A possible link between brittle and ductile failure by viewing fracture as a topological defect

Abstract

A continuum model of fracture that describes, in principle, the propagation and interaction of arbitrary distributions of cracks and voids with evolving topology without a “fracture criterion” is developed. It involves a “law of motion” for crack tips, primarily as a kinematical consequence coupled with thermodynamics. Fundamental kinematics endow the crack tip with a topological charge. This allows the association of a kinematical conservation law for the charge, resulting in a fundamental evolution equation for the crack-tip field and, in turn, the crack field. The vectorial crack field degrades the elastic modulus in a physically justified anisotropic manner. The mathematical structure of this conservation law allows an additive “free” gradient of a scalar field in the evolution of the crack field. We associate this naturally emerging scalar field with the porosity that arises in the modeling of ductile failure. Thus, porosity-rate gradients affect the evolution of the crack field, which then naturally degrades the elastic modulus, and it is through this fundamental mechanism that spatial gradients in porosity growth affect the strain energy density and the stress-carrying capacity of the material and, as a dimensional consequence related to fundamental kinematics, introduce a length scale in the model. A key result of this work is that brittle fracture is energy-driven while ductile fracture is stress-driven. Under overall shear loadings where the mean stress vanishes or is compressive, the shear strain energy can still drive shear fracture in ductile materials.