# Asymptotic analysis of the transient response of a thermoelastic assembly involving a thin layer

Christian Licht, Somsak Orankitjaroen, Thibaut Weller

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Christian Licht, Somsak Orankitjaroen, Thibaut Weller. Asymptotic analysis of the transient response of a thermoelastic assembly involving a thin layer. Comptes Rendus. Mécanique, 2022, 350, pp.27-45. 10.5802/crmeca.101. hal-03011360v2

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# Comptes Rendus
## Mécanique

Christian Licht, Somsak Orankitjaroen and Thibaut Weller

Asymptotic analysis of the transient response of a thermoelastic
assembly involving a thin layer

Volume 350 (2022), p. 27-45

<https://doi.org/10.5802/crmeca.101>

© Académie des sciences, Paris and the authors, 2022.

Some rights reserved.

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This article is licensed under the

CREATIVE COMMONS ATTRIBUTION 4.0 INTERNATIONAL LICENSE.

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Short paper / *Note*

# Asymptotic analysis of the transient response of a thermoelastic assembly involving a thin layer

Christian Licht<sup>a, b, c</sup>, Somsak Orankitjaroen<sup>b, c</sup> and Thibaut Weller<sup>*</sup>,<sup>a</sup>

<sup>a</sup> LMGC, Université de Montpellier, CNRS, Montpellier, France

<sup>b</sup> Department of Mathematics, Faculty of Science, Mahidol University, Bangkok 10400, Thailand

<sup>c</sup> Centre of Excellence in Mathematics, CHE, Bangkok 10400, Thailand

*E-mails*: christian.licht@umontpellier.fr (C. Licht), somsak.ora@mahidol.ac.th (S. Orankitjaroen), thibaut.weller@umontpellier.fr (T. Weller)

**Abstract.** We study the transient response of a thermoelastic structure made of two three-dimensional bodies connected by a thin adhesive layer. Once more we highlight the powerful flexibility of Trotter's theory of approximation of semi-groups of operators acting on variable spaces: considering the geometrical and physical characteristics of the thin layer as *parameters*, we are able to show in a unitary way that this situation leads to a huge variety of limit models the properties of which are detailed. In particular, according to the relative behaviors of the different parameters involved, *new features are evidenced such as the apparition of an added specific heat coefficient for the interface or of additional thermomechanical state variables* defined not only on the limit geometric interface but on its cartesian product by any interval of real numbers.

**Keywords.** Bonding problems, Linearized thermoelasticity, Transient problems, *m*-Dissipative operators, Asymptotic mathematical modeling, Approximation of semi-groups in the sense of Trotter.

Manuscript received 22nd April 2021, accepted 30th November 2021.

## 1. Setting the problem

We pursue our investigations on thin junctions initiated in [1,2], then further developed in [3–11], and hereafter consider the situation of a *transient multi-physical coupling* within the scope of linear thermoelasticity. As in our previous works, we have chosen to use notations that may seem daunting but have the advantage of conveying all the information necessary to express the complexity of the studied problem.

Because interphases play a crucial role in the analysis of structure assemblies, the studies devoted to bonding problems cover a huge landscape. Within the scope of Mechanics, the

<footnote>* Corresponding author.</footnote>
reader may refer to [12] and the references quoted therein for a good introduction. Rigorous
mathematical approach to dynamical situations, however, are scarce. And because steady state
cases should in fact be considered as particular cases of transient phenomena, the theory of
approximation of semi-groups offers, from our point of view, an almost perfect tool dedicated to
the mathematical modeling in Physics of continuous media. The interested reader will find good
presentations of this theory in classical textbooks such as [13, 14], while Trotter's fundamental
contribution [15] is presented and harnessed in various physical applications in [16]. More
recently this theory has been the subject of a revival (see [17]) particularly because of the large
number of problems it can address. Here, the power of this method will appear in three ways:
first, despite the large number of parameters involved, we are able to carry out a rigorous
mathematical study of this transient problem in a unitary manner (Section 2); second, this
unitary study reveals a very wide variety of limit models (Section 3); third, we are able to extract
new thermomechanical features from our models, such as the appearance of an additional
specific heat coefficient for the interface or additional state variables (Section 4).

Let {$e_1, e_2, e_3$} be an orthonormal basis of $\mathbb{R}^3$ assimilated to the physical Euclidean space. For all $\xi = (\xi_1, \xi_2, \xi_3)$ in $\mathbb{R}^3$, $\tilde{\xi}$ stands for $(\xi_1, \xi_2)$. The space of all $(n \times n)$ symmetric matrices is denoted by $\mathbb{S}^n$ and equipped with the usual inner product and norm denoted by $\cdot$ and $||$ (as in $\mathbb{R}^3$). The space of linear symmetric mappings from $\mathbb{S}^n$ into $\mathbb{S}^n$ is denoted by $\text{Lin}(\mathbb{S}^n)$. For all $\eta$ in $\mathbb{S}^3$, $\hat{\eta}$ stands for the matrix $(\eta_{\alpha\beta})_{1\le\alpha,\beta\le2}$ in $\mathbb{S}^2$. We study the dynamic response of a linearly thermoelastic structure consisting of two adhering bodies connected by a thin adhesive layer and subjected to a given loading. Let $\Omega$ be a domain of $\mathbb{R}^3$ with a Lipschitz-continuous boundary $\partial\Omega$. The intersection of $\Omega$ with $\{x_3 = 0\}$ is a domain $S$ of $\mathbb{R}^2$ with positive two-dimensional Hausdorff measure $\mathcal{H}_2(S)$. Let $\varepsilon$ be a positive number and $\Omega^{\pm} := \Omega \cap \{\pm x_3 > 0\}$, then the adhesive and the adhering bodies occupy $B_{\varepsilon} := S \times (-\varepsilon, +\varepsilon)$ and $\Omega_{\varepsilon}^{\pm} := \Omega^{\pm} \pm \varepsilon e_3$, respectively; we define $\Omega_{\varepsilon} := \Omega_{\varepsilon}^{\pm} \cup \Omega_{\varepsilon}^{\mp}$, $S_{\varepsilon}^{\pm} := S \pm \varepsilon e_3$ and $\tilde{\Omega}_{\varepsilon} := \Omega_{\varepsilon} \cup B_{\varepsilon} \cup S_{\varepsilon}^{\pm} \cup S_{\varepsilon}^{-}$. We consider two partitions $(\Gamma^{\mathcal{M}D}, \Gamma^{\mathcal{M}N})$, $(\Gamma^{\mathcal{T}D}, \Gamma^{\mathcal{T}N})$ of $\partial\Omega$, and for all elements $\Gamma$ of these two partitions, the sets $\Gamma^{\pm}, \Gamma_{\varepsilon}^{\pm}$ and $\Gamma_{\varepsilon}$ respectively denote $\Gamma \cap \{\pm x_3 > 0\}$, $\Gamma^{\pm} \pm \varepsilon e_3$ and $\Gamma_{\varepsilon}^{\pm} \cup \Gamma_{\varepsilon}^{-}$. Moreover we assume that $\mathcal{H}_2(\Gamma^{\mathcal{M}D^{+}})$ and $\mathcal{H}_2(\Gamma^{\mathcal{T}D^{+}})$ are positive. The contact between the adhesive and the two adhering bodies is assumed to be perfect from both thermal and mechanical points of view. The structure is clamped on $\Gamma_{\varepsilon}^{\mathcal{M}D}$, subjected to body forces of density $f_{\varepsilon}$ and surface forces of density $g_{\varepsilon}^{\mathcal{M}}$ on $\Gamma_{\varepsilon}^{\mathcal{M}N}$, it is maintained at a uniform temperature $T_0$ on $\Gamma_{\varepsilon}^{\mathcal{T}D} \cup \gamma^{\mathcal{T}D} \times (-\varepsilon, \varepsilon)$ and subjected to a thermal flux $g_{\varepsilon}^{\mathcal{T}}$ on $\Gamma_{\varepsilon}^{\mathcal{T}N} \cup \gamma^{\mathcal{T}N} \times (-\varepsilon, \varepsilon)$, where $(\gamma^{\mathcal{T}D}, \gamma^{\mathcal{T}N})$ is a partition of $\partial S$. The whole structure is modeled as linearly thermoelastic in the following way. Let $(\rho_L, \mu_L, \beta_L, \kappa_L, \alpha_L)$ in $(0, +\infty)^5$, $a_L$ in $\text{Lin}(\mathbb{S}^3)$, $d := (\rho, \beta, \alpha, \kappa, a)$ in $L^{\infty}(\Omega, \mathbb{R} \times \mathbb{R} \times \mathbb{S}^3 \times \mathbb{S}^3 \times \text{Lin}(\mathbb{S}^3))$ satisfying

$$
\left\{
\begin{array}{l}
\alpha(x) \ge 0 \text{ a.e. } x \in \Omega, \\
\\
\exists c > 0 \text{ s.t. } \rho(x), \beta(x) \ge c, \quad \kappa(x)\xi \cdot \xi \ge c|\xi|^2 \quad \forall \xi \in \mathbb{R}^3, \quad a_L e \cdot e, a(x)e \cdot e \ge c|e|^2 \quad \forall e \in \mathbb{S}^3, \quad \text{a.e. } x \in \Omega.
\end{array}
\right.
\tag{1.1}
$$

The symbols $\rho_L, \beta_L, \alpha_L, \kappa_L, \mu_L a_L$ respectively represent the mass density, the specific heat coeffi-
cient, the thermal dilatation, the thermal conductivity, and the elasticity tensor of the adhesive,
while $d_\varepsilon = (\rho_\varepsilon, \beta_\varepsilon, \alpha_\varepsilon, \kappa_\varepsilon, a_\varepsilon)$ denotes the analogous quantities for the adhering bodies with

$$
d_{\varepsilon}(x) := d(x \mp \varepsilon e_3) \quad \text{a.e. } x \in \Omega_{\varepsilon}^{\pm}. \tag{1.2}
$$

Similarly

$$
\left\{
\begin{array}{ll}
\exists(f, g^{\mathcal{M}}, g^{\mathcal{T}}) \in L^2(\Omega, \mathbb{R}^3) \times L^2(\Gamma^{\mathcal{M}N}, \mathbb{R}^3) \times L^2(\Gamma^{\mathcal{T}N}) \text{ s.t.} \\
f_{\varepsilon}(x) = f(x \mp \varepsilon e_3) & \text{a.e. } x \in \Omega_{\varepsilon}^{\pm}, \quad f_{\varepsilon}(x) = 0 \quad \text{a.e. } x \in B_{\varepsilon}, \\
g_{\varepsilon}^{\mathcal{M}}(x) = g^{\mathcal{M}}(x \mp \varepsilon e_3) & \text{a.e. } x \in \Gamma_{\varepsilon}^{\mathcal{M}N}, \quad g_{\varepsilon}^{\mathcal{M}}(x) = 0 \quad \text{a.e. } x \in \partial S \times (-\varepsilon, \varepsilon), \\
g_{\varepsilon}^{\mathcal{T}}(x) = g^{\mathcal{T}}(x \mp \varepsilon e_3) & \text{a.e. } x \in \Gamma_{\varepsilon}^{\mathcal{T}N}, \quad g_{\varepsilon}^{\mathcal{T}}(x) = 0 \quad \text{a.e. } x \in \gamma^{\mathcal{T}N} \times (-\varepsilon, \varepsilon).
\end{array}
\right.
\tag{1.3}
$$
Thus the problem ($\mathcal{P}_s$) of determining the evolution in the framework of small perturbations of the assembly, whose state is denoted by $U_s = (u_s, v_s, \theta_s)$, $u_s, v_s, \theta_s$ being the fields of displacement, velocity and temperature increment with respect to $T_0$, involves a sextuplet $s := (\epsilon, \rho_L, \mu_L, \beta_L, \kappa_L, \alpha_L)$ of data so that all the fields will be thereafter indexed by $s$. If $U_s^0 = (u_s^0, v_s^0, \theta_s^0)$ is the given initial state, a formulation of ($\mathcal{P}_s$) could be:

$$ (\mathcal{P}_s) \quad \left\{ \begin{array}{l} \text{Find } U_s \text{ sufficiently smooth in } \mathcal{O}_\epsilon \times [0, T] \text{ s.t.} \\ \qquad u_s = 0 \text{ on } \Gamma_\epsilon^{\mathcal{M}D}, \theta_s = 0 \text{ on } \Gamma_\epsilon^{\mathcal{T}D} \cup \gamma^{\mathcal{T}D} \times (-\epsilon, \epsilon), U_s(0) = U_s^0 \text{ satisfying:} \\ \qquad \int_{\Omega_\epsilon} \rho_\epsilon \frac{\partial v_s}{\partial t} \cdot v' + a_\epsilon (e(u_s) - \theta_s \alpha_\epsilon) \cdot e(v') \,dx + \int_{B_\epsilon} \rho_L \frac{\partial v_s}{\partial t} \cdot v' + \mu_L a_L (e(u_s) - \alpha_L \theta_s I) \cdot e(v') \,dx \\ = \int_{\Omega_\epsilon} f_\epsilon \cdot v' \,dx + \int_{\Gamma_\epsilon^{\mathcal{M}N}} g_\epsilon^{\mathcal{M}} \cdot v' \,d\mathcal{H}_2, \\ \qquad \int_{\Omega_\epsilon} \beta_\epsilon \frac{\partial \theta_s}{\partial t} \theta' + \kappa_\epsilon \nabla \theta_s \cdot \nabla \theta' + (a_\epsilon \alpha_\epsilon \cdot e(v_s)) \theta' \,dx \\ \qquad + \int_{B_\epsilon} \beta_L \frac{\partial \theta_s}{\partial t} \theta' + \kappa_L \nabla \theta_s \cdot \nabla \theta' + \mu_L \alpha_L (a_L I \cdot e(v_s)) \theta' \,dx = \int_{\Gamma_\epsilon^{\mathcal{T}N}} g_\epsilon^{\mathcal{T}} \theta' \,d\mathcal{H}_2, \\ \text{for all } (v', \theta') \text{ sufficiently smooth in } \mathcal{O}_\epsilon \text{ and vanishing on } \Gamma_\epsilon^{\mathcal{M}D} \times (\Gamma_\epsilon^{\mathcal{T}D} \cup \gamma^{\mathcal{T}D} \times (-\epsilon, \epsilon)), \end{array} \right. $$

where $t$ denotes the time, $e(u)$ is the linearized strain associated with the field of displacement $u$, and $I$ is the identity matrix of $\mathbb{S}^3$.

## 2. Existence and uniqueness of a solution to ($\mathcal{P}_s$)

Assuming

$$ (f, g^{\mathcal{M}}, g^{\mathcal{T}}) \in C^{0,1}([0, T], L^2(\Omega, \mathbb{R}^3)) \times C^{1,1}([0, T], L^2(\Gamma^{\mathcal{M}N}, \mathbb{R}^3)) \times C^{1,1}([0, T], L^2(\Gamma^{\mathcal{T}N})) \quad (H1) $$

we seek $z_s = (u_s, \theta_s)$ in the form

$$ z_s = z_s^e + z_s^r, \quad (2.1) $$

where $z_s^e$ is the unique solution to

$$ z_s^e(t) \in Z_s; \quad \Phi_s(z_s^e(t), z') = L_\epsilon(t)(z') \quad \forall z' \in Z_s, \forall t \in [0, T], \quad (2.2) $$

with

$$ Z_s := H_{\Gamma_\epsilon^{\mathcal{M}D}}^1(\mathcal{O}_\epsilon, \mathbb{R}^3) \times H_{\Gamma_\epsilon^{\mathcal{T}D}}^1(\mathcal{O}_\epsilon), \quad (2.3) $$

where for all open set $G$ of $\mathbb{R}^N$, $1 \le N \le 3$, $H_\gamma^1(G, \mathbb{R}^N)$ denotes the subset of the Sobolev space $H^1(G, \mathbb{R}^N)$ of elements with vanishing trace on $\gamma$ included in $\partial G$. Let

$$ \Phi_s(z, z') := (u, u')_{1,s} + (\theta, \theta')_{4,s} - (u', \theta)_{5,s} + (u, \theta')_{5,s} \quad \forall z = (u, \theta), \forall z' = (u', \theta') \in Z_s, \quad (2.4) $$

$$ \left\{ \begin{array}{ll} (u, u')_{1,s} := \displaystyle\int_{\Omega_{\epsilon}} a_{\epsilon} e(u) \cdot e(u') \,dx + \displaystyle\int_{B_{\epsilon}} \mu_{L} a_{L} e(u) \cdot e(u') \,dx & \forall u, u' \in H_{\Gamma_{\epsilon}}^{1}(\mathcal{O}_{\epsilon}, \mathbb{R}^{3}), \\ (\theta, \theta')_{4,s} := \displaystyle\int_{\Omega_{\epsilon}} \kappa_{\epsilon} \nabla \theta \cdot \nabla \theta' \,dx + \displaystyle\int_{B_{\epsilon}} \kappa_{L} \nabla \theta \cdot \nabla \theta' \,dx & \forall \theta, \theta' \in H_{\Gamma_{\epsilon}}^{1}(\mathcal{O}_{\epsilon}), \\ (u, \theta)_{5,s} := \displaystyle\int_{\Omega_{\epsilon}} (a_{\epsilon} \alpha_{\epsilon} \cdot e(u)) \theta \,dx + \displaystyle\int_{B_{\epsilon}} \mu_{L} \alpha_{L} (a_{L} I \cdot e(u)) \theta \,dx & \forall (u, \theta) \in H_{\Gamma_{\epsilon}}^{1}(\mathcal{O}_{\epsilon}, \mathbb{R}^{3}) \times H_{\Gamma_{\epsilon}}^{1}(\mathcal{O}_{\epsilon}), \end{array} \right. \quad (2.5) $$

and

$$ L_{\epsilon}(t)(z') := \int_{\Gamma_{\epsilon}^{\mathcal{M}N}} g_{\epsilon}^{\mathcal{M}} \cdot u' \, d\mathcal{H}_2 + \int_{\Gamma_{\epsilon}^{\mathcal{T}N}} g_{\epsilon}^{\mathcal{T}} \theta' \, dx \quad \forall z' \in Z_s, \forall t \in [0, T]. \quad (2.6) $$

The bilinear forms $((\cdot), \cdot)_{1,s}$, $((\cdot), \cdot)_{4,s}$ and $((\cdot), \cdot)_{5,s}$ are associated with the strain energy, the thermal dissipation and the thermoelastic coupling respectively.
As $(g^{\mathcal{M}}, g^{\mathcal{T}}) \mapsto z_s^e$ is linear continuous from $L^2(\Gamma^{\mathcal{M}N}, \mathbb{R}^3) \times L^2(\Gamma^{\mathcal{T}N})$ into $Z_s$, we have

$$z_s^e \in C^{1,1}([0, T], Z_s). \qquad (2.7)$$

The remaining part $z_s^r$ of $z_s$ will be involved in an evolution equation governed by a $m$-dissipative operator $A_s$ in a Hilbert space $\mathfrak{H}_s$ of possible states with finite thermomechanical (strain, kinetic, thermal) energy defined by

$$\mathfrak{H}_s := H_{\Gamma^{\mathcal{M}D}}^1(\mathcal{O}_E, \mathbb{R}^3) \times L^2(\mathcal{O}_E, \mathbb{R}^3) \times L^2(\mathcal{O}_E), \qquad (2.8)$$

and endowed with the following inner product and norm:

$$(U, U')_s := (u, u')_{1,s} + K_s((v, \theta), (v', \theta')), \quad |U|_s^2 := (U, U)_s, \qquad (2.9)$$

$$K_s((v, \theta), (v', \theta')) = (v, v')_{2,s} + (\theta, \theta')_{3,s} \quad \forall U = (u, v, \theta), \forall U' = (u', v', \theta') \in \mathfrak{H}_s, \quad \text{with} \quad (2.10)$$

$$\left\{ \begin{array}{l} (v, v')_{2,s} := \int_{\Omega_E} \rho_E v \cdot v' \, dx + \int_{B_E} \rho_L v \cdot v' \, dx, \\ (\theta, \theta')_{3,s} := \int_{\Omega_E} \beta_E \theta \theta' \, dx + \int_{B_E} \beta_L \theta \theta' \, dx. \end{array} \right. \qquad (2.11)$$

Operator $A_s$ is defined by

$$\left\{ \begin{array}{l} D(A_s) := \left\{ U = (u, v, \theta) \in \mathfrak{H}_s; \begin{array}{l} \text{(i)} \quad (v, \theta) \in Z_s \text{ and} \\ \text{(ii)} \quad \exists! (w, \tau) \in L^2(\mathcal{O}_E, \mathbb{R}^3) \times L^2(\mathcal{O}_E) \text{ s.t. } \forall (v', \theta') \in Z_s: \\ \qquad (w, v')_{2,s} + (u, v')_{1,s} - (v', \theta)_{5,s} = 0, \\ \qquad (\tau, \theta')_{3,s} + (\theta, \theta')_{4,s} + (v, \theta')_{5,s} = 0, \end{array} \right\} \\ A_s U = (v, w, \tau). \end{array} \right. \qquad (2.12)$$

It is straightforward to check the following.

**Proposition 2.1.** Operator $A_s$ is $m$-dissipative and, for all $\phi_s = (\phi_s^1, \phi_s^2, \phi_s^3)$ in $\mathfrak{H}_s$,

$$\left\{ \begin{array}{l} \bar{U}_s = (\bar{u}_s, \bar{v}_s, \bar{\theta}_s) \text{ s.t.} \\ \bar{U}_s - A_s \bar{U}_s = \phi_s \end{array} \right. \quad \Longleftrightarrow \quad \left\{ \begin{array}{l} \bar{u}_s = \bar{v}_s + \phi_s^1 \\ \bar{z}_s = (\bar{v}_s, \bar{\theta}_s) \in Z_s; \quad \Psi_s(\bar{z}_s, z) = \mathcal{L}_s(z) \quad \forall z = (v, \theta) \in Z_s \\ \text{with} \\ \Psi_s := \Phi_s + K_s \\ \mathcal{L}_s(z) := -(\phi_s^1, u)_{1,s} + K_s((\phi_s^2, \phi_s^3), z) \quad \forall z = (v, \theta) \in Z_s. \end{array} \right. \qquad (2.13)$$

Then, taking into account (H1), (2.1), (2.2), (2.7), (2.12), it is clear that $(\mathcal{P}_s)$ is “formally equivalent” to

$$\left\{ \begin{array}{l} \displaystyle \frac{dU_s^r}{dt} - A_s U_s^r = F_s := \left( u_s^e - \frac{du_s^e}{dt}, -\frac{du_s^e}{dt} + f_s, -\frac{d\theta_s^e}{dt} \right), \\ U_s^r(0) = U_s^0 - (u_s^e(0), 0, \theta_s^e(0)), \end{array} \right. \qquad (2.14)$$

with $f_s$ equal to $f_E / \rho_E$ in $\Omega_E$ and 0 in $B_E$. So the Hille-Yosida theorem (see [13]) leads to:

**Theorem 2.1.** If $(f, g^{\mathcal{M}}, g^{\mathcal{T}})$ satisfies (H1) and $U_s^0$ belongs to $(u_s^e(0), 0, \theta_s^e(0)) + D(A_s)$, then (2.14) has a unique solution such that $U_s^r$ belongs to $C^1([0, T], \mathfrak{H}_s)$. Hence there exists a unique $(u_s, \theta_s)$ in $(C^1([0, T], H_{\Gamma^{\mathcal{M}D}}^1(\mathcal{O}_E, \mathbb{R}^3)) \cap C^2([0, T], L^2(\mathcal{O}_E, \mathbb{R}^3))) \times (C^1([0, T], L^2(\mathcal{O}_E)) \cap C^0([0, T], H_{\Gamma^{\mathcal{T}D}}^1(\mathcal{O}_E)))$, which does satisfy

$$\left( \begin{array}{l} \displaystyle \left( \frac{d^2 u_s}{dt^2}, u \right)_{2,s} + (u_s, u)_{1,s} - (u, \theta_s)_{5,s} = \int_{\Omega_E} f_E \cdot u \, dx + \int_{\Gamma_E^{\mathcal{M}N}} g_E^{\mathcal{M}} \cdot u \, d\mathcal{H}_2 \quad \forall u \in H_{\Gamma_E^{\mathcal{M}D}}^1(\mathcal{O}_E, \mathbb{R}^3), \\ \displaystyle \left( \frac{d\theta_s}{dt}, \theta \right)_{3,s} + (\theta_s, \theta)_{4,s} + \left( \frac{du_s}{dt}, \theta \right)_{5,s} = \int_{\Gamma_E^{\mathcal{T}N}} g_E^{\mathcal{T}} \theta \, d\mathcal{H}_2 \quad \forall \theta \in H_{\Gamma_E^{\mathcal{T}D}}^1(\mathcal{O}_E). \end{array} \right. \qquad (2.15)$$
We set

$$ U_s^e := (u_s^e, z_s^e). \qquad (2.16) $$

### 3. A mathematical analysis of the asymptotic behavior

Now we regard the sextuplet $s$ of geometrical and thermomechanical data as a sextuplet of parameters taking values in a countable subset of $(0, +\infty)^6$ with a single cluster point $\bar{s}$ in $\{0\} \times [0, +\infty]^5$ and study the asymptotic behavior of $U_s$ in order to suggest a simplified but accurate enough model for the genuine physical situation. We will show that, depending on the relative behavior of $(\rho_L, \mu_L, \beta_L, \kappa_L, \alpha_L)$ with respect to $\epsilon$, numerous (100!) limit models appear. They are indexed by $I = (I_M, I_T) \in (\{0, 1\} \times \{0, 1, 2, 3, 4\})^2$, $I_M = (I_{M1}, I_{M2})$, $I_T = (I_{T1}, I_{T2})$ defined as follows. Let $r_0^M = r_1^M = r_2^M = 1$, $r_3^M = -1$, $r_4^M = -3$, $r_0^T = r_1^T = r_2^T = 1$, $r_3^T = r_4^T = -1$, we assume:

$$ \left\{ \begin{array}{l} \text{There exists } (\bar{\rho}_L, \bar{\beta}_L, \bar{\mu}_L^I, \bar{\kappa}_L^I) \text{ in } [0, +\infty)^2 \times [0, +\infty]^2 \text{ such that:} \\ (\bar{\rho}_L, \bar{\beta}_L) := \lim_{s \to \bar{s}} \epsilon (\rho_L, \beta_L), \\ (\bar{\mu}_L^I, \bar{\kappa}_L^I) := \lim_{s \to \bar{s}} \left( \mu_L / \epsilon r_{I_{M2}}^M, \kappa_L / \epsilon r_{I_{T2}}^T \right) \text{ with } \overline{\lim}_{s \to \bar{s}} \epsilon^2 (\mu_L^{-1}, \kappa_L^{-1}) \in [0, +\infty)^2, \\ \text{and} \\ I_{M1}, I_{T1} = 0 \text{ if } \bar{\rho}_L, \bar{\beta}_L = 0, \quad I_{M1}, I_{T1} = 1 \text{ if } \bar{\rho}_L > 0, \bar{\beta}_L > 0, \\ \bar{\mu}_L^I = 0 \text{ and } \mathcal{H}_2(\Gamma^{\mathcal{H}D^-}) > 0 \text{ when } I_{M2} = 0, \quad \bar{\mu}_L^I \in (0, +\infty) \text{ when } I_{M2} = 1, \\ \bar{\mu}_L^I = +\infty \text{ and } \lim_{s \to \bar{s}} \epsilon \mu_L = 0 \text{ when } I_{M2} = 2, \quad \bar{\mu}_L^I \in (0, +\infty) \text{ when } I_{M2} = 3, 4, \\ \bar{\kappa}_L^I = 0 \text{ and } \mathcal{H}_2(\Gamma^{\mathcal{H}D^-}) > 0 \text{ when } I_{T2} = 0, \quad \bar{\kappa}_L^I \in (0, +\infty) \text{ when } I_{T2} = 1, \\ \bar{\kappa}_L^I = +\infty \text{ and } \lim_{s \to \bar{s}} \epsilon \kappa_L = 0 \text{ when } I_{T2} = 2, \quad \bar{\kappa}_L^I \in (0, +\infty) \text{ when } I_{T2} = 3, \\ \bar{\kappa}_L^I = +\infty \text{ when } I_{T2} = 4. \end{array} \right. \qquad (H2) $$

The physical properties of the adhesive layer corresponding to the various values of $I$ will be conveyed in Section 4 through brief comments.

#### 3.1. A candidate for the limit behavior

From now on, $C$ denotes various constants which may vary from line to line and we use the convention $0 \times \infty = \infty \times 0 = 0$.

##### 3.1.1. The limit space $\mathcal{H}_2^1$

This candidate could be determined by studying the asymptotic behavior of sequences with bounded total thermomechanical energy. For the sake of notation simplicity, and when no confusion ensues, we will use the same symbols $U_s$, $u_s$, $v_s$ and $\theta_s$ to denote both the elements of general sequences and the solution to $(\mathcal{P}_s)$. It will appear that in some cases the thermomechanical state of the “limit structure”, where the three-dimensional adhesive layer is geometrically replaced by the surface $S$ it shrinks to, does not involve the sole state variables of the adhering bodies but additional thermomechanical state variables not necessarily defined on $S$ but in $B := S \times (-1, 1)$ which accounts for the limit behavior of the adhesive layer.

It is convenient to introduce the following “scaling operators” which transform a field $y_s$ defined on $B_\epsilon$ into a field $y_{sB}$ defined on $B$ in such a way that a bounded energy for $y_s$ is equivalent to a bounded “scaled” energy for $y_{sB}$:
Operators $\mathcal{S}_{\varepsilon}^{u,\mathrm{IM2}}$:

$u \in H^1(B_{\varepsilon}, \mathbb{R}^3) \mapsto u_B = \mathcal{S}_{\varepsilon}^{u,\mathrm{IM2}} u \in H^1(B, \mathbb{R}^3)$ defined by

$$
\left\{
\begin{array}{ll}
\mathrm{I}_{\mathrm{M2}} = 0, 1, 2 & u_B(x) = u(\hat{x}, \varepsilon x_3) \quad \text{a.e. } x \in B, \\
\text{and } e_{ij}^{\mathrm{IM2}}(\varepsilon, u_B) := \begin{cases} \varepsilon e_{ij}(u_B), & 1 \le i, j \le 2 \\ (\varepsilon \partial_i u_{B3} + \partial_3 u_{Bi})/2, & i=1,2, j=3 \\ \partial_3 u_{B3}, & i=j=3, \end{cases} \\
\mathrm{I}_{\mathrm{M2}} = 3 & \hat{u}_B(x) = \hat{u}(\hat{x}, \varepsilon x_3), \quad u_{B3}(x) = \varepsilon u_3(\hat{x}, \varepsilon x_3) \quad \text{a.e. } x \in B, \\
\text{and } e_{ij}^3(\varepsilon, u_B) := \begin{cases} e_{ij}(u_B), & 1 \le i, j \le 2 \\ (1/\varepsilon)e_{i3}(u_B), & 1 \le i \le 2, j=3 \\ (1/\varepsilon^2)e_{33}(u_B), & i=j=3, \end{cases} \\
\mathrm{I}_{\mathrm{M2}} = 4 & \hat{u}_B(x) = \hat{u}(\hat{x}, \varepsilon x_3)/\varepsilon, \quad u_{B3}(x) = u_3(\hat{x}, \varepsilon x_3) \quad \text{a.e. } x \in B, \\
& \text{and } e_{ij}^4(\varepsilon, u_B) = e_{ij}^3(\varepsilon, u_B).
\end{array}
\right.
\tag{3.1}
$$

Clearly:

$$
\mu_L \int_{B_\varepsilon} |e(u)|^2 \, dx = \left( \mu_L / \varepsilon^{r_{\mathrm{IM2}}} \right) \int_B |e^{\mathrm{IM2}}(\varepsilon, \mathcal{S}_\varepsilon^{u,\mathrm{IM2}} u)|^2 \, dx. \quad (3.2)
$$

Operators $\mathcal{S}_{\varepsilon}^{v}, \mathcal{S}_{\varepsilon}^{\theta}$:

$(v, \theta) \in L^2(B_{\varepsilon}, \mathbb{R}^3 \times \mathbb{R}) \mapsto (v_B, \theta_B) = (\mathcal{S}_{\varepsilon}^v v, \mathcal{S}_{\varepsilon}^{\theta} \theta) \in L^2(B, \mathbb{R}^3 \times \mathbb{R})$ defined by

$$
(v_B(x), \theta_B(x)) = (v(\hat{x}, \varepsilon x_3), \theta(\hat{x}, \varepsilon x_3)) \quad \text{a.e. } x \in B, \quad (3.3)
$$

which satisfy

$$
\left( \rho_L \int_{B_\epsilon} |v|^2 dx, \beta_L \int_{B_\epsilon} |\theta|^2 dx \right) = \left( \rho_L \epsilon \int_B |\mathcal{S}_\epsilon^v v|^2 dx, \beta_L \epsilon \int_B |\mathcal{S}_\epsilon^\theta \theta|^2 dx \right). \quad (3.4)
$$

In view of following Proposition 3.1, it is natural to recall some classical notions. Let

$$
\Omega^i := \Omega \setminus S \quad \text{if } i=0,1, \quad \Omega^i := \Omega \quad \text{if } i=2,3,4. \tag{3.5}
$$

For an element $y$ of $H^1(\Omega \setminus S, \mathbb{R}^N)$, $1 \le N \le 3$, we will denote its restrictions to the open sets $\Omega^{\pm}$ by $y^{\pm}$ which is an element of $H^1(\Omega^{\pm}, \mathbb{R}^N)$. The symbols $\gamma_S(y^+)$ and $\gamma_S(y^-)$ will denote the trace of $y^+$ and $y^-$, respectively, on $S$. Of course, for $y$ in $H^1(\Omega, \mathbb{R}^N)$, $\gamma_S(y)$ will denote the trace of $y$ on $S$. We also use

$$
\llbracket y \rrbracket := \gamma_S(y^+) - \gamma_S(y^-), \tag{3.6}
$$

$$
H_{\partial_3}(B, \mathbb{R}^N) := \{y \in L^2(B, \mathbb{R}^N) \text{ s.t. } \partial_3 y \in L^2(B, \mathbb{R}^N)\} \quad 1 \le N \le 3, \qquad (3.7)
$$

it is well known that a continuous mapping $\gamma_{S^{\pm}}$ is defined on $H_{\partial_3}(B, \mathbb{R}^N)$ for the traces on $S^{\pm} := S \pm e_3$ with values in $L^2(S^{\pm}, \mathbb{R}^N)$, and, from now on, $\gamma_{S^{\pm}}(y)$ is treated as an element of $L^2(S, \mathbb{R}^N)$,

$$
\begin{align}
V_{\mathrm{KL}}(B) &:= \{u \in H^1(B, \mathbb{R}^3); \exists (u^M, u^F) \in H^1(S, \mathbb{R}^2) \times H^2(S) \text{ s.t.} \nonumber \\
&\qquad \hat{u}(x) = u^M(\hat{x}) - x_3 \hat{V} u^F(\hat{x}), \quad u_3(x) = u^F(\hat{x}) \text{ a.e. } x \in B\} \tag{3.8} \\
&= \{u \in H^1(B, \mathbb{R}^3); e_{i3} = 0 \text{ a.e. in } B, 1 \le i \le 3\}. \nonumber
\end{align}
$$

We will use the following Hilbert spaces and norms:

spaces of displacement fields $\mathcal{H}_u^1$:

$$
\bullet \quad \mathrm{I}_{\mathrm{M2}} = 0 \quad \mathcal{H}_u^1 := \left\{ u = (u_\Omega, e_B^u) \in H_{\Gamma, \mathcal{A}^D}^1 (\Omega^0, \mathbb{R}^3) \times \{0\} \right\},
$$

• $\mathrm{I}_{\mathrm{M2}} = 1$ $\mathcal{H}_u^1 := \{u = (u_\Omega, e_B^u) \in H_{\Gamma, \mathcal{A}^D}^1 (\Omega^1, \mathbb{R}^3) \times L^2(B, \mathbb{S}^3) \text{ s.t.}$

$$
\exists u_B \in H_{\partial_3}(B, \mathbb{R}^3); e_B^u = \partial_3 u_B \otimes_S e_3, \gamma_{S^\pm}(u_B) = \gamma_S(u_\Omega^\pm)
$$

• $\mathrm{I}_{\mathrm{M2}} = 2 \quad \mathcal{H}_u^1 := \{u = (u_\Omega, e_B^u) \in H_{\Gamma, \mathcal{A}^D}^1 (\Omega^1, \mathbb{R}^3) \times L^2(B, \mathbb{S}^3) \text{ s.t.}$

$$
\exists u_B \in H^1(B, \mathbb{R}^3); 0 = e_B^u = \partial_3 u_B \otimes_S e_3, \gamma_S^\pm(u_B) = \gamma_S(u_\Omega)
$$
• $I_{M2} = 3$ $\mathcal{H}_u^1 := \{u = (u_\Omega, e_B^u) \in H_{\Gamma, \mathcal{M}^D}^1(\Omega, \mathbb{R}^3) \times L^2(B, \mathbb{S}^3) \text{ s.t.} \\ \exists u_B \in V_{KL}(B); u_B^M = \overline{\gamma_S(u_\Omega)}, u_B^F = 0, \widehat{e_B^u} := \overline{e(u_B)}\}$,

• $I_{M2} = 4$ $\mathcal{H}_u^1 := \{u = (u_\Omega, e_B^u) \in H_{\Gamma, \mathcal{M}^D}^1(\Omega, \mathbb{R}^3) \times L^2(B, \mathbb{S}^3) \text{ s.t.} \\ \widehat{e}(\overline{\gamma_S(u_\Omega)}) = 0, \exists u_B \in V_{KL}(B); u_B^F = \gamma_S(u_{\Omega 3}), \widehat{e_B^u} := \overline{e(u_B)}\}$,

and they clearly are complete relatively to the inner product

$$ (u, u')_1^1 := \int_{\Omega^{1M2}} ae(u_\Omega) \cdot e(u'_\Omega) \,dx + \bar{\mu}_L^1 \int_B a_L e_B^u \cdot e_B^{u'} \,dx \quad \forall I_{M2} = 0, 1, \dots, 4. \quad (3.9) $$

Note that for a field $y$ in $H^1(S, \mathbb{R}^2)$ we also denote the symmetrized gradient of $y$ by $\hat{e}(y)$.
spaces of velocity fields $\mathcal{H}_v^1$:

• $I_{M1} = 0$ $\mathcal{H}_v^1 := \{v = v_\Omega \in L^2(\Omega, \mathbb{R}^3)\}$, $(v, v')_2^1 := \int_\Omega \rho v_\Omega \cdot v'_\Omega \,dx$,

• $I_{M1} = 1$ $\mathcal{H}_v^1 := \{v = (v_\Omega, v_B) \in L^2(\Omega, \mathbb{R}^3) \times L^2(B, \mathbb{R}^3)\}$, $(v, v')_2^1 := \int_\Omega \rho v_\Omega \cdot v'_\Omega \,dx + \bar{\rho}_L \int_B v_B \cdot v'_B \,dx$,

spaces of temperature fields $\mathcal{H}_\theta^1$:

• $I_{T1} = 0$ $\mathcal{H}_\theta^1 := \{\theta = \theta_\Omega \in L^2(\Omega)\}$, $(\theta, \theta')_3^1 := \int_\Omega \beta \theta_\Omega \theta'_\Omega \,dx$,

• $I_{T1} = 1$ $\mathcal{H}_\theta^1 := \{\theta = (\theta_\Omega, \theta_B) \in L^2(\Omega) \times L^2(B)\}$, $(\theta, \theta')_3^1 := \int_\Omega \beta \theta_\Omega \theta'_\Omega \,dx + \bar{\rho}_L \int_B \theta_B \theta'_B \,dx$,

spaces of limit states $\mathcal{H}^1$:

$$ \begin{gathered} \mathcal{H}^1 := \mathcal{H}_u^1 \times \mathcal{H}_v^1 \times \mathcal{H}_\theta^1, \quad (3.10) \\ (U, U')^1 := (u, u')_1^1 + (v, v')_2^1 + (\theta, \theta')_3^1, \quad |U|^1 := [(U, U)^1]^{1/2}. \end{gathered} $$

So, if $T_\epsilon$ is the operator from $L^2(\Omega_\epsilon, \mathbb{R}^N)$ into $L^2(\Omega, \mathbb{R}^N)$, $N=1$ or $3$, defined by

$$ (T_\epsilon y)(x) := y(x \pm \epsilon e_3), \quad \forall x \in \Omega^\pm, \quad \forall y \in L^2(\Omega_\epsilon, \mathbb{R}^N), \quad (3.11) $$

we have

**Proposition 3.1.** For all sequences $U_s = (u_s, v_s, \theta_s)$ in $\mathcal{H}_s$ such that $|U_s|_s$ is bounded, there exist $U = (u, v, \theta)$ in $\mathcal{H}^1$ and a not relabeled subsequence such that

(i) $(T_\epsilon u_s, T_\epsilon v_s, T_\epsilon \theta_s)$ weakly converges in $H_{\Gamma, \mathcal{M}^D}^1(\Omega \setminus S, \mathbb{R}^3) \times L^2(\Omega, \mathbb{R}^3) \times L^2(\Omega)$ toward $(u_\Omega, v_\Omega, \theta_\Omega)$;

(ii) $\mathcal{S}_\epsilon^\nu v_s$ weakly converges in $L^2(B, \mathbb{R}^3)$ toward $v_B$ if $I_{M1} = 1$, $\mathcal{S}_\epsilon^\theta \theta_s$ weakly converges in $L^2(B)$ toward $\theta_B$ if $I_{T1} = 1$;

(iii) $e^{I_{M2}}(\epsilon, \mathcal{S}_\epsilon^{u, I_{M2}} u_s)$ weakly converges in $L^2(B, \mathbb{R}^3)$ toward $e_B^u$ when $I_{M2}$ is positive;

(iv) $|U|^1 \le \lim_{s \to \bar{s}} |U_s|_s$.

**Proof.** As

$$ \begin{aligned} |U_s|^2 = & \int_{\Omega} ae(T_{\epsilon} u_s) \cdot e(T_{\epsilon} u_s) + \beta |T_{\epsilon} v_s|^2 + \rho (T_{\epsilon} \theta_s)^2 \,dx \\ & + \int_B (\mu_L / r_{I_{M2}}^M) a_L e^{I_{M2}} (\epsilon, \mathcal{S}_\epsilon^{u, I_{M2}} u_s) \cdot e^{I_{M2}} (\epsilon, \mathcal{S}_\epsilon^{u, I_{M2}} u_s) + \epsilon \rho_L |\mathcal{S}_\epsilon^\nu v_s|^2 + \epsilon \beta_L |\mathcal{S}_\epsilon^\theta \theta_s|^2 \,dx, \end{aligned} \quad (3.12) $$

point (ii), and point (i) when $\mathcal{H}_2(\Gamma^{\mathcal{M}^D}) > 0$ are obvious.

Anyway the boundedness of $(u_s, u_s)_{1,s}$ implies that there exists $u_\Omega^+$ in $H_{\Gamma, \mathcal{M}^D}^1(\Omega^+, \mathbb{R}^3)$ and a not relabeled subsequence such that $((T_\epsilon u_s)^+, \gamma_S((T_\epsilon u_s)^+))$ converges weakly in $H^1(\Omega^+, \mathbb{R}^3)$ and strongly in $L^2(S, \mathbb{R}^3)$ toward $(u_\Omega^+, \gamma_S(u_\Omega^+))$, respectively. By using Korn inequality and a cutoff
function $\eta$ such that $\eta(x_3) = 1$ if $0 \le x_3 \le L/3$, $\eta(x_3) = 0$ if $x_3 \ge 2L/3$, with $L = \max\{x_3; (\hat{x}, x_3) \in \partial\Omega^+\}$, one has:

$$ \|\llbracket T_\epsilon u_s \rrbracket\|_{L^2(S, \mathbb{R}^3)}^2 \le C \left( \frac{2\epsilon}{\mu_L} \mu_L \int_{B_\epsilon} |e(u_s)|^2 dx + \epsilon \right), \quad (3.13) $$

$$ \int_{B_\epsilon} |u_s|^2 dx \le C \left( \epsilon |\gamma_S((T_\epsilon u_s)^+)|_{L^2(S, \mathbb{R}^3)}^2 + \frac{\epsilon^2}{\mu_L} \mu_L \int_{B_\epsilon} |e(u_s)|^2 dx + \epsilon^2 \right). \quad (3.14) $$

Hence assumption (H2) implies that there exist $u_{\Omega}^{-}$ in $H_{\Gamma, \mathcal{D}^{-}}^{1}(\Omega^{-}, \mathbb{R}^{3})$ and a not relabeled subsequence such that $(T_{\epsilon}u_{s})^{-}$ converges weakly in $H^{1}(\Omega^{-}, \mathbb{R}^{3})$ toward $u_{\Omega}^{-}$ and

$$ \int_{B_{\epsilon}} |u_s|^2 dx \le C \left( \epsilon + \frac{\epsilon^2}{\mu_L} \right). \quad (3.15) $$

So, if $\mathrm{I}_{\mathrm{M2}} = 1, 2$, $(\mathcal{S}_{\epsilon}^{u, \mathrm{I}_{\mathrm{M2}}} u_s, (\mu_L / \epsilon^{\mathrm{I}_{\mathrm{M2}}}) e^{\mathrm{I}_{\mathrm{M2}}} (\epsilon, \mathcal{S}_{\epsilon}^{u, \mathrm{I}_{\mathrm{M2}}} u_s))$ is bounded in $L^2(B, \mathbb{R}^3 \times \mathbb{S}^3)$ so that the convergence up to a not relabeled subsequence of $\mathcal{S}_{\epsilon}^{u, \mathrm{I}_{\mathrm{M2}}} u_s$ in the sense of distributions on $B$ yields that there exists a unique $u_B$ in $H_{\partial_3}(B, \mathbb{R}^3)$ such that $(\mathcal{S}_{\epsilon}^{u, \mathrm{I}_{\mathrm{M2}}} u_s, (\mathcal{S}_{\epsilon}^{u, \mathrm{I}_{\mathrm{M2}}} u_s)_3, e^{\mathrm{I}_{\mathrm{M2}}} (\epsilon, \mathcal{S}_{\epsilon}^{u, \mathrm{I}_{\mathrm{M2}}} u_s))$ weakly converges to $(\hat{u}_B, u_{B3}, \partial_3 u_B \otimes_S e_3)$ in $L^2(B, \mathbb{R}^2) \times H_{\partial_3}(B) \times L^2(B, \mathbb{S}^3)$ with $\gamma_{S^{\pm}}(u_B) = \gamma_S(u_{\Omega}^{\pm})$ because $\gamma_{S^{\pm}}(\mathcal{S}_{\epsilon}^{u, \mathrm{I}_{\mathrm{M2}}} u_s) = \gamma_S((T_\epsilon u_s)^\pm)$. Moreover, when $\mathrm{I}_{\mathrm{M2}} = 2$, one has $e_B^u = 0$ and $u_\Omega$ belongs to $H^1(\Omega, \mathbb{R}^3)$.

When $\mathrm{I}_{\mathrm{M2}} = 3$, (3.2) implies $\int_B |e^3(\epsilon, \mathcal{S}_{\epsilon}^{u,3}) u_s|^2 dx + \int_{S^+} |\mathcal{S}_{\epsilon}^{u,3} u_s|^2 + ((\mathcal{S}_{\epsilon}^{u,3} u_s)_3^2 / \epsilon^2) d\mathcal{H}_2 \le C$ so that there exist a not relabeled subsequence, $u_B = (\widehat{u}_B, 0)$ in $V_{KL}(B)$ and $e_B^u$ in $L^2(B, \mathbb{S}^3)$ such that $(\mathcal{S}_{\epsilon}^{u,3} u_s, e^3(\epsilon, \mathcal{S}_{\epsilon}^{u,3} u_s))$ weakly converges in $H^1(B, \mathbb{R}^3) \times L^2(B, \mathbb{S}^3)$ toward $(u_B, e_B^u)$ with $\widehat{e}_B^u = \widehat{e}(u_B)$. Moreover $\gamma_{S^{\pm}}(\mathcal{S}_{\epsilon}^{u,3} u_s) = \gamma_S((T_\epsilon u_s)^\pm)$ and (3.13) yield $\|u_\Omega\| = 0$, $\widehat{\gamma}_{S^{\pm}}(u_B) = \gamma_S(u_{\Omega}^{\pm}) = \widehat{\gamma}_S(u_{\Omega})$, which imply $u_\Omega \in H^1(\Omega, \mathbb{R}^3)$, $u_B^M = \widehat{\gamma}_S(u_\Omega)$, $u_B^F = 0$, $\widehat{\gamma}_S(u_\Omega) \in H^1(S, \mathbb{R}^2)$.

When $\mathrm{I}_{\mathrm{M2}} = 4$ one only has $(\gamma_{S^{\pm}}(\mathcal{S}_{\epsilon}^{u,4} u_s)_3, e^4(\epsilon, \mathcal{S}_{\epsilon}^{u,4} u_s))$ bounded in $L^2(S^{\pm}) \times L^2(B, \mathbb{S}^3)$, so that, by using the space $\mathcal{R}$ of infinitesimal rigid displacements and the weak convergence of $(\mathcal{S}_{\epsilon}^{u,4} u_s)_3$ in $H_{\partial_3}(B)$ toward $\gamma_S(u_{\Omega}^{\pm})$, it is routine to establish that there exist a not relabeled subsequence and some $(u_B, e_B^u)$ in $V_{KL}(B) \times L^2(B, \mathbb{S}^3)$ such that $(\mathcal{S}_{\epsilon}^{u,4} u_s, e^4(\epsilon, \mathcal{S}_{\epsilon}^{u,4} u_s))$ weakly converges toward $(u_B, e_B^u)$ in $(H^1(B, \mathbb{R}^2)/\widehat{\mathcal{R}} \times H^1(B)) \times L^2(B, \mathbb{S}^3)$ with $\widehat{e}_B^u = \widehat{e}(u_B)$. Then $\gamma_{S^{\pm}}(\mathcal{S}_{\epsilon}^{u,4} u_s)_3 = \gamma_S((T_\epsilon u_s)_3^{\pm})$ and (3.13) yield $\|u_s\| = 0$, $\gamma_{S^{\pm}}(u_{B3}) = \gamma_S(u_{\Omega3})$ which implies $u_\Omega \in H^1(\Omega, \mathbb{R}^3)$ and $\gamma_S(u_{\Omega3}) \in H^2(S)$. Moreover as there exists an infinitesimal rigid displacement $\rho_s$ such that $\gamma_{S^+}(T_\epsilon u_s / \epsilon) + \gamma_{S^+}(\rho_s)$ strongly converges in $L^2(S, \mathbb{R}^2)$ toward $\widehat{\gamma}_{S^+}(u_B)$, one deduces that $\widehat{e}(\widehat{\gamma}_S(u_\Omega)) = 0$. $\square$

### 3.1.2. The limit operator $A^1$

According to Trotter's theory of approximation of semi-groups of linear operators acting on sequences of variable Hilbert spaces [15, 16], we examine the asymptotic behavior of the resolvent $(I - A_s)^{-1}$ of $A_s$ in order to guess the limit operator $A^1$. Proposition 2.1 implies that a sequence $U_s = (u_s, v_s, \theta_s)$ such that $|U_s|_s + |A_s U_s|_s \le C$ involves $(v_s, v_s)_{1,s} + (\theta_s, \theta_s)_{4,s} \le C$ in addition to $|U_s|_s \le C$ that we already considered. For this purpose we introduce the following spaces $G_{\theta}^{\mathrm{I}2}$ of temperatures and operators $g_B^\theta$:

$$
\left\{
\begin{array}{l}
G_{\theta}^{0} := \{\theta = (\theta_{\Omega}, \theta_{B}) \in H^{1}(\Omega^{0}) \times L^{2}(B)\}, \quad g_{B}^{\theta} := 0; \\
G_{\theta}^{1} := \{\theta = (\theta_{\Omega}, \theta_{B}) \in H^{1}(\Omega^{1}) \times H_{\partial_{3}}(B); \gamma_{S^{\pm}}(\theta_{B}) = \gamma_{S}(\theta_{\Omega}^{\pm})\}, \quad g_{B}^{\theta} := (0, \partial_{3}\theta_{B}); \\
G_{\theta}^{2} := \{\theta = (\theta_{\Omega}, \theta_{B}) \in H^{1}(\Omega^{1}) \times H_{\partial_{3}}(B); \partial_{3}\theta_{B} = 0, \gamma_{S^{\pm}}(\theta_{B}) = \gamma_{S}(\theta_{\Omega})\}, \quad g_{B}^{\theta} := 0; \\
G_{\theta}^{3} := \{\theta = (\theta_{\Omega}, \theta_{B}) \in H^{1}(\Omega^{1}) \times H^{1}(B); \gamma_{S}(\theta_{\Omega}) \in H^{1}(S), \partial_{3}\theta_{B} = 0, \gamma_{S^{\pm}}(\theta_{B}) = \gamma_{S}(\theta_{\Omega})\}, \\
\qquad g_{B}^{\theta} := (\widehat{\nabla}\gamma_{S}\theta_{\Omega}, 0); \\
G_{\theta}^{4} := \{\theta = (\theta_{\Omega}, \theta_{B}) \in H^{1}(\Omega^{1}) \times H^{1}(B); \gamma_{S}(\theta_{\Omega}) = C(= 0 \text{ if } \mathcal{H}_{1}(\gamma^{\mathcal{T}D}) > 0), \\
\qquad \partial_{3}\theta_{B} = 0, \gamma_{S^{\pm}}(\theta_{B}) = \gamma_{S}(\theta_{\Omega})\}, \quad g_{B}^{\theta} := 0.
\end{array}
\right.
\quad (3.16)
$$
Thus, if for all $\theta_B$ in $H^1(B)$

$$ g_{\mathrm{IT2}}^{\mathrm{I}}(\varepsilon, \theta_B) = \begin{cases} (\varepsilon \hat{\nabla} \theta_B, \partial_3 \theta_B) & \text{when } I_{\mathrm{IT2}} \le 2, \\ (\hat{\nabla} \theta_B, \partial_3 \theta_B / \varepsilon) & \text{when } I_{\mathrm{IT2}} > 2, \end{cases} \qquad (3.17) $$

we have the obvious scalar variant of Proposition 3.1:

**Proposition 3.2.** For all sequences such that $(\theta_s, \theta_s)_{4,s} \le C$, there exists $\theta$ in $G_{\theta}^{\mathrm{IT2}}$ such that

(i) $T_{\varepsilon}\theta_s$ weakly converges in $H_{\Gamma\Phi D}^1(\Omega^{\mathrm{IT2}})$ toward $\theta_{\Omega}$;

(ii) $g_{\mathrm{IT2}}^{\mathrm{I}}(\varepsilon, \mathcal{S}_{\varepsilon}^{\theta} \theta_s)$ converges weakly in $L^2(B, \mathbb{R}^3)$ toward $g_B^{\theta}$, when $I_{\mathrm{IT2}} \ge 1$.

Note that it is only when $I_{\mathrm{IT2}} \le 1$, that the “additional” temperature $\theta_B$ depends on $x_3$. We therefore make a supplementary assumption:

$$ \text{there exists } \tilde{\alpha}_L^1 \text{ in } [0, +\infty) \text{ satisfying: } \left\{ \begin{array}{ll} I_{\mathrm{M2}} = 0 & \text{no condition,} \\ I_{\mathrm{M2}} = 1 & \tilde{\alpha}_L^1 := \lim_{S \to \bar{s}} \varepsilon \alpha_L, \\ I_{\mathrm{M2}} = 2 & \text{no condition,} \\ I_{\mathrm{M2}} = 3 & \tilde{\alpha}_L^1 := \lim_{S \to \bar{s}} \alpha_L, \\ I_{\mathrm{M2}} = 4 & \tilde{\alpha}_L^1 := \lim_{S \to \bar{s}} \alpha_L / \varepsilon, \end{array} \right. \qquad (\mathrm{H}3) $$

so that it is reasonable to suggest the following forms $(\cdot, \cdot)_4^1$ and $(\cdot, \cdot)_5^1$ as “potential limits” of $(\cdot, \cdot)_{4,s}$ and $(\cdot, \cdot)_{5,s}$:

$$ \begin{align} (\theta, \theta')_4^1 &:= \int_{\Omega} \kappa \nabla \theta_{\Omega} \cdot \nabla \theta'_{\Omega} dx + \bar{\kappa}_L^1 \int_B g_B^{\theta} \cdot g_B^{\theta'} dx \quad \forall \theta, \theta' \in G_{\theta}^1, \tag{3.18} \\ (u, \theta')_5^1 &:= \int_{\Omega} (ae(u_{\Omega}) \cdot \alpha) \theta_{\Omega} dx + (\mu_L^1 \tilde{\alpha}_L^1) \int_B (a_L e_B^u \cdot I) \theta_B dx \quad \forall u \in \mathfrak{S}_u^1, \forall \theta \in G_{\theta}^1. \nonumber \end{align} $$

It will be convenient to introduce for all $\theta$ in $G_{\theta}^1$

$$ \check{\theta} := \begin{cases} \theta_{\Omega} & \text{if } I_{\mathrm{T1}} = 0, \\ (\theta_{\Omega}, \theta_B) & \text{if } I_{\mathrm{T1}} = 1. \end{cases} \qquad (3.19) $$

The boundedness of both $\rho_L \int_{B_\varepsilon} |v_s|^2 dx$ and $\mu_L \int_{B_\varepsilon} ae(v_s) \cdot e(v_s) dx$ leads us, when $I_{\mathrm{M1}} = 1$ and $I_{\mathrm{M2}} = 3, 4$, to introduce a special space $\mathfrak{S}\mathfrak{S}_v^1$ for velocities and consequently a special space $\mathfrak{S}\mathfrak{S}_v^1$ of limit possible states with finite energy.

Let

$$ \begin{align} L_{\mathrm{KL}}^{M}(B) &:= \{v \in L^2(B, \mathbb{R}^3) \text{ s.t. } v(x) = (\hat{v}(\hat{x}), 0), \hat{v} \in L^2(S, \mathbb{R}^2)\}, \tag{3.20} \\ L_{\mathrm{KL}}^{F}(B) &:= \{v \in H^{-1}(B, \mathbb{R}^3) \text{ s.t. } v(x) = (-x_3 \hat{\nabla} v_3, v_3), v_3 \in L^2(S)\}, \nonumber \end{align} $$

then

$$ \mathfrak{S}\mathfrak{S}_v^1 := \begin{cases} \mathfrak{S}_v^1 & \text{if } I_{\mathrm{M1}} = 0 \text{ or } I_{\mathrm{M2}} \le 2, \\ \{(v_{\Omega}, v_B) \in L^2(\Omega, \mathbb{R}^3) \times L_{\mathrm{KL}}^M(B)\} & \text{if } I_{\mathrm{M1}} = 1 \text{ and } I_{\mathrm{M2}} = 3, \\ \{(v_{\Omega}, v_B) \in L^2(\Omega, \mathbb{R}^3) \times L_{\mathrm{KL}}^F(B)\} & \text{if } I_{\mathrm{M1}} = 1 \text{ and } I_{\mathrm{M2}} = 4, \end{cases} \qquad (3.21) $$

$$ (v, v')_2^1 := (\hat{v}, \hat{v}')_2^1 \quad \forall v, v' \in \mathfrak{S}\mathfrak{S}_v^1, \qquad (3.22) $$

where

$$ \hat{v} = \begin{cases} v_{\Omega} & \text{if } I_{\mathrm{M1}} = 0, \\ (v_{\Omega}, v_B) & \text{if } I_{\mathrm{M1}} = 1 \text{ and } I_{\mathrm{M2}} \le 3, \\ (v_{\Omega}, (0, v_{B3})) & \text{if } I_{\mathrm{M1}} = 1 \text{ and } I_{\mathrm{M2}} = 4, \end{cases} \qquad (3.23) $$

and

$$ \mathfrak{S}\mathfrak{S}_v^1 := \mathfrak{S}_u^1 \times \mathfrak{S}\mathfrak{S}_v^1 \times \mathfrak{S}_\theta^1 \qquad (3.24) $$
equipped with the norm $|\cdot|^{1}$. Thus, in order to proceed in a unitary manner, we denote the space of admissible virtual generalized velocities and temperatures by $Z^{1}$:

$$
Z^1 := \{(v, \theta) \in \mathfrak{S}\mathfrak{H}_u^I \times G_{\theta}^{I\mathrm{T}2}\}, \qquad (3.25)
$$

$$
\mathfrak{S}\mathfrak{H}_u^I := \begin{cases} \{u \in \mathfrak{H}_u^I \text{ with } u_B \text{ any element of } L^2(B, \mathbb{R}^3)\} & \text{when } I_{M2} = 0, \\ \mathfrak{H}_u^I & \text{when } I_{M2} > 0, \end{cases} \quad (3.26)
$$

where, for all elements of $\mathfrak{S}\mathfrak{H}_u^I$, $\tilde{u}$ is defined by:

$$
\tilde{u} := \begin{cases} u_{\Omega} & \text{if } I_{M1} = 0; \\ (u_{\Omega}, u_{B}) & \text{if } I_{M1} = 1, \end{cases} \qquad (3.27)
$$

we are in a position to define operator $A^I$ in $\mathfrak{S}\mathfrak{H}^I$ by:

$$
\left\{
\begin{array}{l}
D(A^I) := \left\{ U = (u, v, \theta) \in \mathfrak{S}\mathfrak{H}^I; \begin{array}{@{}l@{}}
\left\{
\begin{array}{@{}l@{}}
\text{(i)} \\
\text{(ii)}
\end{array}
\right.
\end{array} \right\} \\
\qquad \left.
\begin{array}{@{}l@{}}
\phantom{\text{(i)}} (\tilde{u}, \tilde{\theta}) \in Z^I \text{ s.t. } (\tilde{v}, \tilde{\theta}) = (v, \theta), \\
\phantom{\text{(ii)}} \exists! (w, \tau) \in \mathfrak{S}\mathfrak{H}_v^I \times \mathfrak{S}\mathfrak{H}_{\theta}^I \text{ s.t. } \forall (v', \theta') \in Z^I: \\
\phantom{\text{(ii)}} (\tilde{w}, \tilde{v}')_2^I + (u, v')_1^I - (v', \tilde{\theta})_5^I = 0, \\
\phantom{\text{(ii)}} (\tau, \tilde{\theta}')_3^I + (\tilde{\theta}, \theta')_4^I + (\tilde{v}, \theta')_5^I = 0,
\end{array}
\right\} \\
A^I U = (\tilde{v}, w, \tau).
\end{array}
\right.
\qquad (3.28)
$$

Similar to the case of $A_s$, it can be checked easily that $A^I$ is $m$-dissipative and, more specifically,
that for all $\phi = (\phi^1, \phi^2, \phi^3)$ in $\mathfrak{S}\mathfrak{H}^I$:

$$
\left\{
\begin{array}{l}
\displaystyle \overline{U}^I = (\overline{u}^I, \overline{v}^I, \overline{\theta}^I) \text{ s.t.} \\
\displaystyle \overline{U}^I - A^I \overline{U}^I = \phi \\
\hfill \Leftrightarrow \hfill \\
\displaystyle \begin{cases}
\overline{u}^I = \tilde{v}^I + \phi^1, \\
(\overline{v}^I, \overline{\theta}^I) = (\tilde{v}^I, \tilde{\theta}^I), \\
\overline{z}^I = (\tilde{v}^I, \tilde{\theta}^I) \in Z^I; \quad \Psi^I(\overline{z}^I, z) = \mathcal{L}^I(z) \quad \forall z \in Z^I \text{ with} \\
\Psi^I := \Phi^I + K^I, \\
\Phi^I(z, z') := (v, v')_1^I + (\theta, \theta')_4^I - (v', \theta')_5^I + (v, \theta')_5^I \\
K^I(z, z') := (\tilde{v}, \tilde{v}')_2^I + (\tilde{\theta}, \tilde{\theta}')_3^I, \quad \forall z = (v, \theta), \forall z' = (v', \theta') \in Z^I, \\
\mathcal{L}^I(z) := -(\phi^1, v')_1^I + K^I((\phi^2, \phi^3), z).
\end{cases}
\end{array}
\right.
\qquad (3.29)
$$

Consequently, the same statement as that of Theorem 2.1 is valid for the following equation,
which will be shown to describe the asymptotic behavior of the solution to (P<sub>s</sub>):

$$
\left\{
\begin{array}{l}
\displaystyle \frac{dU^{1r}}{dt} - A^I U^{1r} = F^I := \left( u^{1e} - \frac{du^{1e}}{dt}, -\frac{du^e}{dt} + \frac{f^I}{\rho}, -\frac{d\theta^{1e}}{dt} \right), \\
\\
\displaystyle U^{1r}(0) = U^{1r,0},
\end{array}
\right.
\qquad (3.30)
$$

where $f^1 = f$ if $I_{M1} = 0$, $f^1 = (f, 0)$ if $I_{M1} = 1$ with

$$
\begin{align}
& z^{1e} = (u^{1e}, \theta^{1e}) \in Z^I; \quad \Phi^I(z^{1e}(t), z) = L(t)(z) \quad \forall z \in Z^I \forall t \in [0, T], \nonumber \\
& L(t)(z) := \int_{\Gamma \setminus \mathcal{N}} g^{-\mathcal{M}} \cdot v_{\Omega} \, d\mathcal{H}_2 + \int_{\Gamma \setminus \mathcal{N}} g^{\mathcal{T}} \theta_{\Omega} \, d\mathcal{H}_2. \tag{3.31}
\end{align}
$$

We set

$$
U^{1e} := (u^{1e}, z^{1e}), \quad U^{I} := U^{1e} + U^{1r}. \qquad (3.32)
$$
3.2. Convergence

To prove the convergence of the solution $U_s$ to ($\mathcal{P}_s$) toward $U^1$, we use the framework of Trotter's theory of approximation of semi-groups of linear operators acting on variable spaces (see [15, 16]) because $U_s$ and $U^1$ do not inhabit the same space.

First let the representation operator $P_s^1$ defined by

$$ U \in \mathfrak{SH}^1 \to P_s^1 U = (u_s^*, v_s^*, \theta_s^*) \in \mathfrak{H}_s \quad (3.33) $$

with

$$ \bullet \ u_s^* \in H_{\Gamma_{\mathcal{E}}}^1(\mathcal{O}_{\mathcal{E}}, \mathbb{R}^3); $$

$$ (u_s^*, u_s')_{1,s} = \int_{\Omega} ae(u_{\Omega}) \cdot e(T_{\mathcal{E}} u_s') dx + (\mu_L / \epsilon^{r_{\Gamma_{\mathcal{E}}^M}}) \int_B a_L e_B^u \cdot e(\epsilon, \mathcal{S}_{\mathcal{E}}^{u, \mathrm{IM2}} u_s') dx \quad \forall u_s' \in H_{\Gamma_{\mathcal{E}}}^1(\mathcal{O}_{\mathcal{E}}, \mathbb{R}^3), \quad (3.34) $$

$$ \bullet \ (v_s^*, \theta_s^*) \in L^2(\mathcal{O}_{\mathcal{E}}, \mathbb{R}^3) \times L^2(\mathcal{O}_{\mathcal{E}}); $$

$$ (v_s^*, v_s')_{2,s} := \int_{\Omega} \rho v_{\Omega} \cdot T_{\mathcal{E}} v_s' dx + \epsilon \rho_L \int_B v_B \cdot \mathcal{S}_{\mathcal{E}}^v v_s' dx \quad \forall v_s' \in L^2(\mathcal{O}_{\mathcal{E}}, \mathbb{R}^3), \quad (3.35) $$

$$ (\theta_s^*, \theta_s')_{3,s} := \int_{\Omega} \beta \theta_{\Omega} T_{\mathcal{E}} \theta_s' dx + \epsilon \beta_L \int_B \theta_B \mathcal{S}_{\mathcal{E}}^{\theta} \theta_s' dx \quad \forall \theta_s' \in L^2(\mathcal{O}_{\mathcal{E}}), \quad (3.36) $$

(of course when $I_{\mathrm{M1}} = 0$ or $I_{\mathrm{T1}} = 0$, we set $v_B = 0$ or $\theta_B = 0$) and which satisfies

**Proposition 3.3.**

(i) There exists a positive constant $C$ such that $|P_s^1 U|_s \le C|U|^1$ for all $U$ in $\mathfrak{SH}^1$ and all $s$;

(ii) When $s$ tends to $\bar{s}$, $|P_s^1 U|_s$ converges toward $|U|^1$ for all $U$ in $\mathfrak{SH}^1$.

**Proof.** The part of the result concerning $u_s^*$ when $I_{\mathrm{M2}} = 0$, $v_s^*$ and $\theta_s^*$ is obvious. By choosing $u_s' = u_s^*$ and arguing as in the proof of Proposition 3.1, there exist $(u_{\Omega}^*, e_B^{u*})$ in $\mathfrak{H}_u^1$ such that $(T_{\mathcal{E}} u_s^*, e^{\mathrm{IM2}}(\epsilon, \mathcal{S}_{\mathcal{E}}^{u, \mathrm{IM2}} u_s^*))$ weakly converges (up to a not relabeled subsequence) toward $(u_{\Omega}^*, e_B^{u*})$ in $H_{\Gamma_{\mathcal{E}}}^1(\Omega^{\mathrm{IM2}}, \mathbb{R}^3) \times L^2(B, \mathbb{S}^3)$. To identify $u^* := (u_{\Omega}^*, e_B^{u*})$ as $u := (u_{\Omega}, e_B^u)$, it suffices, for all $(u_{\Omega}', e_B^{u'})$ in $\mathfrak{H}_u^1$ with $e_B^{u'}$ in $C^{\infty}(\bar{B}, \mathbb{S}^3)$, to build a sequence $u_s'$ in $H_{\Gamma_{\mathcal{E}}}^1(\mathcal{O}_{\mathcal{E}}, \mathbb{R}^3)$ such that:

$$ \begin{aligned} & \int_{\Omega_{\mathcal{E}}} a_{\mathcal{E}} e(u_s^*) \cdot e(u_s') dx \to \int_{\Omega} ae(u_{\Omega}^*) \cdot e(u_{\Omega}') dx, \\ & \int_{\Omega} ae(u_{\Omega}) \cdot e(T_{\mathcal{E}} u_s') dx \to \int_{\Omega} ae(u_{\Omega}) \cdot e(u_{\Omega}') dx, \\ & \mu_L \int_{B_{\mathcal{E}}} a_L e(u_s^*) \cdot e(u_s') dx \to \bar{\mu}_L \int_B a_L e_B^{u*} \cdot e_B^{u'} dx, \\ & (\mu_L / \epsilon^{r_{\Gamma_{\mathcal{E}}^M}}) \int_B a_L e_B^u \cdot e^{\mathrm{IM2}}(\epsilon, \mathcal{S}_{\mathcal{E}}^{u, \mathrm{IM2}} u_s') dx \to \bar{\mu}_L \int_B a_L e_B^u \cdot e_B^{u'} dx. \end{aligned} \quad (3.37) $$

Clearly, when $I_{\mathrm{M2}} = 1, 2$, $u_s' := \begin{cases} (\mathcal{S}_{\mathcal{E}}^{u, \mathrm{IM2}})^{-1} u'_B & \text{in } B_{\mathcal{E}} \\ u'_\Omega (\cdot \mp \epsilon e_3) & \text{in } \Omega_{\mathcal{E}}^{\pm} \end{cases}$ satisfies (3.37). When $I_{\mathrm{M2}} = 3, 4$, we use a trick of the mathematical derivation of Kirchhoff-Love theory of plates [18, 19]. Let $\varphi_s, \psi_s$ defined as follows:

$$ \begin{aligned} \varphi_s &:= \left( \overline{\gamma_S(u_{\Omega}') - x_3 \hat{\nabla} \gamma_S(u_{\Omega_3}')} , \gamma_S(u_{\Omega_3}') \right) \\ \psi_s &:= (\mathcal{S}_{\mathcal{E}}^{u, \mathrm{IM2}})^{-1} \psi_{B_{\mathcal{E}}} \end{aligned} \quad (3.38) $$

$$ \psi_{B_{\mathcal{E}}}(x) := \epsilon \int_{0}^{x_3} \left( 2 \hat{w}(\hat{x}, \zeta) - \epsilon \int_{0}^{\zeta} \hat{\nabla} w_3(\hat{x}, \xi) d\xi, \epsilon w_3(\hat{x}, \zeta) \right) d\zeta \quad \text{a.e. } x \in B, $$

where $w_i := (e_B^{u'})_{i3}, 1 \le i \le 3$, then the field $u_s' := \varphi_s + \psi_s$ is such that $e^{\mathrm{IM2}}(\epsilon, \mathcal{S}_{\mathcal{E}}^{u, \mathrm{IM2}}) u_s'$ converges strongly in $L^2(B, \mathbb{S}^3)$ toward $e_B^{u'}$ and $|\gamma_{S_{\mathcal{E}}^\pm}(u_s' - u_{\Omega'}(\cdot \mp \epsilon e_3))|_{H^1(S_{\mathcal{E}}^\pm, \mathbb{R}^3)} = O(\epsilon^{\mathrm{IM2}-2})$. Hence $u_s'$ may be extended into $\Omega_{\mathcal{E}}$ to an element of $H_{\Gamma_{\mathcal{E}}}^1(\mathcal{O}_{\mathcal{E}}, \mathbb{R}^3)$ still denoted $u_s'$ which satisfies (3.37). Lastly by choosing $u_s' = u_s^*$ in (3.34), one obtains that $(u_s^*, u_s^*)_{1,s}$ tends to $(u, u)^1_1$. $\square$
Next we state that:

$$ U_s \text{ in } \mathfrak{H}_s \text{ converges in the sense of Trotter toward } U \text{ in } \mathfrak{H}^1 \text{ if } \lim_{s \to \bar{s}} |P_s^1 U - U_s|_s = 0. \quad (3.39) $$

**Remark 3.1.** Note that $U_s$ converges in the sense of Trotter toward $U$ is equivalent to $T_\epsilon(u_s, v_s, \theta_s)$ converges strongly toward $(u_\Omega, v_\Omega, \theta_\Omega)$ in $H^1(\Omega^{1M_2}, \mathbb{R}^3) \times L^2(\Omega, \mathbb{R}^3) \times L^2(\Omega)$ and $((\mu_L/\epsilon)^{1/2} e^{r_{M_2}^M} u_\Omega, \mathcal{S}_\epsilon^{u,M_2} u_s), (\rho_L\epsilon)^{1/2} \mathcal{S}_\epsilon^v v_s, (\beta_L\epsilon)^{1/2} \mathcal{S}_\epsilon^\theta \theta_s)$ converges strongly toward $(\bar{\mu}_L^{1/2} e_B^u, \bar{\rho}_L^{1/2} v_B, \bar{\beta}_L^{1/2} \theta_B)$ in $L^2(B, \mathbb{S}^3 \times \mathbb{R}^3)$ which, by (H2), (3.2) and (3.4), is equivalent to:

$$ \lim_{s \to \bar{s}} \left\{ \begin{aligned} & \int_{\Omega_\epsilon} a_\epsilon e(u_s - (T_\epsilon)^{-1} u_\Omega) \cdot e(u_s - (T_\epsilon)^{-1} u_\Omega) \,dx \\ & + \mu_L \int_{B_\epsilon} a_L(e(u_s)(x) - \epsilon^{-(1+r_{M_2}^M)^{1/2}} e_B^u(\hat{x}, x_3/\epsilon)) \cdot (e(u_s)(x) - \epsilon^{-(1+r_{M_2}^M)^{1/2}} e_B^u(\hat{x}, x_3/\epsilon)) \,dx \\ & + \rho_\epsilon \int_{\Omega_\epsilon} |v_s - (T_\epsilon)^{-1} v_\Omega|^2 \,dx \\ & + \rho_L \int_{B_\epsilon} |v_s(x) - v_B(\hat{x}, x_3/\epsilon)|^2 \,dx \\ & + \beta_\epsilon \int_{\Omega_\epsilon} |\theta_s - (T_\epsilon)^{-1} \theta_\Omega|^2 \,dx \\ & + \beta_L \int_{B_\epsilon} |\theta_s(x) - \theta_B(\hat{x}, x_3/\epsilon)|^2 \,dx \end{aligned} \right\} = 0 \quad (3.40) $$

which expresses that the energetic gap between the state $U_s$ and the image on the initial physical configuration $\mathcal{O}_\epsilon$ of the limit state $U$ tends to zero when $s$ goes to $\bar{s}$!

Lastly we conclude by making an additional assumption (H4) about the initial state and establishing the

**Proposition 3.4.** *There hold*

$$ (i) \quad \forall \phi \in \mathfrak{H}^1, \lim_{s \to \bar{s}} |P_s^1(I - A^1)^{-1}\phi - (I - A_s)^{-1} P_s^1\phi|_s = 0, $$

$$ (ii) \quad \lim_{s \to \bar{s}} |P_s^1 U^{le}(t) - U_s^e(t)|_s = 0 \text{ uniformly on } [0, T], $$

$$ (iii) \quad \lim_{s \to \bar{s}} \int_0^T |P_s^1 F^1(t) - F_s(t)|_s dt = 0. $$

**Proof.** Regarding (i), by taking advantage of Proposition 2.1 and (3.29) it first suffices to build for all $z' = (v', \theta')$ in a dense subspace of $Z^1$ a sequence $z'_s$ in $\mathfrak{H}_s$ such that

$$ \begin{aligned} & \lim_{s \to \bar{s}} (\bar{v}_s, v'_s)_{1,s} = (\bar{v}, v')_1, \quad \lim_{s \to \bar{s}} (\bar{\theta}_s, \theta'_s)_{4,s} = (\bar{\theta}, \theta')_4, \\ & \lim_{s \to \bar{s}} (v'_s, \bar{\theta}_s)_{5,s} = (v', \bar{\theta})_5, \quad \lim_{s \to \bar{s}} (\bar{v}_s, \theta'_s)_{5,s} = (\bar{v}, \theta')_5, \\ & \lim_{s \to \bar{s}} K_s((\bar{v}_s, \bar{\theta}_s), (v'_s, \theta'_s)) = K^1((\bar{v}, \bar{\theta}), (v', \theta')), \end{aligned} \quad (3.40) $$

where $\bar{z} := (\bar{v}, \bar{\theta})$ in $\mathfrak{H}_u^1 \times G_\theta^{1T_2}$ is the limit supplied by Propositions 3.1 and 3.2 of $\bar{z}_s = (\bar{v}_s, \bar{\theta}_s)$, as $\bar{z}_s$ satisfies $(\bar{v}_s, \bar{v}_s)_1^1 + (\bar{v}_s, \bar{v}_s)_2^1 + (\bar{\theta}_s, \bar{\theta}_s)_3^1 + (\bar{\theta}_s, \bar{\theta}_s)_4^1 \le C$. From Proposition 3.3, the choice $v'_s = v'_s^*$ according to (3.34) is in order, while we use $\theta'_s$ defined by

$$ \theta'_s(x) = \begin{cases} \theta'_\Omega(x \mp \epsilon e_3) & \text{a.e. } x \in \Omega_\epsilon^I, \\ (\mathcal{S}^\epsilon \theta)^{-1}(\theta'_B) & \text{a.e. } x \in B_\epsilon, \end{cases} \quad (3.41) $$

for any $(\theta'_\Omega, \theta'_B)$ in $G_\theta^{1T_2}$ such that $\theta'_B$ belongs to $H^1(B)$, because $(T_\epsilon \theta'_s, (\kappa_L/\epsilon)^{r_{1T_2}^T} g^{1T_2}(\epsilon, \mathcal{S}_\epsilon^\theta \theta'_s))$ strongly converges in $H^1(\Omega^{1T_2}) \times L^2(B, \mathbb{R}^3)$ toward $(\theta'_\Omega, \bar{\kappa}_L^1 g'_B)$. This allows us to claim that $\bar{z} = \bar{z}^1$. Next by choosing $z = \bar{z}_s$ in (2.13), we deduce that $(\bar{v}_s, \bar{v}_s, \bar{\theta}_s)$ converges in the sense of Trotter toward $(\bar{v}^1, \bar{v}^1, \bar{\theta}^1)$ by due account of the remark following the definition (3.39).

Points (ii) and (iii) are established by similar arguments simply taking into account (2.2), (3.31) and (H1). $\square$
Thus the convergence result can be stated as:

**Theorem 3.1.** Under assumptions (H1)–(H3) and

$$ \exists U^{10} \in U^{1e}(0) + D(A^1) \text{ s.t. } \lim_{s \to \bar{s}} |P_s^1 U^{10} - U_s^0|_s = 0 \quad (H4) $$

the solution to

$$ \frac{dU_s}{dt} - A_s(U_s - U_s^e) = (0, f_s, 0), \quad U_s(0) = U_s^0 \quad (3.42) $$

converges toward the solution to

$$ \frac{dU^1}{dt} - A^1(U^1 - U^1_e) = (0, f^1/\rho, 0), \quad U^1(0) = U^10 \quad (3.43) $$

in the sense $\lim_{s \to \bar{s}} |P_s^1 U^1(t) - U_s(t)|_s^1 = 0$, $\lim_{s \to \bar{s}} |U_s(t)|_s^1 = |U^1(t)|^1$ uniformly on $[0, T]$.

## 4. A thermomechanical presentation of the results

Here we intend to make more explicit the formulation (3.43) of the limit behavior of the structure. To lighten notations we skip the superscript I in $(u^1, \theta^1)$ and write $(u, \theta)$ instead; an over dot '·' denoting differentiation with respect to time, the motion equation reads as:

$$ \begin{aligned} & \int_{\Omega} \rho \ddot{u}_{\Omega} \cdot v'_{\Omega} + a(e(u_{\Omega}) - \theta_{\Omega} \alpha I) \cdot e(v'_{\Omega}) \, dx + \int_{B} \bar{\rho}_{L} \ddot{u}_{B} \cdot v'_{B} + \bar{\mu}_{L}^{1} a_{L}(e_{B}^{u} - \theta_{B} \bar{\alpha}_{L}^{I} I) \cdot e_{B}^{v'} \, dx \\ & = \int_{\Omega} f \cdot v'_{\Omega} \, dx + \int_{\Gamma_{\mathcal{M}N}} g^{\mathcal{M}} \cdot v'_{\Omega} \, d\mathcal{H}_{2}, \quad \forall v' \in \mathcal{S}\mathcal{H}_{v}^{I} \end{aligned} \quad (4.1) $$

while the “energy” equation reads as:

$$ \begin{aligned} & \int_{\Omega} \dot{\beta} \dot{\theta}_{\Omega} \theta'_{\Omega} + \kappa \nabla \theta_{\Omega} \cdot \nabla \theta'_{\Omega} + (ae(\dot{u}_{\Omega}) \cdot \alpha I) \theta'_{\Omega} \, dx + \int_{B} \bar{\beta}_{L} \dot{\theta}_{B} \theta'_{B} + \bar{\kappa}_{L}^{I} g_{B}^{\theta} \cdot g_{B}^{\theta'} + \bar{\mu}_{L}^{I} \bar{\alpha}_{L}^{I} (a_{L}(e_{B}^{\dot{u}}) \cdot I) \theta'_{B} \, dx \\ & = \int_{\Gamma_{\mathcal{F}N}} g^{\mathcal{F}} \theta'_{\Omega} \, d\mathcal{H}_{2}, \quad \forall \theta' \in G_{\theta}^{1T2}. \end{aligned} \quad (4.2) $$

Clearly the fields of stress $\sigma_{\Omega}^{\pm}$, thermal flux $q_{\Omega}^{\pm}$, displacement and temperature in the adhering bodies that occupy $\Omega^{+}$ and $\Omega^{-}$ satisfy the following relations written in strong form:

$$ \left\{ \begin{array}{l} \rho \ddot{u}_{\Omega}^{\pm} - \operatorname{div} \sigma_{\Omega}^{\pm} = f \text{ in } \Omega^{\pm}, \quad \sigma_{\Omega}^{\pm} = g^{\mathcal{M}^{\pm}} \text{ on } \Gamma^{\mathcal{M}N^{\pm}} \\ \sigma_{\Omega}^{\pm} = a(e(u_{\Omega}^{\pm}) - \theta_{\Omega}^{\pm} \alpha I) \text{ in } \Omega^{\pm} \\ \beta \dot{\theta}_{\Omega}^{\pm} - \operatorname{div} q_{\Omega}^{\pm} + ae(\dot{u}_{\Omega}^{\pm}) \cdot \alpha I = 0 \text{ in } \Omega^{\pm}, \quad q_{\Omega}^{\pm} = g^{\mathcal{F}^{\pm}} \text{ on } \Gamma^{\mathcal{F}N^{\pm}} \\ q_{\Omega}^{\pm} = \kappa \nabla \theta_{\Omega}^{\pm} \text{ in } \Omega^{\pm} \end{array} \right. \quad (4.3) $$

where n denotes the outward normal to $\Omega$, together with a thermomechanical contact condition along S, the common boundary of $\Omega^{+}$ and $\Omega^{-}$. This corresponds to the transient response to the loading $(f, g^{\mathcal{M}^{\pm}}, g^{\mathcal{F}^{\pm}})$ of each adhering body clamped on $\Gamma^{\mathcal{M}D^{\pm}}$ maintained at a uniform temperature $T_0$ on $\Gamma^{\mathcal{F}D^{\pm}}$ and thermomechanically linked along S. These contact conditions, which stem from the limit behavior of the adhesive layer, can be deduced from the various expressions of the two integrals on B in (4.1)–(4.2). The motion and “energy” equations will be formulated in the form:

$$ \mathcal{M}_{\Omega}(v'_{\Omega}) + \mathcal{M}_{B}(v'_{B}) = \int_{\Omega} f \cdot v'_{\Omega} \, dx + \int_{\Gamma_{\mathcal{M}N}} g^{\mathcal{M}} \cdot v'_{\Omega} \, d\mathcal{H}_{2}, \quad \forall v' \in V^{1M2} \quad (4.4) $$

$$ \mathcal{T}_{\Omega}(\theta'_{\Omega}) + \mathcal{T}_{B}(\theta'_{B}) = \int_{\Gamma_{\mathcal{F}N}} g^{\mathcal{T}} \cdot \theta'_{\Omega} \, d\mathcal{H}_{2}, \quad \forall \theta' \in G^{1T2}. \quad (4.5) $$

Besides the singular case $I_{M2} = 0$ (very soft adhesive), we may distinguish two main cases depending on whether $e_B^v$ depends explicitly on $v_B$ ($I_{M2} = 1, 2$, i.e. soft, not stiff adhesive) or not ($I_{M2} = 3, 4$, i.e. stiff, very stiff adhesive).
✓ When $I_{M2} = 1, 2$, as $e_B^v = \partial_3 v_B \otimes_S e_3$, one has:

$$
\begin{equation}
\begin{cases}
I_{M2} = 1, & V^{I_{M2}} := \{(v_\Omega, v_B) \in H_{\Gamma, \mathcal{L}\Omega}^1 (\Omega \setminus S, \mathbb{R}^3) \times H_{\partial_3}(B, \mathbb{R}^3) \text{ s.t. } \gamma_S(v_\Omega^\pm) = \gamma_{S^\pm}(v_B)\} \\
I_{M2} = 2, & V^{I_{M2}} := \{(v_\Omega, v_B) \in H_{\Gamma, \mathcal{L}\Omega}^1 (\Omega \setminus S, \mathbb{R}^3) \times H_{\partial_3}(B, \mathbb{R}^3) \text{ s.t. } \partial_3 v_B = 0, \gamma_S(v_\Omega) = \gamma_{S^\pm}(v_B)\}
\end{cases}
\tag{4.6}
\end{equation}
$$

$$
\mathcal{M}_{\Omega}(v'_{\Omega}) := \int_{\Omega \setminus S} \rho \ddot{u}_{\Omega} \cdot v'_{\Omega} + a(e(u_{\Omega}) - \theta_{\Omega} \alpha I) \cdot e(v'_{\Omega}) \, dx \quad (4.7)
$$

$$
\mathcal{M}_B(v'_B) := \int_B \bar{\rho}_L \ddot{u}_B \cdot v'_B + \bar{\mu}_L^1 a_L (\partial_3 u_B \otimes_S e_3 - \bar{\alpha}_L^1 \theta_B I) \cdot \partial_3 v'_B \otimes_S e_3 \, dx. \quad (4.8)
$$

When $I_{M2} = 2$, $\mathcal{M}_B(v'_B)$ is equal to $2 \int_S \bar{\rho}_L \ddot{u}_\Omega \cdot v'_\Omega \, d\mathcal{H}_2$ and vanishes when $\bar{\rho}_L = 0$. One deduces:
$$ -[\sigma e_3] = 2\bar{\rho}_L \gamma_S(\ddot{u}_\Omega) \quad (4.9) $$

so that the two adhering bodies are stuck together (i.e. $[\![u_\Omega]\!] = 0$) when $\bar{\rho}_L > 0$ and perfectly stuck together (i.e. $[\![\sigma_\Omega e_3]\!] = 0$) when $\bar{\rho}_L = 0$.
When $I_{M2} = 1$, one has:

$$
\mathcal{M}_B(v'_B) = \int_B \bar{\rho}_L \ddot{u}_B \cdot v'_B + \bar{\mu}_L^1 a_L (\partial_3 u_B \otimes_S e_3 - \bar{\alpha}_L^1 \theta_B I) \cdot \partial_3 v'_B \otimes_S e_3 \, dx \quad (4.10)
$$

and the mechanical contact condition reads as:

$$
\mp \sigma_{\Omega}^{\pm} e_3 = \frac{1}{2} \int_{-1}^{1} \bar{\rho}_L (1 + x_3) \ddot{u}_B \, dx_3 + \bar{\mu}_L^1 a_L \left( [\![u_\Omega]\!] \otimes_S e_3 - \bar{\alpha}_L^1 (\frac{1}{2} \int_{-1}^{1} \theta_B \, dx_3) I \right) e_3. \quad (4.11)
$$

Therefore, if $\bar{\rho}_L = 0$ and $\bar{\alpha}_L^1 = 0$ or $\theta_B$ is independent of $x_3$ (i.e. $I_{T2} > 1$, which means an adhesive with a not too weak conductance), one deduces:

$$
u_B = \mathrm{Aff}(u_\Omega) \tag{4.12}
$$

$$
\text{Aff}(u_{\Omega}) := \langle u_{\Omega} \rangle + \frac{1}{2} x_3 [\![u_{\Omega}]\!], \quad \langle u_{\Omega} \rangle := \frac{1}{2} (\gamma_S(u_{\Omega}^{\dagger}) + \gamma_S(u_{\bar{\Omega}}^{\dagger})) \quad (4.13)
$$

$$
\begin{align}
\mathcal{M}_B(v'_B) &= 2\bar{\mu}_L^1 \int_S a_L (\llbracket u_\Omega \rrbracket \otimes_S e_3) \cdot \llbracket v'_\Omega \rrbracket \otimes_S e_3 \, d\hat{x}, \quad \text{or} \nonumber \\
\mathcal{M}_B(v'_B) &= 2\bar{\mu}_L^1 \int_S a_L (\llbracket u_\Omega \rrbracket \otimes_S e_3 - \bar{\alpha}_L^1 \gamma_S(\theta_\Omega) I) \cdot \llbracket v'_\Omega \rrbracket \otimes_S e_3 \, d\hat{x} \tag{4.14}
\end{align}
$$

$$
\mp \sigma_{\Omega}^{\pm} e_3 = \bar{\mu}_L^1 a_L (\llbracket u_{\Omega} \rrbracket \otimes_S e_3) e_3, \quad \text{or} \quad \mp \sigma_{\Omega}^{\pm} e_3 = \bar{\mu}_L^1 a_L (\llbracket u_{\Omega} \rrbracket \otimes_S e_3 - \bar{\alpha}_L^1 \gamma_S(\theta_{\Omega}) I) e_3. \quad (4.15)
$$

There is an elastic pull-back with a residual term between the two adhering bodies. The other cases for the limit mechanical behavior of the adhesive layer, which clearly appears as a continuous distribution of thermoelastic strings orthogonal to S, will be discussed further.

As regards to the thermal behavior, one has:

$$
\mathcal{T}_{\Omega}(\theta'_{\Omega}) = \int_{\Omega \setminus S} \beta \theta_{\Omega} \theta'_{\Omega} + \kappa \nabla \theta_{\Omega} \cdot \nabla \theta'_{\Omega} + (ae(\dot{u}_{\Omega}) \cdot \alpha I) \theta'_{\Omega} \, dx \quad (4.16)
$$

$$
\mathcal{T}_B(\theta'_B) = \int_B \bar{\rho}_L \theta_B \theta'_B + \bar{\kappa}_L^1 g_B^\theta \cdot g_B^{\theta'} + \bar{\mu}_L^1 \bar{\alpha}_L^1 (a_L(\partial_3 \dot{u}_B \otimes_S e_3) \cdot I) \theta'_B \, dx. \quad (4.17)
$$

We may therefore distinguish two main cases: $I_{T2} \le 2$ (i.e. an adhesive with a very weak or weak heat conductance) when $g_B^\theta = 0$ or $(0, \partial_3 \theta_B)$ and $I_{T2} = 3, 4$ (adhesive with high or very high heat conductance) when $g_B^\theta = (\bar{\nabla} \theta_B, 0)$ or 0.

If $I_{T2} \le 2$, the limit thermal behavior of the adhesive could be considered as the one of a continuous distribution of strings orthogonal to $S$ with specific heat coefficient $\bar{\rho}_L$, thermal conductivity $\bar{\kappa}_L^1$ (thus insulating strings when $I_{T2} = 0$ whereas perfectly conducting ones when $I_{T2} = 2$) subjected to heat sources of lineic density $\bar{\mu}_L^1 \bar{\alpha}_L^1 a_L (\partial_3 \dot{u}_B \otimes_S e_3) \cdot I$ (vanishing when $I_{M2} = 0$ or 2). Therefore the thermal contact condition between the adherent bodies reads as:
• $I_{T2} = 0$, $[\theta_{\Omega}] \neq 0$, $q_{\Omega}^{\pm} \cdot e_3 = 0$: perfectly insulating interface,

• $I_{T2} = 2$, $[\theta_{\Omega}] = 0$, $-[q_{\Omega} \cdot e_3] = 2\bar{\beta}_L \gamma_S(\dot{\theta}_{\Omega}) + \bar{\mu}_L^1 \bar{\alpha}_L^1([\dot{u}_{\Omega}] \otimes_S e_3) \cdot a_L I$: imperfect thermal contact, perfect when $I_{T1} = 0$ (adhesive with a low specific heat coefficient) and $I_{M2} = 0$ or 2.

When $I_{T2} = 1$ we have:

$$ \mp (q_{\Omega}^{\pm} \cdot e_3) = \frac{1}{2} \int_{-1}^{1} (1 + x_3)(\bar{\beta}_L \dot{\theta}_B + \bar{\mu}_L^1 \bar{\alpha}_L^1 a_L (\partial_3 \dot{u}_B \otimes_S e_3) \cdot I) \, dx_3 + \bar{\kappa}_L^1 [\theta_{\Omega}] \quad (4.18) $$

thus, when $\bar{\beta}_L = 0$, there exists a contact conduction whose contact conductance is $\bar{\kappa}_L^1$ and a source $\bar{\mu}_L^1 \bar{\alpha}_L^1 a_L [\dot{u}_{\Omega}] \otimes_S e_3 \cdot I$; the other cases will be treated further.

When $I_{T2} = 3, 4$, we are indeed dealing with a material surface with a specific heat coefficient $\bar{\beta}_L$, a thermal conductivity $\bar{\kappa}_L^1$ (thus perfectly conducting when $I_{T2} = 4$) subjected to a heat source $\bar{\mu}_L^1 \bar{\alpha}_L^1 a_L [\dot{u}_{\Omega}] \otimes_S e_3 \cdot I$. So, the imperfect thermal contact condition reads as (the symbol $\hat{\Delta}$ denoting the Laplacian with respect to the sole coordinates $x_1$ and $x_2$):

• $I_{T2} = 3$, $[\theta_{\Omega}] = 0$, $-[q_{\Omega} \cdot e_3] = 2\bar{\beta}_L \gamma_S(\dot{\theta}_{\Omega}) - \bar{\kappa}_L^1 \hat{\Delta} \gamma_S(\theta_s) + \bar{\mu}_L^1 \bar{\alpha}_L^1 [\dot{u}_{\Omega}] \otimes_S e_3 \cdot a_L I$,

• $I_{T2} = 4$, $[\theta_{\Omega}] = 0$, $\gamma_S(\theta_{\Omega}) = C$, possibly $T_0$ if $\mathcal{H}_1(\gamma^{\mathcal{T}D}) > 0$, $-[q_{\Omega} \cdot e_3] = 2\bar{\beta}_L \gamma_S(\dot{\theta}_{\Omega}) + \bar{\mu}_L^1 \bar{\alpha}_L^1 [\dot{u}_{\Omega}] \otimes_S e_3 \cdot a_L I$,

the thermal contact being perfect when $I_{T1} = 0$ and $I_{M2} = 2$.

✓ When $I_{M2} = 0$, $\mathcal{M}_B(v'_B)$ reduces to $\int_B \bar{\rho}_L \ddot{u}_B \cdot v'_B dx$ which implies $\sigma_{\Omega}^1 e_3 = 0$: the two adhering bodies are free to separate. The thermal contact condition was detailed in the previous study of the case $I_{M2} = 1$.

✓ When $I_{M2} = 3, 4$, by using $e_B^{\Psi} = \Psi \otimes_S e_3$, $\Psi$ arbitrary in $C_0^{\infty}(B, \mathbb{R}^3)$, (4.1) yields $(a_L(e_B^u - \alpha_L^1 \hat{\theta}_B I))_{i3} = 0, 1 \le i \le 3$. Then, if $\tilde{a}_L$ is the element of $\text{Lin}(\mathbb{S}^2)$ defined by:

$$ \tilde{a}_L q \cdot q := \text{Inf} \{ a_L e \cdot e; e \in \mathbb{S}^3 \text{ s.t. } \hat{e} = q \}, \quad \forall q \in \mathbb{S}^2 \quad (4.19) $$

one has $\overline{a_L(e_B^u - \tilde{a}_L^1 \theta_B I)} = \tilde{a}_L(e(\overline{u_B}) - \tilde{a}_L^1 \theta_B \hat{I})$ and consequently:

$$ \mathcal{M}_B(v'_B) = \int_B \bar{\rho}_L \ddot{u}_B \cdot \dot{v}'_B + \bar{\mu}_L^1 \tilde{a}_L (\overline{e(u_B)} - \tilde{a}_L^1 \theta_B \hat{I}) \cdot \overline{e(v'_B)} dx, \forall (v'_B, v'_B) \in V^{1M2} \quad (4.20) $$

$$ \begin{aligned} \mathcal{T}_B(\theta'_B) = \int_B (\bar{\rho}_L + \bar{\mu}_L^1(\tilde{a}_L^1)^2(a_L I \cdot I - \tilde{a}_L \hat{I} \cdot \hat{I})) \dot{\theta}_B \theta'_B + \bar{\kappa}_L^1 g_B^\theta \cdot g_B^{\theta'} \\ + \bar{\mu}_L^1 \tilde{a}_L^1 (\tilde{a}_L \hat{e}(\dot{u}_B) \cdot \hat{I}) \theta'_B dx, \forall (\theta'_\Omega, \theta'_B) \in G_\theta^{1M2} \end{aligned} \quad (4.21) $$

$$ V^3 := \{ v = (v_\Omega, v_B) \in H_{\Gamma, \nabla D}^1(\Omega, \mathbb{R}^3) \times V_{KL}(B); v_B^M = \gamma_S(\overline{v_\Omega}), v_B^F = 0 \} \quad (4.22) $$

$$ V^4 := \{ v = (v_\Omega, v_B) \in H_{\Gamma, \nabla D}^1(\Omega, \mathbb{R}^3) \times V_{KL}(B); \hat{e}(\gamma_S(\overline{v_\Omega})) = 0, v_B^F = \gamma_S(v_{\Omega 3}) \}. \quad (4.23) $$

An interesting phenomena is highlighted: *the appearance of an added specific heat coefficient*

$$ \bar{\beta}_L^{\text{add}} := \bar{\mu}_L^1 (\bar{\alpha}_L^1)^2 (a_L I \cdot I - \tilde{a}_L \hat{I} \cdot \hat{I}) \quad (4.24) $$

is always positive unless $\tilde{a}_L^1(a_L I)_{i3} = 0, 1 \le i \le 3$, in the limit behavior of the adhesive layer, while the stiffness involves $\tilde{a}_L$ in place of $a_L$ as in the Kirchhoff-Love anisotropic plates theory (see [10, 19, 20]).

So, when $I_{M2} = 3$, we have

$$ \mathcal{M}_B(v'_B) = 2 \int_S \bar{\rho}_L \widehat{\gamma_S(u_\Omega)} \cdot \widehat{\gamma_S(v'_\Omega)} + \bar{\mu}_L^1 \tilde{a}_L \left( \hat{e}(\widehat{\gamma_S(u_\Omega)}) - \frac{1}{2} \bar{\alpha}_L^1 \int_{-1}^1 \theta_B \, dx_3 \hat{I} \right) \cdot \hat{e}(\widehat{\gamma_S(v'_\Omega)}) \, dx \quad (4.25) $$

and the mechanical contact condition between the adhering bodies reads as:

• $[u_\Omega] = 0$

• $[\sigma_\Omega e_3]_{I3} = 0$

• $-[\sigma_\Omega e_3] = 2 \left( \bar{\rho}_L \widehat{\gamma_S(u_\Omega)} - \bar{\mu}_L^1 \text{div}(\tilde{a}_L(\hat{e}(\widehat{\gamma_S(u_\Omega)}))) - \frac{1}{2} \bar{\alpha}_L^1 \int_{-1}^1 \theta_B \, dx_3 \hat{I} \right)$.
It looks like a deformable material surface which is stuck between the adhering bodies and enjoys
only in-plane strains.

The nature of the thermal behavior of the adhesive was already examined but here we can
make more explicit the thermal contact conditions:

• $I_{T2} = 0$, $\ll \theta_{\Omega} \gg \neq 0$, $\mp(q_{\Omega}^{\pm} \cdot e_3) = 0$: perfectly insulating interface,

• $I_{T2} = 1$, $\ll \theta_{\Omega} \gg \neq 0$, $\mp(q_{\Omega}^{\pm} \cdot e_3) = \frac{1}{2}(\bar{\beta}_L + \bar{\beta}_L^{\text{add}}) \int_{-1}^{1} (1+x_3) \dot{\theta}_B dx_3 + \bar{\mu}_L^I \bar{\alpha}_L^I \bar{a}_L \hat{e}(\gamma_S(\dot{u}_{\Omega})) \cdot \hat{I} + \bar{\kappa}_L^I \ll \theta_{\Omega} \gg$: contact conduction,

• $I_{T2} = 2$, $\ll \theta_{\Omega} \gg = 0$, $-\ll q_{\Omega} \cdot e_3 \gg = (\bar{\beta}_L + \bar{\beta}_L^{\text{add}}) \gamma_S(\dot{\theta}_{\Omega}) + \bar{\mu}_L^I \bar{\alpha}_L^I \bar{a}_L \hat{e}(\gamma_S(\dot{u}_{\Omega})) \cdot \hat{I}$: imperfect thermal contact,

• $I_{T2} = 3$, $\ll \theta_{\Omega} \gg = 0$, $-\ll q_{\Omega} \cdot e_3 \gg = (\bar{\beta}_L + \bar{\beta}_L^{\text{add}}) \gamma_S(\dot{\theta}_{\Omega}) - \bar{\kappa}_L^I \hat{\Delta} \gamma_S(\theta_{\Omega}) + \bar{\mu}_L^I \bar{\alpha}_L^I \bar{a}_L \hat{e}(\gamma_S(\dot{u}_{\Omega})) \cdot \hat{I}$: imperfect thermal contact,

• $I_{T2} = 4$, $\ll \theta_{\Omega} \gg = 0$, $\gamma_S(\theta_{\Omega})$ constant on S, possibly equal to $T_0$ if $\mathcal{H}_1(\gamma^{\mathcal{T}D}) > 0$, $-\ll q_{\Omega} \cdot e_3 \gg = (\bar{\beta}_L + \bar{\beta}_L^{\text{add}}) \gamma_S(\dot{\theta}_{\Omega}) + \bar{\mu}_L^I \bar{\alpha}_L^I \bar{a}_L \hat{e}(\gamma_S(\dot{u}_{\Omega})) \cdot \hat{I}$: imperfect thermal contact.

When $I_{M2} = 4$ one has:

$$
\begin{equation}
\begin{split}
\mathcal{H}_B(v'_B) = 2 \int_S \bar{\rho}_L \ddot{u}_{\Omega 3} v'_{\Omega 3} + \bar{\mu}_L^I \left( \tilde{a}_L \hat{e}(u_B^M) \cdot \hat{e}(v'_B^M) + \frac{1}{3} \tilde{a}_L \hat{D}^2 \gamma_S(u_{\Omega 3}) \cdot \hat{D}^2 \gamma_S(v'_{\Omega 3}) \right. \\
\left. - \bar{\alpha}_L^I \left( \frac{1}{2} \int_{-1}^1 \theta_B dx_3 \right) \tilde{a}_L \hat{I} \cdot \hat{e}(v'_M B) + \int_{-1}^1 x_3 \theta_B dx_3 \tilde{a}_L \hat{I} \cdot \hat{D}^2 \gamma_S(v'_{\Omega 3}) \right) d\mathcal{H}_2
\end{split}
\tag{4.26}
\end{equation}
$$

$$
\mathcal{H}_B(\theta'_B) = \int_B (\bar{\beta}_L + \bar{\beta}_L^{\text{add}}) \dot{\theta}_B \theta'_B + \bar{\kappa}_L^I g_B^\theta \cdot g_B^{\theta'} + \bar{\mu}_L^I \bar{\alpha}_L^I (\tilde{a}_L (\hat{e}(u_B^M) - x_3 \hat{D}^2 \gamma_S(u_{\Omega 3})) \cdot \hat{I}) \theta'_B dx. \quad (4.27)
$$

Because $g_B^{\mathcal{T}} = (\hat{V}\mathcal{H}_B, 0)$ when $I_{T2} = 3$, the limit behavior of the adhesive layer is then similar to the one observed by [21] for thin linearly thermoelastic plates: a flexural problem for the component of the displacement field normal to S with a coupled membrane-thermal problem for the in-plane component of the displacement and the temperature.

The mechanical contact condition between the adhering bodies reads as:

$$
\begin{align}
& [u_{\Omega}] = 0 \notag \\
& \hat{e}(\widehat{\gamma_S(u_{\Omega})}) = 0 \tag{4.28}
\end{align}
$$

$$
- \ll [\sigma_{\Omega} e_3] \cdot e_3 \gg = 2 \bar{\rho}_L \gamma_S(\ddot{u}_{\Omega 3}) + \frac{1}{3} \hat{D}^2 (\tilde{a}_L \hat{D}^2 \gamma_S(u_{\Omega 3}))
$$

The material surface inserted between the two adhering bodies may be considered as a second-
grade elastic one, enjoying only a motion orthogonal to S (a flexural problem ...).

On the other hand, the thermal contact condition reads as:

$$
\ll \theta_{\Omega} \gg = 0, \quad -\ll q_{\Omega} \cdot e_3 \gg = 2 \left[ (\bar{\beta}_L + \bar{\beta}_L^{\text{add}}) \dot{\theta}_{\Omega} - \bar{\kappa}_L^I \hat{\Delta} \gamma_S(\theta_{\Omega}) + \bar{\mu}_L^I \bar{\alpha}_L^I \bar{a}_L \hat{e}(u_B^M) \cdot \hat{I} \right] \quad (4.29)
$$

it involves the additional variable $u_B^M$ defined on S with values in $\mathbb{R}^2$ satisfying

$$
\hat{\sigma}_B = \bar{\mu}_L^I \bar{a}_L (\hat{e}(u_B^M) - \bar{\alpha}_L^I \gamma_S(\theta_\Omega) \hat{I}), \quad \hat{\text{div}} \hat{\sigma}_B = 0. \quad (4.30)
$$

All this corresponds to a thermomechanical material surface occupying S whose material
constants are given by $\bar{\beta} + \bar{\beta}_L^{\text{add}}$, $\bar{\kappa}_L^I$, $\bar{\alpha}_L^I$, $\bar{\mu}_L^I \tilde{a}_L$ subjected to an inner heat source and free of
mechanical loading. Of course $u_B^M$ may be eliminated and consequently the thermal contact
condition along S is a nonlocal relation (in time, only) between the normal flux ($q_\Omega^\pm \cdot e_3)(\hat{x}, t)$ at
the courant time $t$ and the whole history of $\gamma_S(\theta_\Omega)(\hat{x}, \tau)$, $0 \le \tau \le t$.

When $I_{T2} = 4$, $\bar{\kappa}_L^I = \infty$ compels $\hat{V}\gamma_S(\theta_\Omega) = 0$, so that S is an isothermal surface possibly at $T_0$ when $\mathcal{H}_1(\gamma^{\mathcal{T}D}) > 0$, while the mechanical contact is similar to (4.28), (4.30).

This is also the case when $I_{T2} = 0$ or 2 but with thermal conditions like:

• $I_{T2} = 0$, $\ll \theta_{\Omega} \gg \neq 0$, $\mp q_{\Omega}^{\pm} \cdot e_3 = 0$: perfectly insulating wall,

• $I_{T2} = 2$, $\ll \theta_{\Omega} \gg = 0$, $-\ll q_{\Omega} \cdot e_3 \gg = 2(\bar{\beta}_L + \bar{\beta}_L^{\text{add}})\gamma_S(\dot{\theta}_{\Omega}) + \bar{\mu}_L^I \bar{\alpha}_L^I \bar{a}_L \hat{e}(u_B^M) \cdot \hat{I}$.
When $I_{T1} = 1$ (high specific heat coefficient), as the additional temperature variable $\theta_B$ does depend on $x_3$, the limit thermoelastic behavior of the adhesive layer cannot be interpreted in terms of a material surface.

Thus, in order to clarify the thermomechanical condition between the two adhering bodies, difficulties occur when $I_{M2} = 1$ or 4 and/or $I_{T2} = 1$, which correspond to the cases when the additional state variable $u_B$ depends on $x_3$ ($I_{M2} = 1$), does not explicitly depends on the traces on $S$ of the displacements of the adhering bodies ($I_{M2} = 4$) and/or the additional state variable $\theta_B$ depends on $x_3$ ($I_{T2} = 1$). In some of these cases, by adding a condition like $I_{M1} = 0$ (light adhesive layer) or $I_{T1} = 0$ (low specific heat coefficient) we again meet thermomechanical contact conditions involving the traces on $S$ of the state variables of the adhering bodies only as in the cases we listed previously.

When $I_{M2} = 1$, $u_B^0 := u_B - \text{Aff}(u_\Omega)$ satisfies

$$ egin{cases} u_B^0 \in H_{\partial_3, S^+ \cup S^-}(B, \mathbb{R}^3) := \{u \in H_{\partial_3}(B, \mathbb{R}^3) \text{ s.t. } \gamma_{S^\pm}(u) = 0\}, \\ \int_B \bar{\rho}_L \ddot{u}_B^0 \cdot v' + \bar{\mu}_L^I a_L(\partial_3 u_B^0 \otimes_S e_3) \cdot (\partial_3 v' \otimes_S e_3) \,dx \\ = - \int_B \rho_L \text{Aff}(u_\Omega) \cdot v' + \theta_B \bar{\mu}_L^I \bar{\alpha}_L^I a_L \partial_3 v' \otimes_S e_3 \cdot I \,dx, \quad \forall v' \in H_{\partial_3, S^+ \cup S^-}(B, \mathbb{R}^3) \end{cases} \quad (4.31) $$

so that, except when $I_{M1} = 0$ and $I_{T2} \neq 0$, 1 or $\bar{\alpha}_L^I = 0$, $u_B$ differs from $\text{Aff}(u_\Omega)$ and the *mechanical contact condition (4.11)* does not involve the sole instantaneous values of the traces on $S$ of $u_\Omega^\pm$ and $\theta_\Omega^\pm$. Of course, as the equations governing the evolutions of $u_B$ and $\theta_B$ can be solved in terms of the whole history of the traces on $S$ of $u_\Omega^\pm$ and $\theta_\Omega^\pm$, the contact condition at $(\hat{x}, t)$ is a rather complex function of the history of $\gamma_S(u_\Omega^\pm)(\hat{x}, \cdot)$ and $\gamma_S(\theta_\Omega^\pm)(\hat{x}, \cdot)$ and not only of the history of the jumps $\llbracket u_\Omega \rrbracket(\hat{x}, \cdot)$, $\llbracket \theta_\Omega \rrbracket(\hat{x}, \cdot)$.

When $I_{T2} = 1$, if $\text{Aff}(\theta_\Omega)$ is defined similarly as $\text{Aff}(u_\Omega)$ (see (4.13)), $\theta_B^0 := \theta_B - \text{Aff}(\theta_\Omega)$ satisfies:

$$ egin{cases} \theta_B^0 \in H_{\partial_3, S^+ \cup S^-}(B), \\ \int_B (\bar{\rho}_L \dot{\theta}_B^0 \theta' + \bar{\kappa}_L^I \partial_3 \theta_B^0 \partial_3 \theta' \,dx = - \int_B (\bar{\rho}_L \text{Aff}(\theta_\Omega) + \bar{\mu}_L^I \bar{\alpha}_L^I a_L(\partial_3 \dot{u}_B \otimes_S e_3) \cdot I) \theta' \,dx, & \text{if } I_{M2} = 1, 2 \\ \int_B (\bar{\rho}_L + \bar{\rho}_L^{\text{add}}) \dot{\theta}_B^0 \theta' + \bar{\kappa}_L^I \partial_3 \theta_B^0 \partial_3 \theta' \,dx = \\ = \begin{cases} - \int_B (\bar{\rho}_L + \bar{\rho}_L^{\text{add}}) \text{Aff}(\theta_\Omega) + \bar{\mu}_L^I \bar{\alpha}_L^I \tilde{a}_L \hat{e}(\gamma_S(u_\Omega)) \cdot \hat{I}) \theta' \,dx, & \text{if } I_{M2} = 3 \\ - \int_B (\bar{\rho}_L + \bar{\rho}_L^{\text{add}}) \text{Aff}(\theta_\Omega) - x_3 \bar{\mu}_L^I \bar{\alpha}_L^I \tilde{a}_L \hat{D}^2 \gamma_S(u_\Omega 3) \cdot \hat{I}) \theta' \,dx, & \text{if } I_{M2} = 4. \end{cases} \end{cases} \quad (4.32) $$

Except when $(I_{M2}, I_{T1}) = (2, 0)$ or $(\bar{\alpha}_L^I, I_{T1}) = (0, 0)$, $\text{Aff}(\theta_\Omega)$ does not solve (4.32), the thermal contact condition which at time $t$ reads as:

$$ \mp q_\Omega^\pm \cdot e_3 = \begin{cases} \frac{1}{2} \int_{-1}^{1} (1+x_3) \bar{\rho}_L \dot{\theta}_B + \bar{\mu}_L^I \bar{\alpha}_L^I a_L(\partial_3 \dot{u}_B \otimes_S e_3) \cdot I \,dx_3 + \bar{\kappa}_L^I \llbracket \theta_\Omega \rrbracket, & \text{if } I_{M2} = 1, 2 \\ \frac{1}{2} \int_{-1}^{1} (1+x_3) (\bar{\rho}_L + \bar{\rho}_L^{\text{add}}) \dot{\theta}_B \,dx_3 + \bar{\mu}_L^I \bar{\alpha}_L^I \tilde{a}_L \hat{e}(\gamma_S(\dot{u}_\Omega)) \cdot \hat{I} + \bar{\kappa}_L^I \llbracket \theta_\Omega \rrbracket, & \text{if } I_{M2} = 2, 3 \\ \frac{1}{2} \int_{-1}^{1} (1+x_3) (\bar{\rho}_L + \bar{\rho}_L^{\text{add}}) \dot{\theta}_B + \bar{\mu}_L^I \bar{\alpha}_L^I \tilde{a}_L \hat{D}^2 \gamma_S(u_\Omega 3) \cdot \hat{I} x_3 \,dx_3 + \bar{\kappa}_L^I \llbracket \theta_\Omega \rrbracket, & \text{if } I_{M2} = 4 \end{cases} \quad (4.33) $$

involves the whole history of $\gamma_S(u_\Omega^\pm)$ and $\gamma_S(\theta_\Omega^\pm)$.

So, in every cases, the limit thermomechanical behavior of the two adhering bodies and of the adhesive layer are of the same (thermoelastic) type as that of the original situation. But, of course, peculiarities of the limit behavior of the layer and the thermomechanical contact condition which replaces it strongly depend on the relative behaviors of the geometric and thermomechanical
parameters. *The thermomechanical coupling perpetuates* when $\bar{\mu}_L^1 \bar{\alpha}_L^1 e_B^u$ does not vanish which is the case when $I_{M2}$ differs from 0 or 2 with $\bar{\alpha}_L^1$ positive.

## 5. Concluding remarks

This rather lengthy and complex thermomechanical presentation of the results of our mathematical analysis exemplifies the flexibility of use but also the power of Trotter's theory of approximation of semi-groups of operators acting on variable spaces: it permits a unitary treatment with very few technicalities.

Our proposal of simplified but accurate enough models for the behavior of the structure made of the two adhering bodies and the thin adhesive layer, which has to be formulated on the genuine reference configurations $\Omega_\epsilon^\pm$ and $B_\epsilon$ is of course obtained through the Trotter representant $P_s^I U^1$ of the solution $U^1$ of the limit problem (3.43), *s taking the values of the original data*. When $I_{M2}$ differs from 0, a variant of $P_s^I U^1$ may be used through the construct detailed in the proof of Proposition 3.3. Thus, from a computational and practical point of view, a finite element approximation can be implemented without meshing the thin layer occupied by the adhesive!

It should be noted that, contrary to the cumbersome method—frequent in the literature— consisting of firstly switching to a fixed abstract domain through a “scaling” (change of coordinates and unknowns), abstract domain where the convergence is formally or rigorously studied, and secondly returning—but not always—to the initial physical domain, we have hereby treated *directly* through the representation operator $P_s^I$ the convergence of the initial problem where, obviously, the limit can be, according to index I, expressed in a fixed abstract domain defined through the “scaling” outlined above *but* which is used only when it is necessary to *refine* the determination of the asymptotic behavior of sequences of thermomechanical states with bounded energies.

To reduce the weight of our already copious study we have not detailed the cases $I_{M2} = 3 + \alpha$ and $I_{M2} = 4 + \beta$ corresponding to:

$$ 
\begin{align} 
\bar{\mu}_L^1 &:= \lim (\mu_L / \varepsilon^{1+2\alpha}) = +\infty, \quad \alpha \in (0, 1) \tag{5.1} \\ 
\bar{\mu}_L^1 &:= \lim (\mu_L / \varepsilon^{3+\beta}) = +\infty, \quad \beta > 0 \nonumber
\end{align} 
$$

for which a straightforward comparison argument leads to the mechanical spaces defined by:

$$ 
\begin{align} 
(u_\Omega, u_B) &\in \{H_{\Gamma, u\Phi(\Omega, \mathbb{R}^3)}^1; \hat{e}(\gamma_S(u_\Omega)) = 0\} \times \{0\}, \quad I_{M2} = 3 + \alpha \tag{5.2} \\ 
(u_\Omega, u_B) &\in \{H_{\Gamma, u\Phi(\Omega, \mathbb{R}^3)}^1; e(\gamma_S(u_\Omega)) = 0\} \times \{0\}, \quad I_{M2} = 4 + \beta \nonumber
\end{align} 
$$

endowed by the norm defined in (3.9) together with the convention $\infty \times 0 = 0$.

Eventually the present study which corrects and improves [6] may be considered as a framework to assess the formal and partial modelings proposed in [22], concerning poroelasticity as it is well known that the equations involved in linear poroelasticity are the same as those in linear thermoelasticity, and [23].

## Conflicts of interest

Authors have no conflict of interest to declare.

## References

[1] C. Licht, "Comportement asymptotique d'une bande dissipative mince de faible rigidité", C. R. Acad. Sci. Paris, Sér.-I 317 (1993), p. 429-433.

[2] C. Licht, G. Michaille, "A modelling of elastic adhesive bonded joints", Adv. Math. Sci. Appl. 7 (1997), p. 711-740.
[3] C. Licht, F. Lebon, A. Léger, "Dynamics of elastic bodies connected by a thin adhesive layer", in *Ultrasonic Wave Propagation in Non Homogeneous Media* (A. Léger, M. Deschamps, eds.), Springer Proceedings in Physics, vol. 128, Springer, Berlin, Heidelberg, 2009.

[4] C. Licht, A. Léger, S. Orankitjaroen, A. Ould Khaoua, "Dynamics of elastic bodies connected by a thin soft viscoelastic layer", *J. Math. Pures Appl.* **99** (2013), p. 685-703.

[5] C. Licht, S. Orankitjaroen, "Dynamics of elastic bodies connected by a thin soft inelastic layer", *C. R. Méc.* **341** (2013), p. 323-332.

[6] C. Licht, A. Ould Khaoua, T. Weller, "Transient response of thermoelastic bodies linked by a thin layer of low stiffness and high thermal resistivity", *C. R. Méc.* **343** (2015), p. 18-26.

[7] C. Licht, S. Orankitjaroen, P. Viriyasrisuwattana, T. Weller, "Thin linearly piezoelectric junctions", *C. R. Méc.* **343** (2015), p. 282-288.

[8] C. Licht, G. Michaille, P. Juntharee, "An asymptotic model for a thin, soft and imperfectly bonded elastic joint", *Math. Methods Appl. Sci.* **39** (2016), p. 981-997.

[9] P. Viriyasrisuwattana, C. Licht, S. Orankitjaroen, T. Weller, "Thin hybrid linearly piezoelectric junctions", *C. R. Méc.* **344** (2016), p. 128-135.

[10] C. Licht, S. Orankitjaroen, A. Ould Khaoua, T. Weller, "Transient response of elastic bodies connected by a thin stiff viscoelastic layer with evanescent mass", *C. R. Méc.* **344** (2016), p. 736-742.

[11] E. Bonetti, G. Bonfanti, C. Licht, R. Rossi, "Dynamics of two linearly elastic bodies connected by a heavy thin soft viscoelastic layer", *J. Elast.* **141** (2020), p. 75-107.

[12] F. Lebon, R. Rizzoni, "Asymptotic analysis of a thin interface: the case involving similar rigidity", *Int. J. Eng. Sci.* **48** (2010), p. 473-486.

[13] H. Brézis, *Functional Analysis, Sobolev Spaces and Partial Differential Equations*, Springer, New York, NY, 2011.

[14] E. Sanchez-Palencia, *Non Homogeneous Media and Vibration Theory*, Springer, Berlin, Heidelberg, 1980.

[15] H. F. Trotter, "Approximation of semi-groups of operators", *Pac. J. Math.* **28** (1958), p. 897-919.

[16] C. Licht, T. Weller, "Approximation of semi-groups in the sense of Trotter and asymptotic mathematical modeling in physics of continuous media", *Discrete Contin. Dyn. Syst.*—S **12** (2019), no. 6, p. 1709-1741.

[17] R. Nagel, A. Rhandi, "Theme issue "Semigroup applications everywhere"", *Phil. Trans. R. Soc.* **378** (2019), no. 2185.

[18] P. G. Ciarlet, *Mathematical Elasticity*, vol. II, North-Holland, Amsterdam, The Netherlands, 1997.

[19] C. Licht, "Asymptotic modeling of assemblies of thin linearly elastic plates", *C. R. Méc.* **335** (2007), p. 775-780.

[20] O. Iosifescu, C. Licht, G. Michaille, "Nonlinear boundary conditions in Kirchhoff-Love plate theory", *J. Elast.* **96** (2009), p. 57-79.

[21] D. Blanchard, G. A. Francfort, "Asymptotic thermoelastic behavior of flat plates", *Q. Appl. Math.* **45** (1987), p. 645-667.

[22] M. Serpilli, "Classical and higher order interface conditions in poroelasticity", *Ann. Solid Struct. Mech.* **11** (2019), p. 1-10.

[23] M. Serpilli, S. Dumont, R. Rizzoni, F. Lebon, "Interface models in coupled thermoelasticity", *Special Issue "Advances in Multiscale and Multifield Solid Material Interfaces", Technologies* **9** (2021), no. 1, p. 17.