An introduction to stochastic finite element method analysis of hyperelastic structures

Abstract

The main idea of this work is to demonstrate an application of the generalized iterative stochastic perturbation technique to numerical analysis of the hyperelastic materials and structures with Gaussian random parameter, where the input random variable is a magnitude of the vertical uniformly distributed load. Theoretical apparatus is connected with the general order Taylor expansion of both input and state parameters with random coefficients and analytical derivation of their first four probabilistic moments and coefficients. Our computational implementation is released with the Response Function Method having polynomial basis of the order minimizing variance and maximizing correlation of the least squares fitting to the series of numerical experiments. Computational experiment concerns the hyperelastic rubber-like prismatic beam under three-point bending discretized in the FEM system ABAQUS with the use of various 3D brick finite elements. Large deformations in the vertical symmetry plane of this structure are analyzed in the stochastic context - by determination of their expectations, coefficients of variations, skewness and kurtosis for different increments of the external load. It enables also to recover the basic probabilistic characteristics of the stress-strain curve of such a material, whose further comparison with the experiments will allow a full validation of such a probabilistic model. The entire probabilistic algorithm together with statistically optimized Weighted Least Squares Method fitting are implemented in the symbolic algebra package MAPLE. The proposed scheme of the Stochastic Finite Element Method is contrasted with the crude Monte-Carlo scheme and also with the semi-analytical calculations of the same probabilistic characteristics by direct integration of the response functions.