A refined dynamic finite-strain shell theory for incompressible hyperelastic materials: equations and two-dimensional shell virtual work principle

TL;DR

This paper develops a refined dynamic finite-strain shell theory for incompressible hyperelastic materials, reducing the number of constitutive relations and deriving a 2D virtual work principle, validated through arterial deformation and vibration studies.

Contribution

It introduces a simplified shell theory with only two constitutive relations derived from 3D strain energy, enhancing modeling efficiency for hyperelastic shells.

Findings

Good agreement with 3D solutions for arterial extension and inflation

Effective analysis of arterial vibrations considering pre-stretch, pressure, and fiber angle

Refined theory simplifies shell modeling while maintaining accuracy

Abstract

Based on previous work for the static problem, in this paper we first derive one form of dynamic finite-strain shell equations for incompressible hyperelastic materials that involve three shell constitutive relations. In order to single out the bending effect as well as to reduce the number of shell constitutive relations, a further refinement is performed, which leads to a refined dynamic finite-strain shell theory with only two shell constitutive relations (deducible from the given three-dimensional (3D) strain energy function) and some new insights are also deduced. By using the weak formulation of the shell equations and the variation of the 3D Lagrange functional, boundary conditions and the two-dimensional (2D) shell virtual work principle are derived. As a benchmark problem, we consider the extension and inflation of an arterial segment. The good agreement between the asymptotic solution based on the shell equations and that from the 3D exact one gives verification of the former. The refined shell theory is also applied to study the plane-strain vibrations of a pressurized artery, and the effects of the axial pre-stretch, pressure and fibre angle on the vibration frequencies are investigated in detail.