Asymptotic interface models in magneto-electro-thermo-elastic composites

Abstract

We study the quasi-static behavior of a magneto-electro-thermo-elastic composite constituted by a thin magneto-electro-thermo-elastic plate-like layer inserted between two generic magneto-electro-thermo-elastic bodies by means of an asymptotic analysis. After defining a small dimensionless parameter $\varepsilon$, which will tend to zero, we characterize two different limit models and their associated limit problems, the so-called weak and strong magneto-electro-thermo-elastic interface models, respectively. Moreover, we identify the non classical magneto-electro-thermo-elastic transmission conditions at the interface between the two three-dimensional bodies and we prove a weak convergence result.

Appendix

In the sequel we define the reduced constitutive coefficients characterizing the strong magneto-electro-thermo-elastic interface model. We recall that $(d_{ij}):=(C_{i3j3})^{-1}$.

$$\begin{aligned}&{C}'_{i\beta j\alpha }:={C}^m_{i\beta j\alpha }-d_{pq}{C}^m_{p3 j\alpha }{C}^m_{q3 i\beta }, P'_{i\alpha j}=P^m_{i\alpha j}-d_{pq}P^m_{ip3}C^m_{j3q\alpha }, \\&R'_{i\alpha j}=R^m_{i\alpha j}-d_{pq}R^m_{ip3}C^m_{j3q\alpha }, {\beta }'_{j\alpha }:= {\beta }^m_{j\alpha } -d_{pq}{C}^m_{p3 j\alpha }{\beta }^m_{ q 3}\\&X'_{ij}:=X^m_{ij}+d_{pq}P^m_{i p3}P^m_{j q3}, M'_{ij}:=M^m_{ij}+d_{pq}R^m_{i p3}R^m_{j q3}, \\&{\alpha }'_{ij} :={\alpha }_{ij}^m +d_{pq}P_{i p3}R_{j q3}, \alpha ''_{\alpha 3}:=\alpha ^m_{\alpha 3}+d_{pq}R^m_{3p3}P^m_{\alpha q3},\\&p'_i:= p_i^m+d_{pq}P^m_{ip3}\beta _{q3}^m, m'_i:= m_i^m+d_{pq}R^m_{ip3}\beta _{q3}^m,\\&{c}'_{v}:=c_v^m+d_{pq}\beta _{p3}^m\beta _{q3}^m, k':=\frac{1}{X'_{33}M'_{33}-(\alpha '_{33})^2},\\&{\tilde{C}}^m_{i\beta j\alpha }:={C}'_{i\beta j\alpha }+k' (M'_{33}P'_{3\beta i}-\alpha '_{33}R'_{3\beta i})P'_{3 \alpha j}+k' (X'_{33}R'_{3\beta i}-\alpha '_{33}P'_{3\beta i})R'_{3 \alpha j},\\&{\tilde{P}}^m_{\beta \alpha j}:={P}'_{\beta \alpha j}-k' (M'_{33}X'_{\beta 3}-\alpha '_{33}\alpha ''_{\beta 3})P'_{3 \alpha j}-k'(X'_{33}\alpha ''_{\beta 3}-\alpha '_{33}X_{\beta 3})R'_{3 \alpha j},\\&{\tilde{R}}^m_{\beta \alpha j}:={R}'_{\beta \alpha j}-k'(M'_{33}\alpha '_{\beta 3}-\alpha '_{33}M'_{\beta 3})P'_{3 \alpha j}-k'(X'_{33}M'_{\beta 3}-\alpha '_{33}\alpha '_{\beta 3})R'_{3 \alpha j},\\&{\tilde{\beta }}^m_{j\alpha }:= {\beta }'_{j\alpha }-k' (M'_{33}p'_{3}-\alpha '_{33}m'_{3})P'_{3 \alpha j}-k'(X'_{33} m'_3-\alpha '_{33}p'_3)R'_{3 \alpha j},\\&{\tilde{X}}^m_{\alpha \beta }:= {X}'_{\alpha \beta }-k' (M'_{33}X'_{\beta 3}-\alpha '_{33}\alpha ''_{\beta 3})X'_{\alpha 3} - k'(X'_{33}\alpha ''_{\beta 3}-\alpha '_{33}X'_{\beta 3})\alpha ''_{\alpha 3}\\&{\tilde{\alpha }}^m_{\alpha \beta }:= {\alpha }'_{\alpha \beta }-k' (M'_{33}\alpha '_{\beta 3}-\alpha '_{33}M'_{\beta 3})X'_{\alpha 3}-k' (X'_{33}M'_{\beta 3}-\alpha '_{33}\alpha '_{\beta 3})\alpha ''_{\alpha 3},\\&{\tilde{M}}^m_{\alpha \beta }:= {M}'_{\alpha \beta }-k' (M'_{33}\alpha '_{\beta 3}-\alpha '_{33}M'_{\beta 3})\alpha '_{\alpha 3} - k'(X'_{33}M'_{\beta 3}-\alpha '_{33}\alpha '_{\beta 3})M'_{\alpha 3},\\&{\tilde{p}}^m_{\alpha }:= {p}'_{\alpha } -k'(M'_{33} p'_3-\alpha '_{33}m'_3)X'_{\alpha 3}-k'(X'_{33} m'_3-\alpha '_{33}p'_3)\alpha ''_{\alpha 3},\\&{\tilde{m}}^m_{\alpha }:= {m}'_{\alpha } -k'(M'_{33} p'_3-\alpha '_{33}m'_3)\alpha '_{\alpha 3}-k'(X'_{33} m'_3-\alpha '_{33}p'_3)M'_{\alpha 3},\\&{\tilde{c}}^m_{v}:= {c}'_{v} -k'(M'_{33} p'_3-\alpha '_{33}m'_3)p'_{3}-k'(X'_{33} m'_3-\alpha '_{33}p'_3)m'_{3},\\&{\tilde{K}}^m_{\alpha \beta } := {K}^m_{\alpha \beta }+\frac{K^m_{\alpha 3}K^m_{\beta 3}}{K^m_{33}}. \end{aligned}$$