A framework for model base hyper-elastic material simulation
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ORIGINAL PAPER
A framework for model base hyper‑elastic material simulation Amirheshmat Khedmati Bazkiaei1 · Kourosh Heidari Shirazi1 · Mohammad Shishesaz1 Received: 26 February 2020 / Accepted: 16 September 2020 © The Malaysian Rubber Board 2020
Abstract In this research, a framework for modelling and simulation of hyper-elastic materials is proposed. The framework explains how to employ strain energy functions as a constitutive model, standard loading test data, and a powerful optimisation method to determine a mathematical function for explaining the mechanical behaviour of a hyper-elastic material using minimum types of loading test data. In the first part, a survey on hyper-elastic constitutive models is presented. Fifty models are collected and classified into six categories. Thereafter, five types of standard loading tests including uniaxial, biaxial, equi-biaxial, pure shear, and simple shear are introduced. It is shown that depending on the loading type, physical parameters, Cauchy, and nominal stress tensors, each constitutive model possesses a particular function. The genetic algorithm as a powerful optimisation method is used to determine the most accurate function for each type of loading test data. It is presented that based on the selected constitutive model and regardless of a number of existing loading types test data, a unique function can be determined for expressing and simulating the mechanical behaviour of the considered hyper-elastic material. Keywords Hyper-elastic models · Genetic algorithm · Constitutive model · Mooney–Rivlin · Ogden Abbreviations w Stored strain energy function Sij Piolla–Kirchhof second stress tensor components Eij Green–Lagrange strain tensor components Cij Right Cauchy-Green deformation tensor components δij Kronecker delta Fij Deformation gradient tensor components Xi Non-deformed body Ui Displacement field λi2 Eigenvalues of right Cauchy–Green tensor λi Eigenvalues of deformation gradient tensor J Jacobian Ii Invariants of Cauchy-Green strain tensor Hij Components of Hessian matrix of stored energy function Cpq Model parameters αi Model parameters μi Model parameters Im Limiting value of 1st invariant Jm Parameter of finite chain extensibility * Kourosh Heidari Shirazi [email protected] 1
Department of Mechanical Engineering, Shahid Chamran University of Ahvaz, Ahvaz 61357‑43337, Iran
μ Model parameters n Chain density per unit of volume k Boltzman constant T Absolute temperature L−1 Langevin function I* (α) First invariant of the generalised 𝛼-order strain tensor Bi Model parameters Ai Model parameters P Nominal stress p Hydrostatic pressure
Introduction Rubbers are categorised among nonlinear elastic or hyperelastic materials. The molecular structure of hyper-elastic materials permits high flexibility in room temperature as well as high reversibility against deformation. The most important property of these materials is incompressibility, which is the reason for having Poisson’s ratio near 0.5. This causes the complexity of
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