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

机译：基于状态性逆骨折模型的有限差分近似的数值趋同

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摘要

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 peridynamic 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 Holder continuous functions C-0,C-gamma with Holder exponent gamma is an element of (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 Holder 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 pre-cracked samples subject to a bending load. (C) 2019 Elsevier B.V. All rights reserved.
机译：在这项工作中，我们研究了一类非局部裂缝模型的有限差分近似。非局部模型最初是弹性的，但除了临界应变之外，材料软化随着菌株的增加。该模型用两个电位作为状态性逆动脉模型制定：与静液压应变相关的一种，与拉伸应变相关联。我们表明，动态进化在保持器连续功能C-0的空间中井井有条，C-GAMMA带有支架指数伽玛是（0,1]的元素。在这里，非光性的长度等级是epsilon，尺寸时间步长是ΔT，网格尺寸是h。可以看到有限差分近似，以在常数CT和C的速率CTΔT+ C（伽马）/ epsilon（2）处收敛到保持器解决方案（s）独立于离散化。发现半离散近似随时间稳定。我们呈现了用于计算地验证理论上预测的收敛速率的裂缝传播的数值模拟。我们还呈现了预破裂中的裂纹传播的数值模拟样品受弯曲负荷。（c）2019 Elsevier BV保留所有权利。

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