A note about hardening-free viscoelastic models in Maxwellian-type rheologies at large strains

Abstract

Maxwellian-type rheological models of inelastic effects of creep type at large strains are revisited in relation to inelastic strain gradient theories. In particular, we observe that a dependence of the stored energy density on inelastic strain gradients may lead to spurious hardening effects, preventing the model from accommodating large inelastic slips. The main result of this paper is an alternative inelastic model of creep type, where a higher-order energy contribution is provided by the gradients of the elastic strain and of the plastic strain rate, thus preventing the onset of spurious hardening under large slips. The combination of Kelvin–Voigt damping and Maxwellian creep results in a Jeffreys-type rheological model. The existence of weak solutions is proved by way of a Faedo–Galerkin approximation.

Quasistatic hypoplasticity at large strains Eulerian

“…that φ(det F p ) not only blows up to ∞ if det F p → 0+ but is it very large if det F p = 1; cf. [10,26,36,49] for a Lagrangian formulation. One advantage of the rate-formulation in Section 3 below will be that we can involve the isochoric constraint det F p = 1 into the model exactly and thus will be able to eliminate φ from the model completely.…”

“…This causes a certain "dynamical" hardening involves a certain length scale to the plastic distortion but, does not make spurious hardening effects during long lasting plastification or creep, unlike the conventional kinematic or isotropic hardening in the conservative part. Similarly, in Lagrangian formulation, [10] used the plastic distortion rate in ∆ ∂ ∂t F p .…”

“…Any reference configuration (i.e. the Lagrangian approach as in [10,26,32,36,49]) is thus eliminated from the formulation of the problem. This is very natural especially for materials where such a reference configuration cannot be identified naturally, as e.g.…”

Visco-elastodynamics at large strains Eulerian

“…In the visco-elastic case, there are naturally two basic options how to impose some higher-order terms: to the conservative (elastic) part or to the dissipative (viscous) part, or to both. In the Lagrangian deformation description, the higher-order deformation gradients are analyzed in [28, or [11]. In the Eulerian v/F -formulation, there are even three options how to involve higher gradients: in addition to two mentioned options in the Lagrangian variant, one can also regularize the transport equation (1.1) like (2.7b) or (3.3) below.…”

Thermodynamically consistent model for poroelastic rocks towards tectonic and volcanic processes and earthquakes

“…It should also be mentioned an alternative referencecoordinate (Lagrangian) formulation at large strains in [48], using an artificial reference configuration and quite heavy push-forward and pull-back machinery as well as a hidden spurious hardening in large slips, as articulated later in [13]. Let us still remark that an attempt to formulate the model [31] thermodynamically consistently has been done in [45] in terms of displacements, but not all time-derivatives have been objective and the energy balance was thus corrupted.…”