Numerical convergence of nonlinear nonlocal continuum models to local elastodynamics

We quantify the numerical error and modeling error associated with replacing a nonlinear nonlocal bond-based peridynamic model with a local elasticity model or a linearized peridynamics model away from the fracture set. The nonlocal model treated here is characterized by a double well potential and is a smooth version of the peridynamic model introduced in n Silling (J Mech Phys Solids 48(1), 2000). The solutions of nonlinear peridynamics are shown to converge to the solution of linear elastodynamics at a rate linear with respect to the length scale $\epsilon$ of non local interaction. This rate also holds for the convergence of solutions of the linearized peridynamic model to the solution of the local elastodynamic model. For local linear Lagrange interpolation the consistency error for the numerical approximation is found to depend on the ratio between mesh size $h$ and $\epsilon$. More generally for local Lagrange interpolation of order $p\geq 1$ the consistency error is of order $h^p/\epsilon$. A new stability theory for the time discretization is provided and an explicit generalization of the CFL condition on the time step and its relation to mesh size $h$ is given. Numerical simulations are provided illustrating the consistency error associated with the convergence of nonlinear and linearized peridynamics to linear elastodynamics

Numerical convergence of finite difference approximations for state based peridynamic fracture models

In this work, we study the finite difference approximation for a class of nonlocal fracture models. The nonlocal model is initially elastic but beyond a critical strain the material softens with increasing strain. This model is formulated as a state-based perydynamic model using two potentials: one associated with hydrostatic strain and the other associated with tensile strain. We show that the dynamic evolution is well-posed in the space of H\"older continuous functions $C^{0,\gamma}$ with H\"older exponent $\gamma \in (0,1]$. Here the length scale of nonlocality is $\epsilon$, the size of time step is $\Delta t$ and the mesh size is $h$. The finite difference approximations are seen to converge to the H\"older solution at the rate $C_t \Delta t + C_s h^\gamma/\epsilon^2$ where the constants $C_t$ and $C_s$ are independent of the discretization. The semi-discrete approximations are found to be stable with time. We present numerical simulations for crack propagation that computationally verify the theoretically predicted convergence rate. We also present numerical simulations for crack propagation in precracked samples subject to a bending load.Comment: 42 pages, 11 figure

Solvability of nonlocal systems related to peridynamics

Kaßmann M, Mengesha T, Scott J. Solvability of nonlocal systems related to peridynamics. Communications on Pure and Applied Analysis . 2019;18(3):1303-1332.In this work, we study the Dirichlet problem associated with a strongly coupled system of nonlocal equations. The system of equations comes from a linearization of a model of peridynamics, a nonlocal model of elasticity. It is a nonlocal analogue of the Navier-Lame system of classical elasticity. The leading operator is an integro-differential operator characterized by a distinctive matrix kernel which is used to couple differences of components of a vector field. The paper's main contributions are proving well-posedness of the system of equations and demonstrating optimal local Sobolev regularity of solutions. We apply Hilbert space techniques for well-posedness. The result holds for systems associated with kernels that give rise to non-symmetric bilinear forms. The regularity result holds for systems with symmetric kernels that may be supported only on a cone. For some specific kernels associated energy spaces are shown to coincide with standard fractional Sobolev spaces

Mini-Workshop: Mathematical Analysis for Peridynamics

A mathematical analysis for peridynamics, a nonlocal elastic theory, is the subject of the mini-workshop. Peridynamics is a novel multiscale mechanical model where the canonical divergence of the stress tensor is replaced by an integral operator that sums forces at a finite distance. As such, the underlying regularity assumptions are more general, for instance, allowing discontinuous and non-differentiable displacement fields. Although the theoretical mechanical formulation of peridynamics is well understood, the mathematical and numerical analyses are in their early stages. The mini-workshop proved to be a catalyst for the emerging mathematical analyses among an international group of mathematicians