Integral-based averaging with spatial symmetries for non-local damage modelling

Simulations utilizing local constitutive equations for strain-softening materials are known to be pathologically mesh-dependent. The rational solution to this problem is to use nonlocal material models. In this paper, we discuss the application of the integral-based approach to the simulation of non-local damage accumulation and fracture. Various combinations of non-localities with common spatial symmetries are analysed analytically and numerically. The symmetries include the plane strain and the axisymmetric cases, as well as the presence of symmetry planes and cyclic symmetries. Although not a symmetry, the practically important case of thin plates is also analysed. We show that the delocalization procedure should be executed using symmetry-adapted averaging kernels. For the considered spatial symmetries, analytical closed-form expressions are obtained. Moreover, a new easy-to-use averaging kernel is suggested, the same for 3D, plane strain and plane stress applications. To showcase the delocalization procedures, we consider a ductile damage model, based on the multiplicative decomposition of the deformation gradient, as well as hyperelastic relations between stresses and strains. FEM solutions for a series of problems are presented, including graduate damage accumulation, crack initiation and fracture.