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Stochastically-constrained Koiter shell models

Prince Romeo Mensah, Pierre Marie Ngougoue Ngougoue

Abstract

We derive a stochastically-constrained Koiter shell model 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 nonlinearly- and linearly-elastic Koiter shell models with their abstract functional derivatives for their corresponding membrane and flexural energies. We then present a prototype for the stochastically-constrained (simplified) linearised Koiter shell models that encodes all necessary information on stiffness due to shell curvature, bending stress, membrane stress, interior and surface forces, and more generally, stochastic buckling.

Stochastically-constrained Koiter shell models

Abstract

We derive a stochastically-constrained Koiter shell model 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 nonlinearly- and linearly-elastic Koiter shell models with their abstract functional derivatives for their corresponding membrane and flexural energies. We then present a prototype for the stochastically-constrained (simplified) linearised Koiter shell models that encodes all necessary information on stiffness due to shell curvature, bending stress, membrane stress, interior and surface forces, and more generally, stochastic buckling.

Paper Structure

This paper contains 6 sections, 2 theorems, 66 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Modelling
  3. The nonlinear shell model
  4. The linearized shell model
  5. Prototype for the simplified linear model.
  6. Conclusion

Key Result

Theorem 2.1

Stationarity zeroAction of the action mainAction-ltk subject to t0t1zero result in the following stochastic Euler-Lagrange equation where $\frac{\partial f}{\partial \eta}$ is the functional derivative of $f$ with respect to $\eta$. $\blacktriangleleft$$\blacktriangleleft$

Figures (2)

  • Figure 1: Left: A thin elastic cylinder (e.g. blood vessel) with an imperfection characterised by a slight deviation in the diameter at point $\mathbf{x}$. Middle: A thin cylinder with varying shell thickness from right to left. Right: A thin shell subject to load.
  • Figure 2: Three snapshots for $\eta(T,\mathbf{y})$ solving \ref{['shellVelocity']} for $N = 2$. The stochastic forcing is defined by vector fileds $\bm{\sigma}_1=2(\sin(y_1),-\cos(y_2))^\top$, $\bm{\sigma}_2=2(-\cos(y_1),\sin(y_2))^\top$ on the torus $\Gamma=[-2\pi,2\pi]^2$ with initial condition $\eta(0,\mathbf{y})=\exp(-((y_1-\pi)^2+(y_2-\pi)^2))$.

Theorems & Definitions (3)

  • Theorem 2.1
  • proof
  • Corollary 2.2