Crack Tip and Associated Domain Integrals

Engineering Fracture Mechanics Vol. 27. No. b, pp. 61.5442, 1987 Printed in Great Britain.

w13-7944/87 $3.00 + 0.00

0 1987 Pergamon Journals Ltd.

CRACK TIP AND ASSOCIATED DOMAIN INTEGRALS FROM MOMENTUM AND ENERGY BALANCE

B. MORAN and C. F. SHIH Division of Engineering, Brown University, Providence, RI 02912, U.S.A.

Abstract-A unified derivation of crack tip flux integrals and their associated domain representations is laid out in this paper. Using a general balance statement as the starting point, crack tip integrals and complementary integrals which are valid for general material response and arbitrary crack tip motion are obtained. Our derivation emphasizes the viewpoint that crack tip integrals are direct consequences of momentum balance. Invoking appropriate restrictions on material response and crack tip motion leads directly to integrals which are in use in crack analysis. Additional crack tip integrals which are direct consequences of total energy and momentum balance are obtained in a similar manner. Some results on dual (or complementary) integrals are discussed. The study provides a framework for the derivation of crack tip integrals and allows them to be viewed from a common perspective. In fact, it will be easy to recognize that every crack tip integral under discussion can be obtained immediately from the general result by appropriately identifying the terms in the general flux tensor. The evaluation of crack tip contour integrals in numerical studies is a potential source of inaccuracy. With the help of weighting functions these integrals are recast into finite domain integrals. The latter integrals are naturally compatible with the finite element method and can be shown to be ideally suited for numerical studies of cracked bodies and the accurate calculation of pointwise energy release rates along a curvilinear three-dimensional crack front. The value of the domain integral does not depend on domain size and shape ~ this property provides an independent check on the consistency and quality of the numerical calculation. The success of the J-based fracture mechanics approach has led to much literature on path- independent integrals. It will be shown that various so-called path-independent integrals (including path and area integrals) are but alternate forms of the general result referred to above and do not provide any additional information which is not already contained in the general result. Recent attempts to apply these ‘newer’ integrals to crack growth problems are discussed.

1. TNTRODUCTION

WIDOW doubt the J-integral is the most used crack tip integral in fracture mechanics. Its role in nonlinear fracture mechanics was introduced by Rice[l, 21 who provided the interpretation of .I as a measure of the intensity of the deformation at a notch or crack tip and, in the context of nonlinear elasticity, as the energy release rate. Within a thermodynamics framework, the integral was introduced by Cherepanov[3] as an extension of Griffith’s approach to inelastic solids. Subsequently, the connection between J and Eshelby’s energy momentum tensor was noted (cf. Eshelby[ct, 51). Translational and non-translational conservation integrals have been presented by Knowles and Sternberg[6]; the energetic force interpretations were made by Budiansky and Rice[7]. In this context the J-integral is a member of the translational conservation integrals. However the J-integral possesses additional features which make it unique amongst the conserved integrals. The integrand is divergence free for a material which admits a (nonlinear) strain energy function. Furthermore the J-integral has the same value for all open paths beginning on one face of the crack and ending on the opposite face (this assumes that there are no contributions to the integral from the crack faces, a condition which is usually met in most crack problems). This rather special path-independent property under fairly general conditions has been advantageously exploited in the development of nonlinear fracture mechanics. For example, the above noted path-independence, which we shall refer to as global path-independence, allows a direct computation of the strength of crack tip singularities by evaluating the line integral in regions remote from the crack tip or along remote boundaries.

To the extent that the HHR singularity fields (Hutchinson[8], Rice and Rosengren[9]) prevail over a distance which is larger than the fracture process zone and the zone of finite strains, the crack tip fields can be said to be characterized by the value of the J-integral and the onset of crack growth can be correlated with a critical value of .I. The J value referred to here is evaluated on a contour placed within the region dominated by the HRR singularity. It is in this context

615

616 B. MORAN and C. F. SHIH

that J is referred to as a crack tip parameter. In other words the concept of a crack tip parameter is not tied to the property of global path-independence. Nevertheless global path-independence has far-reaching consequences, e.g. the value of the crack tip parameter can be determined from remote fields. Within deformation plasticity theory, J also has an energy-based definition which has been used to derive useful formulae for its determination directly from the load~isplacement record of a cracked body. In other words while global path-independence and the energy-based definition are not essential to the concept of J as a parameter characterizing the crack tip field, these features play important roles in an engineering fracture mechanics methodology. The J-based phenomenological approach to the initiation of crack growth and small amounts of subsequent quasi-static crack growth has been reviewed by Hutchinson[lO].

The success of a J-based fracture mechanics approach has stimulated an enormous activity directed to finding ‘newer’ path-independent integrals. Many of these works are motivated by the belief that new path-independent integrals will lead to further advances in fracture mechanics. In this paper we will show that various so-called path-independent integrals are but alternate forms or manifestations of a general result. Stated another way, each of the seemingly different path- independent integrals can be extracted from the general result by invoking appropriate restrictions on material response and crack tip motion. Furthermore a variety of path and area integrals can be obtained from the general crack tip integral by application of the divergence theorem over some part of the fracturing solid. More precisely, all recently proposed path-area integrals (inappropriately referred to as path-independent integrals), are merely alternative representations of specialized crack tip integrals.

The general result we referred to has its origin in a crack tip integral expression for elastodynamic energy release rate proposed by Atkinson and Eshelby[l I] and derived from the field equations by Kostrov and Nikitin[l2] and Freund[l3]. It has since been recognised that the result is also valid for general material response (e.g. Willis[ 141, Nguyen[lSJ and Nakamura et al.[16]). In Section 2 a general balance equation and a general crack tip flux integral are derived. Under certain conditions the crack tip flux and energy release rate integrals are path-independent in the crack tip region r -+ 0’. We refer to this as local path-inde~ndence. The derivation will show that crack tip integrals, rate integrals, and their complementary counterparts are direct consequences of the variational form of linear momentum balance; a priori restrictions on material response and crack tip motion are not invoked in the derivations. The J-integral, C-integral and lesser known integrals (including a useful steady state result) follow immediately by appropriate restrictions on the material response and on crack tip motion.

Riedel and Rice[20], Ohji et a/.[211 and Bassani and McClintock[22] have analysed the creep relaxation of crack tip stresses in an elastic-nonlinear viscous solid subject to a (quasi-static) step load at time t=O. At short times they showed that the amplitude of the HRR-type fields denoted by C(t) is related to the elastic stress intensity factor. At long times C(t) approaches its steady state value given by a path-independent integral (Goldman and Hutchinson[23]). The latter, which is the creep analog of the J-integral, is designated by C* and was proposed by Landes and Begley[24] for correlating creep crack growth rates under extensive creep conditions. In [22] the value of C(r) was extracted from the numerical fields with the help of the C-integral. In the latter part of Section 2 the C-integral is formally derived using the adopted variational approach. Its interpretation as a dissipation integral is given by Moran and Shih[19].

The mechanical energy integrals mentioned above are also valid under general thermomechanical loading. Nevertheless in some problems additional i~ormation (including the asymptotic form of the temperature fields) may be extracted from the total energy flux to the crack tip which explicitly accounts for the heat flux and temperature effects (Willis[l4] and Nguyen[15]). In Section 3 we derive the total energy flux integral using the general approach. The complementary form of the integral is obtained in an analogous fashion. It is also clear from the derivations that crack tip integrals specific to particular material response can be obtained from the general result by appropriately identifying the fluxes in the general flux tensor.

Three-dimensional (3-D) forms of crack tip integrals and various specializations are discussed in Section 4. We make contact with integrals in use in fracture analysis and discuss some further possible applications. Next we point out an essentially negative result. Bui[17] has shown that energy and complementary energy release rates are equal for nonlinear elastostatics. We establish

Integrals from momentum and energy balance 617

in a direct manner that Bui’s observation is valid for quite general material response and for all dual crack tip integrals under discussion. The implication is that no new information can be extracted from complementary integrals which is not already provided by their counterparts. However, in certain circumstances, the pair of integrals, referred to as dual integrals, may be employed to obtain a bound for the energy release rate (or crack tip flux).

The conditions for global path-independence and some special integrals are presented in Section 5. In particular the role of J as a parameter characterizing the strength of singular crack tip fields in elastic-plastic solids and the analogous role of the C (or C* ) parameter for elastic- viscoplastic solids are discussed.

The evaluation of crack tip contour integrals in numerical studies is a potential source of inaccuracy. To circumvent these numerical difficulties, Kishimoto, Aoki and Sakata[27-291 and Atluri, Nishioka, Brust and their co-workers[3&36] have restated crack tip integrals as path and area integrals. These so-called path-independent integrals contain two terms - a path integral evaluated along a remote contour and an area integral evaluated within the area enclosed by the remote contour. In three-dimensional crack problems these path and area integrals generalize to surface and volume integrals respectively. These alternate representations of crack tip contour integrals are not particularly advantageous. We present a versatile and practical alternative to the evaluation of crack tip integrals for two and three-dimensional problems in Sections 6 and 7. With the help of weighting functions, crack tip integrals are recast into finite domain forms. These domain forms are naturally compatible with the finite element solution process. Furthermore a property of the domain integral provides a useful independent check on the consistency and quality of the numerical calculation. In several applications, accurate pointwise values of J along a 3-D crack front have been obtained (Shih, Moran and Nakamura[37] and Nakamura, Shih and Freund[39]). Li, Shih and Needleman[40] have interpreted the method as an application of the principle of virtual work and made the connection to the virtual crack extension (VCE) technique of Parks[41,42] and Hellen[43]. Under more restrictive conditions, a similar domain form was obtained by deLorenzi[44].

In recent years many studies on applications of various path and area integrals to the analysis of growing cracks have appeared in the literature. In Section 8, we discuss these studies in light of the results presented in the earlier sections of the paper. It is easily demonstrated that these path and area integrals are essentially alternative representations of the crack tip integrals J and C. The assertions that extended crack growth under general loading conditions and material response can be correlated by these path-area integrals are reviewed. It is shown that one of the so-called path-independent integrals is not path-independent in any acceptable sense. Concluding remarks are made in Section 9.

2. MOMENTUM BALANCE - CRACK TIP INTEGRALS

A general treatment of crack tip integrals based on a variational form of momentum (or incremental momentum) balance is presented in this section. To help the development of the ideas we illustrate the approach by reproducing the crack tip energy flux derivation of Nakamara, Shih and Freund[16] which is an extension of an earlier work by Freund[13]. In this familiar setting we introduce a general notation in anticipation of applying the method to the derivation of other useful crack tip integrals by appropriate identification of the various field quantities.

Attention is restricted to small strains and the strain displacement relation takes the usual form

&u = (Uij + Uj,i)/2 (2.1)

In the absence of body forces the equation of motion (or balance of linear momentum) is written as

. . Ojij = PM i (2.2)

where p is the mass density, oY = oji is the Cauchy stress and a superposed dot denotes the

618 8. MORAN and C. F. SHIH

(material) time derivative. We take the inner product of (2.2) with the velocity field, iii, and rearrange the resulting expression to give

(Oji tii)j = @ ilii + Ojiil(j

(2.3) = (i + IV)

where the stress work density, W, and kinetic energy density, L, at a material particle are given

by

I f

w= CT~~~ dt, L = ‘piiiLi dt. 0 I 0

(2.4)

Equation (2.3) is a differential form of mechanical energy balance and is valid for any material response.

For purposes of the subsequent development we introduce the following fluxes. Let

qj = OjiUi , lj/= (WV i). (2.5)

Here ‘pj is identified as the mechanical work flux vector and $V is the time rate of change of the mechancial energy density. (In subsequent sections, the fluxes q and ly will be identified with other physical fields.) The balance law (2.3) can therefore be written as

(2.6)

For reasons which will become obvious, we will call (2.6) the general balance equation. Integrating this expression over an arbitrary volume V and applying the divergence theorem to the term on the left hand side yields

(2.7) “kV

where nlj is the outward unit normal to the surface aV. We have assumed that the fields are sufficiently smooth so that the divergence theorem may be applied. In particular the fields must be free of shock-like discontinuities which can develop in wave propagation problems and dynamic crack propagation problems. Similarly the fields will be assumed to be smooth enough to allow the usual application of the transport theorem. The expression (2.7) can be viewed as a weak or variational form of linear momentum balance. Since the variational field is the velocity, it is also an integral form of mechanical energy balance.

Let vi be the instantaneous velocity of the surface aV. Using Reynolds Transport theorem on the right hand side of (2.7) yields

PV

The result (2.8) is simply a representation of mechanical energy balance for the time dependent volume, V. We now specialize this result to the case of crack propagation and, for purposes of clarity, focus our attention on the planar crack problem.

Consider a two-dimensional body with an extending crack oriented along the xl axis of a rectangular Cartesian coordinate system. The cracked body lies in the x1-x2 plane with the crack plane given by x2=0_ The crack extends in its own plane along the xl axis with speed v. We isolate the vanishingly small crack tip region with a small contour I which is fixed in size and orientation with respect to the crack tip and is translating with the crack tip at speed v. The area bounded by the fixed material curve C and the curve I is denoted by A (t) and is free of singuIarities (see Fig. 1). Without loss of generality, the crack faces are taken to be traction free. The expression


Fig. 1. Conventions at crack tip. Domain A is enclosed by r, C,, C_, and C,,. Unit normal q=nj on C,, C_ and CO, and mi= -nj on r.

(2.8) therefore becomes

It is noted that the only non-vanishing velocity on the boundary of A(t) of consequence in (2.9) is vI = v on r. The term on the left hand side of (2.9) is the rate at which energy is being input into the body; the first term on the right hand side is the rate of increase of internal (mechanical) energy and consequently the last term is the instantaneous rate at which energy is being lost from the body due to flux through r. We denote this quantity F and write, for nj = -mj on r,

There are two contributions to this flux integral. The first term represents the flux across r due to the material outside of r working on the material inside it. If r were a material curve this would be the only contribution. The second term represents the cont~bution due to the flux of the material across r. We define the quantity 57 as the limiting value of F(T)/v as the contour r is shrunk onto the crack tip, i.e. 57 is the energy released from the body per unit crack advance (per unit thickness). For this concept to have physical significance, the limiting value, C??, must be independent of the actual shape of r in the limit lr -+ 0. In other words, the value of 5!? must be path-independent in the limit I” -+ 0 (i.e. in the crack tip region). We now consider the conditions

under which 57 is finite and path-independent in the limiting sense described. Consider the closed path formed by two crack tip contours l-i and Tz and the crack face

segments which connect the ends of the two contours. Application of the divergence theorem to the closed contour integral yields

F(b) - F(rl) = !

(q + v/VS,j)j dA (2.11) A12

where At2 is the area enclosed by the closed contour. On carrying out the differentiation in (2.11) and using (2.6) we obtain

F(h) - f’m = / Ck + V V-9) dA . ‘An

(2.12)

If the integrand in (2.12) is o(l/r2)t then F(r2)- F(T,) = 0 as Fi, r2 + 0 and the path-independence of F in the crack tip region (l? -+ 0”) is established. We note that this condition is satisfied if any

t More precisely, the integrand must be o(l/ra) where fl< 2. We assume however that the crack tip fields are separable, i.e. f - B(B)P + o(P) and furthermore that the derivative off with respect to r is of order rum’.

620 B. MORAN and

field quantity f satisfies the following relation

C. F. SHIH

as r -+ 0’ (2.13)

where r is the radial distance from the moving crack tip. The condition (2.13) can be interpreted as a condition for locally steady state behaviour. Indeed known solutions for growing cracks in elastic and inelastic solids confirm such steady conditions are approached asymptotically at the crack tip [4652]. In particular, if v satisfies (2.13) then for v of order l/r we have

@ + V~,I = 0(1/r*) as r -b 0’ (2.14)

and path-independence in the crack tip region is assured. With local path-independence assured, it follows that on choosing I to be a circular contour, (2.10) yields a finite value of F if the integrand of (2.10) is of order l/r, i.e.

(fl + Vv61j) - Aj(B)/Y + 0(1/r) as r + 0’ . (2.15)

for some A,{ 0). Condition (2.14) may also be viewed as an integrability condition. We reiterate that for field variables, v and p, satisfying a balance law of the form (2.6) and

for which the local conditions (2.13) and (2.15) hold, the integral given by (2.10) is path-independent within the local crack tip region. A special result which has important applications is easily extracted from the general derivation. We note that @ + vl’) is the time derivative off with respect to a coordinate system which translates with the propagating crack and is zero for a steady state problem. Thus the right hand side of (2.14) vanishes identically under steady state crack propagation conditions and (2.10) provides a globally path-independent integral for this class of problems for any material response.

With the understanding that the limiting value of the flux F(T), denoted by S, is independent of the shape of the contour I- as it is shrunk onto the crack tip, we write

(2.16)

We will call .F the general crack tip flux integral and will show that the identification of q and w with relevant field quantities will lead directly to explicit representations for crack tip integrals. To develop the subject matter in a manner that will help us view the specific crack tip integrals from a common perspective we introduce a general representation for the energy release rate using the general flux tensor Hkl. With its x, component denoted by H,j we write the general representation for the energy release rate as

F? =c = lim I

Hljnj dI’ (2.17) r-0 r

It follows from (2.16) that HIj is defined by the asymptotic relation

Hlj - (q + Vyl~,j)/V as r-+0+ . (2.18)

The above relation is not written as an equality since the local steady state condition (2.13) will necessarily be invoked in defining Hli Assuming that the flux fields in Hljare sufficiently smooth, and noting (2.6) and the asymptotic connections (2.14) and (2.18), the conditions for 5 to be finite (but non-vanishing) can be restated as

Hrj N +0)/r + 0(1/r) and 61 E Hlj,j = o(l/r2) as r -+ 0’ (2.19)

We emphasize that with (2.6) as the starting point, no additional physical principle has been


invoked to obtain the result embodied in (2.16) and (2.17). In the next section, the well known result for L!Y will be obtained directly by replacing the field &j in H, by its asymptotic equivalent - VUjJ.

Mechanical energy integral When pj and @ are identified by (2.5) the flux integral (2.16) has the explicit form

[(W + L)V6,j + o&]ni dI. (2.20)

The above result is the well-known integral expression for the crack tip mechanical energy tlux. To obtain the corresponding result for the energy release per unit crack advance 9?, commonly referred to as the energy release rate, we make the substitution ici - - VUi,l (i.e. the local steady state condition (2.13) has been applied to the velocity field) and rearrange (2.20) to get

By formulation (and necessity) the loop I translates with the moving crack tip[l3]. Under transient conditions or for general material response, the energy release rate is given by the crack tip limit in (2.21), although the shape of the contour as shrunk onto the crack tip is arbitrary.

Comparing (2.21) with expression (2.17), the integrand may be written more compactly as the xi component of the general energy tensor,

HIj = (W + L)6lj - C#Ui,I . (2.22)

The result in (2.21) was proposed in [l 1] and subsequently derived from the field equations for elastic solids[12, 131. The generalization to general material response has appeared in [14-161.t

Complementary mechanical energy integral As has already been noted, the derivation which leads to the crack tip flux integral (2.16) is

valid for any balance law which can be written in the form (2.6) and for which the conditions (2.13) and (2.15) are met. For example, the complementary crack tip energy flux integral (Bui[17] and Carlsson[lS]) can be obtained by first taking the inner product of the rate form of momentum balance with the displacement field, ui and then carrying out steps similar to those taken in (2.3) to (2.5). It is straightforward to show that the differential balance law is again given by (2.6) and the associated fluxes are now defined by

e = 6ijUf f li/ = (Fe + tc> (2.23)

where Le and WC the complementary stress work density are given by

wc = i fej&o dt , Lc = .f~~iUi dt (2.24)

.a -0

t For conciseness and clarity. the crack tip integral in (2.21) has been written in a particularly convenient Cartesian coordinate system with the crack plane given by .x2 = 0. Should another Cartesian system (axes arbitrarily oriented with respect to the crack line or plane) be preferred, the appropriate integrand can be obtained by the standard tensor transformations. Alternatively, 5’ in (2.21) could be interpreted as the X, component, 9’,, of the crack tip force vector

5Yk. The latter is given by the integral in (2.21) where the integrand H,, is replaced by Hk, defined by

Hk, = (w + L)6k, - q,,v.k.

The energy released by the body due to the crack extending in its own plane at velocity v is then given by W = ViVi/lVl.


and where Ui = iii is the acceleration. Using the fluxes identified in (2.23) in the integral (2.16) ieads to an explicit expression for the complementary energy flux integral

(2.25)

The negative sign in (2.25) has been introduced for convenience (the reason for the negative sign will become clear in Section 4). Using the local steady state condition (2.13) for the stress rate, o0 _ - v~Q,, in (2.25) leads immediately to the following expression for the complementary energy release rate:

5” =~1 - lim [(W” + L’)&ij - Og,lUi]ni dl? ! r-o r

(2.26)

where the terms within [ ] constitute the x1 component of the general tensor Hkj

Dissipation integral The energy integral (2.21) and integrals which follow from (2.21) by appropriate restrictions

on crack tip motion and material response, have led to many important results in fracture mechanics, e.g. the form of crack tip singularities including the means to calculate strength of the singularity fields and on energy release rate for elastic, elastic-plastic and sufficiently rate-sensitive inelastic solids. These successful applications share a common feature, namely, the behavior at the crack tip is dominated by the instantaneous response of the material. In problems where the effects of material rate sensitivity affect the form of the near tip fields (e.g. viscoplastic response including power law creep) the integrals (2.21) and (2.26) do not appear to be adequate. To capture some aspects of the natural/characteristic time of inelastic material response, we consider a general rate- integral in which the tensor Hq involves an additional time derivative. The C(t) and the C* integrals whch are discussed in Section 5 follow immediately from the general result. The general rate integral is derived in the manner laid out in Section 2. Its interpretation as a dissipation integral is discussed in 1191.

We take the inner product of the momentum equation with the particle acceleration +j, and follow the identical manipulations to reach (2.6) as discussed in Section 2. For this case, the fluxes associated with the balance equation (2.6) are now defined by

@ = qj+i, +f.$+i (2.27)

where

I$ = L” = ! .tpiiiGi dt . 0

(2.28)

and dii is the symmetric gradient of vj, i.e. dv = $. With the fluxes so identified, the crack tip flux integral (2.16) has the explicit form

(2.29)

By comparison with (2.20) the flux ._Y has dimensions of Fdivided by time, hence we will loosely refer to 9 as a dissipation integral. To obtain a dissipation release rate integral GY’, defined by %Y = P/v, we invoke the local steady state condition (2.13) for t, to get

(2.30)


With the terms enclosed within [ J interpreted as the xl component of the general tensor H&, (2.30) conforms to the general representation (2.17).

Complementary dissipation integral Complementary or dual forms of the preceding integrals can be derived similarly. We begin

with the rate form of momentum balance and take the inner product with vj. Again the balance equation (2.6) is obtained when these identifications are made:

where

tic = I ‘di,b, dt , Lc= 'pirividta 0 I 0

(2.3 1)

(2.32)

Using these fluxes in (2.16) we obtain the complementary dissipation flux integral

(2.33)

Again a minus sign has been introduced into the definition for convenience. To obtain the rate integral defined by %’ = Ye/v, we invoke the local steady state assumption for the stress rate, to get

(2.34)

Again if the terms enclosed within [ ] are interpreted as the xl component of the general tensor Hv, (2.34) conforms to the general representation (2.17).

3. ENERGY BALANCE - CRACK TIP INTEGRALS

The integrals in the previous section are viewed as direct consequences of the balance of linear and angular momentum, or their rate forms and we refer to them as mechanical energy balance relations. We emphasize that these integrals remain valid within the framework of a general thermodynamic approach. Under non-isothermal conditions additional information concerning crack tip fields may be obtained from the balance of total energy (First Law of Thermodynamics). In the following subsection we derive the total energy flux to the crack tip, starting with the total energy balance relation. A complementary integral is derived in an analogous fashion.

Total energy integral Let U(Q, S, ai) denote the internal energy per unit volume, where S is the entropy per unit

volume and ai are the internal variables. The First Law of Thermodynamics states that

0 = S&j - hj, j. (3.1)

where hj is the heat flux vector. We use (2.3) in (3.1) and rearrange to yield the differential balance law

(OeGi - hi> qi = (0 + L) (3.2)

The machinery of Section 2 is employed to obtain the total energy flux to the crack tip. Comparing (3.2) with the general balance statement (2.6) we make the identifications

9 = Oui[i - hj ) b= <e+ L) (3.3)


where now qj is the total energy flux vector and rj~ is the time rate of change of the total energy density (internal plus kinetic). Additional energy fluxes can be incorporated into these definitions where relevant (distributed heat sources for example). With the identifications thus made, the total energy balance law is again given by (2.4) and the crack tip flux is given by (2.16). Using the identifi~tions in (3.3) in (2.16), the explicit expression for the crack tip total energy flux is

+ L)vc?,~ + O&i - hj]nj dT. (3.4)

The total energy release rate defined by ‘Z =F/v fohows directly from (3.4) by invoking the local steady state condition iri m - vubl. The result is

g = lim r_O * I

The above expression for the energy flow to the crack tip has appeared in [12,14,15,51]. With the imposition of additional restrictions on material response, Nguyen[lS] used the integral in (3.5) to extract the form of the near tip temperature field for coupled thermomechanical problems. For the case of a stationary crack tip, it can be easily shown that L and hi are 0(1/r) as r -+ 0 and therefore these fluxes make no contribution to (3.5). With the fields within [ ] interpreted as the x1 component of I-IQ, (3.5) conforms to the general energy release rate representation in (2.17).

Complementary total energy integral The complementary integrals are obtained with the help of a Legendre transformation. Let

H(Q S, ai) denote the enthalpy per unit volume, defined by

H = u - c&j. (3.6)

Substituting (3.6) into the First Law (3.1), yields the following expression for the time rate of change of enthalpy,

h = -tyQ - hij. (3.7)

We use (2.3) in (3.7) and rearrange to get

(bqui + h,)j = pdiui - If (3.8)

where, as before, ai is the acceleration vector. We compare the specific balance statement (3.8) with the general balance equation (2.6). These identifications follow:

q = 6uUi + h’ J’

tj/ = (iC - if) , (3.9)

Using (3.9) in (2.16) leads to the complementary energy flux to the crack tip

The complementary energy release rate V”’ = Y-‘/v, follows directly from (3.10) by invoking the local steady state condition (2.13) for the stress rate,

(3.11)

Under certain conditions the integrals (3.4), (3.5) and (3.10), (3.11) can be explicitly written in


terms of other thermodynamic potentials. In the crack tip integral (3.5) the internal energy, U, can be replaced by the Helmholtz free energy Y = U - TS for material response where the product of the absolute temperature, T, and the entropy S, is less singular than U. Similarly the enthalpy in (3.11) can be replaced by the Gibbs function CD = H - TS.

4. 3-D FORMS, SPECIALIZATIONS AND A NOTE ON DUAL INTEGRALS

For 3-D crack problems it is advantageous to employ the vector form of the crack tip integral. Such forms and their interpretation can be found in the papers by Knowles and Sternberg[li] and Budiansky and Rice[7]. Consider a 3-D crack front (a line formed by the intersection of two crack faces) and let the parameter s(x,, x2, xs) denote the arclength along the crack front measured from some arbitrary point. With respect to a fixed coordinate system of arbitrary orientation, the integral in (2.17) can be restated as a vector integral. Specifically, the vector $?k at a point s on the crack front is

(4.1)

Here F(s) is a unit strip of the tubular surface (strip of unit width) which surrounds the crack front about the point s as depicted in Fig. 2. Let v&) denote a unit vector which lies in the plane of the crack and is normal to the crack front (pointing away from the crack). The energy released by the body due to a unit growth of the crack front (in the plane of the crack) at the point s can be obtained by projecting the vector gk(s) onto the vector v&s). Thus the local/pointwise energy release rate is given by

(4.2)

For the mechanical energy integral (see 2.22) Hkj is given by

Hkj = (W + L)6kj - oijui,k s (4.3)

The fluxes in the Hkj tensor associated with the other energy/rate integrals can be found in Sections 2 and 3. Specialized forms of (4.2) were employed by Shih et a/.[371 and Nakamura et al.[39] for 3-D analysis of thermally and dynamically loaded cracked bodies.

Some general comments on the integrals presented in Section 2 and 3 are in order. Under quasi-static conditions or for the case of a stationary crack tip, the terms L, Lc, L” and g c are bounded at the crack tip and consequently make no contribution to the crack tip integrals in (2.21), (2.26), (2.30), (2.34), (3.5) and (3.1 I) - these terms can be deleted directly from the integrals. Alternatively, the same results can be obtained by recognizing that the term t can be omitted from the flux iu (see 2.5) in the general balance law (2.6).

Imposing the restriction that the crack tip is stationary, i.e. v = 0, the above integrals (with the L and L” omitted) are still meaningfully defined. In the case of a stationary crack in a d~ami~ally loaded body, the quantity V must be interpreted as a virtual energy release rate, that is, %’ is the energy that would be released if the mechanical fields were frozen at any instant and the crack tip were given an increment of extension in the x1 direction. With the additional restriction that the fields are quasi-static I;p is the actual energy release rate if the material is (nonlinear) elastic; for more general material response g must still be interpreted as a virtual energy release rate[49, 161. In passing it is worth noting that 9 is non-zero only if Hq is of order r-’ as r + 0. For material response which can be characterized as highly nonlinear, HG is less singular than r-’ and therefore the energy flow to the crack tip V is necessarily zero.

Dual integrals for arbitrary constitutive response have been discussed by Carlsson[ 18) who restricted attention to steady state formulations. Under the assumption of nonlinear elastic material response, Bui[17] has shown that the energy release rate and complementary energy release rate


are equal. In the subsequent paragraphs we discuss an essentially negative result. We will show that Bui’s observation also holds under quite general conditions.

To demonstrate the equality, we consider first the energy release rate integral (2.21) and its complementary counterpart (2.26). We write the difference between dual integrals as

Using the equation of motion (2.2) as well as symmetry of qj we rearrange (4.4) as

Letting& = o+ui, (4.5) can be written as

Z = lim [(V*f)ni - n+f,J df’. r-0 _,‘

(4.6)

The above result (4.6) is also obtained on subtracting the complementary total energy integral (3.11) from the total energy integral (3.5). The difference between the rate integrals (2.30) and (2.34) has exactly the same form upon setting fi = o& in (4.6). We note that the integrand of (4.6) is divergence free which would imply global path-independence. However we wilt now show that under quite general conditions Z is identically zero. We write f in the form

f = j;e, + foe, (4.7)

and take I to be a circular contour of radius r with outward unit normal n = e, (r is the radial distance from the crack tip and 0 is the angular coordinate measured from the xl-axis). Noting that

a a sin e a -=cose---- 8x, dr I ae

we can write (4.6) as

I 7T

z= [V; sin e)’ + If@ cos e)q de .--II

=f,sin@j:, -f,cose/:,.

From (4.8) we have the result Z = 0 if f;( + n) is bounded and f0(7t) = fO( - x). Indeed the result Z = 0 can be expected for most crack problems. For example, if the crack faces are traction free, or if the crack face loading is bounded, we havef,( + K) = 0. Thus we have established the equality of dual integrals under quite general conditions. The ~mpiication of the result is that no new information can be extracted from the complementary integrals that is not already given by the more standard integrals. Nevertheless for certain problems and formulations, it may be more convenient to use the complementary integral. As noted in [17], when a variational principle exists, dual integrals may be employed in conjunction with approximate and numerical field solutions to provide bounding values for the crack tip parameter.

5. (GLOBAL) PATH-INDEPENDENT INTEGRALS AND APPLICATIONS

The distinction between local and global path-independence was discussed in the introduction. It was noted in Section 2 that for the energy release rate 5Y to have fundamental significance, it


is necessary that the value of g be independent of the actual shape of I as its size becomes vanishingly small, i.e. local path-independence must be satisfied. In contrast global path- independence (while certainly a desirable property) is not a necessary property with respect to the energy release rate concept (or crack tip parameter concept). The conditions which ensure local path-independence of the integral in (2.17) have been stated in Section 2. For global path- independence, two conditions have to be met. First, the divergence of Hij must vanish identically. The flux is then said to be conserved. Secondly there must be no flux across the crack faces. In subsequent discussions, we will assume that the latter condition is always met and will also make the additional assumption that material properties do not vary in the x1 direction. Having imposed the preceding rest~ctions, we can focus our attention on conditions under which the tensor field Hij is conserved. In the interest of clarity and conciseness we will work with the integral in (2.21). This will also allow us to make contact with certain known results. It is easy to recognise that the conditions to be discussed below when appropriately interpreted, are also applicable to other identifications of H,j made in Sections 2 and 3.

Steady state crack propagation In general rather strong restrictions on material response are required for Hij to be a conserved

field. However there is an important exception. For steady-state crack growth problems the integral (2.21) is path-independent for any type of material response so that

has the same value for any open path I connecting any points on opposite sides of the crack. The global path-independence is easily seen by noting that the right side of (2.12) vanishes identically by definition for steady-state crack growth problems. This result has been exploited by Freund and Hutchinson[48] for the analysis of a steadily growing crack in an elastic/rate-dependent plastic solid. It can be inferred from the results of Hui and Reidel[46] and Lo[47] that if rate sensitivity is sufficiently pronounced, the near tip fields are sufficiently singular to result in a finite value for $Y’ in (2.21). Although solutions for growing cracks in elastic/rate-independent plastic solids are incomplete, they indicate that the near tip fields are less singular than that required for a finite value of %’ (see 2.19). Thus (5.1) does not appear to be particufarly useful for the problems governed by the latter type of material response. Asymptotic fields for continuously growing cracks in elastic-plastic and inelastic solids are discussed in a number of review articles[49-521.

Conditions for H,j to he divergence free Under the condition that crack faces are free of traction (no flux across the crack faces in

the more general case), the necessary condition for global path-independence is that H,j be divergence free. As discussed in Section 2 the preceding two conditions are necessary and sufficient for path-independence if the fields are sufficiently smooth so that the divergence theorem and transport theorem can be applied to the cracked body. Using the convention employed in Section 2, and for Hij defined by (2.22) we write

For a quasi-statically advancing crack or a stationary crack tip, the kinetic energy term L makes no contribution to the integral in (2.21) so that L and its derivatives can be omitted from (5.1) and (5.2). Thus

The stress work density in (2.4) is written in the more familiar form

‘4 W= cfii d.s@ (5.4)

.O


it being understood that the integration follows the actual strain trajectory of a material point. Now suppose that the stress work density W is a single-valued function of the strain. For an

elastic-plastic material such a special relation exists only if each material point undergoes proportional stressing. Under general loading histories, W is a single-valued function of strain only if the material is nonlinear elastic (hyperelastic). If such a relation holds then

w,l = oijE$,l + w,l Implicit . (5.5)

The latter term vanishes identically since W for a translationally homogeneous material (with respect to the xl direction) does not depend explicitly on xl. We use (5.5) in (5.3) to get

Now assume that the fields are in equilibrium, i.e. qj,:,i = 0, then bl vanishes identically and global path-independence is established. It has been assumed in the proof of global path-independence that body forces and body force like terms (e.g. residual stresses, thermal stresses etc.) are negligible. If such body like forces are important, the integrand H,j is not divergence free.

Constitutive specializations Linear and nonlinear elasticity. Under quasi-static conditions (the crack tip could be stationary

or advancing) and for nonlinear material response where W is identified as the strain energy density we have

N,j = W6lj - C~l4i,l . (5.7)

For this case, H,,is the negative of the xl component of Eshelby’s energy momentum tensor[4,5]. The resulting integral based on (5.7) has the same value for all open paths r connecting any points on opposite sides of the crack and we write

In other words, the integral in (5.8) is path-independent in the global sense. The connection between (5.8) and the energy released from the body per unit quasi-static crack advance (per unit width) and the interpretation of the value of the integral as a measure of the intensity of near tip deformation was made by Rice[l, 21. Eshelby[4,5] introduced the conservation integral (5.8) in the context of the energetic force on a dislocation or point defect; in the present context g is the crack tip force conjugate to the crack length. In this paper we have reserved the use of the symbol J for applications where the intended emphasis and interpretation is its role as the parameter characterizing the strength of crack tip deformation. The symbol I% will be employed under more general circumstances as in (2.17) and when its application has an energy release rate interpretation as in (5.8).

The integral in (5.8) cannot be applied to a stationary crack in a thermally stressed body or in a dynamically loaded body. If thermal stress or inertial effects must be taken into consideration, the inclusion of the limit (r -P 0) in the definition of @’ is necessary[l l-16,37-39].

Rate independent plasticity. For a stationary crack in a quasi-statically loaded body characterized by an elastic/rate-independent incremental plastic response, the interpretation of the integral in (5.8) as the energy release rate cannot be made[lO, 491. Nevertheless the integral may still be used to some advantage. Strain hardening plasticity solutions which elucidate the behavior near the tip of a stationary crack have been obtained by Hutchinson[8] and Rice and Rosengren[9]. While their asymptotic analysis was based on the J2 deformation theory of plasticity, the resulting singular fields satisfy proportional stressing exactly. In other words the HRR fields are also solutions to the corresponding J2 flow (incremental) theory. In particular the amplitude of the HRR fields


is given by the value of the J-integral[i, 21,

(5.9)

It is the above interpretation (valid for both deformation and flow theories of plasticity under the stated conditions) which is pertinent to nonlinear fracture mechanics.

If the load is quasi-statically applied and if each material point is proportionally stressed, the integral in (5.9) is globally path-independent and it is not necessary to shrink the contour r onto the crack tip, i.e. a remote path may be chosen for evaluating the value of J. Stringent as these conditions may be, global path-independence of J has been observed in many full-field numerical studies. Specifically, these full-field finite element solutions of stationary cracks subject to monotonically increasing remote load were obtained using J2 flow theory of plasticity. The path independence of the line-integral in (5.9) strongly indicates that proportional stressing is nearly satisfied everywhere in the cracked body. In addition, detailed numerical studies have revealed that within an annular region which surrounds the zone of finite strain, the full-field solutions are very accurately approximated by the HRR fields when certain size requirements are met (McMeeking and Parks[53], Shih and German[54] and Needleman and Tvergaard[%J). The above studies on J-dominance, and a non-linear fracture mechanics approach to the initiation of crack growth, subsequent quasi-static crack growth and loss of stability has been reviewed by HutchinsonCiO].

Power law creep. The rate integral (2.30) can be applied to an elastic-viscoplastic material where the natural time of the material response has a role in determining the nature of the crack tip fields. It is well known that for a material deforming according to an elastic-power law creep relation the creep strain rate necessarily dominates at the tip of a stationary crack when the power law exponent n is greater than unity[2&22]. Under this condition I,? in (2.30) is asymptotically equivalent to the creep potential W*, i.e.

w N wt/‘* = n fl..k..

n+l Vu asr-+O+. (5.10)

In writing (5.10) we have made use of the result kq IV 2; is the creep strain rate. If inertial contributions are negligible and the conditions leading to (5.10) are applicable, the integral in (2.30) reduces to the so-called C(t) integral for power law creeping solids,

(5.11)

The above integral was employed by Bassani and Mc~lintock[22] to investigate the creep relaxation of crack tip stresses in an elastic-nonlinear viscous solid subject to a (quasi-static) step loading at t = 0. Under steady state conditions or at long times (t + co), the creep potential W*

is applicable everywhere, and therefore the integral in (5.11) is independent of the contour path. The steady-state or long-time value of C(t) has been designated as C* and is the creep analog of the path-independent J_integral[23,24]. In the works mentioned above, C or C* is interpreted as the amplitude of the HRR-type singularity fields. In other words the role of C and C* in creep crack growth is analogous to the role of J as the characterizing parameter for crack initiation and subsequent quasi-static growth in an elastic-plastic solid. Further results on correlation of creep crack growth rates under small scale and extensive creep by K and C* parameters are discussed by Saxena et aI.[25] and Riedel[26).

6. FINITE DOMAIN FORMS FOR Z-D PROBLEMS

In the analysis of crack problems by means of computational methods, such as the finite element method, a fundamental difficulty is encountered in efforts to compute values of the crack tip energy flux versus time or amount of crack growth. The difficulty arises from the fact that, on


the one hand, the crack tip energy flux is defined in terms of the values of field quantities for points arbitrarily close to the crack tip while on the other hand, it is precisely for points near to the crack tip that accurate calculation of the field quantities is most difficult. In applications where the crack tip integral is path-independent (e.g. J-integral, C*-integral etc., which have been discussed in Section 5), it can be evaluated along contours remote from the crack tip where presumably the numerical fields are more accurate. In an effort to circumvent the difficulties arising from near tip integration, Kishimoto, Aoki and Sakata[27-291 and Atluri, Nishioka, Brust and co-workers[3& 361 have restated crack tip integrals in terms of equivalent path and area integrals. They have labeled the latter as path-independent integrals because the value of these integrals is independent of the domain bounded by an arbitrary ‘outer’ path. This is an unfortunate use of the term since any line integral (including those with non-divergence free integrand) can always be restated as a path-area integral which has the property of being domain independent in the sense described above.

In the following section, we present a versatile method whereby crack tip integrals are recast as integrals over finite domains around the crack tip. To keep the presentation concise, the domain forms will be detailed for the integral (2.21). Following exactly the same steps, it is straightforward to derive the domain forms for the other crack tip integrals. It will be demonstrated that the path and area integrals referred to above are non-advantageous specializations of domain integrals.

We introduce weighting functions qi which are defined over the domain of interest[37-40]. With attention confined to two-dimensional fields and the crack advancing in the x,-direction, q1 is the only non-zero function. Consider the simply connected curve C = Cc, + C, + C_ f I- as shown in Fig. 1 where C, and C_ are physical boundaries and C, is an arbitrary interior boundary. The function ql has a value of unity on the vanishingly small inner contour I, and zero on the outer contour C,,. Within the area A enclosed by C, ql is an arbitrary smooth function of xl and x2 with values ranging from zero to one. The function ql (x,, x2) may be interpreted as the virtual translation of material point (x,, x2) due to a unit extension of the crack in the xl-direction[40]. With this interpretation, the connection to the virtual crack extension (VCE) method can be madeC41-441. We use the function ql to rewrite (2.21) in the form

F = I

[-(W + L)6lj + Ogui,l]mjq1 dC - I

qIui,rm2ql dC . (6.1) c c++c

In the above integral we have used mi in the place of ni where m, = ni on Co, mi = - ni on I-, and ml = 0, m2 = f 1 on the crack faces. The last integral in (6.1) vanishes for traction free crack faces. Applying the divergence theorem to the closed integral, we obtain

W = - ! ([(W + L )6lj - ~~~i,lIQl,j + [(I+’ + L ),I - (ogui,l),j] 41) dA

A (6.2)

- qIui,lmzql dC. .c+ +c-

The above is the finite domain representation of the crack tip integral in (2.21). The surface integral term in (6.2) arises because of non-vanishing traction (or fluxes) across actual physical boundaries. It is clear that the value of the integral in (6.2) is independent of the size and shape of the domain. Simpler specialized domain integrals can be obtained from (6.2) by imposing restrictions on crack tip motion (e.g. quasi-statically advancing crack) and on material response through W(e.g. nonlinear elasticity). These aspects will be discussed shortly. It may be noted that the form of the integral in (6.2) is similar to the variational statement of the field equation and that furthermore the domain integral in (6.2) can be evaluated by the integration schemes employed for computation of the stiffness matrix and the boundary loads in the finite element method. In this sense, the domain integral representation of ‘Z is more naturally tied to the variational statement of the problem and naturally suited for the finite element method. Calculations to date have convincingly demonstrated that very accurate values of SY (and J) can be extracted from the domain integral. The noted invariance of the domain integral representation of SY’, J, C(t) with respect to variation in domain size and shape provides an independent check on the consistency and quality of the


numerical calculations. This is another attractive feature of the domain integral. In the same manner, finite domain integral representations of crack tip integrals in (2.26),

(2.30), (2.34), (3.5) and (3.11) can be obtained. In fact a general form of the domain integral can

be written for the general form of Y. First we rewrite (2.17) as

F = - Hljmjql dC + I

H12m2ql dC. (6.3) c .c+ +c

Applying the divergence theorem to the closed integral yields

v = - 1 (Hl,ql, + Hyjq1) dA + / ffe2ql dC. (6.4) .A .c, +c

The integral in (6.2) for the mechanical energy release rate can be recovered from (6.4) by using the identification for Hlj in (2.22). Similarly explicit domain integral representations associated with the other crack tip integrals can be obtained by the appropriate identification of H,j.t

Linearly and nonlinearly elastic materials To get directly to the essence of domain forms, the boundary terms, i.e. the integral over the

crack faces C, and C will be omitted in the subsequent discussion. These boundary terms can be appended to the domain integrals when the need arises. For linearly elastic material response, W = oVsij/2, and (6.2) (and (6.4)) reduces to [38,39]

57 = 1 [- (W + Lkl,l + qqqlj + /$iiiui,l - ic;tii,l)ql] dA JA

The above result neglects any contributions from crack face traction and body force like terms. If crack face traction is not negligible, the integral along the crack faces C, and C_ in (6.2) must be appended to (6.5). With a simple reinterpretation, the above integral can be rewritten to account for thermal and/or residual stresses[37]. The boundedness of the integral in (6.5) follows directly from (2.19). In [16] the boundedness of the integral was established by means of a direct observation. Two other path-area integral representations for @’ (see [16]) can be rewritten in strictly domain forms with the help of the ql function.

We note that the path-area integral representations for the elastodynamic J given by eq. (20) in [33] and by eq. (3.1) in [16] can be regarded as one form of (6.5) where the qi function is taken to be unity within the domain A. It must be pointed out that the path-area integrals advocated by Kishimoto and co-workers, and by Atluri and co-workers which include path integrals on contours C, in the interior of the body are non-advantageous specializations of domain integrals. In particular, the generalization of their path and area integral method for the calculation of pointwise energy release rates on three-dimensional crack fronts is rather cumbersome. In contrast, the present formulation which is based on a weighting function, is readily generalized to treat three-dimensional crack problems; these aspects will be taken up in Section 7.

Under quasi-static crack advance or for a stationary crack in a quasi-statically loaded body, a further simplification of (6.5) is possible (deLorenzi[44], Li et al.[40])

v = I- Wql.1 + q~i,m,l dA . .A

(6.6)

tThe path and area integrals introduced in 127-361 are special cases of the domain integral given above. Referring to (6.4) we choose q = 1 in A and (I = 0 on C,, to get

!?Y = / H,,n, dC - 1 H,,, dA + 1 H,gn2 dC -c. .A -c+ +c-

Such path-area integral expressions are not particularly advantageous forms for the evaluation of crack tip integrals.


In this case the corresponding crack tip integral is globally path-independent (see (5.8)). Therefore any annular strip surrounding the crack tip can serve as the domain of integration A for the integral in (6.6). In the case of a stationary crack in a dynamically loaded body the additional term piii ui, q1 must be added to the integral in (6.6) and the domain A must include the crack tip region since the corresponding crack tip integral is defined only in the limit of I -+ 0[16, 37- 39]. The interpretation of 5Y for the case of a stationary crack in a dynamically loaded body is given in [16].

All of the energy release rate expressions derived in the preceding section also apply for nonlinearly elastic material response. These integrals do not appear to be useful since it can be easily shown that a separable crack tip asymptotic field cannot be constructed for dynamic crack growth in a nonlinear elastic solid (to see this write L as pV2Ui,lUi,I in (2.21)). A discussion of some computational aspects of near tip fields appropriate to linear and nonlinear elastic bodies has been given in Section 3 of [16].

Elastic-plastic material response For a stationary crack in a quasi-statically loaded body which is characterized as elastic-

plastic, the integral in (6.2) simplifies to

(6.7)

If the body is loaded dynamically the additional term pi&,q, must be added to the integral in (6.7). In the above integral W is the strain history dependent stress work density. Under certain conditions a further simplification of the integral in (6.7) is possible. Consider the following argument. If singularity fields of the HRR-type dominate in the crack tip region, then proportional stressing of each material point in the crack tip region is assured. In addition suppose that the body is nearly proportionally stressed. Under these conditions the stress work density is a single-valued function of the strain (or stress), i.e. W is a potential for a+ and the simpler domain form (6.6) can be employed in place of (6.7) even in an incremental plasticity analysis.

Evidence for the above observations can be found in several recent full-field numerical calculations, two of which will be mentioned. Nakamura et al.[38, 393 investigated the response of a dynamically loaded three-point-bend ductile fracture specimen by a finite element calculation using a J2 flow theory of plasticity. They extracted the value of dynamic J using the dynamic version of (6.6) (see eq. 2.5 and 2.13 in [38, 391 respectively where the strain history dependent stress work density is denoted by U) and found that its value did not depend on the size and shape of the domain. The implication is that to a good approximation W is a potential for the stress oV in the case of the stationary crack problem.

For the case of dynamic crack growth in an elastic-plastic material the general result (6.2) can still be applied as long as W is the total density of the accumulated stress crV working on the strain cii at each material point (xi, x2). Similarly the integral in (6.7) is applicable to quasi-static crack advance in a quasi-statically loaded body. However as noted previously, the value of the energy release rate is zero for growing cracks and therefore (6.2) or (6.7) is not particularly useful.?

Visco-plastic material response To obtain the domain integral (2.30) for the elastic-viscoplastic material, we simply replace

the quantities W, L and u,,, in (6.2) by I@, L and Vi,i. Domain forms for specialized applications are easily obtained. For example, carrying out the above noted substitutions in (6.7) leads immediately to the domain form for a stationary (or growing crack) in a quasi-statically loaded body:

w = ( [(- m,, + qjhi,l)qlj + (- @.I + ~~Q,l)q11 dA . (6.8) .A

t Unlike the stationary crack problem, there is no basis for applying the integral in (6.6) to a growing crack in an elastic-plastic solid since material points will undergo distinctly non-proportional stressing as the crack tip advances by the points.


Under the stronger restriction that the crack tip is stationary, and for a material which deforms according to an elastic-power law creep relation, the result (5.10) holds. In this case the relevant crack tip integral is given by (5.11). The domain form for (5.11) is

C = i

[(- W*S, + qjti,l)qld] dA . (6.9) A

The latter domain integral is the creep counterpart of the domain form for J in (4.6). A finite deformation version of the integral in (6.9) has been employed by Li et a1.[56] to characterize the fields around a finitely deforming crack in an elastic-nonlinear viscous solid.

7. FINITE DOMAIN FORMS FOR 3-D CRACK PROBLEMS

In three dimensional problems the pointwise value of the crack tip parameter along the crack front is often required, For illustrative purposes the presentation in this and the following section will be for the crack tip energy release rate integral (2.21). We emphasize that the approach is valid for any of the crack tip integrals we have considered in the previous section.

The derivation to follow will be concise since the details have already appeared in two publications (Shih et al.[37) and Nakamura et al.[39]). Consider a global coordinate system such that the crack lies in the xl-x3 plane with the x2-axis perpendicular to the plane of the crack. If an alternative coordinate system is preferred the usual transformations can be applied to the results of the subsequent development. Asymptotically, as r -+ O', plane strain conditions prevail so that the three-dimensional fields approach the (plane strain) two-dimensional fields at the crack front. Thus (2.21) defines the pointwise energy release rate. The pointwise energy release rate $Y(.s) at the point s of the curved crack front with in-plane unit normal v&) is therefore given by (4.2) which is rewritten as

(7.1)

where mj is the normal to I pointing towards the crack front, i.e. mj = - nj on I (see Fig. 2a). The energy released when a finite segment, Lc (s, < s < .Q), of the crack front advances an

amount Aalk(s) (see Fig. 2b) is given by

PAa = ! F(s)SI(s) ds = Aa ??(S)&Y) t+(s) ds (7.2) Lc Lc

where

i%(s) = A&(s) v&s)

is the local extension normal to crack front. On employing (7.1) in (7.2) we obtain

=- ~kH~im~ dS .

Here, S, is the ‘tubular’ surface enclosing the crack front segment and the limiting process consists of shrinking the ‘tube’ radius to zero (see Fig. 2~). It is noted that the so called end-cap contributions (O(g)) vanish in comparison to the contributions from the curved surface (O(r)) as r -+ 0 in the limiting process. To a first approximation @(,s) may be assumed constant over the length L, and


fb)

Fig. Z(a). Conventions at curvilinear crack front. (b) Virtual crack advance between s, and s,. (c) Inner tubular surface S, and outer arbitrary surface S,.

brought outside the integral sign in (7.2) to yield

57 ‘F(S) ! M)Vk(S) ds * LC

Using (7.3) and (7.4) above we obtain the following expression for P(s), i.e.

F?(S) = - lim 1kHkjmj dS / I

&(s)v~(s) ds . r-0 .s, i .Lc

(7.4)

(7.5)

A more precise procedure for calculating the pointwise value, ?Y(.s), which is naturally suited for finite element computations is discussed in the next section.

Domain integral representation The value of V (for example energy release rate) along a three-dimensional crack front is

given by the limiting contour integral (7.5). There are obvious difficulties with evaluating such an expression in any numerical scheme, since the limit cannot be taken exactly due to the finite nature of the discretization and, in addition, the near crack tip fields are usually least accurate. Alternative representation of the crack tip integrals have been derived in Section 6 for 2-D problems. Here we generalize the derivation to 3-D problems. The resulting domain expression is ideally suited for numerical calculation as it involves only a volume integral and possibly surface integration on the crack faces (when the flux across the crack faces is not zero).

In general, the energy flux tensor Hki is not divergence free and we write

HkjJ E bk in V (7.6)

where V is any simply connected sub region of the body not containing a singularity and the exact form of bk depends on the form of IikPt

We now present a general method for deriving domain integrals based on the approach adopted in Section 2. Following the same procedure as in Section 2 we take the inner product of (7.6) with an arbitrary vector field qk, integrate over the volume V and apply the divergence

?I& can be taken to be the integrand of any of the crack tip integrals discussed in Sections 2 and 3.

theorem to give the weak or

Integrals from momentum and energy balance

variational form of (7.6) as

635

P

qkHkjmj dS = I v(Hkj9k~ + b&k) dV (7.7)

s

where it has been assumed that qk is smooth enough for the indicated operations to be carried out. Now if we identify the closed surface S = S, + S, +S+ + S_ (see Fig. 2c) where S, is an arbitrary outer surface (including the end-caps) and S, and S_ are the crack faces connecting S, to S,. We choose

i

lk on S,

qk= ’ on S,

orthogonal to mi onS, andS_ otherwise arbitrary.

With the help of {7.7) and (7.8), we write (7.3) in the equivalent domain form,

izll = - i

(Hkjf&~ + b,&) dV + Hkjmjq~ dS . V

(7.8)

(7.9)

In the absence of flux across crack faces (e.g. crack face traction in mechancial loading problems) the surface integral uver physical boundaries vanishes identically and the expression (7.9) is a volume integral representation of the limiting surface integral (7.3). The representation (7.9) is naturally suited for numerical computations. For crack advance in the xl-x3 plane, l, = 0 on S, and hence we can choose qz = 0 in V. From the result (7.9) above and noting (7.3) and (7.4) we obtain the following expression for the pointwise value of V (which we write in invariant notation)

t%(s) = -.fV(Vq:H + b*q) dV

jkl*v ds ’ (7.10)

The value of F?(s) does not depend on the domain in the sense that with an appropriate weighting function the choice of the volume V is arbitrary. By choosing a number of domains and q functions the property of ‘domain independence’ provides a useful check on the consistency of the results in finite element calculations. The finite element implementation of the above domain integrals in conjunction with piecewise basis functions to represent V(s) and 61(s) along the crack front is discussed in detail in [37,39,40]. Results for V(s) along a three-dimensional crack front in a thermally stressed body and a dynamically loaded body are presented in the latter two publications. In the context of mechanical energy release rate evaluation, relations similar to (7.5) and (7.10) have appeared in [4144].

Axi~ymme~r~c form The appropriate form of ‘Z(s) for axisymmetric problems can be derived from the three-

dimensional result (7.10). For an axisymmetric crack configuration where R is the radial distance of the crack front from an origin on the axis of symmetry and in the plane of the crack, we define a local set of cylindrical coordinates, (e,, eP, eZ) on the circular crack front. For uniform radiai expansion of the crack I = e,, and a suitable choice of q is q = qr(r, z)e, where q, is unity on the crack front and vanishes on the outer arbitrary curve C,,, which bounds the domain A in the r-z plane. The normal to the crack front is given by v = e, and therefore JL, i*v ds = 2zR where L, denotes the crack front. On substituting the above values into (7.10) we obtain the following expression for the pointwise value of Y

(7.11)

where /I ranges over the coordinates r and z (no sum on r and 9). Details of the above derivation


are given in [37] with an application to thermal crack problems,

Two-dimensional form Now consider a line crack oriented along the x1 axis of a Cartesian coordinate system at

outlined in Section 2. Here f = el, and a suitable choice of the weighting function is q = q1 (x,, x2)el where qI is unity on the crack front and vanishes on the material curve C,, which bounds the domain A in the xl-x2 plane. The normal to the crack front is given by v = el and therefore SL., 1-v & = L where L is the length of the crack front segment being considered. As for the axisymmetric case we substitute these relations into (7.10) to yield the expression for 9 for the two-dimensional problem, i.e.

(7.12)

Thus we have recovered the 2-D domain integral in (6.4) from the 3-D result.

8. CRACK TIP INTEGRALS FOR GROWING CRACKS

The success of the J-based nonlinear fracture mechanics approach has stirred considerable interest in the subject of path-independent integrals. We have stressed that the pertinent role of J in nonlinear fracture mechanics is its interpretation as the parameter characterizing the strength of the crack tip fields and that path-independent per se has little to do with the applicability of J as a characterizing parameter. However carefully laid out arguments on the basis of the J- approach (e.g. [lo]) appeared to have escaped attention of some investigators. An abundance of literature on a variety of so-called path-area independent integrals has appeared in recent years purporting to address unresolved problems in crack mechanics. A number of such recently proposed path-area integrals have been discussed in a review paper by Kim[57]. These ‘newer’ integrals are very similar in many respects. We will therefore confine the discussion to the integrals proposed by Atluri and co-workers.

Elastic-plastic response In a series of papers Brust et aZ.[3436] have proposed that their revised (improved) T* and

i‘* integrals are suitable for correlating crack growth under fairly general loading conditions in materials which can be characterized as elastic-plastic and elastic-viscoplastic respectively (these papers contain additional references to Atluri and co-workers’ publications on earlier versions of these integrals).

To make contact with their representations of J, we employ their notation and designate a vanishingly small loop I by I,. We now restate (5.9) as the difference between an integral over the closed loop C(C = C, + C, + C_ + IJ and an integral over the open loop C, + C, + C_. Noting that mj = - nj on I, (see Fig. l), we have

Now apply the divergence theorem to the closed integral to get


where it may be noted that the term aU~~i,l vanishes identically for an equilibrium field. (With respect to the domain integral presented in Section 6, the representation in (8.2) can be obtained by assigning q1 to be unity in the domain A enclosed by C.) It is clear from the above manipulations that the integrals in (8.2) are merely alternate representations (path and area integral forms) of (5.9), and the latter integral follows immediately from (2.21) under the assumption that L is bounded as r + 0.

We are now ready to impose restrictions on constitutive response (recall that only momentum balance was invoked in the derivation of (2.21)). By requiring W to be the strain energy density, the integrals in (8.2) define the so-called deformation plasticity J. If W is defined as the total accumulated stress work density then the integrals in (8.2) pertain to the incremental plasticity J. It has always been understood that in an incremental plasticity analysis, W is necessarily identified with the total accumulated stress work density and its integration follows the strain trajectory of a material point. This point appeared to have eluded some investigators. While the integrals in (8.2) can be applied to both stationary and growing crack problems some remarks are in order. In the case of a stationary crack tip, the loop r8 is fixed with respect to material points. In the analysis of a growing crack, the loop rE translates with the moving crack tip, i.e. different material points will be advancing through the loop lr, and the boundaries C, and C_. Thus due care must be exercised if the integrals in (8.2) are employed to extract a value of J. Various shapes for the

loop F, have been discussed in the literature (e.g. Aoki er a!.[291 and Brust and Atluri[36]). In the context of a deformation theory solid and for fields which are in equilibrium, the two

expressions in eq. (4) for J, in Brust et al.[34] are precisely the two integrals in (8.2). For an incremental plasticity solid, the path integral, and the path-area integral for z in eq. (7) of [34] and in eq. (5) of [35] are identical to the integrals in (8.2). We again note that the integrals in (8.2) are merely another representation of (5.9) and the basic result in (2.21).

In computational studies, including finite element calculations, it can be convenient to work with incremental quantities. To this end, the first expression in (8.2) can be written in incremental form as

AJ zz Jn+’ - J” (8.3)

where A W is defined by

‘&;+I AW=

! q dgY = (Ok + +AQA+ (8.4)

6;

In (8.3) and (8.4) AJ, A W, AC+ AQ and AZ+, are the increments from step n to step y1 + 1 and o# and Ui,l are the stresses and displacement gradients at step n. To make contact with proposed ‘incremental integrals in path and area form, (8.3) is restated as the difference between a closed integral and outer path integrals. We write

AJ = I

AH,jnj dC - co+c+ +c_ I

AHijm/ dC a

C (8.5)

where the terms within the parentheses in (8.3) are collectively represented by AHIp Assuming that cr@ and Ace are equilib~um fields, the application of the divergence theorem to the closed integral Ieads to the following alternate representation for (8.3)

AJ = I G+C+ +c_

(AW6il - UgAUi,I - Aoi/Ui,l - ADvAu;,l)nj dC

fAq,dl dx.4 .

(8.6)


Other forms of the incremental path-area integrai can be similarly obtained. Such forms are also obvious from the results given in Section 6. A more advantageous incremental representation for AJ can be obtained by differencing the domain integral obtained with the use of the weighting function (see (6.7)). With the latter representation, the path integral on an arbitrary interior boundary C, vanishes identically. We add that

J=cAJ. (8.7)

follows by definition. The revised (or improved) AT; in eq. (3) of Brust and AtluriC36-j and in eq. (6) of [35] and

AT; in eq. (3) of [34] (after a straightforward coordinate transformation) is precisely the integral in (8.3) or (8.6). To support their claim that there is merit to the T* (or AT*) as a crack growth parameter, they pointed to their finite element calculations which showed that the value of T* (defined on a finite length contour which is more amenable to evaluation during the process of simulated crack growth) saturated at a constant value after some amount of crack growth. This observation is not new. Shih et al.[58] and Shih and Dean[59] have carried out finite element modeling of crack growth in ductile solids using an incremental plasticity theory. They evaluated J on contours close to the advancing crack tip and on contours remote from the crack tip, the values so calculated were designated by Jnf and Jf respectively. They reported that Jnr remained essentially constant after some amount of crack growth while .I$ continued to increase. Shih and co-workers noted that the constancy of Jnr does not imply that the local J is an appropriate characterizing parameter for extended crack growth in ductile solids (since the calculated Jnfduring crack growth appeared to be mesh dependent). There is no evidence (that we are aware of) that in a differently sized specimen and/or different crack configuration, the local J value will saturate at the same constant value after the same amount of crack growth.

Brickstad[60] carried out an analysis of rapid crack propagation in a rate independent elastic- plastic solid using a finite element method. He employed uniformly sized elements along the path of crack propagation, and reported that after some amount of crack propagation, the energy flow to the crack tip 4p (Brickstad’s %’ is equivalent to J in (8.2), Shih et al’s Jnr, and to Brust et al.3 T*) remained essentially constant with further crack advance. More significantly, his numerical simulations revealed a definite trend -V saturated at lower and lower values as the element size decreased. In an earlier finite element study, Kfouri and RiceE6I-j convincingly demonstrated that the value of %’ depended on the size of crack growth step (which is controlled by the element length along the crack propagation path) relative to the plastic zone size. These carefully executed numerical studies support the known theoretical result that V = 0 for the growing crack[49] (the result dates back to 1966, see [49, 611). These findings lead to the conclusion that a non-zero value of T* is an artifact of a finite mesh length and that the value itself depends on the choice of a finite path size employed in the computation of 7 ‘*. In light of these results and the noted equivalence between J in (8.2) and Brust et al.3 T *, the usefulness of T* (or AT*) is very much in question.

Visco-pkastie material response

For applications where the kinetic energy L at the crack tip is bounded, (2.30) reduces to

Under the stronger restriction that the crack tip is stationary and for a material which deforms according to an elastic-power law creep relation, we have


and the asymptotic connection between @ and the creep potential W*,

fj? N w* ;= n a&..

n+l @@ as r --, 0’. (8.10)

The integrals in (8.8) and (8.9) satisfy local path-independence. Under steady state conditions (extensive creep), W* is applicable to the entire body. Consequently the integral in (8.9) is globally path-inde~ndent and has been referred to as the C*-integral[2&26].

To establish the connection between (8.3) and recently proposed rate integrals, we consider the 1 integral defined by

(8.11)

where I%’ = a& is the density of the stress work rate. Similarly an equivalent representation of (8.11) in terms of path and area integrals, can be obtained by carrying through the above manipulations on (8.6). The c integral given by eq. (6) in Brust and Atluri[36] is identical to (8.11). The steady-state integral given by eq. (7) in [36] is defined by

To make a point, we take the difference between the integrals in (8.9) and (8.12). This gives

c - (i’~,ss = (W* - @‘)n, dl- . (8.13) i-.

It can be easily shown that the right hand side of (8.13) is not divergence free. Under conditions where I? and W* are asymptotically equivalent, the above observation also holds if (8.8) is employed in the place of (8.9). It may be recalled that the integrals in (8.8) and in (8.9) are locally path-independent. The proof of local path-indepenence under fairly general conditions has been given in Section 2. It follows that the value of (?;>ss is dependent on the shape of the contour Is Thus its significance is in question. This fundamental deficiency can be made more palatable by the inclusion of a prescribed path in the definition (i-;)ss. For example, by choosing a circular contour of vanishingly small radius centered about the crack tip, a particular connection between (p;)ss and C can be made using (8.13). For an HRR-type near tip field and with I, prescribed to be a circular contour, the right hand side of (8.13) is readily evaluated.

Brust and Atluri speculate that p? and (i-T)” parameters are applicable to creep crack growth problems under non-steady state and steady state conditions respectively, but offer little evidence to support their claim. On the basis of the concepts discussed in Sections 2 and 5, it would appear that C and its long-time counterpart C * are the natural candidate parameters for correlating creep crack growth rates. Of course the case for using C and/or C* rests on much more substantial grounds, e.g. the theoretical studies presented in [20-22,261 and the experimental results reported in [24-261. Additional references on theoretical and experimental studies of crack growth in various creep regimes and under fairly general loading conditions are given in the book by RiedeI[26].

9. CONCLUDING REMARKS

A unified approach to the derivation of crack tip integrals and associated finite domain integrals has been presented. The results obtained are consistent with our viewpoint that crack tip integrals based on mechanical fluxes are direct consequences of momentum balance. Similarly crack tip integrals based on ‘total’ fluxes are consequences of ‘total’ energy balance. We have also provided advantageous finite domain forms for the evaluation of crack tip integrals in 2-D and 3-D crack problems. Such representations of crack tip parameters are naturally suited for the finite


element method and similar numerical methods. With the usual reinterpretations, crack tip integrals and their domain forms are also applicable

to deformations of arbitrary magnitude. In this case, the contours/surfaces and associated normals, and areas/volumes are defined in the reference (undeformed) configuration. The Cauchy stresses are replaced by the nominal (transpose of the first Piola-Kirchhoff) stress tensor, displacement gradients are expressed in terms of the deformation gradient, W is interpreted as the stress work density per unit reference volume, and the heat flux is measured per unit reference area (cf. Eshelby[4,5], Knowles and Sternberg[6] and Rice et al.[45].)

The disproportionate attention (and in many cases unwarranted) attached to crack tip integrals and path-independence has detracted attention from the fundamental ideas that are the foundations of nonlinear fracture mechanics. The strength (magnitude) of deformation fields close to the crack tip can always be defined by a line integral, placed appropriately with respect to the crack tip, and where the integrand comprises of deformation quantities etc. In this sense any sensible combination of deformation, product of stresses and strains, etc. which is integrated along a loop surrounding the crack tip could serve the purpose of a crack tip parameter. The important question is whether crack configuration, crack length and applied load level enter into the description of the near fields through that single scalar parameter and if that particular description of deformation (characterized by a scalar parameter) is accurate over a region which is larger than the fracture process zone. To begin to address these issues, the nature of the crack tip fields must be known in some detail. A better understanding of the competing separation mechanisms under various load-temperature regimes and the coupling between progressive failure processes at the crack tip and the surrounding stress and deformation fields is also essential. Until more is known about crack tip fields of growing cracks and the interaction between the relevant micromechanics and continuum near tip fields, the application of crack tip integrals to complex problems (where nonlinear effects are important) is an arbitrary exercise.

Acknowledgemen&--We gratefully acknowledge the research support of the Naval Sea Systems Command (SEA 05R15) through ONR grant NOGO1485-K-0365, and the NSF Materials Research Laboratory at Brown University through Grant DMR83316893.

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REFERENCES

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Hutchinson, Crack-tip singularity fields in nonlinear fracture mechanics: a survey of current status, in Advances in Fracture Research (Edited by D. Francois et al.), Vol. 6, pp. 2669-2684, Pergamon Press (1981). F. Nilsson, Crack growth initiation and propagation under dynamic loading, in Proc. 3rd Conference on MechanicaZ Properties al High Rates of Strain, pp. 185-204. Institute of Physics (1984). L. B. Freund, The mechanics of dynamic fracture. Brown University Report, to appear in Proc. Tenrh U.S. National Congress of Applied Mechanics (1986). R. M. McMeeking and D. M. Parks, On criteria for J-dominance of crack-tip fields in large-scale yielding, in Elastic- PZasfic Fracture, ASTM STP 668 (Edited by J. D. Landes et al.). 175-194 (1979). C. F. Shih and M. D. German, Requirements for a one paramerer ~hara~t~~~at~on of crack tip fields by the HRR singularity. Znr. J. Fracture 17, 27-43 (1981).

[55] A. Needleman and V. Tvergaard, Crack tip stress and deformation fields in a solid with a vertex on its yield surface, in Elastic--Plastic Fracture, ASTM STP 803 (Edited by C. F. Shih and J. P. Gudas), Vol. I, pp. SC115 (1983).

[56] F. Z. Li, A. Needleman and C. F. Shih, Characterizations of near-tip stress and deformation fields in creeping solids, in preparation (1986).

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[57] K. S. Kim, A review of path-independent integrals in elastic-plastic fracture mechanics. General Electric Company Interim Report to NASA (1985).

[SS] C. F. Shih, H. G. deLorenzi and W. R. Andrews, Studies on crack initiation and stable crack growth, in Elastic- Plastic Fracture, ASTM STP 668 (Edited by J. D. Landes et al.), pp. 655120 (1979).

[59] C. F. Shih and R. H. Dean, On .I-controlled-crack growth: evidence: requirements and applications, in Mefhodology for Plastic Fracfure. Combined Seventh-Eighth Ouarterlv Renort. General Electric Comnanv. SRD-78-xx (1978).

[60] B. Brickstad, A viscoplastic analysis of rapid crack propagation experiments in steel. J. heck.’ Phys. Solids !41, 307- 327 (1983).

[61] A. P. Kfouri and J. R. Rice, Elastic/plastic separation energy rate for crack advance in finite growth steps, in Fracture 1977 (Edited by D. M. R. Taplin), Vol. 1, pp. 43-59. University of Waterloo Press (1977).

(Received 21 October 1986)

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Crack Tip and Associated Domain Integrals

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