2017/10/31 by Castiñeira, Gonzalo, Rodríguez-Arós, Ángel
#34E05 #34E10 #34K25 #35J15 #35Q74 #41A60 #74D05 #74K25 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1711.00731
We consider a family of linearly viscoelastic shells with thickness 2ε, clamped along a portion of their lateral face, all having the same middle surface S=\mathbfθ(ω)⊂ℝ3, where ω⊂ℝ2 is a bounded and connected open set with a Lipschitz-continuous boundary γ. We show that, if the applied body force density is O(ε2) with respect to ε and surface tractions density is O(ε3), the solution of the scaled variational problem in curvilinear coordinates, u(ε), defined over the fixed domain Ω=ω×(-1,1), converges to a limit u in H1(0,T;[H1(Ω)]3) as ε→ 0. Moreover, we prove that this limit is independent of the transverse variable. Furthermore, the average u= \frac12∫-11u dx3, which belongs to the space H1(0,T; VF(ω)), where VF(ω):= \ \mathbfη=(ηi)∈ H1(ω)× H1(ω)× H2(ω) ; ηi=∂νη3=0 \textrmon γ0, γαβ(\mathbfη)=0 \textrm in ω\, satisfies what we have identified as (scaled) two-dimensional equations of a viscoelastic flexural shell, which includes a long-term memory that takes into account previous deformations. We finally provide convergence results which justify those equations.