2018/07/31 by G. Castiñeira, Á. Rodŕıguez-Arós, Castiñeira, Gonzalo +1
Computer Science · Engineering · #34E05 #34E10 #34K25 #35J15 #35O30 #35Q74 #41A60 #74D05 #74K15 #74K25 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Contact Mechanics and Variational Inequalities #Elasticity and Material Modeling #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.1808.00543
openalex publication_date 2018/07/31 · openalex created_date 2020/03/13 · openalex updated_date 2026/07/28
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(1) with respect to ε and surface tractions density is O(ε), the solution of the scaled variational problem in curvilinear coordinates, defined over the fixed domain Ω=ω×(-1,1), converges in ad hoc functional spaces as ε→ 0 to a limit u. Furthermore, the average u(ε)= \frac12∫-11u (ε) dx3, converges in an ad hoc space to the unique solution of what we have identified as (scaled) two-dimensional equations of a viscoelastic generalized membrane shell, which includes a long-term memory that takes into account previous deformations. We finally provide convergence results which justify those equations.