Stochastically-constrained Koiter shell models

TL;DR

This paper derives stochastic Koiter shell models using the SALT approach, capturing effects like buckling and stiffness, and shows how solutions relate to deterministic viscoelastic models.

Contribution

It introduces stochastically-constrained Koiter shell models based on SALT, including a prototype that accounts for curvature, stresses, and stochastic buckling effects.

Findings

Derived stochastic PDEs for nonlinear and linear Koiter shell models.

Presented a prototype capturing shell stiffness and buckling effects.

Showed that solutions relate to deterministic viscoelastic models in the limit.

Abstract

We derive stochastically-constrained Koiter shell models in line with the SALT (Stochastic Advection by Lie Transport) approach introduced by Holm [Proc. A. 471 (2015)]. First, we deduce the stochastic partial differential equations for the generalised nonlinear elastic and linear elastic Koiter shell models with abstract functional derivatives of their corresponding membrane and flexural energies. We then present a prototype for a stochastically-constrained (simplified) linearised Koiter shell model that captures stiffness effects arising from shell curvature, bending and membrane stresses, interior and surface forces, and, more generally, stochastic buckling. Finally, we show that if a weak pathwise solution of this prototype is parametrised by a suitably chosen family of noise coefficients, we obtain in the parameter limit, the deterministic viscoelastic shell model with viscous damping.