The time-dependent von K\'arm\'an shell equation as a limit of three-dimensional nonlinear elasticity

TL;DR

This paper investigates how solutions of 3D nonlinear elastodynamics in thin shells behave as the shell's thickness approaches zero, showing convergence to the time-dependent von Kármán or linear shell equations.

Contribution

It establishes the rigorous limit process from 3D nonlinear elasticity to 2D shell equations for arbitrary shell geometries.

Findings

3D solutions converge to von Kármán equations as thickness tends to zero.

The convergence holds under appropriate scalings of forces and initial data.

Results apply to shells of arbitrary geometry.

Abstract

The asymptotic behaviour of solutions of three-dimensional nonlinear elastodynamics in a thin shell is considered, as the thickness $h$ of the shell tends to zero. Given the appropriate scalings of the applied force and of the initial data in terms of $h,$ it's verified that three-dimensional solutions of the nonlinear elastodynamic equations converge to solutions of the time-dependent von K\'arm\'an equations or dynamic linear equations for shell of arbitrary geometry.