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Incrementally objective implicit integration of hypoelastic–viscoplastic constitutive equations based on the mechanical threshold strength model


The present paper focuses on the development of a fully implicit, incrementally objective integration algorithm for a hypoelastic formulation of \(J_{2}\)-viscoplasticity, which employs the mechanical threshold strength model to compute the material’s flow stress, taking into account its dependence on strain rate and temperature. Heat generation due to high-rate viscoplastic deformation is accounted for, assuming adiabatic conditions. The implementation of the algorithm is discussed, and its performance is assessed in the contexts of implicit and explicit dynamic finite element analysis, with the aid of example problems involving a wide range of loading rates. Computational results are compared to experimental data, showing very good agreement.

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Helpful discussions over the course of this work with Drs. Rick Rauenzahn, Bradford Clements, and Jason Mayeur, of Los Alamos National Laboratory, are greatly appreciated. Funding for this work was provided by the Joint DoD/DOE Munitions Technology Development Program, the Advanced Simulation and Computing program (ASC), the NNSA Science Campaign 2—Dynamic Materials Properties, and the Laboratory Directed Research and Development program (LDRD) at Los Alamos National Laboratory. The authors gratefully acknowledge this support.

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Corresponding author

Correspondence to Hashem M. Mourad.

Appendix: Flow stress calculation and linearization

Appendix: Flow stress calculation and linearization

The purpose of the procedure described in this appendix is to calculate \({{}{\hat{\tau }}}^{(k)}\) and \(\frac{{\mathrm {d}}\,{{}{\hat{\tau }}}^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}\), given \({{}\Delta \lambda }^{(k)}\). It is invoked at the end of the \(k\)th iteration of the Newton–Raphson scheme used to effect radial return mapping, cf. Eqs. (53)–(55). The procedure consists of six main, consecutive steps, listed below.

Step 1: Equivalent viscoplastic strain rate

We begin by using Eq. (46) to approximate the equivalent viscoplastic strain rate:

$$\begin{aligned} {{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}=\min \left( \;\max \left( \sqrt{\tfrac{2}{3}}\;\frac{{{}\Delta \lambda }^{(k)}}{\Delta t},{\dot{\bar{\varepsilon }}}^{\text {vp}}_{\text {min}}\right) ,\;{\dot{\bar{\varepsilon }}}^{\text {vp}}_{\text {max}}\right) . \end{aligned}$$

Consequently, we have

$$\begin{aligned} \frac{{\mathrm {d}}\,{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}=\left\{ \begin{array}{ll} \frac{\sqrt{2/3}}{\Delta t}\qquad \quad &{} \text {if }{\dot{\bar{\varepsilon }}}^{\text {vp}}_{\text {min}}\le \sqrt{\frac{2}{3}}\frac{{{}\Delta \lambda }^{(k)}}{\Delta t}\le {\dot{\bar{\varepsilon }}}^{\text {vp}}_{\text {max}},\\ 0 &{} \text {otherwise}. \end{array}\right. \end{aligned}$$

Step 2: Temperature and specific heat

The temperature and specific heat are determined with the aid of a one-point iteration based on Eqs. (47)–(48). In this iterative setup, we set

$$\begin{aligned}&{{}C_{p}}^{(k,l+1)}=C_{0}+C_{1}\,{{}\theta }^{(k,l)}+C_{2}\,\left( {{}\theta }^{(k,l)}\right) ^{-2}, \end{aligned}$$
$$\begin{aligned}&{{}\theta }^{(k,l+1)}={{}\theta }_{(n)}+\frac{\varPsi }{\rho \,{{}C_{p}}^{(k,l+1)}}\nonumber \\&\quad \quad \quad \quad \quad \times \left( \Vert {{}\mathbf{{S}}}^{(0)}\Vert -2G\,{{}\Delta \lambda }^{(k)}\right) \,{{}\Delta \lambda }^{(k)}, \end{aligned}$$

where \({{}(\cdot )}^{(k,l)}\) denotes the \(l\)th iterate (with \(l=0,1,\dots \)). The iterative procedure is initialized by setting \({{}\theta }^{(k,0)}={{}\theta }_{(n)}\), and continues until \([\,{{}\theta }^{(k,l+1)}-{{}\theta }^{(k,l)}\,]\) becomes smaller than a tolerance, indicating convergence. At that stage, we set \({{}\theta }^{(k)}=\min (\,{{}\theta }^{(k,l+1)},\theta _{\text {melt}}\,)\), preventing the temperature from exceeding its melting point, \(\theta _{\text {melt}}\), of the material. It can also be seen from Eqs. (47)–(48), or equivalently from (73)–(74), that

$$\begin{aligned} {\mathrm {d}}{{}\theta }^{(k)}=\frac{\partial \,{{}\theta }^{(k)}}{\partial \,{{}C_{p}}^{(k)}}{\mathrm {d}}{{}C_{p}}^{(k)}+\frac{\partial \,{{}\theta }^{(k)}}{\partial \,{{}\Delta \lambda }^{(k)}}{\mathrm {d}}{{}\Delta \lambda }^{(k)}, \end{aligned}$$

where \({\mathrm {d}}{{}C_{p}}^{(k)}=\left[ C_{1}-2C_{2}\left( {{}\theta }^{(k)}\right) ^{-3}\right] {\mathrm {d}}{{}\theta }^{(k)}\). Thus,

$$\begin{aligned} \frac{{\mathrm {d}}\,{{}\theta }^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}= \left\{ \begin{array}{ll} 0&{}\text {if }{{}\theta }^{(k)}=\theta _{\text {melt}},\\ \frac{\frac{\partial \,{{}\theta }^{(k)}}{\partial \,{{}\Delta \lambda }^{(k)}}}{1-\frac{\partial \,{{}\theta }^{(k)}}{\partial \,{{}C_{p}}^{(k)}}\left[ C_{1}-2C_{2}\left( {{}\theta }^{(k)}\right) ^{-3}\right] }\qquad &{}\text {otherwise}, \end{array}\right. \end{aligned}$$


$$\begin{aligned} \frac{\partial \,{{}\theta }^{(k)}}{\partial \,{{}C_{p}}^{(k)}}&= \frac{-\varPsi }{\rho \left( {{}C_{p}}^{(k)}\right) ^{2}}\big (\Vert {{}\mathbf{{S}}}^{(0)}\Vert -2G\,{{}\Delta \lambda }^{(k)}\big )\,{{}\Delta \lambda }^{(k)}, \end{aligned}$$
$$\begin{aligned} \frac{\partial \,{{}\theta }^{(k)}}{\partial \,{{}\Delta \lambda }^{(k)}}&= \frac{\varPsi }{\rho \,{{}C_{p}}^{(k)}}\big (\Vert {{}\mathbf{{S}}}^{(0)}\Vert -4G\,{{}\Delta \lambda }^{(k)}\big ). \end{aligned}$$

Step 3: Shear modulus

At this stage, the shear modulus is computed by substituting the known temperature, \({{}\theta }^{(k)}\), into Eq. (22):

$$\begin{aligned} {{}\mu }^{(k)}=\mu _{0}-\frac{D_{0}}{\exp \left( \frac{\theta _{0}}{{{}\theta }^{(k)}}\right) -1}. \end{aligned}$$


$$\begin{aligned} \frac{{\mathrm {d}}\,{{}\mu }^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}=\frac{{\mathrm {d}}\,{{}\mu }^{(k)}}{{\mathrm {d}}\,{{}\theta }^{(k)}}\frac{{\mathrm {d}}\,{{}\theta }^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}, \end{aligned}$$


$$\begin{aligned} \frac{{\mathrm {d}}\,{{}\mu }^{(k)}}{{\mathrm {d}}\,{{}\theta }^{(k)}}=\frac{-D_{0}\theta _{0}\exp \left( \frac{\theta _{0}}{{{}\theta }^{(k)}}\right) }{\left( {{}\theta }^{(k)}\right) ^{2}\left[ \exp \left( \frac{\theta _{0}}{{{}\theta }^{(k)}}\right) -1\right] ^{2}}. \end{aligned}$$

Step 4: Rate- and temperature-dependent pre-multipliers

The pre-multiplying term, \({{}S_{i}}^{(k)}\), is calculated by substituting \({{}\theta }^{(k)}, {{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}\), and \({{}\mu }^{(k)}\) into Eq. (23), i.e.

$$\begin{aligned} {{}S_{i}}^{(k)}=\left( 1-\left[ \frac{k{{}\theta }^{(k)}}{{{}\mu }^{(k)} b^{3} g_{0i}}\log \left( \frac{{\dot{\varepsilon }}_{0i}}{{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}\right) \right] ^{\frac{1}{q_{i}}}\right) ^{\frac{1}{p_{i}}}. \end{aligned}$$

Differentiation and application of the chain rule yield

$$\begin{aligned}&\frac{{\mathrm {d}}\,{{}S_{i}}^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}=\frac{\partial \,{{}S_{i}}^{(k)}}{\partial \,{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}\frac{{\mathrm {d}}\,{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}+\frac{\partial \,{{}S_{i}}^{(k)}}{\partial \,{{}\theta }^{(k)}}\frac{{\mathrm {d}}\,{{}\theta }^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}\nonumber \\&\qquad \qquad \qquad +\frac{\partial \,{{}S_{i}}^{(k)}}{\partial \,{{}\mu }^{(k)}}\frac{{\mathrm {d}}\,{{}\mu }^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}, \end{aligned}$$

where the partial derivatives are given by

$$\begin{aligned}&\frac{\partial \,{{}S_{i}}^{(k)}}{\partial \,{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}=\frac{{{}\fancyscript{D}_{i}}^{(k)}}{\log \left( \frac{{\dot{\varepsilon }}_{0i}}{{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}\right) {{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}, \end{aligned}$$
$$\begin{aligned}&\frac{\partial \,{{}S_{i}}^{(k)}}{\partial \,{{}\theta }^{(k)}}=\frac{-{{}\fancyscript{D}_{i}}^{(k)}}{{{}\theta }^{(k)}}, \end{aligned}$$
$$\begin{aligned}&\frac{\partial \,{{}S_{i}}^{(k)}}{\partial \,{{}\mu }^{(k)}}=\frac{{{}\fancyscript{D}_{i}}^{(k)}}{{{}\mu }^{(k)}}, \end{aligned}$$

in terms of

$$\begin{aligned} {{}\fancyscript{D}_{i}}^{(k)}{{}:={}}\frac{{{}S_{i}}^{(k)}\left[ \frac{k{{}\theta }^{(k)}}{{{}\mu }^{(k)}b^{3} g_{0i}}\log \left( \frac{{\dot{\varepsilon }}_{0i}}{{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}\right) \right] ^{\frac{1}{q_{i}}}}{p_{i}q_{i}\left( 1-\left[ \frac{k{{}\theta }^{(k)}}{{{}\mu }^{(k)}b^{3} g_{0i}}\log \left( \frac{{\dot{\varepsilon }}_{0i}}{{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}\right) \right] ^{\frac{1}{q_{i}}}\right) }. \end{aligned}$$

Similarly for \({{}S_{\varepsilon }}^{(k)}\), using Eq. (24):

$$\begin{aligned} {{}S_{\varepsilon }}^{(k)}=\left( 1-\left[ \frac{k{{}\theta }^{(k)}}{{{}\mu }^{(k)} b^{3} g_{0\varepsilon }}\log \left( \frac{{\dot{\varepsilon }}_{0\varepsilon }}{{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}\right) \right] ^{\frac{1}{q_{\varepsilon }}}\right) ^{\frac{1}{p_{\varepsilon }}}, \end{aligned}$$

and we have

$$\begin{aligned}&\frac{{\mathrm {d}}\,{{}S_{\varepsilon }}^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}=\frac{\partial \,{{}S_{\varepsilon }}^{(k)}}{\partial \,{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}\frac{{\mathrm {d}}\,{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}+\frac{\partial \,{{}S_{\varepsilon }}^{(k)}}{\partial \,{{}\theta }^{(k)}}\frac{{\mathrm {d}}\,{{}\theta }^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}\nonumber \\&\qquad \qquad \qquad +\frac{\partial \,{{}S_{\varepsilon }}^{(k)}}{\partial \,{{}\mu }^{(k)}}\frac{{\mathrm {d}}\,{{}\mu }^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}, \end{aligned}$$


$$\begin{aligned}&\frac{\partial \,{{}S_{\varepsilon }}^{(k)}}{\partial \,{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}=\frac{{{}\fancyscript{D}_{\varepsilon }}^{(k)}}{\log \left( \frac{{\dot{\varepsilon }}_{0\varepsilon }}{{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}\right) {{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}, \end{aligned}$$
$$\begin{aligned}&\frac{\partial \,{{}S_{\varepsilon }}^{(k)}}{\partial \,{{}\theta }^{(k)}}=\frac{-{{}\fancyscript{D}_{\varepsilon }}^{(k)}}{{{}\theta }^{(k)}}, \end{aligned}$$
$$\begin{aligned}&\frac{\partial \,{{}S_{\varepsilon }}^{(k)}}{\partial \,{{}\mu }^{(k)}}=\frac{{{}\fancyscript{D}_{\varepsilon }}^{(k)}}{{{}\mu }^{(k)}}, \end{aligned}$$


$$\begin{aligned} {{}\fancyscript{D}_{\varepsilon }}^{(k)}{{}:={}}\frac{{{}S_{\varepsilon }}^{(k)}\left[ \frac{k{{}\theta }^{(k)}}{{{}\mu }^{(k)}b^{3} g_{0\varepsilon }}\log \left( \frac{{\dot{\varepsilon }}_{0\varepsilon }}{{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}\right) \right] ^{\frac{1}{q_{\varepsilon }}}}{p_{\varepsilon }q_{\varepsilon }\left( 1-\left[ \frac{k{{}\theta }^{(k)}}{{{}\mu }^{(k)}b^{3} g_{0\varepsilon }}\log \left( \frac{{\dot{\varepsilon }}_{0\varepsilon }}{{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}\right) \right] ^{\frac{1}{q_{\varepsilon }}}\right) }. \end{aligned}$$

Step 5: Structure-dependent MTS

In order to calculate the structure-dependent MTS, \({{}{\hat{\tau }}_{\varepsilon }}^{(k)}\), we begin by substituting \({{}\theta }^{(k)}, {{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}\) into (51), which gives

$$\begin{aligned} {{}h_{0}}^{(k)}&= A_{0}+A_{1}\log \left( {{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}\right) +A_{2}\sqrt{{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}\nonumber \\&-A_{3}{{}\theta }^{(k)}+A_{4}\left( {{}\theta }^{(k)}\right) ^{-A_{5}}, \end{aligned}$$


$$\begin{aligned} \frac{{\mathrm {d}}\,{{}h_{0}}^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}=\frac{\partial \,{{}h_{0}}^{(k)}}{\partial \,{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}\,\frac{{\mathrm {d}}\,{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}+\frac{\partial \,{{}h_{0}}^{(k)}}{\partial \,{{}\theta }^{(k)}}\,\frac{{\mathrm {d}}\,{{}\theta }^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}, \end{aligned}$$


$$\begin{aligned} \frac{\partial \,{{}h_{0}}^{(k)}}{\partial \,{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}&= \frac{A_{1}}{{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}+\frac{1}{2}\frac{A_{2}}{\sqrt{{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}},\end{aligned}$$
$$\begin{aligned} \frac{\partial \,{{}h_{0}}^{(k)}}{\partial \,{{}\theta }^{(k)}}&= -A_{3}-A_{4}A_{5}\left( {{}\theta }^{(k)}\right) ^{-(A_{5}+1)}. \end{aligned}$$

Similarly, using (52), we evaluate

$$\begin{aligned} {{}{\hat{\tau }}_{\varepsilon s}}^{(k)}={\hat{\tau }}_{\varepsilon s0}\left( \frac{{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}{{\dot{\varepsilon }}_{0\varepsilon s}}\right) ^{\frac{k{{}\theta }^{(k)}}{{{}\mu }^{(k)} b^{3}g_{0\varepsilon s}}}, \end{aligned}$$


$$\begin{aligned} \frac{{\mathrm {d}}\,{{}{\hat{\tau }}_{\varepsilon s}}^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}&= \frac{\partial \,{{}{\hat{\tau }}_{\varepsilon s}}^{(k)}}{\partial \,{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}\,\frac{{\mathrm {d}}\,{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}+\frac{\partial \,{{}{\hat{\tau }}_{\varepsilon s}}^{(k)}}{\partial \,{{}\theta }^{(k)}}\,\frac{{\mathrm {d}}\,{{}\theta }^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}\nonumber \\&+\frac{\partial \,{{}{\hat{\tau }}_{\varepsilon s}}^{(k)}}{\partial \,{{}\mu }^{(k)}}\,\frac{{\mathrm {d}}\,{{}\mu }^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}, \end{aligned}$$


$$\begin{aligned} \frac{\partial \,{{}{\hat{\tau }}_{\varepsilon s}}^{(k)}}{\partial \,{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}&= {{}{\hat{\tau }}_{\varepsilon s}}^{(k)}\frac{k{{}\theta }^{(k)}}{{{}\mu }^{(k)}b^{3}g_{0\varepsilon s}}\frac{1}{{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}},\end{aligned}$$
$$\begin{aligned} \frac{\partial \,{{}{\hat{\tau }}_{\varepsilon s}}^{(k)}}{\partial \,{{}\theta }^{(k)}}&= {{}{\hat{\tau }}_{\varepsilon s}}^{(k)}\frac{k}{{{}\mu }^{(k)}b^{3}g_{0\varepsilon s}}\log \left( \frac{{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}{{\dot{\varepsilon }}_{0\varepsilon s}}\right) ,\end{aligned}$$
$$\begin{aligned} \frac{\partial \,{{}{\hat{\tau }}_{\varepsilon s}}^{(k)}}{\partial \,{{}\mu }^{(k)}}&= {{}{\hat{\tau }}_{\varepsilon s}}^{(k)}\frac{-k{{}\theta }^{(k)}}{\left( {{}\mu }^{(k)}\right) ^{2}b^{3}g_{0\varepsilon s}}\log \left( \frac{{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}{{\dot{\varepsilon }}_{0\varepsilon s}}\right) . \end{aligned}$$

Then, \({{}{\hat{\tau }}_{\varepsilon }}^{(k)}\) is determined iteratively using Newton’s method. In this context, the solution is updated via

$$\begin{aligned} {{}{\hat{\tau }}_{\varepsilon }}^{(k,l+1)}={{}{\hat{\tau }}_{\varepsilon }}^{(k,l)}-\frac{{{}\fancyscript{R}_{\varepsilon }}^{(k,l)}}{{{}\fancyscript{R}_{\varepsilon }^{\prime }}^{(k,l)}}, \end{aligned}$$

where \({{}(\cdot )}^{(k,l)}\) denotes the \(l\)th iterate, \(l=0,1,\dots \), and in light of (49)–(50), the residual is defined as

$$\begin{aligned} {{}\fancyscript{R}_{\varepsilon }}^{(k,l)}{{}:={}}{{}{\hat{\tau }}_{\varepsilon }}^{(k,l)}-\left[ {{}{\hat{\tau }}_{\varepsilon }}_{(n)}+{{}\left[ \frac{{\mathrm {d}}\,{\hat{\tau }}_{\varepsilon }}{{\mathrm {d}}\,{\bar{\varepsilon }}^{\text {vp}}}\right] }^{(k,l)}\left( \sqrt{\tfrac{2}{3}}\,{{}\Delta \lambda }^{(k)}\right) \right] ,\nonumber \\ \end{aligned}$$

in terms of

$$\begin{aligned} {{}\left[ \frac{{\mathrm {d}}\,{\hat{\tau }}_{\varepsilon }}{{\mathrm {d}}\,{\bar{\varepsilon }}^{\text {vp}}}\right] }^{(k,l)}={{}h_{0}}^{(k)}\left( 1-\frac{{{}{\hat{\tau }}_{\varepsilon }}^{(k,l)}}{{{}{\hat{\tau }}_{\varepsilon s}}^{(k)}}\right) ^{\kappa }. \end{aligned}$$

Thus, we also have

$$\begin{aligned}&{{}\fancyscript{R}_{\varepsilon }^{\prime }}^{(k,l)}{{}:={}}\,\frac{{\mathrm {d}}\,{{}\fancyscript{R}_{\varepsilon }}^{(k,l)}}{{\mathrm {d}}\,{{}{\hat{\tau }}_{\varepsilon }}^{(k,l)}} =\,1+\bigg [\frac{{{}h_{0}}^{(k)}\,\kappa }{{{}{\hat{\tau }}_{\varepsilon s}}^{(k)}}\left( 1-\frac{{{}{\hat{\tau }}_{\varepsilon }}^{(k,l)}}{{{}{\hat{\tau }}_{\varepsilon s}}^{(k)}}\right) ^{(\kappa -1)}\nonumber \\&\qquad \qquad \qquad \times \left( \sqrt{\tfrac{2}{3}}\,{{}\Delta \lambda }^{(k)}\right) \bigg ]. \end{aligned}$$

Initialization is performed by setting \({{}{\hat{\tau }}_{\varepsilon }}^{(k,0)}={{}{\hat{\tau }}_{\varepsilon }}_{(n)}\), and the iterative process is terminated when \(|{{}\fancyscript{R}_{\varepsilon }}^{(k,l)}|\) becomes smaller than a tolerance. At that point, we set \({{}{\hat{\tau }}_{\varepsilon }}^{(k)}={{}{\hat{\tau }}_{\varepsilon }}^{(k,l)}\) and \({{}\left[ \frac{{\mathrm {d}}\,{\hat{\tau }}_{\varepsilon }}{{\mathrm {d}}\,{\bar{\varepsilon }}^{\text {vp}}}\right] }^{(k)}={{}\left[ \frac{{\mathrm {d}}\,{\hat{\tau }}_{\varepsilon }}{{\mathrm {d}}\,{\bar{\varepsilon }}^{\text {vp}}}\right] }^{(k,l)}\). It can also be seen from Eq. (49) that

$$\begin{aligned} \frac{{\mathrm {d}}\,{{}{\hat{\tau }}_{\varepsilon }}^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}=\frac{{\mathrm {d}}\,{{}\left[ \frac{{\mathrm {d}}\,{\hat{\tau }}_{\varepsilon }}{{\mathrm {d}}\,{\bar{\varepsilon }}^{\text {vp}}}\right] }^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}\sqrt{\tfrac{2}{3}}\,{{}\Delta \lambda }^{(k)}+{{}\left[ \frac{{\mathrm {d}}\,{\hat{\tau }}_{\varepsilon }}{{\mathrm {d}}\,{\bar{\varepsilon }}^{\text {vp}}}\right] }^{(k)}\sqrt{\tfrac{2}{3}}, \end{aligned}$$

and from (50),

$$\begin{aligned} \frac{{\mathrm {d}}\,{{}\left[ \frac{{\mathrm {d}}\,{\hat{\tau }}_{\varepsilon }}{{\mathrm {d}}\,{\bar{\varepsilon }}^{\text {vp}}}\right] }^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}&= \left( 1-\frac{{{}{\hat{\tau }}_{\varepsilon }}^{(k)}}{{{}{\hat{\tau }}_{\varepsilon s}}^{(k)}}\right) ^{\kappa }\frac{{\mathrm {d}}\,{{}h_{0}}^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}\nonumber \\&+{{}\fancyscript{D}}^{(k)}_{{\hat{\tau }}_{\varepsilon }}\left( \!\frac{{{}{\hat{\tau }}_{\varepsilon }}^{(k)}}{{{}{\hat{\tau }}_{\varepsilon s}}^{(k)}}\frac{{\mathrm {d}}\,{{}{\hat{\tau }}_{\varepsilon s}}^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}-\frac{{\mathrm {d}}\,{{}{\hat{\tau }}_{\varepsilon }}^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}\!\right) , \end{aligned}$$


$$\begin{aligned} {{}\fancyscript{D}}^{(k)}_{{\hat{\tau }}_{\varepsilon }}=\frac{{{}h_{0}}^{(k)}\,\kappa }{{{}{\hat{\tau }}_{\varepsilon s}}^{(k)}}\left( 1-\frac{{{}{\hat{\tau }}_{\varepsilon }}^{(k)}}{{{}{\hat{\tau }}_{\varepsilon s}}^{(k)}}\right) ^{(\kappa -1)}. \end{aligned}$$


$$\begin{aligned}&\frac{{\mathrm {d}}\,{{}{\hat{\tau }}_{\varepsilon }}^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}\nonumber \\&\quad =\left( \left[ \left( 1-\frac{{{}{\hat{\tau }}_{\varepsilon }}^{(k)}}{{{}{\hat{\tau }}_{\varepsilon s}}^{(k)}}\right) ^{\kappa }\frac{{\mathrm {d}}\,{{}h_{0}}^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}+{{}\fancyscript{D}}^{(k)}_{{\hat{\tau }}_{\varepsilon }}\frac{{{}{\hat{\tau }}_{\varepsilon }}^{(k)}}{{{}{\hat{\tau }}_{\varepsilon s}}^{(k)}}\frac{{\mathrm {d}}\,{{}{\hat{\tau }}_{\varepsilon s}}^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}\right] \right. \nonumber \\&\qquad \left. \times \sqrt{\tfrac{2}{3}}{{}\Delta \lambda }^{(k)}\!+\!{{}\left[ \frac{{\mathrm {d}}\,{\hat{\tau }}_{\varepsilon }}{{\mathrm {d}}\,{\bar{\varepsilon }}^{\text {vp}}}\right] }^{(k)}\sqrt{\tfrac{2}{3}}\right) \!\bigg /\!\left( 1\!+\!{{}\fancyscript{D}}^{(k)}_{{\hat{\tau }}_{\varepsilon }}\sqrt{\tfrac{2}{3}}{{}\Delta \lambda }^{(k)}\!\right) .\nonumber \\ \end{aligned}$$

Step 6: Flow stress

Finally, the flow stress is evaluated using [cf. Eqs. (21), (45)]:

$$\begin{aligned} {{}{\hat{\tau }}}^{(k)}={\hat{\tau }}_{a}+\frac{{{}\mu }^{(k)}}{\mu _{0}}\left[ {{}S_{i}}^{(k)}{\hat{\tau }}_{i}+{{}S_{\varepsilon }}^{(k)}{{}{\hat{\tau }}_{\varepsilon }}^{(k)}\right] . \end{aligned}$$

Differentiation and use of the chain rule lead to:

$$\begin{aligned} \frac{{\mathrm {d}}\,{{}{\hat{\tau }}}^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}&= \frac{\partial \,{{}{\hat{\tau }}}^{(k)}}{\partial \,{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}\,\frac{{\mathrm {d}}\,{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}\nonumber \\&+\frac{\partial \,{{}{\hat{\tau }}}^{(k)}}{\partial \,{{}\theta }^{(k)}}\,\frac{{\mathrm {d}}\,{{}\theta }^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}\nonumber \\&+\frac{\partial \,{{}{\hat{\tau }}}^{(k)}}{\partial \,{{}\mu }^{(k)}}\,\frac{{\mathrm {d}}\,{{}\mu }^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}} +\frac{\partial \,{{}{\hat{\tau }}}^{(k)}}{\partial \,{{}{\hat{\tau }}_{\varepsilon }}^{(k)}}\,\frac{{\mathrm {d}}\,{{}{\hat{\tau }}_{\varepsilon }}^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}, \end{aligned}$$


$$\begin{aligned}&\frac{\partial \,{{}{\hat{\tau }}}^{(k)}}{\partial \,{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}=\frac{{{}\mu }^{(k)}}{\mu _{0}}\left[ \frac{\partial \,{{}S_{i}}^{(k)}}{\partial \,{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}\,{\hat{\tau }}_{i}+\frac{\partial \,{{}S_{\varepsilon }}^{(k)}}{\partial \,{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}{{}{\hat{\tau }}_{\varepsilon }}^{(k)}\right] , \end{aligned}$$
$$\begin{aligned}&\frac{\partial \,{{}{\hat{\tau }}}^{(k)}}{\partial \,{{}\theta }^{(k)}}=\frac{{{}\mu }^{(k)}}{\mu _{0}}\left[ \frac{\partial \,{{}S_{i}}^{(k)}}{\partial \,{{}\theta }^{(k)}}\,{\hat{\tau }}_{i}+\frac{\partial \,{{}S_{\varepsilon }}^{(k)}}{\partial \,{{}\theta }^{(k)}}{{}{\hat{\tau }}_{\varepsilon }}^{(k)}\right] , \end{aligned}$$
$$\begin{aligned}&\frac{\partial \,{{}{\hat{\tau }}}^{(k)}}{\partial \,{{}\mu }^{(k)}}=\frac{1}{\mu _{0}}\left[ {{}S_{i}}^{(k)}\,{\hat{\tau }}_{i}+{{}S_{\varepsilon }}^{(k)}{{}{\hat{\tau }}_{\varepsilon }}^{(k)}\right] \end{aligned}$$
$$\begin{aligned}&\frac{\partial \,{{}{\hat{\tau }}}^{(k)}}{\partial \,{{}{\hat{\tau }}_{\varepsilon }}^{(k)}}=\frac{{{}\mu }^{(k)}}{\mu _{0}}\,{{}S_{\varepsilon }}^{(k)}. \end{aligned}$$

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Mourad, H.M., Bronkhorst, C.A., Addessio, F.L. et al. Incrementally objective implicit integration of hypoelastic–viscoplastic constitutive equations based on the mechanical threshold strength model. Comput Mech 53, 941–955 (2014).

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  • Viscoplasticity
  • Mechanical threshold strength
  • MTS model
  • Finite deformation
  • Objective integration
  • Radial return mapping
  • Finite element method