The Equilibrium Field Near the Tip of a Crack for Finite Plane Strain of Incompressible Elastic Materials Citation Stephenson, Rodney Alan (1981) The Equilibrium Field Near the Tip of a Crack for Finite Plane Strain of Incompressible Elastic Materials. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/y00g-0984. https://resolver.caltech.edu/CaltechETD:etd-10052006-094802 Abstract This investigation is concerned with the deformations and stresses in a slab of all-around infinite extent containing a traction-free plane crack, under conditions of plane strain. The analysis is carried out within the framework of the fully nonlinear equilibrium theory of homogeneous and isotropic incompressible elastic solids. For a fairly wide class of such materials and general loading conditions at infinity, asymptotic estimates appropriate to the various field quantities near the crack-tips are deduced. For a subclass of the materials considered, these results -- in contrast to the analogous predictions of the linearized theory -- lead to the conclusion that the crack opens up in a neighborhood of its tips even if the applied loading is antisymmetric about the plane of the crack, (e.g., Mode II loading). It is shown further that the non-linear global crack problem corresponding to such a loading in general cannot admit an antisymmetric solution. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Applied Mechanics) Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Applied Mechanics Thesis Availability: Public (worldwide access) Research Advisor(s): Sternberg, Eli Thesis Committee: Unknown, Unknown Defense Date: 1 July 1980 Record Number: CaltechETD:etd-10052006-094802 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-10052006-094802 DOI: 10.7907/y00g-0984 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 3931 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 13 Oct 2006 Last Modified: 26 Feb 2025 20:05 Thesis Files Preview PDF (Stephenson_ra_1981.pdf) - Final Version See Usage Policy. 1MB Repository Staff Only: item control page

The equilibrium field near the tip of a crack for finite plane strain of incompressible elastic materials Thesis by Rodney Alan Stephenson In Partial Fulfillment of the Requirements for the degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1981 (Submitted July, 1980) ii Acknowledgments I wish to express my sincere appreciation for the encouragement and guidance provided by my advisor Dr.Eli Sternberg. I am also pleased to acknowledge the interest shown in my work by other members of the faculty of the California Institute of Technology, and in particular thank Dr.J.K.Knowles for many useful discussions. Ms.Susan Berkley deserves special thanks for the typing of the manuscript. T am grateful to Caltech for the award of tuition scholarships and research and teaching assistantships and finally acknowledge the financial aid provided by the Office of Naval Research. iii Abstract This investigation is concerned with the deformations and stresses in a slab of all-around infinite extent containing a traction-free plane crack, under conditions of plane strain. The analysis is carried out within the framework of the fully nonlinear equilibrium theory of homo- geneous and isotropic incompressible elastic solids. For a fairly wide class of such materials and general loading conditions at infinity, asymptotic estimates appropriate to the various field quantities near the crack-tips are deduced. For a subclass of the materials considered, these results — in contrast to the analogous predictions of the linearized theory —~ lead to the conclusion that the crack opens up in a neighborhood of its tips even if the applied loading is antisymmetric about the plane of the crack, (e.g., Mode II loading). It is shown further that the non- linear global crack problem corresponding to such a loading in general cannot admit an antisymmetric solution. iv Table of Contents Acknowledgments Abstract Table of Contents Introduction i. Preliminaries from the finite theory of plane strain for incompressible elastic solids. 2. Formulation of the global crack problem. Global consider- ations. 3. Asymptotic analysis of the elastostatic field near the tips of the crack. 4, The transition case n=3/2. Higher-order considerations for the Mooney~Rivlin material. 5. Discussion of the deformation and stresses near the crack- tips. Appendix: Representation of functions biharmonic in a neighbor- hood of infinity. References ii iii iv 17 42 Sl 63 Introduction Early investigations of crack problems beyond the scope of the clas- sical theory of elasticity typically retain the kinematic assumption of infinitesimal deformations while relinquishing the linear stress-strain law in favor of nonlinear constitutive relations of one kind or another.* Apparently the first investigation of a crack problem within the fully nonlinear equilibrium theory of elastic materials is due to Wong and Shield [2]. In [2] an approximative global solution is obtained to the problem of a finite crack in an all-around infinite thin incompressible sheet of a neo-Hookean material, subjected to biaxial tension at infinity, on the assumption that the deformations are large throughout the sheet. Knowles and Sternberg [3], [4] deal with the problem of an infinite slab of a compressible elastic material containing a plane traction-free crack, within the equilibrium theory of finite plane strain; they restrict their attention to a loading of uniaxial tension at right angles to the plane of the crack (Mode 1 loading) and analyze the structure of the en- suing elastostatic field near the crack-tips. The Mode III crack prob- lem, in which the loading at infinity is one of longitudinal shear parallel to the edges of the crack, is explored asymptotically in the finite theory for a class of incompressible elastic materials by Knowles [5]. Additional work on the nonlinear Mode III problem may be found in [6], [7], [8]. The investigations referred to above, as well as some related studies in finite elastostatics, are reviewed in several survey papers [9], [10], [11], [12]. 1560, for example, a comprehensive survey by Rice [1]. -2- The present study, which is also set within finite elastostatics, retains the body geometry underlying [3], [4], as well as the assumption of plane strain, but supposes the material to be incompressible. The particular constitutive assumptions introduced here allow for both harden- ing and softening behavior in simple shear, but preclude a loss of ellip- ticity of the governing displacement equations of equilibrium. In ad- dition, more general loading conditions at infinity are considered, so as to encompass in particular the Mode II loading of simple shear parallel to the crack faces. Section 1 contains a review of some prerequisites from the finite theory of plane strain for homogeneous, isotropic incompressible elastic solids and introduces a particular class of materials underlying the sub- sequent analysis. In Section 2 we formulate the global crack problem with which we are concerned and recall certain properties of the solution to the analogous crack problem in the linearized theory. We then show that, at least for the Mooney-Rivlin material, the global nonlinear Mode IT problem, in contrast to its counterpart in the linear theory, does not admit a solution antisymmetric about the plane of the crack. Section 3 and Section 4 are devoted to an asymptotic analysis of the elastostatic field near the crack-tips. Finally Section 5 contains a discussion of the main conclusions reached concerning the ensuing local deformation and stresses. Ty potential loss of ellipticity in the context of the Mode ITI prob- lem is provided for in [7], [8]. 1. Preliminaries from the finite theory of plane strain for incompres- sible elastic solids Throughout this paper we shall be concerned with plane finite elastostatics of homogeneous and isotropic incompressible materials, in the absence of body forces. Consider a body which in an undeformed configuration occupies a cylindrical region of space. Denote by I the open projection of this region onto a plane that is perpendicular to the generators of the lateral boundary. Introduce a two-dimensional rectangular cartesian coordinate system in this plane and thus let (x, Xo) be coordinates associated with the position vector! x. A plane deformation of the body parallel to the (x)> x,)-plane is characterized by the mapping y=y¥(x)=xtu(x) on I. (1.1) Thus y is the position vector of points in the deformation image * na I =y(I), while u is the displacement field. We suppose the function y to be twice continuously differentiable on il, and the mapping (1.1) to be invertible. Let F be the associated deformation-gradient field, so that F=Vy on Il, (1.2) ~ Since the body is incompressible, the deformation (1.1) must be locally L : : : : Here and in what follows, letters underlined by a tilde designate vectors and second-order tensors in two dimensions. volume~preserving and hence J=detF=1 on To. (1.3) Further, let [=trPF on Il. (1.4)4 With the aid of (1.3), the scalar deformation invariant I is readily found to obey I22 on TT , (1.5) and I=2 only in an undeformed configuration. Next, let t stand for the two-dimensional Cauchy (true) stress- tensor field, regarded as a function of position on 1”. For an equi- librium deformation, in the absence of body forces, divt=0, tet on I” . (1.6) If o designates the associated Piola (nominal) stress field on Il, one has in view of (1.3), “ -T . 2 o(x)=t(y(x))F (x) for all x in I. (1.7) In terms of the nominal stresses, the equilibrium equations (1.6) become T divo=0, oF =Fo on fl. (1.8) 1 : a The superscript T indicates transposition. Here ro denotes the inverse of EF. -5- For future reference we also recall the following result. Suppose T is a regular arc in Il, and r= y¥(T) its image in nr, Next let n and n* be the oriented unit normal vectors of T and [%*, respectively, and denote by s and t the Piola and Cauchy traction vectors defined by s=on on TF ,t=tn* on [* , (1.9) we Then, s=Q on T if and only if t=Q on I* . (1.10) Moreover, if the deformation and the nominal stress field are suitably regular in the closure of II, (1.10) continues to hold true for an arc T on the boundary of Il. The foregoing result thus enables one to im- pose boundary conditions at a traction-free edge without involving the unknown deformation image of such a part of the boundary. Suppose now the elastic solid under consideration possesses an elastic potential, and let W stand for the strain-energy density per unit undeformed volume, regarded as a function of position on JI. For plane deformations of a homogeneous and isotropic incompressible elastic solid, W depends on the material position vector exclusively through the deformation invariant I. Hence, Wx) =W@)) for all x in I, (1.11) te] where W(I) is the plane~strain elastic potential. The appropriate constitutive law, as far as the in-plane stress response is concerned, now becomes ~6- o=2W (I)F-pF oon I, (1.12) in which Ww is the derivative of W, while p is an arbitrary hydro- static pressure field arising from the constraint of incompressibility. In view of (1.12), the second of the equilibrium equations (1.8) is satisfied a priori. We now turn to certain restrictions to which the elastic potential W(1) is subject. First, we shall take for granted that ° cs] W(I)20 for I>2, W(2)=0 , (1.13) so that in particular the strain-energy density vanishes in the undeformed state. Next, according to the relevant Baker-Ericksen inequality, Ct W (I1)>0 for I>2 ., (1.14) This inequality admits a simple physical interpretation in the context of a plane homogeneous deformation corresponding to simple shear. In this case, the components of the deformation-gradient tensor are given by 1 k [F ]- , (1.15)7 aB 0 1 in which k is the amount of shear. It follows from (1.12), (1.7) that for such a deformation the true shearing-stress component Ti obeys Typ = t(k) = 2kW (24K), (1.16) soreek subscripts have the range (1,2) and summation over repeated sub- scripts is implied. -7~ and we henceforth refer to the graph of t(k) as the response curve for simple shear. The inequality (1.14) thus holds provided t(k) and k have the same sign, as demanded by obvious physical considerations. Finally, we recall from Abeyaratne [13] that the pertinent displace- ment equations of equilibrium — obtained by elimination of G between (1.8), (1.12) and subject to the constraint (1.3) — are elliptic at a solution u and at a material point x if and only if Wonydo, MO G-asiso , (1.17) w’ (1) where I= I(x) is the associated value of the deformation invariant de- fined in (1.4). The first of these inequalities is assured by (1.14), while the second is readily seen to be equivalent to the requirement that the slope of the response curve for simple shear be non-negative at k=VI-2., At this point we introduce a particular class of incompressible elastic materials, hereafter referred to as power-law materials, which will play an important part in the analysis to follow. This class of materials is characterized by n-1 ° _ wT) =at@+ Br ts4 oc as t-0 , (1.18) in which A, B and n are constants with A and n positive. We shall assume further that the asymptotic identities obtained by two suc- cessive formal differentiations of (1.18) are also valid. It should be emphasized that (1.18) merely stipulates the asymptotic behavior of the elastic potential at large values of I, and does not otherwise restrict -8- its specific form. A special case of an incompressible material con- forming to (1.18) is supplied by the Mooney~Rivlin material, with the complete plane-strain elastic potential wi) =" (1-2). (1.19) This particular potential evidently satisfies (1.18) with n=1, A=u/2, BF~w. According to (1.16), the response of a power~law material in simple shear obeys 2n-1 | ( 2n-1 t(k) = 2nAk o(k ) as ko . (1.20) The corresponding response curve, for a range of values of the parameter n, is indicated in Figure 1. For n>1 the material hardens under simple shear, for n<1l it softens, while for n=1 the response function t(k) is asymptotically linear. For this reason we shall from here on refer to n as the "hardening parameter." When n=1/2, t(k)7A as k7o; when n<1/2, t(k) is bound to reach a maximum beyond which it monoton- ically declines to zero. Such a "collapse under shear" is, as previously remarked, associated with a possible loss of ellipticity of the governing field equations. In contrast, if n>1/2, the ellipticity conditions are necessarily met at all sufficiently large values of I. 2. Formulation of the global crack problem. Global considerations. We proceed here to the global formulation of the problem with which we are concerned. Let &£ be the straight-line segment i= {x]x, = 0, “bsx, <b}, (2.1) and suppose I] to be the region exterior to &, as indicated in Figure 2. Accordingly, the undeformed body is an all-around infinite slab containing a plane crack of length 2b. We shall assume that the faces of the crack are traction-free and that a known homogeneous state of plane deformation prevails at infinity. The problem to be considered may be stated as follows. Given an elastic potential W(T), we seek a suitably regular plane deformation y on Il, subject to (1.3), as well as a nominal stress field g and a pres- sure field p such that (1.12) holds, while o satisfies the equilibrium equations (1.8); in addition y is to be consistent with the pre-assigned deformation at infinity, whereas 6 should conform to the boundary condi- tion for the crack faces. The latter, in view of (1.10), are satisfied provided Tyo (%, 0+) = 0, Oyo (X41 29-) = 0 (-b<x,<b) . (2.2) 1 Finally, we stipulate that 3 (x) = Fx+0(1) as xf + x57 00 ; det F =1 > (2.3) -~10~ co where FE is a constant tensor. The function y should be twice continuously differentiable on Il, as required before, and continuous up to £, while F is to be con- tinuous up to the interior of ££. Further, we assume that W is twice continuously differentiable on [2,00) and throughout its interval of definition meets the ellipticity conditions (1.17). Thus, taking for granted the existence of a solution to the global problem at hand, el- lipticity is bound to prevail on Tl, and 4 is assured to have continuous partial derivatives of all orders on [Tl. Let ®& be the class of all ty, 9, p} that satisfy the field equa- tions referred to above, as well as the boundary conditions (2.2). Clearly, a solution to the complete problem formulated above belongs to &; con- versely anymember of & that meets condition (2.3) is a solution to this problem. For future purposes it is essential to note that if iy, 0, p} is in &, the same is true of iQy, Qo, p} for every proper orthogonal tensor Q. This claim is readily confirmed because of the objectivity of the constitutive law (1.12) and by virtue of the form of the boundary conditions (2.2). At this stage we recall certain properties of the familiar solutions to the analogous crack problem within the linearized equilibrium theory of plane strain, which encompass both compressible and incompressible elas- tic solids.* In view of the principle of superposition it is sufficient in this connection to consider separately the following two loading modes at infinity. See for example Rice [1]. -ll~ . ° _ ° _ 2 2, 1 Mode I: 05,700), Fon = 9 +01) as x +x, co, (2.4) Mode II: 6,,=6,,=0(1), &,,=0+0(1) or (2.5) ode 2 044 = 5% =0 » O97 re) as x, +x,700 . Thus Mode I corresponds to uniform uniaxial tension perpendicular to the crack faces, while Mode II corresponds to simple shear parallel to the crack faces. For the Mode I loading the resulting displacement field is symmetric about the X-axis, so that uy Cx, »X) =u, (x) 5- Xo), Uy (XX = Uy OX) »-X,) . (2.6) On the other hand, the displacement field associated with Mode II is anti- symmetric, whence 5 ° fo) U, (xy »X_) = -u, (x, Xy), Uy CX] »X_) = Uy (XK) . (2.7) Further, the crack faces in Mode II undergo an in-plane gliding deforma- tion and do not separate. We now cite the asymptotic behavior of various field quantities near the crack-tips. To this end let (r,8) be the local polar coordi- nates (see Figure 2) defined by x,-b=rcos6@, x,=rsin6d, (O0<r<0o0o, -7<6<m7) . (2.8) 1 2 Accordingly, the right-hand crack-tip corresponds to r=0, while the crack faces are located at 9=-n and 6=7, respectively. 1 . . Ug and Sy are the components of the displacement and stress fields in the linearized theory. -12- For the Mode I loading, as r-0: K eo I 1/2, 87, 28 1/2 uw, 7 T cosy (1 2v+ sin x) +o(r ) , (2.7) o _ tf 1/2, 8 | 28] 1/2, , any r sins 2(1-v)- cos 3 -+otr ) 3 6..=K pile cos (1~ sin” sin 2”) to(r t/2) > ll I 2 2 2 ° 64> K, pile cos 2 cos 22 si 8 toe l/4y ; \ (2.10) o _, 1/2 6 8. 36. -1/2 Fo 9* Kr cos 5 (1+ sinz sin a) + o(r ) a, Here wu is the shear modulus for infinitesimal deformations, v is Poisson's ratio, while K is the Mode I stress~intensity factor, given T by K = 0(b/2)1/? (2.11) For the Mode II loading, as r-70: o _ Str a/2, 8 26 1/2 u =r sin’ [2(1-v) +08 2] sor ) , 1 u 2 2 (2.12) Biz 4/2 : ° > 8] in? 2] 1 9 7 r cos 5 (1 - 2v) + sin 9 +o(r ~13- ° -1/2_, 6 9 36 ~1/2 = 2 (~2-cos 2 cos > C14 Kit sins ( 2 cos 5 COS“ Yto(r ) , 0 | ~1/26.,_ , 68, 38 1/2 S10= 8d x cos 5 (1 - sin 7 Sinz J+to(r d) , >» (2.13) o _, 1/2. 6, 8 36 -1/2 Cn o* Rey r cos sin cos~- +o(r yy J with K 2 o(b/2)/? (2.14) T The displacements appropriate to the special case of an incompressible linearly-elastic body are obtained by setting v=1/2 in the formulas cited above. Although superposition no longer holds in the nonlinear theory, it is still useful to consider separately the kinematic counterparts of the Mode I and Mode II loadings. Corresponding to the symmetric loading case we now require the deformation at infinity to be one of uniaxial in which % and yt are the principal stretch ratios at infinity. The analogue of the antisymmetric loading case is a state of simple shear governed by Fal = (2.16) -14- and k is the amount of shear at infinity. Since the governing field equations in the finite theory do not remain invariant if U(X Xo)» Uy (x, Xo) are replaced by TU, (54 -X5) uy (x, »- xy), one is led to wonder whether or not the nonlinear Mode II problem admits a solution that is antisymmetric about the plane of the crack. We shall now show that, at least for the Mooney-Rivlin material, a solution with this parity cannot possibly exist. To this end suppose now that W is given by (1.19). Eliminating ey between (1.12) and (1.8), and invoking (1.2), (1.3), (1.4), (1.19), we find that 1 ~~ ~ a Tea Yaa, 88 on IT. (2.17) Equation (1.3), in view of (1.1), becomes ltu +u 1 on FT. (2.18) 1,17 Y2,27 %1,1%2,27 M1, 242,17 Now assume wu has the parity of a in (2.7). Then (2.18) implies Loup p78, ot 4, 4B 97 By 9% 17 1 on fl . (2.19) Therefore Wy ptt, 270 on Il , (2.20) =Q on I. (2.21) 4741°2,27 “1,2%2,1 L : ‘ so 4s : Here and in the sequel subscripts preceded by a comma indicate partial differentiation with respect to the appropriate cartesian coordinate. -15- Next, p=p+p” on IT, provided p’= Spx, 5x5) - PGs x], (2.22) p"= Zp (x, ,x,) + p(x, ,-x,)] 2 1°" 2 1’? “2 By virtue of the assumed parity of u, (2.17) and (1.1) give iy =u on JI (2.23) _ Pa 0,88 Pk, . From (2.23), (2.20) it follows that p’ is harmonic and uy is biharmonic on ITI: Vp=0, Vu =0 on Il. (2.24) For the Mode II problem one draws from (2.16), (2.3), (1.1) that u, =kx, +01), u,=0(1) as pa val +x, 7 @ . (2.25) We shall now prove that (2.20), (2.21), (2.24), (2.25) imply u, =kx,, u,=e on fi, c=constant , (2.26) so that the entire deformation field is one of homogeneous simple shear, which contradicts the boundary condition (2.2) for traction-free faces of the crack. With a view to establishing this claim, we observe first that u, — kx and u 1 2 2 hood of infinity defined by = {x|p >b}, 2b being the crack length; are both biharmonic on II, and hence in the neighbor- ~16- further both of these functions are bounded as p~oo. As shown in the Appendix, uy then admits an expansion of the form oO . =-Lé (a) u, Ox, +X) k6,y%ot 2} la, y= (a) (a) cos jy+b, sin jetc, cos(j - 2) (a) +4, sin(j-2)g]o ? on % , (2.27) where g is the polar angle defined by x,=pcosy, x,=psiny. The 1 2 series in (2.27) is absolutely and uniformly convergent on every closed subset of 2%. Substitution from (2.27) into (2.20), (2.21), in view of the parity of use confirms that these equations cannot hold unless all 2 : : : : : ‘ ) in the above series vanish. Continuation coefficients apart from a of the biharmonic functions us from 7 onto TI now yields the desired conclusion (2.26). In view of the above result it is a plausible conjecture that the Mode II global crack problem under consideration fails to admit an anti- symmetric solution for every legitimate choice of the plane-strain elastic potential W. -17- 3. Asymptotic analysis of the elastostatic field near the tips of the crack. In this section we aim at the local structure near one of the crack-tips of the solution to the global problem stated in Section 2. For this purpose it is sufficient to consider the right-hand crack-tip, situated at x, =b, Xo potential of the power-law type (1.18). Further, if (r,8) are the local =O. We suppose that the material has an elastic polar coordinates introduced earlier (see Figure 2), we assume that the deformation field of the solution admits an asymptotic representation of the form m “a 1o4 1 ¥ 684 2%) WE U, (8) as r70O (no sum on a) . (3.1) Here m, and m, are originally unknown constants obeying O<m=minim,,m,}<1 > (3.2) whereas UL are as yet undetermined functions that fail to vanish iden- tically on [-1,7]. In addition we take for granted the validity of the asymptotic equalities obtained by two successive formal differentiations of (3.1). Note that (3.2) implies that the deformation is continuous at the crack-tip, and that at least one of the deformation gradient components F becomes unbounded there. Observe also that according to (3.1), (3.2) eas) the deformation image of the crack-tip has for convenience been placed tthe asymptotic equality symbol "~" is used in its standard connotation. Thus, f(r)~g(r) abbreviates f(r)=g(r)+o(g(r)) as r-0. -~18- at the origin of the cartesian coordinates, which can always be ar- ranged by means of a suitable translation. Finally, we suppose that the pressure field associated with the global solution satisfies LR P(x) .X,) ~4 P(6) as r-7o , (3.3) 2 being another constant exponent, and assume that this asymptotic identity may be formally differentiated at least once. As pointed out in Section 2, the unknown global solution LY. 5P} necessarily belongs to the class & introduced there, regardless of the particular prescription (2.3) at infinity. It follows that the determina- tion of Vn and p in accordance with (3.1) and (3.3) for an arbitrary member of & is bound to encompass the asymptotic behavior of the solu- tion to the complete crack problem at hand. We now show that for the present purpose (3.1) may — without loss of generality — be replaced by n ; mh ¥, XyX_) wr (8) as r7oO . (3.4) With this aim in mind we first note on the basis of the discussion in Section 2 that whenever {y,o,p} is in &, the same is true of L¥*,0%,p*} , where ye=Q9, of =Qo, p¥=p , (3.5) and Q is an arbitrary proper orthogonal tensor. The components of every such tensor in the underlying coordinate frame admit the representa- tion -19- lose _ cosy -sing (3.6) sing cos © for some ¢ in the interval [0,27). From (3.6), (3.5), (3.1) one finds mh m nan m m year *y, (cos p= x *u, (8) sin pt o( 1) +o(r ?) 5 (3.7) m, m, m) m, ys=r U,()singtr Uy (8)cos p+ o(r ) + o(s ) 5 whence there are functions ux such that F* (x, 5X5) = r'U* (8) +o(r"). (3.8) Further, since UN is not permitted to vanish identically, it is always possible to choose so as to assure that Ux fails to vanish identi- cally. Now, dropping the asterisk in (3.8) we arrive at (3.4). Note that the transformation (3.5) represents a rigid rotation of the deformed body about the origin, determined by the rotation tensor Q; further, the same rotation is applied to the nominal stress field g as- sociated with the original deformation. Consequently the asymptotic analy- sis to follow can determine the local behavior of the global solution at most to within an unknown rigid rotation, which would ultimately depend on the particular prescription at infinity. In this connection it should be emphasized, however, that most of the desired local field quantities are in fact invariant under such a rotation. For the special case of the Mode I loading the local solution is bound to be symmetric about the x, ~axis and this additional restriction supplies essential information -20- relevant to the determination of the rotation tensor Q. As is clear from the discussion at the end of Section 2, analogous parity restric- tions are not availabie for the Mode II loading. We now seek to determine the smallest exponent m in (0,1) and the functions UO appearing in (3.4) consistent with the governing field equations and boundary conditions. This objective is most easily reached by recourse to the polar coordinates (r,6). Writing y, (e598) in place of ¥, (658), x, (r,8)) one finds that the incompressibility condition (1.3} becomes sal 8y4 8Y5 - 8¥5 8Yy 1 (3.9) r\ or 38 ar 96 . . Further (1.3), (1.8), (1.12) lead to S SP = of Cat ol, i ot or 2W (T)H + 2M (1) rr or 2 r6 36 ’ _ _ > (3.10) op = a] vf or i ol 397 2W CI)H, + 2Wo (1) Je, ax 2 Beg ae] ° ZS and (1.4) furnishes = i T=8., 779 896 ° (3.11) with ~ ay oy He ~ or v You? Hy ~ 936 V You? Ss (3.12) Srr ~ or or ° Sre dor 90 ° See 36 2 -21- The boundary conditions (2.2) take the form °; ov °F 5 ) 2W (1) ra + rp 5 =O, > (3.13) oy ay oy 2 1 ——— —_——_ = = + 2W (T) 50 rp Dn O on 6=H7 / With the aid of (3.12) and (3.9), equations (3.13) are readily found to be equivalent to p= 2it’ (1)/e_. = 20 (Dg, /x > 8.5 = 0 for 6=in . (3.14) We thus need to determine m and UN in (3.4) consistent with (3.9), (3.10) and (3.14). Equation (3.11), together with (3.4), gives pear ard) (Uf (e) +02 (e) +motuy(e) +u5(@)13 (3.15) where the dot denotes differentiation with respect to 6. We shall hence- forth take for granted that uD and Uy do not have a common multiple zero on [-n,7] so that the coefficient of 2 ml) in (3.15) does not vanish on [-n,7]. Next (3.4), (3.9) lead to se or?) Ty, (yb, (8) - u,@E, CT +0(e MY )=1 2 (3.16) 2(m-1) Dividing this identity by fr > proceeding to the limit as r-0, and bearing in mind that m<1, we obtain U,U,- 0,0, = 0 on f[-t,7] . (3.17) -22— Thus, U=avU on ([-n,7] , a #0 , (3.18) a Oo a in which ay and ay are constants and U an unknown function. Equations (3.18), (3.15), (3.12), (3.10), (1.18) eventually lead to ~\ SP ~ 2naPe? @DPrdiy (ez (8) : > (3.19) ae a. fm 7 Yn? 3 ~ daar dy? fm */T(6)2 (8) ; provided n-2- 21, 2 _° Z=G {G(U+ m°U) + (n-1) [GU + 2m(m-1)GU]} on [-n,n] , (3.20) c=t?+m°v" on f[-r,7], a= vay + as On the other hand, the boundary conditions (3.14) yield UU=0 at 6=in , (3.21) p~2Aana ne? Om Dagn-1i2 at 6=in , (3.22) We now show on the basis of (3.22), (3.21), (3.19) that Z=Q on ([-1,7], U(-1) = U(r) = 0 . (3.23) For this purpose we integrate the first of (3.19) to obtain mw IAaroy? mr mU(6)2Z(6) ; (3.24) P 2(m-1)n ~23- According to (3.21) either U(m) or U(n) must vanish. By assumption U has no multiple zeros on [-1,7] sothat U(r) and U(n) cannot vanish together. Suppose U(m)=0, U(r) #0. Then (3.24) gives 2(m-1) p(r,7) =o(r ) as r-0 , (3.25) contradicting (3.22). Thus U(r) #0, U(n) =0 and similarly U(-1r) #0, U(-1) =Q. On comparing (3.24) and (3.22) we infer that Z(m) =Z(-1) =O. (3.26) Next, we differentiate (3.24) with respect to © and use the second of (3.19) to confirm vis [14 2Gcmalr§ 0 on f[-n,7] . (3.27) Equations (3.27) and (3.26) necessitate Z=Q on [-n,7] , (3.28) which proves (3.23). Thus G(ii+ m2) + (n-1) [GU + 2m(m-1) GU] = 0, U(-m) =U) =0. (3.29) Equations (3.29) constitute a nonlinear eigenvalue problem that arose also in connection with asymptotic analyses of other crack prob- lems in finite elasticity [3], [4], [5] and was solved in [3]. The unique eigenvalue in the range (0,1) is given by m=1-1/2n, 1/2<n<o , ,(3.30) ~24- while the associated eigenfunction U is supplied by u(@) = ein(@/2)|- Tr at@n) [w(8,n) +k cos 6] » (3.31) where 2.2 1/2 w(6,n) = [l-«sin’é] > k=(n-1)/n . (3.32) The representation of the local deformation field furnished by (3.4), (3.18), (3.31) yields only the weak estimates g=0(r2 hy | p=o(r) as r70 , (3.33) and is therefore inadequate. Furthermore, the Jacobian determinant of the dominant terms in (3.4) is found to vanish, which reflects the de- generate character of the asymptotic approximation established so far. With a view towards refining this approximation we now replace (3.4) by é ‘ eit! Th TAY ‘ y, (7.8) =ar u(s)4+r V (8) + o(t ) > m>m i, (3.34) with m’ another unknown exponent and Vv O8) unknown functions that are not permitted to vanish identically, while m and U _ are now given by (3.30), (3.31). We suppose that (3.34) may be formally differentiated at least twice. Again, by virtue of the discussion that led to the adop- tion of (3.4) in place of (3.1), no generality is lost in assuming equal exponents in the second term of (3.34). From (3.34), (3.9) one now draws -25- p= 2mm 2 gut my) + o(e™™ ~2) = 1 > (3.35) where ¥=a,Vo- a,Vy on [-nt,7] . (3.36) Consequently mtm’-2<0 , (3.37) and further mUY - m’ UY = 0 on [-1,7] if m<m/<2-m , (3.38) mUY~m/UY=1 on [-7, 7] if m=2-m . (3.39) Next, combining (3.34), (3.10), (3.11), bearing in mind (1.18), and re- calling that U satisfies (3.29), one confirms with the aid of arguments similar to those used in deriving (3.28) that Y=0 on [-n,7] if m<m’<2-m , (3.40) B os 2 . ' Y=- nAtUtm U) on [-t,7] if m=2-m , (3.41) where Y is an auxiliary function to be defined presently. In fact, Y=G(x+m‘ “x) + 2K(t + m0) + (n-1)xG +2(n-1) [m’ (u- 1) yG + m(m’+m- 2)UK+ UR], (3.42) in which X= a,V, taVo, K= x0 + mm’ yv on [-7,7] . (3.43) ~26- Finally, the boundary conditions (3.14) lead to y(tn) =0 if m<m’s<2-m ; (3.44) Y(4n)=0 if m<m/<2-m , (3.45) (rn) = 1/mU(r) 5 ¥(-1) =1/mU(-7) if m/=2-m . (3.46) In view of (3.42), equation (3.40) together with (3.44) constitute an eigenvalue problem for xy with m’ as eigenvalue parameter, On the other hand, (3.41) is an inhomogeneous differential equation for yx, to be solved subject to the boundary conditions (3.44). Consider next the characterization of through the differential equations (3.38), (3.39) together with (3.45), (3.46). Since these boundary conditions are readily seen to be implied by (3.38), (3.39), additional information is needed to determine the appropriate solutions. Such information is supplied by the assumption that the solution ¥Y to (3.38) or (3.39) possesses con- tinuous derivatives of all orders on [-1,7]. Equation (3.38) now poses an unconventional eigenvalue problem: one is to determine the values of m’ on the interval (m,2-m) for which the solution to (3.38) is infi- nitely many times continuously differentiable on [-n,7]. We turn at present to the determination of the relevant eigenvalues of (3.46), (3.44) or (3.38). If at least one of the foregoing problems has eigenvalues in the appropriate range, we choose for m’ the smallest such eigenvalue. The following three possibilities now arise. In case m’ so determined is an eigenvalue for (3.40), (3.44) but not for (3.38), xX must be the eigenfunction associated with m, while ¥=0 on [-n,7]. ~27- Next, if m is an eigenvalue for (3.38) but not for (3.40), (3.44), Y has to be the eigenfunction associated with m, while x=0Q on [-1,7]. Further, if m’ is an eigenvalue common to both problems, Y and yx are the eigenfunctions associated with this value of m’. Finally, suppose neither of the eigenvalue problems considered above has an eigenvalue in (m,2-m). Then evidently m’=2-m_ and the inhomogeneous equations (3.39) and (3.41), (3.44) are to be solved for y and ¥, which in turn determine Vy and V,: At this stage we investigate the solutions to (3.38) and (3.39). Consider first (3.38). In view of (3.31), we find mn’ /m m /m -U) on [-1,0) , (3.47) Y=a.U on (O,7], Y=a 3 4 § in which ay and a, are constants. If ¥ is to possess unlimited é : . ‘ smoothness on [-1,7], m /m must be a positive integer. Thus the smallest eigenvalue n’>m is given by m=2m=2-1/n , (3.48) and the associated eigenfunction by Y=b_U ; (3.49) where by # 0 is a constant. In addition m defined by (3.48) conforms to (3.37), provided n<3/2. (3.50) In the event that n23/2, (3.38) does not possess an eigenvalue in -28— (m,2-m). With a view to the determination of ¥ when n23/2 we con- sider (3.39) merely on the interval [0,1], which will presently be seen to be sufficient. Evidently wew +y" on [-n,7] , (3.51) if one sets nie v’ (0) =$1¥(8) - ¥(-9)], vw" (6) ==[¥(6) +¥(-6)], -1s6sn . (3.52) Moreover, ¥’ and ¥” are found to satisfy mUY’-m/ Uv’ =O on [O,7] , (3.53) ri mnUY” —-m’ ty” =1 on [O,7] . (3.54) Tf n23/2, the only solution to (3.53) having an infinitely smooth odd extension to [-1,7] is vY=0 on [0,7] . (3.55) Further, (3.54) gives n’ / 1 {* d wy’ (ayeu /™ Jb -= | ——S- 7, m’=2-m, (05657), (3.56) Pd 7 / Ike with by an arbitrary constant. We now change the variable of integra- tion in (3.56} from » to & by means of the one-to-one mapping 1 [1+ xsin“o - w(e,n)cos el 1/2 cosé = (OSE <3) , (3.57) nv2 wlp,n) +kcos 9 ~29- w and « being given by (3.32). This change of variable carries (3.56) into 2 1+ 2(n-l)cos’& Tm «| 5 (3.58) 3 ai-l/m re 2m 6 [cos —] “ _ m /m _ vy" (8) =U I», where 1 [1+ sin’ 6 -~w(6,n)cos g}t/2 nV2 w6,n) +Kcos 6 cos Ey = (0s657) . (3.59) The integral in (3.58) may be evaluated in terms of hypergeometric functions, provided n>3/2. Following an analysis, the details of which will be omit- ted here, one is led to the representation y" = bo U +H on ([0,r], m=2-m . (3.60) Here b =b ven /™ P(1/2-1/m) (3.62) 2 "1 2 T(l-i/m) ”’ , 2m P denoting the Gamma function, while 5/2 mn! 5 fw(6,n)+xKcos 6] H(e) =-= Th x |= F(1/2 = 1/m,1/233/2 - 1/mzcosé )-xsin& . (3.62) m 0 0 -30- in which F stands for the hypergeometric function.+ Now if ¥” is to possess continuous derivatives of all orders on [0,7] then b, must vanish.” From (3.51), (3.55), (3.60) one infers ¥(6) = H(6) (-71 S687) . (3.63) One draws from (3.62), (3.63) and familiar properties of the hypergeo- metric function that ¥(0) = ~2nt!? (2m) 7™ fan! $ | mm (3.64) _ non 1(3/2-1/m) Wm) = TG dim)” J the first of which is also a direct consequence of the differential equation (3.39). The second of (3.64) will be used later in discussing the shape of the deformed crack. In case n= 3/2, (3.39) fails to admit a solution of the requisite smoothness. We postpone consideration of this special case, which requires separate attention, until Section 4 and turn to the determination of x. Consider first (3.40) and set é “i X=x +x on [-1,7] , (3.65) where hole x! (8) = Flx(0) - x(-8) J, x (8) =5lx(@) + x(-8)] Grsesr) . (3.66) Ie e6 for example [15], p.556. *Note that H(8) is an even function of 6 with continuous derivatives of all orders on [-1,7] and recall that U(6) has a simple zero at 8=0. ~31- From (3.65), (3.66), (3.40), (3.42) we have, bearing in mind the parity of U, G(y/+m! 7x) + 2K’ (i+m-v) +(n- 1y’C+ 2(n-1) [m’ (m-1)x‘G +m(m’+m- 2)UK’ + UK’ ] =0 on [0,7] , (3.67) CX" +m’ ay + 2K” (i+ mv) +(n-1) x"G+2(-1)[m’ (m-1) x"G +m(m’+m-2) UK” + UK” | =O on [0O,7n] , (3.68) where Ki=y/Utmn’y/t, KY = y"U4mm’ y“0_. (3.69) Next, (3.66), (3.44) furnish x (m)=0, x/(0)=0 , (3.70) x" (m)=0 , x” (O)=0 (3.71) The differential equation (3.67) and the boundary conditions (3.70) are also encountered in [4]. As indicated there, the transformations 2 1/2 cos & = nv2 w(6,n)+kKcos 8 (0s057) , (3.72) W, C4) ={(wte,n)+xcos6] ’ (8) (O<T0S7) , (3.73) reduce (3.67), (3.76) to (2) +27, =0 (OSE<0/2), W,(0)=W,(n/2)=0 5 (3.74) ~32- where 2 = 4nm’ (nm’-n+1)/(2n-1) The transformed eigenvalue problem (3.74) has the solutions W(E) =e cosh, A=j (J 1,3,55-04) 1 in which cy is a constant. Under the transformation W, (E) = [w(8,n) + kcos 6]™ x” (8) (OSO6sm) , together with (3.72), equations (3.68), (3.71) become i,(E) +2°W,(E)=0 (OSE Sn/2), W,(0)=Wy(n/2)=0 and the solution to (3.78) is given by W,(€) =cycoshé, A= jj (7 =2,4,6,...) Thus the eigenvalues of the original problem (3.40), (3.44) the solutions for m’ of dnm! (nm! -n+1)/(2n-1) =4° (j=1,2,3,...) For j an odd positive integer, yy’ =0 on [-n,7] and y‘(@) nished by (3.76), (3.73), (3.72), while if j is even, x‘(8) = [-",7] and (6) is furnished by (3.79), (3.77), (3.72). On discarding negative roots m’ of (3.80) one arrives at é K m, . meat + 503 (j=1,2,3,...) (3. (3. (3. (3. (3. are (3. 75) 76) 77) 78) 79) 80) is fur- on (3. 81) ~33~ Further, m=1-1/2n=m if j=l , so that j22 since m’>m. The minimum admissible eigenvalue eccurs at j=2 and is given by [e+ H+ 8m/n |, e=(n-1)/n 3; f nh = Ne the corresponding eigenfunction is readily shown to be [w(8,n) +Kkcos a) on? (3.82) / Tm (3.83) / 2 | x(8) = c,[w(@,n) +kcos 8)” { ssi sawteangsg| (-7 S087). (3.84) The eigenvalue (3.83) conforms to (3.37) as long as n<7/2 (3.85) Finally suppose that n27/2, in which case (3.40), (3.44) do not possess an eigenvalue on (m,2-m). Equations (3.41), (3.44), (3.65), (3.66) in conjunction with the transformations (3.72), (3.73), (3.77) lead to -- 2 Beyn . Wy (E) + 9°, (E) =- =F cose, W,(0)=W(n/2)=0 , mA oe 2. _ o _ ° _ WCE) +2 Wy) =0, W, (0) = W, (1/2) =O , where 7 = 3(2n¢1)/(2n-1), k= (n-1)/n (3.86) (3.87) ~34- Thus, ~Benvn W, CE) = 1 4am? (14) cos—& if n27/2 , (3.88) W, (é) = ¢,cos 2& if n=7/2 (c, = constant) > (3.89) W, CE) = 0 if n>7/2 Equations (3.88), (3.89), together with (3.77), (3.73), (3.72), in turn furnish é x(6) = - —Bev2n__ [w(6,n)+kKkcos e]™ -t Jie sin’ 6 -w(6,n)cos § 4Am™” (1+n) t _ 2 +c, [w(6,n) + kcos 6}™ it+ksin fw (Sn) cose -1 (-1 567), [w(6,n)+Kcos 6] n for n27/2, with c= 0 when n>7/2. At this point we summarize the various possibilities that arise in the determination of the second term in (3.34). First, if n27/2, there are no eigenvalues of (3.38) or (3.40), (3.44) on the interval (m,2-m). Thus m’=2-m in this instance and y, ¥ are given by (3.90), (3.63). On solving (3.36) and the first of (3.43) for Ve one is led to 2 2 v= (a,x-ay¥)/a ; Vo= (a,¥+ayx)/a on [-n,7] . (3.91) ~35- Next, suppose 3/2<n<7/2. In this case only the y-problem admits an eigenvalue on (m,2-m), which is given by (3.83), whereas yy is sup- plied by (3.84). vo is now found by setting ¥=0 in (3.91), which gives , 2 2 yr ayX/a s Vo= ayx/a on f-n,7] . (3.92) Finally, for the range 1/2<n<3/2, in which case both the y-problem and the Y-problem admit eigenvalues on the appropriate range, we seek the smallest such eigenvalue. This criterion, along with (3.48), (3.83), ‘leads to ~ n= 5 [et ve? + 8n/n | if i<n<3/2 , m=1 if n=1 , > (3.93) m=2-1/n if 1/2<n<1l S Further, ¥=0Q on [-r,7] for 1<n<3/2, y=0 on [-1,7] for 1/2<n<1 . (3.94) For 1/2<n<1, Vo is supplied by Vi=-a v/a> Vi=a var on [-7,7] (3.95) i 2 > "2 71 , , where ¥ is given by (3.49). When n=i1, both problems have the common eigenvalue m=1, In the following section we shall explore in greater detail the Mooney-Rivlin material, which is a power-law material with n=l. ~36- It is convenient, for reasons that will be made clear presently, to apply an additional rigid rotation to the deformation field charac- terized by (3.34). In particular, consider the rotation tensor whose component matrix in the underlying frame is given by a,/a ~a,/a [Pag] = ; ) a=va+a- . (3.96) 1 2 ;/4 a/a Calculating y* in the first of (3.5) from (3.34), (3.96), bearing in mind (3.36), (3.43) and dropping the asterisk from the resulting estimates for the components yx » one arrives at 1m’ y,(r,8) =- Sr" ¥(8) +0(r™)? (3.97) yo (x48) = ar"U(6) +ar™ x(6) to(r™) The sienificance of (3.97) is as follows. Suppose, in particular, that n>7/2 and consider the deformation-image of the extended crack axis 6=0. From (3.97), (3.90), (3.64), (3.31) fellows ont! ? (2m) 7™ é é é cr to(x™), ya(r,0)=o0(r") , (3.98) ma 2 y, (F590) = and thus the ray @=0 after deformation is tangent to the x, axis at the origin. In case 1<n<7/2 (n#3/2), ¥=0 on [-n,n] and the first of é (3.97) yields merely the weak estimate yz o(r™ ). There is, however, no difficulty in establishing a non-degenerate estimate for Yq" Indeed, -37- let a “ty 1 y,(r,8)=- 52" X(6)+o0(r" ), m’>m , (3.99) feb) ‘and suppose that Yo is given by the second of (3.97). Entering (3.9) with this Ansatz, one has mU-m’xU=0 if m<m’<2-m : (3.100) hd a hd . af mxU-m XU=1 if m =2-m These equations are simply (3.38), (3.39) with m’, ¥ replaced by m”, X, and hence nm” =2-2, X=b,U° on [-1,7] for 1<n<3/2 , (3.101) mn =14+2, X=H on [-1,7] for n>3/2 Thus for 1/2<n< 3/2, . 2 2 2 y,(r,8) =bor [U(8)] For), (3.102) _ Ms 1_m’ m y(r,6) = ar (8) +=r x(6)+o0(r) , while for 3/2<n<7/2, yz (r,0)=- Sr MH(8) tole ™) (3.103) yy (r,6) = ar"U(8) +2ym x(6) $0(2™) , m, x and H being given by (3.83), (3.84), (3.62). Incase n27/2, -38- y, (58) =+ 21° ™H(e) to(r2 a) : (3.104) Yo (rs 8) = ar U(6) tir" ~™ (8) to(r *) 3 with y now furnished by (3.90). U and m are supplied in all cases by (3.30), (3.31). As mentioned earlier, the special case n=3/2 will be ‘dealt with in the next section. We now turn to the determination of the local structure of the pres- sure field, which has been assumed to admit the representation (3.3). From (3.9), (3.10), (3.12) follows oy oy ay oy mctyyey 4 28" zat, 1 "1 at\_1(ap 2%2_ ap 3%2 2W (1) Voy, + 2W ols ar 2 36 | (x 56736 or} 7° (63-405) Suppose first 1/2<n<3/2, in which case Vy is given by (3.102). Sub- stituting from (3.102) and (3.3) into (3.105), one is led to arn n -l 1 mrt 2 2-2 A4nAb Oe oe ty = are (opt - mbu) tor 7) | (3.106) 0 where G is given by (3.20). By virtue of the boundary conditions (3.14), along with (3.102), p(x,tm) ~2nA [mat (n) }2 08-2), (2) /n ; (3.107) whereas from (3.3), p(r,tn) ~r’P(+n) (3.108) Since by #0 one has ~39- $sl-m=1/2n . (3.109) From (3.107), (3.108), (3.109) one now draws P(im)=0 . (3.110) Suppose 2<1/2n. Then (3.106) gives 2PU-mPU=0 on f-r,7], (3.111) which together with (3.110) yields P=0 on [-n,7] . (3.112) Therefore Q=1/2n , (3.113) opt - mBU = 4nAb ae? 0" 0 on [-7,7], P(t7)=0 . (3.114) The solution to (3.114) is found to be given by P(8) = 4b ha n/n? La (8,n) + Keos 011/27) - cose, : (3.115) with cas Bq and w(@,n) given by (3.59), (3.32). When n>3/2, substitution from (3.103) er (3.104) into (3.105) eventually yields Q=a-1 , (3.116) 2n-4 .n-2, mPU - 2PU = 2nAa on [-7,7] , (3.117) ~40- where S=Gli+ (2-m)"H] + (n-1)(GH4+2(2-m)(m-1)CH] (3.118) The boundary conditions (3.107), (3.108) give P(tn) = 2nAlamU(n) 12274 . (3.119) The boundary-value problem consisting of (3.117), (3.119) fails to admit a solution with continuous derivatives of all orders at 6=0, and con- sequently the Ansatz (3.3) cannot possibly be consistent with the field equations when n> 3/2. In summary, the behavior of the pressure near the crack-tip is governed by o(r,6) = 4 P(e) to0(re/ 2%) for 1/2<n<3/2 , (3.120) ~l+i1/n p(r,8) =o(r ) for n>3/2 , with P furnished by (3.15); the weak estimate for p appropriate to n>3/2 follows from (3.10) together with (3.103) or (3.104). A non- degenerate pressure estimate valid for n>3/2 would presumably neces- sitate a higher-order asymptotic analysis of the deformation field. We note that if n>3/2 the dominant terms in (3.103), (3.104) have a Jacobian determinant J~1 and hence constitute a locally one-to- one mapping. In contrast, J computed from the dominant terms in (3.102), 1-3/2n r which apply to 1/2<n<3/2, is merely of ). The discussion in the next section of the Mooney-Rivlin material (n=1) and of the special ~41- case n=3/2 will, however, lead to an improved approximation to the deformation at least when l1sn< 3/2. -42- 4, The transition case n=3/2. Higher-order analysis for the Mooney- Rivlin material. We first investigate the modifications necessary in the asymptotic structure of the deformation and pressure fields when n= 3/2. The fol- lowing analysis parallels that in [4] for the derivation of the logarithmic term in the deformation field when the hardening parameter there takes the value 7/6. Recall from the first of (3.103) that y,(x,@) =-44™H(6,n) to(r™), (n>3/2) (4.17 and note on the basis of (3.62) that 2 c,[U(6, 3/2) ] H(6,n) = 0 na 372 +0(1) as na, Co= 2B . (4.2) Thus y, (4,8) -@ as n-~3/2 from above for all 6#0 and every suf- ficiently small r>0. Consequently the preceding estimate for vy is inadequate near n= 3/2. We shall now show that in a deleted neighborhood of n=3/2, - m yz Ce.) =F ren) tbr (ende PB to(e *) 5 (4.3) with m,=max(i+1/2n, 2-1/n) and by? 0 a constant. Suppose first that n>3/2 and replace (4.1) by ie now write U(6,n), H(8,n) in place of U(®8), H(8) in order to em- phasize the dependence of these functions upon the hardening parameter. -43- a “ y, (1,0) =-2 27 MH(8,n) +2 T(6,n)+o(r™ ), mé>2-m . (4.4) Substitution from (4.4) and the second of (3.103) into (3.9) leads to mUT-m’TU=0 on [-7,7] , (4.5) which is merely (3.38) with m, ¥ replaced by m’, T. Thus m’=2m=2-1/n, T=b,U . (4.6) In case 1<n<3/2, a similar procedure yields y4(r,8) = bor MU (@,n) -S 4° ™H(Oyn) to(r™) (4.7) which confirms (4.3). It should be noted that the first term of (4.3) dominates as r70O for n>3/2, while for n<3/2 the second term dom- inates; in either case the respective dominant term agrees with our pre- vious results. The two terms trade the dominant role as n passes through 3/2, the respective powers of r having the common exponent 4/3 at n= 3/2. From (4.2), (4.3) now follows o pltl/an 7 5 _|_i 0 2~1/n ~ i+] /2n y, (758) a yD +bor [vce,3/2) | +4 n, (8) + by (n- 3/2)4° Pa, (8) $o(r *) as n73/2 . (4.8) where n, ©) and ng (8) are functions that stay bounded as n-~3/2. Thus if y, (*,8) is to remain bounded as n7~3/2 one must have -44- 1 ‘0 bo= a aoa tO) as 1-3/2 . (4.9) Proceeding to the limit in (4.8) as n-3/2 one arrives at 3ce y, (7.89 = 5 [0(6,3/2) 1°r*/F10g rer 2an */30 (6) 4 0(r4/3) | (4.10) in which n(6) is bounded. This result motivates the following modifi- cation of (4.3) in case n=3/2. We suppose yy (rye) =r tog rv, (e) +28! By, (8) to(r*/?) ; (4.11) and, recalling (3.102), (3.103), assume y5 (58) = ar~/7u(6,3/2) +0(r/ 8), (4.12)? where 6>0. Substitution from (4.11), (4.12) into (3.9) yields . ; = " ; -3 i-+) ~ Uv, — 2Uv, = 0, uv, 2Uv., a(v,0 a, on [-1,7] . (4.13) The first of (4.13) gives v,=c ue on [-n,T] (4.14) 1 1 , > . so that the second of (4.13) may now be written as es e _3 e 2-1 _ vw, - abv, =F(c, bu =) on [-t,7]. (4.15) Merely requiring the error to be o(r2/3) purpose. is inadequate for the present ~45- If (4.15) is to possess a solution V5 {-1,7], then c, = 3¢y/2an” in agreement with (4.10). of unrestricted smoothness on Next, observe that in view of (4.9), (3.115), P becomes unbounded as n73/2 for -1<@<9n, The foregoing analysis suggests that we re- place (3.3) by 1/3 p(r,o8)~r logrP(6) for n=3/2 Substitution from (4.11), (4.12), (4.16) into (3.105) gives Loe 2s 3 PU~ZPU = 6Ac GC on [-1,7] Moreover, from the boundary conditions (3.14), P(tn) =0 Consequently, (4.16) (4.17) (4.18) P(@) = 4c pAlw(®,3/2) +4 cos e}i/3 VL ~ cos“Ep for n=3/2 , (4.19) in which cos"Ey is given by (3.59) with n= 3/2. Equations (4.16), (4.19) supply a non-degenerate estimate for the pressure in case n=3/2. In addition, (4.3) together with the second of (3.102) yield a locally one-to-one approximation of the deformation if 1<n< 3/2, We turn now to the Mooney-Rivlin material (n=1), which has the complete elastic potential (1.19), and seek improved estimates for the corresponding deformation and pressure fields. The equilibrium equations (3.10) in this instance reduce to -46- (4.20) cle OP _ i op _ or Hs py 36 =H, ° with A and Hy supplied by (3.12). The boundary conditions (3.14), in turn, at present become =Q at 6@=tn7 . (4.21) _ Be9 pe 2 ele 1% Ser r ré Setting n=1 in (3.30), (3.31) one finds that m=1/2, U(6)=sin(0/2) . (4.22) We recall also that the eigenvalue problems determining y, ¥Y — and hence Vo in (3.34) — admit the common eigenvalue m‘’=1. Upon substituting n=i1 in the corresponding solutions (3.49), (3.84) one arrives at cos@, ¥=b sin’ 2 5 (4.23) xX=b 2 1 where bj and by are constants. Finally, (3.113), (3.115) supply a first estimate for the pressure: 2ub zoos ek (4.24) g=1/2, P(6)=- Equations (3.3), (3.34), (3.91) together with (4.22), (4.23), (4.24) now give -47~ 9g el/2.,, 84 1 abi ein2® y,(%,0) =a,r sing +9 [a,b cos 6 aybosin zir + o(r) _. 1/2... 8, 1 28 yo (1,8) =ayr sing +3 [a,b sin 7 + ayb, cos O@]lr+o(r) , >(4.25) 2ub _ 21/2 8 1/2 P(r, 8) == > r cos zt o(r ) . J With a view to refining these estimates we first replace (3.34) by é é y (r,6) =a ruc) + rv (8) +r *R (oe) tes (8) +0(r™) . (4.26) a Qa a a a where m,>m,>1 and Re oo are as yet undetermined, whereas the func- tions U, vy are already known. Combining (4.26) with (3.9), (4.20) and invoking the boundary conditions (4.21) one finds after considerable com- putations that m,= 3/2, m=2 , (4.27) -1 _ =i _ ora a,¥iI, Ry 2t80%q * ay YI) on [-a,7] , (4.28) -t -t _ 817 ala 1X. ay¥ol, S27 7lapX # a,¥o] on [-7,7] , (4.29) with Ye Xy> Pos Xo supplied by ¥, (8) = i 30, PiP2 sin’ -4sin? 2 cos 2-4 cos? 2 1 nn ¥ 27 82D ZOOS ZF 2° (4.30) be (@A=c sino’ -—2, sin® at a rt ha -48- 3 a2 ) b. 8b*b _ 46, 1 Po. 2P1°2 26 ¥, (6) =d,sin at 5 (2, at 5 +6, oi 5 a a a 2 8; Pre Sr 8 38 Pr 38648 4 5} COS St 3 5 cos 2 2 5 a a a > (4.31) bo (6) =—=[2sin 6 - sin 26] + d,cos 26 a b. [2b.b Sey] b | bb | Po [Prb2, 32] ogg Pa [7 Pubes 9 a zZ 4 | |" Z Lo “| a a a a , d d are constants. Here Cys Cos yo do In order to calculate an additional term for the pressure field, we apply the rotation with the component matrix (3.96) to the deformation field (4.26).7 This yields y, (r,0) = - Arye) -2x°/?y, (8) -Er7y,(8) tol?) , (4.32) 1/2 1 ro 2. 1 2 2 yo (r,6) = ar / u(s) += = rx (6) += / X, (8) tor xy (8) + o(r ). Next, for the Mooney-Rivlin material, (3.105) reduces to dy oy 2. if sp 2 _ 9p “2 Woy Ty (2 30. 90 OF (4.33) We now suppose that the pressure field conforms to TRecall that the pressure field is not affected by such a rotation. -49- q ‘ p(r,6) =x P(e) +r° P, (8) +0(r™ ), 2 >e , (4.34) with 2%, P supplied by (4.24) and 2’, Po unknown. Equations (4.32), (4.33), (4.34) and the boundary conditions lead to 2 2 2 2 3c 5b_b v=1, £00) =25 fa natn? ( Bs i 2 ase . (4.35) a a Finally, suppose that the Mode I loading (2.15) prevails at infinity, in which case the local deformation field (4.26) is to be symmetric about the x ,~axis. In this case, =Q, a,=a, b, =0, cy = 0, d,=0 , (4.36) ay 2 2 and (4.26) reduces to ley cin? 24) y3/2 ltcin? 2 cos 2 tb coed e y, (1,9) =->rbysin atgr 4sin 7 COS 5 +3 Cos 5 2 c I». / 71 a2 le sant fh aa) 3 _ > 2 ene | + (r?) r'jdjsin’5 x (3c, 7 )sin’ 5 Oo 5 a 2a . (4.37) be __1/2., 6,1. 3/2 36 2 8 y,(r,0)=ar “sing t+ or 1 2 one a b +2 rf 2sin @- sin 26] + o(r’) 4 a Further, the associated pressure field conforms to -50- 2ub, p(r,9) =- 5 cos $4 2h r (3-cos6)+o(r) . (4.38) a a The above results for the symmetric loading mode are identical with those arrived at in an unpublished study by Knowles and Sternberg, which is sum- marized in [9]. In contrast, it is not possible to specialize the con- stants in (4.26) so that the displacement field WV 7X has the parity fe] : of u, in (2.7). This fact is consistent with the conclusion reached at the end of material fails to admit an antisymmetric global solution. -51- 5. Discussion of the deformation and stresses near the crack-tip. Our initial objective here is to explore the asymptotic results for the deformation field established in the preceeding two sections. Of particular concern is the deformation image of the crack faces 6=tn. Suppose first that n>3/2 and consider (3.103). Then, to leading order, 1+1/2 1+ yy (x.9) =— 30H CG) + o(e tt Pmy G.at yy(r,0) = ar /79y(9) + o(r 1/2) . | In view of (3.31), (3.64), one has in particular ~ __ 1 1+1/2n von T(3/2 -1/m) yy (r,tm) =- or 2, 1/2n T(2-1/m) 2m mn 141/2n —, oa. tote ), mal-5F > (5.2) yy (rym) = art 1AM K/2 4 g(t t/2ny ; yy (ej-m) sear RK? 4 g(t t/2ny Thus, to dominant order, the two crack faces are carried into the curve represented by 1 : . . : The shape of the deformation image of the crack faces is of course in- variant under a rotation such as (3,96). yy = 3K(n)a , <0, Kia) #0. (5.3)? From (5.1) it follows that points near the crack-tip in the undeformed body lie to the right of this curve after deformation. We note on the basis of (5.3)? and (3.34) that the crack is bound to open when n>3/2, regardless of the magnitude and nature of the particular loading at in- finity. This conclusion, which is readily shown to hold also for 1<ns3/2, is in marked contrast to the predictions of the linearized theory for a Mode II loading. It should be emphasized, however, that the foregoing result is con- tingent upon the applicability of the Ansatz (3.4), inwhich the functions UL are not permitted to vanish identically. For example, in the presence of a loading of uniform uniaxial tension parallel to the crack faces, the ensuing deformation field is homogeneous and the crack fails to open. In this instance the deformation field is non-singular at the crack-tips, so that (3.4) cannot possibly apply. We shall remark on the question of the scale of the crack opening later in this section. Note that the local deformation field (3.34) may be regarded as the result of two successive mappings: a deformation of the undeformed body, in which the crack opens, at least when n->1, followed by a rigid rota- tion, with Que given by (3.96). In view of the principle of objectivity we shall therefore be entitled to base the computation of the local stress 1 oes ' . : : : The specific form of K(n), which is expressible in terms of the Gamma function, is of no particular interest here. 2Recall from (3.18), (3.20) that the amplitude parameter "a" cannot vanish in this instance. ~53- distribution on (3.102), (3.103), (3.104). We cite next the symmetric deformation fields? appropriate to a Mode I loading at infinity. Since U in (3.34) is an odd function, one has a,=0, a,=a. This specialization leads to the following results. 1 2 For 1/2<n<3/2, ~. 22-1 2-1/n r (5.4) yy (x56) = at (er Pg (p/P) while for n>3/2, yy (rye) =F Ha) oe TM), (5.5) yo (r,8) _ art t/2ny (9) + (ort 2m yg ¢ptth/2ny in which yxy is furnished by (3.90) with c, = 0. Finally, for n= 3/2, 131 06 rt o(r*/3y > y, (1,8) = dU (0 ,3/2)x" (5.6) /3 y,(r,8) = ar?! 7u(0,3/2) + o(r? ) It is not possible, however, to specialize the asymptotic results for the deformation so as to arrive at an antisymmetric local displace- ment field, at least when n>1. To see this, recall that WRT ¥y and observe that for n>1 one has m‘>1, whence (3.34) give 1 observe that the associated displacement field Uy =Vg7Xq is also bound to be symmetric by virtue of the parity of Xo -~54- u, (r, 8) =a pivl/2n4 (9) -reos@t+o(r) , 1 (5.7) 1-1/2n u,(r,6) = ayr ucé) -rsin6+o0(r) The constants in (5.7) evidently cannot be chosen to yield a displace- ment field of the desired parity. As we observed in the previous section, the local asymptotic solu- tion obtained for the Mooney-Rivlin material (n=1) also fails to admit an antisymmetric displacement field. The foregoing conclusions concern- ing n21 lend support to the conjecture that there does not exist an antisymmetric global solution for all values of the hardening parameter. At this stage we calculate the associated local true-stress field, as well as the strain~energy density. On account of (1.7), (1.12) one has T = 20’ (1)F OF - pé (5.8) a8 vy BY aB Suppose first that 1/2<n<3/2. In this case, (3.102) and the first of (3.120) ,together with (5.8), yield , £8) 20/2, (yn) toe) ; ty) (ree) =e PD (8,n) + oe t/2M) ; ~ (5.9)4 Ty9 (E48) =F Ty5(8,n) +0(r-) If n>3/2, the appropriate stresses obey 1 : Observe that the true stresses here are referred to the material polar coordinates (r,6). -55- ~ _ -1+1/n T1,(4,8) = o(r ) ’ -141/n « - t,,(r,9)=r 1/n 6 (6,n) +o(r 1+1/n) ; ~(5.10) 12 12 ton(t,6) =r T,.(8,n) t0(r +) To5(r,8) =F T,5(8,n) + o(r ; The auxiliary functions T appearing in (5.9), (5.10) are defined by a8 2 2n-2 2n 2-n l-1l/n 2 *) T,,(6,n) = 8Ab_a m on [w(6,n) + Kcos 6] cos & 3 il 0 0 Ty (65n) = bab a mn fu (8 yn) +kKcos pg} i/2n cos EO »~ (5.11) 2n 2n 1-n m on To5(8,n) = 2Aa [w(@,n) + Kcos ey , J with cos EQ given by (3.59), while Ty, involves the hypergeometric function and will not be cited explicitly. In the transition case n=3/2, equations (4.11), (4.12), (4.16) may be used to deduce the corresponding stress field. We note that in this instance Too is given by the last of (5.9) or (5.10) with n= 3/2 and thus the representation ToT, 9) erty (e,n) to(r}) ; (5.12) 22 is valid for 1/2<n<o. The components of stress in (5.9), (5.10) are those associated with the rotated deformation fields (3.102), (3.103). If Te are the true-stress components associated with the original field (3.34), one has ~56- * = TAB Qty Tuy , (5.13) in which Que are the components (3.96) of the rotation tensor Q. Observe that a more satisfactory estimate for 1 than that in 11 the first of (5.10) is precluded by the weak estimate for the pressure in (3.120). It is interesting to recall in this connection that the asymptotic solution in [3], [4] of the Mode I crack problem for a class of compressible elastic solids also fails to yield a non-degenerate estimate for the stress T 1d at sufficiently large values of the cor- responding hardening parameter. The most singular of the stress com- ponents in (5.9), (5.10) is 9? which becomes infinite at the crack-tip "2 like rt, According to (5.11), T4579 and hence Too is tensile for all smail enough values of r. We now determine the dominant character of the true stresses when the latter are referred to the spatial coordinates Yat Since such a representation of the stresses depends on the availability of an in- vertible estimate for the local deformationwe confine our attention at present to n21. One draws from (3.34) that wp 2n-L/an , 2.2, 2 * if Te=¥, ty - (5.14) Equations (5.9), (5.14) imply that for 1lsn<3/2, 2(n-1) -1 -2n 2n-1 2n-1 _ 2n-1 T1470 Ty > T,,=O0lr > Ton = Olr ' (5.15) For n>3/2, (5.10) yield -57- ~2(n-1) ~2(n-1) -2n 2n-1 - 2n-1 t..=olr 2n-1 12 x > T22 x . (5.16) We observe that the singularity of 1 in (5.15), (5.16) becomes pro- 22 gressively more severe with increasing values of the hardening parameter and — for the range of n under consideration — is always stronger than that predicted by the linear theory. From (1.18), (3.34) we gather that 2n W(T) = Aare (er tto(r), 4-0, n>1/2 , (5.17) so that the strain-energy density has a singularity near the crack-tips which is of the same order as its counterpart in the linear theory. Finally we supply an estimate for the amplitude parameter "a", valid for small loads at infinity. In this connection we recall a conser- vation law appropriate to the equilibrium theory of finite plane strain: for every curve C that is the boundary of a finite regular subregion of TI, | (in-F's)a2=0 , (5.18)" provided W and F are the strain-energy density and the deformation gradient tensor, whereas n is the unit outward normal of C and s is the Piola traction, 1 . : . 1: (5.18) is an adaptation to plane strain of a conservation law originally due to Eshelby [16]. The importance of this conservation law in fracture mechanics was first recognized by Rice [17]. s=on on C . (5.19) The conservation law (5.18) holds for both compressible and incompressible materials, and is readily confirmed with the aid of the divergence theorem and the appropriate field equations. For the present purpose we consider first the Mode I loading case and assume that the solution to the nonlinear problem for sufficiently small loads is uniformly approximated by the corresponding solution from the linear theory, at all material points a finite distance from the crack- tips. A limit process depending on this assumption enables one to deduce from (5.18) and from the available solution of the linearized problem the de- sired small-load amplitude estimate. Since the required calculation is strictly parallel to one spelled out in detail in [3], we cite the en- suing estimate directly: a Pas +o(o ) as o~0 . (5.20) Here o is the applied normal stress at infinity, 2b is the crack length, while wu is the shear modulus at infinitesimal deformations. According to a conclusion reached earlier in this section, the finite theory ~ in contrast to linearized elastostatics — predicts that the crack opens in the vicinity of its tips also for a Mode II loading, at least for a certain range of the hardening parameter. Clearly, if the solution to the linearized Mode II problem is to approximate its counter- part in the finite theory uniformly at small loads on every bounded and closed material point set that excludes the crack-tips, the lengths of the opening crack-segments would have to tend to zero with the load -59- intensity. Making this assumption, invoking the conservation law (5.18) once again, and appealing to the Mode II results of the linear theory summarized in (2.12), (2.13), (2.14), one is led to precisely the esti- mate obtained for Mode I: 2n_ obn?? 2 = +olo') , (5.21) 4uame™ 1 " Lea in which o is now the shear stress applied at infinity. The length parameter L, in view of (5.3), governs the scale of the crack-opening, which is thus found to be a second-order effect at small shearing loads. -60- Appendix: Representation of functions biharmonic in a neighborhood of infinity. In this paper a function U is called biharmonic on an open plane region &, which need not be simply connected, if it is four times continuously differentiable on & and vu =0 on &. Let 7 be a neighborhood of infinity in the (x, »x5)-plane, so that 7% is the exterior of some circle centered at the origin, and let N" be the simply~connected domain obtained by deleting from % its intersection with the positive x ,~axis. As shown by Muskhelishvili [14] in his analysis of the structure of the Airy stress function on multiply connected domains, any function U biharmonic on %' , whose partial derivatives of the second and higher orders admit continuous ex- tensions to %, may be represented in the form U(x, »¥5) = Re (z¥(z) + x(z)} ; (1) where ¥(z) = Azlogz+Blogz+¥*(z) , (2) x(z) = Czlog z+Dlog z+ y*(z) Here z=x,+ix,=pe* » logz refers to the principal branch of the logarithm, while Y*(2) and y*(z) are functions analytic on 7%. Let Aj» Bi> C,> Dy and Ans Bo; Co» dD, denote the real and imaginary parts of A, B, C, D, respectively. Invoking the Laurent ex- pansions on 2 of ¥*(z) and y*(z), one readily deduces from (1) and -61- (2) the representation 2 2 UCx, »X_) = Ale loge - A, + [(B, +C, )cos p+ (B, +C,)sin g]plogo + [(B, - C,)sing - (B, +C,)cos plog +D logo ~ Dog + ) [a,cos jp+b sin jy jz-@ + e,cos(j ~ 2@+d,sin(j - Delos , (3) valid on vi , the coefficients ass ba Cy qd. being real constants. Suppose now U is biharmonic on 2 and hence continuous on 7 together with all its partial derivatives up to the fourth order. Then (3) demands that A,=0; B,-C,=0; B,+C,=0, D,=0 . (4) Suppose further that U(x, s%5) = OC4) as p~a . (5) This requirement evidently yields the additional restrictions A,=0, B,+C,=0, B,+C,=0, D,=0, a,=bo=c,=d = 0 for 4<0O . (6) In view of (4), (6), the representation (3) finally reduces to ~62- to*] U(x, +X) = )} [a,cos jg+b,sin jot c,cos(j - 2)o+ d,sin(3 - elo 2 . (7) j=0 This expression is valid on 7, the infinite series being uniformly and absolutely convergent on every closed subset of 7. Thus any function biharmonic on 7 and bounded at infinity necessarily admits the re- presentation (7). -63- References [1] [2] [3] [4] [5] [9] [10] [11] Rice, J.R., Mathematical analysis in the mechanics of fracture, Fracture, Vol.II, Chapter 3, Academic Press, New York, 1968. Wong, F.S. and Shield, R.T., Large plane deformations of thin elastic sheets of neo-Hookean material, Zeitschrift ftir angewandte Mathematik und Physik, 20 (1969) 2, p.176. Knowles, J.K. and Sternberg, E., An asymptotic finite-deformation analysis of the elastostatic field near the tip of a crack, Journal of Elasticity, 3 (1973) 2, p.67. Knowles, J.K. and Sternberg, E., Finite-deformation analysis of the elastostatic field near the tip of a crack: Reconsideration and higher order results, Journal of Elasticity, 4 (1974) 3, p.201. Knowles, J.K., The finite anti-~plane shear field near the tip of a crack for a class of incompressible materials, International Journal of Fracture, 13 (1977) 5, p.61. Lo, K.K., Finite-deformation crack in an infinite body under anti- plane simple shear, International Journal of Solids and Structures, 13 (1977), p.1219. Knowles, J.K. and Sternberg, E., Discontinuous deformation gradients near the tip of a crack in finite anti~plane shear: an example, Journal of Elasticity, 10 (1980) 1, p.81. Knowles, J.K. and Sternberg, E., Anti-plane shear fields with dis- continuous deformation gradients near the tip of a crack in finite elastostatics, Technical Report No.43, ONR Contract NO0Q014-75-C-0196, September 1979. To appear in Journal of Elasticity. Knowles, J.K., Finite elastostatic fields with unbounded deformation gradients, in Finite Elasticity, Applied Mechanics Division, ASME, Yol.27 (1977), p.23. Knowles, J.K., Crack problems in finite elastostatics, Proceedings, Symposium on Mathematical Problems in Fracture, AMS and SIAM, New York, March 1978. Knowles, J.K., On some inherently nonlinear singular problems in finite elastostatics, Proceedings, Eighth U.S.National Congress of Applied Mechanics, UCLA, June 1978. [12] [13] [15] [16] [17] -64- Sternberg, E., Some recent advances in the application of nonlinear elastostatics to singular problems, in Trends in Solid Mechanics 1979, Proceedings of the Symposium dedicated to the 65th Birthday of W.T.Koiter, Delft University Press, Sijthoff and Noordhoff International Publishers, 1979. Abeyaratne, R.C., Discontinuous deformation gradients in plane finite elastostatics of incompressible materials, Technical Report No.41, ONR Contract NO0014-75-C-0196, June 1979. To appear in Journal of Elasticity. Muskhelishvili, N.I., Some basic problems of the mathematical theory of elasticity, Second English edition, P.Noordhoff Ltd., Groningen, 1963. Abromowitz, M. and Stegun, I.A., eds., Handbook of mathematical functions, National Bureau of Standards, Applied Mathematics Series = oS 55, 1964. Eshelby, J.D., The continuum theory of lattice defects, Solid State Physics, Vol.3, Academic Press, New York, 1956. Rice, J.R., A path-independent integral and the approximate analysis of stress concentration by notches and cracks, Journal of Applied Mechanics, 35 (1968) 2, p.379. ~65~ tT(k)/A | n> /2<n<l a Y~ FIGURE |. POWER-LAW MATERIAL RESPONSE CURVES FOR SIMPLE SHEAR. n<l/2 Oo -66- REGION TI (X1Xo) o (-b,O) ane S \ (b,0) SEGMENT & — FIGURE 2. SLAB WITH CRACK AND COORDINATES.