A nonlocal energy-informed neural network based on peridynamics for elastic solids with discontinuities

In this paper, a nonlocal energy-informed neural network is proposed to deal with elastic solids containing discontinuities by considering the long-range interactions of material points. First, the solution to peridynamic equilibrium equation is converted to an variationalenergy minimization problem based on principle of virtual work, which automatically satisfies the zero-traction boundary conditions and avoids the introduction of artificial damping. Furthermore, the energetic representation of behavior of physical system can be tractable as the loss function for deep learning neural network. This allows to approximate the solution of the system by the active machine learning community. As the basic technique in deep learning, automatic differentiation is capable to evaluate derivatives for smooth functions, but it is prone to singularity due to the presence of discontinuities. To address this limitation, spatial integration is employed in the proposed neural network to evaluate the strain energy of the system rather than the spatial derivatives of displacement fields calculated by automatic differentiation. Moreover, the initial cracks can be directly introduced in the constitutive model without the explicit definition of crack surfaces. The accuracy and efficiency of the proposed neural network is validated by conducting several mechanical problems with or without discontinuities. More importantly, the proposed neural network is capable to capture the jump discontinuities at the crack surfaces, where the neural network with automatic differentiation is hard to address. Additionally, the convergence of the proposed neural network and the comparison of two widely used activation functions are investigated.