On a consistent finite-strain plate theory based on 3-D energy principle

TL;DR

This paper develops a finite-strain plate theory derived from 3-D nonlinear elasticity principles, providing a consistent, accurate, and general approach for analyzing large deformation problems with a third-order error.

Contribution

It introduces a new 3-D energy-based plate theory with exact recursion relations, improving accuracy and applicability over existing models.

Findings

The theory achieves second-order accuracy in pure bending problems.

It maintains consistency with 3-D energy principles under general loadings.

The approach avoids unphysical quantities and extends to finite-strain scenarios.

Abstract

This paper derives a finite-strain plate theory consistent with the principle of stationary three-dimensional (3-D) potential energy under general loadings with a third-order error. Staring from the 3-D nonlinear elasticity (with both geometrical and material nonlinearity) and by a series expansion, we deduce a vector plate equation with three unknowns, which exhibits the local force-balance structure. The success relies on using the 3-D field equations and bottom traction condition to derive exact recursion relations for the coefficients. Associated weak formulations are considered, leading to a 2-D virtual work principle. An alternative approach based on a 2-D truncated energy is also provided, which is less consistent than the first plate theory but has the advantage of the existence of a 2-D energy function. As an example, we consider the pure bending problem of a hyperelastic block. The comparison between the analytical plate solution and available exact one shows that the plate theory gives second-order correct results. Comparing with existing plate theories, it appears that the present one has a number of advantages, including the consistency, order of correctness, generality of the loadings, applicability to finite-strain problems and no involvement of unphysical quantities.