Approximate impedance of a planar thin layer in couple stress elasticity

Impedance operator of a curved thin layer in linear elasticity with voids

“…The asymptotic technique is used in vast literature for studying the asymptotic behavior in thin layers; see, for instance, earlier studies [12][13][14][15]. This paper is a continuation of the researches of the first author in the modeling of the impedance of thin layers of elastic deformable bodies; see, for instance, prior research [16][17][18][19], where the authors have derived first-order approximations of the impedance in asymmetric elasticity. To begin with, we consider a two-dimensional transmission model problem ( P 𝛿 ) of linear microdilatation elasticity in a bounded domain Ω 𝛿 (Ω 𝛿 ⊂ R 2 ) consisting of two smooth subdomains: an open bounded subset Ω − with disjoint regular boundaries Σ and Γ − , an exterior domain Ω 𝛿 + with disjoint regular boundaries Σ and Γ 𝛿 + (see Figure 1).…”

“…The asymptotic technique is used in vast literature for studying the asymptotic behavior in thin layers; see, for instance, earlier studies [12–15]. This paper is a continuation of the researches of the first author in the modeling of the impedance of thin layers of elastic deformable bodies; see, for instance, prior research [16–19], where the authors have derived first‐order approximations of the impedance in asymmetric elasticity. To begin with, we consider a two‐dimensional transmission model problem $$$ \left({P}^{\delta}\right) $$$ of linear microdilatation elasticity in a bounded domain $$$ {\Omega}^{\delta } $$$ ( $$$ {\Omega}^{\delta}\subset {\mathrm{\mathbb{R}}}^2\Big) $$$ consisting of two smooth subdomains: an open bounded subset $$$ {\Omega}_{-} $$$ with disjoint regular boundaries $$$ \Sigma $$$ and $$$ {\Gamma}_{-} $$$, an exterior domain $$$ {\Omega}_{+}^{\delta } $$$ with disjoint regular boundaries $$$ \Sigma $$$ and $$$ {\Gamma}_{+}^{\delta } $$$ (see Figure 1).…”