On the path independent integral-Ĵ

En~inwinr,Fmc~unMechanicsVol. 13.pp.841-81 0 Pergamon Press Ltd.. l960. Printed in GreatBrilain

ONTHEPATHINDEPENDENTINTEGRAL-.i K. KISHIMOTO, Department

S. AOKI

and M. SAKATA

of Physical Engineering, Tokyo Institute of Technology, Japan

Ookayama,

Meguro-ku,

Tokyo

152,

Ah&act-The path independent integral, _( is presented as the rate of energy flux during crack extension. This integral is an extension of the J-integral proposed by Rice and includes the existence of a fracture process region and the effect of plastic deformations, body forces, thermal strains and inertial of material. It isshown that the j-integral can include as special cases other fracture mechanics parameters such as J by Rice, G by Eftis et al., /* by Blackburn or f by Strifors. A definition of the j-integral in a three-dimensional problem is presented and possibility of applying the ./-integral as fracture criterion is discussed.

1. INTRODUCTION of materials plastic deformation occurs at the crack tip. Linear elastic fracture mechanics is effective if the plastic zone is limited in size compared with crack and structure dimensions. Elastic-plastic treatment is necessary if the plastic zone is of considerable scale. The path independent J-integral has been proposed by Rice [ 11and it provides a means of extending fracture mechanics concepts from linear elastic behavior to plastic behavior[2]. The J-integral is truly path independent for linear or nonlinear elastic stress-strain laws but loses its physical significance as a crack driving force when deformation is not reversible. It seems necessary to extend it as to be path independent for the Prandtl-Reuss relation. It has been discussed by Broberg[3,4] that a fracture process region exists near the crack tip and it plays an important role in real fracture. The fracture process region is thought to be a small region where fracture process such as initiation of voids or micro-cracks, their growth and coalescence, take place and continuum mechanics does not work. However, it is apparent that system energy balance has to be satisfied even if the fracture process region exists and release of reversible energy may play a central role in crack growth problems. In this paper the path independent j-integral, which is an extension of the J-integral by Rice, is presented taking into account of the existence of a fracture process region and the effects of plastic deformations, body forces, thermal strains and inertia of material. A definition of j-integral in three-dimensional problem is presented and the possibility of applying the j-integral as a fracture criterion is discussed.

IN ACTUAL

FRACTURE

2. PATH INDEPENDENT ~-INTEGRAL AS ENERGY RELEASE RATE (1) Energy balance equation We first consider the equation of motion of a solid Uij,j

+

Fi

=

piii

(1)

where Uij is the stress tensor, Fi the body forces, p the density and Ui the displacements. By multiplying both sides of the equation by iii and integrating with respect to volume element d v of the body, we have (Uij,j+ e)rii d V =

I V

piiiliid V,

The volume integral in the left-hand side is transformed as

(3)

K. KISHIMOTO et al.

842

where S is the surface of the body and Ti = gijnj is the surface traction with nj the external normal of the surface S. Equation (2) is written as (4) where the left-hand side represents work done by the surface tractions and the body forces. while the right-hand side is the rate of kinetic energy of the volume Vof the body and work done by the stress tensor. (2) Energy balance during crack extension We consider a plate containing a crack as shown in Fig. 1 and assume that the crack tip moves virtually from the initial location 0 to the final location 0’ with the distance 00 being infinitesimally small. Here, 0 - XI. Xz is fixed frame and O’- x1, x2 the moving frame whose origin O’iscoincident with the tip of the extending crack. The direction of X, and x2is perpendicular to the crack surface. It is assumed, following Broberg, that fracture occurs in a fracture process region, or end region denoted by Aendin which continuum mechanics does not work effectively. As illustrated in Fig, 1, Iendis the contour surrounding Aend,r any contour surrounding the end region, f,Ycurves along the crack surfaces, and A the region surrounded by these curves. We now consider the energy balance of the material in A + Aendduring virtual crack extension. In so doing we assume all changes of the field quantities with time are attributable to changes in crack length, and take the length I of a propagating crack as the time-like variable. With reference to eqn (4) we obtain .. dui Puqj- dA +

dui T,dr+

$hdA+j p d/

It is noted that the rate of the energy change of the material in the fracture process region is denoted by j, since it could not be represented in terms of continuum mechanics. By using Gauss’ theorem and the equation of motion given by eqns (I) and (5) reduces to

f ZZ

I

rend

T.dui dr ’ dl

(4

-

pii,)% - qj$]

dA +

dr j” T.dui

I*end’ d1

*

Fig. 1. Configuration of crack tip; Aend:fracture process region, rend: boundary of Aend,l? arbitrary curve surrounding A and lYs:curves along crack surfaces, 00: direction of infinitesimal crack extension.

(6)

843

On the path independent integral-.!

With reference to Fig. 1, transformation between the fixed and moving frame is given by (7) and the displacements are represented in terms of the fixed frame 0 - X, ,X, as

whence we obtain the following relation:

(9) Substitution of eqn (9) into eqn (6) yields

(10) Since in deriving eqn {lo) no restriction has been imposed on the stress-strain relation of the material, the j-integral is thought to be an energy release rate of an arbitrary material during crack extension and has physical significance as a crack driving force.

3. FRACTURE CRITERION Fracture process occurring in the fracture process region will be different for different materials and enviromental conditions. For example, initiation of voids or micro-cracks, their growth and their coalescence will take place in metallic materials. We assume that the fracture process region is autonomous in the sense that it does not depend upon geometry of body or crack or upon load conditions[3, 41. Thus, the process region is assumed to be constant in dimensions and moving along with the same speed as the crack tip, and the relation ~~i~a~= 0 holds on rend, We obtain from eqn (10)

(11) Let j, denote the rate of energy dissipated in the fracture process region co~espon~ng to the crack growth per unit area. Then, the fracture criterion in terms of energy balance is written as .i = j,.

(12)

We again point out that in deriving eqns (IO) or (11) the stress-strain relation has been arbitrary so that eqn (12) may be employed as a fracture criterion for a wide range of materials.

4. j AS PATH INDEPENDENT INTEGRAL We obtain from eqns (7) and (8)

!$ =

cos

1

&$s ax + sin &-$ 1

(13)

2

and eqn (11) is written as j-_....

z

f rend f

cose($+ sin f&,2 1

3

dr.

(14)

K. KISHIMOTO

844

el al.

We decompose j into the components & and .& as j = .&cos do+ & sin 8”

(15)

where .&k = 1,2) are defined by jk = f I-end

Tiau,dT,

(k = 1,2).

axk

(16)

In order to obtain a path independent representation which has the same value as eqn (16), we consider of the following expression j;=

-

I

T-$ dT + Fk(A)

i-i-I‘,

(17)

k

where Fk(A) are quantities determined when the area A surrounded by r,, rend and r is specified, where r is an arbitrary contour surrounding the process region Aendand the crack tip. From eqns (16) and (17) we obtain j;-j,=f

r* T,-$dr

(18)

t F,(A)

k

where r* = r t I’, - rend denotes the contour which surrounds the area A. In order that & be equal to &, the following relation has to be satisfied

h(A) = r*T,gdI’=//A(cij-$),idA f

(19)

and we obtain the path independent expression of jk as

(20) Furthermore, using the equation of motion (eqn I) we obtain (21) We decompose strain tensor Eii into elastic strain components EC and ponents E$ and write as

eigen

strain com-

Eii=e$tE$

(22)

where the eigen strains are inelastic, stress-free strains typically exemplified by thermal strains or plastic components of strains. In case the elastic strain energy density function, WJe&), is defined by (23)t and Wc(ETj)does not explicitly depend on xk, introducing

.fk=

T.au’ dr + wenk dr I r+r, ’ax, f r+r,-rCnd

eqn

(23) into

eqn

(21) leads to

If Iuijg

t (,& - Fig

A

dA.

(24)

k

tThe elastic strain energy density function satisfying eqn (23) can be defined even if material behavior an elastic-plastic material described by the Prandtl-Reuss constitutive relation (see Appendix 1).

is not elastic,

e.g.

845

On the part independent integral-j

We also consider the case where the area of the fracture process region is diminishing and the contour integral along rend is identical to zero.? Equation (24) reduces to

&=

f r+r,

c

dA.

(25)

5.sPEcIALCASES 0F ~-~EGRAL AND ITSFLAGON TO OTHER FRACTURE MECHANICS PARAMETERS We will give our attention to the case where the fracture process region is ignored and the crack extends in a direction parallel to the crack surfaces, i.e. 8 = 0 in eqn (8) or in Fig. 1. Relation to the J-integral by Rice We assume that the stress-strain relation of the material considered is elastic, and body forces, inertia of mate~als and crack surface tractions are absent. Equation (25) reduces to

(26) which is in conformance with the J-integral proposed by Rice, so that j = .I. In the case where body forces exist, we have j = j, =

Wen, -

T*

(27)

'8X,

which has been used by the authors in analyzing cracked rotating disks [S, 61. j-integral in thermal stress field Let &ii denote the coefficient of thermal expansion and 8 the temperature increment from the natural state. Thermal strains are represented by eij* =

(28)

fXij0

and eqn (25) leads to j=&

cy..P.*

Wet11 - Ti$ I

”

dA

“ax,

(2%

Furthermore, for isotropic materials, eqn (29) reduces to j=j,= f{f

Welt, -

amae dA “8x1 (30)

where Sii is Kronecker’s delta. For a uniform temperature field, we have

(31) It is noted that when the temperature is uniform but different from that of the normal state the elastic strain energy function does not coincide with the strain energy function, i.e.

(32) See Appendix 2.

K.KISHIMOTOetul.

846 J-integral with inertia eflect

Retaining of the inertia effect in eqn (25) yields j=j,=

Supposing that A and r are moving with the tip of the extending crack, eqn (33) is written as

where v denotes the crack extension velocity in the X, direction. This equation coincides with the formula proposed by Bui[7], and is written as

Since the relation iii = - vui,, is thought to be satisfied in the vicinity

of the crack

tip. we obtain

(36) which coincides with the formula given by Freund[g]. Relation to C?. the energy release rate proposed by Eftis. Liebowitz and Jones Eftis et al. proposed a fracture criterion under a large scale yielding

where W is the work of the applied forces. U’ the elastic strain energy and U” the plastic strain energy, and 6 is interpreted as the excess of the rate of work of the applied forces over the rate which is observed in elastic and plastic deformation[9, lo]. We compare eqn (37) with eqn (5). The left-hand side of eqn (5) represents the work rate of the applied forces and is equal to (a W/al). The second term of the right-hand side of eqn (5) represents the work rate of the internal forces and is equal to (~?/ar)(U’+ U”) for elasticplastic deformations. Thus, j is equivalent to (I? when the deformation process is quasi-static. We again use eqn (25) neglecting inertia of material and considering, that the eigen strains ~1; are plastic strains E; and we then have (38) This equation provides a representation for I!? if the conditions on the end region stated in this paper are satisfied. Relation to J* proposed

by Blackbum

Blackburn has proposed the parameter J* which is a measure of the characteristic of a crack tip elastic-plastic field [ 111,i.e.

(39)

where r, is a circle with center at the crack tip and radius rn. He also has derived a path

847

On the path independent integral-j

independent expression which is written, for two-dimensional problem, as (40) where the contour for integration is arbitrary provided that it surrounds the crack tip. In the case where the limiting process “(rijUi,ij - uij,lUi,j)T dr d0 = 0

(41)

holds, where r and 0 denote the polar coordinate with the origin at the crack tip, eqn (40) reduces to J* = -

(42)

which is in conformity with & without body forces and inertia of material. It is noted that eqn (41) is not generally satisfied and J* is not always equivalent to j,. Relation to f, the crack extension force by Strifors

Strifors has proposed an apparent crack extension force within the framework of continuum mechanics. The principle of virtual work has been applied to an arbitrary part 9 of a body containing a crack which is represented by a singular surface with a finite cohesive zone. The apparent crack extension force in the direction IR, where IR is an arbitrary but fixed unit vector in the reference configuration, is defined by fI,@)

=

IR ’

I “R(9)

([Ts]VxF - HT(b -a)&)

dUR- IR .

I J"R(B)

HTtR dSR

(43)

where TR is the first Piola-Kirchhoff tensor, F = V,X the deformation gradient, H = V,u the displacement gradient with u the displacement vector, b the body force, a the acceleration, pR the mass density in the material configuration, and tR the stress vector[l2]. In the case of plane infinitesimal deformation and the situation of crack growth as shown in Fig. 1, eqn (43) may be written as f,,(A) = fl cos f3,+ f2 sin $

(44)

where

k

f =/II A

qjz

-

(E

-

&I-$}

dA - I,,,

Tz

dI’, (k = 1,2).

This equation is equivalent to eqn (21). It is shown that within the infinitesimal deformation, is equal to 1 provided that the fracture process region is substituted by the cohesive zone.

(45)

fIR

6. THREE-DIMENSIONAL FORMULATION We consider a crack extension problem in a three-dimensional configuration as shown in Fig. 2, where the solid curve represents the crack front contour in the initial state, while the dashed curve is the final crack front contour after a virtual crack extension has taken place. Let O-X,, X,, X, denote the reference frame with the origin 0 being at an arbitrary point along the crack front and the direction & X,, X, and X, taken to coincide with the normal, the binormal and the tangent of the crack front contour at 0. Let O’-xl, x2, x3 be the moving frame with the origin 0’ being at the point on the dashed curve, i.e. the contour of extended crack front, where the increment 00’ coincides with the xl axis and the distance 00’ is infinitesimal. We may further

K. KISHIMOTO ef a!.

848

$

_--_-- ----...

x3

Fig. 2. Schematic representation for 3-dimensional crack front.

Fig. 3. Illustration for 3-dimensional crack front, fracture process region and plate with arbitrary contour i for integration.

Fig. 4. Circular process region and integration contour.

On the path independent integral-j

849

assume that 0’ is unique for a given 0, and at 0 and 0’ the osculating planes coincide with the tangent planes of the crack surfaces. We next consider a plate of infinitesimal thickness as shown in Fig. 3 and assume that the fracture process region is autonomous. The energy balance of the material in the process region is given by, with reference to eqn (1 l),

(W whence a local fracture criterion may be presented as j = j,. Equation (46) is expressed in terms of 0 - X,, X2, X3 as ~=~,cose~t~*coscp~ts,cos~~

(47)

where jk = -

TaU’dT (k=123) ‘ax, ’ ”

and &,, cpoand $. are defined as illustrated in Fig. 2. We further consider a plate element of arbitrary contour as shown in Fig. 3. The energy release rate, .&,is defined as

where p denotes a unit vector normal to the surface of the plate element, and pj its Xj component.

REFERENCES R. Rice, A path independent integral and the approximate analysis of strain concentration by notches and cracks. 111J. Trans. ASiUE .I. ADDI. Mech. 90.379-386 (1968).

PI J. A. Begley and J.-c. Landes, ‘fhe J-inte& ai a fracture criterion. ASTM STP 514, 1-23 (1972). K. B. Broberg, Crack-growth criteria and non-linear fracture mechanics. J. Mech. Phys. Solids 19,407-418 (1971). Mat. 10(2), 33-U (1975). PI M. Sakata, S. Aoki and K. Ishii, J-integral analysis for rotating disk. Pm. Int. Conf. Fract. Mech. Tech. (Ed. G. C. Sih and C. L. Chow), Vol. 1, pp. 515-523,Sijthoff & Noordhoff, Leyden (1977). [61M. Sakata ef al., J-integral approach to fracture of rotating disk. Trans. ASME 1. Engng. Mat. Tech. 100, 128-133 (1978). [71 H. D. Bui, Stress and crack-displacement intensity factors in elastodynamics. ICF4, Waterloo, Canada 3,91-95 (1977). 181L. B. Freund, Energy flux into the tip of an extending crack in an elastic solid. I. EIasGcity 2,341-349 (1972). [91 J. Eftis and H. Liebowitz, On fracture toughness evaluation for semi-brittle fracture. Engng Fracture Mech. 7, 101-135 (1975). m J. Eftis, D. L. Jones and H. Liebowitz, On fracture toughness in the nonlinear range. Engng Fracture Mech. 7,491-503 (1975). [111W. S. Blackbum, Path independent integrals to predict onset of crack instability in an elastic plastic material. Inr. J. Fracture Mech. 8, 343-346(1972). H. C. Strifors, A generalized force measure of conditions at crack tips. Int. I. Solids Struct. 10, 1389-1404(1974). :1:; G. C. Sih, Some basic problems in fracture mechanics and new concepts. Engng Fracture Mech. $365-377 (1973). U41 J. D. Achenbach, Dynamic effects in brittle fracture. Mechanics Today (Ed. S. Nemat-Nasser), Vol. 1, pp. l-57, Pergamon Press, New York (1972).

::; K. B. Broberg, Energy methods in statics and dynamics of fracture. J. Japan Sot. Sfreng. Fracl.

(Received 12 May 1979;received for publicafion 29 February 1980)

APPENDIX 1 For example, the Prandtl-Reuss constitutive relation has a form as dq = A d+& + 2s(dcii -de&)

(Al)

where A and p are Lamb’s constants and dc$ the increments of plastic strains. The above relation can be rewritten as drii = A d
(A2)

K. KISHIMOTO

850

et af.

Thus, we can define the elastic strain energy density function satisfying eqn (23) as

APPENDIX 2 It is assumed no material is included in the fracture process region in deriving the J-integral. In order to caicufate the J-integral by using eqn (24). the definition of the j-integral, the contour Pcnd must not enclose a region of finite area. To depict this, we consider of a circular process region as shown in Fig. 4, where the contour along the process region is broken into two parts: Tend = ACB = the contour around the small circle of radius E and T = AOCOB = the contour along the xi axis. We assumed, for the sake of simplicity, that neither body force, inertia of material nor surface traction is acting. The line integral on the closed contour is written as

where Aend denotes the area enclosed by the path rend and f, i.e. the fracture process region. The term rir(&ijaX,) has, in general, a singularity and the right-hand side of the above equation does not always tend to zero in the limiting case t +O However, the right-hand side does disappear when the area A,,, is identical to zero, i.e. when no material is included in the process region. Following are illustrative examples. where Mode I deformation problems of linear elastic materials are considered. Limiting cases of circular process region The strain energy density function for plane strain is given by[l3]

where p and Y are the shear modulus and Poisson’s ratio, respectively, and I and 8 denotes the polar c~rdinate origin at the crack tip. Let the radius of the process region he e. We then have the following relation. ~(3 - 4~

cos 8)( I

t cos8)cos0 d0 +0(&r)

with the

I -2v

=__ *~ K,’ + qrq

which does not tend to zero when E -+O. This is the natural consequence of the fact that the process region was of finite area before the limiting process and contains the stress and strain singularjty. Cohesive process region We assume that cohesive forces act ahead of the crack

The j-integral

in this case is given by

Substituting the linear elastic solution of the crack tip stress and strain field and also utilizing formula[l4] (AR) where H is the unit step-function,

we obtain the final result

where K=

3-4u (3 - u)/(I

for plane strain

t v) for plane stress.

It is shown that .f in this case is identical to J since no material is included in the cohesive process region.

(AIO)