A singularly perturbed nonlinear traction problem in a periodically perforated domain: A functional analytic approach

M. Dalla Riva, P. Musolino*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

9 Citations (SciVal)

Abstract

We consider a periodically perforated domain obtained by making in Rn a periodic set of holes, each of them of size proportional to ε. Then, we introduce a nonlinear boundary value problem for the Lamé equations in such a periodically perforated domain. The unknown of the problem is a vector-valued function u, which represents the displacement attained in the equilibrium configuration by the points of a periodic linearly elastic matrix with a hole of size ε contained in each periodic cell. We assume that the traction exerted by the matrix on the boundary of each hole depends (nonlinearly) on the displacement attained by the points of the boundary of the hole. Then, our aim is to describe what happens to the displacement vector function u when ε tends to 0. Under suitable assumptions, we prove the existence of a family of solutions {u(ε, ×)} ε â̂̂ ]0,ε ′ [ with a prescribed limiting behavior when ε approaches 0. Moreover, the family {u(ε, ×)}ε â̂̂ ]0,ε ′ [ is in a sense locally unique and can be continued real analytically for negative values of ε.

Original languageEnglish
Pages (from-to)106-122
Number of pages17
JournalMathematical Methods in the Applied Sciences
Volume37
Issue number1
Early online date28 May 2013
DOIs
Publication statusPublished - 15 Jan 2014
Externally publishedYes

Keywords

  • integral representations, integral operators, integral equation methods
  • linearized elastostatics
  • nonlinear boundary value problems for linear elliptic equations
  • periodically perforated domain
  • real analytic continuation in Banach space
  • singularly perturbed domain

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