Boundary formulation for three-dimensional anisotropic crack problems

Boundary formulation for three-dimensional anisotropic crack problems

A. Le Van and J. Royer

Laboratoire de M&anique des Structures et des Mat&aux, Ecole Centrale de Nantes, Nantes,

France

Boundary integro-differential equations for three-dimensional anisotropic cracked bodies are deriued. Both the cases of the infinite body (with an embedded crack) and a finite body with an embedded or surface crack are considered. Detailed mathematical conditions for the results to be valid are specified throughout.

Keywords: anisotropic elasticity, crack, integral equation

1. Introduction

In the field of numerical techniques, the most widely known is undoubtedly the domain finite element method (FEM). In recent years, the structural components engi- neers have to cope with become more and more complex, and the number of degrees of freedom entailed by a sufficiently refined mesh is continually increasing to obtain more and more accurate results. It was precisely for this reason that the advent of the boundary integral equation method (BIEM) is of great significance since it allows the structure to be meshed over the boundary only, at least when the body forces are absent. In any case, the BIEM allows the solving of problems with smaller matri- ces because the unknowns are exclusively boundary quantities.

The pioneering works in the BIEM were those of Rizzo,’ Cruse and Rizzo,2 and Cruse.3,4 The regularized expression of the BIE was first given by Rizzo and Shippy.5 Since then the boundary method has been extensively developed showing its ability to deal with various types of mechanical problems. In the field of fracture mechanics, the BIE, or more precisely the boundary integro-differential equation (BIDE), has been formulated for elastostatic, thermoelastic as well as elastodynamic problems6-lo Reg- ularized BIDES have also been given elsewhere.“-‘4 A complete review of different works in the BIE field can be found in the paper of Tanaka et al.,” though related to regularization techniques.

Address reprint requests to A. Le Van at the Labortoire de Mecan-

ique des Structures et des Materiaux, Ecole Centrale de Nantes, 1, rue de la Noe, Nantes 44072, Cedex 03 France.

Received 23 October 1995; accepted 11 March 1996

Appl. Math. Modelling 1996, Vol. 20, September 0 1996 by Elsevier Science Inc. 655 Avenue of the Americas, New York, NY 10010

As a matter of fact, whereas most of the materials are more or less anisotropic, the BIEM was mainly developed for isotropic materials. This can be accounted for by two reasons. First, equations for the isotropic case are simpler to solve from the theoretical and numerical standpoints alike. Second, the elastic material properties are much more difficult to be determined experimentally in anisotropy. Nevertheless, increasing use is being made of anisotropic structural components, and this requires more efficient studies for this class of materials. Using the decomposition of the Dirac function into plane waves, Vogel and Rizzo I6 derived the integral representation of the fundamental solution for a general anisotropic elastic three-dimensional (3-D) continuum. Later on, an efficient numerical implementation for anisotropic problems was proposed by Wilson and Cruse.” In fracture mechanics, Sladek and Sladek” and Balas et alI9 (pp. 50-52) have discussed the boundary formulation for anisotropic cracked bodies, and the corresponding BIDE has been proposed, conjecturing that some results in isotropy can be extended to anisotropy.

In the first part of this paper, the integral representation of the fundamental solution for an anisotropic elastic medium derived by Vogel and Rizzo16 is briefly reviewed and its basic properties are investigated. In particular, relations describing the limit behavior of the fundamental solution near a closed or open surface are presented. All the results obtained are generalizations of the well-known ones in the isotropic case. As an application, in the second part of the paper, the BIE and BIDE are derived for the problem of anisotropic cracked bodies. The boundary formulation includes both the cases of the infinite body (with an embedded crack) and a finite body with an embedded or surface crack. Throughout the paper, em- phasis is made on the mathematical conditions for the results to be obtained.

0307-904X/96/$15.00 PII SO307-904X(96)00047-3

2. The fundamental solution and its basic properties

In this section, the fundamental solution for a homoge- neous anisotropic linear elastic medium is recalled and its basic properties are given. The components of the fourth- order elastic tensor C written in a fixed base (e1e2e3) of the 3-D space 8 are material constants verifying the usual symmetries:

Cijk, = Cjikl = Cij& = Cklij (1)

Let us introduce the following definition where the summation convention is implied over repeated subscripts, which all have the range (1,2,3).

Definition

Given a unit vector e, of the base (e1e2e3), the fundamental solution related to e, and denoted by U(e,, x, y) is the solution of the partial differential equation in the infinite space E:

div[ C: grad U(e,, x,y)l + S(y -x)e, = 0

where S(y -x) is the Dirac function. The corresponding stress tensor is defined as:

(2)

Xe,, X,Y) =~t~~~,,~,~~(Y~n~ =ej) @ ej

=C: gradU(e,,x,y) (3)

where the tensor product a 8 b is defined by (a 0 blij = ajbj. The differentiation in equation (9) is performed with respect to variable y, and tUCe,,x,yj(y,Ity) denotes the stress vector at point y with respect to normal nY and corresponding to the displacement field U(e,, x, y).

Now the fundamental displacement tensor is defined by:

E(x,y) = U(e,,x,y) Bee,, i.e., Eij(X, Y) = V,(ej, XT _Y) (4)

This, in turn, gives rise to the third-order tensor of the fundamental stress:

D(x,y) =Z(e,,x,y) Bee,, i.e.,

Dijk(X, Y) = xij(ek, XT Y) = Cijpq- (5)

Eventually, the Kupradze tensor is defined by*’ (p. 99):

T(x,y,n,) =tU~~,,x,yj(y,ny) Bee,, i.e.,

Tk(X, Y, ny) = Cijpqnj(Y)-

Anisotropic crack problems: A. Le Van and J. Royer

The relationships between the above-defined tensors are straightforward:

T,k(x,y,n,)=Zjj(ek,x,y)nj(y)=Djjk(x,y)nj(y) (7)

Conversely:

xij(ek, X, y> = Dijk(X, Y) = Tikcx> Y? ny = ej) (8)

Because tensors Z and T are functions of D, any equation in the sequel can be expressed in terms of D alone. In practice, however, the simultaneous use of notations T and D proves to be more convenient.

The equilibrium equation in terms of the fundamental solution reads

Vk, div C(e,,x,y) + 6(y -x>‘e, = 0, i.e.,

dDilk tli, k,-- dy, (x,Y> + s(Y -X)‘& = 0

I

which gives rise to the so-called rigid body identity:

VXEfi\S, / s

D,,,,k(% y)n,,(J’) d,S = -s,,,,

(10)

where Sz is any bounded region and S its boundary. We now come to the expression of the fundamental

solution and its derivatives. For this purpose, let us introduce the following notations. Given two points x and y of space 8, let (a,,a,,a) be an orthogonal base with the third vector a equal to e, = (y -x>/lly -x11, a, and u2 being arbitrary. This based allows us to define the spherical coordinates ( x, $) E [O, n[ x [0,27~[ such that the line x = 0 coincides with e,, whereas the origin for $ is arbitrary.

Let us define the tensor Q as Qik(l > = Cijk)$&, I& is the j-component of vector 5 in the global fixed base (e,, e2, e3). The inverse of Q is denoted by P:

P(5)=Q-‘(l), i.e.,

1 Pik( 5) = - &kpq sirs Qpr Qqs 2det Q

where ckpq is the permutation symbol. Both P and Q are symmetric tensors. Since a unit vector 5 can be entirely determined by its spherical coordinates ( x, $1 in the base (a,, u2, a), tensors P(l) and Q( 5) are also written as P(u, x, $1 and Q<u, X, $).

The following theorem was proved by Vogel and Rizzo” by decomposing the Dirac function into plane waves.

Appl. Math. Modelling, 1996, Vol. 20, September 663


Theorem l The asymptotic behavior at infinity:

The fundamental solution for an anisotropic elastic medium is given by:

1 Vm, U(e,,x,y) = -

/ &r2r 0 2nPP(u, +> d+e,

1 -E(x,y) = -

/ &r2r 0 2%u, ICI) d9 (11)

in which r = Ily -XII and Ha, $,> stands for P(a, x= v/2, *j.

The integral is taken along the unit circle in the plane normal to a = e, and passing through x.

The derivatives of tensor E(x, y) can be computed by

‘fk,p,q, JEpk 1

ay(x7y)= -- 4 8rr2r2 i r.q o / 2TP&, $r> drCr

+(S,, -r,,r,q) j2TPpI(a, ICI) 0

~QI, X- du (a, ICI)PAa, $1 dG

i (12)

n

where r,i = dr/dy, = e;ej and Q<u, I/J) stands for Q<u, x =

n/2, *). The following properties result from the above theo-

rem.

l The symmetry of E:

E(x, y) = E=(x, y) (13)

l By applying the same reasoning made for E(x,y) to E(y, x), we obtain the variable interchange properties:

E(x,y) =E(y,x) (14)

(15)

l Using relation (5) and the foregoing properties, we obtain similar relations for Z and D:

%e,,x,y) = -Z(e,,y,x) and

D(x, y) = -my, x) (16)

gc ax ek,x,y) = - -_(e,,x,y) and

1 dYi

ZCX,Y) = - -$X,YI 1

(17)

l The singularity of the fundamental solution:

as r = I/y --xl1 -+ 0

(18)

T(x,y,n,) =o f ( 1 as r = Ily --XII 4 02

(19)

Of course, in isotropic or transversely isotropic cases, relations (11) and (12) simplify appreciably, yielding well- known expressions. 21 The fundamental solution for a general anisotropic medium is known only by the integral representation (111, but the expression in closed form is not available in general since the actual integration is not possible for any time of anisotropy. Nevertheless, the above properties prove to be sufficient to establish the so-called limit theorems in what follows, without the knowledge of any closed form expressions for E and T whatever.

Limit theorems

The purpose of this section is to present some results about the limit behavior of the fundamental solution when the load point x approaches a point y, belonging to a given surface S. These results hold in the case of general anisotropy, thus constituting the generalization of the well-known ones in the isotropic case. They all are based on the lemma below, which alone exploits the integral representation (12) of the fundamental solution.

First let us introduce some notations. Given a point y, and a unit vector nYo, let II be the plane passing through y0 and perpendicular to nyo. The normal nyO defines two

sides of plane TI, which will be recognized by the signs + and -, the + side being the half-space containing the point y. + nyO.

Let E be an arbitrary positive number, B(y,, E) the ball centered at y,, and with radius E. The plane IT divides the boundary aB(ya, E) of the ball into two parts denoted by S&y,, E) and S,(y,, E), situated in the sides + and -, respectively.

Lemma

/ JEpk -(y,,,y)n,(y) d,S

S,(YO> E) JYq

/ qdc = --(Yo,~)n,(y) d,S

S*(Y,, 8) JY,

where the normal rr,, is outward to B(y,, E).

664 Appl. Math. Modelling, 1996, Vol. 20, September

Proofi Regarding the integral over S,(y,, E), relation (12)

yields:

XP,,(a, t+b) dll, I

sin f3 d0dq (21)

in which r = lly -yolI, r,, = (yq -y,,)/r, e, = (y -yJ/r, the spherical coordinates (0, ~0) E [O, rr[x[O, 27r[ are so defined that the line 8 = 0 coincides with nyO whereas the origin for p is arbitrary. As for the integral over S,(y,, E), it is identical to that of (21), only the bounds for 8 are different, since now 8 varies from rr/2 to r. Using the variable change 0’ = r - 0, 40’ = rr - q and introducing new notations: y’ denoting the symmetrical point of y with respect to y,, r’ = lly’ -yOll, r,> = Jr’/ayi = <yi - yoq)/r’ and a’ = -a, we have:

/ d-$k -(Y,,Y)~,(Y) d,S

S,(y,, E) dY,

x c-u’, $)Pmk(-a’, $1 de sin 8’ de’ dq’

(22)

Eventually, by comparing relations (21) and (22), and by noting that:

we obtain the proof of the lemma. Applying relations (5) and (6) to relation (20), we

directly deduce the following propositions.

Proposition

Vy,,Vn,,,V&,Vi,k,

/ SLY,, 8) ~,(Yo,Y,n,)d,S

= / S,(Yo, E)

&(YO,Y,ny)d,S (23)


Proposition

/ S,(Y,, El Dijkbb y)n,(y) d,S

= / Dijk(YO,Y)n,(Y)dyS (24)

S*(Yo, E)

It should be noted that the above propositions involve the plane II passing through y, and normal to vector ny,,. Moreover, the value of E is arbitrary, not necessarily small. In the subsequent applications where y, belongs to an arbitrarily shaped surface S, the neighborhood of y, in S is not plane, this is why we shall have to take the limit & + 0.

Relations (23) and (24) allow us to prove the so-called limit theorems below. Let us first agree about notations for the orientation of a surface. Of course, any surface S (closed or open) considered here is assumed to be ori- entable. This implies that, for any point y, E S \ dS, we can locally define two sides of S which we label side + and side -, all the normals to S being directed from side - to side +.

Theorem

Let S be a surface (closed or open) and u a vector field defined on S. If:

(i) S is a Lyapunov surface: S E C1,a, 0 < (Y I 1, which means that

3C > 0, Vy, y’ E S, IIn,, - nyll 5 Clly’ -yll”

(ii) u satisfies the Holder condition on S: IJ E C”,p(S>, 0 < p < 1, i.e.,

then

VYOES

lim / x’yd s

T(x,y,n,My) d,S

= k ;u(y,) +pu jT(y,, y,nyMy) dyS s

(25)

where by x + yo* are meant the limits as x approaches y,, x belonging to the side + and the side -, respectively. The symbol pv denotes a Cauchy principal value integral. Relation (25) also holds if tensor T is replaced by its transpose TT.


Anisotropic crack problems: A. Le Van and J. Row

Proofi Solving equations (27) to (29) gives By denoting S(y,, E) = S n B(y,, E), the left-hand side

of (25) can be recast as

lim / x-y; s

T(x, y,n,My) d,S

= lim / *+y& S\S(Y,,E)

T(x, y,n,My) d,S

+ lim / x-y& .s(y*, E)

T(x, y, n,)[u(y) - u(y,)l dYS

+ lim / T(x, y, nY) d,S~u(y,) (26)

x-y& SOJ,, 8)

limA*= + L E’O -2’

The theorem is proved.

Theorem

l The first integral is continuous at y. when y -+yo. It tends to the Cauchy principal value integral appearing in the right-hand side of (25) as E + 0.

l Let us denote r = Ily -XII, r. = lly, --XII and p = Ily - yell. From (18) and hypothesis (ii), we have: T(x, y,n,)

= 0(1/r’) = O[l/( p2 + r$l, u(y) - u(y,> = O( pp>, and d,S = 0( p). Hence:

Let S be a surface (closed or open) and t a vector field defined on S. If:

(i) SEC1,a,O<~I1 (ii) t E C”,pLS), 0 < p 5 1

then

tfY,ES

lim / x-y”* s

D(Y, x)t(y,n,) d,S.n,,

lim / x-y& sty,, e)

T(x,y,n,)[u(y) - u(y,)l dyS = O(sP)

l It remains to investigate the last integral in equation (26). For brevity, let us denote:

1 = T Zt(yo,nyO)

+PU D(y,yo)t(y,ny)dyS~nyo / (30) s

A*= lim / x+yJ sty,, 8)

T(x,y,n,) d,S

I? = lim / x-y; S,(Y”, 8)

T(x, y, n,> d,S

where nyo is the normal vector at y, to S and the product of the third-order tensor D and vector t is the second-order tensor defined by (D.t)ij = Dijmt,.

Proofi

= / S,(Y”, &f

T(y,,y,ny)d,S The proof is similar to that of equation (25) bearing in

mind that, in view of relations (7) and (16):

C = lim / x-y: S,(y,, E)

T(x,y,n,)d,S [ D(Y, xMy,,x,,)] .nY = -T(x,y,n,My,,n,,)

The following theorem requires a somewhat stronger condition for the function U.

= / S,(Y", 6)

T(yo,y,ny)dyS

where the normal ny for y ~Si(y~, &)US2(y0, E) is outward to the ball B(y,, E) while we recall that ny for y E S(y,, F) is directed from side - to side + .

The rigid body identity (10) yields:

A++B=o, A-+B= -z

where Z is the unit tensor. Similarly, we have:

(27)

-A++C= -I, -A-+C=O (28)

On the other hand, from hypothesis (i), S,(y,, E) and S,(y,, E) tend to two symmetrical hemispheres as E + 0. Therefore, by applying equation (23), we get

lim B = lim C (29) E’O E’O

Theorem

Assuming that

(i) S E C1bn, 0 < (Y I 1 (ii) u E C’,p(S), 0 < p I 1, i.e., all the derivatives of u

belong to the class C ‘9 p(S) we have the property of continuity across the boundary:

VYOES

lim / ~(dy,x,y,n,)u(y)d,S.n,,

x-Y& 3s

=puj~(dy,y,,y,n,)u(y)d~S.n,, (31) s


where the symbol 9’ represents the differential operator defined as:

[NdY,x,Y,n,)u(yl]ij

'~ijm(dy,x,y,ny)u,(y)

=C;jklDmnk(x,Y)'~~[(ay,n,)u,(y)

(32)

L3,Jdy,nJ is the tangential differential operator defined as

9Jdy,n,) =nnCyl-& -n,(g)-& (33) n

(there is no possible confusion of the normal ny with the subscript n E {1,2,3}), the symbol dy recalls that the differentiation is carried out with respect to variable y.

ProoJ Invoking arguments similar to those in the proof of

(2.9, we can write:

lim / S(dy, x,y,n,My) d,S.n,,

vy(f s

= lim lim E’O x+y,: / Nay, x, y, n,My,> d,S

s(y,,,&)

~~yo+~~~j~~~y,~o,y,~y~~~y~~y~~~yo s

The main work is the study of

A*= lim / ~ijrn(dY,X,Y,nY)u,(Yo)dyS'njo

x-y& SY,,,&)

[njo Enj(Y()>]

Let us also denote

B = lim / ~~jm(dY,~,Y,ny)u,(yo)dyS'njo

*+y$ S,(Y,, &)

= j

~ijm(aY,Yo,Y,n,)u,(yo)d,S'njo SAY,,,&)

C = lim X-Y&

/ ~jjm(dY,x,Y,ny)u,(yo)d,S.njo SZ(Y",E)

= /

~ijjm(dY,Yo,Y,“y)Um(Yo)dyS’njo S,(Yl,, E)

where the normal ny on different surfaces is defined as in the proof of (2.5).


l First let S,(y,, E) be involved by writing

A *+ B = lim Cijk,/ %&PY) J--yoi S(y(J,E)US,(y~,F)

x bw,,l(YO) - nP,.,(Yo)J d,S’njo

(34)

where n, = n,(y). For the sake of brevity, the right-hand side of the previous equation will be written in the form of a difference

A*+B=B*-G’ (35)

In virtue of the rigid body identity (101, we have

Ff= 0, F-z -Cijkr~k,,(yo)njo (36)

Moreover, G * can be transformed successively as follows. Using equation (51,

G *= lim CijklCmnpqj JEpk -_(x,y)

X’YO’ .s(y,, E)US~~YO,~) JYq

Now, since S(y,, E) U S&y,, E) is a closed surface, we have

/ dE,k -n, d,S

S(y,, F)US~CY"'E) JYq

Then, owing to the symmetry of E, relation (141, and relation (5):

G *= lim Cm.,,/ X-+Yi s(y,,~)us,(y”,~)

= lim Cmnpq *‘Yo’ /

sty,, E)US,oJ”, 8)

+ lim Cmnpq X*YO*

j sty,, &)US,oJ”, E)

Using equation (10) and the symmetry C,,,,ij = Cijmn relation (l), we get

G+= lim CmNpql -Y$ SoJ”,E)US,(yo,E)

XDijp(njon, -njn,o)dyS.u,,,(Ya) (37)



and so that equation (24) holds, yielding

lim B = lim C (40) E-+0 E’O

and

lim D = lim E (41) E'O E-+0

l Eventually, relations (38)-(41) form an underdetermined algebraic system of four equations for five unknowns, A, B, C, D, and E, from which we derive a definite result for A: lim, -t 0A * = 0.

The theorem is proved.

G-= lim Cmnpq x+Yd

/ S(y(),E)US1(yO,E)

X Dijp(njonq - njn,o) d,S*u,,,(yo)

- Cijmnum,n(YO)njo

Then, substituting equations (36) and (37) into (35) gives:

A* + B = lim Cmnpqj x-x0’ s(yij,&)Us,(y”,&)

X DijP(x, Y)(-njon, + njn,,) d,S

%,AYJ

=D + lim Cmnpqj X+Y$ S(Yo,E)

Dijp(x, Y)

where

x( -njon, + njn,,) d,S.u,,,(Y,)

D = Cmnpq / Dijp(Yoy Y) S,(YO> E)

XC -njon, + njn,o) d,S~u,,,(y~)

But, from equation (18) and hypothesis (i):

/ S(y,, 6) Dijp(x, y)( -ajon, + njn,o) d,S

= / Dijp(X,Y)[(nj-njo)n,o Sy,, E)

-njo(n, - nqo)] d,S = O(E~)

Thus,

A*+B=D+O(E~) (38)

l The same treatment can be applied to S,(y,, E), with special attention to the normal orientation. The corresponding results for the f cases are interchanged with respect to those obtained with S,(y,, E), and we are led to

-A*+C=E+O(E~)

where

(39)

E = Cmnpq I SAY,, E) D,jp(YotY)

XC-njon, +njn,o)d,S*u,,.(yo)

l Furthermore, according to hypothesis (i>, S&y,, E) and S,(y,, E) tend to two symmetrical hemispheres as E + 0,

Notes

(a)

(b)

Relations (25), (30), and (311, established in the anisotropic case, constitute the generalization of well- known results in isotropy where they can be directly verified using the closed form expressions available for T, D, and LZ (Ref. 19, pp. 29 and 43). The hypothesis (ii) for equation (31), u E C1*p(S), is stronger than that for equation (2.5) because of the derivatives involved by the differential operator 2.

3. Application to fracture mechanics: Integral representation of the displacement in a cracked body

Consider a linear elastic anisotropic, finite or infinite, body 1R containing a crack. If the body is finite, its outer boundary is denoted S, and the crack can be either an embedded one or a surface one. The crack surface S,, is made up of two faces S,: and S, which coincide in the undeformed state. To each point y E S,, correspond two points y+ and y- belonging, respectively, to SG and S,. The respective normal vectors are opposite, i.e., nc= n;, n; being directed from S, to S& thus defined every-

where as outward with respect to the body considered, in accordance with the usual convention. In the sequel, all the equations will be written using S,, so that normal n; is taken as the reference one.

This section gives the displacement at any interior point of the body in terms of boundary quantities. Two cases are considered: the infinite body (with an embedded crack) and a finite body with an embedded or surface crack.

3.1 Infinite body

Consider the infinite body Ll containing the crack S,, = s,u s+. The following hypotheses referred to as the regula%y conditions are assumed.

Regularity conditions

(i) The radiation condition for unknown displacement and stress is assumed as: u(y) = o(l), a(y) = 0(1/r),


(ii)

i.e., t(y, n,) = o(l/r), when r = I/y --XII + m, x being a Note that the regularity conditions are useless in the finite fiied point. body case. The radiation condition for the body forces is assumed as: f(y) = O(r-‘- ‘) when r -+ m, where 6 is a positive constant smaller than 1.

Note that condition (ii) is identically verified if the body forces f is confined to a finite region. Given any point x, let B(x, R) denote the ball centered at x with radius R large enough for B(x, R) to include the crack. Let S, be the exterior surface of B(x, R) and OR, be the bounded region within S,U S$ and S,, with boundary S,u S,:U S,. The second condition (ii) ensures that lim R +Jn, E(x, y)fCy) d,V is bounded.

The following theorem holds.

Theorem

Provided the regularity conditions are fulfilled, we can write the integral representation of the displacement as:

u(x) = / [ E(x,y)Wy) SC;

+TT(n,y,n,)Au(y)] d,S

+ * /

EC-x, y)f(y) d,V n

(42)

in which Ct(y) is the sum of the stress vectors on the crack faces: Xt(y) = t(y+,nl) + t(y-,n;) (if the crack is loaded symmetrically, i.e., t(y+, n,f> = - t(y-, nil, then

Xt(y) = 0) Au is the displacement jump through the crack: Au(y)=u(y+) -u(y-), and the asterisk denotes an improper integral.

The proof of the theorem does not formally differ from that in the isotropic case.

3.2 Finite body

Consider a finite body R with outer boundary S,, containing a crack S,,. The following theorem can be proved in a similar way as in the infinite body case.

Theorem

The integral representation of the displacement reads:

vx E R \ (S, u S,,),

u(x) = /,_ [ E(x,y)Xt(y)

+kTCx,y,n,) Au(y)] d,S

+ Ir E(x, y)t(y, nY)

Sk’(x,y,n,Myl] d,S

+ * / E(x, y)f(y) d,V n

(43)


4. Integral representation of the stress

In practical purposes, it is often necessary to compute the complete stress tensor at any point inside the body. This section is thus devoted to the integral representation of the internal stress.

4.1 Infinite body

Theorem

Assuming that:

(i) the regularity conditions are fulfilled. (ii) S&E C’x =, 0 < (Y I 1

(iii) AU E C’(S,)

we have the integral representation of the stress:

vxps,,, a(x) = i,_ tDtY,x)xy) cr

+ * / Dty, n)f(y> d,V (44) n

Proofi The integral representation of the displacement (42)

holds because of hypothesis (9. Given a point x E S,,, the derivative duk(x)/dx, will be investigated to obtain the stress U(X). All integrals in equation (42) can be differen- tiated behind the integral sign. Indeed, since x is interior to fl the surface integral is regular. Moreover, relation (18) implies that the kernel of the volume integral behaves as l/r as r + 0. Thus:

JL, +- dx (x2 y, n,) Au,(y)

1

+* / *tx, y)f,,,(y) d,V

n Jx, (45)

The first and third integrals containing dE,,Jax, do not require any further transformations. As regards the second integral, exploiting hypotheses (ii) and (iii) it can be transformed by means of the so-called regularization the-

Appt. Math. Modelling, 1996, Vol. 20, September 669


orem (see Appendix):

/ at, -(x,y,n,)Au,(y)d,s

SC; dx,

= / s,

Qn,,(x>~)%z~(dy, n,) Au,(y) d,S

(46)

In writing equation (46), use has been made of the boundary condition along the crack front: Au(y) = 0 for y E dS,,. The theorem is proved.

Note the order of variables x and y in the D-terms of equation (44): D(y, x> instead of D(x,yl. On the other hand, the volume integral in equation (44) is improper convergent on account of the regularity condition f~ 0(r-2-6).

4.2 Finite body Theorem

Assuming that:

(i) S,E C’, a, S, E CATS’, 0 < (Y, CX’ 5 1 (ii) Au E C’(S,), u E Cl@,)

we have the integral representation of the stress for a finite cracked body:

Vx E n \ (S, u S,,),

a(x) = / uxy, x)Zt(y) KY

+ s,(D(Y, XMY, “J /

+ * / D(y, x)f(y) d,l/ (47) R

In the case of a surface crack, S, must be replaced by S, \ L throughout, where L = as&n S, is referred to as the surface line.

Proof: Only the case of the surface crack, which needs special

attention, is investigated; the proof for the embedded crack can be deduced in an obvious way. The reasoning is essentially the same as in the infinite body case, the difference being in that here there are two surfaces: the closed surface Sa upon which the displacement u and the stress vector t are defined and the open surface S, upon which the displacement jump Au and the stress sum Zt are defined.

The surface line L being positively oriented with respect to the normal n; of S,, let us introduce the closed contour L-U L + made up to two arcs L - and L + such that both of them coincide with L whereas the orientation of L- (respectively L+) is the opposite to (respectively the same as) that of L (Figure I).

Assume for definiteness that the contour L-U L + is oriented negatively with respect to the outward normal to the exterior boundary S,, as shown in Figure 1. In fact, it can be easily verified that the final result does not actually depend on this assumption.

Hypotheses (i) and (ii) make it possible to apply the regularization theorem (see Appendix) successively to surfaces S, and S, \ L the boundary of which is L-U L +:

d -/ TT(x,y,n,)Au(y)d,S dx, s,

= surface integral over S&

_ / q,l,. AU,(Y) %,,Jx~ Y) dy, (481

L

and

d --

/ dx, s, TT(x, y, n,)u(y) d,S

d Ez --

/ dx, S,\L TT(x, y, n,)u(y) d,S

= surface integral over S, \ L

+ / E,I~u,(Y)D,,~(x,Y)~Y~ (49)

L-uL+

n” J

Figure 1. Definition of the closed contour L-U L’ in the case

of a surface crack.


in which, by noting that D,,,,Jx, y E L-1 = D,,,,,&x, y E L+) and that the orientation of L- and L+ are opposite

/ L_ULf E,,,u,(Y) %,&,Y) dy,

From equations (48) and (49), it turns out that, although line integrals appear in the case of a surface crack, their very sum is zero. The theorem is proved.

Notes

(a>

(b)

In the case of a surface crack, it has been assumed that u E C’(S, \ L), not u E C’(S,). Indeed, the surface line L corresponds to an incision in the boundary S,, giving rise to a displacement discontinuity on S, along L and making the hypothesis u E C’(S,) im- possible. The theorem cannot be proved by considering S, U S,, as a single closed surface. The crack geometry clearly indicates that, contrary to S,, S,, = s;U SC& cannot be assumed to belong to the class C’,“.

5. Boundary integro-differential equation

It is well-known in isotropy that the BIE obtained in usual way by taking the limit R \ S,, 3x +y, E S,, in equation (42) is nonunique and insufficient for the determination of the unknowns on the crack. Indeed, the loading on the crack faces is involved in the BIE only through the sum Ct so that, if the stress vector is continuous, i.e., Xt = 0, there is no information about the loading at all. The purpose of this section is to overcome this deficiency by obtaining an adequate BIDE, possibly combined with a BIE.

5.1 Infinite body Theorem

Under the following assumptions:

6) (ii) (iii)

the regularity conditions are fulfilled S,E C’v” ,O<crll Au E C’,p(Sc;), 0 < p I 1 (from hypotheses (ii) and (iii>, it follows that Ct = -[C: (grad Au)].n;E C”,a’(S&), 0 < /3’ I 1) the BIDE writes

VY* E s,,

t[t<y,,n;o) - t(y,‘,n:,)]

=pr!/ {D(y, y,)%(y) s,

+~(dy,y,,y,n,)Au(y)}d,S.n,,

+ * / D(y, y,)f(y) d,v.nFo n

(50)


Proo$ Equations (30) and (31) are valid in view of hypotheses

(ii> and (iii), yielding

lim /

X-‘Y$ SC;

D(y, x>%(y) d,S.nio

= T iCt(y,) +pu/ D(y, y,)X.t(y) dyS.nco SC;

and

lim /

x+y: SC;

W(dy,x,y,ny)Au(y)dyS.n~o

=pvj 9(dy,y,,y,n,) Au(y) dyS*njYo &I

where by x +y: are meant the upper and lower limits with respect to normal n;,,, respectively. According to the hypotheses, the integral representation (44) of the stress holds. Then multiplying it by nio and taking the limit as x +yz gives (50). The theorem is proved.

If the loads t+ and t- are specified, the solution of the BIDE (50) gives the displacement jump Au, and then u+ and u- using the integral representation of the displacement (42). Conversely, if Au is prescribed, the BIDE (50) does not allow to obtain tc and t- separately, unless an additional information is supplied, e.g., t+= -t-. This is better accounted for by the particular case of the plane crack considered below.

Plane crack in the isotropic medium

From equation (50) immediately results the following corollary.

Corollary

Consider a plane crack imbedded in the infinite medium and

(i) (ii) (iii)

lying in-the plane yj = 0. Assuming that:

the medium is isotropic the regularity conditions are fulfilled Au E C1(S,) we have the BIDE for a plane crack:

l-2v

= 8&l - v) PVL_ $Wy)r,, d,S ET

I-L +

877(1- v)

XPU / s_ ${(I - 2v)[r,p Au,,, -r+ Au~,~I ET

+3r,,r,pr, Au,,~I~~S + W,(y,) (51a)



iIt&;, e,) - t,<y,‘, -e,)l

1-2v =-- P”L_ -&,(y)r,, d,S

87r(l- v) cr

+ W,(Y,) (51b)

where

r = Ily -y,ll,

r,i = (Yi -Yail/r,

MY,> = * ( D(y, y,Jf(y) d,V*e, n

There is no coupling between (t,, Au,>, E t,,2J and (t3, AU,) if and only if the loading on the crack is symmetrical, i.e., 2 = 0.

The case of the plane crack allows us to go deeper into the question of the aforesaid indetermination of tf and t-. For simplicity, let us consider a circular crack free of body forces (IV = 0). The crack is symmetrical with respect to its plane y, = 0. The prescription of Au, can be real- ized by inserting a rigid wedge, axisymmetrical and d&y- metrical with respect to the plane y, = 0, between the crack faces. It is assumed that there is no friction between the wedge and the crack faces, then Zt, = 0, (Y E {1,2} and equation (51b) can be written in the abbreviated form (1/2Xt:- t;) = function of AU,.

The knowledge of Au, implies that the difference tl- t; is known, but not tl and t; separately. This is confirmed when the wedge is turned upside down: the same quantity Au, is prescribed, the difference t:- t; remains the same, as can be easily verified, but t: and t; are modified individually. This simple example shows that knowing Au in equation (50) is not enough to determine the loads t + and t-.

5.2 Finite body Theorem

Consider a finite body Q containing an embedded or surface crack S,,. Assuming that:

(i) S,E C1sol, S, E C1,a’, 0 < LY, (Y’ I 1 (ii) Au E C1*p(Sc;), u E C’(S,), 0 < j3 I 1

we have the following system of BIE and BIDE:

VY,=Sn,

/,_ [ Hy,,,y)Ct(y) + TT(yo,y,n,) Au(y)] d,S cr

+* /{ SB

Hh,y)t(y,n,)

-TT(yo,y,n,Nu(y) -u(y,)l}d,S

+ * / E(y,, y)f(y) d,V= 0 (52) n

+[t(y;,ng) - t(y,+,n$)]

=pupy,y,My) CT

+~(dy,y,,y,n,)Au(y)}d,S.n,,

+ s,(Ny,yoMy,nY) /

-~(dy,y,,y,n,)u(y)}d,S.n,,

+ * / n ZXy, y,Jf(y) d,V.ng, (53)

where in the case of a surface crack, S, must be replaced by S, \ L throughout.

Proof Equation (53) is obtained in a similar way as equation

(50). To prove equation (521, let us transform the integral representation of the displacement (43) by means of the rigid body identity (10):

/ SC;

[ E(x, y)Ct(y) + T’(x, y,n,) Au(y)] d,S

+ /I SE E(x,yMy,n,)

- TT(x, y,n,>[u(y> - u(x>l} d,S

+ * / E(x,y)f(y)d,V=O (54)

n

Now, we proceed to the limit in equation (54) as R \ (S, u S,,) 3 X -+y, E s,. From hypothesis (ii) and relation (181, the integral over S, is weakly singular while that over SC; is regular. Thus, the limit is performed by replac- ing x by y, in equation (54). The theorem is proved.

The solution of the system (52) and (53) give (u, t) on S, and (Au,21 on S,,. The coupling between these equations accounts for the interaction between the outer boundary and the crack. In the case of a surface crack, the system provides no equations along the surface line, since y, GL L. However, the lacking equations are compensated for by expressing the displacement compatibility at the surface line. With notations introduce in equation (491, we can write at every point on L:

Au(y, E L) + u(y,~ L-) - u(y,+~ L+) = 0

Also, it should be noted that the principal value integral in the BIDE (50) or (53) can be numerically transformed in the manner indicated by Balas et al. (Ref. 19, p. 166) into a regular one which in turn can be easily computed.



6. Conclusions

The specific feature general anisotropy of sisted of:

of the formulation has been the the medium. The results have con-

(1)

(2)

the limit theorems (251, (30), and (311, which describe the limit behavior of the fundamental solution when the load point x approaches a point y, belonging to a given surface S, closed or open. These theorems are generalization of the well-known ones in the isotropic case. It is noteworthy that they have been obtained without even knowing the closed form expression of the fundamental solution. A minimum amount of basic properties, relations (11) to (20), has been enough to complete the proofs.

the BIDES for crack problems, equation (50) for the infinite body with an embedded crack, and the system of coupled equations, (52) and (531, for a finite body with an embedded or surface crack, which clearly shows the interaction between the outer boundary and the crack.

From the numerical points of view, the success of the boundary formulation in anisotropic problems depends on an efficient computation of the integral representations (11) and (12) of the fundamental solution. Wilson and Crusei7 have shown that the anisotropic solution can be numerically evaluated with essentially arbitrary accuracy. Further improvements in the numerical scheme should allow to more precisely calculate the fundamental solution as well as to significantly save the CPU time.

Nomenclature

C

em t nY

E D R S II

Y

SFI S CI

fourth-order elasticity tensor a unit vector stress vector normal vector in the y direction

the fundamental displacement tensor the third-order tensor of the fundamental stress bounded region boundary tangent plane passing through y, displacement vector identity tensor outer boundary crack surface

Appendix

Regulatization theorem

Let S be a surface (open or closed) with edge dS. If:

(i) S E C’v u, 0 < (Y < 1

(ii) u E C’(S) then Vx $! S, tlk, 1,

/ v?lk -(x,y,ny>u,(y> d,S

s dx,

where s,,/ is the tangent differential operator defined by equation (33). If S is a closed surface, the line integral along aS is zero.

Proofi Relations (7) and (17) give

/ %zk -_(x, y,n,h,(y) d,S

s ax,

I aDmnk =- sdy,(x>~M~)~m(~)dyS

Now, it is easy to verify the following formula of com- pound derivatives: for any differentiable functions cp and 9, _?~J(P. I/J> = qGnr3/+ (cIGfi,(p. This enables us to transform the kernel as:

D mnk,,(X, y)n,(y)u,(y)

==%@mnk%z) -Dmnk~nPm

+Dmnk,nntUm (i = d/dyi)

Hence, invoking the Stokes theorem (Ref. 20, p. 282), which is valid owing to hypotheses (i) and (ii):

‘fk,l> j%,,(D,,,u,) d,S = j %,rDmnkUm ‘JY, S JS

and using the equilibrium equation (9) with x #y E S, the theorem is proved.

References

1. Rizzo, F. R. An integral equation approach to boundary value problems of classical elastostatics, Q. Appl. Math. 1967, XXV(l), 83-95

2. Cruse, T. A. and Rizzo, F. J. A direct formulation and numerical solution of the general transient elastodynamic problems, J. Math. Anal. Appl. 1968, I, 244-259

3. Cruse, T. A. A direct formulation and numerical solution of the general transient elastodynamic problems. J. Math. Am!. Appl. 1968, II, 341-355

4. Cruse, T. A. Numerical solutions in three-dimensional elastostat- its. Int. J. Sol. Struct. 1969, 5, 1259-1274

5. Rizzo, F. I. and Shippy, D. J. An advanced boundary integral equation for three-dimensional thermoelasticity. Int. J. Num. Meth. Eng. 1977, 11, 1753-1768

6. Bui, H. D. Applications des potentiels Clastiques a l’etude des fissures planes de forme arbitraire en milieu tridimensionel. Comptes Rendus des AcadPmies des Sciences (Paris) 1975, 280(A), 1157-1160

7. Weaver, J. Three-dimensional crack analysis, Int. J. Sol. Struct. 1977, 13, 321-320



8.

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14.

Sladek, J and Sladek, V. Three-dimensional curved crack in an elastic body. Int. .I. Sol. Struct. 1983, 19, 425-436 Sladek, J. and Sladek, V. Dynamic stress intensity factors studied by boundary integro-differential equations. Int. J. Num. Metk. Eng. 1986, 23, 919-928 Le van, A. and Royer, J. Integral equations for three-dimensional problems. ht. J. Fract. 1986, 31, 125-142 Bui, H. D., Loret, B., and Bonnet, M. Regularisation des tqua- tions integrales de l’elastostatique et de l’blastodynamique. Comptes Rendus des Acadhnies des Sciences (Paris) 1985, 3OOH1, 141, 633-636 Bui, H. D. and Bonnet, M. Regular BIE for three-dimensional cracks in elastodynamics. Proceedings of the IUTAM Symposium, San Antonio, TX, April 1987 Polch, E. Z., Cruse, T. A., Huang, C.-J. Buried crack analysis with an advanced traction BIE algorithm. Proceedings of the Winter Annual Meeting for the ASME, Miami Beach, FL, Novem- ber 1985 Le van, A. and Royer, J. Theoretical basis of regularized integral equations for elastostatic crack problems. ht. J. Fract. 1990, 44, 155-166

15. Tanaka, M., Sladek, V., and Sladek, J. Regularization techniques applied to boundary integral equation methods. Appl. Meek. Rev. 1994,47(101, 457-499

16. Vogel, S. M. and Rizzo, F. J. An integral formulation of three-dimensional anisotropic elastostatic boundary value problems. J. Elasticity, 1973, 3(3), 203-216

17. Wilson, R. B. and Cruse, T. A. Efficient implementation of anisotropic three-dimensional boundary integral equation stress analysis. Int. J. Num. Metk. Eng. 1978, 12, 1383-1397

18. Sladek, J. and Sladek, V. Three-dimensional crack analysis for an anisotropic body. Appl. Math. Modelling 1982, 6, 374-380

19. Balas, J., Sladek, J., and Sladek, V. Stress Analysis by Boundary Equation Methods, Studied in Applied Mechanics 23, Elsevier, New York, 1989

20. Kupradze, V. D., Gegelia, T. G., Basheleishvili, M. O., and Burchuladze, T. V. Three-Dimensional Problems of the Matkemati- cal Theory of Elasticity and Tkermoelasticity, Series in Applied Mathematics and Mechanics, North Holland, Amsterdam, 1979

21. Willis, J. R. The elastic interaction energy of dislocation loops in anisotropic media. Q. J. Meek. Appl. Math. 1965, 17, 157-174


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Boundary formulation for three-dimensional anisotropic crack problems