Astm stp 1359 1999

The ASTM E08-sponsored Symposium on Mixed-Mode Crack Behavior was held in At- lanta, GA on May 6-7, 1998, and gave rise to this STR The conference was international and balanced in scope

STP 1359

Mixed-Mode Crack Behavior

K J Miller and D L McDowell, Editors

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Includes bibliographical references and index

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1 Fracture mechanics Mathematical models Congresses

2 Materials Fatigue Mathematical models Congresses

I Miller, K J (Keith John) I1 McDowell, David L., 1956-

III Symposium on Mixed-Mode Crack Behavior

(1998 : Atlanta, Ga.) IV Series: ASTM special technical

publication ; 1359)

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Printed in Philadelphia November 1999

Foreword

The Symposium on Mixed-Mode Crack Behavior was held 6-7 May 1998 in Atlanta,

GA The symposium was sponsored by ASTM Committee E8 on Fatigue and Fracture and its Subcommittee E08.01 on Research and Education

The symposium was chaired by Keith J Miller, of the University of Sheffield, and David

L McDowell, of the Georgia Institute of Technology These men also served as editors for this resulting publication

Overview vii

C R A C K EXTENSION IN DUCTILE M E T A L S U N D E R M I X E D - M O D E LOADING

Evaluation of the Effects of Mixed Mode I-II Loading to Elastic-Plastic

Ductile Fracture of Metallic Materials A LAUKKANEN, K WALLIN AND

R RINTIMAA

The Crack Tip Displacement Vector Approach to Mixed-Mode Fracture

C DALLE DONNE

A Simple Theory for Describing the Transition Between Tensile and Shear

Mechanisms in Mode I, II, III, and Mixed-Mode Fracture Y.-J CHAO

AND X.-K ZHU

Further Studies on T* Integral for Curved Crack Growth e w LAM,

A S KOBAYASHI~ S N ATLURI AND P W TAN

Recommendations for the Determination of Valid Mode II Fracture

Toughnesses K n c - - w m n s E AND J F KnLTHOF~

A CTOD-Based Mixed-Mode Fracture Criterion F MA, X DENG,

M A SUTTON AND J C NEWMAN, JR

A Software Framework for Two-Dimensional Mixed Mode-I/II Elastic-Plastic

Fracture M A JAMES AND D SWENSON

Mixed-Mode Fracture Behavior of Silica Particulate Filled Epoxide Resin

T KAWAKAMI

Mixed-Mode Fracture Mechanics Parameters of Elliptical Interface Cracks in

Anisotropic Bimaterials Y XUE AND J QU

129

143

Microtexture, Asperities and Crack Deflection in AI-Li 2090 T8E41m

J D HAASE~ A G U V E N I L I R , J R W I T T , M A L A N G O Y , AND S R STOCK

Micromechanical Modeling of Mixed-Mode Crack Growth in Ceramic

Composites J Z H A I AND M Z H O U

160

174

F A T I G U E C R A C K G R O W T H U N D E R M I X E D - M O D E L O A D I N G

Polycrystal Orientation Effects on Microslip and Mixed-Mode Behavior of

Microstructurally Small Cracks v BENNETT AND D L McDOWELL

Some Observations on Mixed-Mode Fatigue Behavior of Polycrystalline

Metals K J MILLER, M W BROWN, AND J, R YATES

A Fractographic Study of Load-Sequence-Induced Mixed-Mode Fatigue Crack

Growth in an AI-Cu Alloy N E A S H B A U G H , W J PORTER, R, V P R A K A S H

AND R SUNDER

Mixed-Mode Static and Fatigue Crack Growth in Central Notched and

Compact Tension Shear Specimens v N SHLYANNIKOV

The Propagation of a Circumferential Fatigue Crack in Medium-Carbon Steel

Bars Under Combined Torsional and Axial Loadings K TANAKA,

Y A K I N I W A AND H YU

Near-Threshold Crack Growth Behavior of a Single Crystal NilBase

Superalloy Subjected to Mixed-Mode Loading R JOHN, D DELUCA,

Engineering components and structures necessarily involve the introduction of defects, including holes, grooves, welds, and joints The materials from which they are made have smaller imperfections, such as surface scratches, inclusions, precipitates, and grain bounda- ries All of these defects range in size from sub-microns to many millimeters Engineers who design such components or structures must be fully cognizant of the level and orientation

of the applied loading (whether static or dynamic, of constant or variable amplitude, or proportional or nonproportional) and the density, size, shape, and orientation of the defects Under combined loading, or even remote Mode I loading, effective strain or strain energy density approaches can lead to dangerously nonconservative predictions of fatigue life, and similarly the opening mode stress-intensity factor, K~, is seldom appropriate for describing local mixed-mode crack growth

For mixed-mode conditions, the crack growth direction does not follow a universal tra- jectory along a path in the orthogonal mixed-mode KI-KH-KHI space Under cyclic loading,

a surface in this space can be defined as representing an envelope of constant crack growth rate that tends towards zero for the threshold state In general, this envelope depends inti- mately on the crack driving and resisting forces The application of linear elastic fracture mechanics (LEFM), elastic-plastic fracture mechanics (EPFM), or microstructural fracture mechanics (MFM) is dictated by the scale of plasticity or material heterogeneity relative to the crack length, component dimension, and damage process zone All of these features come into play during mixed-mode loading and mixed-mode crack growth

ASTM special technical publications (STPs) have a rich history of considering complex aspects of fracture such as effects of mixed-mode loading This subject has been couched under various labels such as multiaxial fatigue, 3-D crack growth, and microstmcturally sensitive crack growth, among others From previous symposia and related STPs, we have gained understanding of the physics of these phenomena and have developed appropriate experimental techniques, yet our understanding is far from complete There is still a struggle

to identify the role of material resistance in establishing the growth path for the mixed-mode propagation of cracks Consequently, industrial practice, codes, and standards have not been updated in the face of this uncertainty

The ASTM E08-sponsored Symposium on Mixed-Mode Crack Behavior was held in At- lanta, GA on May 6-7, 1998, and gave rise to this STR The conference was international and balanced in scope, as witnessed by the presentation of over 20 papers addressing the following topics:

9 Elastic-Plastic Fracture

9 Three-Dimensional Cracks

9 Anisotropic Fracture and Applications

9 Fracture of Composite Materials

9 Mixed-Mode Fracture Toughness

9 Mixed-Mode Fatigue Crack Growth

9 Experimental Studies in Mixed-Mode Fatigue and Fracture

In practice, cracks that are confined to follow weak paths of material resistance along weld fusion lines or relatively weak material orientations due to process history, composite

viii MIXED-MODE CRACK BEHAVIOR

reinforcement, or interfaces will often be subject to local mixed-mode crack driving forces

One of the more difficult challenges facing treatment of mixed-mode effects is the difference

between global (apparent) mode-mixity and local (crack tip) mode-mixity due to micro-

structure heterogeneity, for example, at the tip of small fatigue cracks or within damage

process zones at the tips of longer cracks Although a number of technologies have already

benefitted from an enhanced understanding of mixed-mode fatigue and fracture, much design

today is performed assuming local Mode I conditions even when this assumption is not

applicable Briefly stated, too much focus is placed on the crack driving force and too little

on micromechanisms of damage that lead to crack advance

This STP is intended to contribute to a deeper understanding of these issues Among the

authors of this volume are some of the leaders in the disparate and far-reaching field of

mixed-mode fracture Consequently the papers contained herein span the range of experi-

mental, computational/theoretical, and physical approaches to advance our understanding of

the various aspects of mixed-mode fracture problems, and are organized into several cate-

gories The first set of papers deals with experimental observations and modeling of crack

extension in ductile metals under mixed-mode loading conditions The paper by Laukkanen

and colleagues is selected to lead off this STP because it offers a fairly comprehensive

evaluation of the effects of mixed Mode I-II loading on elastic-plastic fracture of metals and

provides experimental data for a range of alloys as well as taking an, in-depth look at failure

mechanisms ahead of the crack This paper was recognized as the outstanding presentation

at the symposium The paper by Dalle Donne approaches the same class of problems using

the crack tip opening displacementapproach Ma and colleagues apply computational meth-

ods to predict the crack growth path for mixed Mode I-II behavior of 2024-T3 A1 Chao and

Zhu develop an engineering approach to problems of mixed-mode growth to consider ex-

perimental observations of crack path in terms of a plastic fracture criterion based on crack

tip fields Lam et al employ the T* integral to model crack growth by computational means

along curved paths Hiese and Kalthoff present a study that considers the determination of

valid mode II fracture toughness, an essential parameter in any practical mixed-mode law

The work of Deng et al suggests that a critical level of the generalized crack tip opening

displacement (CTOD) at a fixed distance behind the crack tip dictates the onset of crack

extension, while the direction of the crack path is determined by maximizing either the

opening or shearing component of the CTOD Since the crack path is a prior unknown in

complex components, computational fracture approaches must be flexible and adaptive, per-

mitting re-meshing to account for the evolution of the crack; James and Swenson discuss

recent developments in two-dimensional modeling of mixed Mode I-II elastic-plastic crack

growth using boundary element and re-meshing techniques

The next set of papers considers the growth of cracks in materials with a strongly defined

mesostructure that controls mixed-mode fracture Kishimoto and colleagues provide a de-

tailed experimental study of the mixed-mode fracture behavior of silica particulate-filled

epoxide resin that is used in electronic packaging applications The driving force for cracks

between layers of material in composites or lying within bimaterial interfaces between ani-

sotropic materials is of fundamental importance to fracture analysis; in this volume Xue and

Qu present the first analytical solution ever obtained for the mixed-mode stress intensity

factors and crack opening displacement fields for an arbitrary elliptical interface crack be-

tween two distinct, anisotropic, linear-elastic half spaces In an experimental study employing

computed microtomography to quantify closure of deflected fatigue cracks in highly aniso-

tropic A1-Li 2090, Stock presents a means to study highly complex crack opening and sliding

fields in anisotropic materials having, in this case, mesostructure and mesotexture Zhai and

Zhou employ a novel local mixed-mode interface separation law for all interfaces (and

elements) within a finite element mesh to predict crack paths in ceramic composites under

dynamic loading conditions as a function of interface strength and phase properties; this

approach is not of the classical singularity type, but rather can be categorized as a cohesive

zone approach

The final set of papers deals primarily with various aspects of fatigue crack growth under

mixed-mode loading conditions Bennett and McDowell conduct computational studies using

two-dimensional crystal plasticity to shed light on the influence of intergranular interactions

on driving forces for the formation and early growth of fatigue cracks in polycrystals, as

well as discrete orientation effects of neighboring grains and free surface influences on the

crack tip displacements for microstructurally small surface cracks in polycrystals The paper

by Miller and colleagues raises a number of stimulating issues for further consideration, it

also highlights the classification of crack growth behavior as belonging principally to either

normal stress- or shear stress-dominated categories Ashbaugh et al report on a detailed

fractographic study of crack growth behavior under variable amplitude, mixed-mode loading

conditions Shlyannikov provides experimental data regarding mixed crack growth in cdnter

cracked and compact tension shear specimens Tanaka and associates report on their axial-

torsional studies of propagating and nonpropagating fatigue cracks in notched steel bars,

with emphasis on the dependence of the fatigue limit on notch root radius and mixity of

applied loading John and colleagues consider the fatigue threshold for a single crystal Ni-

Base superalloy under mixed-mode loading, a problem of great relevance to fatigue limits

in the design of gas turbine engine components, for example

One of the important points of convergence of this Symposium was the realization that,

for a large number of mixed-mode crack growth problems of which we are aware, there are

two fundamentally distinct classes of growth: maximum principal stress-dominated and

shear-dominated This is true regardless of whether we consider static or cyclic loading

conditions This observation is likely to enable the development of certain very robust, sim-

plified, material-dependent design approaches for cracks in components and structures An-

other point, emphasized in several papers, is the intimate connection of the crack tip dis-

placement concept to mixed-mode elastic-plastic fracture mad fatigue processes

As coeditors of this publication, we are greatly indebted to the host of international re-

viewers who are essential when bringing a publication of this nature to press We can claim

that this volume follows in the proud tradition of the thorough peer-review system that is

characteristic of ASTM STPs in fracture and fatigue We trust that this STP will give valuable

insight into various aspects of mixed-mode fracture, as well as provide substantial inroads

to resolving some characteristic, yet fundamental mixed-mode behavioral problems fre-

quently observed in engineering materials, components, and structures

Keith J Miller

SIRIUS The University of Sheffield Sheffield, UK

Symposium cochairman and coeditor

Crack Extension in Ductile Metals Under

Mixed-Mode Loading

Evaluation of the Effects of Mixed Mode I-II Loading on Elastic-Plastic Ductile Fracture

of Metallic Materials

REFERENCE: Laukkanen, A., Wallin, K., and Rintamaa, R., "Evaluation of the Effects of

Mixed Mode I-II Loading on Elastic-Plastic Ductile Fracture of Metallic Materials,"

Mixed-Mode Crack Behavior, ASTM STP 1359, K J Miller and D L McDowell, Eds., Amer-

ican Society for Testing and Materials, West Conshohocken, PA, 1999, pp 3-20

ABSTRACT: In order to evaluate the mixed-mode fracture behavior of elastic-plastic metallic

materials, experimental tests and numerical calculations were carried out Since the transition

of fracture toughness between opening and in-plane shear modes with ductile materials is a

question of controversy, single-edge notched bend (SENB) specimens were subjected to asym-

metric four-point bending (ASFPB) to provide various mode portions using four materials:

A533B pressure vessel steel, F82H ferritic stainless steel, sensitized AISI 304 austenitic stain-

less steel, and CuA125 copper alloy Fracture resistance curves were determined and fracto-

graphical studies performed Numerical studies focused on determining the J-integral and stress

intensity factor (StF) solutions for the experimental program and the Gurson-Tvergaard con-

stitutive model was used to simulate continuum features of the fracture process The results

demonstrate that Mode II fracture toughness of ductile metallic materials can be significantly

lower than Mode I fracture toughness Studies of the micromechanical aspects of fracture

demonstrate the factors and variables responsible for the behavior noted in this investigation

KEYWORDS: ductile fracture, mixed-mode, Mode I, Mode II, fracture toughness, fractog-

raphy, shear fracture, J-integral, Gurson-Tvergaard model

Mixed-mode fracture research has traditionally dealt with brittle materials behaving in a

Mode II fracture toughness is usually close to or larger than the Mode I fracture toughness, indicating that the Mode I fracture toughness is a conservative estimate of the fracture re- sistance of the material W h e n considering ductile materials and their mixed-mode fracture toughness, the results are not as unequivocal Different researchers with different materials

as well as experimental setups have obtained opposite and controversial results Some re-

other researchers have obtained inverse results suggesting that in Mode II fracture toughness

is lower than in Mode I [6-7] The area of elastic-plastic mixed-mode fracture toughness suffers also from lack of studies, meaning that relatively few studies have been published One reason for this is the difficulty associated with controlling nonlinear elastic-plastic two- dimensional situations, both in numerical simulations and in experimental work

The basic idea and background for the question why mixed-mode fracture and fracture toughness can not be taken as conservative with respect to Mode I stems from the basic

1 Research scientist, research professor, and research manager, respectively, VTT Manufactaxring Tech- nology, P O Box 1704, 02044 VTT, Finland

4 MIXED-MODE CRACK BEHAVIOR

thinking in Mode I, which typically neglects differences in fracture micromechanisms Since

it appears that the Mode II brittle fracture toughness is higher than the Mode I toughness,

we can think that Mode II ductile fracture toughness would be higher than Mode I, with the

same simple analogy This reasoning and other reasoning like it, on the other hand, lacks

the information regarding the differences in fracture micromechanisms and, thus, is not cor-

rect The right approach for brittle mixed-mode and Mode II fracture is obtained when

starting from the simplified result that brittle fracture is controlled by stresses, usually the

hydrostatic stress or the first principal stress ahead the crack When introducing a shear

component to the crack loading, this decreases the value of hydrostatic tension and as a

consequence causes an increase in macroscopic fracture toughness But when considering

ductile fracture, we are faced with a situation where the fracture micromechanisms are con-

trolled by mainly strains When introducing a shear-component to the crack loading we at

the same time increase the values of strain when considering J2-plasticity Because of this

general and simple result, the macroscopic fracture toughness should be lower in ductile

fracture and the situation has a principal difference compared to brittle material behavior

Experimental work in the field of mixed-mode fracture has generally been quite extensive

for the past few decades Yet, several issues still remain open, and when considering ductile

materials behaving in an elastic-plastic manner the results currently available are pretty

scarce Generally, several studies with ductile materials suffer from weaknesses associated

with analysis of results, meaning that very few studies have focused on characterizing the

mixed-mode fracture toughness in terms of J-integral or other associated parameters Con-

centrating on studies related to ductile behavior of metallic materials, Maccagno and Knott

[4] used the asymmetric four-point bend (ASFPB) setup in determining the fracture tough-

ness transition of HY130 pressure vessel steel The study recorded the modes of fracture as

well as the ductile fracture transition The transition in micromechanical terms refers to a

shear-type of crack nucleation in comparison to more typical, Mode I fibrous crack extension

In a revised study Bhattacharjee and Knott [8] focused on micromechanical changes asso-

ciated with different degrees of shear loading Both studies suffered from inadequate analysis

of results, the results presented mostly in terms of load-displacement curves Shi et al [5]

and Shi and Zhou [9] examined the fracture toughness of HT100, HT80 and A36 steels in

Modes I and II They found differences in micromechanical features, as well as that in their

test series the fracture toughness in Mode II was higher than in Mode I Several studies

suffer from uncertainties related to experimental setups (instrumentation, friction, measure-

ment of crack length) in addition to the other weakness, analysis of results

Numerical analysis of mixed-Mode I - I I crack behavior has mainly dealt with using the

Gurson-Tvergaard constitutive model in simulating the effects of shear-stresses on crack

nucleation behavior, if we neglect the numerous driving force solutions for different specimen

geometries Tohgo et al [7] used the original Gurson's model and were able to demonstrate

the competition between two different nucleation processes depending on the degree of shear-

loading, referring to crack nucleation from the blunted side of the notch and from the sharp-

ened tip Aoki et al [10] continued along the same lines and focused on the crack tip

deformation behavior with different mode proportions Ghosal and Narasimhan [11,I2] fo-

cused on determining the fields of equivalent plastic strain, hydrostatic tension, and void

volume fraction with the Gurson-Tvergaard model including nucleation and accelerated void

growth after certain critical void volume fraction They found the same results as before but

most of all, they were able to present their results with better correspondence to microme-

chanics of fracture, priming their consideration on typical Mode I type of fracture process

consisting of nucleation, growth and coalescence of voids Ghosal and Narasimhan [11,12]

used different initial void populations, mainly simulating a situation where a large void

existed ahead of the crack and the ligament failed according to porous failure criterion of

the Gurson-Tvergaard model They were able to determine the simulated fracture nucleation toughness envelope between Modes I and II, and found that when the nucleation is taken to

be strain controlled, the fracture toughness had a decreasing value when moving towards Mode II, but near Mode II it had again a rising trend due to transition to pure shear fracture Mode II fracture toughness as given by their simulations was lower than Mode I fracture toughness

This work focuses on determining the micromechanical aspects of mixed-mode fracture, the transition of fracture toughness between Modes I and II, and using numerical simulations

in interpreting different aspects of the fracture process Elastic-plastic ductile materials were studied, because earlier work has provided some controversial results and, in addition, the background in form of micromechanical features remains unknown

Numerical Simulations

SIF- and J-Integral Solutions

Linear-elastic two-dimensional plane strain finite element (FE) modeling was utilized in order to determine the SIF-solutions for the ASFPB-configuration When comparing SIF-

solutions available in the literature, large differences were noted such as [2] contra [13] and

since the range of applicability of the results was somewhat unclear, it was found that specific analyses for the current work were required The ASFPB-setup was chosen because of the simplicity of a bend-type specimen and is presented with its characteristic dimensions in Fig 1 The variable ~ controls mode mixity, meaning ~ = 0 refers to Mode II loading and

= ~ to Mode I Because measures A and B presented in Fig 1 do not have any influence

on the mode mixity, they were chosen based on suitability for experimental purposes J-

integral was calculated following the domain integral routine presented by Li et al [14]

Because the mode mixity under different loading conditions is of interest, the J-integral must

be partitioned to Mode I and II contributions This was achieved by using the filtering method

presented by Mattheck and Moldenhauer [15] The idea of the filtering technique consists

of applying suitable constraint equations to reduce the situation back to either Mode I or Mode II loading This is achieved by restraining the displacements either symmetrically or antimetrically, depending on whether Mode I or Mode II contribution is to be filtered

6 MIXED-MODE CRACK BEHAVIOR

from the total J-integral A typical FE-mesh used in the calculations is presented in Fig 2a Three-dimensional calculations were performed to determine the variations of equivalent and hydrostatic stresses in the thickness direction with different values of ~, and a deformed mesh from these calculations is presented in Fig 2b

In order to produce the results as a function of a single parameter depending on proportions

of Mode I and Mode II loading, an equivalent mode angle is presented:

where Ki denote the corresponding SIFs The results of the linear-elastic calculations were fitted to polynomial form and are presented in Fig 3a The equivalent mode angle of Eq 1 can be given for the ASFPB configuration as

which is a necessity in controlling the experimental tests and is presented with different values of a / W and ~ in Fig 3b

The J-integral solutions were determined according to the formalism presented by Rice et

al [16] The "qi-factors for Modes I and II were determined based on an ideal-plastic material model and are presented in Fig 4 The calculations required great concern and exact inter- pretation of results, because of the two-dimensionality of the deformation field Since the behavior under mixed-mode loading is neither symmetric nor antimetric, effects such as friction must be considered when the solution is compared to realistic behavior These ad- ditional boundary conditions need to be examined during calculations to form physically sound solutions The assumptions made regarding the ideal-plastic material behavior were verified using incremental plasticity analysis and the assumptions were found valid within the range of observation Three-dimensional results presented the uniform decay in the state

of hydrostatic tension ahead the crack front while the deviatoric stress state remained in proportion nearly constant at a fixed observation point ahead the crack tip

(b) (a)

[ L [ L I [[1[

FIG 2 Finite element meshes; (a) two-dimensional mesh and (b) deformed three-dimensional mesh

7~ 9 mode II, finite element / ,-" 6- o mode I, finite element /

Simulations with the Gurson-Tvergaard Constitutive Model

The Gurson-Tvergaard model was used to simulate the ductile fracture process in order

to provide numerical background for describing the micromechanical features of the fracture

process The results presented here are a part of a wider modeling effort related to numerical

modeling of the ductile fracture process, but only some of the results important for this study

will be presented here It is to be remembered that the Gurson-Tvergaard model does have

severe limitations with respect to practical use even in Mode I, and in mixed-mode and Mode

II these features surface even more vividly The theoretical background is quite lengthy and

because several good presentations already exist, such as in Refs 17 and 18, where the

features of the model are under closer examination, is provided The results presented here

pertain to pure Mode I and Mode II Because the changes associated with the continuum

fields under observation are continuous and monotonic, we can assess the general trends

without requiring to present a huge number of contour plots

The simulations were performed with a two-dimensional boundary-layer model The ma-

trix material followed Jz-theory of plasticity and finite strains The Gurson-Tvergaard model

correction constants were given values q1 = 1.5, qz = 1 and q3 = q~ In these calculations

8 MIXED-MODE CRACK BEHAVIOR

FIG 4 ~lcsolutions for J-integral determination

we will consider a situation where no initial void distribution nor density is given Nucleation

is taken to be strain controlled according to the presentation of Chu and Needleman [19]:

I-ibm' -

f,,.d = A~# = IN e L 2 \ s N ] j I~ m.pl

where ~,P~ is the equivalent plastic strain rate The parameters are chosen followingly: f u =

0.1, SN = 0.1 and eN = 0.3 The selection of parameters was performed according to tradi- tional values used in literature, because within the contents of this presentation the features

we are looking for are not dependent on the numerical values of the parameters as long as they are within reasonable limits The results of crack tip deformation, distributions of hy- drostatic stress, equivalent plastic strain and void volume fraction are presented in Figs 5 and 6 Figure 5 demonstrates that in Mode I the crack tip experiences a typical opening de-

FIG 5 Results of numerical simulations with the Gurson-Tvergaard model Crack tip deformation under (a) Mode I and (b) Mode II

10 MIXED-MODE CRACK BEHAVIOR

FIG 6 Continued

(e)

(0

formation pattern, while in Mode II the crack tip sharpens due to extensive shearing At the

same time from Fig 6 we note that the maximum value of hydrostatic tension decreases and

rotates clockwise, while the values of equivalent plastic strain increase tremendously and

localize on the sharpened tip A formation of a slip-band of intense shearing is visible from

Mode II calculations When observing the damage formation with the void volume fraction,

a transition in fracture mechanisms can be found In near Mode I situations and thereof the

crack tip deforms in a way that the other side is blunted while the other tip sharpens At

near Mode I the damage formation is strongest at the blunted side, due to nucleation of

voids as a consequence of plasticity and growth of existing voids because of hydrostatic

tension, indicating crack nucleation from the blunted side When approaching Mode II and

in Mode II, the damage formation is more rapid in the sharpening tip due to an increase in

plastic straining and a decrease in hydrostatic tension on the blunted side, causing the crack

to nucleate from the sharpened tip These features will be considered in more detail in the

discussion section

Experimental Work

Materials and Specimens

The configuration chosen for the experimental tests was the ASFPB-setup first presented

by Gao [3], where the equivalent mode angle can be adjusted continuously starting from

Mode II The SENB-specimens were either Charpy-size, or following the current Mode I

fracture toughness testing standards, sizes with cross-sections of 10 by 20 rnm ~ (thickness

by width) or 15 by 30 mm 2 Orientation was for A533B, F82H and AISI 304 specimens T-

L and for CuA125 specimens L-T The basic mechanical properties of the materials tested

are presented in Table 1 Experiments were performed under displacement control measuring

TABLE 1 Properties of experimental materials

Material

the global force-displacement curve, which decomposes to the strain energy of the loading

rolls depending on the choice of A and B Instrumentation was found more accurate and

less prone to rotational errors in this method of measurement when compared to measuring

the local displacement variables PD-method was utilized following current procedures pro-

vided by several Mode I fracture resistance testing standards

Several different experimental configurations for mixed-mode testing have been presented

and a general consensus regarding the most suitable choice has not been achieved The

deficiencies of different setups can be divided in to three categories: instrumentation, mea-

surement of crack growth, and friction Instrumentation deficiencies are related to the fact

that mechanical gages, etc are very prone to errors when the deformation field is two-

dimensional, meaning that different types of corrections are needed, and on the other hand,

the correct measurement of the displacement variables under mixed-mode loading is under

question Additional difficulty in instrumentation is a consequence of large displacements,

which are often encountered in mixed-mode testing Measurement of crack growth is another

problem Compliance solutions do not exist and even so, the stiffness of the specimen will

be dependent on the mode angle and makes the arrangement susceptible to additional errors

Potential drop (PD) measurements can be affected by the shearing of the crack front during

loading, resulting in more significant geometry changes than in Mode I testing, and cause

the voltage signal to have a drop of unknown quantity related to current deformation state

Multiple specimen methods are naturally available, but demand many specimens In this

work the PD-method was used with partial success The third problem is friction, because,

when Mode II is approached it is most likely that the crack faces will experience additional

contact, making the results depend on the current crack length with an additional frictional

component doing the work as well An easy way around the problems associated with friction

is to avoid testing in Mode II and to perform the tests near Mode II where the crack faces

separate due to a small opening component

Fracture Resistance Curves

Fracture resistance curves were determined for all materials as a function of the equivalent

mode angle and are presented in Fig 7 In Fig 7a the resistance curves for F82H are shown,

which demonstrate a trend of decreasing fracture resistance In near Mode II (near 90 ~ the

curves are very fiat, indicating that the tearing modulus is very small Similar results are

presented for A533B in Fig 7b, while in Fig 7c the results of AISI 304 present an even

more dramatic decrease of fracture toughness, which will be referred to microstructural

features in the discussion part of the work The fracture resistance curves of CuA125 alloy

indicate a drop in fracture resistance at a certain discrete mode portion rather than a contin-

uous drop, as presented in Fig 7d This effect is most likely related to nficrostructural

orientation effects and anisotropy and is a subject of further studies

FIG 7 Fracture resistance curves as a function of the equivalent mode angle (a) F82H, (b) A533B,

(c) AISt 304, and (d) CuA125

Fractographical Results

The fracture surfaces were investigated with a scanning electron microscope (SEM) and

energy dispersitive X-ray (EDS) analyses were used to study the crack formation micro-

mechanics The results were basically similar in all materials studied with some different

details related to microstructural factors, which will be referred to later In Fig 8 the fracture

surfaces of F82H steel in Mode I and with a modal angle o f 76.2 ~ are presented The fracture

surface o f Fig 8a is a typical surface of Mode I dimple fracture Figure 8b presents the

morphology of a fracture surface near Mode II The differences between the fracture surfaces

are clear: the near Mode II surface is usually characterized as being macroscopically flat,

which is not the case in microscopical terms The morphology of the fracture surface formed

at mode angle o f 76.2 ~ contains areas o f extremely small dimples formed around second

phase particles and the areas are connected to each other through deviations in the macro-

scopic fracture plane, which can be characterized as asperities The dimple size decreases

and becomes more sheared consistently when moving from Mode I towards Mode II The

dimple size experiences a large drop at the beginning stages of the mode locus

500 400-

300- a=

250 -

200 - 150- 100-

Fracture Nucleation Angles and Modes

Crack nucleation angles followed similar trends with all materials with respect to the

equivalent mode angle In Fig 9 the nucleation angles are presented as a function of/3eq for

F82H steel The difference compared to typical linear elastic results is drastic Based on

linear-elastic treatments it has generally been accepted that the crack nucleation angle in

Mode I1 is approximately 70 ~ while based on these results nucleation even in Mode II occurs

nearly self-similarly and between the modes a nearly quadratic variation is observed

The crack nucleation process in elastic-plastic ductile mixed-mode propagation pertains

to the competition between Mode I and Mode II type of crack nucleation and growth Crack

nucleation with these materials was found to change from Mode I to Mode II type of crack

growth with an equivalent mode angle of approximately 40 to 60 ~ This observation was

made based on transitions in the nucleation angles and nucleation values of fracture tough-

ness The macroscopic crack growth, on the other hand, was found to alter its appearance

closer to the Mode II end, when the zigzags of a Mode I crack diminished and the crack

propagated macroscopically like a shear crack This fact is most likely related to local con-

ditions since nucleation and propagation are influenced by the near crack tip material prop-

14 MIXED-MODE CRACK BEHAVIOR

FIG 8 Fracture surface morphology of F82H steel under (a) Mode I and (b) equivalent mode angle

of 76.2 ~

erties, the mode of crack growth near the transition of first nucleation may not be stable with respect to propagation and different modes can exist at different stages Macroscopically Mode II crack growth was observed in tests where the mode angle was 76.2 ~

Micromechanics of Mixed Mode I-II Fracture

The numerical calculations demonstrated the decrease of hydrostatic tension as the loading

was altered from Mode I towards Mode II At the initial stages of Mode II loading the rate

of decrease of hydrostatic tension is high Naturally, if we consider an infinitesimal situation,

under Mode II the crack front would not experience a hydrostatic stress state at all With

finite strains, it is noticed that the maximum of the hydrostatic tension rotates clockwise and

decreases in value Also, at Mode II the sharpened tip of the crack experiences hydrostatic

compression, while the weak peak of hydrostatic tension is far from the area of crack

propagation

The deviatoric stress state and thus the plasticity experienced by the near crack tip region

is enhanced by the introduction of the shear loading component The maximum values at

Mode II are found from the sharpened crack tip and as known even from the basic linear-

elastic crack stress field solutions, the extent of the plastic region is several times larger in

comparison to Mode I loading This feature can also be understood as an expansion of the

process zone of fracture

Numerical simulations also reveal the modes of crack nucleation, which have been verified

experimentally by several researchers see Refs 7, 10-12 The calculations demonstrate that

near Mode I the rate of damage formation is highest at the blunted side of the initial notch,

indicating crack nucleation from the blunted side, while on the sharpened tip at these mode

angles the void formation is less severe When enhancing the Mode II loading component,

it is found that as the hydrostatic tension stress state decreases and the plasticity localizes

in a more volatile manner to the sharpened tip, the damage accumulation of the sharpened

tip overcomes that of the blunted side In Mode II, the lack of hydrostatic tension in the

blunted side impedes void growth and because the plastic strain concentrations are extremely

strong at the sharpened tip, crack propagation from the blunted side is unfavorable The

damage formation at the sharpened tip is extremely strong causing the crack to nucleate as

a thin shear crack through an intensive plastic localization, the process usually referred to

as Mode II type of crack nucleation

16 MIXED-MODE CRACK BEHAVIOR

The differences caused by the previous factors to micromechanisms of fracture under Mode II loading are presented in Fig 10 Figure 10a presents a typical Mode I dimple fracture, which is divided into stages of nucleation, growth and coalescence for reference The first abnormality when comparing to the Mode II fracture of Fig 10b is related to the nucleation process Typically in Mode I the situation is such that the nucleation of voids from large particles is stress controlled, while smaller particles of secondary populations nucleate with a strain controlled mechanism (stress controlled [20], strain controlled [21]) The plastic strains experienced by the near crack tip areas in Mode II are large enough to cause nucleation in smaller particles as well fairly early in the rupture process, because otherwise the dimple sheets as seen in SEM studies could not have been formed The nu-

cleation process takes place throughout the crack front at near tip regions, which is referred

to as the large plastic zone and strain controlled nucleation of different particle populations,

because the interface stress controlled nucleation can be considered a milder criterion com-

pared to nucleation through plasticity In the Mode I type of fracture the strain controlled

nucleation of smaller particles is related to the coalescence stage, where the localization

caused by the grown large voids initiates a formation of void sheets through smaller particles

contributing to the coalescence and final fracture At the next stage in Mode II type of

fracture, due to the lack of hydrostatic tension, the voids that have formed as a result of the

nucleation process do not have any prospects of growing, but will remain near theii" initial

forms that are in relation to the original inclusion sizes In contrast, the strong influence of

void growth on Mode I fracture and the typically exponential relation between void growth

rate and stress triaxiality has been demonstrated in several studies [22] At this stage we

have a crack tip with a large plastic zone, which, when considering nucleation, can be

considered as the process zone of fracture, and a fine distribution of voids The next stage

of fracture is the coalescence of voids, which occurs as a local rupture between the small

voids It can be argued that this stage can occur with much smaller energy consumption than

in a fibrous crack extension even if considering microscopically rectilinear crack growth,

because we are considering a sheet of very fine voids connected with small ligaments and

the loading with reference to localized plasticity is strong The asperities connecting the void

sheets are related to the process zone through rnicrostructurat inhomogeneities and the frac-

ture process Comparing to a traditional Mode I crack propagation, the Mode II crack pos-

sesses more degrees of freedom The micromechanical level at which the fracture process

occurs is smaller due to the lack of void growth, and it causes a situation where the mi-

crostructural inhomogeneities, such as particle distribution and matrix properties and their

anisotropy, have an effect on the end result The larger process zone provides the crack with

degrees of freedom to propagate in the intense slip-zone with respect to non-continuum

properties, causing the asperite surface to form during crack growth The fracture resistance

curves presented previously support the concepts of micromechanical observations and nu-

merical simulations The drastic decrease of the tearing modulus with all materials is a direct

result of the increase in plastic zone with Mode I1 loading leading to a larger process zone,

which can be interpreted as easing the fracture process and decreasing the energy associated

with plastic dissipation Thus, the crack has several possible paths to advance, from which

it selects the one of lowest resistance, which on the other hand is formed as a result of grain

orientation and other effects causing anisotropy

The mixed-mode fracture surfaces are formed with mechanisms that are between both far

ends The decrease of hydrostatic tension is quite rapid at small values of the mode angle,

which is reflected as a decrease of void growth rate and the formation of smaller dimples

even at small values of the equivalent mode angle Otherwise the asperity formation, etc

follow intermediate values when encompassing between Modes I and II

Material Characteristics of the Fracture Micromechanics

The materials considered in this study naturally possessed some characteristic properties

with response to mixed-mode loading The F82H steel is relatively clean in microstructural

terms, and the sparsity of second phase particles is in Mode I reflected as large dimples

surrounded by void sheets of smaller particles This is reflected to a mixed-mode situation

in a sense that in near Mode I the differences in dimple size are larger between different

populations, and larger dimples exist among sheets of smaller dimples, the A533B steel

studied has a very fine particle distribution, which is reflected as sheets filled with small

dimples even at near Mode I situations AISI 304 was found to contain additional impurities

18 MIXED-MODE CRACK BEHAVIOR

that had formed copper-sulfide and the initial void volume fraction was very high This

reflected as a drastic decrease in fracture toughness, even at the early stages of the loading

spectrum near Mode I The CuA125 alloy had a strong texture due to a complex manufac-

turing process (internally oxidized, rolling and hot-isostatic pressing), which was found to

cause a discontinuity type of decrease in fracture toughness This effect was considered to

be a consequence of anisotropic material behavior associated with different crack nucleation

mechanisms, but the studies did not proceed further

Modeling Considerations with Respect to Ductile Mixed Mode I-II Fracture

The general idea of simulating mixed-mode ductile rupture with the Gurson-Tvergaard

model is related to finding the deficiencies of the model, which can then be better understood

for Mode I and general loading It can not be pursued that using the model for mixed-mode

or Mode II fracture analysis is entirely a valid effort, because several features of the model

have been committed to more or less strictly Mode I type of loading and as such we are

definitely not discussing a general fracture model

Because the model basically defines a v o n Mises material enhanced with dilatation effects,

in principle there are no basic features limiting its use The limitations and questions are

more related to the latter variants, the damage evolution equation and the dilatational part

At first, the qi factors were chosen by Tvergaard [23] in order to present interaction and

coalescence effects for a Mode I type of ductile failure and are unlikely to be material

parameters in terms of different loading modes Additionally, the basic model is formed for

a spherical void in a characteristic unit cube, which may not reflect the size scale consistently

Since in the case of Mode II or mixed-mode fracture, the interaction between voids is likely

to be more severe and certainly the concept of unit cube containing a void of cylindrical

geometry forming typically under high stress triaxiality is under question Also, local ani-

sotropy due to changes in length-scale is to be considered Similar arguments can be given

for the general two-dimensional modeling of crack nucleation and propagation, but unfor-

tunately three-dimensional numerical modeling is still an obstacle in many cases Similar

comments can be given for the choice of nucleation functions and the acceleration of void

growth due to coalescence Taking into account the previous factors more general models

for simulating ductile fracture should be considered and some of the shortcuts taken in

present modeling identified

Parameters and Criteria

Based on results for the nucleation angle and associated behavior, the work done within

linear-elastic mixed-mode criteria for predicting crack nucleation direction and fracture

toughness locus lacks in background Criteria related to plastic strain, shear stress, and de-

viatoric stresses should be used instead and some first steps have been taken Naturally,

criteria based on energy release rate could be considered as the most feasible ones Another

issue is the changes in crack nucleation and growth modes, which cause an entirely different

situation to characterize Also, because the transition between brittle and ductile mixed-mode

fracture behavior is dependent on, for example microstructural features, the modeling efforts

are more complicated Behavior of two materials very close to each other in terms of com-

position and properties can differ with respect to mixed-mode brittle and ductile, for example,

and great care must be taken

A similar question arises when considering parameters for assessing ductile mixed-mode

fracture This study used J-integral to characterize the behavior, but it is to be questioned

whether some other parameter should be used In a mixed-mode situation, for example, both

components of the J-vector have nonzero values and neglecting the second component does not have any grounds Several conservation integrals have been presented, but their suitability has not been assessed with detailed accuracy containing numerical and experimental work Finding the correct parameter would solve several, even most, of the problems facing the understanding of mixed-mode fracture

Conclusions

Mixed-mode fracture resistance curves were determined for four metallic elastic-plastic materials experiencing a ductile fracture mechanism Results involving transitions of fracture toughness, and concerning crack nucleation angle and fracture morphology together with numerical modeling were used to describe the micromechanics of the fracture process and some comments for modeling were provided with respect to mixed-mode loading

The conclusions from this work are:

9 Fracture toughness in ductile materials can be lower under mixed-mode or Mode II loading than in Mode I The effect appears most pronounced with materials of great practical importance, such as structural and stainless steels

9 The different micromechanical features of ductile failure under mixed Mode I - I I loading have been demonstrated to be due to differences in the continuum fields characterizing the fracture process The stages of fracture are different because of a lack of hydrostatic tension and an increase in quantitative values of plastic strain when the shear loading component is introduced

9 In terms of mixed-mode behavior, linear-elastic and elastic-plastic materials differ in many fundamental ways and theories and criteria for brittle materials are unable to assess the behavior of ductile materials

Acknowledgments

This work is a part of the Nuclear Power Plant Structural Safety Program performed at the Technical Research Centre of Finland (VTT) and the European Fusion Program by the Association Euratom-TEKES The work was financed by the Ministry of Trade and Industry

in Finland, the Finnish Centre for Radiation and Nuclear Safety (STUK), the Technical Research Centre of Finland, the Finnish Fusion Research Program FFUSION, and the Euro- pean Fusion Program

References

[1] Maccagno, T M and Knott, J F., "The Fracture Behavior of PMMA in Mixed Mode I and II,"

Engineering Fracture Mechanics, Vol 34, 1989, pp 65-86

[2] Suresh, S., Shih, C E, Morrone, A., and O'Dowd, N P., "Mixed-Mode Fracture Toughness of

Ceramic Materials," Journal of the American Ceramic Society, Vol 73, 1990, pp 1257-1267

[3] Gao, H., Zwang, Z., Tang, C., and Zhou, A., "An Investigation on the Brittle Fracture of K,-KH Composite Mode Cracks," ACTA Metallurgica Sinica, Vol 15, 1979, pp 380-391

[4] Maccagno, T M and Knott, J E, "The Mixed Mode I/II Fracture Behavior of Lightly Tempered

HY130 Steel at Room Temperature," Engineering Fracture Mechanics, Vol 41, 1992, pp 805-

820

[5] Shi, 5( W., Zhou, N N., and Zhang, J X., "Comparison of Mode I and Mode II Elastic-Plastic

Fracture Toughness for Two Low Alloyed High Strength Steels," International Journal of Fracture,

Vol 68, 1994, pp 89-97

20 MIXED-MODE CRACK BEHAVIOR

[6] Aoki, S., Kishimoto, K., Yoshida, T., Sakata, M., and Richard, H A., :'Elastic-Plastic Fracture

[7] Tohgo, K., Otsuka, A., and Gao, H W., "The Behavior of Ductile Crack Initiation from a Notch

Under Mixed Mode Loading," Proceedings of Far East Fracture Group Workshop, M Sakata, Ed.,

Tokyo Institute of Technology, 1988, pp 101-108

[8] Bhattacharjee, D and Knott, J F., "Ductile Fracture in HY100 Steel Under Mixed Mode I/II

[9] Shi, Y W and Zhou, N N,, "Comparison of Microshear Toughness and Mode II Fracture Tough-

Dual Population of Second-Phase Particles," Materials Science and Engineering A, Vol 211, 1996,

pp 117-127

Initiation Under Mixed-Mode Loading," International Journal of Fracture, Vol 77, 1996, pp 281-

304

[16] Rice, J R., Paris, R C., and Merkle, J G., "Some Further Results of J-integral Analysis and

Estimates," Progress in Flaw Growth and Fracture Toughness Testing, ASTM STP 536, D T Read

and R R Reed, Ed., American Society for Testing and Materials, Philadelphia, 1973, pp 118-133

Yield Criteria and Flow Rules for Porous Ductile Materials," Journal of Engineering Materials and

Technology, Vol 99, 1977, pp 2-15

Vol 27, 1990, pp 83-15t

gineering Materials and Technology, Vol 102, 1980, pp 249-256

[20] Argon, A S., Ira, J., and Safoglu, R., "Cavity Formation from Inclusions in Ductile Fracture,"

[22] Rice, J R and Tracey, D M., "On the Ductile Enlargement of Voids in Triaxial Stress Fields,"

International Journal of Fracture, Vol 17, 1981, pp 389-407

The Crack Tip Displacement Vector

Approach to Mixed-Mode Fracture

Mixed-Mode Fracture," Mixed-Mode Crack Behavior, ASTM STP 1359, K J Miller and

D L McDowell, Eds., American Society for Testing and Materials, West Conshohocken, PA,

1999, pp 21-40

ABSTRACT: A method for assessing the ductile failure of thin structures containing arbitrarily

oriented cracks is presented The crack tip displacement vector 8~ is used as fracture parameter

Experiments carried out on 4- to 6-mm-thick steel StE 550 and aluminum alloy A12024-T3

sheets with various mixed-mode specimens demonstrate that g~ is more appropriate to char-

The Mode I crack resistance curves of standardized C(T)-type specimens give a conservative

estimate of crack initiation and resistance to stable growth in the range of near Mode II loading

to pure Mode I loading The gv-parameter of a cracked component is evaluated with Engi-

neering Treatment Model (ETM) The analytical ETM method requires only the stress intensity

factor and plastic limit load solutions of the considered structure as well as the material stress

and strain power law as input parameters Close agreement of ETM predictions to the exper-

imental load versus gv relationships or load-displacement curves can be achieved, if the proper

limit load solution of the cracked structure is available

KEYWORDS: mixed-mode fracture, Mode II fracture, crack tip opening displacement, crack

tip sliding displacement, stable crack growth, flaw assessment, Engineering Treatment Model

Crack mouth displacement vector, Mode I and Mode I1 components Crack mouth displacement vector at F = Fy

Load, load at net section yielding Crack initiation values of J-integral under mixed-mode loading Mode I crack initiation values of J-integral

Mode I and II stress intensity factors Mode I and II plasticity-corrected stress intensity factors Elastic mixed-mode parameter (equals 0 for Mode II and 1 for Mode I)

Plastic mixed-mode parameter (equals 0 for Mode II and 1 for Mode I)

Strain hardening exponent, N > l

~Research engineer, Institute of Materials Research, German Aerospace Center DLR, Linder tt6he,

D-51147 Cologne, Germany

22 MIXED-MODE CRACK BEHAVIOR

nodv, nodt, nod~

Crack deflection angle (positive for tensile crack growth) Crack tip opening displacement measured at fatigue crack tip of Mode I

Specimens, 5-ram gage length Crack initiation values of g5 Crack tip displacements at F = F r

Stable crack propagation (measured in crack propagation direction) Crack tip displacement vector, Mode I and II displacements mea- sured at the fatigue crack tip (subscripts 5 and opt indicate the mea- surement technique)

Strain, strain at (r r Global load biaxiality ratio Ligament length, measured in crack growth direction and normalized

by the width in crack growth direction Stress, general yield strength

The assessment of cracked structures against ductile crack initiation and growth is usually performed by comparing a crack loading parameter or driving force to the crack growth resistance of the material In this work, an approach to the assessment of thin (B -< 6 ram, ASTM B-646-87) structures containing arbitrarily oriented cracks is presented The relevant fracture parameter is the crack tip displacement vector go and is defined by

~ and ~H are the Mode I and Mode II components of crack opening and shearing displacements

In the first part of the paper, the materials characterization side is addressed In the case

of ductile materials, the crack growth resistance is expressed in terms of a crack initiation value and a crack resistance curve (R-curve) The crack initiation value and R-curve of a material should be reasonably independent of the test specimen and the applied mixed-mode ratio, allowing thus the utilization of standardized Mode I laboratory tests for fracture tough- ness evaluation Since the problems of transferability (so called "constraint effect") of ductile Mode I fracture data are well known and difficult to overcome [1,2], a minimal requirement

to the mixed-mode fracture parameter is that the Mode I initiation value and R-curve obtained from the standardized bending specimens are at least a conservative estimate of fracture toughness Within the framework of an engineering approach the measurement of the fracture parameter should not be too difficult either On the basis of our experiments and literature data it will be shown that the crack tip displacement vector ~v measured at fixed location near the fatigue crack tip fulfills the requirements previously mentioned Moreover, it is demonstrated that ~ is more appropriate to characterize mixed-mode ductile fracture than the usually used J-integral

In the second part of the paper a simple method for estimating the 8 v driving force under mixed Mode I / I I loading conditions is presented It relies on Schwalbe's Engineering Treat- ment Model (ETM) [3] It requires only the stress intensity factor and plastic limit load

solution of the considered structure as well as the material stress and strain power law as input parameters The ETM predictions are validated by means of the experimental results

of mixed-mode tests performed on biaxially loaded cruciform specimens of different mate- rials The final part of the paper mentions the limits and unresolved problems of this approach,

Experimental Procedure

The crack tip displacement approach is validated by experiments carried out on biaxially loaded cruciform specimens and compact tension shear (C(TS)) shown in Fig 1 and Fig 2, respectively Two materials, a fine-grained structural steel StE 550 and a high-strength alu- minum alloy A12024-T3 were tested to sample fully plastic fracture (StE 550) and fracture under small-scale-yielding and contained yielding conditions (A12024-T3) The thickness of the steel specimens was 5 m m in the cruciform specimens and 4 nun in the C(TS) specimens, whereas the aluminum alloy specimens always had a thickness of 6 ram The tensile test results and Mode I initiation values are given in Table 1 The yield strength of A12024-T3 averages over a 10% variability due to anisotropy [4] The Mode I initiation values, denoted Ji.0.2 for the J-integral and ~5.o.e for the crack tip opening displacement, were obtained from the intersection of the line corresponding to a constant total crack growth of 0.2 rnm with the R-curve following the ESIS-procedure [5] This definition of crack initiation, validated

in [6], is particularly convenient for mixed-mode crack initiation, because especially in near Mode II loading a clear distinction between blunting and crack growth is often lacking [7] The initiation values were measured with different sizes of C(T) (compact tension) and M(T) (middle cracked tension) specimens and were independent of specimen ranges [4]

The "cracked" cruciform specimen (Fig 1, left side) contained a through-thickness crack, whereas the "notched" cruciform specimen was distinguished by 10-ram-long cracks ema- nating from a hole (Fig 1, right side) In both specimens the fatigue pre-cracks were inclined

by an angle of 45 ~ to the loading directions The experiments were carried out on a biaxial test rig The applied elastic mixed-mode parameter M e

24 MIXED-MODE CRACK BEHAVIOR

FIG 2 C(TS) specimen and loading device

was varied by changing the biaxial load ratio k, defined in Fig 1 A biaxial load ratio X =

- 1 corresponded to pure Mode II loading (Me = 0) in both specimen types Increasing X to +0.5 gave M r = 0.78 for the cracked cruciform specimen [7] In the Appendix it is shown how the elastic mixed-mode parameter is linked to the plastic mixed-mode parameter, which

is a function of the crack tip opening to crack tip sliding displacement ratio,

Fracture tests at 15, 45, 60, 75, and 90 ~ loading angles were carried out with fatigue pre- cracked C(TS) specimens [8] mounted in a special fixture shown in Fig 2 The loading angle

of 90 ~ corresponded to pure Mode II loading The other Me-factors were calculated from the stress intensity factor solutions given in Ref 8

On the cruciform specimens the technical crack tip displacement was measured at the

original fatigue crack tip with a specially designed ~5 clip gage [7,9], which allowed a decomposition of the measured displacements in the sliding and opening mode (~5,1 and

~5,n) As its Mode I counterpart [I0] it measures the relative displacement of two points

TABLE 1 Engineering tensile behavior and Mode I initiation (measured at Aa = 0.2 mm) values of

StE 550 and A12024-T3

580 N/ram 2 Ltiders yield strength, ReL

0.2% offset yield strength, Rpo.2

Ultimate strength

Total strain at ultimate strength (50-ram gage length)

Total strain at fracture (50-ram gage length)

Initiation d~integral, Ji,o.2

Initiation crack tip opening displacement, ~5,o.a

located in a distance of 2.5 mm on either side of the fatigue crack tip A simpler procedure

was used on the C(TS) specimens, where bopt.~ and gop,.H were measured optically on the

surfaces 0.5 mm behind the fatigue crack tip through a microscope It was demonstrated

experimentally in Refs 4, 7, and I1 and by finite element calculations in Ref 12 that both

definitions of the crack tip displacement vector are equivalent in an engineering sense, even

though in the limiting case of non-hardening plasticity different slip-line fans with different

displacement fields exist in the fans between 90 and 180 ~ from the crack line In the following

sections the crack tip displacement vector is always denominated as 8~ independently of the

measurement technique

Further details of the experimental procedure, specifically the measurement of the crack

mouth displacements (cmod in Fig 1 and Fig 2) in the cracked cruciform specimens and

in the C(TS) specimens, as well as the displacements measured in the notch of the notched

cruciform specimen (nod in Fig 1) are given in Refs 4, 7, and 13

More information on the validation of 85 as a fracture parameter and the limits of its

application domain may be found in Refs 11 and 14

6v as a Resistance to Crack Initiation and Stable Crack Growth

Correlation Parameter

In this section, the results of experiments on biaxially loaded cruciform specimens and

C(TS) specimens, partially published recently [4,7,13] are summarized and compared to

mixed-mode data taken from open literature

The problem of crack growth direction is addressed in the Appendix Here it is important

to know that under predominant Mode II loading cracks due to shear-type fracture grew

approximately in the maximum shear strain direction This means that the stable cracks

remained almost parallel to the original fatigue pre-crack Relatively high (case of steel) or

moderate (case of aluminum alloy) Mode I crack tip opening components caused a crack

path deviation, meaning the stable crack grew normal to the maximum uncracked tensile

stress (the main loading axis) as a Mode I crack

Each data point in the following diagrams corresponds to a single mixed-mode experiment

(multiple specimen technique) Also, for comparison the Mode I 85-Aa curves of center

cracked M(T) specimens (2w = 250 mm and ao/w = 0.3) and C(T) specimens (W = 50

mm and ao/W = 0.6) are plotted in the diagrams

Figure 3 shows the magnitude of the crack tip displacement vector of the steel C(TS)

specimens as a function of stable crack growth All points are positioned between the Mode

1 85-R-curves of the C(T) and M(T) specimens and at least in the initial part of the resistance

curves no particular effect of mixed-mode loading is discernible In Ref 13 it is shown that

the ~v initiation values measured at ha = 0.2 mm of StE 550 are independent of applied

mixed-mode ratio over almost the entire mixed-mode range Only under pure Mode II load-

ing a 30% decrease of the initiation 8o is observed

For longer shear crack growth, the R-curve already measured with crack and notched

cruciform specimens under mixed-mode loading was obtained This suggests a minimal

influence of specimen geometry and mixed-mode ratio on the shear crack 8~-ha-curve In

case of tensile (Mode I) crack growth (loading angle + -< 60 ~ the behavior can be explained

with the constraint considerations known from Mode I [1,2] Apparently the combination of

tensile and bending loading exerted a constraint of the plastic deformation ahead of the

kinked crack which laid in between C(T) and M(T) specimens [15] Increasing bending

components (decreasing d~) increased constraint and therefore the gv Aa data points were

shifted towards the C(T) R-curve (qb = 15 ~ in Fig 3) The ~5 R-curve of the highly con-

strained C(T) specimens seems to be a conservative estimate of fracture resistance

2 6 MIXED-MODE CRACK BEHAVIOR

The C(TS) results are compared to the shear crack R-curve fitted through the data of the cracked and the notched cruciform specimens and the Mode I R-curves

It is well known that the J-integral does not describe the stress and strain state at the crack tip uniquely and accurately and that the foundation of J is even less sound for growing

cracks in ductile materials (see Ref 16 for an example) Nevertheless most of the data on

ductile mixed-mode fracture is published in terms of J-initiation values and J-crack-resistance curves [17-25] To compare the Bo-approach to the J approach a J estimation formula for

C(TS) specimen was developed and checked with finite element calculations in Ref 13 The

FIG 4 J-integral versus stable crack growth of the C(TS) steel specimens compared to the Mode I

J-R-curves of C(T) and M(T) specimens

striking feature of the J-integral crack resistance curves is that the R-curves decrease with increasing Mode lI loading components For fixed 2~a, the Mode II fracture resistance in terms of a J-R-curve was approximately only one half of the Mode I J-Aa-curve of a C(T)

specimen This finding is typical for ductile ferritic steels and seems to be independent of the thickness B of the material [10,21,26] The reason for this behavior is connected with

the strong influence of the global specimen deformation on the mixed-mode J-R curves

[L13J

Figure 5 highlights the decrease of the mixed-mode J-initiation values Jc of ductile steels

[20,21,24,25,27] and a finite element calculation based on damage mechanics [28] with

increasing Mode II load components The mixed-mode fracture data is normalized by the Mode I J~c or Jr.o.2 Davenport's [21,27] results for A-508-3 steel are related to J~.oe values

obtained from the mixed-mode single-edge-notched (SEN) specimens tested in Mode I The lines in Fig 5 show the Jc/J~c-M e relationship of the plane strain, mixed-mode HRR-

field [29] derived under the av = constant assumption for two work hardening exponents N [30] At least the trend of the initiation-J-decrease with increasing Mode II components is

predicted correctly with the av = constant assumption and the HRR fields, although the strict validity limits mentioned above for the J-integral apply also for the HRR-field In this cir- cumstance it should be noted that the influence of out-of-plane (thickness) and in-plane (geometry of loading) constraint on crack tip fields decreases with increasing Mode II com- ponents [31-33] Therefore the range of HRR-dominance increases as the loading approaches

Mode U

In the aluminum alloy A12024-T3, which initiated under small-scale-yielding and frac- tured under contained yielding conditions, the toughness for shear crack growth (near Mode

II loading) was higher than that for tensile tearing (near Mode I), as shown in Fig 6 Most

of the A12024-T3 specimens, however, failed by tensile fracture The respective 8~-Aa data pairs lie in the common Mode I scatterband of C(T) and M(T) specimens since the stable crack grew as a Mode I crack In small-scale and contained yielding the crack growth process

is controlled by the local constraint ahead of the crack tip Increased Mode II R-curves are

FIG 5 - - T h e decrease o f the normalized mixed-mode J-initiation values Jc with increasing Mode II components in ductile steels [20,21,24,25,27l and a finite element calculation based on damage me- chanics [28] is qualitatively predicted by imposing a constant ~v in the HRR field [30]

28 MIXED-MODE CRACK BEHAVIOR

0.8t AI2024-T3B=6mm 0.6-

FIG, 6 Crack tip displacement vector versus stable crack growth of cracked cruciform and C(TS)

specimens of A12024-T3 compared to the Mode I g5-scatterband of C(T) and M(T) specimens

obtained in case of predominant shear loading components, since the triaxial stress states

ahead of the crack tip are reduced compared to the Mode I loading case [28,34,35] These

findings agree with other investigations on aluminum alloys using J-integral [18,19,36] and

crack tip displacement [37] criteria It should be noted, however that in case of Mode I crack

growth, the ~5 R-curves resulted to be less affected by geometry and loading conditions than

the J-R-curves [10,38]

Within the context of this work, an estimation of mixed-mode crack resistance based on

the g~-concept and the 85 R-curve of C(T) specimens seems to be conservative (only excep-

tion is Mode II initiation in StE 550) The experimental simplicity of measuring the crack

tip displacements at a fixed position has the disadvantage of introducing some geometry and

loading dependence in the R-curves at higher Aa-values, where g~ is increasingly influenced

by the global deformation behavior of the specimen

~v Crack Loading Parameter

The Engineering Treatment Model has been set up by researchers at the GKSS-Research

Center in Geesthacht, Germany to provide a framework for a quick estimation of the Mode

I ~5 loading parameter in function of the applied load In the following, the basic version of

ETM [3], which has been validated for a number practical Mode I loading cases [11,14] is

extended to mixed Mode I / I I loading conditions without taking account of the more so-

phisticated ETM-procedures developed recently [39]

The basic principle for Mode I loading is presented in Fig 7 The cracked structure is

assumed to deform in a state of prevailing plane stress, which implies that the predictions

are more accurate for thin sections The material's engineering stress and strain curve is

approximated with a piece-wise power law:

e y

where gs,Y is constant in Eq 6 and evaluated with Eq 4 at the plastic limit load F r

As might be expected from the results presented in the previous section, in case of mixed- mode loading 85 in Eq 6 is just replaced by the crack tip displacement vector 8 v giving

This approach is corroborated by the slip-line analysis of Saka and Tanaka [24], which showed that the magnitude of 8o is closely related to the maximum equivalent plastic strain ahead of the crack tip

According m the Mode I estimation [41], the contained yielding solution (see Eq 4) is generalized for mixed-mode loading by evaluating the crack tip displacement vector at the physical crack tip assmning an effective crack length of [42]:

30 MIXED-MODE CRACK BEHAVIOR

Eq 9

The problem of crack propagation direction is addressed in the Appendix The assumed crack growth path influences the limit load F r and the contained yielding solutions previously mentioned, which should be applied for standing cracks or cracks propagating coplanar to the fatigue pre-crack If the stable crack kinks immediately after initiation and propagates

as a Mode I crack before the plastic limit load is reached, the Mode I ETM should be utilized In general it is recommended that two analyses be performed, assuming shear crack growth and tensile crack growth and to opt for the conservative result

The ETM procedure is designated for estimation of crack initiation load or R-curve anal- ysis If the deformation behavior of a cracked component is evaluated, crack growth is accounted through the R-curve in a single step iteration procedure [39] Similar to 85 or 8 v also load point displacements can then be related to the applied load, if appropriate solutions for the contained yielding approach are available

The purpose of the following examples is to demonstrate the efficiency of ETM To sup- press unwanted geometry effects of the R-curve, the amount of stable crack propagation was extracted directly from the experimental results The crack propagation direction was pos- tulated as well

An average piece wise power law was fitted to the engineering stress-strain curves of tensile specimens machined in various directions to the rolling direction of the slightly an- isotropic A12024-T3 giving a strain hardening exponent of N = 9.6 Because of the discon- tinuous stress and strain curve of StE 550 steel, the strain hardening exponents were extracted directly from available Mode I fracture experiments Equation 6 was adapted to the experi- mental data of the M(T) specimens with N = 17 The load-displacement data of C(T) spec- imens could be best fitted with N = 12 The difference in strain hardening is attributed to the inhomogeneous stress distribution in the bent ligament of the C(T) specimens, which wipes off the Ltiders yield plateau The N of the C(T) specimens was used for the near Mode I loaded C(TS) specimens All other subsequent examples were calculated with N =

17

In Fig 8, the ETM predictions are compared to the measured crack tip displacement vectors of two cruciform specimens loaded at a negative biaxiality ratio and thus displaying shear controlled fracture The small scale yielding 8~ was calculated with the mixed-mode Dugdate model previously mentioned The limit load solution was obtained assuming con- stant yon Mises equivalent stresses in the slanted ligament:

FIG 8 Comparison of the ETM prediction with two experiments on cruciform specimens having

experiments

4Bw cry

~ (1 -~- •)2 (1 + N) 2 (1 - k ) 2 y2 + (1 + h ) 2 y + 3 3'z

where 3' corresponds to the remaining normalized ligament inclined by 45 ~ , Fig 1:

The ETM prediction is very close to the experimental results The initiation load (which

could not be determined exactly in the biaxial experiments) is evidently slightly

overestimated

In case of the aluminum alloy specimens, the ETM-estimation is less accurate, Fig 9

The stable crack initiated well before the plastic limit load was attained and compared to

the steel specimens much more stable crack propagation was achieved during the tests The

inaccuracy is therefore connected with the limit load and the contained yielding solutions,

which for very long cracks do not account for the finite dimensions of the specimens correctly

(or at all in case of the Dugdale model)

The contained yielding solution (Eq 9) of the notched cruciform specimen was calculated

from the superposition of the Kr-factor of a Mode I cracked notched cruciform specimen

A load ratio of X = +0.5 caused a crack path deviation in both materials The tensile

crack propagated normally to the main loading axis Therefore the limit load was estimated

imposing constant v Mises stress in the ligament parallel to the crack propagation direction

(meaning the shortest distance between the crack tip and the slits of the loading arms):

32 MIXED-MODE CRACK BEHAVIOR

shear crack growth The shaded area indicates the approximate initiation (Aa = 0.2 mm) region of the experiments

The stable cracks propagated as shear cracks