Narrow Results

In this paper we describe the asymptotic behavior of a problem depending on a small parameter ε>0 and modelling the stationary heat diffusion in a two-component conductor. The flow of heat is proportional to the jump of the temperature field, due to a contact resistance on the interface.

More precisely, we give an homogenization result for the stationary heat equation with oscillating coefficients in a domain of ℝⁿ, where is connected and is union of ε-periodic disconnected inclusions of size ε. These two sub-domains of Ω are separated by a contact surface Γ^ε, on which we prescribe the continuity of the conormal derivatives and a jump of the solution proportional to the conormal derivative, by means of a function of order ε^γ.

We describe the limit problem for γ>-1. The two cases -1<γ≤1 (Theorem 2.1) and γ>1 (Theorem 2.2) need to be treated separately, because of different a priori estimates.

In this study, the distributions of microscopic stresses at a free edge of a unidirectional carbon fibre-reinforced plastic laminate (CFRP laminate) are analysed three-dimensionally, based on a homogenisation theory. For this, the mathematical homogenisation theory for elastic materials is reconstructed for unidirectional CFRP laminates with free edges using a traction-free boundary condition. The present theory is then applied to the analysis of microscopic stress distributions at a free edge of a unidirectional carbon fibre/epoxy laminate subjected to in-plane uniaxial tension. It is shown that microscopic shear stress noticeably occurs at the free edge, especially at the interface region between a carbon fibre and its surrounding epoxy matrix.