Abstract
The theory of quasi-linear viscoelasticity (QLV) is modified and developed for transversely isotropic (TI) materials under finite deformation. For the first time, distinct relaxation responses are incorporated into an integral formulation of nonlinear viscoelasticity, according to the physical mode of deformation. The theory is consistent with linear viscoelasticity in the small strain limit and makes use of relaxation functions that can be determined from small-strain experiments, given the time/deformation separability assumption. After considering the general constitutive form applicable to compressible materials, attention is restricted to incompressible media. This enables a compact form for the constitutive relation to be derived, which is used to illustrate the behaviour of the model under three key deformations: uniaxial extension, transverse shear and longitudinal shear. Finally, it is demonstrated that the Poynting effect is present in TI, neo-Hookean, modified QLV materials under transverse shear, in contrast to neo-Hookean elastic materials subjected to the same deformation. Its presence is explained by the anisotropic relaxation response of the medium.
Highlights
The ability to predict time-dependent deformation in soft, compliant solids is important when modelling a diverse range of materials, such as reinforced polymers, elastomers and rubbers, and is of increasing importance in the context of soft tissue mechanics
The review by Wineman [1] provides a comprehensive overview of the current state of the art of the field. In this Introduction, we provide a summary of some details of existing models in order to provide context and to motivate the present study, with respect to the study of viscoelastic anisotropy
A large body of work that implicitly incorporates more than one relaxation function in anisotropic models is that associated with internal variable viscoelasticity theory, which was motivated by some of the earliest work on finite strain in viscoelastic isotropic solids [28,29,30]
Summary
The ability to predict time-dependent deformation in soft, compliant solids is important when modelling a diverse range of materials, such as reinforced polymers, elastomers and rubbers, and is of increasing importance in the context of soft tissue mechanics. A large body of work that implicitly incorporates more than one relaxation function in anisotropic models is that associated with internal variable viscoelasticity theory, which was motivated by some of the earliest work on finite strain in viscoelastic isotropic solids [28,29,30]. This framework employs the uncoupled volumetric/deviatoric elasticity split dating back to Flory [31] and associates the time-dependent viscoelastic response to the deviatoric part only.
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