A stabilized, symmetric Nitsche method for spatially localized plasticity

Abstract

A heterogeneous interface method is developed for combining primal displacement and mixed displacement-pressure formulations across nonconforming finite element meshes to treat volume-preserving plastic flow. When the zone of inelastic response is localized within a larger domain, significant computational savings can be achieved by confining the mixed formulation solely to the localized region. The method’s distinguishing feature is that the coupling terms for joining dissimilar element types are derived from a time-discrete free energy functional, which is based on a Lagrange multiplier formulation of the interface constraints. Incorporating residual-based stabilizing terms at the interface enables the condensation of the multiplier field, leading to a symmetric Nitsche formulation in which the interface operators respect the differing character of the governing equations in each region. In a series of numerical problems, the heterogeneous interface method achieved comparable results on coarser meshes as those obtained from applying the mixed formulation throughout the domain.

Appendices

Appendix 1: Stationary conditions

A sketch of a proof is provided to show that the stationary conditions of the discrete total free energy functional (25) are equivalent to the standard Galerkin weak form (19)–(21) combined with the constitutive laws (22)–(24). Note that the non-smooth nature of the consistency condition (9) suggests that a rigorous proof requires techniques for variational inequalities or constrained optimization; see [2].

These stationary conditions are defined through the variation of $\widehat{\mathcal {P}}_n$ with respect to each primary field:

$$\begin{aligned} \hbox {D}_u \widehat{\mathcal {P}}_n \cdot {\varvec{\eta }} ^{h}+\hbox {D}_p \widehat{\mathcal {P}}_n \cdot q^{h}+\hbox {D}_\lambda \widehat{\mathcal {P}}_n \cdot {\varvec{\mu }} ^{h}=0 \end{aligned}$$

(65)

The variations of (26)–(29) with respect to the displacement field are obtained as follows, accounting for the symmetry of $\overline{\mathbf{C}}$ and $\mathbf{D}$:

$$\begin{aligned} \hbox {D}_u \mathcal {P}_{\mathrm{int}} \cdot {\varvec{\eta }}^{h}= & {} \int _\Omega \left[ {{\varvec{\varepsilon }}\left( {{\varvec{\eta }}^{h}} \right) -\hbox {D}_u \left( {\Delta \gamma _{n+1} {\varvec{r}}_{n+1} } \right) \cdot {\varvec{\eta }} ^{h}} \right] :\nonumber \\&\overline{\mathbf{C}}:\left[ {{\varvec{\varepsilon }} \left( {{\varvec{u}}_{n+1}^h } \right) -{\varvec{\varepsilon }}_n^p -\Delta \gamma _{n+1} {\varvec{r}}_{n+1} } \right] \;\hbox {d}V\nonumber \\&+\int _\Omega {{\varvec{\varepsilon }} \left( {{\varvec{\eta }} ^{h}} \right) :p_{n+1}^h {\varvec{1}}\;\hbox {d}V} \nonumber \\&+\int _\Omega {\left[ {-\hbox {D}_u \left( {\Delta \gamma _{n+1} {\varvec{h}}_{n+1} } \right) \cdot {\varvec{\eta }}^{h}} \right] \cdot \mathbf{D}^{-1} }\nonumber \\&\ \cdot \left[ {{\varvec{q}}_n -\Delta \gamma _{n+1} {\varvec{h}}_{n+1} } \right] \;\hbox {d}V \end{aligned}$$

(66)

$$\begin{aligned} \hbox {D}_u {\mathcal {P}}_{\mathrm{ext}} \cdot {\varvec{\eta }}^{h}=-\int _\Omega {{\varvec{\eta }} ^{h}\cdot {\varvec{b}}_{n+1}^h \;\hbox {d}V} -\int _{\Gamma _h } {{\varvec{\eta }} ^{h}\cdot {\varvec{t}}_{n+1}^h \;\hbox {d}V} \end{aligned}$$

(67)

$$\begin{aligned} \hbox {D}_u \mathcal {P}_I \cdot {\varvec{\eta }} ^{h}= & {} -\int _{\Gamma _I } {{\varvec{\lambda }} _{n+1}^h \cdot \llbracket {{\varvec{\eta }}^{h}} \rrbracket \;\hbox {d}A} \end{aligned}$$

(68)

$$\begin{aligned}&\hbox {D}_u \Delta \mathcal{L}^{p}\cdot {\varvec{\eta }}^{h}\nonumber \\&\quad =\int _\Omega \left[ {\hbox {D}_u \left( {\Delta \gamma _{n+1} {\varvec{r}}_{n+1} } \right) \cdot {\varvec{\eta }}^{h}} \right] :\overline{\mathbf{C}}:\nonumber \\&\qquad \left[ {{\varvec{\varepsilon }} \left( {{\varvec{u}}_{n+1}^h } \right) -{\varvec{\varepsilon }}_n^p -\Delta \gamma _{n+1} {\varvec{r}}_{n+1} } \right] \;\hbox {d}V\nonumber \\&\qquad +\int _\Omega \left[ {\Delta \gamma _{n+1} {\varvec{r}}_{n+1} } \right] :\overline{\mathbf{C}}:\nonumber \\&\qquad \left[ {{\varvec{\varepsilon }} \left( {{\varvec{\eta }}^{h}} \right) -\hbox {D}_u \left( {\Delta \gamma _{n+1} {\varvec{r}}_{n+1} } \right) \cdot {\varvec{\eta }}^{h}} \right] \;\hbox {d}V \nonumber \\&\qquad -\int _\Omega \left[ {-\hbox {D}_u \left( {\Delta \gamma _{n+1} {\varvec{h}}_{n+1} } \right) \cdot {\varvec{\eta }}^{h}} \right] \cdot \mathbf{D}^{-1} \nonumber \\&\qquad \cdot \left[ {-\Delta \gamma _{n+1} {\varvec{h}}_{n+1} } \right] \;\hbox {d}V-\int _\Omega \left[ {{\varvec{q}}_n -\Delta \gamma _{n+1} {\varvec{h}}_{n+1} } \right] \cdot \mathbf{D}^{-1}\nonumber \\&\qquad \cdot \left[ -\hbox {D}_u \left( {\Delta \gamma _{n+1} {\varvec{h}}_{n+1} } \right) \cdot {\varvec{\eta }}^{h} \right] \;\hbox {d}V \nonumber \\&\qquad -\int _\Omega {\Delta \gamma _{n+1} \left( {\hbox {D}_u f_{n+1} \cdot {\varvec{\eta }}^{h}} \right) \;\hbox {d}V}\nonumber \\&\qquad -\int _\Omega {\left( {\hbox {D}_u \Delta \gamma _{n+1} \cdot {\varvec{\eta }}^{h}} \right) f_{n+1} \;\hbox {d}V} \end{aligned}$$

(69)

The variation of (26) with respect to the pressure field is given by:

$$\begin{aligned} \hbox {D}_p \mathcal {P}_{\mathrm{int}} \cdot q^{h}=\int _\Omega {{\varvec{\varepsilon }} \left( {{\varvec{u}}_{n+1}^h } \right) :q^{h}{\varvec{1}}\;\hbox {d}V} -\int _\Omega {{q^{h}p_{n+1}^h }/K\;\hbox {d}V}\nonumber \\ \end{aligned}$$

(70)

Lastly, the variation of (28) with respect to the Lagrange multiplier field follows as:

$$\begin{aligned} \hbox {D}_\lambda \mathcal {P}_I \cdot {\varvec{\mu }}^{h}=-\int _{\Gamma _I } {{\varvec{\mu }}^{h}\cdot \llbracket {{\varvec{u}}_{n+1}^h } \rrbracket \;\hbox {d}A} \end{aligned}$$

(71)

Focusing on the variation of the time-discrete yield function $f_{n+1}$ in (69), applying the chain rule to definition (9) and accounting for the flow rule and hardening law (22)–(24) along with the associate plasticity conditions (10)–(11) results in the expression below:

$$\begin{aligned} \hbox {D}_u f_{n+1} \cdot {\varvec{\eta }} ^{h}= & {} \frac{\partial f_{n+1} }{\partial {\varvec{\sigma }} _{n+1} }:\hbox {D}_u {\varvec{\sigma }}_{n+1} \cdot {\varvec{\eta }} ^{h}\!+\!\frac{\partial f_{n+1} }{\partial {\varvec{q}}_{n+1} }\cdot \hbox {D}_u {\varvec{q}}_{n+1} \cdot {\varvec{\eta }}^{h} \nonumber \\= & {} {\varvec{r}}_{n+1} :\overline{\mathbf{C}}:\left[ {{\varvec{\varepsilon }} \left( {{\varvec{\eta }} ^{h}} \right) -\hbox {D}_u \left( {\Delta \gamma _{n+1} {\varvec{r}}_{n+1} } \right) \cdot {\varvec{\eta }}^{h}} \right] \nonumber \\&-\,{\varvec{h}}_{n+1} \cdot \mathbf{D}^{-1}\cdot \left[ {\hbox {D}_u \left( {\Delta \gamma _{n+1} {\varvec{h}}_{n+1} } \right) \cdot {\varvec{\eta }}^{h}} \right] \end{aligned}$$

(72)

Substituting (72) into (69) and cancelling like terms yields:

$$\begin{aligned} \hbox {D}_u \Delta \mathcal{L}^{p}\cdot {\varvec{\eta }}^{h}= & {} \int _\Omega \left[ \hbox {D}_u \left( {\Delta \gamma _{n+1} {\varvec{r}}_{n+1} } \right) \cdot {\varvec{\eta }}^{h}\right] :\overline{\mathbf{C}}:\nonumber \\&\left[ {{\varvec{\varepsilon }} \left( {{\varvec{u}}_{n+1}^h } \right) -{\varvec{\varepsilon }}_n^p -\Delta \gamma _{n+1} {\varvec{r}}_{n+1} } \right] \;\hbox {d}V \nonumber \\&-\int _\Omega \left[ {{\varvec{q}}_n -\Delta \gamma _{n+1} {\varvec{h}}_{n+1} } \right] \cdot \mathbf{D}^{-1}\nonumber \\&\cdot \left[ {-\hbox {D}_u \left( {\Delta \gamma _{n+1} {\varvec{h}}_{n+1} } \right) \cdot {\varvec{\eta }}^{h}} \right] \;\hbox {d}V \nonumber \\&-\int _\Omega {\left( {\hbox {D}_u \Delta \gamma _{n+1} \cdot {\varvec{\eta }} ^{h}} \right) f_{n+1} \;\hbox {d}V} \end{aligned}$$

(73)

Combining (73) with (66)–(68) provides the variation of $\widehat{\mathcal {P}}_n$ with respect ${\varvec{u}}_{n+1}^h$:

$$\begin{aligned}&\hbox {D}_u \left( {\mathcal {P}_{\mathrm{int}} +\mathcal {P}_{\mathrm{ext}} +\mathcal {P}_I +\Delta \mathcal{L}^{p}} \right) \cdot {\varvec{\eta }}^{h}=\int _\Omega \left[ {{\varvec{\varepsilon }} \left( {{\varvec{\eta }}^{h}} \right) } \right] :\nonumber \\&\qquad \overline{\mathbf{C}}:\left[ {{\varvec{\varepsilon }} \left( {{\varvec{u}}_{n+1}^h } \right) -{\varvec{\varepsilon }}_n^p -\Delta \gamma _{n+1} {\varvec{r}}_{n+1} } \right] \;\hbox {d}V \nonumber \\&\quad +\int _\Omega {{\varvec{\varepsilon }} \left( {{\varvec{\eta }}^{h}} \right) :p_{n+1}^h {\varvec{1}}\;\hbox {d}V} -\int _\Omega {{\varvec{\eta }}^{h}\cdot {\varvec{b}}_{n+1}^h \;\hbox {d}V} \nonumber \\&\quad -\int _{\Gamma _h } {{\varvec{\eta }}^{h}\cdot {\varvec{t}}_{n+1}^h \;\hbox {d}V} -\int _{\Gamma _I } {{\varvec{\lambda }}_{n+1}^h \cdot \llbracket {{\varvec{\eta }}^{h}} \rrbracket \;\hbox {d}A} \nonumber \\&\quad -\int _\Omega {\left( {\hbox {D}_u \Delta \gamma _{n+1} \cdot {\varvec{\eta }}^{h}} \right) f_{n+1} \;\hbox {d}V} \end{aligned}$$

(74)

Finally, taking note that either $f_{n+1} =0$ or $\Delta \gamma _{n+1} =\hbox {D}_u \Delta \gamma _{n+1} \cdot {\varvec{\eta }}^{h}=0$ according to the consistency condition (9), the last term in (74) vanishes identically.

Hence, the stationary conditions of $\widehat{\mathcal {P}}_n$ subject to the constraints (22)–(24) are given by (74), (70), and (71), which are identical to the weak form (19), (20), and (21), thereby completing the proof.

Appendix 2: Consistent linearization

We briefly summarize the consistent linearization of the elasto-plastic interface method (39)–(40) for use in nonlinear solution strategies such as the Newton–Raphson algorithm. Taking the variational derivative of (39) yields the following expression:

$$\begin{aligned}&K_{uu} \left( {{\varvec{\eta }} ^{h},\Delta {\varvec{u}}^{h};{\varvec{u}}_{n+1}^h } \right) :=\hbox {D}_u \left[ {R_u \left( {{\varvec{\eta }}^{h};{\varvec{u}}_{n+1}^h ,p_{n+1}^h } \right) } \right] \cdot \Delta {\varvec{u}}^{h} \nonumber \\&\quad =\sum _{\alpha =1}^2 {\int _{\Omega ^{(\alpha )}} {{\varvec{\varepsilon }} \left( {{\varvec{\eta }}^{(\alpha )}} \right) : \overline{\mathbf{C}}_{n+1}^{(\alpha )} : {\varvec{\varepsilon }} \left( {\Delta {\varvec{u}}^{(\alpha )}} \right) \;\hbox {d}V} } \nonumber \\&\qquad -\sum _{\alpha =1}^2 {\int _{\Gamma _I } {\llbracket {{\varvec{\eta }}^{h}} \rrbracket \cdot \hbox {D}_u \left[ {\left\{ {{\varvec{\sigma }}_{n+1} \cdot {\varvec{n}}} \right\} } \right] \cdot \Delta {\varvec{u}}^{(\alpha )}\;\hbox {d}A} } \nonumber \\&\qquad -\sum _{\alpha =1}^2 \int _{\Gamma _I } \left\{ {\left( {\overline{\mathbf{C}}_{n+1} : {\varvec{\varepsilon }} \left( {{\varvec{\eta }}^{h}} \right) } \right) \cdot {\varvec{n}}} \right\} \cdot \hbox {D}_u \left[ {\llbracket {{\varvec{u}}_{n+1}^h } \rrbracket } \right] \nonumber \\&\qquad \cdot \, \Delta {\varvec{u}}^{(\alpha )}\;\hbox {d}A \nonumber \\&\qquad -\sum _{\alpha =1}^2 \int _{\Gamma _I } \left\{ {\left( {\left( {\hbox {D}_u \left[ {\overline{\mathbf{C}}_{n+1} } \right] \cdot \Delta {\varvec{u}}^{(\alpha )}} \right) : {\varvec{\varepsilon }} \left( {{\varvec{\eta }}^{h}} \right) } \right) \cdot {\varvec{n}}} \right\} \nonumber \\&\qquad \cdot \, \llbracket {{\varvec{u}}_{n+1}^h } \rrbracket \;\hbox {d}A \nonumber \\&\qquad +\sum _{\alpha =1}^2 {\int _{\Gamma _I } {\llbracket {{\varvec{\eta }}^{h}} \rrbracket \cdot {\varvec{\tau }}_s \cdot \hbox {D}_u \left[ {\llbracket {{\varvec{u}}_{n+1}^h } \rrbracket } \right] \cdot \Delta {\varvec{u}}^{(\alpha )}\;\hbox {d}A} } \end{aligned}$$

(75)

The linearization of the average stress terms mirrors (36) by replacing ${\varvec{\eta }}^{h}$ with $\Delta {\varvec{u}}^{h}$, and the jump term follows simply by replacing ${\varvec{u}}_{n+1}^h$ with $\Delta {\varvec{u}}^{h}$ due to linearity. However, the last term in (75) involves the linearization of the algorithmic tangent moduli $\overline{\mathbf{C}}_{n+1}$. Proceeding as in [24], this contribution yields the 6th order tensor of algorithmic tangent moduli denoted as ${\overline{\varvec{\Xi }}}_{n+1}$:

$$\begin{aligned}&\hbox {D}_u \left[ {\overline{\mathbf{C}}_{n+1}^{(\alpha )} } \right] \cdot \Delta {\varvec{u}}^{(\alpha )}={\overline{\varvec{\Xi }}}_{n+1}^{(\alpha )} : {\varvec{\varepsilon }} \left( {\Delta {\varvec{u}}^{(\alpha )}} \right) \end{aligned}$$

(76)

$$\begin{aligned}&{\overline{\varvec{\Xi }}}_{n+1}^{(\alpha )} =\frac{d^{2}{\varvec{\sigma }} _{n+1}^{(\alpha )} }{d{\varvec{\varepsilon }} _{n+1}^{(\alpha )} d{\varvec{\varepsilon }} _{n+1}^{(\alpha )} } \end{aligned}$$

(77)

This tensor in general possesses minor symmetries as well as major symmetry within the last two pairs of indices. For associative flow rules, the other possible major symmetries are also retained. Since this algorithmic tensor can be difficult to derive in closed-form, a numerical approximation procedure based on finite differences [30] is provided in Appendix 3. Recently, other approaches have been proposed [29] for evaluating tangent matrices that use complex variables to avoid numerical cancellation errors.

Making these substitutions into (75) yields the final linearized form:

$$\begin{aligned}&K_{uu} \left( {{\varvec{\eta }}^{h},\Delta {\varvec{u}}^{h};{\varvec{u}}_{n+1}^h } \right) =\int _\Omega {{\varvec{\varepsilon }} \left( {{\varvec{\eta }}^{h}} \right) :\overline{\mathbf{C}}_{n+1} :{\varvec{\varepsilon }} \left( {\Delta {\varvec{u}}^{h}} \right) \;\hbox {d}V} \nonumber \\&\quad -\int _{\Gamma _I } {\llbracket {{\varvec{\eta }}^{h}} \rrbracket \cdot \left\{ {\left[ {\overline{\mathbf{C}}_{n+1} :{\varvec{\varepsilon }} \left( {\Delta {\varvec{u}}^{h}} \right) } \right] \cdot {\varvec{n}}} \right\} \;\hbox {d}A} \nonumber \\&\quad -\int _{\Gamma _I } {\left\{ {\left[ {\overline{\mathbf{C}}_{n+1} :{\varvec{\varepsilon }} \left( {{\varvec{\eta }}^{h}} \right) } \right] \cdot {\varvec{n}}} \right\} \cdot \llbracket {\Delta {\varvec{u}}^{h}} \rrbracket \;\hbox {d}A} \nonumber \\&\quad -\int _{\Gamma _I } {\left\{ {\left[ {\left( {{\overline{\varvec{\Xi }}}_{n+1} : {\varvec{\varepsilon }} \left( {\Delta {\varvec{u}}^{h}} \right) } \right) : {\varvec{\varepsilon }} \left( {{\varvec{\eta }}^{h}} \right) } \right] \cdot {\varvec{n}}} \right\} \cdot \llbracket {{\varvec{u}}_{n+1}^h } \rrbracket \;\hbox {d}A} \nonumber \\&\quad +\int _{\Gamma _I } {\llbracket {{\varvec{\eta }}^{h}} \rrbracket \cdot {\varvec{\tau }}_s \cdot \llbracket {\Delta {\varvec{u}}^{h}} \rrbracket \;\hbox {d}A} \end{aligned}$$

(78)

Upon inspection, the bilinear form (78) is symmetric in the arguments ${\varvec{\eta }}^{h}$ and $\Delta {\varvec{u}}^{h}$ by taking account of the assumed associative flow rule. As remarked in [23], the higher-order moduli term can typically be neglected without significant impact to the Newton–Raphson convergence rate due to the presence of the displacement jump $\llbracket {{\varvec{u}}_{n+1}^h} \rrbracket \approx \mathbf{0}$.

The contributions from the pressure field are also straightforward due to linearity:

(79)

(80)

(81)

In conclusion, the consistent linearization (78)–(81) is symmetric, which is a direct consequence of the existence of a potential functional (30) and implies that the VMDG formulation (39)–(40) possesses variational and adjoint consistency. These consistency properties are numerically verified in Sect. 5.1; the correctness of (78)–(81) is attested by the quadratic rate of convergence of the Newton–Raphson algorithm in Sect. 5.2.

Appendix 3: Numerical evaluation of ${\overline{\varvec{\Xi }}}_{n+1}$

Herein, we outline a numerical procedure for evaluating the 6th order tensor of material moduli ${\overline{\varvec{\Xi }}}_{n+1}$. Similarly, computational methods have been proposed for evaluating the algorithmic tangent moduli $\overline{\mathbf{C}}_{n+1}$ as an alternative to analytical methods; see for example [29, 30]. In the former, a numerical differentiation scheme is advocated that consists of multiple evaluations of the stress update routine with perturbed strain increments. Namely, each “column” of $\overline{\mathbf{C}}_{n+1}$ is computed from differencing the algorithmic stress resulting from the actual strain ${\varvec{\varepsilon }}$ and the perturbed strain $\tilde{{\varvec{\varepsilon }} }$. We extend the concept to compute each generalized “column” of ${\overline{\varvec{\Xi }}}_{n+1}$ by differencing the perturbed algorithmic tangent moduli $\overline{\mathbf{C}}_{n+1}$. For three dimensional analyses, the cost of this procedure is effectively six additional function-calls to the stress-update subroutine $\mathbf{C}^{\mathrm{algo}}$. The algorithm, presented in Table 6, is quite general and can be extended to the geometrically nonlinear regime using ideas from [30]. As demonstrated in Sect. 5.2, the numerical version of ${\overline{\varvec{\Xi }}}_{n+1}$ only slightly reduces the quadratic rate of the Newton–Raphson method. Herein, the perturbation size $\epsilon =10^{-8}\varepsilon _{\left( {mn} \right) }$ as recommended by [30].

Table 6 Numerical computation of 6th order material moduli tensor

Appendix 4: Constitutive tensors for von Mises plasticity

We provide a complete characterization of the material tensors ${\varvec{\sigma }}_{n+1}$, $\overline{\mathbf{C}}_{n+1}$, ${\overline{\varvec{\Xi }}}_{n+1}$ corresponding to the von Mises model with kinematic hardening from Sect. 5.2. Isotropic elastic response is assumed according to the model given in Sect. 2. The elastic predictor–plastic corrector scheme is adopted from [2]. The trial yield function and flow stress at step $n+1$ are as follows:

$$\begin{aligned} f_{n+1}^{tr}= & {} \left\| {{\varvec{\eta }}_{n+1}^{tr} } \right\| -\sqrt{\frac{2}{3}}\sigma _Y , \nonumber \\ {\varvec{\eta }} _{n+1}^{tr}= & {} \hbox {dev}\left[ {{\varvec{\sigma }}_n } \right] +2G\left( {\hbox {dev}\left[ {{\varvec{\varepsilon }}_{n+1} -{\varvec{\varepsilon }}_n } \right] } \right) -\overline{\varvec{\beta }}_n \end{aligned}$$

(82)

The unit normal tensor to the yield surface is given by:

$$\begin{aligned} {\varvec{N}}={{\varvec{\eta }} _{n+1}^{tr} }\big /{\left\| {{\varvec{\eta }}_{n+1}^{tr} } \right\| } \end{aligned}$$

(83)

Assuming plastic flow $f_{n+1}^{tr} >0$, the incremental consistency parameter follows from [2] as:

$$\begin{aligned} \Delta \gamma =\frac{f_{n+1}^{tr} }{2G\left( {1+H/{3G}} \right) } \end{aligned}$$

(84)

Thus, the algorithmic stress at step $n+1$ takes the following form:

$$\begin{aligned} {\varvec{\sigma }}_{n+1}= & {} p_{n+1} {\varvec{I}}+\hbox {dev}\left[ {{\varvec{\sigma }}_n } \right] \nonumber \\&+\,2G\left( {\hbox {dev}\left[ {{\varvec{\varepsilon }}_{n+1} -{\varvec{\varepsilon }} _n } \right] } \right) -2G\Delta \gamma {\varvec{N}} \end{aligned}$$

(85)

We record the results from [2] for the derivatives of the unit normal (83) and consistency parameter (84):

$$\begin{aligned} \frac{\partial N_{ij} }{\partial \varepsilon _{kl} }= & {} \frac{\partial N_{ij} }{\partial \eta _{mn}^{tr} }\frac{\partial \eta _{mn}^{tr} }{\partial \varepsilon _{kl} }={2G}/{\left\| {{\varvec{\eta }}_{n+1}^{tr} } \right\| } \nonumber \\&\left[ {\frac{1}{2}\left( {\delta _{ik} \delta _{jl} +\delta _{il} \delta _{jk} } \right) -\frac{1}{3}\delta _{ij} \delta _{kl} -N_{ij} N_{kl} } \right] \end{aligned}$$

(86)

$$\begin{aligned} \frac{\partial \Delta \gamma }{\partial \varepsilon _{ij} }= & {} \left( {1+H/{3G}} \right) ^{-1}N_{ij} \end{aligned}$$

(87)

The algorithmic tangent moduli is obtained by directly differentiating the algorithmic stress (85) accounting for (86) and (87):

$$\begin{aligned} \frac{d\sigma _{ij} }{d\varepsilon _{kl} }\,:=\,\overline{\mathrm{C}}_{ijkl}= & {} 2G\left[ {\frac{1}{2}\left( {\delta _{ik} \delta _{jl} +\delta _{il} \delta _{jk} } \right) -\frac{1}{3}\delta _{ij} \delta _{kl} } \right] \nonumber \\&-\,2G\left[ {\frac{\partial \Delta \gamma }{\partial \varepsilon _{kl} }N_{ij} +\Delta \gamma \frac{\partial N_{ij} }{\partial \varepsilon _{kl} }} \right] \end{aligned}$$

(88)

Similarly, the 6th order material tensor ${\overline{\varvec{\Xi }}}_{n+1}$ is derived by differentiating (88). To do so, the second derivatives of the unit normal tensor and consistency parameter are required. Following directly from (86) and (87):

$$\begin{aligned} \frac{\partial ^{2}N_{ij} }{\partial \varepsilon _{kl} \partial \varepsilon _{mn} }= & {} -2G\left[ {{N_{ij} }/{\left\| {{\varvec{\eta }}_{n+1}^{tr} } \right\| }\frac{\partial N_{kl} }{\partial \varepsilon _{mn} }+{N_{kl} }/{\left\| {{\varvec{\eta }} _{n+1}^{tr} } \right\| }\frac{\partial N_{ij} }{\partial \varepsilon _{mn} } } \right. \nonumber \\&\left. +\,{N_{mn} }/{\left\| {{\varvec{\eta }} _{n+1}^{tr} } \right\| } \frac{\partial N_{ij} }{\partial \varepsilon _{kl} } \right] \end{aligned}$$

(89)

$$\begin{aligned} \frac{\partial ^{2}\Delta \gamma }{\partial \varepsilon _{ij} \partial \varepsilon _{kl} }= & {} \left( {1+H/{3G}} \right) ^{-1}\frac{\partial N_{ij} }{\partial \varepsilon _{kl} } \end{aligned}$$

(90)

Observe that expressions (86), (87), and (90) imply that $N_{ij} \frac{\partial ^{2}\Delta \gamma }{\partial \varepsilon _{kl} \partial \varepsilon _{mn} }=\frac{\partial N_{kl} }{\partial \varepsilon _{mn} }\frac{\partial \Delta \gamma }{\partial \varepsilon _{ij} }$. Thus, we employ the results from (89) and (90) to obtain the definition for the 6th order tensor by differentiating (88):

$$\begin{aligned} \frac{\partial ^{2}\sigma _{ij} }{\partial \varepsilon _{kl} \partial \varepsilon _{mn} }\equiv & {} \overline{{\Xi }}_{ijklmn} \nonumber \\= & {} -2G\left[ \Delta \gamma \frac{\partial ^{2}N_{ij} }{\partial \varepsilon _{kl} \partial \varepsilon _{mn} }+\frac{\partial N_{kl} }{\partial \varepsilon _{mn} }\frac{\partial \Delta \gamma }{\partial \varepsilon _{ij} }+\frac{\partial N_{ij} }{\partial \varepsilon _{mn} }\frac{\partial \Delta \gamma }{\partial \varepsilon _{kl} }\right. \nonumber \\&\left. +\frac{\partial N_{ij} }{\partial \varepsilon _{mn} }\frac{\partial \Delta \gamma }{\partial \varepsilon _{kl} } +\frac{\partial N_{ij} }{\partial \varepsilon _{kl} }\frac{\partial \Delta \gamma }{\partial \varepsilon _{mn} } \right] \end{aligned}$$

(91)

Note that ${\overline{\varvec{\Xi }}}_{n+1}$ possesses all possible major symmetries due to the associative nature of the flow rule and minor symmetries due to the symmetry of the strain tensor.