Application of Objective Rates in Mechanical Modeling of Solids

Abstract

A unified formulation is developed for deformation-related spins, and for objective rates based on them. The approach generalizes the underlying concepts, and allows new rates to be constructed. Mathematical and thermodynamic^ restrictions on these are shown. As a result, it can be demonstrated that the Eulerian strain rate is an objective rate of logarithmic strain, based on a spin easily derivable from the general form. Interrelations between other known spins and objective rates emerge very clearly. Consequences of the proposed formalism are explored in hypoelastic and in rigid-plastic constitutive relations, the latter involving purely isotropic and purely kinematic hardening. The application of the resulting models to the simple shear deformation is shown.

Numerical study of consistency of rate constitutive equations with elasticity at finite deformation

“…From Equations (5), (8) and (9) we know that these formulae are extremely dependent on the eigenvalues of B (or C). For keeping the completeness we record here a closedform solution for these eigenvalues (including the case that the eigenvalues are multiple).…”

“…Since Truesdell [1][2][3] developed the hypoelasticity theory, the rate constitutive equations have been widely used in inelastic constitutive theory, see for example Green [4], Dafalias [5], Lee et al [6], Hughes [7] and Reinhardt and Dubey [8]. In many general purpose ÿnite element codes the algorithms based on the rate equations occupy an important position.…”

Discontinuous modelling of shear bands using adaptive meshfree methods

“…We employ the logarithmic spin and the Hencky srain, which assure perfect reversibility and therefore zero dissipation in a closed cycle for elastic-only processes (c.f. [55,76]). We assume that material stability is not an issue in the elastic-only regime (this issue is discussed in reference [23]).…”

A Lagrangian model for hardening behaviour of materials at finite deformation based on the right plastic stretch tensor

“…For an Eulerian rate formulation of elastoplasticity, the strain rate tensor can be additively decomposed into its elastic and inelastic parts [4,7,8,10,18]:…”

“…Sources of such issues have been associated with hypoelastic model non-integrability as a Cauchy and Green elastic material (see for example references [11,15,16]). To resolve such issues, following the work of Lehmann et al [17], Reinhardt and Dubey [18], and Xiao et al [19], a new objective rate of stress, called the D or logarithmic rate, was introduced. Furthermore, Bruhns et al [10] investigated the integrability conditions of the classical hypoelastic model and showed that the grade zero hypoelastic model is unconditionally integrable as a Cauchy and Green elastic material when the logarithmic (D) rate is used in the model.…”