Vol.:(0123456789) 1 3 Journal of Rubber Research https://doi.org/10.1007/s42464-020-00057-5 ORIGINAL PAPER A framework for model base hyper‑elastic material simulation Amirheshmat Khedmati Bazkiaei 1  · Kourosh Heidari Shirazi 1  · Mohammad Shishesaz 1 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 frst part, a survey on hyper-elastic constitutive models is presented. Fifty models are collected and classifed into six categories. Thereafter, fve 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 param- eters, 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 pre- sented 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 S ij Piolla–Kirchhof second stress tensor components E ij Green–Lagrange strain tensor components C ij Right Cauchy-Green deformation tensor components δ ij Kronecker delta F ij Deformation gradient tensor components X i Non-deformed body U i Displacement feld λ i 2 Eigenvalues of right Cauchy–Green tensor λ i Eigenvalues of deformation gradient tensor J Jacobian I i Invariants of Cauchy-Green strain tensor H ij Components of Hessian matrix of stored energy function C pq Model parameters α i Model parameters μ i Model parameters I m Limiting value of 1st invariant J m Parameter of fnite chain extensibility 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 -order strain tensor B i Model parameters A i Model parameters P Nominal stress p Hydrostatic pressure Introduction Rubbers are categorised among nonlinear elastic or hyper- elastic materials. The molecular structure of hyper-elastic materials permits high fexibility 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 numerical calculations, especially in three-dimensional analysis. The unique properties of these materials make them highly applicable in everyday life and science. In addition to the wide application of these materi- als in the automotive industry, aerospace, and tires [1–5], * Kourosh Heidari Shirazi k.shirazi@scu.ac.ir 1 Department of Mechanical Engineering, Shahid Chamran University of Ahvaz, Ahvaz 61357-43337, Iran