Higher-Order Peridynamic Material Correspondence Models for Elasticity

Abstract

Higher-order peridynamic material correspondence model can be developed based on the formulation of higher-order deformation gradient and constitutive correspondence with generalized continuum theories. In this paper, we present formulations of higher-order peridynamic material correspondence models adopting the material constitutive relations from the strain gradient theories. Similar to the formulation of the first-order deformation gradient, the weighted least squares technique is employed to construct the second-order and the third-order deformation gradients. Force density states are then derived as the Fréchet derivatives of the free energy density with respect to the deformation states. Connections to the second-order and the third-order strain gradient elasticity theories are established by realizing the relationships between the energy conjugate stresses of the higher-order deformation gradients in peridynamics and the stress measures in strain gradient theories. In addition to the horizon, length-scale parameters from strain gradient theories are explicitly incorporated into the higher-order peridynamic material correspondence models, which enables application of peridynamics theory to materials at micron and sub-micron scales where length-scale effects are significant.

Appendix A: Derivation of the Deformation Gradients and Force Density State for the Third-Order Material Correspondence Formulation

1.1 A.1 Deformation Gradients up to the Third Order

In the limit of finite bond length, the Taylor series expansion of deformation state $\underline{\mathbf{y}}$ by considering the differential terms up to the third-order can be written as

$$\begin{aligned} \underline{y}_{i} = F_{iI}^{\boldsymbol{\xi }} \underline{X}_{I} + \frac{1}{2} F_{\mathit{iIM}}^{(1), \boldsymbol{\xi }} \underline{X}_{I} \underline{X}_{M} + \frac{1}{6} F_{\mathit{iIMN}}^{(2), \boldsymbol{\xi }} \underline{X}_{I} \underline{X}_{M} \underline{X}_{N}, \end{aligned}$$

(A.1)

where $F_{iI}^{\boldsymbol{\xi }}$ and $F_{\mathit{iIM}}^{(1),\boldsymbol{\xi }}$ are given by, respectively, Eqs. (12) and (27), and

$$\begin{aligned} F_{\mathit{iIMN}}^{(2),\boldsymbol{\xi }} := \frac{\partial ^{3} \underline{y}_{i}}{\partial \underline{X}_{I} \underline{X}_{M} \underline{X}_{N}} \end{aligned}$$

(A.2)

is the mixed third-order derivative of the deformation state.

Here, the weighted squares of the errors are given by

$$\begin{aligned} \mathscr{E} = \int _{\mathscr{H}_{\mathbf{X}}}{\omega \Big( \underline{y}_{i} - F_{iI} \underline{X}_{I} - \frac{1}{2} F_{\mathit{iIM}}^{(1)} \underline{X}_{I} \underline{X}_{M} - \frac{1}{6} F_{\mathit{iIMN}}^{(2)} \underline{X}_{I} \underline{X}_{M} \underline{X}_{N} \Big)^{2}dV_{ \mathbf{X}^{\prime }}}. \end{aligned}$$

(A.3)

Minimizing the weighted squares of errors yields

$$\begin{aligned} \frac{\partial \mathscr{E}}{\partial F_{iI}} &= 0, \end{aligned}$$

(A.4a)

$$\begin{aligned} \frac{\partial \mathscr{E}}{\partial F_{\mathit{iIM}}^{(1)}}& = 0, \end{aligned}$$

(A.4b)

$$\begin{aligned} \frac{\partial \mathscr{E}}{\partial F_{\mathit{iIMN}}^{(2)}} &= 0, \end{aligned}$$

(A.4c)

for all $i$, $I$, $M$ and $N \in$ [1, 2, 3]. These necessary conditions for a minimum yield the following equations

$$\begin{aligned} \big( L_{2} \big)_{\mathit{iQ}} - F_{iI} \big( K_{2} \big)_{\mathit{IQ}} - \frac{1}{2} F_{\mathit{iIM}}^{(1)} \big( K_{3} \big)_{\mathit{IMQ}} - \frac{1}{6} F_{\mathit{iIMN}}^{(2)} \big( K_{4} \big)_{\mathit{IMNQ}} &= 0, \end{aligned}$$

(A.5a)

$$\begin{aligned} \big( L_{3} \big)_{\mathit{iQO}} - F_{iI} \big( K_{3} \big)_{\mathit{IQO}} - \frac{1}{2} F_{\mathit{iIM}}^{(1)} \big( K_{4} \big)_{\mathit{IMQO}} - \frac{1}{6} F_{\mathit{iIMN}}^{(2)} \big( K_{5} \big)_{\mathit{IMNQO}} &= 0, \end{aligned}$$

(A.5b)

$$\begin{aligned} \big( L_{4} \big)_{\mathit{iQOR}} - F_{iI} \big( K_{4} \big)_{\mathit{IQOR}} - \frac{1}{2} F_{\mathit{iIM}}^{(1)} \big( K_{5} \big)_{\mathit{IMQOR}} - \frac{1}{6} F_{\mathit{iIMN}}^{(2)} \big( K_{6} \big)_{\mathit{IMNQOR}}& = 0, \end{aligned}$$

(A.5c)

where

$$\begin{aligned} \big( L_{4} \big)_{\mathit{iQOR}} &:= \int _{\mathscr{H}_{\mathbf{X}}}{\omega \underline{y}_{i} \underline{X}_{Q} \underline{X}_{O} \underline{X}_{R} dV_{\mathbf{X}^{\prime }}}, \end{aligned}$$

(A.6a)

$$\begin{aligned} \big( K_{5} \big)_{\mathit{IMQOR}}& := \int _{\mathscr{H}_{\mathbf{X}}}{\omega \underline{X}_{I} \underline{X}_{M} \underline{X}_{Q} \underline{X}_{O} \underline{X}_{R} dV_{\mathbf{X}^{\prime }}}, \end{aligned}$$

(A.6b)

$$\begin{aligned} \big( K_{6} \big)_{\mathit{IMNQOR}} &:= \int _{\mathscr{H}_{\mathbf{X}}}{ \omega \underline{X}_{I} \underline{X}_{M} \underline{X}_{N} \underline{X}_{Q} \underline{X}_{O} \underline{X}_{R} dV_{\mathbf{X}^{ \prime }}}. \end{aligned}$$

(A.6c)

The deformation gradient can be found from Eq. (A.5a) in terms of $\mathbf{F}^{(1)}$ and $\mathbf{F}^{(2)}$ as

$$\begin{aligned} F_{iJ} &= \big( L_{2} \big)_{\mathit{iQ}} \big( K_{2}^{-1} \big)_{\mathit{QJ}} - \frac{1}{2} F_{\mathit{iIM}}^{(1)} \big( K_{3} \big)_{\mathit{IMQ}} \big( K_{2}^{-1} \big)_{\mathit{QJ}} \\ &- \frac{1}{6} F_{\mathit{iIMN}}^{(2)} \big( K_{4} \big)_{\mathit{IMNQ}} \big( K_{2}^{-1} \big)_{\mathit{QJ}}. \end{aligned}$$

(A.7)

Substituting Eq. (A.7) into Eq. (A.5b), one obtains

$$\begin{aligned} \big( \widetilde{L}_{3} \big)_{\mathit{iQO}} - \frac{1}{2} F_{\mathit{iJM}}^{(1)} \big( \widetilde{K}_{4} \big)_{\mathit{JMQO}} - \frac{1}{6} F_{\mathit{iJMN}}^{(2)} \big( \widetilde{K}_{5} \big)_{\mathit{JMNQO}} = 0, \end{aligned}$$

(A.8)

with

$$\begin{aligned} \big( \widetilde{K}_{5} \big)_{\mathit{JMNQO}} := \big( K_{5} \big)_{\mathit{JMNQO}} - \big( K_{4} \big)_{\mathit{JMNR}} \big( K_{2}^{-1} \big)_{\mathit{RS}} \big( K_{3} \big)_{\mathit{SQO}}. \end{aligned}$$

(A.9)

It should be noted that following symmetric relationship exists for $\widetilde{\mathbf{K}}_{5}$ as

$$\begin{aligned} \widetilde{\mathbf{K}}_{5} = \mathbf{K}_{5} - \mathbf{K}_{4} \mathbf{K}_{2}^{-1} \mathbf{K}_{3} = \mathbf{K}_{5} - \mathbf{K}_{3} \mathbf{K}_{2}^{-1} \mathbf{K}_{4}. \end{aligned}$$

(A.10)

Appropriate products between tensors of different ranks are implied in Eq. (A.10).

The Hessian $\mathbf{F}^{(1)}$ can then be found from Eq. (A.8) in terms of $\mathbf{F}^{(2)}$ as

$$\begin{aligned} F_{\mathit{iJM}}^{(1)} = 2 \; \big( \widetilde{L}_{3} \big)_{\mathit{iQO}} \big( \widetilde{K}_{4}^{-1} \big)_{\mathit{QOJM}} - \frac{1}{3} F_{\mathit{iINR}}^{(2)} \big( \widetilde{K}_{5} \big)_{\mathit{INRQO}} \big( \widetilde{K}_{4}^{-1} \big)_{\mathit{QOJM}}. \end{aligned}$$

(A.11)

Substituting Eqs. (A.7) and (A.11) into Eq. (A.5c) yields

$$\begin{aligned} \big( \widetilde{L}_{4} \big)_{\mathit{iQOR}} - \frac{1}{6} F_{\mathit{iJMN}}^{(2)} \big( \widetilde{K}_{6} \big)_{\mathit{JMNQOR}} = 0, \end{aligned}$$

(A.12)

with

$$\begin{aligned} \big( \widetilde{L}_{4} \big)_{\mathit{iQOR}} &:= \big( L_{4} \big)_{\mathit{iQOR}} - \big( L_{2} \big)_{iI} \big( K_{2}^{-1} \big)_{\mathit{IJ}} \big( K_{4} \big)_{\mathit{JQOR}} \\ &- \big( \widetilde{L}_{3} \big)_{\mathit{iIS}} \big( \widetilde{K}_{4}^{-1} \big)_{\mathit{ISTV}} \big( \widetilde{K}_{5} \big)_{\mathit{TVQOR}}, \end{aligned}$$

(A.13)

$$\begin{aligned} \big( \widetilde{K}_{6} \big)_{\mathit{JMNQOR}} &:= \big( K_{6} \big)_{\mathit{JMNQOR}} - \big( K_{4} \big)_{JMNS} \big( K_{2}^{-1} \big)_{\mathit{ST}} \big( K_{4} \big)_{\mathit{TQOR}} \\ &- \big( \widetilde{K}_{5} \big)_{\mathit{JMNST}} \big( \widetilde{K}_{4}^{-1} \big)_{\mathit{STVW}} \big( \widetilde{K}_{5} \big)_{\mathit{VWQOR}}. \end{aligned}$$

(A.14)

Therefore, the mixed third-order derivative $\mathbf{F}^{(2)}$ can be found from Eq. (A.12) as

$$\begin{aligned} F_{\mathit{iJMN}}^{(2)} = 6 \; \big( \widetilde{L}_{4} \big)_{\mathit{iQOR}} \big( \widetilde{K}_{6}^{-1} \big)_{\mathit{QORJMN}}. \end{aligned}$$

(A.15)

Substituting Eq. (A.15) into Eq. (A.11), one obtains the final form of $\mathbf{F}^{(1)}$ as

$$\begin{aligned} F_{\mathit{iJM}}^{(1)} &= 2 \; \big( \widetilde{L}_{3} \big)_{\mathit{iQO}} \big( \widetilde{K}_{4}^{-1} \big)_{\mathit{QOJM}} \\ &- 2 \; \big( \widetilde{L}_{4} \big)_{\mathit{iSTV}} \big( \widetilde{K}_{6}^{-1} \big)_{\mathit{STVINR}} \big( \widetilde{K}_{5} \big)_{\mathit{INRQO}} \big( \widetilde{K}_{4}^{-1} \big)_{\mathit{QOJM}}. \end{aligned}$$

(A.16)

Substituting Eqs. (A.15) and (A.16) into Eq. (A.7), we obtain the final form of $\mathbf{F}$ as

$$\begin{aligned} F_{iJ} &= \big( L_{2} \big)_{iM} \big( K_{2}^{-1} \big)_{\mathit{MJ}} - \big( \widetilde{L}_{3} \big)_{\mathit{iQO}} \big( \widetilde{K}_{4}^{-1} \big)_{\mathit{QOIN}} \big( K_{3} \big)_{\mathit{INM}} \big( K_{2}^{-1} \big)_{\mathit{MJ}} \\ &- \big( \widetilde{L}_{4} \big)_{\mathit{iQOR}} \big( \widetilde{K}_{6}^{-1} \big)_{\mathit{QORINS}} \Big[ \big( K_{4} \big)_{\mathit{INSM}} \\ &- \big( \widetilde{K}_{5} \big)_{\mathit{INSTU}} \big( \widetilde{K}_{4}^{-1} \big)_{\mathit{TUVW}} \big( K_{3} \big)_{\mathit{VWM}} \Big] \big( K_{2}^{-1} \big)_{\mathit{MJ}}. \end{aligned}$$

(A.17)

1.2 A.2 Force Density State Based on Deformation Gradients up to the Third Order

The changes of the deformation gradient $\Delta F_{iJ}$, the Hessian $\Delta F_{\mathit{iJM}}^{(1)}$, and the mixed third-order derivative $\Delta F_{iJMN}^{(2)}$ resulted from increment in the deformation state $\Delta \underline{y}_{i}$ can be found as

$$\begin{aligned} \Delta F_{iJ} &= \Big( \int _{\mathscr{H}_{\mathbf{X}}}{\omega \Delta \underline{y}_{i} \underline{X}_{M}}dV_{\mathbf{X}^{\prime }} \Big) \big( K_{2}^{-1} \big)_{\mathit{MJ}} \\ &- \Big( \int _{\mathscr{H}_{\mathbf{X}}}{\omega \Delta \underline{y}_{i} \underline{X}_{Q} \underline{X}_{O}}dV_{\mathbf{X}^{\prime }} \Big) \big( \widetilde{K}_{4}^{-1} \big)_{\mathit{QOIN}} \big( K_{3} \big)_{\mathit{INM}} \big( K_{2}^{-1} \big)_{\mathit{MJ}} \\ &+ \Big( \int _{\mathscr{H}_{\mathbf{X}}}{\omega \Delta \underline{y}_{i} \underline{X}_{R}}dV_{\mathbf{X}^{\prime }} \Big) \big( K_{2}^{-1} \big)_{\mathit{RS}} \big( K_{3} \big)_{\mathit{SQO}} \big( \widetilde{K}_{4}^{-1} \big)_{\mathit{QOIN}} \big( K_{3} \big)_{\mathit{INM}} \big( K_{2}^{-1} \big)_{\mathit{MJ}} \\ &- \big(\Delta \widetilde{L}_{4} \big)_{\mathit{iQOR}} \Big( \widetilde{K}_{6}^{-1} \Big)_{\mathit{QORINS}} \Big[ \big( K_{4} \big)_{\mathit{INSM}} \\ &- \big( \widetilde{K}_{5} \big)_{\mathit{INSTU}} \big( \widetilde{K}_{4}^{-1} \big)_{\mathit{TUVW}} \big( K_{3} \big)_{\mathit{VWM}} \Big] \big( K_{2}^{-1} \big)_{\mathit{MJ}} \\ &= \nabla _{j} F_{iJ} \bullet \Delta \underline{y}_{j}, \end{aligned}$$

(A.18)

$$\begin{aligned} \Delta F_{\mathit{iJM}}^{(1)} &= 2 \Big( \int _{\mathscr{H}_{\mathbf{X}}}{ \omega \Delta \underline{y}_{i} \underline{X}_{Q} \underline{X}_{O}}dV_{ \mathbf{X}^{\prime }} \Big) \big( \widetilde{K}_{4}^{-1} \big)_{\mathit{QOJM}} \\ &- 2 \Big( \int _{\mathscr{H}_{\mathbf{X}}}{\omega \Delta \underline{y}_{i} \underline{X}_{I}}dV_{\mathbf{X}^{\prime }} \Big) \big( K_{2} \big)^{-1}_{\mathit{IS}} \big( K_{3} \big)_{\mathit{SQO}} \big( \widetilde{K}_{4}^{-1} \big)_{\mathit{QOJM}} \\ &- 2 \big( \Delta \widetilde{L}_{4} \big)_{\mathit{iQOR}} \Big( \widetilde{K}_{6}^{-1} \Big)_{\mathit{QORSIN}} \big( \widetilde{K}_{5} \big)_{\mathit{SINTU}} \big( \widetilde{K}_{4}^{-1} \big)_{\mathit{TUJM}} \\ &= \nabla _{j} F_{\mathit{iJM}}^{(1)} \bullet \Delta \underline{y}_{j}, \end{aligned}$$

(A.19)

$$\begin{aligned} \Delta F_{\mathit{iJMN}}^{(2)} &= 6 \big( \Delta \widetilde{L}_{4} \big)_{\mathit{iQOR}} \Big( \widetilde{K}_{6}^{-1} \Big)_{\mathit{QORJMN}} = \nabla _{j} F_{\mathit{iJMN}}^{(2)} \bullet \Delta \underline{y}_{j}, \end{aligned}$$

(A.20)

with

$$\begin{aligned} \big( \Delta \widetilde{L}_{4} \big)_{\mathit{iQOR}} &:= \int _{\mathscr{H}_{ \mathbf{X}}}{\omega \Delta \underline{y}_{i} \underline{X}_{Q} \underline{X}_{O} \underline{X}_{R}}dV_{\mathbf{X}^{\prime }} \\ &- \Big( \int _{\mathscr{H}_{\mathbf{X}}}{\omega \Delta \underline{y}_{i} \underline{X}_{I}}dV_{\mathbf{X}^{\prime }} \Big) \big( K_{2}^{-1} \big)_{\mathit{IJ}} \big( K_{4} \big)_{\mathit{JQOR}} \\ &- \Big( \int _{\mathscr{H}_{\mathbf{X}}}{\omega \Delta \underline{y}_{i} \underline{X}_{I} \underline{X}_{S}}dV_{\mathbf{X}^{\prime }} \Big) \big( \widetilde{K}_{4}^{-1} \big)_{\mathit{ISTV}} \big( \widetilde{K}_{5} \big)_{\mathit{TVQOR}} \\ &+ \Big( \int _{\mathscr{H}_{\mathbf{X}}}{\omega \Delta \underline{y}_{i} \underline{X}_{M}}dV_{\mathbf{X}^{\prime }} \Big) \big( K_{2}^{-1} \big)_{\mathit{MN}} \big( K_{3} \big)_{\mathit{NIS}} \big( \widetilde{K}_{4}^{-1} \big)_{\mathit{ISTV}} \big( \widetilde{K}_{5} \big)_{\mathit{TVQOR}}. \end{aligned}$$

(A.21)

Therefore, the Fréchet derivatives of these three deformation gradients can be identified as

$$\begin{aligned} \nabla _{j} F_{iJ} &= \omega \delta _{ij} \underline{X}_{M} \big( K_{2}^{-1} \big)_{\mathit{MJ}} \\ &- \omega \delta _{ij} \underline{X}_{Q} \underline{X}_{O} \big( \widetilde{K}_{4}^{-1} \big)_{\mathit{QOIN}} \big( K_{3} \big)_{\mathit{INM}} \big( K_{2}^{-1} \big)_{\mathit{MJ}} \\ &+ \omega \delta _{ij} \underline{X}_{R} \big( K_{2}^{-1} \big)_{\mathit{RS}} \big( K_{3} \big)_{\mathit{SQO}} \big( \widetilde{K}_{4}^{-1} \big)_{\mathit{QOIN}} \big( K_{3} \big)_{\mathit{INM}} \big( K_{2}^{-1} \big)_{\mathit{MJ}} \\ &- \big( \nabla _{j} \widetilde{L}_{4} \big)_{\mathit{iQOR}} \Big( \widetilde{K}_{6}^{-1} \Big)_{\mathit{QORINS}} \Big[ \big( K_{4} \big)_{\mathit{INSM}} \\ &- \big( \widetilde{K}_{5} \big)_{\mathit{INSTU}} \big( \widetilde{K}_{4}^{-1} \big)_{\mathit{TUVW}} \big( K_{3} \big)_{\mathit{VWM}} \Big] \big( K_{2}^{-1} \big)_{\mathit{MJ}}, \end{aligned}$$

(A.22)

$$\begin{aligned} \nabla _{j} F_{\mathit{iJM}}^{(1)} &= 2 \omega \delta _{ij} \Big( \underline{X}_{Q} \underline{X}_{O} - \underline{X}_{I} \big( K_{2} \big)^{-1}_{\mathit{IS}} \big( K_{3} \big)_{\mathit{SQO}} \Big) \big( \widetilde{K}_{4}^{-1} \big)_{\mathit{QOJM}} \\ &- 2 \big( \nabla _{j} \widetilde{L}_{4} \big)_{\mathit{iQOR}} \Big( \widetilde{K}_{6}^{-1} \Big)_{\mathit{QORSIN}} \big( \widetilde{K}_{5} \big)_{\mathit{SINTU}} \big( \widetilde{K}_{4}^{-1} \big)_{\mathit{TUJM}}, \end{aligned}$$

(A.23)

$$\begin{aligned} \nabla _{j} F_{\mathit{iJMN}}^{(2)} = 6 \big( \nabla _{j} \widetilde{L}_{4} \big)_{\mathit{iQOR}} \Big( \widetilde{K}_{6}^{-1} \Big)_{\mathit{QORJMN}}, \end{aligned}$$

(A.24)

with

$$\begin{aligned} \big( \nabla _{j} \widetilde{L}_{4} \big)_{\mathit{iQOR}} &:= \omega \delta _{ij} \Big[ \underline{X}_{Q} \underline{X}_{O} \underline{X}_{R} - \underline{X}_{I} \big( K_{2}^{-1} \big)_{\mathit{IJ}} \big( K_{4} \big)_{\mathit{JQOR}} \\ &- \underline{X}_{I} \underline{X}_{S} \big( \widetilde{K}_{4}^{-1} \big)_{\mathit{ISTV}} \big( \widetilde{K}_{5} \big)_{\mathit{TVQOR}} \\ &+ \underline{X}_{M} \big( K_{2}^{-1} \big)_{\mathit{MN}} \big( K_{3} \big)_{\mathit{NIS}} \big( \widetilde{K}_{4}^{-1} \big)_{\mathit{ISTV}} \big( \widetilde{K}_{5} \big)_{\mathit{TVQOR}} \Big]. \end{aligned}$$

(A.25)

Following the same idea by equating the energy of a peridynamic material point with its continuum counterpart given by Eq. (22), we have

$$\begin{aligned} \Delta \mathscr{W} \big( \Delta \underline{y}_{j} \big) &= \frac{\partial \mathscr{W}^{(2)} \big( F_{iJ}; F_{\mathit{iJM}}^{(1)}; F_{\mathit{iJMN}}^{(2)} \big)}{\partial F_{iJ}} \Delta F_{iJ} \\ &+ \frac{\partial \mathscr{W}^{(2)} \big( F_{iJ}; F_{\mathit{iJM}}^{(1)}; F_{\mathit{iJMN}}^{(2)} \big)}{\partial F_{\mathit{iJM}}^{(1)}} \Delta F_{\mathit{iJM}}^{(1)} \\ &+ \frac{\partial \mathscr{W}^{(2)} \big( F_{iJ}; F_{\mathit{iJM}}^{(1)}; F_{\mathit{iJMN}}^{(2)} \big) }{\partial F_{\mathit{iJMN}}^{(2)}} \Delta F_{\mathit{iJMN}}^{(2)} \\ &= P_{iJ} \Big( \nabla _{j} F_{iJ} \bullet \Delta \underline{y}_{j} \Big) + P_{\mathit{iJM}}^{(1)} \Big( \nabla _{j} F_{\mathit{iJM}}^{(1)} \bullet \Delta \underline{y}_{j} \Big) \\ &+ P_{\mathit{iJMN}}^{(2)} \Big( \nabla _{j} F_{\mathit{iJMN}}^{(2)} \bullet \Delta \underline{y}_{j} \Big) \\ &= \Big[ P_{iJ} \nabla _{j} F_{iJ} + P_{\mathit{iJM}}^{(1)} \nabla _{j} F_{\mathit{iJM}}^{(1)} + P_{\mathit{iJMN}}^{(2)} \nabla _{j} F_{\mathit{iJMN}}^{(2)} \Big] \bullet \Delta \underline{y}_{j}, \end{aligned}$$

(A.26)

where $P_{\mathit{iJMN}}^{(2)}$ is the second gradient of the first Piola-Kirchhoff stress.

Substituting the Fréchet derivatives in Eqs. (A.22)–(A.24) into the above equation, the force density state based on deformation gradients up to the third-order can be obtained as

$$\begin{aligned} \underline{T}_{j} &= \frac{ \Delta \mathscr{W}}{ \Delta \underline{y}_{j}} = P_{iJ} \nabla _{j} F_{iJ} + P_{\mathit{iJM}}^{(1)} \nabla _{j} F_{\mathit{iJM}}^{(1)} + P_{\mathit{iJMN}}^{(2)} \nabla _{j} F_{\mathit{iJMN}}^{(2)} \\ &= \omega P_{jJ} \underline{X}_{M} \big( K_{2}^{-1} \big)_{\mathit{MJ}} \\ &- \omega P_{jJ} \Big( \underline{X}_{Q} \underline{X}_{O} - \underline{X}_{R} \big( K_{2}^{-1} \big)_{\mathit{RS}} \big( K_{3} \big)_{\mathit{SQO}} \Big) \big( \widetilde{K}_{4}^{-1} \big)_{\mathit{QOIN}} \big( K_{3} \big)_{\mathit{INM}} \big( K_{2}^{-1} \big)_{\mathit{MJ}} \\ &- P_{iJ} \big( \nabla _{j} \widetilde{L}_{4} \big)_{\mathit{iQOR}} \Big( \widetilde{K}_{6}^{-1} \Big)_{\mathit{QORINS}} \Big[ \big( K_{4} \big)_{\mathit{INSM}} \\ &- \big( \widetilde{K}_{5} \big)_{\mathit{INSTU}} \big( \widetilde{K}_{4}^{-1} \big)_{\mathit{TUVW}} \big( K_{3} \big)_{\mathit{VWM}} \Big] \big( K_{2}^{-1} \big)_{\mathit{MJ}} \\ &+ 2 \omega P_{\mathit{jJM}}^{(1)} \Big( \underline{X}_{Q} \underline{X}_{O} - \underline{X}_{I} \big( K_{2} \big)^{-1}_{\mathit{IS}} \big( K_{3} \big)_{\mathit{SQO}} \Big) \big( \widetilde{K}_{4}^{-1} \big)_{\mathit{QOJM}} \\ &- 2 P_{\mathit{iJM}}^{(1)} \big( \nabla _{j} \widetilde{L}_{4} \big)_{\mathit{iQOR}} \Big( \widetilde{K}_{6}^{-1} \Big)_{\mathit{QORSIN}} \big( \widetilde{K}_{5} \big)_{\mathit{SINTU}} \big( \widetilde{K}_{4}^{-1} \big)_{\mathit{TUJM}} \\ &+ 6 P_{\mathit{iJMN}}^{(2)} \big( \nabla _{j} \widetilde{L}_{4} \big)_{\mathit{iQOR}} \Big( \widetilde{K}_{6}^{-1} \Big)_{\mathit{QORJMN}}, \end{aligned}$$

(A.27)

with $\big ( \nabla _{j} \widetilde{L}_{4} \big )_{\mathit{iQOR}}$ given in Eq. (A.25).