Abstract

The peridinamics theory, proposed by Silling in 2000, is a nonlocal theory of continuum mechanics based on an integro-differential equation without spatial derivatives and may cope with discontinuous displacement fields commonly occurring in fracture mechanics. Instead of spatial differential operators, integration over differences of the displacement field is used to describe the existing, possibly nonlinear, forces between particles of the solid body.Beside an overview of the peridynamics modelling and its application, results concerning the mathematical solution of the governing equation, which is a partial integro-differential equation with second-order time derivative, are presented.In this paper we consider well-posed Cauchy problem for the singular periodic integro-differential equation of peridinamics. In case of two-dimensional space the existence and uniqueness of solution are proved in Sobolev spaces. In multidimensional space the unique solvability of the problem in logarithmic scale of Hilbert spaces is proved.

中文翻译：

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弹性理论中的线性周线模型

摘要

Silling在2000年提出的周向动力学理论是一种基于不带空间导数的积分微分方程的连续介质力学的非局部理论，可以应对断裂力学中常见的不连续位移场。代替空间微分算子，而是使用位移场差异上的积分来描述固体粒子之间现有的，可能是非线性的力。提出了具有二阶时间导数的偏积分微分方程。本文考虑周长奇异周期积分微分方程的适定柯西问题。在二维空间的情况下，在Sobolev空间中证明了解的存在性和唯一性。在多维空间中，证明了希尔伯特空间对数尺度上问题的独特可解性。