On the volumetric part of strain-energy functions used in the constitutive modeling of slightly compressible solid rubbers

Abstract

Experimental data for compression and simple tension Finite element simulations Volumetric term in strain-energy density a b s t r a c t A popular model for the finite element simulation of slightly compressible solid rubber-like materials assumes that the strain-energy function can be additively decomposed into a volumetric part and a deviatoric part. Based on mathematical convenience, the volumetric part is usually assumed to be a finite polynomial in the volume change. Experimental evidence suggests that for solid rubbers in compression, this polynomial can be taken to be a simple quadratic for moderate deformations and that this function also adequately models the volume change and the stress/stretch relation for materials in simple tension, up to stretches of order 100%. For larger tensile deformations, however, experimental data suggest that the Cauchy stress-volume change relation has an increasingly large slope and therefore a truncated Taylor series expansion is not the most appropriate. A rational function approach is proposed here as an alternative.

Simple Shearing of Incompressible and Slightly Compressible Isotropic Nonlinearly Elastic Materials

“…It was shown that (6.5) is equivalent to assuming that hydrostatic pressure/tension causes only a change in volume. In another recent paper [14], the authors also showed that assuming a simple quadratic form for F (i 3 ), as is usually done in Finite Element implementations, provides an excellent agreement with the experimental data for moderate deformations.…”

“…We refer to our recent papers [12][13][14][15] for background and references to the pertinent literature. Here we briefly describe the usual form of such slightly compressible (or almost incompressible) strain-energy functions used in the Finite Element simulations of rubbers.…”

“…Although the slightly compressible form (6.5) is by far the most common method of including slight compressibility effects for a given incompressible material, many other methods have been proposed (see e.g., [12][13][14][15] for a discussion of some of these). Different generalizations of a given incompressible material to include slight compressibility could result in different values of the pressure chosen to ensure identification of the two stress fields.…”

Volume growth during uniaxial tension of particle-filled elastomers at various temperatures – Experiments and modelling

“…However, different potential functions have been proposed and a discussion of those is included in the review by Boyce and Arruda [44], and in the works of Bischoff et al [43], Doll and Schweizerhof [55], and Horgan and Murphy [56]. Generally, the proposed models for the volumetric response are purely phenomenological, being mathematical expressions satisfying the physical limitations of a potential function (e.g.…”

Experimental evidence of the compressibility of arteries

“…As an example, slight compressibility (volumechange of up to roughly 3%) resulted in circumferential stresses in the arterial wall up to 100% higher compared to the incompressible case for the physiological pressure of 100 mmHg as documented in [21,8]. For almost-incompressible materials ε = µ/κ is usually a small parameter, but not zero, and the interested reader is referenced to a detailed analysis of such cases for ε ≪ 1 in [14,15].…”