A solid–shell formulation based on the assumed natural inhomogeneous strains for modeling the viscoelastic response of electro-active polymers

Abstract

In this paper, an advanced low-order solid–shell formulation is presented for modeling electro-active polymers (EAPs). This advanced finite element is of great importance due to the fact that EAPs actuators are typically designed as shell-like formations, in which the application of standard finite element formulation will lead to various locking pathologies (e.g. shear locking, trapezoidal locking, volumetric locking, etc.). Thus, for alleviating the various locking pathologies, both the assumed natural inhomogeneous strains (ANIS) and the enhanced assumed strain (EAS) methods are adopted for modifying the strain measure. Within the modified kinematics, a strain energy function that accounts for the elastic and the viscoelastic response as well as the electromechanical coupling is adopted. The developed formulation is implemented in the finite-element software Abaqus for further numerical applications, in which the developed ANIS solid–shell is compared with the classical assumed natural strains solid–shell and the mixed finite element formulation.

Appendices

Appendix A: details on the transformation matrices $\underline{\underline{\mathbf {T}}}$ and $\widetilde{\widetilde{\underline{\underline{\mathbf {T}}}}}$, and the interpolation matrix $\underline{\underline{\mathbf {M}}}$

The matrix $\underline{\underline{\mathbf {T}}}$ in (53) is a $6 \times 6$ transformation matrix with entries $T_{i}^{j}=\mathbf {e}_{i}\cdot \mathbf {G}^{j}$ and it is reported by

$$\begin{aligned} \begin{aligned}&\underline{\underline{\mathbf {T}}}=\left[ \begin{array}{l l l} T_{1}^{1}T_{1}^{1} &{} \quad T_{1}^{2}T_{1}^{2} &{} \quad T_{1}^{3}T_{1}^{3}\\ T_{2}^{1}T_{2}^{1} &{} \quad T_{2}^{2}T_{2}^{2} &{} \quad T_{2}^{3}T_{2}^{3}\\ T_{3}^{1}T_{3}^{1} &{} \quad T_{3}^{2}T_{3}^{2} &{} \quad T_{3}^{3}T_{3}^{3}\\ 2T_{1}^{1}T_{2}^{1} &{} \quad 2T_{1}^{2}T_{2}^{2} &{} \quad 2T_{1}^{3}T_{2}^{3}\\ 2T_{1}^{1}T_{3}^{1} &{} \quad 2T_{1}^{2}T_{3}^{2} &{} \quad 2T_{1}^{3}T_{3}^{3}\\ 2T_{2}^{1}T_{3}^{1} &{} \quad 2T_{2}^{2}T_{3}^{2} &{} \quad 2T_{2}^{3}T_{3}^{3} \end{array} \right. \\&\quad \left. \begin{array}{l l l} T_{1}^{1}T_{1}^{2} &{} \quad T_{1}^{1}T_{1}^{3} &{} \quad T_{1}^{2}T_{1}^{3}\\ T_{2}^{1}T_{2}^{2} &{} \quad T_{2}^{1}T_{2}^{3} &{} \quad T_{2}^{2}T_{2}^{3}\\ T_{3}^{1}T_{3}^{2} &{} \quad T_{3}^{1}T_{3}^{3} &{} \quad T_{3}^{2}T_{3}^{3}\\ T_{1}^{1}T_{2}^{2}+T_{2}^{1}T_{1}^{2} &{} \quad T_{2}^{1}T_{1}^{3}+T_{2}^{3}T_{1}^{1} &{} \quad T_{1}^{2}T_{2}^{3}+T_{2}^{2}T_{1}^{3}\\ T_{1}^{1}T_{3}^{2}+T_{3}^{1}T_{1}^{2} &{} \quad T_{1}^{1}T_{3}^{3}+T_{1}^{3}T_{3}^{1} &{} \quad T_{1}^{2}T_{3}^{3}+T_{3}^{2}T_{1}^{3}\\ T_{2}^{1}T_{3}^{2}+T_{3}^{1}T_{2}^{2} &{} \quad T_{2}^{1}T_{3}^{3}+T_{2}^{3}T_{3}^{1} &{} \quad T_{2}^{2}T_{3}^{3}+T_{3}^{2}T_{2}^{3} \end{array}\right] .\!\!\! \end{aligned} \end{aligned}$$

(97)

Also, the transformation matrix $\underline{\underline{{\widetilde{\mathbf {T}}}}}$ in (53) is defined by

$$\begin{aligned} \underline{\underline{\widetilde{\mathbf {T}}}}=\underline{\underline{\mathbf {T}}}\,\underline{\underline{\widetilde{\widetilde{\mathbf {T}}}}}, \end{aligned}$$

(98)

where the transformation matrix $\underline{\underline{\widetilde{\widetilde{\mathbf {T}}}}}$ consists of the differences between the constants ${G}_{ij}^{mn}$ and their interpolated counterparts $\widetilde{G}_{ij}^{mn}$. Thus the rows of the in-plane components consists of zeros and it reads

$$\begin{aligned} \begin{aligned} \underline{\underline{\widetilde{\widetilde{\mathbf {T}}}}}&=\\&\begin{bmatrix} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ \widetilde{\widetilde{G}}_{33}^{11} &{} \widetilde{\widetilde{G}}_{33}^{22} &{} \widetilde{\widetilde{G}}_{33}^{33} &{} \widetilde{\widetilde{G}}_{33}^{12} &{} \widetilde{\widetilde{G}}_{33}^{13} &{} \widetilde{\widetilde{G}}_{33}^{23}\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ 2\widetilde{\widetilde{G}}_{13}^{11} &{} 2\widetilde{\widetilde{G}}_{13}^{22} &{} 2\widetilde{\widetilde{G}}_{13}^{33} &{} \widetilde{\widetilde{G}}_{13}^{12}+\widetilde{\widetilde{G}}_{13}^{21} &{} \widetilde{\widetilde{G}}_{13}^{13}+\widetilde{\widetilde{G}}_{13}^{31} &{} \widetilde{\widetilde{G}}_{13}^{23}+\widetilde{\widetilde{G}}_{13}^{32}\\ 2\widetilde{\widetilde{G}}_{23}^{11} &{} 2\widetilde{\widetilde{G}}_{23}^{22} &{} 2\widetilde{\widetilde{G}}_{23}^{33} &{} \widetilde{\widetilde{G}}_{23}^{12}+\widetilde{\widetilde{G}}_{23}^{21} &{} \widetilde{\widetilde{G}}_{23}^{13}+\widetilde{\widetilde{G}}_{23}^{31} &{} \widetilde{\widetilde{G}}_{23}^{23}+\widetilde{\widetilde{G}}_{23}^{32} \end{bmatrix}. \end{aligned} \end{aligned}$$

(99)

The components $\widetilde{\widetilde{G}}_{ij}^{mn}$ are defined by

$$\begin{aligned} \widetilde{\widetilde{G}}_{ij}^{mn}={G}_{ij}^{mn}- \widetilde{G}_{ij}^{mn}\, , \end{aligned}$$

(100)

The elements $G_{33}^{mn}$, $G_{13}^{mn}$ and $G_{23}^{mn}$ of the matrix $\underline{\underline{\widetilde{\widetilde{\mathbf {T}}}}}$ are defined by (37), while the elements $\widetilde{G}_{33}^{mn}$, $\widetilde{G}_{13}^{mn}$ and $\widetilde{G}_{23}^{mn}$ are defined by (44).

Finally, the interpolation matrix $\underline{\underline{\mathbf {M}}}$, which consists of the enhanced modes, is given by

$$\begin{aligned} \begin{aligned}&\underline{\underline{\mathbf {M}}}=\left[ \begin{array}{l l l l l l l l l l} \xi _{1} &{} \, \xi _{1}\xi _{2} &{} \, \xi _{1}\xi _{3} &{} \, 0 &{} \, 0 &{} \, 0 &{} \, 0 &{} \, 0 &{} \, 0 &{} \, 0\\ 0 &{} \, 0 &{} \, 0 &{} \, \xi _{2} &{} \, \xi _{1}\xi _{2} &{} \, \xi _{2}\xi _{3} &{} \, 0 &{} \, 0 &{} \, 0 &{} \, 0\\ 0 &{} \, 0 &{} \, 0 &{} \, 0 &{} \, 0 &{} \, 0 &{} \, \xi _{3} &{} \, \xi _{1}\xi _{3} &{} \, \xi _{2}\xi _{3} &{} \, \xi _{1}\xi _{2}\xi _{3}\!\!\!\!\!\\ 0 &{} \, 0 &{} \, 0 &{} \, 0 &{} \, 0 &{} \, 0 &{} \, 0 &{} \, 0 &{} \, 0 &{} \, 0\\ 0 &{} \, 0 &{} \, 0 &{} \, 0 &{} \, 0 &{} \, 0 &{} \, 0 &{} \, 0 &{} \, 0 &{} \, 0\\ 0 &{} \, 0 &{} \, 0 &{} \, 0 &{} \, 0 &{} \, 0 &{} \, 0 &{} \, 0 &{} \, 0 &{} \, 0 \end{array} \right. \\&\quad \left. \begin{array}{l l l l l l l l l l l} 0 &{} \, 0 &{} \, 0 &{} \, 0 &{} \, 0 &{} \, 0 &{} \, 0 &{} \, 0 &{} \, 0 &{} \, 0 &{} \, 0\\ 0 &{} \, 0 &{} \, 0 &{} \, 0 &{} \, 0 &{} \, 0 &{} \, 0 &{} \, 0 &{} \, 0 &{} \, 0 &{} \, 0\\ 0 &{} \, 0 &{} \, 0 &{} \, 0 &{} \, 0 &{} \, 0 &{} \, 0 &{} \, 0 &{} \, 0 &{} \, 0 &{} \, 0\\ \xi _{1} &{} \, \xi _{2} &{} \, \xi _{1}\xi _{2} &{} \, \xi _{1}\xi _{3} &{} \, \xi _{2}\xi _{3} &{} \, 0 &{} \, 0 &{} \, 0 &{} \, 0 &{} \, 0 &{} \, 0\\ 0 &{} \, 0 &{} \, 0 &{} \, 0 &{} \, 0 &{} \, \xi _{3} &{} \, \xi _{1}\xi _{3} &{} \, \xi _{2}\xi _{3} &{} \, 0 &{} \, 0 &{} \, 0\\ 0 &{} \, 0 &{} \, 0 &{} \, 0 &{} \, 0 &{} \, 0 &{} \, 0 &{} \, 0 &{} \, \xi _{3} &{} \, \xi _{1}\xi _{3} &{} \, \xi _{2}\xi _{3} \end{array} \right] . \end{aligned} \end{aligned}$$

(101)

Appendix B: developing the geometrical stiffness matrix

In this “Appendix”, the geometric matrix is derived. Thus, the dot product between the stress measure $\overline{\mathbf {S}}$ and the linearization of the variation of the modified strain measure $\varDelta \delta \mathbf {E}^{\textsf {Comp}}$ is rewritten using the Voigt notation as follows

$$\begin{aligned} \overline{\mathbf {S}}:\varDelta \delta \mathbf {E}^{\textsf {Comp}}=\varDelta \delta \underline{\mathbf {E}}^{\textsf {Comp}\,\text {T}}\underline{\overline{\mathbf {S}}}, \end{aligned}$$

(102)

where $\underline{\overline{\mathbf {S}}}$ is a $6 \times 1$ vector consisting of the modified second Piola-Kircchoff stress components defined by (87), and ${\varDelta \delta \underline{\mathbf {E}}^{\textsf {Comp}}}$ consists of the components of the linearized variation of the modified strain field, and it is defined by

$$\begin{aligned} \begin{aligned} \varDelta \delta \underline{\mathbf {E}}^{\textsf {Comp}}&=\left\{ \varDelta \delta {{E}}_{11}^{\textsf {Comp}},\,\varDelta \delta {{E}}_{22}^{\textsf {Comp}},\,\varDelta \delta {{E}}_{33}^{\textsf {Comp}},\right. \\&\quad \left. 2\varDelta \delta {{E}}_{12}^{\textsf {Comp}},\,2\varDelta \delta {{E}}_{13}^{\textsf {Comp}},\,2\varDelta \delta {{E}}_{23}^{\textsf {Comp}}\right\} ^{\text {T}}. \end{aligned} \end{aligned}$$

(103)

Using (53), the vector ${\varDelta \delta \underline{\mathbf {E}}^{\textsf {Comp}}}$ is defined by

$$\begin{aligned} \varDelta \delta \underline{\mathbf {E}}^{\textsf {Comp}}=\underline{\underline{\mathbf {T}}}\,\varDelta \delta \underline{\widetilde{\mathbf {E}}}^{\textsf {ANS}}+\widetilde{{\underline{\underline{\mathbf {T}}}}}\,\varDelta \delta \underline{\widetilde{\mathbf {E}}}^{0}, \end{aligned}$$

(104)

where the vectors $\varDelta \delta \underline{\widetilde{\mathbf {E}}}^{\textsf {ANS}}$ and $\varDelta \delta \underline{\widetilde{\mathbf {E}}}^{0}$ are defined by the following vectors

$$\begin{aligned} \begin{aligned} \varDelta \delta \underline{\widetilde{\mathbf {E}}}^{\textsf {ANS}}&=\left\{ \varDelta \delta \mathcal {E}_{11},\,\varDelta \delta \mathcal {E}_{22},\,\varDelta \delta \mathcal {E}_{33}^{\textsf {ANS}},\right. \\&\quad \left. 2\varDelta \delta \mathcal {E}_{12},\,2\varDelta \delta \mathcal {E}_{13}^{\textsf {ANS}},\,2\varDelta \delta \mathcal {E}_{23}^{\textsf {ANS}}\right\} ^{\text {T}},\\ \varDelta \delta \underline{\widetilde{\mathbf {E}}}^{0}&=\left\{ \varDelta \delta \mathcal {E}_{11}^{0},\,\varDelta \delta \mathcal {E}_{22}^{0},\,\varDelta \delta \mathcal {E}_{33}^{0},\right. \\&\quad \left. 2\varDelta \delta \mathcal {E}_{12}^{0},\,2\varDelta \delta \mathcal {E}_{13}^{0},\,2\varDelta \delta \mathcal {E}_{23}^{0}\right\} ^{\text {T}}. \end{aligned} \end{aligned}$$

(105)

Substituting (104) into (102), it is possible to show that

$$\begin{aligned} \begin{aligned}&\varDelta \delta \underline{\mathbf {E}}^{\textsf {Comp}\,\text {T}}\underline{\overline{\mathbf {S}}}\\&\quad =\varDelta \delta \underline{\widetilde{\mathbf {E}}}^{\textsf {ANS}\,\text {T}}\widetilde{\underline{\mathbf {S}}}+\varDelta \delta \underline{\widetilde{\mathbf {E}}}^{0\,\text {T}}\widetilde{\widetilde{\underline{\mathbf {S}}}}\\&\quad =\varDelta \delta \mathcal {E}_{11}\widetilde{S}_{11}+\varDelta \delta \mathcal {E}_{22}\widetilde{S}_{22}+\varDelta \delta \mathcal {E}_{33}^{\textsf {ANS}}\widetilde{S}_{33}\\&\qquad +2\varDelta \delta \mathcal {E}_{12}\widetilde{S}_{12}+2\varDelta \delta \mathcal {E}_{13}^{\textsf {ANS}}\widetilde{S}_{13}+2\varDelta \delta \mathcal {E}_{23}^{\textsf {ANS}}\widetilde{S}_{23}\\&\qquad +\varDelta \delta \mathcal {E}_{11}^{0}\widetilde{\widetilde{S}}_{11}+\varDelta \delta {\mathcal {E}}_{22}^{0}\widetilde{\widetilde{S}}_{22}+\varDelta \delta {\mathcal {E}}_{33}^{0}\widetilde{\widetilde{S}}_{33}\\&\qquad +2\varDelta \delta {\mathcal {E}}_{12}^{0}\widetilde{\widetilde{S}}_{12}+2\varDelta \delta {\mathcal {E}}_{13}^{0}\widetilde{\widetilde{S}}_{13}+2\varDelta \delta {\mathcal {E}}_{23}^{0}\widetilde{\widetilde{S}}_{23}, \end{aligned} \end{aligned}$$

(106)

where the two stress vectors $\widetilde{\underline{\mathbf {S}}}$ and $\widetilde{\widetilde{\underline{\mathbf {S}}}}$ are, respectively, given by

$$\begin{aligned} \begin{aligned} \widetilde{\underline{\mathbf {S}}}&=\left\{ \widetilde{S}_{11},\,\widetilde{S}_{22},\,\widetilde{S}_{33},\,\widetilde{S}_{12},\,\widetilde{S}_{13},\,\widetilde{S}_{23}\right\} ^{\text {T}}=\underline{\underline{\mathbf {T}}}^{\text {T}}\underline{\overline{\mathbf {S}}},\\ \widetilde{\widetilde{\underline{\mathbf {S}}}}&=\left\{ \widetilde{\widetilde{S}}_{11},\,\widetilde{\widetilde{S}}_{22},\,\widetilde{\widetilde{S}}_{33},\,\widetilde{\widetilde{S}}_{12},\,\widetilde{\widetilde{S}}_{13},\,\widetilde{\widetilde{S}}_{23}\right\} ^{\text {T}}=\widetilde{{\underline{\underline{\mathbf {T}}}}}^{\text {T}}\underline{\overline{\mathbf {S}}}. \end{aligned} \end{aligned}$$

(107)

Applying the linearization operator on (67) and (73) yields

$$\begin{aligned} \begin{aligned} \varDelta \delta \mathcal {E}_{11}&=\sum _{I=1}^{8}\sum _{J=1}^{8}\sum _{i=1}^{3}\delta \widehat{u}_{i}^{I}\left[ \frac{\partial N_{I}}{\partial \xi _{1}}\frac{\partial N_{J}}{\partial \xi _{1}}\right] \varDelta \widehat{{u}}_{i}^{J},\\ \varDelta \delta \mathcal {E}_{22}&=\sum _{I=1}^{8}\sum _{J=1}^{8}\sum _{i=1}^{3}\delta \widehat{u}_{i}^{I}\left[ \frac{\partial N_{I}}{\partial \xi _{2}}\frac{\partial N_{J}}{\partial \xi _{2}}\right] \varDelta \widehat{{u}}_{i}^{J},\\ \varDelta \delta \mathcal {E}_{33}^{\textsf {ANS}}&=\sum _{I=1}^{8}\sum _{J=1}^{8}\sum _{i=1}^{3}\sum _{K=A}^{D}\delta \widehat{u}_{i}^{I}\left[ {\widetilde{N}}_{K}\left( \frac{\partial N_{I}^{K}}{\partial \xi _{3}}\frac{\partial N_{J}^{K}}{\partial \xi _{3}}\right) \right] \varDelta \widehat{u}_{i}^{J},\\ \varDelta \delta \mathcal {E}_{12}&=\frac{1}{2}\sum _{I=1}^{8}\sum _{J=1}^{8}\sum _{i=1}^{3}\delta \widehat{u}_{i}^{I}\left[ \frac{\partial N_{I}}{\partial \xi _{1}}\frac{\partial N_{J}}{\partial \xi _{2}}+\frac{\partial N_{I}}{\partial \xi _{2}}\frac{\partial N_{J}}{\partial \xi _{1}}\right] \varDelta \widehat{{u}}_{i}^{J},\\ \varDelta \delta \mathcal {E}_{13}^{\textsf {ANS}}&=\frac{1}{2}\sum _{I=1}^{8}\sum _{J=1}^{8}\sum _{i=1}^{3}\sum _{K=E}^{F}\delta \widehat{u}_{i}^{I}\left[ {\widetilde{N}}_{K}\left( \frac{\partial N_{I}^{K}}{\partial \xi _{1}}\frac{\partial N_{J}^{K}}{\partial \xi _{3}}\right. \right. \\&\quad \left. \left. +\frac{\partial N_{I}^{K}}{\partial \xi _{3}}\frac{\partial N_{J}^{K}}{\partial \xi _{1}}\right) \right] \varDelta \widehat{u}_{i}^{J},\\ \varDelta \delta \mathcal {E}_{23}^{\textsf {ANS}}&=\frac{1}{2}\sum _{I=1}^{8}\sum _{J=1}^{8}\sum _{i=1}^{3}\sum _{K=G}^{H}\delta \widehat{u}_{i}^{I}\left[ {\widetilde{N}}_{K}\left( \frac{\partial N_{I}^{K}}{\partial \xi _{2}}\frac{\partial N_{J}^{K}}{\partial \xi _{3}}\right. \right. \\&\quad \left. \left. +\,\frac{\partial N_{I}^{K}}{\partial \xi _{3}}\frac{\partial N_{J}^{K}}{\partial \xi _{2}}\right) \right] \varDelta \widehat{u}_{i}^{J}. \end{aligned} \end{aligned}$$

(108)

Moreover, the linearization of 68 takes the form,

$$\begin{aligned} \begin{aligned} \varDelta \delta \mathcal {E}_{mn}^{0}&= \frac{1}{2}\sum _{I=1}^{8}\sum _{J=1}^{8}\sum _{i=1}^{3}\delta \widehat{u}_{i}^{I}\left[ \frac{\partial N^{0}_{I}}{\partial \xi _{m}}\frac{\partial N^{0}_{J}}{\partial \xi _{n}}+\frac{\partial N^{0}_{I}}{\partial \xi _{n}}\frac{\partial N^{0}_{J}}{\partial \xi _{m}}\right] \delta \widehat{u}_{i}^{J}. \end{aligned} \end{aligned}$$

(109)

Substituting (108) and (109) into (106) yields,

$$\begin{aligned} \varDelta \delta \underline{\mathbf {E}}^{\textsf {Comp}\,\text {T}}\underline{\overline{\mathbf {S}}}=\sum _{I=1}^{8}\sum _{J=1}^{8}\sum _{i=1}^{3}\delta \widehat{u}_{i}^{I}G_{IJ}\varDelta \widehat{u}_{i}^{J}, \end{aligned}$$

(110)

where the set of the coefficients $G_{IJ}$ are given by

$$\begin{aligned} G_{IJ}= & {} \frac{\partial N_{I}}{\partial \xi _{1}}{\widetilde{S}}_{11}\frac{\partial N_{J}}{\partial \xi _{1}}+\frac{\partial N_{I}}{\partial \xi _{1}}\widetilde{S}_{12}\frac{\partial N_{J}}{\partial \xi _{2}}+\frac{\partial N_{I}}{\partial \xi _{2}}\widetilde{S}_{12}\frac{\partial N_{J}}{\partial \xi _{1}}\nonumber \\&+\frac{\partial N_{I}}{\partial \xi _{2}}\widetilde{S}_{22}\frac{\partial N_{J}}{\partial \xi _{2}}+\sum _{K=A}^{D}\widetilde{N}_{K}\frac{\partial N^{K}_{I}}{\partial \xi _{3}}\widetilde{S}_{33}\frac{\partial N^{K}_{J}}{\partial \xi _{3}}\nonumber \\&+\sum _{K=E}^{F}\widetilde{N}_{K}\left( \frac{\partial N^{K}_{I}}{\partial \xi _{1}}\widetilde{S}_{13}\frac{\partial N^{K}_{J}}{\partial \xi _{3}}+\frac{\partial N^{K}_{I}}{\partial \xi _{3}}\widetilde{S}_{13}\frac{\partial N^{K}_{J}}{\partial \xi _{1}}\right) \nonumber \\&+\sum _{K=G}^{H}\widetilde{N}_{K}\left( \frac{\partial N^{K}_{I}}{\partial \xi _{2}}\widetilde{S}_{23}\frac{\partial N^{K}_{J}}{\partial \xi _{3}}+\frac{\partial N^{K}_{I}}{\partial \xi _{3}}\widetilde{S}_{23}\frac{\partial N^{K}_{J}}{\partial \xi _{2}}\right) \nonumber \\&-\left( \frac{\partial N^{0}_{I}}{\partial \xi _{1}}\widetilde{\widetilde{S}}_{11}\frac{\partial N^{0}_{J}}{\partial \xi _{1}}+\frac{\partial N^{0}_{I}}{\partial \xi _{1}}\widetilde{\widetilde{S}}_{12}\frac{\partial N^{0}_{J}}{\partial \xi _{2}}+\frac{\partial N^{0}_{I}}{\partial \xi _{2}}\widetilde{\widetilde{S}}_{12}\frac{\partial N^{0}_{J}}{\partial \xi _{1}}\right. \nonumber \\&+\frac{\partial N^{0}_{I}}{\partial \xi _{1}}\widetilde{\widetilde{S}}_{13}\frac{\partial N^{0}_{J}}{\partial \xi _{3}}+\frac{\partial N^{0}_{I}}{\partial \xi _{3}}\widetilde{\widetilde{S}}_{13}\frac{\partial N^{0}_{J}}{\partial \xi _{1}}+\frac{\partial N^{0}_{I}}{\partial \xi _{2}}\widetilde{\widetilde{S}}_{22}\frac{\partial N^{0}_{J}}{\partial \xi _{2}}\nonumber \\&+\left. \frac{\partial N^{0}_{I}}{\partial \xi _{2}}\widetilde{\widetilde{S}}_{23}\frac{\partial N^{0}_{J}}{\partial \xi _{3}}+\frac{\partial N^{0}_{I}}{\partial \xi _{3}}\widetilde{\widetilde{S}}_{23}\frac{\partial N^{0}_{J}}{\partial \xi _{2}}+\frac{\partial N^{0}_{I}}{\partial \xi _{3}}\widetilde{\widetilde{S}}_{33}\frac{\partial N^{0}_{J}}{\partial \xi _{3}}\right) .\nonumber \\ \end{aligned}$$

(111)

Finally, using (92), it can be shown that

$$\begin{aligned} \int _{\varOmega _{0}^{e}}\overline{\mathbf {S}}:\varDelta \delta \mathbf {E}^{\textsf {Comp}}\,d\varOmega _{0}^{e}=\int _{\varOmega _{0}^{e}}\varDelta \delta \underline{\mathbf {E}}^{\textsf {Comp}\,\text {T}}\underline{\overline{\mathbf {S}}}\,d\varOmega _{0}^{e}=\delta \underline{\widehat{\mathbf {u}}}\,\underline{\underline{\mathbf {K}}}_{\text {G}}\varDelta \underline{\widehat{\mathbf {u}}}, \end{aligned}$$

(112)

where $\underline{\underline{\mathbf {K}}}_{\text {G}}$ is the geometric stiffness matrix and takes the form

$$\begin{aligned} \underline{\underline{\mathbf {K}}}_{\text {G}}= \begin{bmatrix} \overline{G}_{11}\underline{\underline{\mathbf {I}}}\quad &{} \overline{G}_{12}\underline{\underline{\mathbf {I}}}\quad &{} \ldots \quad &{} \overline{G}_{18}\underline{\underline{\mathbf {I}}}\\ \overline{G}_{21}\underline{\underline{\mathbf {I}}}\quad &{} \overline{G}_{22}\underline{\underline{\mathbf {I}}}\quad &{} \ldots \quad &{} \overline{G}_{28}\underline{\underline{\mathbf {I}}}\\ \vdots \quad &{} \vdots \quad &{} \ddots \quad &{} \vdots \\ \overline{G}_{81}\underline{\underline{\mathbf {I}}}\quad &{} \overline{G}_{82}\underline{\underline{\mathbf {I}}}\quad &{} \ldots \quad &{} \overline{G}_{88}\underline{\underline{\mathbf {I}}} \end{bmatrix}. \end{aligned}$$

(113)

In which $\underline{\underline{\mathbf {I}}}$ is the $3\times 3$ identity matrix, and the set of the coefficients $\overline{G}_{IJ}$ are defined by

$$\begin{aligned} \overline{G}_{IJ}=\int _{\varOmega _{0}^{e}}G_{IJ}\,d\varOmega _{0}^{e}. \end{aligned}$$

(114)