Engineering Transactions, 56, 2, pp. 117–157, 2008

Constitutive Modelling and Parameter Identification for Rubber-Like Materials

Polish Academy of Sciences, Institute of Fundamental Technological Research Department of Mechanics of Materials

The aim of the paper is to determine the phenomenological model to characterize the
stress-strain relation and to simulate the behaviour of solid polyurethane (PUR) rubbers used
in civil engineering, as well as to present the process of identification of model parameters for
such materials. For the material studied the strain energy density function was established
and a general constitutive relationship for the second-order tensor of Piola–Kirchhoff stress for
elasticity is determined. Constitutive relationships for engineering stress in terms of the principal
stretches are also specified. The paper presents the method of identification of parameters
for constitutive models of hyperelasticity and hypoelasticity for the accessible experimental
data. The applied identification procedure is based on the feature of two-phase structure of
polyurethane material and is supported by the experimental data from uniaxial quasi-static
tension and compression tests. In the analysis, the material behaviour was considered both
for the case of incompressible deformation and also for the case of slightly compressible, nonlinearly
elastic materials that are homogeneous and isotropic. The change of volume was admitted
too, in range of large deformations in a tension and compression test. The attempt of
description of stress-softening phenomenon was undertaken in rubber-like materials, for a given
level of strain, under unloading (the Mullins effect) caused by the damage of microstructure
of this material. Different descriptions of the stress-softening phenomenon were already proposed
in the literature but they fail to give fully satisfactory conformity of experimental data
with theoretical predictions. The phenomenological model by Elias–Zúñiga and Beatty,
A new phenomenological model for stress-softening in elastomers, ZAMP, 53, 794–814, 2002,
for such materials was modified by different softening functions and a simplified version of this
model was identified, based on the experimental data. In the proposed model, the damage of
microstructure was described by a new exponential function, which depends on the current
magnitude of intensity of strain and its earlier maximum value during the process of material
loading. In this paper, a suitable analysis of existent models and their verification based on
experimental data for polyurethane rubber is presented for uniaxial experiments. It is shown
that the magnitude of stress-softening varies with strain and this phenomenon increases with
the magnitude of the pre-strain and the type of loading: monotonic tension, compression or
cyclic loading. The obtained results are presented graphically for uniaxial tension and compression.
Keywords: rubber-like material, hyperelastic constitutive model, damage of polyurethane rubber
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