A model-reduction approach to the micromechanical analysis of polycrystalline materials

Abstract

The present study is devoted to the extension to polycrystals of a model-reduction technique introduced by the authors, called the nonuniform transformation field analysis (NTFA). This new reduced model is obtained in two steps. First the local fields of internal variables are decomposed on a reduced basis of modes as in the NTFA. Second the dissipation potential of the phases is replaced by its tangent second-order (TSO) expansion. The reduced evolution equations of the model can be entirely expressed in terms of quantities which can be pre-computed once for all. Roughly speaking, these pre-computed quantities depend only on the average and fluctuations per phase of the modes and of the associated stress fields. The accuracy of the new NTFA-TSO model is assessed by comparison with full-field simulations on two specific applications, creep of polycrystalline ice and response of polycrystalline copper to a cyclic tension-compression test. The new reduced evolution equations is faster than the full-field computations by two orders of magnitude in the two examples.

Appendices

Appendix 1: Alternative writing of the constitutive relations for single crystals

The aim of this section is to show that the two crystal plasticity models presented in Sects. 2.2.1 and 2.2.2 can written in the form

$$\begin{aligned} \dot{{\varvec{\alpha }}} = {\varvec{\mathcal {F}}}({\varvec{\mathcal {A}}}), \end{aligned}$$

(58)

where the thermodynamic forces associated with the state variables are ${\varvec{\mathcal {A}}}=({\varvec{\mathcal {A}}}_{{\textsf {v}}}, {\varvec{\mathcal {A}}}_\beta ,{\varvec{\mathcal {A}}}_{{\textsf {p}}})$ given in (20). In both models, the evolution of the viscoplastic strain-rate is governed by a potential $\psi$

$$\begin{aligned} \dot{{\varvec{\varepsilon }}}_{{\textsf {v}}}= \frac{\partial \psi }{\partial {\varvec{\mathcal {A}}}_{{\textsf {v}}}}({\varvec{\mathcal {A}}}), \end{aligned}$$

(59)

where the potential $\psi$ is the combination of individual potentials $\psi _s$ on the different slip systems

$$\begin{aligned} \psi ({\varvec{\mathcal {A}}})= \sum _{s=1}^S \psi _s({\varvec{\mathcal {A}}}_{{\textsf {v}}},{\varvec{\mathcal {A}}}_\beta ,{\varvec{\mathcal {A}}}_{{\textsf {p}}}). \end{aligned}$$

(60)

The $\psi _s$ take the form (61) and (62) for the model of Méric-Cailletaud and for ice, respectively,

$$\begin{aligned}&\psi _s({\varvec{\mathcal {A}}}_{{\textsf {v}}},{\varvec{\mathcal {A}}}_\beta ,{\varvec{\mathcal {A}}}_{{\textsf {p}}})\nonumber \\&\quad = \frac{K_s \dot{\gamma }_{0,s}}{n_s+1} \left( \frac{\left( \left| {{\varvec{\mathcal {A}}}_{{\textsf {v}}}:{\varvec{m}}_s + \mathcal {A}_{\beta _s}}\right| +\mathcal {A}_{{p}_s}\right) ^+}{K_s} \right) ^{n_s+1}, \end{aligned}$$

(61)

$$\begin{aligned}&\psi _s({\varvec{\mathcal {A}}}_{{\textsf {v}}},{\varvec{\mathcal {A}}}_\beta ,{\varvec{\mathcal {A}}}_{{\textsf {p}}}) = \frac{\dot{\gamma }_{0,s}}{n_s+1} \frac{\left| {{\varvec{\mathcal {A}}}_{{\textsf {v}}}:{\varvec{m}}_s + \mathcal {A}_{\beta _s}}\right| ^{n_s+1}}{\left| {\mathcal {A}_{{p}_s}}\right| ^{n_s}}. \end{aligned}$$

(62)

The other evolution equations for $\beta _s$ and $p_s$ have to be considered separately for the two models.

Méric-Cailletaud crystal plasticity model Note that (10) can be re-written as (recall that $x_{\beta _s}= c_s \beta _s=- \mathcal {A}_{\beta _s}$ and divide the two sides of (10) by $c_s$)

$$\begin{aligned} \dot{\beta }_s = \dot{\gamma }_s + \frac{d_s}{c_s} \mathcal {A}_{\beta _s}\left| { \dot{\gamma }_s}\right| . \end{aligned}$$

(63)

Furthermore

$$\begin{aligned} \dot{\gamma }_s = \frac{\partial \psi }{\partial {\mathcal {A}}_{\beta _s}}({\varvec{\mathcal {A}}}), \quad \left| {\dot{\gamma }}\right| _s = \left| {\frac{\partial \psi }{\partial {\mathcal {A}}_{\beta _s}}({\varvec{\mathcal {A}}})}\right| . \end{aligned}$$

Eq. (63) can be re-written as

$$\begin{aligned} \dot{\beta }_s = \frac{\partial \psi }{\partial {\mathcal {A}}_{\beta _s}}({\varvec{\mathcal {A}}}) + \frac{d_s}{c_s} \left| {\frac{\partial \psi }{\partial {\mathcal {A}}_{\beta _s}}({\varvec{\mathcal {A}}})}\right| \mathcal {A}_{\beta _s}. \end{aligned}$$

The first relation in (11) can be inverted as

$$\begin{aligned} p_s = \frac{1}{Qb} \sum _{s'=1}^S (h^{-1})_{s,s'}(r_{s'}-r_{ini,s'}), \end{aligned}$$

and the differential equation for $p_s$ reads as

$$\begin{aligned} \dot{p}_s = \left[ 1+ \frac{1}{Q} \sum _{s'=1}^S (h^{-1})_{s,s'} (\mathcal {A}_{{p}_{s'}}- \mathcal {A}_{ini,{p}_{s'}})\right] \left| { \frac{\partial \psi }{\partial {\mathcal {A}}_{\beta _s}}({\varvec{\mathcal {A}}})}\right| , \end{aligned}$$

where $\mathcal {A}_{ini,{p}_{s'}}=-r_{ini,s'}$. Finally the constitutive relations (8), (9), (10) and (11) can be written as $\dot{{\varvec{\alpha }}}= {\varvec{\mathcal {F}}}({\varvec{\mathcal {A}}})$ where ${\varvec{\alpha }}$ and ${\varvec{\mathcal {A}}}$ are given by (16) and (20) respectively and

$$\begin{aligned} {\varvec{\mathcal {F}}}({\varvec{\mathcal {A}}})&= \left( \begin{array}{c} {\varvec{\mathcal {F}}}_{{\textsf {v}}}({\varvec{\mathcal {A}}}) \\ \\ {\varvec{\mathcal {F}}}_{\beta _s}({\varvec{\mathcal {A}}}) \\ \\ {\varvec{\mathcal {F}}}_{{p}_s}({\varvec{\mathcal {A}}}) \end{array} \right) \nonumber \\&= \left( \begin{array}{c} \displaystyle \frac{\partial \psi }{\partial {\varvec{\mathcal {A}}}_{{\textsf {v}}}}({\varvec{\mathcal {A}}}) \\ \displaystyle \frac{\partial \psi }{\partial {\mathcal {A}}_{\beta _s}}({\varvec{\mathcal {A}}}) + \frac{d_s}{c_s} \left| {\frac{\partial \psi }{\partial {\mathcal {A}}_{\beta _s}}({\varvec{\mathcal {A}}})}\right| \mathcal {A}_{\beta _s}\\ \displaystyle \left[ 1+ \frac{1}{Q} \sum _{s'=1}^S (h^{-1})_{s,s'} (\mathcal {A}_{{p}_{s'}}- \mathcal {A}_{ini,{p}_{s'}}) \right] \left| { \frac{\partial \psi }{\partial {\mathcal {A}}_{\beta _s}}({\varvec{\mathcal {A}}})}\right| \end{array} \right) . \end{aligned}$$

(64)

$I_h$ ice. Note that (13) can be re-written as (recall that $x_{\beta _s}= c_s \beta _s=- \mathcal {A}_{\beta _s}$ and divide the two sides of (13) by $c_s$)

$$\begin{aligned} \dot{\beta }_s = \dot{\gamma }_s + \frac{d_s}{c_s} \mathcal {A}_{\beta _s}\left| {\dot{\gamma }_s}\right| + \frac{e_s}{c_s} \mathcal {A}_{\beta _s}. \end{aligned}$$

(65)

Furthermore

$$\begin{aligned} \dot{\gamma }_s = \frac{\partial \psi }{\partial {\mathcal {A}}_{\beta _s}}({\varvec{\mathcal {A}}}),\quad \left| {\dot{\gamma }_s}\right| = \left| {\frac{\partial \psi }{\partial {\mathcal {A}}_{\beta _s}}({\varvec{\mathcal {A}}})}\right| , \end{aligned}$$

where $\psi$ is the potential (62) for ice. Eq. (65) can be re-written as

$$\begin{aligned} \dot{\beta }_s = \frac{\partial \psi }{\partial {\mathcal {A}}_{\beta _s}}({\varvec{\mathcal {A}}}) + \frac{d_s}{c_s} \left| {\frac{\partial \psi }{\partial {\mathcal {A}}_{\beta _s}}({\varvec{\mathcal {A}}})}\right| \mathcal {A}_{\beta _s}+ \frac{e_s}{c_s} \mathcal {A}_{\beta _s}. \end{aligned}$$

Eq. (14) is simply

$$\begin{aligned} \dot{p}_s = \sum _{s'=1}^S h_{s,s'} \left| { \frac{\partial \psi }{\partial {\mathcal {A}}_{\beta _{s'}}}({\varvec{\mathcal {A}}})}\right| . \end{aligned}$$

Finally the constitutive relations (8), (12), (13) and (14) can be written in the form (3) with ${\varvec{\alpha }}$ and ${\varvec{\mathcal {A}}}$ given by (16) and (20) respectively and

$$\begin{aligned}&{\varvec{\mathcal {F}}}({\varvec{\mathcal {A}}}) = \left( \begin{array}{c} {\varvec{\mathcal {F}}}_{{\textsf {v}}}({\varvec{\mathcal {A}}}) \\ \\ {\varvec{\mathcal {F}}}_{\beta _s}({\varvec{\mathcal {A}}}) \\ \\ {\varvec{\mathcal {F}}}_{{p}_s}({\varvec{\mathcal {A}}}) \end{array} \right) \nonumber \\&\quad = \left( \begin{array}{c} \displaystyle \frac{\partial \psi }{\partial {\varvec{\mathcal {A}}}_{{\textsf {v}}}}({\varvec{\mathcal {A}}}) \\ \displaystyle \frac{\partial \psi }{\partial {\mathcal {A}}_{\beta _s}}({\varvec{\mathcal {A}}}) + \frac{d_s}{c_s} \left| {\frac{\partial \psi }{\partial {\mathcal {A}}_{\beta _s}}({\varvec{\mathcal {A}}})}\right| \mathcal {A}_{\beta _s}+ \frac{e_s}{c_s} \mathcal {A}_{\beta _s}\\ \displaystyle \sum _{s'=1}^S h_{s,s'} \left| { \frac{\partial \psi }{\partial {\mathcal {A}}_{\beta _{s'}}}({\varvec{\mathcal {A}}})}\right| . \end{array} \right) . \end{aligned}$$

(66)

Appendix 2: Effective potentials

In order to check that (25) and (26) define the effective response of the polycrystal (the correct average stress history), note first that the solution ${\varvec{\varepsilon }}$ of the variational problem in (26) is precisely the solution of the local problem (23). It remains to prove that the force associated with $\overline{{\varvec{\varepsilon }}}$ is precisely the average of the stress solution of (23) and that the differential equation satisfied by $\widetilde{{\varvec{\alpha }}}$ is consistent with (24). To this end, one has to compute the thermodynamic forces associated with $\overline{{\varvec{\varepsilon }}}$ and $\widetilde{{\varvec{\alpha }}}$,

$$\begin{aligned}&\frac{\partial \widetilde{w}}{\partial \overline{{\varvec{\varepsilon }}}} \left( \overline{{\varvec{\varepsilon }}},\{{\varvec{\alpha }}({\varvec{x}}) \}_{{\varvec{x}} \in V} \right) , \quad \widetilde{{\varvec{{\mathcal {A}}}}} = \{ \widetilde{{\varvec{{\mathcal {A}}}}}_{{\varvec{x}}} \}_{{\varvec{x}} \in V},\quad \\&\widetilde{{\varvec{{\mathcal {A}}}}}_{{\varvec{x}}}= - \frac{\partial \widetilde{w}}{\partial {\varvec{\alpha }}({\varvec{x}})} \left( \overline{{\varvec{\varepsilon }}},\{{\varvec{\alpha }}({\varvec{x}}) \}_{{\varvec{x}} \in V} \right) . \end{aligned}$$

The first force can be computed using Hill’s lemma

$$\begin{aligned} \frac{\partial \widetilde{w}}{\partial \overline{{\varvec{\varepsilon }}}} \left( \overline{{\varvec{\varepsilon }}},\widetilde{{\varvec{\alpha }}} \right)&=\left\langle \frac{\partial w}{\partial {\varvec{\varepsilon }}}({\varvec{\varepsilon }},{\varvec{\alpha }}): \frac{\partial {\varvec{\varepsilon }}}{\partial \overline{{\varvec{\varepsilon }}}} \right\rangle =\left\langle {\varvec{\sigma }} : \frac{\partial {\varvec{\varepsilon }}}{\partial \overline{{\varvec{\varepsilon }}}} \right\rangle \\&= \left\langle {\varvec{\sigma }} \right\rangle :\left\langle \frac{\partial {\varvec{\varepsilon }}}{\partial \overline{{\varvec{\varepsilon }}}} \right\rangle = \overline{{\varvec{\sigma }}}, \end{aligned}$$

since $\left\langle \frac{\partial {\varvec{\varepsilon }}}{\partial \overline{{\varvec{\varepsilon }}}} \right\rangle = {\varvec{I}}$. The first force is therefore the average stress associated with the solution of (23). The second force is computed by taking the Frechet derivative of $\widetilde{w}$

$$\begin{aligned} - \widetilde{{\varvec{\mathcal {A}}}} \cdot \widetilde{{\varvec{\alpha }}}^*&= \lim _{t \rightarrow 0} \frac{\widetilde{w} \left( \overline{{\varvec{\varepsilon }}},\widetilde{{\varvec{\alpha }}}+ t \widetilde{{\varvec{\alpha }}}^* \right) - \widetilde{w} \left( \overline{{\varvec{\varepsilon }}},\widetilde{{\varvec{\alpha }}} \right) }{t} \nonumber \\&= \lim _{t \rightarrow 0} \left\langle \frac{ {w} \left( {{\varvec{\varepsilon }}},{{\varvec{\alpha }}}+ t \widetilde{{\varvec{\alpha }}}^* \right) - {w} \left( {{\varvec{\varepsilon }}},{{\varvec{\alpha }}}\right) }{t} \right\rangle \nonumber \\&= \left\langle \frac{\partial w}{\partial {\varvec{\alpha }}}({\varvec{x}}). {{\varvec{\alpha }}}^*({\varvec{x}}) \right\rangle , \end{aligned}$$

(67)

which implies that $\widetilde{{\varvec{{\mathcal {A}}}}}_{{\varvec{x}}}$ coincides with the local force ${\varvec{\mathcal {A}}}({\varvec{x}})$ at point ${\varvec{x}}$. Following an identical procedure it can be shown that

$$\begin{aligned} \frac{\partial \widetilde{\varphi }}{\partial \dot{\widetilde{{\varvec{\alpha }}}}}(\dot{\widetilde{{\varvec{\alpha }}}})\cdot \widetilde{{\varvec{\alpha }}}^* = \left\langle \frac{\partial {\varphi }}{\partial \dot{{{\varvec{\alpha }}}}}(\dot{{{\varvec{\alpha }}}}). {{\varvec{\alpha }}}^*({\varvec{x}}) \right\rangle . \end{aligned}$$

(68)

The evolution equations for the standard model corresponding to the potentials (26) read as

$$\begin{aligned} \frac{\partial \widetilde{w}}{\partial \widetilde{{\varvec{\alpha }}}}(\overline{{\varvec{\varepsilon }}},\widetilde{{\varvec{\alpha }}})+ \frac{\partial \widetilde{\varphi }}{\partial \dot{\widetilde{{\varvec{\alpha }}}}}(\dot{\widetilde{{\varvec{\alpha }}}}) = 0. \end{aligned}$$

(69)

which, by virtue of (67) and (68) shows that the evolution equations for the field of internal variables ${\varvec{\alpha }}({\varvec{x}})$ are precisely (24).

Appendix 3: TSO approximation for the two crystal plasticity models

1.1 C.1 Evolution equation for ${\varvec{\mathfrak {a}}}_\beta$

For the non-standard variable $\beta _s$ (and similarly for $p_s$) the TSO approximation of the evolution equations for the generalized forces consists in making use of a second-order expansion of ${\mathcal {F}}_{\beta _s}$ in (53), where

$$\begin{aligned} {\varvec{\mathcal {F}}} _{\beta _s}({{{\varvec{\mathcal {A}}}}}) = \frac{\partial \psi }{\partial {\mathcal {A}}_{\beta _s}}({\varvec{\mathcal {A}}}) + \frac{d_s}{c_s} \left| {\frac{\partial \psi }{\partial {\mathcal {A}}_{\beta _s}}({\varvec{\mathcal {A}}})}\right| {\mathcal {A}}_{\beta _s}+ \frac{e_s}{c_s} {\mathcal {A}}_{\beta _s}, \end{aligned}$$

(70)

with ${\mathcal {A}}_{\beta _s}=-x_s$ and $e_s=0$ in the model of Méric and Cailletaud. The first term in the expression of $\displaystyle \left\langle {\varvec{\mathcal {F}}}_{\beta _s}({\varvec{\mathcal {A}}})\frac{\partial \mathcal {A}_{\beta _s}}{\partial \xi _{\beta _s}^{(\ell )}} \right\rangle$ is standard and the TSO approximation of this is

$$\begin{aligned}&\displaystyle \frac{\partial \widetilde{\psi }_{TSO}}{\partial \xi ^{(\ell )}_{\beta _s}} ({\varvec{\mathcal {A}}}) = \displaystyle \sum _{g=1}^N c^{(g)} \left[ - \frac{\partial f_s}{\partial \tau }\left( \overline{\tau }^{(g)}_s-\overline{x}_s^{(g)},r_s^{(g)}\right) \frac{\partial \overline{x}_s^{(g)} }{\partial \xi ^{(\ell )}_{\beta _s}} \right. \nonumber \\&\qquad + \displaystyle \frac{1}{2} \frac{\partial ^2 f_s}{\partial \tau ^2}\left( \overline{\tau }^{(g)}_s-\overline{x}_s^{(g)},r_s^{(g)}\right) \frac{\partial C^{(g)}(\tau _s-x_s)}{\partial \xi ^{(\ell )}_{\beta _s}} \nonumber \\&\qquad - \displaystyle \left. \frac{1}{2} \frac{\partial ^3 f_s}{\partial \tau ^3} \left( \overline{\tau }^{(g)}_s-\overline{x}_s^{(g)},r_s^{(g)}\right) C^{(g)}(\tau _s-x_s) \frac{\partial \overline{x}_s^{(g)} }{\partial \xi ^{(\ell )}_{\beta _s}} \right] . \end{aligned}$$

(71)

The TSO approximation of the second term in $\left\langle {\varvec{\mathcal {F}}}_{\beta _s}({\varvec{\mathcal {A}}})\frac{\partial \mathcal {A}_{\beta _s}}{\partial \xi _{\beta _s}^{(\ell )}} \right\rangle$ is

$$\begin{aligned}&\displaystyle \sum _{g=1}^N c^{(g)} \left\langle \frac{d_s}{c_s} \left| {\frac{\partial \psi }{\partial {\mathcal {A}}_{\beta _s}}({\varvec{\mathcal {A}}})}\right| x_s \frac{\partial x_s }{\partial \xi ^{(\ell )}_{\beta _s}} \right\rangle ^{(g)}\nonumber \\&\quad = \displaystyle \sum _{g=1}^N c^{(g)} \left\{ \frac{d_s}{c_s} \left[ \frac{\partial f_s}{\partial \tau }\left( \overline{\tau }^{(g)}_s-\overline{x}_s^{(g)},r_s^{(g)}\right) \right. \right. \nonumber \\&\qquad \times \left( \overline{x}_s^{(g)} \frac{\partial \overline{x}_s^{(g)} }{\partial \xi ^{(\ell )}_{\beta _s}} + \frac{1}{2} \frac{\partial C^{(g)}(x_s)}{\partial \xi ^{(\ell )}_{\beta _s}} \right) \nonumber \\&\qquad + \displaystyle \frac{\partial ^2 f_s}{\partial \tau ^2}\left( \overline{\tau }^{(g)}_s-\overline{x}_s^{(g)},r_s^{(g)}\right) \left( C^{(g)}(\tau _s-x_s,x_s) \frac{\partial \overline{x}_s^{(g)} }{\partial \xi ^{(\ell )}_{\beta _s}} \right. \nonumber \\&\qquad \left. - \displaystyle \frac{1}{2} \frac{\partial C^{(g)}(\tau _s-x_s)}{\partial \xi ^{(\ell )}_{\beta _s}} \overline{x}_s^{(g)} \right) \nonumber \\&\qquad \left. + \displaystyle \frac{1}{2} \frac{\partial ^3 f_s}{\partial \tau ^3} \left( \overline{\tau }^{(g)}_s-\overline{x}_s^{(g)},r_s^{(g)}\right) C^{(g)}(\tau _s-x_s) \overline{x}_s^{(g)} \frac{\partial \overline{x}_s^{(g)} }{\partial \xi ^{(\ell )}_{\beta _s}} \right] \nonumber \\&\qquad \times \left. \text {sign}\left( {\frac{\partial f_s}{\partial \tau }\left( \overline{\tau }^{(g)}_s-\overline{x}_s^{(g)},r_s^{(g)}\right) }\right) \right\} , \end{aligned}$$

(72)

where

$$\begin{aligned} C^{(g)}(\tau _s-x_s,x_s) = \left\langle (\tau _s-\overline{\tau }^{(g)}_s-x_s+\overline{x}_s^{(g)} ) (x_s-\overline{x}_s^{(g)}) \right\rangle ^{(g)}. \end{aligned}$$

The third term in $\displaystyle \left\langle {\varvec{\mathcal {F}}}_{\beta _s}({\varvec{\mathcal {A}}})\frac{\partial \mathcal {A}_{\beta _s}}{\partial \xi _{\beta _s}^{(\ell )}} \right\rangle$ does not require any approximation and is simply

$$\begin{aligned}&\sum _{g=1}^N c^{(g)} \frac{e_s}{c_s} \left\langle x_s \frac{\partial x_s }{\partial \xi ^{(\ell )}_{\beta _s}} \right\rangle ^{(g)} \nonumber \\&\quad = \sum _{g=1}^N c^{(g)} \frac{e_s}{c_s} \left( \overline{x}_s^{(g)} \frac{\partial \overline{x}_s^{(g)} }{\partial \xi ^{(\ell )}_{\beta _s}} + \frac{1}{2} \frac{\partial C^{(g)}(x_s)}{\partial \xi ^{(\ell )}_{\beta _s}} \right) . \end{aligned}$$

(73)

Finally the evolution equation for $\mathfrak {a}^{(\ell )}_{\beta _s}$ is obtained by equalling $\dot{\mathfrak {a}}_{\beta _s}^{(\ell )}$ to the sum of the right-hand sides of (71), (72) and (73).

1.2 C.2 Evolution equation for the isotropic hardening variables

According to (46), the evolution equation for the reduced variable $p_s^{(g)}$ is

$$\begin{aligned} \dot{p}_s^{(g)} = \left\langle \left( \mathcal {F}({\varvec{\mathcal {A}}}) \frac{\partial {\varvec{\mathcal {A}}}}{\partial \mathfrak {a}_{{p}_s}^{(g)}}\right) _{TSO} \right\rangle . \end{aligned}$$

According to (34) only $\mathcal {A}_{{p}_s}=-r_s$ depends on $\mathfrak {a}_{p_s}^{(g)}$ with

$$\begin{aligned} \mathcal {A}_{{p}_s}({\varvec{x}}) = \sum _{g=1}^N \frac{1}{c^{(g)}} \mathfrak {a}_{p_s}^{(g)} \chi ^{(g)}({\varvec{x}}), \quad \frac{\partial \mathcal {A}_{{p}_s}}{\partial \mathfrak {a}_{p_s}^{(g)}} = \frac{1}{c^{(g)}} \chi ^{(g)}({\varvec{x}}) . \end{aligned}$$

Finally the differential equation for $p_s^{(g)}$ reduces to

$$\begin{aligned} \dot{p}_s^{(g)} = \left\langle \mathcal {F}_{{p}_s}({\varvec{\mathcal {A}}}) \right\rangle ^{(g)}. \end{aligned}$$

(74)

It remains to expand $\left\langle \mathcal {F}_{{\textsf {p}}_s}({\varvec{\mathcal {A}}}) \right\rangle ^{(g)}$ to second-order in the fluctuations of ${\varvec{\mathcal {A}}}$. This expansion depends on the constitutive relations under consideration.

Méric-Cailletaud The differential equation takes the form

$$\begin{aligned} \dot{p}_s^{(g)}&= \left\langle \left( 1 -bp_s^{(g)} \right) \left| {\frac{\partial \psi }{\partial \mathcal {A}_{\beta _s}}({\varvec{\mathcal {A}}})}\right| \right\rangle ^{(g)} \nonumber \\&= \left( 1 -bp_s^{(g)} \right) \left\langle \left| {\frac{\partial \psi }{\partial \mathcal {A}_{\beta _s}}({\varvec{\mathcal {A}}})}\right| \right\rangle ^{(g)}. \end{aligned}$$

(75)

Table 3 Ice single crystal. a Slip systems. b Hardening matrix

It results from (48) that up to second-order in the fluctuations of $\tau _s$ and $x_s$ ($p_s$ has no fluctuations),

$$\begin{aligned}&\displaystyle \left| {\frac{\partial \psi }{\partial {\mathcal {A}}_{\beta _s}}({\varvec{\mathcal {A}}})}\right| \simeq \displaystyle \Biggl [ {\frac{\partial f_s}{\partial \tau }\left( \overline{\tau }^{(g)}_s-\overline{x}_s^{(g)},r_s^{(g)}\right) } \nonumber \\&\qquad + \displaystyle \frac{\partial ^2 f_s}{\partial \tau ^2}\left( \overline{\tau }^{(g)}_s-\overline{x}_s^{(g)},r_s^{(g)}\right) (\tau _s-x_s- \overline{\tau }_s^{(g)} + \overline{x}_s^{(g)}) \nonumber \\&\qquad + \displaystyle \frac{1}{2} \frac{\partial ^3 f_s}{\partial \tau ^3} \left( \overline{\tau }^{(g)}_s-\overline{x}_s^{(g)},r_s^{(g)}\right) \left( \tau _s-x_s- \overline{\tau }_s^{(g)} + \overline{x}_s^{(g)}\right) ^2 \Biggr ] \nonumber \\&\qquad \times \text {sign}\left( {\frac{\partial f_s}{\partial \tau }\left( \overline{\tau }^{(g)}_s-\overline{x}_s^{(g)},r_s^{(g)}\right) }\right) , \end{aligned}$$

(76)

Therefore

$$\begin{aligned} \dot{p}_s^{(g)}= & {} \left( 1 -bp_s^{(g)} \right) \epsilon \Biggl [ {\frac{\partial f_s}{\partial \tau }\left( \overline{\tau }^{(g)}_s-\overline{x}_s^{(g)},r_s^{(g)}\right) } \nonumber \\&+ \frac{1}{2} \frac{\partial ^3 f_s}{\partial \tau ^3} \left( \overline{\tau }^{(g)}_s-\overline{x}_s^{(g)},r_s^{(g)}\right) C^{(g)}(\tau _s-x_s) \Biggr ] , \end{aligned}$$

(77)

where

$$\begin{aligned} \epsilon =\text {sign}\left( {\frac{\partial f_s}{\partial \tau }\left( \overline{\tau }^{(g)}_s-\overline{x}_s^{(g)},r_s^{(g)}\right) }\right) . \end{aligned}$$

Ice. For ice, the differential equation (74) takes the form

$$\begin{aligned} \dot{p}_s^{(g)} = \left\langle \sum _{s'=1}^S h_{s,s'} \left| {\frac{\partial \psi }{\partial \mathcal {A}_{{\beta }_{s'}}}({\varvec{\mathcal {A}}})}\right| \right\rangle ^{(g)}. \end{aligned}$$

(78)

Using (76) the following differential equation is obtained

$$\begin{aligned} \dot{p}_s^{(g)}= & {} \sum _{s'=1}^S h_{s,s'} \epsilon ' \Biggl [ {\frac{\partial f_{s'}}{\partial \tau }\left( \overline{\tau }^{(g)}_{s'}-\overline{x}_{s'}^{(g)},r_{s'}\right) } \nonumber \\&+ \frac{1}{2} \frac{\partial ^3 f_s}{\partial \tau ^3} \left( \overline{\tau }^{(g)}_{s'}-\overline{x}_{s'}^{(g)},r_{s'}^{(g)}\right) C^{(g)}(\tau _{s'}-x_{s'}) \Biggr ], \end{aligned}$$

(79)

where

$$\begin{aligned} \epsilon '=\text {sign}\left( {\frac{\partial f_{s'}}{\partial \tau }\left( \overline{\tau }^{(g)}_{s'}-\overline{x}_{s'}^{(g)},r_{s'}^{(g)}\right) }\right) . \end{aligned}$$

Alternatively a differential equation for $r_s^{(g)}$ can be obtained. The evolution equation (53) for the reduced force $\mathfrak {a}_{{p}_s}^{(g)}$ reads

$$\begin{aligned} \dot{\mathfrak {a}}_s^{(g)}&= \sum _{s'=1}^S \sum _{g'=1}^N c^{(g')} \left\langle \mathcal {F}_{p_{s'}}({\varvec{\mathcal {A}}}) \frac{\partial \mathcal {A}_{p_{s'}}}{\partial p_s^{(g)}} \right\rangle ^{(g')}\\&= c^{(g)} \left\langle \mathcal {F}_{{p}_s}({\varvec{\mathcal {A}}}) \right\rangle ^{(g)} \frac{\partial \mathcal {A}_{{p}_s}^{(g)}}{\partial p_s^{(g)}}, \end{aligned}$$

where the last equality has been obtained by noting that only $\mathcal {A}_{{p}_s}$ in grain g depends on $p_s^{(g)}$. Using the relations

$$\begin{aligned}&\mathfrak {a}_{{p}_s}^{(g)} = c^{(g)} \mathcal {A}_{{p}_s}^{(g)}, \quad r_s^{(g)} = - \mathcal {A}_{{p}_s}^{(g)}, \\&r_{s}(p) = \tau _{sta,s}+ \left( \tau _{ini,s}- \tau _{sta,s}\right) e^{-p}, \end{aligned}$$

one finds a differential equation for $r_{s}^{(g)}$

$$\begin{aligned} \dot{r}_{s}^{(g)}= & {} \displaystyle \left( r_{sta,s} -{r}_{s}^{(g)}\right) \sum _{s'=1}^S h_{s,s'} \Biggl [ \left| {\frac{\partial f_{s'}}{\partial \tau }\left( \overline{\tau }^{(g)}_{s'}-\overline{x}^{(g)}_{\beta _{s'}},{r}_{s'}^{(g)}\right) }\right| \\&+ \displaystyle \frac{1}{2} \frac{\partial ^3 f_s}{\partial \tau ^3} \left( \overline{\tau }^{(g)}_{s'}-\overline{x}^{(g)}_{{s'}} ,{r}_{s'}^{(g)}\right) \,\\&\times \, \text {sign}\left( {\frac{\partial f_{s'}}{\partial \tau }\left( \overline{\tau }^{(g)}_{s'}-\overline{x}_{s'}^{(g)},{r}_{s'}^{(g)}\right) }\right) C^{(g)}(\tau _{s'}-x_{{s'}}) \Biggr ] . \end{aligned}$$

Table 4 Material parameters used in the full-field simulations for single crystals of ice at $-{1}0\,^{\circ }\mathrm{C}$. Units are MPa and s$^{-{1}}$. After Suquet et al. [43]

Table 5 Copper single cristal. a Slip systems. b Hardening matrix

Table 6 Méric-Cailletaud model. Material parameters for copper. After [16]

1.3 C.3 Useful relations

The following definitions and relations can be useful to the reader:

$$\begin{aligned}&\displaystyle {\varvec{m}}_s({\varvec{x}}) = \sum _{g=1}^N {\varvec{m}}^{(g)}_s \chi ^{(g)}({\varvec{x}}), \\&\displaystyle {\tau }_s ({\varvec{x}}) = {\varvec{L}}({\varvec{x}}):{\varvec{A}}({\varvec{x}})::\overline{{\varvec{\varepsilon }}}\otimes {\varvec{m}}_s({\varvec{x}}) + \sum _{k=1}^{M_{{\textsf {v}}}} \xi _{{\textsf {v}}}^{(k)} {\rho }_s^{(k)}({\varvec{x}}),\\&{\rho }^{(k)}_s({\varvec{x}}) = {\varvec{\rho }}^{(k)}({\varvec{x}}):{\varvec{m}}_s({\varvec{x}}),\quad \displaystyle x_s({\varvec{x}}) = c_s \sum _{\ell =1}^{M_{\beta _s}} \xi _{\beta _s}^{(\ell )} \mu _{\beta _s}^{(\ell )}({\varvec{x}}), \\&r_s({\varvec{x}}) = \sum _{g=1}^{N} r_{s}^{(g)} \chi ^{(g)}({\varvec{x}}), \quad r_s^{(g)}=r_s({\varvec{p}}^{(g)}), \\&\displaystyle \overline{\tau }^{(g)}_s = \left\langle \tau _s \right\rangle ^{(g)} = \left\langle {\varvec{L}}:{\varvec{A}} \right\rangle ^{(g)}::\overline{{\varvec{\varepsilon }}}\otimes {\varvec{m}}_s^{(g)} + \sum _{k=1}^{M_{{\textsf {v}}}} \xi _{{\textsf {v}}}^{(k)} \overline{\rho }_s^{(k,g)},\\&\overline{\rho }^{(k,g)}_s = \left\langle {\rho }^{(k)}_s \right\rangle ^{(g)},\quad \displaystyle \overline{x}_s^{(g)} = c_s \sum _{\ell =1}^{M_{\beta _s}} \xi _{\beta _s}^{(\ell )} \overline{\mu }_{\beta _s}^{(\ell ,g)}, \quad \\&\displaystyle \overline{\mu }_{\beta _s}^{(\ell ,g)} = \left\langle \mu _{\beta _s}^{(\ell )} \right\rangle ^{(g)},\quad \frac{\partial \overline{\tau }^{(g)}_s }{\partial \xi ^{(k)}_{{\textsf {v}}}} = \overline{\rho }^{(k,g)}_s, \quad \frac{\partial \overline{x}_s^{(g)} }{\partial \xi ^{(\ell )}_{\beta _s}} = c_s \overline{\mu }^{(\ell ,g)}_{\beta _s},\\&\displaystyle \frac{\partial C^{(g)}(\tau _s-x_s)}{\partial \xi ^{(k)}_{{\textsf {v}}}} = 2 C^{(g)}\left( \tau _s-x_s, {\rho }_s^{(k)}\right) \\&\qquad = 2 \left\langle \left( \tau _s - \overline{\tau }^{(g)}_s -x_s+\overline{x}^{(g)}_s\right) \left( {\rho }_s^{(k)} - \overline{\rho }_s^{(k,g)}\right) \right\rangle ^{(g)},\\&\displaystyle \frac{\partial C^{(g)}(\tau _s-x_s)}{\partial \xi ^{(\ell )}_{\beta _s}} = -2c_s C^{(g)}\left( \tau _s-x_s, \mu _{\beta _s}^{(\ell )}\right) \\&\qquad = \displaystyle - 2 c_s \left\langle \left( \tau _s - \overline{\tau }^{(g)}_s -x_s+\overline{x}^{(g)}_s \right) \left( {\mu }_{\beta _s}^{(\ell )}-\overline{\mu }_{\beta _s}^{(\ell ,g)}\right) \right\rangle ^{(g)}. \end{aligned}$$

In the above, the symbol  :  :  is to be understood as the full contraction of two fourth-order tensors over their four indices.

Appendix 4: Material data for ice

Ice $I_h$ is an HCP material with 12 slip systems. The list of the slip sytems can be found in Table 3. The hardening matrix is taken in the simple form shown in Table 3. It depends on 4 independent parameters. The hardening coefficients identified in Suquet et al. [43] are

$$\begin{aligned} h_1=70 \ \text {MPa},\quad h_2=110 \ \text {MPa},\quad h_3=125 \ \text {MPa},\quad h_4=0. \end{aligned}$$

(80)

The elastic moduli are (in MPa),

$$\begin{aligned}&L_{1111}=L_{2222}=13 930,\ L_{1122}=7 082,\ L_{1133}=5 765,\\&L_{2323}=L_{1313}=3 014,\ L_{1212}=3 424. \end{aligned}$$

The remaining material parameters are identical within the same family of slip systems, but differ from one family to the other (see Table 4).

Appendix 5: Material data for copper

Copper is a FCC material with twelve slip systems. The list of the slip sytems can be found in Table 5.

The cubic symmetries can be used to reduce the hardening matrix to 6 parameters $h_i$: self-hardening is expressed through $h_0$, whereas latent hardening is expressed by the other 5 $h_i$’s.

As suggested by [15] (chapter 6), the elastic moduli are taken to be isotropic (although this is probably a crude approximation) with Young’s modulus $E=120{,}000$ MPa and Poisson ratio $\nu =1/3$. Following [16] the four coefficients $c_s$, $d_s$, $r_{0,s}$, $b_s$ were taken to be the same for all systems (no dependence on s) (see Table 6).