A study on the mechanical response of magnesium using an anisotropic elasticity twinning CP FEM

Abstract

Magnesium and its alloys are used in many applications due to their high strength-to-weight ratios. The poor formability which is caused by lack of available slip systems and the existence of twinning has been the major hindrance in expanding their usage to other new applications. In order to improve the formability of magnesium and its alloys, it is critical to understand the characteristics of the available slip and twin systems and the interactions among them. The slip activities occur due to dislocation motions, and twinning and dislocation motions interact with each other in order to reduce the overall plastic dissipation energy in deformation. Since the complex interaction hardening among slip and twin modes is not automatically realized by conventional standard CP (crystal plasticity) theories, additional measures must be included in the crystal plasticity constitutive theory to accurately represent the mechanical behavior of magnesium and its alloys. This paper takes into account interaction hardening among slip and twin modes by employing an interaction hardening model based on a physical property (saturation strength), which reduces trials and errors significantly in the stress–strain data fitting process. Taking into account elastic anisotropy, a twinning CP theory is proposed and an implicit time integration scheme for the proposed anisotropic elasticity twinning CP theory is derived in this study. The derived CP theory and the implicit time integration scheme are implemented into a large deformation FE code, and the single crystal channel-die compression tests and polycrystal uniaxial tension/compression tests of magnesium done in [1] are successfully reproduced by simulations. Using the anisotropic elasticity twinning CP FE code, the effects of strong elastic anisotropy on the convergence and stability of CP FE codes are investigated. Strong elastic anisotropy turns out to lower the stability and accuracy of CP FE codes, and the proposed implicit time integration scheme successfully overcomes these difficulties caused by strong elastic anisotropy.

Appendix A. Jacobian matrix

The expressions for components of the Jacobian matrix $\bigg [ \frac{\partial [{{\mathbf {R}}}]}{\partial [{{\mathbf {V}}}]} \bigg ]$ are provided as follows:

$$\begin{aligned} \frac{\partial {\mathbf {R}}_{\varvec{\varepsilon }^*}}{\partial \varvec{\varepsilon }^*} = \frac{1}{\Delta t }\mathbb {S} + \frac{\partial \hat{{{\mathbf {D}}}}^{\mathrm{p}}}{\partial \varvec{\varepsilon }^*} +{\mathcal {N}} \end{aligned}$$

(A.1)

where $\mathbb {S}$ denotes the fourth order identity matrix, i.e., $\mathbb {S}_{ijkl} = \frac{1}{2}(\delta _{ik} \delta _{jl} + \delta _{il} \delta _{jk} )$,⁴ and the fourth-order tensor $\frac{\partial \hat{{{\mathbf {D}}}}^\mathrm{p}}{\partial \varvec{\varepsilon }^*}$ is defined as

$$\begin{aligned} \frac{\partial {\hat{D}}^{\mathrm{p}}_{ij}}{\partial {\varepsilon }^*_{kl}} = \left\{ \begin{aligned}&\sum _{\alpha =1}^{N_s} n {\dot{\gamma }}^0_{sl} \bigg | \frac{\mathbb {C}^{\mathrm{e}}:\varvec{\varepsilon }^*:\hat{{{\mathbf {P}}}}^{\alpha }}{\tau ^{(i)}_0} \bigg |^{n-1} \frac{1}{\tau ^{(i)}_0} {\hat{P}}^{\alpha }_{ij} \mathbb {C}^{\mathrm{e}}_{pqrs} \mathbb {S}_{rskl} {\hat{P}}^{\alpha }_{pq} \;+ \\&\sum _{\beta =1}^{N_{tw}}n {\dot{\gamma }}^0_{tw} \bigg <\frac{\mathbb {C}^{\mathrm{e}}:\varvec{\varepsilon }^*:\hat{{\mathbf {P}}}^{\beta }}{\tau ^{(i)}_0} \bigg >^{n-1} \frac{1}{\tau ^{(i)}_0} {\hat{P}}^{\beta }_{ij} \mathbb {C}^{\mathrm{e}}_{pqrs} \mathbb {S}_{rskl} {\hat{P}}^{\beta }_{pq} \end{aligned} \right. \end{aligned}$$

(A.3)

and the fourth-order tensor ${\mathcal {N}}$ is defined as:

$$\begin{aligned}&{\mathcal {N}}_{ijkl} = \mathbb {S}_{imkl}{\hat{W}}^{\mathrm{p}}_{mj} +\varepsilon ^*_{im}\frac{\partial {\hat{W}}^{\mathrm{p}}_{mj}}{\partial \varepsilon ^*_{kl}} -\bigg [ \frac{\partial {\hat{W}}^\mathrm{p}_{im}}{\partial \varepsilon ^*_{kl}} \varepsilon ^*_{mj} +{\hat{W}}^{\mathrm{p}}_{im} \mathbb {S}_{mjkl} \bigg ] \end{aligned}$$

(A.4)

$$\begin{aligned}&\frac{\partial {\mathbf {R}}_{\varvec{\varepsilon }^*}}{\partial f^{\alpha }} = {\mathbf {0}}_{( 9 \mathrm{x} N_{tw})} \end{aligned}$$

(A.5)

where ${\mathbf {0}}_{( 9 \mathrm{x} N_{tw})}$ denotes a zero tensor of which dimension is 9 by $N_{tw}$ and components are all zero.

$$\begin{aligned} \frac{\partial {\mathbf {R}}_{\varvec{\varepsilon }^*}}{\partial \tau _0^{(i)}}= & {} \frac{\partial \hat{{\mathbf {D}}}^{\mathrm{p}}}{\partial \tau _0^{(i)}} + \bigg (\varvec{\varepsilon }^*\frac{\partial \hat{{\mathbf {W}}}^{\mathrm{p}}}{\partial \tau _0^{(i)}} - \frac{\partial \hat{{\mathbf {W}}}^{\mathrm{p}}}{\partial \tau _0^{(i)}} \varvec{\varepsilon }^* \bigg ) \end{aligned}$$

(A.6)

$$\begin{aligned} \frac{\partial \hat{{\mathbf {D}}}^{\mathrm{p}}}{\partial \tau _0^{(i)}}= & {} \sum _{\alpha =1}^{N_s} \frac{\partial {\dot{\gamma }}^{\alpha }}{\partial \tau _0^{(i)}} \hat{{\mathbf {P}}}^{\alpha } + \sum _{\beta =1}^{N_{tw}} \frac{\partial {\dot{\gamma }}^{\beta }}{\partial \tau _0^{(i)}} \hat{{\mathbf {P}}}^{\beta } \end{aligned}$$

(A.7)

$$\begin{aligned} \frac{\partial {\dot{\gamma }}^{\alpha }}{\partial \tau _0^{(i)}}= & {} n {\dot{\gamma }}^0_{sl} \bigg | \frac{\mathbb {C}^\mathrm{e}:\varvec{\varepsilon }^*:\hat{{\mathbf {P}}}^{\alpha }}{\tau _0^{(i)}} \bigg |^{(n-1)} (\mathbb {C}^\mathrm{e}:\varvec{\varepsilon }^*:\hat{{\mathbf {P}}}^{\alpha }) (-1) (\tau _0^{(i)})^{-2} \end{aligned}$$

(A.8)

$$\begin{aligned} \frac{\partial {\dot{\gamma }}^{\beta }}{\partial \tau _0^{(i)}}= & {} n {\dot{\gamma }}^0_{tw} \bigg < \frac{\mathbb {C}^\mathrm{e}:\varvec{\varepsilon }^*:\hat{{\mathbf {P}}}^{\beta }}{\tau _0^{(i)}} \bigg >^{(n-1)} (\mathbb {C}^\mathrm{e}:\varvec{\varepsilon }^*:\hat{{\mathbf {P}}}^{\beta }) (-1) (\tau _0^{(i)})^{-2} \end{aligned}$$

(A.9)

$$\begin{aligned} \frac{\partial {R}_{f^{\alpha }}}{\partial \varvec{\varepsilon }^*}= & {} \left\{ \begin{array}{l l l} \displaystyle { - n\frac{{\dot{\gamma }}^0_{tw}}{\gamma ^{tw}} \bigg< \frac{\mathbb {C}^{\mathrm{e}}:\varvec{\varepsilon }^*:\hat{{\mathbf {P}}}^{\alpha }}{{\tau _0^{(i)}}} \bigg>^{n-1} \frac{1}{\tau _0^{(i)}} \mathbb {C}^{\mathrm{e}}:\mathbb {S}:\hat{{\mathbf {P}}}^{\alpha } , \;\; \mathrm{for} \;\; f^{\alpha }< f_{\mathrm{crit}} \;\mathrm{and} \;\;{\bar{r}}^{(i)}< {\bar{r}}_{\mathrm{crit}}} \\ \displaystyle { - n\frac{{\dot{\gamma }}^0_{tw}}{\gamma ^{tw}} \bigg < \frac{\mathbb {C}^\mathrm{e}:\varvec{\varepsilon }^*:\hat{{\mathbf {P}}}^{\alpha }}{{\tau _0^{(i)}}} \bigg >^{n-1} \frac{1}{\tau _0^{(i)}} \mathbb {C}^\mathrm{e}:\mathbb {S}:\hat{{\mathbf {P}}}^{\alpha }{\bar{r}}^{(i)} , \;\; \mathrm{for} \;\; f^{\alpha } \ge f_{\mathrm{crit}} \;\mathrm{or} \;\;{\bar{r}}^{(i)} \ge {\bar{r}}_{\mathrm{crit}}} \end{array} \right. \end{aligned}$$

(A.10)

$$\begin{aligned}&\frac{\partial {R}_{f^{\alpha }}}{\partial f^{\beta }} = \frac{1}{\Delta t} \delta _{\alpha \beta } \end{aligned}$$

(A.11)

$$\begin{aligned}&\begin{aligned}&\frac{\partial {R}_{f^{\alpha }}}{\partial \tau _0^{(i)}} = \left\{ \begin{array}{l l l l l} \displaystyle { 0, \;\; \mathrm{for } \; \mathrm{i=1}\;\mathrm{to}\;5 }\\ \\ \displaystyle { - n\frac{{\dot{\gamma }}^0_{tw}}{\gamma ^{tw}} \bigg< \frac{\mathbb {C}^\mathrm{e}:\varvec{\varepsilon }^*:\hat{{\mathbf {P}}}^{\alpha }}{{\tau _0^{(i)}}} \bigg>^{n-1} (\mathbb {C}^\mathrm{e}:\varvec{\varepsilon }^*:\hat{{\mathbf {P}}}^{\alpha }) (-1)(\tau _0^{(i)})^{-2}, \; \mathrm{for} \;\; f^{\alpha }< f_{\mathrm{crit}} \;\mathrm{and} \;\;{\bar{r}}^{(i)}< {\bar{r}}_{\mathrm{crit}}} \\ \displaystyle { - n\frac{{\dot{\gamma }}^0_{tw}}{\gamma ^{tw}} \bigg < \frac{\mathbb {C}^\mathrm{e}:\varvec{\varepsilon }^*:\hat{{\mathbf {P}}}^{\alpha }}{{\tau _0^{(i)}}} \bigg >^{n-1} (\mathbb {C}^\mathrm{e}:\varvec{\varepsilon }^*:\hat{{\mathbf {P}}}^{\alpha }) (-1)(\tau _0^{(i)})^{-2} {\bar{r}}^{(i)}, \; \mathrm{for} \;\; f^{\alpha } \ge f_{\mathrm{crit}} \;\mathrm{or} \;\;{\bar{r}}^{(i)} \ge {\bar{r}}_\mathrm{crit}} \end{array} \right. \end{aligned}\qquad \end{aligned}$$

(A.12)

$$\begin{aligned}&\begin{aligned}&h^{(i)}(\bar{\gamma }) = \left\{ \begin{array}{l l l l l} \displaystyle { h_0 \bigg ( 1- \frac{\tau _0}{\zeta ^{(i)}\kappa ^{(i)}(\tau _{\infty })_0^{(i)}} \bigg ) \exp {\bigg ( - \frac{h_0(\bar{\gamma }_0+\sum _{i=1}^{N_\mathrm{modes}}\Delta \bar{\gamma }^{(i)})}{\zeta ^{(i)}\kappa ^{(i)}(\tau _{\infty })_0^{(i)}} \bigg )}, \;\; \text {for i=1 to 5} } \\ \\ \left\{ \begin{array}{l l l} h_0 \times \eta , \;\;\;\mathrm{for}\; \bar{\gamma }^{(6)}\le \gamma _{\mathrm{ref}} \\ \displaystyle { h_0 \left( \frac{\bar{\gamma }^{(6)}}{\gamma _\mathrm{ref}} \right) ^{m-1} \times \eta }, \;\;\;\mathrm{for}\; \bar{\gamma }^{(6)} > \gamma _{\mathrm{ref}} \end{array} \right\} \; \text {for i=6 } \\ \displaystyle { h_0}, \; \;\; \text { for i=7} \end{array} \right. \end{aligned} \end{aligned}$$

(A.13)

$$\begin{aligned}&\zeta ^{(i)}=[ \xi , \zeta , 1, 1, 1]^T \end{aligned}$$

(A.14)

$$\begin{aligned}&\kappa ^{(i)}=[ 1, 1, 1, \kappa , \kappa ]^T \end{aligned}$$

(A.15)

$$\begin{aligned}&(\tau _{\infty })_0^{(i)}=[ 3, 73.44, 86.4, 198, 165]^T \end{aligned}$$

(A.16)

$$\begin{aligned}&\Delta \bar{\gamma }^{(i)} = \left\{ \begin{array}{l l l} \displaystyle { \sum _{\alpha =1}^{N^{(i)}}{\dot{\gamma }}^0_{sl} \bigg | \frac{\mathbb {C}^{\mathrm{e}}:\varvec{\varepsilon }^*:\hat{{\mathbf {P}}}^{\alpha }}{{\tau _0^{(i)}}} \bigg |^{n} \Delta t , \;\; \mathrm{for } \; \mathrm{i=1}\;\mathrm{to}\;5} \\ \displaystyle { \sum _{\beta =1}^{N^{(i)}}{\dot{\gamma }}^0_{tw} \bigg < \frac{\mathbb {C}^\mathrm{e}:\varvec{\varepsilon }^*:\hat{{\mathbf {P}}}^{\beta }}{{\tau _0^{(i)}}} \bigg >^{n} \Delta t , \;\; \mathrm{for } \; \mathrm{i=6}\; \mathrm{and} \;7} \end{array} \right. \end{aligned}$$

(A.17)

Note that $N^{(i)}$ is the number of systems in a slip or twin mode.

$$\begin{aligned}&\frac{\partial {\mathbf {R}}_{\tau _0^{(i)}}}{\partial \varvec{\varepsilon }^*} = -\sum _{j=1}^{N_{\mathrm{modes}}}\frac{\partial h^{(i)}}{\partial \Delta \bar{\gamma }^{(j)}} \frac{\partial \Delta \bar{\gamma }^{(j)}}{\partial \varvec{\varepsilon }^*} \Delta \bar{\gamma }^{(i)} -h^{(i)}\frac{\partial \Delta \bar{\gamma }^{(i)}}{\partial \varvec{\varepsilon }^*}\nonumber \\&\quad -\sum _{j=1}^{N_{\mathrm{modes}}} q_{(i)(j)} \frac{\partial \Delta \bar{\gamma }^{(j)}}{\partial \varvec{\varepsilon }^*}, \;\; \text { no summation on (i)} \end{aligned}$$

(A.18)

$$\begin{aligned}&\begin{aligned}&\frac{\partial \Delta \bar{\gamma }^{(i)}}{\partial \varvec{\varepsilon }^*} = \left\{ \begin{array}{l l l} \displaystyle { \sum _{\alpha =1}^{N^{(i)}} n {\dot{\gamma }}^0_{sl} \bigg | \frac{\mathbb {C}^{\mathrm{e}}:\varvec{\varepsilon }^*:\hat{{\mathbf {P}}}^{\alpha }}{{\tau _0^{(i)}}} \bigg |^{n-1} \mathrm{sgn}(\mathbb {C}^{\mathrm{e}}:\varvec{\varepsilon }^*:\hat{{\mathbf {P}}}^{\alpha }) \frac{1}{\tau _0^{(i)}} (\mathbb {C}^{\mathrm{e}}:\mathbb {S}:\hat{{\mathbf {P}}}^{\alpha }) \Delta t , \;\; \mathrm{for } \; \mathrm{i=1}\;\mathrm{to}\;5 }\\ \displaystyle { \sum _{\beta =1}^{N^{(i)}} n {\dot{\gamma }}^0_{tw} \bigg < \frac{\mathbb {C}^\mathrm{e}:\varvec{\varepsilon }^*:\hat{{\mathbf {P}}}^{\beta }}{{\tau _0^{(i)}}} \bigg >^{n-1} \frac{1}{\tau _0^{(i)}} (\mathbb {C}^\mathrm{e}:\mathbb {S}:\hat{{\mathbf {P}}}^{\beta }) \Delta t, \;\; \mathrm{for} \; \mathrm{i=6}\;\mathrm{and}\;7} \end{array} \right. \end{aligned} \end{aligned}$$

(A.19)

$$\begin{aligned}&\begin{aligned}&\frac{\partial h^{(i)}}{\partial \Delta \bar{\gamma }^{(j)}} = \left\{ \begin{array}{l l l | |} \left. \begin{aligned} &{}\displaystyle { h_0 \bigg ( 1- \frac{\tau _0}{\zeta ^{(i)}\kappa ^{(i)}(\tau _{\infty })_0^{(i)}} \bigg ) \exp {\bigg ( - \frac{h_0 \bar{\gamma }}{\zeta ^{(i)}\kappa ^{(i)}(\tau _{\infty })_0^{(i)}} \bigg ) } \bigg ( -\frac{h_0}{\zeta ^{(i)}\kappa ^{(i)}(\tau _{\infty })_0^{(i)}} \bigg )} \\ &{}\displaystyle { + h_0 \bigg ( - \frac{\tau _0}{\kappa ^{(i)}(\tau _{\infty })_0^{(i)}} \bigg ) (-1)(\zeta ^{(i)})^{-2} \frac{\partial \zeta ^{(i)}}{\partial \Delta {\bar{\gamma }}^{(j)}} \exp {\bigg ( - \frac{h_0 \bar{\gamma }}{\zeta ^{(i)}\kappa ^{(i)}(\tau _{\infty })_0^{(i)}} \bigg ) } \delta _{i2} } \\ &{}\displaystyle { + h_0 \bigg ( 1- \frac{\tau _0}{\zeta ^{(i)}\kappa ^{(i)}(\tau _{\infty })_0^{(i)}} \bigg ) \exp {\bigg ( - \frac{h_0 \bar{\gamma }}{\zeta ^{(i)}\kappa ^{(i)}(\tau _{\infty })_0^{(i)}} \bigg ) } \bigg ( -\frac{h_0 {\bar{\gamma }}}{\kappa ^{(i)}(\tau _{\infty })_0^{(i)}} \bigg )*} \\ &{}\displaystyle {(-1)(\zeta ^{(i)})^{-2} \frac{\partial \zeta ^{(i)}}{\partial \Delta {\bar{\gamma }}^{(j)}}\delta _{i2} } \end{aligned} \;\; \right\} \mathrm{for } \; \mathrm{i=1}\;\mathrm{to}\;5\\ \left\{ \begin{aligned} &{}\displaystyle { \delta _{1j} h_0 \frac{\partial \eta }{\partial {\bar{\gamma }}^{(1)}}, \;\; \mathrm{for}\;\; {\bar{\gamma }}^{(6)} \le \gamma _{\mathrm{ref}} } \\ &{}\displaystyle { \delta _{6j}h_0(m-1)\bigg (\frac{{\bar{\gamma }}^{(6)}}{\gamma _\mathrm{ref}}\bigg )^{m-2}\frac{ \eta }{\gamma _{\mathrm{ref}}} + \delta _{1j} h_0 \bigg (\frac{{\bar{\gamma }}^{(6)}}{\gamma _{\mathrm{ref}}}\bigg )^{m-1} \frac{\partial \eta }{\partial {\bar{\gamma }}^{(1)}}, \;\; \mathrm{for}\;\; {\bar{\gamma }}^{(6)} > \gamma _{\mathrm{ref}}} \end{aligned} \;\; \right\} \mathrm{for} \; \mathrm{i=6} \\ \left. \displaystyle { 0 } \;\; \right\} \mathrm{for} \; \mathrm{i=7} \end{array} \right. \end{aligned} \end{aligned}$$

(A.20)

$$\begin{aligned}&\left\{ \begin{array}{l} \begin{aligned} &{}\frac{\partial \zeta ^{(i)}}{\partial \Delta {\bar{\gamma }}^{(j)}} = 0, \;\; \mathrm{except} \\ &{} \frac{\partial \zeta ^{(2)}}{\partial \Delta {\bar{\gamma }}^{(1)}} = \left\{ \begin{array}{l l l} 0, \;\; \mathrm{for} \; {\bar{\gamma }}^{(1)} \le {\bar{\gamma }}^{(1)}_{\mathrm{crit}} \\ 2.7563*0.2160*({\bar{\gamma }}^{(1)})^{(0.2160-1)}, \;\; \mathrm{for} \; {\bar{\gamma }}^{(1)}> {\bar{\gamma }}^{(1)}_{\mathrm{crit}} \end{array} \right. \\ &{} \frac{\partial \zeta ^{(2)}}{\partial \Delta {\bar{\gamma }}^{(5)}} = \left\{ \begin{array}{l l l} 0, \;\; \mathrm{for} \; {\bar{\gamma }}^{(5)} \le {\bar{\gamma }}^{(5)}_{\mathrm{crit}} \\ 1.559*0.0558*({\bar{\gamma }}^{(5)})^{(0.0558-1)}, \;\; \mathrm{for} \; {\bar{\gamma }}^{(5)} > {\bar{\gamma }}^{(5)}_{\mathrm{crit}} \end{array} \right. \end{aligned} \end{array} \right\} \end{aligned}$$

(A.21)

$$\begin{aligned}&\frac{\partial \eta }{\partial {\bar{\gamma }}^{(1)}} = 0.1 \times 10.5 \times (\gamma ^{(1)})^{-0.9} \end{aligned}$$

(A.22)

$$\begin{aligned}&\frac{\partial {\mathbf {R}}_{\tau _0^{(i)}}}{\partial f^{\alpha }} = -\frac{\partial h^{(i)}}{\partial f^{\alpha }} \Delta \bar{\gamma }^{(i)},\; \text {no summation on (i)} \end{aligned}$$

(A.23)

$$\begin{aligned}&\begin{aligned}&\frac{\partial h^{(i)}}{\partial f^{\alpha }}= \left\{ \begin{array}{l l l} \left. \begin{array}{l l l | |} \displaystyle { h_0 \bigg ( - \frac{\tau _0}{\zeta ^{(i)}(\tau _{\infty })_0^{(i)}} \bigg ) (-1)(\kappa ^{(i)})^{-2} \frac{\partial \kappa ^{(i)}}{\partial f^{\alpha }} \exp {\bigg ( - \frac{h_0 \bar{\gamma }}{\zeta ^{(i)}\kappa ^{(i)}(\tau _{\infty })_0^{(i)}} \bigg ) } } \\ \displaystyle { + h_0 \bigg ( 1 - \frac{\tau _0}{\zeta ^{(i)}\kappa ^{(i)}(\tau _{\infty })_0^{(i)}} \bigg ) \exp {\bigg ( - \frac{h_0 \bar{\gamma }}{\zeta ^{(i)}\kappa ^{(i)}(\tau _{\infty })_0^{(i)}} \bigg ) } \times }\\ \displaystyle { \bigg \{ - \frac{h_0 \bar{\gamma }}{\zeta ^{(i)}(\tau _{\infty })_0^{(i)}} \bigg \} (-1)(\kappa ^{(i)})^{-2} \frac{\partial \kappa ^{(i)}}{\partial f^{\alpha }} } \\ \displaystyle {+ h_0 \bigg ( - \frac{\tau _0}{\kappa ^{(i)}(\tau _{\infty })_0^{(i)}} \bigg ) (-1)(\zeta ^{(i)})^{-2} \frac{\partial \zeta ^{(i)}}{\partial f^{\alpha }} \exp {\bigg ( - \frac{h_0 \bar{\gamma }}{\zeta ^{(i)}\kappa ^{(i)}(\tau _{\infty })_0^{(i)}} \bigg ) } } \\ \displaystyle { + h_0 \bigg ( 1 - \frac{\tau _0}{\zeta ^{(i)}\kappa ^{(i)}(\tau _{\infty })_0^{(i)}} \bigg ) \exp {\bigg ( - \frac{h_0 \bar{\gamma }}{\zeta ^{(i)}\kappa ^{(i)}(\tau _{\infty })_0^{(i)}} \bigg ) } \times }\\ \displaystyle { \bigg \{ - \frac{h_0 \bar{\gamma }}{\kappa ^{(i)}(\tau _{\infty })_0^{(i)}} \bigg \} (-1)(\zeta ^{(i)})^{-2} \frac{\partial \zeta ^{(i)}}{\partial f^{\alpha }} } \end{array} \right\} \;\;\mathrm{for } \; \text { i=1 to 5}\\ \displaystyle {0, \;\; \text {for i=6 and 7}} \end{array} \right. \end{aligned} \end{aligned}$$

(A.24)

$$\begin{aligned}&\begin{aligned}&\frac{\partial \kappa ^{(i)}}{\partial f^{\alpha }}= \\&\left\{ \begin{array}{l l l} \displaystyle {0, \;\; \text {for i=1 to 3}}\\ \left\{ \begin{array}{l l l} \displaystyle { 3.3892\times 0.5956\times f_{TT}^{(0.5956-1)}\times \delta _{1-6,\alpha }+7.5\times \delta _{7-12,\alpha }, \text { for } ;f_{TT} \le f_{\mathrm{crit}}} \\ \left\{ \begin{array}{l l l} \displaystyle { 4.2128\times 10^{-11}\times \exp {(41.9051\times f_{TT})} } \\ \displaystyle { \times 41.9051\times \delta _{1-6,\alpha }+7.5\times \delta _{7-12,\alpha }} \end{array} \right\} , \;\; \mathrm{for}\;f_{TT} > f_{\mathrm{crit}} \end{array} \right\} \;\; \mathrm{for } \; \mathrm{i=4}\; \mathrm{and } \;5 \end{array} \right. \end{aligned} \end{aligned}$$

(A.25)

$$\begin{aligned}&\frac{\partial \zeta ^{(i)}}{\partial f^{\alpha }}= \left\{ \begin{aligned}&\displaystyle {0.1\times 13.49 \times f_{TT}^{-0.9}\times \delta _{1-6,\alpha }, \;\; \mathrm{for} \; \mathrm{i=1}} \\&0, \;\;\mathrm{for } \; \mathrm{i=2} \;\mathrm{to}\;5 \end{aligned} \right. \end{aligned}$$

(A.26)

where $\delta _{1-6,\alpha }$ and $\delta _{7-12,\alpha }$ are defined as follows:

$$\begin{aligned}&\delta _{1-6,\alpha } = \left\{ \begin{array}{l l l} 1, \;\; if\;\alpha =1 \;to\; 6 \\ 0, \;\; if\;\alpha =7 \;to\; 12 \end{array} \right. \end{aligned}$$

(A.27)

$$\begin{aligned}&\delta _{7-12,\alpha } = \left\{ \begin{array}{l l l} 0, \;\; if\;\alpha =1 \;to\; 6 \\ 1, \;\; if\;\alpha =7 \;to\; 12 \end{array} \right. \end{aligned}$$

(A.28)

$$\begin{aligned}&\begin{aligned} \frac{\partial {\mathbf {R}}_{\tau _0^{(i)}}}{\partial \tau _0^{(j)}}=\delta _{ij} -\bigg ( \frac{\partial h^{(i)}}{\partial \Delta {\bar{\gamma }}^{(j)}} \frac{\partial \Delta {\bar{\gamma }}^{(j)}}{\partial \tau _0^{(j)}} \Delta {\bar{\gamma }}^{(i)} + h^{(i)} \delta _{ij} \frac{\partial \Delta {\bar{\gamma }}^{(i)}}{\partial \tau _0^{(j)}} \bigg ) \\ -q_{(i)(j)}\frac{\partial \Delta {\bar{\gamma }}^{(j)}}{\partial \tau _0^{(j)}},\;\;\mathrm{no\;summation\;on\;}(i)\;\mathrm{and}\; (j) \end{aligned} \end{aligned}$$

(A.29)

$$\begin{aligned}&\begin{aligned}&\frac{\partial \Delta {\bar{\gamma }}^{(j)}}{\partial \tau _0^{(j)}} = \left\{ \begin{array}{l l l} \displaystyle { \delta _{ij}\sum _{\alpha =1}^{N^{(i)}} n {\dot{\gamma }}^0_{sl} \bigg | \frac{\mathbb {C}^{\mathrm{e}}:\varvec{\varepsilon }^*:\hat{{\mathbf {P}}}^{\alpha }}{{\tau _0^{(i)}}} \bigg |^{n-1} \mathrm{sgn}(\mathbb {C}^{\mathrm{e}}:\varvec{\varepsilon }^*:\hat{{\mathbf {P}}}^{\alpha }) (\mathbb {C}^{\mathrm{e}}:\varvec{\varepsilon }^*:\hat{{\mathbf {P}}}^{\alpha }) (-1)( \tau _0^{(i)})^{-2} \Delta t , \;\; \mathrm{for} \; \mathrm{i=1}\;\mathrm{to}\;5 }\\ \displaystyle { \delta _{ij}\sum _{\beta =1}^{N^{(i)}} n {\dot{\gamma }}^0_{tw} \bigg < \frac{\mathbb {C}^\mathrm{e}:\varvec{\varepsilon }^*:\hat{{\mathbf {P}}}^{\beta }}{{\tau _0^{(i)}}} \bigg >^{n-1} (\mathbb {C}^\mathrm{e}:\varvec{\varepsilon }^*:\hat{{\mathbf {P}}}^{\beta }) (-1)( \tau _0^{(i)})^{-2} \Delta t, \;\; \mathrm{for} \; \mathrm{i=6}\;\mathrm{and}\;7} \end{array} \right. \end{aligned} \end{aligned}$$

(A.30)