A Two-scale High-order Damaged Elasticity Theory and Solution Procedure for Quasi-brittle Fracture

To rigorously capture the tight coupling between elastic deformation and damage initiation/propagation in quasi-brittle fracture processes, a two-scale damaged elasticity theory is proposed that accounts for the meso-scale inhomogeneity of both the strain and the damage fields. It formulates a degraded strain energy density to capture size effects and localized damage initiation and propagation through a homogenization operation. This approach takes a simplified and unified physics and offers a consistent treatment of higher-order deformation and damage, enabling natural incorporation of size effects on fracture strength. No additional regularization is needed to maintain damage localization. Structural deformation is solved using the principle of minimum potential energy, where the Augmented Lagrangian Method (ALM) reduces the order of gradient operators. All boundary conditions are free of high-order terms and can be applied in the conventional manner. Numerical investigations demonstrate the theory's capability to predict non-singular deformation at crack tips, accurately model size-dependent fracture in perforated brittle plates, and achieve mesh-independent failure predictions in benchmark problems.