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Effective thermal conduction of a Signorini-type problem in composites with rough interface

Sara Monsurrò, Carmen Perugia, Federica Raimondi

TL;DR

This paper analyzes heat conduction in a two-component cylinder separated by a rapidly oscillating rough interface under Signorini-type transmission that couples the jump in temperature to the interface flux via a parameter $\gamma$. It develops a homogenization framework for $\varepsilon$-periodic coefficients and an oscillating interface with amplitude $\varepsilon^k$, casting the problem as a variational inequality on a split domain and deriving uniform a priori bounds. The main result identifies three regimes for $\varepsilon\to 0$ depending on $(k,\gamma)$: (A) a Signorini-type homogenized problem on a flat interface with an effective term $h_{\gamma,k}$, (B) a semi-conductive interface of negligible thickness, and (C) a homogenized Dirichlet problem on the whole cylinder; the homogenized tensor $A^0$ is obtained from a standard cell problem. The analysis hinges on carefully chosen test functions to pass to the limit and on handling products of weakly convergent sequences via lower semicontinuity, providing a rigorous link between micro-roughness and macroscopic thermal properties.

Abstract

We analyse the effect of a Signorini-type interface condition on the asymptotic behaviour, as ε tends to zero, of a problem posed in an open bounded cylinder of {R^N}, {N\geq 2}, divided in two connected components by an imperfect rough surface. The Signorini-type condition is expressed by means of two complementary equalities involving the jump of the solution on the interface and its conormal derivative via a parameter γ. Different limit problems are obtained according to the values of γ and the amplitude of the interface oscillations.

Effective thermal conduction of a Signorini-type problem in composites with rough interface

TL;DR

This paper analyzes heat conduction in a two-component cylinder separated by a rapidly oscillating rough interface under Signorini-type transmission that couples the jump in temperature to the interface flux via a parameter . It develops a homogenization framework for -periodic coefficients and an oscillating interface with amplitude , casting the problem as a variational inequality on a split domain and deriving uniform a priori bounds. The main result identifies three regimes for depending on : (A) a Signorini-type homogenized problem on a flat interface with an effective term , (B) a semi-conductive interface of negligible thickness, and (C) a homogenized Dirichlet problem on the whole cylinder; the homogenized tensor is obtained from a standard cell problem. The analysis hinges on carefully chosen test functions to pass to the limit and on handling products of weakly convergent sequences via lower semicontinuity, providing a rigorous link between micro-roughness and macroscopic thermal properties.

Abstract

We analyse the effect of a Signorini-type interface condition on the asymptotic behaviour, as ε tends to zero, of a problem posed in an open bounded cylinder of {R^N}, {N\geq 2}, divided in two connected components by an imperfect rough surface. The Signorini-type condition is expressed by means of two complementary equalities involving the jump of the solution on the interface and its conormal derivative via a parameter γ. Different limit problems are obtained according to the values of γ and the amplitude of the interface oscillations.
Paper Structure (5 sections, 5 theorems, 50 equations, 2 figures)

This paper contains 5 sections, 5 theorems, 50 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Geometrical setting and functional framework
  3. The $\varepsilon$-problem and its solvability
  4. Existence, uniqueness and uniform a priori estimates
  5. The homogenization result

Key Result

Proposition 2.1

Under hypothesis (A$_g$), let $\{w_\varepsilon \}$ be a family of functions in $W_0^\varepsilon$ such that with $C$ positive constant independent of $\varepsilon$. Then, there exist a subsequence (still denoted $\varepsilon$) and a function $w$ in $W_0^0$ such that

Figures (2)

  • Figure 1: The domain $Q$ with the oscillating interface $\Gamma_\varepsilon$
  • Figure 2: The domain $Q$ with the flat interface $\Gamma_0$

Theorems & Definitions (10)

  • Proposition 2.1: donpiat
  • Proposition 2.2: MPR1
  • Definition 3.1
  • Theorem 3.2
  • proof
  • Proposition 4.1
  • Theorem 4.2
  • proof
  • Remark 4.3
  • Remark 4.4