Modeling a smooth elastic–inelastic transition with a strongly objective numerical integrator needing no iteration

Abstract

Large deformation evolution equations for elastic distortional deformation and isotropic hardening/softening have been developed that model a smooth elastic–inelastic transition for both rate-independent and rate-dependent response with no need for loading–unloading conditions. A novel special case is a rate-independent overstress model. Specific simplified constitutive equations are proposed that capture the main effects of elastic-plastic and elastic-viscoplastic materials with only a few material parameters. Moreover, a robust and strongly objective numerical integrator for these simplified evolution equations has been developed which needs no iteration. Examples show the influence of the various parameters on the predicted material response. The smoothness of the elastic–inelastic transition in the proposed model, with the associated overstress, tends to spread the inelastic region. This side effect prevents severe deformation from being localized in an element region that continues to reduce in size with mesh refinement. However, preliminary calculations indicate the need for additional modeling of a material characteristic length that independently controls the size of a localized severely deformed region.

Appendix: Details of the numerical implementation

The constitutive equations described in the above sections were implemented into the commercial finite element package—ABAQUS [1] for the numerical investigation of realistic problems. Specifically, ABAQUS allows users to program a user subroutine called UMAT, where it is possible to implement general material constitutive equations. In the UMAT user subroutine, ABAQUS provides the subroutine with the deformation gradient F and the UMAT provides ABAQUS with the Cauchy stress T and the local tangent stiffness. Specifically, with the help of (3.1) and (6.2) the Cauchy stress T and the Kirchhoff stress $\varvec{\uptau }$ are given by

$$\begin{aligned} \mathbf{T}=\text{ K}\left({\text{ J}-1}\right)\mathbf{I}+\text{ J}^{-1}\upmu \mathbf{B}_\mathrm{e}^{\prime \prime }, \varvec{\uptau } =\text{ J}\mathbf{T}. \end{aligned}$$

(10.1)

Taking the material derivative of $\varvec{\uptau }$ it can be shown that

$$\begin{aligned} \text{ J}^{-1}\left[\varvec{{\dot{\uptau }}-\mathbf{W}\varvec{\uptau }-\varvec{\uptau } \mathbf{W}^\mathrm{{T}}}\right]=\mathbf{K}\cdot \mathbf{D}-\left[{\text{ a}_0 +\text{ a}_{1}\left\langle \text{ g}\right\rangle }\right]\mathbf{T}^{\prime }, \end{aligned}$$

(10.2)

where the stiffness tensor K based on the Jauman–Zaremba stress rate is defined so that

$$\begin{aligned} \mathbf{K}\cdot \mathbf{D}&= \mathbf{DT}+\mathbf{TD}+\text{ K}\left({\text{2J}-1} \right)\left({\mathbf{D}\cdot \mathbf{I}}\right)\mathbf{I}-\text{2K}\left({\text{ J}-1}\right)\mathbf{D}\nonumber \\&+\frac{2}{3}\text{ J}^{-1}\upmu \left({\mathbf{B}_{\mathrm{e}}^{\prime } \cdot \mathbf{I}} \right)\mathbf{D}^{\prime }-\frac{2}{3}\left({\mathbf{T}^{\prime }\cdot \mathbf{D}}\right)\mathbf{I}-\frac{2}{3}\left({\mathbf{D}\cdot \mathbf{I}} \right)\mathbf{T}^{\prime }\nonumber \\&-\left[{\left({\frac{2}{3{\dot{\upvarepsilon }}}} \right)\left({\text{ b}_0 +\text{ b}_1 \left\langle \text{ g} \right\rangle }\right)\mathbf{D}^{\prime }\cdot \mathbf{D}}\right]\mathbf{T}^{\prime }. \end{aligned}$$

(10.3)

Moreover, the rectangular Cartesian component $\text{ K}_\mathrm{ijmn}$ of K can be written in terms of the rectangular Cartesian component $\left\{ {\text{ D}_\mathrm{{ij}},\text{ T}_\mathrm{{ij}}, \text{ T}_\mathrm{{ij}}^{\prime }}\right\}$ of $\left\{ {\mathbf{D, T, T}^{{\prime }}}\right\}$ and the Kronecker delta symbol $\updelta _\mathrm{{ij}}$ in the form

$$\begin{aligned} \text{ K}_\mathrm{{ijmn}}&= {\updelta }_\mathrm{{im}} \text{ T}_\mathrm{{nj}} +\text{ T}_\mathrm{{im}} {\updelta }_\mathrm{{nj}} +\text{ K}\left({\text{2J}-1}\right){\updelta }_\mathrm{{ij}}{\updelta }_\mathrm{{mn}}\nonumber \\&-\mathrm{2K}\left({\text{ J}-1}\right){\updelta }_\mathrm{{im}} {\updelta }_\mathrm{{jn}} -\frac{2}{3}\left({{\updelta }_\mathrm{{ij}} \text{ T}_\mathrm{{mn}}^\prime +\text{ T}_\mathrm{{ij}}^\prime {\updelta }_\mathrm{{mn}}}\right)\nonumber \\&+\frac{2}{3}\text{ J}^{-1}\upmu {\alpha }_{1} \left({{\updelta }_\mathrm{{im}} {\updelta }_\mathrm{{jn}} -\frac{1}{3}{\updelta }_\mathrm{{ij}} {\updelta }_\mathrm{{mn}}} \right)\nonumber \\&-\left(\frac{2}{3{\dot{\upvarepsilon }}}\right)\left({\text{ b}_0 +\text{ b}_1 \left\langle \text{ g} \right\rangle } \right)\text{ T}_\mathrm{{ij}}^\prime \left({\text{ D}_\mathrm{{mn}} -\frac{1}{3}\text{ D}_\mathrm{{rr}} {\updelta }_\mathrm{{mn}}}\right).\nonumber \\ \end{aligned}$$

(10.4)

This expression for the stiffness tensor K was motivated by the work in [14] where different approximations of K on the numerical convergence rate were analyzed. Moreover, it is noted that this expression for K does not represent a consistent tangent.