A quadrature element formulation of geometrically nonlinear laminated composite shells incorporating thickness stretch and drilling rotation

Abstract

In this paper, a weak form quadrature element formulation of a geometrically nonlinear shell model is proposed and applied for analysis of laminated composite shell structures. Thickness stretch parameters of the shell are incorporated for introducing 3D constitutive relations in the formulation. A drilling rotation constraint on the basis of polar decomposition of a modified deformation gradient is enforced by the Lagrange multiplier method and employed for implementing spatial finite rotations. The present formulation is shown to be feasible to model complex structures and circumvent locking problems naturally. A series of numerical benchmark examples are presented to demonstrate the validity of the formulation.

Appendix: Expression of element tangent stiffness matrix

The element tangent stiffness matrix $\mathbf{K }^{(e)}$ consists of three parts corresponding to $\mathbf {G}_{\mathrm{int}}^{(e)}$, $\mathbf {G}_{\mathrm{ext}}^{(e)}$ and $\mathbf {G}_{c}^{(e)}$ in the element residual force vector as

$$\begin{aligned} \mathbf{K }^{(e)}=\mathbf{K }_{\mathrm{int}}^{(e)} +\mathbf{K }_{\mathrm{ext}}^{(e)} +\mathbf{K }_{c}^{(e)} . \end{aligned}$$

(A.1)

The element internal tangent stiffness matrix can be derived from Eq. (56) as

$$\begin{aligned} \mathbf{K }_{{\mathrm{int}} }^{(e)} = \sum \limits _{i = 1}^m {\sum \limits _{j = 1}^n {{w_i}{w_j}\left| {\mathbf{J _{ij}}} \right| \sum \limits _{p = 1}^{nl} {\sum \limits _{k = 0}^2 {\sum \limits _{l = 0}^2 {{a_{(k + l)(p)\hbox {ij}}}\left[ {\tilde{\mathbf{C }}_{ij}^T{\bar{\mathbf{J }}}_{ij}^T{{\left( {\mathbf{A ^{(l)\mathrm{T}}}\mathbf{C _{(p)}}\mathbf{A ^{(k)}} + {\varvec{\Xi }} _{(p)}^{(k + l)}} \right) }_{ij}}{{{\bar{\mathbf{J }}}}_{ij}}{{{\tilde{\mathbf{C }}}}_{ij}} + {\varvec{\Gamma }} _{(p)ij}^{(k + l)}} \right] .} } } } }\nonumber \\ \end{aligned}$$

(A.2)

The expressions of matrices $\varvec{\Xi }^{(k+l)}$ are

$$\begin{aligned} \varvec{\Xi }^{(0+k)}=\left[ {{\begin{array}{*{20}c} {s^{11(k)}\mathbf {I}_{3\times 3} } &{} {\frac{s^{12(k)}\mathbf {I}_{3\times 3} }{2}} &{} {\frac{us^{13(k)}\mathbf {I}_{3\times 3} }{2}} &{} {\mathbf{0 }_{3\times 6} } &{} {\frac{s^{13(k)}\mathbf {t}}{2}} &{} {\mathbf{0 }_{3\times 5} } \\ &{} {s^{22(k)}\mathbf {I}_{3\times 3} } &{} {\frac{us^{23(k)}\mathbf {I}_{3\times 3} }{2}} &{} {\mathbf{0 }_{3\times 6} } &{} {\frac{s^{23(k)}\mathbf {t}}{2}} &{} {\mathbf {0}_{3\times 5} } \\ &{} &{} {s^{33(k)}u^{2}\mathbf {I}_{3\times 3} } &{} {\mathbf{0 }_{3\times 6} } &{} {\frac{s^{13(k)}\mathbf {r},_{1} + s^{23(k)}\mathbf {r},_{2} }{2}+ 2s^{33(k)}\varvec{\upphi }} &{} {\mathbf{0 }_{3\times 5} } \\ &{} &{} &{} {\mathbf{0 }_{6\times 6} } &{} {\mathbf{0 }_{6\times 1} } &{} {\mathbf{0 }_{6\times 5} } \\ &{} {\mathrm{sym}} &{} &{} &{} {s^{33(k)}\left\| {\mathbf {t}} \right\| ^{2}} &{} {\mathbf{0 }_{3\times 5} } \\ &{} &{} &{} &{} &{} {\mathbf{0 }_{5\times 5} } \\ \end{array} }} \right] \end{aligned}$$

(A.3)

and

(A.4)

with the sub-matrices

$$\begin{aligned} \varvec{\Xi }_{11}^{(1+k)}= & {} \left[ {{\begin{array}{*{20}c} {\mathbf {0}_{3\times 3} } &{} {\mathbf {0}_{3\times 3} } &{} {\left( {s^{11(k)}u,_{1} + s^{13(k)}q+ \frac{s^{12(k)}u,_{2} }{2}} \right) \mathbf {I}_{3\times 3} } &{} {us^{11(k)}\mathbf {I}_{3\times 3} } &{} {\frac{us^{12(k)}}{2}\mathbf {I}_{3\times 3} } \\ &{} {\mathbf {0}_{3\times 3} } &{} {\left( {s^{22(k)}u,_{2} + s^{23(k)}q+ \frac{s^{12(k)}u,_{1} }{2}} \right) \mathbf {I}_{3\times 3} } &{} {\frac{us^{12(k)}}{2}\mathbf {I}_{3\times 3} } &{} {us^{22(k)}\mathbf {I}_{3\times 3} } \\ &{} &{} {\left( {s^{13(k)}uu,_{1} + s^{23(k)}uu,_{2} + 4s^{33(k)}uq} \right) \mathbf {I}_{3\times 3} } &{} {\frac{s^{13(k)}u^{2}}{2}\mathbf {I}_{3\times 3} } &{} {\frac{s^{23(k)}u^{2}}{2}\mathbf {I}_{3\times 3} } \\ &{} {\mathrm{sym}} &{} &{} {\mathbf {0}_{3\times 3} } &{} {\mathbf {0}_{3\times 3} } \\ &{} &{} &{} &{} {\mathbf {0}_{3\times 3} } \\ \end{array} }} \right] , \end{aligned}$$

(A.5)

$$\begin{aligned} \varvec{\Xi }_{12}^{(1+ k)}= & {} \left[ {{\begin{array}{*{20}c} {s^{11(k)}\mathbf {t},_{1} + \frac{s^{12(k)}\mathbf {t},_{2} }{2}} &{} {s^{11(k)}\mathbf {t}} &{} {\frac{s^{12(k)}\mathbf {t}}{2}} &{} {s^{13(k)}\mathbf {t}} &{} {\mathbf {0}_{3\times 2} } \\ {s^{22(k)}\mathbf {t},_{2} + \frac{s^{12(k)}\mathbf {t},_{1} }{2}} &{} {\frac{s^{12(k)}\mathbf {t}}{2}} &{} {s^{22(k)}\mathbf {t}} &{} {s^{23(k)}\mathbf {t}} &{} {\mathbf {0}_{3\times 2} } \\ {s^{13(k)}\mathbf {d},_{1} + s^{23(k)}\mathbf {d},_{2} + 4s^{33(k)}\varvec{\upphi }} &{} {s^{11(k)}\mathbf {r},_{1} + s^{13(k)}\mathbf {d}+ \frac{s^{12(k)}\mathbf {r},_{2} }{2}} &{} {s^{22(k)}\mathbf {r},_{2} + s^{23(k)}\mathbf {d}+ \frac{s^{12(k)}\mathbf {r},_{1} }{2}} &{} {s^{13(k)}\mathbf {r},_{1} + s^{23(k)}\mathbf {r},_{2} + 4s^{33(k)}\mathbf {d}} &{} {\mathbf {0}_{3\times 2} } \\ {s^{11(k)}\mathbf {r},_{1} + s^{13(k)}\mathbf {d}+ \frac{s^{12(k)}\mathbf {r},_{2} }{2}} &{} {\mathbf {0}_{3\times 1} } &{} {\mathbf {0}_{3\times 1} } &{} {\mathbf {0}_{3\times 1} } &{} {\mathbf {0}_{3\times 2} } \\ {s^{22(k)}\mathbf {r},_{2} + s^{23(k)}\mathbf {d}+ \frac{s^{12(k)}\mathbf {r},_{1} }{2}} &{} {\mathbf {0}_{3\times 1} } &{} {\mathbf {0}_{3\times 1} } &{} {\mathbf {0}_{3\times 1} } &{} {\mathbf {0}_{3\times 2} } \\ \end{array} }} \right] , \nonumber \\ \end{aligned}$$

(A.6)

$$\begin{aligned} \varvec{\Xi }_{22}^{(1+k)}= & {} \left[ {{\begin{array}{*{20}c} {s^{13(k)}\mathbf {t},_{1}^{T} \mathbf {t}+s^{23(k)}\mathbf {t},_{2}^{T} \mathbf {t}} &{} {\frac{s^{13(k)}\mathbf {t}^{T}\mathbf {t}}{2}} &{} {\frac{s^{23(k)}\mathbf {t}^{T}\mathbf {t}}{2}} &{} {2s^{33(k)}\mathbf {t}^{T}\mathbf {t}} &{} {\mathbf {0}_{3\times 2} } \\ &{} 0 &{} 0 &{} 0 &{} {\mathbf {0}_{1\times 2} } \\ &{} &{} 0 &{} 0 &{} {\mathbf {0}_{1\times 2} } \\ &{} {\mathrm{sym}} &{} &{} 0 &{} {\mathbf {0}_{1\times 2} } \\ &{} &{} &{} &{} {\mathbf {0}_{1\times 2} } \\ \end{array} }} \right] , \end{aligned}$$

(A.7)

and

$$\begin{aligned}&\varvec{\Xi }_{11}^{(2+k)}\nonumber \\&=\left[ {{\begin{array}{*{20}c} {\mathbf {0}_{3\times 3} } &{} {\mathbf {0}_{3\times 3} } &{} {\left( {s^{11(k)}q,_{1} + \frac{s^{22(k)}q,_{2} }{2}} \right) \mathbf {I}_{3\times 3} } &{} {s^{11(k)}q\mathbf {I}_{3\times 3} } &{} {\frac{s^{12(k)}q\mathbf {I}_{3\times 3} }{2}} &{} {\mathbf {0}_{3\times 1} } \\ &{} {\mathbf {0}_{3\times 3} } &{} {\left( {s^{22(k)}q,_{2} + \frac{s^{12(k)}q,_{1} }{2}} \right) \mathbf {I}_{3\times 3} } &{} {\frac{s^{12(k)}q}{2}\mathbf {I}_{3\times 3} } &{} {s^{22(k)}q\mathbf {I}_{3\times 3} } &{} {\mathbf {0}_{3\times 1} } \\ &{} &{} {\left( {\begin{array}{l} s^{11(k)}u,_{1}^{2} + s^{22(k)}u,_{2}^{2} \\ + 4s^{33(k)}q^{2}+ s^{12(k)}u,_{1} u,_{2} \\ + s^{13(k)}q,_{1} u,_{1} + s^{23(k)}q,_{2} u,_{2} \\ \end{array}} \right) \mathbf {I}_{3\times 3} } &{} {\left( {\begin{array}{l} s^{11(k)}uu,_{1} + \frac{s^{12(k)}uu,_{2} }{2} \\ + \frac{3s^{13(k)}qu}{2} \\ \end{array}} \right) \mathbf {I}_{3\times 3} } &{} {\left( {\begin{array}{l} s^{22(k)}uu,_{2} + \frac{s^{12(k)}uu,_{1} }{2} \\ + \frac{3s^{23(k)}qu}{2} \\ \end{array}} \right) \mathbf {I}_{3\times 3} } &{} {\left( {\begin{array}{l} s^{11(k)}\mathbf {d},_{1} + s^{22(k)}\mathbf {d},_{2} \\ + \frac{u,_{1} \mathbf {t},_{2} + u,_{2} \mathbf {t},_{1} }{2}s^{12(k)} \\ + s^{13(k)}\varvec{\upphi },_{1} + s^{23(k)}\varvec{\upphi },_{2} \\ \end{array}} \right) } \\ &{} &{} &{} {s^{11(k)}u^{2}\mathbf {I}_{3\times 3} } &{} {\frac{s^{12(k)}u^{2}\mathbf {I}_{3\times 3} }{2}} &{} {\left( {\begin{array}{l} s^{11(k)}\mathbf {d},_{1} + s^{12(k)}\mathbf {d},_{2} \\ + \frac{3}{2}s^{13(k)}\mathbf {d} \\ \end{array}} \right) } \\ &{} &{} {\mathrm{sym}} &{} &{} {s^{22(k)}u^{2}\mathbf {I}_{3\times 3} } &{} {\left( {\begin{array}{l} s^{22(k)}\mathbf {d},_{2} + s^{12(k)}\mathbf {d},_{1} \\ + \frac{3}{2}s^{23(k)}\mathbf {d} \\ \end{array}} \right) } \\ &{} &{} &{} &{} &{} {\left( {\begin{array}{l} s^{11(k)}\left\| {\mathbf {t},_{1} } \right\| ^{2}+ s^{22(k)}\left\| {\mathbf {t},_{2} } \right\| ^{2} \\ + s^{12(k)}\mathbf {t},_{1}^{T} \mathbf {t},_{2} \\ \end{array}} \right) } \\ \end{array} }} \right] , \nonumber \\ \end{aligned}$$

(A.8)

$$\begin{aligned}&\varvec{\Xi }_{12}^{(2+ k)} =\left[ {{\begin{array}{*{20}c} {\mathbf {0}_{3\times 1} } &{} {\mathbf {0}_{3\times 1} } &{} {s^{11(k)}\mathbf {t},_{1} + \frac{s^{12(k)}\mathbf {t},_{2} }{2}} &{} {s^{11(k)}\mathbf {t}} &{} {\frac{s^{12(k)}\mathbf {t}}{2}} \\ {\mathbf {0}_{3\times 1} } &{} {\mathbf {0}_{3\times 1} } &{} {s^{22(k)}\mathbf {t},_{1} + \frac{s^{12(k)}\mathbf {t},_{2} }{2}} &{} {\frac{s^{12(k)}\mathbf {t}}{2}} &{} {s^{22(k)}\mathbf {t}} \\ {\left( {\begin{array}{l} s^{11(k)}\mathbf {d},_{1} + \frac{s^{12(k)}\mathbf {d},_{2} }{2} \\ + 2s^{13(k)}\varvec{\upphi } \\ \end{array}} \right) } &{} {\left( {\begin{array}{l} s^{22(k)}\mathbf {d},_{2} + \frac{s^{12(k)}\mathbf {d},_{1} }{2} \\ + 2s^{23(k)}\varvec{\upphi } \\ \end{array}} \right) } &{} {\left( {\begin{array}{l} s^{13(k)}\mathbf {d},_{1} + s^{23(k)}\mathbf {d},_{2} \\ + 8s^{33(k)}\varvec{\upphi } \\ \end{array}} \right) } &{} {\left( {\begin{array}{l} s^{11(k)}\mathbf {r},_{1} + \frac{s^{12(k)}\mathbf {r},_{2} }{2} \\ + s^{13(k)}\mathbf {d} \\ \end{array}} \right) } &{} {\left( {\begin{array}{l} s^{22(k)}\mathbf {r},_{2} + \frac{s^{12(k)}\mathbf {r},_{1} }{2} \\ + s^{23(k)}\mathbf {d} \\ \end{array}} \right) } \\ {s^{11(k)}\mathbf {d}} &{} {\frac{s^{12(k)}\mathbf {d}}{2}} &{} {\left( {\begin{array}{l} s^{11(k)}\mathbf {r},_{1} + \frac{s^{12(k)}\mathbf {r},_{2} }{2} \\ + \frac{3}{2}s^{13(k)}\mathbf {d} \\ \end{array}} \right) } &{} {\mathbf {0}_{3\times 1} } &{} {\mathbf {0}_{3\times 1} } \\ {\frac{s^{12(k)}\mathbf {d}}{2}} &{} {s^{22(k)}\mathbf {d}} &{} {\left( {\begin{array}{l} s^{22(k)}\mathbf {r},_{2} + \frac{s^{12(k)}\mathbf {r},_{1} }{2} \\ + \frac{3}{2}s^{23(k)}\mathbf {d} \\ \end{array}} \right) } &{} {\mathbf {0}_{3\times 1} } &{} {\mathbf {0}_{3\times 1} } \\ {s^{11(k)}\mathbf {t},_{1}^{T} \mathbf {t}+ \frac{s^{12(k)}\mathbf {t},_{2}^{T} \mathbf {t}}{2}} &{} {s^{22(k)}\mathbf {t},_{2}^{T} \mathbf {t}+ \frac{s^{12(k)}\mathbf {t},_{1}^{T} \mathbf {t}}{2}} &{} {\frac{3}{2}\left( {s^{13(k)}\mathbf {t},_{1}^{T} \mathbf {t+ }s^{23(k)}\mathbf {t},_{2}^{T} \mathbf {t}} \right) } &{} {\frac{s^{13(k)}\left\| {\mathbf {t}} \right\| ^{2}}{2}} &{} {\frac{s^{23(k)}\left\| {\mathbf {t}} \right\| ^{2}}{2}} \\ \end{array} }} \right] , \end{aligned}$$

(A.9)

$$\begin{aligned}&\varvec{\Xi }_{22}^{(2+k)} =\left[ {{\begin{array}{*{20}c} {s^{11(k)}\left\| {\mathbf {t}} \right\| ^{2}} &{} {\frac{s^{12(k)}\left\| {\mathbf {t}} \right\| ^{2}}{2}} &{} {s^{13(k)}\left\| {\mathbf {t}} \right\| ^{2}} &{} {\mathbf {0}_{1\times 2} } \\ &{} {s^{22(k)}\left\| {\mathbf {t}} \right\| ^{2}} &{} {s^{23(k)}\left\| {\mathbf {t}} \right\| ^{2}} &{} {\mathbf {0}_{1\times 2} } \\ &{} &{} {4s^{33(k)}\left\| {\mathbf {t}} \right\| ^{2}} &{} {\mathbf {0}_{1\times 2} } \\ {\mathrm{sym}} &{} &{} &{} {\mathbf {0}_{2\times 2} } \\ \end{array} }} \right] . \end{aligned}$$

(A.10)

With the definition of the vector

$$\begin{aligned} \mathbf {v}^{(k+l)}={\bar{\mathbf {J}}}^{T}\mathbf {A}^{(k)T}\mathbf {C}_{(p)} \varvec{\upvarepsilon }^{(l)}, \end{aligned}$$

(A.11)

the matrix $\varvec{\Gamma }_{ij}^{(k+l)}$ is written as

$$\begin{aligned} \varvec{\Gamma }_{ij}^{(k+l)} =\hbox {diag}\left[ {{\begin{array}{*{20}c} {\varvec{\Gamma }_{(ij)(11)}^{(k+l)} } &{} \cdots &{} {\varvec{\Gamma }_{(ij)(st)}^{(k+l)} } &{} \cdots &{} {\varvec{\Gamma }_{(ij)(mn)}^{(k+l)} } \\ \end{array} }} \right] , \end{aligned}$$

(A.12)

where

$$\begin{aligned} \varvec{\Gamma }_{(ij)(st)}^{(k+l)} =\left[ {{\begin{array}{*{20}c} {\mathbf {0}_{3\times 3} } &{} {\mathbf {0}_{3\times 3} } &{} {\mathbf {0}_{3\times 3} } \\ {\mathbf {0}_{3\times 3} } &{} {\left[ {\begin{array}{l} \delta _{is} \delta _{jt} \left( {\mathbf {t}_{st} \otimes \mathbf {v}_{ij(3)}^{(k+l)} -\hbox {t}_{kl}^{T} \mathbf {v}_{ij(3)}^{(k+l)} \mathbf {I}_{3\times 3} } \right) \\ +\delta _{jt} C_{is}^{(1,m)} \left( {\mathbf {t}_{st} \otimes \mathbf {v}_{ij(4)}^{(k+l)} -\hbox {t}_{kl}^{T} \mathbf {v}_{ij(4)}^{(k+l)} \mathbf {I}_{3\times 3} } \right) \\ +\delta _{is} C_{jt}^{(1,n)} \left( {\mathbf {t}_{st} \otimes \mathbf {v}_{ij(5)}^{(k+l)} -\hbox {t}_{kl}^{T} \mathbf {v}_{ij(5)}^{(k+l)} \mathbf {I}_{3\times 3} } \right) \\ \end{array}} \right] } &{} {\mathbf {0}_{3\times 3} } \\ {\mathbf {0}_{3\times 3} } &{} {\mathbf {0}_{3\times 3} } &{} {\mathbf {0}_{3\times 3} } \\ \end{array} }} \right] , \end{aligned}$$

(A.13)

with $\mathbf {v}_{(i)}^{(k+l)}$ being the $i\hbox {th}$ term of $\mathbf {v}^{(k+l)}$.

By defining the following matrices

$$\begin{aligned} \mathbf{H }_{v}= & {} \left[ {{\begin{array}{*{20}c} {\mathbf{0 }_{3\times 3} } &{} {\mathbf{0 }_{3\times 3} } &{} {\mathbf{0 }_{3\times 1} } &{} {\mathbf{0 }_{3\times 1} } \\ &{} {\left( {a_{1} u+a_{2} q} \right) \left( {\mathbf {t}\otimes \mathbf {p}-\mathbf {p}^{T}\mathbf {tI}_{3\times 3} } \right) } &{} {a_{1} {\hat{\mathbf {t}}\mathbf{p }}} &{} {a_{2} {\hat{\mathbf {t}}\mathbf{p }}} \\ &{} &{} 0 &{} 0 \\ {\mathrm{sym}} &{} &{} &{} 0 \\ \end{array} }} \right] ,\nonumber \\ {\bar{\mathbf{H }}}_{a}^{\pm }= & {} \left[ {{\begin{array}{*{20}c} {\mathbf {0}_{3\times 3} } &{} {\mathbf {0}_{3\times 3} } &{} {\mathbf {0}_{3\times 1} } &{} {\mathbf {0}_{3\times 1} } \\ &{} {\left[ {h_{0}^{\pm } u+\left( {h_{0}^{\pm } } \right) ^{2}q} \right] \left( {\mathbf {t}\otimes {\bar{\mathbf {p}}}^{\pm }-\mathbf {t}^{T}{\bar{\mathbf {p}}}^{\pm }\mathbf {I}_{3\times 3} } \right) } &{} {h_{0}^{\pm } {\hat{\mathbf {t}}\bar{{\mathbf{p }}}}^{\pm }} &{} {\left( {h_{0}^{\pm } } \right) ^{2}{\hat{\mathbf {t}}\bar{{\mathbf{p }}}}^{\pm }} \\ &{} &{} 0 &{} 0 \\ {\mathrm{sym}} &{} &{} &{} 0 \\ \end{array} }} \right] ,\nonumber \\ {\tilde{\mathbf{H }}}_{e}^{\pm }= & {} \left[ {{\begin{array}{*{20}c} {\mathbf {0}_{3\times 3} } &{} {\mathbf {0}_{3\times 3} } &{} {\mathbf {0}_{3\times 1} } &{} {\mathbf {0}_{3\times 1} } \\ &{} {\left[ {h_{0}^{\pm } u+\left( {h_{0}^{\pm } } \right) ^{2}q} \right] \left( {\mathbf {t}\otimes {\tilde{\mathbf {p}}}^{\pm }-\mathbf {t}^{T}{\tilde{\mathbf {p}}}^{\pm }\mathbf {I}_{3\times 3} } \right) } &{} {h_{0}^{\pm } {\hat{\mathbf {t}}\tilde{{\mathbf{p }}}}^{\pm }} &{} {\left( {h_{0}^{\pm } } \right) ^{2}{\hat{\mathbf {t}}\tilde{{\mathbf{p }}}}^{\pm }} \\ &{} &{} 0 &{} 0 \\ {\mathrm{sym}} &{} &{} &{} 0 \\ \end{array} }} \right] , \end{aligned}$$

(A.14)

the element external tangent stiffness matrix has the form

$$\begin{aligned} \mathbf {K}_{\mathrm{ext}}^{(e)}= & {} \sum \limits _{i=1}^m {\sum \limits _{j=1}^n {w_{i} w_{j} \left| {\mathbf {J}} \right| _{ij} \mathbf {M}_{ij}^{T} \left( {\mathbf {H}_{vij} +\mu _{ij}^{+} {\bar{\mathbf {H}}}_{aij}^{+} +\mu _{ij}^{-} {\bar{\mathbf {H}}}_{aij}^{-} } \right) \mathbf {M}_{ij} } } \nonumber \\&+\,\sum \limits _k {\sum \limits _{i=1}^m {w_{i} \mathbf {M}_{ik}^{T} \left( {j_{1ik}^{+} {\tilde{\mathbf {H}}}_{eik}^{+} +j_{1ik}^{-} {\tilde{\mathbf {H}}}_{eil}^{-} } \right) \mathbf {M}_{ik} } }\nonumber \\&+\,\sum \limits _l {\sum \limits _{j=1}^m {w_{j} \mathbf {M}_{lj}^{T} \left( {j_{2lj}^{+} {\tilde{\mathbf {H}}}_{elj}^{+} +j_{2lj}^{-} {\tilde{\mathbf {H}}}_{elj}^{-} } \right) \mathbf {M}_{lj}. } } \end{aligned}$$

(A.15)

The element tangent stiffness derived from the kinematic constraint term is presented as

$$\begin{aligned} \mathbf {K}_{c}^{(e)} =\sum \limits _{i=1}^m {\sum \limits _{j=1}^n {w_{i} w_{j} j_{0ij} \left| {\mathbf {J}} \right| _{ij} {\tilde{\mathbf {B}}}_{ij}^{T} \mathbf {H}_{cij} {\tilde{\mathbf {B}}}_{ij} } } \end{aligned}$$

(A.16)

with the matrix

$$\begin{aligned} \mathbf {H}_{cij} =\left[ {{\begin{array}{*{20}c} {\mathbf {H}_{c11} } &{} \cdots &{} {\mathbf {H}_{c14} } \\ \vdots &{} \ddots &{} \vdots \\ {\mathbf {H}_{c41} } &{} \cdots &{} {\mathbf {H}_{c44} } \\ \end{array} }} \right] _{ij} . \end{aligned}$$

(A.17)

The sub-matrices in Eq. (A.17) are

$$\begin{aligned} \mathbf {H}_{c11}= & {} \lambda \varvec{\Phi \Omega }_{1} \varvec{\Phi };\nonumber \\ \mathbf {H}_{c12}= & {} \mathbf {H}_{c21} =\mathbf {0}_{3\times 3} ;\nonumber \\ \mathbf {H}_{c13}= & {} \mathbf {H}_{c31}^{T} =\lambda \varvec{\Phi \Omega }_{1} \varvec{\Upsilon }_{1} {\hat{\mathbf {t}}}+\lambda \mathbf {t}^{T}\varvec{\Psi }_{1} {\varvec{\Lambda }}{\bar{\varvec{\uptau }}}_{0} {\hat{\mathbf {t}}}-\lambda \mathbf {t}\otimes {\hat{\mathbf {t}}\Psi }_{1} {\varvec{\Lambda }}{\bar{\varvec{\uptau }}}_{0} -\lambda \varvec{\Phi }\varvec{\Psi }_{1} {\varvec{\Lambda }}{\hat{{\bar{\varvec{\uptau }}}}}_{0} {\varvec{\Lambda }}^{T};\nonumber \\ \mathbf {H}_{c14}= & {} \mathbf {H}_{c41}^{T} =\varvec{\Phi }\varvec{\Psi }_{1} {\varvec{\Lambda }}{\bar{\varvec{\uptau }}}_{0}, \end{aligned}$$

(A.18)

$$\begin{aligned} \mathbf {H}_{c22}= & {} \lambda \varvec{\Phi \Omega }_{2} \varvec{\Phi };\nonumber \\ \mathbf {H}_{c23}= & {} \mathbf {H}_{c32}^{T} =\lambda \varvec{\Phi }\varvec{\Psi }_{2} {\varvec{\Lambda }}{\hat{{\bar{\varvec{\uptau }}}}}_{0} \varvec{\Lambda }^{T}-\lambda \varvec{\Phi \Omega }_{2} \varvec{\Upsilon }_{2} {\hat{\mathbf {t}}}-\lambda \mathbf {t}^{T}\varvec{\Psi }_{2} {\varvec{\Lambda }}{\bar{\varvec{\uptau }}}_{0} {\hat{\mathbf {t}}}-\lambda \left( {\mathbf {t}\otimes \varvec{\Psi }_{2} {\varvec{\Lambda }}{\bar{\varvec{\uptau }}}_{0} } \right) {\hat{\mathbf {t}}};\nonumber \\ \mathbf {H}_{c24}= & {} \mathbf {H}_{c42}^{T} =-\varvec{\Phi }\varvec{\Psi }_{2} \varvec{\Lambda \bar{{\uptau }}}_{0}, \end{aligned}$$

(A.19)

$$\begin{aligned} \mathbf {H}_{c33}= & {} \lambda \varvec{\Lambda \hat{{\bar{{\uptau }}}}}_{0} {\varvec{\Lambda }}^{T}\left( {\varvec{\Psi }_{1} \varvec{\Upsilon }_{1} -\varvec{\Psi }_{2} \varvec{\Upsilon }_{2} } \right) {\hat{\mathbf {t}}+} {\varvec{\lambda }} {\hat{{\bar{\varvec{\uptau }}}}\Lambda \hat{{\bar{{\uptau }}}}}_{0} {\varvec{\Lambda }}^{T} \nonumber \\&+\,\lambda \left[ {\mathbf {t}\otimes \left( {\varvec{\Upsilon }_{2}^{T} \varvec{\Psi }_{2} -\varvec{\Upsilon }_{1}^{T} \varvec{\Psi }_{1} } \right) {\varvec{\Lambda }}{\bar{\varvec{\uptau }}}_{0} -\mathbf {t}^{T}\left( {\varvec{\Upsilon }_{2}^{T} \varvec{\Psi }_{2} -\varvec{\Upsilon }_{1}^{T} \varvec{\Psi }_{1} } \right) {\varvec{\Lambda }}{\bar{\varvec{\uptau }}}_{0} \mathbf {I}_{3\times 3} } \right] \nonumber \\&-\,\lambda {\hat{\mathbf {t}}}\left( {\mathbf {r},_{2} \otimes \varvec{\Psi }_{2} {\varvec{\Lambda }}{\bar{\varvec{\uptau }}}_{0} +\varvec{\Psi }_{2} {\varvec{\Lambda }}{\bar{\varvec{\uptau }}}_{0} \otimes \mathbf {r},_{2} +\mathbf {r},_{1} \otimes \varvec{\Psi }_{1} {\varvec{\Lambda }}{\bar{\varvec{\uptau }}}_{0} +\varvec{\Psi }_{1} {\varvec{\Lambda }}{\bar{\varvec{\uptau }}}_{0} \otimes \mathbf {r},_{1} } \right) {\hat{\mathbf {t}}} \nonumber \\&+\,\lambda {\hat{\mathbf {t}}}\left( {\varvec{\Upsilon }_{2}^{T} \varvec{\Omega }_{2} \varvec{\Upsilon }_{2} -\,\varvec{\Upsilon }_{1}^{T} \varvec{\Omega }_{1} \varvec{\Upsilon }_{1} } \right) {\hat{\mathbf {t}}}-\lambda {\hat{\mathbf {t}}}\left[ {\varvec{\Upsilon }_{2}^{T} \varvec{\Psi }_{2} -\varvec{\Upsilon }_{1}^{T} \varvec{\Psi }_{1} } \right] {\varvec{\Lambda }}{\hat{{\bar{\varvec{\uptau }}}}}_{0} {\varvec{\Lambda }}^{T}; \end{aligned}$$

(A.20)

$$\begin{aligned} \mathbf {H}_{c34}= & {} \mathbf {H}_{c43}^{T} =-\,{\hat{{\bar{\varvec{\uptau }}}}}{\varvec{\Lambda }}{\bar{\varvec{\uptau }}}_{0} +\,{\hat{\mathbf {t}}}\left[ {\varvec{\Upsilon }_{2}^{T} \varvec{\Psi }_{2} -\varvec{\Upsilon }_{1}^{T} \varvec{\Psi }_{1} } \right] {\varvec{\Lambda }}{\bar{\varvec{\uptau }}}_{0} ;\nonumber \\ \mathbf {H}_{c44}= & {} 0, \end{aligned}$$

(A.21)

where

$$\begin{aligned} \varvec{\Omega }_{\alpha } =\frac{3\varvec{\uptau }_{\alpha }^{T} {\varvec{\Lambda }}{\bar{\varvec{\uptau }}}_{0} \left( {\varvec{\uptau }_{\alpha } \otimes \varvec{\uptau }_{\alpha } } \right) -\left\| {\varvec{\uptau }_{\alpha } } \right\| ^{2}\left( {{\varvec{\Lambda }}{\bar{\varvec{\uptau }}}_{0} \otimes \varvec{\uptau }_{\alpha } +\varvec{\uptau }_{\alpha } \otimes {\varvec{\Lambda }}{\bar{\varvec{\uptau }}}_{0} +\varvec{\uptau }_{\alpha }^{T} {\varvec{\Lambda }}{\bar{\varvec{\uptau }}}_{0} \mathbf {I}_{3\times 3} } \right) }{\left\| {\varvec{\uptau }_{\alpha } } \right\| ^{5}}. \end{aligned}$$

(A.22)