A gauge theory for brittle damage in solids and a peridynamics implementation

Abstract

We propose a gauge theoretic model for quantifying and evolving spatial defects in the form of micro-cracks in deforming solids. A non-trivial affine connection — the gauge connection, is introduced to accommodate local changes in the configuration due to spatial defects. The gauge connection enables defining the covariant derivative, a procedure known as the minimal replacement. The configuration gradients in the Lagrangian are then determined using covariant derivatives, instead of partial derivatives, thereby incorporating kinematic information pertaining to the spatial defects. Minimal replacement ensures that the invariance of the Lagrangian under the local action of the symmetry group, which in this case is the scaling of the deformed coordinates, is restored. Introduction of the gauge field Lagrangian via yet another construct — the minimal coupling, obtains the additional energy contribution pertaining to defect evolution. The resulting Euler–Lagrange (EL) equations thus describe the coupled motion of the solid and the evolving defects. The EL equation for defect evolution presently accounts for microscopic inertia. Different features of brittle damage, viz. tension–compression asymmetry, stiffness degradation and an energy functional including the contribution from defects in the form of cracks, are described within the gauge theoretic model considering the kinematic aspects of deformation and damage. The usefulness of the model and its advantages over a phase-field based damage model are assessed through peridynamics-based numerical simulations on high-speed oscillatory crack-tip instabilities as observed experimentally during dynamic crack propagation in a pre-notched polymethyl methacrylate (PMMA) plate. Simulations are also carried out for dynamic crack propagation in maraging steel subjected to impulsive loading and dynamic crack branching in a glass type material. We claim that the model offers a physically transparent and accurate tool to understand nonlinearities in the fracture process zone.