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Green’s functions and integral representation of generalized continua: the case of orthogonal pantographic lattices
Zeitschrift für angewandte Mathematik und Physik ( IF 2 ) Pub Date : 2021-03-01 , DOI: 10.1007/s00033-021-01480-3
Claude Boutin , Francesco dell’Isola

This paper shows how the classical representation techniques for the solution of elasticity problems, based on the Green’s functions, can be generalized to second-gradient continua focusing on the specific case of pantographic lattices. As these last are strongly anisotropic, the fundamental solutions of isotropic second-gradient continua involving bi-Helmholtz-type operators are not applicable. More specifically we establish the analytical fundamental solution for the linearized equations governing the equilibrium of pantographic 2D continua in the neighbourhood of the reference configuration. Moreover, by means of found novel Green’s functions, it is shown that it is possible to solve aforesaid equilibrium equations by using Fredholm integral equations. It is seen that an approximated analytical solution for the standard bias test for pantographic 2D continua can be found by using judiciously the found analytical fundamental solutions. The micro-macro-asymptotic identification allows for a clear and satisfactory physical interpretation of the obtained analytical results.



中文翻译:

Green的功能和广义连续性的整体表示:正交全景图格的情况

本文展示了如何基于格林函数将用于解决弹性问题的经典表示技术推广到针对梯度格的特殊情况的第二梯度连续体。由于这些最后一个是强各向异性的,因此涉及bi-Helmholtz型算子的各向同性第二梯度连续体的基本解不适用。更具体地说,我们为控制参考配置附近二维全景连续体平衡的线性化方程式建立了解析基本解。此外,通过发现新颖的格林函数,表明可以通过使用弗雷德霍尔姆积分方程来求解上述平衡方程。可以看出,通过明智地使用发现的基本分析解决方案,可以找到全景2D连续体标准偏差测试的近似分析解决方案。微观宏观渐近识别可以对获得的分析结果进行清晰而令人满意的物理解释。

更新日期:2021-03-02
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