Optical Measurements, Modeling, and Metrology, Volume 5

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

Optical Measurements, Modeling, and Metrology, Volume 5 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 978-1-4614-0227-5 e-ISBN 978-1-4614-0228-2 DOI 10.1007/978-1-4614-0228-2 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011929640 © 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

Optical Measurements, Modeling, and Metrology 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 Dynamic Behavior of Materials, Mechanics of Biological Systems and Materials, Mechanics of Time-Dependent Materials and Processes in Conventional and Multifunctional Materials; MEMS and Nanotechnology; 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 papers comprising Optical Measurements, Modeling and, Metrology were taken from the general call for papers as well as sessions organized by: E. Maire, MATEIS-INSA, S. Yoshida, Southeastern Louisiana University and C.A. Sciammarella, Illinois Institute of Technology/Northern Illinois University; R. Rodriguez-Vera, Centro de Investigaciones en Optica A.C. Among the topics included in this volume are: 3D Imaging Applied to Experimental Mechanics Modeling and Numerical Analysis in Optical Methods Identification from Full-field Measurements Recent Advances in Displacement-Metrology Methods Phase Unwrapping, Phase Stepping, and High Speed Camera Calibration Dynamic and Quasi Dynamic Measurements Digital Image Correlation The Society thanks the authors, presenters, organizers and session chairs for their participation and contribution to this volume. 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

3D Structures of Alloys and Nanoparticles Observed by Electron Tomography K. Sato, K. Aoyagi, T.J. Konno, Tohoku University

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Damage Characterization in Dual-phase Steels Using X-ray Tomography C. Landron, E. Maire, J. Adrien, INSA-Lyon, MATEIS; O. Bouaziz, ArcelorMittal Research

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In-situ Synchrotron-radiation Computed Laminography Observation of Ductile Fracture T.F. Morgeneyer, Mines ParisTech; L. Helfen, ANKA/Institute for Synchrotron Radiation/European Synchrotron Radiation Facility; I. Sinclair, University of Southampton; F. Hild, LMT-Cachan; H. Proudhon, Mines ParisTech; F. Xu, T. Baumbach, ANKA/Institute for Synchrotron Radiation; J. Besson, Mines Paris Tech

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Understanding the Mechanical Behaviour of a High Manganese TWIP Steel by the Means of in Situ 3D X ray Tomography D. Fabrègue, C. Landron, Université de Lyon, CNRS/INSA-Lyon; C. Béal, Université de Lyon, CNRS/INSA-Lyon/ArcelorMittal Research; X. Kleber, E. Maire, Université de Lyon, CNRS/INSA-Lyon; M. Bouzekri, ArcelorMittal Research

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Mechanical Properties of Monofilament Entangled Materials L. Courtois, E. Maire, M. Perez, MATEIS UMR 5510 - INSA Lyon; Y. Brechet, D. Rodney, Domaine Universitaire

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Characterisation of Mechanical Properties of Cellular Ceramic Materials Using X-ray Computed Tomography O. Caty, F. Gaubert, Laboratoire des Composites Thermostructuraux/LCTS; G. Hauss, Institut de Chimie et de la Matière Condensée de Bordeaux; G. Chollon, Laboratoire des Composites Thermostructuraux/LCTS

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Multiaxial Stress State Assessed by 3D X-ray Tomography on Semi-crystalline Polymers L. Laiarinandrasana, T.F. Morgeneyer, H. Proudhon, Mines Paris Tech

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Effect of Porosity on the Fatigue Life of a Cast al Alloy N. Vanderesse, J.-Y. Buffiere, E. Maire, Université de Lyon – INSA; A. Chabod, Centre Technique des Industries de la Fonderie

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Fatigue Mechanisms of Brazed Al-Mn Alloys Used in Heat Exchangers A. Buteri, Université de Lyon/Alcan CRV; J. Réthoré, J-Y. Buffière, D. Fabrègue, Université de Lyon; E. Perrin, S. Henry, Alcan CRV

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Three Dimensional Confocal Microscopy Study of Boundaries between Colloidal Crystals E. Maire, INSA-Lyon; M. Persson Gulda, N. Nakamura, K. Jensen, E. Margolis, C. Friedsam F. Spaepen, Harvard University

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Scale Independent Fracture Mechanics S. Yoshida, D. Bhattarai, T. Okiyama, K. Ichinose, Southeastern Louisiana University

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Consistent Embedding: A Theoretical Framework for Multiscale Modelling K. Runge, University of Florida

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Analysis of Crystal Rotation by Taylor Theory M. Morita, O. Umezawa, Yokohama National University

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Numerical Solution of the Walgraef-aifantis Model for Simulation of Dislocation Dynamics in Materials Subjected to Cyclic Loading J. Pontes, Federal University of Rio de Janeiro; D. Walgraef, Université Libre de Bruxelles; C.I. Christov, University of Louisiana at Lafayette

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Photoelastic Determination of Boundary Condition for Finite Element Analysis S. Yoneyama, S. Arikawa, Y. Kobayashi, Aoyama Gakuin University

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Discussion on Hybrid Approach to Determination of Cell Elastic Properties M.C. Frassanito, L. Lamberti, A. Boccaccio, C. Pappalettere, Politecnico di Bari

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Mesh Refinement for Inverse Problems with Finite Element Models A.H. Huhtala, S. Bossuyt, Aalto University

125

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Assessment of Inverse Procedures for the Identification of Hyperelastic Material Parameters M. Sasso, G. Chiappini, Università Politecnica delle Marche; M. Rossi, Arts et Métiers ParisTech; G. Palmieri, Università degli Studi e-Campus

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Digital Image Correlation Through a Rigid Borescope P.L. Reu, Sandia National Laboratories

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Scale Independent Approach to Strength Physics and Optical Interferometry S. Yoshida, Southeastern Louisiana University

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Optical Techniques That Measure Displacements: A Review of the Basic Principles C.A. Sciammarella, Northern Illinois University

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Studying Phase Transformations in a Shape Memory Alloy with Full-field Measurement Techniques D. Delpueyo, M. Grédiac, X. Balandraud, C. Badulescu, Clermont Université

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Correlation Between Mechanical Strength and Surface Conditions of Laser Assisted Machined Silicon Nitride F.M. Sciammarella, M.J. Matusky, Northern Illinois University

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Analysis of Speckle Photographs by Subtracting Phase Functions of Digital Fourier Transforms K.A. Stetson, Karl Stetson Associates, LLC

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Measurement of Residual Stresses in Diamond Coated Substrates Utilizing Coherent Light Projection Moiré Interferometry C.A. Sciammarella, Northern Illinois University; A. Boccaccio, M.C. Frassanito, L. Lamberti, C. Pappalettere, Politecnico di Bari

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Automatic Acquisition and Processing of Large Sets of Holographic Measurements in Medical Research E. Harrington, Worcester Polytechnic Institute; C. Furlong, Worcester Polytechnic Institute/Massachusetts Eye and Ear Infirmary/Harvard Medical School; J.J. Rosowski, Massachusetts Eye and Ear Infirmary/Harvard Medical School/MIT-Harvard Division of Health Sciences and Technology; J.T. Cheng, Massachusetts Eye and Ear Infirmary/Harvard Medical School

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Adaptative Reconstruction Distance in a Lensless Digital Holographic Otoscope J.M. Flores-Moreno, Worcester Polytechnic Institute/Centro de Investigaciones en Optica A. C.; C. Furlong, Worcester Polytechnic Institute/Massachusetts Eye and Ear Infirmary/MIT-Harvard Division of Health Sciences and Technology; J.J. Rosowski, Massachusetts Eye and Ear Infirmary

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3D Shape Measurements With High-speed Fringe Projection and Temporal Phase Unwrapping M. Zervas, C. Furlong, E. Harrington, I. Dobrev, Worcester Polytechnic Institute

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Measuring Local Mechanical Properties of Membranes Applying Coherent Light Projection Moiré Interferometry F.M. Sciammarella, C.A. Sciammarella, Northern Illinois University; L. Lamberti, Politecnico di Bari

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Experimental Analysis of Foam Sandwich Panels With Projection Moiré A. Boccaccio, C. Casavola, L. Lamberti, C. Pappalettere, Politecnico di Bari

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Panoramic Stereo DIC-based Strain Measurement on Submerged Objects K. Genovese, L. Casaletto, Università degli Studi della Basilicata; Y.-U. Lee, J.D. Humphrey, Yale University

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Advances in the Measurement of Surfaces Properties Utilizing Illumination at Angles Beyond Total Reflection C.A. Sciammarella, F.M. Sciammarella, Northern Illinois University; L. Lamberti, Politecnico di Bari

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Filters with Noise/Phase Jump Detection Scheme for Image Reconstruction J.-F. Weng, Y.-L. Lo, National Cheng Kung University

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An Instantaneous Phase Shifting ESPI System for Dynamic Deformation Measurement T.Y. Chen, C.H. Chen, National Cheng Kung University

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Development of Linear LED Device for Shape Measurement by Light Source Stepping Method Y. Oura, M. Fujigaki, A. Masaya, Wakayama University; Y. Morimoto, Moire Institute Inc.

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Calibration Method for Strain Measurement Using Multiple Cameras in Digital Holography M. Fujigaki, R. Nishitani, Wakayama University

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Performance Assessment of Strain Measurement With an Ultra High Speed Camera M. Rossi, Arts et Métiers ParisTech; R. Cheriguene, Université Paul Verlaine de Metz; F. Pierron, Arts et Métiers ParisTech; P. Forquin, Université Paul Verlaine de Metz

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Rigid Body Correction Using 3D Digital Photogrammetry for Rotating Structures T. Lundstrom,, C. Niezrecki, P. Avitabile, University of Massachusetts Lowell

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Development of Sampling Moire Camera for Landslide Prediction by Small Displacement Measurement M. Nakabo, M. Fujigaki, Wakayama University; Y. Morimoto, Moire Institute Inc.; Y. Sasatani, H. Kondo, T. Hara, Wakayama University

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Energy Dissipation in Impact Absorber S. Ekwaro-Osire, I. Durukan, F.M. Alemayehu, Texas Tech University; J.F. Cardenas-García, United States Patent and Trademark Office

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Mechanics Behind 4D Interferometric Measurement of Biofilm Mediated Tooth Decay M.S. Waters, National Institute of Standards and Technology; B. Yang, American Denal Association Foundation; N.J. Lin, S. Lin-Gibson, National Institute of Standards and Technology

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Validating Road Profile Reconstruction Methodology Using ANN Simulation on Experimental Data H.M. Ngwangwa, University of South Africa; P.S. Heyns, H.G.A. Breytenbach, P.S. Els, University of Pretoria

43

Electro-optical Property of Sol-gel-derived PLZT7/30/70 Thin Films J.-F. Lin, J.-S. Jeng, W.-R. Chen, Far East University

44

Decoupling Six Effective Parameters of Anisotropic Optical Materials Using Stokes Polarimetry T.-T.-H. Pham, Y.-L. Lo, National Cheng Kung University

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Measurement of Creep Deformation in Stainless Steel Welded Joints Y. Sakanashi, S. Gungor, P.J. Bouchard, The Open University

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Thermal Deformation Measurement in Thermoelectric Coolers by ESPI and DIC Method W.-C. Wang, T.-Y. Wu, National Tsing Hua University

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Structural Health Monitoring Using Digital Speckle Photography F.-P. Chiang, J.-D. Yu, Stony Brook University

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Determining the Strain Distribution in Bonded and Bolted/Bonded Composite Butt Joints Using the Digital Image Correlation Technique and Finite Element Methods D. Backman, G. Li, T. Sears, National Research Council Canada

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Improved Spectral Approach for Continuous Displacement Measurements From Digital Images F. Mortazavi, M. Lévesque, École Polytechnique de Montréal; I. Villemure, École Polytechnique de Montréal/Sainte-Justine University Hospital Center

50

Experimental Testing (2D DIC) and FE Modelling of T-stub Model D. Carazo Alvarez, University of Jaén; M. Haq, Michigan State University; J.D. Carazo Alvarez, University of Jaén; E. Patterson, Michigan State University

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3D structures of alloys and nanoparticles observed by electron tomography Kazuhisa Sato*, Kenta Aoyagi, and Toyohiko J. Konno Institute for Materials Research, Tohoku University 2-1-1 Katahira, Aoba, Sendai, Miyagi 980-8577, Japan *E-mail address: [email protected]

ABSTRACT 3D structures of bulk alloys and nanoparticles have been studied by means of electron tomography using scanning transmission electron microscopy (STEM). In the case of nanoparticles of Fe-Pd alloy, particle size, shape, and locations were reconstructed by weighted backprojection (WBP), as well as by simultaneous iterative reconstruction technique (SIRT). We have also estimated the particle size by simple extrapolation of tilt-series original data sets, which proved to be quite powerful. We demonstrate that WBP yields a better estimation of the particle size in the z direction than SIRT does, while the latter algorithm is superior to the former from the viewpoints of surface roughness and dot-like artifacts. In contrast, SIRT gives a better result than WBP for the reconstruction of plate-like precipitates in Mg-Dy-Nd alloys, in respect of the plate thickness perpendicular to the z direction. We also show our recent results on the 3D-tomographic observations of microstructures in Ti-V-Al, Ti-Nb, Cu-Ag, and Co-Ni-Cr-Mo alloys obtained by STEM tomography. 1.

Introduction

Our understanding on the microstructure in metals and alloys has been advanced with the progress of transmission electron microscopy (TEM) and electron diffraction. Thus, for example in the fifties, dislocation theories were directly confirmed by electron imaging based on the diffraction contrasts [1]. This technique immediately found enormous areas of applications in materials science, and utilized for instance to clarify the phase transformation behavior in a number of alloy systems. Other applications of TEM include of course high-resolution transmission electron microscopy (HRTEM) and scanning transmission electron microscopy (STEM) [2]. The images obtained by these techniques are projections of three dimensional (3D) objects; and in order to better understand the nature of phase transformation behavior, for example, a direct 3D observation is much needed. In this respect, recent advance in 3D-tomography (x-ray, electron, and atom probe tomography) has opened a new prospect. Electron tomography, especially its applications to materials science, is a novel technique, which can retrieve 3D structural information usually missing in TEM and STEM. A 3D structure can be reconstructed by processing a tilt-series of electron micrographs with mass-thickness contrasts, formed by several different imaging techniques: bright-field (BF) TEM [3], dark-field (DF) TEM [4, 5], atomic number (Z) contrast of STEM [6], energy-filtered TEM [6] and electron holography [7]. The recent progress in this field has been summarized in review articles [8, 9]. In all the techniques, acquisition of clear contrast images and accurate alignments of the tilt axis are essential for subsequent 3D reconstruction. Some model simulations on the accuracy of reconstruction have been presented in detail [6]. An investigation for a novel method to quantify 3D reconstructed structures is one of the fundamental interests in the electron tomography [10-13]. Here, we report some of our recent studies on magnetic T. Proulx (ed.), Optical Measurements, Modeling, and Metrology, Volume 5, Conference Proceedings of the Society for Experimental Mechanics Series 9999999, DOI 10.1007/978-1-4614-0228-2_1, © The Society for Experimental Mechanics, Inc. 2011

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nanoparticles and bulk alloys, where electron tomography has played an important role in identifying the 3D structures and spatial distribution of nanocrystals, dislocations, and precipitates. 2.

Experimental Procedures

We employed BF and high-angle annular dark-field (HAADF) imaging modes of STEM for the tilt-series acquisition using an FEI Titan 80-300 (S)TEM operating at 300 kV with a field emission gun. We set the beam convergence to be 10-14mrad in half-angle, taking into account the spherical aberration coefficient (1.2 mm) of the pre-field of objective lens. The Xplore3D software (FEI Co. Ltd) was used for data sets acquisition taking the dynamic focus into consideration. A single-tilt holder (Fischione model 2020) and a triple-axes holder (Mel-Build model HATA-8075) were used for the tilt series acquisition with the maximum tilt angle of 70º. Alignment of the tilt axis for the obtained data set by an iterative cross-correlation technique and subsequent 3D reconstruction were performed by using the Inspect3D software package (FEI Co. Ltd). As for the algorithm for 3D reconstruction, we employed weighted back-projection (WBP) [14], as well as simultaneous iterative reconstruction technique (SIRT) [15]. The reconstructed 3D density data were then visualized using the AMIRA 4.1 software (VISAGE IMAGING). 3.

Results and Discussion

3.1 Shapes and distribution of FePd nanoparticles Figure 1a shows a series of STEM-HAADF images taken at different tilt angles with the detector inner half angle of 60mrad. The tilt-series was observed sequentially from 0 to -70º and then 0 to +70 º. The tilt angle increments were set 2º for angle ranges of 0 to |50|º, and 1º for |50| to |70|º. Out of this data set, we employed, by careful inspection of contrasts, images taken at tilt angles between -66 and +64º for later 3D reconstruction. As seen, the apparent particle length in the y-direction becomes shorter as the tilting angle increases. A nanoparticle enclosed by the circle in the figure is one of the examples to demonstrate the reduction of the particle image in the y-direction. To examine an accuracy of a reconstructed particle height in the z-direction, we therefore measured projected particle length in the y-direction as a function of tilt-angle, and deduced the particle height by extrapolating the projected length to the value expected at the tilt-angle of 90º. The results are plotted in Fig.1b. The projected length clearly decreases with tilting, which indicates that the particle height is actually shorter than the in-plane diameter. Here, the extrapolation was performed by fitting the data points at angles higher than 40º using cosine of the tilt angle (α), because of the fact that the projected y-length is proportional to cosα at high angles when the particle height is shorter than the diameter. Using the aforementioned procedure, here termed “tilt-series extrapolation (TSE) method”, we obtained a relation, which summarizes the relation between particle diameter and thickness estimated by using several different techniques (Fig.2a). Solid triangles and solid squares indicate the results obtained from the reconstructed images based on SIRT and WBP, respectively. In the present study, 20 iterations were carried out in SIRT to minimize the differences between the original projected series and the calculated ones. The large error bar for WBP indicates a possible elongation of dz = 4.1 nm [13]. Therefore, we divided the apparent particle thickness (tz), which was deduced from the 3D volumes based on WBP, by the elongation factor (eyz=1.42) [12] for the present experimental condition. The results, tz / eyz, are indicated by open squares. Solid circles denote the deduced particle thickness measured from the TSE method. A solid curve indicates the previous result based on the electron holography [16]. Note that the deduced thicknesses obtained by the TSE agree well

3

with those obtained by WBP (tz / eyz) as well as those by electron holography. On the other hand, the thicknesses suggested by SIRT are much larger than the values deduced by the TSE method or electron holography. The apparent thickness predicted by WBP (tz) is close to the deduced values with an error of about 1-4 nm in thickness, without taking the elongation factor into consideration. Therefore, within a framework of single-axis tilt geometry, it is demonstrated in a semi-quantitative manner that the WBP gives a better result in terms of the accuracy of the particle length in the z-direction than that predicted by SIRT, despite the fact that the latter algorithm is superior to the former from the viewpoints of artifacts.

Fig. 1 (a) A series of Z-contrast images taken at different tilt angles. (b) The analyzed particle length in the y-direction as a function of the tilt-angle. The particle length decreases as the tilt-angle increases towards 90º, indicating the fact that the particle height is shorter than the diameter. Extrapolation of particle length in the y-direction to the value expected at the tilt-angle α = 90º leads to an elucidation of the true particle height. Here, the extrapolation was performed by fitting the data points at angles higher than 40º using cosine of the tilt angle [13].

Fig. 2 (a) The relation between particle diameter and thickness (height) for the FePd nanoparticles estimated by using several different techniques. (b) Oblique-view of the reconstructed volume processed by SIRT (upper) and (c) WBP (lower). Large discrepancy in particle thickness (height) is apparent. The reconstructed volume is 75 × 75 × 36 nm3 [13].

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The difference in the particle height of the reconstructed results is pronounced when viewed from an oblique direction as shown in Fig.2b. Indeed SIRT gave particle heights, which are almost comparable to or even longer than the particle diameter (Fig.2b), while rather flat 3D shapes can be seen in the result by WBP (Fig.2c). Nanoparticles in the upper image (SIRT) show prolate 3D-shapes, i.e., elongated in the z-direction. The reason for this artifact is not clear at this moment. To reduce the artifacts, a minimization of the missing wedge will be most effective, which can be attained by increasing the maximum tilt-angle together with number of 2D-slice images as possible. Using the same experimental setup, recently we succeeded in reconstructing double-layer of 2nm-sized CoPt nanoparticles separated by a thin amorphous carbon film [18]. 3.2 Phase separation in Ti-V-Al alloy We have examined 3D morphology of α (hcp) and β (bcc) dual phase structure of a Ti-12mass%V-2mass%Al alloy after aging for 24 h at 500oC by means of STEM-HAADF tomography. In the present study, we set the inner half angle of the HAADF detector to be 30mrad to ensure a clear contrast during the tilt-series acquisition. This setting of a rather low angle may break the simple Z2 dependence of the HAADF-STEM images to some extent due to possible diffraction contrasts during the tilting. The tilt-series was obtained sequentially from 0 to -70° and then 0 to +70°. The angular tilt angle increments were set 2°. Out of this data set, we employed images taken at tilt angles between -62 and +62° for subsequent 3D reconstruction. Figure 3 shows one of the original images (Fig.3a) and corresponding reconstructed images processed by WBP (Fig.3b). Here, the x-axis is the tilt-axis, about which the specimen film is sequentially tilted towards the y-direction; while the primary beam incidence direction is parallel to the z-axis. Bright contrast region corresponds to the precipitated V-rich β phase (�#� J6�' /;;;�/7;; .650 ��? //;;�/4;; .670 /6;;�6;;; .68�690 @�'�

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As pointed out in [5], void nucleation in DP steels occurs during the entire deformation process, i.e. each different single interface probably exhibits a different value of  and each interface is possibly subjected to a scattered value of χ. Interface decohesion is thus a progressive phenomenon, starting for a strain of 0.18 (in smooth specimens) but continuing after this value of strain and the evolution of the cavity density has to be modeled as a function of the local strain. Fig.5 shows that the studied material exhibits two different nucleation regimes at low and at high strain. Firstly, the number of cavities increases slowly and linearly. In a second regime many voids appear exponentially. This experimental observation leads us to propose the following empirical equation based on the local criterion of decohesion and involving the parameters χ and σC. dN χ N =A 1 dεloc σC N 0 (9)





A and N0 being two constants (expressed in the same unit as N, for instance in mm-3). The two extreme regimes are well described by this empirical expression. When N ≪ N 0 the following approximation can be done: dN χ ≈A dεloc σ C (10) The interface decohesion is then only linearly controlled by the local stress χ which increases with the applied strain. In the second regime, when N ≫ N 0 the approximation becomes : dN χ N ≈A (11) dεloc σC N 0

The evolution rate of N with strain is proposed to depend on N itself, transcribing a self catalytic effect and thus the exponential acceleration of the number of cavities. We then now have a mean to integrate the value of N, by accounting for the local triaxiality at the interface. The assessment of the model is firstly done using experimental data from smooth specimens. The values of the two constants A and N0 giving the best fit between modeling and experimental data for the smooth sample are A=4500mm-3 and N0=1250mm-3. These values, when used in the framework of the notched samples, also show a satisfactory agreement as shown in Fig.8. This validates that using a local value of the triaxiality as a driving force in an interface fracture criterion is a reasonable procedure.

Fig.8 Comparison of the prediction of the nucleation model and experimental data [21]. The constants of the model are fitted to reproduce the experimental evolution in the case of the smooth sample and are then used “as-is” to calculate the evolution of the notched sample. 4 Conclusions and perspective Using in-situ tensile tests during X-ray tomography, the present study has shown that it is possible to obtain quantitative information about damage. Concerning the sites of nucleation, optical micrographs of fractured samples have shown that most cavities appear by decohesion of the ferrite/martensite interface. A value of the critical interface strength (1100 MPa)

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has been estimated for the onset of nucleation. The evolution of the void density has then been modeled according to an analytical approach based on a local version of the Argon decohesion criterion and accounting for the triaxiality. The model has been fitted with the experimental data on the smooth samples. The identified parameters were then used for the notched sample and also lead to a satisfactory agreement of the predicted evolution of the number of nucleated cavities. Some improvements could be foreseen in the present approach, particularly concerning the value of the interface strength in DP steels. This strength probably depends on the carbon content in the martensite and on an eventual tempering. These effects have to be investigated in more details before being modeled. Acknowledgments The authors would like to thank the ESRF for the provision of synchrotron radiation at the ID15 beamline through the ma560 long term project. References [1] Argon AS, Im J, Safoglu R, Cavity formation from inclusions in ductile fracture, Metallurgical Transactions A, Volume 6, Issue 4, pp 825-837, 1975. [2] Goods SH, Brown LM, Nucleation of cavities by plastic-deformation – Overview, Acta Metallurgica,Volume 27, Issue 1, pp 1-15,1979. [3] Beremin FM,Cavity formation from inclusions in ductile fracture of A508 steel, Metallurgical and Materials Transactions A, Volume 12, Issue 5, pp 723-731,1981. [4] Steinbrunner DL, Matlock DK, Krauss G, Void formation during tensile testing of dual phase steels, Metallurgical Transactions A, Volume 19, Issue 3, pp 579-589,1988. [5] Avramovic-Cingara G, Saleh CAR, Jain M, Wilkinson DS, Void Nucleation and Growth in Dual-Phase Steel 600 during Uniaxial Tensile Testing, Metallurgical and Materials Transactions A, Vlume 40, pp 3117-3127, 2009. [6] Tanaka K, Mori T, Nakamura T, Cavity formation at the interface of a spherical inclusion in a plastically deformed matrix, Philosophical Magazine, Volume 21, Issue 170, pp. 267–279, 1970. [7] Thomason PF, Ductile Fracture of Metals, Pergamon Press, Oxford, 1990. [8] Kwon D, Asaro RJ, A study of void nucleation, growth, and coalescence in spheroidized-1518 steel, Metallurgical Transactions, Volume 21, Issue 1, pp 91-101, 1990. [9] Walsh JA, Jata KV, Starke EA, The influence of Mn dispersoid content and stress state on ductile fracture of 2134 type Al-alloys, Acta Metallurgica, Volume 37, Issue 11, pp 2861-2871, 1989. [10] Bugat S, Besson J, PineauA, Micromechanical modeling of the behavior of duplex stainless steels, Computational Materials Science, Volume 16, Issue 1-4, 158-166, 1999. [11] Needleman A, A continuum model for void nucleation by inclusion debonding, Journal of Applied Mechanics, Volume 54, pp 525-531, 1987. [12] Needleman A, Tvergaard V, An analysis of ductile rupture in notched bars, Journal of the Mechanics and Physics of Solids, Volume 32, Issue 6, pp 461-490, 1984. [13] Nutt SR, Needleman A, Void nucleation at fiber ends in Al-SiC composites, Scripta Materialia, Volume 21, Issue 5, pp 705-710, 1987. [14] Buffiere JY, Maire E, Cloetens P, Lormand G, Fougères R, Characterization of internal damage in a MMCp using x-ray synchrotron phase contrast microtomography, Acta Materialia, Volume 47, Issue 5, pp 1613-1625, 1999. [15] Martin CF, Josserond C, Salvo L, Blandin JJ, Cloetens P, Boller E, Characterisation by X-ray micro-tomography of cavity coalescence during superplastic deformation, Scripta Materialia, Volume 42, Issue 4, pp 375-381, 2004. [16] Babout L, Maire E, Fougeres R, Damage initiation in model metallic materials: X-ray tomography and modelling, Acta Materialia, Volume 52, Issue 8, pp 2475-2487, 2004. [17] Maire E, Bouaziz O, Di Michiel M, Verdu C, Initiation and growth of damage in a dual-phase steel observed by X-ray microtomography, Acta Materialia, Volume 56, Issue 18, pp 4954-4964, 2008. [18] Bron F, Besson J, Pineau A, Ductile rupture in thin sheets of two grades of 2024 aluminum alloy, Materials Science and Engineering A, Volume 380, Issue 1-2, pp 356-364, 2004. [19] Abramoff MD, Magelhaes PJ, Ram SJ, Image Processing with ImageJ, Biophotonics International, Volume 11, Issue 7, pp 36-42, 2004. [20] Bridgman PW, Effects of High Hydrostatic Pressure on the Plastic Properties of Metals, Revue of Modern Physics, Volume 17, Issue 1, pp 3-14, 1945. [21] Landron C, Bouaziz O, Maire E, Characterization and modeling of void nucleation by interface decohesion in dual phase steel, Scripta Materialia, Volume 63, Issue 10, pp 973-976, 2010. [22] Helbert AL, Feaugas X, Clavel M, Effects of microstructural parameters and back stress on damage mechanisms in

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alpha/beta titanium alloys, Acta Metallurgica, Volume 46, Issue 3, 939-951, 1998. [23] Allain S., Bouaziz O., Microstructure based modeling for the mechanical behavior of ferrite-pearlite steels suitable to capture isotropic and kinematic hardening, Materials Science and Engineering A, Volume496, Issue 1-2, pp 329-336, 2008. [24] Grange RA, Hribal CR, Porter LF, Hardness of tempered martensite in carbon and low-alloy steels, Metallurgical Transactions A, Volume 8, Issue 11, pp 1775-1787, 1977. [25] Kosco JB, Koss DA, Ductile fracture of mechanically alloyed iron-yttria alloys Metallurgical Transactions A, Volume 24, Issue 3, pp 681-687, 1993. [26] Qiu H, Mori H, Enoki M, Kishi T, Development of A Three-dimensional Model for Void Coalescence in Materials Containing Two Types of Microvoids,ISIJ International, Volume 39, Issue 4, pp 358-364, 1999. [27] LeRoy G, Embury JD, Edwards G, Ashby MF, A model of ductile fracture based on the nucleation and growth of voids, Acta Metallurgica, Volume 29, Issue 8, pp 1509-1522, 1981. [28] Kwon D, Interfacial decohesion around spheroidal carbide particles, Scripta Metallurgica, Volume 22, Issue 7, pp 11611164, 1988.

In-situ synchrotron-radiation computed laminography observation of ductile fracture

T.F. Morgeneyer a, L. Helfenb,d, I. Sinclairc, F. Hildd, H. Proudhona, F. Xub, T. Baumbachb, J. Bessona a

Mines ParisTech, Centre des Matériaux, CNRS UMR 7633, BP87 91003 Evry Cedex, France b ANKA/Institute for Synchrotron Radiation, Karlsruhe Institute of Technology, Germany c Materials Research Group, School of Engineering Sciences, University of Southampton, Southampton SO17 1BJ, UK d European Synchrotron Radiation Facility/Experimental Division (ESRF), BP 220, 6 rue J. Horowitz, F-38043 Grenoble Cedex, France e LMT-Cachan, ENS Cachan/CNRS/UPMC/PRES UniverSud Paris, 61 avenue du Président Wilson, 94235 Cachan Cedex, France Synchrotron-radiation computed laminography (SRCL) allows for imaging at high resolution (~ 1 µm) and in three dimensions objects that are thin (~ 1 mm) but extended laterally in two dimensions. This represents a major advantage over computed tomography in terms of loading conditions that can typically only investigate samples elongated in one direction. Here SRCL is used to observe ductile crack initiation and propagation in high strength aluminium alloy sheet for aerospace applications. Several load steps are applied and permit us to follow the changes of damage and crack path. An attempt is made to measure strains via a digital volume correlation technique. Introduction Fracture resistance and toughness are critical design criteria for thin sheet materials in aerospace applications and require in-depth understanding of the underlying physical mechanisms to enhance material performance [1]. In the past 2D observation techniques and SEM post mortem fractography have mainly been used to assess fracture mechanisms [2] Synchrotron tomography is increasingly used to assess fracture mechanisms by observing arrested cracks at the initiation [3] and propagation [4] stages. Unprecedented insights into void growth and also intergranular ductile fracture could be gained [5]. In-situ ductile crack growth has been observed [6] via in situ loading of a 2024 aluminium alloy sample with a 1.3 mm × 1.0 mm cross section that permitted to confirm fracture mechanics models. The sample geometry had however dimensions that were far from engineering conditions. With the progress made with synchrotron laminography there is a clear opportunity to observe the damage mechanisms during crack propagation in advanced engineering materials [7], which allows for unprecedented insights into damage mechanisms for sample geometries that can reproduce stress states similar to those experienced in service. In situ loading of a notched sample fracture mechanisms in carbon fibre-epoxy laminates provided insights into delamination processes. This technique is applied in the present study to assess ductile fracture initiation and propagation in a ductile 2XXX alloy for aerospace applications. T. Proulx (ed.), Optical Measurements, Modeling, and Metrology, Volume 5, Conference Proceedings of the Society for Experimental Mechanics Series 9999999, DOI 10.1007/978-1-4614-0228-2_3, © The Society for Experimental Mechanics, Inc. 2011

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20

With the recent progress in digital volume correlation (DVC) it has become possible to measure strains in 3 dimensions thanks to natural markers / speckle caused by attenuation differences between phases present in the material [8]. DVC has successfully been performed on tomography data of fatigue crack propagation in nodular cast iron. An attempt is made in the present work to apply this technique to in situ synchrotron laminography data to assess the feasibility of the measurement, to obtain insights into strain / damage relationships, and to validate models [9]. Experimental For the experiment a commercial ductile 2139 Al-Cu alloy in T3 condition for aerospace application has been machined symmetrically from 3.2 mm to 1 mm thickness. Testing has been performed in the T(long transverse)-L(rolling direction) configuration. The material exhibits an initial void volume fraction of ~ 0.3%. For further details on material microstructure and mechanical properties the reader is referred to [3,4]. The sample geometry shown in figure 1a has been used. The notch has been machined via electron discharge (EDMC) machining resulting in a notch diameter close to the EDMC wire diameter of 0.3 mm. The loading has been achieved via a displacement-controlled wedging device that prescribes a specimen crack mouth opening displacement (CMOD) similar to the one used in [7] (see Figure 1). Machined notch Loading device

70mm

CMOD Figure 1: In situ loading device and 1 mm thick notched sample An anti-buckling device was used to hamper the sample from significant buckling and out-ofplane motion. The entire rig was mounted in a dedicated plate that has been removed from the SRCL disc for every loading step. The loading has been applied via stepwise opening of the wedge. Before every loading step a scan has been performed. The scanned regions of interest (ROI) have been moved with the propagating crack to image its tip and the damaged material ahead of the notch / crack tip. Twenty scans have been carried out but only results from 6 scans will be shown hereafter. Laminography is an alternative approach to standard tomography imaging that overcomes the aforementioned sample size / shape problems. The technique is fundamentally similar to tomography except that the sample is imaged at an inclined angle relative to the beam [7]. Previous laminography work at ESRF has yielded very satisfactory results [10]. The further development of the method exploiting propagation-based phase-contrast imaging [11] opens the way for new applications related to weakly absorbing / weakly contrasted structures [12,13].

21

Imaging was performed on beamline ID19 at ESRF using a monochromatic X-ray beam for an energy of 25 keV. Volumes were reconstructed using an in-house software from 1500 angularly equidistant radiographs; the exposure time of each projection was 250 ms. Due to the experimental set-up the sample to detector distance was 70 mm leading to strong phase contrast. The scanned volume was ~ 1mm3 with a voxel size of 0.7 µm. For 3D void representation a simple grey value threshold has been used to segment voids; a VTK software rendering is used for the 3D image. Results and discussion Figure 2 shows 2D sections of laminography data at different openings (CMOD). The section is taken at mid-specimen thickness for every scan. In figure 2a the machined notch can be observed. Initial porosity can clearly be seen in black, intermetallics in white and the aluminium matrix in analogy observations of this or similar material via tomography [3-5]. White edges around porosities are attributed to strong phase contrast. Some ring artefacts can be seen. Figures 2b-f show the change of damage that can clearly be observed. The feasibility of the in-situ experiment is proven, namely, SRCLaminography allows for unprecedented observation of ductile crack initiation and propagation in industrial grade materials at realistic length scales compared to engineering applications. Substantial void growth can be seen ~ 50µm ahead of the notch (figure 2b). The coalescence of the two voids can be seen in figure 2c. The narrow coalescence band is oriented ~ 45 ° with respect to the loading direction. This result shows that crack initiation does not start immediately from the notch but somewhat (50 µm in this case) ahead of the notch. At a CMOD of 1.875 mm coalescence of the crack with the notch and further crack propagation in propagation direction occurs. It can be seen that both large voids have changed shape in this coalescence step with the notch. The crack plane orientation is not normal to the loading direction but inclined. The coalescence mechanisms appear to occur in shear bands that is consistent with findings made on the sample surface and theoretical considerations [14]. Figure 2f shows the crack at CMOD = 2.375 mm, which has propagated and reached a length of ~ 1mm. The void growth ahead of the propagating crack appears substantially less than at crack initiation. The 2 dimensional representation may only be adequate and show the same voids and particles for the mid-thickness plane. This plane can be assumed to show the change of the same microstructural features at least during the first loading steps. Once the fracture process becomes asymmetric, e.g. due to slant fracture, one plane will not show the same material features for all loading steps. As laminography provides 3D data it offers the opportunity to show the change of voids in 3D. This is shown in figure 3 where voids from a 105 µm thick slice symmetric to the mid plane are represented. Only voids are made visible here. (The thickness is not corrected for transverse contraction as the latter is inhomogeneous in the plane). The same loading steps as in figure 2 are shown. Between the initial microstructure (figure 3a) and the deformed voids it can be seen that void growth takes place around the entire notch. A lot more voids can be observed than for the 2D representation. The void growth and orientation appear highly directional. The voids change orientation around the notch during loading and orient themselves tangential to the notch circumference. The 3D representation also confirms the crack initiation from the interior of the material. However, in figure 3d it can be seen that the two large voids are already connected with the notch that was not discernable on the 2D sections at the mid-plane.

22

(a)

(b)

Notch (c)

(d)

(e)

(f)

T S

1500 µm

L

Figure 2: 2D section at sheet mid-thickness from in-situ SRCLaminography data for a) CMOD = 0.5 mm b) CMOD = 1.5 mm c) CMOD = 1.625 mm d) CMOD = 1.75 mm e) CMOD = 1.875 mm f) CMOD = 2.375 mm

23

(a)

(b)

(c)

(d)

(e)

(f)

T S

400 µm

L

Figure 3. 3D representation of voids and the crack in a numerically cut 105 µm thick slice at sample centre for a) CMOD = 0.5mm b) CMOD = 1.5 mm c) CMOD = 1.625 mm d) CMOD = 1.75 mm e) CMOD = 1.875 mm f) CMOD = 2.375mm Figure 4 shows the displacement field measured by DVC with same the code as that used in [8] on the present laminography data by registering 2 volumes located in the squared box indicated in figure 4a. The reconstructed volumes corresponding to CMOD = 0.5 mm and CMOD = 0.625mm are compared. The volume of the compared cubes is taken symmetrically to the sheet mid-thickness. The measured displacement field is qualitatively correct, namely, an expansion in T direction is detected that is consistent with the loading along the same direction. This preliminary result prompts us to continue the analysis for further loading steps, and to derive the strain fields. These measurements may then open the possibility to obtain insights into deformation field and strain localization mechanisms leading to crack

24

bifurcation. A relationship between measurements of void growth at different levels of stress triaxiality and strain may become accessible in that case. (c)

(b) (~350µm)3

T S

L

200µm

S L

T

Figure 4: a) 2D section of the scan when CMOD = 0.5 mm indicating the region of interest for DVC b) Measured displacement field between CMOD = 0.5mm and 0.625 mm, colorbar indicating displacements in T direction expressed in voxels Conclusion Ductile fracture of thin (but laterally extended) sheet aluminium alloy has been successfully observed for the first time thanks to the use of Synchrotron Radiation Laminography. The artefacts introduced by this technique are small enough to clearly observe ductile fracture mechanisms involving void nucleation, growth and coalescence in conditions that are realistic for engineering applications. Damage can easily be extracted via simple grey value thresholding. Fracture revealed to initiate ahead of the notch and only after substantial void growth to join the notch. The crack path is not normal to the loading direction but inclined. The obtained data can be used for in-situ void growth measurements. An initial attempt to perform digital volume correlation on the laminography data led to qualitatively successful measurements of the displacement field, which encourages further strain analyses. References [1] Bron F, Besson J, Pineau A. Ductile rupture in thin sheets of two grades of 2024 aluminum alloy, Mater Sci Eng A 2004;A 380:356–64. [2] Lautridou JC, Pineau A, Crack initiation and stable crack growth resistance in A508 steels in relation to inclusion distribution , Eng Fract Mech 1981, 15, 55-71 [3] Morgeneyer T.F., Besson J., Proudhon H., Starink M.J., Sinclair I., Experimental and numerical analysis of toughness anisotropy in AA2139 Al-alloy sheet, Acta Mater. 2009;57:3902–3915.

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[4] Morgeneyer TF, Starink MJ, Sinclair I. Evolution of voids during ductile crack propagation in an aluminium alloy sheet toughness test studied by synchrotron radiation computed tomography, Acta Mater 2008;56:1671–9. [5] Morgeneyer TF, Starink MJ, Wang SC, Sinclair I. Quench sensitivity of toughness in an Al alloy: Direct observation and analysis of failure initiation at the precipitate-free zone, Acta Mater 2008;56:2872–84. [6] Toda H., Maire E., Yamauchi S., Tsuruta H., Hiramatsu T., Kobayashi M., In situ observation of ductile fracture using X-ray tomography technique, Acta Maert 2011; 59:1995-2008 [7] Moffat, A. J., Wright, P, Helfen, L., Baumbach, T., Johnson, G., Spearing, S. M., Sinclair, I. In situ synchrotron computed laminography of damage in carbon fibre–epoxy [90/0]s laminates, Scripta Mater, 2010;62: 97-100 [8] Limodin N., Rethore J., Buffiere J.Y., Hild F., Roux S., Ludwig W., Rannou J., Gravouil A. Influence of closure on the 3D propagation of fatigue cracks in a nodular cast iron investigated by X-ray tomography and 3D volume correlation, Acta Mater 2010;58: 2957–2967 [9] McMeeking R.M., Finite deformation analysis of crack-tip opening in elastic-plastic materials and implications for fracture, J. Mech. Phys. Solids 1977;25:357-81 [10] Helfen L., Myagotin A., Rack A., Pernot P., Mikulik P., Di Michiel M., Baumbach T., Synchrotron-radiation computed laminography for high-resolution three-dimensional imaging of flat devices, Phys. Status Solidi A 2007; 204: 2760. [11] Cloetens P., Pateyron-Salome M. , Buffiere J.Y., Peix G., Baruchel J., Peyrin F., Schlenker M., Observation of microstructure and damage in materials by phase sensitive radiography and tomography. J. Appl. Phys. 1997 ;81 :5878–5885. [12] Krug K., Porra L., Coan P., Tauber G., Wallert A., Dik J., Coerdt A., Bravin A., Elyyan M., Helfen L., Baumbach T., Relics in Medieval Altarpieces. Combining X-ray tomographic, laminographic and Phase-Contrast Imaging to Visualize Thin Organic Objects in Paintings, J. Synchrotron Rad. 2008 ;15: 55–61. [13]Helfen L., Baumbach T., Cloetens P., Baruchel J., Phase-contrast and holographic computed laminography, Appl. Phys. Lett. 2009;94:104103. [14]Russo VJ, Chakrabarti AK, Spretnak JW; The role of pure shear strain on the site of crack initiation in notches, Metal Transaction A 1977;8A:729-40

Understanding the mechanical behaviour of a high manganese TWIP steel by the means of in situ 3D X ray tomography

D. Fabrègue(1,2), C. Landron(1,2), C. Béal(1,2,3), X. Kleber(1,2), E. Maire(1,2), M. Bouzekri(3) (1) Université de Lyon, CNRS (2) INSA-Lyon, MATEIS UMR5510, F-69621 Villeurbanne, France (3) ArcelorMittal Research, Voie Romaine-BP30320, F-57283 Maizières les Metz, France ABSTRACT The high manganese TWIP (twinning induced plasticity) steels exhibit very high mechanical properties compared to others grades. Indeed they have a mechanical strength that can attain 1.5 GPa and fracture strain that can go up to 60%. However the governing damage mechanisms that maximize their ductility are not well understood. To have a better understanding of these mechanisms, in situ tensile tests have been carried out at the European Synchrotron Radiation Facility. During tensile test no necking can be observed which has already been observed on this type of steel. Moreover, the number of cavities in a given volume do not deeply evolve during the deformation meaning that nucleation of voids is weak in the TWIP steel considered. This leads to a maximal number at fracture very low compared to other steels (interstitial free steel, dual phase,…). Morever, the growth of cavities according to local strain seems to be equivalent to other austenitic or ferritic steels. At last, shear bands can be observed in the sample which seems to be correlated with some of the cavities. 1. Introduction High manganese austenitic steels are promising candidates in automotive industry due to their excellent mechanical properties. In effect, they exhibit at the same time very high strength (higher than 1000 MPa in tension) and high ductility (about 50% at room temperature). Thus they can absorb a lot of plastic energy before failure due to their unusual work-hardening capacity. This comes from the fact that their stacking fault energy is low enough to permit at the same time deformation by twining and by dislocation gliding without any martensitic transformation that could be detrimental for the ductility. Concerning their fracture behaviour, these alloys do not usually exhibit necking at room temperature but slant fracture because they seem to be sensitive to shearing. Even if the Fe-Mn-C alloys have been developed for a long time now (Hadfield in 1880 already presented the Fe-Mn-C alloys), their deformation and fracture behaviour are still not well understood and there is a real need to investigate them with novel techniques such 3D X Ray tomography permitting to observe in 3 dimensions the damage mechanisms. Recently, the same type of steel has been investigated by post mortem 3D tomography but on a shearing deformation mode [1]. This study is more dedicated to tensile tension and the evolution of the damage is quantitatively carried out for different strains. 2. Experimental Procedure The TWIP steel was cut from a 1mm thick sheet obtained by hot rolling and annealing thermal treatment. The designation of the steel is Fe-22Mn–0.6C (composition in mass percent, balance iron). This steel if fully austenitic at room temperature with an average grain size diameter of about 2-3 µm. Micro tensile specimens have been machine by spark discharge according to the Figure 1.

T. Proulx (ed.), Optical Measurements, Modeling, and Metrology, Volume 5, Conference Proceedings of the Society for Experimental Mechanics Series 9999999, DOI 10.1007/978-1-4614-0228-2_4, © The Society for Experimental Mechanics, Inc. 2011

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28

Fig 1 Specimen dimensions for in situ X ray tomography X ray microtomography has been used to quantify damage during in situ tensile tests. The tomography set up is the one located at the ID15 beam line at the European Synchrotron Radiation Facility (ESRF) in Grenoble, France. Tomography acquisition was realised with a voxel size of 1.6 µm3. Initial reconstructed volumes were median filtered and simply thresholded to differentiate by absorption difference the material from the voids. In order to correlate the distribution of voids with the strain, local values of the strain is obtain by considering the minimal section area S and using the relation:

 S0   S 

 loc  ln 

With S0 the initial section of the sample. The local strain is then calculated at each step. Using this equation implies that the volume fraction of the voids considered is small enough to keep the total volume unchanged. The beginning of the tensile test is carried out with a stress triaxiality of 0.33. Anyway, after some deformation, the triaxiality can evolve. This is considered by using the Bridgman formula [2] considering the curvature radius of the surface of the sample RS:

T

 1 a   ln  1   3  2 RS 

a being the radius of the minimal section. 3. Results and discussion The tensile curve of the sample submitted to tensile test is given in Figure 2. As it can be seen, the mechanical properties of the steel are outstanding with a yield strength of about 460 MPa, an ultimate tensile strength of about 920MPa and a fracture strain of about 0.55. One can notice also the high hardening due to the twinning phenomenon. These values are in accordance with the supplier specifications and explain their promising use in safety parts in automotive industry.

29 1000

Stress (MPa)

800

600

400

200

0

0

0,1

0,2

0,3

0,4

0,5

0,6

Strain

Fig. 2 Stress/strain curve obtain during the in situ tensile test Anyway, figure 3 shows the 3D tomography of the TWIP steel investigated just before the final fracture. For the sake of comparison, the same state for a DP steel from [3] is also shown and the voids are underline by a red color. The difference in fracture behavior between steel with a ferritic matrix and the TWIP austenitic steel is then obvious. When the first one exhibits an important necking and a very high density of voids before fracture, the TWIP steel shows no localization of the deformation and only a small density of voids. This qualitative observation is checked by plotting the number of voids as a function of the local strain in figure 4. This number is compared with the same experiment on an interstitial free steel. This steel has been chosen for comparison in order to have two monophased steels. The different damage behavior is here straight forward. The IF steel exhibits a three steps damage with a high density of cavities for small strains then a step increase for larger strains and finally a saturation of the number of cavities before fracture. This last step is due to coalescence phenomenon leading to the fracture of the sample. The austenitic TWIP steel exhibits a very different behavior: it contains only a small number of voids and shows only a small increase of this number up to the fracture of the sample. It is worth noticing that the number of voids at fracture is quite different between the two steels: around 10000 per mm3 in the case of the IF and 1000 per mm3 for the TWIP steel. This density of voids is very low considering all other type of ductile material at fracture [4, 5, 6]. Anyway, looking at the fracture surface explains partly this point. In fact, the fracture surface of this austenitic steel is characterized by dimples exhibiting a very small diameter (figure 5). Thus smallest dimples are not seen by using this spatial resolution of about 1,6 µm3 and the number of cavities must be then larger than the one measured. Experiments with better spatial resolution are thus needed to characterize perfectly the damage behavior of such a steel.

a)

b)

30 Fig. 3 3D views of the damage at the center of the specimen strained at a) εloc=0.5 (just before fracture) b) DP steel just before fracture.

density of voids in XiP(N/mm3)

4

1,2 10

3

density of voids in IF(N/mm )

3

Density of voids(N/mm )

1 104 8000 6000 4000 2000 0 0

0,1

0,2

0,3

0,4

0,5

0,6

local strain Fig. 4 Density of voids as a function of the local strain for the TWIP steel and a ferritic one.

Fig. 5 SEM fractography of the in situ tensile test Anyway, as it could be deduced from the absence of necking even at strain close to fracture, the triaxiality in the TWIP steel remains almost constant as it can be seen in Figure 6. Thus considering the Rice and Tacey law, the cavity growth should be limited in the case of TWIP steel, nucleation should be thus the major phenomenon governing the damage behavior. To check that point the evolution of the cavity size is calculated from the 3D images and plotted as a function of the local strain. It can be seen that if the entire population of cavities is considered, the average diameter of the cavity remains constant. This could be due to the fact that in that case, nucleation is also considered. Thus it is worth considering only the 20 largest voids. It leads to the conclusion that growth is experienced during the deformation. This growth is almost equivalent to the one experienced by other steel grade. Thus either the mechanisms governing the growth of cavities is different from the others steels and then growth is possible for low triaxiality or locally the triaxiality could be higher than the macroscopic one. One more interesting feature about the damage behavior of these TWIP austenitic steel is shown in figure 7. This figure shows a 3D picture of the TWIP steel just before fracture. It is clearly seen some shear bands in the material. This is in accordance with previous study on the same alloy but in a different stress state [1]. Primary

31 observations seem to indicate that voids are preferentially situated on these shear bands. Thus we could think that due to these shear bands, the triaxiality could be locally higher involving some growth of the cavities. 9

0,4

Average equivalent diameter (µm)

0,35 0,3

Triaxiliaty

0,25 0,2 0,15 0,1

total population of voids 20 largest voids 8

7

6

5

0,05 0

4 0

0,1

0,2

0,3 Strain

0,4

0,5

0,6

0

0,1

0,2 0,3 0,4 local strain

0,5

0,6

Fig. 6 a) Evolution of the triaxiality during the tensile test and b) Evolution of the average equivalent diameter with the local strain

Fig 7 3D picture of the TWIP steel just before fracture showing shear bands

4 Conclusions and perspectives 3D X ray tomography has been used to get a better insight of an austenitic TWIP steel. It shows that although the triaxiality remains constant equal to 0.33, some cavity growth is experienced. This could be due to the presence of shear bands involving an increase of the local triaxiality and permitting cavity growth. Moreover, the evolution of the number of cavities is different from other steel. However due to the small size of the cavities, tomography with higher resolution is needed to have a better idea of the real number of voids and to propose a clear scenario for the final fracture.

32 References [1] Lorthios J, Nguyen F, Gourgues AF, Morgeneyer TF, Cugy P, Dalage bservation in a high manganese austenitic steel by synchrotron radiation computed tomography, Scripta Materialia, volume 63, Issue 12, pp 1220-1223, 2010 [2] Bridgman PW, Effects of High Hydrostatic Pressure on the Plastic Properties of Metals, Revue of Modern Physics, Volume 17, Issue 1, pp 3-14, 1945. [3] Landron C, Bouaziz O, Maire E, Characterization and modeling of void nucleation by interface decohesion in dual phase steel, Scripta Materialia, Volume 63, Issue 10, pp 973-976, 2010. [4] Martin CF, Josserond C, Salvo L, Blandin JJ, Cloetens P, Boller E, Characterisation by X-ray microtomography of cavity coalescence during superplastic deformation, Scripta Materialia, Volume 42, Issue 4, pp 375-381, 2004. [5] Babout L, Maire E, Fougeres R, Damage initiation in model metallic materials: X-ray tomography and modelling, Acta Materialia, Volume 52, Issue 8, pp 2475-2487, 2004. [6] Bron F, Besson J, Pineau A, Ductile rupture in thin sheets of two grades of 2024 aluminum alloy, Materials Science and Engineering A, Volume 380, Issue 1-2, pp 356-364, 2004. Acknowledgements The authors would like to thank O. Bouaziz and M. Goune from ArcelorMitall for fruitfull discussions.

Mechanical properties of Monofilament entangled materials

Loïc Courtoisa, Eric Mairea, Michel Pereza, Yves Brechetb, David Rodneyb a

b

Université de Lyon – INSA de Lyon - MATEIS, UMR 5510, Villeurbanne SIMAP-GPM2, Domaine Universitaire BP 46 38402 Saint Martin d’Heres, France

ABSTRACT A new type of architectured materials, namely « monofilament entangled materials », were studied in order to have a better understanding of their behavior under compressive loading and damping. The materials studied in this paper were made of an entanglement of a single steel wire. Their complex internal architecture was investigated using X-ray computed tomography. The evolution of the number of contacts per unit of volume, as well as of the density profile, were followed during the compression test in order to compare it to the mechanical results. Dynamical Mechanical Analysis (DMA) was performed to characterize the evolution of the loss factor of this material with the frequency and the volume fraction. It was shown that this material present an interesting strength/loss factor ratio. A discrete element model was proposed to model the mechanical properties of this material. 1. Introduction Playing with the architecture of a material is a clever way of tailoring its properties for multifunctional applications. A lot of research have been made, in the past few years, on what is now referred to as « architectured materials » (metal foams [1], entangled materials, steel wool [2], etc), mostly for their capacity to be engineered in order to present specific properties, inherent to their architecture. In this context, some studies have been carried out concerning entangled materials [3], but only a few on monofilament entangled materials [4-6]. Such a material, with no filament ends, could exhibit interesting properties for shock absorption, vibration damping and ductility. Because of the complex architecture of these materials, X-ray computed tomography is used in this paper as a main characterization method. This technique enables a 3D non destructive microstructural characterization of the material [2], which can be coupled with an in-situ mechanical characterization. Different parameters can be measured from the acquired 3D data: density profiles, number of contact per unit of volume, volume fraction. From the 3D images, a discrete element model [7] is finally defined in order to be able to model the structural and mechanical behaviors of this material. 2. Samples and procedures In this study, entanglements were manually produced, using different wire diameters and yield strengths (Table 1) and placed into a cylindrical die with a 15 mm diameter. The samples were initially 35 mm high with a 5% volume fraction. They were then submitted to a constrained (inside a PVC die) in-situ compressive test within the laboratory tomography presented in more details [8] and shown in Figure 1-A. Table 1 Properties of the used wires Material Stainless steel 304L Pearlitic steel

Diameters (µm) 127, 200 and 280 120

Yield strength (MPa) 200 4000

The samples were compressed, step by step, with a compression rate of 0.5 mm/s and for each step, samples were unloaded. From the displacement of the grips and the initial length and diameter of the wires, it was possible to calculate the “theoretical” volume fraction of the sample during compression tests (starting at 5%). A 3D volume was acquired and reconstructed (Figure 1-B) for both the loaded and unloaded state. Each image had a 24 µm resolution. During the whole T. Proulx (ed.), Optical Measurements, Modeling, and Metrology, Volume 5, Conference Proceedings of the Society for Experimental Mechanics Series 9999999, DOI 10.1007/978-1-4614-0228-2_5, © The Society for Experimental Mechanics, Inc. 2011

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34 compression test, a stress-strain curve was acquired and the strength of the entangled media was characterized by discharge modulus measurements (slope of the curve at the first moment of the discharge). Dynamic Mechanical Analysis was also performed on a set of sample with various volume fractions, for frequencies ranging from 1 to 100 Hz. We then obtained the loss factor that could be used to calculate the specific strength/loss factor ratio.

Figure 1: A) In-situ experimental compressive device, B) example of a reconstructed 3D volume In order to link, the internal architecture to the global mechanical strength of the samples, a microstructural analysis was performed based on the 3D images. The homogeneity of the sample was first studied by monitoring the evolution of the radial density profile. The 3D data were first treated to obtain a binary image where the wire appeared as white voxels, and the air as black. By measuring a local density, as shown in Figure 2-A, for each consecutive tube with a thickness dr (Figure 2-B), we obtained an indicator of the local volume fraction at a distance r of the axis of the die.

Figure 2: A) Local density equation, where Np is the total number of voxels, B) Principle of the recursive density measurement From the binary 3D images, it was also possible to measure the number of contacts per unit of volume. The data was first reduced to its center-line (skeletonization of the wire architecture). The whole structure then consisted in a list of

35 segments and nodes where one contact corresponded to a H-like structure, an example of which is shown in Figure 3-A. By counting the number of segments shorter than the diameter of the wire (definition of a contact point), we could estimate the number of contacts per unit volume. This count could be refined if one considers that, if the distance between two short segments is smaller than the diameter of the wire, those two segments belong to the same contact point (Figure 3-B). After refinement, this calculation was applied to both the loaded and unloaded states, for each volume fraction.

Figure 3: A) H-like structure corresponding to a simple contact, B) Refinement of multiple contact From those measurements, it was possible to link the evolution of the internal microstructure to the mechanical behavior of this material and thus, have a better understanding of the behavior of the entangled media under compressive loading. In parallel to the experimental testing and microstructural analysis, a model was defined in order to reproduce the experimental process using a discrete element method. In this model, the wire was represented by a succession of spherical elements (bead-like model) with the same diameter as the wire (Figure 4).

Figure 4: Bead-like representation of the wire Numerical sample could be generated (i) from a random walk algorithm, or, (ii) from experimental 3D images. In the latter case, the 3D data could either be discretized numerically, using their skeleton, or manually, following the wire by hand. This way, it was possible to obtain a description of the initial structure that corresponds exactly to the experimental samples. Two consecutives elements along the wire were bonded by a FENE [9] (finite extensible nonlinear elastic) potential, preventing two parts of the wire to cross each other. Friction plays an important role in the comprehension of the

36 behavior of entangled media. It was taken into account via a Coulomb/Hertz interaction [10-12] (Figure 5), allowing to accurately model both the compressive and the hysteretic behavior. The first parenthesized term is the contact force between two particles and the second parenthesized term is the tangential force. At each “Molecular dynamic timestep”, the tangential force, which corresponds to a “history” effect, is updated to account for the tangential displacement between the particles for the duration of the time they are in contact.

Figure 5: Expression of the granular force between two particles in contact with: delta = d - r = overlap distance of 2 particles Kn = elastic constant for normal contact Kt = elastic constant for tangential contact gamma_n = viscoelastic damping constant for normal contact gamma_t = viscoelastic damping constant for tangential contact m_eff = Mi Mj / (Mi + Mj) = effective mass of 2 particles of mass Mi and Mj Delta St = tangential displacement vector between 2 spherical particles which is truncated to satisfy a frictional yield criterion n_ij = unit vector along the line connecting the centers of the 2 particles Vn = normal component of the relative velocity of the 2 particles Vt = tangential component of the relative velocity of the 2 particles In our model configuration, the sample is contained within a cylindrical die like in the experiment. Regarding friction, the die interacts in the same way as the constitutive elements of the wire. Two pistons were then created in order to apply the displacement along the axis of the cylinder. It was finally possible to fit the model’s parameter using the static experimental results, as well as the dynamic ones (application of a sinusoidal displacement). From this model, stress-strain curves could be plotted as well as density profiles in order to compare them to the experimental ones. 3. Results and discussion A qualitative study of the 3D data was performed in order to have a first look at the compressive behavior of monofilament entangled materials. From the tomography data, it was possible to extract a median section of the sample along the axis of compression in order to follow qualitatively the evolution of the microstructure through the mechanical test.

Figure 6: Evolution of the cross section of a monofilament entangled sample (stainless steel, 200µm diameter) as a function of the volume fraction

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Figure 7: Evolution of the cross section of a monofilament entangled sample (stainless steel, 127µm diameter) as a function of the volume fraction In the case of stainless steel wire, a follow-up of the deformation was performed for two diameters (Figure 6 and 7). First, we can notice, for both diameters, that the distribution of the wire through the volume of the sample is heterogeneous. The local density seems higher at the contact with the mold and the pistons and smaller in the center. The increase of the volume fraction does not appear to radically change this heterogeneous distribution. Nevertheless, the sample with the smaller diameter (127µm) seems slightly more homogeneous than the one with a 200µm diameter. This was expected since smaller wire diameter means smaller curvature radius and thus easier arrangement of the wire for the same mold radius.

Figure 8: Evolution of the cross section of a monofilament entangled sample (pearlitic steel, 120µm diameter) as a function of the volume fraction. The volume fractions are different from the one for the stainless steel because of technical limitations. In the case of the pearlitic steel (Figure 6-B), we can notice that the profile is even more heterogeneous. Due to the very high yield strength of the wire, the curvature radius is very large and the wire ends up on the outer volume of the mold. Qualitatively, we can already notice the heterogeneous nature of monofilament entangled materials submitted to a constrained compression test, as well as the influence of the diameter and yield strength of the constitutive wire.

38 4. Conclusion In this study, a set of methods was defined in order to characterize the behavior of entangled media and more particularly, of monofilament entangled materials. The mechanical behavior of such a material can now be linked to its microstructure, which was showed to be more or less heterogeneous, depending on the wire’s characteristics. Both mechanical and microstructural behaviors can also be modeled using a discrete element method, taking into account friction. Acknowledgements National Research Agency (MANSART, ANR-REG-071220-01-01) . References [1] Cathy O., « Fatigue des empilements de spères creuses métalliques », PhD thesis, INSA Lyon, 2008 [2] Masse J.P., « Conception optimale de solutions multimatériaux multifonctionnelles : l’exemple des structures sanwdwiches à peaux en acier – choix des matériaux et développement de nouveaux matériaux de cœur », PhD thesis, INPG, 2009 [3] Rodney D., Fivel M., Dendievel R., « Discrete Modeling of the Mechanics of Entangled Materials », Physical Review Letters, Volume 95, pp 108004, 2005 [4] Liu P., He G., Wu L., « Fabrication of sintered steel wire mesh and its compressive properties », Materials Science and Engineering A., Volume 489, pp 21-28, 2008 [5] Tan Q., Liu P., Du C., Wu L., He G., « Mechanical behaviors of quasi-ordered entangled aluminium alloy wire material », Materials Science and Engineering A., Volume 527, pp 38-44, 2009 [6] Liu P., He G., Wu L., « Uniaxial tensile stress-strain behaviour of entangled steel wire material», Materials Science and Engineering A., Volume 509, pp 69-75, 2009 [7] Barbier C., « Modélisation numérique du comportement mécanique de systèmes enchevêtrés », PhD thesis, INPG, 2008 [8] Buffiere J.Y., Maire E., Adrien J., Masse J.P., Boller E., « In Situ Experiments with X ray Tomography: an Attractive Tool for Experimental Mechanics », Experimental Mechanics, Volume 50, Issue 3, pp 289-305, 2010 [9] Kremer K., Grest G.S., « Dynamics of entangled linear polymer melts:   A molecular-dynamics simulation », J Chem Phys, Volume 92, pp 5057, 1990 [10] Zhang H.P., Makse H.A., « Jamming transition in émulsions and granular materials », Physical Review E., Volume 72, pp 011301, 2005 [11] Silbert L.E., Ertas D., Grest G.S, Halsey T.C., Levine D., Plimpton S.J., « Granular flow down an inclined plane : Bagnold scaling and rheology », Phys Rev E, Volume 64, pp 051302, 2001 [12] Brilliantov N.V, Spahn F., Hertzsch J.M., Poschel T., « Model for collisions in granular gases», Phys Rev E, 53, pp 5382-5392, 1996

Characterisation of mechanical properties of cellular ceramic materials using X-Ray computed tomography O.Caty1, F.Gaubert1, G.Hauss2, G.Chollon1 [email protected] 1 LCTS : Laboratory of Thermostructural Composites - 3 Allée de la Boetie, 2 ICMCB - 87, Avenue du Docteur Schweitzer F33 600 PESSAC, France

Abstract Carbon foams are refractory cellular materials. The material exhibits an open porosity with a tridimensional architecture offering multifunctional properties. These properties can ever be tailored threw post processing in order to use the material in many fields such as energy (fuel cell) or transport (shock absorber). For all applications, the knowledge of the mechanical properties is important. These properties depend on the 3D architecture and the damage kinetics. In this study the computed X-Ray microtomography (µCT) has been used firstly to analyze the damage kinetics during in-situ compression tests and secondly to simulate the behavior. The µCT compression tests lead to the local damages and global deformations. The analyses of these images will be presented to illustrate the potentiality of tomographic investigations for brittle cellular materials. To assess both the stress and strain fields, a model based on the real material was developed. This model consists in meshing the tridimensional images and modeling the behavior of the constitutive material. The fracture of cells is treated using an adapted law and a brittle criterion. The models are also compared with the measured macroscopic mechanical behavior or adapted to simulate numerically-generated materials.

Introduction Ceramic foams are cellular materials which are used in industry in particular for their attractive mechanical properties coupled with other functionalities. Thermal insulation, acoustical absorption properties, low density with large specific surface and electrical conductivity are some examples of these properties [1-2-3]. The main industrial sectors are packaging (as shock absorbers), filtering (through the open cell) or energy (fuel cells). Although the main reason to select this material is not always its mechanical properties, they need to be known and must be understood in regards with the parameters of the 3D structure. The ceramic foams studied were manufactured at the LCTS using vitreous carbon preform provided by the CEA - Le Ripault [1-2]. This material is composed of a solid 3D arrangement of struts and edges delimitating open cells. The solid phase is glassy carbon and the resulting material combines its properties (thermal stability, high strength…) with the properties of the cellular structure (low density, high specific surface, open porosity). The material is macroscopically ductile while the vitreous carbon itself is brittle. This surprising behavior is explained by the 3D structure and its damage kinetics. Macroscopic compression tests were carried on foams having different pore sizes and relative densities by S.Delettrez [1-2]. The results of these tests are not in accordance with the theory of Gibson and Ashby [1–3]. One hypothesis to explain this difference is that the 3D structure of the material is far from the one considered by Gibson and Ashby [3]. In particular one can not define a simple unite cell, the material being rather random. This random 3D structure must be characterized to better understand the mechanical properties. Probably the better way to do is using micro-computed X-Ray tomography (µCT). Firstly the mechanical properties and damage kinetics are analyzed using a compression test coupled with tomography. These observations are necessary to understand the damaged mechanisms responsible of the macroscopic behavior. In addition it would be useful to measure further informations, such as the strain and stress fields in the material. The choice of T. Proulx (ed.), Optical Measurements, Modeling, and Metrology, Volume 5, Conference Proceedings of the Society for Experimental Mechanics Series 9999999, DOI 10.1007/978-1-4614-0228-2_6, © The Society for Experimental Mechanics, Inc. 2011

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calculating these data directly using the µCT images will be presented in this paper. This method based on finite element modeling is not a direct experimental measurement but is derived from the images of the real microstructure.

Material and Methods Materials The material was produced using highly porous vitreous carbon foam (relative density

r 

*  0.98 – with S

ρ* the apparent density of the foam and ρS the density of the constitutive material). This preform was produced from the pyrolysis of a polymer foam. Fig. 1-a and Fig. 1-b presents two secondary electron microscope (SEM) pictures respectively at low and high magnification of the porous vitreous carbon foam. Unfortunately this foam does not have sufficient mechanical and corrosive strength for shock absorber or fuel cell applications. the foam was thus densified by vapour deposition with pyrocarbon (PyC) or silicon carbide (SiC) using respectively propane and methiltrichlorosilane/hydrogen (MTS/H2) precursor [1-2-4-5] (Fig. 1-c). The deposition time was adjusted to obtain the desired thickness and thus the desired relative density. The final material is an open foam characterised by its relative density ρr and the number of pores per inch (ppi). In this paper only a PyC densified foam was analysed with a relative density and a cell size of 0.14 and 60 ppi respectively.

(a)

(b)

(c)

Fig. 1 : MEB pictures of he foam. (a) Magnification x50 to illustrate the mesoscopic morphology of the non densified foam. (b) and (c) Magnification x500 to illustrate the microscopic morphology of a ligament respectively before and after densification. Adapted from [1]. Fig. 2-a presents the stress/strain compression behavior of a 60 ppi foam densified with PyC up to a relative density of 0.14. The sample was placed between two plates. The superior plate is mobile and controlled in displacement. The device records the load (F) and displacement (l-l0), the stress (   (   ln(1 

F ) and the true strain S0

l  l0 ) ) being calculated using S0 as the initial surface (S0=78.5 mm2) and l0 the initial length of the l0

sample (l0=10mm). This curve is in good agreement with the standard compression behavior of foams [3], as presented on Fig. 2-b. The different domains are highlighted with numbers. Firstly (1) the material exhibits a pseudo elastic behavior, in this part of the curve, or during unloading, the Young’s modulus is measured. The value is about 320 MPa. Then a crack appears and the load decreases (2) to reach a stable value often named the plateau (3). In this part the load is almost constant with some fluctuations around an average value. These perturbations are generally attributed to successive brittle failures corresponding to individual pores crushing. The average stress value of the plateau (σpl –estimated by an average of the values in this domain) is about 2 MPa, the plateau appearing for a strain higher than about 0.02 in the present case. In this domain the material

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seems to be ductile although the constitutive material is fragile. (4) When all the open porosity is filled a densification state is observed. Here the densification appears for a strain of 0.7.

(a)

(b)

Fig. 2 : Stress/Strain compression behavior of the foam. (a) Compression test done on a foam with relative density of 14 % and a cell size of 60 PPi, densified with PyC. (b) Scheme reporting the different domains of the curve. Adapted from [1]. To model the ceramic foams Gibson and Ashby [3] have proposed cell models based on elementary cells. They have derived analytical expressions of the Young’s modulus and of the plateau stress. None of these models gives the values measured by [1]. These differences between the experimental and calculated values of the foam’s mechanical properties are probably due to a too simple model. In the Gibson and Ashby model [3], the elementary cell is simply repeated avoiding localization problems. The fact that damage kinetics and the local stress and strain fields are not taken into account in the material could probably explain the differences. A local image analysis during insitu compression test has been used to obtain more information on these points.

Methods X-ray tomography [6] is well-suited to study cellular materials and especially the development of damage [7-89-10]. To scan the material, we used a standard microtomograph from TOMOMAT in Pessac (Nanotom – Phoenix/Xray – see Fig. 3-b) with a voxel size of 6 microns. The tomograph was operated with a Molybden target at 50 keV and 500 µA with an Aluminium filter of 0.2 mm. Compression tests with constant strain were conducted on a specially-designed machine for in-situ compression measurements. The insitu compression device is presented on Fig. 3. Main features are: 1- A quartz pipe which can allow the X-ray to go thought and the visual check of the installation. 2- A manual mechanical actuator using a micrometric screw system for fine displacements. 3- A bearing ball, placed between the screw and the superior plate, avoiding moment transmission to the sample. Foam cylinders (10 mm of diameter and 10 mm length) are analysed at a voxel size of 6µm, resulting in images of about 1600x1600x1600 voxels. The strain is controlled knowing the screw thread (0.5 mm/tr). Eight µCT scans were performed at increasing strained states, the first one corresponding to the initial undeformed state. The true strain (   ln(1  screw thread and confirmed by µCT image analysis.

l  l0 ) ) was calculated from the l0

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(a)

(b)

Fig. 3 : pictures of the insitu compression device used to apply the load to the foam samples installed in the micro-tomograph. (a) Close visualisation of the compression device – the sample is inside the transparent pipe. (b) Compression load cell placed in the laboratory microtomograph – on the right the nanotom Phoenix Xray tube. The solution adopted to determine the stress and strain fields in the foam is a meshing of the µCT images with tetrahedral elements and a finite element calculation ([12] and [13]). The 3D image of the foam was meshed with Avizo software [11] The meshing is done in three steps; the surface is first reconstructed using triangles by a marching cubes algorithm [14], [15]; the meshed surface is then simplified to reduce the number of triangles this is achieved through an edge collapsing algorithm [16]; finally, the tetrahedral mesh is built by an advancing front algorithm [17]. The µCT images of the sample foams being too large to be meshed directly. The image was cropped into a 300x300x300 voxels image (1.8x1.8x1.8 mm3) to get a sub-sample of the foam. The choice of the boundary conditions is very important. We decided to unconstrain the lateral faces, completely constrain the lower face in displacement, and impose a displacement along the compression direction (z direction) on the upper face, the displacements in the perpendicular plane (x and y directions) being blocked on this face (a scheme of the boundary conditions is presented on Fig. 4-b). A parallel study of the modeling of the foam was also executed to determine the ideal parameters of the mesh (convergence study – about 300 000 elements), the constitutive laws (elastic brittle law – E=35 GPa – σy = 300 MPa) for the PyC and the minimum size of sample to be representative. A mesh of the foam is presented on Fig. 4-a.

(a)

(b)

Fig. 4 : (a) mesh of a 3D µCT image of the foam. (b) Scheme of the boundary conditions applied to the mesh.

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Results Damages in the foam were observed at the nine states of compression acquired by µCT. First, the analysis of the global deformations/damages was carried out by observing the outer surface of the foam. These observations were done by choosing the adapted angle of view to see the failure and using a voltex rendering to visualize the foam in 3D (voltex rendering is a function of the Avizo software - see [11] for details). This angle of view is crucial to well analyze the failure and would be hard to be found without the use of in-situ µCT. On Fig.5, three states of deformation are presented – at the initial state (Fig.5-a) – at 6% of strain (Fig.5-b) and at 15 % of strain (Fig.5-c). As indicated by the arrows a macroscopic failure is visible in the middle of the foam. This failure actually consists in a series of individual failures of the ligaments of the cells, these individual failures being connected in a plane presenting an angle at about 30° to 45° with the loading direction (the loading direction is vertical on these pictures).

(a)

(b)

(c)

Fig. 5 : µCT in-situ compression test on a ceramic foam with a relative density of 0.14. Observation of the external surface using a voltex rendering [11]. 3 states of deformation: (a) Initial state, (b) 6% strain – arrows indicate the initiation of failure (c) 15% strain – Arrows indicate the evolution of damage. In a second study we have exploited one of the main assets of µCT : the possibility to visualize the local damages inside the material. These observations were made at different places in the material. The recognition of the different zones on the 3D images of the foam is difficult and was made easier following particular cells or defects. A major failure mode was evidenced in this study, an example being presented on Fig.6. This figure compares the initial state and a damaged state of the same ligament’s cell. As highlighted by the arrows, the failure appears at both ends of a ligament. It is a fragile failure with no plastic deformation.

(a)

(b)

Fig. 6 : Local analysis of a µCT in-situ compression test on a ceramic foam with a relative density of 0.14. Visualisation of a ligament inside the foam using a voltex rendering [11]. Arrows indicate the fracture, numbers are visual marks to highlight the bounds.

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The local behavior observed above by µCT analysis is also observable on the measurement of damage in an isolated ligament. Fig. 7 presents a ligament randomly chosen in the foam with no particular orientation and position. The measures of damage clearly show that the damage concentrates at both ends of the ligament where the thickness of the ligaments changes. This stress concentration zone is also a weak point and the weakness of these ligaments initiates the damage in the foam. Thus if one wanted to simulate the damaging behavior of the foam without taken into account the real microstructure, these singular points in the structure would be missed, leading probably to an erroneous behavior.

Fig. 7 : Calculation of the damage rate field inside a ligament. A 0 damage rate is a material without damage and 1 is a failed material. When the first failures appear on both ends of the ligament, the load is transmitted to the neighbouring ligaments. This creates particular development of the damage. This organisation is well illustrated on Fig.8, where the damage rate in a rather large part of the sample is shown. On Fig.8-b, a damage band orientated at about 30° with the loading direction appears. The damage appears in the ligaments and propagates throughout the structure. When the damage in this band is saturated, other local failures appear (Fig.8-c). These subsequent failures are often organized in smaller bands.

(a)

(b)

(c)

Fig. 8 : Damage rate field inside the cellular structure. FE calculations of different states of compression corresponding to states ref. 1, 2 and 3 in the in-situ µCT observations. (a) Initial state. (b) 3% strain. (c) 6% strain.

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Discussion As explained in the results section, a failure was observed at 30° to 45° with respect to the loading direction on Fig.5. We can make an analogy with ductile metallic materials were failure is often observed at 45°. This angle corresponds to the maximum sheered plane in the sample. Thus, the failure has a quasi-ductile character whereas the constitutive material is intrinsically fragile. This is in good agreement with the macroscopic behavior, as shown on Fig.2-a. In the measure of damage by FE calculation on the structure presented on Fig.8, the failure seems to propagate in a plane oriented at 30°. This plane is not exactly the same as the one described on Fig. 5. Another difference is the presence of local damages in the calculations of Fig. 8. These isolated damages are not observable during experimental investigations of Fig. 5. These differences are probably explained by the difference in the size considered during the experimental tests. Here appears the problem of the representative size. The size of foam used for calculation is probably too small to be representative of the experimental test. Another question appears: the boundary conditions applied were chosen uniformly. This condition is obviously not effective in a foam extracted in the middle of the sample tested. We have chosen these conditions for simplification reasons and because we have estimated that these conditions are rather close to the real ones. Further investigations are needed to validate this choice. A solution would be to apply the displacement directly measured on the µCT images. The comparison of Fig.6 and Fig.7 reveals that the stress/damage concentration zones correspond to the failure zones. Most loaded ligaments are generally broken at their ends as illustrated on Fig.6. These zones are indeed the most stressed/damaged regions in the FE calculation (Fig.7). The final goal of these investigations was to better understand the macroscopic mechanical behavior. The fractures in bands observed explain the perturbations in the plateau domain described in the material section. These perturbations are also well simulated by the model presented. Then, in order to optimize the material, the model is a useful tool. It is rather easy to adjust parameters such as the dimensions, the volume fraction or the constitutive material properties to find the material adapted to a specific application.

Conclusion Using µCT technique, it is possible to visualize the inside of the materials at high resolution. These investigations can also be carried out insitu during mechanical loading. Using the 3D images, it is also possible to calculate several fields like stress, strain or damage. All these microscopic data are very important to understand the macroscopic mechanical behavior. In the case of cellular materials (like the PyC densified porous vitreous carbon foams presented here), these investigations are rather easy. The information obtained is crucial to optimize or understand mechanical behaviour. Another important benefit of these calculations using meshed µCT images is the relevance of the simulated behaviour compared with classical models like those presented in [3]. This benefit is explained by local damages linked to local defects. These defects are singular points that must be considered to simulate the effective mechanical behaviour of such brittle materials. Unfortunately these calculations are heavy. In this study, we have chosen to use only one part of the sample. This choice may be a problem, as explained in the discussion section. A solution would be to use the entire sample for calculations using more powerful computers, or to improve the calculation efficiency. Another improvement would be on the experimental set-up used for in-situ compression tests. A new loading device is under development at the ICMCB in Pessac. This new insitu device is adapted for accurate and small displacements. The main difficulty in the development of such a device is the lack of room available in the laboratory tomograph and the necessity to be X-ray transparent throughout a 360° rotation. Finally, to improve the test and the resolution of 3D images in particular, a solution would be to use a synchrotron like the ESRF in Grenoble [18].

46

Acknowledgments CEA is thanked for providing the starting vitreous carbon foam. The insitu compression tests and µCT acquisitions were carried out at the ICMCB laboratory (TOMOMAT group), we want to thank Ali Chirazi and Dominique Bernard in particular for their collaboration. The “Groupement d’Intérêt Scientifique” (GIS) Advanced Materials in Aquitaine (AMA) is acknowledged for founding the development of the In-Situ device. Florian Canderaz is acknowledged for the model developments.

References [1] S.Delettrez. Elaboration par voie gazeuse et caractérisation de céramiques alvéolaire bas pyrocarbone ou carbure de silicium. Phd thesis. University of Bordeaux1. 2008. [2] G.L.Vignoles, C.Gaboriau, S.Delettrez, G.Chollon, and F.Langlais. Reinforced carbon foams prepared by chemical vapor infiltration : a process modeling approach. Surface and coatings technology. 203 :510–515. 2008. [3] L.J.Gibson, M.F.Ashby. Cellular solids structure and properties - second edition. Pergamon Press, 2001. [4] R. Naslain, F. Langlais. Mater. Sci. Res. 20-145. 1992. [5] R. Naslain, F. Langlais, G.L. Vignoles, R. Pailler. Ceram. Eng. Sci. Proc. 27 (2) 373. 2006. [6] E. Maire, J.-Y. Buffière, L. Salvo, J.J. Blandin, W. Ludwig, J.M. Létang. Advanced Engineering Materials, 3 No 8, 539 - 546. 2001. [7] L. Babout, E. Maire, R. Fougères. Acta materiala 52, 2475-2487. 2004. [8] E. Maire, A. Elmoutaouakkil, A. Fazekas, L. Salvo. MRS Bulletin, 28, 284. 2003. [9] O.Caty, E.Maire, R.Bouchet. Fatigue of Metal Hollow Spheres Structures. Advanced Engineering Materials. 10. 179-184. 2008. [10] O.Caty, E.Maire, S.Youssef, R.Bouchet. Modelling the properties of closed cell cellular materials from tomography images using finite shell elements. Acta Materialia. 56, 5524-5534. 2008. [11] http://www.vsg3d.com/avizo [12] S.Youssef, E.Maire, R.Gaertner. Finite element modelling of the actual structure of cellular materials determined by X-Ray tomography. Acta Materialia. 53. 719-730. 2005. [13] K.Madi, S.Forest, M.Boussuge, S.Gailliere, E.Lataste, J-Y. Buffière, D.Bernard, D.Jeulin. Finite element simulation of the deformation of fused-cast refractories based on X-Ray computed tomography. Computational Materials Science. 39. 224-229. 2007. [14] W.E. Lorensen, H.E. Cline. Marching cubes: high resolution 3-D surface reconstruction algorithm. Proceeding of the 14th annual conference on Computer graphics and interactive techniques, Anaheim, july 2731. 163-169. 1987. [15] D.A. Rajon, W.E. Bolch. Marching cube algorithme : review and trilinear interpolation adaptation for image based dosimetric models. Computerized Medical Imaging and Graphics 47. 411-435. 2003. [16] P.Cignoni, C.Montani, R.Scopigno. A comparison of mesh simplification algorithms. Comput. & Graphics 22. 1. 37-54. 1998. [17] P.J. Frey, H. Borouchaki, P.L. Georges. 3D Delauney mesh generation coupled with an advancing-front approach. Computer methods in applied mechanics and engineering 157. 115-131. 1998. [18] www.esrf.eu/

Multiaxial stress state assessed by 3D X-Ray tomography on semi-crystalline polymers

L. Laiarinandrasana, T.F. Morgeneyer, H. Proudhon MINES ParisTech, MAT-Centre des Matériaux, CNRS UMR 7633, BP87 91003 Evry Cedex, France ABSTRACT This work aims at linking the microstructural evolution of semi-crystalline polymers to the macroscopic material behaviour under multiaxial stress state. Tensile tests on notched round bars, interrupted after different stages of deformation before failure, were supposed to have undergone various states of stress in the vicinity of the net cross-section. They were examined using synchrotron radiation tomography. A sample obtained from a test stopped at the end of stress softening stage showed elongated axi-symmetric columns of voids separated by thin ligaments of material. This special morphology allowed investigating the distributions of voids in terms of both volume fraction and orientation. This distribution was compared with the theoretical multiaxial stress/strain field. The combination of the tomographic images analyses and the continuum mechanics approach results in a determination of the principal mechanical parameter driving voids evolution. 1. Introduction In the last years, more and more researchers used the tomography technique to better understand the deformation mechanisms of various materials. Most of studies were dealing with light metals such as aluminium alloys. For a semi-crystalline polymeric material, Laiarinandrasana et al. [1] reported a specific morphology of voids evolution thanks to 3D X-ray tomography. In an initially necked region, void aspect shows elongated shapes separated by thin walls. Taking advantage of the excellent resolution of the images obtained during this campaign, an attempt was made to observe the voids microstructure within an initially notched round bars used to enhance void growth. Tomographic observations were carried out on several polymers. The present paper focuses on the investigations of PolyAmide 6. Before describing the experimental setup (PA6 material, sample preparation, tomography technique), a theoretical background on the mechanical stress/strain state within a notched round bar is given. Then, the radial distribution of voids is discussed in order to highlight the role of the stress triaxiality ratio on void growth. Further analyses of the tomographic images allowed determining void volume fraction gradient according to the longitudinal direction (axis z). Critical analyses of this gradient were performed with the help of the theoretical results about the stress/strain state within the notch region. The last part of the paper describes the void orientation distribution. It is demonstrated that the trajectories of the largest principal stress coincide with the void orientation. 2. Theoretical background on circumferentially notched specimens Figure 1a describes the circumferentially notched specimen. Conventional cylindrical coordinates (r, θ, z) are used. Rotational symmetry is considered around longitudinal axis z. R and a are respectively the radius of curvature of the notch root and the outside radius of the minimal cross section. Authors like Bridgman [2-3], Kachanov [4], Davidenkov and Spiridonova [5] evaluated the distribution of true stress/strain in a necked tensile specimens made of metallic materials. Beremin [6] followed the approach by using initially notched specimens with machined radius curvature. The general assumptions required were: i) perfectly plastic material ii) isochoric transformation with a homogeneous axial strain in the minimal cross section. It was then demonstrated that the stress and strain tensors in the vicinity of the minimal cross section could be expressed as follows [6-7]:

T. Proulx (ed.), Optical Measurements, Modeling, and Metrology, Volume 5, Conference Proceedings of the Society for Experimental Mechanics Series 9999999, DOI 10.1007/978-1-4614-0228-2_7, © The Society for Experimental Mechanics, Inc. 2011

47

48 2 2   2η − 4η z  − 2η r   a a  ~≅σ  σ 0 eq   4ηrz   a2 

 1  − 2 ηz  −12 2 ~ε ≅ ε e a  0 eq   6ηrz  2  a 2

0 −

1 2

0

4ηrz

0 2

z r 2η − 4η  − 2η  a a 0

a2

2

0 z 1 + 2η − 4η  a

2

      2 r  − 2η    a  

(1)

6ηrz   a2   0   1  

(2)

where, η = 0.5 ln[1+a/(2R)] and ln is the naeperian logarithm, (r, z) are the current coordinates of the considered point in this plane. Note that in equations (1) and (2) the distribution of stress in the minimal cross section is assumed to be parabolic.

z R r

σΙ

Line of iso-principal stress α

+ ω r

θ a

r

z

α

a)

a

R

ω +

b)

Figure 1: Sketches of a circumferentially notched specimen with the characteristic parameters

A lot of papers focused on the minimal cross section (z = 0) because it was the critical zone where damage and fracture occurred. The following equations tractable “by hand” were used in many studies. The radial, hoop and axial stresses are expressed as: 2   σ rr (r, z = 0) = σ θθ (r, z = 0) = σ eq 2η1 −  r    a     (3)  2  r    σ zz ( r, z = 0) = σ eq (1 + 2η)1 −      a     It has to be noticed that any stress component in equation (3) consists of a structural term (function of η) and a constitutive term (the equivalent stress). The multiaxiality of the stress state in the minimal cross section is measured by the stress triaxiality ratio (τσ), defined as the mean stress divided by the equivalent von Mises stress.

49

τ σ ( r , z = 0) =

~ )/3. where σm = trace( σ

  r 2  σm 1 = + 2η1 −     a  σ eq 3  

(4)

In addition, strains are assumed to be homogeneous within the cross section. They can be approximated by: εzz = εeq and

εrr = εθθ = -(1/2)εzz

(5)

Similarly, the stress/strain gradients with respect to z axis (r = 0) can also be expressed as: 2   σ rr ( r = 0, z) = σ θθ (r = 0, z) = σeq 2η1 − 2 z      a     2   1 − 2 z   ( ) σ ( r = 0 , z ) = σ 1 + 2 η zz eq    a    

ε zz (r = 0, z) = ε eq e

−12

(6)

ηz 2 a2

εrr(r=0, z) = εθθ(r=0, z) = -(1/2)εzz(r=0, z)

and

(7)

Equations (6-7) were checked by Beremin [6] by comparing these solutions with finite element results. 100

Test stop for observations

80

Net stress (MPa)

Stress softening

60

Unloading stage

40

20

0 0

0.1

0.2

0.3

Applied displacement (mm)

0.4

0.5

Figure 2: Experimental preparation of the PA6 notched specimen Furthermore, the concept of the trajectories of largest principal stress/strain, discussed in [2-4] has never been exploited since. No experimental verification could be done. Additionally, a true stress cannot be measured and the strain field can be estimated at the surface of the specimen. These trajectories of principal stress were assumed to be circular in a necked sample. Figure 1b illustrates this feature and highlights that at any point the line of iso-principal stress should be

50 perpendicular to the corresponding trajectory of principal stress. Actually, the eigenvector of the principal stress is oriented with an angle α that depends on the coordinates (r,z) of the point of interest. Let ρ be the curvature radius of the trajectory of the largest principal stress at point M(r, z). Two different expressions of ρ are given by the authors. According to Kachanov, Davidenkov and Spiridonova [4-5], ρ = aR/r, whereas ρ = (a2+2aR-r2)/2r for Bridgman [2-3]. One can notice that ρ tends to infinity when r = 0, that is on the axis z whereas ρ = R for r = a on the notch root. In fact, by plotting both expressions, one can realize that the difference is very small. In figure 1b, it can be moticed that :

r = ρ sin α   z = R + a − ρ cos α 

(8)

Equation (8) clearly shows that the orientation of the eigenvector corresponding to the largest principal stress depends on the considered point. Up to now, there was no clear verification of this theory. The present paper aims at showing some features, observed on tomographic images of a polymer that can match the trends implied by the previous equations. 2. Experiments The material under study is a PolyAmide 6 polymer that was selected due to the quality of the images obtained by Synchrotron Radiation Tomography (SRT) carried out at the European Synchrotron Radiation Facilities (ESRF). The physico-chemical properties, as well as the tomography technique description were detailed in Laiarinandrasana et al. [1]. Following this last reference, a series of interrupted tests were carried on circumferentially notched specimens. In this paper, focus is set on a specimen with initial notch root radius R = 3.5mm and a minimum section radius a = 2mm. This specimen was tested using a traction machine with a load cell and the crosshead displacement measurements. The test was stopped at the end of the stress softening (fig.2) to focus on the voids morphology and distribution assumed to be “frozen” at that time. After unloading and specimen removal, deformed samples were, first photographed (fig.2) in order to locate the volume of interest (VOI within the box in fig.2), then, scanned at ESRF in Grenoble (France). Locations of volume scan is depicted in fig.1a, symbolized by small rectangles and circles. SRT was carried out using the ID19 tomograph. The local tomographic setup [8] was used to avoid a cutting of the sample. A tomographic scan corresponded to 1500 radiographs, recorded over a 180° rotation of the sample. A radiograph was constituted of 1024 x 512 pixels with an isotropic pixel size of 0.7µm. Hence, in the following tomographic images, the width corresponding to the diameter of the maximum reconstructed 3D volume was of 716 µm whereas the height was of 358µm. 3. Results and discussions 3.1. Radial distribution of voids

Notch root

a)

b)

Figure 3: Radial distribution and morphology of voids within the neck

Figure 3 describes the morphology and distribution of voids within the minimum cross sections through longitudinal cuts. Voids are observed in black. They exhibit elongated aspect separated by thin walls [1]. This particular morphology allows, at

51 least qualitatively, observations of the variation of the height, the radial expansion and the relative orientation of voids. Indeed, spherical voids would not give such all of these information. Figure 3a observed in a VOI located in the centre of the minimal cross section indicates large void expansion in height (distance between two walls and the full height of elongated voids as described in [1]) but also in radial expansion. Note that some patterns indicate coalescence in both directions (radial = flat ellipse and in column = elongated ellipse). Figure 3b shows image of a VOI located close to the sample boundary. No void could be observed in the vicinity of the notch root. Moreover, it can be observed that there is less void with the mean diameter/”height” of voids smaller than at the centre (figure 3a). It can be concluded that in any cross section within the notched region, the void volume fraction (porosity) is maximum in the centre and gradually decreases towards the surface (notch root).

1.4 σ/σeq

τσ

0.7

σzz/σeq

1.2 1

0.6 0.5

τσ

0.8

0.4

0.6

0.3

0.4

0.2 σrr/σeq

0.2

0.1 0

0 0

0.2

0.4

0.6

0.8

1

r/a

Figure 4: Normalized stress and stress triaxiality ratio versus normalized radial abscissa for z = 0. R = 3.5mm, a = 2mm.

This result, reported for many materials in the literature argues that void growth is driven by the stress triaxiality ratio. Indeed, by recalling in figure 4 the plots of the normalized axial/radial stresses (equation 3) as well as the stress triaxiality ratio (equation 4) respect to the normalized radial abscissa, the distribution of the porosity is consistent with the trend of these plots. Conversely, the strain is homogeneous over the whole cross section. Moreover, from equation (2) it can be demonstrated that for any z, ∂ε/∂r is constant. The contour map of any strain component is “flat” (no gradient). Therefore, the strain cannot be considered as a relevant parameter to be associated with void growth. 3.2. Axial distribution of voids Considering the axial distribution of voids near the axis, from the minimal cross section to the basis of the notch shoulder, figure 5 depicts the tomographic images to highlight the void volume fraction gradient. The same analyses as in the previous subsection are carried out here about the height, width and the amount of voids. All of these characteristics decrease from the centre (minimal cross section) to the notch shoulder.

52

▬

▬

▬

▬ Figure 5: Distribution of voids along z axis

Following Beremin [6], figure 6 shows plots of normalized stress/strain with respect to z/a according to equations (6-7). The first conclusion is that figure 5 constitutes an experimental verification of the theory. This important result –which can be considered as a novelty from experimental viewpoint- is essentially due to the tomographic images. It can be expected to further assess, for a given material, how far from the basis of the notch/neck shoulder the effect of the notch is present. Nevertheless, it is to be mentioned that unlike the radial distribution of voids (figure 4), the axial distribution (figure 6) cannot indicate which of the stress or the strain is the leading parameter for void growth.

53 ε/εeq

1.4 σ/σeq 1.2

1.2 1

σzz/σeq

1

0.8

0.8 0.6 εzz/εeq

0.6

0.4

0.4 σrr/σeq

0.2

0.2 0

0 0

0.2

0.4

0.6

0.8

1

z/a

Figure 6: Normalized stress/strain function of z/a for r = 0. R = 3.5mm, a = 2mm.

3.3. Orientation distribution of voids For the sake of clarity, a zoom of figure 3b is discussed in this section (figure 7). The knowledge of the void morphology [1] enables to draw arrows indicating the orientation of these voids. By superimposing on figure 7 the geometrical construction in figure 1b, three points were considered with respective radii ρ1, ρ2, ρ3. Local voids orientations were symbolized by the corresponding angles β1, β2, β3 respectively measured between the arrows and the vertical z axis. Recall that in figure 1b, the orientation of the largest principal stress is symbolized by the angle α described in equation (8). In figure 7, it turns out that the evolution of angle α is in excellent agreement with the aforementioned β. Indeed, voids angle is null close to the z axis and gradually increases to coincide with the local notch root curvature near the surface. As a matter of fact, these observations were encountered when voids were located outside the minimum cross section (z ≠ 0). To the authors’ knowledge, such investigations dealing with the mechanical parameters state combined with tomographic observations constitute a novel experimental approach. At this stage, the main important conclusions applied to the PA6 under study consist of: - voids orientation parallel to the largest principal stress. Therefore this seems to indicate the component of the stress involved in void stretching; - no relevance of the largest principal strain with voids orientation (flat contour map); - quantification of voids characteristics allowing the local stress measurement provided a relevant stress scaling methodology (e.g. finite element analysis). 5. Conclusion PolyAmide 6 semi-crystalline polymer deformation mechanisms were studied thanks to Synchrotron Radiation Tomography (SRT) carried out at the European Synchrotron Radiation Facilities (ESRF). An initially notched round bar was tested under tension, up to the end of the stress softening. The sample was then released from the traction machine to be observed by SRT. Specific morphology of voids allowed identification of the voids distribution according to the axial/radial direction. Furthermore, voids orientation was observed to be dependent on their location within the notched region. By comparing these features with the theoretical stress/strain fields, it can be concluded that the largest principal stress is the key mechanical parameter that control voids growth. Data collected from image analysis of SRT would be of great importance to be utilised as input in the mechanical analyses (FEA). In particular, material model parameters governing damage evolution should be adjusted to match the experimentally measured void volume fraction distribution.

54

β2

β3 ρ3

ρ2

β1 ρ1

R

Minimal cross section

Figure 7: Distribution of voids orientation along r in a longitudinal cut.

References [1] Laiarinandrasana, L., Morgeneyer, T.F. Proudhon, H., Regrain, C, Damage of semi-crystalline polyamide 6 assessed by 3D X-ray tomography: from microstructural evolution to constitutive modelling. Journal of Polymer Science: Part B: Polymer Physics 48, 1516–1525, 2010. [2] Bridgman, P.W., The stress distribution at the neck of a tension specimen, Transaction ASM. 32, 553-574, 1944. [3] Bridgman, P.W., The effect of nonuniformities of stress at the neck of a tension specimen, in Large plastic flow and fracture, Metallurgy and metallurgical engineering series, First edition, New York, McGraw-Hill book company, Inc., 937. 1952. [4] Kachanov, L.M., The state of stress in the neck of a tension specimen, in Fundamentals of the theory of plasticity, MIR Publisher Moscow, 292-294, 1974. [5] Davidenkov, N.N., Spiridonova, N.I., The analysis of the state of stress in the neck of tension specimen. Proc ASTM, 46, 1-12, 1946. [6] Beremin, F.M., Elastoplastic calculation of circumferentially notched specimens using finite element method, Journal de Mécanique appliquée, 4(3), 307-325. 1980. [7] François, D. Pineau, A., Zaoui, A. Comportement mécanique des matériaux 2nd volume. HERMES edition, Paris. 1993. ISBN 2-86601-348-4. [8] Youssef, S., Maire, E., Gaertner, R. Finite element modelling of the actual structure of cellular materials determined by X ray tomography, Acta Materialia, 53(3), 719-730, 2005.

Effect of Porosity on the Fatigue Life of a Cast Al Alloy

Nicolas Vanderessea, Jean-Yves Buffierea, Eric Mairea, Amaury Chabodb a

Université de Lyon – INSA de Lyon - MATEIS, UMR5510, Bâtiment Saint Exupery 20 Av. A. Enstein 69621 Villeurbanne Cedex France b Centre Technique des Industries de la Fonderie Sèvres France

ABSTRACT A methodology has been developed to investigate the causes of fatigue fracture in the pressure-cast aluminium alloy Al Si9 Cu3 (Fe). Several samples have been tested. In each case porosity was the primary cause for failure. 3D tomographic images of the samples porosity at the initial state have been used as a basis for finite element analysis of the stress concentration around each pore above a minimal given size. Morphological assessment of the pores and Finite Element calculation allow investigating the correlations between, stress concentration and crack initiation. Results indicate that the pores causing failure have sizes located at the end-tail of the size distribution. 1.

Introduction

Cast Al-Si alloys are widely used in the automotive industry to produce block engines, because of their high strength to weight ratio, their low processing cost and their ability to be cast in intricate shapes. However, this type of material contains microstructural defects resulting from the casting process such as porosity, oxides and metallic inclusions which strongly reduce the fatigue life and increase its variability. Close to the fatigue limit, previous studies have shown that fatigue crack nucleation occurs predominantly on pores (see for example [1]) which also have a detrimental effect on static mechanical properties. The volume fraction of porosity in a cast Al alloy typically ranges between 0.1 % and 25 % depending on the casting process used and the shape of the resulting product. A value as low as 1% (volume fraction) of porosity can lead to a reduction of 50% of the fatigue life and 20% of the fatigue limit compared with the same alloy with a similar microstructure but showing no pores [1]. To correlate fatigue properties of cast Al alloys to the presence of porosity, the fracture surfaces of broken fatigue samples have been extensively studied by Scanning Electron Microscopy (SEM), in order to identify which defect(s) is (are) responsible for crack initiation. One major drawback of this type of characterization, however, is that (in the best cases) it only reveals the weak link within the pore distribution. When the tail of the distributions matters, as it is the case in fatigue, comparisons with other samples therefore remain qualitative [2, 3] and statistical models are required to infer the actual pore population and to investigate its relationship with the fatigue life [4]. With the rapid development of X-ray microtomography in the last decade [5] this technique has being increasingly used for the characterization of cast materials because of its ability to provide fast and exhaustive information on porosity [6-10]. In this study, an Al-Si-Cu (A380) cast alloy has been tested in fatigue. SEM characterization of broken samples has been carried to determine the crack initiation sites. The 3D distribution of pores in the fatigue samples was determined by X-ray tomography. Finite element computations, based on the 3D tomographic images, have been used to estimate the stress level around pores taking into account their position in the sample, their size and their shape. Those results were used to investigate the ability of the pores to nucleate fatigue cracks. 2.

Materials and methods

The studied material was a A380 alloy which composition is given in Table 1. The samples were produced by permanent mould casting in an industrial mold using a 630 tons Buhler machine. The metal was injected at 700 °C with a pressure of 800 bars. With an injection speed comprised between 40 and 50 m/s,the mold filling was completed in approximately 15 ms. The specimens were then machined in order to produce samples with a cylindrical gage length ( roughness< 0.8 µm diameter 3.89mm and height ranging between 18 to 24 mm). The 0.2% Yield strength of the Al cast material was found to be 90 MPa and its tensile strength 150 MPa. A value of 70 MPa has been measured for the endurance limit (stair case method, R= -1) with some variation depending on the volume fraction of porosity in the material. T. Proulx (ed.), Optical Measurements, Modeling, and Metrology, Volume 5, Conference Proceedings of the Society for Experimental Mechanics Series 9999999, DOI 10.1007/978-1-4614-0228-2_8, © The Society for Experimental Mechanics, Inc. 2011

55

56

The fatigue tests presented here have been performed at room temperature at constant stress amplitude until failure. A stress ratio of 0.1 has been used in order to provide clearer evidences of crack initiation on the fracture surfaces. Three samples have been mistakenly cycled below the endurance limit for several millions of cycles, before getting tested at a correct level ie slightly above the fatigue limit (max ranging from 140 to 145 MPa). For this reason, the total fatigue life of the five samples has not be considered and the analysis has instead focused on the ability of a pore to initiate a fatigue crack. Table 1 Chemical composition of the studied material. Element Composition (weight %)

Al

Si

Cu

Zn

Mg + Mn

Others

85.8 %

8.9 %

3.1%

0.7 %

0.2 %

1.3%

A laboratory tomograph was used for the 3D characterization of the fatigue samples[11]. For each tomographic scan,720 images were acquired (exposure time 0.5 s) with a voxel size of 5 µm (a voxel is the 3D equivalent to the pixel for a 2D image). After reconstruction, a 16 bit image was obtained which was first down-sampled to 8 bits and further cropped to dimensions 850*850*1300 voxels (i.e. 6.5 mm in height). Depending on the sample, three or four scans were necessary to image the whole sample gage length. Five samples have been exhaustively characterized in several steps (see Table 2 for a list). First a 3D image of the fatigue samples was obtained by tomography from which a statistical analysis of the pores was performed. The samples were subsequently cycled until fracture. Their fracture surfaces were characterized using SEM and a second tomographic image of the broken samples was recorded. From the 3D images a Finite Element (FE) model of a region surrounding the crack was generated allowing to study the stress level in this part of the sample. Image analysis of the stress concentration levels computed by the FE analysis was finally performed. This last step consisted in trying to establish a correlation between the morphological and spatial parameters of the pores, stress concentration around them and the occurrence of crack initiation. For this purpose, the severity of the pores has been estimated by calculating the volume of the matrix stressed above the yield stress. As the FE analysis has been performed in the elastic framework, this quantity does not strictly correspond to a “micro-plastic zone” but rather refers to the influence zone of each pore. This pore influence zone has been analyzed following two approaches: - The first one aimed at examining the links between the morphology of each pore and the size of its own influence zone, and is referred to hereafter as the pore-by-pore, analysis. - The second one investigated the links between the local pore fraction (in slices perpendicular to the tensile axis) and the influence zone density in the same slices. It is referred to hereafter as the slice-by-slice analysis.

Table 2 Summary of the studied samples. Sample Pore volume fraction Equivalent σmax for R=-1 (MPa) Fatigue life (cycles)

36

164

C48

E29

F40

0.60 %

0.62 %

0.46 %

0.57 %

0.20 %

40 and 90

85

80

40 and 85

80 and 85

4.106 and 113 190

132 037

194 583

6.106 and 183 093

4.106 and 437 257

The Avizo software [12] was used to mesh the solid/air interface of the samples. This included the external shape of the specimen as well as the inner shape of the porosity. As each of the specimens contains a few thousands pores having a minimal volume of 403 µm3, it was not possible to mesh a complete sample in a single pass. Due to computation limitations, the FE analysis was restricted to a sub-block of 2 mm in height. This sub volume was cropped from the initial block around the location of the crack. The generation of a 3D mesh in Avizo was performed by an advancing front strategy. The interfaces between the different phases of the model were first triangulated and used as seeds for the iterative generation of tetrahedral elements

57

that eventually meshed the material completely. The number of initial 2D elements had a strong effect on the final number of tetrahedral elements and on the complexity of the subsequent simulation. The pore geometry being much more complex than that of the sample cylindrical surface, both types of interfaces had to be triangulated separately and the corresponding meshes were exported as two separate STL files (text-type format files), which were then concatenated into a single, global file. This later was used to generate the 3D mesh. The average number of triangles required to mesh the sample's surface was about 15 000, whereas it varied between 100 000 and 800 000 for the pore surfaces. The number of 3D elements was further reduced by assigning them a target average size close to the dimensions of the 2D elements at the sample’s surface. The 3D mesh was thus reasonably refined around the pores and its number of elements ranged between 600 000 and 1 500 000 elements (Figure 1)

Fig. 1: 3D rendering of the mesh used for the mask surrounding the region of interest (left image:15000 elements ) and for the pores contained in this region (right image:181000 elements). The FE calculation was performed using ABAQUS Standard. While an accurate simulation of the mechanical behavior of the material should have taken the cyclic plasticity into account, as a firs approach, an elastic calculation was carried out (E=70 GPa and =0.3) in order to test whether such a simple approach could give a valid estimator for the severity of the pores. The computation was static. A monotonic, uniaxial displacement was imposed to the sample upper extremity, while the lower one was kept at a constant height (the nodes were nevertheless free to move laterally in the plane perpendicular to the loading direction). The value of the imposed displacement was fixed in order to match the maximal stress imposed during the fatigue tests. The results of the FE calculation were converted into an 8bit volumetric image [13] where each of the 256 gray levels corresponded to a 3 to 5 MPa range, depending on the maximal value of the von Mises Stress calculated by the simulation. This image was then downsized to match the size of the initial tomography block containing the pores. The stress values were thus averaged on voxels of 5*5*5 µm3. The statistical distribution of the stress within the meshed part of the material could be conveniently evaluated through the gray levels histogram of the 3D image. This histogram always showed some abnormally high values which were the result of numerical singularities at the vicinity of very tortuous pores. In order to avoid this problem the 3D images were binarized with a threshold value corresponding to the material 0.2 % yield stress (185MPa). The resulting block showed white regions surrounding the largest pores (Figure 2). These regions correspond to the pore influence zones. Each pore could thus be characterized by the volume of its own influence zone which was related to a micro-plastic region that was likely to develop at the immediate vicinity of the pore. Since the computation was elastic, the actual size of the highly loaded regions around the pores was probably slightly different from that obtained with the simulation. Still, this approximation had no consequence on the validity of the analysis, which relied uniquely on comparisons between parameters. To complement the above described analysis, the influence zones of each individual pore were also averaged on each slice in order to obtain the distribution of influence zones density (surface fraction) along the sample axis.

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Fig. 2: Left:3D rendering of the FE calculation (Von Mises stress) Right: same image after thresholding above the 0.2% yield stress. 3. Results and analysis The analysis of the reconstructed 3D images provided an exhaustive description of the pores present in the samples in terms of volume, surface, sphericity, (ratio of the two former parameters), projected surface, distance to the surface, size of individual influence zone and surface fraction of influence zone in each slice. All those parameters could be used to try and establish correlations with the ability of a pore to initiate a crack as shown elsewhere [13]. Here we mainly focus on the last two parameters which reflect the propensity of a pore to generate local plasticity either individually or more globally in a slice of the sample perpendicular to the loading axis. It is worth recalling that the local stress level (which determines the influence zone sizes) implicitly takes into account all the above mentioned parameters. Figure 3. shows the volumes of the pore influence zones plotted against the volumes of the pores for two typical samples. As expected, large pores tend to induce large stress concentration. For four of the investigated samples the crack leading to failure was found to initiate at the largest pores (like in the case of sample 48). In the case of sample 36, however, the crack nucleates at a small pore which intersects the surface. The slice-by-slice analysis leads to a similar conclusion as can be seen on Figure 4. On this figure, the surface fraction of matrix (on a surface perpendicular to the loading axis) with a stress higher than the yield stress has been plotted as a function of the fraction of pores for the same samples as those shown in Figure 3. Each point represents a slice of 5 µm in thickness. As the location of the crack initiation could not be exactly determined by fractography or post failure tomography within the sample gage length, the pore leading to failure is shown by several marks. Figure 4 shows that there is a correlation between the local densities of the pores and the density of highly stressed zones. This is the result of the sample being highly loaded in sections where the porosity was higher. Crack initiation seemed to occur in sections where the porosity was high, i.e. where the stress was high, with the exception of samples 164 and, to a less extent, F40 [13] The relationship between the pore influence volume / surface fraction and the event of crack initiation varied according to the sample, as is qualitatively summarized in Table 3. For each sample analysed, the “quality of correlation” relates to the position of the initiating pore within the relevant distribution.

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Fig. 3: Plots of the pore influence zone size as a function of pore volume for samples 48 (top) and 36 (bottom).Pore by pore analysis. Stars indicate the initiating pore. In samples C48, E29 and F40, the crack initiated at a pore which had a wide influence zone and a large volume, i.e. which was located at the tail of both the influence zone and the volume distributions. In these cases, the elastic calculation seems to be accurate enough to predict the location of crack initiation. The same can be said for the slice-byslice analysis of samples 36, C48, E29 and to a lesser extent F40: The crack nucleated at a position rich in pores and showing a high density of influence zones. In other words, in these samples crack initiation occured at a local maximum of the pore density. A poor correlation was found for sample 164 which did not comply with any of the analyses: this sample failed at a small pore with a modest influence zone, albeit located near the surface. These results outline the importance of the pore size distribution and spatial repartition for crack initiation. Indeed, the samples for which both analyses yield satisfying results are those in which the spatial repartition is markedly heterogeneous. Conversely, the samples 164 and F40 were characterized by a rather flat distribution along the principal axis [13]. Interestingly, in sample 36, the crack initiates at a small pore which intersected the sample surface, the influence zone volume was moderate, yet considerably higher than that of internal pores of comparable size (Figure 3). In this case, the pore-by-pore analysis was not fully satisfying. As a whole the proposed methodology allowed to identify the most probable zones for fatigue failure when the pore distribution was spatially heterogeneous. These critical zones were those of high porosity, and were directly related to a high density of stressed matrix.

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Fig. 4: Results of the slice-by-slice analysis. Plots of the density of influence zones against surface fraction of pores along the axis of samples 48 (top) and 36 (bottom). The approximate location of the crack starting point is marked by stars. Table 3 .Correlation between the occurrence of crack initiation and the influence zone of the pores considered either individually or slice by slice. Symbols used for indicating the correlation quality: -: bad, * poor, **: good, ***: excellent Sample

36

164

C48

E29

F40

Pore influence volume

-

*

***

***

***

Pore influence density

**

-

***

***

**

Crack initiation site

Pore intersecting the surface

Internal pore

Internal pore

Internal pore

Internal pore

4. Conclusion A partly automated treatment has been developed to analyse the causes for fatigue failure in cast aluminum samples containing porosity. It relies on the 3D characterization by micro-tomography of the pores, used as an input for finite

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element analysis. A careful methodology for generating a refined mesh has been developed,. The proposed methodologyi s based on the conversion of 3D images of the microstructure in FE data and of FE computation results into a 3D image again. This last step proves to be a fast, easy and reliable method for post-processing the results with powerful techniques from the image analysis field The pore influence zone is proposed as a variable that represents the volume of matrix stressed above the yield stress. It has been treated on an individual, pore-by-pore basis, and on a local, slice-by-slice basis. On average, both give satisfying results for samples with a heterogeneous porosity and predict that cracks nucleate at regions with a high local porosity.

References [1] Ødegard J.A., Pedersen K., Report No. 940811, Society of Automotive Engineers, Warrendale, PA, 1984. [2] Zhu X., Yi J.Z., Jones J.W., Allison J.E., . Metallurgical and Materials Transaction A. Vol 38A. 1111 2007. [3] Yi J.Z., Gao Y.X., Lee P.D., Flower H.M., Lindley T.C., Metallurgical and Materials Transaction A. Vol 34A 1879, 2003. [4] Wang Q, Jones P. Metallurgical and Materials Transaction A Vol. 38A, 615, 2007. [5] Stock S., Int Mater Rev 53(3):129, 2008. [6]Buffiere JY, Savelli S, Jouneau PH, Maire E, Fougeres R. Mater Sci Eng, A Vol. 316, 115, 2001. [7] Lashkari O, Yao L, Cockcroft S, Maijer D. Metallurgical and Materials Transaction A Vol.40, 991, 2009. [8] Hardin RA, Beckermann C. Metallurgical and Materials Transaction A Vol. 40, 581, 2009. [9] Zhang H, Toda H, Hara H, Kobayashi M, Kobayashi T, Sugiyama D, Kuroda N,Uesugi K. Metallurgical and Materials Transaction A Vol. 38, 1774, 2007. [10] Kobayashi M, Toda H, Minami K, Mori T, Uesugi K, Takeuchi A, Suzuki Y. J Jpn Inst Light Met Vol. 59, 5, 2009. [11] Buffiere J.-Y., Maire E.,·Adrien J.,·Masse J.P.,·Boller E., Exp Mech Vol. 50, 289, 2010. [12] www.vsg3d.com. Accessed 05/03/11 [13] Vanderesse N., Maire E., Buffiere J.Y., Chabod A., submitted to Int. Jal of Fatigue under revision Feb.2011.

Fatigue mechanisms of brazed Al-Mn alloys used in heat exchangers Aurélien Buteria,b, Julien Réthoréc, Jean-Yves Buffièrea, Damien Fabrèguea, Elodie Perrinb, Sylvain Henryb a

Université de Lyon – INSA de Lyon – MATEIS, UMR5510, Villeurbanne, France b Alcan CRV (Research Center), Voreppe, France c Université de Lyon – INSA de Lyon – LaMCoS, UMR5259, Villeurbanne, France

ABSTRACT The ratio of aluminium alloys used in the automotive industry tends to increase as a consequence of the enforcement of tougher environmental regulation (minimization of vehicles weight). For example, thanks to their good thermal, corrosion and mechanical properties, aluminium alloys have steadily replaced copper alloys and brass for manufacturing heat exchangers in cars or trucks. Such components have been constantly optimized in terms of exchange surface area and, nowadays, this has led to Al components in heat exchangers with a typical thickness of the order of 0.2 to 1.5 mm. With such small thicknesses, the load levels experienced by heat exchangers components has drastically increased leading to an important research effort in order to improve the resistance to damage development during service life. This paper focuses on the resistance to fatigue damage of thin sheets of brazed co-rolled aluminium alloys used for manufacturing heat exchangers and particularly on the mechanisms of fatigue cracks initiation. Digital Image Correlation (DIC) has been used to monitor damage development during constant amplitude fatigue tests of thin (0.27 mm) samples. Fatigue cracks have been found to initiate from deformation bands which presence can be correlated with solidification drops at the sample’s surface resulting from the brazing process. X-ray tomography has been used to obtain the spatial distribution of drops as well as their characteristics (height, surface...), on the sample gauge length. Those 3D data have been used to produce finite element meshes of the samples in order to assess the influence of the drops on fatigue crack initiation. 1.

Introduction

The small thicknesses of the thermal heat exchangers components improve the thermal performance through the increase of exchange surface area, but it leads to an increase of the in use loads which can be detrimental to the service life duration via for example fatigue damage development. Fatigue damage of brazed thin sheet aluminium alloys for thermal heat exchangers has rarely been considered in the literature [1 - 4] and none of these works have dealt with thicknesses below 1.5mm. The main technical issue for the investigation of damage development in fatigue sample with a sub millimeter thickness is that their surfaces cannot be polished. Thus classical optical/electronic microscopy observations of fatigue damage initiation and development cannot be carried out. It has been suggested to use Transmission Electron Microscopy (TEM) to correlate dislocation structures resulting from mechanical cyclic loading of 3000 series alloy on cylindrical gauge section samples with a diameter of 10mm [4]. TEM preparation is however a time consuming and destructive technique. In this study we present a different approach based on DIC [5-6] and 3D tomographic observations. Those techniques are used to identify fatigue crack initiation sites and perform Finite Element (FE) calculations [7-8-9], which help to elucidate the fatigue mechanisms of the studied material. 2.

Experimental procedure

An industrial material made of 3 co-rolled aluminium alloys (total thickness 0.27 mm) has been studied. In spite of its small thickness, the material exhibits a composite structure comprising a core material (3xxx alloy) and 2 clads (4xx and 7xxx alloys). The lower melting point 4xxx alloy is used for producing the heat exchanger assembly during a brazing process while the 7xxx alloy improves internal corrosion resistance. Industrial brazing conditions have been used to produce flat dog bones samples exhibiting representative microstructures through the thickness and also at their surface.

T. Proulx (ed.), Optical Measurements, Modeling, and Metrology, Volume 5, Conference Proceedings of the Society for Experimental Mechanics Series 9999999, DOI 10.1007/978-1-4614-0228-2_9, © The Society for Experimental Mechanics, Inc. 2011

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Figure 1: Material configuration and compositions of the different aluminium alloys used 2.1 Fatigue tests and digital image correlation The fatigue test samples investigated had a minimal rectangular section of 4.05 mm2 (15*0.27 mm2) according to the layout of figure 2(a). Fatigue tests were carried out in a hydraulic tension-tension fatigue test machine under constant stress-amplitude conditions (σmax = 100 MPa, stress ratio = 0.1 and F = 10 Hz) at room temperature. Damage development at the sample surfaces has been monitored during cycling by DIC. A large angle telecentric lens with focal distance of 200 mm associated to a CCD camera with a 1200*1600 pixels resolution (12*16 mm2) has been used. Pictures of sample’s surface prepared beforehand by applying a speckle are recorded every 150 cycles in constant lighting conditions (exposure time = 15ms). A Matlab® post-treatment [5] of the pictures allows the measurement of the displacement field on the sample surface between two cycling steps and the determination of the equivalent strain and stress fields.

Figure 2: (a): sample geometry. (b): 3D-rendering of the sample surface showing clad solidification drops 2.2 Tomography X-ray tomography is a 3D imaging technique based on the difference of absorption of the various constituents of the material that allows visualizing the inner structure of an object [9]. This technique has been used here to characterize the surface roughness of fatigue sample (Figure 2(b)) and to investigate the probable influence of the local microstructure on crack initiation. The principle of the technique and the experimental setups used here are described elsewhere [7]. Two different voxel size/imaging modes have been used: 13µm/voxel (Lab. X-ray source: absorption mode) and 0.7µm/voxel (Synchrotron X ray source: absorption + phase contrast). 2.3 Finite element meshing and tensile test simulation parameters A surface and volume meshing with quadratic tetrahedrons is created from the reconstructed 3D images of the fatigue samples with Amira® software (Figure 3). The mesh corresponding to the sample studied here contains 200 000 quadratic tetrahedrons. A Java plugin [7-8-9] allows to import the mesh into Abaqus®. The boundary conditions have been chosen to prevent displacements of sample edges except in load direction. Simulation consists in a maximal stress approach in elastic conditions (E = 70 000 MPa, ν = 0.33) by applying a vertical displacement of 0.1mm, which correspond to a deformation of 0.2% as a 1 st approach. The multimaterial aspect of the sample has not been taken into account for the FE calculations.

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Figure 3: 3D view of the volume meshing with quadratic tetrahedrons of the fatigue sample 3.

Results and analysis

The fatigue lives (Wöhler curves) and the damage mechanisms (crack initiation sites and propagation) have been characterized. 3.1 Fatigue mechanisms of brazed material The sample analysed by X-ray tomography and shown in section 2.1 (Figure 2(b)) has been submitted to 933744 fatigue cycles. The test was stopped before fracture when the sample contained a 1mm long fatigue crack. Figures 4(a) and 4(b) present the zone of interest (ZOI) used for DIC at reference (750 cycles) and final states. The strain (Exx) fields for 5 specific steps, respectively 47%, 75%, 97% and 100% of the fatigue lifetime, are presented on figures 5(a-c). Large localized plastic deformation leading to large displacements (figure 5(c)) appears during the last part of the cycling, which probably corresponds to the stable crack propagation. Before the few last cycles (97% of the fatigue life, i.e. 28 000 cycles), no strain heterogeneity can be visualized by DIC. The presence of a crack can be evidence by using a discrepancy map (Figure 6(c)) as described in [10].

Figure 4: Zone Of Interest (ZOI) for the digital image correlation (DIC) at: (a) the initial state (750 cycles) and (b): the final state (933744 cycles). The painting black points (speckles) are used to calculate displacement field by DIC. On fatigue crack can be easily visualized on the final state image (b). The shadow clearly visible near the crack on figure 4(b) highlights the presence of a Clad Solidification Drop (CSD). Systematic fractographic analysis of the fatigue samples confirms the presence of a CSD close to the crack initiation zone in most cases. The stress concentration induced by the CSD roughness of the sample surface is likely to promote local plasticity and induce eventually crack initiation. The influence of CSD on local (elastic) stress distribution during cycling can be studied in detail from the 3D FE meshes generated from tomography; this is described in the next section.

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Figure 5: Strain field Exx (a– c) obtained by DIC at 3 different steps of the fatigue test, respectively 75, 98 and 100% of the lifetime. (Scales - Exx: *100%)

Figure 6: Exx (a), Ux (b) and discrepancy map (c) obtained by DIC at 100% of the lifetime (Scales - Ux: *10µm and Exx: *100%) 3.2 Study of clad solidification drops geometric influence by a maximal stress approach Figure 7 shows the Von Mises stress distribution on the fatigue sample described in the two previous sections. A stress concentration zone can clearly be seen near the different CSDs among which is one responsible for crack initiation. Note that for the model considered here (homogeneous material) the stress concentration zones appear to be localized on the surface. Moreover, the influence CSD’s geometry has been quantified through the important stress concentration induced (figure 5(b)). On figure 7(b) is presented the distribution of the maximum values of stress along the loading direction (σ22) is also shown. It can be seen that the CSD responsible for crack initiation is the one, which induce the maximum value of σ22. This situation has been constituently found for 4 of the 5 samples studied so far.

Figure 7: FE calculation of local stress levels in the fatigue samples based on microstructurally realistic 3D FE mesh generated from tomographic data. (a): Zoom showing the distribution of σ22 stresses in a zone close to the initiation site of the crack, which lead to failure after 934000 cycles with a maximal fatigue stress of 100 MPa. (b): Distribution of σ22 values induced by the different CSD present in the sample gage length; the value corresponding to the initiating CSD is highlighted.

67 In the cases where crack initiation could not be correlated with a local maximum value of the σ 22 stress, inclusions or porosity resulting from the brazing process have been detected at the initiation site by SEM inspection and/or tomographic inspection of the fracture surface (high resolution X-ray tomography - ESRF) as illustrated in figure 8.

Figure 8: Microstructure visualisation around one fatigue crack by high-resolution X-ray tomography – A CSD (Clad Solidification Drop) is observed around fatigue crack as well as some porosity (white arrow). 4.

Conclusions

From these results it can be inferred that in this brazed material, fatigue crack initiation is the result of an interaction between high values of local tensile stresses (resulting from the surface roughness), which induce intense plastic activity as evidenced by DIC measurements, and a local microstructure which further enhance the geometrical stress concentration effect. The tomographic data allows to analyze the scatter in fatigue lives at a given stress level on the basis of the CSD distribution in different samples. Preliminary results confirm that longer (resp. shorter) fatigue lives can be correlated with smaller (resp. larger) CSD/stress levels. The results obtained give clear indication that new alloys and/or brazing fluxes enabling to reduce the presence of CSD are expected to enhance greatly the fatigue resistance of brazed assemblies in heat exchangers. Preliminary results obtained with different brazing conditions confirm this trend.

References [1] X.X.Yao, R.Sandström and T.Stenqvist: Mater. Sci. Eng., A267 (1999) 1-6 [2] J-K.Kim and D-S.Shim: Int. J. Fatigue. 22 (2000) 611-618 [3] U.Zerbst, M.Heinimann, C.Dalle Donne and D.Steglich: Eng. Fract. Mech. 76 (2009) 5-43 [4] H.Yaguchi, H.Mitani, K.Nagano, T.Fujii and M.Kato: Mater. Sci. Eng. A315 (2001) 189-194. [5] Elguedj T., Rethore J., Buteri A. - Isogeometric analysis for strain field measurements. - Comput. Methods Appl. Mech. Eng. 2011; 200: 40-56 [6] J.Rethoré, F.Hild and S.Roux: Comput. Meth. Appl. Mech. Eng. 196 (2007) 5016-5030 [7] J-Y.Buffière, P.Cloetens, W.Ludwig, E.Maire and L.Salvo: M.R.S. Bulletin (2008) 33 – 611-619 [8] O.Caty, E.Maire, S.Youssef and R.Bouchet: Acta Mat. 56 (2008) 19, 5524-5534 [9] O.Caty, « Fatigue des empilements de spères creuses métalliques », PhD thesis, INSA Lyon, 2008 [10] Buteri A., Buffiere J.-Y., Fabregue D., Perrin E., Rethoré J., Havet P. – Fatigue mechanisms of brazed AlMn alloys used in heat exchangers – Proceedings of the 12th International Conference on Aluminium Alloys (2010).

Three Dimensional Confocal Microscopy Study of Boundaries between Colloidal Crystals E. Maire1, M. Persson Gulda, N. Nakamura2, K. Jensen, E. Margolis, C. Friedsam F. Spaepen Harvard University, School of Engineering and Applied Science, Cambridge, MA 02138 ABSTRACT Colloidal crystals were grown on flat or patterned glass slides. The structure of the grains and their defects was first visualized by 3D confocal microscopy and then characterized using simple geometric measurements. Crystals grown on a flat surface maintained a layered structure induced by the closed-packed planes. In the case of the [110] Σ5 grain boundary, the presence of particles in interlayer position was established. 1 Introduction The grain-level microstructure of a material influences a wide range of material properties, including strength, toughness and corrosion resistance. For that reason, understanding and controlling the structure and evolution of grain boundaries is one of the central tasks of materials science. Studying grains at the atomic level moreover, is not an easy task. To aid in this, we used colloidal suspensions as model systems that form crystals. In recent research, colloids have been used to model atomic or molecular systems since they form many of the same phases. They can be used for model glasses as well as crystals [1, 2]. A colloidal system has two distinct phases: a dispersed phase and a continuous one. The dispersed phase consists of small solid particles, on a nano to macro scale, which are dispersed evenly through the continuous fluid phase. The particles used in this study interact as hard spheres. When these particles sediment onto a flat surface, they can form crystals; when they sediment onto an irregular, rough surface, they can form amorphous structures. For the random close-packing, the structural paradigm for the amorphous phase, the volume fraction of solid, f, is about 0.63, and for the close-packed crystals, f is about 0.74. The crystals often contain defects and the focus of the present paper is on grain boundaries. The main purpose of the paper is to show how the crystals can be grown and imaged in three dimensions (3D) using confocal microscopy. 2 Experimental procedure 2.1 Colloidal suspension Silica particles (diameter 1.55 μm, density 2.0 g/cm3, mass 3.9 x 10-15 kg) were suspended in a water - 62.8 vol. % dimethylsulfoxide (DMSO) solution that matched the index of refraction of the silica and had a density of 1.10 g/cm3. The average velocity of Brownian motion is given by: = (3kB/Tm)1/2

(1)

where kB is Boltzmann's constant, T the temperature and m the mass of the particle. For our case, this gives =2 x 10-3 m/s. The gravitational settling velocity of the particle is given by vS = VP Δρ g / 6π r η

(2)

where VP and r are, respectively, the volume and radius of the particle, Δρ is the density difference between particle and fluid, and η is the viscosity of the fluid, which is about 10-2 Pa.s. This gives in our case vS=10-6 m/s, which satisfies the condition for a colloidal systems (vB>>vs). The index match makes the system optically transparent, which allows investigation by optical confocal microscopy at large distances into the sample. Contrast between particle and solution was achieved by adding fluorescein dye to the solution. The index match also minimizes the van der Waals forces between the particles, which interact therefore like hard spheres. 1 Present address : INSA-Lyon, MATEIS UMR5510, 25 av. Capelle, 69621 Villeurbanne, France 2 Present address : Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama, Toyonaka, Osaka 5608531, Japan

T. Proulx (ed.), Optical Measurements, Modeling, and Metrology, Volume 5, Conference Proceedings of the Society for Experimental Mechanics Series 9999999, DOI 10.1007/978-1-4614-0228-2_10, © The Society for Experimental Mechanics, Inc. 2011

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2.2 Template The crystal growth can be controlled by slowing down the sedimentation of the colloids (described above) and the use of a template. The controlled growth of the layers was achieved by diluting the initial suspension by a factor of two. A template is a patterned substrate that directs the setting of the particles in such a way that the structure, orientation, and size of the crystal is pre-determined. This process is called “colloidal epitaxy.”[3] The templates were fabricated in the following way. A positive mask of the targeted pattern (set of holes) was first printed on a chrome-coated substrate (chrome was removed using a HeidelbergTM mask maker). A thin layer of photoresist was then spin coated on a primer-coated glass slide (the subsrate) and polymerized (3 minutes at 110°C). This assembly was then exposed using a mask aligner (SussTM) for 2-2.5 sec. The time varied depending on the size of the holes in the chrome mask, the age of the photoresist, calibration of the mask aligner, and the premixed developer. The slides were then developed for 60 to 90 sec in a photoresist developer (mixture of 1 part of MF351 and 5 parts water). The silica was then etched in a reactive ion etcher (RIE). The plasma needed to be stable before the sample was inserted into the clean chamber. The reactive ion etching ran for about 10 minutes. Finally, to remove the remaining photoresist layer, the sample was exposed tin the same RIE to an oxygen cleaning plasma, which cleans the photoresist without etching the silica. An example of the resulting template, imaged by confocal microscopy, can be seen in Figure 1. In this example template, we have attempted to etch holes with a gradient of sizes.

Figure 1 : Confocal image of a pattern etched in a silica glass microscope slide (the white phase is the glass). The pattern has holes of different diameters. 2.3. Confocal microscope In a laser scanning confocal microscope, [4] light is focused through a microscope objective where it excites fluorescence in the sample. The emitted light is retraced through the microscope and passed through a pinhole in the conjugate focal plane of the lit spot in the sample. This allows only light from that spot to pass; light from all other directions, for example from multiple scattering or fluorescence, is blocked. The light intensity is recorded by a detector and stored as the spot is scanned through volume of the sample. The stored information can then be displayed directly as a three-dimensional image or be processed into a reconstructed image, in which computer graphics is used to redraw the spheres. When the refractive index of the fluid is matched to that of the spheres, light can penetrate quite deeply into the sample with little scattering, so that tens of planes of a crystal can be imaged. The lateral resolution (perpendicular to the optical axis) is about 200 nm, typical for optical microscopy. Because of limitations of the optics, the vertical resolution is only 500 nm. Application of image analysis techniques [5, 6] improves the resolution for the location of the center of the particles by about an order of magnitude. A typical time to scan a stack of planes through a sample is a few seconds.

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Figure 2 (a) and (b) show a crystal imaged using this technique and presented in the form of gray-level confocal images. Fig. 2(a) shows an x-z plane, orthogonal to the x-y plane shown in Fig. 2(b). These images are numerically extracted from the 3D data set recorded in the confocal microscope. Note that this data set is collected by acquiring 2D images (x-y images in the reference of the microscope) such as Figure 2(b). These are roughly parallel to the plane of the glass substrate. The acquisition is repeated by scanning along the third perpendicular direction, z, aligned with gravity. The spacing along the z plane is chosen to be equal to the pixel size in the x-y plane so that the stacking of these images is an isotropic reconstruction of the 3D structure of the crystal. The imaged crystal was grown along the z direction onto an etched glass pattern of a [100] Σ 5 grain boundary. The location of the grain boundary is indicated by a dark line in figure a). In Figure 2(b), extracted right at the middle plane of the first layer of deposited particles, the grain boundary is clearly visible. A white line has been drawn to indicate the location where Figure 2(a) was extracted. Figure 3 is a 3D rendering of the same data set after binarization by thresholding the particles and applying 100% transparency to the voxels located in the liquid phase.

Figure 2 : Two gray-level confocal images (slices) of a [100] Σ 5 bi-crystal grain boundary grown on a etched silica template. (a) x-z slice; (b) x-y slice. The location of the grain boundary is indicated by a dark vertical line in figure (a). It is clearly observable in figure (b), which is taken at the level of the first layer of particles, just above the glass pattern. The light gray line in figure (b) shows the location of (a).

Figure 3 : Grazing incidence view of a 3D rendering of the bi-crystal shown in Figure 2. The diameter of the silica particles is 1.55 µm.

3 Results and discussion 3.1. Structure of columnar polycrystals on a flat glass slide When setting slowly on a featureless flat glass slide, the colloidal particles spontaneously form crystals by successive deposition of hexagonal close-packed planes. Crystals nucleate in several places with random orientations and grow into grains, separated by grain boundaries. Two different polycrystalline samples were grown using this procedure. The first was grown using a small amount of solution in the container, while the other one was grown using three times this amount. This resulted in two samples with different thickness. Figure 4 shows an x-y plane of the thin sample. It reveals the polycrystalline nature of the deposited crystals. The thin sample had only 9 deposited layers and the structure is rather well preserved all through the thickness. Note that the image also reveals vacancies and defects in the form of agglomerated particles. Figure 5 shows a x-z slice of the thick sample. In this direction again the different grains can be observed, with the structure becoming more disordered as the distance from the substrate increases. Figure 6(a) and (b) again, show the typical structure of these two crystals observed in the x-z direction, but to account for the total volume analyzed, the gray level was averaged along the

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entire third perpendicular direction (y) and projected onto the 2D images. This averaging clearly highlights the order of the structure along the z direction, in the form of the successive close-packed planes. It also shows that since the sample is imaged from the bottom, the quality of the images is decreases when the distance to the substrate increases because of increasing scattering of the light perhaps due to imperfect index matching.

Figure 4: Confocal imaging of a layer of a thin polycrystalline sample close to the deposition surface.

Figure 5 : Confocal view of an x-z cut of a thick polycrystalline sample showing the different grains. The structure becomes more disordered at a larger distance from the substrate.

Figure 6 a). x-z view of the thin sample. The intensity is averaged over the entire third perpendicular direction (y), to form a projection.

Figure 6 b). x-z view of the thick sample. The intensity is averaged over the entire third perpendicular direction (y), to form a projection.

3.2. [100] Σ 5 grain boundary The structure of the Σ5 boundary shown in Fig. 2-3, is analyzed in more detail here. The bi-crystal consists of only four deposited layers (see Fig. 2(a)). Figure 7 shows the first layer of this model grain boundary. The two superimposed sets of lines show what the grain boundary structure should be if each hole in the pattern was filled with a particle as described in [7]. The atom on the left marked with an arrow is shared between two possible sites which can be considered as a defect. The set of white arrows point atoms which are present while they shouldn't according to the model created for this grain

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boundary. Figure 8 shows a 3D rendering of the position of these atoms replotted after the determination of the position of their center of gravity. In this figure, the atoms are colored according to the value of their distance to the substrate z. All the small blue points are located in layers. The larger colored points (red, yellow and green) have been determined -thanks to their value of z- as being in interlayer positions. The figure shows that, apart for some defects, a lot of atoms in these interlayer position are located close to the grain boundary which is in the plane located at x=12.5 microns and is indicated by a dashed line in the Figure.

Figure 7 : Structure of the [100] Σ 5 grain boundary. The black arrow on the left indicates a location where a single atom is present where there should be two sites. The white arrows show sites where atoms are present at places they should not be in the ideal Σ 5 hard sphere structure. [7]

Figure 8 : 3D view of the position of atoms as seen from the side of the grain boundary. The large atoms are in interlayer positions. Apart from a few defects, most of the interlayer atoms are located at the grain boundary which is in the plane defined at x=12.5 microns indicated by the dashed black line. 4. Conclusions and perspective This paper shows some 3D images of grain boundaries in colloidal crystals grown on glass microscope slides. The particles are silica spheres, monodisperse with a diameter of 1.55 µm. When deposited on a flat surface, grain boundaries form due to

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the aggregation of the particles in the form of close-packed planes. The successive deposition of close-packed planes is preserved over a long distance from the substrate, but a side view of a tall sample shows that the grain structure become more and more disordered. The paper also shows that colloidal epitaxy of bi-crystals is feasible, as exemplified in the case of a [100] Σ 5 grain boundary between two face-centered cubic crystals. The boundary contains atoms in an interlayer position. Future work will focus on the study of the mobility of the atoms at these grain boundaries. Acknowledgments We thank David Weitz and the Weitz group for much help and many discussions. This work was supported by the National Science Foundation through the MRSEC and REU programs. References [1]. P. Schall, I. Cohen, D.A. Weitz and F. Spaepen, Nature 440 (2006) 319. [2]. P. Schall, D.A. Weitz and F. Spaepen, Science 318 (2007) 1895. [3]. van Blaaderen A., Ruel R., and Wiltzius P. Template-directed colloidal crystallization Nature 385 (1997) 321-324 [4]. V. Prasad and D. Semwogerere and E.R. Weeks, J. Phys.: Condens. Matter 19 (2007) 113102. [5]. J.C. Crocker and D.G. Grier, J. Coll. Int. Sci. 179 (1996) 298. [6]. E.R. Weeks, J. C. Crocker, A.C. Levitt, A. Schofield, D. A. Weitz, Science 287 (2000) 627. [7]. In publication #25 of the web site http://seas.harvard.edu/matsci/people/fspaepen/Complete.html

Scale Independent Fracture Mechanics

Sanichiro Yoshida, Diwas Bhattarai, Tatsuo Okiyama and Kensuke Ichinose 1. Southeastern Louisiana University Department of Chemistry and Physics SLU 10878, Hammond, LA 70402, USA, [email protected] Tokyo Denki University Department of Mechanical Engineering 2-2, Kanda-Nishiki-cho, Chiyoda, Tokyo 101-8475, Japan

ABSTRACT Fracture mechanics is considered from the viewpoint of a field theoretical approach based on the physical principle known as gauge invariance. The advantage of this approach is scale independent and universal. All stages of deformation, from the elastic stage to fracturing stage can be treated on the same theoretical foundation. A quantity identified as the deformation charge is found to play a significant role in transition from plastic deformation to fracture. Theoretical details along with supporting experimental results are discussed. 1. Introduction Fracture is initiated at the atomistic scale and develops to the final, macroscopic failure of the object. By nature, it is an interscale phenomenon. However, most theories available to date are scale dependent; Quantum mechanics, dislocation theories, continuum mechanics and fracture mechanics all work well at each individual scale level but they do not describe the development in the scale level. For full understanding of fracture, it is important to argue the dynamics independent of the scale level. In this respect, a field theoretical approach proposed by Panin et al [1] as part of the general theory of deformation and fracture called physical mesomechanics [2] has great advantage. Based on a fundamental physical principle known as local (gauge) symmetry [3] (Aitchson, 1989), this approach is capable of describing deformation dynamics (i.e., the form of the force) without relying on empirical concepts or phenomenology such as experimentally determined constitutive relations; thus, by nature, the formalism is scale-independent and universal. According to this approach, on entering the plastic regime materials loses longitudinal elasticity but gains transverse elasticity. At the same time, the longitudinal effect becomes energy dissipative [4, 5]. Consequently, the displacement field in the plastic regime is characterized as a decaying transverse wave phenomenon. Previously, we applied this formalism to various aspect of deformation and fracture of solid-state materials, and verified the theory with experiments based on optical interferometry. Transverse decaying wave of displacement field has been experimentally observed and it has been confirmed that fracture indeed occurs when the transverse wave decays completely [6]. In this dynamics, a quantity defined as the deformation charge [7], which is closely related to the charge of symmetry associated with the gauge symmetry [3] plays an important role. In particular, it seems that materials fracture when the deformation charge stops flowing. In this paper, we elaborate on the dynamics associated with the deformation charge. In addition, an attempt is made to explain recent experiment [8] on a notched specimen based on this formalism. Quite interestingly, the experiment clearly shows the deformation charge, and its behavior toward fracture in complete consistence with the theory.

T. Proulx (ed.), Optical Measurements, Modeling, and Metrology, Volume 5, Conference Proceedings of the Society for Experimental Mechanics Series 9999999, DOI 10.1007/978-1-4614-0228-2_11, © The Society for Experimental Mechanics, Inc. 2011

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76 2. Theory 2.1 Field equations The basic postulate of the mesomechanical-approach is that when a material enters the plastic regime under the influence of external load, the deformation is still locally linear-elastic. The formalism resulting from this postulate is described in detail elsewhere 1 . In short, by applying the physical principle known as gauge invariance [3, 5] to the displacement field of materials under plastic deformation, the following field equations are obtained [4].

Here and are the translational and rotational displacement related to each other as indicated by eq. (3). and appearing on the right-hand side of eqs. (1) and (2) are the temporal and spatial components of the so-called charge of symmetry. is the phase velocity of the spatiotemporal variation of the field1.

Eqs. (1) and (2) yield a wave equation of the following form.

In the plastic regime, eq. (4) represents a decaying transverse wave, and the phase velocity can be expressed in terms of the density and shear modulus as follows.

With eq. (5) substituted and rearrangement of the terms, eq. (2) can be put in the following form [4].

The left-hand side of eq. (6) is the product of the mass and acceleration of a unit volume. The right-hand side represents the external force acting on the unit volume where the first term is transverse force and the second term is longitudinal force. In the plastic regime, the transverse force is a restoring force associated with the rotational displacement of the local region and the shear modulus , and the second term is the longitudinal, energy dissipating force [4]. Here the first term is responsible for the transverse wave characteristics of displacement in the plastic regime [6]. It has been shown that eq. (6) is valid for the linear elastic regime [9]; in that case, the first term on the right-hand side becomes null and the second term represents longitudinal elastic force. In addition, the phase velocity (5) becomes the well-known expression of the square root of the ratio of the Young’s modulus to the density. 2.2 Significance of charge in fracture The symmetry charge plays a significant role in fracture mechanics. From the law of mass conservation applied to a unit volume, the temporal change in the density is equal to the divergence of velocity field. 1

Detailed discussion will be found in a paper scheduled to be published in the Journal of Strain Analysis (the volume number has not been assigned).

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With the use of eq. (5), eq. (2) can be written as follows.

Application of divergence to eq. (2) with the use of eq. (5) and the mathematical identity called equation of continuity.

Eq. (9) indicates that

is the flow of

leads to the so-

, allowing us to put it in the following form.

With eq. (7), eq. (10) can further be rewritten as

Here the temporal change in density can be interpreted as the generation of dislocation [10]. Being associated with the net flow into the unit volume , this change in density is in the same direction as the local velocity, or . Thus, the corresponding flow can be put in the following form.

Being proportional to velocity, the longitudinal force

is by nature energy dissipating.

2.3 Transition to fracture The above argument indicates the following scenario of transition from elastic deformation to fracture of initially linearly elastic materials [7,11]. When a material enters the plastic regime, it loses the longitudinal elastic force, and instead, gains the transverse restoring force represented by on the right-hand side of eq. (6). At the same time, the longitudinal force becomes energy dissipative as represented by eq. (12). Based on this observation, the transition from the elastic regime to the plastic can be characterized by and proportional to the local velocity. Note that this is a local effect; even if the stress-strain curve is before the yield point (i.e., in the linear regime) and therefore the specimen is considered to be globally elastic, it is possible that the deformation is locally plastic. There are a number of experimental observations that support this interpretation [12]. While the mechanism of energy dissipation via is effective, the work done by the external force causing the deformation, such as the work done by a tensile machine, is partially dissipated in this fashion and partially stored as the rotational elastic energy associated with the restoring force . As the deformation develops, the material tends to lose these mechanisms. Some theoretical consideration [11] and experimental observations [6] indicate that it is likely that materials loses the restoring mechanism first, causing the transverse wave to decay, and then enters the final phase where the

78 dissipative force becomes null as well. Thus the transition from late plastic deformation to fracture can be characterized as the initial phase and

(13)

and, the final phase and

.

(14)

3. Experimental A number of experiments [4-9, 12] have been conducted to test the theory discussed above. In particular, observation of inplane displacement with the use of an optical interferometric technique known as the Electronic Speckle-Pattern Interferometry (ESPI) provides significant information. In this section, recent results of ESPI applied to tensile analysis of a notched specimen are presented. Figs.1and 2 illustrate the ESPI setup and the dimension of the carbon steel specimen used in the experiment [8]. The ESPI was a dual-beam type setup using a laser source of 532 nm in wavelength and was sensitive to vertical in-plane displacement of the specimen. The interferometric image was recorded with a CCD (Charge-Coupled Device) camera continuously during the tensile experiment at a rate of 100 frame/s. The specimen was loaded vertically at the through holes shown in Fig. 2 until it fractured. Initial crack of about 1.3 mm was used at the tip of the notch prior to the tensile loading. 62.5

13

3

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18.75 18.75

12.5

12.5 Fig. 1 ESPI optical setup [8]

Fig. 2 Carbon steel specimen with a notch [8]

After completion of image recording, interferometric fringe patterns were formed off-line by subtracting the frame taken at a certain time step from the frame taken at another time step. The frame difference for the subtraction, which corresponds to the total deformation that the resultant fringe patter represents, was adjusted so that the total number of the fringes was appropriate for the analysis. Typically, the frame difference of 30 – 40, corresponding to 0.3 – 0.4 s was used.

Fig. 3 ESPI fringes representing vertical displacement and corresponding finite element analysis

79 Fig. 3 is a fringe pattern observed at an early stage of deformation (frame number 8064 minus 8000). At this stage fringes are all continuous although its density is higher near the tip of the notch indicating that the deformation is somewhat concentrated there. The figure shown next to this fringe pattern is the contour of vertical displacement computed with a simple finite element model. The features that the fringes are almost vertical, springing out from the notch tip and that one fringe originating from the notch tip circles around the tip are clearly seen in the computed contours as well. Fig. 4 shows a series of fringes formed 771 frames (or 7.71 s as the frame rate is 100 frame/s) after Fig. 3. The four fringe patterns are created by subtracting the common image (frame number 8835) from images taken after this with an increment of 25 frames (i.e., frame number 8860, 8885, 8910 and 8935 going from the left to right). The frame difference of 25 corresponds to 250 ms, during which the tensile machine’s crosshead moves . While the number of fringes increases during this time but the basic pattern remains the same. Note that at this stage, the fringe patter is vertically symmetric, unlike Fig. 3. Since the difference between Fig. 3 and 4 in time is 7.71 s and the corresponding crosshead’s displacement is over the specimen’s size of 70 mm, the difference in strain is . At this stage, the fringes are still continuous.

Fig. 4 Fringes obtained by subtracting a common frame from various frames to observe increase in displacement When the deformation further develops, the fringes start to show discontinuity. Fig. 5 shows fringe patters formed by subtracting images different by 40 frame numbers (0.4 s). The leftmost pattern is formed by subtracting frame 9600 from 9640, i.e., 765 frames or 7.75 s after the leftmost fringe pattern of Fig. 4. The total strain at this point is approximately . Notice that the discontinuous fringes are divided by circular bright pattern, and the size of the circular pattern increases with time.

(1)

(2)

(3)

(4)

Fig. 5 Discontinuous fringe patterns observed at several time steps This bright circle divides regions of different fringe patterns. From this viewpoint, this bright circular pattern is considered to be equivalent to the bright band pattern (linear bright pattern) shown in Fig. 6, which was observed in our previous tensile experiments [13, 14] on not-notched specimen. Notice that the regions separated by the bright band patterns show completely different fringe patters. Our investigation strongly indicates that this linear bright pattern corresponds to the Lüder’s front, or dislocation line [13]. Thus, it is naturally considered that it represents the flow of deformation charge [eq. (12)].

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Fig. 6 Bright linear band pattern observed in tensile experiments on a non-notched specimen The series of images on the left in Fig. 6 show the appearance of the bright linear pattern as the deformation develops. The numbers under the images are the frame numbers. The rightmost image is from another experiment in which the specimen has a through hole and, a symmetric, X -shaped bright pattern appears around the hole. Previous studies on the linear bright patterns show that when the patterns stop drifting, the specimen fractures. This is in perfect consistence with the above argument of the final phase of transition from deformation to fracture [eq. (14)]. Another interesting aspect of the linear bright pattern in conjunction with the final fracture is that when the linear pattern is in a symmetric X-shape, the specimen fractures horizontally. However, when the linear bright patter appears in only one diagonal direction (i.e., “/” or “\” shaped as opposed to “X” shaped), the fracture is along the bright-line. Commonly to both cases, fracture occurs after the band becomes stationary. The question is now whether a similar argument can be made for the circular bright pattern observed in Fig. 6. Fig. 7 plots the change in the size of the circular bright pattern as a function of time, where time 0 corresponds to the moment when the bright circular pattern appears for the first time. The numbers (1) – (4) inserted in this figure indicate when images (1) – (4) in Fig. 5 are observed. It is interesting to note that the increase in the size of the circular pattern is saturated around 12 s. Viewing this size increase as the circular pattern being drifting in the radial direction, the saturation of increase in its size can be interpreted as the circular pattern ceases drifting, corresponding to the linear bright pattern becoming stationary. In fact, the above FEM modeling indicates that the displacement of the material in region of the circular pattern is approximately radial. Thus the saturation point in Fig. 7 (around 12 s) can be viewed as the beginning of final phase of the transition from plastic deformation to fracture. In addition, the fracture occurs horizontally penetrating the circular bright pattern symmetrically, as is the case of the X-shaped linear bright bands.

Fig. 7 Variation of the size of circular bright pattern 4. Summary Fracture has been viewed based on the field theory of deformation and fracture. It has been found that the deformation charge plays a significant role in transition from the final stage of plastic deformation to fracture. Experimentally, a bright pattern observed in fringe patterns formed by electronic speckle pattern interferometery has been found to visualize the deformation charge, and thereby useful to study fracture process.

81 Acknowledgement The present study was in part supported by Southeastern Alumni Association grant. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

Panin, V. E., Grinaev, Yu. V. Egorushkin, V. E., Buchbinder, I. L., and Kul’kov, S. N. Spectrum of ecxited states and the rotational mechanical field. Sov. Phys. J. 30 , 24-38 (1987). Panin, V. E. Ed. Physical Mesomechanics of Heterogeneous Media and Computer-Aided Design of Materials, vol. 1, Cambridge International Science, Cambridge (1998). Aitchson, I. J. R. and Hey, A. J. G. Gauge theories in particle physics, IOP Publishing, Ltd., Bristol and Philadelphia (1989). Yoshida, S. Dynamics of plastic deformation based on restoring and energy dissipative mechanisms in plasticity. Physical Mesomechanics, 11, 3-1, 140-146 (2008). Yoshida, S. Field theoretical approach to dynamics of plastic deformation and fracture, AIP Conference Proceedings, vol. 1186, pp. 108-119 (2009). Yoshida, S., Siahaan, B., Pardede, M. H., Sijabat, N., Simangunsong, H., Simbolon, T., and Kusnowo, A. Observation of plastic deformation wave in a tensile-loaded aluminum-alloy. Appl. Phys. Lett., A, 251, 54-60 (1999). Yoshida, S. Physical Meaning of Physical-mesomechanical Formulation of Deformation and Fracture, AIP Conference Proceedings, vol. 1301, pp. 146-155 (2010). Okiyama, T, Ichinose, K. and Yoshida, S. Research on evaluation of dynamics fracture characteristics by ESPI, to be presented at 17th Japan Soc. Mech. Eng. Kanto-branch meeting , March 18-19 (2011) Yoshida, S. Physical Meaning of Physical-mesomechanical Formulation of Deformation and Fracture, AIP Conference Proceedings, vol. 1301, pp. 146-155 (2010). Suzuki, T., Takeuchi, S. and Yoshinaga, H. Dislocation dynamics and plasticity, Springer-Verlag, Tokyo (1989). Yoshida, S. Consideration on fracture of solid-state materials. Phys.. Lett. A, 270 , 320-325 (2000). Yoshida, S. Muchiar, Muhamad, I, Widiastuti, R., and Kusnowo, A. Optical interferometric technique for deformation analysis, Optics Express, 2 (focused issue on "Material testing using optical techniques 516 - 530 (1998) Yoshida, S., Ishii, H., Ichinose, K., Gomi, K. and Taniuchi, K., An optical interferometric band as an indicator of plastic deformation front, J. Appl. Mech., 72, 792-794 (2005) B. Hu and S. Yoshida, Stress and strain analysis of metal plates with holes, 2010 SEM Annual Conference, June 7-10, 2010 Indianapolis, IN, USA (2010)

Consistent Embedding: A Theoretical Framework for Multiscale Modeling

Keith Runge Quantum Theory Project University of Florida PO Box 118435 Gainesville, FL 32611-8435

Abstract A fundamental framework for the undertaking of computational science provides clear distinctions between theory, model, and simulation. Consistent embedding provides a set of principles which when appropriately applied can create multi-scale models that capture the physical behavior of more computationally challenging methods within methods that are more easily computed. The consistent embedding methodology is illustrated within the context of brittle fracture for two serial and one concurrent multi-scale modeling examples. The examples demonstrate how predictive modeling hierarchies can be established. Introduction Co-workers and I have previously argued that the fundamental framework for computational science is intrinsically independent of the discipline in which it is applied. [1] In Ref. [1], we argued that process of computational science allows for the clear distinction among theories, models, and simulations. In this framework for the understanding of the computational scientist’s task, theory is taken to be comprised of the axioms and interpretive procedure that construct a mathematical description of the physical world. A model, in our way of thinking, is a chosen physical description of a system or class of systems formulated using the concepts of the theory. One commonly used model is Newtonian dynamics, where the system is modeled as a set of particles that move under the influence of interaction potentials by obeying Newton’s second law. The model can then be computational realized in a simulation using a molecular dynamics computer code where the consequences of the choice of initial conditions and interaction potentials are determined using algorithms and rules. Here an algorithm might solve a differential equation or an eigenvalue problem under the constraining rules applying to boundary condition, number of particles, or a prescribed temperature. In this description of computational

T. Proulx (ed.), Optical Measurements, Modeling, and Metrology, Volume 5, Conference Proceedings of the Society for Experimental Mechanics Series 9999999, DOI 10.1007/978-1-4614-0228-2_12, © The Society for Experimental Mechanics, Inc. 2011

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science, we see that improvements can be made in theory, model, and/or simulation in an attempt to improve the fidelity of the computed result in comparison with experiment. With the forgoing framework for understanding the task of the computational scientist, it may be well to confront a particular physical problem as an exemplar. In order to elucidate the interplay among theory, model, and simulation, let us consider the generic problem of mechanical failure. The phenomenon of mechanical failure presents challenges for the computational scientist at a number of temporal and spatial scales, as it involves the breaking of chemical bonds at the atomistic scale, and perhaps the interaction of grains at a somewhat larger length and time scales and the radical reshaping of overall structure at the macroscopic length scales over times that might be from fraction of seconds to days, weeks, or years. If we restrict ourselves to considering only brittle fracture, which is rapid and abrupt, then it seems clear that the fundamental event is the rupturing of chemical bonds. From a theory perspective, one must choose a level of quantum mechanical theory for the description of the evolution of electrons and nuclei in the stress fields that lead to fracture. While a most fundamental approach might be to attempt a solution of the time-dependent Schrödinger equation for all particles, it would quickly become evident that such an approach is computationally too demanding for current computational resources. One might also realize that simplifying the theory so that more common quantum chemical techniques, which rely on the Born-Oppenheimer approximation, and treating the (slower) nuclei as moving on the potential energy surface of the (faster) electrons seems a reasonable starting point for our choice of theory. We could now try to proceed by applying quantum chemical theory to the electrons and Newtonian dynamics to the nuclei to examine the problem of brittle fracture. Another computational bottleneck would then, no doubt, assail us. The implementation of quantum chemical theory for larger and larger numbers of electrons would quickly become prohibitively expensive. Fortunately, from a theory point of view, only those electrons, and their associated nuclei that are near, that is within a couple of chemical bond lengths or a few Angstoms, of the crack tip, are fully involved in the crack propagation. Atoms, that is, nuclei and their associated electrons, which are more remote are not nearly as perturbed by the crack tip. Hence, one sees the possibility of building a multi-scale model by choosing a set of theories to be applied at various length and time scales as measured from the crack tip. For instance, we might chose Born-Oppenheimer quantum chemistry and Newtonian dynamics in a small region around the crack tip, Newtonian dynamics using atomistic potential in a larger region, and a continuum modeling for the remainder of the structure. Implementing these choices of theory would require a multi-scale model. Multi-scale modeling is frequently divided into concurrent and serial multi-scale modeling. In serial modeling, inputs into models at one scale are generated by computational simulations at another, typically smaller scale. Here I will present an example of training atomistic potentials for Newtonian dynamics from quantum chemical calculations. Concurrent multi-scale modeling

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implements different theories at a number of length scales and then joins them in a single simulation hierarchy. The seamless joining of multi-scale models is generally challenging as the interface between models must pass all appropriate information while avoiding the generation of artifacts. Concurrent multi-scale models must also take care to obey conservation laws, like mass conservation or energy conservation, for the full system. Consistent Embedding The principle that we call consistent embedding dictates that the information that comprises a model at a larger spatial or longer time scale be compatible with the model used at a smaller or shorter scale. This principle is essential to the development of predictive theory and modeling, as the materials that exist on either side of a multi-scale interface must be physically consistent. For example, if we wish to look at the phenomenon of fracture, then the stress-strain relationship for the model at the shorter length scale should, at the very least, display the same small strain behavior, Young’s modulus, as the model used at the longer length scale. Enforcing this kind of constraint, that the Young’s modulus of the quantum chemical and atomistic models be equivalent, up to some controllable error, is an example of consistent embedding. We can develop other criteria, based on the physical properties being modeled, which improve the likelihood that emergent behavior of larger systems is grounded in the theoretical description of the smaller system. We have now established the computational science framework in which to do develop a multiscale theory, model, and simulation for brittle fracture. In this context, we illustrate the implementation on a multi-scale model and simulation based on the principle of consistent embedding. In particular, the choice of quantum chemical theory will be made subject to consistent embedding constraints, where higher level quantum chemical theory will be used to train less computationally demanding semi-empirical quantum chemical forms. Here it is important to note that the form of the quantum chemical Hamiltonian used is known as ‘semiempirical’, but our serial multi-scale training will be based solely on computed results from correlated calculation. A second illustration of consistent embedding principles in a serial multiscale model will be provided by the training of atomistic potentials solely from quantum chemical calculations. Finally, concurrent multi-scale modeling within consistent embedding principles will be demonstrated using a pseudo-atom termination scheme to facilitate the transfer of information across a quantum chemical/ classical mechanical interface. Transfer Hamiltonian The first example of serial multi-scale modeling that we consider is the training of one less computationally intensive quantum chemical method from a more computational intensive method. The name that has been given to this type of quantum chemical training is the Transfer

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Hamiltonian. [2] Silica, in particular amorphous silica, is the system whose brittle fracture will serve as the exemplar for this serial multi-scale model. For our higher level of quantum chemical theory we choose a method that includes the effects of electron correlation, known as coupledcluster theory including single and double excitations (CCSD). This highly accurate level of quantum chemical theory is also, of necessity, very computationally demanding. Hence it is necessary to choose a training molecule which exhibits the chemical bonding of interest in the mechanical failure of amorphous silica, but is limited to a relatively small number of atoms. We choose pyrosilicic acid (H6Si2O7) to create a CCSD training set for the ‘semi-empirical’ Hamiltonian. As seen in Fig. 1, the Si-O bond length is varied through compressions and stretches to generate a training set for the Transfer Hamiltonian. In this case, we have chosen to use a neglect of diatomic differential overlap (NDDO) Hamiltonian as our less computationally demanding quantum chemical model. NDDO is one of a set of approximations collectively referred to as zero differential overlap methods. When computers were much less powerful than they are today, these methods were developed to be computationally tractable and used mathematical forms derived from theory. These forms were parameterized to reproduce certain empirical data, e.g. heats of fusion, for simple molecules and the resulting parameterized Hamiltonians were applied to more complex problems with some success. The empirical parameterization of theoretically derived forms came to be known as ‘semi-empirical’ theory. By choosing to parameterize the NDDO Hamiltonian based on high accuracy, ab initio quantum chemistry, CCSD, we remove the empirical information from the procedure and replace it with a more detailed theoretical model. This substitution of theoretical for empirical information characterizes one possible implementation of the consistent embedding framework for the development of predictive modeling. The training of the Transfer Hamiltonian is accomplished using genetic algorithms, which are tuned to reproduce CCSD forces for the training set of molecular geometries. The choice of training on forces is motivated by our interest in stressstrain relations. In the next section, the implications of this Transfer Hamiltonian will be examined in a somewhat larger system.

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Figure 1 Pyrosilicic acid is used to create a training set of forces from CCSD for the Transfer Hamiltonian Small Strain Potential Pyrosilicic acid was sufficient for the training of the Transfer Hamiltonian, however, to see the effects of using it, we need a somewhat larger model system. Figure 2 shows a silica nanorod which we use to illustrate the next step in our serial multi-scale model. The nanorod is comprised of 108 atoms with two oxygen atoms for each silicon atom, the same ratio as found in silica. Various deformations are used to build a database of forces for training a small strain potential based on Transfer Hamiltonian calculations. Two ionic silica potentials, referred to by their authors’ initials, have found wide use in recent years, BKS [3] and TTAM [4, 5]. These potentials have the same general form and we have chosen to use this form for the parameterization of a small strain potential from Transfer Hamiltonian force data. Again, the parameterization is accomplished using a genetic algorithm. As shown in Table 1, the small strain potential reproduces the Young’s modulus obtained by the quantum chemical Hamiltonian to within a few percent for uniaxial stain along the long axis of the nanorod, while equilibrium

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bond lengths and bond angles are reproduced to within 2.5%. Details of the construction of the small strain potential can be found elsewhere. [6]

Figure 2 A silica nanorod comprised of 108 atoms, oxygen is green and silicon is gray Table 1: Young’s modulus comparison among potential and Transfer Hamiltonian Method Transfer Hamiltonian New potentail BKS TTAM

Young’s Modulus (arbitrary units) 1026 1022 1516 1214

Concurrent Multi-scale Modeling As a last example, we turn to the topic of concurrent multi-scale modeling. Dealing with the details of the interface is essential in this style of multi-scale modeling and as our target properties relate to stress-strain relations, we must concern ourselves with forces on either side of the interface. However, brittle fracture occurs by the rupture for chemical bonds, so it is also essential that the character of the chemical bonding of the system be preserved as well. The small strain potential presented in the previous section assures that the forces across a quantum chemical/classical mechanical interface are in good agreement for small strains. This has been confirmed by the agreement of the Young’s modulus and equilibrium configurations. In this section, we consider the interaction between the classical mechanical part of the system and the part describe by quantum chemistry. We choose to represent the effect of the remainder of the nanorod in the quantum chemical regime by a truncation scheme which we call pseudoatoms. These pseudoatoms replace oxygen atoms in the full system that serve as the interface between the two styles of treatment. For a typical fracture problem, the quantum chemical treatment would be focused around the crack tip, where the most strained chemical bonds are found.

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Psuedoatoms are trained using the pyrosilicic acid molecule shown in Fig. 1. Fig. 3 illustrates that the full system as seen from the quantum chemical viewpoint. In the case of the Transfer Hamiltonian, a fluorine atom has been reparameterized to preserve the equilibrium bond lengths and electron distribution in the remainder of the molecule. More detailed studies of the effect of pseudoatoms in the context of both the Transfer Hamiltonian and Density Functional Theory have been presented elsewhere. [7]

Figure 3 Pseudoatoms (labeled Modified F) are trained to reproduce local effects in the electron density Conclusions The distinction among the roles of theory, model, and simulation provides us with insight into ways one might improve our descriptions of the physical world. Using the conceptual framework of consistent embedding, we are able to pose sharp questions with quantifiable answers that can allow us to assess the quality of serial and current multi-scale models and their simulations. Illustrations of serial multi-scale modeling shown here, indicate that less computationally intensive quantum chemical methods can be developed that reflect the quality of more computationally intensive quantum chemical methods to a few percent for chosen properties, for brittle fracture we have concerned ourselves with forces. Further, small strain potentials can be trained, using forms available in the literature, to capture the behavior of quantum chemical methods. Finally, a strategy for concurrent multi-scale modeling has been provided that allows a system to be separated into classical and quantum domains while preserving the fidelity of each to the full system. These ingredients are essential to a predictive modeling capability. References [1] Trickey, S. B., Yip, S., Cheng, H.-P., Runge, K., and Deymier, P. A. A perspective on multiscale simulation: Toward understanding water-silica, J. Computer-Aided Mat. Design 13, 75 (2006).

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[2] Taylor, C. E., Cory, M. G., Bartlett, R. J. and Thiel, W., The transfer Hamiltonian: a tool for large scale simulations with quantum mechanical forces, Comp. Mater. Sci. 27, 204 (2003). [3] van Beest, B. W. H, Kramer, G. J. and van Santen, R. A., Force fields for silicas and aluminophosphates based on ab initio calculations, Phys. Rev. Lett. 64, 1955 (1990). [4] Tsuneyuki, S., Tsukada, M., Aoki, H. and Matsui, Y., First-Principles Interatomic Potential of Silica Applied to Molecular Dynamics, Phys. Rev. Lett. 61, 869 (1988). [5] Tsuneyuki, S., Tsukada, M., Aoki, H. and Matsui, Y., Molecular-dynamics study of the α to β structural phase transition of quartz, Phys. Rev. Lett. 64, 776 (1990). [6] Mallik, A., Runge, K., Cheng, H.-P. and Dufty, J. W., Constructing a Small Strain Potential for Multi-Scale Modeling, Molecular Simulation 31, 695 (2005). [7] Mallik, A., Taylor, D. E., Runge, K., Dufty, J. W. and Cheng, H.-P., Procedure for building a consistent embedding at the QMCM interface, J. Computer-Aided Mat. Design 13, 45 (2006).

Analysis of Crystal Rotation by Taylor Theory Motoaki Morita Graduate Student, Graduate School of Engineering, Yokohama National University 79-5 Tokiwadai, Hodogaya, Yokohama, 240-8501, Japan Osamu Umezawa Professor, Faculty of Engineering, Yokohama National University 79-5 Tokiwadai, Hodogaya, Yokohama, 240-8501, Japan ABSTRACT Simple shear along specific slip plane in polycrystalline and rotation of grains was discussed. The Taylor theory was applied to bridge between macroscopic deformation behavior and crystal plasticity and to evaluate the orientation distribution. Its theoretical solution can hardly satisfy all of boundary condition and plastic dynamics so that the condition of dynamics was simplified and relaxed in the analysis. The path of crystal rotation due to slip deformation was quantitatively predicted by Taylor theory and gave an advantage on understanding of deformation texture. The analysis method can be applied to polycrystalline materials. Although good evaluation was available in fcc and bcc where the orientation distribution fitted well, no good fitting to experimental result in hcp materials was obtained. 1. Introduction 1.1 Texture The arrangement of lattice is almost the same in a grain, and each grain in polycrystalline is usually distributed in random (Fig. 1(a)). On the other hand, the grains after working and/or heat-treatment reveal an arrangement with almost the same orientation (Fig. 1(b)). The arrangement of “deformation texture” is developed as the slip deformation progresses, because the grains are constrained by surroundings in polycrystalline. The deformation texture as well as “recrystallized texture” by heat treatment accompanies with an anisotropy in microstructure and affects on the properties of materials. The deformation texture mainly results from “crystal rotation” by slip deformation or twinning. Slip deformation occurs on the specific crystal planes and to the specific crystal directions. Each orientation of grain, therefore, changes to the specific one. The specific orientation is called as “preferred orientation”. Since its formation mechanism makes an important role to control their microstructure, quantitative prediction of the deformation texture is needed. To understand the formation mechanism of the deformation texture, the active slip systems and crystal rotation in each of grain should be considered under large deformation. In order to bridge macroscopic deformation behavior and crystal plasticity , Taylor theory which was based on minimum internal work principle has been applied to analyze the slip deformation behavior under large deformation and to evaluate the orientation distribution .[1-3] The theory is applicable to finite element method.[4]

. Fig. 1 Illustration of polycrystalline with random orientation (a) and texture (b).

T. Proulx (ed.), Optical Measurements, Modeling, and Metrology, Volume 5, Conference Proceedings of the Society for Experimental Mechanics Series 9999999, DOI 10.1007/978-1-4614-0228-2_13, © The Society for Experimental Mechanics, Inc. 2011

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1.2 Crystal Rotation Due to Slip Deformation The plastic deformation in materials strain and the rotation is described as velocity gradient tensor Lij (u i / xi ) . The velocity gradient tensor Lnij is described as

Lnij 

u i x j

(1)

where u i is velocity of material point in current configuration xi (i 1, 2, 3) . Strain velocity tensor Dij and total spin Wij are derived as

Dij 

1  u i u j  2  x j xi

   

(2)

Wij 

1  u i u j  2  x j xi

   

(3)

where Wij shows rigid body rotation, and is determined by not only stress but also the constrained geometrical condition. Figure 2 shows that single slip operation cause the crystal rotation under tension mode. When the tensile stress is applied to the body of single crystal, the deformation along a slip direction on a slip plane results in the change of its shape (Fig. 2 (a)). In the crystal coordinate, the slip deformation induces the crystal rotation Wijp in the body. However, the body cannot rotate itself in the specimen coordinate, because the slip plane is invariant (Fig. 2(b)). The proportion of material axis is parallel to the tensile axis so that the body rotates about specimen coordinate as

 ij  Wij  WijP

(4)

where WijP is plastic spin, and lattice spin  ij means the crystal rotation. Lattice spin is dominant in the condition of constraint. When elastic component is ignored during deformation, operating slip system gives the strain rate Dij and the rotation rate

WijP at each point in the body. When n-th slip system of the normal to slip plane n in and slip direction bin operates, Dij and WijP are described as

Dij 

1 N n n n  (mij  m ji ) 2 n 1

(5)

WijP 

1 N n n n  (mij  m ji ) 2 n 1

(6)

 b1n n1n  mijn   b2n n1n  n n  b3 n1

b1n n 2n b2n n 2n b3n n 2n

b1n n3n   b2n n3n   b3n n3n 

(7)

where  n is slip rate from n-th slip system. The equations in above are integrated as N

Lij  Dij  Wij   mijn  n   ij n 1

(8)

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 ij  Wij  WijP  Wij 

1 N n n n  (mij  m ji ) 2 n1

(9)

The lattice spin  ij means the infinitesimal rotation in t . Through the repeated calculation on the lattice spin, the path of crystal rotation can be analyzed.

Fig. 2 Illustration of crystal rotation due to slip deformation under tensile deformation in single crystal.[5] 1.3 Taylor’s Full Constraints Model A polycrystalline body deforms without defects at grain boundary during deformation, and all of grains are compatible each other in their strain. Taylor assumed that all of grains have the same strain. The Taylor model is called as full constraints model. In all of deformation modes, the compatibility in polycrystalline can be achieved by operating five independent slip systems.[5] When an uniaxial strain is parallel to the z-axis in the specimen’s coordinate system, XYZ, grain deformation takes place under axial symmetry at a fixed volume (  xx   yy   zz  0 ):

1 2

 xx   yy    zz

(10)

 xy   yz   zx  0

(11)

where  xx ,  yy and  zz are the plastic strain rate, and  xy ,  yz and  zx are the plastic shear strain rate in a grain. The internal plastic work rate W is the increment of work per volume and it is the sum of the work of five independent slip systems in a grain:

W    n  min n

(12)

where  n is the CRSS (critical resolved shear stress) and  n is the slip rate in the n-th slip system. There are a number of combinations of operating slip systems that satisfy the external work constraint, but only one or few numbers combinations should be chosen. When the minimum rate of internal plastic work W can be obtained, the combination of operating slip systems is selected and the slip rate of their operating slip systems can be analyzed. W depends on the relationship between the tensile axis or compressive one (z-axis) and grain orientation.

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2. Texture Formation in Magnesium-based Solid Solution Taylor theory can hardly satisfy all of boundary condition and plastic dynamics so that the condition of dynamics was simplified and relaxed in the analysis. We have mentioned the problems in the description of slip deformation and the prediction of crystal rotation by Taylor theory. Simple shear along specific slip plane in polycrystalline and rotation of grains in magnesium alloy have been discussed. 2.1 Application of Taylor’s Full Constraints Model Recently activities on research and development for magnesium alloys were very high. However their applications have been mostly for cast parts because of poor workability at low temperature. The poor workability in magnesium alloys intrinsically reflects on the behavior of plastic deformation in the hcp. To understand the deformation in magnesium alloys, deformation texture in magnesium-based solid solution was evaluated by Taylor’s full constraints model. In the texture simulation, the crystal rotations in 300 grains were done. The grains were initially given as random orientation (Fig. 3(a)). In the case of fcc, the  n was taken into account, because only one slip system {111} 110  operates. In the case of hcp, three principal slip systems, {0001}  1120  , {10 1 0}  1120  and {1122}  1123  , were taken into account as deformation mode. However, the CRSSs were installed to the analysis, because they were different from each other (Eq. (13)) [6]:

W   n  n  min

(13)

n

Table 1 represents the conditions of CRSSs for the evaluation. The deformation becomes more homogeneous at higher temperature and all primary slip systems can operate sufficiently as Type 1.[7] In Type 2, the ratio of CRSSs at room  temperature were installed. The primary slip was {0001}  1120  , because the CRSSs of {10 1 0}  1120  and  {1122}  1123  was higher than that of {0001}  1120  .[8-10] In addition, the CRSS of {10 1 0}  1120  in magnesium alloys was lower than that of {1122}  1123  at room temperature.[8] In the present study, no deformation twinning was considered. The solution may be given at the ratios between Types 1 and 2. Table 1 Ratios of critical resolved shear stress of principal slip systems in magnesium alloy.[7-9]

Slip system Type 1 Type 2 (300 K) 

{0001}  1120 

{101 0}  1120 

{1122}  1123 

1

1

1

40

80

1



2.2 Evaluation The calculated deformation texture in magnesium alloy was insensitive to the condition of CRSSs so that the reasonable solution may be given. Figure 3 represents the rotation of the grains in the case of Type 2. The grains rotated to two orientations from ε = 0.0 to 0.5 [-], although some grains still remained around (α, β) = (0, 0) (Fig. 3 (a) and (b)). At ε = 1.0 [-], two preferred orientations at (, ) = (27, 30) and (90, 30) were clear as shown in Fig. 3 (c). The grains around (α, β) = (0, 0) can hardly deformed but their number was lower than that in the preferred orientation. Therefore, (α, β) = (0, 0) direction gives quasi-stable orientation. The orientation distribution with ε = 1.0 [-] at 573 K for AZ61 alloy is shown in Fig. 4.[11] In this experiment the most preferred orientation was detected at (α, β) = (33, 0). It was reported out that the preferred orientation resulted from slip deformation. In the present simulation, however, the preferred orientation due to slip deformation was not at (α, β) = (33, 0) but (α, β) = (27, 30) and (90, 30). The simulation suggested that the deformation texture at (α, β) = (33, 0) did not result from slip deformation. The path of crystal rotation was also shown in Fig. 3(d). According to the simulation, the rotation paths were divided into the regions of (α, β) = (25 ~ 35, 0 ~ 30) which were called as tradition bands [12]. It is pointed out that the recrystallization easily occur at the grains on the tradition band.[11] The recrystallization may cause the stronger texture at grains with (α, β) = (33, 0). The reason why the preferred orientation at (α, β) = (90, 30) was not observed in the experiment [11] may result in twinning.

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Fig. 3 Prediction of inverse pole figure for 300 grains in the Type 2: (a) ε = 0 [-], (b) ε = 0.5 [-], (c) ε = 1.0 [-] and (d) illustration of rotation path. α is defined as the rotation angle from c-axis [0001] to a-axis [1120] and β is defined as the rotation angle about c-axis.

Fig. 4 Contour map of orientation distribution at 573 K for AZ61 alloy (ε = 1.0 [-], ε  1.0  10 4 [/s]).[7] 3. Summary The Taylor theory was applied to bridge between macroscopic deformation behavior and crystal plasticity and to evaluate the orientation distribution. Simple shear along specific slip plane in polycrystalline and rotation of grains was discussed. The path of crystal rotation due to slip deformation was quantitatively predicted by Taylor theory and gave an advantage on

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understanding of deformation texture. The analysis method can be applied to polycrystalline materials. Although good evaluation was available in fcc and bcc where the orientation distribution fitted well, no good fitting to experimental result in hcp materials was obtained. It is still difficult to understand the mechanism of texture formation in hcp metals. 4. Acknowledgement The authors thank Prof. H. Fukutomi and Prof. K. Sekine in Yokohama National University for their valuable discussion. 5. References [1] Taylor G.I., Plastic strain in metals. J. Inst. Metals 62, 307-324, 1938. [2] Bishop, J.F.W. and Hill, R., A theory of the plastic distortion of a polycrystalline aggregate under combined stresses. Phil. Mag. 42, 414, 1951a. [3] Bishop, J.F.W. and Hill, R., A theoretical derivation of the plastic properties of a polycrystalline face centred metal. Phil. Mag. 42, 1298 , 1951b. [4] Houtte P.V. et al., Deformation texture prediction: from the Taylor model to the advanced Lamel model, Int. J. Plasticity, 21, 589, 2005. [5] Hosford, W.F., The mechanics of crystals and textured polycrystals, Oxford Univ. Press, 56, 1993. [6] Morita, M. and Umezawa, O., Slip deformation analysis based on full constraints model for α-type titanium alloy at low temperature, Journal of Japan Institute of Light metals, 2, 60, 61-67, 2010. [7] ION, S.E., Humphreys, F.J. and White, S.H., Dynamic Recrystallization and the development of microstructure during high temperature deformation of magnesium, Acta metal., 30, 1909, 1982. [8] Hutchinson W.B. and M.R. Barnett, Effective values of critical resolved shear stress for slip in polycrystalline magnesium and other hcp metals, Scripta Meter., 63, 737, 2010. [9] Akhtar, A. and Teghtosoonian, E., Solid solution strengthening of magnesium single crystals- I Alloying behavior in basal slip, Acta Metal., 17, 1339, 1969. [10] Akhtar, A. and Teghtosoonian, E., Solid solution strengthening of magnesium single crystals- II The effect of solution on the ease of prismatic slip, Acta Metal., 17, 1351, 1969. [11] Helis, L., Behavior of deformation and texture formation of AZ31 and AZ61 magnesium alloys at high temperatures, Ph.D. Thesis, Yokohama National University, 2006. [12] Dillamore, I.L. and Katoh, H., The mechanisms of recrystallization in cubic metals with particular reference to their orientation-dependence, Mat. Sci.Tech., 8, 73, 1974.

Numerical Solution of the Walgraef-Aifantis Model for Simulation of Dislocation Dynamics in Materials Subjected to Cyclic Loading José Pontes∗ , Daniel Walgraef† and Christo I. Christov∗∗ ∗

Metallurgy and Materials Engineering Department, Federal University of Rio de Janeiro, P.O. Box 68505, 21941-972, Rio de Janeiro, RJ, Brazil † Center for Nonlinear Phenomena and Complex Systems, CP-231, Université Libre de Bruxelles, B-1050, Brussels, Belgium ∗∗ Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA, 70504-1010, USA Abstract. Strain localization and dislocation pattern formation are typical features of plastic deformation in metals and alloys. Glide and climb dislocation motion along with accompanying production/annihilation processes of dislocations lead to the occurrence of instabilities of initially uniform dislocation distributions. These instabilities result into the development of various types of dislocation micro-structures, such as dislocation cells, slip and kink bands, persistent slip bands, labyrinth structures, etc., depending on the externally applied loading and the intrinsic lattice constraints. The Walgraef-Aifantis (WA) (Walgraef and Aifanits, J. Appl. Phys., 58, 668, 1985) model is an example of a reaction-diffusion model of coupled nonlinear equations which describe 0 formation of forest (immobile) and gliding (mobile) dislocation densities in the presence of cyclic loading. This paper discuss two versions of the WA model, the first one comprising linear diffusion of the density of mobile dislocations and the second one, with nonlinear diffusion of said variable. Subsequently, the paper focus on a finite difference, second order in time Cranck-Nicholson semi-implicit scheme, with internal iterations at each time step and a spatial splitting using the Stabilizing, Correction (Christov and Pontes, Mathematical and Computer 0, 35, 87, 2002) for solving the model evolution equations in two dimensions. The discussion on the WA model and on the numerical scheme was already presented on a conference paper by the authors (Pontes et al., AIP Conference Proceedings, Vol. 1301 pp. 511-519, 2010). The first results of four simulations, one with linear diffusion of the mobile dislocations and three with nonlinear diffusion are presented. Several phenomena were observed in the numerical simulations, like the increase of the fundamental wavelength of the structure, the increase of the walls height and the decrease of its thickness. Keywords: Finite differences, pattern formation, dislocation patterns, fatigue PACS: 05.45.-a, 02.70.Bf, 62.20.me, 46.70.-p

THE WALGRAEF-AIFANTIS (WA) MODEL In the spirit of earlier dislocation models derived for example by Ghoniem et al. (1990) [1] for creep, or by Walgraef and Aifantis (1985 [2], 1986 [3], 1997 [4]), by Schiller and Walgraef (1988 [5]), and by Kratochvil (1979) [6], for dislocation microstructures formation in fatigue, the dislocation population is divided into static dislocations, which may result from work hardening and consist in the nearly immobile dislocations of the “forest”, of sub-grains walls or boundaries, etc., and the mobile dislocations which glide between these obstacles. The essential features of the dislocation dynamics in the plastic regime are, on the one

T. Proulx (ed.), Optical Measurements, Modeling, and Metrology, Volume 5, Conference Proceedings of the Society for Experimental Mechanics Series 9999999, DOI 10.1007/978-1-4614-0228-2_14, © The Society for Experimental Mechanics, Inc. 2011

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side, their mobility, dominated by plastic flow, but which also includes thermal diffusion and climb, and their mutual interaction process, the more important being (Mughrabi et al., 1979 [7]): Multiplication of static dislocations within the forest; • Static recovery in the forest via static-static annihilation processes; • Freeing of static dislocations: when the effective stress increases and exceeds some threshold, it disturbs the local structure of the forest and, in particular, destabilize dislocation clusters which decompose into mobile dislocations. The freeing of forest dislocations occurs with a rate β , which depends on the applied stresses and material parameters; • Pinning of mobile dislocation by the forest. Effectively, mobile dislocations may be immobilized by the various dislocation clusters forming the forest. The dynamical contribution of such processes is of the form G(ρs)ρm , where G(ρs) = gn ρsn is the pinning rate of a mobile dislocation by a cluster of n static ones. The WalgraefAifantis (WA) model considers n = 2. •

The resulting dynamical system may then be written as:

∂ ρs = Ds ∇2 ρs + σ − vs dc ρs2 − β ρs + γρs2 ρm ∂t ∂ ρm = Dm ∇2x ρm + β ρs − γρs2 ρm , ∂t

(1) (2)

where time is measured in number of cycles of loading, Ds represents the effective diffusion within the forest resulting from the thermal mobility and climb and Dm represents the effective diffusion resulting from the glide of mobile dislocations between obstacles (Dm ≫ Ds ). The coefficient dc is the characteristic length of spontaneous dipole collapse. β is the rate of dislocation freeing from the forest and is associated with the de-stabilization of dislocation dipoles or clusters under stress. Numerical dislocation dynamics simulations show that in BBC crystals, for 0, there is a critical value of external applied stresses above which dislocation dipoles become unstable. This value is a decreasing function of the distance between dipole slip lines. If the forest may be considered as an ensemble of dipoles with a mean characteristic width, the 0 stress for de-stabilization, or freeing, σ f , could be extracted from such simulations. More extended numerical analysis could include higher order dislocation clusters and provide the dependence of the threshold stress on the forest dislocation 0. The freeing rate should thus be zero below the freeing threshold, and an increasing function of the applied stress above it. Hence, β ≈ β0 (σa − σ f )n for σa > σ f , n being a phenomenological parameter.

THE MODIFIED WA MODEL: EFFECT OF GRADIENT TERMS The approximation of mobile dislocation diffusion is controversial and may be addressed. To do so, the mobile dislocation density, ρm is divided into two 1 representing the dislocation gliding in the direction of the Burgers vector (ρm+) and in the opposite one (ρm− ), with ρm = ρm+ + ρm− .

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For crystals with well-developed forest density, and oriented for single slip, we now write (with vg oriented along the x direction):

∂ ρs = Ds ∇2 ρs + σ − vs dc ρs2 − β ρs + γρs2 ρm ∂t ∂ ρm+ β = −∇x vg ρm+ + ρs − γρs2 ρm+ ∂t 2 ∂ ρm− β = ∇x vg ρm− + ρs − γρs2 ρm− , ∂t 2

(3) (4) (5)

or:

∂ ρs = Ds ∇2 ρs + σ − vs dc ρs2 − β ρs + γρs2 ρm ∂t ∂ ρm β = −∇x vg ρm + ρs − γρs2 ρm ∂t 2 ∂ σm− = −∇x vg ρm − γρs2 σm , ∂t

(6) (7) (8)

where σm = ρm+ − ρm− is the density of geometrically necessary dislocations. This variables evolves faster than the other two and may be adiabatically eliminated, leading to the following system, which includes a nonlinear diffusion term in the equation of ρm :

∂ ρs = Ds ∇2 ρs + σ − vs dc ρs2 − β ρs + γρs2 ρm ∂t vg ∂ ρm = ∇x 2 ∇x vg ρm + β ρs − γρs2 ρm . ∂t γρs

(9) (10)

THE NUMERICAL SCHEME FOR SOLVING THE WA MODEL In order to solve the modified WA model, we use a numerical scheme based on a one proposed by Christov and Pontes (2002). Equations (9) and (10) are solved numerically in two-dimensional rectangular domains, through the finite difference method, using a grid of uniformly spaced points, a second order in time Crank-Nicholson semi-implicit method with internal iterations at each time step, due to the nonlinear nature of the implicit terms. The proposed scheme is splitted in two equations using the Stabilizing Correction scheme (Christov and Pontes, 2002 [8], Yanenko, 1971 [9]). The first halfstep comprises implicit derivatives with respect to x and explicit derivatives with respect to y. In the second half-step,n the derivatives with respect to y are kept implicit and those with respect to x are explicit. The splitting scheme is shown to be equivalent to the original one.

100

The target scheme The target second order in time, Crank-Nicholson semi-implicit scheme is: n+1 + ρ n n+1 + ρ n ρsn+1 − ρsn n+1/2 ρs n+1/2 ρs n+1/2 s s = Λx + Λy + f1 ∆t 2 2 n+1 n ρmn+1 − ρmn n+1/2 ρm + ρm n+1/2 = Λ2 + f2 , ∆t 2

(11) (12)

where n is the number of the time step. Upon including the 1/2 factor in the operators n+1/2 n+1/2 n+1/2 Λx , Λy and Λ2 , we obtain:

ρsn+1 − ρsn n+1/2 n+1/2 n+1/2 = Λx ρsn+1 + ρsn + Λy ρsn+1 + ρsn + f1 ∆t
ρmn+1 − ρmn n+1/2 n+1/2 = Λ2 ρmn+1 + ρmn + f2 . ∆t n+1/2

The operators Λx defined as: n+1/2 Λx n+1/2

Λy

n+1/2

f1

n+1/2

Λ2

n+1/2

f2

= = = = =

n+1/2

, Λy

n+1/2

and Λ2

n+1/2

and the functions f1

n+1/2

and f2

 n+1  Ds ∂ 2 1 ρs + ρsn β − vs dc − 2 2 ∂x 4 2 4  n+1  2 n Ds ∂ 1 ρs + ρs β − − vs dc 2 2 ∂x 4 2 4  n+1  2
γ ρs + ρsn σ+ ρmn+1 + ρmn 2 2 " # 2  n+1 vg 1 ∂ ∂ ρs + ρsn vg − γ 2 ∂ x γ  ρsn+1 + ρ n
/22 ∂ x 2 s  n+1  ρs + ρsn β . 2

(13) (14) are

(15) (16) (17) (18) (19)

Internal iterations n+1/2

n+1/2

n+1/2

n+1/2

Since the operators Λx , Λx and Λ2 , as well as the functions f1 and n+1/2 f2 contain terms in the new stage, we do internal iterations at each time step, according to:     ρsn,k+1 − ρsn n+1/2 n+1/2 n+1/2 = Λx ρsn,k+1 − ρsn + Λy ρsn,k+1 − ρsn + f1 (20) ∆t   ρmn,k+1 − ρmn n+1/2 n+1/2 = Λ2 ρmn,k+1 ρmn + f2 . (21) ∆t

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where the superscript (n, k + 1) identifies the “new” iteration, (n, k) and n stand for the values obtained in the previous iteration and in the previous time step, respectively. The n+1/2 n+1/2 n+1/2 n+1/2 n+1/2 operators Λx , Λy , Λ2 and the functions f1 and f2 are redefined as: n+1/2

Λx

n+1/2

Λy

n+1/2

f1

n+1/2

Λ2

n+1/2

f2

Ds ∂ 2 1 β − vs dc Sn+1/2 − 2 2 ∂x 4 4 2 Ds ∂ 1 β = − vs dc Sn+1/2 − 2 2 ∂x 4 4  γ  n+1/2 2  n,k n = σ+ S ρm + ρm 2 !  2 vg ∂ ∂ n+1/2 =
2 vg − γ S ∂ x 2γ Sn+1/2 ∂ x

(22)

=

= β Sn+1/2 ,

where: Sn+1/2 =

(23) (24) (25)

ρsn,k + ρsn . 2

(26)

The iterations proceed until the following criteria is satisfied: max||ρsn,K+1 − ρsn,K || max||ρsn,K ||

1000oC), and has a diameter of 6 inches producing 150 HP. An initial recording of the blade is made in a holographic recording system. The CCD camera captures the interference fringes between the reference image and the image of the vibrating blade produced by the stroboscopic illumination in real time. The blade is excited with the shaker and the corresponding resonant modes are observed in real time. The process previously outlined to obtain strains from holographic moiré patterns was utilized. Since stroboscopic illumination was utilized sinusoidal fringes where recorded. Displacements and strains were obtained by applying double illumination in two orthogonal directions. Since the determination of the local strains in the blades requires the changes of coordinates, it is necessary to get the profile of the blades. A contouring method based on the rotation of the illumination source [7], [25] was used to get the contour of the blade. From the geometrical information all required direction cosines needed to perform the change of coordinates from the global to the local coordinates were computed. Figure 24 shows the holographic moiré patterns of carrier fringes for both displacements and contours. Figure 25 provides the principal stresses and isostatics caused by blade vibration.

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Figure 24. Vibration pattern carrier fringes recorded with stroboscopic illumination b) Carrier contour fringes used to obtain the geometry of the turbine blades Upper figure, shape of the blade, and global coordinate axis utilized Consequently moiré holography can provide complete information of a general 3-D surface subjected to loading.

Figure 25. Principal stresses and isostatics of the SRB-SPU turbine blade.

9.3 3-D Incoherent illumination

The utilization of double viewing system as shown in Figure 17, can be applied to get 3-D information in a similar way to what was described in the preceding sections. The recording of the double illumination in the unloaded and loaded conditions yields the contour of the deformed surface. Likewise in incoherent illumination the superposition of the recorded pattern in the unloaded and loaded conditions will provide the projected displacements. However the sensitivity vector for the general case will be changing from point to point and it will be necessary to utilize the point wise solution of three different projections to get the displacement vector.

Figure 26. Graphical representation of the process of modulation of a carrier

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Viewing with telecentric lenses produces an effect similar to a collimated illumination reducing the changes of the sensitivity vector to the z-direction when one wants to get the in-plane displacement. 10.0 Process to recover the information from recorded data.

Figure 26 shows a cross-section of the signal along the x-axis to simplify the visualization of the relationships between the different variables that must be considered [26]-[30]. In the theory of communications, a carrier wave, or carrier is a waveform (usually sinusoidal) that is modulated (modified) with an input signal for the purpose of conveying information. The carrier wave is usually of much higher frequency than the input signal. The purpose of the carrier is to encode information to transmit it. Phase modulation and amplitude modulation (AM) are used methods to modulate the carrier. In what follows phase modulation will be described. The carriers in the optical techniques applied to Experimental Mechanics can be gratings printed on the surface under analysis. In the case of surface contouring the carriers are projected lines. In methods like speckle or holography the carrier are extracted from surface features existing on the surfaces of the analyzed bodies. These features can also be artificially created. In what follows the analysis of a sinusoidal carrier will be introduced as a mathematical model for all types of carriers. One must remember that by utilizing Fourier transform methodology all the integrable functions can be represented by their expansion in Fourier integrals. The carrier can be thought of as a sinusoidal function generated by a rotating vector E and the phase of the carrier at a point of coordinate x is defined as the total angle rotated by the vector up to that point. Ψ (x) is the modulation function, a function that encodes the optical paths difference as an angular variable. The total phase of the modulated carrier is the addition of the phase generated by the constant rotation plus the modulation function contribution. The general equation for a modulated carrier is, I ( x ) = I oc + I 1 c cos (2 π f c x + δ ( x )

)

(26)

Where f c is the frequency of the carrier, a known quantity because it was introduced with certain spatial frequency that can be defined as f c = 1 p c . The presence of a carrier is required in certain optical methods. In other methods the introduction of a carrier may be useful because it can greatly simplify the process of data processing. Figure 26 shows that the phase change is a linear function of the coordinate x and the modulation function is added to this function, resulting in a total phase that corresponds to the modulated carrier. Hence from (26) φ( x, y) = ( 2πfc x + δ( x) ) = arccos

I( x, y) − Io ( x.y) I1 ( x, y)

(27)

Knowing φ( x, y ) one can get the modulation function. δ( x ) = φ( x ) − 2πf c x

(28)

From (26) to (28) and looking at Figure 26 it is possible to see that starting from a given sign convention the presence of a carrier defines the corresponding sign of the modulation function δ(x). This means that the carrier provides a reference frequency that removes the need of knowing where the zero reference order is. This is a problem that arises in the interpretation of fringe systems, as in Photoelasticity. There are several ways that one can determine the phase but all of them are based on the utilization of a trigonometric function that limits the phase retrieval from 0 to 2π. This leads to the process of unwrapping that although for smooth functions it works well, in actual applications it can present difficult practical problems of implementation. 11.0 Digital imagine correlation In DIC the optical process to obtain correlation between signals is replaced by digital procedures [31]-[34]. In DIC displacements are directly obtained from point trajectories and the process of fringe unwrapping is bypassed. The understanding and the interpretation of the basic aspects that relate phase and displacements [26]-[30], is a straightforward process. The theory behind DIC to relate displacements and light irradiances is more complex. DIC is a general technique to extract displacement information from recorded irradiance of deformed bodies. DIC is particularly useful when random carriers are utilized. Hence this particular application will be emphasized in this paper.

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In the application of two random carriers with DIC two random signal images are recorded and saved in the memory of a computer. From these two images a small subset is extracted, Figure 27. The subset contains a distribution of gray levels. The cross-correlation between the two subsets is computed. A correlation peak is produced; the position of the peak in the sub-set gives the local displacement of the subset. The height of the peak gives the degree of correlation or similarity of the gray levels of the initial and final configuration of the subset. If the cross-correlation has been normalized to the value of 1, values of the peak near one will indicate a good correlation. In the measure that this peak gets lower values the correlation degrades. Unlike moiré or speckle interferometry based on resolved patterns of irradiance levels, DIC is based on a subset of pixels. As a result, information of displacements inside the subset cannot be obtained. This aspect of DIC poses a problem of spatial resolution that must be considered in actual applications. Therefore the ratio of the pixels subset size to the overall size of the region under observation is a very important quantity that determines the spatial resolution of the obtained results. Summarizing, the measured displacements are the displacements of a subset.

Figure 27. Illustration of the cross-correlation of images. Figure 28 shows a speckle displacement field after all the different subsets of the field have been correlated and merged into trajectories. In one single sentence DIC provides the lines that are tangent to the trajectories of the points of a surface. From the trajectories one can extract the displacement field information. If one performs the above described process of correlation without additional corrections the displacement vectors will have random variations both direction and magnitude from subset-to-subset. In the DIC literature there is a large variety of approaches to the solution of this problem. DIC heavily depends on knowledge based information to introduce corrections to the recovered displacements. These different optimization procedures can be subdivided in two basic groups, methods that operate in the actual space and methods that utilize the FT space, [26]-[30],[35]-[38].

Figure28. Displacement field obtained from a speckle pattern. A certain region of a deformed surface is analyzed; this region has experienced rigid body translations and rotations due to the deformation of the rest of the body that the observed patch belongs to, plus a local deformation; the object of DIC to obtain the local deformation. One has a given surface that for the sake of simplicity is assumed to be a plane and is viewed in the direction normal to the surface. Furthermore it is assumed that a telecentric system is used to get the image of the surface. In this way it is possible to separate the problems that were analyzed in some detail in preceding sections concerning the image formation, from the problem of image correlation. In this surface one has a certain distribution of intensities that it will be assumed corresponds to the random signal incorporated to the surface and is represented by a function Ii(x,y). A

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displacement field is applied to the surface and a final distribution of intensities If(x,y) is obtained. It is assumed that the light intensity changes are only a function of the displacement field and as it is the case in all experimental methods noise is present. Noise is indicated as all the changes of intensity that are not caused by the displacement field. The displacement field is defined by the function [39], [41], ∧

∧

(29)

D( x , y ) = u ( x , y ) i + v ( x , y ) j

From the preceding assumption, (30) In (22) ΔI is the change of intensity caused by the rigid body motion plus the local deformation of the analyzed surface. In (30) the assumption that the light intensity is modified only by the displacements is implicit. The term In refers to all other causes of change of intensity. The validity of (30) boils down to the signal to noise ratio. To develop the model one has to postulate that the signal content of In is small and hence can be neglected. The problem to be solved is to find u (x,y) and v (x,y) knowing Ii(x,y) and If(x+u,y+v). The solution of the above problem requires the regularity of the functions u(x,y) and v(x,y) implicit in the Theory of the Continuum. One can formulate the problem as an optimization problem, that is find the best values of these two functions that minimize or maximize a real function, the objective function of the optimization process. There are many criteria that can be utilized for this purpose. One criterion is the minimum squares; the difference of the intensities of the two images must be minimized as a function of the experienced displacements. Calling Φ (u,v) the optimization function. I f ( x i + u, yi + v) = Ii ( x i , y i ) + ΔI( x i +Δu, y i + Δv) + I n

Φ ( u , v) =

∫∫ [I (x f

i

+ u, y i + v) − I i ( x i , y i )]2 dxdy

(31)

For small u(x,y) and v(x,y) the above expression can be expanded in a Taylor series and limiting the expansion to the first order and using vectorial notation as,

Φ(D) =

∫∫ [I (r) − I (r) + D(r) • ∇I f

i

f

(r )]2 dxdy

(32)

In the above equation r is the spatial coordinate; D(r) is the displacement vector and ∇ is the gradient operator. Equation (32) tells us that the gradient of If provides the following information; the displacement information is associated with the gradient of the intensity distribution. If is a scalar function (light intensity), the gradient is a vector and going back to Figure 28 the vectors displacements are plotted following the vectors joining the centers of correlation peaks of the sub-images. Hence the displacement information can be retrieved following the gradient function of the light intensity.The minimization of the objective functions is then a central problem of the image digital correlation technique. In the technical literature there is a large variety of approaches to this problem. One can utilize criteria other than the minimum squares [42]-[44]. u

(b) (a) Figure 29. Field for the correlation process. (a) Dotted rectangle, NsxNs sub-element, δ mesh of the region of interest. (b) Displacement experienced by the sub-image with components u and v.

174

Let us now look at the overall procedures that are necessary to obtain the displacement field. The region of interest is symbolically represented in Figure 29 by a square region. Figure 29 (a) shows a scheme of computation. There is a region of interest, the big square; in one corner there is a sub-element that has a chosen size of Ns×Ns pixels and the raster of dots indicate the position of the centroids that form a regular mesh of δ×δ pixels. Figure 29(b) shows how the sub-image is displaced and distorted after the deformation of the sample has taken place. By utilizing the model adopted in (32) and operating in the coordinate-space it is possible to define the vectors displacement in the region of interest, as shown in Figure 28 and 29.Two images are being compared. The reference image represented in Figure 29(a), by square image of Ns×Ns pixels, and the second one, called the deformed image, represented by the distorted square. The operator chooses the size of the zones of interest, the sub-sample by setting the size Ns so that Ns×Ns pixels are considered. To map the whole region of interest, the second parameter to choose is the separation δ between two consecutive sub-samples. The parameter δ defines the mesh formed by the centers of each sub-sample used to analyze the displacement field (Figure 29). Different strategies can be applied to retrieve the full field. Let us concentrate in the fundamental operation, the extraction of the information from a sub-sample. This aspect of the problem will be covered by utilizing an approach that is followed by a large number of contributors to this method.

∑∑ [I ( x , y ) − I ( x , y )] m

CN =

m

'

i

i

m

m

j

f

i

i =1 j=1

∑∑ I

2

i

'

2

j

(33)

( x'i , y' j )

i =1 j=1

The process begins with a discrete and normalized version of (32).The deformed coordinates are obtained from the initial coordinates by Taylor’s series expansion, ∂u ∂u dx + dy ∂x ∂y ∂v ∂v y' = y + v + dx + dy ∂x ∂y

x' = x + u +

(34) (35)

For example the Taylor’s series is terminated in the first order. Although higher orders can be introduced, it is easier and more convenient to explain the basic ideas of this particular approach to DIC by utilizing the first order. The meaning of the above equation can be better grasped by looking at Figure 29, where u and v contain components of the rigid body displacement of the sub-sample, and the derivatives express the effect of the local deformations in the displacement field. To make sure that the distribution of intensities in one subset is continuous and has continuous derivatives the light distribution, I(x,y) is interpolated utilizing (i.e. bicubic-spline) an expansion of the light intensity, as chosen by many authors that have contributed to DIC. The relationship between the displacement field and the gradient of the intensity field comes from (32). This equation indicates that the displacement field is associated with the gradient of the intensity field. To get displacement information from the image intensity distribution one replaces the bicubic spline expression in a normalized expression of (33). After this substitution the optimization of (33) requires the solution of a non linear system of equations. This brings additional complications but there are many methods that were developed and can be applied in this case. There is a large variety of software packages for DIC. These packages depend fundamentally on the specific choices of the correlation coefficient C defined in this paper by (33). The next step is to select a function that defines the displacement field in a subset. This function is called the shape function ϕ, and on the optimization algorithms and interpolation functions that are needed to compute sub-pixel displacements from images that were obtained with specific pixel resolutions. One very important aspect that is quite often not referred to in the literature is that no matter how complex your algorithm is no gain of information can be achieved if this information does not already exist in your primary data, the gray levels. These levels depend on satisfying the Nyquist condition in connection to both the frequencies recovered and on the sampling of the gray levels by the camera sensor. Summarizing the first basic concept that is clearly shown in Figure 27, the comparison of the distribution of gray levels coming from two images (initial and final) provides a measure of the mechanical displacements experienced by a surface. The analysis of the intensity distribution is done on sub-set images and following the structure of electronic image sensor these sub-images are squares of Ns× Ns pixels. This is the basic foundation of DIC that separates it from the other methods that measure displacements.

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The second basic development is connected with the description of the displacement field in the sub-set. This second basic aspect in DIC heavily depends on knowledge based information; a function ϕ is introduced that describes the displacement field of the sub-set domain; following the nomenclature of Finite Element, ϕ is called the shape function. There are several shape functions ϕ utilized in DIC: ϕ constant that corresponds to a rigid body motion of the sub-image; ϕ linear or affine transformation, ϕ quadratic and it is possible to include higher orders. The next fundamental development is embodied in (33-35), that relate the optical flow to the kinematic variables that depend on the choice of ϕ and can be represented by a vector P. Having posed the problem in terms of optical and mechanical variables the next step is to relate both set of variables. This is an inverse problem, which says knowing Ii(x,y) and If(x,y) find ϕ, that is determine the vector P that best accounts for the observed optical flow. This connection between the two sets of variables is represented by (32) and is embodied in (33). This equation implies two choices, the first choice is utilization of a truncated Taylor’s expansion of the displacement field to the first order term or higher order terms. The second choice is the selection of minimum squares criterion for the optimization procedure implicit in (31). This is the approach followed by the majority of the authors in the field and for most of the commercial packages that are available. However as pointed out before there are other optimization mechanisms that can be utilized. The theoretical framework described above to connect displacements to light intensity is unique to DIC and separates it from all the other techniques that were previously described. The inverse problem is formulated, the main variables set up, and the next step is to solve the optimization problem. The problem is formulated in terms of minimum squares hence it is a non linear problem. The symbol Φ ( P ) represents the solution of the problem. The solution is the sum of the leading term, plus additional terms: P( l ) indicates a linear approach to the displacement vector, P(q) indicates a quadratic solution and one can utilize successive higher order terms.

Φ( P) = P0 (C) + P(l) + P(q ) + ...

(36)

In (28) P 0 (C) , indicates a constant, P(l) indicates a linear term, P(q) indicates a quadratic term. The higher order terms of the power series become smaller as the order of the terms increase. This optimization is achieved utilizing nonlinear iterative optimization algorithms, such as first gradient descent, Newton-Raphson, or Levenberg-Marquard. To summarize DIC in a few sentences, although the actual approach to the solution of obtaining displacements from light intensity is complex and requires a number of choices, the actual choices are made by the developer of the software. Once a package of software is put together the operation of the software is pretty much automatic. This has made DIC a very popular choice for experimental mechanics. Users should be cautious however that the Nyquist condition must always be satisfied otherwise the results obtained will have no value. 11.1 Displacement and spatial resolution of DIC. There are two important aspects of DIC as applied to what is basically a speckle photography method. These two aspects are not specific to DIC as a method to retrieve displacements but are relevant to the currently prevailing methodology applied to the so called white speckles, the resolution in the measurement of displacements, and the spatial resolution. The resolutions in the measurement of displacements of the other techniques that have been considered in this paper depend on the pitch or equivalent pitch of the basic carrier that records the displacements. The application of the Nyquist condition tells us that the maximum spatial frequency that can be retrieved is half the frequency of sampling carrier.

In moiré the sampling depends on the pitch p of the carrier, in speckle interferometry the sampling frequency is given by the sensitivity equation (10) and in photographic speckle by equation (15) that defines the equivalent pitch. It is also required that these frequencies are recorded by the sensor of the camera that must have a spatial frequency twice the frequency of the carrier. This subject is not addressed in many papers in the DIC literature applied to white light speckles but it is an important parameter in the displacement resolution of DIC as in all other methods utilized in pattern analysis. Figure 32 (a), [40], illustrates the definition of the equivalent of the speckle radius as the distance from the center of the correlation to the point of one half of the intensity called r. These definitions are statistical and give a statistical estimate of the minimal distance between spots that can be considered as measurable in the selected sub-domain. Figure 30 (b) illustrates the definition [40] of the equivalent of the speckle radius as the distance from the center of the correlation peak to the point of one half of the intensity called r. The values of r are utilized to define a fine pattern with r slightly larger than one pixel, medium with the radius r=2 pixels and coarse 4 pixels. If one assumes that the minimum distance that can be measured corresponds to the distance of two points that can be separated for a fine pattern the spatial resolution will be 2 pixels, for a medium pattern 4 pixels and for a coarse pattern 8 pixels. These quantities then provide the

176

maximum displacement resolution that statistically can be achieved for the corresponding patterns. This is a point that should be clearly understood in the application of DIC to white light speckles. Signal processing laws are laws that apply to all the type of signals that are utilized independently of the algorithms that one can introduce. Concerning the spatial resolution studies that are described in [40] show that in all the different program utilized in the DIC studies the displacement field inside the sub-set is not defined. Hence the number of the sub-set pixels related to the total number of pixels of the observed region provides a measure of the effect of the subset size in the spatial resolution

Figure 30. (a) Fine, medium speckle sizes defined as the radius of the autocorrelation factor at 50% of the intensity [40].

12. Discussion and conclusions

All the OTD methods have potentially the same capability to perform the different operations required to measure displacement either in 2-D or in 3-D and to retrieve shape information. Basically the OTD methods can be separated in two basic categories, techniques that utilize deterministic signals and methods that utilize random signals. Within the techniques that utilize random signals there are two basic types: a) Techniques that use random signals produced by the pattern of interference generated by random surface roughness and b) Artificially generated random patterns or random patterns existing already on the surface from sources other than random interference patterns. The basic difference between utilizing deterministic and random signals is the final signal-to-noise ratios and the decorrelation phenomenon that is caused by statistical structure the wavefronts of random signals. At the same time the signals produced by all the techniques depend on whether the light is coherent or incoherent. The relationship between signals and displacements or metrology is independent of the light coherence, only the range of application is affected by the degree of coherence of the light. Both speckle interferometry and moiré interferometry can reach very high accuracy because changes of phase of 2π correspond to the wavelength of light λ. Hence one can arrive to the 10 nm range in the displacement measurement [12]. Table 1 1

2

x

Theory 10-6 166.497

0.250 Δ

3 p μm 0.365 10-6 165.000 1.49

4 p μm 0.413 10-6 163.162 3.35

5 p μm 0.492 10-6 162.681 3.81

6 p μm 0.635 10-6 164.021 2.47

7 p μm 0.925 10-6 166.335 0.162

8 p μm 1.22 10-6 166.396 0.10

Despite the possible sources of errors that speckle interferometry may have, it has been verified that in a disk under diametrical compression the computed strains obtained from speckle interferometry, Table 1, are within an error less of 1 % compared to the theoretically computed values. In a disk under diametrical compression it is known that the strains theoretically computed with an ideal concentrated load and the actual case, a disk with a narrow region of contact stresses, experimental and theoretical strains are approximately equal at points located around ¼ of the diameter, Similar agreement in strain values were observed for holographic moiré, [20], with errors on the order of 1 %.

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Incoherent techniques apply to larger deformations, both moiré and speckle photography can be applied to a very large spectrum of deformations and specimen sizes. In all cases the most important factor is the quality of the signals that encode the displacement information. Utilizing the speckle pattern method, [45] the displacements and strains along the diameter of a disk under diametrical compression were determined. Data were obtained for the same load with six different carrier pitches, from 1.22 to 0.365 microns (Table 1). All these carrier frequencies satisfy the Nyquist condition both for the required sampling frequency of the carrier and the required sampling of the utilized sensor. The final result indicated that the accuracy achieved in displacements and strains is the same when the Nyquist condition is satisfied, regardless of the carrier frequency utilized. These studies resulted in the formulation of the following principle similar to the Heisenberg indetermination principle of signal analysis [46], (37)

ΔI s Δf s = C

In the above equation ΔIs is the minimum detectable gray level calling Is the maximum amplitude of the available gray levels. The gray levels in a CCD camera or similar devices are quantized and the maximum theoretical dynamical range (amplitude of the vector) is one half of the total number of gray levels 2 n (for n=8, I = 128 ). The actual dynamic range is smaller than this quantity and Δf is the maximum detectable sampling frequency. 50 45

gray levels

40

Heisenberg Principle

35

Experimental data

30 25 20 15 10 5 0 0

20

40

60

80

100 120 140 160 180 200 220 240

Δ fs Figure 31.Plot of the experimental data that provides a numerical expression for the Heisenberg equation (37). The quantity Δf is defined as: Δf s =

p Δu m

(38)

The practical question to be answered is: what is the minimum displacement information that can be recovered within fringe spacing δ ? Where δ is a fringe wavelength; it is evident that there is a finite limit to the subdivision of the fringe spacing. The constant C reflects the whole process to obtain displacement information. The constant C is a function of the optical system, the device used to detect the fringes (CCD camera) and the algorithms used to get the displacement information. There are many important practical consequences of the principle formulated in (37). This equation is a valuable tool for planning experiments involving fringe analysis. Once the Whittaker-Shannon theorem is applied and the required minimum frequency of the carrier is computed, the next step is to select the carrier that is going to be used. In order to obtain frequency and displacement information it is necessary to maximize the amount of energy levels to encode this information. This implies that the largest portion of the dynamic range of the encoding system should be used to store useful information. By doing this the amount of noise in the signal is minimized. An immediate consequence is the need to increase the visibility of

178

the fringes within the range of options available. Consequently when selecting a carrier the Optical Transfer Function (OTF) and the (MTF), the modulus of the (OTF), of the whole system used to encode the information needs to be taken into consideration. Figure 31 clearly shows the effect of encoding information in gray levels. If it is possible to detect close to 5 of the available 128 gray levels, it is possible to get displacement information that is 1/200 of the spatial frequency of the signal or fringe pitch p. On the other extreme if the minimum detectable gray levels is 45 out of the 128, only 1/20 of the pitch p can be recovered. This is a basic law in the process of encoding displacement information and it is independent of the particular method utilized for fringe analysis. Consequently whether the illumination is coherent or incoherent the recovery of information is governed by (37). Both incoherent light moiré, speckle photography and the white light speckle are particularly suited for the measurement of large displacements. What is called large is relative to the actual field of view. There is a relationship between the actual physical size of the analyzed area and of the sampling frequency required to observe the displacement field in the area. The smaller the area the higher the required sampling frequency and vice versa. To reduce gray levels intensity there are presently two methods, one method has its foundations on the classical analysis of signals as was developed the Theory of Communications. To avoid problems arising from intensity based analysis and following a common trend in optics the notion of phase is utilized. The other option is the digital image correlation and it utilizes the irradiance as expressed in gray levels in a different form. It relates the displacement vector to the changes in irradiance. To put it in perspective the objective of both methods is to obtain from gray levels a vector field that depends on a tensor field (either the strain tensor or the stress tensor). The classical fringe analysis processing technique operates through projection of the vector displacement in two Cartesian components; let us say u and v which are separately determined although they are components of one entity, the displacement vector. Since the basic selected variable is the concept of phase and hence utilizes trigonometric variables. This particular selection of variables leads to a problem, fringe unwrapping. Fringe unwrapping is based on a simple concept, but the difficulty is in the implementation of this concept, due to presence of noise in the signal that are processed. This is particularly true if one is dealing with random signal as is the case with speckle patterns. Utilizing random signals as a carrier of information another important problem must be faced, the decorrelation problem; DIC bypasses these two obstacles. As shown in Figure 29 DIC searches directly for the displacement vectors in the field and operates directly with intensities as shown in (31) by relating the displacement vector to the changes of the intensity field. In the actual implementation of this approach there are a large number of functions that need to be introduced and optimized and choices must be made in the selection of these functions and in the optimization processes. As said before the choices of functions and optimization processes are made by the software developer; once a package of software is assembled the operation does not require intense involvement of the user. In a few words and utilizing general conclusion presented in [41] DIC is particularly suitable for the observation of displacement fields where the selected pixel size of the subsets Ns is a small quantity compared to the total number of pixels of the observed region, strains are large and low order shape functions can be utilized. It is possible to say that DIC has made a variation of speckle photography, white light speckles, a practical tool in many technical problems of mechanics of materials.

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[19] Gilbert J.A., Sciammarella, C.A.,and Chawla S.K., Extension of Three Dimensions of Holographics-Moiré Technique to Separate Patterns Corresponding to Components of Displacement, Experimental Mechanics, Vol. 18 (9), September 1978. [20] Sciammarella C.A., and Chawla S.K, A lens Holographic-Moiré Technique to Obtain Components of Displacements and Derivatives, Experimental Mechanics, Vol. 18 (10), pg. 373, October 1978. [21] Sciammarella, C.A., Ahmadshahi, M, A .,A Computer Based Holographic Interferometry to Analyze 3-D Surfaces,” Proceedings of IMEKO XI World Congress of the International Measurement Confederation, 167-175, Houston, October 1988. [22] Sciammarella C.A., and R. Naryanan, The Determination of the Components of the Strain Tensor in Holographic Interferometry Experimental Mechanics", Vol. 24, No. 4, December 1984. [23] Sciammarella C.A., and M. Ahmadshahi, Computer Aided Holographic Moire Technique to Determine the Strains of Arbitrary Surfaces Vibrating Resonant Modes”, Proceedings of the 1989 Spring Conf. on Exp. Mech., Boston, Massachusetts, May-June, 1989. [24] Sciammarella C.A., and M. Ahmadshahi, Non-Destructive Evaluation of Turbine Blades Vibrating in Resonant Modes”, Moiré Techniques, Holographic Interferometry, Optical NDT and Application to Fluid Mechanics, Fu-Pen Chiang, Editor, Proceedings of SPIE, Part Two, Vol. 1554B, 1991. [25] Sciammarella. C.A., Computer-aided holographic moiré contouring”. Optical Engineering, Vol. 39, 99-105, 2000. [26] Sciammarella, C.A., "Basic Optical Law in the Interpretation of Moiré Patterns Applied to the Analysis of Strains L" EXPERIMENTAL MECHANICS, 5, 154-160 (1965). [27] Sciammarella, C.A., "A Numerical Technique of Data Retrieval from Moiré or Photoelasticity Patterns," Pattern Recognition Studies, Ptvc. SPIE,18, 92-101 (I969). [28] Sciammarella, C.A., "Moiré Analysis of Displacements and Strain Fields", Applications of holography in mechanics, edited by W. G. Gottenberg, The American Society of Mechanical Engineers 1971. [29] Sciammarella, C.A.,and M.A. Ahmadshahi, Determination of fringe pattern information using a computer based method, Proc.8th International Conference on Experimental Stress analysis, Amsterdam, the Netherlands, H,Weringa, editor, Martinus Nijhoff Publisher, ,pp59-368, 1986 [30] Sciammarella C.,A., Fast Fourier Transform Methods to Process Fringe Data, Basic Metrology and Applications, G.Barbato, editor, Levrotto & Bella,Publishers, Torino, 1994. [31] W. H. Peters and W. F. Ranson, "Digital imaging techniques in experimental mechanics," Opt. Eng. 21, 427-431 (1982). [32] Sutton MA , Wolters WJ, Peters WH , Ranson WF, McNeill SR. Determination of displacements using an improved digital correlation method. Image and Vision Computing,Elsevier; I(3): 133-139, 1983. [33] Sutton MA, McNeill SR, Jang J, Babai M. The effects of subpixel image restoration on digital correlation error estimates. Optical Engineeering ;27(3): 173-175, 1988. [34] Sutton MA, Cheng M, McNeill SR, Chao YJ, Peters W.H., Application of an optimized digital correlation method to planar deformation analysis. Image and Vision Computing,Elsevier, ;4(3):143-150, 1988 [35] Chen, D.J., Chiang, F.P., Tan, Y.S., Don, H.S., Digital Speckle-Displacement Measurement Using a Complex Spectrum Method. Appl. Opt. 32, 1839-1849,1993 [36] Chiang, F.P., Wang, Q., Lehman, F., New Developments in Full-Field Strain Measurements Using Speckles. In Non-Traditional Methods of Sensing Stress, Strain and Damage in Materials and Structures, ASTM, Philadelphia (USA), STP 1318, 156- 169,1997. [37] Chiang, F.P., Wang, Q., Lehman, F., New Developments in Full-Field Strain Measurements Using Speckles. In Non-Traditional Methods of Sensing Stress, Strain and Damage in Materials and Structures, ASTM, Philadelphia (USA), STP 1318, 156- 169,1997. [38] Sjödahl,M. Digital Speckle Photography, Trends in optical non destructive Testing,Rastogi, P.K. and Daniele Inaudi, Editors, Elsevier, 2000. [39] Hild, F., and S. RouX, DIGITAL IMAGE CORRELATION:FROM DISPLACEMENT MEASUREMENT TO IDENTIFICATION OF ELASTIC PROPERTIES, strain, 42, 2 ,69-88,may 2006 [40] S Roux, J Réthoré and F Hild, Recent Progress in Digital Image Correlation: From Measurement to Mechanical Identification, 6th International Conference on Inverse Problems in Engineering: Theory and Practice IOP PublishingJournal of Physics: Conference Series 135 (2008). [41] M. Bornert et al, Assessment of Digital Image Correlation Measurement. Errors: Methodology and Results, Workgroup “Metrology” of the French CNRS research network 2519,“Mesures de Champs et Identification en Mécanique des Solides. November 24, 2008. [42] Hubert, P.J. Robust Statistics. Wiley, New York (USA). 1981. [43] Black, M., Robust Incremental Optical Flow. PhD dissertation, Yale University.1992 [44] Odobez, J.-M., Bouthemy, P. Robust multiresolution estimation of parametric motion models. J. Visual Comm. Image Repres. 6, 348365,1995. [45] Sciammarella C.A., Bhat G. K., and A. Albertazzi, Analysis of the Sensitivity and Accuracy in the Measurement of Displacements by Means of Interferometric Fringes”, Hologram Interferometry and Speckle Metrology, Proceedings of SEM, 1990.

180 [46] Sciammarella C.A., and F, M. Sciammarella, Heisenberg principle applied to the analysis of speckle interferometry fringes”. Optics and Lasers in Engineering, Vol. 40, 573-588, 2003.

Studying phase transformation in a shape memory alloy with full-field measurement techniques D. Delpueyo, M. Grédiac, X. Balandraud, C. Badulescu Clermont Université Université Blaise Pascal & IFMA, EA 3867, Laboratoire de Mécanique et Ingénieries BP 10448, F-63000 Clermont-Ferrand, France

ABSTRACT This paper deals with phase transformations that occur in a Cu-Al-Be single-crystal specimen of shape memory subjected to a tensile test. Two different techniques are used to experimentally evidence these phase transformations: the grid method to obtain strain maps and infrared thermography to deduce heat source distributions from temperature fields. Some typical strain and heat source maps obtained during the loading and unloading phases are discussed and interpreted. INTRODUCTION Many studies aimed at studying martensitic microstructures that appear in shape memory alloys (SMA) are available in the literature. Classic means such as microscopes are generally employed to observe them, but the recent development of full-field measurement techniques has made it possible to observe phase appearance and transformation during mechanical tests. The spatial distribution of the phases on the surface of the specimens can be observed by analyzing the contrast in the strain or in the heat source maps. This is due to the fact that the strain level generally varies from one phase to each other and that first-order phase transformation is a phenomenon that is accompanied by latent heat. In Refs. [1-2] for instance, digital image correlation has been used to observe phase transformation in SMA specimens. Phase transformation is accompanied with latent heat that can be deduced from temperature variation fields measured with infrared cameras. This property has been used in Ref. [3] for instance to study phase transformations by combining infrared thermography with digital image correlation. The aim of the current work is to analyse the response of a SMA specimen subjected to a tensile test using two different full-field measurement techniques: the grid method and infrared thermography. These techniques are complementary since they provide strain and temperature maps, respectively. These maps can be determined at different steps of the load, thus enabling us to analyse the evolution of these quantities when the applied stress increases. These two techniques are described in the first part of the paper. Some typical strain and heat source maps obtained during a tensile test performed on a Cu-Al-Be single-crystal specimen are then shown and discussed. FULL-FIELD MEASUREMENT TECHNIQUES USED Grid method The grid method consists first in depositing a crossed grid on the surface under investigation in order to track the evolution of the grid as loading increases and to deduce the 2D strain fields. The grid is deposited using the procedure described in [4]. The pitch of the grid is equal to 0.2 mm along both directions. Processing images of grids classically provides phase evolution maps of this periodical marking. This phase evolution is then

T. Proulx (ed.), Optical Measurements, Modeling, and Metrology, Volume 5, Conference Proceedings of the Society for Experimental Mechanics Series 9999999, DOI 10.1007/978-1-4614-0228-2_22, © The Society for Experimental Mechanics, Inc. 2011

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182 unwrapped and becomes directly proportional to the displacement [5]. Recently, it has been shown that the metrological performance of this technique could be significantly improved by getting rid of grid marking defects which unavoidably occur when grids are printed on their support [6-7]. In particular, a very good compromise is -4 obtained between resolution in strain and spatial resolution. Typically, the resolution is strain is nearly 10 for a spatial resolution equal to 30 pixels. In addition, calculations are performed pixelwise, thus allowing to detect very localized phenomena. In the current case, a 12-bit/1040x1376 pixel SENSICAM camera connected to its companion software CamWare is employed. The small strain maps are obtained directly from the images of the grids taken by the camera. Full details on small strain calculation can be found in [5][6] for unidirectional and crossed grids, respectively. Since large strains must be measured in the current work, the in-plane Green-Lagrange strain tensor E is calculated. In practice, small strain increments are measured using the procedure described in [7]. Assuming that local rotations are small, the Hencky strain tensor H is then deduced by adding these small strain increments. The Green-Lagrange E is finally deduced using the following relationship between E and H

H=

1 ln(I + 2E) 2

(1)

where I is the second-order unit tensor.

Temperature and heat source field determination with an infrared camera Very small temperature changes on the surface of the specimen under load (painted in black to increase thermal emissivity) are detected since the Noise Equivalent Temperature Difference (NETD) of the camera used in this study (a Cedip Jade III-MWIR featuring a 240x320 IR sensor matrix) is nearly 20 mK. For the thermomechanical analysis of materials, the temperature change is however not really the relevant information since it is the consequence of various phenomena among which phase transformation. Hence heat sources must be determined from these temperature fields using a suitable strategy whose main steps are as follows. The temperature evolution is governed by the bi-dimensional version of the heat diffusion equation, which is suitable for thin flat specimens [8]

⎛ dθ θ ⎞ ρC⎜ + ⎟ − kΔθ = s ⎝ dt τ ⎠

(2)

where θ (x, y, t) is the temperature variation with respect to a reference temperature field measured in practice just before the beginning of the test, s(x, y, t) is the heat source field produced by the material, ρ is the mass per unit volume, C the specific heat and k the thermal conductivity of the material, which is assumed to be isotropic. τ a time constant characterizing the heat exchange with ambient air by convection. This latter quantity is determined experimentally during a simple return to room temperature. In the current study where phase transformations occur, source s is mainly due to phase transformation. The objective here is to retrieve the heat source distribution in order to characterize the phase transformation throughout the specimen. Thermal images are processed using the same procedure as that described in [9]. By integrating in time and dividing by ρC, a field of “heat” expressed in °C is obtained. SPECIMEN PREPARATION AND TESTING CONDITIONS

The specimen under test was made of a Cu-Al-Be single-crystal SMA (dimensions: 0.94x17.78x72mm3). The test consisted of a loading-unloading uniaxial test at room temperature. During loading, the strain rate was equal to 0.064 %/s and the maximum strain reached was equal to 9%. The duration of the loading phase was 141.2 s and

183 the maximum stress reached was 73 MPa. The specimen was then unloaded back to zero stress, with a stress rate equal to -0.50 MPa/s up to 20 N. The experiments were performed two times to check that very similar responses were obtained. A typical stress-strain curve obtained during one of these three tests is shown in Figure 1. TYPICAL RESULTS AND DISCUSSION

Stress-strain curve

A typical stress-strain curve is shown in Figure 1. A classic plateau during which austenite transforms into martensite is clearly visible. A hysteresis loop also occurs. The idea is to observe the strain maps at some points chosen during the loading (points A, B and C in Figure 1) and loading phases (points D, E and F) of the curve in order to analyse the microstructure evolution and its link with the global mechanical response of the specimen. Thermal measurements are also analysed to establish the link that exists between strain and heat source maps.

Figure 1. Typical stress-strain curve obtained Loading phase

Three typical longitudinal strain maps collected during the loading phase are shown in Figure 2. At the beginning of the test, the specimen is completely composed of austenite which only slightly elastically deforms because of the loading which is applied. Some martensite needles first appear at Point A, which is located at the beginning of the plateau of the curve. Several parallel needles can be easily seen at the bottom part of the specimen as well as along its right-hand side. The strain amplitude in these very thin needles that is given in these figures is however probably not the actual one. It is in fact certainly underestimated because of the nature of the image processing which is used here to retrieve the strain maps. This procedure is based on a windowed Fourier analysis which cannot correctly identify very small events [6-7]. These local significant strain level increases are due to austenite/martensite phase transform which is accompanied by a sudden strain change in the maps, thus revealing the event that has occurred. A needle is very clearly visible at the top of the specimen (see Needle 11 on the Figure). The strain level in this needle is much higher than the strain level observed in the other needles. Its thickness is also greater. Point B is located at the middle of the plateau. About one half of the austenite has been transformed at this stage. Martensite clearly appears in the red bands in Figure 2-b. Comparing Needle 11 between Figures 1-a and 1-b clearly shows that the displacement of the austenite/martensite transformation front is greater at the bottom of the band than at its top. The strain level in the bands is not completely homogeneous. For instance, the strain level in Band 17 is greater along its right-hand side, thus illustrating the fact that martensite develops from the right to the left. The lower strain level at the left-hand side of this band means that there is a mixture of martensite and

184 austenite in this region. Some intermediate strain levels can be observed at the very top of the strain map at Point B, thus probably revealing that very thin martensite needles are developing in this zone at this stage of the test. The so-called habit planes correspond to the boundary between austenite and martensite. Most of the habit planes are parallel. The habit plane located at the bottom of band 17 exhibits a different orientation. This is probably due to the more complicated state of stress in this region because of the bottom grip which is very close. Point C is located at the very end of the loading stage. Almost all the austenite has been transformed into martensite at this stage. However, a residual region of pure untransformed austenite still remains in this zone (Band 28). The amplitude of the longitudinal strain is globally the same above and below this austenite band. However, comparing the transverse strain fields in these two regions leads to a significant difference between these two regions (the corresponding figures are not shown in this paper): -4% at the top and -6% at the bottom. This is due to the fact that the martensite variants are certainly not the same from one region to each other. The color shade in the martensitic zone is probably due to the fact that different martensite variants exist and that the twinning plane between these variant is nearly parallel to the habit plane. The austenite/martensite transformation is accompanied by a heat source localized in the region where this transformation occurs. This heat source can be detected by a suitable processing of the temperature variation maps captured by the infrared camera. Comparing strain maps and heat source maps shows that the phenomena are logically linked in the specimen.

a- Point A

b- Point B

c- Point C

Figure 2. Typical longitudinal strain maps measured during the loading phase Unloading phase

Some other strain maps taken during the unloading phase of the test are shown in Figure 3. Point D is located just after the sudden change of slope level observed in the unloading phase of Figure 3-a. Interestingly, the strain pattern is different of that observed during the loading phase. In particular, some X-shaped microstructures are clearly visible (see for instance Region 55). This particular microstructure corresponds to austenite appearance in the martensite block. The strain level is the lowest at the crossing between the two branches. It is approximately the same as the color in pure austenite in Figure 3-a above, thus showing that pure austenite has appeared here. The strain level in the other parts of these branches lies between the strain level in austenite and in martensite. This is certainly due to the fact that both phases are mixed in this zone. The strain level in Region 53 is lower than in the surrounding martensite and much greater than in the branches the X-shaped zone 55. This is due to the

185 fact that the percentage of austenite in this zone is certainly much lower than the percentage of martensite. These results are confirmed by thermal measurements. Figure 4 presents a comparison between the in-plane strain maps and the heat source map near point D. This figure enables us to identify the zones which were subjected to the reverse transformation (from martensite to austenite). Based on both measurements, an interpretation in terms of microstructures is proposed on the right-hand part of Figure 4. The strain map at Point E illustrates the fact that the greatest part of martensite is transformed into austenite at this stage. The size of Region 52 has increased. It is also interesting to note that the strain level is slightly greater near the right-hand side compared to the left-hand side, as in Figure 4-a. However, this region stretches significantly along the horizontal direction in Figure 4-a compared to Figure 4-b. This means that the length of the martensite needles that are mixed with austenite in this zone become shorter at Point E, thus showing that they are withdrawing. Interestingly, some traces of the X-shaped region are also still visible in zones 56 and 57. Point F is located close to the end of the hytheresis loop. A wide austenitic region now clearly appears since it corresponds to the blue zone in the map. It is bordered by martensite bands at the bottom. Some traces of the Xshaped region are still visible at the top right and left corners of the specimen. As in the preceding case, the strain level just above Region 59 and along the right-hand side border is slightly higher than in the other parts of the austenitic zone. Again, it is proposed to interpret this result by the fact that martensite needles progressively withdraw. Finally, the strain pattern is more complex at the bottom of the specimen. This is certainly due to the fact that it is located close to the grips.

a- Point D

b- Point E

c- Point F

Figure 3. Typical longitudinal strain maps measured during the unloading phase

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Figure 4. In-plane strain maps and corresponding heat source map measured during the unloading phase, and possible interpretation in terms of martensitic microstructures

CONCLUSION

The grid method and infrared thermography have been combined to investigate the mechanical response of a CuAl-Be single crystal specimen subjected to a tensile test. Various martensitic microstructures are clearly revealed by these techniques, especially the grid method with which very small details can be distinguished. The difference between the loading and unloading phases of the hysteresis loop of the strain-stress curve is illustrated by different microstructures that occur during these phases. Identifying the martensite variants that appear in during the test will be the next step of this study. REFERENCES

[1] Efstathiou C., Sehitoglu H., Carroll J., Lambros J. and Maier H.J., Full-feld strain evolution during intermartensitic transformations in single-crystal Ni-Fe-Ga, Acta Materialia, 56:3791-3799, 2008 [2] Daly S., Rittel D., Bhattacharya K. and Ravichandran G., Large deformation of nitinol under shear dominant loading, Experimental Mechanics, 49:225-233, 2009 [3] Favier D., Louche H., Schlosser P., Orgeas L., Vacher P. and Debove L., homogeneous and heterogeneous deformation mechanisms in an austenitic polycrystalline Ti-50.8 at. Ni thin tube under tension. Acta Materialia, 55:5310-5322, 2007 [4] Piro J.L. and Grédiac M., Producing and transferring low-spatial-frequency grids for measuring displacement fields with moiré and grid methods, Experimental Techniques, 28(4), 23-26, 2004 [5] Surrel Y., Fringe Analysis, in Photomechanics, Topics Appli. Phys. 77, editor: P.K. Rastogi, 55-102, 2000 [6] Badulescu C., Grédiac M., Mathias J.-D. and Roux D., A procedure for accurate one-dimensional strain measurement using the grid method, Experimental Mechanics, 49(6), 841-854, 2009 [7] Badulescu C., Grédiac M. and Mathias J.-D., Investigation of the grid method for accurate in-plane strain measurement, Measurement Science and Technology, 20(9), 2009 [8] Chrysochoos A. and Louche H., An infrared image processing to analyse the calorific effects accompanying strain localisation, International Journal of Engineering Science, 38: 1759-1788, 2000 [9] Badulescu C., Grédiac M., Haddadi H., Mathias J.D., Balandraud X. and Tran H.S., Applying the grid method and infrared thermography to investigate plastic deformation in aluminium multicrystal, Mechanics of Materials, 43(11): 36-53, 2011

Correlation between Mechanical Strength and Surface Conditions of Laser Assisted Machined Silicon Nitride

F.M. Sciammarella, M.J. Matusky College of Engineering & Engineering Technology Northern Illinois University, DeKalb, IL USA Keywords: Silicon Nitride, Lasers, Machining, Flexure Strength, Surface Roughness

ABSTRACT High power fiber-coupled diode lasers for Laser-Assisted Machining (LAM) of ceramics provides an efficient, cost effective solution for surface finishing of ceramic products. This paper presents experimental evidence of advantages of LAM over the traditional diamond wheel grinding, a standard technique currently utilized in the finishing of ceramic surfaces. LAM, utilizing fiber-coupled diode lasers, also provides advantages over other types of lasers such as CO2 and Nd:YAG lasers. The emphasis of this work is in the evaluation of LAM in the strength of finished products of two different sources of silicon nitride. An optical technique based on evanescent illumination was utilized to measure the Ra of the finished surfaces utilizing LAM, laser glazed, diamond ground, and as-received surface conditions. Four point bending test for specimens of each surface condition were utilized to measure the fracture strength. A correlation was found between the measured Ra and the predicted strengths resulting from Weibull analysis. The correlation shows a decrease of strength with the increase of Ra. The fracture surfaces were observed both optically and with a SEM, and the flaw sizes were measured. The analysis of the fractographs indicated that the flaw sizes are consistent with Fracture Mechanics predictions. Explanations of the correlation between Ra, strength, and flaw sizes require further testing.

INTRODUCTION Over the last three decades, ceramics have moved from low strength applications to high temperature and high strength applications, based on remarkable improvements in strength, fracture toughness, and impact resistance [1, 2]. Demand for advanced ceramics is expected to increase as the infiltrate several applications cutting tools, joint implants, capacitors, military armor, aerospace, and automotive components. Advanced ceramics (silicon nitride, silicon carbide, zirconia, etc) offer higher temperature capability, lower density, higher stiffness, and better wear resistance when compared to metals [3, 4]. Traditional processing of these ceramic materials includes forming, green machining, sintering, and final machining stages. These components typically require very tight dimensional tolerances during shaping and surface machining, generally done by diamond wheel surface grinding (containing coarse, intermediate, and fine grinding stages) due to the high hardness of these ceramics. [5]. Also, conventional single point machining (turning, milling, and drilling) produces brittle failure, excessive surface damage, and excessive tool wear at acceptable machining rates [6, 7]. Therefore, the cost of machining ceramics represents 70% to 90% of the cost of finished parts [8]. Considering silicon nitride (Si3N4) as a baseline material for this study, diamond grinding still results in low material removal rates and is limited to simple contours. Prior academic research on Laser Assisted Machining (LAM) of high strength ceramics (silicon nitride, toughened zirconia, silicon carbide, etc) has included empirical studies [10-17] and numerical modeling [18-21]. This work quantifies the benefits of LAM over grinding including, rapid material removal rates and extended tool life. In LAM, the machining process is simplified and accelerated, leading to reduction in equipment cost, labor, and machining time. Applying fiber-couple diode lasers creates a more robust and industrial rugged system for LAM of ceramics.

T. Proulx (ed.), Optical Measurements, Modeling, and Metrology, Volume 5, Conference Proceedings of the Society for Experimental Mechanics Series 9999999, DOI 10.1007/978-1-4614-0228-2_23, © The Society for Experimental Mechanics, Inc. 2011

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How LAM Works In laser assisted machining (LAM), a high energy laser beam is used to locally heat a small zone, on the workpiece, ahead of a single point tool (diamond or cubic boron nitride (CBN)) and then machined by turning or milling. By preheating a ceramic workpiece in the machining zone ahead of the tool, the local spot temperature rises to a point where quasi-ductile behavior, rather than brittle fracture occurs. Quasi-ductile deformation in the ceramic enables reduced cutting forces, high material removal rates, minimal surface damage, and increased tool life [9-11]. A schematic of the LAM process (with the ceramic rod, cutting tool, laser spot, and material removal plane) is illustrated in Figure 1. A good review of past LAM research on a variety of materials using Nd:YAG and CO2 lasers and the benefits is reported in [9].

Figure 1 Laser assisted machining model illustrating the machining removal plane [11]. In the case of silicon nitride, the intense heating of the surface locally raises the temperature of a glassy grain boundary phase, which is the residual of an oxide sintering aid used during liquid phase sintering at temperatures above 1300°C [4]. The composition of the sintering aid determines the temperature (~600-1200°C) at which the grain boundary phase softens. With sufficient heat, the grain boundary phase will soften and produce the desired ductile deformation. However, local overheating can introduce undesirable thermal damage such as devitrification, melting, sublimation of the grain boundary phase, or oxidation of the silicon nitride. Therefore, it is important to monitor the surface temperature profile in the cutting zone with a thermal (IR) imaging and/or 2-colour pyrometer so that material removal zone remains within the critical minimum and maximum preheat temperatures.

EXPERIMENTAL SETUP AND DISCUSSION Material Selection Commercial high strength silicon nitride from two independent sources was chosen as the baseline material for the LAM study and classified as silicon nitride A and silicon nitride B. Both silicon nitride sources were LAM turned in the form of 25 mm diameter rods, 150 mm long. Some tests were also completed on 13 mm by 100 mm rods of silicon nitride B. An investigation into material properties of a similar material to silicon nitride A was characterized (in a 2002 vintage) in a 2005 Army Research Laboratory study [22]. The microstructure of silicon nitride A is illustrated in Figure 2, from [22]. Note the bimodal grain size with the large acicular silicon nitride grains, the light gray grain boundary phase, and the very small white inclusions.

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Figure 2 Microstructure of silicon nitride A [22] Laboratory & Prototype Production LAM Systems A laboratory scale LAM system, Figure 3, was constructed at Northern Illinois University (NIU). The system setup, containing a 250W fiber-coupled diode laser, has been reported in detail in [23-25]. The system utilizes a FLIR® A325 thermal imaging camera for experimentally measuring thermal fields in the ceramic material, allowing for baseline studies of laser parameters during LAM. This academically focused bench top system is designed to assist a commercial partner in integrating LAM technology into their manufacturing operations. Based on the laboratory LAM system, an industrial scale LAM system was designed and built at the industrial partner location using a 25 HP commercial CNC 5-axis turning center. The system utilizes a custom built multi-beam fibercoupled diode laser system reported in [23-25]. The primary laser processing head is equipped with a built-in, coaxial, 2colour pyrometer system (temperature range 543-1500°C) for process monitoring. Thermal imaging is also utilized for thermal measurement during part production. Dedicated hardware and software are used for data acquisition and process control.

Figure 3 NIU LAM system overview Laser Surface Analysis In order to validate the LAM process as a viable industrial solution, Experimental Mechanics represents a necessary tool to understanding properties and behavior of the ceramic parts after machining. Particularly, it is important to measure the surface finish produced during LAM. A recently developed optical method for characterization of the surface that was presented last year at SEM is known as Advanced Digital Moiré contouring [31]. This technique utilizes evanescent illumination to interact with the surface under inspection. This is achieved through the phenomenon of light generation produced by electromagnetic resonance where the self generation of light is achieved through the use of total internal reflection (TIR). When a plane wave front impinges the surface separating two media such that the index of refraction of medium 1, glass, is higher than the index of refraction of medium 2, air (i.e. n1 > n2 ), at the limit angle, total reflection takes place. Under these circumstances a very interesting phenomenon occurs at the interface (glass-air) and evanescent waves are

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produced. At the same time, scattered waves emanate from the medium 1 (glass). More detailed theory on this optical measurement methodology is described in [26, 30, 31]. To achieve this experimentally, a laser-microscope system, seen in Figure 4, has been constructed with a capacity to resolve vertical differences to 120 nm across a 480 × 480 micron surface area. Essentially, the surface of the ceramic is in contact with a grating that has 400 lines per mm. The surface is illuminated by a HeNe laser at an oblique angle that provides total internal reflectance. This generates a 3-D interference image captured by a CCD camera. Accuracy was calibrated against a NIST traceable surface roughness calibration block (Ra range of 3.018 – 3.079 µm). CCD Camera

Grating HeNe Laser

Calibration Block

Figure 4 Laser Surface Analysis system setup during calibration The interference patterns are analyzed with a fringe analysis software package, Holo Moiré Strain AnalyzerTM (HMSA) Version 2.0, developed by Sciammarella et. al. and supplied by General Stress Optics Inc. (Chicago, IL USA). The fringe analysis uses powerful Fast Fourier Transforms for filtering, carrier modulation, fringe extension, edge detection and masking operations, and removal of discontinuities, etc. The software produces a full statistical analysis of the interference pattern in both spread sheet and graphic form (Figure 5). Different roughness metrics like Ra, Rq, and Rz can be quickly determined and Weibull analysis used to characteristic values.

Figure 5 Laser surface analysis field of view for a) LAM turned Si3N4 sample b) As-received Si3N4 sample

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Flexure Strength Testing The 13 mm diameter rods, of silicon nitride B, were split along their axis into half rounds (5 mm thick, 12.8 mm wide, and 50 mm long), two test specimens per rod, while the 25 mm diameter, 50 mm long rods of both silicon nitride, A and B, were sliced into three arc segments (test specimens) with a larger size (5 mm thick, 19 mm wide and 50 mm long – Figure 6a). These specimens were tested in flexure with the surface condition (curved face) in tension (down) at loads less than 4000 N (880 pounds). This arc segment specimen geometry has been used for other ceramics [22] and the method (specimens, fixturing, calculations) is described in detail by Quinn [27]. The test specimens were tested in 4 point- ¼ point bend test at room temperature in an Instron testing machine, using an articulated fixture (40 mm-20 mm spans) with tool steel roller bearings (Figure 6b). The cross head rate was 0.125 mm/min.

Figure 6 (a) Schematic of arc samples used for 4 point bend testing (b) View of experimental setup

EXPERIMENTAL RESULTS AND DISCUSSION Design of Experiments The room temperature flexural strength of LAM machined Si3N4 specimens were investigated, in both silicon nitrides, against surface conditions as listed in Table 1. All specimens were prepared on the prototype production LAM system. While, process parameters remain proprietary, please note that baseline multi-beam LAM parameters were used, and are not optimized for material removal rates, tool wear, or surface conditions. Ongoing and future work aims at process optimization. A two-parameter Weibull analysis was used to determine variability of strength and surface roughness for each sample. In ceramics, the Weibull distribution is used to characterize strength behavior on the basis that the weakest link in the body will control the strength, as described by Quinn [28]. Table 1 Specimens considered for flexure testing and surface characterization Silicon Nitride Material A Silicon Nitride Material B 25 mm Dia., 50 mm Long Arc Segments 25 mm Dia., 50 mm Long Arc Segments # of Samples # of Samples Surface Condition Surface Condition Tested Tested As-Received 9 As-Received 9 Diamond Ground (100 Grit) 9 Diamond Ground (800 Grit) 9 LAM Turned 9 LAM Turned 9 Laser Glazed 9 13 mm Dia., 50 mm Long Half Rounds As-Received 6 Diamond Ground (800 Grit) 6

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Statistical Static Properties Weibull analysis of flexure strength and surface roughness, relating to the 27 tested specimens in silicon nitride A, was previously reported in [25]. Results indicated that LAM did show an increase in Weibull strength and lower variance over the as-received condition, with the exception of one LAM test rod (three arc segment tests). Optical and SEM fractography was done on the high and low strength rods from Source A and flaw sizes measured. Further investigation into the effect of laser surface heating, laser glazing, is highlighted in Silicon nitride B testing and reported in [32]. Results lead to further confirm an increase in Weibull strength, with lower variance, during LAM, as well as laser glazed conditions over as-received conditions. Surface Roughness Surface measurements, Ra, of the 36 silicon nitride B (from twelve 25 mm dia. rods) and 27 silicon nitride A test specimens consisted of taking three different area measurements per specimen, 27 areas per rod, for a total of 81 area measurements per surface condition. From the Weibull distribution, the characteristic roughness value, Raθ with 95% confidence, along with the upper and lower bound (UB and LB) values of Raθ for the two silicon nitrides is found in Table 2. Table 2 Weibull analysis of surface roughness measurements, Raθ Silicon Nitride B Diamond AsLAM Laser Ground Received Turned Glazed (800 Grit) Raθ (µm) 1.2378 0.9339 0.887 0.9485 UB of Raθ (µm) 1.4509 1.034 1.0407 1.0052 LB of Raθ (µm) 1.0561 0.8434 0.7588 0.8949

Silicon Nitride A Diamond AsLAM Ground Received Turned (100 Grit) 1.260 1.336 0.956 1.619 1.544 1.067 0.981 1.156 0.856

Weibull analysis of LAM surface roughness, for both silicon nitrides, showed improvement over the corresponding as-received conditions, while the coarser 100 grit diamond ground produced the highest roughness measurements of all samples considered. LAM and laser glazed conditions are found to have comparable surface finishes to that of the 800 grit diamond ground. Flexure Strength The silicon nitride A, 27 wide (19 mm), specimens fractured across a range of applied loads - 1905 to 3950 N (427 to 885 lbs). While silicon nitride B, 36 wide (19 mm), specimens fractured across a range of applied loads – 2433 to 3960 N (545 to 887 lbs) and the 12 narrow (12 mm) specimens fractured across a range of applied loads – 1679 to 3504 N (376 to 785 lbs). All of the specimens fractured within the inner span and took generally about 100140 seconds. Tables 3 & 4 shows the characteristic strength, σθ, along with its UB and LB and mean flexure strength along with the coefficient of variation (CoV) and extreme values (high and low) for the four surface conditions and both silicon nitrides. Table 3 Comparison of Weibull strength & statistical strength values in Silicon Nitride B testing 25 mm Dia. Rods 13 mm Dia. Rods Diamond Diamond AsLAM Laser AsGround Ground Received Turned Glazed Received (800 Grit) (800 Grit) σθ (MPa) 452.1 526.5 582.0 547.9 N/A N/A UB, LB of σθ (MPa) 476, 429 575, 483 611, 554 575, 522 N/A N/A Mean Strength (MPa) 436.3 497.9 560.5 528.2 318.1 516.4 CoV – Mean Strength 8.10% 13.20% 9.00% 8.50% 5.00% 18.10% High, Low Strength (MPa) 499, 388 624, 430 623, 469 588, 458 338, 296 620, 388 Number of Samples Tested 9 9 9 9 6 6

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Table 4 Comparison of Weibull strength & statistical strength values in Silicon Nitride A testing 25 mm Dia. Rods Diamond LAM LAM AsGround Turned Turned Received (100 Grit) Rods #5&6 Rod #4 416.35 σθ (MPa) 549.0 488.0 609.8 UB, LB of σθ (MPa) 584, 517 514, 463 623, 597 481, 361 Mean - Strength (MPa) 524.0 469.0 599.2 390 CoV – Strength 11.60% 9.30% 4.9% 16.5% High, Low Strength (MPa) 604, 415 531, 408 623, 541 447, 300 Number of Samples Tested 9 9 6 3 Further analysis of the flexure strength data has shown, experimentally, a correlation with an increase in surface roughness, a decrease in flexure strength occurs in both sources of Si3N4 material, as seen in Figure 7. Along with the observed trend, the laser glazed condition also demonstrates a decrease in surface roughness and an increase in flexure strength over the as-received condition. It would appear, as with the LAM turned specimens, that the laser heating may have a healing effect on surface flaws. More investigation into these trends is required and further discussed in the following sections.

Figure 7 Weibull strength, σθ, vs. Weibull surface roughness, Raθ, for all surface conditions tested Fracture Mechanics NIST recommended practice guide by Quinn [28] for the fractography of ceramics is followed. Images of the fracture surfaces were captured both optically and by SEM (gold coated) for sample specimens selected from silicon nitride A testing. Selected samples highlight the low strength LAM (three test specimens), high strength LAM (three test specimens), and as-received conditions. Selected samples for silicon nitride B testing, highlight the various surface conditions and all values reported are averaged per condition, at least three test specimens. Fractographs are also post processed using the HMSA software package, taking advantage of light contrast over the fractured surface, to aid in visualizing the fracture mirror, Figure 7.

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Fracture Toughness Estimation Approximations of fracture toughness (KIc), given in Tables 5 & 6, are estimated based on a technique and (1) described by Quinn [28]. The flaw size (a and c are shown in Figure 8) and shape are estimated by inspection of optical and SEM fractographs. Additionally, the maximum stress intensity factor, Y, is found as the Newman-Raju Y factor at the surface and deepest part of the crack [28]. The Newman-Raju Y factors are also included in ASTM standard C 1421 for fracture toughness of ceramics. The Y factor at the surface is considered for discussed during comparisons. Values for strength, σf, were measured experimentally during silicon nitride A & B testing. These calculations and interpretations are not conclusive as further investigation of these findings, in more detail, is underway. K Ic = Y × σ f × a

(1)

Table 5 Estimation of KIc for highlighted surface conditions of Si3N4 A by descending strength values Crack KIcd KIics Surface Condition Strength, Ysurface Depth, Ydepth Si3N4 A Mpa*m1/2 Mpa*m1/2 MPa µm LAM (Low Strength) 390 140.3 1.62 7.08 1.25 5.51 As- Received (#7B) 604 63.0 1.55 7.43 1.24 5.92 LAM (High Strength) 619 183.6 1.53 12.82 1.21 10.10 Table 6 Estimation of KIc for Highlighted surface conditions of Si3N4 by descending strength values Crack Strength, KIcd KIcs Surface Condition Ysurface Depth, Ydepth MPa Si3N4 B Mpa*m1/2 Mpa*m1/2 µm As-Received (Ave.) 431 245.7 1.55 10.11 1.17 7.73 800 Grit DG (Ave.) 464 181.6 1.58 9.87 1.17 7.33 Laser Glazed (Ave.) 538 148.7 1.61 10.40 1.16 7.59 LAM Turned (Ave.) 595 213.5 1.59 13.80 1.16 10.01 The manufacturer specified fracture toughness for silicon nitride A and B is reported as 6.3 MPa*m1/2 and 6.1 MPa*m1/2, respectively, which is close to that estimated in the as-received conditions. While low values of KIc are estimated for one LAM case in silicon nitride A, Figure 8; inherent material flaws, machining damage, mishandling, or specimen preparation (cutting of arc segments) may have resulted in undesired strengths. On the other hand, laser effected samples of both materials (higher estimated KIc in LAM) appears to further support the surface healing effect of laser heating. Laser glazing of alumina has been reported to similarly improve Weibull strength characteristics over as-received conditions [29]. These results indicate that fully understanding the mechanism warrants additional in-depth study. Fracture Origin

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Figure 8 Fracture Origin in low strength LAM #4A shown during post processing, highlighting the fracture mirror.

a 2 Figure 9 Fracture origin in low strength LAM #4A a) optical photo of the fracture mirror shows crack of about 136 µm deep, and combined with a 447.2 MPa fracture strength and a surface Y factor of ~1.27 gives KIc ≈ 6.65 MPa*m1/2. b) SEM image of fracture mirror showing LAM turned surface.

CONCLUSIONS Various test specimens prepared by laser-assisted machining, laser glazing, conventional diamond grinding, and as-received surface conditions were evaluated on silicon nitride rods from two different sources. During the LAM process it was found that LAM turned silicon nitride rods had a measured 25%-30% increase in flexure strength, when compared to their as-received counterparts. For test specimens prepared by laser glazing an increase in strength and decrease in surface roughness, over as-received conditions, was also observed. Experimental results from both material sources have demonstrated an apparent trend correlating the surface roughness and flexure strength for all surface conditions tested. Initial fractography of the laser heated specimens along with the as-received condition was completed. In one case, LAM indicates reduced strength and fracture toughness compared to as-received conditions. While this result was uncharacteristic of all other LAM samples tested, care must be taken while machining and handling the ceramic so as to avoid degrading the strength aside from surface induced flaws. Results also indicate laser heating of the ceramic surface may in fact increase the strength of the ceramic material . While further investigation is required to confirm this estimation, the increase in strength may be a result of a modification of the flaw sizes and flaw populations found in as-received counterparts, acting in a beneficial way. LAM provides a very promising, cost effective, improvement over conventional diamond grinding of advanced ceramics.

ACKNOWLEDGMENTS The authors thank Tom Wagner and U.S. Army TARDEC (Contract Number W56HZV-04-C-0783) for support that made this work possible. The authors would like to express their gratitude towards Dr. Richard Johnson, Director of ROCK, Alan Swiglo, Assistant Director of ROCK, and Stefan Kyselica, Senior Systems Engineer of ROCK for their continued technical and management support. Thanks are also due to Richard Roberts, Jeff Staes, and Rick Deleon for their partnership in this project. Special thanks to the graduate students at NIU Mechanical Engineering Department, SriHarsha Panuganti and Vishal Burra for all their efforts on this project. Mechanical flexure testing was done at Illinois Institute of Technology with the assistance of Mr. Russ Janota and Professor Philip Nash. Some fractography was provided by Ceradyne, Inc. Costa Mesa, Bilijana Mikijelj.

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REFERENCES [1] US Advanced Ceramics Market, Fredonia Industrial Market Study, Cincinnati, OH (December 2008). [2] T. Abraham, U.S. Advanced Ceramics Industry—Status and Market Projections, Ind. Ceram., 19 [2] 94–6 (1999). [3] Y. Liang and S. P. Dutta, Application Trend in Advanced Ceramic Technologies, Technovation, 21, 61–5 (2001). [4] F. L. Riley, Silicon Nitride and Related Materials, J. Am. Ceram. Soc., Vol. 83, 245-265 (2000). [5] I. P. Tuersley, A. Jawaid, and I. R. Pashby, Review: Various Methods of Machining Advanced Ceramic Materials,’ J. Mater. Processing Technol., 42, 377–90 (1994). [6] V. Sinhoff, S. Schmidt, and S. Bausch, Machining Components Made of Advanced Ceramics: Prospects and Trends, Ceram. Forum Int./Berichte Der Dkg, 78 [6], E12–8 (2001). [7] W. Konig and A. Wagemann, Machining of Ceramic Components: Process-Technological Potentials, Machining Adv. Mater., NIST Spec. Publ., 847, 3–16 (1993). [8] I. D. Marinescu, Handbook of Advanced Ceramic Machining, CRC Press, (2007). [9] N.B. Dahotre and S.P. Harimkar, Laser Fabrication and Machining of Materials, Springer Science + Business Media (2008). [10] W. Konig and A. K. Zaboklicki, Laser-Assisted Hot Machining of Ceramics and Composite Materials, Int. Conf. Machining Adv. Mater. NIST Spec. Publ., 847, 455–63 (1993). [11] Y. C. Shin, S. Lei, F. E. Pfefferkorn, P. Rebro, and J. C. Rozzi, Laser-Assisted Machining: Its Potential and Future, Machining Technol., 11 [3] 1–6 (2000). [12] F. Klocke and T. Bergs, Laser-Assisted Turning of Advanced Ceramics, Rapid Prototyping Flexible Manuf., Proc. SPIE, 3102, 120–30 (1997). [13] J.C. Rozzi, F.E Pfefferkorn, Y.C. Shin, and F.P. Incropera, Experimental Evaluation of the Laser Assisted Machining of Silicon Nitride Ceramics, J. of Manufacturing Science and Engineering, Vol. 122, 666-670 (Nov. 2000). [14] S. Lei, Y.C. Shin, and F.P. Incropera, Experimental Investigation of Thermo-Mechanical Characteristics in Laser-Assisted Machining of Silicon Nitride Ceramics, J. of Manufacturing Science and Engineering, Vol. 123,639646 (Nov. 2001). [15] S. Lei, Y. Shin, and F. Incropera, Experimental Investigation of Thermo-Mechanical Characteristics in LaserAssisted Machining of Silicon Nitride Ceramics, ASME J. Manuf. Sci. Eng., 123, 639–46 (2001). [16] F. E. Pfefferkorn, Y. C. Shin, Y. Tian, and F. P. Incropera, Laser-Assisted Machining of Magnesia-Partially Stabilized Zirconia, ASME J. Manuf. Sci. Eng., 126, 42–51 (2004). [17] Y. Tian and Y.C. Shin, Laser-Assisted Machining of Damage-Free Silicon Nitride Parts with Complex Geometric Features via In-Process Control of Laser Power, J. Am. Ceram. Soc., 89 [11], 3397–3405 (2006). [18] J. C. Rozzi, M. J. M. Krane, F. P. Incropera, and Y. C. Shin, Numerical Prediction of Three-Dimensional Unsteady Temperatures in a Rotating Cylindrical Workpiece Subjected to Localized Heating by a Translating Laser Source, 1995 ASME Int. Mech. Eng. Conf. Exposition, San Francisco, California, HTD,317 [2] 399–411 (1995). [19] J. C. Rozzi, F. E. Pfefferkorn, F. P. Incropera, and Y. C. Shin, Transient, Three-Dimensional Heat Transfer Model for the Laser Assisted Machining of Silicon Nitride: I. Comparison of Predictions with Measured Surface Temperature Histories, Int J. Heat Mass Transfer, 43, 1409–24 (2000). [20] J. C. Rozzi, F.P Incropera, and Y.C. Shin, Transient, Three-Dimensional Heat Transfer Model for the Laser Assisted Machining of Silicon Nitride: II. Assessment of Parametric Effects, Int. J. of Heat and Mass Transfer 43, 1425-1437 (2000). [21] F. E. Pfefferkorn, F. P. Incropera, and Y. C. Shin, Heat Transfer Model of Semi-Transparent Ceramics Undergoing Laser-Assisted Machining, Int. J. Heat Mass Transfer, 48 [10] 1999–2012 (2005). [22] J. J. Swab, A.A. Wereszczak, J. Tice, R. Caspe, R. H. Kraft, and J. W. Adams, Mechanical and Thermal Properties of Advanced Ceramics for Gun Barrel Applications, Army Research Laboratory Report ARL-TR-3417, February (2005). [23] Panuganti, S., Understanding Fiber-Coupled Diode Laser Superheating in Laser Assisted Machining of Silicon Nitride (Si3N4), Department of Mechanical Engineering, Northern Illinois University (2009). [24] F.M. Sciammarella and M.J. Matusky, Fiber Laser Assisted Machining of Silicon Nitride, Conference Proceedings ICALEO, (2009) (to be published). [25] F.M. Sciammarella, J. Santner, J. Staes, R. Roberts, F. Pfefferkorn, S.T. Gonczy, S. Kyselica, and R. Deleon, Production Environment Laser Assisted Machining of Silicon Nitride, Conference Proceedings ICACC, (2010). [26] C.A. Sciammarella, L. Lamberti, F.M. Sciammarella, G. Demelio, A. Dicuonzo, and A. Boccaccio, Application of Plasmons to the Determination of Surface Profile and Contact Stress Distribution, Strain, (2009).

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Analysis of speckle photographs by subtracting phase functions of digital Fourier transforms Karl A. Stetson Karl Stetson Associates, LLC 2060 South Street Coventry, CT 06238 Abstract This paper presents a method for measuring displacements and strain in digital speckle photography that is an alternative to currently used correlation techniques. The method is analogous to heterodyne speckle photogrammetry wherein optical Fourier transforms are taken of individually recorded specklegrams and combined in a heterodyne interferometer where an electronic phase meter measured the phase differences between the two transforms. Here, digital photographs are recorded and Fourier transformed so that their phase functions can be subtracted and fitted to a linear function of the transform coordinates. The effect of different recording and processing parameters is investigated. It is found that incoherent speckles give better results that those formed by coherent laser light. In addition, image correlation is used to process an identical data set so that comparison of the two methods can be made. 1. Introduction Laser speckle photography1 arose as an alternative to holographic interferometry. When an object is illuminated with laser light, the speckles that form in its image move as if attached to the object itself. If an object moves between two exposures of a photograph, the resultant doubling of the speckle pattern can be observed via an optical Fourier transform created by illuminating a small region of the photograph with a narrow, converging laser beam. The doubling of speckles in the photograph gives rise to linear fringes in the transform plane where the beam comes to focus, and because of the similarity to Young’s experiment, they are often called Young’s fringes. The fringes are normal to the direction of the speckle displacement, and their spacing is inversely proportional to its magnitude. Whereas speckle photographs, or specklegrams, can measure object displacements, they have limitations for measuring strain, which is defined as the change in displacement between two object points divided by their separation. Resistive strain gages can measure strains down to 10-5 over gage lengths as short as one to two mm, and to duplicate this with speckle photography is very difficult. This is especially so in regions of the object where the displacement is so small that no fringes are observable in the transform plane and in regions where the displacement is so large that the fringes are too narrow to observe. To overcome these problems, a technique was developed2-6 for heterodyne readout of halo fringes. In heterodyne speckle photogrammetry, two photographic glass plates are used to record separate specklegrams before and after a stress is applied to the object. These two plates are placed side by side in an interferometer on a common translation stage and aligned so that the same region on each can be illuminated with each of two small, mutually coherent, converging laser beams. The transmitted beams and scattered halos are combined after equal propagation by a set of mirrors and a beamsplitter. Adjustments are provided so that the halo fringes can be minimized, and the two plates aligned to eliminate relative rotation. The interferometer is provided with the capability of shifting the optical frequency of one beam relative to the other to cause the halo fringes to move and generate sinusoidal irradiance fluctuations. These are detected by an array of photodiodes located in the transform plane, processed by a phase meter, and the phases recorded. As the pair of plates is moved, any translation of one speckle pattern relative to the other causes a change in the relative phases in the fluctuating halo fringe pattern. These changes are recorded and used to calculate the relative displacement of the speckle pattern, which, divided by the amount by which the pair of plates is translated, gives the object strain. Strain measurement by this technology requires that the object be photographed along the surface normal by means of a telecentric lens system, i.e., one that has no spherical perspective. The high accuracy available from electronic phase meters, 0.1 degree out of 360, makes it possible to measure strain down to a theoretical level of 10 microstrain (i.e., 10-5) from recordings made with lens systems whose f/numbers are as high as f/10. Furthermore, multiple recordings allow cumulative strain to be measured up to several percent. Although this technique was demonstrated, it has drawbacks. The photographic plates require chemical development and drying before they can be analyzed. The setup and alignment of the interferometer is quite complex and requires considerable space in a darkened laboratory, and the alignment of the photographs is quite critical. T. Proulx (ed.), Optical Measurements, Modeling, and Metrology, Volume 5, Conference Proceedings of the Society for Experimental Mechanics Series 9999999, DOI 10.1007/978-1-4614-0228-2_24, © The Society for Experimental Mechanics, Inc. 2011

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200 They must be placed so that the same regions are at least approximately illuminated in order to obtain halo fringes at all. Finally, the procedure of recording the phases, moving the pairs of plates, and computing the strains is time consuming. For these reasons, it is desirable to reconsider this process from the point of view of modern digital photography and digital image analysis to see which aspects can be improved and which cannot. Digital photography has been used for specklegram analysis for quite some time,8,9 however, speckle displacements have been analyzed by image correlation. (See www.lavision.de, and Aramis at www.trilion.com, for example. ) The purpose of this paper is to present and investigate an alternative method of specklegram analysis based upon the subtraction of the phases of digital Fourier transforms in a manner analogous to heterodyne speckle photogrammetry. It begins with a description of the process, followed by the mathematical analysis and experimental study. Several parameters were investigated – recording lens f/number, 8-bit versus 12-bit digitization, and incoherent versus coherent speckle patterns. Finally, a comparison is made to an image correlation measurement of displacement of the best translation data set. 2 Procedure for digital Fourier processing The procedure begins by photographing the object with a digital camera. Unless a telecentric lens system is used, any translation of the object toward or away from the object will result in a magnification change, and this will give an apparent strain that must be considered in any subsequent analysis. The optical axis of the lens should be aligned along the surface normal and the camera should have black and white pixels with square pixel format. Twelve-bit image digitization should offer increased accuracy; however, shot noise may be expected to restrict the useful digitization to eight bits. Photographs are captured before and after object perturbation and designated as A and B. For simple displacement analysis, an entire camera image may be used; however, for strain analysis it will be necessary to divide the camera images into segments for separate processing. The next step is to compute the digital Fourier transforms FAnm and FBmn for each photograph or segment thereof. A digital Fourier transform has the advantage over an optical Fourier transform that it is possible to calculate the phase of the pixels in the transform. The phase values will range randomly from –π to π; however, a speckle displacement will generate a linear phase change across the transform plane whose slope is proportional to the displacement and whose gradient is in the direction of displacement as described by the shift theorem of Fourier transforms. The phase function can be obtained by subtracting the phases of the two transforms; however, simple subtraction will exhibit what may be called random wrapping. This occurs when the value of a pixel phase before displacement is made to exceed –π or π when the additional phase is added to it. For example, if the phase of a pixel is π - ∆ and the phase change is +2∆, the resulting phase will be −π+∆, and subtracting the two values will give 2∆ − 2π. Wrapping the phase difference from the range of –2π to 2π into the range of –π to π will remove these random wrapping effects.

Figure 1. 1a. Subtraction of two analytically created random 8-bit phase patterns with a linear phase difference between them. 1b. The same data as 1a after wrapping into 8-bit pixels. Note the wrapping removes the effect of random wrapping shown in 1a. [ Figure 1a illustrates the random wrapping that occurs when a phase function is added to a random phase distribution and is attempted to be recovered by simple subtraction. Figure 1b shows the result of wrapping the data of Fig. 1a into the range of –π to π, i.e., the phase difference is recovered without the effects of random wrapping. Of course, the phase difference itself

201 is wrapped every time it exceeds either limit, –π to π, and it must be unwrapped for further data processing. After it is unwrapped, the next step is to fit the unwrapped phase difference to a linear function of the spatial frequencies, ωx and ωy, and these slopes correspond to the x and y translations of the speckles. If strain analysis is being performed, the image will be divided into segments and the slope values from neighboring segments can be used to calculate the average strains experienced between the segments as described below. 3. Mathematical Analysis The digital Fourier transform used for this analysis is described by the following equation:

where: f(m,n) is the discrete function whose transform is being calculated, m and n are the pixel indices in the x and y directions, M and N are the number of pixels in the x and y directions of the image or segment thereof, j and k are the horizontal and vertical indices of the coordinates of the transform, and F(j,k) is the Fourier transform of f(m,n) with respect to the variables j and k. Consistent with the usage in television, we take the y axis, represented by n, to be positive downward. We may calculate the effect of a displacement of one pixel in the x or y directions, dpx and dpy, for the function f(m,n) by substituting m-1 for m or n-1 for n in Eq.(1). This will generate the respective phase functions Φωx or Φωy in the transform plane, where Φωx = exp(i2πj/M), and Φωy = exp(i2πk/N).

(2a) (2b)

The slopes of these phase functions, sj1 and sk1 per incremental values of j or k, are sj1 = 2π/M, and sk1 = 2π/N.

(3a) (3b)

If dx and dy are the actual displacements of the image pattern and sk and sj are the corresponding measured phase changes per pixel of their transforms, then the corresponding fractions of pixel displacements, dx/dpx and dy/dpy, will equal the corresponding fractional changes in slope of the transform function, sk/sj1 and sk/sk1. dx/dpx = sj/sj1, and dy/dpy = sk/sk1

(4)

Substituting from Eqs.(3) gives dx = dpxsjM/2π, and dy = dpyskN/2π,

(5a) (5b)

where the units of sj and sk are radians per pixel in the Fourier transform plane. Equations (5) allow calculation of the object displacement in terms of the pixel spacing, the transform plane slope, and the total number of pixels in the direction considered. The products, dpxM and dpyN, are the physical sizes in x and y of the camera array or the segments of the array. If the object is magnified relative to its image on the camera, then both sides of equations 5 must be multiplied by that magnification to obtain the object displacement. Strain analysis requires measuring the change in displacement between two points on a surface and dividing that by the distance between the two points. Because, in general, surface strain is expressed as a 2x2 matrix, we need to measure the fractional displacements of four points on the surface. Let these four points be the centers of four neighboring segments which we identify by the subscripts shown below.

202 11 21

12 22

The average relative displacements of these segments may be defined as: ∆xx = (dx12 – dx11 + dx22 – dx21)/2, the average x expansion in the x direction.

(6a)

∆ yy = (dy21 – dy11 + dy22 – dy12)/2, is the average y expansion in the y direction.

(6b)

∆ xy = (dx21 – dx11 + dx22 – dx12)/2, is the average x expansion in the y direction.

(6c)

∆ yx = (dy11 – dy12 + dy21 – dy22)/2 , is the average y expansion in the x direction.

(6d)

Given the fact that the M by N pixel segments are separated in x by M pixels and in y by N pixels, the x strain, y strain, and shear are defined as: εxx = ∆xx/dpxM

(7a)

εyy = ∆yy/dpyN

(7b)

εxy = (∆yx/dpxM + ∆xy/dpyN)/2

(7c)

When Eqs. (3) and (4) are substituted into Eqs. 6a-6d, and the results substituted into Eqs. 7a-7c, it is seen that the factors of pxM and pyN cancel from the strain calculations. Equations 7a-7c may be rewritten in terms of the measured Fourier transform plane slopes as εxx = (sj12 – sj11 + sj22 – sj21 )/4π

(8a)

εyy = (sk21 – sk11 + sk22 – sk12)/4π

(8b)

εxy = (sj21 – sj11 + sj22 – sj12 + sk12 – sk11 + sk22 – sk21)/8π

(8c)

For a segment of 256 by 256 pixels, for example, the strain corresponding to a lateral shift of one pixel is (1/256) which equals 3906 microstrain. Measurement of strain to a level of 10-5 would require measurement of subpixel displacement to a level of 1/391 of the pixel spacing. 4. Experimental Study An experimental study was carried out to determine the effectiveness of the process described above and investigate the effect of some experimental parameters. Images were obtained via a Prosilica EC650 monochrome TV camera with a 1/3 inch format sensor (4.7 mm horizontal by 3.5 mm vertical) with a pixel array of 659x493 elements on 7.4 µm centers. Image capture was done via two separate programs: the HoloFringe300K program that provided 8-bit images, 640x480 pixels, in a raw data format, and the Prosilica Viewer program that provided 12-bit images, 659x493, in tiff format. The object, mounted on a translation table, was a flat aluminum bar whose visible surface was whitened to provide a flat, diffuse reflection. The object was translated laterally by amounts that were read from the dial of the micrometer on the translation stage, single divisions of which corresponded to 10 µm each. A 25 mm lens was used on the camera which was set with its entrance pupil 355 mm from the object, which resulted in a demagnification of 12.9. The lens aperture was set to f/10, for which the characteristic speckle size should be 7.6 µm, approximately the size of the pixel spacing. Recordings were made of the object at displacements of 0, -10, -20, -40, -80, -160, and -320 micrometers, each was transformed and the phase functions of all the transforms calculated. The phase of the transform at 0 micrometers was subtracted from the phase of each of the other the transforms and the results wrapped into eight bits obtain the wrapped phase differences. These operations were done using a program called DADiSP, which provides a number of useful functions in addition to the 2DFFT computation. (See www.dadisp.com) These images were unwrapped using the method of calculated wrap regions from the HoloFringe300K electronic holography program7. Figure 2 shows, for the specklegrams at 0 and 320 micrometers, the wrapped and unwrapped phase difference. Note that the quality of the data is poorer at the left and right edges, which leads to errors in the unwrapping for those regions. To eliminate those errors, the unwrapped phase data was win-

203 dowed to remove the left, right, top, and bottom 25% of the pixels. The corners of the window were located at, xmin=160, xmax=480, ymin=120, and ymax=360.

Figures 2. 2a shows the wrapped phase difference between the transforms of the specklegram recorded at 0 micrometers and the one recorded at 320 micrometers, and 2b shows the data of 2a unwrapped. A linear function in x and y was then fitted to each data image for least square error by means of a program written in Liberty Basic. The resulting slope values in ωx and ωy, that is, sj and sk, were substituted into Eqs. 5a and 5b to obtain the x and y image displacements at the camera detector with M taken as 640 and N as 480. The image displacements were then multiplied by the magnification, 12.9, to get displacements of the object. Table 1 presents the measured displacements beside the displacements read from the micrometer dial. x trans. µm

x meas. µm

y meas. µm

-10.0 -20.0 -40.0 -80.0 -160 -320

-10.0 -23.7 -49.3 -90.6 -166 -324

2.57 7.68 8.62 8.06 7.03 6.85

Table 1. Tabulated values of measured displacement versus micrometer dial readings. The measured displacements in the y direction should be zero, and it is not clear why they have a nearly constant value. It is possible that the translation stage had some vertical displacement associated with the first 20 µm of horizontal travel. The RMS error for the x displacements is 5.75 µm and 7.09 µm for the y displacements. The role of lens aperture was investigated by recording another set of translations, 0 to +640 mm, with the lens set to f/2.8. This resulted in a noticeably different image, with a pixel histogram that looked much more like a Gaussian distribution. The results are tabulated in Table 2. The RMS error in the x measurement is 16.8 µm and for the y measurement is 2.33 µm. Clearly, the increase in lens aperture has not helped the measurement accuracy. x trans. µm 10.0 20.0 40.0 80.0 160 320 640

x meas. µm 5.04 30.0 57.5 96.6 181 343 667

y meas. µm 4.51 -1.86 -2.64 -3.02 -2.69 -2.37 -4.68

Table 2. Translations measured with the lens set to f2.8.

204 It was also of interest to learn if increasing the digitization to 12 bits would improve the measurement accuracy. Data was acquired via the Prosilica Viewer program for the same translations, the lens set to f/11 and processed in the same way. The results are presented in Table 3, and these show poorer results than with the 8-bit digitization; the RMS error for the x measurement is 11.9 µm and for the y measurement is 1.14 µm. Because the detectors are small, it is expected that their performance is limited by shot noise so that the difference between 8-bit and 12-bit digitization should not be significant. Why the RMS error for the x measurement is approximately twice that found with 8-bit digitization is not clear. It may be an artifact of the image capture programs which are different for the two cases. x trans. µm 10.0 20.0 40.0 80.0 160 320 640

x meas. µm 16.6 316 56.4 95.8 172 324 527

y meas. µm 1.42 0.494 -0.0219 -0.0142 -1.84 -1.45 -12.8

Table 3. Translations measured with the camera lens set to f/11 and the data digitized to 12 bits. Live observation of the speckle patterns showed that, for continuous translation, the shifted patterns exhibit a periodic change of the speckles themselves. CCD detector arrays, with interline transfer, typically have active sensors that occupy only about 35% of the area of the array itself. Thus, the speckles integrated by the detectors vary periodically with the translation of the object. Also, the object is illuminated spherically, and this causes a shift of the field that actually passes through the entrance aperture of the lens so that the speckles decorrelate slightly with translation. To get a comparison, an incoherent speckle pattern was generated with the DADiSP program as an array of random numbers displayed as pixels. This was printed on a sheet of paper and cemented to the object surface. The recorded speckle pattern and its histogram of pixel values looked very much like those for laser speckles. Images were recorded of this incoherent speckle pattern at the positions of the translation stage that were previously used. The camera lens was set to f/5.8, and only 8 bit images were captured based upon the evidence that 12 bit digitization did not improve the results. The results are shown in Table 4. x trans. µm 10.0 20.0 40.0 80.0 160 320 640

x meas. µm 12.54 21.79 40.90 80.17 161.4 319.7 639.6

y meas. µm 7.940 13.51 14.51 16.31 20.89 15.82 15.31

Table 4. Translations measured using an incoherent speckle pattern. The results obtained using an incoherent speckle pattern are considerably more accurate than any of those obtained using laser speckles. The RMS error for the x measurements is 1.34 µm and for the y measurements is 15.3 µm. Observation of the images obtained by this method showed that the speckles translated with much less change in their pattern than was observed with the laser speckles, which is consistent with the improved measurement results. The large amount of measured displacement in the y direction; however, indicates that the stage has a significant amount of vertical displacement associated with its horizontal travel. For comparison, another stage was used made by New Focus, their model 8095, and the results are presented in Table 5. These results show an RMS error for the x and y displacements of 1.97 µm and 3.22 µm respectively. Clearly, this stage moves with less vertical displacement that the previously used one. It is interesting to note that there is an almost linear increase in vertical displacement for the y displacement as a function of the quadratic increase in x displacement.

205 x trans. µm 8.00 16.0 32.0 64.0 128 256 472

x meas. µm 9.339 16.27 34.14 66.12 131.6 257.2 473.3

y meas. µm -1.043 -0.1983 1.619 3.152 3.351 4.044 4.941

Table 5. Displacement for incoherent speckles with the New Focus 8095 stage. Strain was simulated by reorienting the translation stage so that the surface moved toward the camera. An apparent strain of 2817E-6 was generated between two recordings by moving the object toward the camera by 1 mm at a distance of 355 mm. The camera lens was set to f/5.8, and 8-bit digitization was used. The two recordings were divided into 12 segments of 160x160 pixels each, four horizontal by three vertical. Each segment in one recording was processed with its corresponding segment in the other recording to obtain the slope of the unwrapped phase difference for that segment pair. Sets of slope values for four adjacent segments were used in Eqs 8a and 8b to obtain six values of strain, three horizontal by 2 vertical. Table 6 presents the results. Simulated 2.817E-03

x strain 2.829E-03 2.792E-03

x strain 2.776E-03 2.749E-03

x strain 2.845E-03 2.802E-03

Simulated 2.817E-03

y strain 2.765E-03 2.765E-03

y strain 2.815E-03 2.754E-03

y strain 2.837E-03 2.789E-03

Table 6. Strain measurements using digital FFT specklegram analysis. All six results for x strain and all six for y strain should be equal to 2.817E-3. The average of the six x values is 2.799E-03 and 2.787E-03 for the y values which differ from the correct value by 18E-6 and 30E-6 respectively. The standard deviations are 32E-6 and 30E-6 respectively. 5. Comparison to Image Correlation The data set that generated the data presented in Table 5 was alternatively analyzed by image correlation via the equation, Cab(m,n) = F-1{Fa Fb*},

(9)

Where Cab(m,n) is the correlation of the two specklegram images, a and b, F-1 is the inverse Fourier transform operator, and Fa and Fb are the Fourier transforms of the two specklegrams where the asterisk indicates the complex conjugate. This image correlation results in a peak whose displacement from zero equals the displacement of one image relative to the other. Several issues arise with this method of measurement. First, zero displacement for these operations normally lies in the upper left corner (for the DADisP program), so the quadrants of the correlation array must be re-arranged to center the zero-displacement point, and care must be taken in doing this to keep track of which point actually corresponds to zero displacement. Next, in order to measure the image displacement to subpixel accuracy, the discrete values calculated by Eq. (9) must be modeled by a continuous function and its maximum located between the discrete values. As pointed out by Sjödahl and Benckert10, the function used to do this modeling will influence the results obtained. The model chosen here is a Gaussian function. An array of 10 by 10 values surrounding the peak was selected and the natural logarithms calculated of those values. Ideally, these values should be fitted to a bi-quadratic function, to fit the data to a Gaussian function, but a cubic spline function was used instead, because that was available in the DADisP program, and interpolation was performed to 100th of

206 the spacing between the correlation values. For this data, the magnification was 12.56 so that the pixel spacing on the object was 92.94 µm, and interpolation to 100th of that spacing resulted in a resolution of 0.93 µm. x trans. µm 8.00 16.0 32.0 64.0 128 256 472

Phase diff. x meas. µm 9.339 16.27 34.14 66.12 131.6 257.2 473.3

Phase diff. y meas. µm -1.043 -0.1983 1.619 3.152 3.351 4.044 4.941

Correlation x meas. µm 6.5 13.0 30.7 66.0 128.3 258.4 474.0

Correlation y meas. µm -1.8 -0.93 0.93 1.8 1.8 2.8 2.8

Table 7. Comparison of displacement measurement obtained via the FFT shift theorem and via image correlation using the same data as for table 5. 6. Discussion and Conclusions Based on these results, we may state that the process presented here, using incoherent speckles and 8-bit digitization, can measure displacements to within a few micrometers and strains to approximately 30 microstrain. In this setup, the gage length for the strain measurement is approximately 15 mm owing to the magnification of the lens system. With one-to-one imaging, the gage length could be reduced to about 1.2 mm, and the accuracy should remain the same. The process is considerably less accurate using laser speckles, due probably to speckle decorrelation. The laser speckle decorrelation results, most likely, from the relatively small aperture of the camera lens and the small fill factor of the detector array. The distinction between laser and incoherent speckles observed here should be observable with image correlation as well and should be investigated. The displacement results obtained for image correlation are shown here to be of similar accuracy to the method of transform phase subtraction. Each method has advantages and problems, however, due to the digital nature of the data. With the phase subtraction method, there is a clear connection between the phase difference of Fourier transforms and displacement via the shift theorem, and it is natural to fit the digital values obtained to a linear function of transform coordinates. This method requires applying phase unwrapping to the wrapped phase difference of two transforms; however, the unwrapping method used here, via calculated phase unwrap regions, is extremely fast and robust. Although this whole process has not been integrated into a single program, there is no reason to believe that such an integrated program would not be as rapid and easy to implement as existing correlation programs. The correlation method is straightforward in that it measures the translation needed to get the best correlation between the two images. Subpixel resolution can only be obtained, however, by interpolating between the discrete values obtained, and this is dependent upon the nature of the model of the continuous function used to interpolate between pixels. A Fourier series expansion was shown to work well in Ref. 9 and a cubic spline fit to the natural logarithm of the correlation values was shown to work well here; however, there really is no compelling argument for any model over another beyond its demonstrated accuracy. In any case, the two measurement methods are distinctly different and equivalent only to the extent that they give similar results when analyzing the same data. It is not the intent of this communication to present the FFT phase subtraction method as better than the correlation method but simply to present it as an alternative that, with future investigation, may possibly be shown to have advantages. References 1. 2. 3. 4. 5. 6.

A. E. Ennos, “Speckle Interferometry,” in Progress in Optics Vol. XVI, Emil Wolf, Ed. (North-Holland, Amsterdam, 1978), Chap. 4, pp. 235-290. K. A. Stetson and G. B. Smith, “Heterodyne readout of specklegram halo fringes,” Appl. Opt., 19, 3031-3033 (1980). K. A. Stetson, "Speckle and Its Application to Strain Sensing," Proc. SPIE 353, 12-18 (1983). K. A. Stetson, "The Use of Heterodyne Speckle Photogrammetry to Measure High Temperature Strain Distributions," Proc. SPIE, 370, 46-55 (1983). K. A. Stetson, "The Effect of Scintillation Noise in Heterodyne Photogrammetry," Appl. Opt. 23, 920-923 (1984). K. A. Stetson, "Strain Measurement Using Heterodyne Photogrammetry of White-Light Specklegrams," Exp. Mech., 25, 312-315 (1985).

207 7.

K. A. Stetson, J. Wahid, and P. Gauthier "Noise-immune phase unwrapping by use of calculated wrap regions, Appl. Opt. 36, 4830-4838 (1997). 8. Mikael Sjödahl, “Digital Speckle Photography” in Digital Speckle Pattern Interferometry and Related Techniques, P. K. Rastoji, (Ed.), J. Wiley & Sons, New York, 2001, Chap. 5. 9. J. M. Huntley, “Automated Analysis of Speckle Interferograms” in Digital Speckle Pattern Interferometry and Related Techniques, P. K. Rastoji, (Ed.), J. Wiley & Sons, New York, 2001, Chap. 2, pp. 89-95. 10. M. Sjödahl and L. R. Benckert, “Electronic speckle photography: analysis of an algorithm giving the displacement with subpixel accuracy,” Appl. Opt. 32, 2278-2284 (1993).

MEASUREMENT OF RESIDUAL STRESSES IN DIAMOND COATED SUBSTRATES UTILIZING COHERENT LIGHT PROJECTION MOIRÉ INTERFEROMETRY C.A. Sciammarella*, A. Boccaccio**, M.C. Frassanito**, L. Lamberti** and C. Pappalettere** *

Northern Illinois University, Department of Mechanical Engineering, 590 Garden Road, DeKalb, IL 60115, USA. [email protected] **Politecnico di Bari, Dipartimento di Ingegneria Meccanica e Gestionale, Viale Japigia 182, Bari, 70126, ITALY. E-mail: [email protected] , [email protected], [email protected], [email protected]

Abstract Thin film technology is an area of great importance in current applications of opto-electronics, electronics, MEMS and computer technology. A critical issue in thin film technology is represented by residual stresses that arise when thin films are applied to a substratum. Residual stresses can be very large in magnitude and may result in detrimental effects on the role of the thin film must play. For this reason it is very important to perform “online” measurements in order to control variables influencing residual stress. The research work presented in the paper represents the first step towards the practical solution of such a challenging problem. A methodology to measure residual stresses utilizing reflection/projection moiré interferometry to measure deflections of thin coated specimens is developed. Results are in good agreement with experimental values provided by well established measurement techniques. A special optical circuit for the in situ measurement of residual stresses is designed trying to satisfy the constraints deriving from the tight geometry of the vacuum system utilized to carry out the deposition.

1. INTRODUCTION Thin film technology has great importance in opto-electronics, electronics, MEMS and computer technology applications. A critical issue in thin film technology is represented by the presence of residual stresses that develop when thin films are applied to a substratum. Residual stresses arise because of the existence of discontinuous interfaces, inhomogeneous thermal history during deposition or subsequent fabrication process, and various imperfections by ion bombardment [1]. Residual stresses may affect significantly mechanical properties and reliability of the thin film as well as the performance of thin-film based devices: in particular, high residual stresses will result in detrimental effects on the role that the thin film is designed to play. Experimental techniques for measuring residual stresses in thin films are critically revised in Ref. [2]. In general, there are two possible approaches to this problem: (i) lattice strain based methods, including X-ray diffraction and neutron diffraction; (ii) physical surface curvature-based methods. However, the measured values of residual stress may be quite different because the former methods provide local information while the latter methods provide average values. Lattice-based methods rely on the principle that residual strains and hence residual stresses are caused by the relative displacement between atomic planes: therefore, the variations of lattice spacing are measured. However, these methods are rather expensive in terms of experimental equipment and can be used only if the films are crystalline. Determination of residual stress through curvature measurements is relatively easy to perform by the experimental point of view. The average level of residual stress can be found by using the classical Stoney’s equation [3,4] which is valid when the coating is much thinner than the substrate. Nanoindentation is the most recent approach to the problem of measuring residual stresses in thin films: this is done by observing how the force-indentation curve changes with respect to a stress free surface (see, for example, the discussion presented in Ref. [1] and the references cited in that paper). Among curvature measurement methods, non-contact optical techniques are preferable in view of their high sensitivity and because they do not alter the specimen surface. Classical interferometry (i.e., Newton’s rings) [5] or moiré techniques [6] can be used for measuring curvatures. Reflection moiré allows to measure the slope of a reflective surface and, through differentiation of the fringe pattern in the frequency space, the curvature of the surface. Projection moiré measures the height of a surface with respect to a reference plane. A system of lines is projected onto the specimen surface and modulated by the curved specimen. The topography of the surface can be obtained from the phase difference generated by the modulation of the grating lines due to the surface curvature. The reconstructed surface can be fitted by a mathematical function and then differentiated in order to compute curvature values. Variables influencing residual stress could be controlled by performing in situ measurements of quantities such as deflections and curvatures that are directly related to stress. This work represents a first step towards the practical solution of such a challenging problem. For that purpose, a reflection moiré interferometry setup is developed. The T. Proulx (ed.), Optical Measurements, Modeling, and Metrology, Volume 5, Conference Proceedings of the Society for Experimental Mechanics Series 9999999, DOI 10.1007/978-1-4614-0228-2_25, © The Society for Experimental Mechanics, Inc. 2011

209

210 novelty is in the fact that the optical setup is now utilized in the projection moiré mode: therefore, the experimental setup measures displacements but not slopes. The validity of the optical setup developed in this paper is tested in the measurement of residual stress developed in a diamond-like-carbon (DLC) thin film deposited on a quartz substratum via Plasma Enhanced Chemical Vapor Deposition (PECVD). The value of residual stresses developed in the film and in the substrate are finally derived from the average curvature of the DLC specimen determined via moiré measurements. The optical circuit is then adapted to the vacuum system utilized to make the deposition taking care to satisfy the tight geometry constraints posed by the layout of the deposition reactor.

2. STRESS ANALYSIS OF THE THIN FILM In the Plasma Enhanced Chemical Vapor Deposition (PECVD) process for thin films, an electric discharge generates film precursors, such as neutral radicals and ions, by electron-impact decomposition. Significant residual stresses can develop in the coated specimen because the atoms passing from the gas phase to the substrate during the adsorption process do not reach their correct position in the reticular structure finally formed. The energetic ions cause atoms to be incorporated into spaces in the growing film which are smaller than the usual atomic volume. This leads to expansion of the film outwards from the substrate. In the plane of film, however, the film is not free to expand and the entrapped atoms cause macroscopic compressive stresses. Figure 1 shows the stress distributions developed in the coating and in the substrate by the deposition process. The coating/substrate system can be considered as a plate subject to bending. Since the substrate is much thicker than the coating, it will present the classical bi-triangular stress distribution while the compressive stress field in the coating can be considered uniform.

w(x,y) Y

X

Z

Figure 1. Schematic representation of stresses developed in the coating and in the substrate

Following the Kirchhoff’s plate theory, stress components can be expressed as:

⎧ Esz ⎛ ∂ 2w ∂2w ⎞ ⎜ ⎟ +ν ⎪σ xx = − 2 2 ∂ y 2 ⎟⎠ 1 − ν s ⎜⎝ ∂x ⎪ ⎪ Esz ⎛ ∂ 2w ∂ 2w ⎞ ⎪ ⎜ ⎟ +ν ⎨σ yy = − 2 2 ∂x 2 ⎟⎠ 1 − ν s ⎜⎝ ∂y ⎪ ⎪ E z ∂ 2w ⎪ τ xy = − s ⋅ 1 + ν s ∂x ∂y ⎪⎩

(1)

where: w(x,y) is the out-of-plane displacement experienced by the middle plane of the substrate; z is the distance from the middle plane measured in the direction of the curvature radius; Es and νs are respectively the Young modulus and the Poisson ratio of the substrate material. If the thickness of the deposited film ff is much smaller than the thickness of the substrate ts (i.e. tf