Multiaxial Fatigue and Deformation: Testing and Prediction (ASTM Special Technical Publication, 1387)

STP 1387 Multiaxial Fatigue and Deformation: Testing and Prediction Sreeramesh Kalluri and Peter J. Bonacuse, editors ...

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STP 1387

Multiaxial Fatigue and Deformation: Testing and Prediction Sreeramesh Kalluri and Peter J. Bonacuse, editors

ASTM Stock Number: STP1387

ASTM 100 Barr Harbor Drive P.O. Box C700 West Conshohocken, PA 19428-2959 Printed in the U.S.A.

Library of Congress Cataloging-in-Publication Data Multiaxial fatigue and deformation: testing and prediction/Sreeramesh Kalluri and Peter J. Bonacuse, editors. p. cm.--(STP; 1387) "ASTM stock number: STP 1387." Includes bibliographical references and index. ISBN 0-803-2865-7 1. Materials-Fatigue. 2. Axial loads. 3. Materials-Dynamic testing. 4. Deformations (Mechanics) I. Kalluri, Sreeramesh. I1. Bonacuse, Peter J., 1960TA418.38.M86 2000 620.11126-dc21 00-059407

Copyright 9 2000 AMERICAN SOCIETY FOR TESTING AND MATERIALS, West Conshohocken, PA. All rights reserved. This material may not be reproduced or copied, in whole or in part, in any printed, mechanical, electronic, film, or other distribution and storage media, without the written consent of the publisher.

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Peer Review Policy Each paper published in this volume was evaluated by two peer reviewers and at least one editor. The authors addressed all of the reviewers' comments to the satisfaction of both the technical editor(s) and the ASTM Committee on Publications. The quality of the papers in this publication reflects not only the obvious efforts of the authors and the technical editor(s), but also the work of the peer reviewers. In keeping with long-standing publication practices, ASTM maintains the anonymity of the peer reviewers. The ASTM Committee on Publications acknowledges with appreciation their dedication and contribution of time and effort on behalf of ASTM.

Printed in Philadelphia,PA October2000

Foreword

This publication, Multiaxial Fatigue and Deformation: Testing and Prediction, contains papers presented at the Symposium on Multiaxial Fatigue and Deformation: Testing and Prediction, which was held in Seattle, Washington during 19-20 May 1999. The Symposium was sponsored by the ASTM Committee E-8 on Fatigue and Fracture and its Subcommittee E08.05 on Cyclic Deformation and Fatigue Crack Formation. Sreeramesh Kalluri, Ohio Aerospace Institute, NASA Glenn Research Center at Lewis Field, and Peter J. Bonacuse, Vehicle Technology Directorate, U.S. Army Research Laboratory, NASA Glenn Research Center at Lewis Field, presided as symposium co-chairmen and both were editors of this publication.

Contents Overview

vii MULTIAXIAL STRENGTH OF MATERIALS

Keynote Paper: Strength of a G-10 Composite Laminate Tube Under Multiaxial Loading--D. SOCrEANDJ. WANG Biaxial Strength Testing of Isotropic and Anisotropic Monoliths--J. A. SALE~AND

1 13

M. G. JENKINS

In-Plane Biaxial Failure Surface of Cold-Rolled 304 Stainless Steel Sheets--s. J. COVEY AND P. A. BARTOLOTrA

26

MULTIAXIAL DEFORMATION OF MATERIALS

Analysis of Characterization Methods for Inelastic Composite Material Deformation Under Multiaxial Stresses--J. AHMAD, G. M. NEWAZ, AND T. NICHOLAS Deformation and Fracture of a Particulate MMC Under Nonradial Combined Loadings--D. w. A. REESAND Y. H. J. AU M u l t i a x i a l S t r e s s - S t r a i n N o t c h Analysis--A. BUCZYNSKI AND G. GLINKA Axial-Torsional Load Effects of Haynes 188 at 650 ~ C----c. J. LlSSENDEN,M. a. WALKER, AND B. A. LERCH

A Newton Algorithm for Solving Non-Linear Problems in Mechanics of Structures Under Complex Loading Histories--M. ARZT,W. BROCKS,ANDR. MOHR

41 54 82

99 126

FATIGUE LIFE PREDICTION UNDER GENERIC MULTIAXIAL LOADS

A Numerical Approach for High-Cycle Fatigue Life Prediction with Multiaxial Loading--M. DE FREITAS, B. LI, AND J. L. T. SANTOS Experiences with Lifetime Prediction Under Multlaxial Random Loading--K. POTTER,F. YOUSEFI, AND H. ZENNER

Generalization of Energy-Based Multiaxial Fatigue Criteria to Random Loading--T. LAGODA AND E. MACHA Fatigue Strength of Welded Joints Under Multiaxial Loading: Comparison Between Experiments and Calculations--M. WITT,F. YOUSEFLANDH. ZENNER

139 157 173 191

FATIGUE LIFE PREDICTION UNDER SPECIFIC MULT1AXIAL LOADS

The Effect of Periodic Overloads on Biaxial Fatigue of Normalized SAE 1045 Steel--J. J. F. BONNEN AND T. H. TOPPER Fatigue of the Quenched and Tempered Steel 42CrMo4 (SAE 4140) Under Combined In- and Out-of-Phase Tension and Torsion---a. LOVaSCH,~. BOMAS,AND P. MAYR

In-Phase and Out-of-Phase Combined Bendlng-Torsion Fatigue of a Notched Specimen--J. PARKANDD. V. NELSON

213

232 246

vi

CONTENTS

The Application of a Biaxial Isothermal Fatigue Model to Thermomechanical Loading for Austenitic Stainless Steel--s. v. ZAMRIKANDM.L. RENAULD Cumulative Axial and Torsional Fatigue: An Investigation of Load-Type Sequencing Effects--s. KALLURI AND P. J. BONACUSE

266 281

MULTIAXIAL FATIGUE LIFE AND CRACK GROWTH ESTIMATION

A New Multiaxial Fatigue Life and Crack Growth Rate Model for Various In-Phase and Out-of-Phase Strain Paths--A. VARVANI-FARAHANIANDT. H. TOPPER Modeling of Short Crack Growth Under Biaxial Fatigue: Comparison Between Simulation and Experiment--H.A. SUHARTONO, K. POTTER, A. SCHRAM, AND H. ZENNER

305

323

Micro-Crack Growth Modes and Their Propagation Rate Under Multiaxial Low-Cycle Fatigue at High Temperature--N. ISOBEANDS. SAKURAI

340

MULTIAXIAL EXPERIMENTAL TECHNIQUES

Keynote Paper: System Design for Multiaxial High-Strain Fatigue Testing--R. D. LOHR An In-Plane Biaxial Contact Extensometer--o. L. KRAUSEANDP. A. BARTOLOTTA Design of Specimens and Reusable Fixturing for Testing Advanced Aeropropulsion Materials Under In-Plane Biaxial Loading--J. R. ELLIS,G. S. SANDLASS,AND M. BAYYARI

Cruciform Specimens for In-Plane Biaxiai Fracture, Deformation, and Fatigue Testing----c. DALLE DONNE, K.-H. TRAUTMANN, AND H. AMSTUTZ Development of a True Trlaxlal Testing Facility for Composite Materials--J. s. WELSH AND D. F. ADAMS

Indexes

355 369

382 405 423 439

Overview Engineering materials are subjected to multiaxial loading conditions routinely in aeronautical, astronautical, automotive, chemical, power generation, petroleum, and transportation industries. The extensive use of engineering materials over such a wide range of applications has generated extraordinary interest in the deformation behavior and fatigue durability of these materials under multiaxial loading conditions. Specifically, the technical areas of interest include strength of the materials under multiaxial loading conditions, multiaxial deformation and fatigue of materials, and development of multiaxial experimental capabilities to test materials under controlled prototypical loading conditions. During the last 18 years, the American Society for Testing and Materials (ASTM) has sponsored four symposia to address these technical areas and to disseminate the technical knowledge to the scientific community. Three previously sponsored symposia have yielded the following Special Technical Publications (STPs): (1) Multiaxial Fatigue, ASTM STP 853, (2) Advances in Multiaxial Fatigue, ASTM STP 1191, and (3) Multiaxial Fatigue and Deformation Testing Techniques, ASTM STP 1280. This STP is the result of the fourth ASTM symposium on the multiaxial fatigue and deformation aspects of engineering materials. A symposium entitled "Multiaxial Fatigue and Deformation: Testing and Prediction" was sponsored by ASTM Committee E-8 on Fatigue and Fracture and its Subcommittee E08.05 on Cyclic Deformation and Fatigue Crack Formation. The symposium was held during 19-20 May 1999 in Seattle, Washington. The symposium's focus was primarily on state-of-the-art multiaxial testing techniques and analytical methods for characterizing the fatigue and deformation behaviors of engineering materials. The objectives of the symposium were to foster interaction in the areas of multiaxial fatigue and deformation among researchers from academic institutions, industrial research and development establishments, and government laboratories and to disseminate recent developments in analytical modeling and experimental techniques. All except one of the 25 papers in this publication were presented at the symposium. Technical papers in this publication are broadly classified into the following six groups: (1) Multiaxial Strength of Materials, (2) Multiaxial Deformation of Materials, (3) Fatigue Life Prediction under Generic Multiaxial Loads, (4) Fatigue Life Prediction under Specific Multiaxial Loads, (5) Multiaxial Fatigue Life and Crack Growth Estimation, and (6) Multiaxial Experimental Techniques. This classification is intended to be neither exclusive nor all encompassing for the papers published in this publication. In fact, a few papers overlap two or more of the categories. A brief outline of the papers for each of the six groups is provided in the following sections. Multiaxial Strength of Materials

Multiaxial strengths of metallic and composite materials are commonly investigated with either tubular or cruciform specimens. Three papers in this section address multiaxial strength characterization of materials. The first, and one of the two keynote papers in this publication, describes an experimental study on the strength and failure modes of woven glass fiber/epoxy matrix, laminated composite tubes under several combinations of tensile, compressive, torsional, internal pressure, and external pressure loads. This investigation illustrated the importance of failure modes in addition to the states of stress for determining the failure envelopes for tubular composite materials. The second paper describes a test rig for biaxial flexure strength testing of isotropic and anisotropic materials with the pressure-on-ring approach. The tangential and radial stresses generated in the disk specimens and the strains measured at failure in the experiments are compared with the theoretical predictions. The vii

viii

OVERVIEW

third paper deals with in-plane biaxial testing of cruciform specimens manufactured from thin, coldrolled, 304 stainless steel sheets. In particular the influence of texture, which occurs in the material from the rolling operation, on the effective failure stress is illustrated and some guidelines are proposed to minimize the rejection rates while forming the thin, cold-rolled, stainless steel into components.

Multiaxial Deformation of Materials Constitutive relationships and deformation behavior of materials under multiaxial loading conditions are the subjects of investigation f6r the five papers in this section. The first paper documents detailed analyses of tests performed on off-axis tensile specimens and biaxially loaded cruciform specimens of unidirectional,fiber reinforced, metal matrix composites. The simplicity associated with the off-axis tensile tests to characterize the nonlinear stress-strain behavior of a unidirectional composite under biaxial stress states is illustrated. In addition, the role of theoretical models and biaxial cruciform tests for determining the nonlinear deformation behavior of composites under multiaxial stress states is discussed. Deformation and fracture behaviors of a particulate reinforced metal matrix alloy subjected to non-radial, axial-torsional, cyclic loading paths are described in the second paper. Even though the composite's flow behavior was qualitatively predicted with the application of classical kinematic hardening models to the matrix material, it is pointed out that additional refinements to the model are required to properly characterize the experimentally observed deformation behavior of the composite material. The third paper describes a methodology for calculating the notch tip stresses and strains in materials subjected to cyclic multiaxial loading paths. The Mroz-Garud cyclic plasticity model is used to simulate the stress-strain response of the material and a formulation based on the total distortional strain energy density is employed to estimate the elasto-plastic notch tip stresses and strains. The fourth paper contains experimental results on the elevated temperature flow behavior of a cobalt-base superalloy under both proportional and nonproportional axial and torsional loading paths. The database generated could eventually be used to validate viscoplastic models for predicting the multiaxial deformation behavior of the superalloy. Deformation behavior of a rotating turbine disk is analyzed with an internal variable model and a Newton algorithm in conjunction with a commercial finite element package in the fifth paper. Specifically, the inelastic stress-strain responses at the bore and the neck of the turbine disk and contour plots depicting the variation of hoop stress with the number of cycles are discussed.

Fatigue Life Prediction under Generic Multiaxial Loads Estimation of fatigue life under general multiaxial loads has been a challenging task for many researchers over the last several decades. Four papers in this section address this topic. The first paper proposes a minimum circumscribed ellipse approach to calculate the effective shear stress amplitude and mean value for a complex multiaxial loading cycle. Multiaxial fatigue data with different waveforms, frequencies, out-of-phase conditions, and mean stresses are used to validate the proposed approach. Multiaxial fatigue life predictive capabilities of the integral and critical plane approaches are compared in the second paper for variable amplitude tests conducted under bending and torsion on smooth and notched specimens. Fatigue life predictions by the two approaches are compared with the experimental results for different types of multiaxial tests (pure bending with superimposed mean shear stress; pure torsion with superimposed mean tensile stress; and in-phase, 90 ~ out-of-phase, and noncorrelated bending and torsional loads) and the integral approach has been determined to be better than the critical plane approach. In the third paper, a generalized energy-based criterion that considers both the shear and normal strain energy densities is presented for predicting fatigue life under multiaxial random loading. A successful application of the energy method to estimate the fatigue lives under uniaxial and biaxial nonproportional random loads is illustrated. Estimation of the fatigue lives of welded joints subjected to multiaxial loads is the subject of the fourth paper. Experimental results on flange-tube type welded joints subjected to cyclic bending and torsion are reported and a

OVERVIEW

ix

fatigue lifetime prediction software is used to calculate the fatigue lives under various multiaxial loading conditions.

Fatigue Life Prediction under Specific Multiaxial Loads Biaxial and multiaxial fatigue and life estimation under combinations of cyclic loading conditions such as axial tension/compression, bending, and torsion are routinely investigated to address specific loading conditions. Five papers in this publication address such unique issues and evaluate appropriate life prediction methodologies. The effects of overloads on the fatigue lives of tubular specimens manufactured from normalized SAE 1045 steel are established in the first paper by performing a series of biaxial, in-phase, tension-torsion experiments at five different shear strain to axial strain ratios. The influence of periodic overloads on the endurance limit of the steel, variation of the crack initiation and propagation planes due to changes in the strain amplitudes and strain ratios, and evaluation of commonly used multiaxial damage parameters with the experimental data are reported. Combined in- and out-of-phase tension and torsion fatigue behavior of quenched and tempered SAE 4140 steel is the topic of investigation for the second paper. Cyclic softening of the material, orientation of cracks, and fatigue life estimation under in- and out-of-phase loading conditions, and calculation of fatigue limits in the normal stress and shear stress plane both with and without the consideration of residual stress state are reported. High cycle fatigue behavior of notched 1%Cr-Mo-V steel specimens tested under cyclic bending, torsion, and combined in- and out-of-phase bending and torsion is discussed in the third paper. Three multiaxial fatigue life prediction methods (a von Mises approach, a critical plane method, and an energy-based approach) are evaluated with the experimental data and surface crack growth behavior under the investigated loading conditions is reported. The fourth paper illustrates the development and application of a biaxial, thermomechanical, fatigue life prediction model to 316 stainless steel. The proposed life prediction model extends an isothermal biaxial fatigue model by introducingfrequency and phase factors to address time dependent effects such as creep and oxidation and the effects of cycling under in- and out-of-phase thermomechanical conditions, respectively. Cumulative fatigue behavior of a wrought superalloy subjected to various single step sequences of axial and torsional loading conditions is investigated in the fifth paper. Both high/low load ordering and load-type sequencing effects are investigated and fatigue life predictive capabilities of Miner's linear damage rule and the nonlinear damage curve approach are discussed.

Multiaxial Fatigue Life and Crack Growth Estimation Monitoring crack growth under cyclic rnultiaxial loading conditions and determination of fatigue life can be cumbersome. In general, crack growth monitoring is only possible for certain specimen geometries and test setups. The first paper proposes a multiaxial fatigue parameter that is based on the normal and shear energies on the critical plane and discusses its application to several materials tested under various in- and out-of-phase axial and torsional strain paths. The parameter is also used to derive the range of an effective stress intensity factor that is subsequently used to successfully correlate the closure free crack growth rates under multiple biaxial loading conditions. The second paper on modelling of short crack growth behavior under biaxial fatigue received the Best Presented Paper Award at the symposium. The surface of a polycrystalline material is modeled as hexagonal grains with different crystallographic orientations and both shear (stage I) and normal (stage II) crack growth phases are simulated to determine crack propagation. Distributions of microcracks estimated with the model are compared with experimental results obtained for a ferritic steel and an aluminum alloy subjected various axial and torsional loads. Initiation of fatigue cracks and propagation rates of cracks developed under cyclic axial, torsional, and combined axial-torsional loading conditions are investigated for 316 stainless steel, 1Cr-Mo-V steel, and Hastelloy-X in the third paper. For each material, fatigue microcrack initiation mechanisms are identified and appropriate strain parameters to correlate the fatigue crack growth rates are discussed.

X

OVERVIEW

Multiaxial Experimental Techniques State-of-the-art experimental methods and novel apparati are necessary to generate multiaxial deformation and fatigue data that are necessary to develop and verify both constitutive models for describing the flow behavior of materials and fatigue life estimation models. Five papers in this publication address test systems, extensometers, and design of test specimens and fixtures to facilitate multiaxial testing of engineering materials. The second of the two keynote papers reviews progress made in the design of multiaxial fatigue testing systems over the past five decades. Different types of loading schemes for tubular and planar specimens and the advantages and disadvantages associated with each of those schemes are summarized in the paper. Development of an extensometer system for conducting in-plane biaxial tests at elevated temperatures is described in the second paper. Details on the calibration and verification of the biaxial extensometer system and its operation under cyclic loading conditions at room temperature and static and cyclic loading conditions at elevated temperatures are discussed. Designing reusable fixtures and cruciform specimens for in-plane biaxial testing of advanced aerospace materials is the topic of investigation for the third paper. Feasibility of a fixture arrangement with slots and fingers to load the specimens and optimal specimen designs are established with finite element analyses. Details on three types of cruciform specimens used for biaxial studies involving fracture mechanics, yield surfaces, and fatigue of riveted joints are described in the fourth paper. Methods used for resolving potentially conflicting specimen design requirements such as uniform stress distribution within the test section and low cost of fabrication are discussed for the three types of specimens. The final paper describes the development and evaluation of a computer-controlled, electromechanical test system for characterizing mechanical behavior of composite materials under biaxial and triaxial loading conditions. Verification of the test system with uniaxial and biaxial tests on 6061-T6 aluminum, biaxial and triaxial test results generated on a carbon/epoxy cross-ply laminate, and proposed modifications to the test facility and specimen design to improve the consistency and accuracy of the experimental data are discussed. The papers published in this book provide glimpses into the technical achievements in the areas of multiaxial fatigue and deformation behaviors of engineering materials. It is our sincere belief that the information contained in this book describes state-of-the-art advances in the field and will serve as an invaluable reference material. We would like to thank all the authors for their significant contributions and the reviewers for their critical reviews and constructive suggestions for the papers in this publication. We are grateful to the excellent support received from the staff at ASTM. In particular, we would like to express our gratitude to the following individuals: Ms. Dorothy Fitzpatrick, Ms. Hannah Sparks, and Ms. Helen Mahy for coordinating the symposium in Seattle, Washington; Ms. Monica Siperko for efficiently managing the reviews and revisions for all the papers; and Ms. Susan Sandler and Mr. David Jones for coordinating the compilation and publication of the STP.

Sreeramesh Kalluri Ohio Aerospace Institute NASA Glenn Research Center at Lewis Field Cleveland, Ohio Symposium Co-Chairman and Editor

Peter J. Bonacuse Vehicle Technology Directorate US. Army Research Laboratory NASA Glenn Research Center at Lewis Field Cleveland, Ohio Symposium Co-Chairman and Editor

Multiaxial Strength of Materials

Darell Socie a and Jerry Wang 2

Strength of a G-IO Composite Laminate Tube Under Multiaxial Loading REFERENCE: Socie, D. and Wang, J., "Strength of a G-10 Composite Laminate Tube Under Multiaxial Loading," Multiaxial Fatigue and Deformation: Testing and Prediction, ASTM STP 138Z S. Kalluri and P. J. Bonacuse, Eds., American Society for Testing and Materials, West Conshohocken, PA, 2000, pp. 3-12.

ABSTRACT: An experimental study of the strength and failure behavior of an orthotropic G-10 glass fiber-reinforced epoxy laminate has been conducted. Tubular specimens were loaded in combinations of tension, compression, torsion, internal pressure, and external pressure to produce a variety of stress states. Previous work involved the loading of two simultaneously applied in-plane stresses. This investigation furthers the previous work by simultaneously applying three in-plane stresses. One interesting observation from this work is that combined axial compression and torsion loading results in a much lower failure strength than combined hoop compression and torsion even though the stress state is identical. For the same torsion stress, axial compression is more damaging than hoop compression because torsion loading rotates fibers aligned in the axial direction to accommodate the shear strains. Hoop fibers do not rotate and remain aligned in the compressive loading direction. A simple failure mode dependent maximum stress theory that considers low-energy compressive failure modes such as delamination and fiber buckling provides a reasonable fit to the experimental data. KEYWORDS: composite strength, multiaxial loading, failure theories

High specific strength and stiffness of composite materials make them attractive candidates for replacing metals in many weight-critical applications. Many of these applications involved complicated stress states. Although the behavior of composite materials has been studied for many years, much of the work on multiaxial stress states has been limited to theoretical studies and off-axis testing. Failure of composite materials is more complicated than monolithic materials because: (1) Failure modes of composite materials under a particular stress state are determined not only by their internal properties such as constituent properties and microstructural parameters, but also by geometric variables, loading type, and boundary conditions. (2) Stress caused by applied external loads does not distribute homogeneously between the fiber and matrix because of large differences between their elastic properties. From a strength viewpoint, composite materials cannot be considered as homogeneous anisotropic materials. Failure of composite materials is controlled by either the fiber, matrix, or interface between them, depending on the geometry and external loading. (3) Identical laminae have different behavior in various angle-ply laminates. Laminate failure is difficult to predict with only the lamina properties. Failure of composite laminates can be studied from many different levels: micromechanics, lamina, and laminates. Failure behavior of composite laminates is expected to be predicted by the properties of individual lamina which might be obtained from basic properties of the resin and matrix. Mechanical Engineering Department, University of Illinois at Urbana-Champaign, Urbana, IL. 2 Ford Motor Company, Dearborn, MI.

Copyright9

by ASTM International

3 www.astm.org

4

MULTIAXIALFATIGUE AND DEFORMATION

However, the failure behavior of composite laminates in a structure is much more complicated. This complication is demonstrated in that failure behavior among constituents, lamina, and laminate are quite different. Lamina properties, particularly those involving in-plane shear, are not easily obtained from the properties of the constituents. Interaction between the fiber and resin cannot be predicted from properties of the constituents. In composite structural analysis, laminate properties are frequently obtained from laminate theory with properties of the lamina obtained from experiments. To model complicated behavior of a structure, various anisotropic strength criteria have been developed for both the lamina and laminate level. Anisotropic strength theories may be classified broadly into one of three categories. In the first category, anisotropic strength theories are failure mode dependent. Failure will occur if any or all of the longitudinal, transverse, or shear stresses or strains exceed the limits determined by unidirectional tests. The simplest forms include maximum stress and maximum strain theories. These simple estimates have been shown to overestimate the strength in the comer regions of the failure envelope [1]. Many extensions to these simple ideas have been made to accommodate different failure modes. For example, Hart-Smith [2] advocates cutting off the corners of the failure envelope to account for shear failure modes caused by in-plane principal stresses of opposite signs. In the second category, anisotropic strength theories are failure mode independent and a gradual transition from one failure mode to another is assumed. Although they have been developed many years ago, the Tsai [3] and Tsai-Wu [4] failure theories are still widely used failure criteria. Almost all failure mode independent strength criteria are in the form Fz + (Fijo-i~) '~ = 1, with or without nonlinear terms. For an in-plane loading ~x, ~y and ~-this criterion becomes FIt"x + F2o'y + F6T + (Fll O'2 + F22o-y2 + 2F12o'xO'y +

F66~'2) ~ =

1

(1)

The term F12o'x oy represents the interaction among stress components and is negative to account for shear produced by in-plane loadings of opposite signs. Jiang and Tennyson [5] have added cubic terms to Eq 1 in the form Fi + F i j ~ + Fijko-i~o'k = 1. These types of failure theories contain enough adjustable constants, Fi, to include many failure modes. In the absence of an applied shear stress, these criteria predict that composite laminates are stronger in biaxial tension or biaxial compression loading and are weaker under biaxial tension-compression loading. Although the parameters can be adjusted to fit different sets of test data, physical meaning of the parameters and resulting failure envelope described by these criteria are not very clear and can lead to unrealistic results when extrapolated outside the range of test data. The third group of models includes micromechanical theories where stresses and strains in the matrix and fibers are computed. Ardic et al. [6] use strains computed from classical lamination plate theory for the laminate as input to calculate the strains in each layer using a three-dimensional elasticity approach. Layer strains are then used to compute fiber and matrix stresses and strains. Failure surfaces are then constructed based on the allowable stresses and strains for the fiber, matrix, and lamina. Sun and T a t [7] have computed failure envelopes with linear laminated plate theory using a failure criterion that seperates fiber and matrix failure modes. Many lamina failure criteria and laminate failure analysis methods have been proposed [8]. Soden et al. [9] provides a good review of the predictive capabilities of failure theories for composite laminates. They reported that predicted failure loads for a quasi-isotropic carbon/epoxy laminate varied by as much as 1900% for the various failure theories considered. This paper presents new test results that explore the failure envelope for combined loading experiments utilizing glass fiber-reinforced epoxy G-10 laminate tubes. These results are combined with previous test results [10,11] on the same composite to evaluate the failure envelope for three simultaneously applied in-plane stresses.

SOCIE AND WANG ON MULTIAXIAL LOADING

5

TABLE 1--Loading combinations. 1

Tension Compression Internal pressure External pressure Torsion

2

3

4

5

X

6

7

X

X

X X

X X

X X

8

9

X X

X

10

11

12

13

X

14

X X X

X X

15

X

X

X X

X

X X

X

X

Experiments

In this study, a N E M A / A S T M G-10 epoxy resin reinforced laminate with E-glass plain woven fabric was used. This industrial composite was selected because it is commercially available in both sheet and tubular forms. E-glass plain woven fabric consists of fill and warp yarns crossing alternatively above and below the adjacent yarns along the entire length and width. Fiber volume fractions in the two perpendicular directions are slightly different such that the nominal fiber volume in the fill direction is about 75% of that in the warp direction. This results in a laminate with nearly equal tensile strengths in both directions. The laminates are stacked in plies with fill fibers in the same direction. Crimp angles, a measure of the waviness of the fibers, for both fill and warp yarns were less than l0 ~ Tubular specimens with an inside diameter of 45 mm and length of 300 mm were employed in this study with the fill fibers running along the axis of the tube. Specimens were mechanically ground to reduce the wall thickness from 5 to 3 mm to form a reduced gage section with a length of 100 mm. Specially designed test fixtures were used to achieve tensile or compressive stresses in the warp (hoop) direction. A mandrel was used for internal pressure tests to generate hoop tension. Hoop compression was obtained with an external pressure vessel that used high pressure seals on the grip diameter of the specimens. These fixtures were placed in a conventional tension-torsion servohydraulic testing system to generate the various combinations of in-plane loads given in Table 1. Fifteen different combinations of loading were used in the study. Failure is determined in the pressure loading experiments by a sudden loss of pressure. This corresponds to a longitudinal split in the tube. In torsion, failure is determined by excessive angular deformation which corresponds to a spiral crack around the circumference of the tube. Tension and compression failures are determined by a sudden drop in load. Additional details of the specimen and test system can be found in Ref 10. Results and D i s c u s s i o n

The failure envelope for combinations of biaxial tension and compression loading is shown in Fig. 1. These test results are shown by the open square symbols. The X symbols are the results of threeaxis loading and will be discussed later in the paper. Failure modes were determined by scanning electron microscope (SEM) observations of failed specimens and are indicated in the figure. The tensile strength in both the fill and warp directions is similar. Compressive strength in the warp direction is much lower than that in the fill direction for the tubular specimen. This is caused by a change in failure mode. In the fill direction, the failure mechanism is out-of-plane kinking of the fibers. Figure 2 illustrates the difference between in-plane and out-of-plane shear stresses for the composite laminate. Under this loading condition, the in-plane shear failure stress is twice as large as the outof-plane shear failure stress. Both sets of fill and warp fibers need to be broken for an in-plane failure while only one set of either fill or warp fibers needs to be broken for an out-of-plane or tensile failure. Delamination failures occurred during compressive loading in the warp direction. This is a

6

MULTIAXlAL FATIGUE AND DEFORMATION 400 -

Tensile fiber fracture

Out-of-plane kinking

q

.

j

EL -400 |

T

I

-200

200

I

I I

400

I I

_.~____~I Delamination / ~ -40(1 afu ~ , M P a

FIG. l--Biaxial tension-compression failure envelope.

common failure mode in hoop compression of a tubular specimen. In contrast to tube tests, small coupon specimens cut from fiat plates show the same compressive strength in both fill and warp directions and fail in a mode known as kink buckling. The shear cutoff predicted by many theories for tension-compressionloading is not observed in this material. For a stiff fiber and soft matrix, the interaction between fill and warp fibers will be small. External loads are carried by fibers parallel to the applied loads. Although each fiber is in an in-plane biaxial stress state, the transverse stress on a fiber is small because the more compliant matrix accommodates the transverse strain. A simple rule of mixtures approach based on fiber modulus, matrix modulus, and volume fraction shows that the transverse stresses in the fibers are less than 15% of the longitudinal fiber stress during equibiaxial tensile loading so that little interaction is expected between ten-

.4-----

1 In-plane shear

Out-of-plane shear

FIG. 2--1n-plane and out-of-plane shear stress.

SOCIE AND WANG ON MULTIAXIAL LOADING

7

sile loads in the fill and warp directions. The net result is that there is little interaction between loads in the axial and hoop direction and final composite failure is dictated by the lowest energy failure mode in either direction for all combinations of biaxial tension and compression loading along the fill and warp directions. Compressive loading in the hoop direction is expected to generate delamination failures between the plies. Kachanov [12] employed a simple energy analysis to model the delarnination buckling of composite tubes under external pressure. The critical compressive stress, o'er, of tubular specimens is given by

F(hol '

(R, 1"2 Kr~ho)J

+

~cr-- 0.916Ew I_\Ri)

(2) Kr = 4.77yEwRi where Ew is the composite elastic modulus in the warp direction, ho is the thickness of the buckled layer, Ri is the inner radius, and y is the specific fracture energy according to Griffith. The weak layer can be found by differentiating the above expression with the result ho = (Kr/2) 1/3 Ri. For the epoxy resin, y is about 700 J/m2 [13,14] and the critical failure stress is computed to be 200 MPa. This is about 18% higher than the experimental data. Experimental evidence of delaminationis shown in Fig. 8 of Ref 10. It is worth noting that compression and tension-compression tests of a flat plate G-10 laminate specimen did not show evidence of delamination [15]. For these tests, the compressive strengths in the fill and warp directions were the same and the failure envelope shown in Fig. 1 was a square. This shows the importance of considering the specimen design when evaluating failure criteria for any particular application. Two types of in-plane shear can be applied to a tubular specimen: (1) tension and compression along the fiber directions, or (2) with torsion applied along the tube axis. Under torsion loading, shear stresses act in the direction of the fibers. During torsion loading, most of the in-plane shear stress is first taken by the soft matrix as both fill and warp yarns rotate. Interaction between fill and warp yarns under in-plane shear loading influence the failure strength even though the shear strength is predominantly controlled by the weaker matrix. The failure envelope for hoop stress and torsion is given in Fig. 3 and in Fig. 4 for axial stress and torsion. These test results are shown by the open square sym-

Matrixcracking Interfacedebonding \ Fiber pun-out ~

Tensileliberfracture

,ooX-- "E~I/ / U

Delamination~ [ ~ - - ~ I I I I

!

-400

.

.

.

.

.

.

.

I I I I I

I-~

IITI'I -200

.

I

0

200

IT~

warp I

400

o.. m , MPa FIG. 3--Failure envelope for combined tension~compression and torsion in warp direction.

8


Matrix cracking Interface debonding Fiber puU-out X

,oo \

Fiber buckling ~

I -400

El"

x D, E~Jee"

I -200

Tensile fiber fracture

~

0

ll'-I

I I'lTn I 200 400

~r~, MPa

FIG. 4--Failure

envelope for combined tension~compression and torsion in fill direction.

bols. The X symbols are the results of three axis loading and will be discussed later in the paper. Two distinct types of behavior are observed. Even though the stress states are identical, there is an interaction between axial compression and torsion shown in Fig. 4. No interaction between hoop compression and torsion was observed in the test results shown in Fig. 3. No interaction was observed between tension and torsion loads in either direction. For a combination of tensile stress in the axial direction and in-plane shear stress, the tensile stress is carried by the fill fibers and the in-plane shear stress is carried by the matrix. This laminate should not be affected by the direction of the in-plane shear stress and the failure envelope should be symmetric about the O'fill-O'warp plane. In tension the macroscopic failure surface is perpendicular or 90 ~ to tube axis. Fracture surfaces in torsion are oriented 60 ~ with respect to the tube axis. The combined loading experiments failed on one of these planes. Two different failure modes are found on the fracture surface but there is no observed interaction between the failure mechanisms. Under SEM examination, the failure surface oriented at 90 ~ shows a typical tensile failure mode of fiber fracture while the 60 ~ planes show evidence of matrix cracking, interface debonding, and fiber pull-out typical of the torsion tests. Combined tension in the hoop direction and shear loading resulted in the same failure mechanisms that were observed in the axial direction. The maximum in-plane shear strain is about 20% which corresponds to a 10 ~ rotation of the fill fibers from the axial direction. While these strains may be considered unreasonably high for a high-performance composite, they could easily occur in a composite pressure vessel and piping system during an overload condition. The combined action of the applied tensile and shear stresses increases the fiber stress about 9% compared to that of uniaxial tension so that a small reduction in the strength may be expected. Scatter in the data was such that this small difference could not be observed and the addition of an in-plane shear stress did not reduce the tensile strength of the laminate. Hoop or warp compression and shear loading results in failures that are caused by either delamination followed by out-of-plane kinking as a result of the hoop compressive stress or by matrix cracking and interface debonding followed by fiber pullout. It might be anticipated that the interface debonding from the torsion stresses would lead to premature delamination from the compressive loads and result in a lower strength. This was not observed in either the experiments or the SEM observations of the fracture surfaces and we conclude that the applied in-plane shear stress does not change the failure mode or strength in combined loading in this direction. Figure 4 shows a substantial interaction between the shear and compressive stress. Fiber buckling is the dominant compression failure mechanism. Fracture surfaces for torsion and combined torsion

SOClE AND WANG ON MULTIAXIALLOADING

9

FIG. 5--Failure surfaces. and compression are compared in Fig. 5. In torsion the final fracture plane was oriented about 30~ to the axial direction and perpendicular to the specimen surface. Evidence of matrix cracking, interface debonding and final fracture by fiber pullout is shown in the SEM micrograph for torsion. Since Gl0 is a woven fabric laminate, fibers pull out in bundles. The addition of axial compression changed the failure mode to in-plane fiber buckling shown in Fig. 5 followed by out-of-plane kinking. The macroscopic fracture surface was oriented 45 ~ to the specimen surface. The difference in behavior of the hoop and fill fibers is illustrated in Fig. 6. Rotated fill fibers lose compression load-carrying ability. These fiber rotations from the in-plane shear loading lead to much lower compressive strengths because it activates a low energy fiber buckling mechanism followed by a kink band failure. Fiber rotations did not affect the tensile load-carrying capability. Hoop fibers do not rotate and the compressive load-carrying capacity is not reduced by an additional in-plane shear loading. Budiansky and Fleck [16] have shown that remote shear stresses activate yielding within a microbuckle band and greatly reduce the compressive strength of unidirectional composites with a remotely applied shear stress, r. The critical compressive stress, ~rcr,is found to be

1.2ry-r ~

FIG. 6--Rotation of fill fibers.

(3)

10

MULTIAXIAL FATIGUE AND DEFORMATION

FIG. 7--Failure envelope.

where ~-yis the shear yield strength of the matrix and th is the initial misalignmentangle of the fibers. Equation 3 suggests that the shear stress will have a large influence on the compressive strength. This work was extended by Jelf and Fleck [17] to include the effects of fiber rotations under combined compression and torsion loading. Their data for unidirectional carbon fiber epoxy tubes follows the same linear reduction in compressive strength with applied torsion that is shown in Fig. 4. The addition of an in-plane shear stress does not affect the delamination failure mode because delamination is not controlled by shear yielding of the matrix. Since no degradation of compressive strength was observed in the hoop direction, we conclude that fiber rotations are more important than shear yielding of the matrix. This failure mode would only be identified by compression-torsion testing of a tubular specimen. Test data from Figs. 1, 3, and 4 are combined into a single failure envelope in Fig. 7. The failure envelope can be described by five material properties: in-plane shear strength, fill tensile strength, fill compressive strength, warp tensile strength, warp compressive strength, and the knowledge of the interaction between compressive and in-plane shear loading in the fill direction. A series of experiments was conducted to probe the extremes of the three-dimensional failure envelope. Five combinations of loading shown in Fig. 8 were selected for testing. Table 2 gives the expected failure stresses normalized with the static strength for each direction. A negative ratio indicates compression. Specimens were loaded in load, torque, and internal pressure control with a ratio between them determined by the expected failure strength. A common command signal was used to control the three loads in the tests and no attempt was made to control the exact phasing between the channels. All of the loads should be in-phase; however, each test took several minutes and that is well

FIG. 8--Experimental load points.

SOCIE AND WANG ON MULTIAXIALLOADING

11

TABLE 2--Experimental results for combined loading. Expected

A-1 A-2 A-3 B-1 C-1 C-2 D-1 E-2 E-2

Observed

O'filI

O'war p

T

O'fil 1

O'wax p

7

1.0 1.0 1.0 -0.3 -0.7 -0.7 -0.3 -0.7 -0.7

1.0 1.0 1.0 1.0 1.0 1.0 0.7 0.7 0.7

1.0 1.0 1.0 0.7 0.3 0.3 0.7 0.3 0.3

0.98 0.88 0.70 -0.32 -0.48 -0.67 -0.40 -0.96 -0.79

0.83 0.88 0.76 0.96 0.71 1.01 0.78 0.95 0.91

0.82 0.90 0.60 0.65 0.24 0.34 0.79 0.48 0.40

within the control capabilities of the servohydraulic system. Failure is expected when any one of the stress components reaches the expected strength. The first series of tests designated A in Fig. 8 was designed so that all three stress components reached a maximum at the same time, There were three repetitions of this test. Macroscopic fracture surfaces were examined and compared to those under uniaxial loading. Specimen A-1 had a fracture surface that closely resembled that of a uniaxial tensile test. Specimens A-2 and A-3 fractured from the hoop tension loading. When a specimen contains a vertical split along the specimen axis, we conclude that internal pressure was the first failure mode. If tension or shear fractures occurred first, the specimen would leak oil and the internal pressure would decrease and not be able to split open the tube. Once a large tensile or hoop crack forms, the specimen loses torsional stiffness and the shear loads lead to final fracture. None of these tests reached the expected failure strengths and one of the tests failed at loads much lower than the other tests. Results of these three tests are plotted in Fig. 1 with the X symbols. These data fall in line with the other data shown in Fig. 1 that do not have shear loading. The dashed lines are drawn through the uniaxial strengths rather than as a best fit to all the data to form the expected failure envelope. For high stresses, the data for tension-tension loading falls inside the failure envelope indicating some interaction between the two stress systems at high loads. Similarly, the test data for compression-compression loading in Fig. l also falls inside the failure envelope. The remainder of the tests, B-E, were conducted in a region where there is interaction between the in-plane shear and normal stresses. The loading was chosen so that none of the specimens would be expected to fail from the torsion loading. Rather, the torsion loads were expected to reduce the compressive load-carrying capacity in the fill direction. All of these tests had fracture surfaces that were similar to uniaxial compression tests in the fill direction. The failure plane was perpendicular to the axial direction and oriented 45 ~ to the specimen surface indicating that the failures were due to outof-plane shear stresses. Results of these three tests are plotted in Fig. 4 with the X symbols. The failure envelope was constructed by drawing a straight line between the shear and compressive strength rather than a fit to the experimental data. All of the data scatter around this line.

Summary Longitudinal and transverse fiber stresses are decoupled in a composite laminate with stiff fibers and a compliant matrix such as the G-IO woven fabric laminate used in this study. As a result, a simple maximum stress theory provides a reasonable fit to the experimental data for combined tensiontension multiaxial loading when low-energy failure mode cutoffs are employed. The in-plane shear cutoff predicted by many of the anisotropic strength criteria for composite laminates under a biaxial

12


tension-compression loading was not observed. More important, tubular specimens have low-energy compressive failure modes such as delamination and fiber buckling that must be considered. Delamination results in a lower compressive strength in the hoop direction when compared to the axial direction, and a delamination cutoff must be added to the maximum stress criterion for hoop compression tests of tubular specimens. The state of stress for axial compression and torsion is identical to that of hoop compression and torsion. The failure modes and resulting strengths are quite different. Fiber rotations in the axial direction lead to fiber buckling and a strong interaction is observed between torsional shear and axial compressive loads. These interactions are not predicted by any of the anisotropic strength theories. Failure mode-dependent theories are required to obtain the failure envelope of this material.

Acknowledgment The three-dimensional loading tests were conducted by Mr. David Waller for a course entitled "Laboratory Investigations in Mechanical Engineering" at the University of Illinois.

References [1] Abu-Farsakh, G. A. and Abdel-Jawad, Y. A., "A New Failure Criterion for Nonlinear Composite Materials," Journal of Composites Technology and Research, JCTRER, Vol. 16, No. 2, 1994, pp. 138-145. [2] Hart-Smith, L. J., "Predictions of a Generalized Maximum Shear Stress Criterion for Certain Fiberous Composite Laminates," Composites Science and Technology, Vol. 58, 1998, pp. 1179-1208. [3] Tsai, S. W., "Strength Characteristics of Composite Materials," NASA CR-224, April, 1965. [4] Tsai, S. W. and Wu, E. M., "A General Theory of Strength for Anisotropic Materials," Journal of Composite Materials, Vol. 5, 1971, pp. 58-80. [5] Jiang, Z. and Tennyson, R. C., "Closure of the Cubic Tensor Polynomial Failure Surface," Journal of Composite Materials, Vol. 23, 1989, pp. 208-231. [6] Ardic, E. S., Anlas, G., and Eraslanoglu, G., "Failure Prediction for Laminated Composites Under Multiaxial Loading," Journal of Reinforced Plastics and Composites, Vol. 18, No. 2, 1999, pp. 138-150. [7] Sun, C. T. and Tao, J., "Prediction of Failure Envelops and Stress/Strain Behavior of Composite Laminates," Composites Science and Technology, Vol. 58, 1998, pp. 1125-1136. [8] Nahas, M. N., "Survey of Failure and Post-Failure Theories of Laminated Fiber-Reinforced Composites," Journal of Composite Technology Research, Vol. 8, 1986, pp. 1138-1153. [9] Soden, P. O., Hinton, M. J., and Kaddour, A. S., "A Comparison of the Predictive Capabilities of Current Failure Theories for Composite Laminates," Composites Science and Technology, Vol. 58, 1998, pp. 1225-1254. [10] Wang, J. Z. and Socie, D. F., "Biaxial Testing and Failure Mechanisms in Tubular G-10 Composite Laminates" ASTMSTP 1206, 1993, pp. 136-149. [11] Socie, D. F. and Wang, Z. Q., "Failure Strength and Mechanisms of a Woven Composite Laminate Under Multiaxial In-Plane Loading," Durability and Damage Tolerance, ASME AD-Vol. 43, 1994, pp. 149-164. [12] Kachanov, L. M., Delamination Buckling of Composite Materials, Kluer Academic Publishers, 1988. [13] Sih, G. C., Hilton, P. D., Badaliance, R., Shenberger, P. S., and Villarreal, G., "Fracture Mechanics for Fibrous Composites," ASTM STP 521, 1973, pp. 98-132. [14] Browning, C. E. and Schwartz, H. S., "Delamination Resistance Composite Concepts," ASTM STP 893, 1986, pp. 256-265. [15] Wang, Z. Q. and Socie, D. F., "A Biaxial Tension-Compression Test Method for Composite Laminates," Journal of Composites Technology and Research, JCTRER, Vol. 16, No. 4, 1994, pp. 336-342. [16] Budiansky, B. and Fleck, N. A., "Compressive Failure of Fiber Composites," Journal of Mechanics and Physics of Solids, Vol. 41, No. 1, 1993, p. 183. [17] Jelf, P. M. and Fleck, N. A., "The Failure of Composite Tubes Due to Combined Compression and Torsion," Journal of Materials Science, Vol. 29, 1994, pp. 3080-3084.

J. A. S a l e m I a n d M. G. Jenkins 2

Biaxial Strength Testing of Isotropic and Anisotropic Monoliths REFERENCE: Salem, J. A. and Jenkins, M. G., "Biaxial Strength Testing of Isotropic and Anisotropic Monoliths," Multiaxial Fatigue and Deformation: Testing and Prediction, ASTM STP 1387, S. Kallnri and P. J. Bonacuse, Eds., American Society for Testing and Materials, West Conshohocken, PA, 2000, pp. 13-25. ABSTRACT: A test apparatus for measuring the multiaxial strength of circular plates was developed and experimentally verified. Contact and frictional stresses were avoided in the highly stressed regions of the test specimen by using fluid pressurization to load the specimen. Both isotropic plates and singlecrystal NiA1 plates were considered, and the necessary strain functions for anisotropic plates were formulated. For isotropic plates and single-crystal NiA1 plates, the maximum stresses generated in the test rig were within 2% of those calculated by plate theory when the support ring was lubricated. KEYWORDS: anisotropy, single crystals, ceramics, composites, multiaxial strength, nickel aluminide, tungsten carbide, displacement, strain, stress

Nomenclature An Bn bq Cz Do D* ep eq ~p ~z ki k* P r Rs RD So

SDxz t v o-q O-p O-q

Series constant in anisotropic, plate displacement solution Series constant in anisotropic, plate displacement solution Reduced elastic stiffness Constants in the anisotropic displacement stress solution Flexural rigidities Effective flexural rigidity of an anisotropic plate Measured major principal strain component Measured minor principal strain component Measured principal strain uncorrected for transverse sensitivity Measured strain uncorrected for transverse sensitivity; i = 1,2,3 Reduced flexnral rigidity Effective, reduced flexural rigidity of an anisotropic plate Pressure Radius Radius of support ring Radius of disk test specimen Elastic compliance Standard deviation of x~ variable Disk test specimen thickness Poisson's ratio Stress component Measured major principal stress Measured minor principal stress

1 NASA Glenn Research Center, MS 49-7, 21000 Brookpark Rd., Cleveland, OH 44135. University of Washington, Box 352600, Seattle, WA 98195.

Copyright9


13 www.astm.org

14


(a)

~\\\\\\\\\\\\\\\\\~

(b)

KN\\\\\\\\\\\\\\\\\~

fT"

P

(c)

[~\\\\\\\\\\\\\\\\\"~1

J_ t

f FIG. 1--Schematic of typical testing configurations used to generate biaxial tensile stresses in plate specimens: (a) ball-on-ring, (b) ring-on-ring, and (c) pressure-on-ring.

O'rr Radial stress o00

Tangential stress

~rO Shear stress o's w x y z

Correction term for effect of lateral stresses on plate deflection Plate deflection in the z-direction Abscissa as measured from plate center Ordinate as measured from plate center Distance from midsurface of plate ranging over +_t/2

The strength of brittle materials such as ceramics, glasses, and semiconductors is a function of the test specimen size and the state of applied stress [1]. Engineering applications of such materials (e.g., ceramics as heat engine components, glasses as insulators, silicon and germanium as semiconductors) involve components with volumes, shapes, and stresses substantially different from those of standard test specimens used to generate design data. Although a variety of models [2] exist that can use conventional test specimen data to estimate the strength of large test specimens or components subjected to multiaxial stresses, it is frequently necessary to measure the strength of a brittle material under multiaxial stresses. Such strength data can be used to verify the applicability of various design models to a particular material or to mimic the multiaxial stress state generated in a component during service. Further, these materials tend to be brittle, and machining and handling of test specimens can lead to spurious chips at the specimen edges which in turn can induce failure not representative of the flaw population distributed through the materials' bulk. In the case of a plate subjected to lateral pressure, the stresses developed are lower at the edges, thereby minimizing spurious failure from damage at the edges.

SALEM

AND

JENKINS

ON

BIAXIAL

STRENGTH

TESTING

15

For components that are subjected to multiaxial bending, three different loading assemblies, shown schematically in Fig. 1, can be used to mimic component conditions by flexing circular or square plates: ball-on-ring, ring-on-ring (R-O-R), or pressure-on-ring (P-O-R). The R-O-R and the P-O-R are preferred because more of the test specimen volume is subjected to larger stresses. However, significant frictional or wedging stresses associated with the loading ring can be developed in the highly stressed regions of the R-O-R specimen [3,4]. These stresses are not generated in the P-O-R configuration. Rickerby [5] developed a P-O-R system that used a neoprene membrane to transmit pressure to a disk test specimen (diameter to thickness ratio of 2Ro/t ~ 17). The reported stresses were in excellent agreement with plate theory at the disk center (< 0.5% difference). At 0.43Rs the differences in radial and tangential stresses were -3.6 and -2.5%, respectively, and at 0.85Rs the differences were ~27 and -2.4%, respectively, where Rs is the support ring radius. The biaxial test rig used by Shetty et al. [6] included a 0.25 mm spring steel membrane between the disk test specimen (2Ro/t ~ 13) compressive surface and the pressurization medium. Despite the steel membrane, the rig was reported to produce stresses in reasonable agreement with plate theory. The measured stresses at the disk center were -3.5% greater than theoretical predictions. The radial and tangential stresses were -1.5 and -1.9% greater, respectively, at 0.25Rs, and at 0.8Rs the radial error was -10%. Reliability calculations are strongly dependent on the peak stress regions, and thus the differences need to be small in the central region of the disk. Although the overall differences are not large, -10% toward the disk edge, they are somewhat greater than Rickerby's at the highly stressed central region. This may be due to the clamped edge of the steel membrane. The objective of this work was to design, build, and experimentally verify a P-O-R biaxial flexure test rig for strength and fatigue testing of both isotropic and anisotropic materials. One goal was to eliminate the membrane between the pressurization medium and the test specimen, thereby eliminating interaction between the test specimen and membrane.

Biaxial Test Apparatus The rigs consist of a pressurization chamber, reaction ring and cap, extensometer, and oil inlet and drain ports, as shown in Fig. 2. The desired pressurization cycle is supplied to the test chamber and

TFRT

c- v ?- E- i~i ~T,r7 iwIF T ['I D

FIG. 2--Schematic of pressure-on-ring assembly and test specimen.

16


specimen via a servohydraulic actuator connected to a closed loop controller. The feedback to the controller is supplied by a commercial pressure transducer connected to the oil inlet line. The test chamber and cap are 304 stainless steel, and the reaction ring is cold rolled, half-hard copper or steel depending on the strength of the material tested. For low strength specimens, minor misalignments or specimen curvatures can be accommodated via the copper support ring. The hydraulic oil is contained on the compressive face of the specimen by a nitrile O-ring retained in a groove. A cross section of the test rig, which can accommodate 38.1 or 50.8 mm diameter disks by using different seals and cap/reaction ring assemblies, is shown in Fig. 2. A similar rig for testing specimens with 25.4 mm diameters was also developed.

Stress Analysis of the P-O-R Test Specimen

Isotropic Materials The radial and tangential stresses generated in a circular, isotropic plate of radius RD and thickness t that is supported on a ring of radius Rs and subjected to a lateral pressure P within the support ring are [7]

O'rr

-

-

o00 = ~

8t 2

(1 - v) R~ + 2(1 + u) - (3 + v)

E

(1 - v) R--~o+ 2(1 + v) - (1 + 3v)

+ ~rs

+ os

(1)

e(3 + v) O ' s - 4(1 - u) where r is the radius of interest. The term O's is a small correction factor to the simple plate theory for the effects of the sheafing stresses and lateral pressure on the plate deflection [8]. Equation 1 is based on small-deflection theory and thus assumes that the plate is thin and deflects little relative to the plate thickness.

Anisotropic Materials The displacement solution for a circular, orthotropic plate of unit radius and thickness subjected to a unit lateral pressure was determined by Okubu [9] in the form of a series solution and as an empirical approximation. Such a solution is useful in the testing and analysis of composite plates and plates made from single crystals such as silicon, germanium, or nickel aluminide. The analysis was based on small-deflection theory and thus assumes that the plate is thin and deflects little relative to the plate thickness (i.e., less than 10%).

Approximate Solution The approximate displacement solution given by Okubu for a plate of unit radius is w ~ ( 1 - rP2 ) ( k

* - r 2)

(2)

where

1

D* = ~ (3Dlt + 2D12 + 4D66 + 3022)

k* =

7Dll + 10D12 + 12D66 + 7D22 2(Dll + 2D12 + D22)

(3)

SALEM AND JENKINS ON BIAXlAL STRENGTH TESTING

17

and Oil

t3

S22

t3

12 SHS12 - $22

Sl 1

D22 = 1-~ $11S12 - $22 (4)

--t 3 S12 D12 = 12 S l l S 1 2 - $22

t3 1 12 $66

D66

where the S o terms are the material compliances or single crystal elastic constants. The plate rigidity terms, Dii, and associated functions are written in the more standard notation used by Hearmon [10] instead of that used by Okubu [9]. For the general case of a plate of variable support radius the displacement becomes w -~ ~

P

(R~ - ? ) ( k * 1 ~ -

,~)

(5)

As the symmetry of cubic crystals and orthotropic composites is orthogonal, the elastic constants are in Cartesian form and the stress and strain solutions are determined in Cartesian coordinates: OZw

02W

ax2

~ 02W Z Oy2 , e66 = - - A Z O ~ y

__ 02__..~ W .

811 = --Z-0-'~-; e22 =

- P [2R2(k * + 1) - 12x2 - 4y 2] 64D* (6)

02W

Oy2

- P [2R~(k* + 1) - 12y 2 - 4x2] 64D* O2w OxOy

P

8D* [xy]

where z is the distance from the midsurface of the plate. The stresses are determined from the strains by [10]

0.11 = --Z~Oll " - ~

L

-I- b12 OY2 /

~w

o2.,]

(7)

where bla = $221(SalS22 - Sa2), b22 = Snl(Sa~S22 - $12), b ~ = 11Sa6, and b12 -- -S121($11S22 - S~2). As the plate is cylindrical, a description of the stresses in polar coordinates is more intuitive, and the Cartesian values at any point in the plate can be converted to polar coordinates with O'rr = Orll COS2 0 q- 0"22 sin 2 0 + 0"a2 sin 20 or00 = 0"22 COS2 0 + O"11sin 2 0 -- 012 sin 20

O'ro

=

(0"22 -- O"11) sin 0 cos 0 + o'12 cos 20

where 0 is the angle counterclockwise from the x-axis.

(8)

18


The Series Solution If the series displacement solution given by Okubu is redetermined for the case of variable radius, thickness, and pressure, the following displacement function results

2(1 - ~) ~ A n n=2

cosh(2n + 2)a' ( 2 - n ~ ( 2 - n ~ ) i~

1 ] cosh2nc~' cos2n/3" + 2)/3' - [ (2n + 1)2n + 2n(2n1)

cosh(2n - 2)a +(~n_--l)-~--_~cos(Zn - 2)/3

P~

+ 2(1 - ~) ~ B.

w = ~-~-

n=2

cosh(2n + 2)a" os(2n i~-(~--~(~-n---~-~) c /

( 1 2 ) i f ' - _ (2n+ 1)2n

2n(2n - 1)-] cosh2nd' cos2n/3"

cosh(2n - 2)a' l + (2nn-- 1)-~----2) cos(2n - 2)if' + {(Cx - C2 + C3)(cos4fl + 3) + 4(Ct

~+

4{(C4

s

C5) cos 2/3 + C4 + C5}

-

r~

C3) COS2/3 + 4C2}

+ 8C6

R~ (9)

The curvatures are 02w _ ax 2

P~J- [~=2 (A. cosh 2ncd cos 2nfl' + B~ cosh 2na" cos2nff') re + 2C4] + (6C1 + C2 + (6C1 - C2) cos 2/3) R--~

02w OY2 - PR~ t3 [ -- .~=2(A.k 2 cosh 2nct' cos 2n13' + B.k~ cosh 2nd' cos 2nff') (10) rEz + 2C5] + (6C3 + C2 - (6C3 - C2)cos 2fl) Rs

02W OxlOx2

ta

-

(Ankl sinh 2ncg sin 2n/3' q .2 + B~kz sinh 2ha" sin 2nil') + 2Cz ~ sin 213/ Rs J

SALEM AND JENKINS ON BIAXIAL STRENGTH

TESTING

19

TABLE 1--Constants ( XlO -6 rn2/MN) for NiAI and graphite/epoxy plates of unit thickness and radius subjected to a unit lateral pressure. NiAI: $22 = SI1 = 1.0428, Sa2 = -0.421, $66 = 0.892 (• 10 -5 m2/MN) [11] Cl

C2

C3

C4

C5

C6

A2

B2

A3

03

1.392

2.009

1.392

-7.253

-7.253

5.958

0.474

-0.105

10 -15

10 -16

Graphite Epoxy: Sll = 0.6667, $22 = 11.11, Sl2 = -0.2000, $66 = 14.08 (• 10-5 m2/MN) [12] C1 2.741

C2 9.046

(73 4.080

C4 -15.52

C5 -16.34

C6 12.24

A2 0.385

B2 0.385

A3 0.079

B3 0.079

where

DI/z

D~a kl = (D2 +

D4

-'F

{(D2

+ D4) 2 -

D1D3}lt2) 112

k2 = (D2 + D4

--

{(D2

+ 04) 2 -

D1D3}1/2) 1/2 (11)

and theAn, Bn, and Ci terms are constants determined from the boundary conditions, and the/3,/3' and /3" terms are functions describing the angular position of interest. The solution converges rapidly for a plate of cubic material in the "standard" orientation and only the constants A2, Bz, and Ci are needed, as shown in Table 1. For an orthotropic material such as graphite-epoxy, the higher order constants are small but significant. The stresses generated in a NiA1 (nickel aluminide) plate of {001 } crystal orientation are shown in polar coordinates in Fig. 3. The stresses are a function of both radius and angle, with the peak stresses being tangential components occurring at the (110) crystal directions. The effect of anisotropy is most

/k

Tangential Stress, r/R= 0.2 Radial Stress, r / R ==0~2 . 9 Tangential Stress, r / R = 0.8 -- Radial Stress, r/R, = 0.8

.....

Q V

---

1.2 1.0 0.8 0.6 0.4 0.2

0.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

Stress/Pressure

FIG. 3--Stresses generated in a NiA1 single crystal plate of unit radius and thickness subjected to a unit pressure.

20

MULTIAXlAL FATIGUE AND DEFORMATION

apparent at the plate edges where the stresses vary with angular position by -45% for r/Rs = 0.8. At r/Rs = 0, the stresses become equibiaxial as in the isotropic case.

Test Rig Verification Isotropic Materials Ideally a test rig will generate stresses described by simple plate theory. A comparison was made between Eq 1 and the stresses measured with stacked, rectangular strain gage rosettes placed at eight radial positions on the tensile surfaces of two 4340 steel disk test specimens and at seven positions on two WC (tungsten carbide) disk test specimens. The strain-gaged specimens were inserted, pressurized, and removed repeatedly while the strain was recorded as a function of pressure. Three supporting conditions were considered: (1) unlubricated, (2) lubricated with hydraulic oil, and (3) lubricated with an anti-seizing compound. The average of at least three slopes, as determined by linear regression of strain as function of pressure, was used to calculate the mean strains and stresses in the usual manner [13,14] at the pressure level of interest. As the calculation of stress from strain via constitutive equations requires the elastic modulus and Poisson's ratio, measurements of the steel were made with biaxial strain gages mounted on tension test specimens, and by ASTM Standard Test Method for Dynamic Young's Modulus, Shear Modulus, and Poisson's Ratio for Advanced Ceramics by Impulse Excitation of Vibration (C 1259-94) on beams fabricated from the same plate of material as the disk test specimens. The elastic modulus as estimated from the strain gage measurements was 209.3 _+ 0.9 GPa and the Poisson's ratio was 0.29, in good agreement with handbook values [15]. The elastic modulus as estimated from ASTM C 1259-94 was 209.9 • 0.5 GPa. The elastic modulus and Poisson's ratio of the WC material were measured by using ASTM C 1259-94 on ten 50.8 mm diameter disk specimens. The elastic modulus was 607 + 3 GPa and Poisson's ratio was 0.22. The stresses generated in the steel specimens with the lubricant on the copper reaction ring were consistently greater than those generated without lubricant. However, the differences were small (4.2 MPa at -400 MPa equibiaxial stress) and approximately one standard deviation of the measurements. For an applied pressure of 3.45 MPa, agreement between plate theory and the measurements on the steel specimens without lubrication on the boundary were within 1% at the disk center, within 2% at 0.49Rs, and within 7 and 8%, respectively, for the radial and tangential components at 0.75Rs. In general, the differences increase with increasing radial position, particularly for the tangential component. In contrast, the WC specimens, which were tested on a steel support due to the large strength, exhibited a substantial effect of friction. The maximum stresses decreased by - 5 % when the specimens were tested without anti-seizing lubricant, and the use of hydraulic oil on the support ring did little to reduce friction. For the WC specimens and anti-seizing lubricant on the boundary, agreement between plate theory and the measurements at a pressure of 8.3 MPa was within 2% at the disk center, within 2% at 0.49Rs, and within 6 and 9%, respectively, for the radial and tangential components at 0.75Rs. The significance of the differences between the plate theory and the measured stresses can be assessed by estimating the standard deviations and confidence bands of the measurements. The standard deviations of the strains and stresses were calculated from the apparent strain variances by applying a truncated Taylor series approximation [16] to the transverse sensitivity correction equations, the strain transformation equations, and the stress-strain relations. For a rectangular strain rosette, the standard deviations of principal stress, principal strain, and principal strain uncorrected for transverse strain errors are

SD~p

_

E N / S D ~ + ~2SD2~ 1 - v2

SALEM AND JENKINS ON BIAXIAL STRENGTH TESTING

SD,~ -

SD~o =

21

E 1 - - v 2 X'/ ~SD2" + SD2~ 1 - vokt 1 -- -s 2 %/SD}" + ~ SD2~ (12) 1 - vokt - ~

SD~q = -f-el_e_2

~,

2 S D 2 + (2~2

.

2

Ve, SDr + SD}q ~1 - ~3)2 SD22 +

+

SD~3

v5 =

(gl

-- ~2) 2 +

(2~2

- - ~1 - - g 3 ) 2

where E and v are the elastic modulus and Poisson' s ratio of the test material, vo is Poisson' s ratio of the strain gage manufacturer's calibration material, kt is the transverse sensitivity of the strain gages, ~1, ~2, ~3 are the apparent strains, and the SDxi terms are the standard deviations of the following xi variables: ~p and ~q being the principal strains uncorrected for transverse effects, ep and eq being the corrected principal strains, and o-p and O'q being the corrected principal stresses. The elastic constants in Eq 12 are assumed to be exact for a single test specimen. The results along with 95% confidence bands are summarized in Tables 2 and 3 and shown in Fig. 4 for the condition of a lubricated boundary. Because the 95% confidence bands of the tangential stress measurements on the 4340 steel specimens do not overlap the theory for radii greater than 0.5Rs, the differences are significant. The radial stresses are in good agreement for all radii. For the WC specimens, overall agreement between theory and the experiment is better than for the steel specimen.

TABLE 2--Measured stresses, standard deviations, and theoretical stresses for a 2.3-mm-thick, 51-mm-diameter 4340 steel plate supported on a 45.6-mm-diameter copper ring and subjected to 3.45 MPa uniform pressure. Radial Position Tangential Stress, MPa

Radial Stress, MPa Percent of Support Radius and Lubrication 0, Dry 17, Dry 33, Dry 49, Dry 61, Dry 74, Dry 75, Dry 76, Dry 0, Anti-seizing I 49, Anti-seizing 1 0, Clamped only 49, Clamped only

Plate Theory 2 403.8 391.5 357.4 302.1 247.9 173.5 168.0 164.1 403.8 302.1

Measured 3 401.3 +- 2.7 389.8 + 2.3 351.6 + 1.0 297.8 -+ 1.3 236.4 - 0.9 164.9 --- 3.6 156.1 - 1.5 ' 165.1 + 3.3 403.7 + 2.9 300.0 + 1.0 5.6 + 2.4 8.0 + 2.4

Percent Difference

Plate Theory2

-0.6 -0.4 -1.6 -1.4 -4.6 -5.0 -7.1 0.6 0.0 -0.7

403.8 396.8 377.6 346.4 315.7 273.8 270.6 268.5 403.8 346.4

Never-Seez, Never-Seez Compound Corp., Broadview, IL. 2 See Ref 7. 3 Mean _+ one standard deviation.

Measured 3 405.4 _+ 4.1 396.8 -+ 1.3 369.7 -+ 2.9 338.2 -+ 1.4 301.8 -+ 1.1 252.4 + 1.1 249.8 _+ 0.5 248.7 _+ 2.2 409.6 _+ 4.1 339.5 _+ 2.1 -2.2 + 1.3 -2.0 _+ 1.3

Percent Difference 0.4 0.0 -2.1 -2.4 -4.4 -7.8 -7.7 -7.4 1.4 -2.0

22

M U L T I A X l A L FATIGUE A N D D E F O R M A T I O N

TABLE 3--Measured stresses, standard deviations, and theoretical stresses for a 2.2-ram-thick, 51-rnm diameter WC plate supported on a 45.4-ram-diameter steel ring and subjected to 8.3 MPa uniform pressure. Radial Position Tangential Stress, MPa

Radial Stress, MPa Percent of Support Radius and Lubrication

Plate Theory 2

0, Dry 16, Dry 32, Dry 49, Dry 72, Dry 73, Dry 83, Dry 0, Hydraulic oil 49, Hydraulic oil 72, Hydraulic oil 73, Hydraulic oil 0, Anti-seizing] 16, Anti-seizing] 32, Anti-seizing ] 49, Anti-seizing ] 72, Anti-seizing] 73, Anti-seizing] 83, Anti-seizing ]

1005.5 977.8 899.1 755.7 466.8 449.7 288.1 1005.5 755.7 466.8 449.7 1005.5 977.8 899.1 755.7 466.8 449.7 288.1

Measured 3 939.0 950.7 878.1 701.4 403.7 349.8 262.4 942.6 709.1 410.3 377.1 983.1 972.7 892.6 756.4 458.5 423.3 271.5

Percent Difference

Plate Theory 2

-6.6 -2.8 -2.3 -7.2 -13.5 -22.2 -8.9 -6.3 -6.2 - 12.1 -16.2 -2.2 -0.5 -0.7 0.1 -1.8 -5.9 -5.8

1005.5 991.2 950.7 876.7 727.8 719.0 635.7 1005.5 876.7 727.8 719.0 1005.5 991.2 950.7 876.7 727.8 719.0 635.7

• 1.9 • 6.3 • 5.1 +_ 4.5 • 5.6 • 8.8 • 2.5 • 17.2 • 9.6 + 6.1 • • 27.5 • 2.7 • 5.3 • 18.3 • 22.9 • 20.4 • 2.6

Percent Difference

Measured 3 968.3 969.4 942.9 829.3 668.0 616.0 777.9 975.9 822.1 671.0 618.5 1019.0 992.2 962.1 866.8 712.7 655.8 789.8

• 3.0 • 8.7 • 8.6 ___6.9 + 8.2 • 11.7 • 9.2 • 10.4 • 15.3 • 3.4 • 15.6 • 19.2 • 5.3 --- 7.3 • 35.2 • 15.3 • 33.4 • 7.2

-3.7 -2.2 -0.8 -5.4 -8.2 -14.3 22.4 -2.9 -6.2 -7.8 -14.0 1.3 0.1 1.2 -1.1 -2.1 -8.8 24.2

1Never-Seez, Never-Seez Compound Corp., Broadview, IL. 2 See Ref 7. 3 Mean • one standard deviation.

T h e forces exerted by the O-ring on the test s p e c i m e n resulted in stresses on the s p e c i m e n surfaces. T h e level a n d c o n s i s t e n c y o f t h e s e stresses were m e a s u r e d at the disk center a n d at 0.49Rs b y repeatedly inserting a n d r e m o v i n g an unlubricated, steel s t r a i n - g a g e d test s p e c i m e n into a n d f r o m the fixture. T h e stresses generated b y c l a m p i n g varied with orientation a n d radial position. 125

125 ungsten Carbide

340 Steel 100

~

100

b-

~

75

n

75

\

ffl

r o9

50

09

25

o o 9 9 ----

Disk 2 Radial Disk 2, Tangential Disk 1, Radial Disk 1, Tangential Theory, Tangential Theory, Radial

\

~

~ \

\

\

N

\

0

so 25

o o ----

&

Measured, Radial Y'\ Measured, Tangential ~ Theory, Radial \ Theory, Tangential

0 0.0

0.2

0.4

0.6

0.8

1.0

Radial Position/Support Radius, r / R s

0.0

0.2

0.4

0.6

0.8

1.0

Radial P o s i t i o n / S u p p o r t radius, r/R s

FIG. 4---Measured and theoretical stresses as a function o f normalized radial position. Error bars indicate the 95% confidence bands: (left) steel disk on a copper support, and (right) tungsten carbide disk on a steel support.

23

S A L E M A N D J E N K I N S ON BIAXIAL S T R E N G T H T E S T I N G

70

70 60

~

5O

~

40

~

30

~

I

{001}

6=

t

50 40

\ ,~

20 10

9

----

Radial, Measured Tangential, Measured

\\

r r

0.2

0.4

0.6

\\ ~9

\

20 10

\ 0.8

\

z,

0

0.0

\

\

30

,,

Radial, Theory Tangential, Theory

{001}

9 ----

\

\

Measured, Radial

Measured, Tangential Theory, Radial Theory, Tangential

olo

1.0

\

o14

\

\

\\

\

o18

1.o

Radial Position/Support Radius, r/Rs

Radial P o s i t i o n / S u p p o r t Radius, r/R s

FIG. 5---Measured and theoretical stresses for a [001} NiA1 disk as a function o f normalized radial position. Error bars indicate the 95% confidence bands." (left) (100) direction and (right) (110) direction.

During seven clampings, the mean principal stresses ( + one standard deviation) were 5.6 -+ 2.4 and - 2 . 2 _+ 1.3 MPa, respectively, at the disk center, and 8.0 + 2.4 and - 2 . 0 + 1.3 MPa, respectively, at 0.49Rs. The maximum principal stresses observed during a clamping were 9.5 and 3.8 MPa at the disk center. Anisotropic Materials To compare the stresses generated in the test rig with the solutions of Okubu, single crystal NiA1 disk test specimens were machined with face of the disk corresponding to the {001 }. One specimen was strain gaged at four locations and pressurized to 4.8 MPa in the rig with anti-seizing lubricant on the steel support. The resulting stresses are shown in Fig. 5 and summarized in Table 4. The stresses calculated with the series solution are within 2% of the measured stresses at the plate center and within 7% at radii less than 50% of the support radius.

TABLE ~-Measured stresses, standard deviations, and theoretical stresses for a 1.55-ram-thick, 25.4-mm-diameter [001} NiAl single crystal plate supported on a 23.1-mm-diameter lubricated steel ring and subjected to a 4.8 MPa uniform pressure. Radial Position Tangenital Stress, MPa

Radial Stress, MPa Percent of Support Radius and Angular Position

Plate Theory 1

2, center 44, < 100 > 51, < 500> 50, < 110 >

305.7 259.8 234.2 239.8

] See Ref 9. 2 Mean • one standard deviation.

Measured 2 300.1 251.3 232.9 223.7

• 1.0 _+ 3.1 • 1.0 __+1.0

Percent Difference

Plate Theory 1

-1.8 -3.3 -5.6 -6.7

305.7 272.2 274.8 275.6

Measured 2 311.2 264.4 262.8 288.8

• • + •

1.2 1.7 1.0 1.0

Percent Difference +1.8 -2.9 -4.4 +4.8

24


1.10

1.05

| O

!

n,, 1.00

9

| 0.95

9

|

0.90

Measured

Approximate

Measured

Exact

FIG. 6--Measured and theoretical strains at failure for {001} NiA1 disk test specimens. The measured strains are normalized with Okubu's approximate and series solutions [9].

Additionally, disk test specimens were strain gaged at the center and pressurized to failure. The strain at failure is compared to those calculated with Eqs 6 and 10 in Fig. 6. The strains generated in the rig lie between those of the solutions, with the approximate solution overestimating the strains by - 5 % and the series solution underestimating the rig data by -3%. However, neither the approximate or series solutions consider the effect of lateral pressure and shear on the strains and stresses. If the isotropic correction term, os, in Eq 1 is used with the Poisson's ratio of polycrystalline NiA1 (~0.31 [17]) to approximate the error, an additional strain of -1.7% is expected, implying that the bending stresses generated by the test rig closely approximate the series solution.

Summary A test apparatus for measuring the multiaxial strength of brittle materials was developed and experimentally verified. Contact and frictional stresses were avoided in the highly stressed regions of the test specimen by using fluid pressurization to load the specimen. For isotropic plates, the experimental differences relative to plate theory were functions of radial position with the maximum differences occurring toward the seal where the stresses are the least. The maximum stresses generated in the test rig were within 2% of those calculated by plate theory when the support ring was lubricated. The effects of friction and the clamping forces due to the seal were typically less than 2% of the equibiaxial (maximum) applied stress when an unlubricated copper support ring was used. When an unlubricated steel ring was used, the effect of friction on lapped tungsten carbide was approximately 5% of the maximum stress. Application of a lubricant to the support eliminated the detectable effects of friction. For a single-crystal NiA1 plate, the maximum stresses generated in the test rig were within 2% of those calculated by plate theory when the support ring was lubricated. For radial positions of less than 50% of the support radius, the calculated and measured stresses were within 7%. The stress distribution in a single-crystal plate of cubic symmetry is a function of both radial position and orientation. The maximum stresses at any radius are tangential and occur at (110) orientations.

References [1] Weibull, W., "A Statistical Theory of the Strength of Materials," Ingeniors Vetenskaps Akademien Handlinger, No. 151, 1939. [2] Batdorf, S. B. and Crose, J. G., "A Statistical Theory for the Fracture of Brittle Structures Subjected to Nonuniform Polyaxial Stresses," Journal of Applied Mechanics, Vol. 41, No. 2, June 1974, pp. 459~-64.

SALEM AND JENKINS ON BIAXIAL STRENGTH TESTING

25

[3] Adler, W. F. and Mihora, D. J., "Biaxial Flexure Testing: Analysis and Experimental Results," Fracture Mechanics of Ceramics, Vol. 10, R. C. Bradt, D. P. H. Hasselman, D. Munz, M. Sakai, and V. Shevchenko, Eds., Plenum Press, New York, 1991, pp. 227-246. [4] Fessler, H. and Fricker, D. C., "A Theoretical Analysis of the Ring-On-Ring Loading Disk Tests," Journal American Ceramic Society, Vol. 67, No. 9, 1984, pp. 582-588. [5] Rickerby, D. G., "Weibull Statistics for Biaxial Strength Testing," Fracture 1977, Vol. 2, ICF4, Waterloo, Canada, 19-24 June 1977, pp. 1133-1141. [6] Shetty, D. K., Rosenfield, A. R., Duckworth, W. H., and Held, P. R., "A Biaxial Test for Evaluating Ceramic Strengths," Journal of the American Ceramic Society, Vol. 66, No. 1, Jan. 1983, pp. 36-42. [7] Szilard, R., Theory and Analysis of Plates, Classical and Numerical Methods, Prentice-Hall, Englewood Cliffs, NJ, 1974, p. 628. [8] Timoshenko, S. P. and Woinowsky-Krieger, S., Theory of Plates and Shells, 2nd ed., McGraw-Hill, New York, NY, 1959, p. 72. [9] Okubu, H., "Bending of a Thin Circular Plate of an Aeolotropic Material Under Uniform Lateral Load (Supported Edge)," Journal of Applied Physics, Vol. 20, Dec., 1949, pp. 1151-1154. [10] Hearmon, R. F. S., An Introduction to Applied Anisotropic Elasticity, Oxford University Press, 1961. [11] Wasilewski, R. J., "Elastic Constants and Young's Modulus of NiAI," Transactions of the Metallurgical Society ofAIME, Vol. 236, 1966, pp. 455-456. [12] Lee, H. J. and Saravanos, D. A., "Generalized Finite Element Formulation for Smart Multilayered Thermal Piezoelectric Composite Plates," International Journal of Solids Structures, Vol. 34, No. 26, 1997, pp. 3355-3371. [13] "Errors Due to Transverse Sensitivity in Strain Gages," Measurements Group Tech Note TN-509, Measurements Group, Raleigh, NC. [14] "Strain Gage Rosettes--Selection, Application and Data Reduction," Measurements Group Tech Note TN515, Measurements Group, Raleigh, NC. [15] Aerospace Structural Metals Handbook, CINDAS/USAF CRDA Handbook Operations, West Lafayette, IN, Vol. 1, 1997, p. 41. [16] Hangen, E. B., Probabilistic Mechanical Design, Wiley, New York, 1980. [17] Noebe, R. D, Bowman, R. R., and Nathal, M. V., "Physical and Mechanical Properties of the B2 Compound NiAI," International Materials Reviews, Vol. 38, No. 4, 1993, pp. 193-232.

Steven J. Covey I and PauI A. Bartolotta 2

In-Plane Biaxial Failure Surface of Cold-Rolled 304 Stainless Steel Sheets REFERENCE: Covey, S. J. and Bartolotta, P. A., "In-Plane Biaxial Failure Surface of ColdRolled 304 Stainless Steel Sheets," Multiaxial Fatigue and Deformation: Testing and Prediction,

ASTM STP 1387, S. Kalluri and P. J. Bonacuse, Eds., American Society for Testing and Materials, 2000, pp. 26--37. ABSTRACT: Cold forming of thin metallic plates and sheets is a common inexpensive manufacturing

process for many thin lightweight components. Unfortunately, part rejection rates of cold (or warm) rolled sheet metals are high. This is especially true for materials that have a texture (i.e., cold-rolled stainless steel sheets) and are being cold-formed into geometrically complex parts. To obtain an understanding on how cold forming affects behavior and subsequent high rejection rates, a series of in-plane biaxial tests was conducted on thin 0.l-ram (0.004-in.) fully cold-rolled 304 stainless steel sheets. The sheets were tested using an in-plane biaxial test system with acoustic emission. A failure surface was mapped out for the 304 stainless steel sheet. Results from this study indicated that an angle of 72 ~ from the transverse orientation for the peak strain direction during forming should be avoided. This was microstructurally related to the length-to-width ratio of the elongated 304 stainless steel grains. Thus on rejected parts, it is expected that a high number of cracks will be located in the plastic deformation regions of cold-formed details with the same orientation. KEYWORDS: in-plane biaxial failure surfaces, stainless steel, texture, cold forming, equivalent stress,

failure loads

Metals are among the most common manufacturing materials in the world. Unless cast to shape, metals are typically solidified in large billets and then subsequently processed via cold (or warm) working into near final shape. This cold working of a material into the final shape changes the material's microstructure and associated properties. In fact, the metal's grains take on a preferred orientation (or texturing) which aligns the crystal structure differently in the direction of rolling (longitudinal) than in the direction perpendicular to rolling (transverse). Texturing can transform a material with similar properties in all directions (isotropic) to one with substantial variations in material properties with direction (nonisotropic). In most cases, yield strength is higher in the rolling direction while strain-to-failure is higher in the transverse direction. Tensile strength, strength coefficient (K) and strain hardening exponent (n) values (as defined in ASTM E 646) and other mechanical properties can also he affected. For manufacturing facilities which utilize many rolling or forming operations, it is important to understand how the material properties may be evolving in each direction from one forming process to the next. During the forming of sheet metal components, a biaxial stress state is encountered by the material. Biaxial stress states can result in a much different stress-strain behavior than observed under uniaxial loading conditions. Generally, the strength, and associated forming forces, can increase by up to 30% depending on the biaxiality of the stress state as discussed by Shiratori and Ikegami [1] and Kreibig and Schindler [2]. Strain-to-failure also depends on states of stress. Another point of interest is how subsequent material behavior is affected by a substantial inelastic strain. For example, a sheet 1 St. Cloud State University, St. Cloud, MN. 2 NASA Glenn Research Center, Cleveland, OH.

Copyright9


26 www.astm.org

COVEY AND BARTOLO-I-I'AON STAINLESSSTEEL

27

metal may be plastically deformed in one manufacturing process and then subsequently deformed in another operation. It is hypothesized that these types of complex processing are typically the cause for high part rejection rates in sheet metal components. Consequently, an understanding of material behavior under complex stress states is essential for detailed tool and process design. To investigate the intricacies of the sheet metal forming process, a series of in-plane biaxial tests was conducted on thin 0.1 rnm (0.004 in.) fully cold-rolled 304 stainless steel sheets. This paper discusses the results of the study describes briefly the unique capabilities of the biaxial test system that was used to generate the failure surface data.

Material Details The material used in this study was a fully cold-rolled 304 stainless steel sheet 0.1 mm (0.004 in.) thick. Using a standard etching solution (10 mL HNO3, 10 mL acetic acid, 15 mL HCL, and 5 mL glycerol), the textured microstructure of the 304 stainless steel is clearly visible (Fig. 1). The grain length is three times longer than its width indicating the rolling direction of the material. Initial uniaxial static tests were conducted on coupon samples. These samples were cut from the same lot of 304 stainless steel as used in the subsequent biaxial tests. The test specimens were machined in two orientations: longitudinal (parallel with the rolling direction) and transverse (perpendicular with the rolling direction). The specimens were 12 mm wide by 0.1 mm thick with a 114.3mm-long test section. The extensometer gage length was 50.9 mm. The specimens were tested in displacement control at a rate of 0.5 mm/min up to 0.75 mm displacement and then at a faster displacement rate of 5 mm/s until failure.

FIG. 1--Photomicrograph of the 304 stainless steel grain structure showing that the rolling direction grain size is three times that of the transverse direction (original magnification m400, electropolished).

28

MULTIAXIAL FATIGUE AND DEFORMATION TABLE 1--Uniaxial tensile properties of 304 stainless steel sheet.

Orientation

Modulus, GPa

0.2% Yield Stress, MPa

Ultimate Tensile Strength, MPa

Failure Strain, %

n

K, MPa

Longitudinal Transverse

160 183

1225 1181

1343 1409

2.48 4.73

0.285 0.137

4437 2477

Uniaxial longitudinal and transverse properties are summarized in Table 1. The data are averages from 12 tests for each direction. Standard deviations on stress and elastic modulus values are less than 0.5%. Note that the elastic modulus values differ by almost 15% and the strain-to-failure by nearly a factor of two for this "homogeneous" material. Experimental Details

Specimen Geometry The specimens were machined from 300 m m (12-in.) square plates with geometry based on the work of Shiratori and Ikegami [1] and Kreibig and Schindler [2]. These specimens had a reduced width gage section with a double reduction of radius of curvature from about 11 m m (0.43 in.) to about half that at the comer root (Fig. 2). The intent of the specimen geometry was to induce a true uniform biaxial stress state over as much of the gage section as possible, without a large stress concentration within the comer root. Shiratori and Ikegami [1] and Kreibig and Schindler [2] report a fairly uniform stress distribution as defined by numerical, strain gage, photoelastic, and failure results. The specimen geometry used here should provide useful results even though fabrication of these thin specimens required some minor changes from those in the references. Generally, verification of stress state quality in the gage section of cruciform test specimens requires extensive finite-element analysis and utilizes a reduced thickness for optimization among the relevant parameters.

5.5 mm

f~

11 mm "-4 ~ / - - F ~ d i u s

150 m m

r

50

~_ 2; 6

tso

50

-50 -0.1

I

i

i

i

i

0.0

0.1

0.2

0.3

0.4

E,%

FIG. 8--Flow curves and strain path for stress path lb.

(b)

65

66


0.5 0.4 0.3 0.2 0.1

0.0 .0.1 0.2 .0.3 ,0.4 0.5 0.6

0.7 -0.1

i

i

i

i

l

0.0

0.1

0.2

0.3

0.4

,,.

(c)

350

250

~.

~so

50

-50 i

i

i

.0.7

.0,6

,

i

.0.5

i

t

I

-0.4

-0.3

i

i

i

-0.2

-0.1

i

i

i

0.0

0.1

7.%

FIG. 8---(Continued.)

,

i

i

0.2

0.3

,

i

i

0.4

0.5

,

(d)

67

REES AND AU ON DEFORMATION AND FRACTURE

0

9

L~

o

|

I

0 0

"-

I

o o

-7

0

~,

.~

~

.~

e d IAI 'o

I

O o

,

O o

oO

oO

o

~dlAI '~-

O

o

oo

o o

f

oo

,/

L'~ /

68


o

o o

o

o

r

o

,

.._t

,

o

i

o

c9

co

,

,

f

,

o

~-

,

o

.

1

i

o

~

~T

L.

~

co,

i

_

o ~t

0 ,13

T o

o,

o,

o

"T

tt3

co

""

T

o

%%

~,

69

REES AND AU O N D E F O R M A T I O N A N D F R A C T U R E

y ,

o co

o o

f

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,

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70


Fig. 8c. If it is to remain consistent within the positive strain quadrant in Fig. 8c, then a rotation or a local distortion of the subsequent yield locus should occur. The cross-plot in Fig. 8d shows the dependence of y upon o. The near verticals apply to tension and the horizontals to torsion. This shows that the air cylinder is capable of sustaining tension while the test piece undergoes torsion but there is some axial stress variation between cycles.

Test 3 In Fig. lc, forward and reversed torsion is superimposed upon an elastic compressive stress o- = - 2 5 0 MPa. The specimen failed in cycle 10 and Figs. 9a, b show that far greater stress and strain levels (~- = 350 MPa, y = -L-_2%,6 = - 1 . 2 % ) were achieved than in Test 2 (see Figs. 8a, b). The elastic moduli in compression were E = 100 MPa and in shear G = 38 GPa. The effect of increasing --+~'m~x within each cycle is shown in the stress-strain plots of Figs. 9a, b. These show that the width of the hysteresis loop increases approximately symmetrically by 26y p about its origin while the axial plastic compressive strain, 6e e, increments by between 0.1 and 0.2% per cycle. In a model of kinematic hardening (mirrored about the z-axis in Fig. 2b) the yield locus is raised and lowered by -+ rmax. The inelastic strains 6e P and gyp are the components of the plastic strain increment vector and the plastic strain path, 3,e versus eP, is proportional to the path traced by the center of the translating locus as shown. In this, as with all cycles involving a stress reversal, the model will describe the Bauschinger and ratcheting behavior. The deviations and irregularities observed may be attributed to asymmetries in yield locus motion arising from progressive strain damage to the material. The plot between the two total strains (Fig. 9c) shows the growth in strain within each cycle. The horizontal limbs show the elastic strain arising and recovered from the compression. By taking a gradient to this plot we see how the direction 6y/re gradually changes as the yield locus is carried to the stress point with the progress of plasticity. Most axial strain is accumulated from forward torsion which is broadly consistent with the model's prediction that path 012 produces e p and where only yP is found following the reversal at point 2. A similar observation was made for Test 2 but now the extent of compressive ratchet strain is greater than its tensile counterpart (compare Fig. 9b with Fig. 8b). In all the present tests the principal axes of stress and strain rotate. When shear stress is absent no such rotation occurs. This is the fundamental difference between conducting nonradial load tests with and without shear stress. The plot in Fig. 9d reveals how the principal direction of stress and total strain (Eqs 3b and 4b) alternate to either side of the specimen axis with increasing stress and strain in these cycles. Figures 9e a n d f show that as these axes rotate they do not remain coincident. The principal strain and stress ratios are calculated from Eqs 3a and 4a. If we were to subtract the elastic component of strain from Fig. 9e then the principal planes of plastic strain become more nearly aligned with the principal stress planes, which is an assumption made in the classical theory of isotropic plasticity.

Test 4 The loading sequence Fig. l d was preceded by two cycles of "elastic" loading. Shown in Fig. 10a is the complete strain history, which was apparently important to the integrity of this specimen. A precompression OA and superimposed forward torsion AB are sensibly elastic since their strains are recovered when these loads are removed. A further compression-tension cycle OCDO and a forwardreversed torsion cycle OEOFO also appear elastic since very little plastic strain remains with their removal. Then the first cycle 012 (see Fig. ld) was applied for which the material failed at point 2. The cross-plot Fig. 10b shows that there clearly had been some plastic shear strain from the branch 01. If the strains were wholly elastic then Figs. lOa, b would appear geometrically similar, with G the multiplying factor between their ordinates. It is suspected that a defect existed in this test piece that rendered it unable to sustain the reversal through tension to point 2.

REES A N D AU ON D E F O R M A T I O N A N D F R A C T U R E

0.5

E

0.4 0.3 0.2 0.1 0.0 0.1

p:

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i

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FIG. lO--Cross plots f o r a premature failure under stress path ld.

(b)

71

72


The path EF shows the manner in which axial strain can arise under pure torsion. It may be tensile or compressive depending upon the sense of the torque and appears with a hysteresis accompanying load-unload. This is not attributed to gage misalignment or rotation since these influences have been removed. The literature has attributed other examples of this phenomenon to anisotropy in wrought metals [20]. Here it is believed to arise from the mismatch in principal strains for a damaged composite with alternating tension and compression along its • ~ principal planes. If torsion produces axial plastic strain then there would be an increase in the axial yield stress given that a material hardens. Consequently, following a torsional prestrain path, the subsequent yield locus in or, ~"axes would show a cross-effect, i.e., a widening in a direction parallel to the ~ axis.

Test 5 The stress-strain plots shown in Figs. 1 la, b reveal further information on how a rotation might accompany the translation in the yield locus. The asymmetry in the hysteresis loops (Fig. 1 la) is due to a greater amount of plastic shear strain arising from reversed torsion. Axial strains arise with the translations to points 1 and 3, etc. (Fig. le) but are more dominant on the compressive side (2,4, etc.) which lead to a ratcheting along the negative c-axis (see Fig. 1 lb). We may interpret these results when the kinematic motion within each cycle is as shown in Fig. 2d. Greater amounts of each component strain would appear consistent within the observed strain path if we were to allow successive rotations as the yield locus translates into its subsequent positions 3 and 4, etc. These rotations can serve to accommodate either a steady or an accelerating axial ratchet strain within the normal vector as the locus is dragged back along the shear stress axis. It appears from Fig. l i b that the tensile branches to points 2, 4, etc., lie within this locus and remain essentially elastic. In returning to point 3, 5, etc., the unloading is elastic within this locus before carrying it back in the +T-direction. The rotation should only admit further plastic strain within each compressive limb to account for the ratcheting seen in Fig. 1 lb. The total strain trajectory in Fig. 1 lc again shows the inelastic strain increasing in the compressive direction while attaining a fixed strain on the tensile side. Increases in the negative shear strain account for the progressive widening of the loop in Fig. 1 la. Thus, while axial compressive ratchet strains apply to points 1, 3, etc., shear ratchet strain applies to points 2, 4, etc. This is consistent with the change to the normal gradient that should accompany a translation and rotation of the yield locus at these points. Figure 1 ld shows the alternation in the principal planes of strain following the cyclic application of shear strain.

Test 6 The specimen endured 23 cycles of the type shown in Fig. lf. Within a cycle the shear stress arising from forward torsion was increased to a given value and then the axial stress was alternated between fixed limits: from compression 1, 3, 5, etc., to tension 2, 4, 6, etc., before unloading. Figures 12a and b show the component stress-strain plots in which failure occurred when y and e reached 3.5% and - 0 . 4 % respectively at corresponding stress levels ~-= 525 MPa and o- = - 2 0 0 MPa. Figure 12a is typical of the 2124 alloy matrix elastic-plastic response to incremental torsional loading despite a slight irregularity at a peak shear stress arising from the application of compression/tension. The hysteresis is narrow, thus preserving the elasticity in shear between loading and unloading. The axial stress-strain response (see Fig. 12b) reveals that the particulates are more effective in inhibiting tensile flow, where c < 0.1%. With advanced cycling the compression branch leads to a net compressive strain with progressive ratcheting. Inelastic axial strain behavior shows that the loops widen from the origin and cross over on the compressive side thus incrementing the plastic strain. The total strain path (Fig. 12c) reveals the manner in which c accumulates with + 3,. Compressive strain dominates and is only partly recovered by the application of tension. The gradient, 6Tire, is an approximate indicator of how a rotation in the yield surface should modify the kinematic translation shown

FIG. 11--Flow curves and strain path showing compressive ratcheting under stress path le. Ca~

..-t c m

71

z

z

m 71 0

c 0 z

z

m m (~

74


C I d


75

in Fig. I f Here, if the center traces the plastic strain path, then, with the addition of elastic strains: e e = ~/E, ,/e = T/G, the total strain path may be predicted. The cross plots in Figs. 12d and e show how e and y depend, respectively, upon an alternating ~" and or. They reveal again the bias for negative axial strain ratcheting. Cycling between points l and 2 (see Fig. le) remains elastic. This is confirmed from Fig. 12b if we ignore the through-zero machine irregularity. Thus the current yield locus at point 1 will contain point 2 within its interior. Most inelastic shear strain arises with the loading from the origin to point 1 and in unloading to the origin from point 2, since here the stress path crosses the boundary. Figure 12fshows the rotation in the principal axes of stress for this test as calculated from Eq 4b. The rotations lie to either side of the 45 ~ orientations and decrease with increasing shear stress. Ratch-

FIG. 12--Flow curves, strain path, and orientations showing ratcheting for stress path I f

76


FIG.

12----(Continued.)

eting increases as the rotation decreases. With different rotations per cycle a new plane in the section is placed under maximum tension, compression, and shear. The damage arising from both modes would thus be spread over a wider area of material than under torsion alone and may serve to prolong the strain to failure. Plastic Strain Trajectories

The elastic strain components are removed from the total strains by applying: 7P = T -

~'IG a n d ~ e = ~ -

tr/E

(5a,b)


77

FIG. 12--(Continued.)

where E and G are the elastic moduli given in Table 1. The plot of 3,p versus e e defines the plastic strain trajectory in a given test. Figure 13 gives an example of the trajectory derived from applying Eqs 5a,b to the corresponding total strain plots in Fig. 9c (Test 3) for which plasticity was significant. The bias for compressive ratcheting is clearly evident as is the shift and widening of the torsional hysteresis loop. Shift of the hysteresis loop occurs negatively with accompanying widening. We have seen that according to Prager's kinematic hardening rule [17] the trajectory: (1) is directly proportional to the motion of the center, and (2) ties normal to the boundary of the current yield locus. In (1) a single work hardening constant c must connect the plastic strains (e p, TP) to the center coordinates (a,/3) of the current yield locus in an incremental manner for a nonradial path. That is: da = c de e

78


1.0

0.5

-0.0

-0.6

-1.0

-1.5

-2.0

'

'

I

i

I

I

~

l

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

-0.0

0.2

EP, %

FIG. 13--Plastic strain trajectory for Test 3.

and dfl = c d'gP. The present tests show that linear hardening cannot be assumed. Thus c is not constant; this violates the assumption of a rigid translation made within Prager's rule. An appropriate function for c for this material will be examined in a future paper. Gage Misalignment and Rotation The short gage length of 2 mm resulted in a slight misalignment 0o with the test piece axis during bonding (see Fig. 6). Once known, 0o was used to correct rosette strains for the true axial and shear strains, i.e., e and 3', aligned with the test piece axis. However, 0o is continually altered with the application of shear strain and it becomes necessary to upgrade 0o for the calculation of axis strains at each load step. For this, it is necessary to determine the shear strain lying in the direction defined by 0o. Assuming pure shear we can ignore rigid body rotation and take one half the shear strain increment d3'~ to upgrade 0o by addition or subtraction depending upon the sense of the torque. This procedure was programmed so that the dependence of 0o upon 3' could be monitored throughout each test. Figure 14 shows the worst case of Test 6, where shear strain was not reversed (see Fig. 12a). Here 0o grows l!nearly with 3' within each cycle but is disrupted by the alternating axial stress imposed at peak shear stress. The result is that 0o varies from near zero initially to - 4 ~ at fracture ( - v e means thatx lies on the opposite side of the x-axis in Fig. 6). Correspondingly, the rosette strains were eA = 1.873%, en = --0.139% and ec = -1.816%. From Eqs 2a-c the x, y coordinate strain values are ex = -0.139%, ey = 0.196%, and 3'xy = 3.689%. When the rotation effect is ignored the axial and shear strains are the ex and 3'xyvalues given above. Compare these to the test piece axis strain components: -0.392% and 3.606%, as calculated from Eqs la,b. Clearly an unacceptable error arises in the estimation of the small axial strain value despite a relatively insignificant 2% error in a far greater shear strain.

Fracture Finally, Fig. 15 shows that failure surfaces were aligned with the planes of maximum shear. As the shear stress in each test increases so these planes become more closely aligned with the axial and

REES AND AU ON DEFORMATION AND FRACTURE I

I

I

I

I

I

I

1

-1

o

-3

-4 I

I

I

I

I

I

~

l

0.0

0.5

1.0

1.5

2.0

2,5

3.0

3.5

FIG. 14--Effect o f shear strain upon gage misalignment f o r Test 6.

FIG. 15--MMC specimens showing shear failures.

79

80


transverse directions of the specimen. With the exception of an explosive fracture under compression, which fragmented the test section, crack paths lay in these directions. The specimen ends were kept in line by the grips and so torsion alone permits a relative sliding between the adjacent faces of the shear planes. Some shear cracks formed within the lead-in to the fillet radii and were accompanied by secondary cracks that ran helically into the gage section. Unlike the case of pure compression where sliding along 45 ~ planes can accommodate an axial displacement, here the axial ratcheting is due to compressive plasticity of matrix material on transverse sections. Broadly, the results of the present tests show that the flow behavior of this MMC may be understood from a knowledge of the plasticity behavior of its aluminum matrix under similar load paths. The concept of a hardening rule involving a translating yield locus has long been applied to metals and is particularly useful here to provide a qualitative description of the results obtained. This may not be surprising since the SiC particles remain brittle and do not themselves contribute to plastic strain. However, these particles impede the flow to promote a semi-brittle behavior. This is seen in the low ductility of a MMC composite compared to its matrix metal/alloy. Despite its limited strain range, the MMC composite permits cyclic applications of nonradial loads to high stress levels. Initially, the usual features of cyclic elasto-plasticity for metals also appear in the composite. These are linear elasticity, strain hardening, the Bauschinger effect, creep, ratcheting, and elastic recovery. With continued cycling some of these features become less clear, this being most likely due to an accumulation of damage under tensile stressing, where the material remains essentially brittle. That is, it does not flow plastically when tension is applied either monotonically or in a repeated manner. In contrast, compression induces a plastic ratcheting mechanism than enables it to sustain cyclic loading. It is believed that an advantageous interplay between compressive ratchet strain, residual stress, and bond strength permits repeated tensile cycling in the absence of plasticity. A possible description of the complex behavior observed may follow from assuming that each particulate acts as a metallurgical notch around which a stress concentration exists. This should be combined, say, from using the rule of mixtures, with the features of traditional matrix plasticity reported here. Conclusion The various responses of a particulate MMC to combined cyclic loading paths appear complex but are not wholly unpredictable. It has been shown how classical kinematic hardening model predictions, as applied to the matrix material, are in qualitative agreement with the composite flow behavior. These experimental results show, however, that certain refinements would be necessary to model some of the more unusual features of this material. In particular is its capability to sustain a greater degree of compressive flow in combination with essentially brittle tensile behavior. Crucial to maintaining integrity of this composite are the plastic strains from compression and the damage from tension branches of a given cycle. The material will undergo an axial compressive ratchet strain for many cycles and this appears to prolong the ability to bear tension. Microscopically, there appears to be an advantageous interplay between existing residual stress in the matrix and the subsequent strain from load cycling. Macroscopically, a continually changing internal stress can be identified with the center coordinates of a translating yield locus. The preference for matrix compressive flow suggests, within the rule of normality, that a rotation and possibly a distortion will accompany the translation. References [1] Ikegami, K., "An Historical Perspective of the Experimental Study of Subsequent Yield Surfaces for Metals," Parts 1 and 2, Brit. Ind. & Sci. Int Trans Ser., BISITS 14420, Sept 1976, The Metals Society, London. [2] Ikegami, K., "Experimental Plasticity on the Anisotropy of Metals," Proceedings, Euromech Col1115, Mechanical Behaviour of Anisotropic Solids, J-P. Boehler, Ed., No. 295, CNRS 1982, pp. 201-242. [3] Rees, D. W. A., "A Survey of Hardening in Metallic Materials," Failure Criteria of Structured Media, JP. Boehler, Ed., Balkema, 1993, pp. 69-97.

REES AND AU ON DEFORMATION AND FRACTURE [4] [5] [6] [7] [8] [9]

[10] [11] [12] [13] [14] [15] [16] [17]

[18] [19] [20]

81

Rees, D. W. A., "Deformation and Fracture of Metal Matrix Particulate Composites Under Combined Loadings," Composites PartA, Vol. 29A, 1998, pp. 171-182. Rees, D. W. A. and Liddiard, M., "Elasticity and Flow Behaviour of a Metal Matrix Composite," Key Engineering Materials, Trans Tech., Vol. 118-119, 1996, pp. 179-185. Majumdar, B. S., Yegneswaran, A. H., and Rohatgi, P. K., "Strength and Fracture Behaviour of Metal Matrix Particulate Composites," Mat Sci and Engng, Vol. 68, 1984, pp. 85-95. Everett, R. K. and Arsenault, R. J., Metal Matrix Composites: Mechanisms and Properties, Academic Press Ltd, 1991. Rees, D. W. A., "Applications of Classical Plasticity Theory to Non-Radial Loading Paths," Proceedings, Royal Society, Vol. A410, 1987, pp. 443--475. Rees, D. W. A., "A Review of Stress-Strain Relations and Constitutive Relations in the Plastic Range," Journal of Strain Analysis, Vol. 16, No. 4, 1981, pp. 235-249. Chaboche, J. L., "Time-Independent Constitutive Theories for Cyclic Plasticity, International Journal of Plasticity, Vol. 2, No. 2, 1986, pp. 149-188. Lamba, H. S. and Sidebottom, O. M., "Cyclic Plasticity for Non-Proportional Load Paths, Parts 1 and 2, J1 Engng Mat Tech, Vol. 100, 1978, pp. 96-103, pp. 104--111. McDowell, D. L., "Experimental Study on Structure of Constitutive Equations for Non-Proportional Cyclic Plasticity," Jl Engng Mater Technol., Vol. 107, 1985, pp. 307-315. Abdul-Latif, A., Clavel, M., Feruey, V., and Saanouni, K., "Modelling of Non-Proportional Cyclic Plasticity of Waspalloy," Jl Engng Mat Tech, VoL 116, No. 1, 1994, pp. 35-44. Beruallal, A., Cailletaud, G., Chaboche, J. L., Marquis, D., Nouailhas, D., and Rousser, M., "Description and Modelling of Non-Proportional Effects in Cyclic Plasticity," Proceedings: Biaxial andMulti-Axial Fatigue, M. W. Brown and K. J. Miller, Eds., EG3 Pub. 3, 1989, pp. 107-129. Armstrong, P. J. and Frederick, C. O., "A Mathematical Representation of the Multi-Axial Bauschinger Effect," CEGB Rpt. RD/B/N/731. Chaboche, J. L., "Constitutive Equations for Cyclic Plasticity and Cyclic Visco-Plasticity," International Journal of Plasticity, Vol. 5, No. 3, 1989, pp. 247-302. Prager, W., "A New Method of Analyzing Stresses and Strains in Work Hardening Plastic Solids," Journal of Applied Mechanics, Vol. 23, 1956, pp. 483-496. Phillips, A., "A Review of Quasistatic Experimental Plasticity and Viscoplasticity," International Journal of Plasticity, Vol. 4, No. 2, 1986, pp. 315-328. Wu, H. C. and Yeh, W. C., "Experimental Determination of Yield Surfaces and Some Results of Annealed Stainless Steel," International Journal of Plasticity, Vol. 7, No. 8, 1991, pp. 803-826. Billington, E. W., "Non-Linear Response of Various Metals: Permanent Length Changes in Twisted Tubes," Jl Phys D: Appl Physics, Vol. 10, 1977, pp. 533-545.

A. Buczynski I and G. Glinka 2

Multiaxial Stress-Strain Notch Analysis REFERENCE: Buczynski, A. and Glinka, G., "Multiaxial Stress-Strain Notch Analysis," Multiaxial Fatigue and Deformation: Testing and Prediction, ASTM STP 1387, S. Kalluri and P. J. Bonacuse, Eds., American Society for Testing and Materials, West Conshohocken, PA, 2000, pp. 82-98.

ABSTRACT: Fatigue and durability analyses require the use of analytical and/or numerical methods for calculating elastic-plastic notch tip stresses and strains in bodies subjected to nonproportional loading sequences. The method discussed in the paper is based on the incremental relationships, which relate the elastic and elastic-plastic strain energy densities at the notch tip and the material stress-strain behavior, simulated according to the Mroz-Garud cyclic plasticity model. The formulation described below is based on the equivalence of the total distortional strain energy density, which appears to give the upper-bound estimations for the elastic-plastic notch tip strains and stresses. The formulation consists of a set of algebraic incremental equations that can easily be solved for elasticplastic stress and strain increments, based on the increments of the hypothetical elastic notch tip stress history and the material stress-strain curve. The validation of the proposed model against the experimental and numerical data includes several nonproportional loading histories. The basic equations involving the equivalence of the strain energy density are carefully examined and discussed. Finally, the numerical procedure for solving the two sets of equations is briefly described. The method is particularly suitable for fatigue life analyses of notched bodies subjected to cyclic multiaxial loading paths. KEYWORDS: notches, multiaxial stress state, elastic-plastic strain analysis

Nomenclature Coordinates of center of mth (fro) plasticity surface Modulus of elasticity Equivalent strain energy density e~ Actual elastic-plastic strains at notch tip e~j Hypothetical elastic strains at notch tip G Shear modulus of elasticity K' Cyclic strength coefficient Kr Stress concentration factor due to axial load Kr Stress concentration factor due to torsional load k,n Load increment n u m b e r 11p Cyclic strain hardening exponent 8ii Kronecker delta, 6ij 1 for i = j and 6ij = 0 for i :~ j Plastic strain increments Elastic strain increments Actual elastic-plastic strain increments aSe~ Equivalent plastic strain increment Pseudo-elastic stress increments

E ESED

=

i Institute of Heavy Machinery Engineering, Warsaw University of Technology, ul. Narbutta 85, 02-524 Warsaw, Poland. 2 Department of Mechanical Engineering, University of Waterloo, Ontario N2L 3G1, Canada.

Copyright9


82 www.astm.org

BUCZYNSKI AND GLINKA ON NOTCH ANALYSIS

83

A,~ Actual stress increments Ao-eaq Actual equivalent stress increment s~ Deviatoric stresses of elastic input s~ Actual deviatoric stresses pa Actual equivalent plastic strain ~'eq Actual elasto-plastic notch-tip strains Elastic notch tip strain components Nominal strain F.n Poisson' s ratio O-~q Actual equivalent stress at notch tip a Size parameter of the mth (fm) plasticity surface O'eq, m Actual stress tensor components in notch tip Notch tip stress tensor components of elastic input o-o Parameter of the material stress-strain curve P Axial load T Torque R Radius of the cylindrical specimen Notches and other geometrical irregularities cause significant stress concentration. Such an increase of stresses results often in localized plastic deformation, leading to premature initiation of fatigue cracks. Therefore, the fatigue strength and durability estimations of notched components require detailed knowledge of stresses and strains in such regions. The stress state in the notch tip region is in most cases multiaxial in nature. Axles and shafts may experience, for example, combined outof-phase torsion and bending loads. Although modem finite-element commercial software packages make it possible to determine notch tip stresses in elastic and elastic plastic bodies with a high accuracy for short loading histories, such methods are still impractical in the case of long loading histories experienced by machines in service. A representative cyclic loading history may contain from a few thousands to a few millions of cycles. Therefore, incremental elastic-plastic finite-element analysis of such a history would require prohibitively long computing time. For this reason more efficient methods of elastic-plastic stress analysis are necessary in the case of fatigue life estimations of notched bodies subjected to lengthy cyclic stress histories. One such method, suitable for calculating multiaxial elastic-plastic stresses and strains in notched bodies subjected to proportional and nonproportional loading histories, is discussed in the paper.

Loading Histories The notch tip stresses and strains are dependent on the notch geometry, material properties and the loading history applied to the body. If all components of a stress tensor change proportionally, the loading is called proportional. When the applied load causes the directions of the principal stresses and the ratio of the principal stress magnitudes to change after each load increment, the loading is termed nonproportional. If plastic yielding takes place at the notch tip then almost always the stress path in the notch tip region is nonproportional regardless of whether the remote loading is proportional or not. The nonproportional loading/stress paths are usually defined by successive increments of load/stress parameters and all calculations have to be carried out incrementally. In addition the material stress-strain response to nonproportional cyclic loading paths has to be simulated, including the material memory effects.

Stress State at the Notch Tip For the case of general multiaxial loading applied to a notched body, the state of stress near the notch tip is triaxial. However, the stress state at the notch tip is biaxial because of the notch-tip stress

84


FIG. 1--Stress state at a notch tip (notation).

free surface (Fig. 1). Since equilibrium of the element at the notch tip must be maintained, i.e., ~r23 = 0"32 and 823 = 832, there are three nonzero stress components and four nonzero strain components. Therefore, there are seven unknowns all together and a set of seven independent equations is required for the determination of all stress and strain components at the notch tip

O'i=

0"~2 0"23

aWaf

(42)

S(To)=[i~=l0 Ny~ forWaeqi,

7--,

~- - - -',- - ~ -~,k'~- .~ ~ . . . . . . . . . ~ --,,,v,,,,,,~l~-I - - -~- ~ - ~,g.~- " , . . . .~ 7 - - I ]- 7 - ,T 7 ,, ' - ~ , , , , , , , , , ~ / . ~ . ~ , r-' ~ , , -~ r7

:::-::::]:-

o

~-r.,

T---~ --~--,--I-;-q-I T - - - , - - q - - C q

. . . . 7 - - -p- - q - -,- -, r -,- - - | b e n d i n g R = - I ( B F 1 ) Y -,~-,~ - F : bendJtor. ~: ~ - ,~ ~ - ~B . . . . . . . . . . ",~ . . . . . :.,, . . . . . .

~

bendJt~176 R=-I IBF3,

:

",

l~:~ I, , ,

-.,-:]

I-,---~--~--~--,-,-~ 1 , . . . . . . I- - ,- - - ~ - - - - ~ - . - ~ . . . . I L

~

:

1,E+06 C y c l e s to failure; e x p e r i m e n t

FIG. 24--'Ncazc. versus Nexp.for variable-amplitude tests: bending, torsion, combined.

1 ,E+07

210


Acknowledgment The authors would like to thank the German Research Foundation (DFG) for the financial support of this research program (contract ZE 248/8-1 and ZE 248/8-2).

References [1] Sonsino, C. M., "Fatigue Life to Crack Initiation of Welded Components under Complex Elasto-Plastic Multiaxial Deformation," Forschungsvorhaben der Europ~tischen Gemeinschaft fiir Kohle und Stahl (EGKS), EUR-Report No. 16024, Luxemburg, 1997. [2] Witt, M. and Zenner, H., "Multiaxial Fatigue Behavior of Welded Flange-Tube Connections under Combined Loading. Experiments and Lifetime Prediction," Proceedings, 5th International Conference on Biaxial/Multiaxial Fatigue & Fracture, Vol. 1, Cracow, Poland, Sept. 1997, pp. 421-434. [3] Pfeiffer, J. and Witt, M., "Model Based Control of Hydraulic Test Benches," Proceedings, 8th Congreso Latinoamericano de ControlAutomatico, Vol. 1, Vina del Mar, Chile, Nov. 1998, pp. 189-193. [4] EUROCODE Nr. 3, "Gemeinsame einheitliche Regeln ftir Stahlbanten," Kommission der Europiiischen Gemeinschaft, Bericht Nr. EUR 8849, DE, EN, FR, Stahlbau-Verlagsgesellschaft mbH, Cologne, 1984. [5] AD-Merkblatt $2, "Berechnung gegen Schwingbeansprnchung," Beuth-Verlag, Berlin / Cologne, 1982. [6] ASME Boiler and Pressure Vessel Code, Section III, Division 1, Subsection NA, Article XIV.1212, Subsection NB-3352.4, New York, 1984. [7] Hobbacher, A., "Recommendations on Fatigue of Welded Joints and Components," International Institute of Welding, doc. XIII-1539-96/XV-845-96, Paris, 1996. [8] British Standard BS 5400, "Steel, Concrete and Composite Bridges," Part 10 Code of Practice for Fatigue, British Standard Institution, 1980. [9] Esderts, A., "Betriebsfestigkeit bei mehrachsiger Biege- und Torsionsbeanspruchung," Dissertation TUClausthal, 1995. [10] Amstutz, H., Seeger, T., Ktippers, M., Sonsino, C. M., Witt, M., Yousefi, F., and Zenner, H., "Schwingfestigkeit yon Schweiflverbindangen bei mehrachsiger Beanspruchung," Abschlu[3bericht DFG_Forschungsvorhaben, Darmstadt, Clausthal, (anticipated date of publication Sept. 2000).

Fatigue Life Prediction Under Specific Multiaxial Loads

John J. F. Bonnen I and T. H. Topper 2

The Effect of Periodic Overloads on Biaxial Fatigue of Normalized SAE 1045 Steel REFERENCE: Bonnen, J. J. F. and Topper, T. H., "Effect of Periodic Overloads on Biaxiai Fatigue of Normalized SAE 1045 Steel," Multiaxial Fatigue and Deformation: Testing and Prediction, ASTM STP 1387, S. Kalluri and P. J. Bonacuse, Eds., American Society for Testing and Materials, West Conshohocken, PA, 2000, pp. 213-231. ABSTRACT: During the past decade it has been observed that periodically applied overloads of yield stress magnitude can significantly reduce or eliminate crack closure under uniaxial or Mode I loading. This paper reports the results of a series of biaxial in-phase tension-torsion experiments that were performed to evaluate the effects of overloads on the fatigue life of smooth tubes constructed of normalized SAE 1045 steel. Five strain ratios were investigated, including uniaxial ()t = e ~ / e = = 0), pure torsion (A = co), and three intermediate ratios (A = 3/4, 3/2, and 3). Periodically applied overloads of yield stress magnitude allowed cracks to grow under crack face interference-free conditions. Strain-life curves were developed by computationally removing the overload cycle damage from test results and calculating equivalent fatigue lives. A factor of two reduction in the fatigue limit was found at all ratios when these results were compared with constant-amplitude results. Cracking behavior was observed and it was noted that for strain ratios greater than one, cracks initiated along the rolling direction (longitudinally); otherwise, the cracks initiated on maximum shear planes. This observation was used to help explain the similarity in fatigue life results for all strain ratios for both constant-amplitude and overload data. Parameter-life curves were developed using the equivalent fatigue life data and several common multiaxial damage parameters, and the damage parameters were evaluated. It was found that the simple maximum shear strain criterion together with uniaxial overload data provided a good estimate of the fatigue behavior for all strain ratios. KEYWORDS: multiaxial fatigue, biaxial fatigue, fatigue (materials), fracture (materials), steels, overloads, sequence effects, tension-torsion loading, axial torsion loading, in-phase loading, proportional loading, testing, crack closure, crack face interference, mean stresses, cracking behavior Nomenclature

D e e~y y r/ A ni N NI N/ P

D a m a g e to c o m p o n e n t subjected to fatigue Nominal strain Local strain Tensorial shear strain Engineering shear strain N u m b e r o f small cycles b e t w e e n overloads Biaxial strain ratio, e ~ / e = N u m b e r o f cycles applied at amplitude i N u m b e r o f cycles applied N u m b e r o f cycles to failure N u m b e r o f cycles to failure at amplitude i A damage parameter

1 Research staff, Manufacturing Systems Dept., Ford Motor Co., 3135 MD3135 SRL, P. O. Box 2053, Dearborn MI 48121-2053. 2 Professor, Civil Engineering Department, University of Waterloo, Ontario, Canada, N2L3G 1.

Copyright9


213 www.astm.org

214


R~ Strain ratio, emin/emax S Nominal stress Sop Crack opening stress o" Local stress ~y Cyclic yield stress ~" Local shear stress

Subscripts, Superscripts and Operators x,y,z 1,2,3 a max min rn

op ol sc n A

Orthogonal specimen coordinates Orthogonal principal stress/strain coordinates Amplitude of variable during load cycle Maximum value of variable in load cycle Minimum value of variable in load cycle Mean value of variable during load cycle Value of variable when crack is just fully open Value of variable for overload cycle Value of variable for small cycle Normal to crack plane Range of variable during loading cycle

Introduction The fatigue analysis of typical vehicle structures is greatly complicated by the variable-amplitude loads to which most of these structures are subjected. The analysis is further complicated by multiaxial loading. Recent investigations indicate that, while uniaxial fatigue analysis is satisfactory in most cases, it is estimated that it produces seriously nonconservative life estimates 5 to 10% of the time because of multiaxial loading [1]. A large analytical infrastructure has been built up around variable-amplitude loading, but even with these tools a life estimation error of as much as a factor of 20 [2] is possible. One of the problems with estimating the damage resulting from variable-amplitude loading is that the mean stress rules used are typically developed using constant-amplitudefatigue data for which crack closure levels are higher than for variable-amplitude loading. In the past two decades, several researchers have linked mean stress effects to crack closure levels [3-5]. It has long been known that overloads can alter the crack closure level and radically change experimental fatigue life. Overload studies, usually performed with compact tension specimens, focus on a stress range limited to much less than half of the net section yield stress. Unfortunately, the combination of severe service loading and notched engineering components can often lead to stresses well beyond the range normally investigated in conventionalfatigue crack growth studies. Only in the past decade [5, 6] was it determined that yield stress level overloads, whether compressive or tensile, substantially accelerated short fatigue crack growth rates by reducing closure levels. For the purposes of this paper the term overload will be used to refer to these large yield stress level excursions. DuQuesnay [5] determined that overloads which are of the order of the yield stress significantly reduce crack closure. Compressive overloads drastically reduce closure by flattening asperities and eliminating the interaction between crack faces. Tension overloads stretch the crack mouth open-separating the crack faces and eliminating closure. He was able to determine, with the aid of acetate replicas and with strain gages laid across the crack that overloads of sufficient amplitude and frequency can maintain a fully open crack throughout the life of a specimen. Later researchers [7,8] were able to corroborate these observations via direct optical measurements. Varvani-Farahani et al. [9] obtained similar results for Stage I cracks using a confocal scanning laser microscope.

BONNEN AND TOPPER ON PERIODICOVERLOADS

215

The concept of closure does not work well for cases involvingmultiaxial loading. Instead, the term "crack face interference" is used to extend the closure analogy into combined Mode I/II/III loading systems in which a crack may not "open" or "close." Various researchers [10-14], have found that crack growth rates could be strongly influenced by secondary stresses. Tschegg [15] observed that as crack lengths increased in AISI C1018 steel, crack face interference under Mode III loading consumed an increasing part of the crack driving force, especially at low crack driving force levels. Crack-face interference is also reduced by applying Mode III overloads during Mode III crack growth. Ritchie et al. [16] applied Mode III overloads during a Mode III crack growth test and found that overloads accelerated the smaller cycle crack growth. One group of researchers [17] applied bending overloads to a notched shaft during torsion cycling and found that the overloads not only increased smaller cycle damage but substantially lowered the torsional fatigue limit. Varvani-Farahani [18] removed crack face interference by applying large compressive (Mode I) overloads across shear cracks growing under various strain ratios and obtained accelerated crack growth rates. The current work explores the interactions of overloads with smaller cycles over a range of five different in-phase biaxial strain ratios that varied from pure uniaxial loading to pure torsional loading. Crack face interference-free strain life diagrams were developed for each ratio, and observations were made of the cracking behavior in each strain ratio and used to help explain the fatigue behavior. Finally, the effectiveness of different multiaxial parameters in consolidating test data for different biaxial strain ratios was examined. Materials and Procedures The material used in this study was an SAE 1045 steel in the normalized condition. The steel was hot rolled into a 63.5-ram-diameter bar and normalized to produce a Brinell hardness of 203 BHN. A synopsis of the history of the material, monotonic and cyclic material properties, and the chemical composition may be found in Ref 17. Figure 1 presents the microstructure of the steel in the longitudinal-transverse (L-T) orientations. It has a pearlitic/ferritic microstructure and, because of the nor-

FIG. 1--Microstructure of normalized SAE 1045 steel ( X400), L-T orientation.

216


malizatiou procedure, has equiaxed grains of roughly 25/xm. Also observable in Fig. 1 is the banding characteristic in this lot of material, but this banding is not evident in the short-transverse orientation. MnS stringers are present in this steel and are aligned with the rolling direction. They range in length from 0.1 to 2 mm.

Testing Techniques Uniaxial Tests--The techniques, methods, and specimen used in uniaxial testing are discussed in Ref 17. Uniaxial Periodic Overload Tests--Overload tests were conducted in strain control. The histories used in the uniaxial overload tests, shown in Fig. 2, consisted of a single compression-tension overload cycle followed by a number of smaller cycles whose peak tensile strains were the same as the peak tensile strain of the overload cycle. The amplitude of the overload cycle itself was selected to be 0.48% strain, which corresponded roughly to 10 000 cycles to failure under conventional constantamplitude testing. A number of smaller constant-amplitude cycles (7) followed the overload cycle, and the smaller cycle amplitude was set, depending on the test, at a value between 0.2% strain and 0.06% strain. Finally, the number of smaller cycles, r/, placed between the overload cycles was chosen such that the overload cycles constituted no more than approximately 25% of the total damage. In this paper the term "small cycles" (sc) is used when referring to overload tests to indicate the smaller cycles in the overload history. Axial-Torsion Tests--A tension-torsion machine performed multiaxial fatigue life tests on the tubular specimens shown in Fig. 3. Rough machining of the specimen consisted of lathe turning on the outer surface and boring the inside. Finish machining consisted of low-stress grinding and sanding on the outer surface and honing the inner surface with successively finer stones. The outer surface was polished to 1 /xm for the purpose of observing cracking behavior. The inner diameter received a 5/zm finish.

1]

caLi ~min

-0

FIG. 2--Uniaxial overload history.

BONNEN ANDTOPPER ON PERIODIC OVERLOADS =_

- 52+0.1/-0.00

217

200+1.0/-0.1

, 33.0-+03,30.0+0~I-0.00

~45+0.0/-1.O R75_+1

\

\

FIG. 3--Axial-torsion specimen. All dimensions in mm.

The axial-torsional tests were conducted using an axial-torsion load frame capable of exerting a 250 kN axial force and a 2250 Nm torque on test specimens. The specimens were mounted with hydraulic grips developed at the University of Illinois [19]. Strains were measured with an axialtorsion extensometer that had the advantage of sensing directly the shear strain at the specimen surface rather than measuring specimen twist. Tests were conducted in strain control, and strains were controlled to an accuracy of 1% by the adaptive parametric control program described in Ref 20. A maximum test frequency of 40 Hz was employed for some high cycle tests while slower frequencies were used for shorter lives. Frequencies above 8 Hz were used only when specimen stress-strain response was elastic and load control was then used on both axes. Specimen failure was defined as the first discernible compliance change. This technique resulted in an estimated failure crack length of 1 to 3 ram. Five strain ratios (A) were used in this research: ~, 3, 3/2, 3/4, and 0 (uniaxial). For tests performed under A = ~ loading the axial actuator is left in load control at 0 load for the duration of the test. Biaxial Periodic Overload Tests--Just as in the uniaxial periodic overload tests a large, fully reversed, overload cycle was applied and followed by 7/smaller cycles, as in Fig. 2. The overload cycle was set such that it alone would cause specimen failure in 10 000 cycles, and it was applied inphase with the subsequent smaller cycles, while the smaller cycles were set such that they shared the same peak strain as the overload cycle.

Biaxial Fatigue Results Uniaxial Fatigue Behavior, A = 0 As discussed in the introduction, the presumption that the mean stress effect observed in fatigue is a direct result of fatigue crack closure implies that if a cycle is free of closure, then the mean stress of that cycle will not affect the crack growth rate--except where the maximum stress intensity approaches the critical stress intensity. The Palmgren-Miner rule [21,22] was used to calculate the equivalent number of cycles to failure for the small cycles in the overload histories. This equivalent life was then paired with the small cycle amplitude, and the crack-closure-free result was plotted (along with conventional fully reversed

218


fatigue data). For the history shown in Fig. 2, the equivalent number of cycles to failure for the small cycles, N~, is given [17] by ni

Z~//=

1

(1)

= 1

(2)

i

nol

nsc

No--~t+ ~

N~c-

1 nsc

1

l

(3)

~lNol

where ni is the number of cycles at amplitude i, N~ is the expected constant-amplitude life at amplitude i, "oF' indicates overload cycle, "sc" indicates the small cycle, and 7/is the ratio of the total number of small cycles applied to the total number of overload cycles applied (~/= n~c/nol). The life for the overload cycle, No~, is taken from the constant amplitude strain life curve. Figure 4 depicts the constant-amplitude (R~ = - 1, plotted as open circles) and overload data (plotted as triangles). The fatigue limit was reduced from a strain amplitude of 0.0017 to 0.00072--a reduction to two-fifths of its original value by the overloads. A curve which approximates fully open fatigue crack growth is also presented in Fig. 4 and labeled as Ae~g/2. The underlying theory and the development of this curve is described in Refs 17 and 23. Crack Closure-Free Fatigue Life Testing--In the tests in this study the method used to determine whether the cracks were growing under fully open conditions involved performing an overload test

10 -1 "'E

~ &

EE 102

~,'0.

9 Constant amplitude Equivalent. small cycles (z~~ = 0.0048)

i"~/~eff/2

= (~i +/~*)/2

O "O

< C 9

lo

.3

..........

_0_:9_0_

O3 '"~,

~~

= 0.0065

~1;'12 = 0.50(2N~)"~

10 4

........ , ........ , ........ , ....... . ....... . ....... . ........ , ........ , 104 102 103 104 10 s 106 107 108 109 Reversals to Failure (2Nf)

FIG. 4---Uniaxial (~ = g~ ~" = O) periodic overload and effective strain curves for normalized SAE 1045 steel.

219

BONNEN AND TOPPER ON PERIODIC OVERLOADS

1 0 -2 '

X=oo

E

_.E E E

"--" 10-3

Periodic Overload

(~ty). = 0.0035 11

10 -4

' ' .,,,,.| 10 ~

10 2

,,

,,,,,,i

Constant

...,1,,,|

10 3

10 4

..

.,,.,! 10 5

. ,,,,,l|

,

Amplitude

. ...,i,

10 6

I

10 7

,

. IH...

I

I

,

.

10 e

Equivalent Cycles to Failure FIG. 5 - - F a t i g u e response o f normalized S A E 1045 under torsional loading, )t -~ ~ = o~ Circled datapoint indicates special overload level (exy)a ol = 0.00425, (exy)a sc = 0.00125, and ~ = 50.

with an overload amplitude considerably larger than the one used in the standard overload tests. This type of test is termed a "companion test." The circled data point in Fig. 4 gives the life of the companion test for )t = 0. This specimen was subjected to an overload strain amplitude e ~ 0.0065, a small cycle amplitude e sc = 0.00125, and r / = 250. With the exception of the overload amplitude, the adjacent point and the the circled data point share the same test conditions. Since a larger overload is expected to reduce the crack closure stress, the failure of the larger overload to further reduce the fatigue life suggests that, at the lower overload level, the crack was already fully open, i.e., the opening stress was below the minimum stress (Sop 1 is illustrated in Fig. 10 which shows typical initiation cracks for a specimen subjected to A = 3/2. Initiation predominantly takes place on maximum shear strain planes (as marked by solid lines in the figure) and along the longitudinal direction. These longitudinal cracks became dominant and grew to failure. At the lowest strain levels cracks would eventually switch over to growth on maximum tensile strain planes regardless of strain ratio. As the strain level was increased for A > 1, the length of the initial longitudinal shear crack increased until the shear crack would grow the length of the specimen. Cracking for specimens subjected to A < 1 can be seen in Fig. 11 where a specimen subjected to A = 3/4 loading is presented. In these tests cracks also initiated in ferrite-rich regions, but after growing a short distance (-50/xm) they switched over to growth on maximum tensile strain planes and grew to failure. As in A > 1 tests the initial shear crack length would lengthen with increasing strain amplitude, but as A increased the strength of this effect was reduced.

FIG. 9--Preferential cracking through ferrite grains and along the ferrite/pearlite grain boundaries. Specimen subjected to (exy)a ox = 0.00425, (/3xy)a sc = 0.00125, and r1 = 50, and photo taken at Nf = 108 540.

FIG. l O ~ T y p i c a l small crack growth under ~ = ~-~ = 3/2 loading o f normalized SAE 1045 steel. Specimen subjected to (e~ax)a = 0.0027 and (s~xay)a = 0.004. Photo taken at Nf = 6930. 223

224


FIG. 11--Small cracks developed under in-phase A = 3/4 loading. Specimen subjected to (eca)a = 0.004 and (e ~x~y)a= 0.003. Photo taken at Nf = 6470.

Crack Face Interactions It was observed that, for all values of A, shear cracks initiated on shear planes. Under A < 1 conditions cracks soon moved onto maximum tensile strain planes, while for A > 1, cracks mostly grew along the longitudinal shear plane. Shear Crack Growth--At crack lengths of 0.5 mm and greater, shear-crack faces are observed to slide back and forth across each other, and it is believed that this is also true for shorter crack lengths. Under this motion (Modes II and III crack growth) the energy available at the crack tip is reduced by the interaction of crack-face asperities. After being smeared by the overload the asperities offer much less resistance to crack-face motion and the crack driving force is increased at the crack tip. Lateral smearing of asperities consistent with this model was observed on fracture surfaces of specimens subjected to A > 1 straining. Tensile Crack Growth--Under tensile (Mode I) loading asperities contribute significantly to crack face interference (closure) [5]. Compressive overloads applied across the crack face have been shown to flatten crack face asperities [18]. The crack is open during more of the loading cycle and the effective stress intensity range is increased. The difference between the tensile and shear mode overloads is that the asperities are crushed rather than smeared. Crushed asperities were visible in the ten-

225


sile growth regions of fracture surfaces when A < 1 (and on those for A > 1 which also demonstrated tensile growth).

Consolidation of Fatigue Life Results Constant-Amplitude Tests Cracking in the constant amplitude tests as in the periodic overload tests was dominated by longitudinal anisotropy and cracks were aligned with the longitudinal axis. Presumably, these cracks are not greatly influenced by tension loading and it follows that the life of these tests was controlled by the applied shear strain. Figure 12 shows all of the constant-amplitude data from tube tests and some additional fatigue life data for this material from notched shafts [17] plotted against the applied shear strain (ex~y)a. The figure clearly demonstrates that, in all regions except for data at the endurance limit, the applied shear strain as a fatigue parameter reduces data for A > 1 into a single characteristic curve. The data for A = 3/4 falls on a separate but parallel curve below that of the other data. At the fatigue limit the A = ~ (open squares), A = 3 (dot-center squares), and notched shaft (filled squares) data fall into a single narrow band. The notched shaft data fall in the gap between the runout and regular fatigue data, and, while A = 3/2 data (bar-center squares) initially follows the trend of the other data, it then departs from the trend at the A = ~ endurance limit and proceeds along an extension of the trend curve to a lower level.

10 -2

-~'~,.O'~.

Initiation T y p e ~

-

E

E

E 10 .3

E

k---~

=

[] 9

Tension-Torsion (I) Notched Shaft (i)

;L=3/2 = ;L=3/4

Tension-Torsion

10 .4

[]

X=3

(I)

[] []

~,=3/2 X=3/4

(I) (/)

'

02

'

' ' ' ' " !

'

10 3

'

' ' ' ' " 1

10 4

Initiation Key (I) = longitudinal ( / ) = inclined

'

'

' ' ' ' " 1

'

'

' ' ' ' " 1

10 8

10 8

'

'

' ' ' ' " 1

10 7

'

'

''

....

I

10 8

Cycles to Failure

FIG. 12--Constant-amplitude curves for tension-torsion and notched shaft tests [ 17] plotted on the basis of applied shear strain amplitude ((eC~,)a).

226


Also shown in the figure is the initiation angle observed for each strain ratio. For biaxial strain ratios of 0% 3, and 3/2 initiation is aligned with the longitudinal axis as discussed in "Cracking Behavior." A single failure at A = 3/2 lies below the A = oo and 3 endurance limit, and this failure did not initiate longitudinally; its initiation was along the plane of maximum tensile strain. It is assumed that this change in initiation type is responsible for the endurance limit shift in the ,~ = 3/2 data. The A = 3/4 data lie below all of the rest of the data in this figure. This is presumably because the initiation plane is aligned with the maximum shear planes. It seems that the influence of strains applied on the tension axis was small since, as seen in Fig. 12, there was essentially no difference in the fatigue response between ~ = 0% 3 and 3/2 loading--the life is the same for a given applied torsional strain. Under A = 3/4 loading an influence of tension loading is seen in the fatigue response of the material--for a given applied torsional strain amplitude the )t = 3/4 life is shorter than that of the other strain ratios. Since the bulk of the life in A = 3/4 and ~ = 0 loading is spent initiating and growing a crack along planes of maximum shear strain these data are plotted in Fig. 13 against the maximum shear strain amplitude, (e 12)a, while the rest of the data is plotted against the applied torsional shear strain amplitude (resolved shear strain on along the x-axis), (exy)a. When the data of Fig. 13 are presented in terms of maximum shear strain amplitude, (e 12)a, in Fig. 14 there is only a small shift in the h = oo and 3. = 3 strain ratios (increases of 3 and 10%, respectively). The correlation of the data based on maximum shear strain amplitude is almost as good as that given by using the shear strain on the initiation plane. As a predictive tool its advantage is that it does not require tests to determine the initiation angle.

10 2

E E

Torsion, ;L=oo (~=~12) [] Tension-Torsion " ~ 1 0 -3. 9 Notched Shaft #

I~

=

Tension-Torsion []

X=3

[]

X=3/2 ;L=3/4

[]

(~,~) (Zxy) (s12)

Uniaxial, ;L=O 104

........ 10 2

i 10 3

........

i 10 4

........

!

........

10 6

i 10 6

........

i 10 z

........

i 10 8

Cycles to Failure FIG. 13--Constant-amplitude tension-torsion and notched shaft tests [ 17] considered on the basis of initiation plane.


227

1 0 .2

E

Torsion, X=oo

E

E 104 ,~E

[] 9

,~~1~[3----~

Tension-Torsion Notched Shaft

N

Tension-Torsion []

;L=3

[]

;L=3/2

[]

X=3/4

Uniaxial, X=O 10.4 10 2

10 3

10 4

10 5

10 6

10 7

10 8

Cycles to Failure FIG. 14---Constant-amplitude tension-torsion and notched shaft tests [17] plotted using maximum shear strain amplitude.

Overload Tests The mean stress corrections found in most damage criteria are corrections for the presence of closure/crack face interference. When an increasing tensile mean stress is applied to a growing crack, the actual stress range which reaches the crack tip increases because the crack opening stress increases less rapidly than the maximum stress. In other words, Smax - Sop increases because Smax increases more than Sop. The increased effective stress range causes the crack to grow faster. A graph of fatigue life versus maximum shear strain for the periodic overload data is presented in Fig. 15. As observed for the constant-amplitude data, there were no large differences between the resuits when viewed in terms of the resolved shear on the initiation plane, as shown in Fig. 16, and the results when viewed in terms of the maximum shear. In Fig. 15 about two thirds of the data fall within the 2x bands, and most of the data are contained by the 5x bands, while in Fig. 16 the result is quite similar. The Brown and Miller parameter [25], which can be expressed [26,27] as PsM = (exy)~ + 0.5(exx)a when applied to the experimental data results in the plot in Fig. 17. This parameter provides a good condensation of the overload data. Most of the data points fall within the 2x bands, and the 5x bands contain all of the rest of the tension-torsion data. The parameter-life plot for the Fatemi-Socie-Kurath parameter [28,29], as expressed [26,27] by PF = Ya (1 + KF ~ ) , where KF is taken to be 0.3, is presented in Fig. 18. Under this parameter the data fall in much the same fashion as in Brown and Miller's parameter, with somewhat more of the data falling outside the 2x bands. The parameter-life diagram which demonstrates the least scatter is the

1 0 -2 "

5x

2x

2x

"',,

5x

\

E

",,

,

E l 0 "3

V

~.= O, u n i a x i a l

0

Z=3/4

~ > V

..

".

E

,

\\

~

x\

x

@,

.

~ v ~ ....

Z=3/2

10 . 4

'

~.=3

[]

;~=oo. t o r s i o n

'''""I

'

102

10 ~

- -

'''""I

'

'''""I

103

L=O (uniaxial) e f f e c t i v e c u r v e

' ''""I

104

'

105

'''""I

'

'''""I

10 s

'

107

' ''""I

'

' ''""I

10 s

109

E q u i v a l e n t C y c l e s to Failure

FIG. 15--Maximum shear strain amplitude (e ]2)a curves for periodic overload tests.

10

"2 '

5x 2x

2x

5x

E E vE r 10 .3

v

10 .4

[]

~=~

(%)

0

;~=3

(%)

z~ o

k=3/2 Z=3/4

(%) (~12)

- -

v X=O (~12) (uniaxial) . . . . . . . . i . .......i . . .... ..i . . . . . . . . i 101 102 103 104 10 s

X=O (uniaxial) e f f e c t i v e c u r v e

.......i . ....... i . . . . . ...i . . ...... i 106 10 z 108 109


FIG. 16---Periodic overload data considered on the basis of initiation plane. 228

10 .2

5x

2x

2x

5x

"'.. \Q

.''.

o_==10-s

1

0 -4

'

v

3,=0, uniaxial

o

3,=3/4

A

3,=3/2

O []

3,=3 3,=00, torsion

''''"'l

101

'

' ''""I

1 02

'

- -

''"~"I

1 03

'

Z=0 (uniaxial) effective c u r v e

'''"'q

1 04

'

'''""l

10 s

'

'''""I

1 06

'

''''"'l

107

'

~ ''""I

1 08

1 09


FIG. 17--Brown and Miller [25] parameter-life plot for periodic overload tests. 1 0 -2 -

5x

2•

2x

5x

"',

u. D_

"..

10 -3

'.

10 "4 10 ~

v

3,=0, uniaxiat

o

3.=3/4

A 0

X=3/2 3,=3

[]

3,=00, torsion

........ | 1 02

,

......

- -

,i

1 03

,

",Q

,

"..

k--o (uniaxial) effective curve

,, ..... ! . . . . . . . . i 10" 10s

,

,

......

i

. . . . . .

10 s

,,i

1 07

,

,, ..... i 1 08

,

,

......

I

1 09


FIG. t 8--Fatemi-Socie-Kurath [28,29] parameter-life plot for periodic overload tests. 229

230


Brown and Miller parameter. However, the simpler maximum shear parameter provides very nearly as good a consolidation of the data.

Summary and Conclusions The purpose of this investigation was to observe the effect of overloads in multiaxial fatigue. Inphase strain controlled constant-amplitude and periodic overload tests were conducted on tubular specimens, and the tension-torsion strain ratios (A = ~ x y / ~ ) were selected to be % 3, 3/2, 3/4, and 0. Both the overloads and small cycles shared the same strain ratio. 9 Periodic overloads reduced the 107 cycle endurance limit of normalized SAE 1045 steel to about one-half the constant-amplitude value for all strain ratios. Experiments in which cycling was continued to l0 s cycles exhibited a further endurance limit reduction to 2/5 of the constant-amplitude value. 9 It was found that for h = E~y > 1 cracks initiated and grew along the specimen longitudinal ~ axis. For h < 1 cracks tended to initiate on planes of maximum shear strain, and eventually move onto planes of maximum tensile strain. Another trend was also noted where, at low strain amplitudes, long cracks would tend to grow on maximum tensile planes but, at high strain amplitudes, long shear cracks dominated. 9 Companion tests with overloads higher than those used in the standard test series were performed on one specimen at each strain ratio in order to determine whether the overload level used in the regular tests was large enough to produce crack-face interference-free conditions. These tests indicated that the overloads used did produce a maximum fatigue life reduction, and it should follow that for small cycle amplitudes below that employed in the companion test, the fatigue cracks would grow under crack-face interference-free conditions. Simple models, supported by fractographic evidence, are used to describe the nature of crack-face interference and explain how it was reduced by overloads. 9 A series of multiaxial damage parameters that correlate fatigue data for different strain ratios was examined. For constant-amplitude data it was found that plotting the resolved shear strain on the initiation plane against fatigue life provided good data consolidation. Maximum shear strain also gave good consolidation of the constant-amplitude data. For the periodic overload fatigue data the Brown and Miller parameter gave the best consolidation. However, the maximum shear strain parameter also provided good consolidation of the data and is simpler to implement.

References [1] Chu, C.-C. and Htibner, A., Personal communication, 1997. [2] Bannantine, J., Comer, J., and Handrock, J., Fundamentals of Metal Fatigue Analysis, Prentice-Hall, 1990. [3] Vosikovski, O., "The Effect of Stress Ratio on Fatigue Crack Growth Rates in Steels," Engineering Fracture Mechanics, Vol. 11, 1979, pp. 595-602. [4] Yu, M. and Topper, T., "The Effects of Material Strength, Stress Ratio and Compressive Overloads on the Threshold Behavior of a SAE 1045 Steel," Journal of Engineering Materials and Technology, Vol. 107, 1985, pp. 19-25. [5] DuQuesnay, D., "Fatigue Damage Accumulation in Metals Subjected to High Mean Stress and Overload Cycles," Ph.D. thesis, University of Waterloo, Waterloo, Ontario, 1991. [6] Jurcevic, R., DuQuesnay, D., Topper, T., and Pompetzki, M., "Fatigue Damage Accumulation in 2024T351 Aluminium Subjected to Periodic Reversed Overloads," International Journal of Fatigue, Vol. 12, No. 4, 1990, pp. 259-266. [7] Vormvald, M. and Seeger, T., "The Consequences of Short Crack Closure on Fatigue Crack Growth under Variable Amplitude Loading," Fatigue and Fracture of Engineering Materials and Structures, Vol. 14, No. 2/3, 1991, pp. 205-225. [8] Dabayeh, A. and Topper, T., "Changes in Crack Opening Stress After Overloads in 2024-T351 Aluminum Alloy," International Journal of Fatigue, Vol. 17, No. 4, 1995, pp. 261-269.


231

[9] Varvani-Farahani, A., Topper, T., and Plumtree, A., "Confocal Scanning Laser Microscopy Measurements of the Growth and Morphology of Microstructurally Short Fatigue Cracks in A12024-T351 Alloy," Fatigue and Fracture of Engineering Materials and Structures, Vol. 19, No. 9, 1996, pp. 1153-1159. [10] Brown, M. and Miller, K., "Mode I Fatigue Crack Growth Under Biaxial Stress at Room and Elevated Temperature," Multiaxial Fatigue, ASTM STP 853, American Society for Testing and Materials, pp. 135-152. [11] Youshi, H., Brown, M., and Miller, K., "Fatigue Crack Growth from a Circular Notch under High Levels of Biaxial Stress," Fatigue and Fracture of Engineering Materials and Structures, Vol. 15, No. 12, 1992, pp. 1185-1197. [12] Hourlier, F. and Pineau, A., "Fatigue Crack Propagation Behavior Under Complex Mode Loading," Advances in Fracture Research (Fracture 81), D. Francois, Ed., Vol. 4, Oxford, Pergamon Press, 1982. Held in Cannes, March 29-April 3, 1981, pp. 1833-1840. [13] Brown, M., Hay, E., and Miller, K., "Fatigue at Notches Subjected to Reversed Torsion and Static Axial Loads," Fatigue and Fracture of Engineering Materials and Structures, Vol. 8, No. 3, 1985, pp. 243-258. [14] Fatemi, A. and Socie, D., "A Critical Plane Approach to Multiaxial Fatigue Damage Including Out-ofPhase Loading," Fatigue and Fracture of Engineering Materials and Structures, Vol. 11, No. 3, 1988, pp. 149-169. [15] Tschegg, E., "Sliding Mode Crack Closure and Mode III Fatigue Crack Growth in Mild Steel," Acta Metallurgica, Vol. 31, No. 9, 1983, pp. 1323-1330. [16] Ritchie, R., McClintock, F., Tschegg, E., and Nayeb-Hashemi, H., "Mode III Fatigue Crack Growth under Combined Torsional and Axial Loading," Multiaxial Fatigue, ASTM STP 853, American Society for Testing and Materials, pp. 203-227. [17] Bonnen, J. and Topper, T., "The Effect of Bending Overloads on Torsional Fatigue in Normalized SAE 1045 Steel," International Journal of Fatigue, Vol. 21, No. 1, January 1999, pp. 23-33. [18] Varvani-Farahani, A., "Biaxial Fatigue Crack Growth and Crack Closure under Constant Amplitude and Periodic Compressive Overload Histories in 1045 Steel," Ph.D. thesis, University of Waterloo, Waterloo, Ontario, 1998. [19] Kurath, P., Personal communication, 1994. [20] Bonnen, J. and Conle, F., "An Adaptable, Multichannel, Multiaxial Control System," Technical Paper #950703, Society of Automotive Engineers, 1995. Also in Recent Developments in Fatigue Technology, SAE PT-67, Warrendale, PA, 1997. [21] Palmgren, A., "Die Lebensdauer yon Kugellagern (Fatigue Life of Ball Bearings)," Zeitschrift Verein Deutscherlngenieure, Vol. 68, No. 14, 1924, pp. 339-34l. In German. [22] Miner, M., "Cumulative Damage in Fatigue," Journal of Applied Mechanics, Vol. 67, September 1945, pp. A159-A164. [23] Topper, T. and Lam, T., "Effective-Strain Fatigue Life Data for Variable Amplitude Fatigue," International Journal of Fatigue, Vol. 19, Supplement No. 1, 1997, pp. S137-S143. [24] Socie, D., "Critical Plane Approaches for Multiaxial Fatigue Damage Assessment," Advances in Multiaxial Fatigue, ASTMSTP 1191, American Society for Testing and Materials, 1993, pp. 7-36. [25] Brown, M. and Miller, K., "A Theory for Fatigue Failure under Multiaxial Stress-Strain Conditions," The Institution of Mechanical Engineers Proceedings, Vol. 187, No. 65, 1973, pp. 745-755. [26] Chu, C.-C., "Fatigue Damage Calculation Using the Critical Plane Approach," Journal of Engineering Materials and Technology, Vol. 117, 1995, pp. 41-49. [27] Chu, C.-C., "Critical Plane Fatigue Analysis of Various Constant Amplitude Tests for SAE 1045 Steels," Technical Paper #940246, Society of Automotive Engineers, 1994. [28] Fatemi, A. and Kurath, P., "Multiaxial Fatigue Life Predictions Under the Influence of Mean-Stresses," Journal of Engineering Materials and Technology, Vol. 110, October 1988, pp. 380-388. [29] Socie, D., Waill, L., and Dittmer, D., "Biaxial Fatigue of Inconel 718 Including Mean Stress Effects," in Multiaxial Fatigue, STP 853, American Society for Testing and Materials, pp. 463-481. [30] American Society for Testing and Materials, Multiaxial Fatigue, ASTM STP 853, 1985.

Giinther Lgwisch, 1 Hubert Bomas, 1 and Peter Mayr 1

Fatigue of the Quenched and Tempered Steel 42CrMo4 (SAE 4140) under Combined In- and Out-of-Phase Tension and Torsion REFERENCE: L6wisch, G., Bomas, H., and Mayr, P., "Fatigue of the Quenched and Tempered Steel 42CrMo4 (SAE 4140) under Combined In- and Out-of Phase Tension and Torsion," Multiaxial Fatigue and Deformation: Testing and Prediction, ASTM STP 1387, S. Kalluri and P. J. Bonacuse, Eds., American Society for Testing and Materials, West Conshohocken, PA, 2000, pp. 232-245. ABSTRACT: Two types of specimens of the quenched and hardened steel 42CrMo4 (similar to SAE 4140) that differed in their residual stress state were tested by combined tension-torsion in- and out-ofphase loading. Under cyclic, stress controlled loading an elastic behavior is registered until 50% of the lifetime. Then a continuous softening occurs, the velocity of which correlates with the von-Mises equivalent stress in the case of in-phase loading. The residual stresses have no influence on the lifetime when cyclic softening occurs. Analytically, the lifetime is best described by the fatigue criterion of Zenner which considers the integral average of the stress state in every plane. This stress state is described by a function of the shear stress amplitude and the normal stress amplitude. Below the cyclic yield strength, the residual stresses must be taken into account as static stresses. The comparison of the local residual stress distributions is possible by using the weakest link model of Heckel.

KEYWORDS: quenched and hardened steel, multiaxial fatigue, residual stresses, fatigue criteria, weakest-link model

Nomenclature A A0 M ma my N No Nf PE V Vo

Surface o f a specimen, m m 2 Reference surface o f a specimen, I n . l I l 2 M e a n stress sensivity Weibull exponent for surface Weibull e x p o n e n t for volume N u m b e r o f cycles N u m b e r o f cycles at the beginning o f plastic softening N u m b e r o f cycles to failure Probability for endurance o f a specimen V o l u m e o f a specimen, m m 3 Reference volume o f a specimen, m m 3 Normal strain ~'pa Equivalent plastic strain amplitude 3' Shear strain A z~/~ra; loading ratio for c o m b i n e d loading O'a Normal stress amplitude, N / m m 2 cra,~q Equivalent stress amplitude, N / m m 2

1 Research engineer, senior research engineer, and managing director, respectively, Stiftung Institut fuer Werkstofflechnik, Badgasteiner Stra[3e 3, D-29359 Bremen, Germany. 232

Copyright9

by ASTM lntcrnational

www.astm.org

LOWlSCH ET AL. ON QUENCHED AND TEMPERED STEEL O'a,eq,A O'a,eq, V O'WAO O'WV0

,ra

233

Equivalent stress amplitude at the surface, N/mm 2 Equivalent stress amplitude in the volume, N/ram 2 50% endurance limit of the reference surface At N/mm 2 50% endurance limit of the reference volume Vo, N/mm 2 Shear stress amplitude, N/mm 2

High Cycle Fatigue Criteria The influence of multiaxial load and mean stresses on high cycle fatigue lifetime and endurance limit is usually described by high cycle fatigue criteria, which define a scalar equivalent value that allows a comparison of the actual cyclic load with a uniaxial mean stress free push-pull load. The equivalent stress amplitude is deduced from the equivalent value as that push-pull stress amplitude that generates the same equivalent value as the actual load. Many high cycle fatigue criteria can be assigned to four classes which are characterized by different methods of building the equivalent value: 9 The equivalent value is a linear combination of the maximum shear stress amplitude that arises in a critical plane and a normal stress in this plane, 9 The equivalent value is a linear combination of the maximum shear stress amplitude that arises in a critical plane and a hydrostatic stress. 9 The equivalent value is a linear combination oftbe octahedral shear stress amplitude and a hydrostatic stress. 9 The equivalent value is a mean over all planes of a function of the shear stresses and normal stresses in these planes Table 1 shows the fatigue criteria that were examined in this work.

Inhomogeneous Stress States In components, the local stresses that are originated by an outside load usually are inhomogeneously distributed at the surface and in the volume. This may be due to the component geometry or the kind of load. Residual stresses are also inhomogeneously distributed, which is due to the balance of the residual forces. For higher strength materials, the influence of such stress states on the fatigue limit is well described by the weakest-link model. This model was developed by Weibull to describe the scattering static strength of brittle materials [10]. Later, it was transferred by Heckel and his group to cyclic loads [11-13]. Different authors have made observations that are in good agreement with the weakest link concept [14-16]. The basis of the model is the assumption that surface or volume defects are equally distributed, and that the worst defect initiates a fatigue crack which leads to failure. In the opinion of the Heckel group, there exists a fracture mechanics relation between the defect size and a threshold stress amplitude which is identical with the fatigue limit. Since crack propagation is not considered, the application of this model is restricted to materials or conditions where crack propagation is of less importance. With respect to different crack initiation mechanisms at the surface and in the volume of a material, a distinction has to be made between surface crack initiation and volume crack initiation. In the first step, crack initiation in the volume shall be considered. Proceeding from a reference volume V0, the probability for endurance of this volume is described as a function of the stress amplitude:

=

"v

Present value

Mean value and amplitude

Maximum value

Ta,ma x ~- O/ On,ma x

McDiarmid [2]:

Ta,ma x Jr- O/ O"n

Findley [1 ]:

Type of Extensions

Mean value

Extensions of Tresca Criterion with Normal Stresses

~'a,max+ CrPm + /3 pa

Dang Van [3]: ~a,max -I- a Prnax Bomas, Linkewitz and Mayr [4]:

Extensions of Tresca Criterion with Hydrostatic Stresses

l ~ a ~ + b,~) 9 (1 + d,~..)2d,p

~,oct + a Pm + ~ Pa

S. = f ('ra, o'a, ~'m, or,.)

~s~dV

Simburger [9]:

Zenner [8]:

Criteria with Averaged Stress Functions

Kakuno and Kawada [7]:

7a,oc t + Ot Pmax

Crossland [6]:

Ta oct + Ot Pm

Sines [5]:

Extensions of von-Mises Criterion with Hydrostatic Stresses

TABLE l--Critical values of some high cycle fatigue criteria.

z

5

m -I1 O

z

c m

--rl --4

v-

x

r'-

C

4~

fad

LOWlSCH ET AL. ON QUENCHED AND TEMPERED STEEL

235

Regarding the more general case of an inhomogeneous stress distribution in a part or specimen, the whole volume has to be divided into volume elements with the following probability of endurance derived from Eq 1 by the rules of probability calculation:

dr( ~o )"v

e~(dV) = 2-v~ ~

(2)

The endurance probability of the total stressed volume can be calculated by multiplying the probabilities of all volume elements; this means an integration in the exponent:

Pe(V) = 2-f~ (~@o) "v d-Evv~

(3)

In the case of multiaxial stresses or mean stresses the stress amplitude o-~ has to be replaced by an equivalent stress amplitude. The volume considerations are only sufficient if subsurface crack initiation is the failure process. If surface crack initiation is the origin of failure analogous derivations have to be made with replacement of the volume V by the surface A. The total endurance probability of a part with both, surface and subsurface crack initiation, is the product of the probabilities for surface and volume:

Pe(V)=2

[ [ ....... Ite~AdA_[ ( ...... ImVdw JA\r At Jv~O'wvo] Vo

(4)

For this general case, two reference fatigue limits, trwvo and OrWAO,tWO Weibull exponents m A and my, as well as two fatigue criteria for surface and volume have to be distinguished.

Experimental The high-cycle fatigue behavior of the steel, 42CrMo4 (SAE 4140), under combined tension and torsion was examined at tubular specimens. The material came from one batch which was continuously cast under air exclusion and magnetic stirring. The chemical composition which fulfills the demands of EN 10083-1 is shown in Table 2. The specimen fabrication and its geometry is shown in Fig. 1. The ultimate tensile strength of the specimen was 950 N/mm 2. In order to remove the distortion caused by heat treatment, the specimens of a first series, which will be called "A," have been ground outside and honed inside. Due to the honing, compressive residual stresses were introduced in the inner surface. A part of the specimens was annealed after honing so that the residual stresses were reduced. This series will be called "B". Figure 2 shows the residual stresses which were measured across the wall of the specimens. The load was exercised force- and moment-controlled with a PTT-type machine built by SCHENCK which has both a longitudinally and torsionally working cylinder with servohydraulic

TABLE 2--Chemical composition of the material. Element Mass, % Element Mass-%

C 0.43 Mo 0.22

Si 0.21 Ni 0.23

Mn 0.77 Cu 0.13

P 0.010 A1 0.029

S 0.030 N 0.008

Cr 103 Ti 0.004

236


Heat treatment Drilling, |850~ 30 min / ~I~1~ Tuming ..,~ Oil, 60~ + . Cutting | 650~ 2h / | Water

Honing Grinding Polishing I ~

2'~ identical

Heat treatment

olishing

FIG. 1--Specimen geometry and manufacturing procedure, dimensions in ram.

drives. The tube form of the specimens avoids stress gradients under torsion. The ratio between shear stresses and normal stresses, A = ~-~/o-~, varied from 0.5 to 1. The phase shift, ~, between shear stresses and normal stresses varied between 0 and 90 ~ Table 4 shows the load variants which were free of mean stresses. The load frequency was 10 Hz at the fatigue limit and 1 Hz at a lifetime N s = 10 000. Between these loads, it was interpolated linearly.

\~-~-%'~\\'~\\~- 9.~\~\'~,\\~.\%%\'~ t ~ ~,~

E E

-loo

~~

u

Z o~

r

-200 (3

(D

-g_

-300

u)

-400 I

,

-200

0

200

1

1400

Distance to the outer surface [~n] FIG. 2--Residual stresses across the specimen wall.

1600

LOWlSCH

ET AL. O N Q U E N C H E D

AND TEMPERED

STEEL

237

TABLE 3--Load variants. Short Description

Normal Stress Amplitude o-a

Shear Stress Amplitude r,

Phase Shift 6

TC AT 0.5, 0~

oa o o-~

0

...

I-.

o;

% = o-J2

0.5, 90 ~ 1, 0~

o-. o-.

~'. =

o-~,

90 ~ 0~

1, 90~

o-~

r~ = or.

90 ~

r = o-,,/2

The strain measurement was achieved by a clip device that measured axial shear, 1/, and axial strain, e. The strain amplitude was determined by drawing the y - e pairs of one cycle in a e - X / ~ y coordinate system and taking the radius of the circumscribing circle as the total strain amplitude. An analogous procedure was exercised for the determination of the von-Mises equivalent stress amplitude and the von-Mises equivalent plastic strain amplitude. The actual plastic strains were calculated with Hooke's law.

Cyclic Strain Response Under loads in the finite lifetime region first, the material deforms elastically, but later exhibits a progressive plastic behavior (Fig. 3). The plastic strain amplitude is a linear function of the logarithm of the number of cycles and can be described as

I?,pa :

rap"

log

N)

(5)

The number of cycles, No, at the beginning of plastic softening is about half of the number of cycles to failure. Figure 4 shows an analysis of this relation. The increase, mp, can be described as a function of the von-Mises stress amplitude. All proportional loads have the same function, whereas the non-proportional loads have extra functions (Fig. 5). In the latter cases the softening is more pronounced, especially in the case of A = 0.5, where the highest softening rate is observed. It is assumed that this is due to the constant maximum shear stress which rotates over all planes perpendicular to the specimen surface, and allows the movement of dislocations in many gliding systems.

Crack Initiation The specimens were loaded until a crack of at least lO-mm surface length occurred. Usually, this crack changed its direction after a surface length of some mm. Only the cracks that were initiated un-

TABLE 4--Model parameters for fatigue limit calculation. O'WAON/mm~

Criterion Findley Zenner

Ao = 1 mma

mA

~rwvoN/mm2 Vo = 1 mm 3

mv

M

a a

b

d

619 625

13 13

980 972

10 10

0.4 0.4

0.33 0.33

0.22

1.22 10 3

200

400

C O >

R

600

9~

1

_'%'

i

s

l

i

I

I

Illlll

100

I

III1||

1000

|

I

I

10000

I I l | l l

cycles to failure Nf

l

/ ~ 8 I

I

|

100000

IIII1|

=0";Nt =92.730

/ % = 230 N/mm2; x = 230 N/mm2;

FIG. 3--Cyclic deformation behavior of two specimens.

10

I l l l l l

both specimen: eq(von Mises)..._. ~ = 508 N/mm 2_

z

0 3o

-11

m

z

cm

/

C) ,~ 800 % = 450 N/mm2;~, = 225 N/mm2; 8=90";Nf=8.515

-n

,..~1000

x

cI-"

r~ Go co

L(SWISCH ET AL. ON QUENCHED AND TEMPERED STEEL

239

100000 o

Z

~o

10000

o0 ~ / /

[]

-~

O

go

1000

9 9

13

J~

N, = 2,15 NoI'~ 100 1000

.

.

,

,

,

,

,.J

,

10000

,

t

i

o []

tension torsion

~x

;~=0,5; ~5=0"

o

~=0,5; 5=90*

9

~=1;(5=0"

9

7~=1; 5 = 9 0 *

11111

,

,

=

=

,

100000

,

||1

1000000

cycles to failure Nf FIG. 4--Correlation between lifetime NF and numbers of cycles No to the beginning of cyclic soft-

ening.

540

D

Q

~176176 I

A

520

E

500

..

.z. ~- 480 b [] *-'...... LU

0

O

.--'=

; proportional:

9

9

o

tension

.O, . . . . . . . .

n n

torsion ~.=0,5; 5=0 ~

9

;~=1; 5=0 ~

"

,'"

~o-~" I

O____O..

440 420

0

~. mp

400 0

d~pa/dIOgN with N>N o

I 2000

o .

.....

L=0,5; 8=90 ~

........ L = I ; 5=90 ~ I 40O0

slope m r [10 "6] FIG. 5--Slope of the plastic strain amplitude versus the von-Mises equivalent stress.

240


der push-pull conditions did not change their direction of propagation. In all cases, the crack that led to failure was the only one that could be detected in the specimen. The crack orientations before the change of direction are shown in Fig. 6 and Fig. 7 for the loads with the stress ratio, A = 1. The vertical axis has no meaning for the experimental points. Their shift on this axis is just to make the individual points visible. In case of the proportional load, most of the crack orientations lie between - 15 and + 30 ~ In case of the non-proportional load, most crack orientations lie at 0 ~ The crack orientations were observed only at the specimens surfaces. This means that it is the orientation of the axis of intersection between surface and crack face. A true crack face orientation measurement is very expensive and could not be realized within the project. However, a relation was searched between the frequency of crack initiation at a certain intersection axis, and the maximum shear stress and normal stress amplitude that can be found in all planes that have this intersection axis common. These stresses are also shown in the figures. The relation between the experimental frequency and the curves of the stresses supports the idea that both shear stresses and normal stresses are enhancing crack initiation, which is in accordance to the fatigue criteria presented in Table 1.

Lifetimeand FatigueLimit Lifetime and fatigue limit were predicted with all fatigue criteria shown in Table 1. The best predictions were achieved with the Zenner criterion. The following two examples, the Findley prediction, the Zenner prediction and their results are described in detail. Since in the region of limited lifetime the residual stresses had little influence, the lifetime was predicted without taking them into account. The specific parameters, c~, a and b, of the criteria were determined as functions of the number of cycles to failure by using push-pull and the alternating torsion results as reference. The comparison of the S-N curves is shown in Fig. 8 and Fig. 9.

1,2

1,0

Z=I

; 8=90

~

-.

..

"~0,8 ~ 0,6

3 e" 0,4 0,2 ..... 0,0 "90 ~

=

I

I

-60 ~

I -30 ~

i

I 0~

shear =

I

stress I

30 ~

amplitude I = 60 ~

I 90 ~

apearance of the crack at the surface FIG. 6---Crack orientation and stresses on the crack planes under combined in-phase loading with A=I.

241

L(~WISCH ET AL. ON QUENCHED AND TEMPERED STEEL

1,8

Z = I ; 8=0 ~

1,6 1,4 i~| 1 , 2 1,0

~ 0,8 ~

0,6 0,4 0,2

0,0 ,

I

.90 ~

,

I

_60 ~

,

I

.30 ~

I

,

0~

,

I

30 ~

,

60 ~

I

90 ~

a p p e a r a n c e of the crack at the surface FIG. 7 - - C r a c k orientation and stresses on the crack planes under combined out-of-phase loading vith A = 1.

600

A con

~

~

o

o []

9

A 0 []

O

9

0

A

9

9

0

9 o@ 9 O O O

[]

O O

O 0

400

,

O 9

Findley: "~am.x + ~ a . ,

i

i

,=1

10000

i

=

9

0

0

,

A

0

[]

m

]

torsion ~,=0,5; 8=0 ~ ~,=0,5; 6=90 ~ ~=1; 6 = 0 ~ ~,=1; 6=90 ~

o

A

ooooo 500

tension

[] A

9

E E Z

o

=

,

=

i , i i

i =

~

9

o o i

100000

,

""

,

,

,

=,1

1000000

cycles to failure Nf FIG. 8--Lifetime as a function o f the Findley equivalent stress.

"

242

MULTIAXIAL FATIGUE AND DEFORMATION O 600

22

o ~~

Zenner: l(a% +b% ) dm

I

9 Z~O~ A

9

9

O

500

"o :h

E Z

t)"

400

300

,

i

o

tension

[] zx

torsion ;~=0,5; 5=0 ~

0

~.=0,5; 5=90 ~

9 9

L=I; 5=0 ~ L=I; 5=90 ~

1 1 , 1 , 1

A@

O

o

,

,

10000

|

i

I

,

,11

I

i

i

1 , 1 , , 1

100000

1000000

cycles to failure Nf

FIG. 9~Lifetime as a function of the Zenner equivalent stress.

The residual stresses can not be neglected at near fatigue limit loads. Figure 10 shows this for the push-pull S-N curves of the specimen series A and B which are different in residual stress state. At stress amplitudes near the fatigue limit, the lifetimes differ from series to series, whereas at higher stress amplitudes where plastic deformation is observed no difference

regression ' cyclic

5O0

"E E Z

series A

O

550

Nf = 440 184 (ca/424 N/mm2) ls'a

0o

softening elastic

O O~.

()

behaviour 450

o=I..%

9

o"2

400

9

series B

........ regression 350

Nf = 785 772 (cr=/357 N/mm 2) -8,e I

'

'

'

'

I I I

10000

I

I

I

I

'

j

' ' l

=

100000

t

I

|

i

|

i

.l

1000000

cycles to failure Nf FIG. I O---S-N diagram of the series A and B under tension-compression load.

L(gWlSCH ET AL. ON QUENCHED AND TEMPERED STEEL

243

can be detected. This can be explained by the declining of the residual stresses by plastic deformation. At the fatigue limit a comparison of the specimen series could only be achieved by applying the weakest-link model where the local residual stresses were introduced as mean stresses in the local equivalent stress amplitudes. According to Eq 2, 3 and 4 the total endurance probability was calculated by variation of the stress amplitude until the total endurance probability is 0.5 the fatigue limit is gained. The model parameters were determined on the base of reference variants. These were the specimens of series A and B under tension compression and the alternating torsion variant. The surface reference fatigue limit, trwao, and the Weibull exponent, mA, were taken from the specimens of series B under push-pull. The volume reference fatigue limit, O-wvo, and the Weibull exponent mv were taken from the specimens of series A under push-pull. A mean stress sensitivity defined by Schuetz [17] of M = 0.4 was assumed, according to results from literature [18]. The parameters a, a and b which describe the damaging effect of the normal stress against the shear stress were gained with the help of the torsion fatigue limit. The Zenner parameter, d, can be calculated from the mean stress sensitivity M [8]. Table 4 gives a survey over the calculation parameters. Figures 11 and 12 show the results of the calculations in a o-~-~-~diagram. The open squares indicate the reference variants. The filled symbols show the experimental fatigue limits of the combined loads. The lines show the calculations without considering the residual stresses, and the open symbols show the predictions which include the residual stress influence. Generally, the Findley criterion predicts large differences between in- and out-of-phase loading, whereas the Zenner criterion predicts no differences for these loads.

3O0

i

i

Findley:

(NP'--I

E E z

q

1;a,max "k 0,33 % @

i

200

13

. O

~

E ~

t-

100

O

[] reference ecperiment 9 in-phase 9 out-pf-phase .calculation O in-phase O out-pf-phase calculation without residual stresses . . . . . in-phase out-pf-phase I

0

I

100

i

I

200

~

~ %

I

In

300

400

normal stress amplitude a [N/mm 2] FIG. l l--Measured fatigue limits in a t r a - ra plane in comparison with the prediction using the Findley criterion.

244

MULTIAXIALFATIGUE AND DEFORMATION 300

i

z

Zenner:

E E

(a%2+bo" 2)(1 +d~m )2

d~

Z

i

~ 200 "o Q..

E 100 I,.,.

t~ (lJ t,-

--

[] reference experiment 9 in-phase 9 out-of-phase calculation O in-phaseand out-of-phase calculationwithout residualstresses in-phase and out-of-phase

0

i

I

1O0

v

I

200

300

400

normal stress amplitude % [N/mm 2] FIG. 12--Measured fatigue limits in a ~a - "l'aplane in comparison with the prediction using the Zenner criterion.

Conclusion The presented experiments on tubular specimens of the steel 42CrMo4 (SAE 4140) show that under combined constant amplitude stress-controlled fatigue loads with longitudinal forces and torsion moments the plastic strain response under proportional loading can be described by the von-Mises equivalent stress. The cyclic softening that starts at about 50% of the lifetime increases when the loads are non-proportional. The observed crack orientations support the idea that initiation of the cracks is controlled by shear stresses and normal stresses. This is reflected in the fatigue criteria that were tested. A satisfying prediction of the lifetime and fatigue limit is possible with the Zenner criterion. At non-proportional loading, the equivalent stress amplitude is close to the maximum value for a large numbers of planes. Therefore, the probability of crack initiation is higher than under an in-phase loading with the same equivalent stress in the critical plane. The Zenner criterion considers the stresses in the material in every plane, which is obviously a better approach to fatigue damage than restriction to a critical plane. For fatigue limit prediction it is necessary to take account of the local residual stresses. This was achieved with the weakest-link concept where the residual stresses were handled like local mean stresses. Acknowledgement The work that is presented here was supported by the German Bundesministerium fuer Wirtschaft and the Arbeitsgemeinschaft Industrieller Forschungsvereinigungen under contract number AiF 10 058. The authors are grateful for this.

LOWlSCH ET AL. ON QUENCHED AND TEMPERED STEEL

245

References [1] Findley, W. N., "Effect of Stress on Fatigue of 76S-T61 Aluminium Alloy under Combined Stresses Which Produce Yielding," Journal of Applied Mechanics, Vol. 75, 1953, pp. 365-374. [2] McDiarrnid, D. L., "A General Criterion for High Cycle Multiaxial Fatigue Failure," Fatigue and Fracture of Engineering Materials and Structures, Vol. 14, 1990, pp. 429--454. [3] Dang Van, K., Griveau, B., Message, O., "On a New Multiaxial Fatigue Criterion: Theory and Application," Brown, M. W. and Miller, K. J., Eds. Biaxial andMultiaxial Fatigue, EGF 3, 1989, p. 459. [4] Bomas, H., Linkewitz, T., Mayr, P., "Application of a Weakest-Link Concept to the Fatigue Limit of the Beating Steel SAE 52 100 in a Bainitic Condition," Fatigue and Fracture of Engineering Materials and Structures, Vol. 22, 1999, pp. 738-741. [5] Sines, G, "Behaviour of Metals under Complex Static and Alternating Stresses," Metal Fatigue, Herausgegeben von Sines, G., und Waismann, J. L., Eds., McGraw Hill, New York, 1959. [6] Crossland, B., "Effect of Large Hydrostatic Pressure on the Torsional Fatigue Strength of an Alloy Steel," Proceedings of the International Conference on the Fatigue of Metals, Institute of Mechanical Engineers, London, 1956, pp. 138-149. [7] Kakuno, K. and Kawada, Y., "A New Criterion of Fatigue Strength of a Round Bar Subjected to Combined Static and Repeated Bending and Torsion," Fatigue of Engineering Materials and Structures, Vol. 2, 1979, pp. 229-236. [8] Zenner, H., Heidenreich, R., and Richter, I., "Schubspannungsintensit~itshypothese-Erweiterung und experimentelle Absttitzung einer neuen Festigkeitshypothese fiir schwingende Beanspruchung," Konstruktion, Vol. 32, 1980, pp. 143-152. [9] Simbiirger, A., "Festigkeitsverhalten ZSher Werkstoffe bei Einer Mehrachsigen, Phasenverschobenen Schwingbeanspruchung mit K~Srperfestenund Verg.nderlichen," Hauptspannungsrichtungen. LBF Darmstadt. Bericht Nr. FB-121, 1975. [10] Weibull, W., "Zur Abh~ingigkeit der Festigkeit vonder Probengrrge," Ingenieur-Archiv, Vol. 28, 1959, pp. 360-362. [i1] Brhm, J. and Heckel, K., "Experimentelle Dauerschwingfestigkeit unter Be~cksichtigung des Statistischen GrSl3eneinflusses," Zeitschrififiir Werkstofftechnik, Vol. 13, 1982, pp. 120-128. [12] Heckel, K. and KOhler, J., "Experimentelle Untersuchung des Statistischen Grrfleneinflusses irn Dauerschwingversuch an Ungekerbten Stahlproben. Zeitschrift ftir Werkstofftechnick" Vol. 6, 1975, pp. 52-54. [13] Kra, C., "Beschreibung des Lebensdauerverhaltens Gekerbter Proben unter Betriebsbelastung anf der Basis des Statistischen Grrfleneinflusses," Dissertation, M~inchen, 1988. [14] Kuguel, R., "A Relation between Theoretical Stress Concentration Factor and Fatigue Notch Factor Deduced from the Concept of Highly Stressed Volume," ASTM Proceedings 61, 1961, pp. 732-744. [15] Liu, J. and Zenner, H., "Berechnung von BauteilwShlerlinien unter Berticksichtigung der Statistischen und Spannungsmechanischen Sttitzziffer," Materialwissenschaft und Werkstofftechnik, Vol. 26, 1995, 25-33. [16] Sonsino, C. M., "Zur Bewertung des Schwingfestigkeitsverhaltens von Bauteilen mit Hilfe 13rtlicher Beansprnchungen," Konstruktion, Vol. 45, 1993, pp. 25-33. [17] Schtitz, W., Ober eine Beziehung Zwischen der Lebensdaner bei Konstanter und bei Veranderlicher Beanspruchungsamplitude und Ober Ihre Anwendbarkeit anf die Bemessung von Flugzeugbauteilen," Zeitschriftfiir Flugwissenschafien, Vol. 15, 1967, pp. 407--419. [18] Macherauch, E. and Wohlfahrt, H., "Eigenspannung und Ermiidung, Ermtidungsverhalten MetaUischer Werkstoffe," DGM-lnformationsgesellschafi, D. Munz (Hrsg.), 1985, pp. 237-283.

Jinsoo P a r k 1 and D r e w V. Nelson 2

In-Phase and Out-of-Phase Combined Bending-Torsion Fatigue of a Notched Specimen REFERENCES: Park, J. and Nelson, D. V., "In-Phase and Out-of-Phase Combined Bending-Torsion Fatigue of a Notched Specimen," Multiaxial Fatigue and Deformation: Testing and Prediction, ASTM STP 1387, S. KaUuri and P. J. Bonacuse, Eds., American Society for Testing and Materials, West Conshohocken, PA, 2000, pp. 246-265. ABSTRACT: An experimental study of the high-cycle biaxial fatigue behavior of notched specimens is reported. Solid round bars of I%Cr-Mo-V steel having a circumferential semicircular groove were tested under fully reversed constant-amplitude bending, torsion, and combined bending and torsion, inphase, and 90 ~ out-of-phase. Smooth specimens of the same material were also tested under bending as well as torsion to provide baseline data. Observed fatigue life data are used to evaluate several multiaxial fatigue life prediction models, including a critical plane method, a yon Mises approach, and an energy-based approach. Crack growth behavior as observed on the surface in smooth and notched specimens is presented and discussed.

KEYWORDS: fatigue, biaxial, multiaxial, crack growth, stress concentration, notch, crack initiation Nomenclature

a, A b B d D E Ae,~ ~eq e,,om eiz el,lit G "Yeq Y, om k Ktb Ktt h MSE

Slope and intercept in fatigue life relation Shear fatigue strength exponent Bending moment Diameter of notched cross section Diameter of unnotched cross section Modulus of elasticity Deviatoric elastic strain ranges Equivalent nominal bending strain amplitude Nominal bending strain amplitude Strain measured along specimen axis Strains measured at • ~ to specimen axis Shear modulus Equivalent nominal torsional shear strain amplitude Nominal torsional shear strain amplitude Constant used to merge bending and torsional data Elastic stress concentration factor for bending Elastic stress concentration factor for torsion Ratio of 'Yeq to F.eq Mean squared error

1 Senior researcher, Hyundai Heavy Industries Co., Ltd., 1 Cheonha-Dong, Dong-Gu, Ulsan, Korea 682-792. : Professor, Mechanical Engineering Department, Stanford University, Stanford, CA 94305-4021. 246

Copyright9


www.astm.org

PARK AND NELSON ON BENDING-TORSION FATIGUE n

2N

Ni No P tO

4, SeQA As o O-b Crx,~ry,~rz Orxa, O'ya or1 On,max

O'eq A Oemax t T ";a

# % Txya A'Tmax

0

we

247

Number of data points Number of reversals Cycles to crack initiation (1.0 mm crack) Cycles to 10% load drop Observed life Predicted life Poisson's ratio Frequency Phase angle between normal and shear stresses Angle relative to specimen axis Equivalent stress amplitude Deviatoric stress ranges Bending stress amplitude Normal stress components Amplitudes of o'x, Cry Maximum normal stress amplitude Maximum normal stress on plane of % yon Mises equivalent stress amplitude Maximum principal stress range Time Torque Maximum amplitude of shear stress Shear fatigue strength coefficient Torsional shear stress amplitude Amplitude of ~'~ Shear stress components Maximum range of shear stress Phase angle between normal stress components Elastic distortion energy fatigue damage parameter

Despite its importance in mechanical design, experimental research on the topic of multiaxial fatigue of specimens with stress concentrations (notches) has been relatively sparse. In the forties, Gough [1] investigated fatigue limits of specimens with four types of notches (e.g., V-type groove, oil hole, transition fillet, and longitudinal spline) subjected to in-phase bending and torsion. Test resuits were correlated by the so-called ellipse arc and quadrant based on nominal stress amplitudes. Several decades later, steel shaft specimens with a shoulder fillet were investigated by Simburger [2] and in an SAE biaxial fatigue test program [3]. Combined bending and torsion was applied, both inphase and out-of-phase, producing lives over the range of 103 to 106 cycles. Correlations by various life prediction models for those tests are presented in Refs 2, 4, and 5. During the late eighties, studies of the low-cycle, proportional, axial-torsional, elevated temperature fatigue behavior of 304 stainless steel specimens with a hole or V-notch were reported [6, 7]. More recently, Yip and Jen [8] reported studies of crack initiation at the edge of a hole in 1045 steel solid round bars for low-cycle, proportional, axial-torsional loadings. Subsequently, those authors used nonproportional, axial-torsional loadings in low-cycle tests of AIS1 316 stainless steel round bars with a semicircular circumferential notch [9]. Many structural members and machine parts contain various kinds of stress concentrations and operate in the high-cycle regime, experiencing macroscopically elastic or small elastic-plastic deformation at the stress concentrations. As indicated by the review above, published research in this cycle range has been limited, especially for nonproportional stresses. A main objective of this experimental program was to investigate effects of a stress concentration on high-cycle biaxial fa-

248


tigue behavior in the range of approximately 105 to 2 • 106 cycles. A circumferential semicircular notch in a solid round bar specimen was chosen as a stress concentration and specimens tested under combined bending and torsion, in-phase, and 90 ~ out-of-phase. The selection of 90 ~ out-of-phase loading was motivated by the desire to investigate the effect of nonproportional cyclic stresses. Also, 90 ~ out-of-phase cyclic bending and torsion simulates the stresses felt by a surface element on a rotating shaft experiencing a steady torque and bending moment at a given cross section, a situation commonly encountered in turbines and other machinery. A further objective of these tests was to make observations of crack initiation and small crack growth in the notch, rather than just recording life to fracture, as was the practice in earlier studies such as that of Gough. In addition, results of the test program will be used to evaluate three life prediction approaches: a critical plane method, a version of the von Mises criterion, and a new energy-based method.

Life Prediction Approaches Critical Plane Method The following critical plane approach suggested by Socie [10] for high-cycle multiaxial fatigue will be considered in this paper

ra + kon,max = r}(2N) b

(1)

where ra is the maximum value of shear stress amplitude, On.max is the maximum normal stress on the plane of ra, and r} and b are the shear fatigue strength coefficient and exponent, respectively. The value of k may be determined from two different sets of test data, for example, from axial (or bending) and torsional test data, as a value merging the two sets of data into a line on a plot of ra + ko'n.max versus 2N.

Von Mises Criterion The von Mises criterion has been widely used for correlating high-cycle multiaxial fatigue life under proportional cyclic stresses, when ratios of principal stresses and their directions remain fixed during cycling. For nonproportional cases, a stress-based version of the ASME Boiler and Pressure Vessel Code Case N-47-23 [11] may be used as an extension of the von Mises criterion, in which an equivalent stress amplitude parameter, SEQA, is defined from stress ranges &rx, Aoy, Aoz, Ar~y, Aryz, Arzx in the form

SEQA = ~

[(Am

-

AO'y)2

-{- ( A O ' y

-- aO'z) 2

- - 2 1112 + (Ao. - A~z)2 + 6Ar 2 + 6Areyz + ~ozarLd

(2)

where Aox = o'x(t0 - ~x(t2), A% = ~y(tl) -- Oy(t2), etc. SEQA is maximized with respect to two arbitrary instants, tl and t2, during a fatigue loading cycle. For constant amplitude bending and torsional stresses such as ox = ob sin(ogt) and z~ = rtsin(~ot - q~), where 4' is the phase angle between ~rx and r.y, Eq 2 becomes

SEQA where K = 2rt/o'b.

=o[ -~

I+~-K

+

1

1-~K 4

]1,2

(3)

PARK AND NELSON ON BENDING-TORSION FATIGUE

249

Elastic Distortion Strain Energy The SEQA approach can be considered a form of a distortion energy criterion. A different distortion energy approach can be derived based on the ranges of deviatoric stress and strain, Asij, and Ae~, for a load cycle. An elastic distortion energy fatigue damage parameter, We, can be defined as

From the relations si/= 2Ge~j = Ee~jl(1 + v) in the elastic range, Eq 4 can be expressed in terms of deviatoric stresses

We = 1 + v (AsL + AsZy + Ask + 2As 2 + 2Adz + 2As~)

4E

(5)

For biaxial fatigue with two normal stress components and a shear stress component, ~rx = O'xa sin(wt), % = O'yasin(wt - 0), rxy = rxya sin(oJt - 05), where 0 is the phase angle between crx and ~y, and 05 is the phase angle between Cxyand o-x, Eq 5 becomes

We= 2(1 q- lJ) [~

~176176176 3

2

1

+ rZxya

(6)

For in-phase stress, Eq 6 reduces to

We =

2(1 + 3E

O-e2q

(7)

where OTqis the yon Mises equivalent stress amplitude = ((3/2)$2).1/2It is of interest to note that Eq 6 has the same mathematical form as the average resolved shear stress amplitude for all of the planes in stress space as derived by Papadapolous [12], who also showed that this approach successfully correlated high-cycle biaxial fatigue data generated with (a) out-of-phase combined axial-torsional loading, or (b) out-of-phase normal stress components.

Experimental Program Material The test material used for this investigation was a hot-rolled I%Cr-Mo-V steel, which is used for bolts, nuts, and pins in turbines and many other machine parts. The steel was quenched in oil after a solution heat treatment at 930~ for 2 h and then tempered at 680~ for 3 h. Prior to being machined into specimens, solid round bars with a diameter of 40 mm were stress-relieved at 650~ for 3 h and cooled in air. The chemical composition and mechanical properties of the material are summarized in Tables 1 and 2, respectively.

TABLE 1--Chemical composition of 1% Cr-Mo-V steel (weight %). C 0.42~0.50

Si 0.20-0.35

Mn 0.45~0.70

P Max. 0.025

S Max. 0.025

Ni Max. 0.25

Cr 0.80-1.15

Mo 0.45-0.65

V 0.25~0.35

A1 Max. 0.015

250


TABLE 2--Mechanical properties of l % Cr-Mo-V steel. Modulus of elasticity, E Poisson's ratio, v Ultimate strength, o-. Yield strength (0.2% offset), try Total elongation, ef Reduction in area, RA Brinell hardness, HB Cyclic strength coefficient, K' Cyclic yield strength (0.2% offset), try

211000 MPa 0.29 828 MPa (min) 725 MPa (min) 18% (min) 50% (rain) 302 (max) 1442 MPa 515 MPa

Specimens

The geometries of the smooth and notched specimens used in the tests are shown in Fig. 1. Smooth solid round bar specimens had an hourglass test section with a minimum diameter of 16.5 mm. Notched specimens had a circumferential semicircular groove of 1.5 mm radius with an inner diameter of 14 mm at the section of the notch. Theoretical elastic stress concentration factors (SCF) for the notch are 1.95 for bending and 1.49 for torsion from Peterson's charts [13], 1.97 for bending and 1.48 for torsion from a closed-form analysis [14], and 1.98 for bending and 1.50 for torsion from a finite-element analysis (FEA) of the specimen. The values of SCFs obtained from the FEA were used in this investigation. Both ends of the specimens were designed to have fiats for gripping. Surfaces in the notch root and the smooth specimen test section were polished with diamond paste, ending with paste of approximately 1/xm particle size. (Attempts to electropolish the specimens led to rapid formarion of ferric oxides.) Test Procedure

All of the tests were conducted under fully reversed, constant-amplitude bending and torsion using a machine described in Ref 15 that applied desired angles of twist and/or bending deflections to specimens. Smooth specimens were first tested under bending and then torsion to obtain baseline data. Strain amplitudes were chosen to result in fatigue lives ranging between about 105 and 2 x 106 cycles. Strain amplitudes applied in the smooth specimen tests are listed in Table 3, as determined from strain gage rosettes attached on the top and bottom of the test section of a sample specimen (Fig. 2a). Use of the sample specimen allowed angle of twist and bending deflection applied by the test machine to be adjusted prior to each test to produce the desired strains in the specimens used for fatigue testing. Measurements of specimen diameters showed that they differed by less than 0.06% from the diameter of the sample specimen, so that strains in the specimens used for fatigue testing would be expected to be within 0.2% of those in the sample specimen (with strain varying as the cube of diameter). Fully reversed cycling was confirmed using the top and bottom rosettes of the sample specimen. For the notched specimens, four fixed ratios A of equivalent nominal torsional strain amplitude ')teq to equivalent nominal bending strain amplitude eeq, A 0 (bending), 1, 2, and ~ (torsion), were used for in-phase and 90 ~ out-of-phase bending and torsion =

')teq a

-

eeq

~//3~tno m -

2(I + v)e,,om

(8)

where 3'eq = N/3T,,oml2(l + v) and eeq = e,,om based on the von Mises criteria, and e.om and 3'no,~ are, respectively, nominal bending and torsional strain amplitudes, and v is elastic Poisson's ratio.

I_

105

---0.5

1.6/

t

105 •176

1,6/ ~,,

I 28,5(~ ='~

I

i 28,5(~

"-.05

Prior to

+--I

I_ '-

-, (b)

375

•

0.~//(IJ.m] Priorto 20~1/1.5R=.o2 polishing 25R-+1

(a)

375

t

0'4X//[I-tm] polishing

105 •176

105 •176

-1

-I

_1

21

TESTSECTION

" ~J~28.,5

FIG. 1--Geometries of(a) smooth and (b) notched specimens (dimensions in ram).

--

-r-"

=!-

+02/+

%

r •176

O"l

c m

z

0

4

z 0

0 Z 0 z W m z

Z m i-"

Z

~D

"o

252

MULTIAXIAL FATIGUE AND DEFORMATION TABLE 3--Smooth specimen bending and torsionalfatigue data. Strain Amplitude

Fatigue Life

Test No.

e (%)

,~1 (%)

N~ (cycles)

N~(cycles)

S1 $2 $3 $4 $5 $6 $7 $8 $9 $10 Sll S12 S13 S14 $15

0.262 0.257 0.246 0.236 0.224 0.210 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0.423 0.415 0.409 0.409 0.396 0.381 0.371 0.365 0.351

84 200 138 000 394 000 595 000 1 198 000 >3 400 000 72 300 89 000 140 0013 306 000 232 000 >1 500 000 371 000 >1 582 000 >2 912 000

110 000 161 000 416 000 633 000 1 249 000 102"000-123 000 181 000 389 000 275 000 ---449"000 .....

1Engineering shear strain. 2 Number of cycles to 1.0 mm surface crack length. 3 Number of cycles to 10% load drop.

(a)

Rosette

MovingClamp

Fixed

Clamp Displacement

Rosette

(b) Fixed Clamp

lo.51 =1=2o.I ~

MovingClamp

(dimensions : mm)

!

o0,e of

I TW's' Displacement

Rosette

Top View of Rosette

q,y/~4 E;[I[ 5o

SpecimenAxis

45

8i FIG. 2--Schematic showing deflection-controlled loadings and arrangement of strain rosettes in (a) smooth and (b) notched specimens.


253

The enom and "Ynomvalues were calculated from the bending and torsional moments, B and T, at the notch cross section

enom

32B ETrd 3

and

16T "Y,,o,,, - Gird 3

(9)

where E and G are, respectively, the elastic and shear moduli, and d is the diameter of the notch cross section. B was assumed to vary linearly between the two strain gage rosettes installed on the top of a sample notched specimen (Fig. 2b), which was used in the same manner as the sample smooth specimen to set up desired strains. The values of B at the positions of the rosettes were computed as et1E~rD3/32, where D is the diameter of the unnotched section and Su is the strain value of the second element (in direction of specimen axis) of a rosette. The value of B at the notch cross section was determined using the two values o r B computed from the two rosettes based on the aforementioned assumption of a linear distribution of B between the two rosettes. This assumption was confirmed from the linear distribution of bending stress along the direction of the specimen axis (except near the notch) obtained from the finite element analysis. Twas assumed to be constant along the direction of the specimen axis and computed as (el - elzl)GcrD3/16 where ~:1and si11(~I > F'Ill) are the strain values of the first and third elements (45 ~ with respect to the specimen axis) of the rosette. The value of T at the notch cross section was taken as the average of the two values of T (which were very close) determined from the two top rosettes. The bottom rosette was used to confirm fully-reversed cycling. FEA of the notched specimen showed that stresses at the positions of the rosettes were not affected by the notch geometry; in other words, the rosettes were located sufficiently far from the shoulder of the notch so that measured strains were nominal strains. The small size of the rosettes (2 mm gage length) minimized effects of the curvature of the specimens on strain measurements. The ability of the test machine to produce desired in-phase and 90 ~ out-of-phase loadings was verified by examining bending and torsional strain histories measured from the rosettes for different loading (in-phase and 90 ~ out-of-phase) conditions. All of the strain amplitudes applied to the notched specimens were selected to result in approximately the same range of fatigue lives as the smooth specimen tests. The nominal strain amplitudes and phase differences applied are listed in Table 4. Crack initiation was defined as a 1.0-ram-long surface crack and final failure as a 10% drop in load monitored by a load cell in the test machine fixture holding the clamped end of a specimen. The drop corresponded to a comparable decrease in bending moment or torque for the bending or torsion tests, respectively. For combined bending and torsion, the drop was created by a loss of both bending stiffness and torsional rigidity. Crack formation and growth behavior were observed through a microscope installed vertically over the specimen. (Attempts to take surface replicas of notches were unsuccessful due to the small radii and double curvature of the notches.) All of the tests were performed at a frequency of about 2.5 ~ 3.5 Hz in air at room temperature. Crack initiation and final failure are further discussed in a later section. Results

Smooth Specimens In bending tests of smooth specimens, all of the cracks that initiated propagated on the plane of maximum principal stress range Atrm~x, i.e., along the direction perpendicular to the specimen axis (Fig. 3). At the higher strain amplitudes, it appeared that cracks initiated on the plane of maximum shear stress range A~-. . . . grew to a length of approximately 10 - 50/xm, then proceeded in a zigzag manner perpendicular to Ao'r~a~. At the lower strain amplitudes, initiation by shear was not clear. Only one crack occurred on the top or bottom of a specimen and propagated to final failure. In the torsion tests, cracks always formed and grew on the plane of A~'maxalong the specimen axis to a length of approximately 50 - 400/zm; then the cracks changed direction onto the plane of Ao'max

254

MULTIAXlAL FATIGUE AND DEFORMATION TABLE 4 ~ N o t c h e d specimen bending, torsional, and combined bending-torsional fatigue data. Strain Amplitude

Test No. N1 N2 N3 N4 N5 N6 N7 N8 N9 N10 Nil N12 N13 N14 N15 N16 N17 N18 N19 N20 N21 N22 N23 N24 N25 N26 N27 N28

8no ml

(%)

0.133 0.128 0.124 0.116 0 0 0 0 0 0 0.110 0.106 0.103 0.100 0.079 0.078 0.077 0.075 0.073 0.125 0.122 0.116 0.112 0.089 0.089 0.089 0.083 0.079

Phase

Fatigue Life 3

3~o,~ (%)

~0(deg.)

N i (cycles)

0 0 0 0 0.304 0.304 0.293 0.290 0.275 0.252 0.161 0.158 0.153 0.149 0.235 0.231 0.227 0.222 0.218 0.184 0.179 0.171 0.166 0.265 0.265 0.265 0.246 0.235

... ... ... ... ... ... ... ... ... ...

130 000 208 000 >1 700 000 >1 700 000 65 000 142 000 97 000 700 000 N.M. 5 260 000 232 000 233 000 165 000 >1 300 000 157 000 385 000 165 000 > 1 300 000 >1 300 000 197 000 151 000 353 000 >1 300 000 125 000 130 000 289 000 N.M. > 1 300 000

0 0 0 0 0 0 0 0 0 90 90 90 90 90 90 90 90 90

Nominal bending strain amplitude at the notch section, e,,om --

4

N f (cycles)

262 000 380 000 ... .H

110 000 420 000 164 000 1 230 000 184 000 870 000 453 000 524 000 334 000 315"000-725 000 309 000 ... 236"000-185 000 440 000 160000 213 000 440 000 246 000 ...

32B E ~rd~ "

16T 2 Nominal torsional strain amplitude at the notch section, Y, om= Gird 3. 3 Number of cycles to 1.0 m m surface crack length. 4 Number of cycles to 10% load drop. 5 Not measured.

(Fig. 4). In t h o s e tests, f r o m one to a few cracks occurred a r o u n d the c i r c u m f e r e n c e o f a s p e c i m e n at the m i n i m u m section, o n e or two o f t h o s e cracks leading to final failure. Elastically calculated strain a m p l i t u d e s applied to the s m o o t h s p e c i m e n s a n d r e s u l t i n g fatigue lives, N / ( 1 . 0 m m surface crack) a n d N s (10% load drop), are listed in T a b l e 3. M o s t o f the life w a s s p e n t in the crack initiation stage, f o r m i n g m i l l i m e t e r - s i z e d cracks; for instance, N i / N f w a s 75 ~ 9 5 % for b e n d i n g a n d 70 - 85% for torsion. Figure 5 represents the b e n d i n g a n d torsional fatigue data u s i n g the s h e a r stress-based critical p l a n e p a r a m e t e r o f Eq 1, w h e r e the solid line w a s d r a w n f r o m the 10% load drop data u s i n g the relation: ~'a + 0.23o',,m~x = 7 2 3 N ] -~176

(mPa)

(10)

T h e c o n s t a n t s in Eq 10 were d e t e r m i n e d b y c o m p u t i n g ~'a for the torsion data points a n d ~'a a n d On,max for the b e n d i n g points. T h e n a least-squares r e g r e s s i o n w a s p e r f o r m e d for the data set o f the torsion


255

FIG. 3--Crack growth in a smooth specimen tested in bending. A~mox and A'rmaxdenote ranges of maximum principal stress and maximum shear stress, respectively. plus bending points using log (i-~ + ko-. . . . . ) = logA + a log Nf

(11)

with k = 0 for the torsion points and k a variable for the bending points. The value of k was varied from 0 to 1 in steps of 0.01 to see which value maximized the sample correlation coefficient, also al-

FIG. 4---Crack growth in a smooth specimen tested in torsion. A~r,~ and A'Cmaxdenote ranges of maximum principal stress and maximum shear stress, respectively.

256


500

1%Cr-Mo-V Steel Smooth Specimen []

1.0 mm crack Bending Torsion

|

Bending

[]

Torsion

0 (2.

10% load drop

400 x E C

b 03

350

....

C)

I-I --i.--

"'"-..

300 0"--~

25O i

i

i

i

i

ii

i

i

i

~

10 s

i

i

i

i]

i

106

I

i

i

i

i

i

i~

107

N (cycles) FIG. 5--Correlation of smooth specimen bending and torsion fatigue data by the critical plane approach. The dashed line is based on the 1.0 mm crack data and the solid line on the 10% load drop data.

lowing determination of the values of A and a. The same procedure was applied separately for the lives to 1.0 mm cracking. The k value again turned out to be 0.23, but the right-hand side of Eq 10 became 661Ni -~176 as given by the dashed line in Fig. 5. The elastic distortion strain energy parameter We of Eq 6 was also used to represent the smooth specimen data, as shown in Fig. 6, where the solid line is

We = 5.50Ni ~

(MJ/m 3)

(12)

from the bending test data for final failure and the dashed line is We = 4.72Ni-~ from the bending data for crack initiation. Note that lives in torsion were somewhat greater than those in bending for the same value of We since We does not use a parameter such as k to merge bending and torsional data. Figure 7 shows correlations by the SEQA parameter of Eq 3, where the solid line is from the bending data for final failure

SEQA = 1159 N f 0"063

(MPa)

(13)

and the dashed line is SEQA = 1074N~ -~176 from the bending data for crack initiation. The above baseline equations will be used in the next section for comparison with the notched specimen fatigue test results.

Notched Specimens In bending tests, cracks initiated on the top or bottom of a notch and grew along the notch root to final failure. Characteristics of cracking behavior were almost the same as that in smooth specimens. In torsion tests, cracks initiated on the plane of maximum shear stress range Aq'max either along the


1%Cr-Mo-V Steel Smooth S p e c i m e n

E

2 1.5

O []


| []

10% load d r o p Bending Torsion

257

[]

0--'~

.6

,

,

i

,

, , i

i

i

i

i

10 2

i

[

i

i [

[

i

i

i

i

10 8

N

i

i

ii

10 7

(cycles)

FIG. 6--Correlation of smooth specimen bending and torsion fatigue data by the elastic distortion energy parameter We. The dashed line is based on the 1.0 mm crack data and the solid line on the 10% load drop data.

1000

1%Cr-Mo-V Steel Smooth S p e c i m e n O []


| []

10% load d r o p Bending Torsion

800 t~ (2_

< (2 UJ 09

[]

600

...

E]

"-'11~

500 0---~ 400

......

I

,

,

, ,,

105

,,,I

106 N

I

I

I

I

, ,,,I

107

(cycles)

FIG. 7--Correlation of smooth specimen bending and torsion fatigue data by the equivalent stress

parameter SEQA. The dashed line is based on the 1.0 mm crack data and the solid line on the 10% load drop data.

258


FIG. 8--Crack growth in a notched specimen under torsion. A0-,no_~and A.r,~, denote ranges of maximum principal stress and maximum shear stress, respectively.

direction of the specimen axis or along the circumferential notch root. The surface lengths over which this shear mode cracking occurred, however, were less than that in the smooth specimens, for instance, approximately 50-200/xm, which may be due to the effect of geometry of the notch. Then, the cracks turned into the tensile mode (Fig. 8) and propagated on the plane of maximum principal stress range A0"maxto final failure. Under in-phase bending and torsion, cracks formed on the plane of ATmax and grew to a length of about 10~50/~m and then turned onto the A0-maxplane for both cases of A = 1 and 2 (Fig. 9). Under 90 ~ out-of-phase combined bending and torsion, the two cases of A = 1 and 2 showed different cracking behavior. For A = 1, cracks initiated on the A~'m~xplane but propagated on the AO'maxplane (Fig. 10) as with the in-phase cases. For A = 2, cracks initiated on the A ~'maxplane as for other load cases; however, the cracks did not follow the Ao'max plane exactly (Fig. 11) but rather grew in a direction between the planes of Arm,x and AO'max. The difference in crack growth behavior for the case of A = 2 with a phase difference of 90 ~ might be explained by examining the magnitude of maximum normal stress amplitude o"1 (= A0-m,x/2) on planes whose normals are at angle 0 with respect to the specimen axis 0-1 = maxt [0-~ COS2 ~/ -~ Txy sin(20)]

(14)

where o'x is the notch root normal stress given as gtbEeno m sin(tot), ~'xyis the notch root shear stress given as gttG'Ynom sin(tot - th), Ktb and Kit are, respectively, theoretical stress concentration factors for bending and torsion, and th is the phase difference between bending and torsion. The value of o1 in Eq 14 is maximized with respect to the time t. Figure 12 shows the variation of 0-1 versus the angle of plane 0, where it can be seen that for the case of 90 ~ out-of-phase loading with A = 2, the maximum value of 0-1 changes little (less than about 5%) in the range of 0 = 0~176 and 140~ ~ Therefore, cracks initiated on the Armax plane seemed to have followed a plane in that range. On the other hand, the 0-1 values for the other loading cases drop abruptly from the maximum point and all of the cracks propagated on the trl planes.


259

FIG. 9--Crack growth in a notched specimen under in-phase bending and torsion. A~mo~and A'cm~x denote ranges of maximum principal stress and maximum shear stress, respectively.

FIG. lO--Crack growth in a notched specimen under 90 ~ out-of-phase bending and torsion ( A = 1). A~m~ and A'r,,~ denote ranges of maximum principal stress and maximum shear stress, respectively.

260


FIG. 11--Crack growth in a notched specimen under 90 ~ out-of-phase bending and torsion ( A = 2). AO'ma x and A ~'m~xdenote ranges of maximum principal stress and maximum shear stress, respectively.

-

Bending - In-Phase (X=I)

-

---------. . . . . . . . .

.... 1.0

0.8

0.6

0.4

7

In-Phase (~.=2) 90* (~,=1) 90* (~=2) Torsion

,,,, .~~'~.'~\-.~..

/...c;'~.. i:~,-~"f-

/"'~\,Q.\X'\"\ ,?.;',.

/./I"" ..i/.\.X'~.I

'_ / \"\.\'Q

,......,.,i-\/ ..

-/ _!

/ / 7 /I7.,"\

\ -~,;..!,/>-/,

0.2 9

0.0 0

.

,

,

30

,

,

,

,

60

."-C'k/~,./,

90

T

,",,,/, ',~/. 120

150

,

'

180

(degree)

FIG. 12--Maximum normal stress amplitude 0 1 vs. angle of plane ~ with respect to the specimen axis under combined bending and torsion.

PARK A N D NELSON ON B E N D I N G - T O R S I O N FATIGUE

261

To evaluate results of the tests with notched specimens, nominal elastic stresses at the notch were multiplied by Ktb and K , to obtain notch stresses. Multiaxial cyclic elastic-plastic analyses of notch strains were not attempted because of the significant computational uncertainties that would be involved, especially for nonproportional stresses, and because notch plastic strains were relatively small in any case. The nominal strain amplitudes applied and corresponding fatigue lives are listed in Table 4. Figures 13 to 15 show correlations by the parameters ~'a + ktrn,m,x, We, and SEQA, respectively, based on elastically calculated notch stresses. Solid and dashed lines in the figures were obtained based on the smooth bending and/or torsional test data, as described in the previous section. The parameters ~-a + ko'~,m~• and We resulted in conservative correlations. The SEQA parameter correlated the test data conservatively except for 90 ~ out-of-phase loadings with h = 1. As a measure of the relative performance of the different life prediction approaches, mean squared errors (MSE) were computed from

MSE =

_1 _s ~.~ (log Np -

(15)

log No)2i

n i=l

where n is the number of data points, Np is predicted life, and No is observed life. MSE for the parameters ~'a + k~ We, and SEQA in Figs. 13 to 15 were computed to be 0.72, 1.23, and 0.52 for the 1.0 mm surface crack data, and 0.76, 1.29, and 0.64 for the 10% load drop data, respectively.

Crack Initiation and Final Failure In order to compare lives spent in the crack initiation and propagation stages, numbers of cycles to a 0.1 mm surface crack, to a 1.0 mm surface crack, and to a 10% load drop are compared in

800 1%Cr-Mo-V Steel Notched Specimen ~3

600

O [] A

empty symbol : 1.0 mm crack marked symbol: 10% load drop

x

E cO co oq o 4-

~|174

400

Bending Torsion In-Phase (;L=I)

v

In-Phase (X=2)

0

90- (~=1)

0

90 ~ (;L=2)

|

300 250

......

r

. . . . . . . .

105

r

106

N

,

. . . . . . .

107

(cycles)

FIG. 13--Correlation of notched specimen bending and torsion fatigue data by the critical plane parameter. The dashed line is based on the smooth specimen 1.0 mm crack data and the solid line on the 10% load drop data.

262


1%Cr-Mo-V Steel Notched Specimen

2

O

90 o (x=2)

[]

Z~ V

E v

O

Bending Torsion In-Phase (2~=1) In-Phase (X=2) 90 ~ (Z=l)

O

empty symbol : 1.0 mm crack marked symbol: 10% load drop

~ | 1 7 4

|

1.5

0 . 7

r

r

i

i

i i i

i

I

I

i

r

i

105

,

,r

i

,

,

,

,

108 N

,

,

,

107

(cycles)

FIG. 14---Correlation of notched specimen bending and torsion fatigue data by the elastic distortion energy parameter We. The dashed line is based on the smooth specimen 1.0 mm crack data and the solid line on the 10% load drop data.

1000

1%Cr-Mo-V Steel Notched Specimen empty symbol : 1.0 mm crack marked symbol: 10% load drop

O [] L& v O O

n

Bending Torsion In-Phase (;L=I) In-Phase (;L=2) 90 ~ (~=1) 90 ~ (X=2)

v

< O

700 []

[] [] []

LU 03

[] []

i~

[]

500

400

......

, 105

. . . . . . . . N

i 106

....... 107

(cycles)

FIG. 15--Correlation of notched specimen bending and torsion fatigue data by the equivalent stress parameter SEQA. The dashed line is based on the smooth specimen 1.0 mm crack data and the solid line on the 10% load drop data.

PARK AND NELSON ON BENDING-TORSIONFATIGUE

263

Fig. 16, where it can be seen that for smooth specimens most of the life was spent in forming a crack of 1 mm length, and slightly more so for bending. On the other hand, notched specimens spent more of their lives after cracks reached 1 ram, for instance, about 40-50% of total lives for bending, torsion, and in-phase bending and torsion. Under 90 ~ out-of-phase loadings, remaining lives were reduced to about 20~30%, especially for the case of A = 1. It is also of interest to note that most of the life was spent in forming a 0.1 mm surface crack under 90 ~ out-of-phase loadings.

Discussion The scatter in fatigue lives of notched specimens as correlated by the critical plane, We and SEQA parameters in Figs. 13-15 is likely associated with the stress levels used in testing being close to a fatigue limit, a situation that tends to increase scatter [16]. The visual appearance of the scatter is also somewhat exaggerated in those figures by the difference in log coordinates used for the abscissae and

FIG. 16---Comparison of fatigue lives to crack initiation and final failure.

264


ordinates. The correlations by the SEQA approach follow the trends observed in other studies [5,15] of being nonconservative for out-of-phase combined bending and torsion, but the somewhat lower mean squared error of the correlations by that parameter compared to the We or critical plane parameters was unexpected. The three fatigue damage parameters considered in this paper were evaluated with elastically calculated stresses. Over the years, elastic stresses have often been used to correlate high-cycle fatigue data from notched specimens, even when there was some plastic straining at notch roots. In future work, it might be of interest to evaluate the test results in this paper using notch strains estimated by various multiaxial cyclic elastic-plastic notch analyses that have been under development in recent years [17]. In high-cycle fatigue of lab specimens, it is generally assumed based on numerous empirical observations that most of the fatigue life (to 10% load drop) is spent in forming millimeter-sized cracks [18]. Such is the case in tests of smooth specimens reported here. However, in most of the tests of the notched specimens, roughly half of the life was spent in initiating cracks of that size and the remainder in crack propagation, The role of crack growth in high-cycle multiaxial fatigue of notched specimens may be even more significant for other types of notches where the geometry of the notch offers greater resistance to crack growth than the one used in these tests. Such geometries might include splines, keyways, or circumferential V-grooves. Thus, the multiaxial fatigue life of notched specimens or components should depend not only on surface stresses in the notch but also on notch geometry. The test results reported here also showed that, in many cases, small cracks initiated by shear grew a small distance (-- 1 (4)

TF=

X/(o-i - o'11)2 + (o'ii - o'1.) 2 + (o'm - o'i)2

v~ This approach was previously used to estimate torsional fatigue behavior from axial fatigue life relationships o f Haynes 188 at 316 and 760~ [24,25]. For torsion (o"i = -O'ni and oil = 0; TF = 0; and ,10-1

104 102

=

,

,,,,=,1

,

,

,,,,l,I

,

103

,

,,,,,,I

t

104

,

,,,,

105

10 e

Cyclic Life, Nf FIG. 3--Estimation of torsional fatigue lives with modified multiaxiality factor approach.

MF = 0.5), the estimated torsional fatigue life relation from Eqs 3 and 4 forms an upper bound to the experimentally observed torsional fatigue data at 538~ (Fig. 3). Four nominal strain ranges, two each for axial (Ae 1 = 0.02 and Ae2 = 0.0067) and torsional (A3'1 = 0.035 and AT2 = 0.012) loading conditions, were selected for the subsequent cumulative fatigue tests. Duplicate tests were conducted in the baseline test program to evaluate repeatability of the cyclic deformation behavior and to provide a more accurate estimate of the fatigue life for each test condition. The evolution of cyclic axial and shear stresses are plotted in Fig. 4 for the baseline axial and torsional tests. In each of these tests, Haynes 188 exhibited cyclic hardening for a majority of the life with a slight softening towards the end of the test. No significant differences were observed be-

1400 D. b e2 > e3) Elastic strain and plastic strain, respectively Phase delay between strains on the axial and torsional axes Shape factor for a semielliptical opening mode crack, and for a semielliptical shear crack, respectively Biaxial strain ratio (A = e3/el) Equivalent number of small cycles to failure Elastic, plastic, and effective Poisson's ratios, respectively Angles during loading and unloading parts of a cycle, respectively, at which the Mohr's circles are the largest Principal stresses (0-1 > 0"2 > 0"3) Mean normal stress Maximum normal stress Minimum normal stress Axial fatigue strength coefficient and axial fatigue ductility coefficient, respectively Shear fatigue strength coefficient and shear fatigue ductility coefficient, respectively

Many engineering components that undergo fatigue loading experience multiaxial stresses in which two or three principal stresses fluctuate with time; i.e., the corresponding principal stresses are out-of-phase or the principal directions change during a cycle of loading. Extensive reviews of multiaxial fatigue life prediction methods are presented by Garud [1], Brown and Miller [2], and You and Lee [3]. Methods for predicting multiaxial fatigue life will be briefly reviewed. Equivalent Stress/Strain Approach The most commonly used method of correlating uniaxial and multiaxial fatigue life transforms cyclic multiaxial stresses into an equivalent uniaxial stress amplitude thought to produce the same fatigue life as the multiaxial stresses. The most popular methods for making the transformation are extensions of the von Mises yield criterion in which static values of principal stress are replaced by amplitudes and the yield strength is replaced by the uniaxial fatigue strength. Jordan [4] and Garud [5] showed that the von Mises criterion successfully correlates multiaxial life data only under proportional loading in the high-cycle fatigue regime. Energy Approach Fatigue is generally believed to involve cyclic plastic deformations which are dependent on the stress-strain path. Garud [6] applied this approach in conjunction with incremental plasticity theory to predict fatigue crack initiation life under complex nonproportional multiaxial loading conditions. Ellyin et al. [7,8] tried to correlate uniaxial and torsional data using the total strain energy density. They proposed that the durability of components should be characterized by the quantity of energy that a material could absorb. Critical Plane Approach Fatigue analysis using the concept of a critical plane of maximum shear strain is very effective because the critical plane concept is based on the fracture mode or the initiation mechanism of cracks.

VARVANI-FARAHANIAND TOPPER ON STRAIN PATHS

307

In the critical plane concept, after determining the maximum shear strain plane, many researchers define fatigue parameters as combinations of the maximum shear strain (or stress) and normal strain (or stress) on that plane to explain multiaxial fatigue behavior [5,9-11]. Strain terms are used in the region of low-cycle fatigue (LCF) and stress terms are used in the high-cycle fatigue (HCF) region in these critical plane approaches to multiaxial fatigue analysis. Brown and Miller [9] tried to analyze multiaxial fatigue in the low-cycle fatigue region by using the state of strain on the plane where the maximum shear strain occurred, while Findley [10] and Stulen and Commings [12] used stress terms in the high-cycle fatigue region. Combined Energy~Critical Plane Approach

Critical plane parameters have been criticized for lack of adherence to rigorous continuum mechanics fundamentals. To compensate for this lack, Liu [13], Chu et al. [14], and Glinka et al. [15] used the energy criterion in conjunction with the critical plane approach. Liu [13] calculated the virtual strain energy (VSE) in the critical plane by the use of crack initiation modes and stress-strain Mohr's circles. In the calculation of VSE, Liu included both elastic energy and plastic energy while the elastic energy was not considered in Garud's model [6]. Chu et al. [14] formulated normal and shear energy components based on the Smith-Watson-Topper parameter. They determined the critical plane and the largest damage parameter from the transformation of strains and stresses onto planes spaced at equal increments using a generalized Mroz model. Glinka et al. [15] proposed a multiaxial life parameter based on the summation of the products of normal and shear strains and stresses on the critical shear plane. In the present study, a multiaxial fatigue parameter for various in-phase and out-of-phase strain paths is proposed. The parameter is given by the sum of the normal energy range and the shear energy range calculated for the critical plane at which the stress and strain Mohr's circles are the largest during the loading and unloading parts of a cycle. The normal and shear energies in this parameter have been weighted by the tensile and shear fatigue properties, respectively, and the parameter requires no empirical fitting factor. This parameter takes into account the effect of the mean stress applied normal to the maximum shear plane. The proposed parameter also increases when there is additional hardening caused by out-of-phase straining, while strain-based parameters fail to take into account this effect. The proposed parameter gives a good correlation of multiaxial fatigue lives and crack growth rates for various in-phase and out-of-phase straining conditions.

Materials and Multiaxial Fatigue Data Table 1 lists the references for in-phase and out-of-phase multiaxial fatigue data used in this study and tabulates the fatigue properties of the materials used. Fatigue coefficients tr} and e~ are the axial

TABLE 1--Fatigue properties of materials used in this study. Materials and Fatigue Data

E, GPa

~

o-j MPa

G, GPa

yj

~'~MPa

Ni-Cr-Mo-V steel* [16] 1 1045 steel [17-19] 1 Incone1718 [20]1 Haynes 188 [21]1 Waspaloy [22, 23] 2 Mild steel [24]2 Stainless steel [25]1

200 206 208.5 170.2 362 210 185

1.14 0.26 2.67 0.489 0.381 0.1516 0.171

680 948 1640 823 2610 1009 1000

77 79.2 80.2 65.5 139.2 80.8 71

1.69 0.413 3.62 1.78 0.516 0.322 0.413

444 505 1030 635 1640 431 709

* Ni-Cr-Mo-V steel is known as rotor steel. 1Fatigue properties are given by referenced papers. 2 Fatigue properties are calculated from uniaxial and torsional fatigue life-strain data.

308


" [ ~axial

l

(a)

-- ~

fatigue curve

~,.

Log (fatigue life-cycles) 107

'I•\

(b)

i

'atigue

1 Log (fatigue life-cycles) 107

FIG. 1--Schematic presentation of fatigue life-strain curves for (a) uniaxial loading, and (b) torsional loading.

fatigue strength coefficient and axial fatigue ductility coefficient, respectively, and ~-}and y~ are the shear fatigue strength coefficient and shear fatigue ductility coefficient, respectively. These coefficients are illustrated in Fig. 1 for: (a) uniaxial, and (b) torsional fatigue loading life-strain curves. In-Phase and Out-of-Phase Strain Paths

In this study, for convenience of presentation, in-phase and out-of-phase strain paths have been categorized into three kinds: (a) in-phase strain paths (strain histories A1, A2, A3, A4, A5, and A6), (b) linear out-of-phase strain paths (strain histories B1, B2,), and finally (c) nonlinear out-of-phase strain paths (C1, C2, C3, C4, and C5). In-Phase Strain Paths In in-phase straining, both axial and shear strain cycles are alternating with no phase difference. Strain paths in in-phase straining are linear. For the in-phase straining data used in this study, the strain histories, strain paths, and strain and stress Mohr's circles are presented in Fig. 2a and Fig. 2b. The largest stress and strain Mohr's circles during the loading part (at 0a) and unloading part (at 02) of a cycle for which the maximum shear stress and strain and corresponding normal stress and normal strain values are calculated are illustrated in Fig. 2. In this figure, the light Mohr's circles are the largest during the loading part, and the dark Mohr's circles are the largest during the unloading part of a cycle. The strain histories A1, A2, and A3 correspond to uniaxial straining, torsional straining, and inphase combined axial and torsional straining, respectively. The linear in-phase strain paths shown in Fig. 2b have mean values. Strain history A4 is a combined axial and torsional strain path with an axial mean strain. Strain history A5 has a torsional mean strain, and finally, strain history A6 has both axial and torsional mean strains. Linear Out-of-Phase Strain Paths In out-of-phase alternating straining there is a phase difference between the axial and shear strain cycles. Strain history B 1 (Box) and strain history B2 (Two-Box) shown in Fig. 2c, are linear out-ofphase strain histories. Nonlinear Out-of-Phase Strain Paths In the nonlinear out-of-phase strain histories examined there is a phase delay between the axial strain and torsional strain. Strain paths are elliptical and as the phase difference increases the ellipti-

VARVANI-FARAHANIANDTOPPERONSTRAINPATHS Strain Path

~/~History A 1

02=270~

v

I

,I

4 90~

0o at 270 ~

q

,

-! 02=270~

'

~

~'i

'

F)o

7~.

f

,

at 270

I x ~,,rz/

k_LJi ' ' '"Lo

270o~

I 7/q3 ~

r o1~0o~Q/ .-

Ii

Stress Mohr's Circle ~

t90~

.._

270~

.~

/~

17 / "/'~

Strain Mohr's Circle ~ 7/21" , / 7 ~ / 2 i at90 ~

309

f at 25o ~

t ACt .

I

~'m:

Y)r r

Q'

A^o

p,

~Ae,,~ Ao n

FIG. 2a--Strain history, strain path, and Mohr's circle presentation for in-phase strain paths.

~istoryA4

Strain Path

Strain Mohr's Circle

270~ ~

Q ' Ae,

Q

!

Stress Molar's Circle

270~ t~al. ~Q'

l

/r,,r--X..b/.

. ~

~

,

~

I

-

-

,.~., 7/21 ~./90~

x! ,D/90o

/ 2 7 0 ~

90~

I

Ao- u

FIG. 2b----In-phase strain histories, paths, and Mohr's circle presentations for in-phase paths containing mean strain values.

310


I 4Strai~a History B 1 02

Strain Path

Strain Mohr's Circle

Stress Mohr's Circle

t

Ae~

AO"n

AE n

AtT~

01

\ T

lil

I I

i/

FIG. 2c--Linear out-of-phase strain history, path, and strain-stress Mohr' s circles.

cal path becomes larger in its minor diameter, and finally, at a 90 ~ phase difference the strain path becomes circular. Strain histories C1, C2, C3, and C4 present out-of-phase axial and torsional straining with phase delays of 30, 45, 60, and 90 ~ respectively. Strain history C5 corresponds to a 90 ~ out-ofphase strain path containing an axial mean strain value. Figure 2d presents nonlinear out-of-phase strain paths, strain histories, and strain and stress Molar's circles. The maximum shear strains for in-phase and out-of-phase strain paths were numerically calculated at 10~ increments through a cycle and are presented in Figs. 3a and b, respectively. Proposed Parameter and Analysis

Figure 4 illustrates a thin-walled tubular specimen subjected to combined axial and torsional fatigue. The strain and stress tensors for a thin-walled tubular specimen subjected to axial and torsional fatigue are given by Eq I and Eq 2, respectively f

- - ~,effA~ap

Aeap

(1)

0

(L

(2) 0

where axial and shear strain ranges Aeap, A(Tap[2), respectively, are given by Eq 3 and Eq 4 as Aeap = Aea sinO

(3)

311

VARVANI-FARAHANI AND TOPPER ON STRAIN PATHS

Strain History C 1 I,/~ 02~300~

strain Path

Strain Mohr's Circle yI2[ y,.=/2 at 120~

e

Stress Mohr's Circle v~ ~ at 120~

AT

Am~

at 300~ al~-e. I

300" I/'~

310; 9

Y[ 1 2 0 o ~

\

~..~'vrg'

i

Va~3()0~ ] ~ a .

A~e.[

3~-A~Ja "

Y/2~ ~.-/13(P

"r I

3l O ~ [ - ~ e J

~

I

I

Ig ~11

1 II

~/130 ~

31( W ' 4 " ~

~.3a:r~

m

27 o 0 ~ , , ' ~

l / r'-,,

n

FIG. 2d--Strain histories, paths, and Mohr' s circle presentations for nonlinear out-of-phase strain paths without and with mean strain values.

9~ 1.2

--~

o~!

1.0 - m- HistoryAS]

i.,--,,,'~

(a)

B

'~

i

~0.8 E.. 8--g~--~...~. "~0.6 ~0.4 " i~9 . . . . . --~ , "~ 0.2 r"..~--" ! "'l ~o.o 20 40 60 80 100 120 140 160 0 (Degrees)

~'0.40

"go.35 0.30 0.25 .~ 0.20 ~0.15 ~0.10 "~ 0.05 0.00

.. ~ . i ~ - - ~ , ~ . o x "~,

z-v.. 9

---'l'--Hi~o~

C1

(~=30,

el~1201

-,~-aistory c2 (~-~51m=12o~I --A--History

C3

0~'60,

01ffi130~

--o--Historye4 (~--9o.ol--9o)| , i . , , 1 1 . ,

60

80

i,**

i,

,.i

ii

ii

i i I

I00 120 140 160 180 0 (Degrees) FIG. 3--Maximum shear strain through loading part of a cycle for various (a) in-phase loading, and (b) out-of-phase loading conditions.

312


I.

x

zr (a)

(b)

(c)

FIG. 4 ~ ( a ) Thin-walled tubular specimen subjected to combined axial and torsional fatigue, (b) 3-D presentation of strain state, and (c) stress state.

(4) where "sa and ya/2 are the applied axial and shear amplitude strains, respectively. The angle 0 is the angle during a cycle of straining at which the Mohr's circle is the largest and has the maximum value of shear strain. Angle th corresponds to the phase delay between strains on the axial and torsional axes. In Eq 2 AO'aand hra are the ranges of axial and shear stresses, respectively. In Eq 1 veff is the effective Poisson's ratio which is given by

l)eff =

(5)

llee e + 1.'pSp "se -~- "sp

where Ve = 0.3 is the elastic Poisson's ratio and Vp = 0.5 is the plastic Poisson's ratio. The axial elastic and plastic strains are given by Eq 6a and Eq 6b, respectively o'a

"sp z ,Sap

(6a)

oa E

(6b)

The range of maximum shear strain and the corresponding normal strain range on the critical plane at which both strain and stress Mohr's circles are the largest during loading (at the angle 01) and unloading (at the angle 02) of a cycle (see Fig. 2) are calculated as

\2}

\

2

= ('$1 +'s,) A's.

\

2

,]Ol

(7a)

\~]o2

('s1 +'$3) J01 - \

2

,/ee

(7b)

where el, e2, and "$3are the principal strain values @1 > "$2> "$3)which are calculated from the strain Mohr's circle (see Fig. 5a) as: 81 = ( 1 - -

1-'eft)

+ ~

"sap(1 + b'eff) 2 +

(8a)


~max

313

max

(a)

(b)

FIG. 5--(a) Strain M o h r ' s circle, a n d (b) stress M o h r ' s circle.

e3 = (1 - Ve,~)

8"2 = -- Peffgap

(8b)

- - ~1 eap 2

(8c)

(1 + veff)2 +

Similarly, the range of maximum shear stress and the corresponding normal stress range are calculated from the largest stress Mohr's circle during loading (at the angle 01) and unloading (at the angle 02) of a cycle as:

(0-1 -- 0-3~

( 0-1 -- 0-3~

Agmax = \ ~ ] 0 1

-- \

(0"1 +

A0"n = \

2

2

(0"1

i]01 --

t~

7o2

(ha)

]oz

(9b)

where 0-1, (re, and 0-3 are the principal stress values (0-1 > 0-2 > 0-3) and they are calculated from the stress Molar's circle (see Fig. 5b) as: OVa 0-1 = ~-+ ~1 [(~,2 + 4,/.211/2

(lOa)

0"2 = 0

(lOb)

03 - 0-a 2

21 [o-.2 + 4r 2] 1/2

(10c)

In strain paths with no mean strain, the largest strain and stress Mohr's circles, obtained during loading (at 01) and unloading (at 02) in a cycle, have equal diameters. In these strain paths, to achieve the plane of maximum shear strain, the plane P (obtained at 01) and plane Q (obtained at 02) should rotate counterclockwise with the angle of c~ = tan-

1

[A~alA~#a~

on the Mohr's circles (see Fig.

2a--history A3). For strain paths having a mean strain, the largest Molar's circles obtained at 01 and 02 do not have equal diameters (see Fig. 2b). To achieve the same critical plane, both planes P and Q on Mohr's circles have to rotate through an angle a (see Fig. 2b--histories A4 and A5). The ranges of shear strain and normal strain for strain histories containing axial and shear mean strains are shown in Fig. 2b. For the strain history A4 which has an axial mean strain, the ranges of shear strains and

314


stresses are calculated by multiplying the second terms of Eqs 7a and 9a by cosc~ and the ranges of normal strains and stresses are calculated by multiplying the second terms of Eqs 7b and 9b by sina. For strain history A5 which has a mean shear strain, the ranges of shear strains and stresses are calculated by multiplying the second term of Eqs 7a and 9a by sin~ which, in calculating the ranges of normal strains and stresses, the second terms of Eq 7b and 9b are multiplied by 1 + cosa. For strain history A5, containing both axial and shear mean strains, the second terms of Eqs 7 and 9 become zero. The range of maximum shear stress A~'ma~and shear strain A (~-~-~) obtained from the largest stress and strain Mohr's circles at angles 0j and 02 during the loading and unloading parts of a cycle and the corresponding normal stress range A~r, and the normal strain range mE n o n that plane are the components of the proposed parameter. Both the normal and shear strain energies are weighted by the axial and shear fatigue properties, respectively:

(o-~ ~)

(~s ~'~) \

where o-} and ~} are the axial fatigue strength coefficient and axial fatigue ductility coefficient, respectively, and ~-~and ~ are the shear fatigue strength coefficient and shear fatigue ductility coefficient, respectively. Multiaxial fatigue energy based models have been long discussed in terms of normal and shear energy weights. In Garud's approach [6] he found that an empirical weighting factor of C = 0.5 in the shear energy part of his model (Eq 12) gave a good correlation of multiaxial fatigue results for 1% Cr-Mo-V steel for both in-phase and out-of-phase loading conditions. AeA~r + CA~,AT = I(Ns)

(12)

Tipton [34] found that a good multiaxial fatigue life correlation was obtained for 1045 steel with a scaling factor C of 0.90. Andrews [35] found that a C factor of 0.30 yielded the best correlation of multiaxial life data for AISI 316 stainless steel. Chu et al. [14] weighted the shear energy part of their formulation by a factor of C = 2 to obtain a good correlation of fatigue results. Liu's [13] and Glinka et al.'s [15] formulations provided an equal weight of normal and shear energies. The empirical factors (C) suggested by each of the above authors gave a good fatigue life correlation for a specific material which suggests that the empirical weighting factor C is material dependent. In the present study, the proposed model correlates rnultiaxial fatigue lives by normalizing the normal and shear energies using the axial and shear material fatigue properties, respectively, and hence the parameter uses no empirical weighting factor.

Out-of-Phase Strain Hardening Under out-of-phase loading, the principal stress and strain axes rotate during fatigue loading often causing additional cyclic hardening of materials. A change of loading direction allows more grains to undergo their most favorable orientation for slip, and leads to more active slip systems in producing dislocation interactions and dislocation tangles to form dislocation cells. Interactions strongly affect the hardening behavior and as the degree of out-of-phase increases, the number of active slip systems increases. Socie et al. [25] performed in-phase and 90 ~ out-of-phase fatigue tests with the same shear strain range on 304 stainless steel. Even though both loading histories had the same shear strain range, cyclic stabilized stress-strain hysteresis loops in the 90 ~ out-of-phase tests had stress ranges twice as large as those of the in-phase tests. They concluded that the higher magnitude of strain and stress ranges in the out-of-phase tests was due to the effect of an additional strain hardening in the material [26]. During out-of-phase straining, the magnitude of the normal strain and stress ranges is larger than that for in-phase straining with the same applied shear strain ranges per cycle. The proposed param-


315

eter via its stress range term increases with the additional hardening caused by out-of-phase tests, whereas critical plane models that include only strain terms do not change when there is strain path dependent hardening. To calculate the additional hardening for out-of-phase fatigue tests, these approaches may be modified by a proportionality factor like the one proposed by Kanazawa et al. [27].

Mean Stress Correction Under multiaxial fatigue loading mean tensile and compressive stresses have a substantial effect on fatigue life. Sines [28] showed compressive mean stresses are beneficial to the fatigue life while tensile mean stresses are detrimental. He also showed that a mean axial tensile stress superimposed on torsional loading has a significant effect on the fatigue life. In 1942 Smith [29] reported experimental results for 27 different materials from which it was concluded that mean shear stresses have very little effect on fatigue life and endurance limit. Sines [28] reported his findings and Smith's results by plotting mean stress normalized by monotonic yield stress versus the amplitude of alternating stress normalized by fatigue limit (R = - 1) values (see Fig. 6). Figures 6a and 6b show the ef-

1

i .... ' .... ' .....

1.5

'''' ';+cli'c'str" i'a)

...a

I

9

O [] 9

0.5

5 approaches a nominal plane strain condition, th = 0, and a stress ratio approaching 2:1 for the fully plastic condition. At Lehigh University, Gross and Stout [1] adopted this approach for several pressure vessel steels using constant deflection and constant load cycling. Later, Sachs et al. [2] varied the width-to-thickness ratio achieving strain ratios - v -< ~b < O. Repeated pressurization of rectangular plate specimens was developed by Blaser et al. [3] at Babcock and Wilcox, achieving a stress ratio of 2:1. The technique was improved considerably in the 1960s by Ives et al. [4] through adopting a circular specimen, simply supported at its edges and alternately pressurizing each face to provide a fully reversed equibiaxial stress and strain field, ~b = + 1, at the center. Zamrik at Pennsylvania State University [5] introduced different specimen shapes from oval to circular providing a range of multiaxiality +0.5 --< ~ - 1. Anticlastic bending of rhombic plates, by applying equal and opposite point loads at adjacent corners, generates a relatively large region over which a given strain field is produced. By varying the ratio of the diagonals, strain ratios - 1 ~ ~b -< 0.5 can be generated. First reported by Zamrik [6] in the late 1960s, an improved system was described more recently by Zamrik and Davis [7]. The above bending techniques are all relevant to plate material and there are numerous references in the literature for studies of steels and light alloys. Table 1, which provides the format used throughout the paper to discuss multiaxial systems, shows that each bending technique is generally limited to a relatively small range of biaxiality. Perhaps, more importantly, the specimen geometries have to be varied in order to change the strain ratio. Benefits are the rig simplicity, specimen resistance to buckling, and in the case of anticlastic bending, an essentially constant strain field over a significant area which, together with ease of observation, still make it a useful technique for crack growth studies.

Torsion Pure torsion provides a simple technique for generating shear strain, 4) = - 1, albeit with the principal axes inclined at 45 ~ to the specimen axes. However the analysis of data from solid specimens is complicated by the reduction in strain with decreasing radius, especially in plastic cycling when the surface layers yield first. In the early 1960s, Halford and Morrow [8] published LCF data for torsion of thin-walled tubular specimens. The removal of core material and a mean diameter to wall thickness ratio of 10 or higher enable a much simpler stress/strain system to be realized. Subsequently, Miller and Chandler [9] demonstrated a progressive reduction in torsional fatigue life with reducing wall thickness which they attributed to the removal of elastic constraint. This emphasizes the need to maintain constant geometry in multiaxial studies if the understanding of constitutive laws is to be achieved.

Axial + Torsion At Tohoku, in 1965, Yokobori et al. [10] reported torsional and uniaxial LCF data, derived from separate machines but with identical gage length geometry. At Kyoto, Taira et al. [11] performed

LOHR ON MULTIAXIAL HIGH-STRAIN FATIGUE TESTING

[..

357

358

MULTIAXlALFATIGUE AND DEFORMATION

combined axial + torsional fatigue at ambient and elevated temperature for in-phase, and subsequently, out-of phase cycling. From the 1970s onwards the axial + torsion system provided by closed loop servohydraulic testing machines has been widely used. As Table 2 shows, while there are numerous benefits, the achievable range of surface strains is restricted to - 1 -< ~b -< - ~,, i.e., no positive strain ratios or even plane strain can be realized. In addition, buckling, a problem in many LCF specimens, can be present both axially and torsionally. It is also important to consider the effect of rotation of principal strain axes (as the applied strains change from pure axial to pure torsion) when testing anisotropic materials. Axial 4- Torsion 4- Internal Pressure Results at elevated temperature for creep fatigue tests employing cyclic axial + constant internal pressure and separately cyclic torsion were reported in 1963 by Kennedy [12] at Oak Ridge. Subsequently Crosby et al. [13] utilized combined cyclic axial and internal pressure, although plastic buckling instability was a reported problem. Commercial biaxial fatigue systems providing axial + internal pressure at elevated temperature and axial + torsion + constant internal pressure at ambient were reported by the author previously [14]. Reference to Table 2 shows that the full range of strain ratios is achievable with this triple loading configuration, however, because it is not possible to apply negative internal pressure, tests are not possible in the lower left quadrant. The consequence of repeated internal pressure cycling, above the yield stress, is ratcheting on the circumferential axis. The cycle may become elastic after the first cycle as a result of monotonic strain hardening or somewhat later due to cyclic hardening. The conclusion is that fully reversed biaxial tests are not possible without the addition of external pressure,

Design Review--2 Earlier work had shown that a single geometry specimen capable of being loaded in two orthogonal directions, under fully reversed strain-controlled cycling, was the key to progress in high-strain multiaxial fatigue. Two distinct approaches developed, which to this day have distinct advantages and disadvantages. Thin-Walled Tube Tubular specimens have the advantage that all axial forces, pressures, and torques are fully carried by the gage length. Stresses and plastic strains can be determined during the test unambiguously to the clear benefit of studies aimed at modeling material behavior. Axial 4- Differential Pressure Previous studies had shown that adding external pressure to axial force + internal pressure would permit testing in all four principal strain quadrants. At Waterloo, Havard [15] reported in 1968 on a rig which potentially provided these facilities; however, the use of a single hydraulic supply for force and pressure required the specimen design to be varied to change the strain ratio. Closed Loop Control In Bristol, during the 1970s, Andrews and Ellison [16], followed by Lohr and Ellison [17], developed a system which achieved the design goals of a single specimen capable of fully reversed cycles for all - 1 -< 4>--< + 1 with the ability to monitor stress, strain, and plastic strain continuously on both axes (Fig. 1). By using one closed loop actuator for axial force ( • 90 kN), a second to drive an intensifier for internal pressure (• 110 MPa), and adopting a fixed external pressure adjustable up to 55 MPa so that a controllable differential pressure could be achieved, the solution was also economic.

LOHR ON MULTIAXlAL HIGH-STRAIN FATIGUE TESTING

9

< [...,

359

360


FIG. 1--Bristol biaxial specimen and pressure vessel.

Finite-element analysis and experimental optimization led to specimen dimensions of 25.4 mm bore, 0.8 mm wall thickness, 9.5 mm parallel length with 25.4 mm fillet radii for tests on RR58 A1 alloy and 1Cr-Mo-V steel. The wall thickness was subsequently increased to 1 mm by Shatil et al. [18] to enable testing at higher plastic strain ranges without buckling on EN15R steel. Averaging axial and diametral capacitive extensometry enabled strain control throughout all tests. The advantages of this approach are identified in Table 2 and include the potential for an elevated temperature version using inert gas as the pressurizing medium since the internal volume is small and the external volume, at constant pressure, could readily be maintained at constant temperature. However, the pressurizing system does result in the hydrostatic component varying cyclicly which may have a second-order influence on fatigue life. Axial + Differential Pressure + Torsion The final evolution in mechanical design took place during the 1980s at Sheffield by Found et al. [19] and Fernando et al. [20] where four independent closed loops provide control of axial force (-+400 kN), internal and external pressure (160 MPa) plus torsion (-+ 1 kNm). The high pressures are required to be able to plastically deform steel specimens of 16 mm bore, 2 mm wall thickness, and 20 mm parallel length with 25 mm fillet radii. In such a system, torsion provides the ability to rotate the principal applied strain axes with respect to the specimen and investigate anisotropic effects. The system is large and complex, but fully comprehensive in its ability to command not only strain ratio but also the direction of the principal strain axes. However, this approach is not suitable for high temperatures since pumping and compressing large gas volumes for external pressure would make it impossible to stabilize temperature.


361

Composite Specimens At Alberta, Ellyin and Wolodko [21] recently reported on a new system for testing tubular samples with capabilities similar to the Sheffield machine. Axial forces of (_+260 kN) plus independent internal (82 MPa) and external (41 MPa) pressure plus torsion (_+2.7 kNm) are achievable. Structurally, with the actuator assembly directly bolted to the pressure vessel, the system is similar in concept to the Bristol machine. Modern PC computer control and software provide improved data acquisition and test flexibility. Of particular interest are the specimens of 38.2 mm bore, 1.4 mm wall thickness, and 102 mm parallel length manufactured from glass fiber/epoxy composite with bonded segmented aluminum tab ends. Initial hiaxial results were given for monotonic tensile behavior under stress control.

Cruciform Systems A specimen lending itself directly to biaxial testing is a cross-shaped plate, or cruciform, loaded in-plane by four orthogonal actuators. A higher tendency to buckle in compression than a tube, and a gage area that does not react all the load (some is shunted around its periphery) means that, in fatigue studies, stresses and plastic strains cannot be directly measured.

Tension In the early 1960s, at the Chance Vought Corporation, a rig was developed capable of applying tensile loads to a cruciform specimen. Initially, biaxial monotonic, and subsequently,biaxial fatigue tests were reported by McClaren and Terry [22] for equibiaxial and 2: I stress ratios on plate specimens with no reduced central section.

Open to Closed Loop Control At Cambridge, the development and application of cruciform testing spanned a period from the early 1960s to the mid-1980s. Pascoe and de Villiers [23] reported on the first practical rig based on a stiff octagonal frame carrying four 200 kN double acting actuators. Specimen development resulted in a design with central spherical recesses of 76 mm radius and a minimum thickness of 1 mm; a flat bottom was rejected despite favorable finite-element analysis because of premature failure at the fillet. Pressure limit cycling enabled tension-compression fatigue tests under 1:1 (equibiaxial), 1:-1 (pure shear), and uniaxial conditions to be performed. Strain gages enabled the total strain ranges to be measured; however, because of the ring reinforcement around the gage area, stresses and plastic strains remained indeterminate. Considerable development resulting in full servo-control of the actuators, and the ability to perform fully reversed biaxial fatigue tests at any strain ratio, was reported in 1975 by Parsons and Pascoe [24]. They found it necessary [25] to modify the specimen geometry by halving the spherical radius and doubling the wall thickness to obviate buckling at shorter lives on QT 35 ferritic and AISI 304 austenitic steels.

Specimen Development From the early 1980s, further development of the cruciform testing technique took place at Sheffield. In 1985, Brown and Miller [26] described a new specimen featuring a recessed flat-bottomed square gage area (100 mm X 100 mm • 4 ram) connected to the loading arms by sets of fingers created by slotting the arms (Fig. 2). This geometry effectively decouples the adjacent loading arms and means that the majority of the force on either axis is carried by the gage area over which the strain field is substantially uniform. The only restrictions are that high compressive forces, generating plastic specimen deformation, will result in buckling, while fatigue tests with an unnotched spec-

362


FIG. 2--Sheffield cruciform specimen. imen may result in first cracks initiating from the slot roots. For crack propagation studies, however, where plasticity is essentially a crack tip phenomenon, this new geometry represented a major step forward and has been influential in subsequent research worldwide.

Center Control A particular problem with the operation of cruciform systems has been controlling to a minimum the movement of the specimen center. Such unwanted motion generates specimen side forces, and hence bending, and is problematic for dynamic crack observation. The principal cause is the 100% cross coupling between opposing actuators which results in a "fight" if each actuator is separately controlled both for deformation and center position. The complete solution was developed by McA1lister for JUTEM [27] when, for each axis, the deformation and center position control loops were made independent of each other (Fig. 3). Considering one axis in displacement control, deformation is provided by the sum of the position transducer signals, while the center position is given by their semi-difference. The same principles enable load control and strain control to be realized (Fig. 4). A further useful consequence is the ability to simultaneously apply strain control for deformation and load control for zero side force. These are examples of "modal control," when two or more actuators are each driven by more than one control loop. The end result was that for cycling in strain control at 1 Hz the center position could be held stationary to circa _+1/xm.

New Materials The JUTEM system described by Masumoto and Tanaka [28], utilizes radio frequency (RF) heating plus susceptor in vacuum to enable temperatures up to 1800~ for testing structural composites at up to - 100 kN over the full range of biaxiality under strain control with crack observation by laser scanning microscope. At NASA Lewis, Bartolotta et al. [29] reported on a + 500 kN system designed

LOHR ON MULTIAXIALHIGH-STRAINFATIGUE TESTING

363

FIG. 3--Cruciform: modal control of deformation and center position.

DEFORMATION CONTROL

TRANSLATION CONTROL

9 Sum of LVDT readings

9 Half-Difference of LVDT readings

9 Average of LoadceU readings

9 Difference of Loadcell readings

9 Extensometer FIG. 4---Cruciform: multiple control modes.

for testing CMCs, intermetallics, and other advanced aerospace materials. Modified Sheffield type specimens, with central flat gage area 95 turn square x 2mm thick, can be heated up to 1500~ using an advanced quartz lamp radiant furnace. These, and similar systems, demonstrate the advantages of the cruciform solution for biaxial fatigue and crack growth studies of materials whose received form is sheet or thin plate (see Table 1).

Thermomechanical Fatigue High-strain thermomechanical fatigue of uniaxial specimens can trace its history back to at least to the mid-1970s, Taira [30] and Hopkins [31], However, the problems associated with the test under multiaxial conditions have only been addressed in the 1990s following the development of multiaxis digital closed loop controllers with high-speed data acquisition and software enabling flexibility in test design and data analysis.

Axial-Torsion The system initially selected for TMF studies has been the thin-walled tube under axial + torsional loading. This provides deterministic stress/strain relationships along with the practical benefits of "relatively simple to mechanically load and heat" and the possibility of"blowdown cooling" through the center. However, the range of biaxiality is a limitation.

364


System Description Kalluri and Bonacuse [32] reported on the development, at NASA Lewis, of four basic TMF test sequences derived from the traditional cases of 0 ~ and 90 ~ phasing between axial and torsional mechanical cycles and 0 ~ and 180 ~ phasing between mechanical and thermal cycles. Using a commercial machine providing axial force (-+220 kN) and torque (-+2.2 kNm), they developed a specimen of 22 mm bore with 2 mm wall thickness, a parallel length of 41 mm and fillet radii of 86 mm. Biaxial extensometry locates in a pair of dimples impressed within the parallel length of the specimen 25 mm apart. Heating is provided by audio frequency induction (50 kW, 10 kHz) with three independently adjustable coils. During TMF testing, real time thermal strain compensation is provided by "learned" polynomial relations for heating and cooling. Hysteresis loops and fatigue endurance data were reported for Haynes 188 superalloy for strain ranges of _+0.4% axial and _+0.7% shear over the temperature range 316~ to 760~ In order to prevent local buckling, temperature deviations in the parallel length were held to -+-I~ which resulted in a cycle time of 10 min and hence heating and cooling rates of 1.5~ s -I.

Complex Cycles At BAM, Bedim using similar servohydraulic hardware, Meersman et al. [33] reported an extended program of tests for nickel-based superalloys IN 738 LC and SC 16. Simple TMF implied linear, diamond, and sinusoidal cycles, whereas complex TMF referred to simulation of a strain-time history representative of the leading edge of a first stage "bucket" in service. Tests were performed within the range 450~ to 950~ at rates up to 4.2~ s -~ for equivalent strain ranges between 0.6% and 1.24% and at strain rates 10 5s 1 and 10-4s -1.

Failure Criteria The definition of failure in uniaxial low-cycle fatigue is often characterized by a specified reduction in tensile stress amplitude measured in relation to the current half-life value or to the stabilized trend value after the end of cyclic hardening or softening. The percentage drop can vary between 2 and 50% in different studies. Another technique compares the unloading moduli from the peaks of the hysteresis loop, the lower value being associated with unloading from tension. Both approaches reflect cumulative cracking damage which, because of crack closure, is not registered in compression. However, in multiaxial fatigue studies the specimen geometry and the method of loading can often make other criteria more relevant or even unavoidable. Cruciform fatigue specimens, which generally employ dished center sections, may show little loss of peak tensile force, even with quite large cracks, because load is shunted around the gage area by thicker material. As a result, specified surface crack length has often been used as the endof-test criterion. In contrast, thin-walled tubes, loaded by axial force and differential pressure, will invariably require the test to be terminated soon after the fatigue crack has penetrated through the wall resulting in a coupling between the internal and external pressure systems and subsequent loss of control. Crack morphology and rate of crack growth has been shown to vary with strain ratio and with the plane in which the maximum shear strain lies [18,34]. Although the various theories and microstructural mechanisms governing crack propagation are outside the scope of this paper, it is instructive to look at the three-dimensional Mohr's strain circles (Fig. 5) in which four biaxial strain ratios (~b = - 1, - u, 0, + 1) are depicted for el = constant and u = 0.5 (fully plastic). At q5 = - 1, maximum shear strain YzO,given by the diameter of the largest circle (ez - e0), lies in the surface plane and long shallow surface cracks are observed for all test geometries and loading systems. As a result, in a cruciform, the crack grows more quickly to a specified surface length, whereas, in a pressurized thin-walled tube, penetration of the thickness is delayed because of the shallow ha-


365

do=-1 ~__..[~___~ Uniax 1 3 3 ~ 1 3 t 13o

13z

~13

133

131

8r

13z

do=01~ 1

133

131

z

13

13z

FIG. 5--Three-dimensional Mohr's circles f o r v = 0.5.

ture of the crack. At ~b = + 1, maximum shear strains Yzr and "YOrlie in the planes which intersect the surface plane and short deep surface cracks are observed for both cruciforms and thin-walled tubes. This time in a cruciform the growth of surface crack length is retarded, while in the case of the thinwalled tube, penetration is accelerated. These observations can help to explain why, for pure shear, cruciforms (with surface crack length failure criteria) may give relatively shorter lives than pressurized thin-walled tubes (with penetration failure criteria) and why, for equibiaxial straining, cruciform lives may be relatively extended. Similar arguments can be applied in the case of plane strain, ~b = 0, where the Mohr's circle is geometrically identical with pure shear but the maximum shear strain is Tzr, not ~tzO. In summary, different failure criteria can be expected to modify the relative fatigue lives measured. It would be interesting to establish the effect in a cruciform test series when crack penetration of the specimen thickness is the failure criterion rather than surface crack length.

System Selection In this final section the author offers a process for system specification. In conjunction, Table 1 provides information for systems required to test plate and sheet materials, while Table 2 addresses tubular specimen test systems for which thicker material must be available. Research Purpose

It is important to first decide whether the system is for fundamental materials properties determination (e.g., inputs to constitutive equations), crack growth studies, or component simulation. These considerations should decide the range of biaxiality to be provided by the system. The proposed environment (ambient, elevated temperature, or TMF) should then identify a particular scheme. The capital budget is an issue here since generally speaking the cost of a system is related to the number of actuators and the complexity of the environment.

366


Specimen Geometry Specimen definition forms the cornerstone of the subsequent design process. Cruciform optimization has been discussed in some detail in Design Review - 2; however, optimization of the gage length for thin-walled tubular specimens is worthy of a summary. Tube mean diameter and wall thickness, together with material strength, determine the axial force, differential pressure, and torque requirements. In uniaxial LCF (solid samples) it is normal to have a parallel length of at least twice the gage diameter and a large fillet radius to achieve a low-strain concentration at the fillet runout. For axial-torsion (thin-walled tubes) only a modest reduction in parallel length may be necessary to maintain geometric stability. However, specimens subject to the full biaxial range of - 1 --< th --< + 1 may need parallel length and fillet radius reduced to the mean diameter or less in order to avoid buckling under plastic equibiaxial conditions. The ratio of mean diameter to wall thickness should approach 20:1 to minimize through-thickness strain gradients under pressure and .torsion. However, buckling considerations often reduce this ratio nearer to 10:1. Finite-element analysis is recommended for specimen geometry optimization.

Loadstring With the specimen defined, loadcells, axial and torsional actuators, and pressure intensifiers should be sized to at least 110% of the maximum required to break, in each mode, the strongest material envisaged for testing. Grips or pullrods should be specified in terms of capacity, operating temperature, and hydraulic or manual clamping.

Environment and Extensometry Sizing of specimen and grips together with maximum temperature and heating rates enables the furnace type and power rating to be defined along with any requirements for an enclosure such as a vacuum chamber. Extensometry can now be specified for operating environment, number of axes, averaging or not, strain ranges, and performance class.

Reaction Frame With force capacities and working space (crosshead to table, and between columns) defined, the frame can be specified, All low-cycle fatigue frames and loadstrings require high lateral stiffness as well as axial stiffness to ensure minimum specimen bending during plastic deformation. Under multiaxial conditions, when torsion may also be applied, it is advisable to specify larger diameter columns to ensure adequate lateral and torsional frame stiffness.

Controller and Software A modem digital closed loop multiaxis controller should provide the ability to generate and control complex mechanical and thermal waveforms. The latest graphical user interface (GUI) based software running in an open architecture system should provide flexibility and forward compatibility. Conclusions

1. Techniques for multiaxial testing derived from half a century of research work have been reviewed with special focus on those systems that broke new ground. 2. Thin-walled tubes under axial + differential pressure permit a two actuator economic design achieving full biaxiality with the unambiguous determination of all stresses and strains, essential for materials modeling. Adding torsion enables rotation of the principal axes.


367

3. Cruciform systems have been confirmed as the only approach for testing materials in plate and sheet form over the full biaxial range - 1 C 6O m

09

r

%-

0

1000

2000

3000

4000

5000

6000

8000 70O0

9000

10000

O

9j

O O

O O

(C)

O O

Ave:,

iert. Axis Extens.

~//

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KRAUSE AND BARTOLOTTA ON EXTENSOMETERS

381

in the horizontal. This strain "wander" may be explained by specimen thermal drift, as the two strain oscillations were roughly in phase and had a long period (approximately 10 h). Strains were very sensitive to temperature; for example, a 4~ change produced 68 p,m/m thermal strain in this material.

Conclusions A mechanism to measure strains in flat cruciform specimens at room and elevated temperatures has been developed from substantially off-the-shelf components to produce a reliable, accurate, low-cost in-plane biaxial contact extensometer. The device attaches two standard axial extensometers to the test system support structure using adjustable alignment fixtures. Final calibration produced a maximum relative error of 0.8%, resulting in the extensometer designation of Class C according to ASTM Practice E 83. An extensive series of tests were performed to validate operation of the biaxial extensometer. Static and dynamic tests at room and elevated temperatures to 600~ established accurate strain measurement, good specimen coupling, and excellent load signal tracking. Using a different probe tip configuration could alleviate a small amount of offset discovered upon unloading in hot cyclic testing. Strain "wander" in long duration testing is thought to be a real measurement, possibly related to slight specimen temperature fluctuations. Acknowledgments The valuable technical assistance and mechanical skills provided during extensometer development by Stephen J. Smith (Gilcrest Electric and Supply Company), and insights provided into the fine points of extensometry by Christopher S. Burke (Dynacs Engineering Inc.) are gratefully acknowledged.

References [1] Ginty, C. A., "Overview of NASA's Advanced High Temperature Engine Materials Technology Program," HITEMP Review 1997: Advanced High Temperature Engine Materials Technology Program Volume L NASA Conference Publication 10192, 1997, pp. 2-1 to 2-19. [2] Ellyin, F. and Wolodko, J. D, "Testing Facilities for Multiaxial Loading of Tubular Specimens," Multiaxial Fatigue and Deformation Techniques, STP 1280, S. Kalluri and P. J. Bonacuse, Eds., American Society for Testing and Materials, West Conshohochen, PA, 1997, pp. 7-24. [3] Lefebvre, D. F., Ameziane-Hassani, H., and Neale, K. W., "Accuracy of Multiaxial Fatigue Testing with Thin-Walled Tubular Specimens," Factors that Affect the Precision of Mechanical Tests, STP 1025, R. Papimo and H. C. Weiss, Eds., American Society for Testing and Materials, West Conshohochen, PA, 1989, pp. 103-114. [4] Makinde, A., Thibodeau, L., and Neale, K. W., "Development of an Apparatus for Biaxial Testing Using Cruciform Specimens," ExperimentalMechanics, Vol. 32, No. 2, 1992, pp. 138-144. [5] Boehler, J. P., Demmerle, S., and Koss, S., "A New Direct Biaxial Testing Machine for Anisotropic Materials," Experimental Mechanics, Vol. 34, No. 1, 1994, pp. 1-9. [6] Demmerle, S. and Boekler, J. P., "Optimal Design of Biaxial Tensile Cruciform Specimens," Journal of the Mechanics and Physics of Solids, Vol. 41, No. 1, 1993, pp. 143-181. [7] Popov, E. P., Mechanics of Materials, 2nded., Prentice-Hall Inc., Englewood Cliffs, NJ, 1976, pp. 235-265. [8] Bhat, G. K., "Electronic Speckle Pattern Interferometry Applied to the Characterization of Materials at Elevated Temperature," NDT Solution, American Society for Nondestructive Testing Inc., January 1998. [9] Vishay Measurements Group, "Strain Gage Selection: Criteria, Procedures, Recommendations, TN-5054," Measurements Group Tech Note, Measurements Group Inc., 1989, pp. 1-15. [10] Makinde, A., Thibodeau, L., Neale, K. W., and Lefebvre, D., "Design of a Biaxial Extensometer for Measuring Strains in Cruciform Specimens," ExperimentalMechanics, Vol. 32, No. 2, 1992, pp. 132-137. [11] Bartolotta, P. A., Ellis, J. R., and Abdul-Aziz, A., "A Structural Test Facility for In-Plane Biaxial Testing of Advanced Materials," Multiaxial Fatigue and Deformation Techniques, STP 1280, S. Kalluri and P. J. Bonacuse, Eds., American Society for Testing and Materials, 1997, pp. 25-42. [12] MTS Systems Corporation, "Series 632 High Temperature Extensometers," Product Manual, MTS Systems Corporation, 1976. [13] MTS Systems Corporation, "Model 407 Controller, Version 3.0," Product Manual, MTS Systems Corporation, 1995.

J. R. Ellis, l G. S. Sandlass, 2 and M. Bayyari 3

Design of Specimens and Reusable Fixturing for Testing Advanced Aeropropulsion Materials Under In-Plane Biaxial Loading REFERENCE: Ellis, J. R., Sandlass, G. S., and Bayyari, M., "Design of Specimens and Reusable Fixturing for Testing Advanced Aeropropulsion Materials Under In-Plane Biaxial Loading," Multiaxial Fatigue and Deformation: Testing and Prediction, ASTM STP 1387, S. Kalluri and P. J. Bonacuse, Eds., American Society for Testing and Materials, West Conshohocken, PA, 2000, pp. 382-404. ABSTRACT: A design study was undertaken to investigate the feasibility of using simple specimen designs and reusable fixturing for in-plane biaxial tests planned for advanced aeropropulsion materials. Materials of interest in this work include: advanced metallics, polymeric matrix composites, metal and intermetallic matrix composites, and ceramic matrix composites. Early experience with advanced metallics showed that the cruciform specimen design typically used in this type of testing was impractical for these materials, primarily because of concerns regarding complexity and cost. The objective of this research was to develop specimen designs, fixturing, and procedures that would allow in-plane biaxial tests to be conducted on a wide range of aeropropulsion materials while at the same time keeping costs within acceptable limits. With this goal in mind, a conceptual design was developed centered on a specimen incorporating a relatively simple arrangement of slots and fingers for attachment and loading purposes. The ANSYS finite-element code was used to demonstrate the feasibility of the approach and also to develop a number of optimized specimen designs. The same computer code was used to develop the reusable fixturing needed to position and grip the specimens in the load frame. The design adopted uses an assembly of slotted fingers which can be reconfigured as necessary to obtain optimum biaxial stress states in the specimen gage area. Most recently, prototype fixturing was manufactured and is being evaluated over a range of uniaxial and biaxial loading conditions. KEYWORDS: in-plane biaxial testing, advanced aeropropulsion materials, cruciform specimen design, reusable fixturing, finite-element analysis, optimization techniques, attachment methods, prototype fixturing

One technique for investigating material behavior under complex stress states is to use in-plane biaxial loading. Using this approach, cruciform specimens fabricated from plate or sheet material are gripped at four locations and loaded along two orthogonal axes. Servo-hydraulic loading systems are used in this application which are similar to those used for uniaxial testing. Thus, the technique has the advantage that the loading arrangement is relatively straightforward and uses equipment which has seen extensive development over the past 30 years. Also, the test method allows a wide range of biaxial stress states to be investigated with minimum complication from the load application viewpoint. For these reasons, the test method has been used to generate a sizable body of biaxial test data for both monolithic and composite materials [1-29]. l Senior research engineer, NASA Glenn Research Center, Cleveland, OH 44135. 2 Structural analyst, MTS Systems Corp., Eden Prairie, MN 55344. 3 Principal research engineer, Research Applications, Inc., San Diego, CA 92121. 382

Copyright9


www.astm.org

ELLIS ET AL. ON AEROPROPULSIONMATERIALS

383

One difficulty facing these investigations has been the selection and/or development of the most suitable specimen design for the particular program. It should be noted that consensus standards do not exist for this method of testing, and so the experimentalist is faced with a wide range of possibilities. A major complication here is that use of the cruciform specimen configuration and associated gripping fixtures results in "coupling" between the two loading directions. In the present research, specimens are positioned in the load frame using four hydraulic grips which rigidly constrain the specimen over the gripped regions. It follows that loading applied in one direction is partially reacted by the specimen and partially by the grips associated with loading in the second direction. One method of minimizing this effect is to use specimen designs which incorporate fairly complicated arrangements of flexures as illustrated in Fig. 1. It has been demonstrated that flexures with low bending stiffness in the plane of loading can be used to minimize the constraint imposed by specimen gripping. Also, it has been shown that the geometry of the flexures can be optimized and tailored to give near-uniform stress/strain conditions in the gage area for specific biaxial loading conditions. One obvious disadvantage of using flexures is that regions of high stress concentration can be introduced into specimens in close proximity to the gage area. Of particular concern are stress concentrations at the ends and intersection points of the flexures. This raises the possibility that failure can be initiated outside of the gage area in regions where stress/strain conditions are ill-defined. Traditionally, this problem has been addressed by incorporating a gage area within which specimen thickness is reduced significantly from the value in the gripped regions. In the case of plate specimens incorporating flexures, experience has shown that thickness reduction factors as high as ten are needed to achieve acceptable performance. That is failure initiating within the gage area where stress/strain conditions are both relatively uniform and relatively well defined. Although the above approach has been used effectively in the case of conventional structural alloys, it has proved impractical for the materials of interest in this work including: advanced metallics, polymeric matrix composites, metal and intermetallic matrix composites, and ceramic matrix composites. Problems include the unavailability of material in "large" product forms and also the diffi-

FIG. 1--Current NASA specimen design.

384

MULTIAXIALFATIGUEAND DEFORMATION

culties associated with machining complex three-dimensional geometries in complex multi-phase materials. The aim of the present work was to develop an alternative approach involving use of a simplified specimen design and use of reusable fixtures incorporating the design features needed to decouple the applied biaxial loading. A further goal was to develop fixturing and procedures which would allow in-plane biaxial tests to be conducted on a wide range of advanced materials while at the same time keeping costs within acceptable limits.

Specimen Design and Analysis The current cruciform specimen design (Fig. 1) developed at NASA for testing conventional structural alloys served as a starting point for this work. In the new approach, the gripped regions of the current design are replaced by four individual fixtures which incorporate slotted fingers to decouple the biaxial loading. The gage section of the current design is replaced by a specimen incorporating a reduced gage area and four sets of slots and fingers for attachment purposes. The focus of the preliminary design and analysis work was on determining whether a specimen design with this configuration would meet two straightforward design requirements. These were: (1) that the maximum stress in the part should occur within the gage area, and (2) that the stress/strain distribution in the gage area should be reasonably uniform, say, within +-5% of the mean. Details of the conceptual design and the results of specimen design and analysis work are described in the following.

Conceptual Design A conceptual design for the new approach is shown in Fig. 2. One constraint on the overall size of the assembly was that a 432 X 432 mm envelope is available within the load frame for installation and gripping purposes. Four slotted finger fixtures are shown attached to a specimen fabricated from a 229 X 229 • 6 mm plate. These dimensions were selected to give a relatively large gage area, 76 mm outside diameter in the case of specimens with circular gage areas. This approach was adopted primarily with instrumentation requirements in mind. The attachment method is not shown in Fig. 2 for simplicity of drawing. Details of a slotted finger attachment are given in Fig. 3. The design shown is idealized in that no stress relieving blend radii were included so as to simplify finite-element analysis of the complete assembly. This approach was acceptable because the focus of initial work was on the performance of the specimen rather than on the performance of the fixturing. The slotted finger attachments are assumed to be gripped over 152 • 38 mm 2 areas on both top and bottom surfaces. Earlier experiments using the current NASA cruciform specimen design had shown that this arrangement met the loading requirements of planned test programs. Experience gained in these earlier experiments was also used to size the finger and flexure configuration shown in Fig. 3. The four finger arrangement was an obvious choice given the need to locate a slot on the fixtures centerline and also given the need to maintain symmetry about the fixtures centerline. Details of the initial specimen design are given in Fig. 4. As noted earlier, it is assumed to be fabricated from a 229 x 229 X 6 mm plate. Perhaps the most important design feature is the arrangement of slots and fingers used for attachment and loading purposes. It was recognized at the outset that the slot configuration would play a key role in obtaining an optimized specimen design. As indicated in Fig. 4, a slot width of 10 mm was selected for initial feasibility studies. Other dimensions shown in symbolic form were treated as variables in subsequent optimization analyses.

Stress Analysis Details The ANSYS finite-element code, version 5.4, was selected for this work, primarily because it features an optimization package. The plan was to model '/8of the complete assembly with the six specimen dimensions shown in Fig. 4 expressed as variables and to perform fully automated analyses un-

385

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til an optimum set of specimen dimensions had been obtained. To facilitate this process, a command input file was created using the ANSYS Parametric Design Language (APDL). This provides the means to create finite-element models in terms of variables, which in turn allows for easy and rapid design changes. Approximately 1000 lines of ANSYS commands were used to define parameters, generate the finite-element model, solve, evaluate results, and begin optimization looping. As indicated earlier, external loading was applied to the assembly over 152 • 38 mm 2 areas. An equibiaxial stress state was introduced into the finite-element model by constraining all surface nodes within the gripped regions to displace 0.127 mm in the positive sense in the two loading directions. Similarly, clamping within the specimen grips was simulated by constraining all surface nodes within the gripped region to displace 0.005 mm in the thickness sense. Regarding the attachment method, the specimen and the slotted finger fixtures were modeled as a single unit in the early analyses which focused on specimen performance. Note that the material properties used in this work were handbook values for Inconel 718 and that the results of stress analyses are expressed in the form of von Mises equivalent stress throughout. Stress Analysis Results

The results shown in Figs. 5 and 6 were obtained in one of a large number of stress analyses performed during the initial stages of the research. One interesting feature of the overall deformation behavior of the assembly shown in Fig. 5 is the ability of the finger and flexure arrangement to accommodate large overall displacements and rotations without becoming overstressed. This flexibility is, of course, the mechanism by which loading in the two directions is partially decoupled. As might be

ELLIS ET AL. ON AEROPROPULSION MATERIALS

387

expected, analysis of the results showed that stress concentrations occurred at three locations outside of the gage area. As indicated in Fig. 6, these locations were the center slot, the outer slot, and the fillet region. Simply stated, the optimization process involved varying specimen dimensions in a systematic manner until the maximum stress values at these locations were less than the average stress in the gage area, say, with a 20% margin of conservatism. Analysis of the data shown in Figs. 5 and 6 indicated that this condition had partially been achieved with the particular specimen design shown. The average and maximum stresses in the gage area were 392 MPa and 397 MPa, respectively. The maximum stress levels in the center and the outer slots were 367 MPa and 352 MPa, and the maximum stress in the fillet region was 382 MPa. Thus, this design met the design requirement that the maximum stress should occur within the gage area. However, it was apparent that additional work was needed to achieve the 20% margin of conservatism. The second design requirement was that the stress distribution in the gage area should be uniform within -+5% of the mean. Analysis of the data shown in Fig. 6 showed that the stress distribution in the gage area fell within ---2.4% of the mean, easily meeting the target value.

Optimization Method The combination of specimen dimensions used in obtaining the above result was as follows: 9 9 9 9

Gage section radius (R1) = 41.28 m m Center slot length (SLA) = 35.50 m m Outer slot length (SLB) = 37.00 m m Fillet radius (FR) = 27.94 m m

FIG. 5--Stress distribution in assembly.

388


FIG. 6---Stress distribution in specimen.

9 Gage area thickness (T1) = 1.25 m m 9 Thickness transition radius (TR1) = 12.70 m m The optimization process followed in obtaining these results was not straightforward and proved to be extremely time-consuming. Evaluation of the various ANSYS optimization routines showed that in this application, the routines had difficulty converging on optimum sets of values. The factorial routine was found to be most useful for the present work as it allowed predetermined combinations of specimen dimensions to be investigated in a straightforward manner. As described in the following, the "automatic" process was supplemented by presenting the results of stress analysis in graphical form and by analyzing the results by hand. This approach effectively narrowed the design space and allowed optimum data sets to be determined more efficiently. As a first step, fully automated stress analyses were conducted for up to 36 combinations of center and outer slot lengths and gage section thicknesses. As illustrated in Fig. 7, these data were plotted to establish the optimum combination of center and outer slot length. This was defined as the intersection point giving the minimum stress condition at the two slot locations. It can be seen in Fig. 7 that the optimum values of center and outer slot length established in this manner were 35.50 m m and 37.00 mm. One important result was that the curves representing stress conditions at the two slot locations and at the fillet region were found to be unaffected for the most part by changing the gage area thickness. The most important effect of changing this variable was to shift the position of the gage area curve

ELLIS ET AL. ON A E R O P R O P U L S I O N M A T E R I A L S

389

in the vertical sense relative to the stress axis. Thus, selecting the optimum gage area thickness simply involved identifying the curve that fell above the intersection point with some reasonable level of conservatism. The optimum value of gage area thickness selected in this manner was 1.25 mm. Additional stress analyses were then conducted to determine the optimum value of fillet radius. This approach was possible because changing this variable did not have a major effect on the stress states at the slots and within the gage area. As illustrated in Fig. 8, these results were also plotted to determine the minimum feasible value of fillet radius which was 26.9 mm for the case shown. A value of 27.94 mm was selected as being optimum for the particular specimen design as it provided some margin of conservatism.

Final Specimen Designs Given this encouraging result, attention was shifted to the design of gripping and attachment methods and to modifying the initial specimen design as found to be necessary. By way of background, it was anticipated that load levels as high as _+222 kN would be needed in tests planned for aeropropulsion materials of interest. Relatively simple attachment methods using bolts, for example, as the primary means of load transfer proved inadequate, primarily because of the previously noted size constraints. The approach adopted to resolve this difficulty was to incorporate tapers on the specimen fingers to allow load transfer by means of shear. This change was made reluctantly as it was viewed as introducing a major element of complexity into the specimen design. A further change was that the width of the slots was increased to reduce stress levels at the root locations and to increase margins of conservatism. Details of the final specimen design featuring a circular gage area are given in Figs. 9 and 10. One change from the initial design is that the width of the fingers is reduced from 19 mm to 16 mm. This

O Center slot A Outer slot [] Fillet radius

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allowed the slot width to be increased from 10 m m to 14 mm. Further, an 8~ ' --- 15' taper was incorporated on the gripped section to facilitate load transfer into the specimen (Fig. 10). Optimization exercises similar to those described above were conducted for two values of overall plate thickness, 19 nun and 25.4 mm. The optimized set of dimensions for the two thicknesses are summarized in Table 1, and the results of stress analyses are summarized in Table 2. The relative merits of the designs will be discussed later in the paper. TABLE 1--Summary of specimen types and optimized specimen designs. Optimized Specimen Dimensions (mm) Specimen Type

TI

T2

SLA

SLB

FR

R1

Design with 19 mm overall thickness, and 14 mm slot width

2.1 2.2 2.3

1.524 1.524 2.032

6.35 6.35 6.35

35.56 30.48 30.48

36.45 30.78 30.78

40.64 50.80 50.80

44.45 44.45 44.45

Design with 25.4 mm overall thickness, and 14 mm slot width

3.1 3.2 3.3

2.032 2.032 2.540

12.70 12.70 12.70

35.56 30.48 30.48

36.45 30.78 30.78

40.64 45.72 45.72

48.46 48.46 48.46

Specimen Details

NOTE: L1 = 80.65 mm and TR1 = 25.4 mm throughout.

3.3

3.2

3.1

2.3

3l 6 (_+2.6%) 305 (_+2.4%) 283 (_+2.8%)

361 (_+2.7%) 352 (_+ 1.9%) 317 (_+2.0%)

MPa

1

1

1

1

1

1

Normalized

241

253

293

268

288

326

MPa

0.85

0.83

0.93

0.85

0.82

0.90

Normalized

Center Slot (maximum)

248

248

293

270

271

330

MPa

0.88

0.81

0.93

0.85

0.77

0.92

Normalized

Outer Slot (maximum)

248

249

272

297

299

346

MPa

0.88

0.82

0.86

0.94

0.85

0.96

Normalized

Fillet Radius (maximum)

NOTES: Values in brackets are % deviations about the mean. Normalization of von Mises equivalent stress obtained using average gage section stress.

Design with 25.4 mm overall thickness, and 14 mm slot width

2.1

Design with 19 mm overall thickness, and 14 mm slot width

2.2

Specimen Type

Specimen Details

Gage Section (average)

Von Mises Equivalent Stress at Location Shown

TABLE 2--Summary of stress results at critical locations for optimized specimen designs.

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393

Reusable Fixturing Design and Analysis The first objective of the fixturing design work was to identify the optimum attachment method for specimens with the slot and finger configuration described above. A number of attachment methods were considered during initial design studies and all but two were rejected, primarily because of concerns regarding complexity and cost. Work on the two most promising concepts was continued through detailed design and in one case through manufacture. The preferred approach uses a yoke arrangement which has the important advantage of not requiring any further modifications to the specimen designs described earlier. The design and analysis process followed in achieving a final design is outlined as follows.

Design Details As a first step, a number of changes were made to the slotted finger attachment (Fig. 3) which served as a starting point for this work. It was recognized that the design in its original form was going to be difficult and expensive to manufacture and, with this in mind, the original unit design was broken down into nine subcomponents. These were eight fingers and a single mounting plate for assembly purposes. Details of this assembly are shown in Figs. 11 and 12. Here it can be seen that the mounting plate incorporates eight slots to accommodate the fingers and that setscrews are used for

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assembly purposes. The finger design includes tapers on two surfaces to match the corresponding tapers on the specimen and the yoke. A further design feature worth noting is the fiats which were provided to ensure positive location between the end of the specimen and the fingers. An undercut was included at the same location to ensure proper specimen seating and also to provide some flexibility for specimen gripping. The important features of the yoke gripping arrangement are shown in Figs. 13, 14, and 15 where, for simplicity, a single pair of fingers is shown attached to a rectangular block. The plan was to first subject the fixturing to detailed evaluation under uniaxial loading. The prototype fixturing shown in these figures was manufactured specifically for this purpose. The primary function of the yoke fixture is to prevent the fingers from separating under load and to prevent relative motion between the specimen and the fingers. Stated differently, the yoke's function is to ensure that the end of the specimen stays in full contact with the mating surfaces of the fingers under both tensile and compressive loadings. This condition was to be achieved by effectively clamping the ends of the specimen between finger pairs by applying suitable preload to the assembly. This preloading was to be achieved using the two preload bolts in conjunction with the tapers on the mating surfaces of the fingers and the yoke. The first goal of analysis was to determine whether effective clamping could be obtained without overstressing either the bolts or the yoke. Also, the stiffness and load transfer characteristics of the attachment method were investigated to establish the useful loading range available with the design.

ELLIS ET A L ON AEROPROPULSION MATERIALS

395

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13--Yoke gripping arrangement and setup for evaluation under uniaxial loading.

FIG. 14--Prototype fixturing in partially disassembled form.

396


FIG. 15--Prototype fixturing in assembled form.

Stress Analysis Details The ANSYS finite-element code, version 5.4, was again used for this work. The plan was to model one half of the assembly shown in Fig. 13 and to investigate the characteristics of the attachment method over a range of uniaxial loading conditions. The effect of simply preloading the assembly was the first loading case considered. Bolt preloading in these analyses was obtained by applying known displacements to the underside of the bolt head. Further, it was of interest to investigate how the stress state resulting from preloading was modified by the superimposition of external loading. As in earlier analyses, external loading was applied by constraining surface nodes in the gripped region to displace predetermined amounts in the loading direction. The magnitudes of these displacements were varied to simulate strength tests under both tensile and compressive loading. As with most general purpose finite-element codes, contact elements are available within ANSYS for analyzing joints and attachments. Line contact elements were used in the present work to model the contact surfaces between the mating components. One complication with this approach is that a value of contact stiffness has to be specified up front for these nonlinear analyses. Also, it is well known that the correct choice of contact stiffness is critical in regard to the accuracy of the solution and also in regard to the time taken to converge on a solution. To address this issue, a series of preliminary analyses was conducted investigating the effect of varying contact stiffness over the range 1.776 x 101 to 1.776 • 106 kN/mm. The highest value was selected for the final analyses as it was shown to provide reasonably accurate solutions within acceptable time periods. As would be expected, a coefficient of friction value has to be specified which is judged typical for the surface condition of the mating parts under consideration. In the absence of experimental data, a value of 0.1 was assumed for the majority of analyses and a limited number of spot checks were conducted using a value of 0.2. As in the case of the earlier work, elastic constants used in these analyses were handbook values for Inconel 718.

Stress Analysis Results The results shown in Figs. 16 and 17 were obtained in one of a large number of stress analyses performed on the yoke assembly. These data were obtained using a contact stiffness of 1.776 X l06 kN/mm, a coefficient of friction of 0.2, and a bolt preload of 31.6 kN. Analysis of the results showed that the maximum stress in the yoke, 692 MPa, occurred at location (2) in Fig. 16. The maximum stress in the finger, 1792 MPa, occurred at location (4) in Fig. 17. Clearly, the preload assumed in this


397

FIG. 16~Stress distribution in central section of yoke for 31.6 kN bolt preload.

analysis was less than ideal since it resulted in unacceptably high stresses in both the yoke and the finger. This situation was corrected in subsequent analyses by using more realistic values of bolt preload. One result of changing this variable was to change the location of maximum stress. For example, in analyses using a bolt preload of 5.816 kN and a coefficient of friction of 0.1, the maximum stress condition occurred at location (1) in Fig. 16 and at location (3) in Fig. 17. The results of all analyses performed using a contact stiffness of 1.776 • l 0 6 kN/mm are summarized in Table 3.

Stiffness and Load Transfer Characteristics The efficiency of the yoke attachment method was investigated by generating plots of applied grip displacement versus the corresponding specimen load. Such a curve is shown in Fig. 18 for a bolt preload of 5.816 kN and a coefficient of friction of 0.1. As would be expected, the performance of the fixture under tensile loading is significantly different from that under compressive loading. This is because the load path between the specimen and the fingers is completely different for the two loading cases. Under tension, a minor change in slope occurs at point (A) and a major change occurs at point (B) where slippage apparently occurs. Under compression, behavior is more straightforward with a minor change in slope occurring at point (C). The above data are summarized in more quantitative form in Table 4 along with results obtained for other combinations of preload and coefficient of friction.

398


FIG. 17--Stress distribution in tapered section of finger for 31.6 kN bolt preload.

TABLE 3--Stress analysis resultsfor the yoke gripping arrangement. Coefficient of Friction

Bolt Preload (kN)

0.1

5.816 (0.381m m) 5

0.2

31.60 (0.762 ram)

Grip Displacement (ram)

Bolt Load (kN)

Specimen Force (kN)

0

5.816

0

0.152

5.790

12.725

0.254

6.016

17.298

-0.152

6.966

-14.140

-0.254

6.962

-23.000

0

31.600

0

0.152

31.620

12.721

0.254

31.620

21.760

Maximum Yoke Stress (MPa)

Maximum Finger Stress (MPa)

450 (1) 450 (1) 519 (1) 411 (1) 394 (1) 692 (2) 704 (2) 709 (2)

718 (3) 763 (3) 830 (3) 826 (3) 909 (3) 1792 (4) 1772 (4) 1765 (4)

NOTE: (1) and (2) see Fig. 15 for maximum stress locations in yoke; (3) and (4) see Fig. 16 for maximum stress locations in finger; and (5) corresponding bolt head displacement in mm.


399

22.5 18.0 - -

13.5 -z -~

9.0 --

4.5O '0 9 E -4.5 "~. r -9.0 --

_

/

13.29~ // i 0.1 50 // / / - - Secant i ~p ~" modulus Secant ~ , = 88.6 modulus /,, "~- Initial slope = 90.33 --~TZ I", = 91.8 yC '"~ Initial slope 0.150/ = 92.8 /

-13.5

-18.0 ~-- /

~ ] 1/.55 3 I

J

I

I

I

-22.5

-25 -20 - 1 5 - 1 0 -5 0 5 10 15 20 25 Grip displacement, m m x l 0 -2

Note: Assumed coefficient of friction = 0.1 and bolt preload = 5.816 kN. FIG. 18--Stiffness and load transfer characteristics under tensile and compressive loading.

An attempt was made to investigate the cause of the slope changes using the contact status option available with ANSYS. Using this option, the status of contacting surfaces is given in three categories: 1. 2. 3.

Gap closed, no sliding. Gap closed, sliding. Gap open.

TABLE 4---Stiffness and load transfer characteristics of the yoke gripping arrangement.

Coefficient of Friction

Bolt Preload (kN)

0.1

5.816 (0.381/rnm) 2 11.62 (0.762 mm) 31.64 (0.762 mm)

0.2

Maximum Grip Displacement (mm) 0.254 -0.254 0.381 -0.351 0.254

Initial Slope (kN/mm)

0.150 Secant Modulus (kN/mm)

Limit of Proportionality Load (kN)/Displacement (nun)

91.8 92.8 96.4 91.44 93.93

88.6 90.3 91.26 (I) 92.15

5.19/0.0559 5.36/0.0584 12.36/0.130 (1) 11.04/0.119

NOTE: (1) Discontinuities in the load versus displacement data prevented determination of these values. (2) Corresponding bolt head displacements in ram.

400


Data of this type were determined at frequent intervals during simulated loading for both the finger/yoke interface and the specimen/finger interface. The approach adopted in analyzing the data was simply to identify any significant changes in contact status that occurred during the various stages of loading. Overall, the data did not appear reliable and proved difficult to analyze. For example, there were no obvious changes in contact status at load/displacement combinations corresponding to points (A), (B), and (C) in Fig. 18. One expected result was that the status at both interfaces was predominantly category (1) for the case of preload only. Also, as expected, the status at the specimen/finger interface changed to mixed (2) and (3) almost immediately on load application as a result of specimen straining. Beyond this, the data were judged to have little value and will not be discussed further.

Discussion As noted at the beginning of this paper, the aim of this research was to develop specimen designs and fixturing which would allow in-plane biaxial tests to be conducted on a wide range of aeropropulsion materials including advanced metallics and composites. The plan was to develop optimized specimen designs with relatively simple geometries to facilitate manufacture and to keep costs within reasonable limits. Further, reusable fixturing was to be developed for specimen gripping and loading purposes that would incorporate the flexures needed to decouple the applied biaxial loading. The fixturing was to be manufactured using conventional structural alloys, again with the goal of keeping costs within acceptable limits. Inspection of Figs. 9 and 10 and Tables 1 and 2 shows that the goal of developing a relatively simple specimen design was met as a result of this research. The design feature which enabled this result was the arrangement of slots and fingers used for attachment and loading purposes. As expected, the geometry of the slots played a key role in obtaining optimized specimen designs. Over the course of the design process, the width of the slots was progressively increased to a final value of 14 mm with the aim of reducing stress levels at the ends of the slots. Thus far, the slots have been configured using a simple circular detail at the root location with ease of manufacture in mind. Clearly, the possibility exists that noncircular geometries could be used to give improved results. Slot length also played an important role in the optimization process. It was shown that best results in terms of stress distribution at the root locations were obtained when the shorter slot was located on the specimen's centerline. It was also shown that increasing slot length caused increased stress levels at the ends of the slots and reduced stress levels at the fillet radius. Best results were obtained for slot lengths in the range 30.48 to 38.10 mm and fillet radii in the range 40.64 to 50.80 ram. Interestingly, these results applied for all the various specimen thicknesses considered. The importance of these results is that it allows some flexibility in the choice of gage area thickness (T1). For example, increased values of this variable might be preferred in test programs investigating the effects of preexisting notches or defects. One advantage offered by the slotted finger attachment is that the flexures are located some distance from the specimen gage area. This allows the use of generous fillet radii at the specimen corners to reduce the stress concentrations at this location. As indicated in Table 1, fillet radii as high as 50.80 mm were used to obtain feasible specimen designs. Such an approach is not possible with the current NASA cruciform specimen design shown in Fig. 1. In this case, use of large fillet radii increases the stiffness of the outermost flexures resulting in unacceptably high stress concentrations at the corner locations. The results summarized in Table 1 show that it is possible to develop feasible specimen designs for a range of specimen dimensions. Six designs are shown in this table with overall thicknesses of 19 mm and 25.4 mm and with gage area thicknesses ranging from 1.524 mm to 2.540 ram. One method of evaluating the various designs is to normalize the maximum stresses at the slot and fillet radius locations using the average gage area stress. This procedure was followed with the results shown in


401

Table 2. The normalized values give an indication of "margin of conservatism" and can be used to rank the various designs. Adopting this approach, review of the data showed that most favorable results were obtained for center and outer slot lengths of 30.48 mm and 30.78 mm. With one exception, the normalized values were 0.88 or less, giving about a 10% design margin. The ranking process was carried one stage further by evaluating the results for various gage area thicknesses. Best results were obtained for a 1.524 mm gage area thickness in the case of the 19 mm specimen, and for a gage area thickness of 2.032 mm in the case of the 25.4 nun specimen. The margin of conservatism for these particular designs was about 20%. The above result illustrates that the subject specimen design offers some flexibility in meeting the particular size requirements of test programs investigating the behavior of advanced materials. Turning to the design of reusable fixturing, the slotted finger attachment in its final form is shown in Figs. 11 and 12. One important modification was that the original unit design was broken down into nine subcomponents with the primary aim of simplifying manufacture. Subsequent to the design study, eight slotted fingers were fabricated from Inconel 718 using the electrical discharge machining (EDM) method. The wire EDM method was used to machine the somewhat complicated finger profile shown in Fig. 12 in a single part about 32 mm wide. This part was then cut into two 16 mm widths to form a matched pair of fingers. This approach ensured the symmetry of the finger pair about the central plane of the fixture. The obvious concern here was load alignment and the need to minimize the effects of variability in machining. A similar issue addressed during the manufacturing exercise concerned the tolerances specified for the various tapered surfaces. As noted earlier, one important goal of the fixturing design was to ensure that the end of the specimen stays in full contact with the fingers under both tensile and compressive loading. For this condition to be achieved, analysis showed that careful attention had to be given to the taper angles specified for the mating parts. Assuming a less than perfect machining job, it was shown that the taper angle on the finger should be less than that on the specimen and greater than that on the yoke. For simplicity, an angle of 7030 ' --_ 15' was selected for both taper angles on the slotted fingers. The taper angle selected for the specimen was 8000 ' -+ 15' and that selected for the yoke was 7000 ' • 15'. It remains to experimentally verify that this approach will provide the required results in terms of gripping efficiency and effective load transfer under both tensile and compressive loading. Turning to the yoke attachment, the final design is shown in Figs. 13, 14, and 15 and the results of stress analysis are shown in Figs. 16 and 17 and in Table 3. It should be noted that a single value of contact stiffness, 1.776 • 106 kN/mm, was used throughout this work. Initial analysis focused on investigating the effect of varying bolt preload and coefficient of friction on the stress distribution in the assembly. Stress analysis results are summarized in Table 3 for two combinations of bolt preload and coefficient of friction (/x). Here, it can be seen that for/z = 0.2 and bolt preload = 31.6 kN, the maximum stress in the yoke is 692 MPa and that in the finger is 1792 MPa. Since the ultimate tensile strength for Inconel 718 in an aged condition is about 1448 MPa, the latter stress was known to be unrealistic. However, data for this particular combination were retained as it provided useful insight regarding the load transfer characteristics of the assembly for higher values of coefficient of friction and bolt preload. More reasonable results were obtained for/z = 0.1 and bolt preload = 5.816 kN. For this combination, the maximum stress in the yoke was 450 MPa and that in the finger was 718 MPa. These values were judged acceptable as the 0.2% yield strength for Inconel 718 in an aged condition is about 1172 MPa. The analyses were carried one stage further by simulating tile effect of specimen loading. This was done by applying a range of grip displacements in both the tensile and compressive senses. One interesting result was that superimposition of tensile loading had little effect on the bolt preload, whereas compressive loading caused the bolt preload to increase by about 20%. Stresses in the yoke were found to increase by about 15% under tensile loading and to decrease by about 13% under compressive loading. In contrast, stress levels in the finger increased by about 22% for both tensile and

402


compressive loading. The important result here is that the major component of stress in the assembly resulted from bolt preloading and that subsequent specimen loading had a relatively minor effect. Data regarding the stiffness and load transfer characteristics of the assembly are shown in Fig. 18 and in Table 4. Note that the data shown in this figure were determined for a coefficient of friction of 0.1 and for a bolt preload of 5.816 kN. It can be seen in Fig. 18 that at relatively low load levels, the initial slope for tensile loading is 91.8 kN/mm and that for compressive loading is 92.8 kN/mm. Relatively small changes in slope occurred at load (kN)/displacement (mm) combinations of 5.19/0.0559 in the case of tensile loading and 5.36/0.0584 in the case of compressive loading. The 0.150 secant modulii for the two loading directions were 88.6 kN/mm and 90.33 kN/mm. Thus, the magnitude of the slope changes at points (A) and (C) are relatively small, about 5% on average. Under tensile loading, a major change in slope occurred at point (B) which corresponds to a load level of about 14.20 kN. Load-carrying capability was lost at this point giving an upper limit on the useful range of the fixture for the particular combination of bolt preload and coefficient of friction considered. Under compressive loading, behavior was better behaved with near-linear response extending to at least 22.50 kN. Turning to the results summarized in Table 4, similar behavior as that described above was observed for higher values of coefficient of friction and preload. More specifically, for a coefficient of friction --- 0.1 and a bolt preload = 11.62 kN, the initial slope in tension was 96.41 kN/mm and that in compression was 91.44 kN/mm. Further, the 0.150 secant modulus for tensile loading was 91.26 kN/mm. The magnitude of these values was very close to those determined for a bolt preload = 5.816 kN indicating that the value of initial slope is not a function of this variable. Under tensile loading, the change in slope occurred at a load (kN)/displacement (mm) combination of 12.36/0.130. Apparently, increasing bolt preload by a factor of two effectively doubled the initial linear range of the fixture. Inspection of Table 4 shows that this trend was not continued when bolt preload was increased to 31.64 kN for a coefficient of friction of 0.2. In this case, increasing preload did not result in any increase in the linear range. One important result here is that the stiffness characteristics of the fixturing are not a function of coefficient of friction or bolt preload. Had this been the case, the design would not have been useful for in-plane biaxial testing. Conclusions The following conclusions were drawn from this design study aimed at developing improved specimen designs and fixturing for in-plane biaxial testing: 1. The feasibility of using specimen designs incorporating relatively simple arrangements of slots and fingers for loading purposes was demonstrated by analysis for conventional structural alloys. 2. A number of optimized specimen designs were developed with gage area thicknesses ranging from 1.524 to 2.032 ram. These designs were suitable for investigating material behavior under equibiaxial stress states. 3. Reusable fixturing was developed incorporating an assembly of slotted fingers which provide the flexibility needed to decouple the applied biaxial loading. This assembly can be reconfigured as necessary to obtain optimum biaxial stress states in the specimen's gage area. 4. A yoke gripping arrangement was developed which facilitates specimen loading while avoiding the need for holes or other forms of discontinuity in the specimen. Future Work The slotted finger and yoke fixtures will be subjected to detailed experimental evaluation under uniaxial loading with the focus on stiffness and load transfer characteristics. Given a positive result, a second series of experiments will investigate the performance of the fixturing under in-plane biaxial loading.


403

References [1] Johnson, A. E., "Creep Under Complex Stress Systems at Elevated Temperatures," Proceedings, Institution of Mechanical Engineers, Vol. 164, No. 4, 1951, pp. 432---447. [2] Mtnch, E. and Galster, D., "A Method for Producing a Defined Uniform Biaxial Tensile Stress Field," British Journal of Applied Physics, Vol. 14, 1963, pp. 810-812. [3] Pascoe, K. J. and DeVillers, J. W. R., "Low-Cycle Fatigue of Steels Under Biaxial Straining," Journal of Strain Analysis, Vol. 2, No. 2, 1967, pp. 117-126. [4] Wilson, I. H. and White, D. J., "Cruciform Specimens for Biaxial Fatigue Tests: An Investigation Using Finite Element Analysis and Photoelastic Coating Techniques," Journal of Strain Analysis, Vol. 6, 1971, pp. 27-37. [5] Pascoe, K. J., "Low Cycle Biaxial Fatigue Testing at Elevated Temperatures," Proceedings, 3rd International Conference on Fracture, Munich, Verein Deutscher Eisenhuttenleiite, Dusseldorf, 1973, Vol. 6, paper V-524/A. [6] Hayhurst, D. R., "A Biaxial-tension Creep Rupture Testing Machine," Journal of Strain Analysis, Vol. 8, No. 2, 1973, pp. 119-123. [7] Parsons, M. W. and Pascoe, K. J., "Low Cycle Fatigue Under Biaxial Stress," Proceedings, Institution of Mechanical Engineers, Vol. 188, 1974, pp. 657~571. [8] Mon-ison, C. J., "Development of a High Temperature Biaxial Testing Machine," Leicester University report, Vol. 71, No. 13, 1974. [9] Odqvist, F. K. G., Mathematical Theory of Creep Rupture, Oxford Mathematical Monographs, 2nd ed., Clarendon Press, Oxford, 1974. [10] Weerasooriya, T., "Fatigue Under Biaxial Loading at 565~ and Deformation Characteristics of 2 1/4% Cr1% Mo Steel," Ph.D. Thesis, University of Cambridge, Jan. 1978. [11] Duggan, M. F., "An Experimental Evaluation of the Slotted-Tension Shear Test for Composite Materials," ExperimentalMechanics, 1980, pp. 233-239. [12] Charvat, I. M. H. and Garrett, G. G., "The Development of Closed Loop Servo-Hydraulic Test System for Direct Stress Monotonic and Cyclic Crack Propagation Studies under Biaxial Loading," Journal of Testing and Evaluation, Vol. 8, 1980, pp. 9-17. [13] Brown, M. W., "Low Cycle Fatigue Testing Under Multiaxial Stresses at Elevated Temperature," Measurement of High Temperature Properties of Materials, M. S. Loveday, M. F. Day, and B. F. Dyson, Eds., HMSO, 1982, pp. 185-203. [14] Henderson, J. and Dyson, B. F., "Multiaxial Creep Testing," Measurement of High Temperature Properties of Materials, M. S. Loveday, M. F. Day, and B. F. Dyson, Eds., HMSO, 1982, pp. 171-184. [15] Jones, D. L., Poulose, P. K., and Liebowitz, H., "Effect of Biaxial Loads on the Static and Fatigue Properties of Composite Materials," Multiaxial Fatigue, ASTM STP 853, K. J. Miller and M. W. Brown, Eds., American Society for Testing and Materials, Philadelphia, 1985, pp. 413-427. [16] Radon, J. C. and Wachnicki, C. R., "Biaxial Fatigue of Glass Fiber Reinforced Polyester Resin," Multiaxial Fatigue, ASTM STP 853, K. J. Miller and M. W. Brown, Eds., American Society for Testing and Materials, Philadelphia, 1985, pp. 396--412. [17] Found, M. S., "A Review of the Multiaxial Fatigue Testing of Fiber Reinforced Plastics," Multiaxial Fatigue, ASTM STP 853, K. J. Miller and M. W. Brown, Eds., American Society for Testing and Materials, Philadelphia, 1985, pp. 381-395. [18] Brown, M. W. and Miller, K. J., "Mode I Fatigue Crack Growth Under Biaxial Stress at Room and Elevated Temperature," Multiaxial Fatigue, ASTM STP 853, K. J. Miller and M. W. Brown, Eds., American Society for Testing and Materials, Philadelphia, 1985, pp. 135-152. [19] Sakane, M. and Ohnami, M., "Creep-Fatigue in Biaxial Stress Using Cruciform Specimens," Third International Conference on Biaxial/Multiaxial Fatigue, Vol. 2, University Stuttgart, Paper No. 46, 1989, pp. 1-18. [20] Susuki, I., "Fatigue Damage of Composite Laminate under Biaxial Loads," Mechanical Behavior of Materials--W, Vol. 2 (ICM 6), Pergamon Press, Oxford, 1991, pp. 543-548. [21] Trautman, K.-H., Dtker, H., and Nowack, H., "Biaxial Testing," Materials Research and Engineering, H. Buhl, Ed., Springer Verlag, Berlin, 1992, pp. 308-319. [22] Makinde, A., Thibodeau, L., and Neale, K. W., "Development of an Apparatus for Biaxial Testing Using Cruciform Specimens," Experimental Mechanics, Vol. 32, 1992, pp. 138-144. [23] Demmerle, J. and Boehler, J. P., "Optimal Design of Biaxial Tensile Cruciform Specimens," Journal of the Mechanics and Physics of Solids, Vol. 41, No. 1, 1993, pp. 143-181. [24] Boehler, J. P., Demmerle, S., and Koss, S., "A New Direct Biaxial Testing Machine for Anisotropic Materials," ExperimentalMechanics, Vol. 34, 1994, pp. 1-9. [25] Wang, J. Z. and Socie, D. F., "A Biaxial Tension-Compression Test Method for Composite Laminates," Journal of Composites Technology & Research, Vol. 16, No. 4, Oct. 1994, pp. 336-342.

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[26] Masumoto, H. and Tanaka, M., "Ultra High Temperature In-Plane Biaxial Fatigue Testing System with InSitu Observation," Ultra High Temperature Mechanical Testing, R. F. Lohr, and M. Steen, Eds., Woodhead Publishing Limited, Cambridge, 1995, pp. 193-207. [27] Bartolotta, P. A., Ellis, J. R., and Abdul-Aziz, A., "A Structural Test Facility for In-Plane Biaxial Testing of Advanced Materials," Multiaxial Fatigue and Deformation Testing Techniques, ASTM STP 1280, S. Kalluri and P. J. Bonacuse, Eds., American Society for Testing and Materials, 1997, pp. 25-42. [28] Trautmann, K.-H., Maldfeld, E., and Nowack, H., "Crack Propagation in Cruciform 1MI 834 Specimens Under Variable Biaxial Loading," Multiaxial Fatigue and Deformation Testing Techniques, ASTM STP 1280, S. Kalluri and P. J. Bonacuse, Eds., American Society for Testing and Materials, 1997, pp. 290-309. [29] Dalle Donne, C. and D6ker, H., "Plane Stress Crack Resistance Curves of an Inclined Crack Under Biaxial Loading," Multiaxial Fatigue and Deformation Testing Techniques, ASTM STP 1280, S. Kalluri and P. J. Bonacuse, Eds., American Society for Testing and Materials, 1997, pp. 243-263.

Claudio Dalle Donne, t Karl-Heinz Trautmann I and Hans Amstutz 2

Cruciform Specimens for In-Plane Biaxial Fracture, Deformation, and Fatigue Testing REFERENCE: Dalle Donne, C., Trautmann, K.-H., and Amstutz, H., "Crueiform Specimens for InPlane Biaxial Fracture, Deformation, and Fatigue Testing," Multiaxial Fatigue and Deformation: Testing and Prediction, ASTM STP 1387, S. Kalluri and P. J. Bonacuse, Eds., American Society for Testing and Materials, West Conshohocken, PA, 2000, pp. 405-422. ABSTRACT: Three different types of cruciform specimens, which have been used successfully on the DLR biaxial test rig for investigations in the fields of fracture mechanics, yield surface evaluation, and fatigue of riveted joints are presented in detail. The following three characteristics were required from these specimens: a uniform stress and strain distribution in the central testing regions, low manufacturing costs, and easy mounting in the DLR biaxial test rig. It is shown that the first and the second requirement are in conflict. Because of the uneven stress distribution in the simple and inexpensive fracture mechanics specimen, additional finite-element calculations were needed for a full fracture mechanical characterization. On the other hand, impelling a very uniform stress distribution in the riveted cruciform specimens increased the manufacturing costs considerably. The specimen which satisfies the demands best is the yield locus evaluation specimen of deep drawing steel. It has low manufacturing costs and a very uniform strain distribution in the central testing section. KEYWORDS: cruciform specimen, biaxial loading, stress intensity factor, plastic limit load. yield surface, deep drawing, compression testing, riveted joint, fiber metal laminate

Nomenclature a

B E F1, F2 Fl,y J

KI N W Y1, Y2 A O'1, 0"2 O'1 ,lig, O-2,1ig O'vM3ig O-y

.y

Half-crack length in center cracked cruciform specimen Thickness Young' s modulus Forces in main and secondary loading axis respectively, F1 --> F2 Plastic limit load Mode I J-integral Mode I stress intensity factor Ramberg-Osgood hardening exponent, N > 1 Half-width of center cracked cruciform specimen Finite width correction factors for KI induced by F1 and F2, respectively Biaxiality ratio, F2/F1 Gross stresses in main and secondary loading axis, respectively, o"1 ~ 0-2 Ligament stresses in main and secondary loading axis, respectively von Mises equivalent ligament stress Yield strength Ligament width normalized by specimen width

1 Research engineer and senior engineer, respectively, Institute of Materials Research, German Aerospace Center DLR, Linder Hthe, D-51147 Cologne, Germany. 2 Senior research engineer, Department of Mechanics of Materials, University of Darmstadt, Petersenstr, 13, D64287 Darmstadt, Germany. 4O5

Copyright9


www.astm.org

406


Introduction The importance of studying the effects of biaxial loading on fracture, fatigue, and deformation behavior of materials and structures has been recognized for many years. The objectives of such biaxial investigations were either the better understanding of biaxial loading effects on the failure or deformation behavior of materials, or a more realistic simulation of the complex loading situation of the structure to be assessed. The most realistic experimental test method to create in-plane biaxial loading in flat sheets is the in-plane biaxial test with cruciform specimens (see review in Ref 1). A uniform stress and strain distribution in the central rectangular or circular testing region of the crossshaped specimens is usually achieved through an array of slots in the loading arms, Fig. 1. The applied load is distributed by the material's "fingers" along the edge of the testing region. Moreover, the slots minimize the specimen strain cross-sensitivity between the two loading axes, since as the load is applied on one axis, the individual fingers on the other loading axis are able to flex relatively freely. As shown in Fig. 1, the thickness in the gage section is sometimes reduced to ensure maximum stresses to occur in this region. In this paper, three different types of cruciform specimens, which have been used successfully on the DLR biaxial test rig [2] for investigations in the fields of fracture mechanics, fatigue, and plastic deformation, are presented in detail. They were optimized for uniform stress and strain distributions in the central testing regions, for easy mounting in the DLR biaxial test rig and for low manufacturing costs.

Fracture Mechanics Cruciform Specimen The fracture mechanics cruciform specimen was used in a basic research program on biaxial load effects on stable crack growth in ductile steel and aluminum alloy sheets [3-7]. To keep manufacturing costs low, the possibility of a thickness reduction of the cruciform specimens was rejected at the beginning of the program. Therefore, only the influence of the slots in the loading arms on the stress intensity factor was investigated in a preliminary finite-element study. The mesh of a Mode I cracked cruciform specimen employed in the plane stress finite-element (FE) calculations is displayed in Fig. 2. Due to symmetry, a quarter mesh was used with the relevant boundary conditions applied to the specimens edges. The mesh consisted of 560 eight-node isoparametric elements with four integration points. These elements were employed through the whole specimen, also in the refined area around the crack tip. The FE program delivered J-integral values ob-

slotted

F-

FIG. 1--Schematic of a typical cruciform specimen for in-plane biaxial testing.

DALLE DONNE ET AL. ON CRUCIFORM SPECIMENS

407

=;LF

FIG. 2--Finite-element model of the cruciform fracture mechanics specimen used for stress intensity factor evaluation and plastic limit load calculations.

tained through the virtual crack extension method of Parks and De Lorenzi [8,9]. Since in linear elasticity the J-integral corresponds to the energy release rate, the stress intensity factor could be calculated from the well known relationship between K~ and the energy release rate [10] K, = x / ~

O)

The notation E indicates the Young's modulus. The high accuracy of this approach (within 1% of the analytical solution of a simple cracked configuration) was proven in a previous study [ll]. The loading arms contained 11 slots, which were modeled by uncoupled knots on both sides of each slot. Figure 3 shows the influence of the slot length on the stress intensity factor KI of a cruciform specimen with a crack to width ratio of a/W = 0.25 at different applied load biaxiality ratios h. Here h is defined by the ratio of the crack parallel load F2 to the load perpendicular to the crack F1. The stress intensity factor is normalized by the KI= value

El

of an infinite plate under the same remote biaxial loading. In this particular case, remote crack parallel loading (/;2) has no influence on KI,= [12]. Very long, open ended slots introduced uniform stress and strain distribution in the specimens. The K1 values are therefore very close to the infinite plate solution for any biaxiality ratio A. Decreasing slot length increased the stiffness of the loading arms. As shown in Fig. 4, the load trajectories are deviated in the adjacent loading arms causing severe notch stresses at the comer fillets of the specimens.

408

MULTIAXlAFATI L GUEANDDEFORMATION 1.1 ~ -I

1

21q 18o /

.

0.8

0

~

~

W=150mm ao/W=0"25

0.7

-1.o

~

0

-

slot length [ram]

I

-o.5

o'.o

;L

1.o

FIG. 3--1nfluence of the slot length on stress intensity factor of a cruciform specimen.

Especially in the case of tensile crack parallel stresses (A > 0) these notch stresses led to an unloading of the cracked region and therefore to K~ values which are considerably lower than the idealized Kr,= values. It should be noted here that the independence of the infinite plate K~,~has often led researchers to the misinterpretation that also in the case of finite specimens the load biaxiality ratio has no influence on Kr. It is clear from Fig. 3 that this assumption is only true in connection with very long slots in the loading arms. The often contradictory experimental results of biaxial load effects on fracture toughness and fatigue crack growth can therefore be explained with a misinterpretation of the data caused by the wrong K1 calibration [13-16]. To prevent buckling problems at negative A, an intermediate value of slot length of 45 mm was chosen for the fracture mechanics specimen. The final shape is shown in Fig. 5. A cutout in the less load carrying area of the loading arms simplified the gripping procedure, since the servohydraulic actuators of the biaxial test rig could remain almost in their final testing position.

FIG. 4---Deviation of force trajectories in adjacent loading arms in cruciform specimens.

DALLE DONNE ET AL. ON CRUCIFORM SPECIMENS

409

FIG. 5--Final shape offracture mechanics cruciform specimen (all dimensions in ram). The gripping procedure was simplified by a cutout in the loading arms.

The stress intensity factor KI is presented as the sum of the components perpendicular (F1) and parallel (F2) to the crack K1 = 2__B~ V~-aa~y1 (_~)

F2

2-gff

a

where Y1 and Y2 are the finite width correction factors for Kr induced by F1 and F2, respectively. The Y1 and Y2values obtained by the FE calculations are displayed in Fig. 6. Since the crack parallel component (Y2) is negative, dominant crack parallel loading could lead to crack face contact and interaction. The FE calculations were performed with and without the central 8-mm-wide hole. It is evident from Fig. 6 that the hole affects the crack parallel correction factor for cracks shorter than a/W = 0.4. The main loading axis correction factor is fitted by the following equation Y~ = 0 . 9 9 1 + 0.786 ( W ) 2532

(4)

The secondary loading axis correction factor for the specimen with the central hole is approximated by Y2 = -0.1 + 0.0544 W

(5)

410

MULTIAXlAL FATIGUE AND DEFORMATION 1.5"

O 1.4-

Y1 with central 8 mm hole Y

1.3> Z 1.21.11.0-

g

w

0.9 0.0 ' 0'.1 ' 012

017

0.8

-0.10 0.0 ' 0'.1 ' 012 ' 0'.3 ' 0'.4 ' 015 ' 0'.6 ' 0:7 a/W

0.8

-0.05|

ix

0.3 ' " 0:4 a/W

0.5 . . . 0.6 .

Y2 with central 8 mm hole

/

-0.061 -0.071 >_~ -0.08-

-0.09-

FIG. 6---Correction factors for stress intensity factor of cruciform fracture mechanics specimen.

whereas without the central hole Y2 is given by

Y2 = -0.04966 - 0.03476 cos 1.1126 ~ ~-

(6)

Taking into account the uncertainties of the FE calculations, these fits should be accurate within 2.5% for 0.2 --< a/W

Multiaxial Fatigue and Deformation: Testing and Prediction (ASTM Special Technical Publication, 1387)

Fatigue Testing and Analysis Under Variable Amplitude Loading Conditions (ASTM special technical publication, 1439)

Fatigue and Fracture of Medical Metallic Materials and Devices (ASTM special technical publication, 1481)

Beryllium: Sampling and Analysis (ASTM special technical publication, 1473)

Fretting Fatigue: Current Technology and Practices (ASTM Special Technical Publication, 1367)

Fretting Fatigue: Advances in Basic Understanding and Applications (ASTM Special Technical Publication, 1425)

Durability 2000: Accelerated and Outdoor Weathering Testing (ASTM Special Technical Publication, 1385)

Testing and Performance of Geosynthetics in Subsurface Drainage (ASTM Special Technical Publication, 1390)

Automotive Lubricant Testing and Advanced Additive Development (ASTM special technical publication 1501)

Tissue Engineered Medical Products (TEMPs) ASTM special technical publication, 1452

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Mechanical, Thermal, and Environmental Testing and Performance of Ceramic Composites and Components (ASTM Special Technical Publication, 1392)

Evaluation and Remediation of Low Permeability and Dual Porosity Environments (ASTM Special Technical Publication, 1415)

Performance and Durability of the Window-Wall Interface (ASTM special technical publication, 1484)

Contaminated Sediments: Evaluation and Remediation Techniques (ASTM special technical publication, 1482)

Acoustic Emission: Standards and Technology Update (ASTM Special Technical Publication, 1353)

Joining And Repair Of Composite Structures (ASTM Special Technical Publication, 1455)

Constructing and Controlling Compaction of Earth Fills (ASTM Special Technical Publication, 1384)

Stainless Steels for Medical and Surgical Applications (ASTM Special Technical Publication, 1438)

Deformation Mechanisms, Rheology and Tectonics (Geological Society Special Publication 54)

Bench Testing of Industrial Fluid Lubrication and Wear Properties Used in Machinery Applications (ASTM Special Technical Publication, 1404)

Isocyanates: Sampling, Analysis, and Health Effects (ASTM Special Technical Publication, 1408)

Geostatistics for Environmental and Geotechnical Applications (Astm Special Technical Publication Stp)

Materials Characterization by Dynamic and Modulated Thermal Analytical Techniques (ASTM special technical publication, 1402)

Technology of Floor Maintenance and Current Trends (ASTM special technical publication, 1448)

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Biaxial Multiaxial Fatigue and Fracture (European Structural Integrity Society)

Thermal Measurements: The Foundation of Fire Standards (ASTM Special Technical Publication, 1427)

Oil Flow Studies at Low Temperatures in Modern Engines (ASTM Special Technical Publication, 1388)

Multiaxial Fatigue and Deformation: Testing and Prediction (ASTM Special Technical Publication, 1387)

Fatigue Testing and Analysis Under Variable Amplitude Loading Conditions (ASTM special technical publication, 1439)

Fatigue and Fracture of Medical Metallic Materials and Devices (ASTM special technical publication, 1481)

Beryllium: Sampling and Analysis (ASTM special technical publication, 1473)

Fretting Fatigue: Current Technology and Practices (ASTM Special Technical Publication, 1367)

Fretting Fatigue: Advances in Basic Understanding and Applications (ASTM Special Technical Publication, 1425)

Durability 2000: Accelerated and Outdoor Weathering Testing (ASTM Special Technical Publication, 1385)

Testing and Performance of Geosynthetics in Subsurface Drainage (ASTM Special Technical Publication, 1390)

Automotive Lubricant Testing and Advanced Additive Development (ASTM special technical publication 1501)

Tissue Engineered Medical Products (TEMPs) ASTM special technical publication, 1452

Multiaxial Notch Fatigue

Mechanical, Thermal, and Environmental Testing and Performance of Ceramic Composites and Components (ASTM Special Technical Publication, 1392)

Evaluation and Remediation of Low Permeability and Dual Porosity Environments (ASTM Special Technical Publication, 1415)

Performance and Durability of the Window-Wall Interface (ASTM special technical publication, 1484)

Contaminated Sediments: Evaluation and Remediation Techniques (ASTM special technical publication, 1482)

Acoustic Emission: Standards and Technology Update (ASTM Special Technical Publication, 1353)

Joining And Repair Of Composite Structures (ASTM Special Technical Publication, 1455)

Constructing and Controlling Compaction of Earth Fills (ASTM Special Technical Publication, 1384)

Stainless Steels for Medical and Surgical Applications (ASTM Special Technical Publication, 1438)

Deformation Mechanisms, Rheology and Tectonics (Geological Society Special Publication 54)

Bench Testing of Industrial Fluid Lubrication and Wear Properties Used in Machinery Applications (ASTM Special Technical Publication, 1404)

Isocyanates: Sampling, Analysis, and Health Effects (ASTM Special Technical Publication, 1408)

Geostatistics for Environmental and Geotechnical Applications (Astm Special Technical Publication Stp)

Materials Characterization by Dynamic and Modulated Thermal Analytical Techniques (ASTM special technical publication, 1402)

Technology of Floor Maintenance and Current Trends (ASTM special technical publication, 1448)

Spectral method in multiaxial random fatigue

Biaxial Multiaxial Fatigue and Fracture (European Structural Integrity Society)

Thermal Measurements: The Foundation of Fire Standards (ASTM Special Technical Publication, 1427)

Oil Flow Studies at Low Temperatures in Modern Engines (ASTM Special Technical Publication, 1388)

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