Description of document versions ==================================== .. csv-table:: "**Version** **Aster**", "**Author (s)** **Organization (s)**", "**Description of changes**" "5", "P.Massin, A.Laulusa EDF -R&D/ MMN ", "Initial text" "7.4", "X.Desroches", "Update: minor changes" Unprogrammed extension to anisotropic materials It is considered that the shell consists of an orthotropic material, orthotropic axes :math:`{\tilde{\tilde{x}}}_{k}` associated with the base :math:`{k}_{k}`. The law of behavior in these axes is written as: :math:`\underset{(6\mathrm{\times }1)}{\tilde{\tilde{\varepsilon }}}\mathrm{=}\underset{(6\mathrm{\times }6)}{{\tilde{\tilde{S}}}_{k}}\underset{(6\mathrm{\times }1)}{\tilde{\tilde{\sigma }}}` where :math:`\tilde{S}` is the flexibility matrix of the component :math:`k`. Let :math:`\tilde{\varepsilon }` and :math:`\tilde{\sigma }` be the deformation and stress tensors in the :math:`{\tilde{x}}_{k}` axes, we have: :math:`\begin{array}{c}\tilde{\sigma }\mathrm{=}{}^{t}\text{}Q\tilde{\tilde{\sigma }}Q\\ \tilde{\varepsilon }\mathrm{=}{}^{t}\text{}Q\tilde{\tilde{\varepsilon }}Q\end{array}` where :math:`Q\mathrm{=}{\left[{T}_{1},{T}_{2},{T}_{3}\right]}_{\mathrm{/}{k}_{k}}` (:math:`{Q}_{\text{ij}}={T}_{i}\text{.}{k}_{j}`) is the direction cosine matrix of :math:`{T}_{k}` in base :math:`{k}_{k}`. In vector form, we have: :math:`\begin{array}{c}\tilde{\sigma }\mathrm{=}\tilde{T}\tilde{\tilde{\sigma }}\\ \tilde{\varepsilon }\mathrm{=}\tilde{T}\tilde{\tilde{\varepsilon }}\end{array}` where the components of :math:`\tilde{T}` are defined according to those of :math:`Q`. Conversely, we have: :math:`\begin{array}{c}\tilde{\tilde{\sigma }}\mathrm{=}{\tilde{T}}^{\text{-1}}\tilde{\sigma }\\ \tilde{\tilde{\varepsilon }}\mathrm{=}{\tilde{T}}^{\text{-1}}\tilde{\varepsilon }\end{array}` so, we get: :math:`\tilde{\varepsilon }\mathrm{=}\tilde{T}{\tilde{\tilde{S}}}_{k}{\tilde{T}}^{\mathrm{-}1}\tilde{\sigma }` that we write: :math:`\tilde{\varepsilon }\mathrm{=}{\tilde{S}}_{k}\tilde{\sigma }` To be consistent with the plane stress hypothesis :math:`{\tilde{\sigma }}_{\text{33}}\mathrm{=}0`, we write: :math:`\underset{(5\mathrm{\times }1)}{{\tilde{\varepsilon }}_{r}}\mathrm{=}\underset{(5\mathrm{\times }5)}{{\tilde{S}}_{\text{kr}}}\underset{(5\mathrm{\times }1)}{{\tilde{\sigma }}_{r}}` with the symbol r as a reduced one, which gives: :math:`{\tilde{\sigma }}_{r}\mathrm{=}{\tilde{C}}_{k}{\tilde{\varepsilon }}_{r},{\tilde{C}}_{k}\mathrm{=}{\tilde{S}}_{\text{kr}}^{\mathrm{-}1}` that we rewrite by omitting the symbol :math:`r`, :math:`\tilde{\sigma }\mathrm{=}{\tilde{C}}_{k}\tilde{\varepsilon }` The elastic deformation energy :math:`{W}^{\text{el}}` is: :math:`{W}^{\text{el}}\mathrm{=}\frac{1}{2}{}^{t}\text{}{\tilde{q}}^{e}\underset{\mathrm{-}1}{\overset{1}{\mathrm{\int }}}\underset{\text{Ar}}{\overset{}{\mathrm{\int }}}{}^{t}\text{}\tilde{B}{\tilde{C}}_{k}\tilde{B}\text{det}Jd{\xi }_{1}d{\xi }_{2}d{\xi }_{3}{\tilde{q}}^{e}` If the shell consists of :math:`\mathit{Nc}` layers, with each layer considered to be a :math:`k` component, then: :math:`{W}^{\text{el}}\mathrm{=}\frac{1}{2}{}^{t}\text{}{\tilde{q}}^{e}\mathrm{\sum }_{k\mathrm{=}1}^{\text{Nc}}\underset{2{e}_{k}^{\mathrm{-}}\mathrm{/}h}{\overset{2{e}_{k}^{+}\mathrm{/}h}{\mathrm{\int }}}\underset{\text{Ar}}{\overset{}{\mathrm{\int }}}{}^{t}\text{}\tilde{B}\tilde{{C}_{k}}\stackrel{}{˜}B\text{det}Jd{\xi }_{1}d{\xi }_{2}d{\xi }_{3}{\stackrel{}{˜}q}^{e}` where :math:`{e}_{k}^{\mathrm{-}}` and :math:`{e}_{k}^{+}` are the abscissa of the lower and upper terminals of the :math:`k` thick layer :math:`{e}_{k}\mathrm{=}{e}_{k}^{+}\mathrm{-}{e}_{k}^{\mathrm{-}}`, with :math:`{e}_{1}^{\mathrm{-}}\mathrm{=}\mathrm{-}h\mathrm{/}2` and :math:`{e}_{\text{Nc}}^{+}\mathrm{=}h\mathrm{/}2`. By posing: :math:`{\xi }_{3}\mathrm{=}\frac{{e}_{k}}{h}{\overline{\xi }}_{3}+\frac{{e}_{k}^{+}+{e}_{k}^{\mathrm{-}}}{h},{\overline{\xi }}_{3}\mathrm{\in }\left[\mathrm{-}\mathrm{1,1}\right]` we have: :math:`{W}^{\text{el}}\mathrm{=}\frac{1}{2}{}^{t}\text{}{\tilde{q}}^{e}\mathrm{\sum }_{k\mathrm{=}1}^{\text{Nc}}\frac{{e}_{k}}{h}\underset{\mathrm{-}1}{\overset{1}{\mathrm{\int }}}\underset{\text{Ar}}{\overset{}{\mathrm{\int }}}{}^{t}\text{}\tilde{B}{\tilde{C}}_{k}\tilde{B}\text{det}J({\xi }_{1},{\xi }_{2},{\overline{\xi }}_{3})d{\xi }_{1}d{\xi }_{2}d{\overline{\xi }}_{3}{\tilde{q}}^{e}` Likewise, for the work due to thermal expansion :math:`{W}^{\text{th}}`, we have: :math:`{\tilde{\tilde{\varepsilon }}}_{\text{th}}^{k}\mathrm{=}({\alpha }_{1}^{k}T,{\alpha }_{2}^{k}T,{\alpha }_{3}^{k}T\mathrm{,0}\mathrm{,0}\mathrm{,0})` where :math:`{\alpha }_{i}^{k}` are the thermal expansion coefficients of layer :math:`k` in the orthotropic axes (:math:`{\tilde{\tilde{\xi }}}_{k}`). With the relationship: :math:`{\tilde{\varepsilon }}_{\text{th}}^{k}\mathrm{=}\tilde{T}{\tilde{\tilde{\varepsilon }}}_{\text{th}}^{k}` we get: :math:`{W}^{\text{th}}\text{=-}{}^{t}\text{}{\tilde{q}}^{e}\underset{\mathrm{-}1}{\overset{1}{\mathrm{\int }}}\underset{\text{Ar}}{\overset{}{\mathrm{\int }}}{}^{t}\text{}\tilde{B}(\mathrm{-}{\tilde{C}}_{k}{\tilde{\varepsilon }}_{\text{th}}^{k})\text{det}Jd{\xi }_{1}d{\xi }_{2}d{\xi }_{3}` Either: :math:`{W}^{\text{th}}\mathrm{=}{}^{t}\text{}{\tilde{q}}^{e}\mathrm{\sum }_{h\mathrm{=}1}^{\text{Nc}}\frac{{e}_{k}}{h}\underset{\mathrm{-}1}{\overset{1}{\mathrm{\int }}}\underset{\text{Ar}}{\overset{}{\mathrm{\int }}}{}^{t}\text{}\tilde{B}{\tilde{C}}_{k}{\tilde{\varepsilon }}_{\text{th}}^{k}\text{det}Jd{\xi }_{1}d{\xi }_{2}d{\overline{\xi }}_{3}` **Shape functions for the Q9H element** These functions are given on page 174 of [:ref:`bib8 `]. **A2.1** **Shape functions for translations** The 8 incomplete Lagrange shape functions of the quadrangle element Q9H [:ref:`Figure A2.2-a
`] for the interpolation of the :math:`{u}_{k}` displacements are: * :math:`{N}_{i}^{(1)}({\xi }_{1},{\xi }_{2})\mathrm{=}\frac{1}{4}(\mathrm{-}1+{\xi }_{\mathrm{1i}}{\xi }_{1}+{\xi }_{\mathrm{2i}}{x}_{2})(1+{\xi }_{\mathrm{1i}}{\xi }_{1})(1+{x}_{\mathrm{2i}}{\xi }_{2})i\mathrm{=}\mathrm{1,2}\mathrm{,3}\mathrm{,4}` * :math:`{N}_{i}^{(1)}({\xi }_{1},{\xi }_{2})\mathrm{=}\frac{1}{2}(1\mathrm{-}{x}_{1}^{2})(1+{\xi }_{\mathrm{2i}}{\xi }_{2})i\mathrm{=}\mathrm{5,7}` * :math:`{N}_{i}^{(1)}({\xi }_{1},{\xi }_{2})\mathrm{=}\frac{1}{2}(1\mathrm{-}{x}_{2}^{2})(1+{\xi }_{\mathrm{1i}}{\xi }_{1})i\mathrm{=}\mathrm{6,8}` with: :math:`\begin{array}{c}{\xi }_{\mathrm{1i}}\text{=-}\mathrm{1i}\mathrm{=}\mathrm{1,8}\mathrm{,4};\\ {\xi }_{\mathrm{1i}}\mathrm{=}\mathrm{0i}\mathrm{=}\mathrm{5,7};\\ {\xi }_{\mathrm{1i}}\text{=+}\mathrm{1i}\mathrm{=}\mathrm{2,6}\mathrm{,3}\text{.}\end{array}` and :math:`\begin{array}{c}{\xi }_{\mathrm{2i}}\text{=-}\mathrm{1i}\mathrm{=}\mathrm{1,5}\mathrm{,2};\\ {\xi }_{\mathrm{2i}}\mathrm{=}\mathrm{0i}\mathrm{=}\mathrm{6,8};\\ {\xi }_{\mathrm{2i}}\text{=+}\mathrm{1i}\mathrm{=}\mathrm{3,7}\mathrm{,4}\text{.}\end{array}`. **A2.2** **Shape functions for rotations** The 9 Lagrange shape functions of the quadrangle element Q9H [:ref:`Figure A2.2-a
`] for the interpolation of the :math:`{\tilde{\theta }}_{\alpha }` rotations are: :math:`{N}_{i}^{(2)}({x}_{1},{\xi }_{2})\mathrm{=}{N}_{i}({\xi }_{1}){N}_{i}({\xi }_{2})` where :math:`{N}_{i}({\xi }_{P})\mathrm{=}\underset{\mathit{r¹i}}{P}\frac{{\xi }_{\text{Pr}}\mathrm{-}{\xi }_{P}}{{\xi }_{\text{Pr}}\mathrm{-}{\xi }_{\text{Pi}}}` for :math:`p\mathrm{=}\mathrm{1,2}` and where :math:`r` describes the set of two nodes aligned with node :math:`i` in the :math:`{\xi }_{P}` direction. We have: :math:`\begin{array}{c}{x}_{\mathrm{1i}}\text{=-}\mathrm{1i}\mathrm{=}\mathrm{1,8}\mathrm{,4};\\ {x}_{\mathrm{1i}}\mathrm{=}\mathrm{0i}\mathrm{=}\mathrm{5,7};\\ {x}_{\mathrm{1i}}\text{=+}\mathrm{1i}\mathrm{=}\mathrm{2,6}\mathrm{,3}\text{.}\end{array}` and :math:`\begin{array}{c}{x}_{\mathrm{2i}}\text{=-}\mathrm{1i}\mathrm{=}\mathrm{1,5}\mathrm{,2};\\ {x}_{\mathrm{2i}}\mathrm{=}\mathrm{0i}\mathrm{=}\mathrm{6,8};\\ {x}_{\mathrm{2i}}\text{=+}\mathrm{1i}\mathrm{=}\mathrm{3,7}\mathrm{,4}\text{.}\end{array}`. .. image:: images/100017CA000069D50000374BFEADD09940653A23.svg :width: 419 :height: 220 .. _RefImage_100017CA000069D50000374BFEADD09940653A23.svg: **Figure A2.2-a: Degrees of freedom for translations and rotations of the quadrangle element Q9H** **Shape functions for the T7H element** **A3.1** **Shape functions for translations** The 6 shape functions of the triangular element T7H [:ref:`Figure A3.2-a
`] for interpolating :math:`{u}_{k}` displacements are given on page 175 of [:ref:`bib8 `]: * :math:`{N}_{1}^{(1)}\left({x}_{1},{x}_{2}\right)=\lambda (2\lambda -1)` * :math:`{N}_{2}^{(1)}\left({x}_{1},{x}_{2}\right)={x}_{1}({\mathrm{2x}}_{1}-1)` * :math:`{N}_{3}^{(1)}\left({x}_{1},{x}_{2}\right)={x}_{2}({\mathrm{2x}}_{2}-1)` * :math:`{N}_{4}^{(1)}\left({x}_{1},{x}_{2}\right)={\mathrm{4x}}_{1}\lambda` * :math:`{N}_{5}^{(1)}\left({x}_{1},{x}_{2}\right)={\mathrm{4x}}_{1}{x}_{2}` * :math:`{N}_{6}^{(1)}\left({x}_{1},{x}_{2}\right)=4\lambda {x}_{2}` where: :math:`\lambda \mathrm{=}1\mathrm{-}{x}_{1}\mathrm{-}{x}_{2}` **A3.2** **Shape functions for rotations** The 7 shape functions of the triangular element T7H [:ref:`Figure A3.2-a
`] for the interpolation of the :math:`{\tilde{q}}_{\alpha }` rotations are: * :math:`{N}_{1}^{(2)}\left({x}_{1},{x}_{2}\right)=\left(1-{x}_{1}-{x}_{2}\right)\left[2\left(1-{x}_{1}-{x}_{2}\right)-1\right]+\frac{1}{9}{N}_{7}^{(2)}` * :math:`{N}_{2}^{(2)}\left({x}_{1},{x}_{2}\right)={x}_{1}\left({\mathrm{2x}}_{1}-1\right)+\frac{1}{9}{N}_{7}^{(2)}` * :math:`{N}_{3}^{(2)}\left({x}_{1},{x}_{2}\right)={x}_{2}\left({\mathrm{2x}}_{2}-1\right)+\frac{1}{9}{N}_{7}^{(2)}` * :math:`{N}_{4}^{(2)}\left({x}_{1},{x}_{2}\right)={\mathrm{4x}}_{1}\left(1-{x}_{1}-{x}_{2}\right)-\frac{4}{9}{N}_{7}^{(2)}` * :math:`{N}_{5}^{(2)}\left({x}_{1},{x}_{2}\right)={\mathrm{4x}}_{1}{x}_{2}-\frac{4}{9}{N}_{7}^{(2)}` * :math:`{N}_{6}^{(2)}\left({x}_{1},{x}_{2}\right)={\mathrm{4x}}_{2}\left(1-{x}_{1}-{x}_{2}\right)-\frac{4}{9}{N}_{7}^{(2)}` with: * :math:`{N}_{7}^{(2)}({x}_{1},{x}_{2})\mathrm{=}\text{27}{x}_{1}{x}_{2}(1\mathrm{-}{x}_{1}\mathrm{-}{x}_{2})` **Figure A3.2-a: Degrees of freedom for translations and rotations of the triangle element T7H** .. image:: images/10000000000002800000019060B96DC8FFE06752.gif :width: 6.5307in :height: 4.0819in .. _RefImage_10000000000002800000019060B96DC8FFE06752.gif: 7 6 3 5 2 4 1 6 5 4 3 2 1 6 4 3 1 5 2