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On Solvability of One Singular Equation of Peridynamics
Lobachevskii Journal of Mathematics Pub Date : 2020-07-18 , DOI: 10.1134/s1995080220060190
A. V. Yuldasheva

Abstract

In the classical theory of solid mechanics, the behavior of solids is described by partial differential equations (PDE) through Newton’s second law of motion. However, when spontaneous cracks and fractures exist, such PDE models are inadequate to characterize the discontinuities of physical quantities such as the displacement field. Recently, a peridynamic continuum model was proposed which only involves the integration over the differences of the displacement field. A linearized peridynamic model can be described by the integro-differential equation with initial values. In this paper, we study the well-posedness and regularity of a linearized peridynamic model with singular kernel. The novelty of the paper is that the singular kernel is represented as the Laplacian of a regular function. This let to convert the model to an operator valued Volterra integral equation. Then the existence and regularity of the solution of the peridynamics problem are established through the study of the Volterra integral equation.


中文翻译:

关于一个奇动力学方程的可解性

摘要

在经典的固体力学理论中,固体的行为通过牛顿第二运动定律由偏微分方程(PDE)描述。但是,当存在自发的裂缝和断裂时,此类PDE模型不足以表征物理量(例如位移场)的不连续性。最近,提出了一种绕动力连续体模型,该模型仅涉及位移场差异的积分。可以用具有初始值的积分微分方程来描述线性化的动力学模型。在本文中,我们研究具有奇异核的线性化动力学模型的适定性和正则性。本文的新颖之处在于,奇异内核被表示为正则函数的拉普拉斯算子。这样就可以将模型转换为运算符值的Volterra积分方程。然后,通过研究沃尔泰拉积分方程,建立了绕动力学问题解的存在性和规律性。
更新日期:2020-07-18
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