AN ANALYSIS OF AN ANTIPLANE SHEAR CRACK IN A

NONHOMOGENEOUS ELASTIC MEDIUM

by

WILLIAM TODD TIMMONS, B.S. in Ch.E.

A THESIS

IN

MATHEMATICS

Submitted to the Graduate Faculty of Texas Tech University in

Partial Fulfillment of the Requirements for

the Degree of

MASTER OF SCIENCE

August, 1988

1

ACKNOWLEDGEMENTS

I would Uke to express my appreciation to my committee chairman. Dr.

Lawrence Schovanec, for his encouragement and direction. I would also like

to thank my other committee members. Dr. Ronald Anderson and Dr. Darell

Johnson. In addition, I would like to thank my wife, Becky, for her help in typ­

ing, proofreading, graphics, and other important details of this thesis. Financial

support from the Air Force Office of Scientific Research, Boiling AFB. DC, is

gratefully acknowledged.

n

CONTENTS

ACKNOWLEDGExMENTS ii

ABSTRACT iv

LIST OF FIGURES v

I. INTRODUCTION ! 1

II. ENERGY RELEASE RATE 17

III. ALTERNATE FRACTURE THEORIES 30

REFERENCES 39

ni

ABSTRACT

In this thesis, a rigorous derivation of the energy release rate based on the

change in potential energy of a body is given for a nonhomogeneous linear elastic

medium. The energy release rate is calculated for an antiplane shear crack whose

shear modulus corresponds to a reduced rigidity about the crack tip. Plastic

zones about the crack tip are calculated based upon the yielding condition of

von Mises, and the effect of decreasing rigidity upon these zones is displayed.

In addition, the crack problem is analyzed within the framework of the strain

energy density theory and the maximum cleavage stress theory.

IV

LIST OF FIGURES

Fig. 1.1. Deformation of a body 3

Fig. 1.2. The stress vector at x acting on the surface S 5

Fig. 1.3. (a.) Physical interpretation of the shear modulus

(b.) Physical interpretation of Poisson's ratio 7

Fig. 1.4. The three modes of crack loading 11

Fig. 1.5. Calculation of Q from work done by the crack tip stress field 13

Fig. 1.6. The crack geometry 15

Fig. 2.1. Comparison of potential energy in two notched elastic bodies 20

Fig. 2.2. The normsdized energy release rate 26

Fig. 2.3. Normalized crack surface displacement versus T/TQ 27

Fig. 2.4. The nondimensionalized shear modulus versus r/r. 28

Fig. 3.1. Plastic zone shapes according to the von Mises criterion 31

Fig. 3.2. The angular variation of the normalized cleavage stress 33

Fig. 3.3. The region of reduced rigidity and the core region around

the tip of a loaded crack 34

Fig. 3.4. The normalized strain energy density for r" = r^ 36

Fig. 3.5. The effect of the size of the core region on the SED when the

crack is embedded in the region of reduced rigidity 37

V

CHAPTER I

INTRODUCTION

Continuum mechanics is concerned with analyzing the behavior of a solid

body, regarded as a continuous meditun, under the influence of applied forces.

The theory of continuum, or solid, mechanics is built upon the basic concepts of

stress, motion and deformation, balance of linear and angular momentum, and

constituitive relations which describe the mechanical properties of a particular

materizd. Fracture mechanics is a branch of soHd mechanics which seeks to

quantify conditions under which a body containing a defect such as a crack would

fail to meet its designed purpose due to enlargement of the defect. The beginning

of the study of crack behavior in solids is marked by the work of A.A. Griffith [Ij

who, by considering a global energy balance, recognized that initiation of crack

growth is possible when energy released due to crack extension is siifficient to

create new crack surface. This theory led to the notion of the so-called energy

release rate (ERR), which is simply the energy available for fracture. In early

applications of Griffith's theory the ERR was calculated from a knowledge of

the stresses acting throughout the entire body. This led Irwin [2] to consider the

singular stress field near the crack tip. He wa.s able to show that the coefficient of

the dominant singular term in the expression for the stress field at the crack tip is

directly related to the energy release rate. Irwin called this coefficient the stress

intensity factor (SIF). The stress intensity factor connects the global concepts

of Griffith to a more calculable crack tip parameter and the energy release rate

can be determined from it in a simple manner. Two important fracture criteria

are based upon the idea that the onset of crack propagation will occur when the

ERR or the SIF reach some critical value.

The Elements of Solid Mechanics

Before addressing the problem of this thesis, a brief review of the concepts

of solid mechanics is given. For a more extensive discussion of these topics one

may wish to consiilt [3] or [4]. Identify the undeformed configuration of a body

with a region B referred to a fixed frame of reference (see Fig. 1.1). Subsequent

configurations of the body may be characterized by a mapping that carries the

point p into a point x = f{p). The displacement vector, n, is defined by

2L = f{p) -p = x-p.

If d£ = Pi — P2 denotes the vector connecting two points pi and p2 in the unde­

formed configuration, then ^ = u{pi) — u{p2) indicates the relative change in

position of pi and p2 before and after deformation. If c = |!Vu||. then to within

an error 0{£^) a.s e —^ 0,

dx = [Vu(pi)]ffp

from which it follows that

|ix||^ - ||Jp||^ = [VU^VTJ ^'ViiVvFidp-dp.

Thus in the case that e is very small the strain tensor, which is a measure of

deformation, is defined by

£ = i ( V a - V a ^ ) . (1.1)

(With but one exception, the word tensor will mean a linear map from R ^ —» R "',

i.e., an element of J L ( R ' ^ , R ^ ) . The exception being the elasticity tensor, which

is a Unear mapping from X(R"^,R^) into itself.) In terms of components

1 dui duj

In the mechanics of continuous bodies, two types of forces must be considered:

1. Body forces are forces acting on an element of volume. A typical example

is gravity.

2. Surface forces, or tractions, are forces acting on units of area within the

body or its boundary. Roughly speaking these forces are thought of as

being exerted on the surface elements of adjacent volume elements.

X = f(p)

fCB)

Fig. 1.1. Deformation of a bodv

4

Consider a small volume V containing a point x e f{B) and a surface 5 passing

through it with normal n at x (Fig. 1.2). The surface 5 partitions T' into the

components V'"*" and V~ with n pointing into V^. The force exerted across 5

by V"^ on V~ is described by the stress vector at x with respect to n, denoted

l„ (x) . Thus a complete description of "stress"' in a body depends on position as

well as orientation.

Suppose a body undergoes a deformation due to various applied forces and

displacements and, after a period of time, achieves a state of equilibrium. This

state of equilibrium is characterized by the laws of balance of angular momentum

/ s^ds - / bdV = 0 Jd(f(B)) - Jf(B)~ fd{f{B)) - Jf{B)

and

/ ( r X i»)<i3 ^ / r X 6 (fV = 0, Jd{f{B)) - Jf{B)

where k denotes the body forces on f{B) and r(x) denotes the position vector

corresponding to a point x e f{B). A major result of continuum mechanics, the

Cauchy-Poisson theory [3], asserts that if the above momentum balance laws are

satisfied then there exists a tensor field S on f{B) called the stress tensor such

that

1. For each unit vector n,

[S{x)]n = i„(x),

2. 5 = 5"^, and

3. S satisfies the equation of equiHbrium

divS-b = 0. (1.2)

If the components of 5 are denoted by 5 = [cij], then divS is the vector

(ciz.5),- = ^ ^ = o-ij,j-

(In the above expression the simimation convention has been introduced. Hence

forth, the repetition of an index in a term will denote a sunmiation with respect

Fig. 1.2. The stress vector at x acting on the surface S

to that index over its range.) One can show that the stress vector acting on an

infinitesimal surface with normal n at the point x is given by

In(a^) = [S{x)]n.

The concepts of stress and strain are valid for all materials. One material

is distinguished from another by its constituitive law, which characterizes the

relation between the stressed state of the body and the corresponding deforma­

tion. A material which exhibits a linear relationship between stress and strain is

said to be a linear elastic material. In a linear elastic material the relationship

between stress and strain is given by a generalized form of Hooke's Law

S(x) = [C{x)]E{x), (1.3)

where C is called the elasticity tensor. When C is independent of position, the

material is said to be homogeneous. For an isotropic material (one in which

the elastic properties of the body do not vary with orientation within the body)

equation (1.3) is given by

2fiu (Tij = 2f££ij + ——^{£kk)Sij,

where fi is the shear modulus ajid v is called Poisson's ratio. The shear modulus

represents the ratio of the shearing stress to the change in the angle associated

with the shearing strain (see Fig 1.3). The shear modulus describes the rigidity

of a material. If an element is subjected to a purely tensile stress in the direction

of £i, the ratio of the extension of a linear element in the direction of e^ to the

contraction of an element perpendicular to it is given by Poisson's ratio (Fig. 1.3).

An elastic material may also be characterized by the existence of a scalar

function W of strain such that the stresses are derivable from it by the formula

For a linear elastic material, W-referred to as the strain energy density (SED)-is

given by

W = lc[E] • E. 2 ^ ^

!a

(a.) 9 i 2 * 2 e i 2 ' y = S i 2 / a i 2

a,. ; ^ 2 2 1 1

1 1

c 1 1

V = | e 2 z | / | G l t i

(b.)

Fig. 1.3. (a.) Physical interpretation of the shear modulus

(b.) Physical interpretation of Poisson's ratio

8

The SED has the physical interpretation of energy of deformation per unit volume

of material. A consideration of the work done by external forces causing the

deformation shows that the mechanical energy stored in the body is given by

U= f WdV, Jf{B)

where U is called the strain energy of the body.

In summary, the system of field equations that describe the static behavior of

an elastic body within the framework of the Hnear theory consist of the strain-

displacement relation (1.1), the equation of equilibrium (1.2), and the stress-strain

relationship (1.3). A list 5 = [u^E.S] of fields which satisfy equations (1.1 -1.3)

is called aji elastic state.

To solve a boundary value problem in elastostatics means to find an elastic

state in a body B when components of the displacement vector or the stress

tensor are prescribed on the boundary of 5 . If the displacement vector u is given

on dB one has the displacement problem. If surface forces are specified on the

boundary of 5 , i.e., Sn = 1 for n a unit exterior normal on the surface of B, the

boundary value problem is referred to as a traction problem. If a combination

of displacements and stresses are specified, then a mixed problem results. It is

important to note that in the linear theory the boundary conditions are always

prescribed with respect to the undeformed configuration and that the governing

equations are solved in this configuration.

Two theorems basic to elastostatics are Betti's Reciprical Theorem and the

Principle of Virtual Work [3]. These theorems are presented here for future

reference.

THEOREM 1 (Betti's Reciprocal Theorem)

Let s = [u, E, S] and s = [u, E, S] be elastic states corresponding to bod^

forces b and 6, respectively. Then

/ Sn • ^ dS -T b • u dV = / Sn • u dS ~r / b - u dV JdB Js" JdB JB"

= J S-E dV = J S-E dV.

A special case of Betti's Reciproccd Theorem is sometimes referred to CLS the

Principle of Virtuzd Work.

THEOREM 2 (Principle of Virtual Work)

Let 5 be a smooth tensor field on B such that divS — 6 = 0. Let u be a

smooth vector field and define E = ^[Vu -r Vu^]. Then

/ Sn-uds^ f b-udV= f S-EdV. JdB JB " JB

Denote the boundary of 5 as the disjoint union of Su and ST and let 5 = [u, E, S]

be an elastic state on B that satisfies the additional boundary conditions u = UQ

on Su and Sn = i on ST- Define u = 6u where ^u = 0 on 5c;, and 6E =

^[Vu -h VyJ']. Within this setting, the Principle of Virtual Work may be stated

as

f t'Sikds^ [ k'Su= I S-SE dV. JST JB JB

Linear Elastic Fracture Mechanics

A special type of boundary value problem arises when a body contains a

crack-like defect. Physically a crack corresponds to a flat cavity in the body.

Mathematically it is described by a surface of discontinuity in the displacement

field and a special set of boundary conditions. There are three fundamental types

of boundary value problems that arise in the study of crack problems. These dif­

ferent types of BVP's correspond to the possible movement of the crack surfaces

and depend on the manner in which extemsd forces are applied to the body. In

this thesis these problems are discussed under the assumption that the body is in

a state of plane strain or antiplane strain. In a state of plane strain or antiplatne

strain all field quantities are assumed to be functions of only two coordinates.

10

say Xi and X2. In a state of plane stredn the displacement component u^ is zero,

while a state of antiplane strain is described by a displacement field with only one

nonzero component, say uz = U3(xi, X2). A mode I crack, or opening mode crack,

occurs when applied loads acting in the X2 direction, 22( 15 0)? tend to open the

crack (Fig. 1.4). In a mode II crack the stresses acting in the Xi direction tend

to "slide" one crack surface over the other. An antiplane shear crack (or mode

III crack) occurs when the stresses acting in the X3 direction, o'23(xi,0), tend to

"tear" the crack surfaces apart.

It has already been mentioned that two methods of predicting failure in linear

elastic fracture mechanics are based upon the notions of the energy release rate

(ERR) and the stress intensity factor (SIF). Griffith employed an energy bal­

ance approach to show that for a body of unit thickness the condition for crack

propagation is d,_ . . . dW

^{F-U) = da. da.

where U is the elastic strain energy of the body, F is the work done on the

body by external forces, W is the energy required for crack growth, and da is

the incremental crack growth. Since the total potential energy of the body is

P = U — F, the energy release rate may be defined as

da

Griffith's fracture criterion for initiation of crack growth is given by ^ = ^c where

Qc is the critical energy release rate required for crack propagation.

For the sake of convenience the concept of the SIF will be illustrated in the

case of a mode I crack. It can be shown [6] that the asymptotic form of the

stresses and the displacement of the crack surface for a mode I crack are given

by

V27rr

and

where r denotes a distance measured from the crack tip and jxj{9) is a function

of the angle made with the crack tip. K[ is called the stress intensity factor and

11

MODE / OPENfNC M O O £

MODE /' SLIDING moos

MODE III TEARING MkO0£

Fig. 1.4. The three modes of crack loading (after [5])

12

is given by

Ki = (TyTra,

where <T is the applied load and a is a length scale associated with the loaded

crack. A formed argument to relate the SIF to the ERR is provided by considering

forces applied to the surface of the crack sufficient to close the crack over an

infinitesimal distance Sa (see Fig. 1.5). The work done by these forces will be

released as energy when the forces are released. It then follows that the energy

release rate is given by

1 /•*» ^7 = lim ~ / o-22(r,0)v(r - ^a,0)<fr. (1.6)

6a—*o oa Jo

It can be shown from equations (1.4 - 1.6) that the relationship between the

energy release rate atnd the stress intensity factor for the mode I problem is

C ^ /

Similar relationships can be derived for modes II and III. In contrast to Griffith's

globsd energy criterion, a local fracture criterion is provided by the SIF if one

postulates that crack growth occurs when Kj attains a critical value, Kic-

Statement of Problem

The specific problem to be studied in this thesis corresponds to a semi-infinite

antiplane shear crack in a nonhomogeneous medium. The crack is assumed to lie

along the negative X2-axis and the body is assumed to be in a state of antiplane

strain. The nonzero components of strain are given by

1 duz

ajid

The nonzero components of

£31 =

£z2 —

stress are

2^X1

1 du2

2 5X2

1

•

duz

d 1 ^2

13

a22

Xi

Fig. 1.5. Calculation of Q from work done by the crack tip stress field

14

and

0-31 = 2fi£2i = M-7—. ox I

For a state of antiplane strain there is but one equation of equilibrium

^^31 _ d€rz2 _ «

5xi 5X2

which, when expressed in terms of displacements becomes

^ / ^^3 >, ^ 5 5U3

axi oxi 0x2 0x2

If the problem is assumed symmetric around the crack plane, it suffices to solve

the equation of equilibrium in the upper half plane subject to the boundary

conditions

ti3(xi,0) = 0 , xi > 0

and

(T23{xi,0) = <r{xi) , Xi < 0,

where <r(xi) is the traction applied to the crack surface. These boundary condi­

tions simply mean that the displacement ahead of the crack tip is zero and the

stress behind the crack tip is equal to the applied load.

The type of material inhomogeneity considered here corresponds to a reduced

rigidity in the vicinity of the crack tip. Consequently, this problem is most con­

veniently analyzed in terms of polar coordinates (r, 6) wdth r = 0 correspond­

ing to the crack tip (see Fig. 1.6). The shear modulus is assumed in the form

/x(r) = fj.Q{r/rc)^, where /3 is called the inhomogeneity paramieter (0 < -5 < 1)

and Tc is a characteristic distance from the crack tip which serves to nondimen-

sionilize the quantity [T/TCY. In terms of polar coordinates, if the displacement

is denoted as a; = u;{r,9), then the nonzero stresses are given by

fi dijj

and OiJ or

15

Fig. 1.6. The crack geometry

16

The equation of equilibrium is

g;ir^r,) + g^(<r,.) = 0,

which expressed in terms of the displacement yields

. d^LJ du.,d(jj iid^uj M ^ H T + / - ^\^ + -^77 = 0.

OT^ or Or T OO^

Finally, the boundary conditions in polar coordinates become

a;(r,0) = 0

and

(yezir^Tr) = <T{r).

An analytic solution for the stress and displacement for this problem was

derived in [7]. One objective of this thesis is to calculate the energy release

rate for this crack problem. This will be achieved by showing that the local

work argument described by (1.6) is valid for nonhomogeneous media. Numerical

illustrations reveal that the ERR is influenced by the inhomogeneity exponent

/? and an additional parameter which relates the size of the zone determined by

Tc to the length of the loaded crack. What emerges from this calculation is that

unlike the case of a homogeneous medium no simple relationship between the

coefficient of the dominant singular term in the stress field and the ERR exists.

With softening of the material around the crack tip, extensive deformation and

crack tip blunting is observed in conjunction with a reduced stress level and

ERR. Consequently, the load required to cause fracture is elevated with the

attendant possibility of substantial yielding or inelastic deformation. In view of

this behavior, plastic zones about the crack tip axe calculated based upon the

yielding condition of von Mises and the effect of decreasing rigidity upon these

zones is displayed. In addition, the crack problem under consideration is also

analyzed within the framework of the strain energy density failure criterion [8i

and the maximum cleavage stress theory utilized in [9].

CHAPTER II

ENERGY RELEASE RATE

For the sake of future reference it is convenient to summarize the relevant

results of [7]. Explicit formulas, valid throughout the body, for the out of plane

displacement a; and the nonzero stresses cg^ and Crz are given in the above paper.

Here only the asymptotic form of the crack surface displacement near the crack

tip and expressions for the stresses when r < r^ are provided in the case of a

crack surface traction

( \ I \ / ^0 0 < r < ro

[ 0 r > ro

The results that are derived here depend only on the asymptotic form of the field

quantities given in [7]. Consequently, the qualitative nature of the results to be

presented are unchanged for any form of applied loading that is bounded near

the crack tip and satisfies <r(r) = 0 (7 ) , r — 00 [7i. For the problem under

consideration, three cases must be distinguished:

case i) 0 < /? < f

. c o s [ ^ ^ / ^ ^ ^

^^'(^'^) = "°cos[7rv/l-/3] ^ ^^ 2a, ^ ( - l )^^^-cos [<?(^ ) ] (^ )^ - - -^

( a\ A s i n j g y T - ^ ] ^ COSlTTVl — pi

2 ^ - (-i)*^M/? - ^J0'' (2A: - ly] sm[dC-^yijy-^^-' ^ fc=i ^/32 ^ (2A: - 1)2;2 - 3 - ^/3'- - (2k - l)-

a;(r.T . ., r = = = y ( —)^( —) ^ (-.3)

17

18

case ii) 13 = -

^,.{r,B) = f » [ ( - 4 1 n ( - ) - ? ) c o s ( 5 ) + 5flsm(5)i (2.4)

^ 2 £ j " _ ( - i ) ' - i c o s [ 9 ( 3 ^ ) ] ( ^ ) ' - " - «

" ^^ Vl + (5jrT)n2 - ? - p ' + {2k - 1)2]

..(r,<») = - i P M ^ ) - g ) s i n ( g ) - . f c o s ( g ) l (2.0)

2<ro C-l)*^'!/? - 7 /3 ' - (2fe - 1)2] s i n j g ( ^ ) ] ( y ) ' ^ " - ^

^ h ^0' + (2A - 1)2(2 - ^ - y/32 + (2* - 1)']

^ , , , ) . ( ^ ) ( ^ ) ^ ( J l ) i l „ ( i L ) . (2.6)

In these expressions

_l3-y//3^^{2k-iy Sjg — ^ , AC — 1 , 2 , O , . . . .

For case iii), /? > 3/4, the stresses are bounded and are given by series expansions

identical in form to those of case i) while the asymptotic form of the crack surface

displacement near the crack tip is given by

/ N f^o^O\/^^\0 sin[7rVl-/3] / M-/3 /O-N

It is important to note that for this problem the stresses do not exhibit a

square root singularity that is normally associated with a crack in a homoge­

neous medium. In fact, the stresses change from an algebraic to a logarithimic

singularity as /? increases, and are bounded for jS > ^.

Derivation of the Energy Release Rate

A heuristic argument for the form of the energy release rate was given in

Chapter I. The ERR was expressed as the work done by the stresses ahead of the

crack tip in moving through displacements behind the tip. By considering the

change in potential energy of a body, a rigorous argument will now be given to

19

establish that this form for the energy release rate is valid for an inhomogeneous

linear elastic material.

Suppose a body of volume V which contains a traction free void or notch is

loaded by surface tractions Tf on a portion of its boundary ST and by prescribed

displacements it° on Su- While holding the loading and displacements on ST and

Su fixed, the notch is enlarged by volume AV with surface AS (see Fig. 2.1).

Assume that the body forces acting on the material are negligible. If £°j represents

the state of strain in the initial state of the body before the void has enlarged,

then the potential energy of the initial state is given by

P" = lyV(el)dV - J^ ifuldS. 'ST

Here W is the strain energy density and is given by W = ^£' • 5 for a hnear

ela.stic material. Denote the displacements, strains, and stresses in the final state

by Uj = uj -h Atii, Cij — (j\^ ~ A<rij, and lij — £\- + Ae^j, respectively, and the

potential energy of the body in the final state by

P = P° + A P = / y^{£ii)dV - f T°u,dS. Jv-^v JST

Then the change in the potential energy from the initial to the final state is given

by

-AP = J W(4)(iV' - J^ TS'dS

- I W(f. )(iV - / TfuidS JV-£^V JST

= / W(c%)dV-l ;w(?i^)-w(£°)!<iv

^ I Jf(ui-u^)dS. (2.8) JST 'ST

By the Principle of Virtual Work

/ ((7°. + AcTij)njAuidS = f ( 4 + Aai^)A£x,dV. Jd{v-Avy ' Jv-Av ^

Since Aui = 0 on 5c;, A(T,jnj = 0 on 5 r , and the void is traction free,

/ (cr°- ^ AcT,j)njAuidS = f JfAuidS. ya(v--AV) ^ J ST

20

^IfA a. .n.=T.

Fig. 2.1. Comparison of potential energy in two notched elastic bodies

21

After substituting the above expression into (2.8), the change in potentieJ energy

may be written as

- A P = / W{£%)dV JAV ^

+ J (4 + A<ri,.)A£o- - [W(£?, - Ae.-,) - W(£»-)]dF JV—AV

Expanding J2, one has

- i[(<r?. + A<rij)(4 + A £ i ^ ) - 4 4 ] < i V

- I [<^A + ^<^iA + 4A^'i - A<ryA£,, - ^^ey,dV

= i.,,^A<r.,A.., + i<.A.<,-iA.,4<iV.

Betti's Reciprocal Theorem gives

/ <.A£i,(fy = / A^,,£°.ciV'. Jv-AV ^ JV-AV ^

Therefore

/2 = / -A(ri^A£ij(iV'. Jv-AV2

Another application of the Principle of Virtual Work yields

I2 = - I AcijnjAuidS 2 Jd{V-AV)

= - I AaijnjAuidS + - / AcTijnjAuidS - - / AaijnjAmdS. 2 JST 2 JSu ^ •'^•5

Again, recalling that Acij and Aui vanish on ST and Su, respectively, one finds

that

I2 = •- I Aa-ijnjAuidS 2 J AS

22

The notch is assumed to be traction free so aijnj = 0 and hence

12 = - - / (T^jnjAuidS. Z J AS

Therefore the change in potential energy of the body in which a notch has un­

dergone an enlargement AV is given by

-AP = /^^W(4)<iV-i/^^<.n,Au..<f5.

In the case that the notch is a crack, AV = 0 and the change in potential energy

becomes

A P = - / a^-njAuidS. 2 J AS "' '

Let a denote the crack length in the initial state and a — Aa denote the crack

length in the final state. Let / (x i ,0~ , / ) denote a field quajitity associated with

a crack of length / where 0^ corresponds to hm^_o= /(xi,X2,/) . Then

A P = - / a'ij{xi,0^ya)nj[ui{xi,0~,a ~ Aa) — Ui{xi,Q~,a)]dS. 2 J AS

The unit normal to the surface of the crack, n^, is zero except when j = 2, so

rij == 712 = — 1 on the upper crack surface and nj = n2 = 1 on the lower crack

surface. In a mode III crack the only nonzero stresses are (T13 and a'23. Therefore

aijnj = — cr32. Thus

"^ /"a-i-Aa A P = -z -<7'32(xi,0^,a)[uJ(xi,0*,a-r Aa)-u^(xi ,0*,a)]<ixi

2 J a 2 /•o+Aa _ _

+ - / (T32(xi,0~,0,a)[uj(xi,0 , a - A a ) - U 3 ( x i , 0 " , a ) ] ( i x i .

2 Ja

If Xi varies between a and a -H Aa, with the crack tip corresponding to Xi = a.

then U3(xi,0'^,a + Aa) - U3(xi,0'=,a) = U3(xi,0*.a ^ Aa).

Therefore,

A P = — / <r32(xi,0,a)[u3(xi,0*,a - A a ) - U 3 ( i i . 0 .a-Aa)]dx^, 2 Ja

23

and since

^3(aJi,0",a-f Aa) = - U 3 ( x i , 0 ^ , a - Aa),

the change in potential energy becomes

ya+Aa A P = / o'32(xi,0,a)[tij(xi,0,a — Aa)<ixi.

Ja

If the energy release rate, Q, is defined to be -dP/da, then

\ ya+Aa

Q= lim -—- / «r32(xi,0.a)u3(xi,0"^,a + Aa)<ixi. (2.9) Aa->0 A a Ja

Note that the derivation just completed for the energy release rate is completely

independent of ajiy material inhomogeneity. Futhermore, if the crack advances in

a self-similar manner (meaning the geometry of the crack is not suddenly altered)

then in the limit as Aa goes to zero the displacement Ui(xi,0,a -r Aa) will be

a translation of the displacement of the crack front. In this case (2.9) may be

written as

Q= lim --— / o'32(xi,0)u3(Aa - xi.O)<ixi. Aa—0 Aa 0

It is assumed here that the form of material inhomogeneity is unchanged during

an infinitesimal extension of the crack and that the crack is symmetric about the

plane of the crack. When Q is expressed in terms of polar coordinates the stress

(732 is replaced by (TQ^ and, if the out-of-plane displacement is denoted by a;, the

energy release rate may be expressed as dP 1 t^°-

G = = lim -— / (rfi,(r,0)u;(Aa-r,7r)<i5. (2.10) dl Aa—o Aa Jo

Calculation of ERR

It is important to notice that only the asymptotic form of the stress (T^,

and displacement as r ^ 0 contribute to the calculation of Q from (2.10). The

asymptotic form of the crack surface displacement u;(r. r ) in • 7 is given by the

term in the series expansion which experiences the smallest order of decay as

r — 0. These forms for the various cases are given in equations (2.3), (2.6),

and (2.7). The asymptotic form of the stress near the crack tip is simply the

24

dominant singular term in the stress field. In case i), the asymptotic form of the

stress is the first term in the infinite sum in (2.1) since it is the only term which

contains a singular power of r. For case ii), the asymptotic form corresponds to

that term in (2.4) which exhibits a logarithimic singularity in r. In case iii) when

the stresses are bounded, the asymptotic form of the stress is the term in (2.1)

that is independent of r . By substituting these asymptotic forms into (2.10), the

energy release rate for each case is found:

case i)

where

case ii)

g = C. lim -i- fy^f-^H^)^^3^Y=^^¥^)ir, (2.11) Aa-*o AaJor^ 7*o

_ Sa-lrp (;;)

' 7 r V o ( l + / ? ^ ) i 2 - / 3 - V r : ^ l ^ '

1 r^» r Aa — r Aa — r i , , g = C , H m „ f - / l n ( - ) l n ( )( )UT, (2.12)

Ao-»0 A a Jo 7*0 7*0 TQ

where

c, = A^(-)'-25TT^ fj.0 To

case iii)

where

e = C, Um - ^ t°{^^-ly-^dr, (2.13) Aa—0 A a Jo 7*0

r - ^o^O/^=^/^ sin(7rVl - / ? ) ^ ' - fio Vo^ v / r ^ c o s 2 ( 7 r y r ^ ) '

Define the nondimensional quantities

and

r, r Aa 7 = — .s = — ,oa =

ro TQ TQ

Then for case i)

g = C,limi- 5^^' ^Sa-sY'^-^Us. (2.14) 6a—'0 oa Jo

25

where

Co = — 8 7^

7 r 2 ( l + / ? 2 ) ( 2 _ / 5 _ v T ^ - ^ ) 2 -

A simple integration shows that the integral in (2.13) is equal to zero, hence

the energy release rate for f < /3 < 1 vanishes. It is now shown that ^ = 0 for

case ii), /? = f. For 6a sufficiently small, \s* \ns\ < \6a^ ln6a\. Then

\T\ = I T - / ln(^a — 5)5* ln5 (i l oa 70 1 i r*°

< -T-|^a* ln^a| / \ln{6a - s)\ds da Jo

< —^6ahn6af''\n{6a-s)ds. (2.15) oa Jo

After integrating the right hand side, one finds

| / | < 6a* ln^a(ln^a - 1).

From an application of L'Hopitals rule it follows that lim^a—0- = 0 and hence

^ = 0 for /? = | . Thus the energy release rate is zero for 3 > ^.

The normalized energy release rate Q was calculated numerically using an

IMSL subroutine, and the results are illustrated in Fig. 2.2. With the exception of

very small values of/3, the effect of a decreasing rigidity is to significantly reduce

the ERR. Q is only illustrated for /? < .5 since Q = 0(10"^) for /? > .5. However,

it should be noted that ^ > 0 for /? < .75. Since Q represents the energy available

to the crack tip for propagating the crack, an effect of material softening near

the crack tip is similar to that provided by a yielded region: energy is dissipated

that would otherwise contribute to the fracture process. In conjunction with a

decreasing rigidity and ERR there is a marked change in the geometry of the crack

profile, as illustrated in Fig. 2.3, accompanied by a large displacement gradient

near the crack front. This is similar to behavior in highly deformable and certain

nonlinear materials in which it is difficult to transmit and concentrate energy

into the region where crack propagation may take place ([10. pp 149-150]). It

is also of interest to note that when 3 is very small, corresponding to a nearly

constant modulus except in a small zone about the crack tip (Fig. 2.4), and -v is

large enough so that the loaded crack may be regarded as being embedded in

26

applied load = IQ j->rQ "

y=yQ(r/r^)

SE:T.=. ' -^

Fig. 2.2. The normalized energy release rate

27

1 . 2 -

Fig. 2.3. Normalized crack surface displacement versus T/TQ (after [7])

28

Po

Fig. 2.4. The nondimensionalized shear modulus versus r/vg

29

the region determined by r^, then Q attains a value greater than that for a homo­

geneous medium with modulus /XQ. For such a scenario, the failure to account for

a reduced rigidity near the crack tip could lead to \'alues of Q that underestimate

the actual energy release rate. However, if the size of the region where mate­

rial softening occurs is much smaller than the length of the loaded crack, then

^ is a monotonically decreasing function of 3. This situation may be viewed

as modelling a small process zone at the crack tip where damage effects have

reduced rigidity and strength [11]. Within this context, the results indicate that

the incorporation of a process zone at the crack tip reduces the energy available

for crack propagation.

When a crack tip stress field does not display the conventional square root

singularity, as is the case for this problem, the usual notion of the SIF does not

lend itself to an obvious physical interpretation. For instance, in studies of cracks

propagating through a bimaterial interface where the stress singularity is of the

form r""', 0 < a < 1, employing the usual definition of the stress intensity

factor may result in a SIF of zero or infinity, depending on whether the power

of the singularity is less or greater than 1/2 (see, for example, [12!, [13]). This

phenomena is due to the fact that the stress state does not remain autonomous

as the crack advances through a material interface. In any case, the significance

of the coefficient of the singular part of the stress field as it relates to the crack

energetics is not always apparent. In the problem studied in this thesis, no simple

relationship exists between the stress coefficient of the dominant term near the

crack tip, say C(/3), and Q. Indeed it is a simple matter to show that C{,3) is

monotoniczdly increasing (decreasing) on the inter\'al 0 < /3 < 3/4 (3/4 < (3 <

1) , tending to infinity at 3/4. Of course, for /? = 0 one does recover the known

relationship that Q = 7rC2(0)/;io [14].

CHAPTER III

ALTERNATE FRACTURE THEORIES

Because the material inhomogeneity considered in this problem may result

in reduced values of the energy release rate, the initiation of crack growth may

be associated with a large applied load and attendant substantial inelastic de­

formation or plastification at the crack tip. Moreover when /3 > 3/4 and the

energy release rate vanishes, Q does not provide zn applicable fracture criterion.

For these reasons it is of interest to investigate the effect of the material inhomo­

geneity upon the plastic zones about the crack tip and to examine two alternative

fracture theories within the context of this problem.

In the case of antiplane strain the von Mises yield criterion may be written

in the form

<^L+^r, =^V. (3.1)

where cTy, is the yield stress in shear. This means that when the stress in the

body reaches this critical value, a linear relationship between stress and strain

is no longer valid and the material may begin to exhibit plastic deformation or

flow. More precisely, when yielding has occured deformations may be permanent

or large strains may result with Httle or no increase in appHed load. A convenient

method for visualizing the regions plastic effects may occur is to represent a^^ =

acTo, where ao is the applied load, and then construct a family of level curves

from (3.1) for various values of the parameter a. If only the singular terms of the

stresses given in (2.2) and (2.3) are utilized, the boundary of the plastic zone as

a function of 9 can be represented as

, . , ,4[cos2(g/2) - iVTTW - ay sin2(9/2)] ,u(2-0-./^» , . ,x

For the sake of illustration, plastic zones as predicted by (3.2) are plotted nondi-

mensionally as T/TQ in Fig. 3.1 for a = 3. For ^ = 0 the usual circular zone

associated with a mode III crack is obtained. When '^ is small and the eff'ects

30

LP

o r/r.

o

31

<re=.45

Fig. 3.1. Plastic zone shapes according to the von Mises criterion

32

of material inhomogeneity are very localized near the crack tip, the size of the

zone is actually reduced but its location is translated to a region ahead of the

crack. As (3 increases and matericd softening becomes more pronounced, the

plastic zones undergo a change in shape and a dramatic increase in size. It should

be noted that although the power of the singularity in the stresses decreases ^-ith

a reduced rigidity, it is the coefficient of the singular terms which contribute to

the enlarged zones of plasticity at the crack tip.

A fracture criterion which has been utilized in problems where the stresses

have an arbitrary power singularity is based on the maximum cleavage stress

theory (see [12],[9], and [15]). This criterion states that propagation will take

place in a direction 9 = 9^ for which the cleavage stress <rgz(7*p, ^c) is maximum

and equal to some critical value. The distance r^ is an experimentally determined

material constant related to the size of the process zone about the crack tip. The

normalized cleavage stress ae^la-Q is illustrated in Fig. 3.2 for T/TQ = 0.001. If

the direction of crack propagation is controlled by the maximum cleavage stress,

then for small (3 the crack path would be the crack axis, 5 = 0. However, with

decreasing rigidity, the stress distribution about the crack tip is nearly uniform

with no preferred direction. This effect becomes even more pronounced if a^^ is

evaluated at a sHghtly larger distance ahead of the crack. Experimental evidence

suggests that the path of crack propagation and local yielding do not coincide as

a crack always tends to bypass the yielded portion of a material [16i. This, in

conjunction with the results of Figures 3.1 and 3.2, suggests that a change in the

direction of crack propagation would occur in a matericd modeled by the shear

modulus adopted in this work. Such behavior was observed in solid propellants

which initially led to a consideration of this model [11].

Finally, the suitability of the strain energy density (SED) as a criterion for

determining failure or yielding for the crack problem considered will be discussed.

In this theory attention is focused on the strain energy density W when evaluated

along a fixed radius r" from the crack tip (see Fig 3.3). Yielding and fracture

are assumed to coincide with locations of maximum and minimum values of the

strain energy density, W^ax or Wmtn- Morever, the onset of yielding or fracture

is initiated when these extrema attain their respective critical values, say W'^ max

33

r/rQ=.001

Fig. 3.2. The angular variation of the normalized cleavage stress

34

Fig. 3.3. The region of reduced rigidity and the core region around the tip of a loaded crack

35

^^ ^min- For this problem, the strain energy density is given by

2fJL

UtiHzing equations (2.2- 2.3) and (2.5- 2.6), W may be calculated exactly. The

issue of approximating the SED by using only singular terms has been addressed

in several studies (see [17] and references contained therein). The conclusion

of these works-that higher order effects are essential in utilizing the SED-is

reinforced by comparing the first order strain energy density approximation with

the exact calculation of W. In particular the approximate SED always exhibits a

minimum in a direction given by 5 = 0** whereas the results here show that this

is not necessarily the case for the exact strain energy density.

Because of the variable shear modulus and the inclusion of higher effects in

calculating W, it is not possible to remove a "strain energy density factor" as in

[8]. Consequently the calculation of W is influenced by the choice of r' as well as

the length scales associated with reduced rigidity and the loaded crack, Vc and ro,

respectively. These choices compHcate the interpretation of results as predicted

by the strain energy density criterion. A normalized strain energy density

was evaluated for a variety of combinations of r ' and 7 = rc/r©. If one identifies

the parameter Vc with the radius r" of the "core region", on whose perimeter

the SED is to be evaluated [8], then r^ = r" and 7 < 1. For this scenario W

is illustrated in Fig. 3.4 with T'/TQ = 0.01 = 7. Fig. 3.5 shows the effect of the

size of the core region on the SED when the crack is embedded in the region

of reduced rigidity. The results illustrate that Wmin is not necessarily located

in the direction 9 = 0°, though this is more pronounced in Fig. 3.5. In general

one finds that as r^ and /3 increase, Wmax increases. For ductile materials W^^^

is always smaller than W^^^. This fact combined with the previous observation

suggests that with an increase in the extent and degree of materisd softening,

failure by material yielding would precede fracture initiation. In view of the

results displayed in Figures 3.4 and 3.5, it is not clear that one could predict.

36

Fig. 3.4. The norm alized strain energy density for V - r.

-ssssaeessSS

( a . ) f l , §-=.01 ^0

Cc.) Y=l, - -=1 ^0

Fig. 3.5. The effect of the size of the core region on the SED when the

crack is embedded in the region of reduced rigidity

38

with any degree of confidence, the direction of crack propagation based on the

strain energy density.

REFERENCES

[1] Griffith, A. A., The phenomena of rupture and flow in soHds, Phil. Trans. Roy. Soc. of London, A 221 (1921) pp. 163-197.

[2] Irwin, G. R., Analysis of stresses and strzdns near the end of a crack travers­ing a plate. Journal of Applied Mechanics, 24 (1957) pp. 361-364.

[3] Gurtin, M. E., An Introduction to Continuum Mechanics. Chicago: Aca­demic Press, 1981.

[4] Fung, Y. C , Foundations of Solid Mechanics. New York: Prentice Hall, 1965.

[5] Ewalds, H. L., Fracture Mechanics. London: Edward Arnold and Delftse Uitgevers Maatschappij, 1986.

[6] Broek, D., Elementary Engineering Fracture Mechanics, Boston: Martinus Nijhoff PubUshers, 1982.

[7] Schovanec, L., An ajitiplane shear crack in a nonhomogeneous elastic mate­rial, to appear in Engng. Fracture Mech.

[8] Sih, G. D. (ed.), A special theory of crack propagation. Methods of Analysis and Solutions to Crack Problems (1973) pp. 17-45.

[9] Erdogan, F., Fracture of composite materials. Proceedings of Prospects of Fracture Mechanics, (edited by G. D. Sih, et al.), (1974) pp. 447-491.

[10] Atkins, A. G., and Y. W. Mai, Elastic and Plastic Fracture. New York: Halstead Press, 1985.

[11] Schovanec, L., Fracture in damaged media: An inhomogeneous material approach. Air Force Office of Scientific Research Technical Report, Air Force Astronautics Lab, Edwards, CA, 1987.

[12] Erdogan, F. and T. S. Cook, Antiplane shear crack terminating at and going through a bimaterial interface. Int. J Fracture. 10 (1974) pp. 227-240.

39

40

[13] Erdogan F., and V. Biricikoglu, Two bonded half planes with a crack going through the interface. Int. J. Engng. Science, 11 (1973) pp. 745-766.

[14] Sih, G. C , and H. Leibowitz, Mathematical theories of brittle fracture. Fracture: An Advanr^H Treatise, (edited by H. Leibowitz), 2 (1968), pp. 68-188.

[15] Fenner, D. N., Stress singularities in composite materials with an arbitrarily oriented crack meeting an interface. Int. J. Fracture. 12 (1976) pp. 705-721.

[16] Sih, G. C. (ed.), Elastodynamic crack problems, Amsterdam: Noordhoff Leyden, 1977.

[17] Smith, R. N. L., Second-order terms and strain energy density for the angled crack problem, Engng. Fracture Mech., 26 (1987) pp. 463-469.

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