A weak form quadrature element formulation of geometrically exact shells incorporating drilling degrees of freedom

Abstract

Geometrically nonlinear analysis of shell structures is conducted using weak form quadrature elements. A new geometrically exact shell formulation incorporating drilling degrees of freedom is established wherein rotation quaternions in combination with a total Lagrange updating scheme are employed for rotation description. An extended kinematic condition to serve as the drilling rotation constraint, derived from polar decomposition of modified mid-surface deformation gradient, is exactly satisfied in the formulation. Several benchmark examples are presented to illustrate the versatility and robustness of the present formulation.

Appendix: Expression of element tangent stiffness matrix

Corresponding to the vectors ${\mathbf{G}}_{int}^{(e)}$ and ${\mathbf{G}}_{c}^{(e)}$ composing the element residual force vector, the element tangent stiffness matrix is written as

$$ {\mathbf{K}}^{(e)} = {\mathbf{K}}_{int}^{(e)} + {\mathbf{K}}_{c}^{(e)} . $$

(71)

It can be derived from Eq. (43) that

$$ {\mathbf{K}}_{int}^{(e)} = \sum\limits_{i = 1}^{m} {\sum\limits_{j = 1}^{n} {w_{i} w_{j} j_{0ij} \left| {{\mathbf{J}}_{ij} } \right|\left( {{\mathbf{B}}_{ij}^{T} {\mathbf{D}}_{ij} {\mathbf{B}}_{ij} + {\mathbf{C}}_{ij}^{T} {\bar{\mathbf{J}}}_{ij}^{T} {\varvec{\Xi}}_{ij} {\mathbf{B}}_{ij} + {\varvec{\Theta}}_{ij} } \right)} } , $$

(72)

where

$$ {\varvec{\Xi}}_{ij} = \left[ {\begin{array}{*{20}c} {n^{11} {\mathbf{I}}_{3 \times 3} } & {n^{12} {\mathbf{I}}_{3 \times 3} } & {q^{1} {\mathbf{I}}_{3 \times 3} } & {m^{11} {\mathbf{I}}_{3 \times 3} } & {m^{12} {\mathbf{I}}_{3 \times 3} } \\ {n^{12} {\mathbf{I}}_{3 \times 3} } & {n^{22} {\mathbf{I}}_{3 \times 3} } & {q^{2} {\mathbf{I}}_{3 \times 3} } & {m^{12} {\mathbf{I}}_{3 \times 3} } & {m^{22} {\mathbf{I}}_{3 \times 3} } \\ {q^{1} {\mathbf{I}}_{3 \times 3} } & {q^{2} {\mathbf{I}}_{3 \times 3} } & {{\mathbf{0}}_{3 \times 3} } & {{\mathbf{0}}_{3 \times 3} } & {{\mathbf{0}}_{3 \times 3} } \\ {m^{11} {\mathbf{I}}_{3 \times 3} } & {m^{12} {\mathbf{I}}_{3 \times 3} } & {{\mathbf{0}}_{3 \times 3} } & {{\mathbf{0}}_{3 \times 3} } & {{\mathbf{0}}_{3 \times 3} } \\ {m^{12} {\mathbf{I}}_{3 \times 3} } & {m^{22} {\mathbf{I}}_{3 \times 3} } & {{\mathbf{0}}_{3 \times 3} } & {{\mathbf{0}}_{3 \times 3} } & {{\mathbf{0}}_{3 \times 3} } \\ \end{array} } \right] $$

(73)

and

$$ {\varvec{\Theta}}_{ij} = diag\left[ {\begin{array}{*{20}c} \cdots & {\left[ {\begin{array}{*{20}c} {{\mathbf{0}}_{3 \times 3} } & {{\mathbf{0}}_{3 \times 3} } & {{\mathbf{0}}_{3 \times 1} } \\ {{\mathbf{0}}_{3 \times 3} } & {{\bar{\varvec{\varTheta }}}} & {{\mathbf{0}}_{3 \times 1} } \\ {{\mathbf{0}}_{1 \times 3} } & {{\mathbf{0}}_{1 \times 3} } & 0 \\ \end{array} } \right]_{(ij)(kl)} } & \cdots \\ \end{array} } \right] $$

(74)

with

$$ \begin{aligned} {\bar{\varvec{\varTheta }}}_{(ij)(kl)} &= \delta_{ik} \delta_{jl} \left( {q^{1} {\hat{\mathbf{r}}},_{1} + q^{2} {\hat{\mathbf{r}}},_{2} } \right)_{ij} {\hat{\mathbf{t}}}_{kl} \\ & \quad + \delta_{jl} C_{ik}^{(m)} \left[ {\left( {\bar{J}_{11} m^{11} + \bar{J}_{12} m^{12} } \right){\hat{\mathbf{r}}},_{1} + \left( {\bar{J}_{11} m^{12} + \bar{J}_{12} m^{22} } \right){\hat{\mathbf{r}}},_{2} } \right]_{ij} {\hat{\mathbf{t}}}_{kl} \\ & \quad + \delta_{ik} C_{jl}^{(n)} \left[ {\left( {\bar{J}_{21} m^{11} + \bar{J}_{22} m^{12} } \right){\hat{\mathbf{r}}},_{1} + \left( {\bar{J}_{21} m^{12} + \bar{J}_{22} m^{22} } \right){\hat{\mathbf{r}}},_{2} } \right]_{ij} {\hat{\mathbf{t}}}_{kl} \\ \end{aligned} . $$

(75)

The element tangent stiffness matrix related to extended kinematics constraint is obtained from Eq. (56) as

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

(76)

with the matrix

$$ {\mathbf{H}}_{ij} = \left[ {\begin{array}{*{20}c} {{\bar{\mathbf{H}}}_{11} } & \cdots & {{\bar{\mathbf{H}}}_{14} } \\ \vdots & \ddots & \vdots \\ {{\bar{\mathbf{H}}}_{41} } & \cdots & {{\bar{\mathbf{H}}}_{44} } \\ \end{array} } \right]_{ij} . $$

(77)

By introducing the matrix

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

(78)

the sub-matrices in Eq. (77) are given by

$$ {\bar{\mathbf{H}}}_{11} = \lambda {\varvec{\varPhi \varOmega }}_{1} {\varvec{\varPhi ;}} $$

$$ {\bar{\mathbf{H}}}_{22} = \lambda {\varvec{\varPhi \varOmega }}_{2} {\varvec{\Phi}}; $$

$$ \begin{aligned} {\bar{\mathbf{H}}}_{33} &= \lambda {\varvec{\varLambda \hat{\bar{\tau }}}}_{0} {\varvec{\Lambda}}^{T} \left( {{\varvec{\Psi}}_{1} {\varvec{\Upsilon}}_{1} - {\varvec{\Psi}}_{2} {\varvec{\Upsilon}}_{2} } \right){\varvec{\hat{t} + }}\lambda {\varvec{\hat{\bar{\tau }}\varLambda \hat{\bar{\tau }}}}_{0} {\varvec{\Lambda}}^{T} \\ & \quad + \lambda \left[ {{\mathbf{t}} \otimes \left( {{\varvec{\Upsilon}}_{2}^{T} {\varvec{\Psi}}_{2} - {\varvec{\Upsilon}}_{1}^{T} {\varvec{\Psi}}_{1} } \right){\varvec{\Lambda}}{\bar{\varvec{\tau }}}_{0} - {\mathbf{t}}^{T} \left( {{\varvec{\Upsilon}}_{2}^{T} {\varvec{\Psi}}_{2} - {\varvec{\Upsilon}}_{1}^{T} {\varvec{\Psi}}_{1} } \right){\varvec{\Lambda}}{\bar{\varvec{\tau }}}_{0} {\mathbf{I}}_{3 \times 3} } \right] \\ & \quad - \lambda {\hat{\mathbf{t}}}\left( {{\mathbf{r}},_{2} \otimes {\varvec{\Psi}}_{2} {\varvec{\Lambda}}{\bar{\varvec{\tau }}}_{0} + {\varvec{\Psi}}_{2} {\varvec{\Lambda}}{\bar{\varvec{\tau }}}_{0} \otimes {\mathbf{r}},_{2} + {\mathbf{r}},_{1} \otimes {\varvec{\Psi}}_{1} {\varvec{\Lambda}}{\bar{\varvec{\tau }}}_{0} + {\varvec{\Psi}}_{1} {\varvec{\Lambda}}{\bar{\varvec{\tau }}}_{0} \otimes {\mathbf{r}},_{1} } \right){\hat{\mathbf{t}}} \\ & \quad + \lambda {\hat{\mathbf{t}}}\left( {{\varvec{\Upsilon}}_{2}^{T} {\varvec{\Omega}}_{2} {\varvec{\Upsilon}}_{2} - {\varvec{\Upsilon}}_{1}^{T} {\varvec{\Omega}}_{1} {\varvec{\Upsilon}}_{1} } \right){\hat{\mathbf{t}}} - \lambda {\hat{\mathbf{t}}}\left[ {{\varvec{\Upsilon}}_{2}^{T} {\varvec{\Psi}}_{2} - {\varvec{\Upsilon}}_{1}^{T} {\varvec{\Psi}}_{1} } \right]{\varvec{\Lambda}}{\varvec{\hat{\bar{\tau }}}}_{0} {\varvec{\Lambda}}^{T} ; \\ \end{aligned} $$

$$ \begin{aligned} & {\bar{\mathbf{H}}}_{44} = 0; \\ & {\bar{\mathbf{H}}}_{12} = {\bar{\mathbf{H}}}_{21} = {\mathbf{0}}_{3 \times 3} ; \\ \end{aligned} $$

(79)

$$ {\bar{\mathbf{H}}}_{13} = {\bar{\mathbf{H}}}_{31}^{T} = \lambda {\varvec{\varPhi \varOmega }}_{1} {\varvec{\Upsilon}}_{1} {\hat{\mathbf{t}}} + \lambda {\mathbf{t}}^{T} {\varvec{\Psi}}_{1} {\varvec{\Lambda}}{{\bar{{\varvec{\tau}}}}}_{0} {\hat{\mathbf{t}}} - \lambda {\mathbf{t}} \otimes {\varvec{\hat{t}\varPsi }}_{1} {\varvec{\Lambda}}{{\bar{{\varvec{\tau}}}}}_{0} - \lambda {\varvec{\varPhi \varPsi }}_{1} {\varvec{\Lambda}}{{\hat{\bar{\varvec{\tau }}}}}_{0} {\varvec{\Lambda}}^{T} ; $$

$$ {\bar{\mathbf{H}}}_{14} = {\bar{\mathbf{H}}}_{41}^{T} = {\varvec{\varPhi \varPsi }}_{1} {\varvec{\Lambda}}{{\bar{{\varvec{\tau}}}}}_{0} ; $$

$$ {\bar{\mathbf{H}}}_{23} = {\bar{\mathbf{H}}}_{32}^{T} = \lambda {\varvec{\varPhi \varPsi }}_{2} {\varvec{\Lambda}}{{\hat{\bar{\varvec{\tau }}}}}_{0} {\varvec{\Lambda}}^{T} - \lambda {\varvec{\varPhi \varOmega }}_{2} {\varvec{\Upsilon}}_{2} {\hat{\mathbf{t}}} - \lambda {\mathbf{t}}^{T} {\varvec{\Psi}}_{2} {\varvec{\Lambda}}{{\bar{{\varvec{\tau}}}}}_{0} {\hat{\mathbf{t}}} - \lambda \left( {{\mathbf{t}} \otimes {\varvec{\Psi}}_{2} {\varvec{\Lambda}}{{\bar{{\varvec{\tau}}}}}_{0} } \right){\hat{\mathbf{t}}}; $$

$$ {\bar{\mathbf{H}}}_{24} = {\bar{\mathbf{H}}}_{42}^{T} = - {\varvec{\varPhi \varPsi }}_{2} {\varvec{\varLambda \bar{\tau }}}_{0} ; $$

$$ {\bar{\mathbf{H}}}_{34} = {\bar{\mathbf{H}}}_{43}^{T} = - {{\hat{\bar{\varvec{\tau }}}}}{\varvec{\Lambda}}{{\bar{{\varvec{\tau}}}}}_{0} + {\hat{\mathbf{t}}}\left[ {{\varvec{\Upsilon}}_{2}^{T} {\varvec{\Psi}}_{2} - {\varvec{\Upsilon}}_{1}^{T} {\varvec{\Psi}}_{1} } \right]{\varvec{\Lambda}}{{\bar{{\varvec{\tau}}}}}_{0}. $$