Alternative finite strain viscoelastic models: constant and strain rate-dependent viscosity (Q6556490)

The authors derive finite strain viscoelastic models and subsequently proposed their FEM implementation. The multiplicative decomposition of the right Cauchy-Green strain tensor, (\(\mathbf{C}\)), into its isochoric and volumetric part, is adopted. A specific representation of the elastic energy density function, written in terms of the first two invariants of the isochoric strain and of the third invariant of (\(\mathbf{C}\)), allows to associate appropriate second (symmetric) Piola-Kirchhoff stress tensors, as well as isochoric and volumetric strain measures. In order to derive the viscoelastic models, the authors perform an analogy with one-dimensional rheological models, namely with Kelvin-Voigt-like and Zener-like models, to establish the viscous behaviour. The symmetric Piola-Kirchhoff stress tensor results as a sum of the so-called viscous and elastic stresses, which are associated with rate of the appropriate isochoric and elastic strains, respectively. There is a similarity with those from the Kelvin-Voigt-like model. When the Zener-like rheological model is considered, the symmetric Piola-Kirchhoff stress tensor results from the presence of the two branches: the one corresponding to the elastic behaviour (in terms of the isochoric strain) and the second, due to the presence of the Maxwell branch. The stress component on the Maxwell branch measures a viscous effect (in terms of the rate of the viscous strain part) and the elastic effect (in terms of the elastic strain part) at the same time. An appropriate relationship between the rate of the viscous strain part and the elastic part follows (they are parts of the isochoric strain on Maxwell branch). In order to avoid the time derivative of the Piola-Kirchhoff stress tensor (which ought to be involved in a viscoelastic model associated with Zener-like rheological model), a ``backward finite difference'' procedure is applied. Consequently, certain finite difference viscoelastic models are derived, but \(1/{\Delta t}\) is present, where \({\Delta t}\) is the time step. In Equation (28) the elastic part of the isochoric strain, which cannot be identified directly from it, is still present. Another versions of the viscoelastic models are derived, when the appropriate rate of the viscous strains are replaced by tensor fields which kept the direction of the isochoric strain, but involve the time derivative of the scalar invariants of the isochoric strain. FEM implementation of the proposed viscoelastic models is based on the weak form of the motion (non-equilibrium) equation written in the reference configuration, in Equation (43). Here the second Piola-Kirchhoff stress tensor appears instead of the first (non-symmetric) Piola-Kirchhoff stress tensor, which ought to be written, see for instance Truesdell (1972), Eringen (1967). Consequently, the proposed FEM implementation works only in the case of small strains, in order to have with necessity the symmetry of Piola-Kirchoff stress tensor, used by the authors.