A Justification of Two-Dimensional Nonlinear Viscoelastic Shells Model

and Applied Analysis 3 then there exists ε0 > 0 such that for all 0 < ε ≤ ε0 the mapping Θ : Ω ε → R3 is an injective mapping and the three vectors gεi x ε : ∂εiΘ x ε ∂α ∂/∂xα, ∂ ε 3 ∂/∂x ε 3 are linear independent for each x ∈ Ωε. The injectivity of the mapping Θ : Ωε → R3 ensures that the physical problem described below is meaningful. The three vectors gεi x ε form the covariant basis at the point Θ x , and the three vectors g x defined by g x · gεi x δ j i form the contravariant. We define the metric tensor g ij or g ij,ε and the Christoffel symbols of the manifold Θ Ω ε by setting we omit the explicit dependence on x g ij g ε i · gεj , g : g · g, Γ ij : g · ∂εi gεj . 2.5 Note the symmetries g ij g ε ji, g ij,ε g, Γ ij Γ k,ε ji . 2.6 The volume element in the set Θ Ω ε is √ gεdx, where g : det g ij . For each 0 < ε ≤ ε0, the set Ω̂ε : Θ Ω ε is the reference configuration of a viscoelastic shell withmiddle surface S θ ω and thickness 2ε. We assume that thematerial constituting the shell is homogeneous isotropic and Θ Ω ε is of a nature state, so that the material is characterized by its two lamé relaxation modules λ t ≥ 0 and μ t > 0 t ≥ 0 . Under the action of forces, the shell undergoes a displacement field. Let û t uεi t g i,ε in terms of curvilinear coordinates x of the reference configuration Θ Ω ε . Then, the covariant displacement field u t uεi t satisfies the following threedimensional equations c.f. 1, 10 : u t ∈ Lc −∞, T ;W Ω with W Ω : { v ∈ W1,4 Ω , v 0 on Γε0 } , ∫ Ωε uεitt t v ε i g ij,ε √ gεdx ∫ Ωε A 0 E k‖l u ε t F i‖j u ε t ,v √ gεdx ∫ Ωε ∫ t −∞ A′ t − τ E k‖l u τ F i‖j u τ ,v √ gεdτ dx ∫ Ωε f i,ε t v i √ gεdx ∫ Γ ∪Γ− h t v i √ gεdΓ, ∀vε ∈ W Ω , 2.7 where the symbol Lc denotes the subspace of L ∞ > 0 such that there exists a constant T such that the functions vanish as s < −T . And, A t : λ t gg μ t ( gg gg ) 2.8 designate the contravariant components of the three-dimensional elasticity tensor, E i‖j v ε : 1 2 ( v i‖j v ε j‖i g v m‖iv ε n‖j ) , where v i‖j : ∂ ε j v ε i − Γ p,ε ij v ε p, 2.9 4 Abstract and Applied Analysis designate the strains in the curvilinear coordinates associated with an arbitrary displacement field v i g i,ε of the manifold Θ Ω , F i‖j u ε t , v : 1 2 ( v i‖j v ε j‖i g mn,ε { uεm‖i t v ε n‖j u ε n‖j t v ε m‖i }) , 2.10 and, finally, f ∈ L∞ 0, T ;L2 Ω and h ∈ L∞ 0, T ; Γ ∪ Γ− denote the contravariant components of the applied body and surface force densities, respectively, applied to the interior Θ Ω of the shell and to its “uper” and “lower” faces Θ Γ and Θ Γ ε − , and designate the area element along ∂Ω. We thus assume that there are no surface forces applied to the portion Θ γ − γ0 × −ε, ε of the lateral face of the shell. We record in passing the symmetries A A A 2.11 and the relation Γ α3 Γ p,ε 33 0, A αβσ3,ε Aα333,ε 0 in Ω ε . 2.12 Our final objective consists in showing, by means of the method of formal asymptotic expansions that, if the data are of an appropriate order with respect to ε as ε → 0, the above three-dimensional problems are “asymptotically equivalent” to a “two-dimensional problem posed over the middle surface of the shell.” This means that the new unknown should be ζ t ζ i t , where ζ ε i t are the covariant components of the displacement ζ ε i t ai y : ω → R3 of the middle surface S θ ω . In other words, ζ i t, y ai y is the displacement of the point θ y ∈ S. “Asymptotic analysis” means that our objective is to study the behavior of the displacement field uεi t g i,ε : Ω ε → R3 as ε → 0, an endeavour that will be a behavior as ε → 0 of the covariant components uεi t : Ω ε → R of the displacement field, that is, the behavior of the unknown u t uεi t : Ω ε → R3 of the three-dimensional shell problem. Since these fields are defined on sets Ω ε that themselves vary with ε, our first task naturally consists in transforming the three-dimensional problems into problems posed over a set that does not depend on ε. Furthermore, we transform problem 2.7 into an equivalent problem independent of ε, posed over the domain. Let Ω : ω × −1, 1 , Γ0 γ0 × −1, 1 , Γ : ω × { 1}, and Γ− : ω × {−1}, and let x xi denote a generic point in Ω. With each point x ∈ Ω, we associate the point x through the bijection π : x x1, x2, x3 ∈ Ω → x x i x1, x2, εx3 ∈ Ω ε ; we thus have ∂α ∂α and ∂ ε 3 1/ε ∂3. Let u ε uεi , Γ ε ij , g , A : Ω ε → R and the vector fields v v i appearing in the three-dimensional problem 2.7 be associated with the functions Γpij ε , g ε , A ijkl ε : Ω → R and the scaled vector fields v vi defined by ui ε t, x uεi t, x ε , vi x v i x ε ∀xε ∈ Ωε, Γpij ε x Γ p,ε ij x ε , g ε x g x , A ε x A x ∀xε ∈ Ω. 2.13 Abstract and Applied Analysis 5 Functions f ε t : Ω → R and h t ε : Γ ∪ Γ− → R are defined by setting f ε t, x f t, x ∀xε ∈ Ω, h ε t, x h t, x ∀xε ∈ Γ ∪ Γ−. 2.14and Applied Analysis 5 Functions f ε t : Ω → R and h t ε : Γ ∪ Γ− → R are defined by setting f ε t, x f t, x ∀xε ∈ Ω, h ε t, x h t, x ∀xε ∈ Γ ∪ Γ−. 2.14 Then the scaled unknown u ε t defined above satisfies c.f. 1 u ε t ∈ Lc −∞, T ;W Ω with W Ω : { v ∈ W1,4 Ω , v 0 on Γ0 } , ∫ Ω uitt ε t vi √ g ε dx ∫ Ω A ε 0 Ek‖l ε;u ε t Fi‖j ε;u ε t ,v √ g ε dx ∫ Ω ∫ t −∞ A′ ε t − τ Ek‖l ε;u ε τ Fi‖j ε;u ε τ ,v √ g ε dτ dx ∫ Ω f ε t vi √ g ε dx 1 ε ∫ Γ ∪Γ− h ε t vi √ g ε dΓ, ∀v ∈ W Ω , 2.15