Analysis of Small Edge Cracks and Its Implications to Multi Axial Fatigue

8/6/2019 Analysis of Small Edge Cracks and Its Implications to Multi Axial Fatigue

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Y. Wang1

e-mail: [email protected]

J. Pane-mail: [email protected]

Department of Mechanical Engineering,

The University of Michigan,

Ann Arbor, MI 48109

Analysis of Small Edge Cracksand Its Implications to MultiaxialFatigue TheoriesThe near-tip fields of small edge (Case B) cracks in power-law hardening materials are

investigated under generalized plane strain, mixed mode, and general yielding conditionsby finite element analyses. The results of the J integral from the finite element analysesare used to correlate to a fatigue crack growth criterion for Case B cracks. The trend of constant J contours on the -plane is compared reasonably well with those of the experi-mental results of constant fatigue life and constant fatigue crack growth rate under multiaxial loading conditions. DOI: 10.1115/1.1342012

Keywords: Fatigue Crack Growth, General Yielding, Generalized Plane Strain, J Inte-gral, Low-Cycle Fatigue, Mixed-Mode Fracture, Multiaxial Fatigue, Power-Law Hard-ening Materials

1 Introduction

Experimental observations of low-cycle fatigue tests suggest

that the fatigue life can be divided into four stages: nucleation of fatigue cracks defined as initiation, growth on the maximumshear planes stage I crack growth 1, propagation normal to themaximum principal stress stage II crack growth 1, and finalfracture of specimens. At low strain amplitudes, most of the fa-tigue life may be taken up with the initiation phase. At high strainamplitudes, the majority of the fatigue life may be spent on propa-gating cracks, and the fracture mechanics approach is thereforeapplicable to estimate the fatigue life due to crack propagation.

Based on the linear elastic fracture mechanics LEFM, thestress intensity factor K has been extensively used to characterizefatigue crack growth. However, when small fatigue cracks aresubject to general yielding conditions, the use of LEFM to modelthe fatigue processes becomes questionable. Therefore, a nonlin-ear or elastic-plastic fracture mechanics concept based on the cy-

clic J integral has been used to correlate fatigue crack growthunder general yielding conditions 2,3. The results of these cor-relations indicate that the crack growth rate can be characterizedby the cyclic J integral under general yielding conditions.

Brown and Miller 4 proposed a multiaxial fatigue theorybased on the physical interpretation of the mechanisms of fatiguecrack growth. Their theory can be represented graphically by con-tours of constant fatigue life on the -plane where the maximum

shear strain12 * is the abscissa and the normal strain n

* on the

maximum shear strain plane is the ordinate. Each constant fatiguelife contour can be expressed mathematically by

1

2 * f n

* (1)

This equation underlines the strain parameters for correlating fa-tigue data, and indicates the importance of both the maximumshear strain and the tensile strain normal to the maximum shearstrain plane. Two types of stage I fatigue cracks are proposed:Case A for cracks propagating along the surface and Case B forcracks propagating away from the surface.

An investigation of Case A cracks has been carried out byWang and Pan 5,6. The results suggest that the growth in the

maximum shear strain direction for Case A fatigue cracks is pos-sibly the consequence of a crack growth criterion controlled bythe maximum effective strain of the near-tip strain fields. In addi-tion, the trend of constant J contours on the -plane is quite simi-lar to that of the constant fatigue life contours for 1 percent Cr-Mo-V steel 7 and that of constant crack growth rate for Type304 stainless steel at 550°C 8. In this paper, we examine thenear-tip fields of Case B cracks under generalized plane strain,mixed-mode, and general yielding conditions. We attempt to cor-relate the near-tip fields of the mixed-mode cracks to a criterion of fatigue crack growth under multiaxial loading conditions. Basedon the concept of the characterization of fatigue crack growth bythe cyclic J integral, we attempt to correlate the constant J of themixed-mode cracks under different strain ratios to the experimen-tal results of constant fatigue life for Case B cracks under multi-axial loading conditions.

2 Governing Equations

To examine the implications of plastic fracture mechanics tomultiaxial fatigue theories, we here present an analysis based on asimple nonlinear material model. We consider a power-law hard-ening material with a uniaxial tensile stress-strain relation as

0

0

n

(2)

where is the tensile strain, is the tensile stress, 0 and 0 arethe reference strain and stress, respectively, is a material con-stant, and n is the hardening exponent. When n is equal to 1, Eq.2 represents a linear elastic material. When n approaches infin-ity, Eq. 2 gives a rigid-perfectly plastic behavior. However, the

Ramberg-Osgood law is usually used to describe the nonlinearstress-strain behavior in uniaxial tension

0

0

0

n

(3)

Here, we usually take 0 0 / E where E is Young’s modulus.The Ramberg-Osgood law can be generalized to multiaxial

stress states. Here, the strains i j can be written as the sum of an

elastic part i je and a plastic part i j

p

i j i je i j

p (4)

with

1Currently at Optimal CAE, Inc., Novi, MI 48377.

Contributed by the Pressure Vessels and Piping Division and presented at the

Pressure Vessels and Piping Conference, Boston, Massachusetts, August 1–5, 1999,

of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS. Manuscript received by

the PVP Division, October 18, 2000; revised manuscript received October 20, 2000.

Editor: S. Y. Zamrik.

2 Õ Vol. 123, FEBRUARY 2001 Copyright © 2001 by ASME Transactions of the ASME

Downloaded 21 May 2011 to 129.2.63.169. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm



i je

1

E s i j

12

3 E kk i j (5)

and

i j p

3

2 e

0

n1 s i j

0(6)

where is Poisson’s ratio, s i j are the deviatoric stresses, which

are defined as s i j i j13 kk i j , and e((

32s i js i j)

1/2) is the ef-

fective stress. Here, i and j have a range of 1 to 3 and the sum-mation convention is adopted for repeated indices. Also, the ef-

fective plastic strain e is defined here as e(23 i j

p i j p

)1/2.

Consider a cracked body where the elastic strain componentsare negligible compared to their plastic counterparts. Equation 6can then be adequately used to describe the stress-strain behaviorfor deformation plasticity theory materials with power-law strainhardening. For power-law strain-hardening materials, when thetractions prescribed on the boundary of a cracked body are scaledby a loading parameter P, the stresses, strains, and displacementscan be scaled by P as 9

i jP i j x i ,n (7)

i j P

0

n

i j x i ,n (8)

where x i is the coordinate of the material element. The quantities i j and i j are functions of x i and n. These quantities are inde-

pendent of P because of the homogeneous nature of the equationsof equilibrium, compatibility, and the constitutive relation. Sincethe stresses and strains increase proportionally at every point, thefully plastic solution based upon deformation plasticity theory co-incides with the solution for the corresponding incremental orflow theory.

Since the integrand of the J integral 10 involves the productsof stresses and displacement gradients, based on Eqs. 7 and 8,

the fully plastic J will be proportional to Pn1. Then, the J inte-gral for power-law strain-hardening materials can be expressed as

J p 0 0ah a

w;n P

P0

n1

(9)

where P0 is a reference load in terms of 0 , a is the crack length,and w is the width of specimen. Here the subscript p is used todesignate the power-law material behavior. The dimensionlessfunction h depends on a / w, n, and possibly other normalizedgeometric parameters, but is independent of P. This equation willbe used later to derive the relationship between J and the bound-ary loading conditions in our finite element analysis.

Hutchinson 11,12, Rice and Rosengren 13, and Shih 14have shown that for power-law strain-hardening materials, thedominant asymptotic crack-tip stress, strain, and displacementfields for planar crack problems can be expressed as

i j 0 J

00 Ir

1/ n1

˜ i j ;n , M p (10)

i j 0J

00 Ir n / n1

˜

i j ;n, M p (11)

where r and are the polar coordinates centered at the crack tip,and I is a dimensionless constant. The dimensionless angularfunctions ˜ i j and i j depend on the hardening exponent n, thestate of plane strain or plane stress, and the mode mixity param-

eter M p 14. The HRR singularity will be used later for normal-izing the stresses and strains of our computational results.

3 Finite Element Model

Metallographic observations of Parsons and Pascoe 15 indi-cate that small fatigue cracks were initiated and propagated in the

direction of maximum shear strain into the specimen at 45 degaway from the specimen surface under low-cycle, biaxial stressingconditions. Based on the observations, we consider an edge-cracked panel under generalized plane strain conditions in theout-of-plane direction as shown in Fig. 1. We select a coordinatesystem such that the x and y-axes are the in-plane coordinates andthe z-axis is the out-of-plane coordinate. As shown in Fig. 1, theedge-cracked panel is subjected to biaxial strains in the y and z

directions. Also, I represents the largest principal strain and II

represents the second largest principal strain. The crack is as-sumed to be oriented in the maximum shear strain direction,which is 45 deg away from the specimen surface. In the figure, arepresents the crack length, and w represents the width of thepanel. Also, represents the crack orientation. Here, we considerthe case of 45 deg. The crack is therefore subject to in-planemixed-mode loading conditions. The strains I and II are appliedby displacing the top surface and the lateral surface uniformly inthe y and z directions.

Figure 2 shows the finite element model of the generalizedplane strain crack problem. Here, 8-node quadrilateral elementsare used. In the immediate crack-tip region, we have a ring of 32wedge-shaped elements. Collapsed nodes resulting in the 1/ r sin-gularity are used for these wedge-shaped elements. The entire

finite element model consists of 766 elements and 3955 nodes. Inthis study, we are interested in the near-tip fields of small cracksunder general yielding conditions. Therefore, the ratio a / w istaken as small and practical as possible to ensure that the effect of the crack tip plasticity on the remote boundary is minimum. Here,the ratio a / w is taken as 0.02.

All our computations are performed using the finite elementcode ABAQUS Version 5.5. We choose the Ramberg-Osgoodstress-strain relation to describe the material deformation plastic-ity behavior. We select n3 to represent high-hardening materi-als, n10 to represent low-hardening materials, and n20 toapproximate perfectly plastic materials. Also, we select the mate-rial constants such that the elastic strains are small and negligible

Fig. 1 An edge-cracked panel subject to biaxial strains

Journal of Pressure Vessel Technology FEBRUARY 2001, Vol. 123 Õ 3




when compared with the plastic strains. Therefore, our solutionscan be used to approximate those for pure power-law hardeningmaterials.

4 Mixed-Mode Crack-Tip Fields

For various combinations of the applied strains along theboundary, the near-tip stress and strain fields are investigated. Tolink our results with the existing multiaxial fatigue theories, wedefine a strain ratio as the ratio of the out-of-plane constraintstrain II in the z direction to the largest applied strain I in the ydirection as

II

I

(12)

We also defined the slope s on the

-plane as the ratio of thenormal strain n* on the maximum shear strain plane to the maxi-

mum shear strain12 *

sn

*

12 *

(13)

Note that the maximum shear strain occurs in the direction of 45deg from the I direction toward the III direction. Here, III

represents the normal strain in the direction perpendicular to thestress-free surface. Note that I II III for Case B cracks.

Therefore, n*

12( I III ) and

12 *

12( I III ). For example,

if incompressibility is assumed, the ratio is equal to 1/2 andthe slope s is equal to 1/3 under uniaxial tension. Under planestrain conditions ( II 0) , and s are both equal to 0. For equal

biaxial loading conditions ( I

II ), the ratio is equal to 1 andthe slope s is equal to 1/3. Consequently, the range of isbetween 1/2 uniaxial tension and 1 equal biaxial tension ands is between 1/3 uniaxial tension and 1/3 equal biaxial ten-sion for Case B cracks when incompressibility is assumed.

In our finite element computations, we set 0.4999, 1,

and E / 01.0104 in the Ramberg-Osgood constitutive law toapproximate the behavior of pure power-law hardening materials.We consider three materials with the hardening exponent n equalto 3, 10, and 20. We also select five applied strain ratios corre-sponding to 1/2, 1/4, 0, 1/3, and 1 or s1/3, 1/7, 0, 1/7and 1/3.

Figure 3 shows the angular variations of the near-tip stresses

normalized by 0( J / 00r )1/ n1 at the radial distance of r 2 J / 0 for n3. Figures 3a–e show the cases of 1/2,1/4, 0, 1/3, and 1 or s1/3, 1/7, 0, 1/7, and 1/3, respec-tively. Here, the polar coordinates r and are centered at the crack tip, and 0 deg represents the crack line direction which is ori-ented at 135 deg clockwise from the y-axis. Note again that for

Fig. 2 „a … The finite element model for an edge-cracked panel;„b … the finite element model of the near-tip region

Fig. 3 The angular variations of the near-tip stresses normal-

ized by 0„J Õ 00r …1 Õn ¿1 at r É2J Õ 0 for n Ä3. „a … ÄÀ1 Õ2, „b …

ÄÀ1 Õ4, „c … Ä0, „d … Ä1 Õ3, „e … Ä1.

4 Õ Vol. 123, FEBRUARY 2001 Transactions of the ASME




incompressible materials, 1/2 represents a uniaxial tensionwith uniform deformation in the z direction, 0 represents aplane strain loading, 1 represents an equal biaxial tensile load-ing, and 1/4 and 1/3 represent two intermediate cases. Thecommon characteristic of the near-tip stress fields for different ’sis that the maximum value of ¯ e is located at 15 deg. Inaddition, there is a second peak of ¯ e at 105 deg.

It should be noted that the normalized in-plane stress fields i.e., ¯ zz excluded in Figs. 3a (s1/3) and 3e (s1/3) havealmost the same distributions. So do the normalized stress fields inFigs. 3b (s1/7) and 3d (s1/7). These results indicate

that the loading conditions which are symmetrical to the12 *-axis

x-axis on the -plane introduce the same normalized in-plane

Fig. 4 The angular variations of the near-tip stresses normal-

ized by 0„J Õ 00r …1 Õn ¿1 at r É2J Õ 0 for n Ä10. „a … ÄÀ 1 Õ2,

„b … ÄÀ1 Õ4, „c … Ä0, „d … Ä1 Õ3, „e … Ä1.

Fig. 5 The angular variations of the near-tip strains normal-

ized by 0„J Õ 00r …n Õn ¿1 at r É2J Õ 0 for n Ä3. „a … ÄÀ1 Õ2,

„b … ÄÀ1 Õ4, „c … Ä0, „d … Ä1 Õ3, „e … Ä1.





stress distributions. Figure 4 shows similar results for n10. Asshown in both Figs. 3 and 4, the angular distributions of the nor-malized in-plane stresses are quite similar. However, as in-creases, the value of ¯ zz increases. This can be explained by aperfectly plastic analysis of singular plastic sectors of the near-tipfields 16, as detailed in Appendix A.

Figure 5 shows the angular variations of the near-tip strains

normalized by 0( J / 00r )n / n1 at the radial distance of r 2 J / 0 for n3. Figures 5a–e show the cases for

1/2, 1/4, 0, 1/3, and 1 or s1/3, 1/7, 0, 1/7, and 1/3,respectively. The common characteristic of the near-tip strainfields for different ’s is that the maximum value of e is locatedat 15deg. Also, there is a second peak of e at

105 deg. It should be noted that the normalized strain fields, eand r , in Figs. 5a and e have the same distributions. So dothe normalized strain fields in Figs. 5b and d . However, thenormalized strain fields, rr , , and zz , in Figs. 5a and e

have the same curve shapes, but with a constant difference be-cause of volume constancy. So do the strain fields in Figs. 5band d . Figure 6 shows similar results for n10. If we select aradial distance much smaller than the value of r 2 J / 0 , thenormalized zz will become smaller closer to zero. Then the near-tip strain fields for various s’s should approach those for s0under plane strain conditions as r decreases.

With the selection of a / w0.02 for the edge-cracked panel, theeffect of the crack-tip plasticity on the stress field of the remoteboundary is shown to be minimum by examining the stresses asfunctions of the radial distances to the tip. Therefore, our compu-tational results can be used to approximate the results for a / wapproaching zero. In this case, the J integral for pure power-lawhardening materials, based on the Il’yushin theory 9, can beexpressed as

J 0 0aH I ,n I

0n

1 / n(14)

or

J 0 0aH II ,n II

0n1 / n

(15)

where H I and H II are dimensionless functions of the hardeningexponent n and the strain ratio . The foregoing equations showthat J is simply expressed as a function of the hardening exponentn, one of the remote applied strains, and the strain ratio . Equa-tions 14 and 15 are used later in this paper to develop theconstant J contours on the -plane for correlation of Case B crack growth rate and fatigue life under multiaxial loading conditions.

5 Characterization of Low-Cycle Multiaxial Fatigue

The crack length that we consider in our modeling effort maybe in the range of a few grain sizes. When we consider the mod-eling of cracks in this length scale, the microstructure of the ma-terial, such as the grain orientation, inclusions, voids, and precipi-tates along the grain boundary should be considered. Other factorssuch as the elastic and plastic anisotropy of the grains, the residualstresses and plastic strains due to manufacturing processes andfatigue crack initiation processes should also be considered. Inaddition, crack closure under cyclic loading conditions should beconsidered, as indicated in Newman 17.

Here we intend to make some simplified assumption so that themixed-mode crack-tip field obtained here can be used to correlateto the fatigue life for Case B cracks. First, we assume that Case B

cracks observed in experiments such as those in Parsons and Pas-coe 15 are noninteracting to each other. Then, our small crack model can be applicable. When the interaction of multiple cracksneeds to be considered, the boundary condition of the edge-cracked panel and the ratio a / w can be modified to reflect theinteraction of the plasticity from the multiple crack tips.

It is well known that surface fatigue cracks are initiated byextrusion and intrusion processes under cyclic loading conditions.The slip processes may occur on the crystallographic planeswhere the resolved shear stresses are largest. From the viewpointof a two-dimensional crystal model, the orientation of the fatiguecracks should be 45 deg inclined to the largest principal stressdirection. However, due to the discreteness nature of available slip

Fig. 6 The angular variations of the near-tip strains normal-

ized by 0„J Õ 00r …n Õn ¿1 at r É2J Õ 0 for n Ä10. „a …

ÄÀ1 Õ2, „b … ÄÀ1 Õ4, „c … Ä0, „d … Ä1 Õ3, „e … Ä1.





planes and directions in each grain, the surface fatigue crack planes may more or less appear to be 45 deg inclined to the largestprincipal stress direction in those grains with favorable slip planesand directions for extrusion and intrusion processes. Therefore,fatigue cracks with crack planes 45 deg inclined to the largestprincipal stress direction seem to be a good starting point forstudying stage I Case B fatigue cracks.

The crack-tip plastic deformation and the closure phenomenondue to unloading are rather complex. If a crack propagates in thesame direction under a quite large number of cyclic loads underlow-cycle fatigue conditions, certain self-similar or steady-state

conditions may prevail near the tip. If stabilized stress-strain hys-teresis loops of the material elements occur near the propagatingcrack tip, the near-tip fields that we obtained for the deformationplasticity materials should shed some light on the mechanics of fatigue crack propagation. As shown in Figs. 5 and 6, the maxi-mum value of the normalized near-tip effective strain e is locatedclose to 15 deg for different ’s. In other words, the crack orientation could be about 30 deg inclined to the stress-free sur-face for this isotropic model. When the crack is reoriented at 30deg from the stress-free surface, our computational results showthat the maximum effective plastic strain is located directly aheadthe tip under various applied strain ratios. If we argue that thecrack growth direction is controlled by the maximum value of theeffective plastic strain e as for Case A cracks, then Case B cracksshould be oriented at 30 deg from the stress-free surface, and this

leads to a conflict with the experimental observations of Case Bcracks 4,15. Therefore, more experimental and analytical worksare needed, or a new criterion could be suggested.

Zheng and Liu 18 argued that under low-cycle fatigue condi-tions, stabilized stress-strain hysteresis loops occur due to the highcyclic strains experienced by the material elements near thepropagating crack tip. Therefore, it is justified to use the cyclicstress-strain relation to calculate the crack-tip stress and strainfields. They further argued that under completely reversed cyclicloads, the tensile half of the hysteresis loop causes the crack toopen and grow. Then it is justified to make a monotonic loadingcalculation for the cyclic crack-tip stress and strain fields.

In order to correlate the J integral and the fatigue life under

multiaxial loading conditions, we plot the computational resultsfor constant J values, marked by symbols, for n3, 10, and 20 onthe -plane in Fig. 7. Note that the constant fatigue life contoursfor several metals were plotted on the -plane in Brown andMiller 4. Since we are working with pure power-law hardeningmaterials, the constant J curves on the plane are self-similar.These constant J contours can be fitted by two linear equations

12 *

A

n*

B1 for the upper half of the -plane (16)

12 *

An*

B1 for the lower half of the -plane (17)

where A and B are constants. Table 1 lists the values of A, B, and B / A for n3, 10, and 20. The fitted curves are also shown as thesolid lines in Fig. 7. It should be noted that the constant J contours

are symmetrical to the12 *-axis on the -plane regardless of the

hardening exponent n. As detailed in Appendix B for the linearelastic case ( n1), the constant J contours are perpendicular to,

and therefore symmetrical to, the12 *-axis. In Fig. 7, several

computational results for n1 are also plotted for comparison.

Fig. 7 The computational results for constant J values, marked by symbols, for n Ä1, 3, 10, and 20 on the -plane. For n Ä3, 10,and 20, the fitted solid lines are represented by Eqs. „16… and „17….

Table 1 Values of A, B , and B ÕA for n Ä1, 3, 10, and 20 for the

fittedJ

contours shown in Fig. 7





The computational results agree well with the theoretical results

which are straight lines perpendicular to the12 *-axis. The trend

of B / A for n20, 10, and 3 seems to agree with the limit of B / A for n1 as listed in Table 1. The general trends of thesecontours in Fig. 7 are similar to those of constant fatigue lifeobtained from experiments for QT 35 steel aluminum alloy 4.As listed in Table 1, as n increases, the ratio B / A representing theslope of the fitted curve decreases, and, consequently, the effect of

n* on fatigue life increases.

6 Conclusions

Based on the experimental observations of Case B fatiguecracks, a small crack model is proposed to investigate low-cyclefatigue under multiaxial loading conditions. We first present theresults of the near-tip stress and strain fields for cracks in power-law hardening materials under generalized plane strain, mixedmode, and general yielding conditions. Based on the crack growthcriterion controlled by the maximum effective strain as for Case Acracks, the results of the near-tip strain fields suggest that thecrack should propagate into the specimen at about 30 deg inclinedto the specimen surface for Case B cracks. Based on the modelwith a crack at 45 deg inclined to the stress-free surface, the trendof constant J contours from the computational results is quitesimilar to that of the constant fatigue life contours on the -planeobtained from experiments.

Here we use a simple nonlinear fracture mechanics method to

model Case B mixed-mode cracks under generalized plane strainconditions. The results correlate well with the corresponding mul-tiaxial fatigue experimental results. Also, a similar nonlinear frac-ture mechanics method was used to model Case A mixed-modecracks under plane stress conditions 6. The results also correlatewell with the corresponding multiaxial fatigue experimental re-sults. If one considers the drastic simplification of Case A and Bsurface cracks to the corresponding two-dimensional idealizedcracks, and our computations under monotonically increasingloading conditions, instead of under cyclic loading conditions, theresults from the use of fracture mechanics for the investigation of the first-order effects of multiaxiality are surprisingly good. Inaddition, for Case A cracks, the use of the concept of constrainteffects of the nonlinear fracture mechanics can tell us more thanwe can normally extract from the multiaxial experimental results

19. Therefore, further use of fracture mechanics to gain moreunderstanding of multiaxial fatigue and apply the results to predictthe durability of structures in conjunction with the critical planeapproach appears to be very promising.

Appendix A

Consider a crack in a perfectly plastic solid. The yield functionof the material can be expressed as

f 3s i js i j /2 020 (18)

where s i j are the deviatoric stresses and 0 is the yield stress. The

associated flow rule gives the in-plane plastic strain rates p

as

ps (19)

where is a proportionality and the subscripts and have arange of 1 to 2. The out-of-plane plastic strain rate zz is ex-pressed as

zz ps zz

132 zz rr (20)

Consider a singular plastic sector near the crack 16 such that

at least one component of p becomes as r →0. Since s are

finite, → as r →0. Also, 0. Since zz is finite, zz→( rr

)/2 as r →0. From Eq. 12, zz / yy under proportionalstraining conditions. For the loading conditions considered here, yy0. When r is not small enough at a finite value, Eq. 20 is

still applicable with being finite, then

0, zz0, zz rr /2 (21)

0, zz0, zz rr /2 (22)

0, zz0, zz rr /2 (23)

As shown in Figs. 3 and 4, when 0, zz( rr )/2 andwhen 0, zz( rr )/2. However, when 0, zz( rr

)/2. Theoretically speaking, as we decrease the value of r further from 2 J / 0 , even for the cases of 0, zz should ap-proach ( rr )/2 in the singular plastic sectors for perfectlyplastic materials.

Appendix B

First we consider an edge-cracked panel as shown in Fig. 1, butwith a crack perpendicular to the free surface ( 90 deg) forsimplicity. Here we consider the linear elastic case ( n1). As theratio a / w approaches 0, the stress intensity factor K I due to theremote normal stress yy , which is perpendicular to the crack, is

K I F I yy a (24)

where F I is a dimensionless constant. F I is equal to 1.12 whena / w→0. Note that a constant out-of-plane normal strain will notaffect the singularity nature of the crack-tip field characterized bythe stress intensity factor K . The energy release rate J is related toK I under plane strain conditions as

J K I

21 2

E (25)

Substituting Eq. 24 into Eq. 25 gives

JE

1 2 aF I

2 yy2

(26)

For the case as shown in Fig. 1, the normal stress yy and thetransverse stress zz are expressed in terms of I and II accord-ing to Hooke’s law as

yy

E

1 2 I

E

1 2 II (27)

zz

E

1 2 I

E

1 2 II (28)

Consequently, III can be expressed as

III

E yy zz (29)

1 I II (30)

Combining Eqs. 26 and 27 gives

J 1 2

E aF I

2 I II 2 (31)

Note that n* and

12 * are defined as

n*

I III

2(32)

1

2 *

I III

2(33)

Substituting Eq. 30 into Eqs. 32 and 33 gives

n*

12 I II

21 (34)

1

2 *

I II

21 (35)





Combining Eqs. 31 and 35 gives J as

J 1 2

E aF I

2 21 1

2 *

2

(36)

Equation 36 clearly indicates that J is only a function of 12 *

and is independent of n* on the -plane for the linear elastic case.

This means that constant J contours on the -plane are verticalstraight lines. However, the value of affects significantly thevalue of J . For example, J increases by 61.5 percent, when

changes from 0.5 to 0.3.For an edge-cracked panel as shown in Fig. 1 with an inclined

crack, the stress intensity factors K I and K II can be expressed as

K I F I yy sin2 a (37)

K II F II yy sin cos a (38)

where represents the crack orientation as shown in Fig. 1, andF I and F II are the dimensionless constants that depend on thecrack orientation . For example, F I is about 1.41 and F II is about0.725 from our linear elastic finite element computational resultsfor 45 deg.

Under plane strain mixed-mode loading conditions, J is relatedto K I and K II as

J 1 2

E K I

2K II

2 (39)

Combining Eqs. 37 to 39 gives the relationship

JE

1 2 a yy

2F I

2sin2 F II

2cos2 sin2 (40)

Note that Eq. 40 reduces to Eq. 24 for 90 deg. Note alsothat the right side of Eq. 40 depends on the crack orientation.Similar to the case of 90 deg, we can derive

J 1 2

E aF I

2sin2 F II

2cos2 sin2 21

1

2 *

2

(41)

for mixed-mode cracks. Therefore, for a mixed-mode crack with a

given crack orientation, J is still only a function of 12 * and is

independent of n* on the -plane for the linear elastic case. Thus,

constant J contours are vertical straight lines on the -plane.However, the crack orientation and Poisson’s ratio can stillaffect the value of J . For example, when changes from 45 to 90deg for a given a and 0.5, J increases by 41 percent.

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PA, pp. 82–103.

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Analysis of Small Edge Cracks and Its Implications to Multi Axial Fatigue