Asymptotic Study on a Soft Thin Layer: The Non-Convex Case

Abstract

It is proposed to model the adhesive bonding of elastic bodies when the adhesive is a phase-transforming material. For this purpose, the (isothermal) Frémond model is adopted, including only two variants of martensite. In the first part of this paper, asymptotic expansions are used to study the asymptotic behavior of the adhesive as its thickness and elastic coefficients tend toward zero. In the second part, the energy minimization approach is used and the equilibrium of a one-dimensional bar is studied in detail. The simplified one-dimensional context adopted here makes it possible to compute contact laws taking nucleation and the kinetics of the phase transformation explicitly into account.

Keywords:

• soft thin layers
• asymptotic studies
• non-convex energy

Notes

¹ Here u ^ϵ (x ₁, (ϵ/2)⁺) (resp. u ^ϵ (x ₁, (ϵ/2)⁻) indicates the limit of u ^ϵ (x ₁,x ₂) when x ₂ tends to (ϵ/2), x ₂ ≥ (ϵ/2) (resp. x ₂ ≤ (ϵ/2)).