A thermoelastoplastic theory for special Cosserat rods

Abstract

A general framework is presented to model coupled thermoelastoplastic deformations in the theory of special Cosserat rods. The use of the one-dimensional form of the energy balance in conjunction with the one-dimensional entropy balance allows us to obtain an additional equation for the evolution of a temperature-like one-dimensional field variable, together with constitutive relations for this theory. Reduction to the case of thermoelasticity leads us to the well-known nonlinear theory of thermoelasticity for special Cosserat rods. Later on, additive decomposition is used to separate the thermoelastic part of the strain measures of the rod from their plastic counterparts. We then present the most general quadratic form of the Helmholtz energy per unit undeformed length for both hemitropic and transversely isotropic rods. We also propose a prototype yield criterion in terms of forces, moments, and hardening stress resultants, as well as associative flow rules for the evolution of plastic strain measures and hardening variables.

A finite element formulation for a direct approach to elastoplasticity in special Cosserat rods

“…A similar direct theory for an arbitrary spatial deformation of beam-like slender bodies based on a one-dimensional continuum theory was missing in the literature. The same was recently proposed by Smriti et al 19 within the framework of the special Cosserat rod theory. In this work, we present a finite element based numerical scheme to implement their direct theory.…”

“…The constitutive equations for the elastoplastic rod are presented next. We refer the readers to Smriti et al 19 for a detailed discussion. Figure 1 shows a typical rod deforming from its reference configuration which we assume to be the straight state configuration.…”

“…Similarly, the first two components of k 0 represent the components of local bending curvature while the third component represents twist. As discussed by Smriti et al, 19 the material strain measures can be additively decomposed into elastic and plastic parts, that is,…”

“…Similarly, the first two components of k 0 represent the components of local bending curvature while the third component represents twist. As discussed by Smriti et al, 19 the material strain measures can be additively decomposed into elastic and plastic parts, that is, Here, the superscript “ e ” is used to denote the elastic part of total strain, whereas the superscript “ p ” is used to denote the plastic part. Their spatial counterparts can simply be obtained through rotational push‐forward.…”

“…The constitutive equations for the elastoplastic rod are presented next. We refer the readers to Smriti et al 19 for a detailed discussion.…”