Computing upper and lower bounds for the J-integral in two-dimensional linear elasticity

We present an a posteriori method for computing rigorous upper and lower bounds for the J-integral in two-dimensional linear elasticity. The J-integral, which is typically expressed as a contour integral, is recast as a quadratic continuous functional of the displacement involving only area integration. By expanding the quadratic output about an approximate finite element solution, the output is expressed as a known computable quantity plus linear and quadratic functionals of the solution error. The quadratic component is bounded by the energy norm of the error scaled by a continuity constant, which is determined explicitly. The linear component is expressed as an inner product of the errors in the displacement and in a computed adjoint solution, and bounded by an appropriate combination of the energy norms of the error in the displacement and the adjoint. Upper bounds for the energy norm of the error are obtained by using a complementary energy approach requiring the computation of equilibrated stress fields. The method is illustrated with two fracture problems in plane strain elasticity. An important feature of the method presented is that the computed bounds are rigorous with respect to the exact weak solution of the elasticity equations.