Green’s functions and integral representation of generalized continua: the case of orthogonal pantographic lattices

Abstract

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.

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Notes

  1. 1.

    Since \(g_x (x,y)\) is odd on x and even on y, \(G_x (x,y)\) is even on both x and y so that \(G_x(x,y)= G_x(|x|,|y|)\). Thus, it is sufficient to focus on the case where consider x and y positive. In that case, setting \(a^2 = \frac{y^2}{4\eta u}\) one has,

    $$\begin{aligned} \int ^{L}_{x}\frac{\exp (\frac{-y^2}{4\eta u})}{\sqrt{4\pi \eta u}}\mathrm {d}u = \frac{-y}{4\eta \sqrt{\pi }}\int ^{\frac{y }{\sqrt{4\eta L}}}_{\frac{y }{\sqrt{4\eta x}}} \frac{\exp (-a^2) }{a^2} \mathrm {d}a = \frac{-y}{4\eta \sqrt{\pi }} \bigg \{\left[ \frac{ -\exp (-a^2) }{a}\right] ^{\frac{y }{\sqrt{4\eta L}}}_{\frac{y }{\sqrt{4\eta x}}} -2\int ^{\frac{y }{\sqrt{4\eta L}}}_{\frac{y }{\sqrt{4\eta x}}} \exp (-a^2) \mathrm {d}a)\bigg \} \end{aligned}$$

    Then taking \(L = 0\), one obtains the following expression when \(x>0\) and \(y>0\):

    $$\begin{aligned}&G_x (x,y) = \frac{-y}{4\eta \sqrt{\pi }} \bigg \{ \frac{ \exp ((\frac{y }{\sqrt{4\eta x}})^2) }{\frac{y }{\sqrt{4\eta x}}} -2\int _{\frac{y }{\sqrt{4\eta x}}}{+\infty } \exp (-a^2) \mathrm {d}a)\bigg \} \\&\quad = - \sqrt{\frac{x}{ 4\pi \eta } }\exp (\frac{-y^2}{4\eta x})+ \frac{y}{4\eta }\bigg (1-\frac{2}{\sqrt{\pi }}\int ^{\frac{y }{\sqrt{4\eta x}}}_{0} \exp (-a^2) \mathrm {d}a \bigg ) \end{aligned}$$

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Claude, B., Francesco, d. Green’s functions and integral representation of generalized continua: the case of orthogonal pantographic lattices. Z. Angew. Math. Phys. 72, 58 (2021). https://doi.org/10.1007/s00033-021-01480-3

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Keywords

  • Green’s functions
  • Integral equation
  • Second-gradient continua
  • Anisotropy
  • Pantographic lattices

Mathematics Subject Classification

  • 41A21
  • 45B05
  • 74A60
  • 74B99
  • 74E10
  • 74G05
  • 74K99
  • 74Q05
  • 74Q15
  • 74S15