On the duality of finite element discretization error control in computational Newtonian and Eshelbian mechanics

In this paper, goal-oriented a posteriori error estimators of the averaging type are presented for the error obtained while approximately evaluating theJ-integral in nonlinear elastic fracture mechanics. Since the value of the J-integral is one component of the material force acting on the crack tip of a pre-cracked elastic body, the appropriate mechanical framework to be chosen is the one named after Eshelby rather than classical Newtonian mechanics. However, in a finite element setting, the discretized Eshelby problem is generally not solved explicitly. Rather, its solution is approximated by the finite element solution of the corresponding discretized dual Newton problem. As a consequence, discrete material forces arise not only at the crack tip but also at other nodes of the current finite element mesh. It is the objective of this paper to establish goal-oriented a posteriori error estimators in both the framework of Eshelbian and Newtonian mechanics and to elaborate their dual relations. This allows to control the error of the J-integral while, at the same time, no further discrete material forces arise during the adaptive mesh refinement process which could lead to misleading mechanical interpretations of the results obtained by the finite element method. The paper is concluded by numerical examples that illustrate our theoretical results.