Hooke matrix and flexibility matrix ======================================== Notations --------- Instead of using the indices 1, 2 and 3 to locate the axes of the Cartesian coordinate system, we will use the corresponding indices :math:`L`, :math:`T` and :math:`N`: 1. * :math:`L` for longitudinal * :math:`T` for transversal * :math:`N` for normal .. _OLE_LINK1: .. image:: images/Object_8.svg :width: 187 :height: 117 .. _RefImage_Object_8.svg: The coefficients that are involved are as follows: +-------------------------------+--------------------------+-----------------------------------------------------------------------------+ |**Keyword** |**Grading** |**meaning** | +-------------------------------+--------------------------+-----------------------------------------------------------------------------+ |E_L |:math:`{E}_{L}` |Longitudinal Young's modulus | +-------------------------------+--------------------------+-----------------------------------------------------------------------------+ |E_T |:math:`{E}_{T}` |Transversal Young's modulus | +-------------------------------+--------------------------+-----------------------------------------------------------------------------+ |E_N |:math:`{E}_{N}` |Normal Young's modulus | +-------------------------------+--------------------------+-----------------------------------------------------------------------------+ |G_LT |:math:`{G}_{\text{LT}}` |Shear modulus in plane :math:`(L,T)` | +-------------------------------+--------------------------+-----------------------------------------------------------------------------+ |G_TN |:math:`{G}_{\text{TN}}` |Shear modulus in plane :math:`(T,N)` | +-------------------------------+--------------------------+-----------------------------------------------------------------------------+ |G_LN |:math:`{G}_{\text{LN}}` |Shear modulus in plane :math:`(L,N)` | +-------------------------------+--------------------------+-----------------------------------------------------------------------------+ |NU_LT |:math:`{\nu }_{\text{LT}}`|Poisson's ratio in plane :math:`(L,T)` | +-------------------------------+--------------------------+-----------------------------------------------------------------------------+ |NU_TN |:math:`{\nu }_{\text{TN}}`|Poisson's ratio in plane :math:`(T,N)` | +-------------------------------+--------------------------+-----------------------------------------------------------------------------+ |NU_LN |:math:`{\nu }_{\text{LN}}`|Poisson's ratio in plane :math:`(L,N)` | +-------------------------------+--------------------------+-----------------------------------------------------------------------------+ | .. code:: |:math:`{\alpha }_{L}` | .. code:: | + + + + | ALPHA_L | | Coefficientde dilatation thermique moyen longitudinal | +-------------------------------+--------------------------+-----------------------------------------------------------------------------+ |ALPHA_T |:math:`{\alpha }_{T}` | .. code:: | + + + + | | | Coefficientde dilatation thermique moyen transversal | +-------------------------------+--------------------------+-----------------------------------------------------------------------------+ |ALPHA_N |:math:`{\alpha }_{N}` | .. code:: | + + + + | | | Coefficientde dilatation thermique moyen normal | +-------------------------------+--------------------------+-----------------------------------------------------------------------------+ **Very important note:** :math:`{\nu }_{\text{LT}}` *is different than* :math:`{\nu }_{\mathit{TL}}` *:* *If traction is applied along the axis* :math:`L` *:* :math:`{\epsilon }_{\text{LL}}=\frac{{\sigma }_{\text{LL}}}{{E}_{L}}` *(Hooke's law in one direction) .* This traction is accompanied, proportionally, by a contraction :math:`-{\nu }_{\text{LT}}\text{.}\frac{{\sigma }_{\text{LL}}}{{E}_{L}}` along the axis :math:`T`, and by a contraction :math:`-{\nu }_{\text{LN}}\text{.}\frac{{\sigma }_{\text{LL}}}{{E}_{L}}` along the axis :math:`N`. The first index indicates the axis where the effect of loading occurs and the second index indicates the direction of loading. Then we exert a pull along the :math:`T` axis, then a pull along :math:`N`, we obtain: :math:`\begin{array}{c}{\epsilon }_{\text{LL}}=\frac{{\sigma }_{\text{LL}}}{{E}_{L}}-{\nu }_{\text{TL}}\frac{{\sigma }_{\text{TT}}}{{E}_{T}}-{\nu }_{\text{NL}}\frac{{\sigma }_{\text{NN}}}{{E}_{N}}\\ {\epsilon }_{\text{TT}}=-{\nu }_{\text{LT}}\frac{{\sigma }_{\text{LL}}}{{E}_{L}}+\frac{{\sigma }_{\text{TT}}}{{E}_{T}}-{\nu }_{\text{NT}}\frac{{\sigma }_{\text{NN}}}{{E}_{N}}\\ {\epsilon }_{\text{NN}}=-{\nu }_{\text{LN}}\frac{{\sigma }_{\text{LL}}}{{E}_{L}}-{\nu }_{\text{TN}}\frac{{\sigma }_{\text{TT}}}{{E}_{T}}+\frac{{\sigma }_{\text{NN}}}{{E}_{N}}\end{array}\}` [:ref:`éq3.1-1 <éq3.1-1>`] The flexibility matrix :math:`{\left[\mathrm{H}\right]}^{-1}` being symmetric; from this we deduce: :math:`\frac{{\nu }_{\text{LT}}}{{E}_{L}}=\frac{{\nu }_{\text{TL}}}{{E}_{T}}`; :math:`\frac{{\nu }_{\text{LN}}}{{E}_{L}}=\frac{{\nu }_{\text{NL}}}{{E}_{N}}`; :math:`\frac{{\nu }_{\text{TN}}}{{E}_{T}}=\frac{{\nu }_{\text{NT}}}{{E}_{N}}` 3D case ------ Orthotropy ~~~~~~~~~~~ Flexibility matrix ^^^^^^^^^^^^^^^^^^^^^^^ :math:`\left[\begin{array}{c}{\epsilon }_{\text{LL}}\\ {\epsilon }_{\text{TT}}\\ {\epsilon }_{\text{NN}}\\ 2{\epsilon }_{\text{LT}}\\ 2{\epsilon }_{\text{LN}}\\ 2{\epsilon }_{\text{TN}}\end{array}\right]=\left[\begin{array}{cccccc}\frac{1}{{E}_{L}}& \frac{-{\nu }_{\text{TL}}}{{E}_{T}}& \frac{-{\nu }_{\text{NL}}}{{E}_{N}}& 0& 0& 0\\ \frac{-{\nu }_{\text{LT}}}{{E}_{L}}& \frac{1}{{E}_{T}}& \frac{-{\nu }_{\text{NT}}}{{E}_{N}}& 0& 0& 0\\ \frac{-{\nu }_{\text{LN}}}{{E}_{L}}& \frac{-{\nu }_{\text{TN}}}{{E}_{T}}& \frac{1}{{E}_{N}}& 0& 0& 0\\ & & & \frac{1}{{G}_{\text{LT}}}& 0& 0\\ & \text{SYM}& & & \frac{1}{{G}_{\text{LN}}}& 0\\ & & & & & \frac{1}{{G}_{\text{TN}}}\phantom{\rule{2em}{0ex}}\end{array}\right]\left[\begin{array}{c}{\sigma }_{\text{LL}}\\ {\sigma }_{\text{TT}}\\ {\sigma }_{\text{NN}}\\ {\sigma }_{\text{LT}}\\ {\sigma }_{\text{LN}}\\ {\sigma }_{\text{TN}}\end{array}\right]` :math:`{\left[\mathrm{H}\right]}^{-1}` — Orthotropy Hooke matrix ^^^^^^^^^^^^^^^^^ :math:`\left[\begin{array}{c}{\sigma }_{\text{LL}}\\ {\sigma }_{\text{LL}}\\ {\sigma }_{\text{NN}}\\ {\sigma }_{\text{LT}}\\ {\sigma }_{\text{LN}}\\ {\sigma }_{\text{TN}}\end{array}\right]=\left[\begin{array}{cccccc}\frac{\left(1-{\nu }_{\text{TN}}{\nu }_{\text{NT}}\right)}{\mathrm{\Delta }\cdot {E}_{T}{E}_{N}}& \frac{\left({\nu }_{\text{TL}}+{\nu }_{\text{NL}}{\nu }_{\text{TN}}\right)}{\mathrm{\Delta }\cdot {E}_{T}\text{.}{E}_{N}}& \frac{\left({\nu }_{\text{NL}}+{\nu }_{\text{TL}}{\nu }_{\text{NT}}\right)}{\mathrm{\Delta }\cdot {E}_{T}\text{.}{E}_{N}}& 0& 0& 0\\ \frac{\left({\nu }_{\text{LT}}+{\nu }_{\text{LN}}{\nu }_{\text{NT}}\right)}{\mathrm{\Delta }\cdot {E}_{L}{E}_{N}}& \frac{\left(1-{\nu }_{\text{NL}}{\nu }_{\text{LN}}\right)}{\mathrm{\Delta }\cdot {E}_{L}\text{.}{E}_{N}}& \frac{\left({\nu }_{\text{NT}}+{\nu }_{\text{NL}}\text{.}{\nu }_{\text{LT}}\right)}{\mathrm{\Delta }\cdot {E}_{L}\text{.}{E}_{N}}& 0& 0& 0\\ \frac{\left({\nu }_{\text{LN}}+{\nu }_{\text{LT}}\text{.}{\nu }_{\text{TN}}\right)}{\mathrm{\Delta }\cdot {E}_{L}\text{.}{E}_{T}}& \frac{\left({\nu }_{\text{TN}}+{\nu }_{\text{TL}}\text{.}{\nu }_{\text{LN}}\right)}{\mathrm{\Delta }\cdot {E}_{L}\text{.}{E}_{T}}& \frac{\left(1-{\nu }_{\text{LT}}\text{.}{\nu }_{\text{TL}}\right)}{\mathrm{\Delta }\cdot {E}_{L}\text{.}{E}_{T}}& 0& 0& 0\\ & & & \begin{array}{c}{G}_{\text{LT}}\end{array}& 0& 0\\ & \text{SYM}& & & {G}_{\text{LN}}& 0\\ & & & & & {G}_{\text{TN}}\end{array}\right]\left[\begin{array}{c}{\epsilon }_{\text{LL}}\\ {\epsilon }_{\text{TT}}\\ {\epsilon }_{\text{NN}}\\ 2{\epsilon }_{\text{LT}}\\ 2{\epsilon }_{\text{LN}}\\ 2{\epsilon }_{\text{TN}}\end{array}\right]` :math:`\left[\mathrm{H}\right]` — Orthotropy with: :math:`\frac{{\nu }_{\text{TL}}}{{E}_{T}}=\frac{{\nu }_{\text{LT}}}{{E}_{L}}\phantom{\rule{2em}{0ex}};\phantom{\rule{2em}{0ex}}\frac{{\nu }_{\text{NL}}}{{E}_{N}}=\frac{{\nu }_{\text{LN}}}{{E}_{L}}\phantom{\rule{2em}{0ex}};\phantom{\rule{2em}{0ex}}\frac{{\nu }_{\text{NT}}}{{E}_{N}}=\frac{{\nu }_{\text{TN}}}{{E}_{T}}` with: :math:`\frac{1}{\mathrm{\Delta }}=\frac{{E}_{L}{E}_{T}{E}_{N}}{1-{\nu }_{\text{TN}}{\nu }_{\text{NT}}-{\nu }_{\text{NL}}{\nu }_{\text{LN}}-{\nu }_{\text{LT}}{\nu }_{\text{TL}}-2{\nu }_{\text{TN}}{\nu }_{\text{NL}}{\nu }_{\text{LT}}}` Transverse isotropy ~~~~~~~~~~~~~~~~~~~~~~ Transverse isotropy is defined here in plane :math:`(L,T)`, and the direction of orthotropy is therefore :math:`N`. The reader's attention may be drawn to the fact that this convention differs from a usual convention which designates by "longitudinal direction" the direction of orthotropy of isotropic transverse materials. .. image:: images/Object_30.svg :width: 187 :height: 117 .. _RefImage_Object_30.svg: Note that the expansion coefficients verify: :math:`{\alpha }_{T}={\alpha }_{L}`. Flexibility matrix ^^^^^^^^^^^^^^^^^^^^^^^ Matrix :math:`{\left[\mathrm{H}\right]}^{-1}` can be deduced directly from matrix :math:`{\left[\mathrm{H}\right]}^{-1}` - Orthotropy using the properties of transverse isotropy. In plan :math:`(L,T)`, see [:ref:`éq2.2-1 <éq2.2-1>`]: :math:`\begin{array}{c}{E}_{L}={E}_{T}\\ {\nu }_{\text{TL}}={\nu }_{\text{LT}}\\ {G}_{\text{LT}}=\frac{{E}_{L}}{2\left(1+{\nu }_{\text{LT}}\right)}\end{array}` In plans :math:`(L,N)` and :math:`(T,N)`: :math:`\begin{array}{c}{\nu }_{\text{NT}}={\nu }_{\text{NL}}\\ {\nu }_{\text{LN}}={\nu }_{\text{TN}}\\ {G}_{\text{TN}}={G}_{\text{LN}}\\ \frac{{\nu }_{\text{NT}}}{{E}_{N}}=\frac{{\nu }_{\text{LN}}}{{E}_{L}}\end{array}` :math:`\left[\begin{array}{c}{\epsilon }_{\text{LL}}\\ {\epsilon }_{\text{TT}}\\ {\epsilon }_{\text{NN}}\\ 2{\epsilon }_{\text{LT}}\\ 2{\epsilon }_{\text{LN}}\\ 2{\epsilon }_{\text{TN}}\end{array}\right]=\left[\begin{array}{cccccc}\frac{1}{{E}_{L}}& \frac{-{\nu }_{\text{LT}}}{{E}_{L}}& \frac{-{\nu }_{\text{NL}}}{{E}_{N}}& 0& 0& 0\\ \frac{-{\nu }_{\text{TL}}}{{E}_{L}}& \frac{1}{{E}_{L}}& \frac{-{\nu }_{\text{NT}}}{{E}_{N}}& 0& 0& 0\\ \frac{-{\nu }_{\text{LN}}}{{E}_{L}}& \frac{-{\nu }_{\text{TN}}}{{E}_{L}}& \frac{1}{{E}_{N}}& 0& 0& 0\\ & & & \frac{2\left(1+{\nu }_{\text{LT}}\right)}{{E}_{L}}& 0& 0\\ & \text{SYM}& & & \frac{1}{{G}_{\text{LN}}}& 0\\ & & & & & \frac{1}{{G}_{\text{TN}}}\phantom{\rule{2em}{0ex}}\end{array}\right]\left[\begin{array}{c}{\sigma }_{\text{LL}}\\ {\sigma }_{\text{TT}}\\ {\sigma }_{\text{NN}}\\ {\sigma }_{\text{LT}}\\ {\sigma }_{\text{LN}}\\ {\sigma }_{\text{TN}}\end{array}\right]` :math:`{\left[\mathrm{H}\right]}^{-1}` - Transverse isotropy Hooke matrix ^^^^^^^^^^^^^^^^^ The :math:`\left[\mathrm{H}\right]` matrix has the same symmetries as :math:`{\left[\mathrm{H}\right]}^{-1}`. :math:`\left[\begin{array}{c}{\sigma }_{\text{LL}}\\ {\sigma }_{\text{TT}}\\ {\sigma }_{\text{NN}}\\ {\sigma }_{\text{LT}}\\ {\sigma }_{\text{LN}}\\ {\sigma }_{\text{TN}}\end{array}\right]=\left[\begin{array}{cccccc}\frac{1-{\nu }_{\text{NL}}\text{.}{\nu }_{\text{LN}}}{\mathrm{\Delta }\text{'}\cdot {E}_{L}\text{.}{E}_{N}}& \frac{{\nu }_{\text{LT}}+{\nu }_{\text{NL}}{\nu }_{\text{LN}}}{\mathrm{\Delta }\text{'}\cdot {E}_{L}\text{.}{E}_{N}}& \frac{{\nu }_{\text{NL}}+{\nu }_{\text{LT}}{\nu }_{\text{NL}}}{\mathrm{\Delta }\text{'}\cdot {E}_{L}\text{.}{E}_{N}}& 0& 0& 0\\ \frac{{\nu }_{\text{TL}}+{\nu }_{\text{NL}}{\nu }_{\text{LN}}}{\mathrm{\Delta }\text{'}\cdot {E}_{L}\text{.}{E}_{N}}& \frac{1-{\nu }_{\text{NL}}\text{.}{\nu }_{\text{LN}}}{\mathrm{\Delta }\text{'}\cdot {E}_{L}\text{.}{E}_{N}}& \frac{{\nu }_{\text{LN}}+{\nu }_{\text{LT}}{\nu }_{\text{LN}}}{\mathrm{\Delta }\text{'}\cdot {E}_{L}\text{.}{E}_{N}}& 0& 0& 0\\ \frac{{\nu }_{\text{LN}}+{\nu }_{\text{LT}}\text{.}{\nu }_{\text{LN}}}{\mathrm{\Delta }\text{'}\cdot {E}_{L}^{2}}& \frac{{\nu }_{\text{TN}}+{\nu }_{\text{LT}}\text{.}{\nu }_{\text{TN}}}{\mathrm{\Delta }\text{'}\cdot {E}_{L}^{2}}& \frac{1-{\nu }_{\text{LT}}^{2}}{\mathrm{\Delta }\text{'}\cdot {E}_{L}^{2}}& 0& 0& 0\\ & & & \frac{{E}_{L}}{2\left(1+{\nu }_{\text{LT}}\right)}& & \\ & & & & {G}_{\text{LN}}& \\ & & & & & {G}_{\text{LN}\phantom{\rule{2em}{0ex}}}\end{array}\right]\left[\begin{array}{c}{\epsilon }_{\text{LL}}\\ {\epsilon }_{\text{TT}}\\ {\epsilon }_{\text{NN}}\\ 2{\epsilon }_{\text{LT}}\\ 2{\epsilon }_{\text{LN}}\\ 2{\epsilon }_{\text{TN}}\end{array}\right]` :math:`\left[\mathrm{H}\right]` — Transverse isotropy with: :math:`\frac{1}{\mathrm{\Delta }\text{'}}=\frac{{E}_{L}^{2}\text{.}{E}_{N}}{1-2{\nu }_{\text{NL}}\cdot {\nu }_{\text{LN}}-{\nu }_{\text{LT}}^{2}-2{\nu }_{\text{NL}}\cdot {\nu }_{\text{LN}}\cdot {\nu }_{\text{LT}}}` Cubic symmetric elasticity ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Cubic-symmetric elasticity occurs when, in addition to the three Cartesian planes of symmetry, the six planes rotated at 45° are also symmetric. We then have 3 independent elastic coefficients. This corresponds to an elasticity matrix of the form: :math:`\left[\begin{array}{cccccc}\phantom{\rule{2em}{0ex}}{H}_{1111}& {H}_{1122}& {H}_{1122}& \phantom{\rule{2em}{0ex}}& \phantom{\rule{2em}{0ex}}& \phantom{\rule{2em}{0ex}}\\ {H}_{1122}& {H}_{1111}& {H}_{1122}& \phantom{\rule{2em}{0ex}}& \phantom{\rule{2em}{0ex}}& \phantom{\rule{2em}{0ex}}\\ {H}_{1122}& {H}_{1122}& {H}_{1111}& \phantom{\rule{2em}{0ex}}& \phantom{\rule{2em}{0ex}}& \phantom{\rule{2em}{0ex}}\\ \phantom{\rule{2em}{0ex}}& \phantom{\rule{2em}{0ex}}& \phantom{\rule{2em}{0ex}}& {H}_{1212}& \phantom{\rule{2em}{0ex}}& \phantom{\rule{2em}{0ex}}\\ \phantom{\rule{2em}{0ex}}& \phantom{\rule{2em}{0ex}}& \phantom{\rule{2em}{0ex}}& \phantom{\rule{2em}{0ex}}& {H}_{1212}& \phantom{\rule{2em}{0ex}}\\ \phantom{\rule{2em}{0ex}}& \phantom{\rule{2em}{0ex}}& \phantom{\rule{2em}{0ex}}& \phantom{\rule{2em}{0ex}}& \phantom{\rule{2em}{0ex}}& {H}_{1212}\end{array}\right]` Given the cubic symmetry, it remains to determine 3 coefficients: :math:`{E}_{L}={E}_{N}={E}_{T}=E,\phantom{\rule{6em}{0ex}}{G}_{\text{LT}}={G}_{\text{LN}}={G}_{\text{TN}}=G,\phantom{\rule{6em}{0ex}}{\nu }_{\text{LN}}={\nu }_{\text{LT}}={\nu }_{\text{LN}}=\nu` Note that the expansion coefficients verify: :math:`{\alpha }_{T}={\alpha }_{L}={\alpha }_{N}`. To reproduce cubic-symmetric elasticity with ELAS_ORTH, simply calculate the orthotropy coefficients such that the elasticity matrix obtained is of the form above: :math:`\begin{array}{c}{H}_{1111}=\frac{E(1-\nu )}{(1+\nu )(1-2\nu )}\\ {H}_{1122}=\frac{\nu E}{(1+\nu )(1-2\nu )}\\ {H}_{1212}={G}_{\text{LT}}={G}_{\text{LN}}={G}_{\text{TN}}\end{array}` so as long as :math:`(1+\nu )(1-2\nu )\ne 0` (i.e. :math:`\nu` different from :math:`0.5`). :math:`\frac{{H}_{1122}}{{H}_{1111}}=\frac{\nu }{1-\nu }` which provides :math:`\nu =\frac{1}{1+\frac{{H}_{1111}}{{H}_{1122}}}` then :math:`E={H}_{1111}\frac{(1+\nu )(1-2\nu )}{(1-\nu )}` Isotropy ~~~~~~~~ Hooke's law takes the following form with Lamé coefficients :math:`\lambda` and :math:`\mu =G`: :math:`{\sigma }_{\text{ij}}=\lambda {\epsilon }_{\text{kk}}{\delta }_{\text{ij}}+2\mu {\epsilon }_{\text{ij}}` Flexibility matrix based on :math:`E` and :math:`\nu` ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ :math:`\left[\begin{array}{c}{\epsilon }_{\text{LL}}\\ {\epsilon }_{\text{TT}}\\ {\epsilon }_{\text{NN}}\\ 2{\epsilon }_{\text{LT}}\\ 2{\epsilon }_{\text{LN}}\\ 2{\epsilon }_{\text{TN}}\end{array}\right]=\left[\begin{array}{cccccc}\frac{1}{E}& \frac{-\nu }{E}& \frac{-\nu }{E}& 0& 0& 0\\ & \frac{1}{E}& \frac{-\nu }{E}& 0& 0& 0\\ & & \frac{1}{E}& 0& 0& 0\\ & & & \frac{1}{G}=\frac{2(1+\nu )}{E}& 0& 0\\ & \text{SYM}& & & \frac{1}{G}=\frac{2\left(1+\nu \right)}{E}& 0\\ & & & & & \frac{1}{G}=\frac{2\left(1+\nu \right)}{E}\end{array}\right]\left[\begin{array}{c}{\sigma }_{\text{LL}}\\ {\sigma }_{\text{TT}}\\ {\sigma }_{\text{NN}}\\ {\sigma }_{\text{LT}}\\ {\sigma }_{\text{LN}}\\ {\sigma }_{\text{TN}}\end{array}\right]` :math:`{\left[\mathrm{H}\right]}^{-1}` — Complete isotropy Hooke matrix as a function of :math:`E` and :math:`\nu` ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ :math:`\left[\begin{array}{c}{\sigma }_{\text{LL}}\\ {\sigma }_{\text{TT}}\\ {\sigma }_{\text{NN}}\\ {\sigma }_{\text{LT}}\\ \\ {\sigma }_{\text{LN}}\\ \\ {\sigma }_{\text{TN}}\end{array}\right]=\frac{E}{\left(1+\nu \right)\left(1-2\nu \right)}\left[\begin{array}{cccccc}1-\nu & \nu & \nu & 0& 0& 0\\ & 1-\nu & \nu & 0& 0& 0\\ & & 1-\nu & 0& 0& 0\\ & \text{SYM}& & \frac{1-2\nu }{2}& 0& 0\\ & & & & \frac{1-2\nu }{2}& 0\\ & & & & & \frac{1-2\nu }{2}\end{array}\right]\left[\begin{array}{c}{\epsilon }_{\text{LL}}\\ {\epsilon }_{\text{TT}}\\ {\epsilon }_{\text{NN}}\\ 2{\epsilon }_{\text{LT}}\\ \\ 2{\epsilon }_{\text{LN}}\\ \\ 2{\epsilon }_{\text{TN}}\end{array}\right]` :math:`\left[\mathrm{H}\right]` — Complete isotropy Flexibility matrix according to Lamé coefficients :math:`\lambda` and :math:`\mu` ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ :math:`\left[\begin{array}{c}{\epsilon }_{\text{LL}}\\ {\epsilon }_{\text{TT}}\\ {\epsilon }_{\text{NN}}\\ 2{\epsilon }_{\text{LT}}\\ 2{\epsilon }_{\text{LN}}\\ 2{\epsilon }_{\text{TN}}\end{array}\right]=\left[\begin{array}{cccccc}\frac{\lambda +\mu }{\mu (3\lambda +2\mu )}& \frac{-\lambda }{2\mu (3\lambda +2\mu )}& \frac{-\lambda }{2\mu (3\lambda +2\mu )}& 0& 0& 0\\ & \frac{\lambda +\mu }{\mu (3\lambda +2\mu )}& \frac{-\lambda }{2\mu (3\lambda +2\mu )}& 0& 0& 0\\ & & \frac{\lambda +\mu }{\mu (3\lambda +2\mu )}& 0& 0& 0\\ & & & \frac{1}{\mu }& 0& 0\\ & \text{SYM}& & & \frac{1}{\mu }& 0\\ & & & & & \frac{1}{\mu }\phantom{\rule{2em}{0ex}}\end{array}\right]\left[\begin{array}{c}{\sigma }_{\text{LL}}\\ {\sigma }_{\text{TT}}\\ {\sigma }_{\text{NN}}\\ {\sigma }_{\text{LT}}\\ {\sigma }_{\text{LN}}\\ {\sigma }_{\text{TN}}\end{array}\right]` :math:`{\left[\mathrm{H}\right]}^{-1}` — Complete isotropy Hooke matrix as a function of Lamé coefficients :math:`\lambda` and :math:`\mu` ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ :math:`\left[\begin{array}{c}{\sigma }_{\text{LL}}\\ {\sigma }_{\text{TT}}\\ {\sigma }_{\text{NN}}\\ {\sigma }_{\text{LN}}\\ {\sigma }_{\text{LT}}\\ {\sigma }_{\text{TN}}\end{array}\right]=\left[\begin{array}{cccccc}\lambda +2\mu & \lambda & \lambda & 0& 0& 0\\ & \lambda +2\mu & \lambda & 0& 0& 0\\ & & \lambda +2\mu & 0& 0& 0\\ & \text{SYM}& & \mu & 0& 0\\ & & & & \mu & 0\\ & & & & & \mu \end{array}\right]\left[\begin{array}{c}{\epsilon }_{\text{LL}}\\ {\epsilon }_{\text{TT}}\\ {\epsilon }_{\text{NN}}\\ 2{\epsilon }_{\text{LN}}\\ 2{\epsilon }_{\text{LT}}\\ 2{\epsilon }_{\text{TN}}\end{array}\right]` :math:`\left[\mathrm{H}\right]` — Complete isotropy with Lamé coefficients 2D orthotropic case in plane and axisymmetric deformations --------------------------------------------------------- Flexibility matrix ~~~~~~~~~~~~~~~~~~~~~~ :math:`\left[\begin{array}{c}{\epsilon }_{\text{LL}}\\ {\epsilon }_{\text{TT}}\\ \\ 0\\ {\mathrm{2\epsilon }}_{\text{LT}}\end{array}\right]=\left[\begin{array}{cccc}\frac{1}{{E}_{L}}& -\frac{{\nu }_{\text{TL}}}{{E}_{T}}& -\frac{{\nu }_{\text{NL}}}{{E}_{N}}& 0\\ -\frac{{\nu }_{\text{LT}}}{{E}_{L}}& \frac{1}{{E}_{T}}& -\frac{{\nu }_{\text{NL}}}{{E}_{N}}& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \frac{1}{{G}_{\text{LT}}}\end{array}\right]\left[\begin{array}{c}{\sigma }_{\text{LL}}\\ {\sigma }_{\text{TT}}\\ \\ {\sigma }_{\text{NN}}\\ {\sigma }_{\text{LT}}\end{array}\right]` :math:`{\left[\mathrm{H}\right]}^{-1}` — Plane orthotropy in plane deformations and axisymmetry Hooke matrix ~~~~~~~~~~~~~~~~~ :math:`\left[\begin{array}{c}{\sigma }_{\text{LL}}\\ {\sigma }_{\text{TT}}\\ {\sigma }_{\text{NN}}\\ {\sigma }_{\text{LT}}\end{array}\right]=\frac{1}{\mathrm{\Delta }}\left[\begin{array}{cccc}\frac{\left(1-{\nu }_{\text{TN}}{\nu }_{\text{NT}}\right)}{{E}_{T}{E}_{N}}& \frac{\left({\nu }_{\text{TL}}+{\nu }_{\text{NL}}{\nu }_{\text{TN}}\right)}{{E}_{T}\text{.}{E}_{N}}& \frac{\left({\nu }_{\text{NL}}+{\nu }_{\text{TL}}{\nu }_{\text{NT}}\right)}{{E}_{T}\text{.}{E}_{N}}& 0\\ \frac{\left({\nu }_{\text{LT}}+{\nu }_{\text{LN}}{\nu }_{\text{NT}}\right)}{{E}_{L}{E}_{N}}& \frac{\left(1-{\nu }_{\text{NL}}{\nu }_{\text{LN}}\right)}{{E}_{L}\text{.}{E}_{N}}& \frac{\left({\nu }_{\text{NT}}+{\nu }_{\text{NL}}\text{.}{\nu }_{\text{LT}}\right)}{{E}_{L}\text{.}{E}_{N}}& 0\\ \frac{\left({\nu }_{\text{LN}}+{\nu }_{\text{LT}}\text{.}{\nu }_{\text{TN}}\right)}{{E}_{L}\text{.}{E}_{T}}& \frac{\left({\nu }_{\text{TN}}+{\nu }_{\text{TL}}\text{.}{\nu }_{\text{LN}}\right)}{{E}_{L}\text{.}{E}_{T}}& \frac{\left(1-{\nu }_{\text{LT}}\text{.}{\nu }_{\text{TL}}\right)}{{E}_{L}\text{.}{E}_{T}}& 0\\ 0& 0& 0& {G}_{\text{LT}}\cdot \mathrm{\Delta }\end{array}\right]\left[\begin{array}{c}{\epsilon }_{\text{LL}}\\ {\epsilon }_{\text{TT}}\\ 0\\ 2{\epsilon }_{\text{LT}}\end{array}\right]` :math:`\left[\mathrm{H}\right]` — Plane orthotropy in plane deformations and axisymmetry with: :math:`\frac{1}{\mathrm{\Delta }}=\frac{{E}_{L}{E}_{T}{E}_{N}}{1-{\nu }_{\text{TN}}{\nu }_{\text{NT}}-{\nu }_{\text{NL}}{\nu }_{\text{LN}}-{\nu }_{\text{LT}}{\nu }_{\text{TL}}-2{\nu }_{\text{TN}}{\nu }_{\text{NL}}{\nu }_{\text{LT}}}` 2D orthotropic case in plane constraints --------------------------------------- Flexibility matrix ~~~~~~~~~~~~~~~~~~~~~~ :math:`\left[\begin{array}{c}{\epsilon }_{\text{LL}}\\ {\epsilon }_{\text{TT}}\\ \\ {\epsilon }_{\text{NN}}\\ 2{\epsilon }_{\text{LT}}\end{array}\right]=\left[\begin{array}{cccc}\frac{1}{{E}_{L}}& -\frac{{\nu }_{\text{TL}}}{{E}_{T}}& 0& 0\\ -\frac{{\nu }_{\text{LT}}}{{E}_{L}}& \frac{1}{{E}_{T}}& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \frac{1}{{G}_{\text{LT}}}\end{array}\right]\left[\begin{array}{c}{\sigma }_{\text{LL}}\\ {\sigma }_{\text{TT}}\\ \\ {\sigma }_{\text{NN}}\\ {\sigma }_{\text{LT}}\end{array}\right]` :math:`{\left[\mathrm{H}\right]}^{-1}` — Plane orthotropy in plane stresses Hooke matrix ~~~~~~~~~~~~~~~~~ Using the system of equations [:ref:`éq3.1-1 <éq3.1-1>`], we get: :math:`\left[\begin{array}{c}{\sigma }_{\text{LL}}\\ {\sigma }_{\text{TT}}\\ 0\\ {\sigma }_{\text{LT}}\end{array}\right]=\frac{1}{1-{\nu }_{\text{LT}}\text{.}{\nu }_{\text{TL}}}\left[\begin{array}{cccc}{E}_{L}& {\nu }_{\text{TL}}{E}_{T}& 0& 0\\ {\nu }_{\text{LT}}{E}_{L}& {E}_{T}& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& {G}_{\text{LT}}\end{array}\right]\left[\begin{array}{c}{\epsilon }_{\text{LL}}\\ {\epsilon }_{\text{TT}}\\ {\epsilon }_{\text{NN}}\\ 2{\epsilon }_{\text{LT}}\end{array}\right]` :math:`\left[\mathrm{H}\right]` — Orthotropy in plane constraints