Nonlinear dynamic formulation for flexible origami-based deployable structures considering self-contact and friction

Abstract

Compared with the conventional rigid origami, the flexible origami has larger deformation and more complicated mechanical property and nonlinear problems due to self-contact and friction. In this paper, the nonlinear dynamic formulation for flexible origami-based deployable structures considering self-contact and friction is investigated. Firstly, a symmetric rigid origami model is presented based on the forward recursive formulation without the inclusion of contact, and then, a discretized dynamic model for flexible origami structures is established by using thin plate element of absolute nodal coordinate formulation. To consider the normal contact, the penalty method is adopted to enforce the nonpenetration condition. In order to improve the precision and applicability, a modified mixed contact method considering the friction effect is developed by integrating the advantages of node-to-surface, edge-to-surface and surface-to-surface contact elements. This proposed method can effectively avoid the mutual penetration of different corner nodes, element edges and contact element surfaces. Moreover, the tangential friction model considering the stick–slip transition and large sliding is established by the regularized Coulomb friction law. A series of numerical examples validate the effectiveness of the proposed mixed contact method considering the friction and show the advantage of the flexible model compared with the rigid origami model. Furthermore, the nonlinear performance of the flexible origami-based deployable structures due to the contact and friction is revealed.

Appendices

Appendix 1

The shape function matrix ${\mathbf{S}}_{A}$ for triangular thin plate element is given by

$$ \begin{aligned}&{\mathbf{S}}_{A} = \left[ \begin{array}{*{20}l} {S_{11} {\mathbf{I}}_{{3}} } & {S_{12} {\mathbf{I}}_{{3}} } & {S_{13} {\mathbf{I}}_{{3}} } & {S_{21} {\mathbf{I}}_{{3}} } & {S_{22} {\mathbf{I}}_{{3}} } & {S_{23} {\mathbf{I}}_{{3}} }\end{array}\right. \\ &\left.\begin{array}{*{20}l}{S_{31} {\mathbf{I}}_{{3}} } & {S_{32} {\mathbf{I}}_{{3}} } & {S_{33} {\mathbf{I}}_{{3}} } \\ \end{array} \right] \in \Re^{{{\kern 1pt} 3{\kern 1pt} {\kern 1pt} \times {\kern 1pt} {\kern 1pt} 27}} . \end{aligned}$$

The shape functions are as follows

$$ \begin{aligned} S_{11} &= \Delta_{1} - \Delta_{1} \Delta_{2} + \Delta_{3} \Delta_{1} + 2\left( {P_{1} - P_{3} } \right) \hfill \\ S_{21} &= \Delta_{2} - \Delta_{2} \Delta_{3} + \Delta_{1} \Delta_{2} + 2\left( {P_{2} - P_{1} } \right) \hfill \\ S_{31} &= \Delta_{3} - \Delta_{3} \Delta_{1} + \Delta_{2} \Delta_{3} + 2\left( {P_{3} - P_{2} } \right) \hfill \\ S_{12} &= c_{22} \left( {P_{3} - \Delta_{3} \Delta_{1} } \right) + c_{32} P_{1} \hfill \\ S_{22} &= c_{32} \left( {P_{1} - \Delta_{1} \Delta_{2} } \right) + c_{12} P_{2} \hfill \\ S_{32} &= c_{12} \left( {P_{2} - \Delta_{2} \Delta_{3} } \right) + c_{22} P_{3} \hfill \\ S_{13} &= - c_{21} \left( {P_{3} - \Delta_{3} \Delta_{1} } \right) - c_{31} P_{1} \hfill \\ S_{23} &= - c_{31} \left( {P_{1} - \Delta_{1} \Delta_{2} } \right) - c_{11} P_{2} \hfill \\ S_{33} &= - c_{11} \left( {P_{2} - \Delta_{2} \Delta_{3} } \right) - c_{21} P_{3} \hfill \\ \end{aligned} $$

where $P_{i} = \Delta_{i}^{2} \Delta_{j} + {{\Delta_{i} \Delta_{j} \Delta_{k} \left[ {3\left( {1 - \mu_{k} } \right)\Delta_{i} + \left( {1 + 3\mu_{k} } \right)\left( {\Delta_{k} - \Delta_{j} } \right)} \right]} \mathord{\left/ {\vphantom {{\Delta_{i} \Delta_{j} \Delta_{k} \left[ {3\left( {1 - \mu_{k} } \right)\Delta_{i} + \left( {1 + 3\mu_{k} } \right)\left( {\Delta_{k} - \Delta_{j} } \right)} \right]} 2}} \right. \kern-\nulldelimiterspace} 2}$, $\mu_{i} = {{\left( {l_{k}^{2} - l_{j}^{2} } \right)} \mathord{\left/ {\vphantom {{\left( {l_{k}^{2} - l_{j}^{2} } \right)} {l_{i}^{2} }}} \right. \kern-\nulldelimiterspace} {l_{i}^{2} }}$, $l_{i}^{2} = c_{i1}^{2} + c_{i2}^{2}$. In addition, $\Delta_{i} {\kern 1pt} {\kern 1pt} (i = 1,2,3)$ and $c_{ij} {\kern 1pt} (i,j = 1,2,3)$ are triangular area coordinates and area coefficients, respectively.

Appendix 2

The shape function matrix ${\mathbf{S}}_{B}$ for rectangular thin plate element is written as

$$ \begin{aligned} &{\mathbf{S}}_{B} = \left[ {\begin{array}{*{20}l} {S_{11} {\mathbf{I}}_{{3}} } & {S_{12} {\mathbf{I}}_{{3}} } & {S_{13} {\mathbf{I}}_{{3}} } & {S_{21} {\mathbf{I}}_{{3}} } & {S_{22} {\mathbf{I}}_{{3}} } & {S_{23} {\mathbf{I}}_{{3}} } \\ \end{array} } \right. \\& \left. {\begin{array}{*{20}l} {S_{31} {\mathbf{I}}_{{3}} } & {S_{32} {\mathbf{I}}_{{3}} } & {S_{33} {\mathbf{I}}_{{3}} } & {S_{41} {\mathbf{I}}_{{3}} } & {S_{42} {\mathbf{I}}_{{3}} } & {S_{43} {\mathbf{I}}_{{3}} } \\ \end{array} } \right] \in \Re {\kern 1pt}^{{3{\kern 1pt} {\kern 1pt} \times {\kern 1pt} {\kern 1pt} 36}} \;. \hfill \\ \end{aligned} $$

The shape functions are as follows

$$ \begin{aligned} S_{11} &= - \left( {\xi - 1} \right)\left( {\eta - 1} \right)\left( {2\eta^{2} - \eta - \xi + 2\xi^{2} - 1} \right)\;, \hfill \\ S_{12} &= - a_{e} \xi \left( {\xi - 1} \right)^{2} \left( {\eta - 1} \right)\;, \hfill \\ S_{13} &= - b_{e} \eta \left( {\eta - 1} \right)^{2} \left( {\xi - 1} \right)\;, \hfill \\ S_{21} &= \xi \left( {\eta - 1} \right)\left( {2\eta^{2} - \eta - 3\xi + 2\xi^{2} } \right)\;, \hfill \\ S_{22} &= - a_{e} \xi^{2} \left( {\xi - 1} \right)\left( {\eta - 1} \right)\;, \hfill \\ S_{23} &= b_{e} \xi \eta \left( {\eta - 1} \right)^{2} \;, \hfill \\ S_{31} &= - \xi \eta \left( {1 + 2\eta^{2} - 3\eta - 3\xi + 2\xi^{2} } \right)\;, \hfill \\ S_{32} &= a_{e} \xi^{2} \eta \left( {\xi - 1} \right)\;, \hfill \\ S_{33} &= b_{e} \xi \eta^{2} \left( {\eta - 1} \right)\;, \hfill \\ S_{41} &= \eta \left( {\xi - 1} \right)\left( {2\eta^{2} - 3\eta - \xi + 2\xi^{2} } \right)\;, \hfill \\ S_{42} &= a_{e} \xi \eta \left( {\xi - 1} \right)^{2} \;, \hfill \\ S_{43} &= - b_{e} \eta^{2} \left( {\xi - 1} \right)\left( {\eta - 1} \right)\;, \hfill \\ \end{aligned} $$

where $\xi = {x \mathord{\left/ {\vphantom {x {a_{e} }}} \right. \kern-\nulldelimiterspace} {a_{e} }},\;\eta = {y \mathord{\left/ {\vphantom {y {b_{e} }}} \right. \kern-\nulldelimiterspace} {b_{e} }}\;\;(0 \le \xi ,\eta \le 1)$. $a_{e}$ and $b_{e}$ are the lengths of the rectangular element in the direction of x and y, respectively.