Astm stp 1387 2000

Generalization of Energy-Based Multiaxial Fatigue Criteria to Random Fatigue Strength of Welded Joints Under Multiaxial Loading: Comparison Between FATIGUE LIFE PREDICTION UNDER SPECIFIC

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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

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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., 1960-

TA418.38.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 October 2000

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Foreword

sented 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 Fa-

tigue Crack Formation Sreeramesh Kalluri, Ohio Aerospace Institute, NASA Glenn Research Cen-

ter at Lewis Field, and Peter J Bonacuse, Vehicle Technology Directorate, U.S Army Research Lab-

oratory, NASA Glenn Research Center at Lewis Field, presided as symposium co-chairmen and both

were editors of this publication

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Contents

MULTIAXIAL STRENGTH OF MATERIALS

Keynote Paper: Strength of a G-10 Composite Laminate Tube Under Multiaxial

Biaxial Strength Testing of Isotropic and Anisotropic Monoliths J A SALE~ AND

Deformation and Fracture of a Particulate MMC Under Nonradial Combined

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 82

Axial-Torsional Load Effects of Haynes 188 at 650 ~ C c J LlSSENDEN, M a WALKER,

A Newton Algorithm for Solving Non-Linear Problems in Mechanics of Structures

FATIGUE LIFE PREDICTION UNDER GENERIC MULTIAXIAL LOADS

A Numerical Approach for High-Cycle Fatigue Life Prediction with Multiaxial

Experiences with Lifetime Prediction Under Multlaxial Random Loading K POTTER, F

Generalization of Energy-Based Multiaxial Fatigue Criteria to Random

Fatigue Strength of Welded Joints Under Multiaxial Loading: Comparison Between

FATIGUE LIFE PREDICTION UNDER SPECIFIC MULT1AXIAL LOADS

The Effect of Periodic Overloads on Biaxial Fatigue of Normalized SAE 1045

Fatigue of the Quenched and Tempered Steel 42CrMo4 (SAE 4140) Under Combined

In- and Out-of-Phase Tension and Torsion -a LOVaSCH, ~ BOMAS, AND

In-Phase and Out-of-Phase Combined Bendlng-Torsion Fatigue of a Notched

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vi CONTENTS

The Application of a Biaxial Isothermal Fatigue Model to Thermomechanical Loading

Cumulative Axial and Torsional Fatigue: An Investigation of Load-Type Sequencing

MULTIAXIAL FATIGUE LIFE AND CRACK GROWTH ESTIMATION

A New Multiaxial Fatigue Life and Crack Growth Rate Model for Various In-Phase

Modeling of Short Crack Growth Under Biaxial Fatigue: Comparison Between

Micro-Crack Growth Modes and Their Propagation Rate Under Multiaxial Low-Cycle

MULTIAXIAL EXPERIMENTAL TECHNIQUES

Keynote Paper: System Design for Multiaxial High-Strain Fatigue

Design of Specimens and Reusable Fixturing for Testing Advanced Aeropropulsion

Materials Under In-Plane Biaxial Loading J R ELLIS, G S SANDLASS, AND

Cruciform Specimens for In-Plane Biaxiai Fracture, Deformation, and Fatigue

Development of a True Trlaxlal Testing Facility for Composite Materials J s WELSH

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Overview

Engineering materials are subjected to multiaxial loading conditions routinely in aeronautical, as- tronautical, automotive, chemical, power generation, petroleum, and transportation industries The extensive use of engineering materials over such a wide range of applications has generated extraor- dinary 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 de-

formation 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 De- formation and Fatigue Crack Formation The symposium was held during 19-20 May 1999 in Seat- tle, 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 Spe- cific 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 encompass- ing 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 characteriza- tion 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

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viii OVERVIEW

third paper deals with in-plane biaxial testing of cruciform specimens manufactured from thin, cold- rolled, 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 to- tal 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 wave- forms, 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

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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 se- quences 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

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X OVERVIEW

Multiaxial Experimental Techniques

State-of-the-art experimental methods and novel apparati are necessary to generate multiaxial de-

formation and fatigue data that are necessary to develop and verify both constitutive models for de-

scribing the flow behavior of materials and fatigue life estimation models Five papers in this publi-

cation 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 load-

ing 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 ad-

vanced aerospace materials is the topic of investigation for the third paper Feasibility of a fixture ar-

rangement 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 uni-

form 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-con-

trolled, electromechanical test system for characterizing mechanical behavior of composite materials

under biaxial and triaxial loading conditions Verification of the test system with uniaxial and biax-

ial 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 consis-

tency 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 contri-

butions 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

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Multiaxial Strength of Materials

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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 Mul- tiaxial 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,

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 re- placing 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 Fail- ure 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 view- point, composite materials cannot be considered as homogeneous anisotropic materials Fail- ure 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

3

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4 MULTIAXIAL FATIGUE 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 non-

linear terms For an in-plane loading ~x, ~y and ~- this criterion becomes

FIt" x + F2o'y + F6T + (Fll O'2 + F22o-y 2 + 2F12o'xO'y + F 6 6 ~ ' 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 ex- trapolated 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 simul-

taneously applied in-plane stresses

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SOCIE AND WANG ON MULTIAXIAL LOADING

TABLE 1 Loading combinations

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

R e s u l t s a n d 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 three- axis 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 Fig- ure 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 out- of-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

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6 MULTIAXlAL FATIGUE AND DEFORMATION

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

I n - p l a n e s h e a r Out-of-plane shear

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

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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

composite tubes under external pressure The critical compressive stress, o'er, of tubular specimens is given by

F(hol ' + (R, 1"2

~cr 0.916Ew I_\Ri) Kr~ho)J

Kr = 4.77yEwRi

(2)

layer, Ri is the inner radius, and y is the specific fracture energy according to Griffith The weak layer

about 18% higher than the experimental data Experimental evidence of delamination is shown in Fig

8 of Ref 10

It is worth noting that compression and tension-compression tests of a flat plate G-10 laminate

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 predom- inantly 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-

Matrix cracking Interface debonding \

FIG 3 Failure envelope for combined tension~compression and torsion in warp direction

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Matrix cracking Interface debonding

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 inter-

action between axial compression and torsion shown in Fig 4 No interaction between hoop com-

pression and torsion was observed in the test results shown in Fig 3 No interaction was observed be-

tween 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 sym-

metric about the O'fill-O'war p 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 frac-

ture surface but there is no observed interaction between the failure mechanisms Under SEM exam-

ination, 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 fail-

ure 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 delami-

nation followed by out-of-plane kinking as a result of the hoop compressive stress or by matrix crack-

ing 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 ob-

servations 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

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SOClE AND WANG ON MULTIAXIAL LOADING 9

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 G- l0 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

(3)

~

FIG 6 Rotation of fill fibers

Trang 19

FIG 7 Failure envelope

where ~-y is the shear yield strength of the matrix and t h is the initial misalignment angle of the fibers Equation 3 suggests that the shear stress will have a large influence on the compressive strength This

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 indi- cates 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

Trang 20

SOCIE AND WANG ON MULTIAXIAL LOADING

TABLE 2 Experimental results for combined loading

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 con-

clude 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 uni-

axial 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 inter-

action between the two stress systems at high loads Similarly, the test data for compression-com-

pression 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 com-

pressive 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 out-

of-plane shear stresses Results of these three tests are plotted in Fig 4 with the X symbols The fail-

ure 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 sim-

ple maximum stress theory provides a reasonable fit to the experimental data for combined tension-

tension 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

Trang 21

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 Delam- ination 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 m a x i m u m 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

[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 Com- posite Materials, Vol 5, 1971, pp 58-80

[5] Jiang, Z and Tennyson, R C., "Closure of the Cubic Tensor Polynomial Failure Surface," Journal of Com- posite Materials, Vol 23, 1989, pp 208-231

[6] Ardic, E S., Anlas, G., and Eraslanoglu, G., "Failure Prediction for Laminated Composites Under Multi- axial 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 Lami- nates," 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 Lam-

inates" 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 Fi-

brous Composites," ASTM STP 521, 1973, pp 98-132

[14] Browning, C E and Schwartz, H S., "Delamination Resistance Composite Concepts," ASTM STP 893,

Trang 22

J A S a l e m I a n d M G J e n k i n s 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 Con-

shohocken, 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 single- crystal NiA1 plates were considered, and the necessary strain functions for anisotropic plates were for- mulated 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 Series constant in anisotropic, plate displacement solution

Bn Series constant in anisotropic, plate displacement solution

bq Reduced elastic stiffness

Cz Constants in the anisotropic displacement stress solution

D o Flexural rigidities

D* Effective flexural rigidity of an anisotropic plate

ep Measured major principal strain component

eq Measured minor principal strain component

~p Measured principal strain uncorrected for transverse sensitivity

~z Measured strain uncorrected for transverse sensitivity; i = 1,2,3

ki Reduced flexnral rigidity

Rs Radius of support ring

RD Radius of disk test specimen

S o Elastic compliance

SDxz Standard deviation of x~ variable

v Poisson's ratio

o-q Stress component

O-p Measured major principal stress

O-q 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

13

Trang 23

FIG 1 Schematic of typical testing configurations used to generate biaxial tensile stresses in

o00 Tangential stress

~rO Shear stress

o's Correction term for effect of lateral stresses on plate deflection

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 con-

ventional 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 mod-

els 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

Trang 24

S A L E M A N D J E N K I N S O N B I A X I A L S T R E N G T H T E S T I N G 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 config-

uration

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 excel-

lent 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 eliminat- ing 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

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

Trang 25

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 mis- alignments 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]

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 empir- ical 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%)

Trang 26

SALEM AND JENKINS ON BIAXlAL STRENGTH TESTING 17

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

P

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:

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

where 0 is the angle counterclockwise from the x-axis

Trang 27

18 MULTIAXIAL FATIGUE AND DEFORMATION

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

02w - PR~ [ ~=2 OY 2 t3 (A.k 2 cosh 2nct' cos 2n13' + B.k~ cosh 2nd' cos 2nff')

rE + 2C5] + (6C3 + C2 - (6C3 - C2)cos 2fl) Rs z

Trang 28

S A L E M A N D J E N K I N S O N B I A X I A L S T R E N G T H T E S T I N G 19

TABLE 1 Constants ( XlO -6 rn2/MN) for NiAI and graphite/epoxy plates of unit thickness and radius sub-

jected to a unit lateral pressure

NiAI: $22 = SI1 = 1.0428, Sa2 = -0.421, $66 = 0.892 (• 10 -5 m2/MN) [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

Q

V

1.2 1.0 0.8 0.6 0.4 0.2 0.0

Tangential Stress, r/R= 0.2 Radial Stress, r / R = =0~2

Trang 29

20 MULTIAXlAL FATIGUE AND DEFORMATION

apparent at the plate edges where the stresses vary with angular position by - 4 5 % 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 in- serted, 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 Ra- tio 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 A S T M C 1259-94

was 209.9 • 0.5 GPa The elastic modulus and Poisson's ratio of the WC material were measured

by using A S T M 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, ex- hibited 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 as- sessed 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 ap-

plying 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

_ E N / S D ~ + ~2SD2~

SD~p 1 - v 2

Trang 30

SALEM AND JENKINS ON BIAXIAL STRENGTH TESTING 21

E SD,~ - 1 - - v 2 X'/ ~SD2" + SD2~

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 o f a lubricated boundary Because the 95% confidence bands o f 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

Percent of

Trang 31

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

ter WC plate supported on a 45.4-ram-diameter steel ring and subjected to 8.3 MPa uniform pressure

Radial Position

Percent of

faces 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 s t r e s s e s w e r e m e a s u r e d at the d i s k c e n t e r a n d at 0 4 9 R s

b y r e p e a t e d l y i n s e r t i n g a n d r e m o v i n g a n u n l u b r i c a t e d , 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 t h e fixture T h e s t r e s s e s g e n e r a t e d b y c l a m p i n g v a r i e d w i t h o r i e n t a t i o n a n d radial position

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

indicate the 95% confidence bands: (left) steel disk on a copper support, a n d (right) tungsten carbide

disk on a steel support

Trang 32

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 o f 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% o f the measured stresses at the plate center and within 7% at radii less than 50% o f 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

Percent of

2 Mean • one standard deviation

Trang 33

FIG 6 Measured and theoretical strains at failure for {001} NiA1 disk test specimens The mea-

sured 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 Han-

dlinger, 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

Trang 34

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,

[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, Mea-

surements Group, Raleigh, NC

[14] "Strain Gage Rosettes Selection, Application and Data Reduction," Measurements Group Tech Note TN-

515, 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 Com-

pound NiAI," International Materials Reviews, Vol 38, No 4, 1993, pp 193-232

Trang 35

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 Cold- Rolled 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 mi- crostructurally 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 c o m m o n 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 A S T M 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 m u c h 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

26

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COVEY AND BARTOLO-I-I'A ON STAINLESS STEEL 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 114.3- mm-long test section The extensometer gage length was 50.9 mm The specimens were tested in displacement control at a rate of 0.5 m m / m i n up to 0.75 mm displacement and then at a faster displacement rate of 5 m m / 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, elec- tropolished)

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TABLE 1 Uniaxial tensile properties of 304 stainless steel sheet

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

E x p e r i m e n t a l 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 m u c h of the gage section as possible, without a large stress concentration within the c o m e r 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

11 mm 5.5 mm "-4 ~ / - - F ~ d i u s

f~ 150 m m

3 0 0 m m

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COVEY AND BARTOLOTTA ON STAINLESS STEEL 29

Demmerle and Boehler [3] give an excellent discussion of these methods and the resulting optimized geometry These reduced section specimens are very expensive and are not applicable for testing of thin sheets However, a uniform stress state free of stress risers is still of concern and is discussed later

Equipment and Test Details

Most biaxial material tests are performed on tubes in an axial/torsion test rig However, actual sheet metal geometries (i.e., thin plates) prohibit such testing NASA Glenn Research Center, in Cleveland, Ohio, has two in-plane biaxial test rigs for this type of testing These rigs are computer- controlled servohydraulic test frames with hydraulic grips Figure 3 shows the grip configuration with

a strain-gaged sample used to verify alignment Even though they have a large force capacity of 500

kN (110 kip), testing of these thin plates was successfully performed Since the grip wedges would not allow testing of such thin sheets, hardened steel shims were glued on each side of the cruciform's arms Then, cardboard was glued onto the shims to provide enough lateral stiffness to allow mount- ing the specimen into the test machine Alignment of the load frame and grips was performed using

a precision steel specimen carefully equipped with 44 strain gages: eleven in each direction on each side Alignment was considered adequate when the strain levels on both sides and at each arm were within 5% of the nominal applied strains for equal X and Y loading with no indications of significant localized bending strains

Due to the thinness and the relative geometry of the specimen, all tests were performed in load control In displacement control, the risk for off-axis and unequal loading is significantly high thereby compromising the stress uniformity in the specimen test section Furthermore, since the 304 stainless steel sheets have a relatively low ductility (<5%), the affects of load control mode on the materials yielding behavior is minimized It is the authors' opinion that the benefits of load control for these tests outweigh its disadvantages The X and Y loads were controlled to provide a constant effective

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stress rate of 3.44 MPa/s (0.5 ksi/s) until the sample failed Only tension-tension (i.e., first quadrant

of biaxial stress space) testing was performed because of this thin material's inability to support com-

pressive stresses Fourteen tests were planned with two each at seven different 0-angles (0 ~ (uniaxial

X, transverse), 18, 36, 45, 54, 72, and 90 ~ (uniaxial Y, longitudinal)) For this study, "0-angle" refers

to the orientation of the maximum principal stress plane with respect to the transverse rolling direc-

tion Test control software and sample damage evaluation techniques are discussed below

Control Software

Data acquisition and waveform generation were performed via a program written using an object-

oriented control software (Labview) This software allows design of virtual instruments using graph-

ical icons that can be wired together Each icon serves a unique purpose not unlike a subroutine of a

lower level programming language The control software ramped up load on both the X- and Y-axis

at any selected 0-angle at any effective stress rate to any maximum effective stress The control soft-

ware requires as input: full-scale load and strain levels, the specimen's cross-sectional area, and the

expected elastic modulus and Poisson's ratio During the test, the control software provides a real-

time effective stress versus effective strain curve and large digital readouts of effective stress and X

and Y load Stress levels in the X and Y directions were determined independently from load via load

cells and strain via strain gages The material elastic modulus was determined in the X and Y direc-

tions using stress via load and strain via strain gage

End of test could be determined via change of elastic modulus by a certain percentage, maximum

effective stress, magnitude of inelastic strain, or increase in acoustic emission signal At any time dur-

ing the test, the user could stop, hold/pause, or return to zero load using three large buttons on the

front panel Once the software received the maximum effective stress or other test-end signal, the pro-

gram returned the load to zero in ten seconds All time, load, strain, and displacement data are writ-

ten in a spreadsheet-usable ASCII format file name selected by the user

The effective stress (and hence effective stress rate) and effective strain are defined by a von Mises

equivalent approach as used by Shiratori and Ikegami [1] with

and

2

where ex and e r were measured using strain gages The stresses, o'x and o- r, were calculated using a

specimen area defined by Kreibig and Schindler's work The areas were calibrated for each specimen

by using strain measurements (at a load level well within the material's elastic region), calculating

the stress using equations of elasticity and compare them to the calculated stresses using Kreibig and

Schindler solution

Results and Discussion

Failure Surface

The first quadrant (tension-tension) failure surface was successfully obtained and is given in Fig

4 Note that failure was defined as complete specimen fracture The data are also given in Table 2

The surface is essentially elliptical with a major axis in the transverse direction The cruciform

specimen geometry was justified because the specimen failed at an applied stress within 10% of the

uniaxial strength data when loaded along that axis only However, when loaded only in the longitu-

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COVEY AND BARTOLO'I-[A ON STAINLESS STEEL 31

Xr:nt~Ss (MPa) ~ Uniaxlal Transverse

UTS = 1410 MPa Strain-to-Failure = 4.7%

FIG 4 In-plane biaxial failure surface o f fully cold-rolled 304 stainless steel sheets 0,1 mm (0.004 in,) thick

dinal direction, the cruciform specimen failed at an applied stress 20% lower than that obtained from uniaxial coupons It is likely that the higher uniaxial strain-to-failure in the transverse direction (4.73%) allowed the specimen comer stress concentration factors to reduce to one via plastic deformation while the more brittle rolling direction (strain-to-failure = 2.48%) maintained a stress concentration factor greater than one until failure

Failed samples were clearly indicative of their 0-angle Figure 5 shows failed specimens for 18 ~ and 45 ~ 0-angle tests Note that the failure planes match the loading 0-angle (i.e., tests with an 18 ~ [from the transverse axis] 0-angle had an 18 ~ [from the transverse axis] failure) Figure 6 shows micrographs of the same two specimens at two magnifications The fracture planes are transgranu- lar (i.e., through the grains) with the plane coinciding with the 0-angle The observations shown in

TABLE 2 O-Angle and associated effective stress at failure

Tiêu đề	Multiaxial Fatigue and Deformation: Testing and Prediction
Tác giả	Sreeramesh Kalluri, Peter J. Bonacuse
Trường học	University of Washington
Thể loại	Bài báo
Năm xuất bản	2000
Thành phố	West Conshohocken

Định dạng
Số trang	444
Dung lượng	11,54 MB