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ORIGINAL CONTRIBUTION

A model for rubber elasticity W. E. VanArsdale 1 Received: 13 February 2020 / Revised: 28 June 2020 / Accepted: 8 July 2020 # Springer-Verlag GmbH Germany, part of Springer Nature 2020

Abstract A constitutive equation for rubber-like materials is developed using the left stretch tensor. This process starts with a model for hyperelastic solids based on a separable energy function. This model accurately fits extensional data for vulcanized natural rubber until the onset of hysteresis at intermediate strains. Better predictions outside the hyperelastic range are obtained by directly modifying this constitutive equation to describe limited extensibility. The resulting model accurately fits biaxial, planar, and uniaxial extension data for a variety of rubber-like materials using three constants. This model also predicts simple shear results derived from planar extension data and characterizes inflation of spherical membranes for elastomers and soft tissue. A final modification accurately describes hardening associated with crystallization at large tensile strains. Keywords Finite strain . Hysteresis . Left stretch tensor . Rubber material . Soft tissue

Introduction Vulcanized rubber is an isotropic material consisting of crosslinked polymers. This material is typically modeled as an incompressible, hyperelastic solid, where deformation is characterized by the left Cauchy–Green tensor B = FFT defined in terms of the deformation gradient F (Gurtin 1981 p. 46). This objective tensor is usually assumed to be consistent with the isochoric constraint det(B) = 1. However, applying tensile stress to vulcanized rubber causes volume to increase until strain-induced crystallization results in a decrease at large strains (Treloar 2009 pp. 295, 20–23). Filled and unfilled natural rubber also exhibit hysteresis at intermediate strains (Omnès et al. 2008, Treloar 2009 pp. 87, 89, 92), implying work in a closed cycle of deformation. While this behavior is inconsistent with predictions for hyperelastic solids (Gurtin 1981 p. 190), these models are still used to fit extensional data at large strains. Constitutive equations for hyperelastic solids derive stress from an energy function. Rivlin (1948) developed models for rubber-like materials based on functions of the invariants tr(B) and tr(B−1). For example, the function

* W. E. VanArsdale

1

Colorado Springs, USA


  w ¼ C1 ½trðBÞ−3 þ C2 tr B−1 −3

ð1Þ

involving constants C1 and C2 determines Cauchy stress T ¼ −pI þ 2C1 B−2C2 B−1

ð2Þ

to within an unspecified pressure p associated with the constraint det(B) = 1. While many models use this approach (Hoss and Marczak 2010), most energy functions depend only on the invariant tr(B). For example, Treloar (1943) used network theory with a Gaussian distribution function to derive the neo-Hookean model T ¼ −pI þ 2C1 B;

ð3Þ

where I is the identity tensor. Theories for non-Gaussian networks of flexible chains lead to similar models (Arruda and Boyce 1993; Horgan et al. 2004), where C1 becomes a function of tr(