2. Benchmark solution#

2.1. 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:

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

With:

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

and

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

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

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

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

2.2. 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.

2.3. Uncertainties about the solution#

No uncertainty about the reference solution because it is analytical.

2.4. Bibliographical references#

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

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

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