physically Associating Gels (PAG), Army research laboratary, 2006.

5.

J. Kwon, G. Subhash, Compressive strain rate sensitivity of ballistic gelatin, Journal of Biomechanics 43, 420-425, 2010.

6.

Yang, Shukui L, Lili Y, Dongmei H and Fuchi W: Dynamic compressive properties and failure mechanism of glass fiber reinforced silica hydrogel, Material Science and Engineering., 824-827, 2010

7.

Song, B., Chen, W.W., Dynamic stress equilibration in split Hopkinson pressure bar tests on soft materials, Experimental Mechanics 44,300–312, 2004.

8.

Liu, Q., Subhash, G., Characterization of viscoelastic properties of polymer bar using iterative deconvolution in the time domain. Mechanics of Materials 38, 1105–1117, 2006.

Ballistic Gelatin Characterization and Constitutive Modeling D. S. Cronin Department of Mechanical Engineering, University of Waterloo, Waterloo, Ontario, Canada ABSTRACT Ballistic gelatin is widely used as a soft tissue simulant for non-penetrating and penetrating, and the mechanical properties of gelatin are known to be highly sensitive to strain rate and temperature. Mechanical compression testing was undertaken across a range of strain rates at constant temperature to evaluate the material response. The material strength and stiffness increased with increasing strain rate, while the strain to failure was relatively constant across a wide range strain rates. The mechanical test data was implemented in two constitutive models: a quasi-linear viscoelastic model, commonly available in explicit finite element codes, and a tabulated hyperelasticity model. The implementations were verified using simulations of the experimental tests and it was found that the quasi-linear viscoelasticity model did not adequately capture the low and high strain rate response across the range of data. The tabulated hyperelasticity model was found to provide accurate representation of the material across the range of strain rates considered, and included a damage function to predict material failure. INTRODUCTION Ballistic gelatin powder is produced from biological materials (skin, bone and tendons) through extraction with hot water in an acidic environment for Type A gelatin (Sellier 1994). This powder is then combined with water, heated and mixed, and conditioned at 4°C for a period of 2-3 days [Jussila 2004]. Type A, 250 Bloom is the most common gelatin formulation used in ballistic testing. The two commonly used mixtures are the Fackler formulation (10%, 4⁰C) (Fackler 1988) and the NATO formulation (20%, 10⁰C). The 10% formulation was considered for this study since it has been shown to better represent the properties of muscle tissue compared to the 20% gelatin 3 (Van Sligtenhorst 2004). Gelatin is known to have a density close to that of soft tissue (approximately 1060 kg/m ) (Sellier 1994) and a an elastic wave speed of approximately 1540 to 1550 m/s Van Bree (1996, 1998, 1999). Previous studies have investigated the characterization of ballistic gelatin, and evaluation as a tissue simulant (Van Sligtenhorst 2004, Cronin 2006, Caillou 1994, Van Bree 1998, Sellier 2004, Salisbury 2009) based on mechanical properties and penetration studies. A recent study on the mechanical properties of 10% gelatin -1 -1 -1 provided mechanical data at strain rates of 0.01 s , 0.1 s , and 1.0 s (Figure 1) (Cronin 2010). Intermediate -1 strain rate data, ~100 s has previously been reported (Cronin 2009) and was augmented for this study with -1 additional test data. High strain rate data on the order of 1000 s (Salisbury 2009) was not considered in this study.

Figure 1: 10% Gelatin compressive mechanical properties at low and intermediate strain rates

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52 It should be noted that all test data considered corresponds to 10% Type A, Bloom 250 Gelatin at a temperature of 4°C, which had been conditioned (aged) for approximately 72 hours. The actual conditioning time varied, and the gelatin was determined acceptable using the standard BB impact test (Jussila 2004). It has been shown that insufficient conditioning time or temperatures exceeding 4°C can significantly change the measured mechanical properties (Cronin 2010). METHODS The material data was recorded in terms of applied compressive force and deformation and was converted to engineering stress and strain based on the initial sample dimensions. The true stress and strain, required for constitutive modeling purposes, were calculated from equations 1 and 2 assuming constancy of volume of the samples. This is a reasonable assumption up to the initiation of material damage. σTrue= σEng(1+e)

Eqn.1

ε=ln(1+e)

Eqn.2

Based on the test data (Figure 1) it was apparent that the mechanical properties of ballistic gelatin were sensitive to deformation rate and this should be considered in any constitutive model. Available constitutive models for viscoelastic materials can be generally classified as linear, quasi-linear, and non-linear viscoelastic. The selection of a model was guided by the material response, and by commonly available models in numerical codes since the goal of this work was to use the properties and models to evaluate gelatin response to impact conditions. As a frame of reference, unpublished relaxation test data on 10% gelatin has shown that under relatively low strains (~20% engineering strain), 10% gelatin samples continue to relax for durations exceeding 15 hours. This -1 suggests that the stress strain curve at 0.01 s (Figure 1) does not represent the instantaneous elastic response or fully relaxed behavior of gelatin. However, this rate is relatively low for impact phenomena and was used to represent the fully relaxed behavior in this study. It should be noted that most constitutive models require complete definition of the material response across the range of strain rates and strains considered, and in the case of a inputting a discrete curve in a model, care must be taken to ensure the curve is defined across the entire range of anticipated strains. For this study, the instantaneous response curve was extended beyond the test data using a hyperelastic curve fit, and the tensile properties were determined by assuming symmetric behavior in terms of true stress and strain. This curve is shown in Figure 2, along with the original compression test data. The quasi-linear viscoelastic model as proposed by Fung (1993) for soft tissues and similar materials is widely implemented in numerical modeling codes and was evaluated for this study. Specifically, the implementation in the explicit finite element code LS-Dyna (LSTC, 2009) was investigated. This model requires a representation of the instantaneous elastic response of the material, typically expressed in terms of a hyperelastic stress (σH(ε)), and predicts strain rates through the addition of a viscoelastic stress component (σV(ε,t)) based on the convolution integral as shown in equations 3a, b and c. σ(ε, t) = σH(ε) + 𝜎𝑣 (ε, t) 𝑡

Eqn. 3a

𝜕𝜀

Eqn. 3b

𝐺(𝑡) = ∑𝑛𝑖=1 𝐺𝑖 𝑒 −𝐵𝑡

Eqn. 3c

𝜎v = ∫0 𝐺(𝑡 − 𝜏) 𝑑𝜏 𝜕𝜏 -1

The mechanical properties at a strain rate of 0.01 s were selected to provide the instantaneous elastic response of the gelatin material. It should be emphasized that this selection is somewhat arbitrary; typically a strain rate below which response will not be evaluated is selected to provide this response. It was found that commonly -1 2 available hyperelastic models (Mooney, Ogden) could represent the stress-strain curve at 0.01 s with r values exceeding 0.999. The specific implementation of the quasi-linear viscoelastic model allowed for an effective stress versus strain expressed as a polynomial, or actual test data to be implemented directly. For the purposes of this study, the properties were implemented directly using the tension and compression response shown in Figure 2.

53

Figure 2: Instantaneous elastic response data plotted with original test data A second constitutive model using tabulated stress-strain and strain rate data (Kolling et al. 2007, LSTC 2009) was investigated. This model uses the same concept of an instantaneous hyperelastic response (shown in Figure 2) and incorporates rate effects through the inclusion of actual stress-strain and strain rate data. Similar to the quasi-linear viscoelastic model, the compression test data was extended to strains beyond the material failure strain, and assumed symmetric in tension based on the true stress-strain data. This data was included in the material model, and material failure was addressed through a damage approach, described below. Following model fitting, the data was implemented in an explicit finite element code (LS-Dyna, LSTC 2009) and evaluated using single element test cases, and multiple element test cases. The element size for the simulations was on the order of 1mm, a typical value used to simulate ballistic gelatin in impact scenarios (Cronin, 2009). Both constitutive models considered incorporate a damage-based failure model that could be used to predict the onset of damage and final failure of the material. In this study, the form of the damage model used was based on the first invariant of stretch (equation 4). The full damage model allows for dependence on the square of the first invariant and the second invariant, but only the first term could be justified based on the available test data. The variable K corresponds to 100% damage (D). A second parameter (h) determines the initiation of damage in the material (equation 5). 𝑓(𝐼1 ) = (𝐼1 − 3) 1

𝐷 = � �1 + 2

𝑐𝑜𝑠�𝜋(𝑓−𝐾) �� ℎ𝐾

0.0 𝑓 ≤ (1 − ℎ)𝐾 (1 − ℎ)𝐾 < 𝑓 < 𝐾 1.0 𝑓 ≥ 𝐾

Eqn. 4

Eqn. 5

Figure 3: Finite element models – 1mm single element [L] and 25 mm diameter cylinder [R] (not to scale)

54 RESULTS AND DISCUSSION The quasi-linear viscoelastic model was evaluated relative to the data presented in Figure 1. Although an -1 2 adequate fit could be achieved with the quasi-linear viscoelastic model for strain rates of 0.1 s (r = 0.993) and -1 2 -1 2 1.0 s (r = 0.980), the fit for the intermediate rate data (105 s , r =0.885) was not satisfactory. Also, the general shape of the curve was not in good agreement with the experimental data. This was verified using single element simulations, but was not pursued further. It is anticipated that this modeling approach would not be able to accurately represent data at higher strain rates. The tabulated hyperelasticity model was evaluated using single element test cases in uniaxial compression, and found to reproduce the material data accurately across the range of strains and strain rates considered. Simulations on a multi-element cylindrical sample (Figure 3) were also undertaken and found to produce consistent results, although some oscillation occurred early in the simulation as the sample achieved equilibrium, and a small radial inertial effect was noted. The damage model was fit to the experimental test data presented in Figure 2 using a spreadsheet calculation -1 (K=5.7 and h=0.2) and incorporated in the material model. The compression test at a strain rate of 0.01 s was simulated and found to be in good agreement with the experimental data (Figure 5).

Figure 4: Tabulated hyperelasticity model predictions

Figure 5: Damage model predictions compared to experimental test data

55 CONCLUSIONS The mechanical properties for 10% Type A, 250 Bloom ballistic gelatin were investigated using two different constitutive models. A quasi-linear viscoelastic model, commonly available in finite element codes, was found to provide reasonable predictions for the low strain rate data, but did not accurately predict the intermediate strain rate data in terms of stress magnitude or the shape of the curve. A tabulated hyperelasticity model, based on the actual stress-strain and strain rate curves from the experimental tests, was found to accurately predict the material response across the range of strains and strain rates considered. In addition, the available material damage model was capable of predicting material damage and ultimate failure. Future work will investigate alternate hyper-viscoelastic models. The benefit of this class of material model is the capability to predict material response without the need or bias for specific test data across the range of strains and strain rates. ACKNOWLEDGEMENTS Natural Sciences and Engineering Research Council of Canada REFERENCES

Caillou J.P., Dannawi M., Dubar L., Wielgosz C. (1994) Dynamic behaviour of a gelatine 20% material numerical simulation. Personal Armour System Symposium, pp. 325-331. Cronin, D.S., Salisbury, C.P., Horst, C., “High Rate Characterization of Low Impedance Materials Using a Polymeric Split Hopkinson Pressure Bar”, SEM 2006, Society for Experimental Mechanics, St. Louis, 2006. Cronin, D.S. and Falzon, C., "“Dynamic Characterization and Simulation of Ballistic Gelatin”, Society for Experimental Mechanics, Albuquerque New Mexico, June 2, 2009. Cronin, D.S., Falzon, C.*, “Characterization Of 10% Ballistic Gelatin To Evaluate Temperature, Aging And Strain Rate Effects”, Journal of Experimental Mechanics, Online November 25, 2010, In press. Fackler, M. L. and Malinowski, J. A. “Ordnance Gelatin for Ballistic Studies,” The American Journal of Foresnic Medicine and Pathology, vol. 9 pp. 218-219, 1988. Fung, Y.C., Biomechanics: Mechanical Properties of Living Tissues, 2nd Edition, Springer-Verlag, 1993. Jussila J. (2004) Preparing ballistic gelatine - review and proposal for a standard method. Forensic Science International 141:91-98. Kolling, S., Du Bois, P., Benson, D., Feng, W., “A tabulated formulation of hyperelasticity with rate effects and damage”, Computational Mechanics, Volume 40, 2007. Kwon and Subhash, "Compressivestrainratesensitivityofballisticgelatin", Journal of Biomechanics, 43 (2010) pp 420-425. LSTC, 'LS-Dyna Theory Manual", LSTC, 2009 LSTC, 'LS-Dyna User's Manual", LSTC, 2009 Salisbury, C.P. and Cronin, D.S., "Mechanical Properties of Ballistic Gelatin at High Deformation Rates", Experimental Mechanics, Volume 49, Number 6 (2009), pp 829–840. Sellier, K.G. and Kneubuehl, B.P., Wound Ballistics and the Scientific Background; Elsevier, 1994, ISBN 0-444-81511-2. van Bree, J. and van der Heiden, N., "Behind armour pressure profiles in tissue simulant", Personal Armour Systems Symposium 96, September, 1996. van Bree, J. and van der Heiden, N., "Behind armour blunt trauma analysis of compression waves", Personal Armour Systems Symposium 98, Colchester, U.K., September, 1998. van Bree, J. and Fairlie, G., "Compression wave experimental and numerical studies in gelatine behind armour", 18th International Symposium on Ballistics, San Antonio Texas, November 15-19, 1999. VanSligentorst, C. "High Strain Rate Compressive Properties of Bovine Muscle Tissue", MASc Thesis, Department of Mechanical Engineering, University of Waterloo, 2004.

Strain Rate Response of Cross-Linked Polymer Epoxies under Uni-Axial Compression Stephen Whittie [email protected] Paul Moy [email protected] Andrew Schoch [email protected] Joseph Lenhart [email protected] Tusit Weerasooriya [email protected] Army Research Laboratory Weapons and Materials Research Directorate Bldg 4600 Deer Creek Loop Aberdeen Proving Ground, MD 21005-5069 ABSTRACT The strain rate responses of several cross-linked polymer epoxy materials were investigated under uni-axial compression at low to high strain rates. The properties of the epoxy were tailored through a variety of monomer choices including aromatic, which provide stiff, high glass transition structural materials, and aliphatic, which can form elastomers. The molecular weight and molecular weight distribution as well as the chemical functionality of the monomers can be varied to provide further control over the mechanical response. High rate experiments (greater than 1/sec rates) were conducted using a modified split-Hopkinson Pressure bar (SHPB) with pulseshaping to ensure that the compressive loading of the specimen was at constant strain rate under dynamic stress equilibrium. In this paper, moduli and yield strengths as a function of strain-rate of the epoxies are presented and compared in an effort to understand the effect of the different tailoring to their mechanical response. INTRODUCTION In general, there are two types of epoxies, the glycidyl and non-glycidyl epoxy resins. Each differs in the way that the epoxies are prepared. Glycidyl epoxies use a condensation reaction of dihydroxy compound, dibasic acid, and or a diamine and epichlorohydrin. Whereas, non-glycidyl epoxies are created using peroxidation of olefinic double bond. Each type of epoxy can be tailored to have a rubbery to a brittle toughness by changing the molecular weight, cross-link density, or adding a dispersed toughener into the cured epoxy. These chemically cross-linked thermosetting polymer networks are used in a wide range of applications including composite laminates, anticorrosive coatings, polymer membranes, and electronics. Epoxies can range from glassy structures to flexible gels [1]. In order for epoxies to be used for composite laminates for armor applications, the material response under a variety of strain rates needs to be fully understood to develop complete material models. As discussed in Rittel [2], as the epoxies undergo deformation majority of the mechanical energy is transformed into heat. The heat causes the specimens to undergo intrinsic strain softening followed by strain hardening. This phenomenon is described by Govaert et al. [3]. In 1995 Arruda et al. used an infrared detector to analyze the temperature effects of PMMA during quasi-static to intermediate strain rate compression testing. Arruda then reported that softening of the material occurring after the yield is a result from the combination of strain hardening/softening and thermal softening [4]. T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_8, © The Society for Experimental Mechanics, Inc. 2011

57

58 In this work, uni-axial compression experiments conducted on two epoxies are used to understand the mechanical responses from low to high strain rates. Quasi-static experiments were conducted using a servohydraulic Instron. High strain rate compression testing was carried out on a Kolsky bar which is also referred to as a Split-Hopkinson Pressure Bar (SHPB). The SHPB technique allows for the study of materials under a dynamic loading with high strain-rate deformation. -3 Jordan et al. [5] investigated the compressive behaviors of Epon 826/DEA epoxy from strain rates of 10 up to 4 -1 10 s . Similarly, Chen et al. [6] used a modified SHPB from high strength aluminum to study the dynamic response of Epon 828/T-403. Moy et al. [7] used the same testing techniques to study the mechanical behavior of PMMA at various strain rates. All showed results describing the effects of increasing yield strength and modulus as a function of strain rate. MATERIAL The epoxies used in this experimental investigation are Di-Glycidyl of Bisphenol A cross-linked with Jeffamine Diamine D400 (DGEBA D400), and Di-Glycidyl of Bisphenol F cross-linked with Jeffamine Diamine D400 (DGEBF D400). The polymers bisphenol (resin) and Jeffamine (curing agent) were acquired from Aldrich Chemical Company. The epoxy resin and curing agent were mixed and cured at the Army Research Laboratory Each epoxy mixture were poured into a custom fabricated stainless steel mold designed to provide a specimen diameter of 6.35 mm. The molded epoxy, right cylinder is about 152.4 mm long. Figure 1 shows a picture of each half of the mold. The bottom of the fixture is completely enclosed and the top has a threaded opening. Prior to the casting, mold release was used to ensure the epoxy can be easily removed from the steel mold. Otherwise forcible removal can damage the relatively thin epoxy rod. All compression experiments were conducted at room temperature. Once the epoxy has completely cured, right cylinder 3.18 mm gage length compression test specimens were then fabricated. Diameter to length ratio of these compression specimens is 2:1. Then, the specimen faces at both ends were machined to be flat and parallel to each other with a smooth surface finish. After the final machining, the test samples for DGEBA D400 and DGEBF D400 were then annealed in an oven at 76°C and 66°C respectively for 4 hours to eliminate any residual stresses caused from the cutting tool during machining. This annealing temperature is 20°C above the glass transition temperature, T g for DGEBA D400 and DGEBF D400 which is 56°C and 46°C, respectively.

Figure 1: Epoxy Molds for Compression Specimens LOW RATE EXPERIMENTS -3 -1 The quasi-static (10 /sec), (10 /sec), and intermediate (1/sec) strain rate experiments were conducted on an Instron 1331 servo-hydraulic test frame. A LabView-based program was used to generate an exponentially decaying waveform through a WaveTek function generator to command the Instron controller. This process

59 allowed for a constant true-strain rate compression experiments to be achieved. The load and displacement data were acquired using a separate LabView data acquisition program. Mineral oil was used as lubrication on the specimen ends to minimize friction during the compression. Hardened steel compression platens with a swivel base were used to compensate for any minor misalignment in the loading train. In order to correct the displacement data for the effect of machine compliance, the compliance of the Instron test machine was measured and the displacement due to machine compliance was subtracted from the measured machine displacement.. HIGH RATE EXPERIMENTS High rate experiments were conducted on a modified SHPB. A SHPB consists of a striker, an incident bar, and a transmission bar as shown in figure 2. The working principle of such setup is well documented [8, 9]. The bars used for the incident bar and transmit bar in the test setup were made from high strength Al 7075 that were specified to be centerless grounded to a diameter of 19 mm. Various annealed copper disks with different diameters and thicknesses were used for pulse shaping.

Transmission bar Strain Gage

Incident bar

Specimen

Striker

Strain Gage Pulse shaper

Power Supply

Power Supply

Digital Oscilloscope

Figure 2: Schematic of pulsed-shaped SHPB Set-up Assumptions were made during the SHPB testing that homogeneous deformation in the specimen occurs, identical incident and transmitted bars, and lastly analysis was based on one-dimensional wave theory [10]. The nominal strain rate described by Kolsky in the specimen is ,

(1)

Where is the elastic bar-wave velocity of the bar material, L is the original specimen gage length, and is the time-resolved strain from the reflective pulse from the incident bar. Integration of equation 1 with respect to time yields the time-resolved axial strain of the specimen. The axial stress, σ, of the specimen is determined using the equation ,

(2)

Where As is the cross-sectional area of the specimen, At is the cross-sectional area, Et is the Young’s modulus, and is the time-resolved axial strain from the transmission bar.

RESULTS AND DISCUSSION Figure 3 shows a typical set of stress pulses from the input and output bars from the SHPB with pulse shaping. The stress waves (incident, reflected, and transmitted) are identified on the graph.

60 0.03 Reflected

0.02

Volt (V)

0.01 Transmitted

0 -0.01

input bar (v)

-0.02

output bar (v)

Incident

-0.03 -0.04 -200

0

200

400

600

800

1000

1200

Time (sec) Figure 3: Typical input and output stress pulses Plotting the true strain as a function of time (Figure 4) for the high rate experiment shows that the epoxy sample is undergoing a near constant strain rate. Since a majority of the curve is linear, a valid constant strain rate compression test has been achieved. 0.5

True Strain

0.4

0.3

0.2

0.1

0

0

50

100

150

200

250

300

Time (sec) Figure 4: Compressive True Strain as a function of Time Stresses in the specimen/bar interface at each ends are shown in Figure 5. This indicates that dynamic stress equilibrium in the high rate experiments is achieved through pulse-shaping of the incident wave. This ensures that uniform loading is accomplished throughout the test without cyclic loading pulses going through the material from the incident stress wave.

61 200 specimen/input bar interface

True Stress (MPa)

150

specimen/output bar interface

100

50

0

0

20

40

60

80

100

Time (sec) -1

Figure 5: Stresses at Specimen/Bar Interfaces for DGEBF D400 at 4200s Strain Rate with Pulse-Shaping Figure 6 summarizes the mechanical response for DGEBF D400 from low to high strain rates. The individual plots with either an “X” or “O” states whether the specimen has failed or not failed, respectively. The “failed” compression test specimens are deemed by evidence of visible cracks. As a result, the epoxy specimens tested at 2000/sec strain rate and higher have various cracks in the direction of loading. In this paper, no high rate experiments were completed for the DGEBA D400. Specimen failure did not occur for the test at 1500/sec. Just like the specimens at the lower rate, this specimen did not have any visible signs of cracks. The specimen’s initial gage length was 3.19 mm. The total measured deformation is about 20%. Eight hours after testing, the specimen recovered to a near 100% of the initial gage length. It is quite evident that adiabatic heating effects are present at this strain rate. When the temperature for this epoxy material reaches above the T g the behavior would change to behaving similarly to an elastomer. Thus, the value for the yield strength at 1500/sec cannot be confirmed.

62 200

X X X True Stress (MPa)

150

X

4200/s

Failed X Did not Fail O

3500/s 2600/s 2000/s

O

100

1500/s 1/s 0.001/s

50

oo 0

0

0.1

0.2

0.3

0.4

0.5

0.6

True Strain

Figure 6.True Stress as a Function of True Strain for DGEBF D400 Figure 7 shows the stress-strain behavior for the epoxies DGEBA D400 and DGEBF D400 at the lower strain rates. The DGEBA D400 has a significant lower flow stress for a given strain rate than the DGEBF. For all quasi and intermediate experiments, strain hardening was evident at strains approximately 0.04. Interestingly, the stress-strain behavior for DGEBF at 0.001/sec is similar to DGEBA at 1/sec.

50 1/s DGEBF D400 0.001/s DGEBF D400

True Stress (MPa)

40

1/s DGEBA D400 0.100/s DGEBA D400

30

oo

0.001/s DGEBA D400

oo

20

10

0

o

0

0.1

0.2

0.3

0.4

0.5

0.6

True Strain Figure 7: True Stress as a Function of True Strain for DGEBA D400 and DGEBF D400 at Quasi-Static through Intermediate Strain Rates To illustrate the effects of strain rate on the different epoxies, the yield strength and modulus as a function of strain rate are shown in Figure 8 and Figure 9, respectively. From both figures, it demonstrates that the DGEBF

63 epoxy is rate sensitive. As stated earlier, high rate experiments were not conducted for DBEBA. Furthermore, these results show a bi-linear behavior for the DGEBF in both plots. 200

Yield Strength (MPa)

DGEBF D400 150 DGEBA D400 100

50

0 0.001

0.01

0.1

1

10

100

1000

4

10

Strain Rate Figure 8: Yield Strength as a Function of Strain Rate for DGEBA D400 and DGEBF D400

5 DGEBF D400

4

Modulus (GPa)

DGEBA D400 3

2

1

0 0.001

0.01

0.1

1

10

100

1000

4

10

Strain Rate Figure 9: Modulus as a Function of Strain Rate for DGEBA D400 and DGEBF D400 Figure 10 shows the pictures of both (a) DGEBA D400 and (b) DGEBF D400 epoxy compression specimens tested at the1/sec intermediate strain rate. In both cases, no signs of cracks were present.

64

(a)

(b)

Figure 10: Photo of the Tested Compression Experiments for (a) DGEBA and (b) DGEBF at 1/sec

(a)

(b)

(c)

Figure 11: Photo of the Tested Compression Epoxy DGEBF at (a) 1500/sec, (b) 2000/sec, and (c) 2600/sec Figure 11a, 11b, and 11c shows the pictures of the compression tested DGEBF D400 specimens tested at 1500/sec, 2000/sec, and 2600/sec, respectively. Since the specimen tested at 1500/sec did not have any cracks, the final gage length was measured at 3.18 mm which is only 0.24% of the initial gage length. Yet, the maximum strain from the SHPB calculations is about 20%. This proves that the epoxy transitioned from a glassy state to a rubbery state. The specimen DGEBF D400 at strain rate of 2000/sec showed cracking in the center of the specimen along with cracks radiating outward. Although the specimen failed, the specimen continued to remain intact. This specimen was tested to strain of 8%. Figure 11c shows DGEBF D400 at a strain rate of 2600/sec. This specimen was tested to strain of 9%. At this strain rate, the specimen showed many more cracks then the previous specimen tested at a strain rate of 2000/sec. The specimen exhibit cracking located in the center of the specimen which is in a circular pattern. The specimen also has radial cracks moving outward towards the specimen edges where the edges start to split apart from the body of the specimen. Although the damage is severe the specimen remains intact. Specimens tested at strain rates of 3500/sec and 4200/sec were pulverized during testing. In fact, at these higher strain rates, some of the recovered pieces show evidence of melting and fusing together of the debris. The specimens tested at the strain rate of 3500/sec had larger fragments compared to the specimens tested at 4200/sec. SUMMARY Compression testing on two epoxies, DGEBA D400 and DGEBF D400, was conducted at quasi-static, intermediate, and high strain rates. A near constant strain rate was achieved for the SHPB experiments by stress pulse-shaping the incident wave. Summary of the stress-strain responses for DGEBF show that the epoxy is rate

65 sensitive. Both the yield strength and modulus increase with the increase in the strain rate. Furthermore, results indicate a bi-linear mode from the yield strength and modulus as the function of strain rate graphs. Adiabatic heating effects are quite evident. In particularly to the compression experiment conducted at 1500/sec. The specimen at this rate recovered to a near 100%. Future efforts will investigate the adiabatic heating effects for the high rate experiments as well as other epoxy groups. References [1] Knox, C.K., Andzelm, J., Lenhart, J.L., High Strain Rate Mechanical Behavior of Epoxy Networks from th Molecular Dynamics Simulations. 27 Army Science Conference (2010). [2] Rittel, D., On the Conversion of Plastic Work to Heat during High Strain Rate Deformation of Glassy Polymers. Mechanics of Materials 31 (1999), pp. 131-139. [3] Govaert, L.E., van Melick, H.G.H., Meijer, H.E.H., Temporary Toughening of Polystyrene through Mechanical Pre-conditioning. Polymer 42 (2001), pp. 1271-1274. [4] Arruda, E.M., Boyce M. C., Jayachandran, R., Effects of strain rate, temperature and thermomechanical coupling on the finite strain deformation of glassy polymers. Mechanics of Materials. Volume 19, Issues 2-3, 1995, pp. 193-212. [5] Jordan, J.L., Foley. J.R., Siviour, C.R., Mechanical Properties of Epon 826/DEA Epoxy. Mechanics of TimeDepend Materials. (2008), pp. 249-272. [6] Chen, W., Zhou, B., Constitutive Behavior of Epon 828/T-403 at Various Strain Rates. Mechanics of TimeDepend Materials. (1998), pp. 103-111. [7] Moy, P., Weerasooriya, T., Chen, W., Hsieh, A., Dynamic Stress-Strain Response and Failure Behavior of PMMA. Conference Proceedings of 2003 ASME IMECE. Washington DC. [8] Davies, E.D.H., Hunter, S.C., The dynamic compression testing of solids by the method of the split Hopkinson pressure bar. Journal of the Mechanics and Physics of Solids. Volume 11, Issue 3 (1963), pp. 155-179. [9] Wu, X.J., Gorham, D.A., Stress Equilibrium in the Split Hopkinson Pressure Bar Test. Journal de Physique IV (1997), 7(C3), pp. 91-96. [10] Kolsky, H., An Investigation of the Mechanical Properties of Materials at very High Rates of Loading. Proc. Roy. Soc. London, B62, (1949), pp. 676-700.

Strength and Failure Energy for Adhesive Interfaces as a Function of Loading Rate

Tusit Weerasooriya1 [email protected] C. Allan Gunnarsson1 [email protected] Robert Jensen1 [email protected] 1

Army Research Laboratory Weapons and Materials Research Directorate Bldg 4600 Deer Creek Loop Aberdeen Proving Ground, MD 21005-5069 Weinong Chen2 [email protected] 2

Department of Aerospace and Mechanical Engineering Purdue University

ABSTRACT Adhesives are used to bond different materials to resist impact and penetration. This adhesive bond layer is a frequent source of failure when subjected to impact loadings. Therefore, it is necessary to measure the cohesive strength and failure behavior, especially at high loading rates. Experimental methods are limited in characterizing the mechanical behavior of adhesive bonds at high loading rate, so a unique experimental method and a specimen geometry was developed to determine the effect of loading rate on the failure (Mode I) of a commercially available adhesive (EPON 828). Four-point bend specimens consisting of two aluminum “wings” bonded together with the adhesive were tested at different loading rates, from quasi-static to high-rate. The high rate loading was performed using a unique modified Split-Hopkinson Pressure Bar (SHPB) setup. Embedded quartz transducers at the loading interfaces were used to measure the applied loads at both sides of the bar-specimen interfaces, which helped to optimize the input stress waves using wave-shapers to conduct valid dynamic experiments. Digital image correlation (DIC) method was used as an optical crack opening displacement (COD) gage to measure the onset of crack opening velocity (COV) to determine the failure initiation point on the loaddisplacement trace. The experimental results are used to obtain the energy required to initiate failure at different rates of loading. In this paper, the experimental methodology is presented, along with the results from these experiments. These data are being used to develop computational simulation methodologies to investigate the failure of bonded structures during dynamic impact loading. They will also be used to compare potential adhesives in determining the most qualified adhesive for various applications under different environmental conditions as well as different surface morphologies. KEYWORDS High rate loading, dynamic properties, bond/adhesive strength and energy, ultra high-speed digital image correlation, EPON 828

T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_9, © The Society for Experimental Mechanics, Inc. 2011

67

68 INTRODUCTION Adhesives are used to bond dissimilar materials to form protective systems. This adhesive bond layer is a frequent source of failure when subjected to impact loadings. To develop simulation methodology to predict the behavior of systems that contain adhesives during impact and penetration, it is important to understand the failure behavior of the adhesive interface layer. Therefore, it is necessary to develop experimental methods to measure the cohesive strength and failure behavior, especially at the high loading rates experienced during impact. Experimental methods do not exist to characterize the fracture behavior of adhesive bonds at high loading rates. By developing this experimental technique, the cohesive strengths can be determined, and these material properties can be used in computer models to predict the fracture process. For fiber-resin based composites, understanding the failure behavior at the micro-structural level is important to develop accurate simulation methodologies. This ensures correct prediction of deformation and failure of these composites during high rate impact loading. During impact, fracture at the interface between fibers and resin is the dominant failure mechanism. Currently, there are no valid high rate experimental methods to investigate this type of interfacial failure. The high rate experimental techniques presented in this paper can be used to study such interfacial failure. The failure mechanisms of the adhesive layer depend on many variables, including the surface roughness of the substrate materials being adhered together and environmental conditions. The interface properties can be controlled by varying the surface treatment at the interface [1-2]. Similarly, the bond strength may depend on the temperature and surface morphology in addition to the mechanical behavior (strength) of the adhesive material. Researchers develop various adhesives to bond different materials with high adhesive strength and toughness, and to withstand different environmental conditions. Currently, there does not exist a way to compare different adhesives in terms of bond strength at high loading rates, or to determine how the bond strength of these different adhesives change due to changing environmental conditions or surface morphology. This research documents a methodology that can be used to determine adhesive bond strengths at high loading rates, and which can be extended in future work to investigate the effect of surface morphology and environmental conditions on bond strength at high rate. Many different materials are used to develop protective systems, all with different material properties. Frequently, disparate materials are bonded together, such as ceramic bonded to polymer composite. During impact loading, a frequent failure mechanism involves adhesive failing at the material interface. Therefore it is important to understand the failure along the interface between adhesive and material. Interface failure of the adhesive occurs one of three ways: mode A, where the crack propagates through the adhesive; mode B, where the crack travels between adhesive and bonded material; and mixed mode, where the crack alternates between propagating through the adhesive and at the interface (combination of mode A and B). Higher levels of adhesion usually cause a higher likelihood of mode A failure, where the crack propagates through the adhesive. These modes are not to be confused with traditional fracture modes (mode I, mode II, mixed mode). In this paper, we propose a novel experimental method to study and quantify failure of adhesives at high loading rate under mode I fracture. This information will be used for the development of cohesive zone based simulation methodologies as well as providing data to be used as an evaluation tool for adhesives that are being designed for Army applications. Only recently have experimental methods been developed to adequately investigate the fracture behavior of materials at high rate loading. Weerasooriya et al [3] investigated the fracture behavior of a SiC-N ceramic as a function of loading rate, including at high loading rates. They used a modified split Hopkinson pressure bar (SHPB) setup to perform four point bending experiments on notched and unnotched SiC-N beam specimens to determine fracture toughness (notched) and flexure strength (unnotched) at dynamic loading rates. They found that both properties increased with increased loading rate. They used the same techniques to characterize the dynamic fracture toughness of PMMA as a function of loading rate [4]. They found that the fracture toughness of PMMA is also directly dependent on the loading rate. A comprehensive review of recent progress in dynamic fracture toughness experimental techniques was performed by Jiang et al [5].

69 The effects of surface morphology on the failure of interfaces have been studied at low loading rates. For a glass alumina interface, Cazzato and Faber [6] determined that the crack path was not restricted to the inter-material interface. The failure properties are thus nearly independent of the alumina surface roughness. Zhang et al. [7] used a bi-layer double cantilever beam specimen to study the fracture behavior of an epoxy-aluminum interface as a function of substrate surface morphology. They showed that the roughness index at a microscopic level is more important that the nano-level features of the aluminum surface. Extending the work of Zhang et al [7], Syn and Chen [8] studied the surface morphology effects on an aluminum-epoxy interface when subjected to high rate loading. They developed a novel butterfly like 4point bend specimen that consisted of aluminum on one side and epoxy on the other. This new specimen design was subjected to high rate loading in a modified compressive split Hopkinson pressure bar (SHPB) apparatus, using the 4-point bend technique similar to those detailed in [3-4]. They found that both fracture toughness and energy dissipated by the fracture process increased with increasing surface roughness for specimens loaded in approximately 10 N/µs range. In this paper, novel adhesive specimens were created by using a thin amount of the adhesive to bond two aluminum “wings” together, creating similar geometry specimens as Syn and Chen [8]. The specimens were then subjected to four point bending experiments as in [3-4], which caused failure of the adhesive. These experiments were performed to obtain the fracture properties of the adhesive bond/interface, including failure load, maximum load, failure energy (energy required to initiate crack propagation), and specimen crack opening velocity. These experiments were conducted over a range of loading rates, from quasi-static to dynamic. For dynamic loading experiments, a modified SHPB setup was used with embedded quartz force transducers to measure load history on the specimens, similar to [3-4,8]. High-speed digital image correlation (DIC) was used during the experiments to determine the onset of fracture and to allow quantification of the failure energy absorbed by the adhesive. This was obtained by measuring the velocity at which the two aluminum wings traveled away from each other after the onset of crack growth; hereafter this is referred to as the crack opening velocity (COV). The COV begins to increase just before maximum load; this indicates the initiation of failure; by determining when the failure began, it is possible to measure the energy needed to cause the initiation of failure of the adhesive. This method is more accurate than the assumption that the failure begins at the maximum load point. MATERIAL The adhesive used in this was diglycidyl ether of bisphenol-A (DGEBA), obtained from Shell under the trademarked name EPON 828. DGEBA is a typical commercial epoxy resin; it was used as a “baseline”, against which, later, high performance adhesives can be compared to. The DGEBA was cured with diethylenetriamine (DETA). This adhesive is widely accepted as a strong, versatile adhesive, used in a wide variety of applications. The behavior of this adhesive has been widely investigated. Chen et al [9] and Chen et al [10] investigated the constitutive compressive response of a similar DGEBA, EPON 828 cured with T-403. They found that the compressive strength increased with increasing strain rate, as did the strain at maximum strength. In addition, Chen et al [11] investigated the tensile behavior of EPON 828 as a function of strain rate. EXPERIMENTS Specimens The “butterfly” specimens were created by bonding two aluminum (7075) wings together with the adhesive. The wing geometry is shown in figure 1(a) (all units are in mm). The specimens are assembled using custom molds, the detail of which is shown in figure 1(b); the wings (blue) are held in place with set screws, and the two halves come together guided by alignment pins (purple). Custom made gauge blocks (green) are used to set the adhesive thickness (red); bolts (tan) are used to tighten the two halves together, which squeezes out excess adhesive, leaving only a layer as thick as desired, set by the gauge block thickness. A small pre-crack is created in the adhesive by embedding a thin layer of Teflon at the

70 loading edge (not shown in figure). This ensures that over the narrow length of the Teflon, the adhesive is bonded to only one of the aluminum wings.

(a) (b) Figure 1 – (a) Aluminum wing geometry and (b) specimen fabrication schematic

The completed specimens are then observed with an optical microscope to measure adhesive thickness and uniformity, and initial crack length. Figure 2 shows a complete, untested specimen and a magnified detail of the cracked-end of the adhesive demonstrating the measurements being obtained for the specimen using an optical microscope.

Figure 2 – Complete specimen and microscopic view of adhesive layer and pre-crack

Loading Fixtures To allow for integration into a compressive SHPB setup, cylindrical aluminum pieces of diameter 31.8 mm (1.25 in) were fabricated to mount a pair of hardened steel loading pins of diameter 2.4 mm (0.094 in) parallel to each other and normal to the cylinder axis. These pieces were bonded to a quartz disc of same diameter and 3.175 mm (0.125 in) thickness using a conducting metallic based epoxy. The other side of the quartz disc was bonded to a 25 mm (1.0 in) long aluminum cylinder. The quartz disc is the key instrumentation of the measurement system; as it is piezo-electric, it develops a small charge as it is subjected to mechanical stress. Two small pinholes in the aluminum pieces allow for electrical connectors to measure the charge developed by the quartz disc during loading. Two of these fixtures were created, one for each side of the specimen. The center-to-center distance for the loading side fixture was 12.20 mm and the support side fixture was 3.00 mm. Shown in Figure 3 are the complete fixtures, with the aluminum pieces bonded to each side of the quartz disc. Masking tape on the end of the fixtures ensures a snug fit with the plastic sleeve which is used to hold the fixture against the Hopkinson bar.

71

Figure 3 – Loading and measurement fixtures

Digital Image Correlation Digital image correlation (DIC) is an optical measurement that allows the user to perform displacement measurements of an object from a series of digital images recorded during the experiment. The images are post processed using specialized software. The DIC software allows measurement of the relative displacement of the two speckled wings. Figure 4 shows an example of an adhesive specimen with a speckle pattern that was used for displacement measurement by the DIC software.

Figure 4 – DIC extensometer and speckle pattern

Figure 5 shows the loading and crack opening displacement (COD) history of a typical specimen obtained from DIC method; the COD starts to increase just prior to the peak load, showing the failure initiation (crack growth start) point in the load history. By computing the relative displacement between wings using the high speed camera images of the speckled wings and DIC, a COV can be calculated from the slope of the COD-time curve. DIC measurements were obtained for experiments at all three loading rates, including ultra high-speed DIC (1 million frames per second) on the dynamic loading rate experiments using the modified Hopkinson bar 4-point fracture set-up.

72

Figure 5 – Load history and COD history during typical adhesive experiment

Quasi-static and Intermediate Loading Rate Experiments Two loading rates, which were several orders of magnitude below the dynamic loading rate, were used to characterize the adhesive at slower rates of loading. The experiments were conducted using an Instron 8871 servo-hydraulic test machine, with a load cell of capacity 5 kN (1124 lb). The same high rate loading fixtures discussed previously were used here for the purpose of using the same four-point loading geometry, thus eliminating any variability that may come from using different loading fixtures. However, the quartz cells were not used for the load measurements. The Instron load cell provided the load history for the lower rate experiments. Figure 6 shows the experimental setup for these two lower rate loading experiments. An alignment fixture, not shown, is used to ensure that the upper and lower fixtures are concentric and that the loading pins are parallel. The upper fixture is allowed to pivot to correct for any slight misalignments that may occur and make certain that the full length of the pins are engaged on the specimen during loading.

Figure 6 – Experimental setup for quasi-static and intermediate loading rates

73 During the experiments, the Instron test machine was controlled in displacement mode. The actuator traveled at constant velocity; due to the linear elastic behavior of the adhesive, this resulted in a constant loading rate. The quasi-static loading rate was 0.0116 kN/s (2.61 lb/s), which corresponded to a actuator velocity of 0.51 µ m/s (20.08 µin/s). The intermediate loading rate was 13.92 kN/s (3.13 kip/s), which corresponded to an actuator velocity of 0.64 mm/s (.0252 in/s). A high-speed camera recorded images of the specimen during testing; at failure, a break load detector initiated by the Instron was used to trigger the camera. The camera used was a Photron APX-RS, recording at 60k fps (16.7 µs/pic) and at a resolution of 128 by 128 pixels. The images were then post-processed using DIC software to determine the rigid body motion of the aluminum wings and to calculate the COV. Dynamic Loading Rate Experiments A modified compressive SHPB setup was used to perform the dynamic loading rate experiments. The incident and transmission bars were both aluminum 7075 with a diameter of 31.8 mm (1.25 in) and length of 3.66 m (12 ft). The striker used was 0.41 m (16 in). Wave shaping was employed on the incident bar to ensure the specimens were in a state of dynamic equilibrium. Semi-conductor strain gauges were mounted at midpoint on both bars, and connected to a high-speed digital oscilloscope to record the strain histories in the bars. The specimen-bar interfaces were modified to include the loading fixtures; the fixtures were held concentric with the bars and in place with a plastic sleeve. A detailed view of the experimental setup can be seen in Figure 7.

Figure 7 – Detail of high loading rate experimental setup

The loading fixtures were connected to a Kistler charge amplifier, and the amplifier output was recorded on the oscilloscope. The quartz gauges were calibrated to the known strain (and therefore stress) history recorded by the strain gauges. The calibration was performed by sending a compressive pulse through the system without a specimen. Mating fixtures were used in place with the loading fixtures to allow for smooth stress pulse transfer between bars and fixtures. Figure 8 (a) shows a typical calibration curve for the quartz gauges. Figure 8 (b) shows the loading history for one typical adhesive experiment. The loads are approximately equal at both incident and transmitted bars for nearly the entire experiment, and the loading rate is near constant. For the dynamic loading experiments, the average specimen gap closure velocity was 888.7 mm/s (35.0 in/s) and the average loading rate was 29,648 kN/s (6670 lb/s).

74

(a) (b) Figure 8 – (a) Calibration of quartz gauge using strain gauge and (b) typical load histories at incident and transmitted bars during adhesive experiment

RESULTS A summary of the adhesive’s mechanical properties that were measured during the experiments at the three different loading rates is tabulated in Table 1. The failure energy is the energy absorbed by the adhesive up to the point of failure (crack growth initiation); the failure initiation time is determined using the DIC technique (an optical COD gage) to measure when the COD begins to increase. Energy to failure is calculated by integrating load with respect to displacement. For the high rate tests, load was measured from the transmission quartz gauge, and displacement was determined from bar end motions measured by the strain gauges. At the quasi-static and intermediate rates, the displacement and load were measured using the test machine’s instrumentation. Failure energy rate is simply the failure energy per unit area of adhesive. Table 1 – Summary of adhesive experimental properties

Adhesive Averages Rate

Disp Rate (mm/s)

Loading Rate (kN/s)

Max Load (kN)

Dynamic Intermediate Quasi-static

888.7 0.64 0.00051

29648.20 13.92 0.01158

3.01 2.18 1.55

COD Velocity Failure Energy Failure Energy Rate (m/s) (mJ) (J/m^2) 9.89 4.38 1.70

206.60 97.11 81.79

1525 717 384

Figure 9 shows the relationship between loading rate and (a) failure load, (b) COV, (c) energy rate. The black markers indicate the average for that loading rate group. There is a strong and direct correlation between loading rate and all three properties. This demonstrates the absolutely necessity to obtain the high loading rate behavior of adhesives that may be subjected to these rates.

75

(a)

(b)

(c) Figure 9 – Effect of loading rate on (a) failure load, (b) crack opening velocity, and (c) energy rate

CONCLUSIONS A methodology for performing high-rate four-point bend strength (mode I fracture) experiments on adhesives has been developed at the Army Research Laboratory’s experimental facilities. A unique butterfly-specimen is used to study the adhesive and interface fracture. A modified compressive SHPB setup dedicated for dynamic fracture experimentation was used with this specimen to study the fracture behavior of a DGEBA epoxy cured with DETA. Digital image correlation was used as an optical COD gauge to measure exactly when the crack opening displacement begins to increase, which is shortly before the maximum load, and provided a precise measurement of the initiation of crack growth. These capabilities were used to characterize the loading rate effects on the strength and energy required to initiate fracture of a typical adhesive (EPON 828) used in Army applications. It was found that the failure strength and energy required for fracture initiation are both directly correlated to loading rate. This technique will be used to characterize other adhesives in the future, and allow for direct comparison of high loading rate properties for different adhesives under different environmental conditions as well under different surface morphologies. These experimental failure criteria will be used in developing simulation methodologies to represent fracture along the adhesive interfaces. Furthermore, these simulation methods will be used to explore other possible failure criteria, such as stress and strain fields around the crack-tip region. Also, these simulation methods can be used to explore the mode mixity, where both mode I and II are present at the crack tip region.

76 Determining the mechanism of failure for the adhesive is an important future step for this work. Investigating whether the failure occurs at the bonded material (here, aluminum) adhesive interface (mode B), or within the adhesive (mode A), or some mixture of the two will help researchers better understand the failure behavior. Additionally, this research is focused on mode I failure of the specimens. Development of experimental techniques to understand the mode II failure of adhesives will be needed. ACKNOWLEDGEMENTS The authors would like to acknowledge and thank Jared Gardner for his dedication and development of specimen fabrication techniques. REFERENCES 1. Jennings, C.W. Surface Roughness and Bond Strength of Adhesives. J. Adhesion, 4, pp. 25-38. 1972. 2. Thouless, M. Fracture Resistance of an Adhesive Interface. Scripta Mater. 26, pp. 949-951. 1992. 3. Weerasooriya, T., Moy, P., Casem, D., Cheng, M., and Chen, W. A Four-Point Bend Technique to Determine Dynamic Fracture Toughness of Ceramics. Journal of American Ceramics Society, 89 [3], pp 990-995, 2006. 4. Weerasooriya, T., Moy, P., Cheng, M., and Chen, W. Dynamic Fracture of PMMA as a Function of Loading Rate. Proceedings of the Society of Experimental Mechanics Annual Conference. 2006. 5. Jiang, F., and Vecchio, K. Hopkinson Bar Loaded Fracture Experimental Technique: A Critical Review of Dynamic Fracture Toughness Tests. Applied Mechanics Reviews, Transactions of the ASME, 62. 2009. 6. Cazzato, A. and Faber, K. Fracture Energy of Glass-Alumina Interfaces via the Biomaterial Bend Test. Journal of American Ceramics Society, 80, pp 181-188. 1997. 7. Zhang, Y. and Spinks, G. An Atomic Force Microscopy Study of the Effect of Surface Roughness on the Fracture Energy of Adhesively Bonded Aluminum. Journal of Adhesion Science and Technology, 11, pp 207-223. 1997. 8. Syn, C., and Chen, W. Surface Morphology Effects on High-Rate Fracture of an Aluminum/Epoxy Interface. Journal of Composite Materials, 42, pp 1639-1658. 2008. 9. Chen, W., and Zhang, X. Dynamic response of Epon 828/t-403 under multiaxial loading at various temperatures. Transactions of the ASME, Journal of Engineering Materials and Technology, 45, pp 1303-1328. 1997. 10. Chen, W., and Zhou, B. Constitutive Behavior of Epon 828/T-403 at Various Strain Rates. Mechanics of Time-Dependent Materials, 2, pp 103-111. 1998 11. Chen, W., Lu, F., and Cheng, M. Tension and Compression Tests of Two Polymers under Quasistatic and Dynamic Loading. Polymer Testing, 21 [2], pp 113-121. 2002.

Fracture in Layered Plates having Property Mismatch across the Crack Front

Umesh H. Bankar, Anil Rajesh and P. Venkitanarayanan* Department of Mechanical Engineering Indian Institute of Technology Kanpur, Kanpur, India 208016 (* Corresponding Author: [email protected])

ABSTRACT Layered structures are used in protection systems such as personal and heavy armor, windshields and also in thermal barriers. Such materials have mismatch in the properties, both elastic and fracture, from layer to layer. The focus of this study is to understand the behavior of cracks in such systems, especially when the crack orientation is such that there are property changes along the crack front. Plates comprising of layers of epoxy and PMMA were prepared by bonding together sheets of 6 mm nominal thickness with an epoxy adhesive. Single edge notched (SEN) specimens were loaded in bending. The thickness averaged stress intensity factor (SIF) was obtained through photoelasticity. Subsequently the behavior of crackpropagation in these materials was also investigated by loading SEN specimen dynamically. The crack tip fields were recorded using high-speed imaging coupled with dynamic photoelasticity, from which the thickness averaged fracture parameters are extracted. 1.0

Introduction

Layered structures are used in many applications including, protection systems, thermal barriers, windshields and heavy armor. A layered architecture offers the scope for choosing the layer material and properties in order to optimize the overall performance of the structure. However this advantage comes with added complexities in terms of material characterization and analysis. Particularly one can expect the fracture oriented failure of such structures to be sensitive to the layer architecture, change of elastic and fracture properties from one layer to the other and also to the type of loading experienced and crack orientation. Without a thorough understanding on these aspects, one cannot successfully garner the full potential of layered architecture for critical applications. Fracture of layered materials has received considerable attention over the years. There have been several investigations attempting to explore the fracture of bi-material systems. A bi-material system consists of two material layers joined together along an interface across which there are property jumps. The behavior of stationary cracks both oriented along the interface and oriented normal to the interface have been investigated by several researchers [1-5]. The behavior of propagating cracks along the interface of a bi-material has also received extensive attention [6-10]. Crack propagation cross the interface in a multilayer system has also been studied [11-12]. In all these investigations, the crack orientation is such that there is no property change across the crack front and therefore a two dimensional approach is applicable. A through thickness edge crack in a layered plate can have both elastic properties and the fracture toughness varying along the crack front. In this situation it is logical to anticipate that the stress intensity factor (SIF) will vary along the crack front and this variation will be sensitive to the variation of the elastic properties along the crack front. Recent studies [13, 14] indicate that if the elastic property variation along the thickness (crack front) in a cracked plate is continuous, then under in plane bending, the SIF variation is coterminous with the elastic modulus. This implies that the stiffer layer will have a higher SIF compared to the compliant layer. Therefore, the critical condition for a crack to become unstable may not be reached simultaneously at all points across the thickness (crack front). This can either lead to an overall increase of the critical load or its decrease depending on the relative variation of the SIF and the fracture toughness across the thickness. The purpose of this study is to bring out the fracture behavior of layered plates having cracks with property gradients along the crack front. To this extent, thin plates made of two different polymers, Poly Methyl Methacrylate (PMMA) and Epoxy (LY556) having both

T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_10, © The Society for Experimental Mechanics, Inc. 2011

77

78 elastic mismatch and fracture toughness mismatch were bonded together using an Epoxy adhesive. An edge crack in this plate was loaded in quasi-static and dynamic three point bending. The thickness averaged SIF is obtained through photoelasticity. 2.0

Experimental details

2.1

Specimen preparation and characterization

PMMA sheet used in this study was a commercial grade sheet of nominal thickness 5.5 mm. The epoxy sheets were cast in house and were of nominal thickness 5.8 mm. The elastic modulus and Poisson’s ratio of the materials were determined though tensile tests performed as per ASTM D638 using a 20kN UTM. The longitudinal and lateral strains were measured using a pair of strain gages. Fracture tests were also conducted using the three point bend specimen geometry following ASTM D5045-99. The fracture specimens had a natural crack which was initiated from a machined notch by tapping with a sharp razor blade. Five samples were tested for obtaining the fracture toughness. The elastic and fracture properties of the sheets are listed in table 1. One can easily observe from table 1 the variation of elastic modulus and fracture toughness from one layer to the other. The fringe constant of the materials was also determined by loading circular discs of the materials under diametrical compression and recording the fringes through a circular polariscope equipped with Tardy compensation. The two materials have different sensitivities, with the Epoxy almost 13 times more sensitive than the PMMA.

Material Epoxy PMMA

Table 1. Properties of materials used Elastic Modulus Poisson’s Fringe constant (GPa) ratio (MPa-m/fr) 3.44 0.34 0.018 2.67 0.34 0.239

Fracture toughness (MPa√m) 0.53 ± 0.04 0.95 ± 0.11

The specimen for fracture testing was prepared by bonding together a PMMA sheet to an Epoxy sheet using the two part epoxy adhesive, Araldite®, which has similar characteristics to the Epoxy sheet. The bonding surfaces were first abraded with fine grit paper and then cleaned with methanol. Then a thin layer of the premixed adhesive was applied with a serrated tool. The two sheets were then assembled and placed in a fixture under slight pressure for overnight curing. After curing the sheets were sized to a length of 200 mm and width of 50 mm. From the measured thickness of the specimen after bonding and the thickness of the individual sheets, it was estimated that the adhesive layer had an average thickness of around 170 micrometers. A notch of the required length was made in the specimen using a band saw and a natural crack was subsequently extended from the notch tip using a sharp razor blade. 2.2

Static testing Flash lamp

Single edged notch (SEN) specimens were subjected to both four-point bending and three-point bending in a UTM. The specimen was placed in a light field circular polariscope during loading and the isochromatic fringes were recorded using a CCD camera for further analysis. 2.3

Dynamic testing

Striker bar

V

Incident bar Strain Gages Specimen

Make trigger Trigger in

Circular Polarizer

SEN specimens were subjected to dynamic three-point Camera loading using a Hopkinson bar. A hollow polymeric bar Trigger out of 3 meter length was used for this purpose. Tests were performed on individual Epoxy and PMMA sheets as Trigger in SG Amp well as the combined PMMA-Epoxy specimens also. A SIM02-16 ultra high speed camera coupled with a circular polarizer was used to capture the isochromatic fringes during the fracture process. Sixteen images were captured at framing rates in the range of 100,000 to 150,000 frames per second. Figure 1 shows the schematic of the experimental setup. A make trigger circuit attached on the impact face of the bar was used Fig. 1 Schematic of the experimental setup for dynamic loading

79 to trigger the camera. The flash lamps and the strain gage data acquisition system was triggered by the camera itself. The captured images were analyzed to determine the crack speed and also the SIF as explained in the next section. 3.0

Analysis of isochromatics

The analysis of the crack-tip isochromatics to extract the fracture parameters (SIF) is a well established procedure for a homogeneous plate [15]. However, in the present study, the sample consists of two materials having different elastic and optical properties. Because of this, the stress field may not be constant through the sample thickness. The optical retardation is a function of the principal stress difference (σ1-σ2) and the fringe constant of the materials; both vary along the optical path (z-axis) in this case. The net relative retardation, ∆, can be written as h

1 (σ − σ 2 ) ∆ dz + =∫ 1 2π 0 fσ 1

h1 + h2

∫

h1

(σ 1 − σ 2 ) dz fσ 2

(1)

where, fσ1 and fσ2 denote respectively the optical fringe constant of material 1 and material 2 and h1 and h2 the respective thicknesses. For a cracked plate with elastic modulus varying along the crack front subjected to in plane bending, it has been shown that the variation in the stresses along the plate thickness is same as the elastic modulus variation, implying more or less an iso-strain type of situation [14]. Using this assumption, we can write the above equation as h h2  ∆ (σ
1 − σ
2 )  1 E E dz + ∫ dz  = ∫ Ee f 2π  0 fσ 1  h1 σ 2

(2)

In equation 2, Ee is an equivalent elastic modulus and (σ
1 − σ
2 ) is the principal stress difference in a homogeneous plate h

having an elastic modulus of Ee, where, Ee =

1 E ( z )dz and h=h1+h2 is the total thickness. In the present case, the elastic h ∫0

modulus and the fringe constant do not vary within a layer, hence we can define an equivalent fringe constant, fσe, for the whole plate as follows.

1 1  E1h1 E2 h2  = +   fσ e Ee h  fσ 1 fσ 2 

(3)

The optical fringe constant calculated using the above equation was used for analyzing the isochromatic fringes in this study, however, it should be pointed out that the usage is valid only for an iso-strain situation. A three-dimensional finite element analysis was carried out to investigate the variation of the stresses along the thickness. The contours of (σ1-σ2)/E for a homogeneous specimen and a layered specimen are shown in figure 2. The crack occupies the negative y axis. Figure 2(a) shows the variation of (σ1-σ2)/E at four locations across the thickness, two near surface (s) and two internal planes (i) for a homogeneous material having elastic modulus same as Ee. One can see that there is some variation of the stresses across the thickness even in a homogeneous plate. Figure 2(b) shows the contours of (σ1-σ2)/E for an Epoxy-PMMA layered plate. Interestingly, the variation of (σ1-σ2)/E across the thickness in the PMMA-Epoxy plate is almost identical to that in a homogeneous material indicating that once the stresses are normalized with the local elastic modulus, their variation is identical to that of a homogeneous plate. The method for extracting fracture parameters, particularly SIF from isochromatics is detailed in [15]. The procedure essentially involves fitting the near-tip stress field expressions to the experimentally observed fringe order using the stress-optic law. Recent investigations report that, continuous variation of elastic properties along the crack front, do not affect the structure of the first three terms, corresponding to r(-1/2), r(0) and r(1/2) in the stress field [13]. In the PMMA-Epoxy plate, the elastic properties are constant within each layer and vary only across the interface. Therefore we will assume the near-tip stress field to be identical to that in a homogeneous material. The SIF was calculated from the isochromatics following the over deterministic non-linear least square method using the equivalent optical fringe constant, fσε, in the stress optic law. The SIF thus obtained will be the thickness averaged SIF, Ke, and the SIF in the individual layers can be calculated as [14]

KP =

EP E Ke , K E = E Ke Ee Ee

(4)

80 where, subscripts P and E refer respectively to PMMA and Epoxy.

(a) (b) Fig. 2 Contours of (σ1-σ2)/E at four different planes along the thickness for (a) homogeneous plate and (b) PMMAEpoxy plate with an edge crack subjected to in plane bending. (s-close to surface, i-internal) 4.0

Results and discussion

4.1

Static tests

K (MP a-m ½ )

The results of the three-point and four-point bend tests will be discussed in this section. As mentioned earlier SENB specimens having a span of 200 mm and width of 50 mm were loaded in four-point bending and three-point bending. The thickness averaged SIF, Ke as a function of the applied moment in the four-point test is shown in figure 3 for two crack lengths. The solid line in the figure is the SIF calculated using 0.6 the analytical solution for an edge crack in a homogeneous plate a= 16.4 mm subjected to four point bending. It can be observed that Ke values in figure 3 are in good agreement with the theoretical values 0.5 a= 12.7 mm validating the use of fσe in the stress optic law. Edge cracks were also subjected to quasi-static three point bend test until failure of 0.4 the specimen. Figure 4 shows the load-displacement curve till failure of the sample. The load increased and close to a load of 0.3 300 N the crack in the epoxy layer jumped with a small drop in the load. The load further increased with stable crack growth in 0.2 the epoxy layer. Simultaneously the crack in the PMMA layer also grew stably; however, the crack tip in PMMA lagged the crack tip in epoxy leading to crack tunneling. Figure 5 shows the 0.1 SIF as a function of applied load. In figure the solid symbols represent the equivalent SIF, Ke, before crack jump and the open 0 symbols are the SIF calculated from the fringes assuming that 0 2 4 6 8 10 the fringes are caused by only the epoxy layer crack. The solid line is the SIF calculated using the theoretical expressions M (N.m ) available for an edge crack subjected to three-point bending in a Fig. 3 Equivalent stress intensity factor, Ke, as a homogeneous plate. Once again close agreement between the function of applied moment theoretical SIF and the experimentally determined SIF can be observed. The isochromatic fringes just before first crack jump, at the end of first crack jump and just before the final failure are shown in figure 6. One can see two crack tips in figure 6(c), one in the epoxy layer and one in the PMMA layer, suggesting crack tunneling. The equivalent SIF at the onset of first crack jump is 0.61 MPa-m1/2 which is about 15% larger than the fracture toughness of epoxy (see table 1). This indicates that the presence of the tougher PMMA layer helps in

81 increasing the fracture resistance of the pate and also delays the onset of final unstable failure. There was no sign of delimitation in the specimen. 0.7

500

0.6 K (MP a-m ½ )

Load (N)

400 300

200

0.5 0.4 0.3 0.2

100

0.1 0

0 0

0.1

0.2

0.3

0.4

0.5

0.6

Displacement (mm)

Fig. 4 Load displacement record for three point bend test a

b

0.7

0

100

200 300 L oa d (N)

400

500

Fig. 5 Stress intensity factor as a function of load in three point bending

c Epoxy crack tip

PMMA crack tip

Fig. 6 Stable crack growth in a PMMA-Epoxy plate under three point bending (a) just before first crack jump (load 310 N) (b) after first crack jump (load 300 N) and (c) just before final unstable failure (load 388 N) 4.2

Dynamic loading

SEN specimens were subjected to dynamic three-point bending as explained in section 2.3. The isochromatics recorded during the experiment were analyzed to determine the crack speed and also the value of the equivalent SIF at the point of crack propagation. Figure 7 shows the isochromatics in a PMMA-Epoxy specimen loaded dynamically. The pictures are separated in time by 10 microseconds. The first three pictures show the development of the opening mode stress field around the crack tip. In the fourth picture the crack has already started moving. From the seventh picture onwards, another butterfly shaped fringe (indicated by arrow in the picture) can be seen behind the crack tip (indicated by the vertical line). This fringe continues to follow the crack tip. From this observation, we can deduce that the crack propagation started only in the epoxy layer in picture four and the crack tip in PMMA starts propagating only in the seventh picture. The crack-tip location history obtained from the photographs of two nominally identical experiments is shown in figure 8. The initiation of the crack in epoxy layer could be recorded only in one case (test 02). In both experiments, the crack propagated with a constant velocity of 240 m/sec. From the plot of the crack-tip location versus time (test 02 in figure 8), the exact crack initiation time was determined as 26 microseconds after the first picture was taken. The crack tip in PMMA initiated about 30 microseconds after the crack tip in epoxy layer initiated. The equivalent SIF as a function of time was extracted from the isochromatic fringes and the variation of Ke with time for PMMA-Epoxy specimen is shown in figure 9. The SIF corresponding to a time of 26 microseconds was estimated as 0.6 MPa-√m, which is close to the dynamic initiation toughness of epoxy layer.

82

Fig. 7 Crack propagation in a PMMA-Epoxy layered plate subjected to dynamic three-point bending. The vertical line indicates the crack tip in Epoxy layer and the arrow indicates the crack tip in PMMA. The time interval between two pictures is 10 microseconds. 35

0.65

0.55

25

K (Mpa-m ½ )

Crack-tip location (mm)

30

20 15 PMMA Epoxy

10

0.35

Test-01

5

Test-02

0

0.25 0

20

40

60

80

100

120

Time (µ µ s)

Fig. 8 Crack-tip location history for dynamic fracture of PMMA-Epoxy layered plates under dynamic loading 5.0

0.45

0

5

10

15

20

25

T im e (µs)

Fig. 9 Stress intensity factor history up to crack initiation in PMMA-Epoxy layered plates under dynamic loading

Conclusions

The fracture behavior of layered plates having an edge crack subjected to quasi-static and dynamic three-point loading is investigated. The layered plate was prepared by bonding together PMMA and Epoxy sheets which have 30% mismatch in elastic modulus and 80% mismatch in the fracture toughness. Particularly the effect of sudden change in elastic modulus and fracture toughness across the crack front on the fracture behavior is investigated. The thickness averaged SIF was determined from the photoelastic fringes recorded during the fracture phenomena. The results of the study indicate that the presence of

83 the relatively tough PMMA layer can increase the fracture resistance of the material and also delay the onset of unstable fracture under quasi-static bending. When subjected to dynamic loading, the crack tips in Epoxy and PMMA propagated at different time instants with the crack tip in the relatively brittle epoxy layer initiating earlier than that in PMMA. Further studies are in progress to understand the beneficial effects of layered structure on their fracture tolerance under dynamic loading. 6.0

Acknowledgements

The authors would like to acknowledge the financial support under the FIST program by Department of Science and Technology, Government of India for the Ultra-high speed camera used in this study through grant number SR/FST/ETII003/2006. 7.0

References

1.

Rice J. R. and Sih G. C., Plane problems of cracks in dissimilar media, Journal of Applied Mechanics, Vol. 32, pp. 418423, 1965.

2.

Xu L. and Tippur H. V., Fracture Parameters for Interfacial Cracks: an Experimental-Finite Element Study of Crack Tip Fields and Crack initiation Toughness, International Journal of Fracture, vol. 71, pp. 345-363, 1995.

3.

Ricci V., Shukla A. and Singh R. P., Evaluation of Fracture Mechanics Parameters in Bimaterial Systems using Strain Gauges, Engineering Fracture Mechanics, Vol. 58, pp. 273-283, 1997.

4.

Erdogan F. and Biricikoglu V., Two Bonded Half Planes with a Crack Going through the interface, International Journal of Engineering Sciences, Vol. 11, pp. 745-766, 1973.

5.

Tippur, H. V. and Rosakis, A. J., Quasi-static and dynamic crack growth along bimaterial interfaces: A note on crack-tip field measurements using coherent gradient sensing, Experimental Mechanics, Vol. 31, pp. 243-251, 1991.

6.

Yang, W., Suo, Z. and Shih, C. H., Mechanics of dynamic debonding, Proceedings of the Royal Society (London), Vol. A433, pp. 679-697, 1991.

7.

Liu, C., Lambros, J. and Rosakis, A. J., Highly transient elastodynamic crack growth in a bimaterial interface: higher order asymptotic analysis and optical experiments, Journal of the Mechanics and Physics of Solids, Vol. 41, No. 12, pp. 1857-1954, 1993.

8.

Singh, R. P. and Shukla, A., Subsonic and intersonic crack growth along a bimaterial interface, Journal of Applied Mechancis, Vol. 63, pp. 919-924, 1996.

9.

Singh, R. P., Kavaturu, M. and Shukla, A. Initiation, propagation and arrest of an interface crack subjected to controlled stress wave loading, International Journal of Fracture, Vol. 83, pp. 291-304, 1997.

10. Shukla, A. and Kavaturu, M. Opening-mode dominated crack growth along inclined interfaces: Experimental observations. International Journal of Solids and Structures, Vol. 35, No. 30, pp. 3961-3975, 1998. 11. Singh, R. P. and Parameswaran, V., An Experimental Investigation of Dynamic Crack Propagation in a Brittle Material Reinforced with a Ductile Layer, Optics and Lasers in Engineering, Vol. 40, No. 4, pp.289-306, 2003. 12. Parameswaran, V. and Shukla, A., Dynamic Fracture of a Functionally Gradient Material Having Discrete Property Variation, Journal of Material Science, Vol. 33, No. 13, pp. 3303-3311, 1998.

13. Wadgaonkar S. C. and Parameswaran V., Structure of near tip stress field and variation of stress intensity factor for a crack in a transversely graded material, Journal of Applied Mechanics, Vol. 76, No. 1, 011014, 2009.

14. Kommana R., Parameswaran V., Experimental and numerical investigation of a cracked transversely graded plate subjected to in plane bending, International Journal of Solids and Structures, Vol. 46, No. 11-12, pp. 2420-2428, 2009. 15. Sanford, R. J. and Dally, J.W, A general method for determining mixed mode stress intensity factors from isochromatic fringe patterns. Engineering Fracture Mechanics, Vol. 11, pp. 621-633, 1979.

Stress Variations and Particle Movements during Penetration into Granular Materials Hwun Park, Weinong W. Chen Schools of Aeronautics/Astronautics and Materials Engineering, Purdue University 701 West Stadium Avenue, West Lafayette, IN 47907-2045 Email: [email protected]

ABSTRACT Granular materials such as sand are crushed and compacted locally when subjected to projectile penetration. It is necessary to obtain more diagnostic information during the experiment to develop a thorough understanding of the deformation mechanisms inside the target. In this study, we embedded piezoelectric film pressure gages at strategically distributed locations inside cylindrical sand targets. The gages are designed to measure only the pressure normal to the gage plane. The pressure gages are calibrated using a Kolsky bar before being embedded into the targets. The instrumented target is then subject to penetration by a cylindrical steel projectile with a semi-spherical nose. The distributed gages measure the pressure histories at the local gage positions. Flash X-rays installed around the target record the instant images of moving projectile and the gages inside the target. It was found that the penetration-induced pressure concentrates locally around the projectile. INTRODUCTION Sand has been widely used as a defensive material in military applications to defeat kinetic energy penetrators. Sand is a typical granular material and has complicated behaviors because of particle interactions [1]. Under dynamic loading condition, sand can be locally compacted and allow stress to propagate through specific chains, which may not be completely modeled by continuum mechanics [1-6]. Penetration into sand has been investigated in many studies [4-8]. However, predictive capabilities remain poor, indicating that improvements in the understanding of mechanisms are necessary. Most experimental investigations on penetration into sand have been focused on the relation between projectile conditions and maximum depth of penetration. In terms of numerical approaches, sand has been modeled using hydrodynamic assumptions that many theories of penetration have been based on [6-9]. These models give detail information on the status of targets such as stress profiles and medium displacements as well as the depth of penetration. However, the predicted quantities, such as stress field around a penetrator, have rarely verified by experiments. To develop more accurate models of sand penetration, obtaining diagnostic information beyond traditional experimental methods from experiments is necessary. The pressure measurement in sand and soil has been conducted with diaphragm gages or pressure transducers with static loading conditions in large scale samples [10]. For small scale samples, the measurement can vary with the ratio of gage size to particle size. Under dynamic loading conditions, sand can be locally deformed and inelastically packed, which makes the measurements more challenging [1,2]. Under projectile penetration conditions, stress propagates in three dimensions with various stress components in the medium. To capture the distribution of pressure inside the target, we use special piezoelectric gages designed to measure only the stress component normal to the gage planes. To validate their response and quantify their sensitivity, the in-house-assembled pressure gages are calibrated with a Kolsky bar before being embedded into the targets. To observe trajectory of projectiles during penetration and to evaluate the motion of sand in the target, we use flash X-rays that pass through the sand but are blocked by high density metals. EXPERIMENTS

T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_11, © The Society for Experimental Mechanics, Inc. 2011

85

86 As a projectile move ahead, a cavity expands in the radial direction, generates stress propagation and transfers media in the radial directions [8,9]. The stress propagation and pressure caused by transferring media are generally three dimensional. We developed a special gage responding to only normal pressure as shown in Fig. 1.

Fig. 1 Design of piezoelectric gage to measure normal pressure The pressure gage consists of a piece of 0.0254 mm-thick piezoelectric film, PVDF, between two thin brass plates serving as electrodes. The gage is covered with Teflon tapes to avoid taking in-plane shear stresses. The brass plates and PVDF film are not bonded for the same purpose. To protect the gage from damage by sharp sand particles, steel or aluminum plates are attached at both sides. Even though the manufacturer provides electro-mechanical conversion factors, the gage needs to be calibrated because the specific arrangement of the gage inside the sand medium may affect the gage sensitivity. If there is no electric field around the gage, the response of piezoelectric film is calculated by [12].

Q   DdA  d15 31A1  d15 23 A2  d31 11  d32 22  d33 33 A3

(1)

where dij is a piezoelectric strain constant for each direction. (See Fig. 1 for the directions). The thickness of PVDF film is very thin, only 0.0254 mm, so the area A1 and A2 are negligible. Even though the PVDF film is not bonded with the brass plates, the film can have σ11 and σ22 because the edges of film are loosely confined with Teflon tapes and in-plane deformation is caused by Poisson effect. The low friction and relatively heavy protection cause little transfer of shear stress to the film. To calibrate the gage, a Kolsky bar was employed. The gages were sandwiched between the incident bar and transmission bar to let uni-axial stress be applied in the orthogonal direction. The sensitivity is obtained by adjusting the charge amplifier until the outputs from gages match the difference between the incident and reflected pulses. After the sensitivity of the gage is determined, a polycarbonate tube filled with sand was inserted between two bars. The gages were embedded in the sand tube to evaluate the gage response when the sand is loaded (Fig. 2).

Fig. 2 Configuration of a piezoelectric gage embedded in a sand column in a Kolsky bar The amount of sand in front and behind the gage affects the response because compacted sand reduces the stress transfer [2]. To simulate the environment inside the target, various amounts of sand were inserted between the gage and the incident bar, whereas the sand amount between the gage and the transmission bar remains constant.

87 Fig. 3 illustrates the experimental setup. A light gas gun was employed to launch projectiles at high velocities. The couples of lasers and sensors detect the passage of the projectile, which gives the flight velocity by the time interval. Projectiles are 50.8-mm long, 12.7-mm diameter cylinder and have semi-spherical heads. They are made with 4340 steel and weighs 46.5 g each.

Fig. 3 Configuration of experimental setup The cross-section of the sand target is 10.2 cm by 10.2 cm, which is the maximum depth of sand that X-ray penetrates and records images with reasonable quality. For consistency, Ottawa 20/30 sand conforming to ASTM C778 was used to fill the target. Density was measured as 1.61 kg/m3. Two types of sand targets were constructed; one having distributed iron balls for observation of particle motion by X-ray and the other having pressure gages for pressure measurement. Four flash X-ray tubes located in serial to record instance image of projectile in the target. The gage positions were decided by preliminary tests. The position of each gage is tabulated in Table 1. Fig. 3 showed the coordinate system and the center of target surface is the origin. The speed of projectile is approximately 180 m/s. Table 1 Lists of test with positions of gages Gage 1 Gage 2 Gage 3 X (cm) Z (cm) X (cm) Z (cm) X (cm) Z (cm) -1.3 5.1 -1.3 12.7 -1.3 25.4

Gage 4 X (cm) Z (cm) -1.3 38.1

1

Velocity (m/s) 151

2

184

-1.3

5.1

-1.3

12.7

-1.3

25.4

-1.3

38.1

3

184

1.3

5.1

1.3

12.7

1.3

25.4

1.3

38.1

No

EXPERIMENTAL RESULTS Fig. 4 shows a set of results from gage calibration tests with the Kolsky bar. The gage was inserted between the incident bar and transmission bar without sand. A small part of incident pulse reflects from the gage because the impedance of gage package containing polymer and aluminum is lower than the steel bar. The sensitivity of the charge amplifier was adjusted to make the plateau of gage output closed to the incident pulse. The sensitivity was found to be -3.4 pC/N. This calibration procedure was repeated with incident pulses of higher amplitudes. All the factors obtained from other tests agree well with 3.4 pC/N.

88

Fig. 4 Comparison of incident, transmitted pulses and adjusted gage output To simulate gage behavior in the sand target, the gage was embedded into a sand sample placed between the bars. To examine the effect of distance from a projectile to gages, the length of sand between the gage and the incident bar was set as 6.4 mm, 19 mm and 25 mm. The gas chamber pressure to drive the striker was set at 207 kPa, 345 kPa and 483 kPa. The corresponding outputs of gage for each case are shown in Fig. 5. (A) shows the transmitted pulses from the incident bar to the sand and was calculated by subtracting the reflected pulse from the incident pulse. (B) and (C) are the outputs from gage, which show the trends according to the impulse strength and the thickness of sand in front of the gage, respectively.

Fig. 5 Transmitted impulse to sand and corresponding outputs from gages Fig. 6 shows X-ray images recording the instants of particle motion in the sand target. Three flash X-rays were emitted on the film. Each number on the figure represents the time of each frame was taken after the projectile struck the surface of the target. The region marked by two black lines is affected by the projectile and narrows as the projectile proceeds and slows down. The maximum diameter of this region is approximately 36 mm, which is three times the projectile diameter. The small diameter of head of the projectile locating at the deepest position in the target indicates that the projectile tuned to left side (in the negative y direction in Fig. 3).

89

Fig. 6 X-ray image of target embedding iron balls (striking velocity: 178 m/s) Fig. 7 shows X-ray images capturing the moments of penetration. All gages were installed at the bottom (in the negative x direction in Fig. 3). The projectile turned to the top side away from gages during penetration. Asymmetric positions of gage may cause the curved trajectory. The projectile changed the direction again to the center as shown in the last shadow image. This may be caused by reflection from the aluminum wall of the target.

Fig. 7 X-ray images of sand target embedded with pressure gages (Test 2) Fig. 8 shows X-ray images capturing the moments of penetration. In this case, all the pressure gages were installed near the top (in the positive x direction) of the target box, opposite to previous set. Consequently, the projectile turned to the bottom side, again away from the gages. This indicates that the gages may indeed interfere with the motion of the projectile inside the target. After learning this lesson, we will install pressure gages at symmetric locations inside the sand targets in the future studies.

90

Fig. 8 X-ray image of sand target embedded pressure gages (Test 3) Fig. 9 shows stress variations at each gage location in Test 1. The output from Gage 1 has a sharp peak initially and then slowly decays. The high speed impact by the projectile induces the rapidly increasing pulse for a short duration. As shown in the X-ray images in Fig. 6, Gage 1 was supposed to be exposed at the cavity completely because the radius of cavity at the mouth of target is expected to exceed the distance from the centerline to Gage 1. It is believed that the slow decay and large pressure were caused by the cavity. Gages 2 and 3 recorded peaks when the projectile passed by the gage locations. Gage 3 has a lower peak value than Gage 2.

Fig. 9 Pressure induced by projectile in Test 1 Fig.10 shows the penetration-induced pressure variation recorded in Test 2. Even though the output from Gage 1 has rough fluctuations, it has a sharp peak and slow decaying, which is similar to the Gage 1 records in Test 1. Gages 2, 3 and 4 also had peaks as the projectile passed by the gage locations in sequence. The peaks value of each gage decreased as projectile traveled further because it moved farther away from the gages, as well as slowing down.

91

Fig.10 Penetration induced pressure variations in Test 2

CONCLUSIONS Piezoelectric gages were designed and fabricated to measure pressure variations in sand targets during projectile penetration. The sensitivity of the pressure gages was determined in the calibration tests with a Kolsky bar. The size of cavity during penetration was imaged with flash X-rays. The maximum size of cavity is three times the diameter of the projectile. The embedded pressure gages measured the pressure variations at different locations in the sand target as the projectile passed by. ACKNOWLEDGEMENT This research was supported by Basic Research Programs of Defense Threat Reduction Agency. REFERENCES Jaeger H. M, Nagle, S. R., Behringer R. P., The Physics of Granular Materials, Physics Today, 49(4), 32-38, 1996 Song B., Chen W., Luk V., Impact Compressive Response of Dry Sand, Mech. Materials, 41, 777-785, 2009. Martin B. E., Chen W., Song B., Akers, S. A., Moisture Effects on the High Strain-Rate Behavior of Sand, Mech. Materials, 41, 786-798, 2009. 4. Allen W. A., Mayfield E. B., Morrison H.L., Dynamics of a Projectile Penetrating Sand, J. App. Phy., 28, 370-374, 1957. 5. Borg J.P., Vogler T. J., An Experimental Investigation of the High Velocity Projectile Penetrating Sand, Proc. 11 th Int. Cong. Exp. Soc. Exp. Mech., 2008. 6. Borg J.P., Vogler T. J., Mesoscale Simulation of a Dart Penetrating Sand, Int. J. Impact Eng., 35, 1435-1440, 2008. 7. Zukas J. A., Impact Dynamics, Krieger, 1992. 8. Backman M. E., Goldsmith W., Mechanics of Penetration of Projectiles into Targets, Int. J. Eng. Sci., 16, 1-99, 1978. 9. Forrestal M. J., Luk V. K, Dynamic Spherical Cavity-Expansion in a Compressible Elastic-Plastic Solid, Trans. ASME, 55, 275-279. 1988. 10. Nichols. T. A., Bailey A. C., Johnson C. E., Grisso R. D., A Stress State Transducer for Soil, Trans. ASAE, 30(5), 1237-1241, 1987. 1. 2. 3.

92 11. Gran J. K., Frew D. J., In-Target Radial Stress Measurements from Penetration Experiments into Concrete by Ogive-Nose Steel Projectiles, Int. J. Impact Eng, 19, 715-726, 1997. 12. Moheimani S. O. R., Fleming A. J., Piezoelectric Transducers for Vibration Control and Damping (Advances in Industrial Control), Springer, 2006.

Sand Particle Breakage under High-Pressure and High-Rate Loading Md. E. Kabir, Weinong Chen Schools of Aeronautics/Astronautics and Materials Engineering, Purdue University. 701 W. Stadium Ave. West Lafayette, IN 47907-2045 Email: [email protected] ABSTRACT Under intensive loading, either hydrostatic or dynamic shear, sand particles are broke up to smaller particles. In this study, we randomly embedded color-coated sand grains of five different initial sizes inside cylindrical sand specimens under dynamic tri-axial loading. The quasi-static pressure was varied from 25 to 150 MPa at 25 MPa intervals and then unloaded. The embedded particles were retrieved and found fractured particles increased with increasing pressure. However, many of the colored particles remained intact. Under 100-MPa -1

hydrostatic pressure, additional axial load was added at three different strain rates, 0.01, 500, and 1000 s . It was found that, under additional dynamic loading nearly all the colored particles were fractured. In addition, to the attention on the embedded colored particles, the overall size change of the sand from the loaded specimens were also quantified. INTRODUCTION Particle fracture plays a major role in the mechanical behavior of sands. When sand particle breaks into a number of pieces, the total surface area of sand increases whereas the number of contacts per particle decreases. Particle size distribution of sand is, therefore, a key characteristic, which influences its properties, handling and domain of application. Loading rate on sand may vary significantly in applications. The quantity and

type of particle breakage at high rates of loading may differ from that at low rates. However, there are very limited experimental results found in the high rate range, late alone particle breakage analysis. In our previous work we investigated the shear behavior of sand at high strain rates [1]. In this study, we investigate the effect of strain rate on sand particle breakage behavior. In addition, the effect of hydrostatic pressure was also investigated. All other factors were kept constant. EXPERIMENTS AND RESULTS In order to study the particle breakage in a sand specimen, five different size sand particles (particles retained on US sieve No. 30-50) were first color-coated with different colors. Twenty sand particles corresponding to each of the five different sizes were randomly dispersed in the specimen. The specimen was then subjected to either isotropic consolidation only or isotropic consolidation followed by triaxial shear. Figure 1 shows all the experimental conditions in terms of void ratio and mean pressure for this study. The isotropically consolidated specimens were taken to 6 different confining pressure levels ranging (as shown by different symbols in Figure 1) from 25 MPa to150 MPa at an interval of 25 MPa. On the other hand, triaxial shear experiments were conducted by first applying a hydrostatic pressure of 100 MPa, followed by axial loads that were applied at -1

constant strain rates of 0.01, 500 and 1000 s (as shown by the arrow in Figure 1). After the loading, the specimens were carefully dismantled and spread on a bench from where the colored particles were removed. Presence and type of failure as well as presence of significant fracture surfaces were recorded for each of the colored particles using an electron microscope. Based on the damage observed, the T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_12, © The Society for Experimental Mechanics, Inc. 2011

93

94 colored sand particles were categorized into three groups. These are ‘Type 1’ if there was no visible damage, ‘Type 2’ if there was only surface abrasion of the particles leaving the parent particle largely intact but usually more rounded, and ‘Type 3’ if there was a major fracture of a particle into two or more pieces. Table 1 tabulates the results.

Figure 1: State paths for compression and shear experiments Table 1: Particle Failure Observed for Isotropic Consolidation Experiments Pressure

Grain size

Level (MPa)

(sieve#)

50

100

150

Grain damage Type 1

Type 2

Type 3

(No damage)

(Attrition)

(Fracture)

30

17

1

2

35

10

7

3

40

9

7

4

45

7

9

4

50

7

8

5

30

8

9

3

35

7

12

3

40

7

7

5

45

6

7

7

50

5

6

8

30

7

8

5

35

6

7

7

40

4

6

8

45

3

8

9

50

2

4

11

Lost

1 2

2 3

REFERENCE: [1] Kabir, Md. E., and Chen, Weinong W. “High Rate Triaxial Experiment on Dry Sand”, Proceedings of SEM Conference, IN (2010).

Experimental and Numerical Study of Wave Propagation in Granular Media

T. On1, K.J. Smith1, P.H. Geubelle1, J. Lambros1, A. Spadoni2, C. Daraio2 1

2

Department of Aerospace Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801

Department of Aeronautics and Applied Physics, California Institute of Technology, Pasadena, CA 91125

ABSTRACT One dimensional stress waves travelling in granular chains exhibit interesting characteristics such as filtering, tunability and wave mitigation because on the formation of solitary waves, and solitary wave trains, within them. An idealized one dimensional granular medium, consisting of a linear array of contacting spherical brass beads, was loaded dynamically in a modified split Hopkinson pressure bar, with loading pulses that span a variety of rates and profiles using pulse shaping techniques. Different chain lengths were studied to determine how solitary waves form in a varying length granular medium, as well as the speed of wave propagation. It is found that the wave speed propagates faster for longer chains of brass beads. The high loading rates of the Hopkinson bar also allowed us to investigate plastic dissipation effects in the granular chain when composed of different types of metals. To further our understanding of wave propagation in ductile ordered granular media, the experimental results are compared with companion numerical simulations based on a particle contact law that accounts for plastic dissipation. Knowing the behavior of a stress wave propagating through such materials can lead to arrangements that can produce desired stress wave mitigation characteristics as the waves travel through the granular chain. INTRODUCTION A granular chain can be characterized as a group of particles which can displace independently of one another and interact only when in contact with an adjacent particle. In such a chain, each element can only transmit information to its neighbors via compression. Contact between two spherical surfaces, even elastic, is a nonlinear process, the best known case of which is the Hertz contact solution developed in the late 1890s (Timoshenko and Goodier [1]). Subsequent efforts pertain to the plastic deformation of contacting spheres, which neglect volume conservation of the plastically deformed sphere, were based on the model of Abbott and Firestone [2]. In addition to contact models, considerable amount of work has been done towards the study of the static response in granular materials by Drescher [3]. Dynamic contact studies have been conducted by Iida [4], Hughes and Kelly [5], Shukla [6, 7] and Xu [8] who investigated the effects of particle size and wave velocity in one and two dimensional granular chains. The current paper focuses on the wave velocity traversing through a one dimensional granular medium consisting of brass beads with varying lengths. EXPERIMENTAL PROCEDURE AND RESULTS This set of experiments was conducted using a split Hopkinson pressure bar (SHPB) for dynamic loading at high strain rates (102– 104 s-1). The SHPB consists of an incident bar and a transmitted bar, with the specimen sandwiched between the two. The bars are long enough to be considered uniaxial and are hardened to remain elastic during the loading process. The loading is generally done via a striker bar, usually made from the same material, impacting the

T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_13, © The Society for Experimental Mechanics, Inc. 2011

95

96 incident end, sending a compressive stress wave to the specimen. A momentum trap, developed by Nasser [9], is added to the bar to provide a single loading wave, followed immediately by an unloading wave. This ensures the specimen undergoes a single impact for the duration of the test. Strain gages are placed along the incident and transmitted bar to record the stress waves traversing along the bar as shown in Figure 1.

Figure 1. Schematic of a split Hopkinson pressure bar with momentum trap. Two strain gages are placed in the middle of each bar, and on opposite ends to cancel strain readings caused by possible bending of the bars. Strain gage (EA-06-250BK-10C) was used for all tests. The strain gages were connected through a signal conditioner which amplifies the strain and sent to an Agilent Technologies Digital Oscilloscope. The specimens used were brass beads (Alloy 260) of 9.525 mm diameter obtained from McMaster-Carr. In order to maintain a one dimensional chain of spheres, a holder, adapted from Spadoni and Daraio [10], was fabricated which consists of a hollow metal tube with threaded end caps which can be placed onto the bar. The holders are sized such that a hemisphere extrudes from the tube and comes in contact with the bar, allowing a point-load contact for the stress wave to travel. The edges of the tube are threaded by an end cap of larger inner diameter to ensure the caps will not interfere with the sphere specimens. A holder setup with one end cap mounted is shown in Figure 2. The holder was tested and showed no alterations to the experimental data. Chains of brass ranging from one sphere to twelve spheres are tested, with different length holder tubes for each length of brass chains.

Figure 2. Image of the one dimensional sphere holder with one end cap in place. Figure 3 shows a typical incident reflected and transmitted signal form the strain gauges (compensated for bending). Among other things, data from the experiment is used to calculate the wave velocity through the brass chain using the equation (1) where V is the wave velocity, N is the number of beads in the chain, d is the diameter of the bead, and is the time duration starting when the incident strain gage receives a signal and ending when the transmitted strain gage receives a signal. The travelling time of the wave within the chain can be calculated by knowing the positions of the strain gages. Since the material of the bar is known, both the wave velocity of the bar and the distance from the

97 strain gage to the specimen can be calculated. This results in the travelling time of the wave within the incident bar and transmitted bar, which are denoted as and , respectively. Thus, the denominator in equation (1) denotes travelling time of the wave within the chain.

Figure 3. Raw incident and transmitted data acquired from an experiment. The wave speeds were calculated for tests of various chain lengths and are plotted in Figure 4.

Figure 4. Wave velocity through one dimensional brass granular medium for varying chain lengths. Note that the force in each case, which is known to control the wave speed of such nonlinear waves, is about the same. Aside from the single bead experiments, the wave speed appears to be similar, although possibly somewhat increasing, for increasing chain lengths. The wave propagation in granular medium is governed by the contact mechanism and the wave speed calculated is much smaller than the dilatational wave or shear wave velocity for the same material, although it does depend on loading amplitude. The wave speed for a single bead has significant variations, more than the experimental error. The reason for this is not clear, but we believe it is associated with progressive yielding of the two contact points of the bead.

98 CONCLUSIONS The experimental data obtained by the strain gages show an increasing wave speed for an increasing distance for the wave to traverse. The wave speed also appears to approach a constant velocity value around 25% of the wave speed for a solid brass bar. This result is similar to that observed by Iida [4], and Xu and Shukla [8]. REFERENCES [1] Timoshenko, S. P. and Goodier, J. N., Theory of Elasticity, 3rd Ed., McGraw-Hill, New York, 1970. [2] Abbott, E.J. and Firestone, F.A., Specifying Surface Quality - A Method Based on Accurate Measurement and Comparison, Mech. Eng. Am. Soc. Mech. Eng., 55, p. 569, 1933. [3] Drescher, A., Application of Photoelasticity to Investigation of Constitutive Laws for Granular Materials, Proc. IUTAM-Symposium on Optical Methods in Solid Mechanics, Poities, France, 1979. [4] Iida, K., Velocity of Elastic Waves in a Granular Substance, Bulletin Earthquake Research Institute, Vol. 17, p. 783-808, 1939. [5] Hughes, D.S. and Kelly, J.L., Variation of Elastic Wave Velocity with Saturation in Sandstone, Geomechanics, Vol. 17, p. 739-752, 1952. [6] Shukla, A. and Zhu, C.Y., Influence of the Microstructure of Granular Media on Wave Propagation and Dynamic Load Transfer, J. of Wave Material Interaction, Vol. 3, No. 3, 1988. [7] Shukla, A. and Damaia, C., Experimental Investigation on Wave Velocity and Dynamic Contact Stresses in an Assembly of Disks, Exp. Mech., Vol 27, No. 3, 1988. [8] Xu, Y. and Shukla, A., Stress Wave Velocity in Granular Medium, Mech. Research Communications, Vol. 17, p. 383-391, 1990. [9] Nasser, S.N., Issacs, J.B, and Starrett, J.E., Hopkinson Techniques for Dynamic Recovery Experiments, Math. And Phy. Sciences, Vol. 435, p. 371-391, 1991. [10] Spadoni A. and Daraio C., Private Communication, 2010.

Communication of Stresses by Chains of Grains in High-Speed Particulate Media Impacts William L. Cooper, Mesoscale Diagnostics Engineer, Air Force Research Laboratory, AFRL/RWMW, 101 W. Eglin Blvd Suite 135, Eglin AFB, FL 32542 ABSTRACT Right-circular (φ 15 mm x 26 mm) projectiles were fired vertically-downward (150-720 m/s) into acrylic containers (φ 80-190 mm) containing quartz Eglin sand. Decreasing container size increased projectile drag and decreased total penetration depth. Thus, the container is within the projectile’s event horizon for at least a portion of penetration path length and some mechanism(s) exists for communication between projectile and container. The particulate media fractured near the projectile nose and created a rigid, conical false nose on the front face of the projectile, but the fractured media domain does not extend beyond 1.5 projectile diameters of the shot line. Jammed grains (i.e. mechanically-compacted, but not fractured) can be found adjacent to the fractured media; surrounded by nominally initial-density grains. It is theorized that the projectile communicates with the container via stress chains in the un-fractured grains which span the distance between the projectile/crushed media and container wall. The stress chain event horizon may be limited by either mechanical decay of forces along the chains or limited stress wave speeds in the particulate media. This paper focuses upon the mechanical force decay limits. Multiple analytical models are presented to illustrate how stress chain curvature and friction can limit the maximum stress chain length and thereby the ability of projectiles to communicate through the particulate media with the container wall. INTRODUCTION Researchers at Osaka University [Watanabe 2010] conducted experiments to observe the high-speed impact of right-circular cylinder projectiles (12.3 g, steel front face, polycarbonate body) with quartz Eglin sand (φ 75-1,400 µm grains, d50=400 µm, as-poured ρ=1.53 g/cc) [Cooper 2010]. Projectiles were launched vertically downward and impacted the sand surface normally (flat face parallel to sand) at 150-720 m/s. Projectile velocities were measured using induction loops and impact attitude was validated with high-speed photography. Two containers were used: φ 80 mm & φ 190 mm internal diameter, 300 mm length. Total penetration depth did not vary appreciably with impact velocity, but was strongly affected by the container size. Decreasing the container size from 190 mm to 80 mm cut the penetration depth in half as shown in Fig. 1. Thus, it can be inferred that the projectile communicates with the container--the stresses at the projectile surface and the drag on the projectile are affected by the container size. The goal of this analysis is to examine the mechanics that enable the projectile to communicate with the container. 200 180

φ 190 mm Container

Penetration depth [mm]

160 140 120 100 80 60 40

φ 80 mm Container

20 0 0

100

200

300

400

500

600

700

800

Impact velocity [m/s]

Figure 1 Projectile penetration depth as function of impact velocity

T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_14, © The Society for Experimental Mechanics, Inc. 2011

99

100 High pressure directly ahead of and beside the projectile nose comminute and fracture the sand media, forming a rigid, conical false nose on the front of the projectile. They also comminute and fracture sand up to 1.5 projectile radii from the projectile centerline. The increased density of the comminuted sand can easily account for the sand volume displaced by the projectile. However, this region spans only a fraction of the sand container diameter and does not fully explain the communication between the projectile and container unless the container is on the order of 3 projectile diameters or less. Outside the comminuted sand region the sand can be mechanically-compacted, but does not fracture; suggesting a maximum density of 1.73 g.cc (the maximum mechanical compaction density achieved previously [Cooper 2010]) which decays with distance to the initial density of 1.53 g/cc. 2-D experimental work [Howell et al. 1999] and 2-D discrete element modeling (DEM) [Peters et al. 2005, Tordesillas et al. 2005] indicate that grain contact mechanics allow formation of stress chains which can propagate stress information over considerable distances. Howell et al. measured stresses in a 2-D Couette flow (inner radius 10.34 cm, outer radius 19.14 cm) using birefringent discs and showed that chains have a mean length of 2 to 5 particles, depending upon the particle packing fraction. Most chains are quite short with only a few percent of the chains exceeding 8 or 10 discs in length. However, visual inspection of experimental images shows chains which bridge the gap between inner and outer radius (roughly 100 discs in width) and much longer chains might be visually constructed because the chains form at angles to oppose rotation of the inner drive wheel. The authors note the difficulty in defining the length of chains. In this case the authors defined the chains “to be any set of nearly co-linear discs carrying stress larger than the mean.” The chains end at branches (which are common) or boundaries. This definition is excellent for assessing the force structure in the particulate material fabric. However, it does not address the primary question of interest to the present effort; How far can chains of grains propagate stresses to facilitate communication between points or surfaces in a particulate media (PM)? This question requires assemblies of chains (as defined by Howell et al.) such that the stress at one end affects the stress at the other. Depending upon interactions with surrounding grains, chains can satisfy this question while loaded with stresses more or less than the mean. Such chains can split or re-combine to produce chain segments with more or less stress than the mean. Peters et al. considers a quasi-static case more analogous to the present projectile impact—the impinging of a square punch upon the upper surface of a 2-D PM bed—and reports average chain lengths of approximately 5 grains with an exponential decay of the chain length PDF. In this case the definition of a chain is more specific than above—a “linear string of at least three rigid particles in point contact that can support loads along its axis, with only small amounts of rotation involved.” [Peters 2005, Cates et al. 1998, 1999] Chains are allowed some maximum degree of curvature. Exceeding this value (e.g. ±45º) breaks the chains into shorter segments in a method similar to Howell’s branching criteria. The stress chains organize to oppose the motion of the punch and roughly 40% of the particles are more highly stressed than the average (the “strong network”). Of these stressed particles only half are involved in chains longer than 2 particles. Again, this chain length definition is effective for assessing the stress-carrying structure in the PM fabric, but does not specifically address the communication question. Consider the question of chain lengths if an international cell phone call is made. Each specific segment (cell phone to tower, fiber optic cable, satellite up/down-link, etc.) may have a specific length, but if connected end to end allows the termination points to communicate--affect each other. Fig. 10 in Peters et al. serves to both illustrate this point and to emphasize the far-from-equilibrium behavior that typifies particulate materials [NRC 2007]. Standalone force chains (greater than average stress) are observed in the middle of grain ensembles with less than average stress. This suggests that stress information is communicated through the region of low stress by multiple low-stress force chains which can converge to load a chain more highly than average, before branching again. Thus a communications-focused chain length definition is needed: Communication chains are any ensemble of grains in contact for which stress changes at one location (chain end) affect the stress at another location (other end of chain). Such chains can incorporate multiple segments at higher/lower-than-average stress levels according to the definitions above. It is also possible, even likely, that stress changes at one grain may affect the stress levels in a variety of locations throughout the particulate material. This introduces the idea of ad hoc communication networks in particulate materials. The present effort focuses on a very narrow subset of the theory required to explain ad hoc communication networks in PM. Several analytical models are developed which help explain the stability of communication chains and to set bounds on their lengths based upon consideration of both the chains’ radius of curvature and friction (inner-chain and inter-chain). ANALYTICAL MODELS Given the communication chain definition, a model particulate chain can be constructed as shown in Fig. 2. The chain has both a beginning & end and all grains are in contact (point-wise for spherical grains considered here). Geometrically, any two neighboring grains have a characteristic radius (rc) of curvature. In practice the origin of each rc can vary wildly in three dimensions, but flattening this to 2-D provides a simpler illustration. Following the example of the projectile impacting the PM, the projectile applies a force to one end of the chain and it is theorized that communication with a confining wall can affect this force. In practice the experimental results indicate that the closer the wall, the larger the applied forces by/on the projectile, the higher the drag, and the shorter the total penetration length.

101

Figure 2 Theoretical particulate material communication chain model Kinematically, the grains must be confined by some forces in order to maintain contact along the full chain length. These confining forces are provided by other grains in the PM as shown in Figs. 2 & 3. Fr,i is the confining force for the ith grain. Fi-1 and Fi are the inter-grain forces. Frs,i, Fs,i-1 and Fs,i are the shearing (frictional) forces between grains. The angles φ and β are defined as follows: φ=2asin(r/rc) and β=(π−φ)/2. Γ is the angle of the confining force; which is limited by the presence of the i-1 and i+1 grains. Note that if Γ is negative then the confining force can amplify the Fi-1 loading force such that the force can increase along the length of the chain. For all calculations r, rc, Γ and friction coefficient values (µ) are assumed constant along the entire chain length, although they would vary in practice from grain to grain.

Figure 3 Communication chain inter-grain forces FRICTIONLESS CASE Inter-grain friction requires both that surface roughness and/or asperities exist and that these are activated by sufficient macroscopic deformation (resulting in relative grain-grain lateral sliding) and/or deformation at the grain contact (due to grain loading forces) [Cole 2008]. There are a number of materials and conditions which fail to satisfy these criteria and are represented reasonably by a frictionless model, e.g. as-poured smooth glass beads. Such contacts are dominated by normal contact forces and result in a conservative system of grains which lacks a dissipative mechanism. Ignoring all interfacial friction forces, it can be shown that the inter-grain forces are governed by:

102

tan β −1 Fi tan Γ =Χ= tan β Fi −1 +1 tan Γ Fr ,i

 sin β 1 =  − 1 Fi  Χ  sin Γ

Eq. 1

Eq. 2

If F0 is the applied force on the initial (i=0) grain then the force propagates as follows:

Fi = Χi F0 Fr ,i F0

(

= Χ i −1 − Χ i

Eq. 3

β ) sin sin Γ

Eq. 4

Plotting X as a function of non-dimensionalized Γ & rc shows that the forces increase along the chain for Γ < 0 and decrease for Γ > 0 (see Fig. 4). As noted previously this effect occurs due to the alignment of the confining force. The radius of curvature accentuates this effect for very small values (rc/r < 10). Small radii of curvature transition more of the chain load to the confining force instead of the adjacent grain; effectively branching the chain for very small radii of curvature.

Figure 4 X (ratio of forces for adjacent grains) plotted as function of Γ/Γmax & rc/r. Plotting the number of grains required to decay the initial force (F0) to 5% of its initial value provides a more intuitive view of the same data. These values are capped at 300 grains both for practical readability reasons and because 300 grains easily span the container radii of interest (~40 – 100 mm for 0.4 mm diameter grains). As expected, Γ=0 is the dividing line between chains whose inter-grain forces decay and those that do not. The total chain length rapidly diminishes for Γ>0. This result helps explain how inter-grain forces in PM can be amplified locally (as noted by Peters 2005) by confining forces with Γ 1 and hence er < 0. From (6) and (7), we see that the axial strain rates and stretch rates are related by

. .εz = − λz = e.z

. . ez = −λz ,

λz

1 − ez

.

(9)

Analogous relations hold for the radial strain and stretch rates. Now consider the stress state. Let p denote the pressure and s the deviatoric part of the Cauchy stress tensor. Then σrr = srr + p ,

σθθ = sθθ + p ,

srr + sθθ + szz = 0 ,

σzz = szz + p ,

p = 13 (σzz + σrr + σθθ ) .

(10) (11)

Since the radial and hoop stretches coincide and the specimen is isotropic, the radial and hoop stresses are equal and the shear stress is zero: σrr = σθθ , σrz = 0 . (12) Thus a homogeneous axisymmetric deformation results in a biaxial stress and strain state; the principal axes of stress and strain are the axis of symmetry and any axes orthogonal to it. In this case, the relations (11) simplify to srr = sθθ = − 12 szz ,

p = 13 (σzz + 2σrr ) .

(13)

On substituting the relation (13)2 for p into the relation (10)3 for σzz , we obtain the equivalent relations σzz = 32 szz + σrr ,

szz = 23 (σzz − σrr ) .

(14)

For axisymmetric deformations of an isotropic specimen, the balance of radial momentum is given by2

..

σrr − σθθ ∂σrz ∂σrr + + = −ρ r , ∂r r ∂z

(15)

where ρ is the density in the deformed state. In view of (12), for homogeneous deformations this reduces to

..

∂σrr = −ρ r . ∂r

(16)

Since the lateral surface of the specimen is stress free, σrr = 0 when r = r0 (t). Then on integrating (16) from an arbitrary radius r to r0 (t) and using (4) and (3), we obtain [ [ )2 ] ( ( )2 ] r R σrr = 2 σ rr (t) 1 − = 2 σ rr (t) 1 − , (17) r0 (t) R0 1 The standard sign convention in the continuum mechanics literature takes stress and strain components as positive in tension. The opposite sign convention is typically used in the compression Hopkinson bar literature, and we have followed that convention here. 2 The minus sign on the right side of (15) is a consequence of the convention that stresses are positive in compression.

262 where

..

ρR02 λ r (t) λ r (t) . (18) 4 Working directly with (17), we find that σ rr (t) is the mean value of σrr over the deformed volume of the specimen at time t. Since the deformation is homogeneous, this is also the mean value of σrr over the undeformed reference configuration. And since σrr is axially uniform, σ rr is also the mean value of σrr over any deformed (z = constant) or undeformed (Z = constant) cross-sectional area. Since σθθ = σrr , we conclude that the radial and hoop stresses vary quadratically with the radius but are axially uniform. They attain a peak absolute value of 2σ rr (t) at the centerline (r = R = 0) and reduce to zero at the lateral surface. They have the same sign as the radial stretch acceleration λ r , but without additional assumptions neither the sign nor the magnitude of λ r can be inferred. In any case, it is clear that the presence of nonzero radial and hoop stresses is a radial inertial effect, that is, a consequence of the fact that the ρ r term in (16) is not necessarily negligible. It follows that in the quasi-static limit, that is, in the limit as r (or λ r ) approaches zero, σ rr (t) = 0. Hence, as expected, in a quasi-static test we have a uniaxial stress state (σrr = σθθ = 0), and by (13)2 and (14)2 the pressure and deviatoric stress are completely determined by the axial stress: p = 13 σzz and szz = 23 σzz . On the other hand, it is clear from (13)2 and (14)2 that these quasi-static estimates for p and szz will be in error if σrr is sufficiently large relative to σzz . σ rr (t) =

..

..

..

.. ..

3. The Incompressibility Approximation The Jacobian of the deformation, denoted by J, is the determinant of the deformation gradient F and represents the local ratio of deformed to undeformed volume. For a general deformation, J is the product of the principal stretches; for a homogeneous axisymmetric deformation this yields J = λ r λ θ λz = λ r2 λz . Since the focus of this paper is on nearly incompressible specimens, we will make the approximation that the deformation is volume-preserving. Then J = 1, which is equivalent to any of the following relations: 1 λr = √ , λz

λz =

1 , λ r2

εr = − 12 εz .

(19)

Thus the radial stretches or strains are determined by axial stretches or strains, and vice versa. In this case the true and nominal stress components are related by3 σzz = λz ΣzZ = (1 − ez )ΣzZ ,

σrr = λ r ΣrR .

(20)

Now recall the relations (9) between the axial components of strain rate and stretch rate (with analogous relations for the radial components). By (19) we obtain the following additional relations between the radial and axial rates:

. −λ.z . −er = λ r =

3/2 2λz

.

=

ez 3/2 2λz

.

ez = , 2(1 − ez )3/2

. .ez = −λ.z = 2λ r ,

.

.

εr = − 12 εz .

λr3

(21)

From the relations on the left in (21), we find that the radial strain and stretch accelerations are given in terms of the axial stretch or strain rates by ( )2 ( )2 ez 1 λz 3 λz 1 3 ez + = + . (22) − er = λ r = − 2 λz3/2 4 λz5/2 2 (1 − ez ) 3/2 4 (1 − ez ) 5/2

.. ..

We also have

.

..

..

1 λr = √ λz

.

..

[

] 1 ( )2 1 εz + εz , 2 4

..

.

(23)

although this relation is less useful than (22). For later use in the discussion of optimal pulse shapes, we note that the nominal axial strain acceleration is given in terms of the radial stretch rates by ( )2 λr λr ez = − λz = 2 3 − 6 . (24) λr λ r4

..

..

..

.

Now consider the radial stress. Since the deformation is volume-preserving, ρ = ρ0 , the density in the undeformed state. On substituting this and the relations (19)1 and (22) for λ r and λ r into (18), we obtain the following

..

3 These

relations follow from the general relation σ = J −1 ΣF T , or from the relations between deformed and undeformed area.

263 7 ´ 107

5000 •

Strain Acceleration Hs-2 L

Strain Rate Hs-1 L

••

Axial strain acceleration ez

6 ´ 107

Axial strain rate ez 4000

•

Radial stretch rate Λr

3000

2000

1000

••

Radial stretch acceleration Λr

5 ´ 107

••

Normalized radial stress Λr Λr 4 ´ 107 3 ´ 107 2 ´ 107 1 ´ 107

0

0

50

100

150

200

250

Time HΜsL

(a)

0

300

0

50

100

150

200

250

300

Time HΜsL

(b)

Figure 1: Axial strain and radial stretch rates (a) and accelerations (b) for a smooth ramp-up to a constant nominal axial strain rate ez of 2500/s after 135 µs. Also shown in (b) is the normalized mean radial stress (dashed line).

.

expressions for the mean radial (and hoop) stress in an isotropic specimen undergoing a homogeneous, volumepreserving, axisymmetric deformation: [ ( )2] ] [ 3 λz ρ0 R02 2 ez ρ0 R02 ρ0 R02 −2 λz 3 (ez )2 = . (25) λ r (t) λ r (t) = σ rr (t) = + + 4 16 λz2 λz3 16 (1 − ez )2 (1 − ez )3

.

..

..

..

.

The radial (and hoop) stress distribution is then given by (17). For a given radial or axial strain history, we see that the mean radial stress at any instant is proportional to ρ0 R02 . Thus radial inertial effects can be reduced by decreasing the radius of the specimen.4 Note that the only material property appearing in (25) is the density ρ0 ; in particular, this estimate for the radial stress is independent of the constitutive relation for the specimen. The relation (17), with σ rr (t) given by the expression on the right in (25), is equivalent to relations in Dharan and Hauser [11], Warren and Forrestal [12], and Scheidler and Kraft [13]. Now consider a conventional smooth loading pulse for an SHPB test with rise time t1 . We take time t = 0 to be the instant at which the loading pulse arrives at the specimen-incident bar interface, so that ez (0) = 0. For a smooth loading pulse the nominal axial strain rate and strain acceleration are initially zero also, ez (0) = 0 and ez (0) = 0; and ez (t) increases smoothly with t up to time t1 , after which ez is remains constant at the test strain rate ez1 = ez (t1 ) > 0.5 At early times, the strain acceleration term in (25), ez , dominates as ez increases from its initial value of zero; ez eventually reaches a peak (positive) value and then decays to zero as ez approaches the plateau strain rate ez1 . This peak in ez results in a corresponding early peak in the radial stress. However, while the radial stress initially decreases after this peak, it does not decay to zero with ez since the (ez )2 term in (25) is positive. In fact, since ez is increasing, the (1 − ez )3 term in the denominator is decreasing, and hence the 3(ez )2 /(1 − ez )3 term is strictly increasing, even in the plateau region when ez (t) = ez1 . Consequently, σ rr (t) begins to increase just prior to time t1 and continues to do so while ez remains constant, i.e., there is a strain amplification effect on the inertially generated radial stress. These features are illustrated in Figure 1, where the rise time t1 = 135 µs and the plateau value of the nominal axial strain rate is ez1 = 2500/s. For this strain rate history, ez (t1 ) = 0.17 and ez (300µs) = 0.58. Note that by (25)1 , the product λ r λ r is the mean radial stress σ rr normalized by the factor ρ0 R02 /4; it is this normalized mean radial stress that is plotted in the figure. Also note that by (25)1 and (20)2 , the radial stretch acceleration λ r is the mean value of the nominal radial stress ΣrR normalized by the same factor.

.. .

.

.

. ..

.

. ..

..

..

.

.

. .

.

.

.

. . ..

..

For sufficiently soft materials and sufficiently high strain rates, the inertial effects discussed above must be taken into account when analyzing the data from SHPB tests. In this regard, the following relations for the mean values of the stress components are useful: σ zz = 32 szz + σ rr ,

szz = 23 (σ zz − σ rr ) ,

σ zz = (1 − ez )ΣzZ .

(26)

4 However, reducing the specimen radius also reduces the signal to the transmission bar, so substantial reduction in specimen size must be accompanied by a corresponding reduction in the diameter of the pressure bars. This is one of the motivations for the use of miniaturized Hopkinson bars. 5 For simplicity, we neglect the subsequent unloading stage in both the analysis and the numerical simulations.

264 4 ´ 107

Strain Acceleration Hs-2 L

Strain Rate Hs-1 L

2500

2000

1500

1000

•

Axial strain rate ez •

Radial stretch rate Λr

500

0

0

50

100

150

200

250

300

Time HΜsL

(a)

••

Radial stretch acceleration Λr

2 ´ 10

7

••

Normalized radial stress Λr Λr

1 ´ 107 0 - 1 ´ 107 - 2 ´ 107

350

••

Axial strain acceleration ez

3 ´ 107

0

50

100

150

200

250

300

350

Time HΜsL

(b)

Figure 2: Axial strain and radial stretch rates (a) and accelerations (b) for a smooth ramp-up to a constant nominal radial stretch rate of 1747/s after 135 µs. Also shown in (b) is the normalized mean radial stress (dashed line). These follow from (14) and (20)1 , and may be regarded either as volumetric averages or as cross-sectional averages; the two are equivalent if, as will be assumed here, the specimen is in dynamic equilibrium so that the axial stresses are axially uniform. The well-known relation on the right expresses the mean axial Cauchy stress in terms of the axial strain and the mean axial nominal stress, both of which are measured (or inferred from measurements) in an SHPB test. The relation on the left in (26) implies that if the early spike in σ rr is sufficiently large relative to szz (a situation that could occur for sufficiently high strain rates and sufficiently soft specimens), then a corresponding spike in the measured axial stress is to be expected. These inertial spikes have indeed been observed in SHPB tests on soft, nearly incompressible materials; cf. [4], [8], [9], and [10]. The middle relation in (26), which is equivalent to the relation on the left, indicates that an “inertial correction” must be applied to the (quasi-static) uniaxial stress relation szz = 23 σzz when σ rr is sufficiently large. Since σ rr can be determined from the measured axial strain rate ez by (25)3 , the relation (26)2 could provide a means to estimate the axial component of the deviatoric stress,6 provided the assumptions on which our analysis is based are approximately valid and the difference between σ zz and σ rr is not so small that it is in the noise level. Then the radial and hoop components of deviatoric stress can be determined from (13)1 .

.

4. Optimal Loading Pulses Since a constant nominal axial strain rate does not eliminate inertially generated radial and hoop stresses, it is reasonable to seek axial strain rate histories that do so. On setting σ rr (t) = 0 in (25), we see that the expression on the right yields an ODE for ez : 2 ez + 3(ez )2 /(1 − ez ) = 0. However, it is simpler to proceed as follows. From (25)1 we see that σ rr = 0 iff λ r = 0 iff λ r is constant iff er is constant. Since the loading pulse arrives at t = 0 and since λ r = 1 in the undeformed state, we take λ r (0) = 1. For a smooth loading pulse we cannot impose the constant radial strain rate condition initially. Instead we want λ r (0) = 0 and λ r (0) = 0, with λ r (t) increasing smoothly and monotonically with t up to some time t1 , after which λ r remains constant:

..

..

.

.

.

. .

.

..

.

λ r (t) = λ r (t1 ) > 0 ,

.

for t ≥ t1 .

.

(27)

..

.

Then λz (t) = 1/λ r2 [cf. (19)2 ], and from λz we can determine ez , ez and ez ; alternatively, we can determine ez and ez directly from λ r and its rates by using (24) and the middle relation on (21). Since λ r (t) is positive for t > 0, so is ez (t) [cf. (21)]. However, λ r (t) = 0 for t ≥ t1 , so by (24) we see that ez (t) < 0 for t ≥ t1 ; hence ez (t) is decreasing for t ≥ t1 . Since the initial condition λ r (0) = 0 implies ez (0) = 0, it follows that ez (t) must increase from zero to a peak value at some time tp < t1 , after which ez decreases. In spite of this non-standard feature, the resulting nominal axial strain rate history ez (t) will have the property that for t ≥ t1 , σ rr (t) = 0 and hence [cf. (17)] σrr (r, z, t) = 0 throughout the specimen.

..

..

.

.

.

.

.

..

.

.

.

6 This is essentially what was done in Sanborn [10] for SHPB tests on various rubbers. His “corrected” axial stress is σ zz − σ rr and hence is equivalent to the 32 szz term in (26)1 .

265

Stress (Pa)

x 10

5

8

σzz

6

σθθ

σrr

1.5 szz

4 2 0 150

200

250

300 350 Time (µs)

400

450

Figure 3: Mean values of the axial, radial and hoop stress and of 3/2 the axial deviatoric stress, for the axial strain rate history in Figure 1. The radial and hoop stresses are indistinguishable. For comparison with the more conventional strain rate history considered in Figure 1, we take the rise time to the constant radial strain rate to be the same as the previous rise time to the constant axial strain rate, namely t1 = 135 µs; and we chose the plateau value λ r (t1 ) = 1747/s, since this results in ez (t1 ) = 2500/s. The resulting axial strain and radial stretch rates are shown in Figure 2.a. The corresponding strain and stretch acceleration histories are shown in Figure 2.b along with the normalized mean radial stress, which is nearly indistinguishable from λ r . For this strain rate history, ez (t1 ) = 0.20 and ez (350µs) = 0.55.

.

.

..

9. Numerical Simulations To test the optimal pulse shaping and the inertial correction theory, we performed numerical simulations of hypothetical SHPB tests on a soft, nearly incompressible, solid specimen using the Lagrangean, 3-D finite element code PRESTO from Sandia Laboratories. The initial radius of the specimen was R0 = 6.35 mm and the initial thickness was L0 = 1.45 mm, giving a length-to-diameter ratio of 0.11. The incident and transmission bars were included in the simulation, and the specimen-bar interfaces were treated as frictionless. We used 1000 mm long aluminum bars with a radius of 19.05 mm. The specimen and bar dimensions (except for the bar lengths) were taken from the experimental study on gelatins by Moy et al. [8]. An isotropic, nonlinear elastic model was used for the specimen (a compressible version of the Mooney-Rivlin model). The model was calibrated to give rough agreement with the large strain, quasi-static, uniaxial compression data on the 20% ballistic gelatin tested in [8].7 This resulted in a small strain shear modulus of 80 kPa. Since 20% ballistic gelatin is 80% water, we used the bulk modulus of water, 2.3 GPa; the ratio of bulk to shear modulus is 2.9 × 104 . The loading wave was generated by an imposed axial velocity history vz (t) at the far end of the incident bar. This velocity was obtained from the desired nominal axial strain rate ez using the approximate relation

.

.

vz (t) ≈ 12 L0 ez (t) ,

(28)

which neglects the motion of the specimen-transmission bar interface and assumes the particle velocity doubles on reflection from the specimen-incident bar interface. The nominal axial strain rate histories used in computing vz (t) were those from Figures 1 and 2, but since the relation (28) is only approximate, the actual strain rates and strain accelerations in the specimen would differ somewhat from those in the figures even if the specimen deformation were approximately homogeneous. Figures 3 and 4 plot the histories of σ zz , σ rr , σ θθ and

3 2 szz .

These are the (volumetric) mean values of the radial,

7 Ballistic gelatin is a viscoelastic material, and the quasi-static tests showed some increase in stress with increasing strain rate. The model was calibrated so that the axial stress-strain curve was slightly above the curve for highest (quasi-static) strain rate of 1/s.

266 5

2.5

x 10

Stress (Pa)

2 1.5

σzz σrr

1

σθθ

0.5

1.5 szz

0 −0.5 150

200

250

300 350 Time (µs)

400

450

Figure 4: Mean values of the axial, radial and hoop stress and of 3/2 the axial deviatoric stress, for the axial strain rate history in Figure 2. The radial and hoop stresses are indistinguishable. hoop and axial stress and 3/2 the mean value of the axial deviatoric stress, as computed in the simulations. Recall that σ zz = 32 szz for a uniaxial stress state; cf. (26)1 with σ rr set to zero. The times in Figures 3 and 4 are shifted relative to those in Figures 1 and 2, since in the simulations t = 0 is the instant at which the velocity is applied at the far end of the incident bar. Figure 3 is for a nominal axial strain rate history given (approximately) by that in Figure 1.a, that is, when ez is eventually constant. The radial and hoop stresses are indistinguishable, and they increase substantially after the inertial spike. The compressed specimen began squeezing out beyond the bars at around 453 µs.

.

.

Figure 4 is for a nominal axial strain rate history given (approximately) by that in Figure 2.a, that is, when λ r is eventually constant (an optimal pulse shape). The radial and hoop stresses are again indistinguishable, but now they drop to nearly zero after the inertial spike, that is, once λ r reaches its plateau value, which occurs at a nominal strain of 0.20. Thereafter, σ zz is very close to 32 szz . Both of these facts indicate that a nearly uniaxial stress state has been achieved. When comparing Figures 3 and 4, keep in mind that the scales on the vertical axis are different. The inertial spike in Figure 4 is only slightly larger than that in Figure 3. For the case considered in Figures 1 and 3,

.

x 10

5

s

Stress (Pa)

8

zz

szz|uniaxial

szz|corrected

6 4 2 0 150

200

250

300 350 Time (µs)

400

450

Figure 5: The computed mean axial stress (szz ) compared with two estimates for it. Figure 5 provides a check on the inertial correction (26)2 for the axial deviatoric stress: szz = 23 (σ zz − σ rr ). In the

267 figure legend, szz denotes the mean axial deviatoric stress szz as computed in the simulation. szz |uniaxial denotes an estimate for the deviatoric stress that is only valid for a uniaxial stress state, namely 23 σ zz . It clearly disagrees with szz and shows the error that would result in neglecting inertial effects in the analysis of the data. Finally, szz |corrected denotes the estimate for the deviatoric stress obtained from the inertial correction above, using σ zz computed in the simulation and σ rr determined from (25)3 . It is quite close to the actual mean value for most of the simulation. Near the end of the simulation the conditions departed substantially from those on which the analysis was based: the specimen bulged and was no longer in dynamic equilibrium. 10. Discussion and Conclusions The results of the analysis and the preliminary numerical simulations indicate that radial inertial effects in SHPB tests on soft, nearly incompressible materials could be eliminated (after a preliminary inertial spike) by tailoring the loading pulse so that the nominal radial strain rate in the specimen is eventually constant. The corresponding axial strain rate history is easily determined analytically and can be easily (though only approximately) imposed computationally. Whether such loading pulses can be generated experimentally with use of pulse shapers remains to be seen. Recently, Casem [14] has developed a technique for tailoring loading pulse shapes by means of graded impedance striker bars. This method, in conjunction with conventional pulse shapers, may possibly provide a means to generate the pulse shapes considered here. References [1] Gray III, G. T., Classic Split-Hopkinson Pressure Bar Testing. In H. Kuhn and D. Medlin, editors, ASM Handbook Vol. 8, Mechanical Testing and Evaluation, pages 462–476. American Society for Metals, Materials Park, Ohio, 2000. [2] Gama, B. G., Lopatnikov, S. L. and Gillespie Jr., J. W., Hopkinson bar experimental technique: A critical review. Appl. Mech. Rev., 57:223–250, 2004. [3] Ramesh, K. T., High strain rate and impact experiments. In W. N. Sharpe, editor, Springer handbook of Experimental Solid mechanics, chapter 33, pages 1–30. Springer, New York, 2009. [4] Chen, W. and Song, B., Split Hopkinson (Kolsky) Bar. Springer, New York, 2011. [5] Gray III, G. T. and Blumenthal, W. R., Split-Hopkinson Pressure Bar Testing of Soft Materials. In ASM Handbook Vol. 8, Mechanical Testing and Evaluation, pages 488–496. American Society for Metals, Materials Park, Ohio, 2000. [6] Chen, W., Lu, F., Frew, D. J. and Forrestal, M. J., Dynamic compression testing of soft materials. Exp. Mech., 69:214–223, 2002. [7] Song, B. and Chen, W., Split Hopkinson pressure bar techniques for characterizing soft materials. Latin Am. J. Solids Struct., 2:113–152, 2005. [8] Moy, P., Weerasooriya, T., Juliano, T. F., VanLandingham, M. R. and Chen, W., Dynamic Response of an Alternative Tissue Simulant, Physically Associating Gels (PAG). In Proc. of the 2006 SEM Annual Conference, 2006. [9] Song, B., Ge, Y., Chen, W. W. and Weerasooriya T., Radial Inertia Effects in Kolsky Bar Testing of Extra-soft Materials. Exp. Mech., 47:659–670, 2007. [10] Sanborn, B., An Experimental Investigation of Radial Deformation of Soft Materials in Kolsky Bar Experiments. Master’s thesis, Purdue Univ., Weast Lafayette, IN, 2010. [11] Dharan, C. K. H. and Hauser, F. E., Determination of stress-strain characteristics at very high strain rates. Exp. Mech., 10:370–376, 1970. [12] Warren, T. L. and Forrestal, M. J., Comments on the effect of radial inertia in the Kolsky bar test for an incompressible material. Exp. Mech., 50:1253–1255, 2010.

268 [13] Scheidler, M. and Kraft, R., Inertial effects in compression Hopkinson bar tests on soft materials. In Proceedings of the 1st ARL Ballistic Protection Technologies Workshop, 2010. [14] Casem, D. T., Hopkinson bar pulse-shaping with variable impedance projectiles—an inverse approach to projectile design. Technical Report ARL-TR-5246, US Army Research Laboratory, 2010.

Dynamic Tensile Characterization of Foam Materials

Bo Song, Helena Jin, Wei-Yang Lu Sandia National Laboratories, Livermore, CA 94551-0969, USA Polymeric foams have been widely utilized in packaging and transportation applications due to their light weight but superior energy absorption capabilities. Here the superior energy absorption is usually recognized when the foams are subjected to compression. The polymeric foams possess load-bearing capability under large deformation in compression; whereas, such a load-bearing capability may be lost when the foams are subjected to tensile loading [1]. Dynamic compressive response of foam materials has been extensively characterized, mostly with Kolsky compression bar techniques. However, dynamic tensile characterization of foam materials has 3 been less touched due to the difficulties in dynamic tension techniques. In this study, we employed a 0.26×10 3 kg/m PMDI foam as an example to explore the dynamic tensile characterization of foam materials with our newly developed Kolsky tension bar [2, 3]. The Kolsky tension bar developed at Sandia National Laboratories, California has been presented at 2010 SEM Annual Conference [2]. This Kolsky tension bar has been utilized to characterize alloys, e.g., 4330-V steel [3]. However, the techniques used for the alloys may not be directly transferred to polymeric foams due to drastic difference in mechanical characteristics between alloys and polymeric foams. Characterization of foam materials always challenges experimental techniques even in dynamic compression tests. Dynamic tensile characterization of foam materials is much more challenging. For example, the foam tensile specimen needs to be sufficiently big. The large diameter requires a Kolsky bar system with a larger diameter, while a longer gage section may not satisfy the uniform deformation requirement in Kolsky bar experiments. Therefore, the specimen dimension needs to be carefully determined. Unlike the button specimen sandwiched between the bars in Kolsky compression bar tests, the foam tensile specimen needs to be firmly attached to the ends of both bars. Stress concentration is another concern in foam tensile characterization. Both require additional attention in specimen design. In this study, we designed the foam specimen shown in Fig. 1 for Kolsky tension bar experiments. It is noted that the diameter for the threads portion of the foam specimen is 25.4 mm, which is larger than the bar diameter of 19.1 mm. We employed a pair of couplers to attach the foam specimen to the bar ends, the photograph of which is shown in Fig. 2.

Fig. 1. Foam tensile specimen

Fig. 2. Photograph of the testing section

T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_38, © The Society for Experimental Mechanics, Inc. 2011

269

270 As shown in Fig. 1, the foam tensile specimen has a gage length of 6.35 mm. This gage length was determined based on the preliminary tests. In preliminary experiments, we applied digital image correlation (DIC) techniques on a 9.53-mm-long foam specimen to estimate the elastic wave speed and to monitor the uniformity of deformation in the foam specimen. DIC analysis yields the elastic wave speed of approximately 525 m/s in the foam specimen. This elastic wave speed is important to synchronize stress and strain histories in data reduction. Figure 3 shows the deformation history of the 9.53-mm-long foam specimen obtained from DIC analysis. The DIC results show that the 9.53-mm-long foam specimen did not achieve uniform deformation until t = 46.2 µs. After 46.2 µs, the specimen is nearly under uniform deformation until macroscopic crack initiation at t = 79.2 µs. To be conservative, we reduced the specimen gage length from 9.53 mm to 6.35 mm, which enables earlier uniform deformation (or stress equilibrium). It is noted that the classic “2-wave” “1-wave” method may not be applicable to verify stress equilibrium in the foam experiment due to the weak transmitted signal and unreliable reflected pulse. The DIC analysis presented here becomes an effective alternative method to monitor the process of deformation uniformity in the foam specimen.

T = 0 µs

T = 19.8 µs

T = 39.6 µs

T = 46.2 µs

T = 72.6 µs

T = 79.2 µs

Fig. 3. Axial Deformation (Exx) History in the 9.53-mm-long Foam Specimen Since the reflected pulse may not be reliable for accurate calculation of specimen strain [1], we employed a laser-beam system, as shown in Fig. 2, to directly measure the incident bar end displacement. The specimen strain can be calculated with the following equation, t

ε (t ) =

c′ ⋅ (∆L(t ) − C0 ∫ ε T (τ )dτ ) Ls

0

(1)

where ΔL is the laser beam output; εT is transmitted signal; C0 is the steel bar elastic wave speed; Ls is the specimen gage length, Ls = 6.35 mm; c’ is correction coefficient. As shown in Fig. 1, the foam specimen has a dumbbell shape consisting of a gage section and a transition (non-gage section) from threads to the gage section. When the specimen is subjected to tension, both gage and non-gage sections are elongated. The term, t

ΔL(t ) − C0 ε T (τ ) dτ , in Eq. (1) describes the overall displacement over both gage and non-gage sections.



0

However, the displacement of the non-gage section should not be accounted for the strain calculation of the specimen. Here, the coefficient, c’, is used to correct this effect. If we assume the foam specimen is always in linear elasticity and in stress equilibrium, the coefficient c’ was approximated as a constant, c’=0.4, for the foam

271

specimen specified in Fig. 1. This means only 40% of the overall deformation contributes to the deformation over the gage section. It is noted that, in order to measure the weak transmitted pulse, we employed a pair of semiconductor strain gages on the transmission bar. Figure 4 shows the oscilloscope records of the incident, reflected, and transmitted signals (Fig. 4(a), as well as the laser-beam system output (Fig. 4(b)).

(a)

(b) Fig. 4. Oscilloscope records

Figure 5 shows the resultant tensile stress3 3 strain curve of the 0.26×10 kg/m PMDI foam -1 specimen at the strain rate of 400 s . The foam specimen exhibits a nearly linear elastic behavior with a small failure strain of about 1.5%. It is clearly shown that the tensile stress-strain response of the foam specimen is quite different from the compressive response. The “classic” compaction response in compression changes to be brittle in tension for the PMDI foam material.

ACKNOWLEDGEMENTS Sandia National Laboratories is a multiprogram laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000.

Fig. 5. Tensile stress-strain curve at 400 s

-1

REFERENCES nd

1. Gibson, L. J., and Ashby, M. F., 1999, Cellular Solids, Structure and Properties, 2 ed, Cambridge. 2. Song, B., Antoun, B. R., Connelly, K., Korellis, J., and Lu, W.-Y., 2010, “A Newly Developed Kolsky Tension Bar,” In: Proceedings of 2010 SEM Annual Conference and Exposition on Experimental and Applied Mechanics, Indianapolis, IN, June 7-10, 2010. 3. Song, B., Antoun, B. R., Connelly, K., Korellis, J., and Lu, W.-Y., 2011, “Improved Kolsky Tension Bar for High-rate Tensile Characterization of Materials,” Measurement Science and Technology, 22, 370304 (7pp), in press

On Measuring the High Frequency Response of Soft Viscoelastic Materials at Finite Strains

Sean Teller, Rod Clifton, and Tong Jiao School of Engineering, Brown University 182 Hope St., Providence, RI 02912 *Corresponding author: [email protected] ABSTRACT In such applications as repairing damaged vocal folds to restore normal phonation, viscoelastic properties of normal tissues and candidate replacement materials should be measured at phonation frequencies (100 – 1000 Hz) and at the large strains (up to 30%) that occur during speech and singing. Previously the authors have developed a torsional wave experiment to measure the complex moduli, in shear, for vocal folds subjected to small strains at phonation frequencies. This method has now been extended to finite deformations by sandwiching a thin disk of vocal fold tissue (lamina propria), or replacement material, between two rigid plates. The lower plate is driven by a galvanometer at phonation frequencies and small rotations. A second stiffer material is placed between the upper plate and a third plate attached to an upper galvanometer that oscillates sinusoidally at low frequency and large rotation. At periodic peak rotations of the upper galvanometer, the lower galvanometer superimposes infinitesimal oscillations at a series of higher frequencies. The magnitude and phase of rotation of the middle plate yield the viscoelastic properties of the test specimen for infinitesimal deformations at high frequency superimposed on finite deformations at low frequency. Preliminary results show th e potential of the new test. INTRODUCTION Vocal folds are a soft, layered, viscoelastic mucous membrane that are stretched horizontally across the larynx. The folds a re composed of a vibratory layer, known as the lamina propria (LP), sandwiched between the vocalis muscle and an epithelium membrane [1,2]. During phonation, the LP are driven into a shear-dominated wavelike motion at frequencies between 75 and 1000 Hz and large strains of up to 30%[3]. Due to overuse, abuse, and surgery, the LP can become damaged, and scar tissue can form hindering the use of the vocal folds and adversely affecting voice quality[4,5]. Tissue engineering holds great promise for the repair and replacement of damaged tissue, but the unique bio chemical and mechanical properties of the vocal folds create challenges in the development of candidate replacement materials and testing of natural tissues. As demonstrated in [6], typical shear rheometry is not applicable for vocal fold tissues (and similar synthetic materials) due to the deformation not remaining uniform through the thickness of the sample at phonation frequencies. The Torsional Wave Experiment (TWE)[6] was developed to overcome these difficulties. It has enabled the authors to measure the complex modulus of soft synthetic materials [6-10] and natural tissues [10] over the lower part of the range of phonation frequencies. Although the TWE has proven useful, to completely characterize natural and synthetic materials the shear modulus at strains comparable to those experienced in situ are needed. To this end, consider the experimental setup shown in Figure 1, the Finite Strain TWE (FSTWE). A thin, cylindrical disk of the viscoelastic sample material, in blue, is sandwiched between two rigid plates. Above the middle plate is another cylindrical disk of a known material, also sandwiched between two rigid plates. Both the top and bottom plates are attached to galvanometers. During a test of a sample, the top galvanometer is slowly rotated through a large angle, rotating the midd le plate and inducing a strain field in both materials. At peak rotation of the top galvanometer, the bottom galvanometer oscillates sinusoidally at phonation frequency (much higher than the top galvanometer) sending a torsional wave through the sample. The upper galvanometer continues its periodic rotation, unloading the sample. Upon return to the peak rotation, the lower galvanometer again oscillates, but at a different frequency. Measurement of the amplitude of rotation of the middle plate, and the phase difference between the middle plate and the bottom plate, yield the viscoelastic tangent modulus in shear at a finite strain.

T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_39, © The Society for Experimental Mechanics, Inc. 2011

273

274

Fig. 1 Schematic for finite strain TWE. Sample material is blue, a known elastic material is brown, and rigid plates are white To simplify the solution to the problem, the known material should possess three qualities: the material should be elastic, significantly stiffer than the sample material, and should be stiff enough in torsion so that the stress distribution is uniform throughout the thickness of the sample. The elastic requirement is needed so that analysis of the proposed test is simplified and tractable. The stiffness requirement is necessary so that the majority of the rotation applied by the upper galvanometer is transferred to the middle plate. These factors are discussed further below in the “Experimental Considerations” section. MATHEMATICAL DESCRIPTION OF FINITE STRAIN TORSIONAL WAVE EXPERIMENT Following the example of [6], the additional torsional moment T1(z,t) due to the shear wave propagating through the sample at any cross section z and time t is a

T1(z, t ) = 2π ∫ r 2 τ1(z, r, t )dr 0

(1)

where a is the radius of the sample, r is the radial coordinate, and τ1(z,r,t) is the shear stress. A subscript ‘1’ indicates that the quantity described is that of the sample material, while a subscript ‘2’ denotes properties and quantities in the known elastic material. Assuming the material is viscoelastic[11], and that the deviation from the large strain can be approximated by a tangent modulus, the shear stress is given as

τ1 (z, r , t ) =

t

∫ 0G1(t − t ')γ1(z, r, t ')dt '

(2)

where G1(t) is the viscoelastic tangent shear modulus, γ1 (r, z, t ) is the shear strain history, and a superposed dot indicates a derivative with respect to time. The deformation applied is assumed to be pure torsion, and the amount of rotation of the cross section is θ1 (z , t ) . From the kinematic constraint of pure torsion, the shear strain history is then given as ∂v (z, r, t ) ∂θ (z , t ) ∂Ω1(z, t ) γ1(z, r, t ) = 1 =r 1 =r (3) ∂z ∂z ∂z where v1(z,r,t) is the circumferential velocity and Ω1(z,t) is the angular velocity. Substituting Equations (2) and (3) into (1), the torque is then t ∂Ω (z, t ) T1 (z, t ) = J 1 ∫ G1(t − t ′) 1 dt ′ (4) 0 ∂z where J 1 =

πa 4 is the polar moment of inertia of the cross section. Considering the balance of angular momentum yields 2

the equation

ρJ 1

∂Ω1 (z , t ) ∂t

=

∂T1 (z , t ) ∂z

Combining Equations (4) and (5), the equation of motion for the sample is

.

(5)

275

ρ1

∂Ω1(z, t ) ∂t

=

t

∫0 G1(t − t ')

∂2Ω1(z, t ) ∂z 2

dt ′.

(6)

The solution of (6) requires two boundary conditions: the first is the imposed rotation at the bottom plate, and the second is due to the inertial force of the rigid plate and the torque applied from the known material. These are described as

θ1 (0, t ) = θ0 exp(i ωt ) ρ0 I 0

∂Ω1(h, t ) ∂t

t

∂Ω1(z , t ′)

0

∂z

= −J 1 ∫ G (t − t ′)

(7)

dt ′ − θ1 (h, t ) z =h1

J 2G2 h2

(8)

where θ0 and ω0 are the amplitude and frequency of the applied rotation; ρ0 is the mass density and I0 the polar moment of inertia of the middle plate; J2, G2, and h2 are the polar moment of inertia, shear modulus, and thickness of the known material, respectively. Equations (6)-(8) are most easily solved in Laplace space. Taking the Laplace transform of these equations yields, respectively, ∂2θ1(z, s ) (9) ρ1s θ1 (z, s ) = G1(s ) ∂z 2 θ0 θ1(0, s ) = (10) s − i ω0 ∂θ (z, s ) JG (11) ρ0I 0s 2 θ1(h1, s ) = −J 1G1(s )s 1 − θ1(h1, s ) 2 2 ∂z z = h h2 1

 has been replaced by sθ . The solution to Equation (9) is where a superposed tilde (~) denotes the Laplace transform, and Ω 1 1 θ1 (z, s ) = A cosh λz + B sinh λz where λ =

(12)

ρ1s . The unknown coefficients A and B are found using the boundary conditions (10) and (11). Utilizing the  G1(s )

boundary condition at the lower galvanometer, one obtains

A(s ) =

θ0 s − i ω0

.

(13)

Substituting (11) into (12) yields

J 2G2 cosh λh1 h2 B = −AB = −A . JG ρ0I 0s 2 sinh λh1 + sJ 1G1 (s )λ cosh λh1 + 2 2 sinh λh1 h2 The solution (12) is then inverted, using the definition of the inverse Laplace transform ε +i ∞ 1 exp(st )θ1 (z, s )ds θ1 (z, t ) = ∫ 2πi ε −i∞ ρ0I 0s 2 cosh λh1 + sJ 1G1 (s )λ sinh λh1 +

(14)

(15)

where ε is a positive real number to the right of the imaginary axis, so that the real part of all the poles are less than ε. The contour can be closed in the left half plane at ∞, and it can be shown that there are no contributions from branch cuts. Then, from the Residue theorem, the integral in (15) is 2πi times the sum of the residue at the pole s = iω0 To simplify the expression for the solution, we use the definitions: G1* (ω0 ) =| G1* (ω0 ) | e i δ(ω0 ) = i ω0G1 (i ω0 )

λ(i ω0 ) = i ω0

ρ1 * G1 (ω0 )

(16) (17)

276

ξ = ω0h1

ρ1 * | G1 (ω0 )

(18)

|

δ z sin h1 2 z δ βˆ = zˆβ = cos h1 2

αˆ = zˆα =

(19)

where G*(ω0) is the complex shear modulus, which can be represented in terms of its magnitude |G*(ω0)| and the phase shift δ(ω0)[12]. The solution can be further simplified by noting that

λz = i ξ ( αˆ − i βˆ ) . λh1 = i ξ ( α − i β )

(20)

θ1(z, t ) = θ0eiω0t ( cosh λz − B sinh λz )

(21)

The solution to Equation (6) can now be written as with the complex constant B B =

N r + iN i Dr + iDi

=

N r Dr + N i Di Dr2

+

Di2

+i

N i Dr − N i Dr Dr2 + Di2

= Br + iBi

(22)

where

N r = −d cos(ξα) cosh(ξβ ) − c β cos(ξα) sinh(ξβ ) − cα sin(ξα) cosh(ξβ ) − k cos(ξα) sinh(ξβ ) N i = −d sin(ξα) sinh(ξβ ) − c β sin(ξα) cosh(ξβ ) + cα cos(ξα) sinh(ξβ ) − k sin(ξα)sinh(ξβ ) Dr = −d cos(ξα) sinh(ξβ ) − c β cos(ξα) cosh(ξβ ) − cα sin(ξα) sinh(ξβ ) + k cos(ξα)sinh(ξβ ) Di = −d sin(ξα) cosh(ξβ ) − c β sin(ξα)sinh(ξβ ) + cα cos(ξα) cosh(ξβ ) + k sin(ξα)cosh(ξβ ).

(23)

are real numbers. The constants c1, d, and k are given by

c = ω0J 1 ρ1 | G * (ω0 ) | d = ρ0I 0 ω02 JG k = 2 2. h2

(24)

With Equation (20) the solution can be written as θ1(z , t ) = θ0 ⎣⎡ E cos ω0t + F sin ω0t ⎦⎤ + i θ0 ⎡⎣ P cos ω0t + Q sin ω0t ⎤⎦

(25)

where (see footnote 1)

E = cos(ξαˆ) cosh(ξβˆ) − Br cos(ξαˆ) sinh(ξβˆ) + Bi sin(ξαˆ) cosh(ξβˆ) F = − sin(ξαˆ) sinh(ξβˆ) + B sin(ξαˆ) cosh(ξβˆ) + B cos(ξαˆ) sinh(ξβˆ) r

i

P = sin(ξαˆ) sinh(ξβˆ) − Br sin(ξαˆ) cosh(ξβˆ) − Bi cos(ξαˆ) sinh(ξβˆ) Q = cos(ξαˆ) cosh(ξβˆ) − Br cos(ξαˆ) sinh(ξβˆ) + Bi sin(ξαˆ) cosh(ξβˆ).

(26)

In considering the full solution, it is helpful to assume that the excitation at the bottom plate is the real part of the harmonic excitation described in (7). Then, the real part of the solution (25) becomes θ1 (z , t ) = θ0M (z , ω0 ) cos ⎡⎣ ω0t − φ(z , ω0 ) ⎤⎦

(27)

where M(z,ω0) and φ(z, ω0) are the amplification factor and phase shift at height z and excitation frequency ω0. They are defined as 1

Note that errors in [6] have been corrected here

277

M (z, ω0 ) =

E2 + F2

(28)

F tan ⎡⎣ φ(z , ω0 ) ⎤⎦ = . E

(29)

From measurements of the amplitude and phase of rotation of the middle plate relative to the bottom plate, the real and imaginary parts of the complex shear modulus can be calculated. DISCUSSION To facilitate the discussion of the above solution for the finite strain torsional wave experiment, we define several new dimensionless variables in addition to the dimensionless frequency ξ defined in (18). Similar to [6], we introduce the parameter μ, a measure of the inertia of the sample relative to that of the middle plate, as ρ0 I 0 ω02 ρI ρ1 d = = 0 0 ω0h1 = μξ. c | G1* (ω0 ) | ω0J 1 ρ1 | G1* (ω0 ) | J 1h1ρ1

(30)

Similarly we introduce κ, a measure of the elastic torsional stiffness of the known material to that of the sample, as JG J 2G2h1 k 1 1 = 2 2 = * c h2 ω J ρ | G * (ω ) | ω J 1 | G1 (ω0 ) | h2 0h1 0 1 1 1 0

| G1* (ω0 ) | ρ1

=

κ . ξ

(31)

From Equations (18), (30), and (31) the final solution (27) can be shown to be a function of only five parameters: ξ, δ, μ, κ, and zˆ . Figure 2 shows the dependence of the amplification factor at the middle plate ( zˆ = 1 ) on the dimensionless frequency ξ. Figure 2a shows how the loss angle affects the amplification factor, while 2b shows the effect of the relative torsional stiffness κ. Here, a value of κ=0 indicates that the known material is not present. Due to the increasing damping and energy loss in the system, increasing loss angles decreases the amplification factor with little change in the resonance frequency. As the relative stiffness of the sample decreases, the resonance frequency increases. Figure 3 shows the phase angle at the middle plate as a function of frequency ξ while varying κ. Similar to Figure 2b, this figure shows that increasing the relative stiffness increases the resonance frequency of the system.

Fig. 2 Amplification factor at the middle plate as a function of frequency ξ and (a) loss parameter δ and (b) parameter κ. For both figures: inertia parameter μ=3.0, sample parameters a1=3 mm, h1=0.5 mm, ρ=1000 kg/m3, |G1*|=1 kPa, middle plate parameters ρ0 =1040 kg/m3, l=2.629 mm, h0=2mm. (a) For the known material: G2 = 10 kPa, a2 = 3 mm, h2 = 0.5 mm. Torsion constant κ=10. (b) δ=0.3

278

Fig. 3 Phase angle φ as a function of frequency ξ and parameter κ. Values are the same as those used in Figure 2(b)

Rotation of the middle plate is measured using an optical lever technique depicted in Figure 4. One surface of a hexagonal plate is polished to a mirror finish and coated with a thin (~200 nm) layer of metal to provide a reflective surface. A laser beam is brought in and reflected off the mirrored surface. This reflection then passes through a spherical lens that first focuses then expands the beam. The beam then passes through a cylindrical lens, which creates a thin vertical laser line at the focal plane of the lens. As the plate oscillates, the laser line moves back and forth over a mask-covered photodiode detector that gives a voltage change proportional to the rotation of the middle plate. In this way, small rotations are measured to accuracies of the order of 3.5 milliradians. To determine the unknown material properties |G1*(ω0)| and δ(ω0), the amplification factor is recorded over a wide range of frequencies that includes the resonance frequency via the optical lever technique described above. The amplification factor is then fit with the linear viscoelastic model analytical solution described by (27)-(29). This method assumes a constant complex modulus over this range of frequencies, but small changes with respect to frequency are expected and have been observed in the current method [6-10]. To vary frequencies at which the moduli are determined, the specimen sizes can be varied to affect the resonant frequencies. EXPERIMENTAL CONSIDERATIONS Returning to the discussion in the “Introduction” of the choice of the known upper material, we first focus on the requirement that the material is sufficiently stiff to allow for the assumption of a uniform strain distribution in the upper material, so that a stress wave analysis is not needed for interpreting the deformation of this material. This condition is typically satisfied in mechanical testing by ensuring that the natural frequency determined by the round trip transit time through the thickness of the specimen is much greater than the driving frequency of the applied loading. For an elastic cylinder in torsion this requirement is [13] c G2 / ρ2 (32) f  s2 = 2h2 2h2 where f is the driving frequency, and the properties of the upper material, denoted by subscript ‘2’ are as follows: cs2 is the shear wave speed, G2 is the shear modulus, ρ2 is the mass density, and h2 is the height. For practical purposes it is also important that the strain in the sample be much greater than in the upper material, otherwise the loading configuration would be too soft to impose the deformation on the sample, i.e. the lower material. At the middle plate, we have two conditions that must be met: the total torque applied by both materials must be equal and opposite, and the angle of twist (denoted as ψ ) of the top material must be much smaller than that of the sample, expressed as ψ2  ψ1 . If the loading of the sample material to finite strain is considered to be done quasi-statically and for simplicity, the sample is modeled as neo-Hookean then one can use Rivlin’s solution [14] for large rotations of a circular cylinder to relate the rotations and impose the constraint on the relative rotations:

ψ2 ψ1

=

4 G1J 1h2 1 9 G2J 2h1

(33)

279

Fig. 4 Optical lever technique schematic. Sample and known material in blue and rigid plates are white where G1 is the neo-Hookean modulus, taken to be equal to the magnitude of the complex modulus, |G1(ω0)|. The quotient (33) is similar to the inverse of the parameter κ defined in (31). Together, Equations (32) and (33) indicate that the upper material should have a high shear modulus and a thin, wide geometry. In addition to these constraints, if κ is too large, the amplification factor (both at the resonance peak and the entire response) at the middle plate is greatly reduced. This makes finding the resonance frequency and measuring the amplification factor difficult. To avoid this difficulty, κ should be below 100. Because viscoelastic properties are sought throughout the entire phonation range, the maximum value for ω is 1000 Hz. Typical vocal fold specimens have values for ρ, |G(ω0)|, h1, and a that are approximately 1000 kg/m3, 1000 Pa, 0.5 mm, and 2.5 mm, respectively. Based on the above limitations on potential values for two materials are shown in Table 1. Table 1 Material properties and geometrical sizes of candidate upper materials. All upper sample materials had a radius of 2.5 mm G2 (kPa) ρ2 (kg/m3) h2 (mm) κ Material 1 10 2000 0.25 20 Material 2 100 2000 0.5 100 In addition to the requirements above, the candidate upper material should be easily made, and tunable based on the size and expected properties of the sample material. Three potential materials that are frequently used in cell cultures are polyacrylamide (PA) gels[15-18], polydimethylsiloxane (PDMS)[19], and hyaluronic acid (HA) based gels[20]. These all have material properties in the ranges that are described in Table 1, and meet the ease of use requirements. CONCLUSION A novel experiment to determine the material properties at finite strains of soft viscoelastic materials has been developed, with the intent to test vocal folds at physiologically relevant strains and frequencies. A brief discussion of the mathematical model shows the relevancy and potential of the test. Briefly, the test allows for the stress-strain curves of a single sample to be measured in one experimental configuration. The frequency dependence can be determined by varying the sample size of the specimen to affect the resonance frequency. Implementation and development of the experimental apparatus is ongoing; preliminary results are expected to be presented at the conference. ACKNOWLEDGEMENTS The work was funded by a sub-contract from the University of Delaware on NIH grant R01 008965.

280

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Gray S. D., 2000, “Cellular Physiology of the Vocal Folds.,” Otolaryngologic Clinics of North America, 33(4), pp. 679-98. Hirano M., 1981, “Structure of the Vocal Fold in Normal and Disease States: Anatomical and Physical Studies,” ASHA Rep. Titze I. R., 1994, Principles of Voice Production, Prentice Hall, New Jersey. Hirano S., 2005, “Current Treatment of Vocal Fold Scarring,” Current Opinion in Otolaryngology & Head and Neck Surgery, 13(3), pp. 143-147. Benninger M. S., Alessi D., Archer S., Bastian R., Ford C., Koufman J., Sataloff R. T., Spiegel J. R., and Woo P., 1996, “Vocal Fold Scarring: Current concepts and management,” Otolaryngology - Head and Neck Surgery, 115(5), pp. 474-482. Jiao T., Farran A., Jia X., and Clifton R. J., 2009, “High Frequency Measurements of Viscoelastic Properties of Hydrogels for Vocal Fold Regeneration.,” Experimental Mechanics, 49(2), pp. 235-246. Grieshaber S. E., Nie T., Yan C., Zhong S., Teller S. S., Clifton R. J., Pochan D. J., Kiick K. L., and Jia X., 2011, “Assembly Properties of an Alanine-Rich, Lysine-Containing Peptide and the Formation of Peptide/Polymer Hybrid Hydrogels,” Macromolecular Chemistry and Physics, 212(3), pp. 229-239. Farran A. J. E., Teller S. S., Jha A. K., Jiao T., Hule R. A., Clifton R. J., Pochan D. P., Duncan R. L., and Jia X., 2010, “Effects of matrix composition, microstructure, and viscoelasticity on the behaviors of vocal fold fibroblasts cultured in three-dimensional hydrogel networks.,” Tissue Engineering. Part A, 16(4), pp. 1247-61. Jha A. K., Hule R. a, Jiao T., Teller S. S., Clifton R. J., Duncan R. L., Pochan D. J., and Jia X., 2009, “Structural Analysis and Mechanical Characterization of Hyaluronic Acid-Based Doubly Cross-Linked Networks.,” Macromolecules, 42(2), pp. 537-546. Jia X., Yeo Y., Clifton R. J., Jiao T., Kohane D. S., Kobler J. B., Zeitels S. M., and Langer R., 2006, “Hyaluronic acid-based microgels and microgel networks for vocal fold regeneration.,” Biomacromolecules, 7(12), pp. 3336-44. Fung Y. C., 1993, Biomechanics: Mechanical Properties of Living Tissues, Springer. Pipkin A. C., 1986, Lectures on Viscoelasticity Theory (Applied Mathematical Sciences), Springer. Clifton R. J., Jia X. Q., Jiao T., Bull C., and Haln M. S., 2006, “Viscoelastic Response of Vocal Fold Tissues and Scaffolds at High Frequencies,” Mechanics of Biological Tissue, G.A. Holzapfel, and R.W. Ogden, eds., Springer, New York, pp. 445-455. Rivlin R. S., 1948, “Large Elastic Deformations of Isotropic Materials. III. Some Simple Problems in Cylindrical Polar Co-Ordinates,” Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 240(823), pp. 509-525. Engler A., Richert L., Wong J., Picart C., and Discher D., 2004, “Surface probe measurements of the elasticity of sectioned tissue, thin gels and polyelectrolyte multilayer films: Correlations between substrate stiffness and cell adhesion,” Surface Science, 570(1-2), pp. 142-154. Beningo K. a, Lo C.-M., and Wang Y.-L., 2002, “Flexible polyacrylamide substrata for the analysis of mechanical interactions at cell-substratum adhesions.,” Methods in cell biology, Elsevier Science, pp. 325-39. Kandow C. E., Georges P. C., Janmey P. A., and Beningo K. A., 2007, “Polyacrylamide Hydrogels for Cell Mechanics,” Methods in Cell Biology, VOL 83., Elsevier, pp. 29-46. Lo C.-M., Wang H.-B., Dembo M., and Wang Y.-L., 2000, “Cell Movement Is Guided by the Rigidity of the Substrate,” Biophysical Journal, 79(1), pp. 144-152. Tanaka Y., Morishima K., Shimizu T., Kikuchi A., Yamato M., Okano T., and Kitamori T., 2006, “Demonstration of a PDMS-based bio-microactuator using cultured cardiomyocytes to drive polymer micropillars.,” Lab on a chip, 6(2), pp. 230-5. Park Y. D., Tirelli N., and Hubbell J. a, 2003, “Photopolymerized hyaluronic acid -based hydrogels and interpenetrating networks.,” Biomaterials, 24(6), pp. 893-900.

The Blast Response of Sandwich Composites With a Graded Core: Equivalent Core Layer Mass vs. Equivalent Core Layer Thickness

Nate Gardner and Arun Shukla Dynamic Photomechanics Laboratory, Dept. of Mechanical, Industrial & Systems Engineering University of Rhode Island, 92 Upper College Road, Kingston, RI 02881, USA [email protected] ABSTRACT In the present study, the dynamic behaviors of two types of sandwich composites made of E-Glass Vinyl-Ester (EVE) face sheets and Corecell TM A-series foam were studied using a shock tube apparatus. The materials, as well as the core layer arrangements, and overall specimen dimensions were identical; the only difference arises in the core layers, where one configuration has equivalent core layer thickness, and the other configuration has equivalent core layer mass. The foam core itself was layered based on monotonically increasing the acoustic wave impedance of the core layers, with the lowest wave impedance facing the shock loading. A high-speed side-view camera system along with a high-speed back-view Digital Image Correlation (DIC) system was utilized to capture the real time deformation process as well as mechanisms of failure. Post mortem analysis was carried out to evaluate the overall blast performance of these two configurations. The results indicated that with a decrease in areal density of ~ 1 kg/m2 (5%) from the sandwich composites with equivalent core layer thickness to the sandwich composites with equivalent core layer mass, an increase in deflection (20%), in-plain strain (8%) and velocity (8%) was observed. 1. INTRODUCTION Sandwich structures have important applications in the naval and aerospace industry. Their high strength/weight ratio, high stiffness/weight ratio and high energy absorption capabilities play a vital role in their applications, especially when they are subjected to high-intensity impulse loadings, such as air blasts. Their properties assist in dispersing the mechanical impulses that are transmitted to the structure, and thus protect anything located behind it [1-3]. Core materials play a crucial role in the dynamic behavior of the sandwich structures when they are subjected blast loadings. Common cores are made of metallic and non-metallic honeycombs, cellular foams, balsa wood, PVC, truss and lattice structures. Extensive research exists in the literature regarding the dynamic response of sandwich structures consisting of the various core materials and geometric arrangements. Dharmasena et al. [3], Zhu et al. [4], and Nurick et al. [5] have tested sandwich structures with a metallic honeycomb core material. Tagarielli et al. [6] has investigated the dynamic response of sandwich beams with PVC and balsa wood cores. Radford el al. [7] has conducted metal foam projectile impact experiments to simulate a blast loading on sandwich structures with metal foam cores. McShane et al. [8, 9] have investigated the underwater blast response of sandwich composites with a prismatic lattice (Y-frame, corrugated), as well as simulated an air blast, using metal foam projectiles, on sandwich composites with a pyramidal lattice cores. These studies have indicated that advanced sandwich structures can potentially have significant advantages over monolithic plates of equivalent mass in absorbing the blast energy, whether in air or underwater. In recent years, functionally graded materials, where the material properties vary gradually or layer by layer within the material itself, have gained much attention. The numerical investigation by Apetre et al. [10] on the impact damage of sandwich structures with a graded core (density) has shown that a reasonable core design can effectively reduce the shear forces and strains within the structures. Consequently, they can mitigate or completely prevent impact damage on sandwich composites. Li et al. [11] numerically examined the impact response of layered and graded metal-ceramic structures. They found that the choice of gradation has a great significance on the impact applications and a particular design can exhibit better energy dissipation properties. In their previous work, the authors experimentally investigated the blast resistance of sandwich composites with stepwise graded foam cores [12, 13]. Results indicated that monotonically increasing the wave impedance of the foam core, thus reducing the wave impedance mismatch between successive foam layers, will introduce a stepwise core compression, greatly enhancing the overall blast resistance of sandwich composites. T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_40, © The Society for Experimental Mechanics, Inc. 2011

281

282 In [12] two types of core configurations were studied and the sandwich composites were layered / graded based on the wave impedance of the given foams, i.e. monotonically and non-monotonically. In [13] four types of core configurations were investigated and the sandwich composites had a monotonically graded core based on increasing wave impedance, and the core gradations consisted of one, two, three and four layers respectively. The specimen dimensions and overall thickness were held constant, and the individual core layers had equivalent core layer thickness, i.e. with two layer core gradation, each core layer was 19 mm, while with four layer core gradation, each core layer was 9.5 mm. The present study is an extension of the author’s previous work and focuses on the blast response of sandwich composites with equivalent core layer mass. By using sandwich composites with equivalent core layer mass, the overall areal density of the specimen is reduced in comparison to its sandwich composite counterpart with equivalent core layer thickness The quasistatic and dynamic constitutive behaviors of the foam core materials were first studied using a modified SHPB device with a hollow transmitted bar. The sandwich composites were then fabricated and subjected to shock wave loading generated by a shock tube. The materials, as well as the core layer arrangements, and overall specimen dimensions were identical; the only difference arises in the core layers, where one configuration has equivalent core layer thickness, and the other configuration has equivalent core layer mass. The shock pressure profiles, real time deflection images, and post mortem images were carefully analyzed to reveal the mechanisms of dynamic failure of these sandwich composites. Digital Image Correlation (DIC) analysis was implemented to investigate the real time deflection, in-plane-strain, and particle velocity. 2. MATERIAL AND SPECIMEN 2.1 SKIN AND CORE MATERIAL The skin materials utilized in this study are E-Glass Vinyl Ester (EVE) composites comprised of E-glass fiber (0.61 kg/m2 areal density) and a vinyl-ester matrix. The plain weave woven roving E-glass fibers of the skin material consisted of 8 layers and were placed in a quasi-isotropic layout [0/45/90/-45]s. The core materials used in the present study are CorecellTM A series styrene foams manufactured by Gurit SP Technologies. The two types of CorecellTM A foam were A300 and A800 with density 58.5 and 150 kg/m3 respectively. The cell structures of the two foams are very similar and the only difference appears in the cell wall thickness and node sizes, which accounts for the different densities of the foams. The SEM images of the cell microstructures can be seen in Fig. 1.

A300

A800

Shock Wave

A300

Shock Wave

A800

100 mm A300

100 mm

A800 Fig. 1 Cell microstructures of core materials (a) Equivalent Thickness (b) Equivalent Mass Fig. 2 Specimen configurations and loading direction

2.2 SANDWICH PANELS WITH STEPWISE GRADED CORE The sandwich panels were produced by VARTM-fabricated process. The panels were 102 mm (4 in) wide, 254 mm (10 in) long with 5 mm (.2 in) front and back skins. The core consisted of two layers of foam. For the sandwich panels with equivalent core layer thickness, the first and second core layers of foam, A300 and A800 respectively, were both 19 mm (.75 in).These panels had an areal density of approximately 18.50 kg/m2. For the sandwich panels with equivalent core layer mass, the first core layer of foam, A300, was 25.4 mm (1 in), while the second core layer, A800, was 12.7 mm (.5 in). These panels had an areal density of approximately 17.60 kg/m2. Fig. 2 shows the specimen configurations for the sandwich composites with a core of (a) Equivalent Thickness and (b) Equivalent Mass.

283 3. EXPERIMENT SETUP AND PROCEDURE 3.1 MODIFIED SPLIT HOPKINSON PRESS BARS WITH HOLLOW TRANSMITTER BAR Due to the low wave impedance of CorecellTM foam materials, core material tests were performed by a modified SHPB device with a hollow transmission bar. It has a 304.8 mm (12 in)-long striker, 1600 mm (63 in)-long incident bar and 1447 mm (57 in)-long transmitter bar. All of the bars are made of a 6061 aluminum alloy. The nominal outer diameters of the solid incident bar and hollow transmission bar are 19.05 mm (0.75 in). The hollow transmission bar has a 16.51 mm (0.65 in) inner diameter. At the front and at the end of the hollow transmission bar, end caps made of the same material as the bar were press fitted into the hollow tube. By applying pulse shapers, the effect of the end caps on the stress waves can be minimized. The details of the analysis and derivation of equations can be found in ref [14]. The cylinderical specimens with a dimension Φ10.2 mm (0.4 in) X 3.8 mm (0.15 in) were used for test. 3.2 SHOCK TUBE Fig. 3 shows the shock tube apparatus with muzzle detail, which was utilized to obtain a controlled blast loading. It has an overall length of 8 m, consisting of a driver, driven and muzzle section. The high-pressure driver section and the low pressure driven section are separated by a diaphragm. By pressurizing the high-pressure section, a pressure difference across the diaphragm is created. When this difference reaches a critical value, the diaphragms rupture. This rapid release of gas creates a pressure wave that develops into a shock wave as it travels down the tube to impart dynamic loading on the specimen. The final muzzle diameter is 76.2 mm (3 in). Two pressure transducers (PCB102A) are mounted at the end of the muzzle section 160 mm apart, with the closest transducer 20 mm away from the specimen. The support fixtures ensure simply supported boundary conditions with a 0.1524 m (6 in) span. In the present study, a simply stacked diaphragm of 5 plies of 10 mil mylar sheets with a total thickness of 1.27 mm was utilized to generate an impulsive loading on the specimen with an incident peak pressure of approximately 1 MPa, a reflected peak pressure of approximately 5 MPa and a wave speed of approximately 1050 m/s. For each configuration, at least three samples were tested. A high-speed side-view camera system along with a high-speed back-view Digital Image Correlation (DIC) system was utilized to capture the real time deformation process as well as mechanisms of failure. Both camera systems were operating at 20,000 fps, with an interframe time of 50 µs and the high speed photography system can be seen in Fig. 4.

Shock tube Muzzle Detail and Specimen Fig. 3 Shock tube apparatus

Fig. 4 Digital Image Correlation (DIC) Set-up

3.3 DIGITAL IMAGE CORRELATION (DIC) Digital Image Correlation (DIC) was utilized to obtain the real time response of the sandwich composites. A speckle pattern was placed on the back face sheet of the specimens. Two high speed digital cameras, Photron SA1, were placed behind the shock tube to capture the real time deformation and displacement of the sandwich composite, along with the speckle pattern. During the blast loading event, as the specimen bends, the cameras track the individual speckles on the back face sheet. Once the event is over, a graphical user interface was utilized to correlate the images from the two cameras and generate real time deflection, strain (in plane and out of plane), and particle velocity.

284 4. EXPERIMENTAL RESULTS AND DISCUSSION 4.1 DYNAMIC CONSTITUTIVE BEHAVIOR OF CORE MATERIALS

Table1. Yield strength of core materials Core Layer

A300

A800

Quasi-Static Yield Stresses (MPa)

0.60

2.46

High Strain-Rate Yield Stresses (MPa)

0.91

4.62

Fig. 5 Quasi-static and high strain-rate behaviors TM of the two types of Corecell A Foams Fig. 5 shows the quasi-static and high strain-rate behavior of the different types of Corecell TM A foams. The quasi-static and dynamic stress-strain responses have an obvious trend for the different types of foams. Lower density foam has a lower strength and stiffness, as well as a larger strain range for the plateau stress. The high strain-rate yield stresses and plateau stresses are much higher than the quasi-static ones for the same type of foams. The dynamic strength of A800 increases approximately 100% in comparison to their quasi-static strength, while A300 increases approximately 50%. The improvement of the mechanical behavior from quasi-static to high strain-rates for all core materials used in the present study signifies their ability to absorb more energy under high strain-rate dynamic loading. Table 1 shows the quasi-static and high strain-rate yield stresses respectively. 4.2 RESPONSE OF SANDWICH COMPOSITES WITH GRADED CORES 4.2.1 REAL TIME DEFORMATION

Equivalent Mass

Equivalent Thickness

The real time side view deformation image series of the sandwich composites with equivalent core layer thickness and equivalent core layer mass under shock wave loading are shown in Fig. 6 respectively. The shock wave propagates from the right side of the image to the left side and some detailed deformation mechanisms are pointed out in the figures.

Fig. 6 Real - time side - view deformation of sandwich composites under shock wave loading

285 For the sandwich composites with equivalent core layer thickness, it can be observed that at t = 100 μs indentation failure of the core has initiated. This means that compression has initiated in the first core layer of foam (A300). By t = 400 μs heavy core compression is evident in the A300 foam core layer and core cracking can be seen in the A800 layer where the supports are located. At t = 700 μs delamination between the front face sheet and the foam core can be observed, both at the top and the bottom of the specimen. Also the core cracks have propagated from the back face sheet to the front face sheet. By t = 1600 μs, heavy core cracking and skin delamination are visible, along with heavy compression in the core (A300 only). For the sandwich composites with equivalent core layer mass, it can be observed that at t = 100 μs indentation failure of the core has initiated. This means that compression has initiated in the first core layer of foam (A300). By t = 400 μs, the A300 layer has continued to compress, and core cracking can be seen in the A800 layer where the supports are located. By t = 700 μs the core cracks have propagated from the back face sheet to the front face sheet. Also at this time, skin delamination between the front skin and foam core has initiated (top and bottom of specimen). By t = 1600 μs, heavy core cracking and skin delamination are visible, along with heavy compression in the core (A300 only). In both core configurations, equivalent core layer thickness and equivalent core layer mass, the deformation mechanisms were identical. Both configurations exhibited a double-winged deformation shape which means both configurations were under shear loading. Indentation failure was followed by compression of the first layer of foam (A300) and core cracking, and finally delamination between the front face sheet and foam core. The extent of the damage mechanisms varies between configurations, but the time at which the damage mechanisms were observed is identical. The mid-point deflections of the front face (front skin), interface 1 (between first and second core layer), and the back face (back skin) for both configurations, directly measured from the real – time side – view deformation images, are shown in Fig. 7 respectively. For the sandwich composites with equivalent core layer thickness (Fig. 7a), it can be seen that at t = 1600 μs the front skin, interface 1 and the back face deflect to approximately 46 mm, 32 mm and 32 mm respectively. Since the A300 foam core layer is located between the front skin and interface 1, the difference in deflection between the front skin and interface 1 shows the amount of compression in the A300 layer. Therefore it is evident that the A300 foam compresses approximately 14 mm, which is 75% of its original thickness (19 mm). Also note that the deflection curves for interface 1 and the back face follow the same trend and deflect to the same value at t = 1600 μs (32 mm). Therefore, no compression was observed in the A500 core layer of foam. For the sandwich composites with equivalent core layer mass, the mid-point deflections are shown in Fig. 7b. It can be seen that at t = 1600 μs, the front skin, interface 1, and the back skin deflect to approximately 60 mm, 41 mm and 41 mm respectively. Since the A300 foam core layer is located between the front skin and interface 1, the difference in deflection between the front skin and interface 1 shows the amount of compression in the A300 layer. Therefore it can be observed that the A300 foam compresses approximately 19 mm, which is 75% of its original thickness (25.4 mm). Also note that the deflection curves for interface 1 and the back face follow the same trend and deflect to the same value at t = 1600 μs (41 mm). Therefore, no compression was observed in the A500 core layer of foam.

(a) Equivalent Thickness

(b) Equivalent Mass Fig. 7 Mid-point deflection curves for both configurations

Fig. 8 Comparison of back face deflections Fig. 8 shows a comparison of the mid-point deflections for the back face of each configuration. It can be seen in the figure that at t = 1600 μs, the back face of the sandwich composites with equivalent core layer thickness deflects approximately 20% less than the back face of the sandwich composites with equivalent core layer mass.

286 4.2.2 DIGITAL IMAGE CORRELATION (DIC)

Equivalent Mass

Equivalent Thickness

The real-time response of the sandwich composites was generated using 3-D Digital Image Correlation and the results are shown in Fig 9 - Fig. 11. Fig. 9 shows the full-field deflection contours for both the sandwich composites with equivalent core layer thickness and the sandwich composites with equivalent core layer mass. It is evident from the figure that the back face deflection along the central region of both configurations is in excellent agreement with the results generated using the high-speed deformation images.

t = 0 μs

t = 100 μs

t = 400 μs

t = 700 μs

t = 1000 μs

t = 1600 μs

Fig. 9 Real time full-field deflection (W) contours for both configurations Through DIC analysis, using the inspection of a single point in the center of the back face sheet, the data for mid-point inplane strain and particle velocity during the entire blast loading event was extracted. The results of the in plane strain and particle velocity are shown in Fig. 10 and Fig. 11 respectively. Fig. 10 shows the in-plane strain of both configurations. It can be seen that the sandwich composite with equivalent core layer thickness exhibits 2.2% strain, while the sandwich composite with equivalent core layer mass exhibits 2.4% strain. The back face particle velocity can be observed in Fig. 11. The back face of the sandwich composites with equivalent core layer thickness reach a maximum mid-point particle velocity of 31 m/s, while the sandwich composites with equivalent core layer thickness reach a maximum back face particle velocity of 34 m/s. Table 2 summarizes the back face DIC results for both configurations.

Fig. 10 In-plane strain (εyy) for both configurations

Fig. 11 Particle velocity (dW/dt) for both configurations

287 Table 2. Summary of DIC results Deflection

In-plane Strain

Velocity

(kg/ m )

(mm)

(%)

(m/s)

Eq. Thickness

18.5

32

2.2

31

Eq. Mass

17.6

41

2.4

34

Difference

5%

22%

8%

8%

Configuration

Areal Density 2

4.2.3 POST MORTEM ANALYS After the shock event occurred, the damage patterns in the sandwich composites were visually examined and recorded using a high resolution digital camera and are shown in Fig.12. For both the sandwich composites with equivalent core layer thickness and equivalent core layer mass, there were two main cracks located at the support position. Delamination is visible between the front face and the foam core, as well as between the bottom layer of foam core and back face sheet. Also compression was observed in the A300 core layer. The locations and damage mechanism were identical for both configurations; the only difference was in the extent of damage observed. Equivalent Thickness

Fiber Delamination

Core cracking

Core Compression

Delamination Core cracking

Equivalent Mass

Fiber Delamination

Core Compression

Delamination

(a) Front face sheet (blast side)

(b) Foam and PU core

(c) Back face sheet

Fig. 12 Visual examination of sandwich composites after being subjected to high intensity blast load 5. Summary The following is the summary of the investigation: (1) The dynamic stress-strain response is significantly higher than the quasi-static response for every type of CorecellTM A foam studied. Both quasi-static and dynamic constitutive behaviors of Corecell TM A series foams (A300 and A800) show an increasing trend. (2) Sandwich composites with two types of monotonically graded cores based on increasing wave impedance were subjected to blast loading. In order to reduce areal density, a sandwich composite with equivalent core layer mass was fabricated and its blast performance was compared to its sandwich composite counterpart with equivalent core layer thickness. Results indicated that with a reduction in areal density of approximately 1 kg/ m 2 (5%), the sandwich composites with equivalent core layer mass exhibit a higher back face deflection (22%), higher in-plane

288 strain (8%) and higher back face velocity (8%) in comparison to the sandwich composites with equivalent core layer thickness Acknowledgement The authors kindly acknowledge the financial support provided by Dr. Yapa D. S. Rajapakse, under Office of Naval Research (ONR) Grant No. N00014-04-1-0268. The authors acknowledge the support provided by the Department of Homeland Security (DHS) under Cooperative Agreement No. 2008-ST-061-ED0002. Authors thank Gurit SP Technology and Specialty Products Incorporated (SPI) for providing the material as well as Dr. Stephen Nolet and TPI Composites for providing the facility for creating the composites used in this study. References [1] Xue, Z. and Hutchinson, J.W., Preliminary assessment of sandwich plates subject to blast loads. International Journal of Mechanical Sciences, 45, 687-705, 2003. [2] Fleck, N.A., Deshpande, V.S., The resistance of clamped sandwich beams to shock loading. Journal of Applied Mechanics, 71, 386-401, 2004. [3] Dharmasena, K.P., Wadley, H.N.G., Xue, Z. and Hutchinson, J.W., Mechanical response of metallic honeycomb sandwich panel structures to high-intensity dynamic loading. International Journal of Impact Engineering, 35 (9), 1063-1074, 2008. [4] Zhu, F., Zhao, L., Lu, G. and Wang, Z. Deformation and failure of blast loaded metallic sandwich panels – Experimental investigations. International Journal of Impact Engineering, 35 (8), 937-951, 2008 [5] Nurick, G.N., Langdon, G.S., Chi, Y. and Jacob, N. Behavior of sandwich panels subjected to intense air blast: part 1Experiments. Composite Structures, 91 (4), 433 – 441, 2009. [6] Tagarielli, V. L., Deshpande, V.S., and Fleck, N.A. The high strain rate response of PVC foams and end-grain balsa wood. Composites: Part B, 39, 83–91, 2008. [7] Radford, D.D., McShane, G.J., Deshpande, V.S. and Fleck, N.A. The response of clamped sandwich plates with metallic foam cores to simulated blast loading. International Journal of Solids and Structures, 44, 6101-6123, 2006. [8] McShane, G. J., Deshpande, V.S., and Fleck, N.A. The underwater blast resistance of metallic sandwich beams with prismatic lattice cores. Journal of Applied Mechanics, 74, 352 – 364, 2007. [9] McShane, G. J., Radford, D.D., Deshpande, V.S., and Fleck, N.A. The response of clamped sandwich plates with lattice cores subjected to shock loading. European Journal of Mechanics – A: Solids, 25, 215-229, 2006. [10] Apetre, N.A., Sankar, B.V. and Ambur, D.R., Low-velocity impact response of sandwich beams with functionally graded core. International Journal of Solids and Structures, 43(9), 2479-2496, 2006. [11] Li, Y., Ramesh, K.T. and Chin, E.S.C., Dynamic characterization of layered and graded structures under impulsive loading. International Journal of Solids and Structures, 38(34-35), 6045-6061, 2001. [12] Wang, E., Gardner, N. and Shukla, A., The blast resistance of sandwich composites with stepwise graded cores. International Journal of Solids and Structures, 46, 3492-3502, 2009. [13] Gardner, N., Wang, E., Kumar, P., and Shukla, A. Performance of graded sandwich composite beams under shock wave loading. In Progress [14] Chen, W., Zhang, B., Forrestal, M.J. A split Hopkinson bar technique for low impedance materials. Experimental Mechanics, 39 (2), 81–85, 1998.

Effects of High and Low Temperature on the Dynamic Performance of the Core Material, Face-sheets and the Sandwich Composite

Sachin Gupta1 and Arun Shukla2 1 Dynamics Photomechanics Laboratory, Department of Mechanical, Industrial & Systems Engineering, University of Rhode Island, Kingston, RI 02881. Email: [email protected] 2 Dynamics Photomechanics Laboratory, Department of Mechanical, Industrial & Systems Engineering, University of Rhode Island, Kingston, RI 02881. Email: [email protected] ABSTRACT The performance of sandwich structures is highly affected by the varying environmental temperature during service, especially when they are subjected to blast loading. Typically, sandwich panels consist of polymer based composites (facesheets) and polymer foams (core material), and the properties of its components change substantially under different temperatures. In this paper, high strain rate constitutive behavior of E-glass Vinyl ester composites and CorecellTM M100 foam at different temperatures has been studied. A special temperature control chamber was designed in order to heat or cool the specimen to an assigned temperature. Once the specimen reached the target temperature, it was subjected to high strain rate loading using a SHPB apparatus. Eight different target temperatures were chosen: -40ºC, -20ºC, 0ºC, 22ºC, 40ºC, 60ºC, 80ºC and 100ºC. The sandwich composites were maintained at the target temperature before being subjected to shock-wave loading using a shock-tube. A high-speed photography system utilizing Digital Image Correlation (DIC) was used to record the real time deformation of the specimen. The results show a significant decrease in flow stress with the increase in the temperature of core material. Significant fiber-matrix delamination was observed in face-sheets at elevated ambient temperatures with little change in the value of compression modulus. For the low temperature environment, the core material shows brittle behavior resulting in core-cracking of the sandwich specimen under blast loading. INTRODUCTION Composite materials have important applications in the marine and aerospace industry. Marine structures undergo mechanical loadings under varying ambient temperatures especially when operating in extreme environments such as the Arctic Ocean and gulf areas. The environmental temperature has an adverse effect on the blast resistance of the structure. As a core material, polymeric foams are extensively used for energy absorption for high strain rate applications against ballistic impacts and blast waves [1-2]. Composite materials, such as sandwich structures, have important applications in ship structures due to their advantages, such as high strength/weight ratio and high stiffness/weight ratio. Tekalur [3] studied the dynamic behavior of sandwich structures with reinforced polymer foam cores. Wang and Gardner [4] analyzed the performance of sandwich composites stepwise graded core under blast loading. Most of the previous research only focuses on the blast resistance of various composite structures at room temperature. Recent studies [5] observe significant and complex effects of environmental temperature on the dynamic compressive behavior of syntactic foams. The numerical study on the response of a large aspect ratio sandwich panel subjected to the elevated temperatures on one of the surfaces showed that the maximum deflection of the panel increases as the surface temperature increases [6]. These studies investigated the performance of sandwich composites under static loading only. Erickson et al. [7] performed low-velocity impact experiments on sandwich composites and both found that the temperature can have a significant effect on the energy absorbed and the peak force endured by the specimen. Aktas [8] studied the impact behavior of glass/epoxy laminated composite plates at high temperatures and concluded that energy absorbing capabilities of the specimen reduces with increasing temperature. The study on the role of temperature on impact properties of Kevlar/fiberglass composite laminates by Salehi-Khojin [9] shows that maximum deflection increases with a corresponding increase in temperature. Dutta [10] tested the energy absorption and brittleness of graphite/epoxy composites at room and low temperature under low velocity impact. To the best of the author’s knowledge, there are no results of the blast performance of composite structures under extreme environments. An in-depth analysis on the effects of temperature is needed.

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The present paper experimentally studies the dynamic behavior of Corecell M100 foam core and E-glass Vinyl Ester composite face-sheets under high temperature environments using a Split Hopkinson Bar apparatus. A controlled temperature environment was designed in order to achieve the target temperatures. The shock-tube facility was used to study the dynamic behavior of sandwich composites under blast loading. A special fixture was designed to heat the sandwich composites to different temperatures. For low temperature experiments, dry ice was utilized to cool down the specimen prior to being subjected to blast loading. A high-speed photography system with three cameras was used to capture real-time motion images. Digital Image Correlation (DIC) technique was utilized to obtain the details of the deformation of the sandwich panels during the blast event. Post mortem visual observations were carefully analyzed to identify the mechanisms of dynamic failure of the sandwich composites under low and elevated temperature environments. 2. MATERIAL SPECIMEN: 2.1 Split Hopkinson Bar Specimen: Cylindrical specimens were cut from the foam material using a die having a diameter of 11.5 mm and a thickness of 3.8 mm. Specimen dimensions were carefully chosen to obtain uniform deformation while minimizing inertia effects and maximizing the number of cells in the specimen. For face-sheets, cylindrical specimens with a thickness of 3.18 mm and a diameter of 10.16 mm were used. 2.2 Sandwich Specimen: The skin material utilized in this study was E-Glass Vinyl Ester (EVE) composite. The woven roving E-glass fibers of the skin material were placed in a quasi-isotropic layout [0/45/90/-45]s. The fibers were made of the 0.61 kg/m2 area density plain weave. The resin system used was Ashland Derakane Momentum 8084 and the front skin and the back skin consisted of identical layup and materials. The core material used in the present study was CorecellTM M100 styrene foam, which was manufactured by Gurit SP Technologies for high temperature marine applications. Table 1 list important the material properties of this foam from the manufacturer’s data [11]. Foam Type

Nominal Density (kg/m3)

Compression Modulus (MPa)

Shear Elongation (%)

Corecell M100

107.5

107

52%

Table1 Material properties of the foam core [11]

The VARTM procedure was carried out to fabricate the sandwich composite panels. The overall dimensions for the specimen were 102 mm wide, 254 mm long and 32 mm thick. The foam core itself was 25.4 mm thick, while the skin thickness was 3.3 mm. The average areal density of the samples was 16.81 kg/m2. Figure 1 shows a real image of a specimen, its dimensions and the speckle patterns on both front and side face. The cell microstructure of M100 foam is shown in fig. 2. 3. EXPERIMENT SETUP AND PROCEDURE 3.1 Quasi-static Characterization:

The quasi-static compression tests were performed using a standard compression test machine (Instron Model 5582). The tests were performed following the ASTM D 1621 – 04a standard [12] using rectangular specimens (50.8 mm×50.8 mm, 19.1 mm thick) at a crosshead speed of 1 mm/min for M100 foam. 3.2 Split Hopkinson Bar Apparatus: A split Hopkinson pressure bar (SHPB) apparatus was used to test the M100 foam at high strain rates of deformation. Due to the low-impedance of CorecellTM foam materials, dynamic experiments for the core materials were performed with a

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modified SHPB setup. A hollow incident bar was used to increase the strain rate to achieve high end-strain values and a hollow transmitted bar was used to increase the transmitted signal intensity. It had a 50.8 mm long steel striker, 1828.8 mmlong incident bar and 1447 mm-long transmitted bar. Incident and transmitted bars were made of a 6061 aluminum alloy. The nominal outer and inside diameters of the both hollow incident/transmitted bar were 19.05 mm and 16.51 mm respectively. At each end of the hollow bars, end caps made of the same material were pressure fitted and pinned using aluminum pins. By using lead pulse shapers, the effect of the end caps on the stress waves was minimized. The details of the analysis and derivation of equations for analysis of experimental data can be found in Ref. [13]. 3.3 Heating Chamber for SHPB: A small heating chamber (220mm x 130 mm, 170 mm high) made of wood with a standard resistance heating wire, NickelChromium Alloy, 60% Ni / 16% Cr, was utilized to heat the chamber. An external DC power supply (0 – 30 V) with a voltage controller was utilized to control the amount of heat supplied. A heat resistant borosilicate glass sheet was used as a transparent removable face for the chamber. The chamber was calibrated using a series of experiments by supplying different voltages to the chamber and the saturation temperature value was estimated by using a K-type thermocouple. 3.4 Cooling Setup for SHPB: A low temperature cooling chamber with polystyrene rigid foam insulation was designed and flexible copper tubing of 9.25 mm outer diameter was wound into 100 mm inner diameter spirals. Cooling chamber was filled with dry ice (CO2) and nitrogen gas was supplied using Nitrogen Deliver System from one end of copper tubing. The other end was fed into same wood chamber used for heating. Different steady state temperature inside the chamber can be achieved by controlling the output pressure of nitrogen gas supplied. Shock tube Specimen

Supports 160 mm

Side-view camera system

Transducers Shock-tube muzzle Specimen SS Support frame

Back-view DIC system

Fig. 4 High-speed photography systems

Fig. 3 Shock tube apparatus

3.5 Shock Tube: The shock tube apparatus was utilized in the present study to develop controlled blast loadings. The details of this apparatus can be found in Ref [4]. Fig. 3 shows the shock tube apparatus with a detailed image of the muzzle. The final muzzle diameter is 76.2 mm. Two pressure transducers (PCB102A) are mounted at the end of the muzzle section with a distance 160 mm. The support fixtures ensured simply supported boundary conditions with a 152.4 mm span. The shock tube has an overall length of 8 m, consisting of a driver, driven and muzzle section. The high-pressure driver section and the low pressure driven section are separated by a diaphragm. By pressurizing the high-pressure section, a pressure difference across the diaphragm is created. When this difference reaches a critical value, the diaphragms rupture. This rapid release of gas creates a shock wave, which travels down the tube to impart dynamic loading on the specimen. In the present study, a simply stacked diaphragm of 4 plies of 10 mil mylar sheets with a total thickness of 1 mm was utilized to generate an impulse loading on the specimen with an incident peak pressure of approximately 0.83 MPa and a wave speed of approximately 970 m/s.

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3.6 High-Speed Photography Systems: Two high-speed photography systems were utilized to capture the real-time 3-D deformation data of the specimen. The experimental setup, shown in fig. 4, consists of a back-view 3-D Digital Image Correlation (DIC) system with two cameras and a side-view camera system with one camera. All cameras are Photron SA1 high-speed digital camera, which have the capability to capture images at a frame-rate of 20,000 fps with an image resolution of 512×512 pixels for one second time duration. These cameras were synchronized to make sure that the images and data can be correlated and compared. The 3-D DIC technique is one of the most recent non-contact methods for analyzing full-field shape, motion and deformation. Two cameras capture two images from different angles at the same time. By correlating these two images, one can obtain the three dimensional shape of the surface. Correlating this deformed shape to a reference (zero-load) shape gives full-field in-plane and out-of-plane deformations. To ensure good image quality, a speckle pattern with good contrast is put on the specimen prior to experiments. 4. EXPERIMENTAL RESULTS AND DISCUSSION:

3.5

3.5

3.0

3.0

2.5

2.5

2.0 1.5 22 C 40 C 60 C 80 C 100 C Quasi-static (22 C)

1.0 0.5 0.0 0.00

0.05

0.10

0.15

0.20

0.25

0.30

Eng Stress (MPa)

Eng Stress (MPa)

4.1 Dynamic Constitutive Behavior of the Core-material:

2.0 1.5 22 C 0C -20 C -40 C Quasi-static (22 C)

1.0 0.5 0.0 0.00

0.05

Eng Strain

0.10

0.15

Eng Strain

(a) High temperature

(b) Low temperature Fig. 5 Quasi-static and dynamic behavior of M100 foam under low and high temperature

The quasi-static and dynamic stress-strain curves for M100 foam at low and high temperatures are obtained. Figure 5 shows the difference in the dynamic behavior and quasi-static behavior for M100 foam. The flow stress value under quasi-static loading is 1.5 MPa, while for dynamic testing, the flow stress value increases approximately by 100% and it shows a flow stress value of 3.1 MPa at a strain rate of 3700/s. The effect of high temperature on the stress-strain behavior of foam under dynamic loading can be observed exclusively in fig. 5(a). The plateau stress drops from 3.1 MPa to 2.3 MPa as temperature increases from 22⁰C to 100⁰C. The cell collapse occurs during the plateau stress region in polymeric foam, which is entirely dependent on the cell material. So, in polymeric foams, a decrease in the plateau stress can be expected due to thermal softening of the base material [14].

4.2 Dynamic Constitutive Behavior of the Face-sheet: Dynamic constitutive behavior of the face-sheet under quasi-static and high strain rate loading is shown in fig. 6. The compressive modulus for quasi-static loading is 2.6 GPa. As the fibers in the specimen are running in transverse direction, the compressive properties are moderated by the resin, which results into a relative lower value of compressive modulus as compared to young modulus of E-glass fibers. As the temperature is increases from -40⁰C to 100⁰C,

700 600

Eng Stress (MPa)

Fig. 5(b) shows the dynamic constitutive behavior under low temperatures. The dynamic experiments were performed at a strain rate of 2200/s. As the temperature decreases from 22⁰C to -40⁰C, the collapse of cells occurs rapidly due to low temperature environments and this result into brittle failure of cells. Thus, M100 foam exhibits brittle behavior, which is clearly demonstrated by the decreasing slope in plateau region.

500 400

300

-40 C -20 C 0C 22 C 40 C 60 C 80 C 100 C Quasi static

Strain rate = 1400/s Strain rate = 1400/s

200 100 0 0.00

0.02

0.04

Eng Strain

0.06

0.08

Fig. 6 Quasi-static and dynamic behavior of face-sheet under low and high temperature

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a decrease in the stress-strain curve is observed. The young modulus of resin is highly dependent on the temperature and a decrease in compressive modulus under dynamic testing can be explained by the decrease in young modulus of the resin used in manufacturing of face-sheets. 4.3 Response of Sandwich Composites under Room, Low and High temperature: 4.3.1 Real Time Deformation

The real time observation of the transient behavior of sandwich composites under shock wave loading at low temperature ( 65⁰C ), room temperature (22⁰C) and high temperature (80⁰C) are shown in figs. 7, 8 and 9 respectively. The shock wave propagated from the right side of the image to the left side and some detailed deformation mechanisms were pointed out in the figures. As shown in fig. 7, at room temperature environment, the sandwich composite exhibits no visible failure during the event. At 200 µs, the stress wave has propagated inside the specimen and both front and back faces of the specimen start to deform under blast loading. At t = 400 µs, the core material starts compressing and by t = 750 µs, specimen core is compressed by 5% in the lower part of the supports. The core material stops compressing after t = 1350 µs and a maximum of 9% strain is observed in the core. The specimen deforms in a double wing shape and no global bending response is observed.

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Fig. 8 shows the blast loading response of a sandwich composite subjected to a low temperature environment. The core material temperature is -65⁰C. At t = 300 µs, primary core cracks are formed around the perimeter of loading area. The lower crack initiates from the front face, while the upper crack is initiated from the back face. The crack formation is followed by the skin delamination from front and back face in the lower and upper part of the specimen respectively. The secondary core cracks are visible at t = 750 µs, and the specimen deformation behavior changes from double wing to global bending. This allows for heavy slippage between the core and face-sheet, which results into the formation of other secondary cracks. At 1600 µs, the skin delamination on the top of the back face can be observed. During the whole blast event, there is no core compression observed, which clearly demonstrates the core-hardening at low temperature environments. The blast response of a sandwich composite with a core temperature of 80ºC is shown in fig. 9. The core compression starts at t = 300 µs and the centre of the core is compressed by 6% by t = 600 µs. The bending of grid lines close to the back face shows the presence of shear stresses in the core materials near the face-sheets. The core compression increases with time and at t = 1350 µs, the core is compressed more than 30% at the center. Following the core compression, local failure of the front face-sheet is observed at t = 1350 µs, which results in the local deformation of the core material in the central area. 4.3.2 Deflection The mid-point deflection in the front and back faces, at the center point under different temperature environments were obtained from high speed side-view images and are shown in fig. 10. Comparing the back face deflections at t = 1200 µs, the sandwich specimen at room temperature deflected 17 mm, while the deflection for the specimen tested under low and high temperature environment is 23 mm and 28 mm respectively. Therefore, it can be observed that the deflection of sandwich panel under low temperature is 35% more than as compared to room temperature experiments. Also, the high temperature specimen has 65% more back face deflection than at room temperature. The front face deflections are also plotted in fig. 10 to compare the amount of core compression observed during the blast testing. The core compression and compressive strain at the center of the specimen at t = 1200 µs are listed in table 2. Under low temperature, the core compression is less as compared to Fig. 10 Front and Back face deflection room temperature, which depicts the increase in yield strength of the core curves for room, low and high temperature material under low temperature environments. At Core Compression Compressive Strain high temperature, the thermal softening of foam (mm) (%) and the local failure of the face-sheets allows for 1.89 mm 7.5% Room Temperature more core compression, which results into a 0.82 mm 3.2% Low Temperature compressive strain of 31.5% at the center of the specimen. High Temperature 8.00 mm 31.5% Table 2 Core compression and compressive strain at time t = 1200 µs

4.3.3 Digital Image Correlation (DIC) Analysis

Fig. 11 Out-of-plane velocity on the back face for room, low and high temperature

Shown in fig. 11 and 12, the real time response of the sandwich composites was generated using Digital Image Correlation (DIC). Through DIC analysis, the out-of-plane deflection and velocity during the entire blast event was extracted. The out-of-plane velocity on the back face of the sandwich specimen is plotted in fig. 11. Under room temperature, at t = 300 µs, the specimen reaches a maximum velocity of 25 m/s and it begins to decelerate later in time. A maximum velocity of 30 m/s and 34 m/s is observed in low and high temperature experiments respectively at approximately t = 350 µs. Therefore, low and high temperature experiments exhibit higher maximum velocities, but the time taken to reach maximum velocity is delayed by 50 µs. At t = 1300 µs, the room temperature experiment reaches the maximum deflection and starts reverberating, which is shown by the negative velocities after this time. Compared to the low and high temperature experiments, the specimen is still undergoing deformation with velocities of 5 m/s and 10 m/s respectively at t = 1300 µs.

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Fig. 12 shows the full-field back face deflection for high temperature experiment. By t = 1800 µs, the deflection at the midpoint of the back face is 35 mm, which is the same as the deflection of the back face calculated from the side-view images. W (mm)

t = 0 µs

t = 400 µs

t = 700 µs

t = 1000 µs

t = 1300 µs

t = 1800 µs

Fig. 12 Full field back view for high temperature experiment

4.3.4 Post-mortem Analysis After the blast event occurred, the damage patterns in the sandwich specimens were visually examined and recorded using a high resolution digital camera and are shown in fig. 13. When the sandwich specimen was subjected to shock wave loading under room temperature, the specimen shows a maximum deflection of 2.5 mm and there is no evidence of permanent damage inside core and face-sheet. When the sandwich specimen was subjected to shock wave loading under low temperature, the damage is confined to the places where the supports are located in the shock loading and a significant amount of core-cracking was observed. The core-cracks propagated completely through the foam core and a large amount of skin delamination between the core and the back face-sheet was observed. No specific damage in face-sheets and core compression is visible, and the specimen shows a deflection of 5.5 mm at center. In the high temperature environment, excessive fiber matrix delamination and fiber breakage was seen confined to the center of the shock wave loading. The fiber matrix delamination can be attributed to low heat distortion temperatures of the resin used in preparation of the face-sheets. The heat distortion temperature for Ashland Derakane Momentum is 82ºC, which is very close to the test temperature of the high temperature experiments performed. The center point of the specimen has a deflection of 17.8 mm and a core compression behind the local failure of face-sheet is also evident from images.

Fig. 13 Visual inspection of sandwich specimens after being subjected to blast loading

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5. SUMMARY a)

Under dynamic testing, M100 foam shows thermal softening and the flow stress value decreases with an increase in temperature upto 100ºC. At low temperature, foam has a brittle nature due to hardening and the plateau region shows a decreasing stress value with increasing strain. b) The sandwich specimens were subjected to blast loading at room temperature (22ºC), low temperature (65ºC) and high temperature (80ºC). The sandwich specimen under low temperature demonstrated around 30% more back-face deflection with respect to room temperature and the main failure mechanisms observed are core-cracking and skin delamination. At high temperature, a significant amount of fiber breakage and fiber matrix delamination occurs due to the low tolerance of polymer resin for high temperatures, used in preparation of the sandwich specimens. ACKNOWLEDGEMENT The authors kindly acknowledge the financial support provided by Dr. Yapa D. S. Rajapakse, under the Office of Naval Research (ONR) Grant No. N00014-10-1-0662. Authors thank Gurit SP Technology for providing materials for the study and also thank Dr. Stephen Nolet and TPI Composites for providing the facility for fabricating the materials. REFERENCES [1] Zhang., J., Kikuchi, N., Li, V., Yee, A. and Nusholtz, G., Constitutive modeling of polymeric foam material subjected to dynamic crash loading, Int. J. Impact Eng., 21 (5), 369–386, 1998. [2] Neremberg, J., Nemes, J.A., Frost, D.L., Makris, A., Blast wave loading of polymeric foam, in: Proceedings of the 21 st International Symposium on Shock Waves 1, 91–96, 1997. [3] Tekalur, S.A., Bogdanovich, A.E., Shukla, A., Shock loading response of sandwich panels with 3-D woven E-glass composite skins and stitched foam core, Composite Science and Technology 69 (6), 736–753, 2009. [4] Wang, E., Gardner, N., Shukla, A., The blast resistance of sandwich composites with stepwise graded cores, International Journal of Solid and Structures 46, 3492-3502, 2009. [5] Song, B., Chen, W., Yanagita, T., Frew, D.J., Temperature Effects on dynamic compressive behavior of an epoxy syntactic foam, Composite Structures 67 (3), 289-298, 2005. [6] Birman, V., Kardomateas, G.A., Simitses, G.J., Li, R., Response of a sandwich panel subject to fire or elevated temperature on one of the surfaces, Composites Part A: Applied Science and Manufacturing 37 (7), 981-988, 2006. [7] Erickson, M.D., Kallmeyer, A.R., Kellogg, K.G., Effect of temperature on the low-velocity impact behavior of composite sandwich panels, Journal of Sandwich Structures and Materials 7, 245-264, 2005. [8] Aktas, M., Karakuzu, R., Icten, B.M., Impact Behavior of Glass/Epoxy Laminated Composite Plates at High Temperatures, Journal of Composite Materials 44 (19), 2289-2299, 2010. [9] Salehi-Khojin, A., Bashirzadeh, R., Mahinfalah, M., Nakhaei-Jazar, R., The role of temperature on impact properties of Kevlar/fiberglass composite laminates, Composites Part B: Engineering 37 (7-8), 593-602, 2006. [10] Dutta, P.K., Low temperature compressive strength of glass fiber reinforced polymer composites, Journal of Offshore Mechanics and Arctick Engineering 116, 167-172, 1994. [11] http://www.gurit.com [12] Standard test method for compressive properties of rigid cellular plastics, ASTM Standard D 1621, 2004. [13] Chen, W., Zhang, B., Forrestal, M.J. A split Hopkinson bar technique for low impedance materials, Experimental Mechanics 39 (2), 81–85, 1998.

[14] Gibson, L.J., Ashby, M.F., Cellular Solids-Structure and Properties, Cambridge University Press, Cambridge, 1997.

INFLUENCE OF TEXTURE AND TEMPERATURE ON THE DYNAMICTENSILE-EXTRUSION RESPONSE OF HIGH-PURITY ZIRCONIUM

Daniel T. Martineza, Carl P. Trujilloa, Ellen K. Cerretaa, Joel D. Montalvoa, Juan P. Escobedo-Diaza, Victoria Websterb, and G.T. Gray IIIa a

Material Science and Technology Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA b

Mechanial Engineering Dept. Case Western Reserve University, Cleveland, Ohio

Abstract To create comprehensive models of mechanical deformation in Zirconium (Zr) it is important to observe the effect of high strain on the material. The mechanical behavior and damage evolution in textured, high-purity zirconium (Zr) is influenced by strain rate, temperature, stress state, grain size, and texture. In particular, texture is known to influence the slip-twinning response of Zr, which directly affects the work hardening behavior at both quasi-static and dynamic strain rates. However, while microstructural and textural evolution of Zr in compression and to relatively low strains in tension has been studied, little is understood about the dynamic, high strain, tensile response of Zr. Here, the influence of texture on the dynamic, tensile, mechanical response of high-purity Zr is correlated with the evolution of the substructure. Experiments will be conducted using dynamic-tensile-extrusion process. A bullet-shaped sample has been impacted into a high-strength steel extrusion die and soft recovered in the Taylor Anvil Facility at Los Alamos National Laboratory. Finite element modeling that employs a continuum level constitutive description of Zr will be performed to provide insight into the dynamic extrusion process. Current experimental findings will be presented.

Introduction The hexagonal close packed (HCP) metal, zirconium (Zr) lacks the symmetry and isotropy commonly exhibited in cubic materials. This anisotropy creates a unique challenge in modeling the deformation in the material. To develop accurate predictive models, extensive characterization of the mechanical response must be performed.

T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_42, © The Society for Experimental Mechanics, Inc. 2011

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298 Statistically understanding the deformation behavior over a wide range of strain rates, temperatures and stress states is key. Zirconium has been the subject of significant research. A great deal of this work has focused on microstructural evolution due to the deformation dominated by a combination of twinning and slip. Additionally many of these same studies have examined the effect of texture on deformation [1]. However much of this work has been conducted under uni-axial stress conditions. While a few studies such as the one performed by Vogel et al. [2] have macroscopically observed the effects of high pressure on Zr, systematic studies on the effects of high strain and high strain rates on zirconium have not been performed. In this study, high strain and high strain rates will be established using the dynamic extrusion process developed at Los Alamos National Laboratory. Dynamic extrusion is a method for imparting relatively high strains to specimens that have been accelerated into a steel die at velocities of up to 640m/s using a modified Taylor Gun Facility built at Los Alamos National Laboratory[3]. The study of dynamic extrusion on other materials such as Cu and Ta, has found that considerable grain elongation as well as shear localization leading to void formation accommodates deformation at these strains (Cao et al. [6]). Instability develops and the fragments are extruded from the die during testing. The first portion of the study will focus on the effects of varying velocity and therefore varying strain and strain rate on ductility of Zr. Given prior understanding of zirconium, both twinning and slip are expected to dominate deformation. Twinning has been found to be vital in deformation of HCP metals as they have fewer slip systems than cubic materials (Song and Gray [4]). Zirconium has three prominent twin systems: (1-102), (11-21), and (11-22), and extensive twinning is expected to be observed in each at high strain rates (Tome et al. [5]). The next portion of the study will focus on the effects of temperature on the dynamic ductility of Zr. Relative differences as a function of temperature and texture will be examined.

Experimental Method This study examines high-purity, alpha zirconium which has been clock rolled. The average grain size is 35 µm, as measured by the Heyn method. Using Electron Back Scatter Diffraction (EBSD) a strong 0001 basal texture was observed as shown in Figure 1, which is a characteristic of the plate having been clock rolled during processing. The plate had been annealed at 550°C for one hour and as such the initial dislocation density is very low. Bullet shaped specimens of approximately 0.299” in diameter and 0.310” in length were obtained from the as-received plate.

299

Figure 1. Pole figure from EBSD scans of asannealed Zr

  Figure 2: Schematic representation of the asreceived Zr plate and the dynamic extrusion specimens sectioned from this plate as a function of orientation. IP and TT directions are identified.  

These specimens were machined in the in-plane (IP) and through thickness (TT) directions (See Figure 2).   The dynamic extrusion tests were preformed at velocities ranging from 400 to 640m/s using the Taylor Anvil Gas Gun Facility at Los Alamos National Laboratory. The samples impacted a high strength steel extrusion die with an extrusion angle of 81 degrees. The dies were made of S-7 tool steel, machined to tolerance, and heat-treated to a Rockwell hardness of 56-58. High-speed photography was used to view the in-siu extrusion process on a macroscopic level (Figure 3). The high-speed photographs were utilized to observe the elongation of the material as it exited the die and to reassemble fully extruded and soft recovered segments for post mortem characterization. Additionally, these images can be utilized to determine the exit velocity of the specimens.

300

A)

B)

C)

Figure 3. High speed photography of Zr specimens extruded in the following test conditions: A) 534 m/s, in-plane direction, 25°C; B) 535 m/s, through thickness direction, 25°C; and C) 602 m/s, through thickness direction, 250°C.

For all tests, the extruded pieces were soft captured and collected. The portion of the bullet that remained caught in the die was removed by cutting the die near the sample to subsequently weaken the die thus allowing the specimen ejection. All the soft recovered pieces as well as to verify that all pieces were captured. Measuring the length of each soft recovered piece and summing these lengths established the total elongation. Three exampled of reassembled, fully extruded segments along with the portion left in the die are shown in Fig. 4. For many of the tests, each extruded segment along with the die left in the die was examined using scanning electron microscopy (SEM). These pieces were then mounted in epoxy, ground to the mid-section, prepared using standard metallographic preparation techniques, and chemically etched with a solution of 45 ml H2O, 45 ml HNO3, and 10 ml HF for 20-25 seconds. Optical microscopy was then performed on the samples using a microscope equipped with polarized light.

a)

b)

301

c)

Figure 4. Reassembled soft captured fragments of extruded Zr in the following test conditions: A) 534 m/s, in-plane direction, 25°C; B) 535 m/s, through thickness direction, 25°C; and C) 602 m/s, through thickness direction, 250°C.

Results and Discussion Dynamic extrusion tests have been performed on both IP and TT specimens at velocities of 400-640 m/s and at room temperature and 250°C. Total elongations for each test are given in Fig. 5.

Figure 5. Elongation (mm) as a function of velocity Although in all tests the metal was not fully extruded, velocity is seen to strongly affect the fragmentation and elongation of the specimen. Total elongation was measured after reassembling the fragments in the extrusion order according to the high-speed images. From Fig. 5 it is evident that IP samples consistently display more elongation than the TT specimens. Additionally, elongation increases with increasing impact velocity regardless of specimen texture and test temperature. Finally, the role of temperature on dynamic extrusion is less clear. However,

Figure 6. SEM image of the fracture surface of an extruded specimen

temperature slightly increased the elongation in the TT specimens. Additionally, more instability is developed with higher velocity. The number of fragments varies depending on the velocity (ex. five segments

302 for IP specimen tested at 479.8 m/s whereas nine segments were recovered for an IP tested at 654.6 m/s). Scanning electron microscopy (SEM) was used to observe the exterior appearance of the soft recovered segments as well as the tips of the first and last extruded segments to examine the fracture surfaces. This analysis revealed that rather than having failed due to shear as seen in previous studies of cubic materials, the end of the fragments displayed a ductile, fracture surface similar to that seen in tensile tests (Figure 6).   Optical microscopy was performed on many of the tested specimens. Rather than the expected grain elongation, all samples experienced significant recrystallization in most of the fully extruded segments (Figure 7). However, the initial fully extruded segment and segment remaining the die displayed a range of microstructures as shown in Fig. 7.

P1 (a)

(b)

P2

P3

303 Figure 7. Optical images of the IP, 421m/s, 25°C extrusion. The microstructure of the (a) segment left in the die and (b) the first fully extruded piece display deformed grains in region 1, elongated grains in region 2, and recrystallization in region 3. Electron back scattered diffraction was utilized to examine differences in twinning and evolving texture in IP and TT specimens as a function of test velocity. Twinning was more significant in the TT specimens than in the IP specimens and more twinning was observed with increasing test velocity. The significant differences in twinning result if difference texture evolutions between the TT and IP specimens, as is shown in Figs. 8 and 9. This difference in active deformation mechanism as a function of specimen texture is also likely the reason for the differences in the development of instability as a function of texture and velocity and this likely directly influences elongation of the specimens.

(a)

(b)

(c)

Figure 8. IP case: (a) the undeformed microstructure and texture, (b) the microstructure and local texture in the segment left in the die tested at 25°C and 460m/s, and (c) the microstructure and local texture in the segment left in the die tested at 25°C and 600m/s.

304

(a)

(b)

(c)

Figure 9. TT case: (a) the undeformed microstructure and texture, (b) the microstructure and local texture in the segment left in the die tested at 25°C and 468m/s, and (c) the microstructure and local texture in the segment left in the die tested at 25°C and 603m/s velocity. Conclusions Thus far, we can conclude from this study the following about the dynamic tensile extrusion response of high purity Zr: 1.

Impact velocity strongly influences the large-strain tensile ductility of Zr and thus elongation.

2.

Impact velocity influences the development of instability and therefore the number of extruded segments during the extrusion process; higher velocities produce more fragments.

3.

Differences in the relative activation of twinning as a function of velocity and texture correlated with the observed differences in the development of instability and the total elongation of specimens.

Acknowledgements This work has been performed under the auspices of the United States Department of Energy and was supported by the Joint DoD/DOE Munitions Technology Development Program.

305 References 1.

Kaschner, G. C., and G. T. Gray III. “The Influence of Crystallagraphic Texture and Interstitial Impurities on the Mechanical Behavior of Zirconium.” Metallurgical and Materials Transactions A 31A (200): 19972003. Print.

2.

Vogel, Sven C., Helmut Reiche, and Donald W. Brown. “High Pressure Deformation Study of Zirconium.” Powder Diffraction 22.2 (2007): 113-17. Print.

3.

Gray III, G. T., E. Cerreta, C. A. Yablinsky, L. B. Addessio, B. L. Henrie, B. H. Sencer, M. Burkett, P. J. Maudlin, S. A. Maloy, C. P. Trujillo, and M. F. Lopez. “Influence of shock Prestraining and grain size on the dynamic-tensile-extrusion response of Copper: Experiments and Simulations.” Shock Compression of Condensed Matter (2006): 725-28. Print.

4.

Song, S. G., and G. T. Gray III. “Influence of Temperature and Strain Rate on Slip and Twinning Behavior of Zr.” Metallurgical and Materials Transactions A 26A (1995): 2665-675. Print.

5.

Tome, C. N., P. J. Maudlin, R. A. Lebensohn, and G. C. Kaschner. Acta Metall. 49 (2001): 3085-096. Print.

6.

Cao, F., E. K. Cerreta, C. P. Trujillo, and G. T. Gray III. “Dynamic Tensile Extrusion Response of Tantalum.” Acta Materialia 56 (2008): 5804-817. Science Direct. Elsevier Ltd, 14 Sept. 2008. Web. .

Modeling and DIC Measurements of Dynamic Compression Tests of a Soft Tissue Simulant Steven P. Mates, Richard Rhorer and Aaron Forster National Institute of Standards and Technology 100 Bureau Drive Stop 8553, Gaithersburg, Maryland 20899 Richard K. Everett, Kirth E. Simmonds and Amit Bagchi Naval Research Laboratory 4555 Overlook Ave SW, Washington, D.C. 20375 ABSTRACT Stereoscopic digital image correlation (DIC) is used to measure the shape evolution of a soft, transparent thermoplastic elastomer subject to a high strain rate compression test performed using a Kolsky bar. Rather than using the usual Kolsky bar wave analysis methods to determine the specimen response, however, the response is instead determined by an inverse method. The test is modeled using finite elements, and the elastomer stiffness giving the best match with the shape and force history data is identified by performing iterative simulations. The advantage of this approach is that force equilibrium in the specimen is not required, and friction effects, which are difficult to eliminate experimentally, can be accounted for. The thermoplastic is modeled as a hyperelastic material, and the identified dynamic compressive (non-linear) stiffness is compared to its quasi-static compressive (non-linear) stiffness to determine rate sensitivity. INTRODUCTION Tissue simulant materials are sought to provide realistic experimental devices to simulate the human body’s response to blast or impact loading that can occur in military scenarios, law enforcement and emergency response events, vehicle accidents or sporting events [1]. This approach is meant to help develop better protective equipment or procedures to prevent serious injury or death. In most practical injury scenarios, tissues are subject to dynamic loading involving large amplitude strains (20 % [2]) to vulnerable soft tissues at strain rates -1 above 10 s [3]. Numerical models of these test devices are crucial to interpreting the measurement data in these complicated tests, and efforts are underway to provide the material data needed to calibrate such models. Simpler uniaxial mechanical tests of soft tissue specimens show that the large-strain response of these materials is generally non-linear and rubber-like, and can be represented by hyperelastic models developed for polymers [2, 4, 5]. Strain rate sensitivity is also generally observed in soft tissues, prompting the use of viscoelastic models to describe the relaxation behavior [2, 6, 7]. High strain rate measurements of actual tissues and tissue simulants are often performed using a Split Hopkinson -1 Pressure Bar, or Kolsky Bar [8, 9], which can achieve large strains at uniform strain rates in excess of 100 s . The difficulties in obtaining valid high strain rate data on soft materials using Kolsky bar techniques are well documented. Soft materials take much longer to achieve mechanical equilibrium when subject to a rapidly changing load. In most practical situations, equilibrium is not established in the sample until the test is nearly over, invalidating most if not all of the test [8]. Careful selection of specimen thickness and the use of pulse shaping to increase the rise time of the load pulse have been effective methods for achieving valid dynamic test results [10]. Measuring the forces on soft samples is also challenging because they are below the typical sensitivity of Kolsky bars designed for testing metals. Special, highly sensitive force transducers placed directly on either side of the sample are required to obtain force signals [11]. Care must be taken to separate out inertial effects in these force signals to obtain the true specimen response [12]. Finally, confinement techniques have been successfully employed to force either hydrostatic or shear loading conditions at high strain rates [13]. T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_43, © The Society for Experimental Mechanics, Inc. 2011

307

308 For extremely soft materials (less than 10 Shore A), such as the material examined in our experiments, achieving force equilibrium in a Kolsky bar test is exceedingly difficult. Further, friction between the sample and the compression platens is difficult to eliminate entirely, even with generous lubrication. Under dynamic loading, these effects will tend to produce non-uniform deformation in the sample in the form of large amplitude surface waves and prominent barreling which invalidates the usual Kolsky bar assumptions of uniform, uniaxial stress and strain throughout the specimen. In low-rate compression tests, the effect of friction on the perceived stress-strain behavior can be significant. This has led some researchers to impose no-slip boundary conditions in order to more accurately determine the material response with a known, rather than ambiguous, state of friction [4, 14]. Recent advances in optical shape measurement using stereoscopic (3D) Digital Image Correlation (DIC) [15] and in high speed digital camera technology has now made it feasible to measure the shape evolution of soft specimens during dynamic testing in a Kolsky bar. Further, by combining this new measurement capability with finite element modeling, one may be able to deduce the constitutive behavior of the material at high rates using so-called inverse methods [16]. Such methods, which include minimizing the error of finite element simulations against experimental data by systematically adjusting the relevant material parameters has been used to identify the properties of metals [17-19], composites [20], ceramics [21], polymers [22] and biomaterials [23] usually under quasi-static test conditions but more recently at high rates of strain [24]. In this paper, high-speed 3D DIC is used to measure the dynamic deformation of a soft tissue simulant material while it is compressed at high strain rate using a modified Kolsky bar technique. The sample is allowed to deform non-uniformly due to specimen ring-up effects and to friction between the sample and the bar. Finite element modeling is employed to deduce the constitutive behavior of the material by systematically varying the physical parameters governing the mechanical behavior of the modeled specimen and the friction coefficient to match the DIC history and force history data. This approach was first described in [25]. In this updated work, commercial software is used to perform model sensitivity analysis and to identify optimal parameter values. The identified optimal parameters are averaged for five individual experiments and the result is then compared to the quasistatic behavior to determine rate sensitivity of the material. Standard deviations of the identified parameters are used to estimate the overall uncertainty of the identified dynamic response of the specimen. EXPERIMENTAL METHODS The elastomer was cast into 9.5 mm diameter by 6.5 mm thick specimens for dynamic compression testing. The 3 material has a density of 870 kg/m , and it is nearly transparent. Tests were conducted using a maraging steel Kolsky bar measuring 15 mm in diameter with bar lengths of 1500 mm. Forces were measured with piezo-electric dynamic force transducers placed on either side of the specimen, as shown in Fig. 1 below. The force transducers have a resolution of ± 0.1 % of full scale, or ± 0.5 N on the incident side and ± 0.25 N on the transmission side. Polished steel platens (5 mm thick by 15 mm diameter) are placed between the transducers and the sample. No lubrication was used in the experiments. Compressive pulses were created by impacting a 250 mm long, 15 mm diameter maraging steel striker bar against the incident bar at 5.5 m/s. An additional difference between these tests and more traditional Kolsky bar testing is that the upstream platen and force transducer are allowed to collectively decouple from the incident bar to become a “flyer plate.” The flyer has the advantage of producing a much larger dynamic strain in the specimen than if the transducer and platen remained fixed to the incident bar, which was desirable given the limited compression pulse width (and thus specimen strain) our Kolsky bar could generate on a sample of this size. The flyer test is accomplished by using a low-tack adhesive at the incident bar-transducer interface that is strong enough to hold the pieces in place prior to the test but too weak to resist the tensile wave reflection off the incident platen as the wave passes through the sample. A thick, soft rubber pulse shaper was employed to produce a long, gradually rising stress pulse, which helped develop a repeatable release of the flyer from the incident bar. DIC measurements of the sample deformation are acquired at 120,000 frames per second with an image size of 1 128 x 208 pixels. Commercial DIC software [26] is used to perform the image correlation measurements. The

309

Fig.1 Axisymmetric finite element model showing the arrangement of the sample, platens, load cells and transmission bar cameras use 90 mm macro (1:1 magnification) lenses with f/5.6 apertures and are placed 30 cm from the sample with a 12.5 º pan angle. The resolution is 18 pixels/mm. A random speckle pattern was created on the samples using a light dusting of flat black spray paint. Attempts were made to achieve a nominal average speckle size of 5 to 7 pixels in diameter and a coverage factor of 50 % following manufacturer’s recommendations. The speckle patterns were backlit using a Teflon reflector placed directly behind the specimen and illuminated by halogen optical fiber lamps, as shown in Fig. 2. Using this optical set-up, rigid body translations were measurable within ± 0.01 mm. Fig. 3 indicates the typical correlation region on a sample. Correlation measurements used the default software analysis settings (21 pixel windowing, 5 pixel overlap, with default smoothing). The DIC measurement data, obtained from about 200 stereo image pairs during the test, are automatically oriented to a reference plane that is defined by fitting the initial shape of the (cylindrical) specimen prior to deformation. Because of this automatic plane fit and the fact that the sample deforms axisymmetrically, the DIC coordinate system could be aligned with the axisymmetric finite element model coordinate system using simple y- and z-translations. FINITE ELEMENT MODEL The finite element model consists of the flyer assembly (force transducer plus steel platen), the sample, the 1 transmitted force transducer and platen, and the transmission elastic bar. ABAQUS/Explicit is used to perform the simulations. A portion of the modeled mesh was shown earlier in Fig. 1. The model uses axisymmetric CAX4R elements, with 352 elements in the sample and much coarser meshes in the steel parts. The maraging steel compression bars as well as the platens are modeled as linear elastic solids with the following properties: 11 3 E = 1.9x10 Pa,  = 0.29, and ρ = 8048 kg/m . The donut-shaped force transducers are modeled with the same 3 stiffness and Poisson ratio as the maraging steel but with a density of 6594 kg/m . The boundary condition for the simulation is set by specifying the velocity history of the flyer, which is obtained directly from DIC measurements on the flyer itself. Simulations are carried out until the flyer velocity vector begins to deviate from the compression axis, or “pitch,” which occurs eventually when the flyer decelerates and comes to rest against the specimen. The elastomer is modeled as an isotropic hyper-elastic material using the Marlow strain energy potential [27]. Hyper-elasticity is characterized by large, recoverable elastic strains that are characteristic of rubbery materials. Rubbery response in polymeric materials occurs in the region nestled between the visco-elastic response and viscous flow region in strain-rate space [28]. Micromechanical behavior associated with rate dependency and dissipation (viscoelastic effects), such as chain slippage and the breakage and reforming of secondary (crosslink) bonds, is relatively unimportant in the rubbery regime.

1

Commercial products are identified in this work to adequately specify certain procedures. In no case does such identification imply recommendation or endorsement by NIST, nor does it imply that the materials or equipment identified are necessarily the best available for the purpose.

310

Fig. 2 Measurement set-up

Fig. 3 Typical correlation region shown on an actual image of a speckled sample from which correlation measurements are obtained The rate sensitivity of the elastomer is examined by attempting to identify a purely hyperelastic dynamic response by inverse methods and comparing this identified response to the quasi-static behavior. If the identified dynamic response is found to be much different than the quasi-static response, rate sensitivity would thereby be indicated. The baseline constitutive response of the material used in the model is derived from uniaxial stress-strain data obtained at a low strain rate (4 x 10 -4 s-1) following ASTM D 575-91 [29]. The quasi-static response, shown in Fig. 4, is characteristic of rubbery materials, showing an upturn in the stress-strain curve due to the finite deformability of even very long chain polymeric molecules. To maintain this basic character while allowing for possible rate effects, the response of the specimen is modified by a stiffness scaling factor, m, as shown in Fig. 4.

311

Fig. 4 Quasi-static compression stress-strain response of tissue simulant with polynomial fit and two alternate models with twice (m = 2) and half (m = 0.5) the stiffness of the quasi-static response (defined as m = 1) Rayliegh damping is employed in the numerical scheme to eliminate unphysical oscillations in the simulations to better mimic the dynamic response of the sample. This method is employed by the finite element code as a generic means to account for dissipation in many different materials [27]. It works by adding a small damping stress, σd, to the stress from the basic hyperelastic response that is proportional to the strain rate, by specifying a positive damping factor β:



d

(1)

Ee

The magnitude of the damping factor must be chosen with caution as it is not intended to simulate the strain rate effects due to the bulk micromechanical response of the specimen. To avoid obscuring gross strain-rate effects from those related to numerical damping, a minimal damping coefficient must be chosen that is just large enough to eliminate non-physical oscillations while not adding significant numerical stiffness to the sample. 1

A commercial software package [30] is used to perform the model sensitivity analysis and to identify the dynamic sample stiffness by comparing the simulation results to the experimental data. The software acts as GUI-driven macro that alters finite element model input data, controls the solver execution and displays and analyses the simulation results. It also approximates the finite element model response over the variable space of interest using interpolation functions, and employs optimization tools to identify optimal parameter values using objective functions describing the agreement between the simulation results and the data. The objective functions are described next. OBJECTIVE FUNCTIONS Objective functions expressing the difference between the finite element model results and the data are built individually for the force history and shape history data. Then, a single “cost” function is assembled to represent the overall agreement between simulation and data for identifying optimum parameter values. DIC shape history data are compared to the simulation nodal displacements as indicated in Fig. 1. Residuals are computed at the DIC measurement locations along the center of the correlation region parallel to the compression (y-)axis at each measurement time point (e.g. for each image pair acquired and analyzed). The modeled surface positions are interpolated to match the DIC measurement positions along the length of the specimen. The shape history objective function, ΦS, is given by:

1 S  M

 1  t 1  N  M



2   FEA DIC  Z i ,t  Z i ,t      Z iDIC   i 1   ,t    N



0.5 

   

(2)

312 In Equation 2, Z is the out-of-plane surface position of the deforming sample. The superscript FEA refers to the finite element model result, while the superscript DIC denotes experimental data (DIC measurement). Subscripts i and t refer to space and time, respectively. The transmitted force history objective function, ΦF, is given by:

F 

P

 ([ F jFEA  F jEXP ]2 ) 0.5

(3)

j 1

F is the transmission force, while EXP refers to the experimental data and FEA is again simulation results. At FEA EXP at the locations where F is available. The each time step, ΦF is calculated by linearly interpolating the F combined objective function, Φ, is assembled as follows:



 F  S SFF SFS

(4)

In the above equation, SFF and SFS are scale factors for the force and shape residuals, respectively. The scale factors are selected to weight each individual objective function such that the order of the scaled objective is 1 near the optimum point. For this study, SFF = 1000 N and SFS = 0.001. RESULTS Force-time data from a typical experiment are shown in Fig. 5 along with a sketch showing the corresponding behavior of the flyer plate during the test. Prominent features of the data are labeled in the graph. A sharp rise in the incident force marks the arrival of the incident compressive strain wave generated by the striker. As this wave reflects from the incident platen-sample interface, a tensile (negative) force is observed. This tensile load causes the bond holding the flyer to fail, releasing it into the specimen. As the specimen compresses further, the transmitted force steadily increases until the flyer arrests, causing the transmitted force to peak. Next, the sample begins to release its stored strain energy, pushing back on the flyer. Soon the flyer is thrown back against the incident bar, which all the while has been advancing forward due to the action of the trapped compression wave. This second impact is marked by the final sudden rise in both force signals. Because the free surface of the flyer carries no stress, the forces measured on either side of the specimen will never be equal. Because of this, the force signals cannot be compared to check sample equilibrium nor can they be relied upon to determine the sample stress in the usual way.

Fig. 5 Force signals recorded on the incident and transmitted side of the specimen during a flyer experiment (left) and a sketch of the flyer experiment as visualized using the axisymmetric finite element simulation (right)

313 Fig. 6 plots the overall sample strain and strain-rate versus time for the same experiment. In the test shown, almost 70 % engineering strain is achieved before symmetry breaks down. A Kolsky bar 10 times as long as the one used here would be needed to achieve this level of strain using the same sample at this strain rate without using the flyer technique. A second observation is that the strain rate is relatively uniform over most of the test. Ordinarily this is critical in normal Kolsky bar tests. Here, however, it is less so in this study because the assumption of uniform strain rate is not used in the inverse analysis to deduce the specimen response. The finite element model parameters governing the mechanical response of the sample are the stiffness factor, m, the friction coefficient, f, Poisson’s ratio, , and the damping factor, β. In principle, all of these factors can be examined simultaneously by conducting a sensitivity analysis using a large Design-of-Experiments (DOE) matrix. However, certain parameters can be specified ahead of time to reduce the number of unknowns in the problem. For example, since very soft polymeric materials are incompressible,  = 0.5 should be prescribed. However, computational stability requires that some compressibility be added, which will affect accuracy of the solution of this highly confined compression test [27]. Computational cost and accuracy were found to be adequate with  = 0.4950. The penalty for allowing compressibility is a systematic under-prediction of the out-of-plane deformation, which introduces a systematic error in the value of ΦS. Another parameter that must be prescribed is the damping factor, β. Selecting β = 0.000025 s provides realistic-looking force signal while having little overall effect on the force levels themselves, and therefore not adding a gross amount of stiffness to the material that would confuse attempts to identify a dynamic value of m. Using Eq. 1, this damping level adds a stress equal to -1 about 1 % of the zero-strain modulus of the material for a global strain rate of 400 s .

Fig. 6 Global strain and strain rate histories for a typical flyer experiment from DIC-measured flyer velocity With  and β fixed, we proceed to investigate the influence of the stiffness factor, m, and the friction coefficient, f. To this end, a Full-Factorial DOE is performed between 0.0 < f < 2.0 and 0.75 < m < 1.25 with 10 levels for each parameter. The influences of stiffness and friction on ΦF and ΦS are indicated in Fig. 7. The friction coefficient has dramatic influences on both ΦF and ΦS at low values, but at higher values the sensitivity to friction is minimal. For ΦF, the sensitivity to friction falls beyond about 0.3, while for ΦS the influence dramatically lessens above 0.5. Due to the lack of sensitivity above f = 0.5, identifying an optimal friction coefficient would be difficult and the result would likely be unreliable. Consequently, the friction coefficient is set to the limiting case of no-slip for the remaining simulations used to identify m. DOEs were executed for each experiment with 0.75 < m < 1.5 to identify the dynamic stiffness of the specimen relative to its quasi-static response. The other parameters are fixed to values discussed previously, namely: β = 0.000025 s,  = 0.495 and no-slip friction. A representative plot of the effect of m on ΦF and ΦS is shown in Fig. 8. Both ΦF and ΦS have distinct minima, though not at the same value of m. A conclusion from the latter observation is that the model is not able to achieve perfect agreement with experimental observation, leading to

314 this tradeoff between shape and force agreement. Since neither objective has a clear precedence over the other, the objectives will maintain approximately equal weighting by using the previously-defined scale factors. The simulation results of Fig. 8 are fit with a Radial Basis Function (RBF) approximation model prior to the identification step to reduce the computation time required to identify an optimum value of m [30]. As Fig. 8 shows, the RBF approximations are an excellent representation of the simulation results. A Downhill Simplex optimization method [30] is used to identify optimum values of the stiffness using the force response function for five independent experiments. The identified optimum values, mopt, are listed in Table 1. The average stiffness identified using the inverse method was m opt = 1.05 ± 0.18 (k = 2). Thus, within the observed repeatability level, the material is not strain rate sensitive, as m = 1.0 represents the quasi-static response. The 17 % uncertainty in mopt reflects random errors due to experimental and modeling approximation factors.

Fig. 7 Effect of friction coefficient, f, and stiffness, m, on (a) ΦF and (b) ΦS

Fig. 8 Typical response surface showing the effect of m on actual ΦF and actual ΦS along with RBF approximations used for calculating the optimal m

315 Table 1. Identified values of m opt Experiment

Average Strain -1 Rate [s ]

mopt

ΦF,opt [N]

ΦS,opt

Φ

1786

340

1.18

104

0.00623

6.33

1791

397

1.04

475

0.00607

6.55

1818

405

1.09

649

0.00573

6.38

1819

429

0.94

1346

0.00686

8.20

1820

422

1.02

897

0.00713

8.05

Average

399

1.05

694

0.0064

7.10

70

0.18

930

0.0012

1.86

U

*

*

U is the expanded uncertainty (k = 2) of the average value based on twice the standard deviation of the individual values in the table.

SUMMARY An inverse method was used to determine the dynamic stiffness of a prospective biomimetic elastomer using a Kolsky bar with the intention to determine whether the strain rate sensitivity mimics real tissues. High-speed digital image correlation (DIC) was used to measure the surface deformation of the sample during the test, which suffered from significant friction effects and non-uniaxial deformation due to the extreme softness of the elastomer. Further, a special flyer technique was used to compress the specimen to larger compressive strains that would otherwise have been possible for the NIST Kolsky bar. A finite element model of the experiment was constructed and the resulting simulations were compared to the DIC shape history data and to force history data. The sensitivity of the model to specimen stiffness, friction, Poisson ratio and damping were examined, and the simulation results were compared to the experimental data to identify the dynamic stiffness using an inverse method. An appropriate numerical damping coefficient was determined that produced realistic-looking forcehistory signals while avoiding adding artificial stiffness to the simulated specimen which would confound attempts to identify the true strain-energy-dependent dynamic stiffness. A Poisson ratio of 0.495 was chosen to achieve reasonable computation times, but this led to a systematic under-prediction of the outer surface displacement of the sample compared to the DIC measurements. Friction in the model was very important for capturing the actual deformation of the specimen. Interestingly, the sensitivity of the simulation to friction above f = 0.5 was minimal, so a no-slip friction condition was used. With the friction and damping conditions established, the stiffness scale factor m was observed to produce distinct minima in the force and shape residual plots in five repeat experiments. The optimal stiffness scale factor for these high strain rate tests was m = 1.05 ± 0.18, indicating very little rate sensitivity in this material up to 60 % compressive strain. Thus this particular elastomer is not strain rate sensitive, unlike what has been reported for actual soft tissues. REFERENCES [1] Roberts, J.C., A.C. Merkle, P.J. Biermann, E.E. Ward, B.G. Carkhuff, R.P. Cain and J.V. O’Connor, “Computational and experimental models of the human torso for non-penetrating ballistic impact,” Journal of Biomechanics, 40 (2007) 125-136. [2] Prange, M.T., and S.S. Margulies, “Regional, Directional and Age-Dependent Properties of the Brain Undergoing Large Deformation,” ASME J. Biomech. Eng., 124, 244-252 (2002) [3] LaPlaca, M.C., D.K. Cullen, J.J. McLoughlin and R.S. Cargill, “High rate shear strain of three-dimensional neural cell cultures: a new in vitro traumatic brain injury model,” J Biomech., 38(5) (2005) 1093-1105. [4] Roan, E., and K. Vemaganti, “The Nonlinear Material Properties of Liver Tissue Determined from No-Slip Uniaxial Compression Experiments,” Journal of Biomechanical Engineering, ASME Transactions, 129 (2007) 450456. [5] Fung, Y.C., Biomechanics:Mechanical Properties of Living Tissues, Springer-Verlag, 1993.

316 [6] Sparks, JL, and R. B. Dupaix, “Constitutive Modeling of Rate-Dependent Stress-Strain Behavior of Human Liver in Blunt Impact Loading,” Annals of Biomedical Engineering 36(11) (2008) 1883-1892. [7] Edsberg, L.E., R.E. Mates, R.E. Baier and M. Lauren, “Mechanical characteristics of human skin subjected to static versus cyclic normal pressures,” Journal of Rehabilitation Research and Development, 36 (1999). [8] Saraf, H., K.T. Ramesh, A.M Lennon, A.C. Merkle and J.C. Roberts, “Mechanical properties of soft human tissues under dynamic loading,” Journal of Biomechanics 40 (2007), pp.1960-1967. [9] Chen, W., F. Lu, D.J. Frew and M.J. Forrestal, “Dynamic Compression Testing of Soft Materials,” ASME Transactions 69 (2002), pp.214-223. [10] Song, B., and W. Chen, “Dynamic stress equilibration in split Hopkinson pressure bar tests on soft materials,” Experimental Mechanics, 44 (2004), pp.300-312. [11] Casem, D., T. Weerasooriya and P. Moy, “Inertial Effects in Quartz Force Transducers Embedded in a Split Hopkinson Pressure Bar,” Experimental Mechanics, 45 (2005), pp.368-376. [12] Song, B., Y. Ge, W.W. Chen and T. Weerasooriya, “Radial Inertia Effects in Kolsky Bar Testing of Extra-soft Specimens,” Experimental Mechanics, 47 (2007), pp.659-670. [13] Saraf, H., K.T. Ramesh, A.M Lennon, A.C. Merkle and J.C. Roberts, “Measurement of Dynamic Bulk and Shear Response of Soft Human Tissues,” Experimental Mechanics 47 (2007), pp.439-449. [14] Miller, K, “Method of testing very soft biological tissues in compression,” Journal of Biomechanics, 38 (2005) 153-158. [15] Sutton, M.A., J.-J. Orteu and H.W. Schreier, Image Correlation for Shape, Motion and Deformation Measurements, Springer, 2009. [16] Avril, S. et al., “Overview of Identification Methods of Mechanical Parameters Based on Full-Field Measurements,” Exp. Mech. 48 (2008) 381-402. [17] Mahnken, R, “A comprehensive study of a multiplicative elastoplasticity model coupled to damage including parameter identification,” Computers and Structures, 74 (2000) 179-200. [18] Hoc, T, J. Crépin, L. Gélébart and A. Zaoui, “A procedure for identifying the plastic behavior of single crystals from the local response of polycrystals,” Acta Materialia 51 (2003) 5477-5488. [19] Cooreman, S. et al., “Identification of Mechanical Material Behavior Through Inverse Modeling and DIC,” Exp. Mech. 48 (2008) 421-433. [20] Roux, S., and F. Hild, “Digital Image Mechanical Identification (DIMI),” Exp. Mech. 48 (2008), 495-508. [21] Robert, L., F. Nazaret, T. Cutard and J.-J. Orteu, “Use of 3D Digital Image Correlation to Characterize the Mechanical Behavior of a Fiber Reinforced Refractory Castable,” Exp. Mech. 47 (2007) 761-773. [22] Giton, M., A.-S. Caro-Bretelle and P. Ienny, “Hyperelastic Behavior Identification by a Forward Problem Resolution: Application to a Tear Test of a Silicone-Rubber,” Strain 42 (2006) 291-297. (it does not use DIC). [23] Kauer, M., V. Vuskovic, J. Dual, G. Szekely and M. Bajka, “Inverse finite element characterization of soft tissues,” Medical Image Analysis 6 (2002) 275-287. [24] Kajberg, J, and B Wikman, “Viscoplastic parameter estimation by high strain-rate experiments and inverse modeling – Speckle measurements and high-speed photography,” Int. J. Solids and Structures, 22 (2007) 145164. [25] Mates, S.P., et al.,”High Strain Rate Tissue Simulant Measurements Using Digital Image Correlation,” Proceedings of the 2009 SEM Annual Meeting, June 1-4, 2009, Albuquerque, NM, USA. [26] Correlated Solutions Inc., Columbia, SC, USA. [27] ABAQUS Documentation, Dassault Systèmes, Vélizy-Villacoublay, France. [28] Ward, I.M., and D.W. Hadley, An Introduction to the Mechanical Properties of Solid Polymers, Wiley, Chichester, England, 1993. [29] ASTM D575-91, 2007, “Standard Test Methods for Rubber Properties in Compression,” ASTM International, West Conshohocken, PA, 2007, DOI: 10.1520/D0575-91R07 [30] Isight Documentation, Dassault Systèmes, Vélizy-Villacoublay, France.

Measurement of R-values at Intermediate Strain Rates using a Digital Speckle Extensometry J. Huh1, Y.J. Kim1, H. Huh1 1

School of Mechanical, Aerospace & System Engineering, Korea Advanced Institute of Science Technology, 291 Daehak-ro, Yuseong-gu, Daejeon, 305-701 Korea

ABSTRACT This paper introduces a method to measure the R-value from a speckled pattern with respect to the strain rate and loading direction. The R-value is calculated from series of images taken by a high speed camera by analyzing the deformation history during static and dynamic tensile tests. A Matlab code to track designated points by digital image correlation algorithm was constructed to measure the longitudinal and transversal strain from the speckled pattern, which was created by spraying black paint on white-paint-coated tensile specimens. Using the suggested method, R-values of TRIP590 1.2t and DP780 1.0t were measured with the variation of the strain rate (0.001/sec ~ 100/sec) and loading direction (0° ~ 90°). INTRODUCTION R-value (Plastic strain ratio) is a parameter that indicates the ability of a sheet metal to resist thinning or thickening when subjected to either tensile or compressive forces in the plane of the sheet. It is a measure of plastic anisotropy and sheet metal drawability. ASTM E517 (American Society for Testing and Materials) standard suggests two procedures to measure the Rvalue, manual procedure and automatic procedure [1]. In manual procedure, the R-value is measured from the displacement between the two gauge marks and change in the specimen width after pulling the specimen axially to a certain elongation. In automatic procedure, the R-value is measured from two extensometers, one attached in the axial and the other in the transverse direction. However, neither procedure can be applied to measure the R-value at intermediate strain rates due to the difficulties in attaching extensometers and in limiting the deformation to a certain range. For this reason, there are few researches on the measurement of R-value at intermediate strain rates. One promising solution is the adoption of digital speckle extensometry, which is widely used recently to determine the strain during deformation [2-3]. This technology allows us to obtain the information on the deformation through digital image correlation of a speckled pattern. In this paper, a new method to measure the R-value at intermediate strain rates is suggested based on a digital speckle extensometry. Based on the suggested method, R-values were measured for two advanced high strength steel sheets, TRIP590 and DP780, at the strain rate ranged from 0.001/sec to 100/sec and at the loading angles from 0o to 90o. MEASUREMENT OF R-VALUES USING A DIGITAL SPECKLE EXTENSOMETRY Uniaxial tensile tests were carried out at the strain rate ranged from 0.001/sec to 100/sec. The tensile tests were carried out with Intron 5583 for the strain rate of 0.001/sec and 0.01/sec, and with HSMTM (High Speed Material Testing Machine) [4] for the higher strain rates. Two advanced high strength steel sheets for auto-body, TRIP590 1.2t and DP780 1.0t, were selected for these experiments. The dimensions of a specimen for uniaxial tensile tests were adopted from the previous research [4] and are shown in Fig. 1. The specimens were extracted at intervals of 15 o from 0o (RD; rolling direction) to 90o (TD; transverse direction). R-value is calculated from the true strains in the transverse and longitudinal direction as in Eq. (1) [5].

r

d w  (d w / d l )  d w   d t d l  d w 1  d w / d l

(1)

where dεl, dεw, dεt denotes incremental longitudinal, transverse, and through-the-thickness true strain, respectively. Incompressible condition was assumed in Eq. (1). T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_44, © The Society for Experimental Mechanics, Inc. 2011

317

318

Fig. 1 Specimen dimensions of dynamic tensile specimen for High Speed Material Testing Machine

Fig. 2 Sample images with speckled pattern taken by high speed camera (TRIP590 1.2t-45o, 10/sec)

(a) INSTRON5583

(b) High Speed Material Testing Machine

Fig. 3 Experimental setup of tensile testing apparatus with a high speed camera.. To measure the true strain in the transverse and longitudinal direction during deformation, a random speckled pattern was generated by applying black spray paint on the white-paint-coated specimen surface. Deformation was recorded by a high speed camera, Phantom v.9.0, with the maximum resolution of 800 x 640 (at 0.001/sec) and the maximum sampling rate of 6400 fps (at 100/sec). Fig. 2 shows sample images of a specimen with the speckled pattern during the deformation taken by the high speed camera, and Fig. 3 shows an experimental setup of a tensile testing apparatus with the high speed camera.

319

dyi

dxi

dyf

dxf

(a)

(b)

Fig. 4 Magnified view of Fig. 2: (a) initial state; (b) during deformation. White and Black dots are to be used to calculate longitudinal and transverse strain, respectively. The longitudinal true strain, εl, and the transverse true strain, εw, can be measured by picking out several points and tracking the displacement of the designated points on the series of images. Four designated points will be used to calculate the longitudinal and transverse true strains by Eq. (2) as in Fig. 4. To automate these procedures, a Matlab code was constructed by modifying the existing Matlab code for Digital Image Correlation and Tracking [6].

ew 

d xf  d xi d xi

, el 

d yf  d yi d yi

,  w  ln(1  ew ),  l  ln(1  el )

(2)

Fig. 5 shows a longitudinal strain versus transverse strain curve of TRIP590 at 0.001/sec. The R-value can be calculated from the linearly fitted slope with Eq. (3) since linear relationships were observed in plots for εl vs. εw at all strain rates and loading angles from RD.

r

 d w  (d w / d l ) -r  , Slope in Fig. 5  d l  d w 1  d w / d l 1 r

(3)

To obtain a reliable R-value, the measurement range was limited to the necking instability strain of the steel sheets, which is 12% for TRIP590 and 10% for DP780. The longitudinal and transverse deformations were measured in the maximum broad region to minimize a measurement error. R-values were measured at least five times for each condition for the reliability of the measurement. The comparison of engineering stress–engineering strain curves with and without applying spray paint verifies that application of spray paint does not affect the tensile properties of the material as in Fig. 6.

Transverse strain, t

0.02 0.00 -0.02 -0.04 -0.06 -0.08 0.00

590TRIP 1.2t-TD 0.001 /s 0.02

0.04

0.06

0.08

0.10

Longitudinal strain, l

Fig. 5 Longitudinal strain versus Transverse strain distribution (TRIP590-90o, 0.001/sec)

320

Engineering Stress [MPa]

1000 590TRIP 1.2t - DD, 100 /s

800 600 400

w/ spraying w/o spraying

200 0 0.0

0.1

0.2

0.3

0.4

Engineering Strain

Fig. 6 Engineering stress–strain graph with and without spraying (TRIP590-45o, 0.001/sec) R-VALUES OF AHSS SHEETS AT VARIOUS STRAIN RATES The R-value of TRIP590 and DP780 steel sheets are tabulated on Table 1 and Table 2, respectively for the given range of strain rates and loading angles. Maximum and minimum values are shown along with the average values since the measured R-values have large variation. The variation of the R-value with respect to the loading angle and strain rate are shown in Fig 7 and Fig 8, respectively to visualize the change in the R-value at various loading angles and strain rates. DP780 1.0t

Strain rate [/s] 0.001 0.01 0.1 1 10 100

r-value

1.2

0.9

1.5

0.6 0.00 0

15

30

45

60

75

TRIP590 1.2t

Strain rate [/s] 0.001 0.01 0.1 1 10 100

1.2

r-value

1.5

0.9

0.6 0.00

90

0

o

Loading angle from RD [ ]

15

30

45

60

75

90

o

Loading angle from RD [ ]

(a)

(b)

Fig. 7 Variation of the R-values with respect to the loading angle (a) TRIP590 1.2t; (b) DP780 1.0t

1.2

r-value

1.5

o

Loading angle [ ] 0 15 30 45 60 75 90

TRIP590 1.2t

0.9

0.6 0.00

o

Loading angle [ ] 0 15 30 45 60 75 90

DP780 1.0t

1.2

r-value

1.5

0.9

0.6 0.00 1E-3 0.01

0.1 1 10 Strain rate [/s]

(a)

100

1E-3 0.01

0.1

1

Strain rate [/s]

(b)

Fig. 8 Variation of the R-values with respect to the strain rate (a) TRIP590 1.2t; (b) DP780 1.0t

10

100

321 Table 1 Average, maximum and minimum values of the measured R-values of 590TRIP 1.2t steel sheets (a) 0° (RD), 45° (DD), 90° (TD); (b) 15°, 30°, 60°, 75° (a) 0° (RD)

45° (DD)

90° (TD)

Strain rate [/sec]

avg

max

min

avg

max

min

avg

max

min

0.001 0.01 0.1 1 10 100

1.02 0.97 0.94 1.02 1.06 1.10

1.05 0.99 0.97 1.05 1.08 1.11

0.99 0.92 0.92 0.98 1.02 1.08

0.76 0.72 0.70 0.79 0.81 0.87

0.81 0.78 0.75 0.80 0.84 0.90

0.73 0.69 0.66 0.77 0.80 0.83

1.06 1.03 0.98 1.06 1.10 1.15

1.09 1.08 1.02 1.09 1.14 1.18

1.03 0.98 0.96 1.01 1.06 1.13

(b) 15°

30°

60°

75°

Strain rate [/sec]

avg

max

min

avg

max

min

avg

max

min

avg

max

min

0.001 0.01 0.1 1 10 100

0.96 0.93 0.90 0.97 1.02 1.03

0.97 0.97 0.91 0.99 1.05 1.06

0.94 0.90 0.89 0.93 0.97 1.00

0.82 0.81 0.79 0.84 0.88 0.94

0.86 0.84 0.86 0.86 0.92 0.97

0.78 0.76 0.73 0.79 0.85 0.91

0.80 0.79 0.77 0.84 0.87 0.93

0.86 0.82 0.81 0.88 0.92 0.97

0.76 0.77 0.73 0.81 0.83 0.89

0.95 0.93 0.87 0.95 0.98 1.03

0.97 0.95 0.89 1.00 1.03 1.05

0.92 0.91 0.84 0.90 0.95 1.01

Table 2 Average, maximum and minimum values of the measured R-values of 780DP 1.0t steel sheets (a) 0° (RD), 45° (DD), 90° (TD); (b) 15°, 30°, 60°, 75° (a) 0° (RD)

45° (DD)

90° (TD)

Strain rate [/sec]

avg

max

min

avg

max

min

avg

max

min

0.001 0.01 0.1 1 10 100

0.79 0.73 0.66 0.81 0.82 0.86

0.81 0.78 0.72 0.83 0.86 0.89

0.76 0.65 0.63 0.78 0.77 0.85

0.99 0.93 0.86 1.02 1.05 1.09

1.07 0.99 0.92 1.07 1.08 1.12

0.96 0.89 0.81 0.99 1.03 1.05

0.80 0.78 0.70 0.84 0.86 0.89

0.81 0.82 0.75 0.87 0.89 0.92

0.79 0.75 0.65 0.82 0.84 0.85

(b) 15°

30°

60°

75°

Strain rate [/sec]

avg

max

min

avg

max

min

avg

max

min

avg

max

min

0.001 0.01 0.1 1 10 100

0.84 0.82 0.75 0.88 0.89 0.94

0.85 0.86 0.79 0.94 0.90 1.04

0.83 0.73 0.73 0.83 0.88 0.89

0.92 0.88 0.82 0.93 0.96 1.00

0.96 0.90 0.86 0.97 0.98 1.05

0.87 0.85 0.76 0.91 0.94 0.94

0.87 0.84 0.80 0.92 0.95 1.00

0.90 0.86 0.83 0.94 1.00 1.09

0.84 0.83 0.79 0.91 0.92 0.95

0.83 0.82 0.75 0.87 0.89 0.94

0.86 0.83 0.77 0.89 0.94 0.96

0.81 0.81 0.69 0.85 0.85 0.93

322 The R-value varies with different loading directions, which indicates that these sheets have in-plane anisotropy. The trends in R-value with respect to the loading angle are different for each material. In case of TRIP590, R45 is the lowest and R90 and R0 are higher. By contrast, R45 is the highest followed by R90 and R0 in case of DP780. The different trend is due to the different texture of the two materials. R-value is highly influenced by preferred crystallographic orientations within a polycrystalline material. It can be also noted that the average R-value of both AHSS sheets is lower than that of mild steel, which could lead to poor formability. For both of the material, the R-value changes with the strain rate. It decreases as the strain rate increases up to the strain rate of 0.1/sec and starts to increase above the strain rate of 1/sec. Change in the R-value has a significant effect on the size and the shape of the yield surface. Based on the experimental result, it can be expected that not only the size but also the shape of the yield surface will change with respect to the strain rate. CONCLUSION This paper introduces a new procedure to measure the R-value at intermediate strain rates based on digital speckle extensometry. Two advanced high strength steel sheets for auto-body, TRIP590 and DP780, are considered to investigate the change of the R-value with the suggested method at the strain rate ranging from 0.001/sec to 100/sec. It is observed that the R-value is strain-rate sensitive and it decreases as the strain rate increases up to the strain rate of 0.1/sec and starts to increase above 1/sec for both materials. Based on the experimental result, it can be expected that the size and the shape of the yield surface will change as the strain rate changes. ACKNOWLEDGEMENT The present work was supported by POSCO research fund for POSCO steel research laboratory. REFERENCE [1] ASTM Standard E517 - 00, 2010, "Standard Test Method for Plastic Strain Ratio r for Sheet Metal," ASTM International, West Conshohocken, PA, 2010, DOI: 10.1520/E0517-00R10, www.astm.org [2] E. Parsons, M.C. Boyce and D.M. Parks, An Experimental Investigation of the Large-Strain Tensile Behavior of Neat and Rubber-Toughened Polycarbonate, Polymer, vol. 45, pp. 2665~2684, 2004 [3] F. Laraba-Abbes, P. Ienny and R. Piques, A New ‘Tailor-Made’ Methodology for the Mechanical Behaviour Anlysis of Rubber-Like Materials: I. Kinematics Measurements Using a Digital Speckle Extensometry, Polymer, vol. 44, pp. 807~820, 2003 [4] H. Huh et al., High Speed Tensile Test of Steel Sheets for the Stress-Strain Curve at the Intermediate Strain Rate, International Journal of Automotive Technology, vol.10, no.2, pp.195~204, 2009 [5] Y. C. Liu, On the Determination of Hill’s Plastic Strain Ratio, Metall. Trans. A, vol. 14A, pp. 2566~2567, 1983 [6] C. Eberl, "Digital Image Correlation and Tracking", http://www.mathworks.com/matlabcentral/fileexchange/12413

Study of Strain Energy in Deformed Insect Wings Hui Wan1, Haibo Dong2, and Yan Ren3 Department of Mechanical and Materials Engineering Wright State University, Dayton,OH, 45435, USA Nomenclature E

e h I

r S T U

= = = = = = = = = =

Young’s module Vector of nodal coordinate Wing thickness Identity matrix Absolute coordinates of any point Shape function Kinetic energy Strain energy due to bending and twisting Curvature Density of wing = Relative variation of kinetic energy over a hovering cycle = Relative variation of strain energy over a hovering cycle

ABSTRACT Wing deformation is almost unavoidable in insect flights. In this paper, an approach is introduced to estimate the strain energy in deformed wings during freely hovering of a dragonfly. First, high-speed photogrammetry and three-dimensional surface reconstruction technique are used to quantify the wing kinematics and wing deformation. Finite elements in the absolute nodal coordinate formulation are used to estimate the strain energy associated with wing deformation. The variations of strain and kinetic energy within a stroke cycle are presented and the implications are discussed. Keywords High speed imaging; Insect wing deformation; Absolute nodal coordinate formulation; Strain energy

1. Introduction Wing flexibility and its implication on aerodynamic performance and structural response are fundamental, but challenging problems in the natural insect flight [1, 2]. The measurement of wing deformation has been extremely difficult if it is not impossible at all. However, with a few advent usages of high-speed photogrammetry in insect flight (e.g. Walter et al. 2008 [3], Dong et al. 2010 [4]), it is now possible to quantify the wing deformation with improved accuracy. In fields of materials and structure, Tiwari et al. [5] using high-speed camera measured the plate deformation and calculated Lagrangian strain field due to blast loading. In this paper, High-Speed Photogrammetry is used to record the hovering of a dragonfly (Anax Junius). The wing kinematics and deformation are obtained by three-dimensional surface reconstruction techniques [6]. The flapping wings of the dragonfly feature large rotation and large deformation. Thus, the absolute nodal coordinate formulation (ANCF) [7] based finite elements are used to estimate the strain energy induced by wing bending and twisting. ANCF is characterized by using of slope coordinates which constrain the element orientation. ANCF was first initiated by Shabana et al. [7, 8] to study the dynamics of flexible multi-body system, then this procedure was used to investigate the complex deformation threedimensional beam[9]. Recently, Dmitrochenko et al. [10, 11] extended ANCF to the plate finite element. In the following of this paper, we will first briefly introduce the high-speed camera system, followed by the fundamental theoretical background

                                                             1

Research Scientist, [email protected] Assistant Professor, [email protected] 3 Graduate student, [email protected] 2

T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_45, © The Society for Experimental Mechanics, Inc. 2011

323

324

of ANCF. Then we will discuss the obtained strain energy in the dragonfly wings. The variation of strain energy during a hovering cycle is compared with that of kinetic energy.

2. Experimental Setup and Surface Reconstruction In our high-speed imaging system, three Photron Fastcam SA3 60k cameras are aligned along the principal coordinate axes, as shown in Fig. 1. Each camera in the system can capture black and white pictures with 1000FPS (frames per second) at full resolution (1024x1024) with a shutter speed of 2 us. The cameras are synchronized using daisy-chain method. The cameras are triggered by an external Transistor-Transistor Logic (TTL) signal. By configuring the cameras in a master-slave arrangement, it is possible to trigger one camera to relay the signal to the other two cameras. By using this method of triggering, we are capable of minimizing camera delay and maximizing the response speed. Our current measurements indicate the delay between cameras at a maximum of 30 nanoseconds, which for our applications can be taken as zero delay. More information about the camera system and data acquisition can be found in Dong et al. [4]. The calibration of high-speed camera data, and the detailed procedure to generate 3D reconstruction of wings and body of a hovering dragonfly are presented in [6], based on data gathered from the aforementioned photogrammetry system. A reconstructed dragonfly body and wing surfaces are shown in Fig. 2.

 

  Fig. 1 Experimental setup

Fig. 2 Reconstructed dragonfly body and wing surfaces

3. Theoretical Background Absolute nodal coordinate formulation is applied in the current study of flapping dragonfly wings, which undertake large displacement, large rotation and large deformation during strokes. The principal idea of ANCF is to use the coordinates referred to the global reference system, i.e., it introduces large displacements of finite elements relative to the global frame without using any local frame. The nodal coordinate vector includes the nodal position vector and nodal slope vector. In current paper, the reconstructed wing surfaces of dragonfly are discretized by triangular elements as shown in Fig. 3. Figure 4 presents a triangular element in a curvilinear coordinate system OP1P2. The global coordinate ( of any point in the element can be written as: (1)

,

in which the vector of nodal coordinate

(Fig. 4) in equation (1) has the form of (2)

In detail, the nodal coordinate vector

can be obtained from

,

∂ / ∂P ,

∂ / ∂P

(3)

As shown in Fig. 4, on each node of the element, there are nine degree of freedom, including three components of nodal position, and six components of nodal slope. Therefore, each element has 27 coordinates.

325

Fig. 3 Discretized wing surface after reconstruction, only the right hand side wings are show The shape function

Fig. 4 Triangular element and nodal coordinates

in equation (1) can be expressed as

S where is the 3 by 3 identity matrix. Thus

S

S

S

S

S

S

S

,

(4)

has a dimension of 3 by 27. The shape function can be given in detail as:

S

L S S

in which

Q

S

LL

LL c Q c Q

1 LLL 3 1 2

L L 2Q 2Q c Q L L c Q , L L µ L

1

3µ

(5)

L

L

(6)

and µ ℓ ℓ /ℓ , ℓ is the length of the i-th triangle edge. L is the triangular coordinates defined in elemental coordinate system. The coefficients c are determined from the local coordinate system OP1P2, their expression and the details of triangular element shape functions can be found in Dmitrochenko [11]. Ignoring the longitudinal and shear deformations in the wing, the strain energy in dragonfly wings induced by bending and twisting can be written as:

U=

,

(7)

where , , is the curvature vector, A is wing surface area. is a 3 by 3 matrix determined by the material properties. Given the normal vector of the element, the curvature components can be calculated through

=

,

(8)

where

c 2A

c ∂ ∂L ∂L 2A

(9)

The kinetic energy of the wings are simply defined as

T=

(10)

326

4. Results and Discussion The strain energy in wings of a dragonfly under hovering is being studied. To simplify the problem, the material properties of the fore-wing and hind-wing are assumed to be isotropic, and the thickness of the wings is constant. Thus, in equation (7) leads to:

1

12 1

0

0 1 0 0 1

(11)

Strain energy

Kinetic energy

The surface averaged Young’s modulus of dragonfly wings are chosen as 4.8GPa (Jongerius and Lentink [12]). The Poisson’s ratio is 0.49, as used by Combers and Daniel [1, 2]. The density of wings is 1200Kg/m3, value from [12] and [13].

a)

Start of upstroke

b) Middle of upstroke

c)

Start of downstroke

d) Middle of downstroke

Strain energy

Kinetic energy

Fig. 5 Kinetic energy and strain energy distribution of fore-wing at four time instants. In each figure above, the wing root is at the leftmost

a)

Start of upstroke

b) Middle of upstroke

c)

Start of downstroke

d) Middle of downstroke

Fig. 6 Kinetic energy and strain energy distribution of hind-wing at four time instants As shown in Fig. 3, only the wings at right hand side are studied. The kinematics and deformation of left hand side wings are assumed to be images of their right side counterparts, since the dragonfly is under hovering with negligible maneuvers. The distributions of kinetic and strain energy of fore-wing have been shown in Fig. 5 for four instants during a stroke cycle. First, wing tip has higher kinetic energy than wing root obviously. This is in contrast to the strain energy, which obtains its

327

higher value in a region surrounding the wing root. However, the region very close to the wing root actually has smaller strain energy. Therefore, the wing root of dragonfly may be more appropriately modeled as a hinged structure, instead of clamped structure. This is more understandable from biological point of view. The dragonfly muscles would have to consume a lot more energy to firmly clamp the wing root, compared to hinge it with certain degree of flexibility and leave the bending and twisting naturally adjusted by the wing structure (e.g., vein corrugation). Furthermore, small environmental variation (e.g. aerodynamic load) caused changing of wing deformation and associated disturbance on strain energy may be naturally absorbed by the wing structure itself, instead of transferring to wing root and sensed by neurons in muscles. In other words, the high strain region surrounding the wing root serves a screen that filters out the small fluctuations in environment. For large environmental variations, the high strain region may extend to the wing root. Thus, the increased strain at wing root can be sensed and actions of muscles may be taken to change flight status if needed. Figure 6 shows the energy contours of hindwing. Similar distribution is obtained compared to those of fore-wing. Figure 7 has shown the variation of integrated kinetic energy (equation (10)) and integrated strain energy (7) over a stroke cycle. To avoid the influence on the absolute value of energy due to chosen of the material parameters, we define the relative variations as follows:

,

(12)

where t is time during a stroke. and are the time averaged kinetic and strain energy over a cycle respectively. The negative and positive value of means the energy is below and above the average respectively. The minimum kinetic energy is at the start of upstroke and start of downstroke. The maximum kinetic energy is obtained at the middle of upstroke or downstroke. The kinetic energy varies from minimum to maximum twice during a flapping cycle. The strain energy, however, varies only once. The strain energy is minimum near the start of upstroke, and increases to its maximum at the beginning of downstroke (end of upstroke). It then shows a quick release during the stroke reversal, which may be used to facilitate the wing pronation.

a. Fore-wing b. Hind-wing Fig. 7 Relative variation of integrated kinetic and strain energy over a flapping cycle. Note the flapping of the fore-wing and hind-wing has certain phase difference. Here we have removed the phase difference by showing each wing starts from the initiation of upstroke Also, we have seen that the time duration of upstroke and downstroke are different for both fore-wing and hind-wing. The downstoke takes less time compared with upstroke, which indicate the dragonfly stroke downwards faster to produce enough lift. This is also confirmed by the higher pick value of during downstroke.

5. Summary High-speed photogrammetry and surface reconstruction are used to quantify the wing deformation of a freely hovering dragonfly. Then the wing surface is discretized by triangular plate elements using absolute nodal coordinate formulation. The strain and kinetic energy of dragonfly wing are then calculated over a stroke cycle. Wing root is found to be surrounded by the high strain region, which may serve as a buffer zone to screen the wing root from the small disturbance of environment. The integrated strain energy shows minimum near the supination, and gains its maximum at the beginning of downstroke to

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facilitate the fast pronation. Both the fore-wing and hind-wing take shorter time in downstroke process, thus higher speed and lift can be generated compared to those during upstroke.

Acknowledgements Financial support from AFOSR DURIP-09 Grant #: FA9550-09-1-0460, Wright State University (WSU) Research Challenge Program, and Center for Micro Air Vehicle Study (CMAVS) at WSU are greatly appreciated.

Reference [1] Combes, S. A. and T. L. Daniel, "Flexural stiffness in insect wings. I. Scaling and the influence of wing venation." J Exp. Biol. 206, pp. 2979-2987, 2003. [2] Combes, S. A. and T. L. Daniel, "Flexural stiffness in insect wings. II. Spatial distribution and dynamic wing bending." J Exp. Biol. 206, pp. 2989-2997, 2003. [3] Walker S.M., Thomas, A.L.R, and Taylor G.K. "Photogrammetric reconstruction of high-resolution surface topographies and deformable wing kinematics of tethered locusts and free-flying hoverflies", Journal of the Royal SocietyInterface, Vol. 6, pp. 351-366, 2009. [4] Dong H.B., Koehler, C., Liang, Z.X., Wan, H., and Gaston, Z., "An integrated analysis of a dragonfly in free flight", 40th AIAA Fluid Dynamics Conference and Exhibit, AIAA2010-4390, 2010. [5] Tiwari, V., Sutton M.A., McNeill S.R., Xu, S.W., Deng, X.M., Fourney, W.L., Bretall D., "Applicatin of 3D image correlation for full-filed transient plate deformation measurement during blast loading", Internatial Journal of Impact Engineering, Vol. 36, pp. 862-874, 2009. [6]  Koehler, C., Liang, Z.X., Gaston, Z., Dong H.B., and Wan, H., "Wing Reconstruction, Deformability and Surface Topography of Free-Flying Dragonflies" to be submitted. [7] Shabana, A. A., Hussien, H and Escalona, J. L. Application of the absolute nodal coordinate formulation to large rotation and large deformation problems. ASME Journal of Mechanical Design. 120(3), 188–195,1998 [8] Shabana, A.A. Computer implementation of the absolute nodal coordinate formulation for flexible multi-body dynamics. Nonlinear Dynamics. 16, 293–306, 1998. [9] Shabana, A.A. and Yakoub, R.Y. Three-dimensional absolute nodal coordinate formulation for beam elements: Theory. ASME Journal of Mechanical Design. 123, 606–613, 2001.  [10] Dmitrochenko, O., and Pogorelov, D.Y., "Generalization of plate finite elements for Aboulute Nodela Coordiante Formulation ", Multibody System Dyanics, Vol. 10, pp. 17-43, 2003. [11] Dmitrochenko, O., and Mikkola A., "Two simple triangular plate elements based on the absolute nodal coordinate formulation", Jounal of Computational and Nonlinear Dynamics, Vol. 3, pp. 041012, 2008. [12] Jongerius, S.R. and Lentink D., "Structural Analysis of a Dragonfly Wing", Experimental Mechanics, Vol. 50, pp 13231334, 2010. [13] Vincent, J.F.V. and Wegst, U.G.K., "Design and mechanical properties of insect cuticle", Arthropod Stru Dev. Vol. 33, pp 187–199, 2004

Experimental Study of Cable Vibration Damping Arup Maji a and Yuanzhong Qiu b a

Professor, Civil Engineering Department, University of New Mexico, CENT, Albuquerque, NM 87131, U.S.A b

Graduate Research Assistant, Civil Engineering Department, University of New Mexico, CENT, Albuquerque, NM 87131, U.S.A

ABSTRACT: Measuring and understanding the damping characteristics of cables is particularly significant for structures deployed in space using cables where vibration damping is critical for structural stability. This paper describes an experimental set-up, and provides results from several tests on steel and carbon fiber cables under varying tensile forces. The natural frequencies and the modal damping ratios under different tensile forces are experimental determined; the Rayleigh damping constants are calculated. The difference of the natural frequency between test results and analytical formula are discussed. It is shown that the damping of the cable decreases as the tension force increases. For the case of the carbon fiber cables, the damping decreases as the number of turns of the carbon fiber tows use to make the cable increases. 1.

Introduction

Cables are commonly used tension members in modern flexible structures, and in many applications it is important to know their dynamic properties. Measuring and understanding the damping properties of cables is particularly significant for structures deployed in space using cables where in the absence of air material-related vibration damping is critical for structural stability. Several researchers have investigated the dynamical behavior of cables. Based on experimental and simulated data, Barbieri et al. [1] establish a procedure to identify damping of transmission line cables. Yamaguchi and Adhikari [2] analytically investigated the modal damping characteristics of a single structural cable. They used the energy based representation of modal damping as the ratio of the modal strain energy to the total potential energy. Huang and Vinogradov [3] proposed a model to consider the inter-wire slip and its influence on the dynamic behaviors of the tension cables. Though lots of researches have been conducted on cable damping, the internal damping mechanism of cables has not been fully understood. This is particularly true for the carbon-fiber cables used in the space structures. Working towards fully understand the cable vibration damping of the deployable structures. A test apparatus was built to allow the length, size and tension of the cable to be varied. Vibration tests of stainless steel and carbon-fiber cables were conducted to investigate the characteristics of cable damping. 2.

Experimental set-up

The experimental set-up shown in Fig.1 consists of a wooden frame with two metal plates that can clamp the cables to maintain the applied tension force. Figure 2 shows the load cell and the data acquisition equipment. Fig.3 shows a close-up of the fixture for applying tension forces. Each cable is first fixed to the clamp at the right end of the wooden frame (Fig.1). This is done by screwing four bolts to tighten the two steel plates that constitute the clamp shown at the right end of Fig.3 (b). The left end of the cable is passed through the clamp on the left (Fig.3b) which is initially not tightened. The left of the cable forms a loop that can be connected to a load cell, shown as Fig.3 (a). The load cell with a digital meter provides the readings of the cable tension force during the test. The other end of the load cell is connected to an all thread stainless steel rod, which

T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_46, © The Society for Experimental Mechanics, Inc. 2011

329

330 is used to adjust the cable tension force by tightening the bolts at the left end, shown in Fig.3(c). Once the appropriate load is applied, the clamp on the left (Figure 3b) is tightened to hold the tension during the tests. Two accelerometers, type PCB Model 352A73, are mounted to the cable at approximately three quarters of the span and one fifth of the span length, respectively, shown as Fig.3 (b). These acceleration locations were chosen to such that the first 3 vibration modes could be detected at either transducer by avoiding the nodal points. Having the two accelerometers located on opposite sides on the cable span aims to minimize the torsion effects caused by small movements of the accelerometers during the cable oscillations.

Fig.1 Experimental set-up

(a) NI USB 9234 data carrier

(b) Load cell Fig.2 Sensor devices

(a)

(b)

(c)

Fig.3 The connections of the experimental setup parts The steel cable been tested is made of several stainless steel wires. The carbon fiber cables are made from IM7 carbon fiber (HERCULES INC, type IM7-W-12K). Seven fiber tows are twisted to form a carbon fiber cable. To fabricate the carbon

331 fiber cables, the untwisted strands are first fixed at one end, the other end of the strands are then twisted for 10, 20 and 30 turns to form three different kinds of carbon fiber cables for testing. This aims to investigate how the number of turns influences the damping of the carbon-fiber cable. The geometric and the mechanical properties of the stainless steel and the carbon fiber cables are summarized in Table. 1. 3.

Testing and data processing

The cable is excited at the center. To investigate the influence of tension on damping, the tests are conducted under different cable tension forces of 25lb, 50lb, 75lb, 100lb and 130lb. Data acquisition hardware with signal conditioning function, type NI USB 9234, is connected to a laptop. It digitizes the incoming signals to the analog output signals. The stored data of the signals is used to obtain the spectrum of the signals via LabVIEW 8.6 with the Fast Fourier Transformation (FFT). The modal damping ratio is calculated using the half-power bandwidth method in the frequency domain and the Logarithmic decrement method in the time domain, respectively. For the in-plane vibrations, five sets of data are recorded for each accelerometer. Table.1 cable properties Mass per unit length Length (in) (lbs/in) 12 0.00122 12 0.000184 12 0.000187 12 0.000195

Type of cable Steel cable Carbon fiber cable-10 turns Carbon fiber cable-20 turns Carbon fiber cable-30 turns 4.

Young’s Modulus (psi) 29×106 40×106 40×106 40×106

Experimental results

Fig.4 shows the time history signals of the carbon-fiber cable with 10 turns under tension of 75lbs, recorded by the two accelerometers at three quarters and at one fifth of the cable length. Fig.5 presents the corresponding frequency domain data. The frequencies obtained by the spectral analysis are showed in Table.2 for different tension forces of the cable. The relation between cable vibration frequency and cable tension force is plotted in Fig.6. As shown by Fig.6, the frequency of the cable vibration increses as the tensile force increases. 1500

1000

1000

Acceleration

500 0

-500

0.6

0.62

0.64

0.66

0.68

500 0 -500

0.6

0.65

-1000

-1000 -1500

0.7

Acceleration

1500

-1500

Time/(s)

Time/(s) (a) From accelerometer A

(b) from accelerometer B

Fig.4 The time-history signals for 10 turns carbon-fiber cable under tension of 75lbs

0.7

4

Acceleration(FFT Peak)

Acceleration(FFT Peak)

332

3 2 1 0

0

500 1000 Frequency/Hz

1500

3 2.5 2 1.5 1 0.5 0

0

(a) From accelerometer A

500 1000 Frequency/Hz

1500

(b) from accelerometer B

Fig.5 The spectral of signals for 10 turns carbon fiber cable under tension of 75lbs

Tension Force (lbs)

Table.2 Frequencies of cable vibration under different tensile force Frequency (Hz) Carbon-fiber cable_10 Carbon-fiber cable_20 Carbon-fiber cable_30 turns turns Frequencies (Hz) turns

25 50 75 100 130

309.87 392.63 458.67 517.74 554.68

280.96Frequencies (Hz) 369.02 430.00 Steel Cable 500.49 534.65

Stainless steel cable

266.82 360.33 418.68 485.62 502.50

Carbon fiber cable_10 turns

Carbon fiber cable_20 turns

Carbon fiber cable_30 turns

Stainless steel cable

185.104 218.252 247.820 275.036 303.326

Frequency/Hz

600 500 400 300 200 100 0

0

20

40

60 80 Tensile force/lbs

100

120

140

Fig.6 The relation curve of cable vibration frequency and tensile force The experimental results of the cable vibration frequency under varying tension force are compared to the theoretical results. The theory based on inextensibility of the cable, suggests that the first symmetric natural frequency is given by[4, 5].

1  2.86

 H l

m

(1)

333 Where: l is the length of the cable, H is the horizontal tension of the cable, and m is the mass per unit length of the cable. Fig.7 shown the plots of the trend curves of the experimental results and the theoretical values obtained by Eq. (1). It is observed from Fig.7 that the experimental results have the same trend as the theoretical results, i.e., the frequency increases as the tensile force increases. However, the measured frequencies for carbon-fiber cables are smaller than the theoretical values. The possible explanation is that for Eq. (1), it is assumed that the cable is inextensible, so the length used for the calculation is unchanged. However, the actual cable is capable of some extension, which reduces stiffness and results in the measured frequencies being smaller than the theoretical ones. Table.3 shows the damping of the cable under different tensile forces. The damping is obtained by the half-power bandwidth method in the frequency domain and the Logarithmic decrement method in the time domain, respectively. The damping values corresponding to different tension forces in Table.3 are obtained by averaging the ten damping values obtained from the data recorded at the two measurement locations during the five separately tests. Fig.8 shows the relationship between cable damping and the tensile force using the frequency domain and time domain analysis, respectively. The results demonstrate that higher tension results in lower damping, and that the results obtained by the two methods are similar. In addition, the damping value is in the range of 1-3 percents for the tested cables. For the carbon-fiber cables twisted by different turns, as the number of twists in the cable increases, the damping of the cable decreases. This could be because when the number of twist increases, the cable becomes tighter, reducing the movement and friction internal to the cable.

700.00 600.00 500.00 400.00 300.00 200.00 100.00 0.00

Theoretical

Experimental

Frequency/Hz

Frequency/Hz

Experimental

0

50

100

150

700.00 600.00 500.00 400.00 300.00 200.00 100.00 0.00

0

50 100 Tensile force/lbs

Tensile force/lbs (a) Carbon-fiber cable twisted 10 turns Experimental

Experimental Frequency/Hz

Frequency/Hz

600.00 400.00 200.00 0.00

0

50 100 Tensile force/lbs

150

(c) Carbon-fiber cable twisted 30 turns

150

(b) Carbon fiber cable twisted 20 turns

Theoretical

800.00

Theoretical

350.00 300.00 250.00 200.00 150.00 100.00 50.00 0.00

0

Theoretical

50 100 Tensile force/lbs

(d) Stainless steel cable

Fig.7 The relation curve of cable vibration frequency with tensile force

150

334 Table.3 Damping ratios (%) of different cables and damping variation (%) between frequency domain and time domain Carbon-fiber cable Carbon-fiber cable Carbon-fiber cable Stainless steel cable damping_10 turns (%) damping_20 turns (%) damping_30 turns (%) damping (%) Tension Force (lbs) Frequency Time Error Frequency Time Error Frequency Time Error Frequency Time Error domain domain (%) domain domain (%) domain domain (%) domain domain (%) 2.42

2.39

-1.240

2.27

2.27

0.000

1.68

1.74

3.571

2.265

2.29

1.104

50

2.11

2.05

-2.844

1.8

1.79

-0.556

1.24

1.32

6.452

2.007

1.96

-2.342

75

1.83

1.85

1.093

1.68

1.75

4.167

1.18

1.24

5.085

1.923

1.83

-4.836

100

1.45

1.48

2.069

1.39

1.38

-0.719

0.97

1.03

6.186

1.403

1.38

-1.639

130

1.4

1.45

3.571

1.36

1.34

-1.471

0.93

0.94

1.075

1.155

1.19

3.030

Carbon fiber cable_10 turns

Carbon fiber cable_10 turns

Carbon fiber cable_20 turns

Carbon fiber cable_20 turns

Carbon fiber cable_30 turns

Carbon fiber cable_30 turns

Stainless steel cable 3 2 1 0

0

50 100 Tensile force/lbs

150

Damping-time domain/%

Damping-frequency domain/%

25

Stainless steel cable 3 2 1 0

0

50 100 Tensile force/lbs

150

Fig.8 Variation of cable dampings with applied tension Meanwhile, in order to evaluate the dependence of damping on the vibration mode, the Rayleigh damping theory is introduced. For the Rayleigh damping theory [6], it is assumed that the damping matrix is proportional to the combination of mass and stiffness matrices as given by the expression of Eq. (2):

C   M    K  (2) Where: C  =damping matrix of the vibration system  M  =mass matrix of the vibration system

 K  =stiffness matrix of the vibration system

 and  are the constants After orthogonal transformation and reduction to n-uncoupled equation, we can obtain the following: 2     2  1 1 1  2 (3) 2        2  2 2

335 From Eq. (3), the following is obtained:  2  (    )   1 2 1 2 2 1  22 12  (4)    2( 22 11 )  2 2 2 1 

Where 1 and

2 can be determined from a spectrum analysis of the vibration of the cable system.  1 and  2 are determined

by the half-power bandwidth method. Before applying those values to Eq. (4), the values are smoothed by quadratic curve fitting. The values of  and  for different tension forces obtained by Eq. (4) are plotted in Fig.9. From the plots, it can be observed that  initially increases as cable tension increases, and then decreases. Fig.9 shows that  for both carbon-fiber cables and the stainless steel cable decreases as tension increases. Carbon fiber cable_10 turns

Carbon fiber cable_10 turns

Carbon fiber cable_20 turns

Carbon fiber cable_20 turns

Carbon fiber cable 30 turns

Carbon fiber cable 30 turns

Stailess steel cable

Stailess steel cable 0.000200

5.000 0.000 -5.000

Beta

Alpha

10.000

0

50

100

150

0.000150 0.000100 0.000050 0.000000

0

Tensile force/lbs

50 100 Tensile force/lbs

150

Fig.9 Relation curves of alpha and beta with tension force Conclusions An experimental setup was designed to measure the damping of the steel cable and carbon fiber cables twisted by different turns. The natural vibration frequency was obtained by spectral analysis, and it was compared to the theoretical solution based on the assumption that the cable is inextensible. In addition, Half-power bandwidth method and the Logarithmic decrement method were employed to obtain the damping ratios in frequency domain and time domain, respectively. The Raleigh damping constants were computed using the derived equations. From the experimental results, the following conclusions can be drawn: The frequency of the cable vibration increases as the tensile force decreases, and for the case of carbon-fiber cables, the frequency decreases as the number of twists in the cable increases; The damping of the cable decreases as tensile force increases, and as the number twists of the carbon-fiber cable increases; Also, the damping obtained from frequency domain method agrees well with the results from the time domain method. Regarding the damping coefficients based on the Rayleigh damping theory, the stiffness damping coefficient  decreases as the tensile force increases while the mass damping coefficient References:

 initially increases followed by a decrease.

336 [1] Nilson Barbieri, Oswaldo Honorato de Souza Junior, Renato Barbieri. ―Dynamical analysis of transmission line cables Part 2—damping estimation.‖ Mechanical Systems and Signal Processing, 18 (2004), pp.671–681. [2] H. Yamaguchi, and R. Adhikari, ―Energy based evaluation of modal damping in structural cables with and without damping treatment. Journal of Sound and Vibration, 181(1), pp.71–83, 1995. [3] X. Huang and O. Vinogradov. ―Inter-wire slip and its influence on the dynamic properties of tension cables.‖ Proceeding of the second (1992) International Offshore and Polar Engineering Conference, San Francisco, USA, 14-19, June, Volume (II), pp.392-396, 1992 [4] K.A. Kashani, ―Vibration of Hanning Cables.‖ Computers & Structures, Vol.31, No.5. pp.699-715, 1989. [5] H.Max Irvine, Cable Structures, 1981, MIT, ISBN 0-262-09023-6. [6] Anil K. Chopra, Dynamics of structures-Theory and Applications to Earthquake Engineering, 3 rd Edition, ISBN-978-81203-3446-5.

Dynamic Thermo-Mechanical Response of Austenite Containing Steels

V-T. Kuokkala1, S. Curtze1,2, M. Isakov1 and M. Hokka1 1

Tampere University of Technology, Department of Materials Science, P.O.B. 589, 33101 Tampere, Finland 2 Oxford Instruments Nano Analysis, Nihtisillankuja 5, 02631 Espoo, Finland

ABSTRACT Austenite can be made thermally stable at room temperature by alloying the steel for example with nickel or manganese, which brings the martensite start temperature of the alloy below RT. Also low alloy steels can contain relatively high amounts of (retained) austenite brought about by appropriate heat treatments, which increase the carbon content of some of the austenite grains to such high levels that thermal martensite transformation does not take place in them. Well known steels containing stable or metastable austenite at room temperature are austenitic and duplex stainless steels, Hadfield manganese steels, low alloy TRIP steels, and high manganese TRIP and TWIP steels. Depending on the deformation temperature and strain rate, deformation induced martensite transformation and/or twinning, or the lack of them, can lead to quite extraordinary behavior and strength and elongation combinations. In this paper, the strain hardening behavior and strain rate sensitivity of several fully or partially austenitic steels are discussed in view of their microstructural development during deformation. The discussion is based on the experiments conducted on these materials in wide ranges of strain rate and temperature with conventional materials testing machines and various Hopkinson Split Bar techniques. INTRODUCTION In their simplest form, steels are solid solutions of iron and carbon only. Because of this, probably the most important phase diagram in materials science is the iron-carbon, or rather the iron-iron carbide (Fe3C), phase diagram. It presents the different equilibrium phases and their relative amounts at different iron-carbon compositions and temperatures. By alloying also other elements in varying amounts, stability of different phases and microstructures can be changed. In plain or low alloy carbon steels, austenite is stable only at relatively high temperatures, but by proper alloying it can be made stable down to very low temperatures, as happens for example with many austenitic stainless steels. The low temperature microstructures and existing phases depend not only on alloying but also on the thermal and mechanical deformation history of the steel, i.e., the cooling rate(s), hold times at different temperatures, and deformation taking place before, during, and after cooling. For example, austenite may be thermally stable (or metastable) at low temperatures (such as room temperature), but it may undergo a phase change when it is sufficiently mechanically deformed, as happens for example with certain austenitic stainless steels, where (non-magnetic) austenite transforms mechanically into (magnetic) martensite. Good practical examples of this are found in every household: many items of cutlery and for example the kitchen sink are made of so-called 18/8 austenitic stainless steels. Their straight parts are non-magnetic, but curved (deformed) parts are clearly attracted by a permanent magnet. The effects of temperature and deformation on the austenite-to-martensite transformation are schematically depicted in Fig. 1. The rate of deformation can have several effects on the behavior and development of the microstructure of austenite containing steels. This can be due to the direct effects of strain rate on the actual deformation and strain hardening mechanisms, or to the indirect effects of deformation induced heating on the mechanical response and evolution of the steel’s microstructure during deformation (general thermal softening or, for example, changes in the material’s phase transformation and/or twinning behavior).

T. Proulx (ed.), Dynamic Behavior of Materials, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series 99, DOI 10.1007/978-1-4614-0216-9_47, © The Society for Experimental Mechanics, Inc. 2011

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338

Strain hardening behavior of crystalline materials depends primarily on the evolution of the dislocation structure, but also on the possible occurrence of twinning and/or phase transformations during deformation. The stacking fault energy γSFE of the austenite phase has a strong influence on the martensite transformation and twinning propensity, and it is generally assumed that twinning occurs when stacking faults energy is in the range 18 ≤ γSFE ≤ 45 mJ/m2 and martensite transformation at γSFE values lower than 18 mJ/m2 . When stacking fault energy exceeds ca. 45 mJ/m2, plasticity and strain hardening are controlled solely by the glide of dislocations [1]. Stacking fault energy can be adjusted by proper alloying: for example manganese and aluminum increase γSFE while silicon normally decreases it. As the stacking fault energy is also a function of temperature, changes in the ambient temperature or in the internal temperature of the deforming material due to adiabatic heating can result in changes in the deformation and strain hardening behavior of the material.

Chemical Free Energy

ΔGMγ →S α '

U’ Stress assisted

MS

Strain induced

M σS

T0 Temperature

(a)

Md

U’

Plastic deformation of   austenite, no martensite 

Austenite

 

U’

Spontaneous thermal   martensite transformation 

ΔG α '→γ

Stress 

ΔGTγ1→α '

σ Yγ Yield strength of   parent phase 

MS

MσS

Md

Temperature 

(b)

Fig. 1 Free energy of martensite and austenite (a), and dependence of the critical stress needed to initiate mechanical martensite transformation on temperature (b) [2, 3] MATERIALS AND EXPERIMENTS In this work, the mechanical behavior and microstructure evolution of several fully or partially austenitic steels were studied at different strain rates and temperatures. The fully austenitic steels were three experimental grades of twinning induced plasticity (TWIP) steels and two commercial grades of austenitic stainless steels. The steel containing initially ca. 12 % retained austenite was a commercial low alloy transformation induced plasticity (TRIP) steel. The chemical compositions of the studied steels are presented in Table 1. The materials were tested both in tension and compression at wide ranges of strain rate and temperature using conventional servo-hydraulic materials testing machines and Hopkinson Split Bar techniques with high and low temperature capabilities. The test procedures have been described in details elsewhere [4-6]. Figures 2-4 show some results relevant to the following discussion.

339

Table 1 Compositions and stacking fault energies of the studied steels Material

Mn

Al

Si

C

Cr + Mo

Ni

Nb

Cu

P

N

Fe

SFE

TWIP 1

28

1.6

0.28

0.08

2 m/s), the strain rates generated in the beams were low, and therefore the quasi- static Young’s moduli of mild steel, aluminum, PMMA, araldite were employed for converting peak- tensile strains to peak- tensile stresses. In the case of Poisson’s ratios of these materials the quasi- static values and dynamic values should not, however, be different. Accordingly, Young’s moduli and Poisson’s ratios for these beam materials were determined by conducting quasi- static uniaxial tension tests on these beam specimen themselves. In case of urethane beam, however, Poisson’s ratio specified by the manufacturer was accepted. In dynamic photoelastic study the dynamic material stress- fringe value by 10% (3). In Table- 1.the various impact parameters for each of five beams are listed:

Beam Materials

Mild steel Aluminum PMMA Araldite Urethane rubber

TABLE-1. SOME OF IMPACT PARAMETERS Striker Diameter Mass of the Beam weight/ (mm), Material of Striker (gm) Striker weight the hemispherical tip 16, mild steel 115 2.492 14, mild steel 41.5 2.468 15, mild steel 19 2.450 15, mild steel 19 2.500 14, mild steel 14 2.675

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Experimental Techniques & Results: (a) Experimental Set Up: A system for generating impact (Fig.1a) was fabricated which essentially consisted of an electromagnet, an aluminum guide pipe and striker. The diameter of the guide pipe was slightly larger so that the striker fell freely. Experiments conducted on mild steel, aluminum, PMMA and araldite beams were essentially due to free fall of strikers and contact velocity in each case was calculated to be 1.86 m/s. However, for urethane rubber beam, experiment was conducted on different set up and actual free fall of striker could not be attended. The contact velocity calculated from experiment was found to be 1.6 m/s. These are shown in Table- 2(b). It appears that the striker somehow interfered with guide pipe and consequently its velocity of free fall was to some extent reduced in this case. A variable DC power supply was connected to the electromagnet which was supported separately over the guide pipe. This supporting system was clamped on fixture such that the centre- line of the guide pipe coincided with the centre- line of the span. After adjusting horizontality and verticality of different components of the supporting each beam specimen was placed over the wedge supports according to the marking for central- impact loading. A clear gap of 10 mm was provided between the guide pipe and beam. Each striker was kept at a height of 176.4 mm by energizing the electromagnet. A low power supply current was maintained to minimize the time delay for de-energizing the electromagnet.

Fig.1a. Legend: 1.Electromagnet (60V DC, 40mA, 1a. Guide Pipe., 2. Vertical mild steel rod for supporting electromagnet,3. Aluminum Plate,4.Fastax, 16mm framing camera,5.Monochromator,6.Optical bench,7.Light source(Sun-gun-II,800W),8.Fresnel lens & diffuser plate,9.Plane Polaroid,10.Quarter-wave plate,11.Signal from Timemarker,12.Goose-control unit

402 Dynamic Photoelasticity to Study Transverse Impact on Beams

Fig.1b. Isochrom atic Fringe Pattern Central Impact Span:120mm, Over-hang Ratio: 0.263 & I.F.T=84.11µs

Fig.1c.

120 mm Beam Span, Frame No.30, Time = 2523µs. Enlarged Photograph

(b) Measurement of Dynamic Strain: For the measurement of dynamic strain type KWR-5 (5 mm gauge length) strain gauges both active and dummy, were employed (Fig.2b) along with a storage oscilloscope. A low sweep rate was used to ensure the recording of the entire response for each beam. Several trials were made in each situation before accepting a reading of peak- tensile strain. Usually the variation was 2 to 3%. In one

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situation, it was about 7%. The peak- tensile strain so recorded were 111µ for mild steel, 128µ for aluminum, 353µ for PMMA and 447µ for araldite beams. These values are also shown in Table- 2(b). (c) Fringe Photography: A portable diffused light polariscope (Fig.1a) was designed. An interference filter (of band pass of less than 100 Ao) was used before a high speed before a high speed camera to render the light monochromatic. A light- field arrangement of optical elements (Fig.1a. and 1b.) was employed which helped recording clearly supports and movements of falling striker on film negative. Wollensak Fastax (16 mm) framing camera operating at 12,000 fps (frames per second) was used for fringe photography. From recorded film negative the contact- velocity of striker (14 gm mass) was determined [1]. The contact velocity so obtained was 1.6 m/s and is shown in Table2(b). Enlarged prints of both continuous and discrete frames were prepared (Fig.1b, 1c) from film negative to study the history of fringe formation and identifying precisely fringe orders respectively. Based on enlarged prints of fringe photographs stress distributions along upper and lower boundaries of the beam were plotted.

+ -

-2 -1

Time from onset of Collision Top Boundary Bottom Boundary + Tensile - Compression

0 +

1

Fringe Order

757µs

2 3 4

(d) boundary Stress Distribution for 120 mm Beam- span under Central Impact Loading Fig.2. Free-

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Fig.2a. Experimental arrangement for Measurement of Contact Force, Strain and Contact velocity

A typical frame identifying maximum fringe order at lower boundary after 757µs from the onset of collision was selected and the corresponding free- boundary stress distribution is presented in Fig.2b. In this figure the numerical value of fringe order corresponding to maximum impact stress at lower fiber, i.e. 675 gm/cm2 has been indicated. The nature of free- boundary stresses has also been indicated. Since stress distribution is symmetrical about the centre- line of the beam, the plot for the left- half of the beam only is shown. The peak stress is indicated in Table- 2(b). (e) Results: Transverse impact on beam of five different materials studied keeping the following impact parameters constant viz. (i) span- depth ratio (4.8), (ii) beam- striker weight ratio (2.5 nominal) and (iii) contact- velocity (1.86 m/s). From strain gauge study peak- tensile strains observed were converted to stresses. The following Young’s modulus (E) for different beams was obtained: 222 kN/mm2 for mild steel, 83.072 kN/mm2 for aluminum14.049 kN/mm2 for PMMA and 17.791 kN/mm2 for araldite. Dynamic photoelastic study indicated the peak- tensile stress, 675 gm/cm2 for urethane beam. From the ‘mass- density’ and ‘Young’s modulus of different materials the ‘rod wave’ velocity 5002.19 m/s, highest, for mild steel beam and is 63.185 m/s , lowest, for the urethane rubber beam. The same for other materials are indicated in Table- 2(a). Realizing the importance of ‘characteristic impedance’ of a material which is defined by the product of ‘mass- density’ (ρ) and ‘rod- wave’ velocity (Co), the same for five different

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materials were determined. The ‘characteristic impedance’ (ρCo) [4] is 3998 gm.s/cm3, highest, for mild steel and is 6.7 gm.s/cm3, lowest, for the urethane rubber. These values are indicated in Table- 2 (b). The intensity of ‘initial stress pulse’ (ρC0v0) as defined by the product of characteristic impedance’ and the ‘collision velocity’ (v0), were calculated for beams studied, and the following values were obtained: 743,674 gm/cm2 for mild steel, 249,928 gm/cm2 for aluminum, 41,757 gm/cm2 for PMMA, 42,185 gm/cm2 for araldite and 1,065 gm/cm2 for urethane rubber. Comparing the peak- tensile stress (σ) with the intensity of ‘initial stress pulse’ (ρC0v0) the following results wee obtained for the five beams studied: peak- tensile stresses for the beams made up of mild steel, aluminum, PMMA, araldite and urethane rubber were 30%, 33%, 34%, 42% and 63%, respectively, of the corresponding ‘initial stress pulse’. These were also mentioned in Table- 2(b). It is interesting to note that the ratios of σ/(ρC0v0) indicated almost one-to-one correspondence with the Poisson’s ratio (Table- 2b) for the mild steel, aluminum, PMMA beams whereas for araldite and urethane rubber beams there was difference. The different values of Poisson’s ratio (ν) and the ratios of σ/(ρC0v0) for five beam materials were plotted and is shown in Fig.3.

Peak-tensile Stress / Initial Stress Pulse Vs. Poisson’s Ratio

Peak Tensile Stress / Initial Stress Pulse Intensity σ/(ρC0v0)

0.7

UR-

0.6 0.5 ARMildsteel

0.4 MS-

0.3

PMMA AL-

Aluminium PMMA ARALDITE Urethane Rubber

0.2

Trendline

0.1 0 0

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Poissons’s Ratio (ν)

Fig.3. Peak- Tensile Stress/Initial Stress Pulse Vs. Poisson’s Ratio

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TABLE-2. (a) DYNAMIC RESPONSE OF BEAMSOF DIFFERENT MATERIALS Beam Materials

Span/Depth

Density,γ (gm/cm3)

Mild steel Aluminum PMMA Araldite Urethane rubber

4.800 4.800 4.800 4.800 4.948

7.834 2.726 1.242 1.267 1.031

Massdensity,ρ, (gm.s2/cm4) x10-3 7.993 2.782 1.266 1.292 1.052

Young’s Modulus, E (gm/cm2) 222 0.649 0.398 0.398 0.042

Rod- wave velocity, C0 =√E/ρ, (m/s) 5002.190 4829.963 1773.065 17755.134 63.185

(b) POISSON’S RATIO, CHARACTERISTIC IMPEDANCE, CONTACT VELOCITY, INITIAL STRESS- PULSE INTENSITY AND σ/(ρC0v0) Beam Materials

Mild steel Aluminum PMMA Araldite Urethane rubber

Poisson’s ratio

‘characteristic impedance’

‘collision velocity’

(ν)

(ρCo) (gm.s/cm3) 3998.0 1343.7 224.5 226.8 6.7

(v0),cm/s

0.285 0.33 0.38 0.34 0.46

186 186 186 186 160

‘initial stress pulse’ (ρC0v0), gm/cm2 743,674 249,928 41,757 42,185 1,085

Peak-tensile Strain,ε (µ) 111 128 353 447 -----

Stress,σ (gm/cm2) 222,000 83,072 14,049 17,791 675

σ/(ρC0v0) Ratio 0.30 0.33 0.34 0.42 0.63

Discussions: The term σ/(ρC0v0) (Table-2b) calls for further scrutiny. The quantity σ/(ρC0) represents particle velocity of the lower- fiber point in the direction of tension (that is, tangential to the boundary). On the other hand, v0 represents the particle velocity of the struck point in the vertical direction (i.e. the direction of the fall of the striker). Therefore, σ/(ρC0v0) also represents the ratio of the peak- particle velocity at the measured point (parallel to the beam axis) and the initial particle velocity at the struck point (perpendicular to the beam axis) (Fig.4). It is to be noted, however, that the two particle velocities mentioned above do not refer to the same point of time, and also refer to two different points of the beam. Again, the two quantities refer to velocities, though in two mutually perpendicular directions, rather than strains. Poisson’s ratio of a material on the other hand is a ratio of lateral strain and longitudinal strain measured at the same point for uniaxial state of stress, and this ratio need not change for quasi- static or dynamic loading. The similarity between the Poisson’s ratios of the three materials mild steel, aluminum and PMMA (Fig.3) with the corresponding σ/(ρC0v0) values, however, suggest that for the impact parameter under consideration both σ/(ρC0v0) and ν follow a similar mechanism in respect of deformation in the molecular scale i.e. on the behavior of materials from the point of

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view of material science. Out of the five materials considered two, namely mild steel and aluminum have well defined atomic lattices and their deformations are essentially controlled by Condon- Morse curve. Their elasticity is essentially controlled by potential energy around the potential energy trough. The third material chosen, viz. PMMA, is long- chain polymer with practically no cross- links and low stresses and quasi- static loading it is viscoelastic in nature. The fourth material chosen, namely araldite, is net- work polymer with heavy three- dimensional cross- linking. It is also a two- phase material characterized by very strong (Infusible) primarybond and weak (Fusible) secondary- bond. For low stress and quasi- static loading at room temperature, its behavior should be essentially isotropic and possibly linearly elastic. The fifth material under consideration, namely urethane rubber, is net- work polymer with coiled- bonds and occasional cross- linking. It is a non- linearly elastic material whose elasticity is controlled by changes in entropy.

2. A comparison of peak- tensile stress and the representative ultimate tensile stress indicated the following values: 5% for mild steel, about 11% for aluminum, about 3% for PMMA and about 4% for araldite. The peak tensile stresses in these four materials were thus quite low and for mild steel and aluminum a linearly elastic response is expected. For PMMA also the stress level is low. However, the loading is dynamic and therefore, the viscoelastic effect would not be predominant; or in other words for impact parameter under consideration the response of PMMA beam can also be expected to be nearly linearly elastic. Therefore, a near one- to- one correspondence between σ/(ρC0v0) and ν for these three materials mild steel, aluminum and PMMA is in order. For araldite beam also the stress level is low, however Fig.3 indicates significant departure from one- to- one correspondence with Poisson’s ratio. This departure is attributed to two- phase nature of araldite. For urethane rubber the departure from one- to- one correspondence with Poisson’s ratio (Fig.3) is rather large.

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3. From the results shown in Table- 2(b) the value of the parameter σ/(ρC0v0) for mild steel, aluminum, PMMA and araldite beams have been calculated and the following values have been obtained: 1.05 for mild steel, 1.00 for aluminum, 0.89 for PMMA and 1.24 for araldite. The present study, therefore, indicates that from a structural designers’ point of view the peak- tensile strain/ stress in PMMA can be converted in respect of other material mentioned above by keeping the parameter σ/(ρC0v0) ( 0.25< ν