Determination of nonlocal kernels in nonlocal models for solving strain localization problems

The strain localization during the damage and failure of materials can be accurately predicted by nonlocal continuum theories. Yet, the results of theoretical predictions are directly affected by the nonlocal kernel formulation. In this study, the correlation between the nonlocal kernel formulation and the strain localization simulation results is revealed. First, based on the nonlocal damage theory, a novel free energy functional was proposed with strain and damage parameters as independent variables, and a constitutive relation considering nonlocality and elastic damage was derived within the framework of continuum thermodynamics. Then, one-dimensional governing equations were obtained through the dual-linearization method. Finally, strain localization predictions obtained with several nonlocal kernel formulations were presented, and the relationship between the nonlocal kernels and the solution was analyzed, thereby determining an appropriate nonlocal kernel function for the strain localization problem.