Derivation of von Kármán Plate Theory in the Framework of Three-Dimensional Viscoelasticity

Derivation of a one-dimensional von Kármán theory for viscoelastic ribbons

“…This complements the static Γ-convergence analysis for elastic materials [25], which has justified the von Kármán plate equations proposed more than 100 years ago [41]. In contrast to previous works on viscoelastic plates [9][10][11]35], where the starting point is already a plate model, [22] constitutes a rigorous derivation by proving that solutions to (1.1) converge in a suitable sense to effective two-dimensional equations. More precisely, there are in-plane displacements u : [0, T ] × S ε → R 2 and out-of-plane displacements v :…”

“…As shown in [22,23], the metric gradient flow of φ ε with respect to the metric D ε , for a given initial datum (u 0 , v 0 ) ∈ W 1,2 (S ε ; R 2 ) × W 2,2 (S ε ), corresponds to the 2D equations for viscoelastic von Kármán plates…”

“…Recently in [22], starting from a version of (1.1) for nonsimple materials, a dimension reduction has been performed to derive a von Kármán-like viscoelastic plate model on a two-dimensional plate S ε = (− l 2 , l 2 ) × (− ε 2 , ε 2 ). This complements the static Γ-convergence analysis for elastic materials [25], which has justified the von Kármán plate equations proposed more than 100 years ago [41].…”

“…, where e(u) := 1 2 (∇u + ∇u T ) denotes the linear strain tensor, and C 2 W as well as C 2 R are tensors of elasticity and viscosity coefficients, respectively, derived suitably from W and R. In addition to the vertical force f 2D , we also consider a horizontal force g 2D that was not included in [22] for simplicity. The first equation features a membrane term both in the elastic and the viscous stress, and the second equation contains also a bending term.…”

“…where Q 2 W (F ) = F : C 2 W F for F ∈ R 2×2 . Indeed, (1.2) can proved to be a (metric) gradient flow for φ ε for a metric suitably related to C 2 W , see again [22]. This approach was additionally exploited in [23] for numerical approximation of the system (1.2) via a minimizing movement scheme.…”