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Orthotropic material models for the nonlinear analysis of structural membranes

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Abstract

In this paper, we revisit two orthotropic material models proposed in previous works and develop their expressions to obtain their elastic parameters via calibration with experimental data. Our purpose is to make these models available to the nonlinear finite element analysis of structural membranes. We describe the membrane kinematics as a geometrically exact thin shell, whose in-plane orthotropic character is introduced solely at the constitutive equation. Accordingly, small bending, transverse shear and compressive stiffnesses are always present due to the shell assumptions. A least-squares minimization problem is formulated and resolved for the obtaining of the elastic parameters. Stress–strain curves of physical stretch tests are used to this end. The two resulting models were implemented into a nonlinear finite element code and numerical simulations of the stretch tests are conducted to assess the performance of the scheme.

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Notes

  1. These results are offered for free, “as a suggestion in any appropriate experimentation you may care to undertake” [3].

References

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Acknowledgments

Authors acknowledge CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico) for the research Grants 136581/2009-9 and 303793/2012-0. Second author also acknowledges FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo) for the Grant 2012/04009-0, which made possible his research stay at the University of California at Berkeley (Department of Mechanical Engineering) on a momentary leave from the University of São Paulo. In addition, authors would like to express their gratitude to the colleagues Paulo M. Pimenta and Ruy M. O. Pauletti, for the many fruitful discussions on this paper.

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Authors and Affiliations

  1. Department of Structural and Geotechnical Engineering, Polytechnic School, University of São Paulo, P.O. Box 61548, São Paulo, SP, Brazil

    Fernando R. Gonçalves & Eduardo M. B. Campello

Authors
  1. Fernando R. Gonçalves
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  2. Eduardo M. B. Campello
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Corresponding author

Correspondence to Eduardo M. B. Campello.

Additional information

Technical Editor: Lavinia Maria Sanabio Alves Borges.

Appendices

Appendix 1

For the Saint-Venant orthotropic material, components of the constitutive tangent matrix \( \user2{D}_{\text{ortho}} \), indicated in (46), result as follows (only the non-zero terms are shown):

$$ \begin{gathered} \frac{{\partial n_{{ 1 {\text{ortho}}_{ 1} }}^{r} }}{{\partial \eta_{11}^{r} }} = \left( {2\alpha_{1} + 4\mu_{1} + \beta_{1} } \right)E_{11}^{m} + \left( {\alpha_{1} + \alpha_{2} + \beta_{12} } \right)E_{22}^{m} , \hfill \\ \frac{{\partial n_{{ 1 {\text{ortho}}_{ 2} }}^{r} }}{{\partial \eta_{12}^{r} }} = \left( {2\alpha_{1} + 4\mu_{1} + \beta_{1} } \right)E_{11}^{m} + \left( {\alpha_{1} + \alpha_{2} + \beta_{12} } \right)E_{22}^{m} , \hfill \\ \frac{{\partial n_{{ 1 {\text{ortho}}_{ 1} }}^{r} }}{{\partial \eta_{21}^{r} }} = 2\left( {\mu_{1} + \mu_{2} } \right)E_{12}^{m} ,\quad \frac{{\partial n_{{ 1 {\text{ortho}}_{ 2} }}^{r} }}{{\partial \eta_{22}^{r} }} = 2\left( {\mu_{1} + \mu_{2} } \right)E_{12}^{m} , \hfill \\ \frac{{\partial n_{{ 2 {\text{ortho}}_{ 1} }}^{r} }}{{\partial \eta_{11}^{r} }} = 2\left( {\mu_{1} + \mu_{2} } \right)E_{12}^{m} ,\quad \frac{{\partial n_{{ 2 {\text{ortho}}_{ 2} }}^{r} }}{{\partial \eta_{12}^{r} }} = 2\left( {\mu_{1} + \mu_{2} } \right)E_{12}^{m} , \hfill \\ \frac{{\partial n_{{ 2 {\text{ortho}}_{ 1} }}^{r} }}{{\partial \eta_{21}^{r} }} = \left( {\alpha_{1} + \alpha_{2} + \beta_{12} } \right)E_{11}^{m} + \left( {2\alpha_{2} + 4\mu_{2} + \beta_{2} } \right)E_{22}^{m} \;{\text{and}} \hfill \\ \frac{{\partial n_{{ 2 {\text{ortho}}_{ 2} }}^{r} }}{{\partial \eta_{22}^{r} }} = \left( {\alpha_{1} + \alpha_{2} + \beta_{12} } \right)E_{11}^{m} + \left( {2\alpha_{2} + 4\mu_{2} + \beta_{2} } \right)E_{22}^{m} , \hfill \\ \end{gathered} $$
(62)

in which \( E_{\alpha \beta }^{m} \) are given by (41).

Appendix 2

For the Oliveira orthotropic material, components of the constitutive tangent matrix \( \user2{D}_{\text{ortho}} \), indicated in (57), result as follows (only the non-zero terms are shown):

$$ \begin{gathered} \frac{{\partial n_{{ 1 {\text{ortho}}_{ 1} }}^{r} }}{{\partial \eta_{11}^{r} }} = \left[ {\frac{{\partial \varphi_{1} }}{{\partial \eta_{11}^{r} }} + 2\bar{\mu }\left( {1 + \eta_{11} } \right)} \right]\left( {1 + \eta_{11} } \right) + \left( {\varphi_{1} + 2\bar{\mu }I_{11} } \right) + \bar{\mu }\eta_{21}^{2} , \hfill \\ \frac{{\partial n_{{ 1 {\text{ortho}}_{ 1} }}^{r} }}{{\partial \eta_{12}^{r} }} = \left[ {\frac{{\partial \varphi_{1} }}{{\partial \eta_{12}^{r} }} + 2\bar{\mu }\eta_{12} } \right]\left( {1 + \eta_{11} } \right) + \bar{\mu }\eta_{21} \left( {1 + \eta_{22} } \right), \hfill \\ \frac{{\partial n_{{ 1 {\text{ortho}}_{ 2} }}^{r} }}{{\partial \eta_{11}^{r} }} = \left[ {\frac{{\partial \varphi_{1} }}{{\partial \eta_{11}^{r} }} + 2\bar{\mu }\left( {1 + \eta_{11} } \right)} \right]\eta_{12} + \bar{\mu }\eta_{21} \left( {1 + \eta_{22} } \right), \hfill \\ \frac{{\partial n_{{ 1 {\text{ortho}}_{ 2} }}^{r} }}{{\partial \eta_{12}^{r} }} = \left[ {\frac{{\partial \varphi_{1} }}{{\partial \eta_{12}^{r} }} + 2\bar{\mu }\eta_{12} } \right]\eta_{12} + \left( {\varphi_{1} + 2\bar{\mu }I_{11} } \right) + \bar{\mu }\left( {1 + \eta_{22} } \right)^{2} , \hfill \\ \frac{{\partial n_{{ 1 {\text{ortho}}_{ 1} }}^{r} }}{{\partial \eta_{21}^{r} }} = \left[ {\frac{{\partial \varphi_{1} }}{{\partial \eta_{21}^{r} }}} \right]\left( {1 + \eta_{11} } \right) + 2\bar{\mu }\eta_{21} \left( {1 + \eta_{11} } \right) + \bar{\mu }\eta_{12} \left( {1 + \eta_{22} } \right), \hfill \\ \frac{{\partial n_{{ 1 {\text{ortho}}_{ 1} }}^{r} }}{{\partial \eta_{22}^{r} }} = \left[ {\frac{{\partial \varphi_{1} }}{{\partial \eta_{22}^{r} }}} \right]\left( {1 + \eta_{22} } \right) + \bar{\mu }\eta_{12} \eta_{21} ,\quad \frac{{\partial n_{{ 1 {\text{ortho}}_{ 2} }}^{r} }}{{\partial \eta_{21}^{r} }} = \left[ {\frac{{\partial \varphi_{1} }}{{\partial \eta_{21}^{r} }}} \right]\eta_{12} + \bar{\mu }\left( {1 + \eta_{11} } \right)\left( {1 + \eta_{22} } \right), \hfill \\ \frac{{\partial n_{{ 1 {\text{ortho}}_{ 2} }}^{r} }}{{\partial \eta_{22}^{r} }} = \left[ {\frac{{\partial \varphi_{1} }}{{\partial \eta_{22}^{r} }}} \right]\eta_{12} + \bar{\mu }\left( {1 + \eta_{11} } \right)\eta_{21} + 2\bar{\mu }\left( {1 + \eta_{22} } \right)\eta_{12} , \hfill \\ \end{gathered} $$
(63)
$$ \begin{gathered} \frac{{\partial n_{{ 2 {\text{ortho}}_{ 1} }}^{r} }}{{\partial \eta_{11}^{r} }} = \left[ {\frac{{\partial \varphi_{2} }}{{\partial \eta_{11}^{r} }}} \right]\eta_{21} + 2\bar{\mu }\eta_{21} \left( {1 + \eta_{11} } \right) + \bar{\mu }\eta_{12} \left( {1 + \eta_{22} } \right), \hfill \\ \frac{{\partial n_{{ 2 {\text{ortho}}_{ 1} }}^{r} }}{{\partial \eta_{12}^{r} }} = \left[ {\frac{{\partial \varphi_{2} }}{{\partial \eta_{12}^{r} }}} \right]\eta_{21} \left( {1 + \eta_{22} } \right) + \bar{\mu }\left( {1 + \eta_{11} } \right)\left( {1 + \eta_{22} } \right), \hfill \\ \frac{{\partial n_{{ 2 {\text{ortho}}_{ 2} }}^{r} }}{{\partial \eta_{11}^{r} }} = \left[ {\frac{{\partial \varphi_{2} }}{{\partial \eta_{11}^{r} }}} \right]\left( {1 + \eta_{22} } \right) + \bar{\mu }\eta_{21} \eta_{12} , \hfill \\ \frac{{\partial n_{{ 2 {\text{ortho}}_{ 2} }}^{r} }}{{\partial \eta_{12}^{r} }} = \left[ {\frac{{\partial \varphi_{2} }}{{\partial \eta_{12}^{r} }}} \right]\left( {1 + \eta_{22} } \right) + \bar{\mu }\left( {1 + \eta_{11} } \right)\eta_{21} + 2\bar{\mu }\left( {1 + \eta_{22} } \right)\eta_{12} , \hfill \\ \end{gathered} $$
(64)
$$ \begin{gathered} \frac{{\partial n_{{ 2 {\text{ortho}}_{ 1} }}^{r} }}{{\partial \eta_{21}^{r} }} = \left[ {\frac{{\partial \varphi_{2} }}{{\partial \eta_{21}^{r} }} + 2\bar{\mu }\eta_{21} } \right]\eta_{21} + \left( {\varphi_{2} + 2\bar{\mu }I_{12} } \right) + \bar{\mu }\left( {1 + \eta_{11} } \right)^{2} , \hfill \\ \frac{{\partial n_{{ 2 {\text{ortho}}_{ 1} }}^{r} }}{{\partial \eta_{22}^{r} }} = \left[ {\frac{{\partial \varphi_{2} }}{{\partial \eta_{22}^{r} }} + 2\bar{\mu }\left( {1 + \eta_{22} } \right)} \right]\eta_{21} + \bar{\mu }\left( {1 + \eta_{11} } \right)\eta_{12} , \hfill \\ \frac{{\partial n_{{ 2 {\text{ortho}}_{ 2} }}^{r} }}{{\partial \eta_{21}^{r} }} = \left[ {\frac{{\partial \varphi_{2} }}{{\partial \eta_{21}^{r} }} + 2\bar{\mu }\eta_{21} } \right]\left( {1 + \eta_{22} } \right) + \bar{\mu }\eta_{12} \left( {1 + \eta_{11} } \right)\;{\text{and}} \hfill \\ \frac{{\partial n_{{ 2 {\text{ortho}}_{ 2} }}^{r} }}{{\partial \eta_{22}^{r} }} = \left[ {\frac{{\partial \varphi_{2} }}{{\partial \eta_{22}^{r} }} + 2\bar{\mu }\left( {1 + \eta_{22} } \right)} \right]\left( {1 + \eta_{22} } \right) + \left( {\varphi_{2} + 2\bar{\mu }I_{12} } \right) + \bar{\mu }\eta_{12}^{2} , \hfill \\ \end{gathered} $$
(65)

in which the derivatives of \( \varphi_{\alpha } \) are given by:

$$ \begin{gathered} \frac{{\partial \varphi_{1} }}{{\partial \eta_{11}^{r} }} = \left( {2a + 12hI_{11}^{2} + 6kI_{11} I_{12} + 2lI_{12} } \right)\left( {1 + \eta_{11} } \right), \hfill \\ \frac{{\partial \varphi_{1} }}{{\partial \eta_{12}^{r} }} = \left( {2a + 12hI_{11}^{2} + 6kI_{11} I_{12} + 2lI_{12} } \right)\eta_{12} , \hfill \\ \frac{{\partial \varphi_{1} }}{{\partial \eta_{21}^{r} }} = \left( {c + 3kI_{11}^{2} + 4lI_{11} I_{12} + 3mI_{12}^{2} } \right)\eta_{21} , \hfill \\ \frac{{\partial \varphi_{1} }}{{\partial \eta_{22}^{r} }} = \left( {c + 3kI_{11}^{2} + 4lI_{11} I_{12} + 3mI_{12}^{2} } \right)\left( {1 + \eta_{22} } \right), \hfill \\ \frac{{\partial \varphi_{2} }}{{\partial \eta_{11}^{r} }} = \left( {c + 3kI_{11}^{2} + 4lI_{11} I_{12} + 3mI_{12}^{2} } \right)\left( {1 + \eta_{11} } \right), \hfill \\ \frac{{\partial \varphi_{2} }}{{\partial \eta_{12}^{r} }} = \left( {c + 3kI_{11}^{2} + 4lI_{11} I_{12} + 3mI_{12}^{2} } \right)\eta_{12} , \hfill \\ \frac{{\partial \varphi_{2} }}{{\partial \eta_{21}^{r} }} = \left( {2b + 2lI_{11}^{2} + 6mI_{11} I_{12} + 12nI_{12}^{2} } \right)\eta_{21} \;{\text{and}} \hfill \\ \frac{{\partial \varphi_{2} }}{{\partial \eta_{21}^{r} }} = \left( {2b + 2lI_{11}^{2} + 6mI_{11} I_{12} + 12nI_{12}^{2} } \right)\left( {1 + \eta_{22} } \right). \hfill \\ \end{gathered} $$
(66)

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Gonçalves, F.R., Campello, E.M.B. Orthotropic material models for the nonlinear analysis of structural membranes. J Braz. Soc. Mech. Sci. Eng. 36, 887–899 (2014). https://doi.org/10.1007/s40430-013-0117-8

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