Peridynamics Implementation of a Gauge Theory for Brittle Damage in Solids and Applications

Abstract

We propose a brittle damage model based on linearized kinematic measures derived using a gauge theory. First, we obtain the fully nonlinear kinematic measures by considering a solid with spatial defects due to micro-cracks. To accomplish this, a non-trivial affine connection, called the gauge connection, is introduced to account for local configurational changes induced by the spatial defects. This procedure, called minimal replacement, leads to the emergence of covariant derivatives and ensures the invariance of the Lagrangian under the local action of the gauge symmetry group. The covariant derivatives furnish the configuration gradients used for constructing the invariant Lagrangian. The geometric foundations and symmetry principles of the gauge theory enable accurate characterizations of the kinematic and other important physical features pertaining to defect-induced brittle damage. Another important construct, called minimal coupling, provides the stored energy corresponding to defects via the gauge field Lagrangian. Using linearization of the fully nonlinear kinematic measures, we obtain an energy functional which is quadratic in the field variables and such an energy is invariant with respect to the local gauge transformations. The resulting Euler–Lagrange equations provide the governing laws for coupled evolutions of deformation and defects. The constitutive equations in the linearized setting are linear in the field variables. A kinematically transparent approach is thus followed for modelling different aspects of brittle damage such as material stiffness degradation, tension-compression asymmetry and energy contribution of defects. We propose different tension-compression asymmetry mechanisms based on spectral decomposition of tensile and compressive deformations, dilatation-compression split of volumetric deformation, and volumetric-deviatoric split of deformation. Towards a stable and accurate numerical implementation of the model, the governing partial differential equations are recast into an integro-differential form via the principles of peridynamics (PD). The efficacy of the model, so discretized, is established through PD-based numerical simulations of tension and shear tests on a single-edge-notched plate. For further exploration, the numerical simulation of blast-induced fracture in rock is also carried out. In this case, account is taken of the physical reaction zone developed during shock wave detonation, and the interaction between gas and solid rock.