Return Mapping Algorithms (RMAs) for Two-Yield Surface Thermoviscoplastic Models Using the Consistent Tangent Operator

The return mapping algorithms (RMAs) presented here are designed for use with pressure-dependent thermoviscoplastic constitutive models involving irreversible effects associated with solid–solid phase transformations. During the volume solid–solid phase transformations occurring under mechanical loads, an “anomalous” plasticity, the so-called “TRansformation Induced Plasticity” (TRIP), is generated at much lower stress levels than those related to the yield stress of the material in the context of the classical plasticity. TRIP mechanisms are superimposed on the classical plasticity which is liable to occur in the case of metallic materials. Based on a non-standard generalized material framework, two different models are presented in which an “associative” plastic flow is introduced in the context of classical plasticity and a “non-associative” flow rule in the context of TRIP-like plasticity. In this paper, a complete algorithmic treatment of these two rate-dependent constitutive models is therefore proposed with the associated consistent tangent operator for dealing the quasi-surface irreversible solid–solid transformations which can appear in metal alloys during specific thermomechanical solicitations. The predictive abilities of the presented numerical procedure for modelling this kind of the irreversible solid–solid transformations involving two plasticity processes are tested and assessed by performing a two-dimensional finite-element analysis on some numerical examples.

The derivative and tangent operators of a motion in Lorentzian space

In this paper, by using Lorentzian matrix multiplication, [Formula: see text]-Tangent operator is obtained in Lorentzian space. The [Formula: see text]-Tangent operators related with planar, spherical and spatial motion are computed via special matrix groups. [Formula: see text]-Tangent operators are related to vectors. Some illustrative examples for applications of [Formula: see text]-Tangent operators are also presented.