Benchmark solution
=====================


Calculation method
------------------

To determine the analytical solution, the equation for the total deformation of a plate is based on the equation for the total deformation of a plate taking into account thermal expansion:

:math:`\epsilon \left(u\right)=\left(\begin{array}{c}{E}_{\text{11}}\\ {E}_{\text{22}}\\ {E}_{\text{12}}\end{array}\right)+{x}_{3}\left(\begin{array}{c}{K}_{\text{11}}\\ {K}_{\text{22}}\\ {K}_{\text{12}}\end{array}\right)+{\epsilon }^{\text{th}}\text{et}{\epsilon }^{\text{th}}=\left(\begin{array}{c}{d}_{\text{11}}\\ {d}_{\text{22}}\\ {d}_{\text{12}}\end{array}\right)\text{.}\left(T\left({x}_{3}\right)-{T}^{\text{réf}}\right)`

With:

:math:`{E}_{ij}=\frac{\partial {U}_{i}}{\partial {X}_{j}};{K}_{ij}=\frac{\partial {\beta }_{i}}{\partial {X}_{j}};i,j=\mathrm{1,2}` :math:`{U}_{i}` being the movement in the direction i and :math:`{X}_{i}` the next coordinate i. :math:`{E}_{ij}` is pure membrane deformation while :math:`{K}_{ij}` represents curvature. Boundary conditions are such as :math:`{E}_{ij}={K}_{ij}=0`.

and

:math:`{d}^{\left(m\right)}=\left(\begin{array}{c}{d}_{\text{11}}\\ {d}_{\text{22}}\\ {d}_{\text{12}}\end{array}\right)={P}^{{m}^{-1}}{\left(\begin{array}{c}{\alpha }_{\text{LL}}\\ {\alpha }_{\text{TT}}\\ 0\end{array}\right)}_{\left(L,T\right)}=\left(\begin{array}{cc}{C}^{2}& {S}^{2}\\ {S}^{2}& {C}^{2}\\ 2\text{CS}& -2\text{CS}\end{array}\right){\left(\begin{array}{c}{\alpha }_{LL}\\ {\alpha }_{\mathit{TT}}\end{array}\right)}_{\left(L,T\right)}`

:math:`{P}^{(m)}=\left[\begin{array}{ccc}{C}^{2}& {S}^{2}& 2\text{CS}\\ {S}^{2}& {C}^{2}& -2\text{CS}\\ -\text{CS}& \text{CS}& {C}^{2}-{S}^{2}\end{array}\right]` with :math:`\begin{array}{c}C=1\\ S=0\end{array}`

In the end, the expression of the deformation is determined analytically:

:math:`\epsilon \left(u\right)={\epsilon }^{\text{th}}=\left(\begin{array}{c}{\alpha }_{\text{LL}}\\ {\alpha }_{\text{TT}}\\ {0}_{\text{}}\end{array}\right)\text{.}1`


Reference quantities and results
-----------------------------------

We test the quantity EPSI_ELGA which represents the value of the deformation at a gauss point inside the plate. It tested on the M1 mesh, point 1 and at sub-point 1.


Uncertainties about the solution
----------------------------

No uncertainty about the reference solution because it is analytical.


Bibliographical references
---------------------------


1. Theoretical documentation R4.01.01, *Pre and post-treatment for thin shells* *made of composite materials*.


2. G. DHATT, G. TOUZOT, "Modeling of finite element structures", volume 2: beams and plates page 238-240, Hermès Paris, 1990.

*V*