Static and Dynamic Green’s Functions in Peridynamics

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• 作者：Linjuan Wang ; Jifeng Xu ; Jianxiang Wang
• 关键词：Green’s function ; Peridynamics ; Nonlocality ; Integro ; differential equation
• 刊名：Journal of Elasticity
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：126
• 期：1
• 页码：95-125
• 全文大小：
• 刊物类别：Physics and Astronomy
• 刊物主题：Classical Mechanics; Automotive Engineering;
• 出版者：Springer Netherlands
• ISSN：1573-2681
• 卷排序：126

文摘

We derive the static and dynamic Green’s functions for one-, two- and three-dimensional infinite domains within the formalism of peridynamics, making use of Fourier transforms and Laplace transforms. Noting that the one-dimensional and three-dimensional cases have been previously studied by other researchers, in this paper, we develop a method to obtain convergent solutions from the divergent integrals, so that the Green’s functions can be uniformly expressed as conventional solutions plus Dirac functions, and convergent nonlocal integrals. Thus, the Green’s functions for the two-dimensional domain are newly obtained, and those for the one and three dimensions are expressed in forms different from the previous expressions in the literature. We also prove that the peridynamic Green’s functions always degenerate into the corresponding classical counterparts of linear elasticity as the nonlocal length tends to zero. The static solutions for a single point load and the dynamic solutions for a time-dependent point load are analyzed. It is analytically shown that for static loading, the nonlocal effect is limited to the neighborhood of the loading point, and the displacement field far away from the loading point approaches the classical solution. For dynamic loading, due to peridynamic nonlinear dispersion relations, the propagation of waves given by the peridynamic solutions is dispersive. The Green’s functions may be used to solve other more complicated problems, and applied to systems that have long-range interactions between material points.