Numerical Comparison of AT1 and AT2 gradient damage models within an incremental variational framework for coupled plasticity

Gradient damage models are widely used to regularize strain localization and fracture in computational mechanics. Among the available formulations, the AT1 and AT2 models differ primarily in the adopted crack-density function, leading to distinct damage initiation and evolution characteristics. Although both formulations have been extensively studied for brittle fracture, their responses under identical constitutive assumptions within a unified incremental variational framework remain of considerable interest. In this work, the AT1 and AT2 gradient damage models are compared numerically within an incremental variational framework for gradient damage coupled with perfect plasticity. The formulation is implemented using the finite element method, where the displacement field, plastic internal variables, and damage field are determined within a unified incremental variational framework using an alternating minimization procedure. Both models are evaluated using identical material parameters, loading conditions, and numerical discretization to ensure a consistent comparison. Numerical simulations of a displacement-controlled tensile benchmark are performed to examine damage evolution, plastic strain localization, and energy dissipation. The results show that the AT2 formulation predicts earlier damage initiation, wider damage localization, and higher local plastic strain, whereas the AT1 formulation delays localization and accumulates greater damage-weighted plastic dissipation. The present study provides a consistent assessment of the influence of the crack-density function on coupled damage–plasticity behaviour and contributes to a better understanding of the characteristics of the AT1 and AT2 formulations.