Thermodynamically consistent nonlinear viscoplastic formulation with well-conditioned recovery of the inviscid solution: Theory and implicit integration algorithm with exact solution for the linear case

TL;DR

This paper introduces a thermodynamically consistent nonlinear viscoplasticity model with an implicit integration algorithm that accurately recovers inviscid solutions and encompasses various material behaviors as special cases.

Contribution

The work presents a novel viscoplasticity formulation based on continuum principles, with an implicit algorithm that ensures well-conditioned inviscid limit recovery and exact linear solutions for proportional loading.

Findings

Inviscid solution is recovered by setting viscosity to zero.

Linear viscoplasticity solution is exact for proportional loading.

The model unifies viscoplastic, viscoelastic, and elastic behaviors as special cases.

Abstract

In this work, a consistent viscoplasticity formulation is derived from thermodynamical principles and employing the concept of continuum elastic corrector rate. The proposed model is developed based on the principle of maximum viscoplastic dissipation for determining the flow direction. The model uses both the equivalent viscoplastic strain and its rate as state variables. Power balance and energy balance give, respectively, separate evolution equations for the equivalent viscoplastic strain rate and the viscoplastic strain, the former written in terms of inviscid rates. Several key points distinguish our formulation from other proposals. First, the viscoplastic strain rate (instead of a yield function) consistently distinguishes conservative from dissipative behaviours during reverse loading; and the discrete implicit integration algorithm is an immediate implementation of the continuum theory based on the mentioned principles. Second, the inviscid solution is recovered in a well-conditioned manner by simply setting the viscosity to zero. Indeed, inviscid plasticity, viscoelasticity and viscoplasticity are particular cases of our formulation and integration algorithm, and are recovered just by setting the corresponding parameters to zero (viscosity or yield stress). Third, the linear viscoplasticity solution is obtained in an exact manner for proportional loading cases, independently of the time step employed. Four, general nonlinear models (Perzyna, Norton, etc) may be immediately incorporated as particular cases both in the theory and the computational implementation.