The time-dependent von K\'arm\'an plate equation as a limit of 3d nonlinear elasticity

TL;DR

This paper demonstrates that solutions of 3D nonlinear elastodynamics in thin plates converge to the time-dependent von Kármán plate equation as the plate's thickness approaches zero, under specific scaling conditions.

Contribution

It establishes a rigorous limit process connecting 3D nonlinear elasticity to the 2D von Kármán plate model for dynamic problems.

Findings

3D solutions converge to 2D von Kármán solutions as thickness tends to zero

Convergence holds under specific force and initial data scalings

Provides a mathematical justification for using von Kármán equations in thin plate dynamics

Abstract

The asymptotic behaviour of the solutions of three-dimensional nonlinear elastodynamics in a thin plate is studied, as the thickness $h$ of the plate tends to zero. Under appropriate scalings of the applied force and of the initial values in terms of $h$, it is shown that three-dimensional solutions of the nonlinear elastodynamic equation converge to solutions of the time-dependent von K\'arm\'an plate equation.