Asymptotic model of linearly visco-elastic Kelvin–Voigt type plates via Trotter theory

We conﬁrm the study (Licht in C. R., Méc. 341:697–700, 2013) devoted to the quasi-static response for a visco-elastic Kelvin–Voigt plate whose thickness goes to zero. For each thickness parameter, the quasi-static response is given by a system of partial diﬀerential equations with initial and boundary conditions. Reformulating scaled systems into a family of evolution equations in Hilbert spaces of possible states with ﬁnite energy, we use Trotter theory of convergence of semi-groups of linear operators to identify the asymptotic behavior of the system. The asymptotic model we obtain and the genuine one have the same structure except an occurrence of a new state variable. Eliminating the new state variable from our asymptotic model leads to the asymptotic model in (Licht in C. R., Méc. 341:697–700, 2013) which involves an integro-diﬀerential system.


Introduction
In a recent study [2], Licht and Weller promoted an old but not so well-known convergence tool, namely Trotter theory of convergence of semi-groups of linear operators acting on variable Hilbert spaces, in determining the asymptotic modeling in physics of continuous media. They provided various asymptotic models through the lens of Trotter theory as a comparison to other classical methods. One of the models mentioned is a reduction of the dimension problem on thin linear visco-elastic Kelvin-Voigt type plates. Licht [1] studied this problem before in 2013 and derived the asymptotic model with Laplace transform technique. He found that the mechanical behavior of the limit model is no longer of Kelvin-Voigt type, because a term of fading memory appears like in the homogenization problem. However, with Trotter theory of convergence, Licht and Weller suggested that the mechanical behaviors of limit and genuine models are the same except for the appearance of a new state variable. It is well known that to have the same structure in both limit and genuine models is useful for numerical computations.
In this study we aim to justify and confirm their suggestion. We reconsider a reduction of the dimension problem of thin linearly visco-elastic Kelvin-Voigt plates in Sect. 2.
By defining a small parameter ε, referred to as the thickness of the plate, each problem is expressed as an initial-boundary value problem (2.1). Under suitable assumptions, we rescale the problems and reformulate them in terms of a family of transient problems. In Sect. 3 we discuss our convergence tool, that is, Trotter theory of the convergence of semigroups of linear operators. We then follow a Trotter theory approach by letting ε tend to zero to derive the limit model in Sect. 4. As will be seen in the last section, the limit model contains an additional state variable but with the structure like that of the original one.
Eliminating the new state variable from the limit model recovers an additional term of fading memory in the limit model derived by the Laplace transform technique, which implies integro-differential equations involving partial derivatives of the field of displacement.

Setting the problem
Customarily, we assimilate the physical Euclidean space to R 3 ; the orthonormal basis of which is denoted by {e 1 , e 2 , e 3 }, and for all ξ = (ξ 1 , ξ 2 , ξ 3 ) in R 3 , ξ := (ξ 1 , ξ 2 ). We will study the quasi-static response of a thin linearly visco-elastic Kelvin-Voigt plate subjected to a given load. Like the problem setting in [1], a thin linearly visco-elastic Kelvin-Voigt plate occupies a domain Ω ε := ω × (-ε, ε), where ω is a bounded domain in R 2 with a Lipschitzcontinuous boundary ∂ω and ε is the small thickness of the plate. The upper, lower surfaces ω × {±ε} and the lateral part of the plate ω × (-ε, ε) are referred to as Γ ε ± and Γ ε lat , respectively. The plate is clamped along a portion of the lateral part Γ ε D := γ D × (-ε, ε) where γ D is of positive length. Moreover, it is subjected to body forces of density f ε and surface forces of density g ε on the upper, lower surfaces together with the rest of its lat- The problem of determining the quasi-static evolution of the plate involves the parameter ε of data and the equations satisfied by the fields of displacement u ε and stress σ ε are: where e(u ε ), n ε , a ε and b ε are the linearized strain tensor, the outward unit normal vector, the elasticity and viscosity tensor fields, respectively, while the upper dot represents the time derivative.
To obtain a simpler but precise enough model we study the quasi-static response for the plate as its thickness tends to zero. Following [1,3], first we rescale the domain Ω ε into a fixed domain Ω := ω × (-1, 1) through the mapping π ε : Next we add two hypotheses to the data. Hypothesis (H 1 ) is on the real loading which has to be connected to fixed quantities defined on Ω, they concern their intensity and horizontality: (H 1 ) : while hypothesis (H 2 ) is on the elasticity and viscosity tensors: where Lin(S 3 ) denotes the space of linear mapping from S 3 into S 3 , S 3 being the space of 3 × 3 symmetric matrices. It will be convenient to write , and to denote the projection of e on S and S ⊥ by e and e ⊥ , respectively. Then one associates a scaled displacement Using hypotheses (H 1 ) and (H 2 ) and the scaling of displacement, we can formulate the initial-boundary value problem (2.1) in terms of the scaled variational problem (P ε ) : where H ε is the subspace of H 1 (Ω, R 3 ) whose elements have a vanishing trace on Γ D , and H 2 is the twodimensional Hausdorff measure. Subspace H ε is equipped with the inner product: To deal with non-vanishing external loadings, it suffices to split u ε into u e ε + u r ε , where u e ε solves a static problem associated with the evolution problem under consideration and involving the surface loading only. Then u r ε does solve an evolution equation with a second member which is a continuous function of u e ε and hence a continuous function of the loading. The static problem is which has a unique solution by the Lax-Milgram lemma. As g → u e ε is linear continuous from L 2 (Γ N ; R 3 ) to H ε , we have u e ε in C 1,1 ([0, T]; H ε ) The evolution problem is which can be reformulated in terms of Clearly A ε is bounded, selfadjoint and m-dissipative so that the evolution equation (2.2) has a unique solution u r ε in C 1,1 ([0, T]; H ε ). The crucial point to prove that the asymptotic model has the same structure as the genuine one is to apply the tool of some convergence which is a not so well-known result, involving two fields, in reduction of dimension similar to two-scale convergence in periodic homogenization. The proposed asymptotic models of (P e ε ) and (P r ε ) will be given by the following problems: is the subspace of H 1 (Ω; R 3 ) whose elements have a vanishing trace on Γ D , and equipped with the inner product: which can be reformulated in terms of Obviously A is bounded, selfadjoint and m-dissipative. Problem (P e ) has a unique solution U e by the Lax-Milgram lemma. and the evolution problem (2.3) has a unique solution U r in C 1,1 ([0, T]; H). At this point, we have solutions (u e ε ), (u r ε ), U e , and U r to the problems (P e ε ), (P r ε ), (P e ), and (P r ), respectively. We claim that the sequences (u e ε ), (u r ε ) converge to U e , U r in the sense of Trotter. Before we prove our claim, let us discuss this type of convergence in detail.

Definition 1
We say that a sequence of Hilbert spaces H n approximates a Hilbert space H in the sense of Trotter, if there exists a representative operator P n ∈ L(H; H n ) satisfying two conditions of uniform continuity (T 1 ) and good energetic representation (T 2 ): (T 1 ) ∃C > 0 such that P n u H n ≤ C u H , ∀u ∈ H, ∀n, (T 2 ) lim n→∞ P n u H n = u H , ∀u ∈ H, where L (H; H n ) is a space of all continuous linear mappings from H into H n . The fundamental result of Trotter theory of convergence of semi-groups of linear operators is the following.
Theorem 1 Let u n , u be the solutions of (P n ) and (P), respectively, if (i) P n u 0u 0 n H n → 0, (ii) T 0 P n q(t)q n (t) H n dt → 0, (iii) ∀y ∈ X, dense in H, P n (I -A) -1 y -(I -A n ) -1 P n y H n → 0 then, uniformly on [0, T], P n u(t)u n (t) H n → 0 and u n (t) H n → u(t) H . Now we are ready to prove our assertion.
That is, operator P ε satisfies (T 2 ). Thus, the sequence of Hilbert spaces H ε approximates the Hilbert space H in the sense of Trotter.
To prove our convergence result, the following lemma is important.

Lemma 1 For all z ε in H ε , Z in H,
Therefore, e(ε, u e ε ) converge strongly to e(u e ) + ∂ 3 u e1 ⊗ s e 3 in L 2 (Ω; S 3 ). Note that this convergence is uniform on [0, T].
To prove that (u r ε ) converges uniformly on [0, T] to U r = (u r , u r1 ) in the sense of Trotter, we have to prove the three conditions of Theorem 1.
Concerning initial data u r0 ε and U r0 , we make an additional assumption on u 0 ε . That is, Hence, The first term on the right hand side of the inequality goes to zero by the additional assumption on u 0 ε , while the second term goes to zero, as previously shown. In regard to the second members q ε =u e ε and Q =U e , we again use Lemma 1 by proving e(ε,u e ε ) → e(u e ) + ∂ 3u e1 ⊗ s e 3 in L 2 (Ω; S 3 ) uniformly on [0, T] because of the smoothness of g with respect to the time t.
In the matter of resolvants (I -A ε ) -1 , (I -A) -1 , we have to show that We set z ε = (I -A ε ) -1 P ε Y and Z = (I -A) -1 Y that satisfy the problems Now we consider P ε Zz ε H ε . They involve a kind of static problem to which the previous two-fields result on reduction of dimension applies. The proof is then complete and we now have the main theorem. = Ω f · v dx + Γ N g · v dH 2 ∀V = (v, v 1 ) ∈ V KL × H 1 (-1, 1; L 2 (ω; R 3 ))/L 2 (ω; R 3 ).
Hence U solves a problem of visco-elasticity with short memory of Kelvin-Voigt type, but involving a couple (u, u 1 ) of state variables. Clearly the field u 1 can be eliminated as in [1] so that u does solve a problem of visco-elasticity with long (but fading) memory.