A geometric internal bond model for fracture of brittle materials

Abstract

A Geometric Internal Bond Model is proposed, emphasizing its geometric attributes of discontinuous deformation of solid materials. This model is motivated by the inherent connection between nonlocal vector calculus and geometric decomposition of virtual internal bond for each material point in continuum. A nonlocal governing equation together with an exact nonlocal force boundary condition is derived from a nonlocal variational principle through nonlocal vector calculus. Based on nonlocal operators in nonlocal vector calculus, a nonlocal deformation gradient tensor is developed to describe the discontinuities that appear in the crack zone of materials. Two micro-modulus, corresponding to the radial and tangential deformations of virtual internal bonds, are proposed to keep consistency of the virtual bond energy in the proposed model and the strain energy in Classical Continuum Mechanics. It is shown that the proposed model is asymptotically compatible with the Classical Continuum Mechanics when the characteristic length of nonlocality approaches zero. Moreover, a geometric crack criterion is proposed that involves radial and tangential deformation of the virtual internal bond to capture the mixed mode fracture. We demonstrate the applicability of the proposed model for capturing the propagation of cracks in brittle materials by several numerical examples. The proposed model shows great potential for numerical simulations of fractures in brittle materials.