Representations for implicit constitutive relations describing non-dissipative response of isotropic materials

Numerical scheme for simulation of transient flows of non-Newtonian fluids characterised by a non-monotone relation between the symmetric part of the velocity gradient and the Cauchy stress tensor

“…Conceptually, constitutive relation (1.5) belongs to the class of implicit constitutive relations, see Rajagopal (2003Rajagopal ( , 2006, Průša and Rajagopal (2012), Perlácová and Průša (2015), Rajagopal and Saccomandi (2016) and Fusi et al (2018) to name a few, which seems to be an interesting approach to the modelling of fluid response. (See also Bustamante (2009); Bustamante and Rajagopal (2011 and Gokulnath et al (2017) for a similar developments in the case of solids.) The presented study opens the possibility to investigate the flows of fluids characterised by implicit constitutive relations in complicated geometries.…”

A thermodynamic basis for implicit rate-type constitutive relations describing the inelastic response of solids undergoing finite deformation

“…In particular, the specific Gibbs free energy might be the thermodynamic potential of choice, see [13] for further discussion. In fact, the Gibbs potential is the natural choice in the case of elastic bodies with constitutive relation in the form B = e ( T ) , where e is a tensor-valued function, see especially [14], [16] and [19], and also [20] for the case of thermoviscoelastic solids. For the sake of simplicity, however, we stick to the Helmholtz free energy, which allows us to utilise or recover some well known formulae for Green elastic (hyperelastic) solids, and document the relation of our work to the classical nonlinear elasticity theory.…”

“…While a generalisation of constitutive relations of the type of equation (4) into the fully three-dimensional setting and to finite deformations has been carried out by Rajagopal and Srinivasa [12], such a generalisation still works with the mechanical variables only, and leaves the problem of establishing a thermodynamic basis for this class of models open. (Several studies, see for example [13][14][15][16], are focused on thermodynamics, but mainly in the context of simpler algebraic implicit constitutive relations (2).) Consequently, the question is whether it is possible to develop a complete thermodynamic basis that allows one to recover the mechanical models introduced by Rajagopal and Srinivasa [12], and that guarantees that models of this type are consistent with the first and the second laws of thermodynamics.…”

A model for a solid undergoing rate-independent dissipative mechanical processes

“…Our formulation does not assume that the magnitude of the displacement gradient’s components are small. Logarithmic or Hencky strain [20] is used as the strain measure for the simplicity that it affords in the computation of the stress power [21, 22]. It would be evident from the formulation that no significant simplification occurs on assuming small deformations, except that the linearized strain can replace the Hencky strain.…”