Asymptotic finite deformation analysis of growing crack fields in elastic-perfectly plastic materials

J. Mwh.

Php

0022-5096/93 $6.00+0.00 ICI1993 Pergamon Press Ltd

So/irk Vol. 41, No. 4, pp. 6X9-723, 1993

Printed in Great Britain.

ASYMPTOTIC FINITE DEFORMATION ANALYSIS OF GROWING CRACK FIELDS IN ELASTIC-PERFECTLY PLASTIC MATERIALS C. Department

of Mechanical

Engineering,

R. REID

University of Wyoming, U.S.A.

P.O. Box 3295, Laramie.

WY 82071,

and W. J. DRUGAN Department

of Engineering

(Rrcrired

Mechanics. University of Wisconsin-Madison, Madison, WI 53706, U.S.A.

6 Jmwq

1992

;h

rerisedfivm

1415 Johnson

Drive.

20 Au,9ust 1992)

ABSTRACT ASYMPTOTIC, finite deformation

solutions for the stress and velocity fields near the tip of a quasi-statically propagating Mode I plane strain crack in an incompressible, elastic perfectly plastic material are derived. An objective form of the PrandtllReuss constitutive relation is employed. The solutions obtained are represented by a singular perturbation series with the singularity described by integer powers of the quantity (~1In r) where p, the perturbation parameter, is of the order (yield stress/elastic modulus) and r is the nondimensional distance from the crack tip. Previously developed small-displacement-gradient solutions are shown to result to leading order. However, the extension of these solutions to higher order is not straightforward; the introduction of strained coordinates and asymptotic modifications of the interfaces joining different near-tip angular sectors are required. The resulting solutions contain an asymptotically indeterminate parameter that is equivalent to the free parameter m in DUGAN and CHEN’S (1989, /. Mm-h. P11y.s. Solids 37, 1) small-displacement-gradient solution family. Indeed, a principal conclusion is that the present solution approaches that of Drugan and Chen outside an extremely small radius (of the order of the microstructural scale of the material), differing significantly only for radii below which the quantity (p In I) becomes significant compared to unity. Thus, at least for the constitutive class analysed, the finite deformation solutions confirm and quantify the remarkable accuracy and range of validity of the previous small-displacement-gradient growing crack solutions.

1.

INTRODUCTION

THE PROCESS of plane strain fracture of ductile materials, a significant phase of stable crack growth often occurs between crack growth initiation and final instability (GREEN and KNOTT, 1975 ; HERMANN and RICE, 1980). Load levels, as measured by such far-field parameters as the J-integral, typically increase substantially during this period of stable crack growth. If one hopes to describe stable crack growth effectively for efficient design and accurate safety assessments, theoretical solutions are necessary. It is the intent of this investigation to elucidate the near-tip solutions for the stress and velocity fields that are valid within a rigorous, finite deformation formulation. DURING

689

690

C.

R. REILIand W. J. DKUGAN

The first complete asymptotic solution for the Mode I growing crack problem was developed independently by SLEPYAN (1974). RICE cutul. (1980), and GAO (1980). The resulting stress and velocity fields are illustrated via their (coincident) characteristics in Fig. I, with (I,, = 45 , 0, = 112.1’ and O2 = 162.1 . all being measured counterclockwise from the crack symmetry plane ahead of the crack tip. This configuration is known as the “modified Prandtl field” ; it consists of “constant stress” plastic sectors A ahead of the crack and C adjacent to the crack flank. a “centered fan” plastic sector B following the leading constant stress plastic sector. and an elastic sector II following the centered fan plastic sector. The solution of Fig. I has a continuous, non-singular stress field, but the velocity field exhibits a logarithmic radial singularity and a discontinuity across the boundary between sectors A and B. This solution is characterized by the physically untenable feature that the dominant near-tip terms of the stress and velocity fields are completely specified by the near-tip analysis. By accounting for finite curvature of the interface between the leading constant stress and centered fan sectors, DRU(;AN and CHEN (1989) derived afirr~i/~, of solutions for the fields near a growing crack tip, termed the “Fjz-family”, in which HT is an asymptotically indeterminate parameter that characterizes how the far-field conditions influence the near-tip solution. Their solution family also has the asymptotic configuration of Fig. I. but now H, and 0, depend upon 1~. Their approach also eliminates the velocity discontinuity of the previously discussed solution. One feature common to these previous solutions is that they are based upon a smalldisplacement-gradient formulation. An approximation for the non-objective material time derivative of the Cauchy stress was used in the Prandtl Reuss constitutive equation. and geometry change effects were ignored. The results obtained are inconsistent with the small-displacement-gradient assumption. since the velocity solution predicts large deformation and rotation in the near-tip region. The present investigation seeks to address this limitation by deriving continuous solutions for the asymptotic stress and velocity fields that are valid within a finite deformation context. Finite strain is accounted for by using an objective constitutive relation and by considering geometry changes caused by the deformation. In developing the solutions. the process is assumed to be steady state with respect to a coordinate system moving with the crack tip. This says, in efrect, that stable crack growth is presumed to occur with an invariant crack tip profile. There exists expcrimental evidence for this assumption [see, for instance, SHIH rt a/. (1979) or KANNINEN and POPELAR (19831 and this assumption can bc subsumed within the equivalent

FIB;. I. Illustration

of the near-tip

stress and velocity characteristics crack.

for a growing

mode

I plant dram

Growing crack fields criteria

for continued

crack growth

specified

(HERMANN and RICE, 1980) or by a critical

691

by a constant crack tip opening angle crack tip opening at a specified distance

behind the crack tip (RICE et al., 1980). A solution for Mode I, plane strain, stable crack growth in an incompressible material is developed by using a combination of asymptotic and perturbation methods. The asymptotic part of the analysis, which is with respect to radius from the crack tip, only quantifies the dominant near-tip terms in the solution (with the exception of the expression for one intersector interface). The perturbation method generates a series solution dependent upon a small parameter, ,u, which is of the order of (yield stress)/(elastic modulus). The combination of the two methods produces a solution that is a singular series containing integer powers of the quantity (p In v), where Y is the non-dimensional radius. While the first terms in the perturbation series solutions are in fact the same as the solutions derived in the previous small-displacementgradient investigation of DRUCAN and CHEN (1989), the extension to higher orders is not straightforward and requires the introduction of strained coordinates to eliminate difficulties encountered near a singular surface in the solution. Asymptotic modifications of the locations of the interfaces between near-tip sectors of different types and of the crack flank are also necessary. Nonetheless, the fact that the smalldisplacement-gradient solution is recovered from the finite strain solution, to leading order in ,LL,is one of the more important results of this investigation.

2. GOVERNING EQUATIONS AND PROBLEM DESCRIPTION 2.1. Governing

equations

Two equations governing continuum response momentum and mass. These are, respectively,

embody

P.a=O 9.v

= tr(D)

conservation

of linear

(2.1) = 0,

(2.2)

due to our assumptions of quasi-static and fully incompressible deformations and negligible body forces. Here, d is the (symmetric) Cauchy stress tensor, v is the material velocity vector, and D is the rate of deformation tensor. The quantity tr (D) represents the trace of the given tensor. The rate of deformation tensor D and the spin tensor W (to be used subsequently) are defined by D = +[(vv) + (liv)]

w = :[(vv) - (Qv)].

(2.3)

In the above, (vs?),, = c~L>~/~x~ and (a~),, = &>,/a~; when written in Cartesian index notation. The constitutive relation for the material is based upon the following kinematic description of the elastic-plastic response of a macroscopically homogeneous material [see NEEDLEMAN (1985) for a concise discussion]. Plastic flow is assumed to occur by crystalline slip only (twinning is thereby excluded), and the remaining response of the

692

C. R. REIII and W. J. DKUGAN

material is by rotation and elastic deformation mation gradient can be viewed as the composite

of the crystalline of two responses

lattice.

F = F**FP. The physically appropriate interpretation of F* is the contribution to the total deformation and distortion caused by the elastic response conditions. FP is therefore the residual (plastic) and rotating the lattice back to the reference gradient [L = (vv)] as well as D and W admit L=fi*F

The defor-

(LEE, 1969) :

(2.4)

this expression (WILLIS, 1969) is that gradient consisting of lattice rotation plus the rotation due to boundary deformation gradient after unloading orientation. From (2.4), the velocity additive decompositions :

’ =~*.F*~~+F*.~P.FP~~~.F*~

1

= I,* + LP = (D*+W*)+(DP+WP).

(2.5)

Here, the superposed dot denotes material time derivative and the superscript - 1 denotes tensor inverse. If the elastic response of the material is assumed to be hyperelastic with the reference state for this part of the deformation defined by the intermediate state described by FP, then one obtains y* s t-w*

*z+z*W*

= 5f’*:D*fD**~+t*D*.

(2.6)

Here, r = (p”/p)a is the Kirchhoff stress, Z* is a symmetric, fourth order stiffness tensor whose ground state moduli have been convected to the current configuration, and 8* is the Jaumann rate of the Kirchhoff stress with respect to axes that spin with the crystal lattice, not with the materiul. If (2.6) is specialized to a non-hardening material whose yield stress is much less than the elastic moduli, as is the case for most metals and alloys, then the last two terms on the right-hand side can be neglected. leaving D = $?*:x*+DP

(2.7)

whereY* = LE?*-‘. The material is assumed to be isotropic with elastic moduli that remain constant irrespective of the plastic deformation. The plastic response is treated phenomenologically and the plastic part of the spin tensor is taken to be identically zero as is commonly done in isotropic metal plasticity theories. The material is assumed to be perfectly plastic and to yield according to the Huber-Mises condition : ,I’=

:(s:s)-P

= 0.

(2.8)

Here, s is the deviatoric part of the Cauchy stress tensor and k is a constant having the value a,/J3 where nY is the yield stress in uniaxial tension. Plasticity occurs when ,f = 0 and df’ = 0; when ,f < 0 or df‘ < 0, there is no plastic response. Taking (2.8) to be the plastic potential (i.e. assuming plastic normality), the last term on the righthand side of (2.7) reduces to As where iL 3 0 is unspecified constitutively. The complete constitutive law thus becomes

Growing

D =

crack fields

693

& [8- 4 tr (ti)I] + is.

(2.9)

In this expression, E is Young’s modulus, I denotes the identity tensor, and elastic incompressibility has been enforced by taking Poisson’s ratio, v, to be one half. Note that the Jaumann stress rate of (2.6) is now with respect to the material spin rate, W, because it has been assumed that Wp = 0. Equation (2.9) is a rigorous, finite deformation generalization of the small-displacement-gradient Prandtl-Reuss equation that follows from the assumptions that the elastic part of the response is hyperelastic and unaffected by plastic deformation, that k/E CC1, and that the material is completely incompressible. Equation (2.9) along with the equilibrium equations (2.1), the incompressibility constraint (2.2), and the kinematic definitions (2.3), form the system of partial differential equations that governs the response of the material. As a matter of notational convenience, the equations will be non-dimensionalized using the substitutions (2.10) where R is an undetermined length scale, and b is the crack tip velocity with respect to a fixed reference frame. It is an easy matter to show that equations (2.1), (2.2) and (2.3) retain their original forms except that the quantities are now non-dimensional. Equation (2.9) is rewritten in terms of non-dimensional variables as D =t[&+tr(&)I]+Is,

(2.11)

where p = 3k/E. From this point on, all expressions form unless spec~~cully indicated othetwise.

are written

in non-dimensional

2.2. Plane struin simplijicution The general three-dimensional equations are simplified by assuming plane strain conditions. This implies that, for any Cartesian coordinate system with the 3-direction being out-of-plane, D13 = 0

and

W,, = 0

except possibly

W,2 # 0.

(2.12)

The indices i and j range from 1 to 3. While the above allow some immediate simplifications, the presence of the non-zero component of the spin tensor in the constitutive law (2.11) makes the implications of plane strain with respect to the outof-plane stresses less than obvious. Although the out-of-plane shear stresses have no effect upon the in-plane components of (2.1 l), they do directly affect the yield condition. Fortunately, the analysis is simplified because even though the spin may be non-zero, we prove in Appendix A that the out-of-plane stress components must be

o11+a22 fJ33

=

2

.

(2.13)

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c‘. R. REID and W. J. DKUGAZ

Using these results, the equations governing plane strain conditions can be made explicit. becomes

the response of the material under Dtlring elustic re.~ponse, the system

(2.14b) (2.14~) where the (r, 0) coordinates lie in-plane and the (?/at terms vanish due to the steadystate assumption. No convenient simplification of this quasi-linear system of partial differential equations was found. During plustic response, the plane strain results (2.13) simplify the yield condition to (2.15)

This allows the three parameters as

in-plane

stress

components

0,) = a-sin

(24).

0 00 - o+sin

(24),

to be written

in terms

of two

(2.16)

0 /o ~ - cos (241).

where (T is the hydrostatic stress and C/Iis the counterclockwise angle from the rdirection to the principal shear direction (see, for example, HILL, 1950). Thus, the stress is statically determinate in regions at yield even during finite deformation and is determined from (2.16) and equilibrium : t ;

-2r

cos(24)

- 2r sin (24)

c’r

(‘4

c’cp 1 A$ 1

;$!I ^ ~ 2 sin (24) Fo+l (I [

io + (‘. + 2 cos (24)

L

=o,

(‘. + 1 = 0.

(2.17)

crack fields

Growing

The velocities are determined by introducing (2.11) and eliminating i.. The result is

695

(2.16) into the constitutive

$(D,, - Don) cos (24) + Dro sin (24) = - p [qg

+ T (‘;;

+l)+

PVC,].

relation

(2.18)

which is a form originally due to HILL (1985). This equation plus the incompressibility constraint (2.2) and the kinematic relations (2.3) yields a pair of linear partial differential equations for the two unknown velocity components. Two other results can be obtained from the governing equations in regions responding plastically (HILL, 1985). An explicit expression for the rate of working per unit volume in the current configuration is given by o:D

= 2).

(= ~/UC’),

(2.19)

where the term in parentheses is the result in dimensional form. This indicates that in plastic regions, there is no contribution to the rate of work from elastic response. Requiring that the plastic work rate be non-negative everywhere yields the condition I. >, 0 which provides a check on the physical validity of the solutions obtained. (This check is applied in Section 4.1 to the solutions obtained in this paper.) The specific form (2.19) is also useful for evaluating 1. when a perturbation approach is used. The second result is that, contrary to small-displacement-gradient formulations, the stress and velocity characteristics are not coincident. If 0, is the angle between the .u-axis and one of the stress characteristics and 8,, is either of the angles to the velocity characteristics. then i sin 0, - 0, =

71 2

‘P

I.-I -2sin

(2.20) p

This theoretical discrepancy between the characteristics is artificially removed by the perturbation analysis. Though it can be retained by using the strained coordinate technique of LIN (1954), it is not incorporated into the solution presented here as noted in Section 3.2. 2.3. Coordinate system und houndmy~ conditiom The analysis of a growing crack is carried out with respect to a coordinate system that moves with the crack tip as depicted in Fig. 2. The material velocities, v, are measured relative to this moving coordinate system. The boundary conditions for the problem are b-n

= 0

v-n=0

on r,

(2.2 I)

on I-?.

(2.22)

and crrll --0 I’,, = 0

C

696

R. REII) and W. J. DKIJGA~

g

HI+---a-----j FIG. 2. Crack

tip coordinate

system and boundary

detinitions.

In these equations, n denotes the unit normal to the crack flank surface. The first set of equations is applied on the crack flank (r,) and defines a traction-free surface whose position is invariant in time with respect to the moving coordinate system. It is shown in Appendix B that a displacement-based definition of position-invariance of the crack flank in the moving systems-which is required by our steady-state assumption--is identical to the velocity boundary condition in (2.21). The conditions on the line ahead of the crack tip (I-?) are symmetry conditions imposed by the Mode I loading. Two general comments concerning the boundary conditions given in (2.21) arc appropriate. First, the solutions to follow contain an asymptotically indeterminate parameter that characterizes the effect of the far-field conditions on the near-tip behavior. Because this parameter may, in general. vary in time due to finite geometric effects, the velocity boundary condition may not be strictly true and should contain a time dependent term on the right-hand side. However, it is assumed that such a term is U( I ; r)t and does not affect the dominant, singular terms in the solution. The second comment is related to the fact that the crack flank represents a free surface. If the crack flank location (I-,) were given, the boundary conditions there would represent an overconstraint for a well-posed problem. This overconstraint is relaxed by allowing the crack flank location to be an unknown in the problem. The unknown boundary is determined by expanding the expression for the surface normal in a perturbation series in ,u.

3.

SOLUTION

In this section, an approximate analytical solution satisfying the proposed boundary value problem is presented. Initially, general solutions for the different near-tip angular sector types that comprise the solution are developed as time-independent regular

t The USC of ordering symbols follows that of VAX DYK~ (1975). Howcvcr, as the present discussion may refer to ordering properties with respect to I( or to gauge functions with respect to Y, the ambiguity in notation when referring to O(i) terms in either cast is eliminated by including the relevant quantity after a semicolon. Thus, O( 1; r) is read “order I in r”.

Growing

perturbation series in the small parameter For example, v(r, 8; p) = v’“‘(r,

691

crack fields

p that are valid asymptotically

0) +pv(“(r,

Q) +p2vC2)(r,

0)+

. . ..

as r -+ 0.

(3.1)

The solution so obtained consists of an assembly of near-tip angular sectors of different types that are, at least to leading order in the perturbation parameter, exactly those derived in DRUGAN and CHEN’S (1989) analysis via the small-displacement-gradient formulation. Unfortunately, a continuous solution at the next higher order in the perturbation parameter cunnot be generated using the straightforward expansions. This is because of a singular surface in the solution for the leading constant stress sector located at 0 = 7r/4 (which is the leading order location between this sector and the centered fan plastic sector). In order to circumvent this difficulty, the solution for the leading constant stress sector is modified by rewriting it in terms of strained coordinates (LIGHTHILL, 1949). Modifications of the asymptotic locations of the interfaces between sectors and of the crack flank are also required to define a wellposed problem and to satisfy the velocity boundary condition (2.21). Using this technique, two terms in the expansions for the stress components and three terms in the expansions for the velocities are quantified.

3.1. Gene&

near-tip

ungulur

sector

solutions

3.1. I. Elastic sector. In a sector with DP = 0, the lowest-order (in p) velocity solution, v((‘),is immediately obtained from the constitutive relation (2.11) with ,IA= 0. In this case, the rate of deformation tensor must be zero, meaning that rigid body motion is the only solution. This result provides the basis for the following argument that is crucial to the initiation of the series solutions for stress, not only in the elastic sector but also in plastic sectors. First note that while the constitutive relation for elastic sectors yields directly the above result for the O(1 ; ,u) velocities, the constitutive relation in plastic sectors is not quite as definitive. However, since v “I is at most rigid body motion in elastic sectors, it is assumed that for functional consistency, v(“I = v’“‘(0) in plastic regions for r -+ 0. Except for the rigid body motion part of this, the 0( 1 ;p) constitutive equations for plastic sectors show that this v (‘I has the same form as the solutions for a stationary crack, which RICE (1968) and RICE and JOHNSON (1970) have shown to be O(p) for contained yielding. Thus, v (“I is at most a rigid body motion at all angles about the crack tip, and the velocities in the entire near-tip region must have, to lowest order, the form P!” ‘I = - cos 0, “‘) = sin 0, Z’o

(3.2)

which describes the (normalized) rigid translation past the moving crack tip (rigid rotations are ruled out by the boundary conditions). Using (3.2) and an expansion for the normal to the crack flank requires, to leading order, that the crack tip be mathematically sharp with straight flanks at 8 = frr. It is noted that while the above

698

C. R. REIII and W. J. DKI (IAN

argument is rigorous for contained yielding, the solutions to follow remain valid for general yielding if the leading order (in ~1) velocity field is in fact described by (3.2). It is now possible to argue what the functional form for the lowest order (in 11) stress field must be, though it cannot yet be made explicit. If the stress field to this order were to contain a radial singularity, the presence of a mathematically sharp tip at this same order would lead to infinite stresses at the crack tip. Except for the hydrostatic component, the non-hardening material being analysed cannot tolerate infinite stress and a pronounced geometric alteration of this situation would be necessary. The possibility of a macroscopically blunt tip is disallowed to 0( 1 ; p) by the derived form for the leading order velocity field and is furthermore at variance with both experimental (GREEN and KNOTT, 1975) and numerical (Lru and DRUMPU’, 1992) results that indicate a macroscopically sharp crack tip. It is therefore argued that the deviatoric stress must be a bounded function of radius to leading order. In this case, the analysis presented by DRIKAN (1985) applies to the 0( I ; p) components of stress so that o”” must be finite and a function of 0 only for I’ + 0. Using the above result, substitution of (3.1) into (2.14) and retention of terms of 0( I : p) in (2.14a) and 0(/l) in (2.14b, c) results in a set of equations governing cr”” and v(” that are identical to the small-displacement-gradient elastic sector governing equations. Thus, the solutions for these terms arc found by the same procedure used in the small-displacement-gradient analysis of RICE (1982). This procedure involves investigation of the consequences imposed by the non-trivial component of the compatibility equation, (9 x D x v),, = 0. The results are the same as those from previous analyses (e.g. SLEPYAN, 1974; RI(‘E rt (I/.. 1980) and are presented in Section 3.3. Before proceeding to the next order solutions for stress and velocity, note that the 0(/l) contribution to stress ccrtvwt be a bounded function of radius. This is not at variance with the results for the previous order, as those results are based upon application of boundary conditions to consistent order. The result follows if it is assumed that cr(‘) = ‘a(H) + ‘~(@rr(r) +

.,

with

H!(Y)--f 0,

I’ y;F + 0

as

,_+o,

(3.3)

where the differential condition makes the physically plausible assumption that IV(Y) is well-behaved near the origin. Given (3.3) and the lower order solutions, equation (2.1 I) shows that the most singular part of D”’ arises from the (v(I), cr(“) combination. and from this, the expressions for the polar components of v’?’ can be specified in terms of the constants D, and O2 contained in the O&) velocity solution [see (3.49)]. Consideration of the constitutive equation for the shear component of the rate of deformation tensor, however. requires that either n, or D2 must be zero if (3.3) is to be satisfied. This is not the case (see Table I at the end of Section 3.3). Since the higher order stress terms must contain a singular part. a consistent perturbation analysis requires, for I’ + 0, the form CT ’ ” = a(O) In r-t b(O).

(3.4)

where a(O) and b(H) are tensor valued functions of 0. Using this, an analysis analogous to that used for the previous order solution is carried out; that is, the general

Growing

form for DC” is derived by investigating component of the compatibility equation. assumed form (3.4) for the stress yield

crack fields

699

the restrictions imposed by the non-trivial These results plus incompressibility and the

L$” =G(lnr)‘+y;lnr+k’(O),

z$” = -1;2’(lnr)‘-(y,+~2)lnr-[k(0)+;ll]+rl(r),

(3.5)

where r’, = 2(05 cos B+D6 sin 0) =(-D,+Dacot8)(0,cos8+D,sin8)+ >I>= ,j {cos Q (CI,,--all(i)+4(0,cote+D,)[(D,--D,)sine+(D,+D,)cos8] + 4( -D,

sin 6’+ D2 cos d)o$’ + sin 0 (hj, - hi,,, - 4h,,)}

(3.6)

and the primes denote differentiation with respect to 0. The factors D, are indeterminate constants of integration appearing in the velocity solution. The functions k(H) and d(r) are undetermined and, as in the first order analysis, it is assumed that d(r) does not contribute spurious, inconsistent singular terms. Furthermore, an analysis of the shear component of the rate of deformation tensor shows that u’(r) cannot contribute to the logarithmic terms. Finally, equilibrium (2.1) and the required form (3.4) for o(‘) yields governing equations for the components of a(H) : 4, - GJ +

40 = 0,

a;,, + 2a,,1 = 0.

(3.7)

The first of equations (3.6) and the above provide a linear system of governing equations for the components of a(Q). The results are presented in Section 3.3. Specification of b(8) is straightforward (REID, 1991) but omitted here. 3.1.2. Plastic sectors. In plastic regions under plane strain conditions, the inherent difficulty in dealing with the quasi-linear stress equations (2.17) is removed because functional consistency (in r) with the results of the elastic sector is imposed to specify the functional form for the higher order terms for 4 and g. A regular perturbation analysis is carried out in this section to establish methods and general results. The required re-analysis of the leading constant stress plastic sector in terms of strained coordinates is deferred until Section 3.2 where assembly of the full, near-tip solution is performed. As argued previously, the O( 1 ; p) velocities in plastic sectors are rigid body translations given by (3.2) and the O(1 ; ,u) stress components are functions of 8 only for r -+ 0. Using the latter fact in (2.17) and extracting the O( 1 ; p) terms yields the following pair of equations :

700

C.

R. REIU and W. .I. DKUGAN

(3.8) The first of these equations

delineates

or

two possibilities

nn

d+“‘=y,

n=O,fl.....

:

(3.9)

The first equation describes a region that is equivalent to, at least to lowest order in ,u, a constant stress (Cartesian components) plastic sector, and the second describes a region that is equivalent to a centered fan plastic sector. Not surprisingly, these results are the same as those of previous small-displacement-gradient analyses in which the leading order radial variation in the stresses is removed by the bounded stress proof of DRUGAN (1985). Because the modified Prandtl field is the lowest order (in p) solution configuration. there exists one centered fan sector and two constant stress sectors. The constant stress sectors are distinguished by calling one “leading” (lying ahead of the crack tip) and the other “trailing” (lying adjacent to the crack flank). They are analysed separately in the following. Lrading constunt stress sector’. The solutions for the leading order terms in the expansions for the shear angle and hydrostatic stress in a constant stress sector, from (3.8) and (3.9), are (b“‘1 = -(jfC, 0 10) _-E*.

(3.10)

where C and E* are constants and recall that 4 is measured from the r-coordinate direction. The stress characteristics for this lowest order solution form a Cartesian grid, so it is convenient to do calculations using Cartesian forms of the governing equations. Redefining d, to be the angle measured from the x-coordinate direction, @(“I becomes a constant. This constant is evaluated by noting that in the leading constant stress region. the shear-free boundary condition on the line ahead of the crack implies that 4(O) IS either n/4 or 37c/4. Using the former so that (T,, > CJ,,there,

g.’ (1) =

E*

(3.1 I)

With the lowest order terms for both the stress parameters and the velocity components specified, the analysis of higher order terms proceeds directly. The solution process is facilitated by using characteristic coordinates r~ and p defined by

Growing

crack fields

701

a = x-y, p = x+y.

(3.12)

The solutions for the higher order terms of both the stress parameters and the velocity components are given in terms of arbitrary functions of these characteristic coordinates. The indeterminacy is removed by requiring functional forms (in r) consistent with those of the elastic sector; i.e. they must contain logarithmic singularities. General results are presented in Appendix C. The final solution for this sector requires strained coordinates as indicated in Section 3.2 and is presented in Section 3.3. Trailing constant stress sector : The analysis in the trailing constant stress sector is very similar to the analysis just presented. In this case, the crack flank is at 0 = 71 to leading order, so the shear traction free condition on this boundary requires that 4 be at 3x/4 with respect to the positive .x-axis to leading order. (The possibility that it is 7c/4 is ruled out by the complete solution assembly process.) The normal traction on this boundary must also be zero to leading order. These conditions define the value of the leading order hydrostatic stress solution and define the appropriate characteristic coordinates to be ji = ~ (.YfJ’), /? = -(.Y-_r),

(3.13)

where the choice of sign eliminates the need for absolute values in the arguments of the logarithms. Furthermore, because the crack flank is a free boundary (at higher order), the constants in the solution remain undetermined initially. Incorporating these features results in the solution shown in Section 3.3. Centered f;ln In the centered fan sector, no transformation of coordinates is necessary as the characteristics generated by the perturbation solution are coincident with the polar coordinate system. The leading order solution for the stress is taken from the second of (3.9) with n = 0 to ensure stress continuity with the leading constant stress sector. Hence,

c+“) = -2(8+F*).

(3.14)

Now, as in the constant stress sectors, the general forms for the higher order solutions in r are difficult to ascertain from the governing equations themselves, but for consistency with the elastic sector results, let

4(‘) =

s, (0) In r+sz(8),

~9’) = t, (0) In rf t,(Q). Using these in (2.17) and matching #I’

terms with similar radial dependencies

(3.15) results in

= F,,

cc’) = 4F, In r+F,.

(3.16)

702

C. R. REII) and

W. J. DKUGAN

The expressions for the stresses follow immediately and are shown in Section Finally, a similar procedure applied to the O(p’) terms provides @(” = F,(ln a(?) = 2[F,(ln

r)‘+Fq r):+Fq

3.3.

In r+s,,(H), In r+t,,(H)],

2F;fI = f,,(H) +s,(fl).

(3.17)

The last equation is a relation between two undetermined functions of H that does not specifically exclude the possibility of singular behavior in the H-coordinate. However. as is evident from (2.16), if s,,(O) did contain a singularity, a rapid oscillation in stress would result as the angle of the singularity were approached. Such a situation is physically unlikely and this possibility is therefore ruled out. This argument is presented to show the luck of H-singularity in the solution for this sector. With the stress solution determined, the velocities are determined from (2.2), (2.3) and (2.18). The leading order velocities reflect the translation of the crack tip and the higher-order velocity terms follow directly. The results are as indicated in Section 3.3.

3.2.1. Initial ussewzhly attempt employing rqulur perturbation vrsults. With the general solutions for the sectors of different types established, assembly of a completely continuous near-tip solution using a straightforward, order-by-order approach may be attempted. Unfortunately, insurmountable difficulties occur and a modified solution must be developed. The difficulties are outlined here and the modification of the solution is carried out in the next subsection. The leading order (in p) velocity solution consists of rigid body motion past a propagating crack tip and the solution to this order is therefore continuous. This implies that the crack flank is at 8 = 71to this order. The process of enforcing full stress continuity (or, equivalently, continuity of the principal shear angle (b and the hydrostatic stress (T) between the different sectors defines the intersector 1ocations.t In particular, the leading order location of the interface between the leading constant stress and centered fan plastic sectors is H = 7114from the results of Appendix C and (3.14). However, problems occur when one attempts to match the O(p) velocities at the intersector boundaries. In fact, as seen from writing the polar components of v”’ using (C. 1b), the radial velocity becomes infinite in the leading constant stress sector as f1 + n/4 and some modification of either the solution, the interface, or both is required to maintain continuity between the velocities in this sector and the velocities in the centered fan sector. One possibility is to account for a higher order (in r) correction to the asymptotic location of the interface as in the small-displacemcntgradient solution of DRUGAN and CHEN (1989). In particular. let the interface be defined by

t Full stress continuity across such a moving surface was proved to be required model at finite strain by DKUCAN and SHEK (1990).

for the present constitutive

Growing

0” = ;

crack fields +q(r),

703

(3.18)

where 8,, is the asymptotic location of the interface and q(r) -+ 0 as Y+ 0. This requires that q(r) = -r”’ where m is an asymptotically indeterminate parameter that defines the “m-family” of solutions of DRUGAN and CHEN (1989). Alternatively, the theoretical discrepancy between the velocity and stress characteristics (which is lost by the straightforward perturbation approach) can be reintroduced using the characteristic straining idea of LIN (1954). The result is that the characteristic upon which the velocity becomes singular is more accurately placed at an angle slightly greater than 0 = 7c/4 and the velocity remains bounded in the leading constant stress sector (REID. 1991). Unfortunately, neither of these modifications allows for a straightforward extension to the next order (in p) solutions. While the velocity solution in each case can be made continuous and non-singular on the interface between these two sectors, the proximity of the interface to the singularity results in either an inconsistency in functional forms on the interface if the m-family concept is retained, or an inconsistency with respect to the perturbation series ordering properties if the characteristic modification approach just mentioned is taken. Specifically, consider Y(” in the leading constant stress sector [equations (C. lc)] and the consequence of the m-family modification. On the interface defined by (3.18), c( + $! * Y”‘+ ‘, /i’ + $ * r and the velocities in this sector contain terms that are U(ln r/Y) when evaluated on this interface. Such terms are more singular than the v (2) terms contained in the solution for the centered fan plastic sector [equations (3.46)] and therefore modification of the interface using the m-family concept by itself is insufficient to produce a continuous solution at this level. If, on the other hand, one accounts for the fact that stress and velocity characteristics are skewed, an aberration in the ordering properties of the perturbation series occurs. In this case, there exist terms in the O(P) solution that become 0( 1/,u) on the interface and these terms therefore contribute to the O@+ ‘) solution (REID, 1991). Hence, this approach is likewise not viable. The difficulty in either case is that, though the radial velocity solution is no longer singular on the modified interface, the interface is asymptotically too near the singular surface to allow for a consistent, order-byorder matching of solutions. Furthermore, the difficulty with the leading constant stress-centered fan interface is compounded because now the stresses become singular as 8 + 7-t/4, as is apparent by writing the stress components using (2.16) and the expressions given in Appendix C. This difficulty with the stresses could be circumvented at O(p) by using an m-family type modification of the interface. However, as indicated in (C~C), exacerbation of the singularity occurs in the 0(p2) solution for stresses, and the same difficulty indicated above for the radial velocity also occurs in the stress solution. It is therefore necessary to modify both the stress and velocity solutions in the leading constant stress sector to eliminate the complications associated with the singular surface. 3.2.2. Mod$cation of leading constant stress sector solution. The difficulties outlined are resolved in this analysis by straining the a-coordinate in the leading constant stress sector. This produces a uniformly convergent solution near the singular surface

704

C.

R. Rem and W. J.DKIJGAN

represented by x = 0. The idea of coordinate straining was introduced by LIGHTHILL (1949).The process involves expanding c( in terms of a strained coordinate Z as indicated : x = :+

i

p”“.f;,(f,p).

(3.19)

,>=I This transformation modifies the sequence of governing partial differential equations generated by the perturbation series. These are then solved with the intent of removing the troublesome singularity by imposing restrictions on the transformed equations and the functions ,f;(?, fl). The process outlined above can be simplified by using PRITULO'S(1962)renormalization method, which eliminates the need for re-solving the differential equations in terms of the strained coordinate. In particular. the expression for a generic field quantity, $, is written in two equivalent forms, the first being a regular perturbation solution expansion. and the second being an expansion in terms of the strained coordinate :

,u’?p(‘%,p) = t, pIJ”“‘(2,p).

I) = i

,/

0

I,

=

(3.20)

0

The functions $““(z, /II) and Y”“(2, p) are in general different functions of their arguments. A direct comparison of the corresponding terms in the perturbation series is made by expanding $‘“‘(x, /j) in a Taylor series about the point (_‘,8) and matching like terms. The results for the first three terms are

where, as indicated, the right-hand sides are evaluated at x = 1. In this process. the troublesome singularities are removed by a suitable choice of,fj(P, [j). Because all asymptotic interfaces between different sector types are defined by enforcing stress continuity, coordinate straining is first applied to the singularity in the stress solution of the leading constant stress sector. In particular, the terms containing l/a as a factor are removed by the procedure outlined above, and the expression for the first term in the strained coordinate series is obtained by using the results of Appendix C and (3.21) : a = f-p/?[E, Thus, using the capitalized coordinate,

In 2-E?(ln

functions

[j-

I)+E,]+O(/~‘).

to denote representations

(3.22) in terms of the strained

crack fields

Growing

@“‘=E,lni-E,Infi+E

705

39

X(0’ = E* C”’ = -2[E,

In i+E,ln

jJ+E4].

(3.23)

The stresses are shown in (3.41) where the stress boundary condition on the line ahead of the crack tip has been used so E, = Ez and E3 = 0. Furthermore, as a consequence of the stress boundary condition, it is possible to write a general expression for the strained coordinate _’in terms of polar coordinates : f = :(r, Q) = C*(@r.

(3.24)

This expression is used subsequently to verify the existence of the straining transformation at 0 = 7~14. The next step in the modification of the leading constant stress sector is the reanalysis of the velocity field. A little care must be exercised because the stress solution, which has a direct influence on the velocity solution, is now in terms of a strained coordinate. To O(U), the analysis is unchanged because the stress solution does not contribute to the governing equations. However, the evaluation of v(I) requires expressions for 4”) and @(“j& Fortunately, from (3.21), it follows that

= a+“(_“,p),

(3.25)

so the functional form of this term is unaltered by the coordinate straining. (Higher order velocity terms would have to include the appropriate transformation during their evaluation.) Hence, the velocity solution given by (C. 1) in terms of the unstrained coordinate remains valid. Straining of the r-coordinate in the velocity solution proceeds in a manner analogous to that of the straining utilized in the stress solution. The strained coordinate for this case is denoted by Z, and the expression for a in terms of this quantity, using (C. lc), is given by (3.26) In terms of 2, the velocities are as shown in (3.40). Incorporated in these expressions is the velocity boundary condition on the line ahead of the crack tip. Like the expression for t m (3.24), i can be written as Z(r, 0) = D*(e)r

(3.27)

which will be used to verify the existence of the straining transformation at 0 = 7r/4. Evaluation of the plastic flow parameter in this sector requires one to account for the strained coordinate when taking derivatives in order to calculate the components

706

C. R. REID and

W. J. DKUGAU

of the rate of deformation tensor. By definition. the components of D are determined by taking derivatives such as (:z$“i?xh where l’,“) is the unstrained expression for the velocity and sL is an unstrained coordinate. Evaluation in terms of the strained coordinate can be accomplished by writing the unstrained velocity in terms of the strained coordinate using (3.21), and by using formal expansions of the differential operator and the implicit function theorem. The result is shown in (3.42) 32.3. Fimd ussrtnhl~~. Assembly of a completely continuous near-tip solution requires the determination of the location of the interfaces between sectors and the evaluation of the arbitrary constants in the general solutions for each sector. The primary issue is the analysis at the interface between the leading constant stress and centered fan plastic sectors. Specifically, a representation for the interface must be found, and because of the proximity of this interface to singular surfaces in the stress and velocity solutions, the existence of the straining transformation also must be verified. The first task is accomplished within a strictly asymptotic (in r) analysis by assuming a form for the interface based solely upon functional consistency in the solutions. This procedure introduces an asymptotically indeterminate parameter, p, into the solution. The second task is accomplished by exhibiting an explicit representation for the transformation on the interface. The analysis of the remainder of the near-tip region proceeds in a rather straightforward manner with some minor augmentation to account for finite geometry change effects. Beginning with the interface between the leading constant stress sector and the centered fan sector, the leading order location of this interface is at I) = 71:‘4 as previously noted. On this surface. 31+ r*O and /I + r* Xi_ so using (3.24), (3.22) becomes

C*(71/4) -/iE, ,i2

0 =

(3.28)

i As the following results show. E, < 0 and the above does in fact have a solution for C* (n/4) [it is O(p) based on sample numerical calculations]. Therefore, neither the strained coordinate ? nor the representations for the stresses in terms of the strained coordinate are singular at 7-c/4. Furthermore, because the transformation is a continuous mapping and non-singular at 7114,the inversion theorem guarantees that the transformation exists in some neighborhood pasf n/4. This result is important because it is subsequently shown that the interface is, for very small radii, at an angle greater than n/4. A similar result holds for D* (n/4). Next, a suitable representation for the interface in terms of the strained coordinate ? is required. Here, a sensible perturbation series &nzun~l.s that the representation of the solution result in asymptotically functionally consistent forms. Since the singularity depends upon the logarithm of the argument, consistency requires the interface to have the form z = CXr”.

(3.29)

In this expression, Ch is a positive constant not necessarily related to C* (7-c/4) shown in (3.28) .“f The following shows that this results in an interface of the form t The constant C: is actually superfluous R. This point is addressed in Section 4.2.

as it can bc incorporated

inlo the undetermined

length KBIC

Growing

707

crack fields

8 (mtcr)= 0; +h(r),

(3.30)

where 0; is a constant and h(r) is “small” in that it consists of terms that are O&u> and o(l;u). With the interface defined, stress continuity is enforced by matching the shear angle and the hydrostatic stress. Redefining Q, to be the shear angle with respect to the Ycoordinate direction and matching the expression for this quantity with the expression in the centered fan on the interface at small radial distances from the origin implies Ox= n/4and

h(r) = p

In r+E,

E, (p-l)

h-r3

-F,

Jz = ,u[N~ In r+Ni]

+O(P’)

+O(,~*)+o(l

+o(l ; y).

The term ~(1 ; r) that could be attached to virtually included here for emphasis. Matching the hydrostatic

;r)

I (3.31)

all equations in this analysis component of stress yields

E, = -F,.

is

(3.32)

With these results, an explicit representation for the interface is deduced using r * h (r) on the interface, the equation for (3.29) and (3.30). Noting that x --+ -42. the strained coordinate (3.22) provides

h(r) = -CXrP1

+,uE,

(p-l)lnr+ln

cX.1 J2

J2

1

+o(,u’).

(3.33)

This expression verifies the existence of the transformation on this interface by explicit representation and delineates the u( 1 ; r) term contained in (3.31). The analysis of the velocity solution near and on the interface between the leading constant stress and centered fan plastic sectors proceeds like that of the stress analysis. It can be shown that the coordinate straining for this case is also valid on the surface 0 = 7c/4 and hence the remaining detail is to determine an expression for the interface in terms of the strained coordinate Z. The interface has already been defined by (3.29) and the value of Z on this interface must be deduced from the straining transformation (3.26). This represents a non-linear problem with no apparent analytic representation for 5. However, Z on the interface can be approximated by Z = DXrq.

(3.34)

Comparison of the resulting expressions for h(r) from (3.33) and the above shows that DX = Cg and p = q for consistency but there results a discrepancy of (,~/2)[1 +(E,/A,)] in the expression for the interface. As this discrepancy does not affect the singular part of h(r), it is ignored and the approximation given in (3.34) is used. With the representation for the interface defined, evaluation of the velocity constants yields

C. R. REIU and

A, = -

B,=-

W. J. DKUGAN

1

37’ ‘-tP_. /3

fk A,

=

P-1 +2PF,m-- ‘1 xp3

B

’

= 4~[8A,~2-2F,(p-2)]+n(l

+/.‘)(I

-2F,)

(3.35)

where the value of the stress constant F, is determined subsequently. These constants (as well as all other constants that can be explicitly evaluated) are compiled in Table I at the end of Section 3.3 for several values of the parameter 17. With the interface location and the matching conditions established between the leading constant stress and centered fan sectors, one can proceed with the analysis of the remainder of the near tip region. The analysis is similar to that used in previous small-displacement-gradient formulations of the problem except that modification of both the crack flank and elastic sector boundaries is necessary to satisfy the boundary conditions and to provide a well-posed problem. To this end, the elastic sector interfaces are expressed as 0, = e~+p/7,(r)+u(~L2), u2 = u~+,Lk2(r)+u(p2).

(3.36)

where 0, denotes the elasticcentered fan interface, H2 denotes the elastic-trailing constant stress interface, and the starred quantities are constants. These forms are suggested by the results of the leading constant stress-centered fan interface; the functions I?,(v) are, for the moment, left arbitrary. Terms that are o(l ; r) could be included in these expressions but because the solutions contain no singularities on these inter-sector boundaries, such higher order terms in Ydo not contribute to the analysis nor are they made explicit. Matching the components of the O(p) velocities and the O(l ; p) stresses at the elasticcplastic interfaces yields solutions for 0,” and the constants in the O(p) velocity solution. The boundary condition at the leading order location of the crack flank establishes the constants pertaining to the hydrostatic stress at leading order. Consideration of the next order (in ,r~)solutions first requires a look at the boundary conditions on the crack flank. It has already been established that to leading order. the crack flank is at 0 = 71. If no higher-order correction to the crack flank were present, then v( ‘) - n = p[(C, + Cl) In r-C,] and the boundary condition (2.21) is no[ satisfied to O(p). To alleviate this, let Ol,;lnk= z-,dq(r)+

0(p2),

h,-(r)

> 0.

(3.37)

where h,(r) is to be determined and the condition on hr(r) eliminates the possibility of crack flank interpenetration. Writing the boundary condition (2.21) in terms of the unit normal derived from the above expression and evaluating the velocities at

Growing 0 = TC- ~/Q(T)

generates

a differential h,(r) =

equation

C” ; -(C,

700

crack fields

for h,-(r) with solution

- C,)(ln

Y- I) + Cd.

(3.38)

The constant Co can be set to zero using the principle of minimum singularity; that is, the solution should not increase the singularities already present (see VAN DYKE, 197.5). From Table 1, (C, - CZ) = -LIZ > 0 so hr(~) > 0 for r < e, where e is the natural logarithm base, as required. The fact that the flank is altered is the reason for including higher order terms for stress and velocity in the solution. With the alteration of the crack flank specified, the constants in the O(p) stress solution for the trailing constant stress sector become determinate. Also, the remaining constants associated with the most singular terms (in r) in the o(‘) and v(‘) expressions can be found. This is accomplished by first matching stress and velocity components at the elastic-plastic interfaces. As in the first order (in ,u) solution, it is stress continuity that defines the interface but the velocity coupling in the elastic sector solution again requires evaluation of the velocity constants. If no corrections to 8, and O2 were included in (3.36), matching the three stress components at these two interfaces would generate six equations for four unknowns (M4, MS, Me and F,). However, two additional parameters can be introduced by assuming that, based on consistency with the crack flank correction and the leading constant stress-centered fan interface results, h,(r) = NI In v+Nf, /z~(Y) = N: In r+ NS.

(3.39)

The additional unknown constants that multiply the logarithmic terms in the interface expressions are sufficient to define a well-posed problem. The system generated to determine the remaining constants consists of two equations for matching velocity components at 0, and six equations for matching three components of stress at 0, and 8,. The eight unknowns are Dg, D,, M4, M5, Me, NI, N: and F,. The equations are lengthy but linear with respect to these coefficients. Once these values are determined, the constants in the most singular terms in the velocity solution in the trailing constant stress region (Cj and C,) are evaluated. 3.3. Compilation

c~f solutions

In this section, the results for each sector of the final solution are compiled. The constants appearing in these expressions are presented in Table 1 for various values of the asymptotically indeterminate parameter p. Leading constunt stress sector. V,(-\-,y:PO = [-l+o(l;Y)l+p[A,

ln(~~)+o(ln~)]+~2(A5[(ln~)2+(ln~)2]

+~~~~(-~:[(ln~2++(In~)2]+2lniln~}+o[(lnr)2]}+0(~3), v,(x,y;P)

= o(l ; r)+po(ln

r)+~2(As[(ln

g’-(ln

+~,~,{t[(~n~))‘-(ln~)2]}+o[(lnr)2])+O(~3),

/I)‘] (3.40)

where

and Z and i are given implicitly

by the following

:

relations

(3.44)

G?n teml,

c,(r,V;/l)

jhnx~ctor : = -[cos8+0(1

;r)]-~[sinOlnr+o(ln

+,k”{{[2 cos CI+B,(l z~,~(~,(f;~) = [sin B+o(l -p’{i[2

; Y)]-,u[(cosO+B,>ln sin d+B,(l-2F,)O+B,](ln

cJ.I.y(r,B;Pj = [--2(H+F*j+o(l ~M(J‘,@;P) = [--2(8+F*)+~~(I; ci~ii(i..H;~O = [I +o(l

-2F,)](ln

; r)]+po(ln

; r>]+4p[F, ujJ+4/I[F,

F)] r)“-i-o[(ln r+o(ln

~)‘]t-tO(~~), r)]

r)‘-i-o[(ln

r)‘]j

In r-t-o(ln

rj]+O($),

In r+o(ln

r)]+U(~t’).

+O(p’),

(3.46)

(3.47)

r)+O(pl’),

+.

(In f)’ + cypil”). r )1

i--

(3.48)

Growing

711

crack fields

Elastic sector. rF(r,fl;p) = [-cos0+o(l;r)]+p[(D,cos8+D2sin~)lnr+o(lnr)] +~~{(DjC0S0+D~sinH)(lnr)*+0[(lnr)’]}+0(/LL3), q,(r,@;p)

= [sinB+o(l;r)]+p[(-D, +,/A’{((-Djsin

Grr

-Ml

(r, H ; p) =

cos

sinO+Dzcos8)Inr+o(lnr)] (3.49)

U+D6cosH)(lnr)‘+o[(lnr)‘])+O(p.“),

; r)

+o(l 1 Ml cos28-M?

co? 0 In(sin

sin 28+Mi+2DzB+4D,

2d+Mz

+p[an.(8) In r+o(ln

0)

r)]+O(p’),

I sin’f3In(sin

o,~,,(r,();p)

=

sin2H+M,+2D20+4D,

a,,,(r,O;p)

- 2”sin 20 +I)( 1 ; v) +p[[a,,,,(N) In r+o(ln r)] + O(p’), 1 I = {M, sin2H+M2cos2H-D2-2D,[sin20ln(sin@-0~0~281 +o(l ; ,-)I +p[al,,(8) In r+o(ln r)] +O(p’),

H)

(3.50)

where u,-,_= -(M~+x~)cos20+(Ms+~2)sin2U+M,+~,, +O = (J-J,+xJcos

20-(Ms+xX2)

a,,, = (M,+xJsin

28+(Mj+~~)cos28,

XI = -2{(D,Dl+Dh) +D,Dl(cot

sin 28+M6+x3,

sin 2O+(Df-Df+2Ds)[j

cos 2Q+ln

(sin 0)]

W+20)$,

Truiling wnstant stress sector. z,,(.u,_~>;p)= [-I

+o(l

; r)]+p[C,

In 2+C2

In /7+o(ln

r)]+p*[Cj(ln

6)’

+C,(ln~))‘+~[C,G,(lncz”)‘-C2G2(ln~)~]+o[(lnr)~]}+0(~1), ~~,(-u.~;~)=o(l;r)+~[-CC,ln3i+C~Inp”+o(lnr)]+~~{-Cj(ln~)’ +C,(ln r~(x,,v;p)

fl)‘)‘+f[C,G,(In = [2+0(1;

r)2]}+O(p’),

@‘+C2G2(ln&*]+o[(ln r)]+2p[G,In

a,.,(x,~>;p) = o(1 ; r)+2p[G2 In B-G, a,,.(s,~%;p) = n(1 ; r)+2p[G,

/f-G,

In i+o(ln

Ind+o(ln

In o?+G? In p+o(ln

(3.52)

r)]+O(,u’),

r)]+O(,u’), r)J+O(p’),

(3.53)

717

C. R. REID and W. J. DKI GAN

where

2 = - (.y+.v),

(3.55)

p = - (.Y-.I.). Intecf:fuc.e and crmck junk

(],,(r)

=

i

;

_

rxprrssions

C‘Zr”’ +o(r” ‘) J2

2.1501 2.6642 0.0 I.3725 221x9 -0.5000 0.2195 - 1.4142 0.2295 -1.1X35 1.7742 -1.9453 -1.0544 0.2954 - I.4788 -0.4680 0.0524 4.1345 -0.X7X1 -2.X526 0.8781 1.6051 4.3149 9.7965 II.774 -9.5X63 -1.9667 0.8X71

I

2.0852 2.7291 -0.7526 1.2577 1.7585 -0.2500 0.1567 - 1.0607 -0.94X4 -0.9234 -1.5219 I.1272 ~ 1.0240 0.2993 -1.2226 -0.3199 -0.2900 4.1274 -0.7526 -2.8491 0.7526 I.1794 3.3290 X.5421 X.X523 -3.7934 1.6379 0.7609

I) In r+o(ln

r)]+U(pL?).

5

IO

50

2.0208 2.7832 -2.8125 1.5039 1.0332 -0.l000 0.0673 -0.8485 -4.2X74 -0.7641 - I.3691 0.2092 _ 1.4619 0.3025 - 1.0666 -0.2596 -0.9370 4.1177 -0.7031 -2.x443 0.7031 0.9243 2.7189 7.7840 X.8214 2.9200 7.0017 0.6845

I.9915 2.X050 -6.7082 2.1961 0.00X7 -0.0500 0.0365 -0.7778 -9.9401 -0.7100 -1.3176 -0.9363 -2.4094 0.303X -1.0138 -0.2654 -1.9731 4.1124 -0.7454 -2.8416 0.7454 0.X395 2.5103 7.5288 Il.448 II.674 14.993 0.6588

1.9638 2.8241 -335.19 64.325 -78.X68 -0.0100 0.067X -0.7212 -472.57 -0.6663 - 1.2762 -87.600 -X4.267 0.3050 -0.9712 -2.5949 -X6.225 4.1070 - 6.8406 -2.83X9 6.8406 0.7719 2.3406 7.3231 276.63 6X5.59 655.35 0.6381

+pE,[(p-

(3.56)

Growing Q,(r)

= [HT+o(l

713

crack fields

; r)]+p[Nt

In r+o(ln

r)]+0(1*2),

(3.57)

O,(r) = [~T+0(1;~)]+~[[N~Inr+o(lnr)l+0(~~),

(3.58)

Ofl,,nk(~)= [n+o(l;r)]+~[(C,-Cz)lnr+o(lnr>l+0(~2).

(3.59)

In the above, 8,, specifies the location of the leading constant stress-centered fan sector interface, N, specifies the location of the centered fan--elastic sector interface, O2 specifies the location of the elastic-trailing constant stress sector interface, and Oflank specifies the location of the crack flank.

4. 4.1. Description

cf the usymptotic

DISCUSSION

solution

The asymptotic, perturbative approach used here yields solutions that include small corrections to the previous small-displacement-gradient near-tip velocity and stress field solutions for growing cracks under plane strain conditions. However, the corrections are not obtained by trivial means and require the use of strained coordinates to develop a continuous solution near a singular surface in the general solution for the constant stress plastic sector adjacent to the crack extension line. Two aspects of the solution process deserve mention. The first is that the effect of the spin terms in the Jaumann rate of the Cauchy stress is minimal for this problem. For instance, the finite rate of spin only produces terms in the second order velocity solution that are 0[p2 In (r)] rather than O([,D In (r)]‘) and therefore these terms do not contribute to the most singular part of the solution. Physically, the material spin is constrained by the presence of the elastic sector which cannot tolerate large rotations. In a situation where this constraint is lacking (say, possibly, in the stationary crack problem), the rotation tensor may play a dominant role. Here, however, the dominant terms in the solution arise as a result of the rigorous determination of the material time derivative of the Cauchy stress rather than the approximate representation for this quantity used in small-displacement-gradient analyses. Secondly, it is the imposition of the velocity boundary condition (2.21) that necessitates the inclusion of higher order terms in the solution. It is the inclusion of these terms that eventually leads to the difficulties requiring the use of strained coordinates and to the asymptotic alteration of the boundaries between the sectors of different types. The analysis also imposes some restrictions on both the outer and inner radii of applicability of the solution as well as on the indeterminate parameter p. Scrutiny of the limiting processes used in this analysis reveals that asymptotically vanishing terms (in r) are neglected relative to terms of U(p) ; e.g. In (,u + Y) + In (p) as r --f 0. Hence, an estimate of the outer limit of applicability is approximately the radius at which r M p. The inner limit for the region of applicability is roughly defined by the requirement that p(ln r( be small compared to unity so the expansions of the trigonometric functions and the solutions themselves are justified. The solution is therefore valid in a nebulous region defined by

714

C. R. REIIIand W. J. DKI:C;AN

p >

I’ >

exp

0.1

( 1 ~

I-1 .

(4.1)

The right-hand side of the inequality presumes that the restriction PIIn 1.1< 0.1 is imposed. This shows explicitly that the physical region of applicability is dependent not only upon the characteristic length R but also upon material properties. In terms of dimensional quantities, if one takes ,Uto have the value 5 x 10 ’ and assumes R to be of the order of the size of the plastic zone (say, R z I cm), the outer limit of validity is approximately 5 x 10 3 cm or roughly 1“h of the plastic zone size. The inner limit on the validity of the solution is approximately 2 x 10 ” cm which is a radius of the order of the atomic structure of the material and therefore below the range of applicability of a continuum theory. Obviously, the actual values may have very large variations dependent upon the material constant ,U and the particular choice for the right-hand side of (4.1). However. these results indicate that the region dominated by finite strain effects is quite small. Concerning the free parameter p, (3.33) shows that it is restricted to be greater than or equal to 1 so no spurious singularities are introduced by the first term on the righthand side for small radial distances. If p = I, Cz must be O(p) so h(v) remains small. Furthermore, p cannot be too large otherwise the perturbation series is corrupted because the implicit assumption that coefficients are O(I) would be violated in such a case. This situation is illustrated in Table I for p = 50, which shows that the values of some of the constants become unreasonable for large values of p. Interestingly. although the small-displacement-gradient solutions of DRUGAN and CHEN (1989) do not themselves exhibit a mathematical constraint on the upper limit of their m parameter. CHEN and DRGGAN (1991) found that a very wide range of far-field loading conditions causes nz to vary only within the approximate range 0 < HI < 4.2. This corresponds to 1 < y < 5.2 (see the next section). which easily meets the restrictions on p just noted. Also, the assumed form for the interface (3.30) produces a correction to the lowest order location of the interface composed of a far-field, indeterminate quantity that approaches zero for small r and an 0(/l) correction that is insignificant for large I (i.e. I^--f 1) but becomes dominant as I’ + 0. In fact. this and all other interfaces are described asymptotically by logarithmic spirals (see Fig. 3) and the lower limit ot applicability of the solution must be such that the interfaces are at physically reasonable locations (i.e. p In r must be small compared to unity). In addition to the restrictions on the region of applicability, the solutions are constrained by other physical requirements. Two of the primary requisites are that the stress in the elastic sector remain below yield and that the rate of plastic work, quantified by 23. in (2.19). must be positive everywhere in the plastic regions. It can be verified that the elastic sector is below yield and that the plastic work is positive everywhere in the leading constant stress sector and the centered fan sector if O(u) terms in the expression for the rate of work are assumed to dominate the O(p’) terms. In the trailing constant stress sector, positive plastic work is done everywhere c’.rcept at very small radii near the elastic sector interface. To quantify this, consider only the expression for L(I’ . The requirement that 1.(‘I be positive can be restated using (3.54) as

Growing

FIG. 3. Illustration lines denote actual

crack fields

of the near-tip locations of the interfaces between sectors and the crack flank (solid locations of the interfaces and dotted lines denote leading order, small-displacementgradient locations ; not drawn to scale).

Cl

cos Bfsin

0 (4.2)

Ci
For two values of ,U and some selected values of the parameter p, Table 2 indicates the minimum angle, t?, for which this is satisfied. The leading order location of the It is trailing constant stress sector boundary, G:, is also included for comparison. obvious that A”’ is positive in a region slightly larger than that defined by 0;. However, the correction to the leading order location of this boundary, as indicated in (3.58), enlarges the angular extent of the trailing constant stress sector and for small enough Y, A(‘) may become negative. The value of r for which this occurs can be approximated using HS+pNJln

Table 2 includes

y = ri

or

the values for these calculated

r = exp

0-0; ( x-

(4.3)

)

radii. Obviously,

the results are very

TABLE 2. Eduation

of the approximate radius at tixhich plastic bvork becomes negutiue in the truiling constant stress sector r

p

o^ (radians)

0: (radians)

1

2.553 2.596 2.633

2.664 2.729 2.783

2 5

/L=lO

3

lo-” ,o-” 10mh4

p=lO

2

6.8x 10 3 5.2 x IO-“ 5.0 x IO-’

716

C.

R. REIDand W. J. DKLGAN

sensitive to the value of p and for p = 10 ‘. none of the results has any physical relevance in a continuum theory. However, for p = 10 ’ and p = 2, the value of I estimated from (4.3) is in fact layer than r = 4.5 x IO ’ estimated from (4.1) as the lower limit of applicability. Although the exact location of the crack flank has not been precisely determined in the r + 0 limit (only the crack flank location in the trailing constant stress sector has been quantified), it is probably safe to conjecture that the exceedingly small radii at which negative plastic work is predicted lie outside the actual material domain. The present analysis also allows at least a qualitative discussion of the applicability of some currently used fracture criteria. The crack growth criterion given by RICE c’f al. (1980) requires the maintenance of a critical crack opening displacement at a specific microstructural distance behind the crack tip. This criterion is equivalent to the velocity boundary condition given in (2.21) which in effect requires an invariant profile with respect to the moving coordinate system (see Appendix B). Alternatively. the crack tip opening displacement can be estimated as a function of radius using (1.59). Incorporating small angle approximations, the opening displacement. ijlllll,,,,~, is

Thus, the crack tip opening displacement is, within the range of validity of the solution. O(;L’) or less and, macroscopically, the crack advances with a sharp tip profile. Another quantity of possible interest is the amount of strain incurred by the material near the crack tip. Such calculations are useful if a critical strain crack growth criterion is used (MCCLINTOCK and IRWIN, 1965). The calculations begin by writing implicit relations between the current and initial positions of a material particle as follows :

9 -

x,, -

s, s

I',(.Y(T).j'(T)) dr

= 0 = .f’(.~-.J’~ xc,.yo.f),

r,a

.I-

Y(, -

z‘, (X(T), J'(T))dz

= 0 = $J(S.I’, x,,, y,,.I).

(4.5)

i,>

These represent a pair of equations that must be solved at some point (.Y,y) and time t in order to determine the antecedent point (X,,, Y,,). From these equations. the components of the spatial deformation gradient, F ‘, can be evaluated using implicit differentiation, with the results given by

Growing

(F-l),,

(F-

=y;

=

‘);3 = 0,

_

crack fields

717

ag’ax_ _I I dc,~:__ dz,

r?s/a Y”

‘0

ax

(F- ‘)33 = 1.

(4.6)

The index ?/ ranges from 1 to 2 in this equation. Evaluation of F- ’ cannot be carried out explicitly at arbitrary locations, but can be done on the line ahead of the crack tip where y = 0, fi = r, Z = r( I+ O(p)), and P, = 0. The last relation allows one to replace the time differential ds with dr/c,. Using these facts and formal expansions for the integrands yields, for example, (F-l),,

= l+p[2A,(lnr-lnR,)+o(lnr)] +~2C(2A,+A,E,+2A:)[(lnr)‘-(ln

r)2]}+0(p3).

R0)2]+o[(ln

(4.7)

Note that the above is strictly valid only for a material point originating in the plastic region. If the material point were to originate in the elastic region ahead of the crack tip, the contribution to the inverse of the deformation gradient would be non-singular and O(p) and would not affect subsequent results. Also, the initial non-dimensional position R, is O(1 ; r) and the logarithm containing this quantity can be absorbed into the indeterminate terms. This leaves (F-I),,

= l+~[2A,lnr+o(lnr)]+~‘{(2A,+A,E,+2A~)(lnr)2 + o[(ln r)‘]} + O(p’),

(Fm’)zz = l-~[2Ai

lnr+o(lnr)]-~2{(2A,+A,E,+2Af)(lnr)’

+ o[(ln r)‘]} + 0(p3), (F

‘J12, W

‘j2, = ,u[o(lnr>l+,u2{o[(ln

r)‘]} +O(p3).

(4.8)

These expressions are used to evaluate the stretch of material elements. Noting that W’),, = [(F~‘),,12+[(F~‘)?,lZwhere B = F* FT is the Finger strain tensor, the stretch 1, = ds/dS (where ds is the current differential length of a material element and dS is the original differential length) of a material element currently oriented parallel to the .u-axis is, by formal expansion, I, =

&ill ~~~

= II-~[2A,lnr+o(lnr)] -~‘([2(A5--A~)+A,E,](lnr)2+o[(lnr)2]}+O(~3)~.

(4.9)

Because I, < 1, this material element experiences a compression. Similarly, for a material element currently oriented parallel to the y-axis, the stretch, 12, is given by /I =i~

= /1+~[2A,lnr+o(lnr)] I +~~Z[2(A,--A:)+A,E,](In

and since l2 > 1, this element

undergoes

elongation.

r)‘+o[(ln Included

r)2]}+O(,u3)l

in the above

(4.10) is the

71x

C.

R. REID and W. .I. DKLGAN

plane strain, incompressibility constraint that /,/, = I. Finally, 2E = I-B’ where E is the Eulerian finite strain tensor, E, , = -11[2A, E,,

In ,+o(ln

= ~1[2A, In r+o(ln

using the expression

Y)]+ 0(p2), r)] +O(p’).

(4. I I)

These quantities are small, confirming that the results of the finite strain analysis in fact a perturbation of small strain results.

arc

The finite strain solution presented here is very closely related to the previously developed small-displacement-gradient solution of DRUGAN and CHEN (1989). In fact. the leading order (in p) terms contained in this solution are identical to their ITI-family solution counterparts as shown by the following. The expression for the leading constant stress-centered fan interface, (3.56), contains singular terms that are derivable only in a finite strain context, but the leading order (in 1~)location of the inter&c as well as the extension of this interface to finite radius is the same as that derived by DRUGAN and CHEN (1989). With regard to the expression for the extension of this particular interface to finite distances, the only difference between the present and the previous result is in the method by which the result was derived. Here. the result follows directly from the coordinate straining required to enforce the continuous stress and velocity requirement. In the previous work (DRUGAN and CHEN, 1989), the finite extension was imposed as a physically reasonable description of far-field effects, and the effect that such a modification had on the asymptotic solution was deduced. The non-dimensional radius I’ occurring in the current expressions is the ratio of the actual radius to some characteristic length R. The factor C’$J2 in the finite radius correction to the interface in (3.56) can be absorbed into this indeterminate length scale so the first term on the right-hand side of this equation becomes simply ~ r”’ ‘I. This modification of the characteristic length only afyects the non-singular, O(,U) terms in the remainder of the interface expression:r The finite strain analysis therefore contains exactly the same asymptotically indeterminate characteristic length and exactly the same indeterminate far-field parameter (m = p- I) as does the smalldisplacement-gradient solution. Furthermore. if a more general representation for the interface location of the form p = t, c;(/J’)’ /- I

(4.12)

precisely the same is assumed and terms containing t for small v are expanded, asymptotic results are obtained but with additional o(1 ; r) terms appearing in (3.56). The general form (4.12) substantiates the use of additional terms in the expressions for the characteristics emanating from the centered fan in the analysis of the global

t Similarly, the dominant,

this modification of the characteristic length in t//l the results of this analysis singular terms and only affects the non-dominant. indeterminate terms.

does not alTect

Growing

crack fields

719

behavior of the plastic region during general yielding performed by CHEN and DRUGAN (1991). As a final note, the regions of applicability of the finite versus the small-displacement-gradient solution of CHEN and DRUGAN (I 99 I) is indicated graphically in Fig. 3. The interface between the leading constant stress and centered fan sectors crosses the line 8 = 7c/4 at some radial distance from the crack tip. At a large enough distance, the far-field ri’-’ correction common to both theories dominates whereas the logarithmic terms of this solution dominate at smaller radii. The point of crossover can be estimated from (3.56) and turns out to be at a radius slightly larger than the value of p. Thus it is evident that the two theories are valid in distinct regions with the line of demarcation occurring at a non-dimensional radius of r z p as noted previously.

ACKNOWLEDGEMENTS We gratefully acknowledge support given by the Solid Mechanics Program of the National Science Foundation under Grant MSM-8552486, together with matching grants from Rockwell International and Ford Motor Company, all through the University of Wisconsin-Madison.

REFERENCES CHEN, X.-Y. and DRUGAX., W. J. DRUGAN, W. J. DRUGAN, W. J. and CHEN, X.-Y. DRUGAN. W. J. and SHEN. Y. GAO. Y.-C. GREEN, G. and KNOTT, J. F. H~RMANN, L. and RICE, J. R. HILL, R.

1991 1985 1989 1990 1980 1975 1980 1950

HILL, R

1985

KANNINEN, M. F. and POPELAR, C. H. LEE, E. H. LIGHTHILL, M. J. LIN, C. C. LIU. N. and DRUGAX, W. J

1985 1969 1949 1954 1992

M~CLINTO~~~, F. A. and

1965

IRWIN,

G. R.

MEYER, R. E

1982

NEEI)LEMAN, A. PRITULO, M. F. REID, C. R.

1985 1962 1991

J. Me&. Phys. Solids 39, 895. J. appl. Mcch. 52, 60 I J. Me&. Phys. Solids 37, I. J. Me&. Phys. Solids 38, 553. 4cta Mechanica Sinica 1, 48. (In Chinese.) J. Mech. Phys. Solia’s 23, 167. Metal Sci. 14, 285. The Mathcwatical Theory of Plasticity. Oxford University Press, NY. Metcrl Forming and Impact Mechanics, William Johnson Commemorative Volume (edited by S. R. REID), p. 3. Pergalnon Press, Oxford. Adwnced Fracture Mwhanics. Oxford University Press, NY. J. appl. Meci~. 36, I Phil. Ma,y. 40, I 179. J. Muth. Phys. 33, I 17. Finite deformation finite element analysis of tensile growing crack fields in elastic-plastic material. ht. J. Fracture, to appear. Fracture Toughness Testing and its Applications, p. 84. ASTM-STP 38 I, American Society for Testing and Materials, Philadelphia. Introduction to Mathematical Fluid Dynamics, p. 9. Dover Publications, NY. Comput. Strut. 20, 247. J. appl. math. Mech. 26, 66 I. Ph.D. dissertation, University of WisconsinMadison, Madison, WI.

C. R. REIL)and W. J. Dsti<;AY

720

Rtce, J. R. RI~L:, J. R.

1968 1982

RICE. J. R., DKUGAN. W. J. and SHAM. T. L.

1980

RICE, J. R. and JOHNSON, M. A.

1970

SHIH. C. F.. DELORENZI. H. G. and ANIIREWS. W. R.

1979

SLEPYAN. L. I.

1974

VAh’ Dul
1975

WILLIS. J. R.

1969

J. uppl. Mrch. 35, 379. Mcchunics of’ Soliu’.r : Tlw R. Hill Sixtic& Annimw~~~ Vol~mzc~ (edited by H. G. HOPKINS

and M. J. SEWI:I.I.). p. 539. Pcrgamon Press. Oxford. Fructwe Mechmic.s, p. 189. ASTM-STP 700. American Society for Testing and Materials. Philadelphia. Irwlrrstic~ Bchnrior of’ Sohis (cditcd by M. F. KANNIMN ct rd.), p. 641. McGraw-Hill, Maidcnhcad. U.K. Elrrstic,~Plastic, Frcrc.tur.e, p. 65. ASTM-STP 66X. American Society for Testing and Materials. Philadelphia. 1~. Akorl. Nauk. SSSR. M~~khunikrr Twr~h~o T& 9, 57. (Translated from Russian.) Pcr~trwhrrtiot~ 34ctlwrl.s it1 Fluid Mcchc~t~ic~s. The Parabolic Press, Stanford. CA. J. hlcch. P/I.I,s. LSolitl.r 17, 359.

APPENDIX A Consider a Cartesian system with the (_v._I$)coordinates lying in-plant. From the plant strain condition that the components of the rate of deformation tensor D,, and n,, are zero for 2111 time. (2.1 I) yields the following pair of diffcrcntial equations for the out-of-plane shear stress components : ri,,+ /iici,, =

W,,(T,,.

ci,,+ri/.a,.

~

=

w'\,rT,z.

(A.11

Hcrc x = 2;‘j~and the superposed dot is the material time derivative. These can be vicwcd as linear. first order ordinary differential equations that describe the state of out-of-plant shcat stress for a material particle as a function of time. Solutions are readily obtained and combining the results yields

cr,,(d = 4,(t)

cr,,(o)+o,,(O)~(f,())~

=
1;k..,iT)[ 1:

c,b2(.s) IV,, (s)o,,(s) ds

‘~(t..~)r/,~(.(.)U:,,(~s)~,_(s)ds

I0

] T] d

(A.?)

whcrc c/),(r) =exp[-tii:i(T)dr].

(A.3) The explicit dcpendencc upon a particular material particle has been suppressed. and the integration follows the given material particle that had initial stresses o,,(O) and g,,(O) at time I = 0. For the special case of a completely unstressed body at t = 0. only the last term on the right-

Growing

721

crack fields

hand side of (A.2) remains. As the material is loaded and the crack tip advances, any material particle undergoes an initially elastic response and then may experience successive intervals of elastic or plastic response depending on its proximity to the crack tip. During the initial elastic response, i = 0 and #I, = $J*E 1. It is assumed that the elastic response results in continuous velocity and velocity gradient fields ; however, as evidenced by the solutions developed, there exists the possibility for singular points in the solution domain. For any particle not passing through a fixed small neighborhood of such a singular point, W,,. remains bounded above (by continuity) for any finite time interval. Furthermore, as noted in the text, W,, is independent of 0,; and 0,: so (A.2) has the form

a,,(t) =

y(s, t)o,,(s) ds,

(A.4)

where I,&, t)I is less than some finite constant for all s and t such that 0 < s d t and t is finite. Equation (A.4) is a linear Volterra integral equation so by Gronwall’s inequality. CJ,, = 0 for all t. When the material response becomes plastic, 1. and hence 4, and ~+5~,contribute to the integrand. Like the velocities, i. is independent of the out-of-plane shear stresses and it also may contain a singularity at some point. Also, the velocity governing equations arc hyperbolic so, in general, there exists the possibility of jumps in v and W. Under the condition that the jumps are of finite magnitude, the integrand remains Lipschitz and by the above reasoning, CT,,= 0 even in plastic sectors. This argument obviously extends to any succession of elastic or plastic responses so Q,._= 0 for all time. An analogous result holds for (T,;. To determine the out-of-plane normal stress, use is made of the fact that D,; = 0. This applied to (2.11) gcncrates the following differential equation (S).. = S_. = - tij.s..

(A.3

with the solution s;,(t) =~lZ(0)exp[-h.~;i(~)d~~] This immediately all time,

shows that s,; = 0 under the initial condition

CT;._ =

o,, -~~

of zero stress. Therefore,

+a,,

for

(A.7)

2

APPENDIX B Here we show that the steady-state crack growth assumption, which requires the position of the crack flank to bc fixed with respect to an observer moving with the tip, leads directly to the velocity boundary condition given in (2.21). The analysis uses three coordinate systems in which the position of a material particle can be described by the Lagrangian coordinates relative to the fixed reference configuration (X), by Eulerian coordinates relative to a stationary (intermediate) current configuration (<), or by Eulerian coordinates relative to a current configuration whose coordinate system origin translates with the crack tip (x). The first and last coordinate systems are those already defined in the main article. If it is assumed that the crack flank in the current configuration (<) is comprised of material points that wet-c originally on the crack symmetry plane (X2 = 0), then the current location of the crack flank with respect to this coordinate system is r, = uz(X, >0, t). where the displacement

vector, II, is written in terms of the Lagrangian

(B.1) coordinates.

To obtain

122

an expression so that

C. R. REID and

W. J. DKLJGAN

for the crack flank in terms of the Eulerian x, = <1-l12(<,.;r.t)

coordinates

5, note that 5 = X+u

=o,

(B.2)

where the initial assumption has been enforced. For each (< ,. 1) in the current contiguration. solution of the implicit equation (B.2) gives the current location
_\-z) = 0.

(B.l)

This represents an implicit equation for the location of the crack flank in terms ofthc translating coordinates. Taking the material time derivative of (B.3) and observing that .?, = I’, yields (B.4) This and (B.3) imply that Vf'.v=O

or

v.n=O.

(B.5)

where the gradient operator is with respect to the translating Eulcrian coordinates x and the velocity Y is relative to the translating system by definition. This indicates that (B.2). when writlcn in terms of translating coordinates. is cquivalcnt to the velocity boundary condition cxpresscd in (B.5) and (2.21) in terms of the relative velocities. This analysis is similar in concept and approach to that presented by MEYER (1982) concerning the boundary condition on a material surface moving with a fluid.

APPENDIX C In this appendix, the solutions for the velocities and stress parameters in the leading conhtanl stress sector obtained by the straightforward perturbation analysis arc presented. These solutions are in terms of the characteristic coordinates s( = .X-J’ and /1 = s+j’. and are not in terms of the strained coordinates that arc necessary to obtain the conGnuous solution. The velocities. at each order of the perturbation expansion, have the form “,“‘1 = -1. {.i”’ = 0,

((‘.la)

~~~“=A,lnx+n,ln~i+A;. r~i” = A,lnY-A21n/,‘+A,. P(,~)= A,(ln n)‘+A,

In r+A,(ln

/{)‘+A8

(C.lb) In /C+ilo

-(~jlln(s,~lfi+I/A,+~,)[!;()+(g)1 In (a/I)

(.i” = A,(ln x)‘+A,In-A,(ln/j)‘-A,ln/j+A,,,

723

Growing crack fields

(C.lc) The stress parameters

have the form

(#01

=

4”’ = E, In a-E,In r#‘) = -2[E, qY” = E,(ln ?)‘+E(,

/Q-E8

/I+E?,

In r+Ezln

(C2b)

p+E4],

In P+E,+E,

0

;

]E,(l ncc-l)-EzInfi+E,]

0;

LEtIna-Ez(ln/j-l)+EJ.

+E, o(‘) = -2[E,(ln

-2E,

In cc-E,@

(C.2a)

E*

a)‘+ E, In a+E,(ln

i

fi)‘+E,

In /l+E,,]+2Ez

nr-I)-EZIn/Y+E,]

[E, lna-E,(lnP-l)+E,].

0 The terms with the coefficients E,-E,,, represent assumed forms for the arbitrary functions contained in the general solution; the (In T)’ singularity is appropriate by extension of the results for the elastic sector. For generality, the boundary conditions at 0 = 0 have not been included. These expressions are used in Section 3.2 to motivate the use of a strained coordinate and to make the form for the strained coordinate explicit.