# Analytical and Finite Element Solution for Functionally Graded Pressure Vessels Subjected to Finite Strain Coupled Axial and Torsional Deformations

## TL;DR

This paper provides an analytical and numerical solution for the mechanical behavior of a hyperelastic cylinder under combined axial and torsional deformations.

## Contribution

The study introduces a novel analytical solution and validates it with finite element simulations for functionally graded hyperelastic materials.

## Key findings

- Analytical and numerical results show strong agreement for various stretch and twist levels.
- Torsion causes complex longitudinal stress distribution with two extrema under large twisting.
- A critical axial stretch determines the transition of axial force from tension to compression during twisting.

## Abstract

This study presents an analytical solution to examine the mechanical behavior of an incompressible, functionally graded hyperelastic cylinder under combined extension and torsion. The exp-exp strain energy density function characterizes the hyperelastic material, with parameters varying exponentially along the radial direction. To validate the solution, finite element simulations using a custom UHYPER in ABAQUS are performed. The analytical and numerical results show strong agreement across different stretch and twist levels. The stress distribution and maximum stress are significantly influenced by the exponential parameter governing material gradients. Unlike axial stretch, torsion induces a more intricate longitudinal stress distribution, with large twisting producing two extrema that shift toward the cylinder’s center and outer surface. Longitudinal stress primarily governs von Mises stress and strain energy density variations across the radial direction. A critical axial stretch is identified, below which torsion-induced axial force transitions to compression, elongating the cylinder during twisting. Beyond this stretch, the axial force shifts from tensile to compressive with increasing twist, causing initial shortening before further elongation.