An Improved Kernel Function in Nonlocal Damage Model with the Boundary Effect

The occurrence of localization resulting from strain softening often gives rise to the size effect when forecasting the damage zone, energy dissipation, and load response using finite element analysis. This owes mostly to the absence of a length scale parameter. The current kernel functions in the integral-type nonlocal damage model do not effectively integrate both the damage and boundary effects. This paper proposes an enhanced formulation of the kernel function in the nonlocal damage model that is represented by two key aspects: 1. by deeming that the length scale parameter steadily drops from a small value to zero gradually as damage evolves, the long-term interaction domain among different material points increases to enhance the share of dissipated energy. However, nonlocality disappears at damage initiation and complete damage. Moreover, the interaction domain is expanded by considering the maximum damage between a receiver point and a source point, and it is assumed that the length scale parameter decreases exponentially with the maximum damage. 2. the interaction domain among material points nearby the boundary also expands to weaken the boundary effect, and recovers to the regular case for material points far from the boundary. However, nonlocality disappears at the boundary. Finite element analysis using ABAQUS subroutines is performed to implement the nonlocal damage model with the improved kernel function. Two numerical examples including the concrete plate under tension and the soil slope under compression are presented to discuss the effects of the length scale parameter, the shapes of kernel function and the mesh sizes on the load curves and the evolution of shear bands.