Large strain constitutive modelling of soft compressible and incompressible solids: Generalised isotropic and anisotropic viscoelasticity

This paper discusses a new phenomenological continuum formulation for the constitutive modelling of viscoelastic materials at large strains. Following pioneering works in Sidoroff (1974), Lubliner (1985), Bergström (1998) and Reese and Govindjee (1997), the formulation shares some common ingredients with other phenomenological approaches, including the multiplicative decomposition of the deformation gradient into viscous and elastic contributions, the additive Maxwell-type decomposition of the strain energy density, and the definition of a set of kinematic internal state variables with their associated evolution laws. Our formulation departs from other state-of-the-art methodologies via three distinct novelties. First, and revisiting previous work by Bonet (2001), the paper introduces a thermodynamically consistent linear rate type evolution law in terms of stress-type variables, which resembles the return mapping algorithm typically used in elastoplasticity, facilitating the modelling link between both inelastic constitutive models. In this sense, the proposed viscoelastic evolution law can be identified with a classical plastic flow rule. Very importantly, the evolution law is shown to be compatible with the second law of thermodynamics by construction and have a closed-form solution in the case of incompressible viscoelasticity when using a prototypical neo-Hookean type of non-equilibrium strain energy density. Moreover, the paper shows how using the concept of a stress-driven dissipative potential, more general non-linear type of stress evolution laws can be straightforwardly constructed. Second, to facilitate the joint consideration of anisotropy and thermodynamic equilibrium, a frame indifferent stress free configuration is introduced which facilitates the definition of objective strain measures. Third, the methodology is extended from isotropy to transverse isotropy via the consideration of the appropriate structural tensor. The formulation is first displayed for the simple case of a single transversely isotropic invariant contribution with corresponding closed-form solution, and then straightforwardly extended to the consideration of the second transversely isotropic invariant, multiple families of fibres, or even more complex symmetry groups. To demonstrate the capability of the new framework, a specialised form of the eight-chain long-term strain energy (long term) and a neo-Hookean strain energy (non-equilibrium) have been adopted for the description of the mechanical behaviour of VHB 4910 polymer, due to its use in current Electro-Active Polymers based soft robotics. Good agreement is found between in silico predictions and available experimental data on various tests, including loading–unloading cyclic tests, single-step relaxation tests and a multi-step relaxation test. Finally, biaxial loading–unloading cyclic and relaxation tests are presented to further showcase performance in anisotropic scenarios.