Green-Naghdi Rate Research Articles

The appropriate corotational rate, exact formula for the plastic spin and constitutive model for finite elastoplasticity

The exact formulae for the plastic and the elastic spin referred to the deformed configuration are derived, where the plastic spin is a function of the plastic strain rate and the elastic spin a function of the elastic strain rate. With these exact formulae we determine the macroscopic substructure spin that allows us to define the appropriate corotational rate for finite elastoplasticity.Plastic, elastic and substructure spin are considered and simplified for various sub-classes of restricted elastic-plastic strains. It is shown that for the special cases of rigid-plasticity and hypoelasticity the proposed corotational rate is identical with the Green-Naghdi rate, while the ZarembaJaumann rate yields a good approximation for moderately large strains.To compare our exact plastic spin formula with the constitutive assumption for the plastic spin introduced by Dafalias and others, we simplify our result for small elastic-moderate plastic strains and introduce a simplest evolution law for kinematic hardening leading to the Dafalias formula and to an exact determination of its unknown coefficient. It is also shown that contrary to statements in the literature the plastic spin is not zero for vanishing kinematic hardening.For isotropic-elastic material with induced plastic flow undergoing isotropic and kinematic hardening constitutive and evolution laws are proposed. Elastic and plastic Lagrangean and Eulerian logarithmic strain measures are introduced and their material time derivatives and corotational rates, respectively, are considered. Finally, the elastic-plastic tangent operator is derived.The presented theory is implemented in a solution algorithm and numerically applied to the simple shear problem for finite elastic-finite plastic strains as well as for sub-classes of restricted strains. The results are compared with those of the literature and with those obtained by using other corotational rates.

Implicit objective integration for sensitivity analysis in non‐linear solid mechanics

AbstractIncrementally objective integration schemes are proposed for the accurate and efficient determination of design sensitivity coefficients (DSCs) for solid mechanics problems with both material and geometrical non‐linearities. The derivation of these schemes are based on the direct differentiation of objective schemes that are used in stress analysis for problems of this class. Two widely used objective stress rates, the Jaumann rate and the Green–Naghdi rate, are considered here within the same setting with only minor changes of the integration scheme. Numerical results are presented for a simple shear problem with different material constitutive laws, including a hypoelastic model and a unified isotropic viscoplastic model, for these two objective rates. The numerical results are compared with analytical solutions or direct integration solutions. The close agreement among these solutions demonstrates the accuracy and efficiency of the proposed scheme.

On consistent stress rates in solid mechanics: Computational implications

AbstractIn the context of general isothermal elastic processes, issues related to work conjugacy and its effect on the choice of a stress rate are discussed for processes involving large strains. It is shown that a unique stress rate consistent with hyperelasticity can be defined for every conjugate pair chosen through the rate form of the associated hyperelastic constitutive equation. That results in non‐constant and generally anisotropic spatial elasticity tensors even for a linear isotropic material. The anisotropic spatial elasticity tensor connected with the Green–Naghdi rate of Kirchhoff stress is evaluated, which provides a basis for the spatial formulation of processes governed by the Hencky's strain energy function.48,49 Notions of equivalent and irreducible form of stress rate are introduced using a ‘workless’ component of the deformation rate. The hyperelastic constitutive models for the conjugate pairs {S, E(2)} and {T, In U}, which are of significance for computation, are stated in their explicit form. Exact rate forms are derived and the order of approximation of currently employed computational algorithms discussed. Finally, computational implications of the work presented are reviewed.