Transient response of a thin linearly elastic plate with Norton or Tresca friction

Abstract

By using a nonlinear version of Trotter’s theory of approximation of semi-groups acting on variable Hilbert spaces, we propose an asymptotic modeling for the behavior of a linearly elastic plate in bilateral contact with a rigid body along part of its lateral boundary with Norton or Tresca friction.

Keywords

Linearly elastic thin plates Norton or Tresca friction dynamics maximal-monotone operators Trotter’s theory of approximation of semi-groups

1. Setting the problem

This article is an extension to the dynamic case of our study [4] devoted to the quasi-static evolution of a linearly elastic thin plate in bilateral contact with a rigid body along part of its lateral boundary with a nonlinear friction. By taking into account both this nonstationary and nonlinear boundary condition and various behaviors of the density of the plate, it is also an extension of the classical study [8] devoted to the dynamics of a plate clamped on its lateral boundary. By using the classical choice to describe the state of the system by the couple displacement/velocity, the problem is formulated as a nonlinear evolution equation governed by a maximal-monotone operator acting in a Hilbert space of possible states with finite energy. Thus to study the asymptotic behavior with respect to geometrical or mechanical parameters as thickness, density, intensity of the friction and external loading, it is natural to use Trotter’s theory of approximation of semi-groups of operators acting on variable spaces, specifically a nonlinear extension which we introduced in [4]. The power and flexibility of this theory, where the nature of the limit space may be completely different from the one of the initial problem, make it possible, in a very simple and unified way (with respect to the various relative behaviors of the parameters), to account for the somewhat singular asymptotic behavior: either dynamic in flexion and quasi-static in membrane or dynamic in membrane and frozen in flexion, with friction condition bearing sometimes on the membrane component sometimes on the flexural one. Moreover by using again the fact that limit and initial spaces may be different, we bypass the classic scaling of the physical problem, and the convergence in a suitable sense (that of Trotter) of the solution is nothing but the convergence of relative energetic gaps measured on the real physical plate (the only ones who have a meaning because the energies are going to zero …). Moreover it is observed that the strain tensor in the physical plate is not equivalent to a Kirchhoff–Love or Reissner–Mindlin one. This second appeal of using Trotter’s theory was not fully exploited in [5] of which the present study is also an extension.

More precisely, the problem may be set as follows. Let ${e_{1}, e_{2}, e_{3}}$ be an orthonormal basis of $R^{3}$ assimilated to the physical Euclidean space. For all $ξ = (ξ_{1}, ξ_{2}, ξ_{3})$ of $R^{3}$ , $\hat{ξ}$ stands for $(ξ_{1}, ξ_{2})$ . The Greek coordinate indices run in ${1, 2}$ whereas the Latin ones run in ${1, 2, 3}$ . Let $S^{n}$ be the space of all $(n \times n)$ symmetric matrices equipped with the usual inner product and norm denoted by · and $| \cdot |$ (as well as in $R^{n}$ ). For all η in $S^{3}$ , we define $\hat{η}$ in $S^{2}$ by ${\hat{η}}_{α β} = η_{α β}$ and $η^{⊥}$ in $S^{3}$ by $η_{α β}^{⊥} = 0$ and $η_{i 3}^{⊥} = η_{i 3}$ . The space of linear mappings from $S^{n}$ into $S^{m}$ is denoted by $Lin (S^{n}, S^{m})$ and we simply write $Lin (S^{n})$ when $n = m$ and the mappings are symmetric. For any open subset G of $R^{n}$ , $H_{Γ}^{1} (G, R^{n})$ stands for the subspace of the Sobolev space $H^{1} (G, R^{n})$ composed of the elements which vanish on a smooth part Γ of the boundary $\partial G$ of G.

A reference configuration of a thin linearly elastic plate is the closure of $Ω^{ε} : = ω \times (- ε, ε)$ whose unit outward normal is $n^{ε}$ . Here ε is a small positive number and ω a bounded domain of $R^{2}$ with a Lipschitz-continous boundary $\partial ω$ . Given a partition $(γ_{D}, γ_{N}, γ_{C})$ of $\partial ω$ , the plate with elasticity tensor $a^{ε}$ and density $ρ δ^{ε}$ is clamped on $Γ_{D}^{ε} : = γ_{D} \times (- ε, ε)$ , subjected to body forces with density $f^{ε}$ and surface forces with density $g^{ε}$ on $Γ_{N}^{ε} : = γ_{N} \times (- ε, ε) \cup Γ_{+}^{ε} \cup Γ_{-}^{ε}$ , where $Γ_{\pm}^{ε} : = ω \times {\pm 1}$ . On the remaining part $Γ_{C}^{ε} : = γ_{C} \times (- ε, ε)$ of $\partial Ω^{ε}$ , the plate is in bilateral contact with a rigid body by Tresca or Norton friction with a viscosity coefficient μ.

So the problem $(P^{s})$ of determining the dynamic evolution of the plate, in the framework of small strains, involves a triplet $s : = (ε, ρ, μ)$ in ${(0, + \infty)}^{3}$ of essential data and the displacement and velocity fields $U^{s} : = (u^{s}, v^{s})$ have to satisfy: $\begin{array}{l} (1) & (P^{s}) \{\begin{matrix} Find (u^{s}, v^{s} = \frac{\partial u^{s}}{\partial t}) sufficiently smooth in Ω^{ε} \times [0, T] such that \\ u^{s} = 0 on Γ_{D}^{ε} \times [0, T], u_{N}^{s} = 0 on Γ_{C}^{ε} \times [0, T] \\ (u^{s} (\cdot, 0), v^{s} (\cdot, 0)) = U^{s 0} : = (u^{s 0}, v^{s 0}) in Ω^{ε} \\ and there exists ξ^{s} in \partial ϕ_{p} (v_{T}^{s}) satisfying \\ \int_{Ω^{ε}} ρ δ^{ε} \frac{\partial v^{s}}{\partial t} \cdot v + a^{ε} e (u^{s}) \cdot e (v) d x^{ε} + μ \int_{Γ_{C}^{ε}} ξ^{s} \cdot v_{T} d H_{2} \\ = \int_{Ω^{ε}} f^{ε} \cdot v d x^{ε} + \int_{Γ_{N}^{ε}} g^{ε} \cdot v d H_{2} \\ for all v sufficiently smooth in Ω^{ε} and such that \\ v = 0 on Γ_{D}^{ε}, v_{N} = 0 on Γ_{C}^{ε}, \end{matrix} \end{array}$ where $ϕ_{p} (ζ) = \frac{1}{p} | ζ |^{p}$ for all ζ in $R^{n}$ and $1 ⩽ p ⩽ 2$ , $\partial ϕ_{p} (ζ)$ is the subdifferential at ζ of $ϕ_{p}$ , $e (u)$ is the linearized strain tensor associated with the diplacement field u (the symmetric part of $\nabla u$ the gradient of u) and $u_{N} : = u \cdot n^{ε}, u_{T} : = u - u_{N} n^{ε}$ are the normal and tangential components of u on $Γ_{C}^{ε}$ , t the time and, lastly, $H_{n}$ denotes the n-dimensional Hausdorff measure.

In the next section, we will show that $(P^{s})$ in a clearly defined sense has a unique solution under a suitable assumption on the initial state $U^{s 0}$ and some physically reasonable assumptions, namely: $\begin{array}{l} (H_{1}) \{\begin{matrix} ∙ δ^{ε} \in L^{\infty} (Ω^{ε}); \exists α > 0 s.t. δ^{ε} (x) ⩾ α a.e. x \in Ω^{ε} \\ ∙ a^{ε} \in L^{\infty} (Ω^{ε}; Lin (S^{3})); α | e |^{2} ⩽ a^{ε} (x) e \cdot e a.e. x \in Ω^{ε}, \forall e \in S^{3}, \\ ∙ H_{1} (γ_{D}) > 0, \\ ∙ f^{ε} \in B V^{1} (0, T; L^{2} (Ω^{ε}, R^{3})), g^{ε} \in B V^{1} (0, T; L^{2} (Γ_{N}^{ε}, R^{3})), \end{matrix} \end{array}$ where, for any Hilbert space $H, B V^{1} (0, T; H)$ denotes the subspace of $B V (0, T; H)$ of all elements whose time-derivative in the sense of distributions belongs to $B V (0, T; H)$ the subspace of $L^{1} (0, T; H)$ of all elements whose time-derivative in the sense of distributions is a H-valued bounded measure.

But the essential task is for various reasons, especially numerical ones, to propose a simplified but accurate enough model. This will be done by considering $s = (ε, ρ, μ)$ as a triplet of parameters taking values in a countable subset of ${(0, \infty)}^{3}$ with a unique cluster point $\overline{s}$ in ${0} \times [0, + \infty) \times [0, + \infty]$ and studying the asymptotic behavior of $U^{s}$ when s goes to $\overline{s}$ . Indeed there are other significant physical data like the stiffness of the plate and the nature of the external loading. To simplify the presentation, we assume that the stiffness of the plate remains constant when s goes to $\overline{s}$ and that the loading is simply described by ε (see assumption $(H_{4})$ in Section 2). A more general case will be the subject of Remark 3 of Section 4.

2. Existence and uniqueness results for $(P^{s})$

Classicaly we seek $U^{s}$ on the form $\begin{array}{l} (2) & U^{s} (t) = U^{s e} (t) + U^{s r} (t) \end{array}$ where $U^{s e} (t) = (u^{s e} (t), 0)$ is defined by $\begin{array}{l} (3) & \{\begin{matrix} u^{s e} (t) \in U^{s} : = {u \in H_{Γ_{D}^{ε}}^{1} (Ω^{ε}); u_{N} = 0 on Γ_{C}^{ε}} \\ φ^{s} (u^{s e} (t), u^{'}) = L^{ε} (t) (u^{'}) \forall u^{'} \in U^{s}, \forall t \in [0, T], \end{matrix} \end{array}$ with $\begin{array}{l} (4) & \begin{aligned} φ^{s} (u, u^{'}) : = \frac{1}{ε^{3}} \int_{Ω^{ε}} a^{ε} e (u) \cdot e (u^{'}) d x^{ε} \forall u, u^{'} \in U^{s} \\ L^{ε} (t) (u^{'}) : = \frac{1}{ε^{3}} (\int_{Ω^{ε}} f^{ε} (x^{ε}, t) \cdot u^{'} (x^{ε}) d x^{ε} + \int_{Γ_{N}^{ε}} g^{ε} (x^{ε}, t) \cdot u^{'} (x^{ε}) d H_{2}) \\ for all (u^{'}, t) in U^{s} \times [0, T] . \end{aligned} \end{array}$ Clearly according to $(H_{1}), u^{s e}$ is well defined and satisfies $\begin{array}{l} (5) & u^{s e} \in B V^{1} (0, T; U^{s}) . \end{array}$ The remaining part $U^{s r}$ of $U^{s}$ will suggest an evolution equation in an Hilbert space $H^{s}$ of possible state with finite mechanical energy governed by a maximal-monotone operator $A^{s}$ . Let $k^{s}$ be the bilinear form associated with the kinetic energy: $\begin{array}{l} (6) & k^{s} (v, v^{'}) : = \frac{1}{ε^{3}} \int_{Ω^{ε}} ρ δ^{ε} v \cdot v^{'} d x^{ε} \forall v, v^{'} \in V^{s} : = L^{2} (Ω^{ε}; R^{3}) . \end{array}$ Then $H^{s}$ is defined by: $\begin{array}{l} (7) & H^{s} : = U^{s} \times V^{s} \end{array}$ and equipped with the following inner product and norm: $\begin{array}{l} (8) & \begin{aligned} {(U, U^{'})}^{s} : = φ^{s} (u, u^{'}) + k^{s} (v, v^{'}) \forall U = (u, v), U^{'} = (u^{'}, v^{'}) \in H^{s} \\ | U |^{s} : = {[{(U, U)}^{s}]}^{1 / 2} \end{aligned} \end{array}$ The reason behind the choice of the normalization factor $ε^{3}$ will clearly appear in the next section. Let $ε^{3} D^{s}$ be the global pseudo-potential of dissipation involved by the friction law: $\begin{array}{l} (9) & v \in U^{s} \mapsto D^{s} (v) : = \frac{μ}{ε^{3}} \int_{Γ_{C}^{ε}} ϕ_{p} (v_{T}) d H_{2} \end{array}$ So if $A^{s}$ is the multi-valued operator in $H^{s}$ such that $\begin{array}{l} (10) & \{\begin{matrix} D (A^{s}) : = \{\begin{matrix} U = (u, v) \in H^{s}; \{\begin{matrix} i) v \in U^{s} \\ ii) \exists w \in V^{s} with \end{matrix} \\ {((u, w), (v^{'}, v^{'}))}^{s} + D^{s} (v + v^{'}) - D^{s} (v) ⩾ 0 \forall v^{'} \in U^{s} \end{matrix} \\ - A^{s} U = {(v, w); w satisfies ii) of definition of D (A^{s})} \end{matrix} \end{array}$ then the very definition of $D^{s}$ implies that $(P^{s})$ is formally equivalent to $\begin{array}{l} (11) & (P^{s}) \{\begin{matrix} \frac{d U^{s}}{d t} + A^{s} (U^{s} - U^{s e}) ∋ 0 \\ U^{s} (0) = U^{s 0} \end{matrix} \end{array}$ and we obviously have:

Proposition 1.
The operator $A^{s}$ is maximal monotone and for all $ψ^{s} = (ψ_{u}^{s}, ψ_{v}^{s})$ in $H^{s}$ $\begin{array}{l} \{\begin{matrix} {\overline{U}}^{s} = ({\overline{u}}^{s}, {\overline{v}}^{s}) s.t. \\ {\overline{U}}^{s} + A^{s} {\overline{U}}^{s} ∋ ψ^{s} \end{matrix} ⟺ \{\begin{matrix} J^{s} ({\overline{v}}^{s}) ⩽ J^{s} (v) \forall v \in U^{s}, \\ J^{s} (v) : = \frac{1}{2} {| (v, v) |^{s}}^{2} + {((ψ_{u}^{s}, - ψ_{v}^{s}), (v, v))}^{s} + D^{s} (v) \\ {\overline{u}}^{s} = {\overline{v}}^{s} + ψ_{u}^{s} \end{matrix} \end{array}$

Therefore a result of [1] yields Theorem 1.
Under assumption $(H_{1})$ and $\begin{array}{l} (H_{2}) U^{s 0} \in U^{s e} (0) + D (A^{s}) \end{array}$ the problem $(P^{s})$ has a unique solution such that $U^{s}$ belongs to $W^{1, \infty} (0, T; H^{s})$ and the first line of $(P^{s})$ is satisfied almost everywhere in $(0, T]$ .

3. Asymptotic behavior

Depending on the behavior of $(ρ, μ)$ with respect to ε when s goes to $\overline{s}$ , we may distinguish six main cases indexed by $I = (I_{1}, I_{2}) \in {1, 2} \times {1, 2, 3}$ defined as follows.

Let $ρ^{⋆ I_{1}} = {\begin{matrix} ρ ε^{- 2} I_{1} = 1 \\ ρ I_{1} = 2 \end{matrix}$ and $μ^{⋆ I_{2}} = {\begin{matrix} μ ε^{- 2} I_{2} = 1 \\ μ ε^{p - 2} I_{2} \in {2, 3} \end{matrix}$ , we assume $\begin{array}{l} (H_{3}) \{\begin{matrix} There exists ({\overline{ρ}}^{I_{1}}, {\overline{μ}}^{I_{2}}) in (0, + \infty) \times [0, + \infty] such that \\ ({\overline{ρ}}^{I_{1}}, {\overline{μ}}^{I_{2}}) = lim_{s \to \bar{s}} (ρ^{⋆ I_{1}}, μ^{⋆ I_{2}}) with {\overline{μ}}^{2} \in [0, + \infty) and {\overline{μ}}^{3} = + \infty . \end{matrix} \end{array}$ We also make a due assumption $(H_{4})$ on the density and the elasticity tensor of the plate, and on the loading $(f^{ε}, g^{ε})$ which express that the previous quantities stems from fixed quantities defined in $\overline{Ω}$ the closure of $Ω : = ω \times (- 1, 1)$ by a change $Π^{ε}$ of coordinates (see [2]): $\begin{array}{l} (12) & x = (\hat{x}, x_{3}) \in \overline{Ω} \mapsto x^{ε} = Π^{ε} (x) : = (\hat{x}, ε x_{3}) \in {\overline{Ω}}^{ε} . \end{array}$ In the following, systematically $x^{ε}$ and x are connected by $x^{ε} = Π^{ε} (x)$ . As for Ω, index ε will be dropped for the images by ${(Π^{ε})}^{- 1}$ of $Γ_{D}^{ε}, Γ_{N}^{ε}, Γ_{C}^{ε}, Γ_{\pm}^{ε}$ , and if α denotes a fixed positive real number $(H_{4})$ reads as: $\begin{array}{l} (H_{4}) \{\begin{matrix} ∙ \exists δ \in L^{\infty} (Ω^{ε}); 0 < α ⩽ δ (x) a.e. x \in Ω \\ ∙ \exists a \in L^{\infty} (Ω; Lin (S^{3})); α | e |^{2} ⩽ a (x) e \cdot e a.e. x \in Ω, \forall e \in S^{3}, \\ s.t. δ^{ε} (x^{ε}) = δ (x), a^{ε} (x^{ε}) = a (x) a.e. x \in Ω \\ ∙ \exists (f, g) \in B V^{1} (0, T; L^{2} (Ω, R^{3}) \times L^{2} (Γ_{N}, R^{3})) s.t. \\ {\hat{f}}^{ε} (x^{ε}) = ε \hat{f} (x), f_{3}^{ε} (x^{ε}) = ε^{2} f_{3} (x), \forall x \in Ω \\ {\hat{g}}^{ε} (x^{ε}) = ε^{2} \hat{g} (x), g_{3}^{ε} (x^{ε}) = ε^{3} g_{3} (x), \forall x \in Γ_{N} \cap Γ_{\pm}, \\ {\hat{g}}^{ε} (x^{ε}) = ε \hat{g} (x), g_{3}^{ε} (x^{ε}) = ε^{2} g_{3} (x), \forall x^{ε} \in Γ_{N} \cap Γ_{lat} \end{matrix} \end{array}$ where $Γ_{lat} : = \partial ω \times (- 1, 1)$ .

From now on, c or C denotes various constants which may vary from line to line.

3.1. The crucial point

To display the asymptotic behavior of $U^{s}$ it is convenient to use this simple scaling mapping: $\begin{array}{l} (13) & \begin{aligned} w \in L^{2} (Ω^{ε}, R^{3}) \mapsto S_{ε} w \in L^{2} (Ω, R^{3}) \\ S_{ε} w (x) : = (\frac{1}{ε} \hat{w} (x^{ε}), w_{3} (x^{ε})) a.e. x \in Ω \end{aligned} \end{array}$ As for all w in $H^{1} (Ω^{ε}; R^{3})$ $\begin{array}{l} (14) & e (w) (x^{ε}) = ε e (ε, S_{ε} w) (x) a.e. x \in Ω \end{array}$ with $\begin{array}{l} (15) & e_{i j} (ε, z) : = \{\begin{matrix} e_{α β} (z) \\ \frac{1}{ε} e_{α 3} (z) = \frac{1}{ε} e_{3 α} (z) \forall z \in H^{1} (Ω; R^{3}) \\ \frac{1}{ε^{2}} e_{33} (z) \end{matrix} \end{array}$ there holds: $\begin{array}{l} (16) & φ^{s} (w, w) = \int_{Ω} a e (ε, S_{ε} w) \cdot e (ε, S_{ε} w) d x \forall w \in U^{s} \\ (17) & k^{s} (w, w) = \int_{Ω} ρ δ (| \hat{S_{ε} w} |^{2} + \frac{1}{ε^{2}} {| {(S_{ε} w)}_{3} |}^{2}) d x \forall w \in V^{s} . \end{array}$

To introduce a space suitable for describing the asymptotic behavior we use some classical spaces and notations useful in the mathematical modeling of linearly elastic plates: $\begin{array}{l} V_{K L} : = & {w \in H_{Γ_{D}}^{1} (Ω, R^{3}); e_{i 3} (v) = 0 on Ω} \\ = & {w \in H_{Γ_{D}}^{1} (Ω, R^{3}); \exists (w^{M}, w^{F}) \in H^{1} (ω; R^{2}) \times H^{2} (ω); \\ (18) & w = (w^{M} - x_{3} \hat{\nabla} w^{F}, w^{F})}, \\ (19) & V_{K L}^{m} = & {w \in V_{K L}; w^{F} = 0}, V_{K L}^{f} : = {w \in V_{K L}; w^{M} = 0}, \\ L_{K L} : = & {w \in H^{- 1} (Ω, R^{3}); \exists (w^{M}, w^{F}) \in L^{2} (ω, R^{3}); \\ (20) & w = (w^{M} - x_{3} \hat{\nabla} w^{F}, w^{F})}, \\ (21) & L_{K L}^{m} : = & {w \in L_{K L}; w^{f} = 0}, L_{K L}^{f} : = {w \in L_{K L}; w^{m} = 0}, \end{array}$ where for all w in $L_{K L}$ $\begin{array}{l} (22) & w^{m} : = (w^{M}, 0), w^{f} : = (- x_{3} \hat{\nabla} w^{F}, w^{F}) . \end{array}$ Let define $\begin{array}{l} (23) & \{\begin{matrix} (\tilde{a}, E) \in Lin (S^{2}) \times Lin (S^{2}, S^{3}); \\ \tilde{a} q \cdot q : = Min {a Q \cdot Q; Q \in S^{3}, \hat{Q} = q} = : a E (q) \cdot E (q) \\ E^{ε} (q) (x^{ε}) : = ε E (q) (x) \forall q \in L^{2} (Ω, S^{2}), \\ φ (w, w^{'}) : = \int_{Ω} \tilde{a} \hat{e (w)} \cdot \hat{e (w^{'})} d x \forall w, w^{'} \in U \\ k^{1} (w, w^{'}) : = {\overline{ρ}}^{1} \int_{Ω} δ w_{3} w_{3}^{'} d x \forall w, w^{'} \in V^{1} \\ k^{2} (w, w^{'}) : = {\overline{ρ}}^{2} \int_{Ω} δ \hat{w} \cdot {\hat{w}}^{'} d x \forall w, w^{'} \in V^{2} \\ U : = {w \in V_{K L}, w_{N} = 0 on Γ_{C}}, V^{1} : = L_{K L}^{f} and V^{2} : = L_{K L}^{m} \\ H^{I} : = U \times V^{I_{1}} \forall I = (I_{1}, I_{2}) \in {1, 2} \times {1, 2, 3} \\ {(U, U^{'})}^{I} : = φ (u, u^{'}) + k^{I_{1}} (v, v^{'}) \forall U = (u, v), U^{'} = (u^{'}, v^{'}) \in H^{I} \\ | U |^{I} : = {{(U, U)}^{I}}^{1 / 2} \end{matrix} \end{array}$ Clearly $H^{I}$ equipped with the inner product ${(\cdot, \cdot)}^{I}$ is a Hilbert space and taking into account the fundamental link between velocity and displacement, we obviously deduce

Proposition 2.
For all sequence $(X^{s}) = (ξ^{s}, ζ^{s})$ in $H^{s}$ such that $| X^{s} |^{s}$ is uniformly bounded, there exist a not relabeled subsequence and $X^{I} = (ξ^{I}, ζ^{I})$ in $H^{I}$ such that
$(ξ^{I}, ζ^{I})$ is the weak limit in $H^{1} (Ω, R^{3}) \times L^{2} (Ω, R^{3})$ of $(ξ^{s}, (0, ζ_{3}^{s}))$ when $I_{1} = 1$ or $(ξ^{s}, ζ^{s})$ when $I_{1} = 2$ ,

$| X^{I} |^{I} ⩽ \underset{s \to \bar{s}}{lim_{_}} | X^{s} |^{s}$ .

Hence $H^{I}$ appears to be suitable for describing the asymptotic behavior. Moreover it is exactly the appropriate space because any element U of $H^{I}$ admits a representative $P^{s I} U$ in $H^{s}$ which is energetically very close to U. As the formula defining the linear form ${(U, \cdot)}^{I}$ also works on $H^{s}$ through the scaling $S_{ε}$ , we have:
Proposition 3.
For all $U = (u, v)$ in $H^{I}$ , let $P^{s I} U = (P_{u}^{s I} u, P_{v}^{s I} v)$ in $H^{s}$ defined by $\begin{array}{l} (24) & {(P^{s I} U, U^{'})}^{s} = {(U, (S_{ε} u^{'}, S_{ε} v^{'}))}^{I} \forall U^{'} = (u^{'}, v^{'}) \in H^{s}, \end{array}$ then $\begin{array}{l} (25) & i) \exists C > 0 s.t. {| P^{s I} U |}^{s} ⩽ C | U |^{I} \forall U \in H^{s}, \forall s \\ (26) & ii) lim_{s \to \bar{s}} {| P^{s I} U |}^{s} = | U |^{I} \forall U \in H^{s}, \\ (27) & iii) lim_{s \to \bar{s}} \frac{1}{ε^{3}} \int_{Ω^{ε}} a^{ε} (e (P_{u}^{s I} u) - E^{ε} (\hat{e (u)}) \cdot (e (P_{u}^{s I} u) - E^{ε} (\hat{e (u)}) d x^{ε} = 0 \\ (28) & \frac{ρ}{ε^{3}} \int_{Ω^{ε}} δ^{ε} {| P_{v}^{s I} v - S_{ε}^{- 1} v |}^{2} d x^{ε} = {(\frac{{\overline{ρ}}^{I_{1}}}{ρ^{⋆ I_{1}}} - 1)}^{2} ρ^{⋆ I_{1}} \int_{Ω} δ | v |^{2} d x \end{array}$
Proof.
As $ξ^{s} : = S_{ε} P_{u}^{s I} u$ satisfies $\begin{array}{l} (29) & \{\begin{matrix} ξ^{s} \in V_{N} : = {ζ \in H_{Γ_{D}}^{1} (Ω, R^{3}), ζ \cdot n = 0 on Γ_{C}} \\ \int_{Ω} a e (ε, ξ^{s}) \cdot e (ε, ξ) d x = \int_{Ω} \tilde{a} \hat{e (u)} \cdot \hat{e (ε, ξ)} d x \forall ξ \in V_{N} \end{matrix} \end{array}$ Korn inequality and $ε < 1$ imply: $\begin{array}{l} c {| ξ^{s} |}_{H^{1} (Ω, R^{3}}^{2}) ⩽ {| e (ξ^{s}) |}_{L^{2} (Ω, S^{3})}^{2} ⩽ {| e (ε, ξ^{s}) |}_{L^{2} (Ω, S^{3})}^{2} ⩽ C . \end{array}$ Hence there exist a not relabeled subsequence and $(\overline{ξ}, \overline{e})$ in $U \times L^{2} (Ω, S^{3})$ such that $\begin{array}{l} (30) & \begin{aligned} (ξ^{s}, e (ε, ξ^{s})) weakly converges in H^{1} (Ω, R^{3}) \times L^{2} (Ω, S^{3}) toward (\overline{ξ}, \overline{e}) \\ \hat{\overline{e}} = \hat{e (\overline{ξ})} . \end{aligned} \end{array}$ First, as in [2] choosing $ξ = ε (\hat{ζ}, ε ζ_{3})$ in (29), with ζ arbitrary in $V_{N}$ , implies ${(a \overline{e})}^{⊥} = 0$ which is equivalent to $\overline{e} = E (\hat{\overline{e}})$ and consequently $\overline{e} = E (\hat{e (\overline{ξ}}))$ . Next choosing ξ arbitrary in $U$ gives $\begin{array}{l} \overline{ξ} \in U; \int_{Ω} \tilde{a} \hat{e (\overline{ξ})} \cdot \hat{e (ξ)} d x = \int_{Ω} \tilde{a} \hat{e (u)} \cdot \hat{e (ξ)} d x \forall ξ \in U \end{array}$ which implies $\overline{ξ} = u$ and that the whole sequence satisfies (30).

Last choosing $ξ = ξ^{s}$ in (29) gives $\begin{array}{l} (31) & lim_{s \to \bar{s}} \int_{Ω} a e (ε, ξ^{s}) \cdot e (ε, ξ^{s}) d x = \int_{Ω} \tilde{a} \hat{\overline{e}} \cdot \hat{\bar{e}} d x = \int_{Ω} a E (\bar{e}) \cdot E (\bar{e}) d x . \end{array}$ Hence $e (ε, ξ^{s})$ converges strongly in $L^{2} (Ω, S^{3})$ toward $E (\hat{e (u)})$ , that is to say (26)–(27) by due account of (17) and the definition of $E^{ε}$ in (23). To end the proof it sufficies to notice that $P_{v}^{s I} v = \frac{{\bar{ρ}}^{I_{1}}}{ρ^{⋆ I_{1}}} S_{ε}^{- 1} v$ □

To complete guessing the asymptotic behavior, according to Proposition 1, it remains to consider sequences $(z^{s})$ with uniformly bounded global potential of dissipation $D^{s} (z^{s})$ and “total energy functional” ${[| (z^{s}, z^{s}) |^{s}]}^{2}$ .

Let $U^{I}$ , $D^{I}$ , ${\overset{˚}{z}}^{I}$ defined by: $\begin{array}{l} (32) & \begin{aligned} U^{1, 1} : = U \\ U^{1, 2} : = {z \in U; z_{3} = 0 on Γ_{C}}; U^{2, I_{2}} : = U^{1, I_{2}} \cap V^{2} \forall I_{2} \in {1, 2, 3}, \\ U^{1, 3} : = {z \in U; z = 0 on Γ_{C}} \end{aligned} \\ (33) & \begin{aligned} D^{I_{1}, 1} (z) : = 2 {\bar{μ}}^{1} \int_{γ_{c}} ϕ_{p} (z^{F}) d H_{1} \\ D^{I_{1}, 2} (z) : = 2 {\bar{μ}}^{2} \int_{γ_{c}} ϕ_{p} (z^{F}) d H_{1} + I_{U^{I_{1}, 2}} (z) \forall I_{1} \in {1, 2}, \\ D^{I_{1}, 3} : = I_{U^{I_{1}, 3}} \end{aligned} \end{array}$ and $\begin{array}{l} (34) & {\overset{˚}{z}}^{I} : = \{\begin{matrix} z^{f} & if I_{1} = 1 \\ z^{m} & if I_{1} = 2 \end{matrix} \forall z \in U^{I} \end{array}$ where $I_{A}$ is the indicator function of a subset A of a set B defined as: $\begin{array}{l} I_{A} (x) : = \{\begin{matrix} 0 & if x \in A \\ + \infty & if x \in B ∖ A . \end{matrix} \end{array}$ Thus a simple argument of lower-semi continuity implies
Proposition 4.
For all sequence $(z^{s})$ in $U^{s}$ such that $D^{s} (z^{s}) + {[| (z^{s}, z^{s}) |^{s}]}^{2} ⩽ C$ , there exist a not relabeled subsequence and z in $U^{I}$ such that $S_{ε} z^{s}$ weakly converges in $H^{1} (Ω, R^{3})$ toward z and $\begin{array}{l} (35) & D^{I} (z) + {[{| (z, \overset{˚}{z}) |}^{I}]}^{2} ⩽ \underset{s \to \bar{s}}{lim_{_}} (D^{s} (z^{s}) + {[{| (z^{s}, z^{s}) |}^{s}]}^{2}) \end{array}$

We now are in a position to establish a convergence result for the solution $U^{s}$ of $(P^{s})$ by using a non linear version of Trotter theory of approximation of semi-groups acting on variable spaces, as developed in the Appendix of [4], which is very efficient in such situations where the functional framework is variable (see [6]).
3.2. Convergence

3.2.1. Recaps on Trotter theory of approximation of semi-groups

Let ${(H_{n})}_{n \in N}$ , respectively H be Hilbert spaces with norms $| \cdot |_{n}$ , respectively $| \cdot |$ and a sequence of linear operators ${(P_{n})}_{n \in N}$ from H into $H_{n}$ satisfying:

There exists $C > 0$ such that $| P_{n} X |_{n} ⩽ C | X |$ for all X in H and n in $N$ ,

$lim_{n \to \infty} | P_{n} X |_{n} = | X |$ for all X in H.

A sequence ${(X_{n})}_{n \in N}$ in $H_{n}$ is said to converge in the sense of Trotter toward X in H iff $\begin{array}{l} (36) & lim_{n \to \infty} | P_{n} X - X_{n} |_{n} = 0 \end{array}$ One has the following convergence result (see [4]):

Theorem 2.
Let $A_{n} : H_{n} ⇉ H_{n}$ , $A : H ⇉ H$ be maximal monotone operators, $F_{n} \in L^{1} (0, T; H_{n})$ , $F \in L^{1} (0, T; H)$ , $X_{n}^{0} \in \overline{D (A_{n})}$ , $X^{0} \in \overline{D (A)}$ and let $X_{n}, X$ be the weak solution of $\begin{array}{l} \{\begin{matrix} \frac{d X_{n}}{d t} + A_{n} X_{n} ∋ F_{n} \\ X_{n} (0) = X_{n}^{0} \end{matrix} \{\begin{matrix} \frac{d X}{d t} + A X ∋ F \\ X (0) = X^{0} \end{matrix} \end{array}$

If
$lim_{n \to \infty} | P_{n} X^{0} - X_{n}^{0} |_{n} = 0, lim_{n \to \infty} \int_{0}^{T} | P_{n} F (t) - F_{n} (t) |_{n} d t = 0$

$lim_{n \to \infty} | {(I + A_{n})}^{- 1} P_{n} z - P_{n} {(I + A)}^{- 1} z |_{n} = 0 \forall z \in H$
where I denotes the identity operator in both $H_{n}$ and H spaces, then $X_{n}$ converges in sense of Trotter toward X uniformly on $[0, T]$ , namely: $\begin{array}{l} lim_{n \to \infty} sup_{t \in [0, T]} | P_{n} X (t) - X_{n} (t) |_{n} = 0 \end{array}$ with moreover $\begin{array}{l} lim_{n \to \infty} sup_{t \in [0, T]} | {| X_{n} (t) |}_{n} - | X (t) | | = 0 \end{array}$

3.2.2. Convergence results

Owing to Propositions 2 and 3, we can now use the Trotter theory and, taking into account (27)–(28), we obtain the following result.

Proposition 5.
The sequence $X^{s} = (ξ^{s}, ζ^{s})$ in $H^{s}$ converges in the sense of Trotter toward $X = (ξ, ζ)$ in $H^{I}$ if and only if both limits are satisfied: $\begin{array}{l} (37) & \begin{aligned} i) lim_{s \to \bar{s}} \frac{1}{ε^{3}} \int_{Ω^{ε}} a^{ε} (e (ξ^{s}) - E^{ε} (\hat{e (ξ)}) \cdot (e (ξ^{s}) - E^{ε} (\hat{e (ξ)}) d x^{ε} = 0 \\ ii) lim_{s \to \bar{s}} \frac{ρ}{ε^{3}} \int_{Ω^{ε}} δ^{ε} {| ζ^{s} - S_{ε}^{- 1} ζ |}^{2} d x^{ε} = 0 \end{aligned} \end{array}$

This result illustrates that this convergence notion is the right one from the mechanical point of view: it quantifies the relative error made by replacing $e (ξ^{s})$ by $E^{ε} (\hat{e (ξ)})$ and $ζ^{s}$ by $S_{ε}^{- 1} ζ$ . This will be further discussed in Section 4.

We introduce a fundamental assumption $\begin{array}{l} (H_{5}) \int_{- 1}^{1} x_{3} \tilde{a} (\hat{x}, x_{3}) d x_{3} = 0 a.e. \hat{x} \in ω \end{array}$ which is fulfilled when a is an even function of $x_{3}$ and which implies a decoupling between membrane and flexural desplacements $\begin{array}{l} (38) & φ (u^{1}, u^{2}) = 0 \forall (u^{1}, u^{2}) \in V_{K L}^{f} \times V_{K L}^{m} \end{array}$ As for $U^{s}$ , we consider $U^{I e} = (u^{I e}, 0)$ where $u^{I e}$ is the solution to $\begin{array}{l} (39) & u^{I e} \in U; φ (u^{I e}, w) = L (w) : = L^{ε} (S_{ε}^{- 1} w) \forall w \in U \end{array}$ Obviously $L (w) = \int_{Ω} f \cdot w d x + \int_{Γ_{N}} g \cdot w d H_{2}$ and assumption $(H_{4})$ implies $\begin{array}{l} (40) & U^{I e} \in B V^{1} (0, T; H^{I}) \end{array}$ Let $A^{I}$ the operator in $H^{I}$ defined by $\begin{array}{l} (41) & \{\begin{matrix} D (A^{I}) : = \{\begin{matrix} U = (u, v) \in H^{I}; \{\begin{matrix} i) v \in U^{I} \\ ii) \exists (z, w) \in U^{I} \times V^{I_{1}}; {\overset{˚}{z}}^{I} = v and \end{matrix} \\ {((u, w), (ζ, {\overset{˚}{ζ}}^{I}))}^{I} + D^{I} (z + ζ) - D^{I} (z) ⩾ 0 \forall ζ \in U^{I} \end{matrix} \\ - A^{I} U = {(z, w); w satisfies ii) of definition of D (A^{I})} \end{matrix} \end{array}$ By due account of assumption $(H_{5})$ , it is straightforward to check
Proposition 6.
The operator $A^{I}$ is a maximal monotone operator and for all $ψ = (ψ_{u}, ψ_{v})$ in $H^{I}$ $\begin{array}{l} \{\begin{matrix} {\overline{U}}^{I} = ({\overline{u}}^{I}, {\overline{v}}^{I}) s.t. \\ {\overline{U}}^{I} + A^{I} {\overline{U}}^{I} ∋ ψ \end{matrix} ⟺ \{\begin{matrix} ({\overline{u}}^{I}, {\overline{v}}^{I}) = (\overline{z} + ψ_{u}, {\overset{˚}{\overline{z}}}^{I}) \\ where \overline{z} is the unique minimizer on U^{I} of J^{I} \\ J^{I} (z) : = \frac{1}{2} {[| (z, {\overset{˚}{z}}^{I}) |^{I}]}^{2} + {((ψ_{u}, - ψ_{v}), (z, {\overset{˚}{z}}^{I}))}^{I} + D^{I} (z) \\ \forall z \in U^{I} \end{matrix} \end{array}$

So we have
Theorem 3.
Under assumptions $(H_{1})$ – $(H_{5})$ and $\begin{array}{l} (H_{6}) U^{I 0} \in U^{I e} (0) + D (A^{I}) \end{array}$ the differential inclusion $\begin{array}{l} (P^{I}) \{\begin{matrix} \frac{d U^{I}}{d t} + A^{I} (U^{I} - U^{I e}) ∋ 0 \\ U^{I} (0) = U^{I 0} \end{matrix} \end{array}$ has a unique solution $U^{I}$ belonging to $W^{1, \infty} (0, T; H^{I})$ and the first line of $(P^{I})$ is satisfied almost everywhere in $(0, T]$ .

To affirm the Trotter convergence of $U^{s} (t)$ toward $U^{I} (t)$ uniformly on $[0, T]$ , according to Theorem 2, the definition (3) of $u^{s e}$ , the definition (39) of $u^{I e}$ and their time regularities, it sufficies to make the additional assumption $\begin{array}{l} (H_{7}) \exists U^{I 0} \in U^{I e} (0) + D (A^{I}); lim_{s \to \bar{s}} {| P^{s I} U^{I 0} - U^{s 0} |}^{s} = 0 \end{array}$ and to establish
Proposition 7.
There hold: $\begin{array}{l} (42) & \begin{aligned} i) lim_{s \to \bar{s}} {| P^{s I} {(I + A^{I})}^{- 1} ψ - {(I + A^{s})}^{- 1} P^{s I} ψ |}^{s} = 0 \forall ψ \in H^{I}, \\ ii) lim_{s \to \bar{s}} {| P^{s I} U^{I e} (t) - U^{s e} (t) |}^{s} = 0 \forall t \in [0, T] \end{aligned} \end{array}$
Proof.
i) Let $ψ = (ψ_{u}, ψ_{v})$ in $H^{I}$ . According to Proposition 1, ${\overline{U}}^{s} = ({\overline{u}}^{s}, {\overline{v}}^{s}) = {(I + A^{s})}^{- 1} P^{s I} ψ$ is such that ${\overline{u}}^{s} = {\overline{v}}^{s} + P_{u}^{s I} ψ_{u}$ and ${\overline{v}}^{s}$ is the unique minimizer on $U^{s}$ of $\tilde{J}$ defined by $\begin{array}{l} (43) & \tilde{J} = \frac{1}{2} {[{| (v, v) |}^{s}]}^{2} + {((ψ_{u}, - ψ_{v}), (S_{ε} v, S_{ε} v))}^{I} + D^{s} (v) \forall v \in U^{s} \end{array}$ So ${\overline{v}}^{s}$ is bounded in $U^{s}$ and $V^{s}$ . Proposition 4 implies that there exist $v^{}$ in $U^{I}$ and a not relabeled subsequence such that $S_{ε} \overline{v}$ weakly converges in $H^{1} (Ω, R^{3})$ toward $v^{}$ and $\begin{array}{l} (44) & J^{I} (v^{}) ⩽ \underset{s \to \bar{s}}{lim_{_}} {\tilde{J}}^{s} ({\overline{v}}^{s}) . \end{array}$ To conclude that the whole sequence converges toward $\overline{z}$ the unique minimizer of $J^{I}$ on $U^{I}$ and $\begin{array}{l} (45) & J^{I} (\overline{z}) = lim_{s \to \bar{s}} {\tilde{J}}^{s} ({\overline{v}}^{s}), {| (\overline{z}, \overset{˚}{\overline{z}}) |}^{I} = lim_{s \to \bar{s}} {| ({\overline{v}}^{s}, {\overline{v}}^{s}) |}^{s} \end{array}$ it remains to show that for all z in $U^{I}$ there exists $v^{s}$ such that $S_{ε} v^{s}$ weakly converges in $H^{1} (Ω, R^{3})$ toward z with $\begin{array}{l} (46) & \begin{aligned} 1) \underset{s \to \bar{s}}{lim^{‾}} | (v^{s}, v^{s}) |^{s} ⩽ {| (z, {\overset{˚}{z}}^{I}) |}^{I}, \\ 2) \underset{s \to \bar{s}}{lim^{‾}} D^{s} (v^{s}) ⩽ D^{I} (z), \\ 3) \underset{s \to \bar{s}}{lim^{‾}} {\tilde{J}}^{s} (v^{s}) ⩽ J^{I} (z) . \end{aligned} \end{array}$ To this end, we use an argument of the proof of Proposition 3 in [4] that we recall here. Let $q = E (\hat{e (z)})$ , there exist $(q^{s})$ in $C_{0}^{\infty} (Ω, S^{3})$ such that $\begin{array}{l} \int_{Ω} a (q^{s} - q) \cdot (q^{s} - q) d x ⩽ C ε^{2} \end{array}$ The field $w^{s}$ defined by $\begin{array}{l} (47) & \begin{aligned} w_{3}^{s} (x) = ε^{2} \int_{0}^{x_{3}} q_{33}^{s} (\hat{x}, τ) d τ \\ w_{α}^{s} (x) = 2 ε \int_{0}^{x_{3}} (q_{α 3}^{s} (\hat{x}, τ) - \frac{ε}{2} \int_{0}^{τ} \partial_{α} q_{33}^{s} (\hat{x}, σ) d σ) d τ \end{aligned} \end{array}$ belongs to $H_{Γ_{D} \cup Γ_{C}}^{1} (Ω, R^{3})$ . Because $| e (ε, z + w^{s}) - q^{s} |_{L^{2} (Ω; S^{3})} ⩽ C ε$ , we obtain that $z + w^{s}$ converges strongly in $H^{1} (Ω, R^{3})$ and the three inequalities of (46) are fulfilled for $\begin{array}{l} (48) & v^{s} : = S_{ε}^{- 1} (z + w^{s}) \end{array}$ Eventually as $\begin{array}{l} φ^{s} (P_{u}^{s I} \overline{z} - {\overline{v}}^{s}, P_{u}^{s I} \overline{z} - {\overline{v}}^{s}) = φ^{s} (P_{u}^{s I} \overline{z}, P_{u}^{s I} \overline{z}) - 2 φ (\overline{z}, S_{ε} {\overline{v}}^{s}) + φ^{s} (v^{s}, v^{s}) \end{array}$ Propositions 2 and 3 and (45) imply that $({\overline{u}}^{s}, {\overline{v}}^{s})$ converges in the sense of Trotter toward $({\overline{u}}^{I}, {\overline{v}}^{I})$ .

ii)* As $U^{s e} (t)$ , $U^{I e} (t)$ are the unique minimizers of $\frac{1}{2} {[| (\cdot, \cdot) |^{s}]}^{2} - L^{ε} (t)$ and $\frac{1}{2} {[| (\cdot, \cdot) |^{I}]}^{2} - L (t)$ , it sufficies to use the previous result i) by simply replacing the linear forms $φ (ψ_{u}, S_{ε} \cdot)$ , $φ (ψ_{u}, \cdot)$ by $L^{ε} (t)$ , $L (t)$ respectively and making $ρ = μ = 0, ψ_{v} = 0$ . □

Thus our main convergence result is: Theorem 4.
Under assumptions $(H_{1})$ – $(H_{7})$ , the solution $U^{s}$ to $(P^{s})$ converges to the solution $U^{I}$ to $(P^{I})$ in the sense $lim_{s \to \bar{s}} | P^{s I} U^{I} (t) - U^{s} (t) |^{s} = 0$ uniformly on $[0, T]$ with in addition $lim_{s \to \bar{s}} | U^{s} (t) |^{s} = | U^{I} (t) |^{I}$ uniformly on $[0, T]$ .

4. Conclusive remarks and proposal of an asymptotic model

First, according to each value of I in ${1, 2} \times {1, 2, 3}$ , we give a more explicit way of writting $(P^{I})$ in the form of the principle of virtual power. To shorten notation we write u in place of $u^{I}$ , which does satisfy $u \in W^{1, \infty} (0, T; H^{1} (Ω, R^{3})) \cap W^{2, \infty} (0, T; L^{2} (Ω, R^{3}))$ and suitable initial conditions given by $(H_{6})$ – $(H_{7})$ . For any function h of $L^{1} (Ω, H)$ or $L^{1} (\partial ω \times (- 1, 1), H)$ where H is $R^{n}$ or $Lin (S^{n})$ , $⟨ h ⟩$ will stands for $\int_{- 1}^{1} h (\hat{x}, x_{3}) d x_{3}$ . Moreover let $L^{m} (t) (v), L^{f} (t) (v)$ denote $L (t) (v)$ for all v in $V_{K L}^{m}, V_{K L}^{f}$ , respectively, and $g_{i}^{\pm} : = g_{i} (\cdot, \pm 1)$ . One has: $\begin{array}{l} L^{m} (t) (v^{m}) = \int_{ω} (⟨ \hat{f} ⟩ + \hat{g^{+}} + \hat{g^{-}}) \cdot v^{M} d \hat{x} \\ L^{f} (t) (v^{f}) = \int_{ω} (⟨ f_{3} ⟩ + g_{3}^{+} + g_{3}^{-}) \cdot v^{F} + (⟨ - x_{3} \hat{f} ⟩ - \hat{g^{+}} + \hat{g^{-}}) \cdot \hat{\nabla} v^{F} d \hat{x} \end{array}$ For all $v^{m}$ in $V_{K L}^{m}$ , we will write $e (v^{M})$ for $\hat{e (v^{m})}$ and for all $v^{f} = (0, v^{F})$ in $V_{K L}^{f}$ we recall that $\hat{e (v^{f})} = - x_{3} \hat{D^{2}} v^{F}$ where $\hat{D^{2}} : = \partial_{α β}^{2}$ .

We recall that $U^{I}$ is defined in (32), for each I in ${1, 2} \times {1, 2, 3}$ . The limit problem $(P^{I})$ is: $\begin{array}{l} for I = (1, 1) \\ \exists ξ \in \partial ϕ_{p} (\frac{\partial u^{F}}{\partial t}); \int_{ω} {\overline{ρ}}^{1} ⟨ δ ⟩ \frac{\partial^{2} u^{F}}{\partial t^{2}} v^{F} + ⟨ x_{3}^{2} \tilde{a} ⟩ \hat{D^{2}} u^{F} \cdot \hat{D^{2}} v^{F} d \hat{x} \\ + 2 {\overline{μ}}^{1} \int_{γ_{c}} ξ v^{F} d H_{1} = L^{f} (t) (v^{f}) \forall v \in U^{1, 1} \\ \int_{ω} ⟨ \tilde{a} ⟩ e (u^{M}) \cdot e (v^{M}) d \hat{x} = L^{m} (t) (v^{m}) (i.e. u^{m} = u^{e m}) \\ for I = (1, 2) \\ \int_{ω} {\overline{ρ}}^{1} ⟨ δ ⟩ \frac{\partial^{2} u^{F}}{\partial t^{2}} v^{F} + ⟨ x_{3}^{2} \tilde{a} ⟩ \hat{D^{2}} u^{F} \cdot \hat{D^{2}} v^{F} d \hat{x} = L^{f} (t) (v^{f}) \\ u^{F} (t) = u^{0 F} on γ_{C} \forall t \in [0, T] \forall v \in U^{1, 2} \\ \exists ξ \in \partial ϕ_{p} (\frac{\partial u_{T}^{M}}{\partial t}); \int_{ω} ⟨ \tilde{a} ⟩ e (u^{M}) \cdot e (v^{M}) d \hat{x} + 2 {\overline{μ}}^{2} \int_{γ_{c}} ξ \cdot v_{T}^{M} d H_{1} = L^{m} (t) (v^{m}) \\ for I = (1, 3) \\ \int_{ω} {\overline{ρ}}^{1} ⟨ δ ⟩ \frac{\partial^{2} u^{F}}{\partial t^{2}} v^{F} + ⟨ x_{3}^{2} \tilde{a} ⟩ \hat{D^{2}} u^{F} \cdot \hat{D^{2}} v^{F} d \hat{x} = L^{f} (t) (v^{f}) \\ \int_{ω} ⟨ \tilde{a} ⟩ e (u^{M}) \cdot e (v^{M}) d \hat{x} = L^{m} (t) (v^{m}) \forall v \in U^{1, 3} \\ u (t) = u^{0} on Γ_{C} \forall t \in [0, T] \\ for I = (2, 1) \\ u^{F} (t) = u^{0 F} \forall t \in [0, T] \\ \int_{ω} {\overline{ρ}}^{2} ⟨ δ ⟩ \frac{\partial^{2} u^{M}}{\partial t^{2}} \cdot v^{M} + ⟨ \tilde{a} ⟩ e (u^{M}) \cdot e (v^{M}) d \hat{x} = L^{m} (t) (v^{m}) \forall v \in U^{2, 1} \\ for I = (2, 2) \\ u^{F} (t) = u^{0 F} \forall t \in [0, T] \\ \exists ξ \in \partial ϕ_{p} (\frac{\partial u_{T}^{M}}{\partial t}); \int_{ω} {\overline{ρ}}^{2} ⟨ δ ⟩ \frac{\partial^{2} u^{M}}{\partial t^{2}} \cdot v^{M} + ⟨ \tilde{a} ⟩ e (u^{M}) \cdot e (v^{M}) d \hat{x} \forall v \in U^{2, 2} \\ + 2 {\overline{μ}}^{2} \int_{γ_{c}} ξ \cdot v_{T}^{M} d H_{1} = L^{m} (t) (v^{m}) \\ for I = (2, 3) \\ u^{F} (t) = u^{0 F} \forall t \in [0, T] \\ \int_{ω} {\overline{ρ}}^{2} ⟨ δ ⟩ \frac{\partial^{2} u^{M}}{\partial t^{2}} \cdot v^{M} + ⟨ \tilde{a} ⟩ e (u^{M}) \cdot e (v^{M}) d \hat{x} = L^{m} (t) (v^{m}) \forall v \in U^{2, 3} \\ u^{M} (t) = u^{0 M} on γ_{C} \forall t \in [0, T] \end{array}$ Hence the problem $(P^{I})$ is a two-dimensional problem set on ω. When $I_{1} = 1$ , the flexural part $u^{f}$ of u does satisfy a dynamic equation whereas the membrane part $u^{m} = (u^{M}, 0)$ is involved in a quasi-static equation. On $Γ_{C}$ , a Norton or Tresca friction condition with bilateral contact is satisfied by $u^{F}$ when $I_{2} = 1$ , by $u^{M}$ when $I_{2} = 2$ while complementary Dirichlet boundary conditions $u^{f} (t) = u^{0 f}$ when $I_{2} = 2$ , $u (t) = u^{0}$ when $I_{2} = 3$ must be fulfilled.

On the contrary, when $I_{1} = 2$ , it is the membrane part which satisfies a dynamic equation with a friction condition when $I_{2} = 2$ , whereas the flexural part is frozen in $u^{0 f}$ .

Next, we propose our simplified but accurate enough asymptotic model not by considering $S_{ε}^{- 1} u^{I} (t)$ but by taking into account our convergence result – Theorem 4 – and the crucial Proposition 5. Therefore (37) implies $\begin{array}{l} lim_{s \to \bar{s}} \frac{1}{ε^{3}} \int_{Ω^{ε}} a^{ε} (e (u^{s}) - E^{ε} (\hat{e (u^{I})}) \cdot (e (u^{s}) - E^{ε} (\hat{e (u^{I})}) d x^{ε} = 0 \end{array}$ where we recall that $\begin{array}{l} E^{ε} (q) (x^{ε}) = ε E (q) (x) \forall q \in L^{2} (Ω, S^{2}), x^{ε} : = (\hat{x}, ε x_{3}) a.e. x \in Ω, \end{array}$ $E (η)$ being the unique element of $S^{3}$ such that $\begin{array}{l} a E (η) \cdot E (η) = Min {a Q \cdot Q; Q \in S^{3}, \hat{Q} = η} \end{array}$ Hence as denoted in [3,6,7,9] $E^{ε} (\hat{e (u^{I})}$ is a good approximation of the strain tensor of $u^{s}$ in the sense that the relative error made by replacing $e (u^{s})$ by $E^{ε} (\hat{e (u^{I})}$ tends to zero! This shows that $e (u^{s})$ it is not close to $e (S_{ε}^{- 1} u^{I})$ but close to $E^{ε} (\hat{e (u^{I})}$ , the terms $e_{i 3} (u^{s})$ being of the same order of magnitude as those of $e (S_{ε}^{- 1} u^{I})$ .

As $E^{ε} (\hat{e (u^{I})}$ is not necessarily the strain tensor of a field of $U^{s}$ we are led to use the construction (47) developed in the proof of Proposition 7 which supplies a field ${\tilde{u}}^{s}$ in $U^{s}$ such that $\begin{array}{l} lim_{s \to \bar{s}} \frac{1}{ε^{3}} \int_{Ω^{ε}} a^{ε} (e ({\tilde{u}}^{s}) - E^{ε} (\hat{e (u^{I})}) \cdot (e ({\tilde{u}}^{s}) - E^{ε} (\hat{e (u^{I})}) d x^{ε} = 0 \end{array}$ Thus ${\tilde{u}}^{s}$ is our proposal of approximation for $u^{s}$ . It is obtained by first solving $(P^{I})$ which actually corresponds to two two-dimensional problems set on ω and second the previous construction of ${\tilde{u}}^{s}$ which also involves a two-dimensional framework. Hence it is easy to implement a numerical method of approximation.

Remark 1.

It is worthwhile to notice that in these questions concerning thin linearly plates the field of displacement in the real plate which occupies $Ω^{ε}$ is far from a Kirchhoff–Love field and even from a Reissner–Mindlin one because ${(E^{ε} (\hat{e (u^{I})}))}_{i 3}$ depends on $x_{3}^{ε}$ even in the case of an homogeneous plate.

Remark 2.

It is also possible to deal with the not too much realistic case $2 < p ⩽ + \infty$ by the same method, the variant being that $D^{s}$ and $D^{I}$ are only lower semi-continuous and some trivial approximation process are in order.

Remark 3.

A more realistic approach is when two other physical data r and ϕ are taken into account. We now assume that the elasticity tensor of the plate is $r a^{ε}$ , while ε has to be replaced by ϕ in the assumption $(H_{4})$ on the loading that is to say: $\begin{array}{l} \{\begin{matrix} \exists (f_{ϕ}, g_{ϕ}) \in B V^{1} (0, T; L^{2} (Ω, R^{3}) \times L^{2} (Γ_{N}, R^{3})) s.t. \\ {\hat{f}}^{ϕ} (x^{ε}) = ϕ {\hat{f}}_{ϕ} (x), f_{3}^{ϕ} (x^{ε}) = ϕ^{2} f_{ϕ 3} (x), \forall x \in Ω \\ {\hat{g}}^{ϕ} (x^{ε}) = ϕ^{2} {\hat{g}}_{ϕ} (x), g_{3}^{ϕ} (x^{ε}) = ϕ^{3} g_{ϕ 3} (x), \forall x \in Γ_{N} \cap Γ_{\pm}, \\ {\hat{g}}^{ϕ} (x^{ε}) = ϕ {\hat{g}}_{ϕ} (x), g_{3}^{ϕ} (x^{ε}) = ϕ^{2} g_{ϕ 3} (x), \forall x^{ε} \in Γ_{N} \cap Γ_{lat} \end{matrix} \end{array}$ So if we consider $σ : = (ε, ρ, μ, r, ϕ)$ as a quintuplet of parameters taking values in a countable subset of ${(0, + \infty)}^{5}$ with a unique cluster point $\overline{σ}$ in ${{0} \times [0, + \infty)}^{2} \times (0, + \infty) \times {0}$ such that $\begin{array}{l} \{\begin{matrix} there exists \overline{λ} in [0, + \infty) such that ϕ / ε r converges to \overline{λ} when σ goes to \overline{σ} and \\ (f_{ϕ}, g_{ϕ}) converges strongly in B V^{1} (0, T; L^{2} (Ω, R^{3}) \times L^{2} (Γ_{N}, R^{3})) to (f, g), \end{matrix} \end{array}$ an analysis similar to the previous one works, where ρ and μ have to be replaced by $ρ / r$ and $μ / r$ in order to define the various cases indexed by I in assumption $(H_{3})$ . In the previous sections, we used $(r, ϕ) = (1, ε)$ in order to simplify the mathematical formulas…

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Transient response of a thin linearly elastic plate with Norton or Tresca friction

Abstract

Keywords

1. Setting the problem

2. Existence and uniqueness results for ( P s )

3.1. The crucial point

3.2.1. Recaps on Trotter theory of approximation of semi-groups

References

2. Existence and uniqueness results for $(P^{s})$