Computing Upper and Lower Bounds for the J-Integral in Two-Dimensional Linear Elasticity ¢¡ ¤£ ¤¥ Z.C. Xuan , K.H. Lee , A.T. Patera , J. Peraire Singapore-MIT Alliance ¡ Department of Mechanical Engineering, National University of Singapore £ Department of Mechanical Engineering, Massachusetts Institute of Technology ¥ Department of Aeronautics and Astronautics, Massachusetts Institute of Technology (This paper is dedicated to the memory of Kwok Hong Lee) Abstract— We present an a-posteriori method for computing derived from Eshelby's [3] energy momentum tensor along the rigorous upper and lower bounds of the J-integral in two direction of the possible crack extension. An alternative form dimensional linear elasticity. The J-integral, which is typically of the J-integral in which the contour integral is transformed expressed as a contour integral, is recast as a surface integral which yields a quadratic continuous functional of the displace- into a domain integral involving a suitably defined weighting ment. By expanding the quadratic output about an approximate function is given in [6]. The expression for the energy release finite element solution, the output is expressed as a known rate given in [6] appears to be very versatile and has an easier computable quantity plus linear and quadratic functionals of and more convenient generalization to three dimensions than the solution error. The quadratic component is bounded by the the original form [13]. energy norm of the error scaled by a continuity constant, which is determined explicitly. The linear component is expressed as an Regardless of the method chosen to evaluate the stress inten- inner product of the errors in the displacement and in a computed sity factor, a good approximation to the solution of the linear adjoint solution, and bounded using standard a-posteriori error elasticity equations is required. Unfortunately, the problems estimation techniques. The method is illustrated with two fracture of interest involve singularities and this makes the task of problems in plane strain elasticity. computing accurate solutions much harder. For instance, it is well known [16] that the convergence rate of energy norm of a I. INTRODUCTION standard finite element solution for a linear elasticity problem ¤ The accurate prediction of stress intensity factors in crack involving a ¦¢§©¨ reentrant corner is no higher than , tips is essential for assessing the strength and life of structures where is the typical mesh size. This problem was soon using linear fracture mechanics theories. A crack is assumed realized and as a consequence a number of mesh adaptive to be stable when the magnitude of the stress concentration algorithms have been proposed [4], [5], [7], [8], [14] which, at its tip is below a critical material dependent value. Stress in general, improve the situation considerably. In some cases intensity factors derived from linearly elastic solutions are [7], [8], the adaptivity is driven by errors in the energy norm widely used in the study of brittle fracture, fatigue, stress of the solution, whereas in some others [4], [5], [14], a more corrosion cracking, and to some extend for creep crack growth. sophisticated goal-oriented approach based on a linearized Since the analytical methods for solving the equations of form of the output is used. elasticity are limited to very simple cases, the finite element In this paper we present a method for computing strict method is commonly used as the alternative to treat the more upper and lower bounds for the value of the J-integral in complicated cases. The methods for extracting stress intensity two dimensional linear fracture mechanics. The J-integral is factors from computed displacement solutions fall into two written as a bounded quadratic functional of the displacement categories: displacement matching methods, and the energy and expanded into computable quantities plus additional linear based methods. In the first case, the form of the local solution and quadratic terms in the error. The linear terms are bounded is assumed, and the value of the displacement near crack tip using our previous work for linear functional outputs [9], [11], is used to determine the magnitude of the coefficients in the [12] and the quadratic term is bounded with the energy norm asymmptotic expansion. In the second case, the strength of of the error scaled by a suitably chosen continuity constant, the singular stress field is related to the energy released rate, which can be determined a priori. The bounds produced are i.e. the sensitivity of the total potential energy to the crack strict with respect to the solution that would be obtained position. An expression for calculating the energy release on a conservatively refined reference mesh. This restriction rate in two dimensional cracks was given in [13] and is however can be eliminated if the more rigorous techniques for known as the J-integral. The J-integral is a path independent bounding the outputs of the exact weak solutions are employed contour integral involving the projection of the material force [10], [15]. Also, not exploited here, but of clear practical interest, is the fact that the bound gap can be decomposed ¢ in its plane, we are interested in determining the energy into a sum of positive elemental contributions thus naturally release rate, s 84m , such that, leading to an adaptive mesh adaptive approach [12]. We think that the algorithm presented is an attractive alternative to the I 84ms%W s 84b¢y existing methods as it guarantees the certainty of the computed For a two-dimensional linear elastic body the energy release bounds. This is particularly important in critical problems x relating to structural failure. The method is illustrated for an 2 open mode and a mixed mode crack examples. G II. PROBLEM FORMULATION We consider a linear elastic body occupying a region . The boundary of , , is assumed to be piecewise d l smooth, and composed of a Dirichlet portion ! , and a x1 $&%'( *)+(" Neumann portion #" , i.e. We assume that ,.-/ 10 ("32 a traction 22 is applied on the Neumann Wc boundary and that the Dirichlet boundary conditions are ho- 7 :-<;>= mogeneous. The displacement field 45%6 87 ?A@ @ ¤9 3T C E-F GHI JKML %ONQPSRQ %B 8C satisfies the followingD9weak form of the elasticity equations Fig. 1. Crack geometry showing coordinate axes and the J-integral contour and domain of integration. @ @ @Z^ @ U V%W JX ZY\[], -`; 84 (1) 9 9 9 96_ 9 rate, 4m , can be calculated as a path independent line in which integral known as the -integral [13]. If we consider the ced @ @g geometry shown in figure 1, the -integral has the following aX b% XHf 9 9 expression, c hSi ceh @Z^ @1g g x4 [], % ,`f s 84bV% Hf 9 9 (6) 9 x t XQ-j 10 GK2 u where is the body force. The bi-linear form @ where is any path beginning at the bottom crack face U k mlD;onp;rq is given by, u %B lwe}D~ 9 and ending at the top crack face, is the ctdvu @ g % @ strain energy density, is the traction given as , U s% 8k`mlSwx zy 8k (2) %¡ 9 and is the outward unit normal to . An D9 @ alternative expression for 4m was proposed in [6], where the Here, w denotes the second order deformation tensor which @ contour integral is transformed to the following area integral u is defined as the symmetric part of the gradient tensor { . @ @ @ @ expression, V% a{ Yj a{ |#}D~ That is, wx . The stress is related c¢d¤£p u g x¥ to the deformation tensor through a linear constitutive relation ¤4 | 3y J{H¥! f s 4bV% (7) u x of the form ¤ @ @ s%\rl©w (3) ¥ G¦¤ Here, the weighting function is any function in that where is the constant fourth-order elasticity tensor. We is equal to one at the crack tip and vanishes on . @ ¥ s 4m l#;q define the total potential energy functional For a given , we observe that is a bounded quadratic as functional of 4 . For our bounding procedure it is convenient @ @ @ @ @(^ ¦ U s% # JX ! M[, (4) to make the quadratic dependence of the output on the solu- 9 9 9 ~ tion more explicit. To this end, we define the bilinear form @ § ¨ 4 lD;on`;©q It is straightforward to see that the solution, , to the problem 8k as, 9 @ £ ctd (1) minimizes the total potential energy, and that u g @ § ¨ | k s% J{H¥! f 8kp ¦ ¦ U 9 ¤ 4 4s% LLL 43LLL y 84V% (5) ctd¤£ 9 u ~ ~ @ g ¦ x¥ U 8k`l©w (8) LLLfLLL©% f f Where denotes energy norm associated with 9 ~ ¤ 9 U f f @ the coercive bilinear form . ¨ 9 mlD;on`;©q and its symmetric part 8k , In fracture mechanics we are often interested in determining 9 @ @ @ ¦ § the strength of the crack tip stress fields. A common way to § ¨ ¨ ¨ s% 8k ZY kpy k (9) 9 9 do that is to relate the so called stress intensity factors to 9 ~ the energy released per unit length of crack advancement (see It is clear from these definitions that, figure 1). If the total potential energy defined by (5), decreases ¨ s 84bb% 84 4 by an amount I 4 when the crack advances by a distance (10) 9 9 and that there exists ª`«­¬ such that, and therefore, @ @ @ @ ¨ ¦ ¦ ¦ ¦ ¨ m®Mª¯LLL LLL -`;+y (11) ÁbLLL À· LLL ® · 4 m® ÁLLL À·ÂY LLL y ° 9 _ 9 (18) ¿ 9 ¿ À À III. BOUNDING PROCEDURE B. Quadratic term Our objective is to compute upper and lower bounds, for 4 4m , where satisfies problem (1). Let us consider a finite In the appendix we show that for two dimensional linear element approximation 4°-`;° satisfying elasticity, a suitable value for the continuity constant in ex- @ @Z^ @ @ pression (11) is given by U 84m° s%W aX ¯YQ[, -`;°±y (12) Á©Í 9 9 9 96_ GËSÌY AL {Î¥mL ª©¦%ÅÄÆIÇ ;°/'; ; Here, is a finite dimensional subspace of .