The nonlocal models of peridynamics have successfully predicted fractures and deformations for a variety of materials. In contrast to local mechanics, peridynamic boundary conditions must be defined on a finite volume region outside the body. Therefore, theoretical and numerical challenges arise in order to properly formulate Dirichlet-type nonlocal boundary conditions, while connecting them to the local counterparts. While a careless imposition of local boundary conditions leads to a smaller effective material stiffness close to the boundary and an artificial softening of the material, several strategies were proposed to avoid this unphysical surface effect. In this work, we study convergence of solutions to nonlocal state-based linear elastic model to their local counterparts as the interaction horizon vanishes, under different formulations and smoothness assumptions for nonlocal Dirichlet-type boundary conditions. Our results provide explicit rates of convergence that are sensitive to the compatibility of the nonlocal boundary data and the extension of the solution for the local model. In particular, under appropriate assumptions, constant extensions yield $\frac{1}{2}$ order convergence rates and linear extensions yield $\frac{3}{2}$ order convergence rates. With smooth extensions, these rates are improved to quadratic convergence. We illustrate the theory for any dimension $d\geq 2$ and numerically verify the convergence rates with a number of two dimensional benchmarks, including linear patch tests, manufactured solutions, and domains with curvilinear surfaces. Numerical results show a first order convergence for constant extensions and second order convergence for linear extensions, which suggests a possible room of improvement in the future convergence analysis.
翻译:远洋动力学的非本地模型成功地预测了各种材料的断裂和变形。 与当地机械学不同, 极地动力的边界条件必须在体外的有限体积区域加以界定。 因此,在适当制定Drichlet型非本地边界条件时,在理论和数字上出现挑战,同时将这些条件与当地对应方联系起来。 虽然不小心地强加当地边界条件导致靠近边界的较弱有效物质僵硬性,人为地软化材料,但提出了几项战略,以避免这种无形的地表趋同效应。 在这项工作中,我们研究如何将解决方案与非基于本地的线性线性弹性模型相趋同于当地对应方,同时根据不同的配方和对非本地的Drichlet型非本地边界条件的平稳假设,将这些结果与非本地边界数据的兼容性以及当地模型的解决方案的扩展。 特别是,在适当假设下, 不变的扩展产值产生美元(frac{1 ⁇ 2美元) 和线性扩展率, 等值的趋同率率, 以及两号的线性标准, 显示: 平地平地平的递化的递化的递化的递化的递增率 。