Astm stp 1387 2000

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 Copyright by ASTM Int'l (all rights reserved); Wed Dec 23 19:59:07 EST 2015 Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reproductions authorize 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 Materials-Fatigue Axial loads Materials-Dynamic testing Deformations (Mechanics) I Kalluri, Sreeramesh I1 Bonacuse, Peter J., 1960TA418.38.M86 2000 620.11126-dc21 00-059407 Copyright 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 Photocopy Rights Authorization to photocopy items for internal, personal, or educational classroom use, or the internal, personal, or educational classroom use of specific clients, is granted by the American Society for Testing and Materials (ASTM) provided that the appropriate fee is paid to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923; Tel: 508-750-8400; online: http://www.copyright.com/ 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 Copyright by ASTM Int'l (all rights reserved); Wed Dec 23 19:59:07 EST 2015 Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reproductions authorized 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 Copyright by ASTM Int'l (all rights reserved); Wed Dec 23 19:59:07 EST 2015 Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reproductions authoriz 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 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 Copyright by ASTM Int'l (all rights reserved); Wed Dec 23 19:59:07 EST 2015 Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reproductions authorized 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 Copyright by ASTM Int'l (all rights reserved); Wed Dec 23 19:59:07 EST 2015 Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reproductions authorized 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 Copyright by ASTM Int'l (all rights reserved); Wed Dec 23 19:59:07 EST 2015 vii Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reproductions authorized 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 Copyright by ASTM Int'l (all rights reserved); Wed Dec 23 19:59:07 EST 2015 Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reproductions authorized 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 Copyright by ASTM Int'l (all rights reserved); Wed Dec 23 19:59:07 EST 2015 Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reproductions authorized 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 Copyright by ASTM Int'l (all rights reserved); Wed Dec 23 19:59:07 EST 2015 Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reproductions authoriz WELSH AND ADAMS ON TRIAXIAL TESTING FACILITY 429 @ (8 places) @ @ (~ 24.7 mm (0.975 in) @ ~ 4.o6 I I I mm (o.~6 in) [ t ~- 2.03 ~i mm ~ 161.4 mm (6.354 ~n) FIG (o.oB in) I I I t I Schematic drawing of the biaxial/triaxial test specimen posing 24.8-mm-wide (0.98 in.) loading arms, each of which contain two 4.76-mm-diameter (0.188 in.) holes used to align each set of wedge grips The gage section detail of this cruciform specimen is shown in Fig All dimensions shown in Fig assume a specimen thickness of 4.06 mm (0.160 in.) and a gage section thickness-taper fillet radius of 12.7 mm (0.50 in.) In this configuration, the gage section consists of a 21.6-mm-square (0.850 in.) test region with 1.59-mm-radius (0.063 in.) loading arm fillets used to merge adjacent loading arms at the corners of the gage section The gage section detail shown in Fig was machined into both surfaces of all test specimens used in the present study to produce the desired symmetric thickness-tapered geometry The specific specimen design shown in Figs and was selected after performing an anisotropic, linear-elastic, finite-element analysis Specific concerns included the location and magnitude of stress concentrations, the interaction of the three normal stress components, and the effect of laminate configuration The finite-element analysis demonstrated the ability of this specimen to produce a nearly homogeneous triaxial stress state in the gage section [11] In addition, the three normal stresses were found to be reasonably independent [11] That is, appreciable normal stresses in one loading direcCopyright by ASTM Int'l (all rights reserved); Wed Dec 23 19:59:07 EST 2015 Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reproductions authorized 430 MULTIAXIAL FATIGUE AND DEFORMATION 15.5 mm(ty~)I / : 0.612 in (typ) "] R3.18 mm (typ) R1/8 in (typ) R1.59 mm (typ) R1/16 in (typ) lg,2z:It;;i 4.98 mm 0.196 in R0.256 R6.50 mm \ i i FIG Detail of the biaxial/triaxial test specimen gage section tion did not extend a significant distance into adjacent loading arms This is critical in minimizing undesirable load sharing effects in cruciform-shaped test specimens One of the primary concerns associated with testing cruciform specimens is the location and magnitude of stress concentrations More specifically, the stress concentration generated at the intersection of two adjacent loading arms is a frequent concern [14-16] The present finite-element analysis indicated that the magnitude of this in-plane stress concentration was approximately 2.5 for a quasiisotropic AS4/3501-6 carbon-epoxy laminate This value increased to approximately 3.0 for a unidirectional laminate of the same material Although not specifically modeled in the present study, it is assumed that similar stress concentrations would be present in a cross-ply laminate One stress concentration that is desired when using thickness-tapered cruciform specimens is a direct result of thinning the gage section Perhaps more accurately described as a stress riser, this geometric feature is exploited in an attempt to increase the stresses, and produce subsequent failures, in the gage section The present finite-element analysis indicated that the specimen geometry shown in Figs and produced a stress riser of approximately 1.25 [11] Unfortunately, this value limits the laminate configurations that can successfully be tested, due to biaxial strengthening effects That is, any degree of biaxial loading results in an increase in the ultimate strength of some laminates compared to uniaxial loading, meaning that the ultimate strength of a biaxially loaded laminate should be higher than a uniaxially loaded specimen This is obviously an undesirable situation for a cruciform specimen that is loaded biaxially in the gage section and uniaxially in each of the four loading arms, as unacceptable failures will occur in the arms of the specimen rather than the gage section As a resuit, it is believed that only cross-ply laminates can overcome this affect and be successfully tested Copyright by ASTM Int'l (all rights reserved); Wed Dec 23 19:59:07 EST 2015 Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reproductions authorized WELSH AND ADAMS ON TRIAXIAL TESTING FACILITY 431 TABLE Average uniaxia16061-T6 aluminum tensile and compressive yield strength, ultimate strength, and modulus results Average Yield Strength Ultimate Strength Average Modulus Axis Test Mode MPa ksi MPa ksi CVa % GPa Msi X Y Z X Y Z T T T C C C 2871 2901 2922 -272 a -2761 -2782 41.611 42.11 42.32 -39.51 -40.01 -40.32 323 324 322 -279 -278 -339 46.8 47.0 46.7 -40.5 -40.4 -49.13 0.1 0.1 0.3 1.1 0.9 0.3 68.21 68.9 a 68.92 -68.21 -68.91 NA 9.91 10.01 10.02 -9.91 -I0.01 NA CV % 0.0 1.0 1.0 0.3 0.4 NA NA - Not Available Average of specimens Average of specimens Specimen buckling Coefficient of variation using a thickness-tapered cruciform test specimen [8,11] For this reason, only cross-ply laminates were tested in the present study Experimental Results 6061-T6 Aluminum As an initial evaluation of the performance of the triaxial testing facility, numerous uniaxial and biaxial tests were performed using 6061-T6 aluminum Specifically, these tests were used to investigate the repeatability of, the accuracy of, and the intra-axis performance of the test machine under uniaxial and biaxial testing conditions Experimental data generated were compared to established handbook values and applicable failure theories A total of 30 uniaxial and 27 biaxial 6061-T6 aluminum specimens were tested Tables and present the average uniaxial and biaxial results, re- TABLE Average biaxia16061-T6 aluminum yield strength results Average X-Direction Yield Strength Average Y-Direction Yield Strength Specimen Group I.D Stress Ratio MPa ksi CV 1% MPa ksi CV 1% JAHll0 JAH320 JAH310 JAH260 JAH160 JAH170 JAH680 JAH780 JAH660 1/1/0 3/2/0 3/1/0 / - 1/0 1/- 1/0 1/-2/0 -1/-3/0 -2/-3/0 -1/-1/0 306 333 317 213 172 109 -141 -261 -318 44.4 48.3 46.0 30.9 24.9 15.8 -20.5 -37.8 -46.1 1.6 1.8 0.8 1.8 0.0 2.5 3.2 6.7 0.9 303 227 112 - 112 - 168 -212 -347 -348 -316 44.0 32.9 16.2 - 16.2 -24.3 -30.7 -50.4 -50.5 -45.9 1.7 1.8 1.l 0.3 0.0 2.5 1.1 3.8 1.1 Coefficient of variation Copyright by ASTM Int'l (all rights reserved); Wed Dec 23 19:59:07 EST 2015 Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reproductions authorized 432 MULTIAXIALFATIGUE AND DEFORMATION spectively Each average value shown in Tables and was obtained from testing five and three individual specimens, respectively, unless otherwise noted Details of specimen fabrication and testing procedures are presented in Ref 11 Discussion of 6061-T6 Aluminum Results One of the most notable features of the data presented in Table is that the average strength and modulus values were very consistent Inspection of Table reveals that the average ultimate tensile strength values for all three loading axes were within 0.4% of the mean value of 323 MPa (46.8 ksi), while the average modulus values were within 1.0% of the mean value of 68.9 GPa (10.0 Msi) Because all of the uniaxial compression specimens failed by gross (Euler) column buckling after initial specimen yielding, an analysis of the ultimate strength average values is of limited use However, the average yield strength for the compression specimens were within 2.5% of the mean value of -275 MPa (-39.9 ksi), and the average compressive modulus values were within 0.7% of the mean value of -68.9 GPa ( - 10.0 Msi), These data indicate that the triaxial testing facility is capable of generating very consistent experimental data Perhaps more important than the consistency of the data presented in Table is the fact that the yield strength, ultimate strength, and modulus values obtained in the present study compare very favorably with accepted handbook values for this material Values of 276 MPa, 310 MPa, and 68.9 GPa (40 ksi, 45 ksi, and 10.0 Msi) for the yield strength, tensile strength, and modulus of elasticity, respectively, are generally accepted handbook values for 6061-T6 aluminum [17] The fact that comparable data were generated in the present study was interpreted as a verification of the numerous fabrication, testing, calibration, and data acquisition and reduction procedures that were developed specifically for the present study In addition, numerous aspects of the triaxial testing facility itself were verified by obtaining consistent, accurate data from each of the three loading axes These aspects included the reaction frame compliance, force actuator alignment, and the assumption that the various linear bearings provide a nearly frictionless interface between the test fixture framework and the tension and compression housings While a comparison of the biaxial yield strength data presented in Table for the various stress ratios is of limited use in form, these data can readily be compared to existing failure theories when plotted in ~rl - ~2 stress space Figure presents such a plot of the data, along with Tresca and von Mises yield criteria predictions One of the most significant features of the data presented in Fig is that they are in close agreement with the von Mises yield criterion in all quadrants The largest difference occurred for the three stress ratios in the C/C quadrant, in which the experimental data exceeded the von Mises prediction by approximately 13% This discrepancy is believed to be primarily a result of the technique used to account for the biaxial load sharing between adjacent loading arms for the present specimen design Very briefly, a biaxial tensile test specimen loaded uniaxially was used to experimentally determine the amount of load sharing between adjacent arms of the cruciform specimen [11] It is believed that a more accurate representation of the tests performed in the C/C quadrant could be achieved by using a uniaxial compression specimen to establish the level of load sharing for specimens tested in this quadrant [11] Another significant feature of Fig is that each group of three specimens tested at a particular stress ratio were very consistent As with the uniaxial test specimen results, this was viewed as a confirmation of the biaxial specimen fabrication and testing procedures developed specifically for the present study In addition, inspection of the individual specimen test results presented in Ref 11 indicates that the desired biaxial stress ratio was achieved at failure for nearly every biaxial test This fact, coupled with the close agreement of the uniaxial and biaxial yield strengths obtained in the present study with established values generated reasonable confidence in the triaxial testing facility Copyright by ASTM Int'l (all rights reserved); Wed Dec 23 19:59:07 EST 2015 Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reproductions authorized WELSH AND ADAMS ON TRIAXIAL TESTING FACILITY 433 (1~) Cry (MPa) Yield Surfaces -"3!- 60 Tresea | yon i~4iscs [ Experimental Data.,/ ~_ 400 Axis o f Symmetry / 200 / / / / / // -40O / / / /-200 200 400 fgP~) r -60 -30 / l 3o/ J / Uniaxial Results 60 (ksi) / / :, / ,,M -30 / 9~ ~/-'~ Uniaxial Results / ++ -~ _6o~- -400 ++ FIG -Biaxial yield envelope for 6061-T6 aluminum AS4/3501-6 Carbon~Epoxy The triaxial test facility had been extensively evaluated using 6061-T6 aluminum uniaxial and biaxial tests The primary objective of these tests was to characterize the biaxial and triaxial response of a composite material It was anticipated that the orthotropic response of such a material would provide an additional challenge to the triaxial testing facility A total of 27 biaxial [0/9019s Hercules AS4/3501-6 carbon/epoxy specimens were tested in o1 - 0"2 stress space More specifically, the same stress ratios used to evaluate the response of the 6061-T6 aluminum specimens were used to evaluate the AS4/3501-6 carbon/epoxy material system in the T/T, T/C, and C/C quadrants In addition, 12 triaxial tests were performed using this composite material to demonstrate the ability of the triaxial testing facility to perform triaxial tests in triaxial stress space Tables and 4, respectively, present the average biaxial and triaxial results for all [0/9019~ AS4/3501-6 carbon/epoxy specimens tested in the present study Each average value shown in Tables and was obtained by testing three and two individual specimens, respectively, unless otherwise noted All specimen fabrication and testing procedures are presented in Ref 11 Discussion ofAS4/3501-6 Carbon~Epoxy Results Although the data presented in Tables and are useful in comparing average results obtained using specific stress ratios, it is difficult to interpret these data in tabular form when evaluating the over- Copyright by ASTM Int'l (all rights reserved); Wed Dec 23 19:59:07 EST 2015 Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reproductions authorize 434 MULTIAXIAL FATIGUE AND DEFORMATION TABLE Average biaxial ultimate strengths obtained from testing a [0/9019sAS4/3501-6 carbon~epoxy laminate Average Y-Direction Average X-Direction Ultimate Strength Ultimate Strength Specimen Group I.D Stress Ratio MPa ksi CV % MPa ksi JSX1101 JSX3201 JSX310 JSX260 JSX160 JSX170 JSX680 JSX780 JSX660 1/1/0 3/2/0 3/1/0 / - 1/0 1/- 1/0 1/-2/0 -1/-3/0 -2/-3/0 -1/-1/0 437 487 465 465 432 252 -210 -428 -519 63.4 70.6 67.5 67.5 62.7 36.6 -30.4 -62.1 -75.3 5.4 12.1 11.5 7.3 4.8 4.2 11.1 3.8 2.6 449 322 157 -232 -433 -507 -633 -654 -519 65.1 46.7 22.7 -33.7 -62.8 -73.5 -91.8 -94.8 -75.3 CV r 3.4 10.1 11.5 7.8 5.7 6.2 10.5 6.7 2.6 Specimens failed in loading arm Coefficient of variation all material response in ~rl - tr2 - t~3 stress space Therefore, Fig is a graphical representation of these data Presenting triaxial data on a two-dimensional plot requires the reader to realize that the actual data lies above or below the plane of the plot, as specified by the magnitude of the Z-direction stress One of the most significant aspects of the biaxial data presented in Fig is that a reasonable amount of material scatter exists among each of the three specimens tested in a particular stress ratio group Although each of the specimens failed very near the desired stress ratio, the magnitude of the applied stresses at failure exhibited differences as large as 17% for certain stress ratios While this level of material scatter is undesirable, similar levels of material scatter for biaxial tests of the same AS4/3501-6 carbon/epoxy material system were obtained in prior studies for a quasi-isotropic laminate configuration [18,19] The data presented in Fig also indicates that the present specimen design was successful in generating independent normal stress components in the gage section of the test specimen Each of the TABLE -Average triaxial ultimate strengths obtained from testing a [0/9019sAS4/3501-6 carbon~epoxy laminate Average X-Direction Average Y-Direction Average Z-Direction Ult Strength Ult Strength Ult Strength Specimen Group I.D Stress Ratio MPa JSXI 11 JSX161 JSX661 JSX116 JSX166 JSX666 1/1/11 1/-1/11 -1/-1/11 1/1/- 12 1/- 1/- 12 -1/-1/-12 467 407 -554 468 405 -561 ksi CV3 % MPa 67.8 59.1 -80.3 67.9 58.8 -81.4 2.5 2.5 12.4 2.2 11.4 4.4 458 -399 -550 463 -403 -571 ksi CV3 % MPa ksi CV3 % 66.4 -57.9 -79.7 67.1 -58.4 -82.8 0.5 2.7 16.3 5.2 11.5 4.0 0 - 146 - 143 -143 0 -21.2 -20.8 -20.8 0.0 0.0 0.0 3.3 3.5 0.1 Z-axis loading attachments debonded prior to ultimate specimen failure Actual stress ratios at failure were 1/1/-0.3, 1/- 1/-0.4, - 1/- 1/-0.3, respectively Coefficient of variation Copyright by ASTM Int'l (all rights reserved); Wed Dec 23 19:59:07 EST 2015 Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reproductions authorized WELSH AND ADAMS ON TRIAXIAL TESTING FACILITY (ksi) O- Y 435 ('~a) 800 ~_ Experimental Dam + 100 Bi~ - Axis of Syrnm~ Z-Axis Tension O / Z-Axis Comp, 400 50 ++ / + / -800 _L -400 t I I f f~Lr if_ / -100 +++ / 400 IL ] i 800 (MPa; ~ ~ 50 -50 i _ (7 ]- X 100 (ksi) / ++Jr / -50 -400 / / / J +% ,+ § + % -100 -800 FIG Experimental biaxial and triaxial data obtained by testing a [0/9019s AS4/3501-6 carben~epoxy laminate biaxial stress ratios was achieved by directly manipulating the applied force to each of the four cruciform specimen loading arms Note also that the stress ratios evaluated in the present study reasonably define the entire o-1 - ~2 stress space for a [019019s AS41350l-6 carbon/epoxy laminate configuration demonstrates the potential of the present triaxial test facility to describe the entire biaxial response of a composite material Unfortunately, an undesirable failure mode was identified for the AS4/3501-6 carbon/epoxy specimens tested in the T/T quadrant of ~1 - ~2 stress space The majority of these specimens exhibited a transverse tensile failure at the inboard wedge grip alignment hole This premature failure prevented the test specimens in the T/T quadrant of ~r~ - ~2 stress space from achieving the maximum biaxial stress state in the gage section As a result, it is recommended that this alignment hole be deleted in future studies to prevent this failure mode Several additional recommendations, including decreasing the wedge grip taper angle to 10 ~ increasing the width of the cruciform loading arms, and possibly using thicker laminates are believed to be potential future improvements The data presented in Tables and and Fig also indicate that the triaxial tension tests were not performed successfully Because the Z-axis attachments used to generate through-the-thickness tensile forces in the gage section debonded at approximately 10 MPa (1.5 ksi), a true triaxial stress state was not achieved in these specimens at failure Obviously, this is unacceptable Damage to the fibers on the surface of the gage section as a result of the thickness-tapering machining procedures are believed to be primarily responsible for this undesirable failure mode [11] As a result, several modifiCopyright by ASTM Int'l (all rights reserved); Wed Dec 23 19:59:07 EST 2015 Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reproductions author 436 MULTIAXlAL FATIGUE AND DEFORMATION cations to this specimen configuration, including using end tabs bonded to a thinner composite laminate, are recommended for future studies Another notable feature of the data presented in Tables and is that the desired stress ratios for the triaxial compression specimens were not achieved The maximum capacity of the Z-axis force actuators was reached prior to ultimate failure That is, the stepper motors used to supply power to the Z-axis actuators were only capable of generating approximately - kN ( - kip) of force before stalling Those actuators then maintained that level of force for the remainder of the test Thus, the actual stress ratios at failure for the JSX116, JSX166, and JSX666 specimen groups were approximately 1/1/-0.3, 1/1 - 1/-0.4, and - / - 1/-0.3, respectively While these tests are believed to be valid and demonstrate the ability of the present test facility to generate triaxial test results, this obviously places limitations on the three-dimensional stress ratios that the present test facility is capable of exploring However, it is believed that this situation should be corrected through specimen modifications as well as by increasing the capacity of the stepper motors used to power these actuators [11] Conclusions A triaxial testing facility that is capable of testing composite materials in both biaxial and triaxial configurations has been developed This facility was evaluated by performing uniaxial and biaxial tests on 6061-T6 aluminum before investigating the biaxial and triaxial response of an AS4/3501-6 carbon/epoxy cross-ply laminate Although difficulties associated with the triaxial testing facility and current cruciform specimen design have been identified, the authors believe that the potential of this facility to successfully perform biaxial tests on composite materials in any region of o"1 - o-2 stress space has been demonstrated The authors believe that a similar statement can be made regarding the potential of the testing facility to successfully perform triaxial tests on composite specimens Several modifications to the triaxial test facility and present thickness-tapered cruciform test specimen geometry have been identified that should reduce the difficulties encountered in the present study The authors believe that once these issues have been properly addressed, the resulting test facility will be capable of generating accurate and consistent biaxial and triaxial experimental data for composite materials Acknowledgments The authors are grateful for the continuing support of the Federal Aviation Administration, Office of Research and Technology Application, through FAA Grant No 94-G-009 The technical direction and encouragement of Dr Donald W Oplinger, Technical Monitor, FAA Technical Center, and Mr Joseph R Soderquist, FAA Headquarters, is sincerely appreciated In addition, the authors are grateful for the support of the University of Wyoming, Major Equipment Grant Program, for funding a portion of this study References [1] Welsh, J S., and Adams, D F., "Unidirectional Composite Compression Strengths Obtained by Testing Mini-Sandwich, Angle-, and Cross-Ply Laminates," Report No UW-CMRG-R-95-106, Composite Materials Research Group, University of Wyoming, Laramie, WY, April 1995 [2] Wegner, P M and Adams, D F., "Composite Lamina Compressive Properties Using the Wyoming Combined Loading Compression Test Method," Report No UW-CMRG-R-98-116, Composite Materials Research Group, University of Wyoming, Laramie, WY, September 1998 [3] Traceski, F T., Specifications & Standards for Plastics and Composites, ASM International, Materials Park, OH, 1990 [4] Hart-Smith, L J., "Some Observations About Test Specimens and Structural Analysis for Fibrous Composites," Proceedings of the 9th ASTM Symposium on Composite Materials: Testing and Design, ASTM STP 1059, S P Garbo, Ed., American Society for Testing and Materials, West Conshohocken PA, 1990, pp 86-120 Copyright by ASTM Int'l (all rights reserved); Wed Dec 23 19:59:07 EST 2015 Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reproductions authorized WELSH AND ADAMS ON TRIAXIAL TESTING FACILITY 437 [5] Camponeschi, E T., Jr., "Compression Responses of Thick-Section Composite Materials," Report DTRCSME-90/60, David Taylor Research Center, Annapolis, MD, October 1990 [6] Chen, A S and Matthews, F L., "A Review of Multiaxial/Biaxial Loading Tests for Composite Materials," Composites, Vol 24, No 5, 1993, pp 395-405 [7] Jones, R F., Jr., Ed., "Report of ONR Workshop on Multiaxial Evaluation of Fibrous Composite Materials," Office of Naval Research/DTRC, November 1990, [8] Hart-Smith, L J., "Use of the Cruciform Sandwich Beam Test to Approximate the Bia• Strengths of 0~ ~ Composite Laminates," Proceedings of the 39th International SAMPE Symposium, K Drake, et al Eds., SAMPE International, April 1994, pp 3248-3259 [9] Arnold, W S., Robb, M D., and Marshall, I H., "Failure Envelopes for Notched CSM Laminates under Biaxial Loading," Composites, Vol 26, No 11, 1995, pp 739-747 [10] Mahishi, J M and Adams, D F., "Three-Dimensional Elastoplastic Stress Analysis of Unidirectional Boron/Aluminum Composites Containing Broken Fibers," Report No UWME-DR-201-107-1, Department of Mechanical Engineering, University of Wyoming, Laramie, WY, October 1982 [11] Welsh, J S and Adams, D F., "Development of a True Triaxial Testing Facility for Composite Materials," Report No UW-CMRG-R-99-102, Composite Materials Research Group, University of Wyoming, Laramie, WY, May 1999 [12] Welsh, J S and Adams, D F., "The Development of an Electromechanical Triaxial Test Facility for Composite Materials," submitted for publication in Experimental Mechanics, July 1999 [13] Welsh, J S and Adams, D F., "Biaxial and Triaxial Failure Strengths of 6061-T6 Aluminum and AS4/3501-6 Carbon/Epoxy Laminates Obtained by Testing Thickness-Tapered Cruciform Specimens," submitted for publication in the Journal of Composites Technology & Research, July 1999 [14] Youssef, Y., Laborite, S., Roy, C., and Lefebvre, D., "Validation of an Effective Flat Cruciform-Shaped Specimen to Study CFRP Composite Laminates under Biaxial Loading," Canadian Aeronautics and Space Journal, Vol 40, No 4, December 1994, pp 158-162 [15] Demmerle, S and Boehler, J P., "Optimal Design of Biaxial Tensile Cruciform Specimens," Journal of Mechanics andPhysics of Solids, Vol 41, No 1, 1993, pp 143-181 [16] 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 [17] ASM Specialty Handbook, J R Davis and Associates, Eds., Aluminum and Aluminum Alloys, ASM International, Materials Park, OH, 1993, p 72 [18] Swanson, S R and Christoforou, A P., "Response of Quasi-Isotropic Carbon/Epoxy Laminates to Biaxial Stress," Journal of Composite Materials, Vol 20, No 5, September 1986, pp 457-471 [19] Swanson, S R and Colvin, G E., Jr., "Compressive Strength of Carbon/Epoxy Laminates under Multiaxial Stress," Report No UCRL-21235, Mechanics of Composites Laboratory, Department of Mechanical Engineering, University of Utah, Salt Lake City, UT, September 1989 Copyright by ASTM Int'l (all rights reserved); Wed Dec 23 19:59:07 EST 2015 Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reproductions authorized STP1387-EB/Oct 2000 Author Index L A Lerch, Bradley A., 99 Lagoda, Tadeusz, 173 Li, B., 139 Lissenden, Cliff J., 99 Lohr, Raymond D., 355 L6wishc, Gtinther, 232 Adams, Donald F., 423 Ahmad, Jalees, 41 Amstutz, Hans, 405 Arzt, Markus, 126 Au, Y H J., 54 Bartolotta, Paul A., 26, 369 Bayyari, M., 382 Bomas, Hubert, 232 Bonacuse, Peter J., vii, 281 Bonnen, John J F., 213 Brocks, Wolfgang, 126 Buczynski, A., 82 M Macha, Ewald, 173 Mayr, Peter, 232 Mohr, Tainer, 126 N C Covey, Steven J., 26 Nelson, Drew V., 246 Newaz, Golam M., 41 Nicholas, Theodore, 41 D de Freitas, M., 139 Donne, Claudio Dalle, 405 P Park, Jinsoo, 246 P6tter, Kurt, 157, 323 E R Ellis, J R., 382 G Rees, David W A., 54 Renauld, Mark L., 266 Glinka, G., 82 I lsobe, Nobuhiro, 340 J Jenkins, M G., 13 Sakurai, Shigeo, 340 Salem, J A., 13 Sandlass, G S., 382 Santos, J L T., 139 Schram, A., 323 Socie, Darell, Suhartono, H A., 323 K T KaUuri, Sreeramesh, vii, 281 Krause, David L., 369 Topper, T H., 213, 305 Trautmann, Karl-Heinz, 405 Copyright by ASTM Int'l (all rights reserved); Wed Dec 23 19:59:07 EST 2015 439 Downloaded/printed by Copyright9 by ASTM lntcrnational www.astm.org University of Washington (University of Washington) pursuant to License Agreement No further reproductions authorized 440 AUTHORINDEX Y V Varvani-Farahani, A., 305 Yousefi, Farhad, 157, 191 W Walker, Mark A., 99 Wang, Jerry, Welsh, Jeffry S., 423 Witt, Mario, 191 Z Zamrik, Sam Y., 266 Zenner, Harold, 157, 191,323 Copyright by ASTM Int'l (all rights reserved); Wed Dec 23 19:59:07 EST 2015 Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reproductions authorized STP1387-EB/Oct 2000 Subject Index A ABAQUS, 126 Aeropropulsion materials, 382 AIS| 1015, 323 2124 aluminum alloy metal matrix composite, 54 ANSYS finite-element code, 382 Attachment methods, 382 Axial fatigue, 281 Axial-torsional load effects, Haynes 188, 99 Critical plane, 305 Critical plane approach, 173, 191 Crossland fatigue criteria, 139 Cruciform, 355,369, 382, 405 thickness-tapered, 423 Crystals, single, 13 Cumulative fatigue, 281 Cyclic hardening, 281 Cyclic loading, 126 Cyclic plasticity, 54 Cyclic testing, 369 D Biaxial fatigue effect of periodic overloads, 213 in- and out-of-phase combined bending-torsion, 246 isothermal model, 266 microcrack growth, 323 Biaxial isothermal fatigue model, 266 Biaxial loading, in-plane, 369, 405 Biaxially loaded cruciform-shaped specimen, 41 Biaxial strain ratio, 355 Biaxiai strength testing, isotropic and ansiotropic monoliths, 13 Biaxial testing, 423 in-plane, 382 C Ceramics, 13 Cobalt-base superalloy, 99, 188, 281 Cold forming, stainless steel, 26 Combined loading, 191 Complex loading, 126 Composite materials, 423 Composite strength, Compression testing, 405 Constitutive equations, 126 Crack closure, effect of periodic overloads, 213 Crack face interface, effect of periodic overloads, 213 Crack growth in- and out-of-phase combined bending-torsion, 246 rate model, 305 Cracking behavior, effect of periodic overloads, 213 Crack initiation high-cycle fatigue, 139 in- and out-of-phase combined bending-torsion, 246 Crack propagation mulfiaxial high-strain fatigue, 355 multiaxial low-cycle fatigue, 340 Damage curve approach, 281 Deep drawing, 405 Deformation, 369, 405 metal matrix composite, 54 Displacement, isotropic and ansiotropic monoliths, 13 Dissipation potential, 99 E Effective fatigue, 305 Effective intensity factor range, 305 Elastic-plastic notch tip stresses, 82 Elastic-plastic strain analysis, 82 Elasto-viscoplastic material, 126 Electromechanical testing facility, 423 Energy, 305 Extensometer, in-plane biaxial contact, 369 External pressure, 355 Failure envelope, 423 Failure loads, 26 Failure surface, 423 Failure theories, Fatigue criteria, 232 Fatigue life prediction, 139 Fatigue lifetime prediction combined tension-torsion in- and out-of-phase, 232 cumulative axial and torsional, 281 effect of periodic overloads, 213 in- and out-of-phase combined bending-torsion, 246 under multiaxial random loading, 157 welded joints, 191 Fiber metal laminate, 405 Copyright by ASTM Int'l (all rights reserved); Wed Dec 23 19:59:07 EST 2015 441 Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reproductions authorized 442 MULTIAXIALFATIGUE AND DEFORMATION Finite-element analysis in-plane biaxial loading, 382 non-linear problems, 126 Fracture, metal matrix composite, 54 G G-10 composite laminate tube, Generalized strain energy density crition, 173 Glass fiber-reinforced epoxy laminate, Grain boundaries, effect on microcrack growth, 323 H Haynes 188, 281 axial-torsional load effects, 99 High-cycle fatigue prediction, 139 Hoop compression, Hypothesis of the integral approach, 157 Inelastic deformation, under multiaxial stress, 41 Influencing parameters, 157 In-plane biaxial contact extensometer, 369 In-plane biaxial failure surfaces, 26 In-plane biaxial loading, 405 In-plane biaxial testing, 382 Integral approach, 173, 191 Internal pressure, 355 K Modeling inelastic deformation under multiaxial stresses, 41 microcrack growth, 323 Monoliths, isotropic and ansiotropic, 13 Multiaxial fatigue, 213 in- and out-of-phase combined bending-torsion, 246 in-plane biaxial contact extensometer, 369 Multiaxial fatigue criteria, 157 energy-based, 173 Multiaxial fatigue life model, 305 Multiaxial high-strain fatigue, 355 Multiaxial loading, 99 G-10 composite laminate tube, high-cycle fatigue prediction, 139 weld joints, 191 Multiaxial low-cycle fatigue, microcrack growth modes and propagation rate, 340 Multiaxial strength, isotropic and ansiotropic monoliths, 13 Multiaxial stress, inelastic deformation, 41 Multiaxial stress-strain notch analysis, 82 Newton algorithm, 126 Nickel aluminide, 13 Non-linear problems, 126 Nonproportional loading, high-cycle fatigue prediction, 139 Nonradial loading metal matrix composite, 54 silicon carbide, 54 Notched specimen, in- and out-of-phase combined bending-torsion, 246 Numerical algorithm, 126 Numerical method, 139 Kinematic hardening, 54 O Laminate tube, strength, Linear damage rule, 281 Load-type sequencing, 281 Low-cycle fatigue, 355 M Mean stress effect, 305 Metal matrix composite characterization methods, 41 deformation and fracture, 54 Microcrack growth modes, 340 modelling, 323 propagation rate, 340 Modal control, 355 Off-axis tension tests, 41 Optimization techniques, 382 Out-of-phase loading effects, 139 Overloads, periodic, 213 Phase difference, 157 Phase factors, 266 Plastic limit load, 405 Prediction software, 191 Proportional loading, 213 Prototype fixturing, 382 R Random load, 157 nonproportional, 173 Copyright by ASTM Int'l (all rights reserved); Wed Dec 23 19:59:07 EST 2015 Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reproductions authorized SUBJECT INDEX Ratcheting, 54 Residual stress, 232 Reusable fixturing, 382 Riveted joint, 405 Sequence effects, 213 Shear energy, 305 Shear plane, multiaxial low-cycle fatigue, 340 Shear stress amplitude, effective, 139 Silicon carbide particulate, 54 Sines fatigue criteria, 139 Stainless steel, 26 austenitic, biaxial isothermal fatigue model, 266 Steel biaxial fatigue, 213 combined tension-torsion in- and out-of-phase, 232 effect of periodic overloads, 213 microcrack growth modes and propagation rate, 340 Strain isotropic and ansiotropic monoliths, 13 principal, 340 Strain hardening, 305 Strain measurement, 369 Strain paths, in- and out-of-phase, 305 Strain rates, equivalent, 99 Strain rate vectors, 99 Stress effect of periodic overloads, 213 equivalent, 26, 99 in- and out-of-phase combined bending-torsion, 246 isotropic and ansiotropic monoliths, 13 superimposed mean, 157 Stress intensity factor, 405 Stress relaxation, 99 Superalloy, 99 System design, multiaxial high-strain fatigue, 355 443 Temperature, elevated, 369 Tension, combined in- and out-of-phase, 232 Tension-torsion loading, 213 Thermomechanical fatigue, 355 Thermomechanical loading, 266 Thin-walled tube, 355 Torsion, 355 combined in- and out-of-phase, 232 in- and out-of-phase, 246 Torsional fatigue, 281 Torsion stress, Triaxiality factor, 266 Triaxial testing facility, 423 Tungsten carbide, 13 U Unidirectional fiber-reinforced metal matrix composites, 41 V Variable-amplitude tests, 157 Viscoplasticity, 99 models, potential-based, 99 W Weakest-link model, 232 Weld joints, under multiaxial loading, 191 Y Yield surface, 405 Z-parameter, 266 Copyright by ASTM Int'l (all rights reserved); Wed Dec 23 19:59:07 EST 2015 Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reproductions authorized ISBN 0-8031-2865-7