2. Benchmark solution#
2.1. Calculation method used for the reference solution#
It is an analytical solution.
In fact, although the law introduced is a law of THM, only the mechanical part is active here because the liquid pressure is imposed at zero everywhere on the domain.
Moreover, as far as the mechanical part is concerned, an elastic law was imposed by choosing KIT_THM ELAS as the mechanical law.
Thus, given the boundary conditions and the loading, we have:
\({\sigma }_{\text{xx}}\mathrm{=}\text{11}\text{MPa}\) all over
\({\sigma }_{\text{yy}}=\text{15}\text{.}4\text{MPa}\) all over
In addition, we are in plane deformations that is \({\varepsilon }_{\text{zz}}=0\)
Gold like \(\varepsilon \mathrm{=}\frac{1+\nu }{E}\sigma \mathrm{-}\frac{\nu }{E}\text{Tr}(\sigma )I\)
Let’s be \({\varepsilon }_{\text{zz}}=0=\frac{{\sigma }_{\text{zz}}}{E}-\frac{\nu }{E}{\sigma }_{\text{xx}}-\frac{\nu }{E}{\sigma }_{\text{yy}}\)
Where \({\sigma }_{\text{zz}}=\nu ({\sigma }_{\text{xx}}+{\sigma }_{\text{yy}})\)
So \({\sigma }_{\text{zz}}=7\text{.}\text{92}\text{MPa}\)
We therefore obtain the values of the deformations thanks to the elastic law
\(\begin{array}{}{\varepsilon }_{\text{xx}}=6\text{.}\text{9034482759}{\text{10}}^{-4}\\ {\varepsilon }_{\text{yy}}=1\text{.}\text{67655172414}{\text{10}}^{-3}\end{array}\)
The other values of the strain (and stress) tensor are zero.
The displacement of the structure is also calculated.
\(\begin{array}{}{\varepsilon }_{\text{xx}}=\frac{\partial {u}_{x}}{\partial x}\\ {\varepsilon }_{\text{yy}}=\frac{\partial {u}_{y}}{\partial y}\end{array}\)
\(\begin{array}{}{u}_{x}(x,y)={\varepsilon }_{\text{xx}}x+{u}_{x}(\mathrm{0,0})\\ {u}_{y}(x,y)={\varepsilon }_{\text{yy}}y+{u}_{y}(\mathrm{0,0})\\ \end{array}\)
Now for reasons of symmetry \({u}_{A}=u(-\mathrm{1,}-1)=0\), so
\(\begin{array}{}{u}_{x}(\mathrm{0,0})={\varepsilon }_{\text{xx}}\\ {u}_{y}(\mathrm{0,0})={\varepsilon }_{\text{yy}}\end{array}\)
We are interested in the displacement at point C, with coordinates \((\mathrm{1,1})\)
So we have:
\(\begin{array}{}{u}_{x}(C)=2{\varepsilon }_{\text{xx}}=1\text{.}\text{3806896551}{\text{10}}^{-3}\\ {u}_{y}(C)=2{\varepsilon }_{\text{yy}}=3\text{.}\text{35310344828}{\text{10}}^{-3}\end{array}\)
2.2. Benchmark results#
Displacements \({u}_{x}\) and \({u}_{y}\) at point \(C\) and deformations \(({\varepsilon }_{\text{xx}}\text{,}{\varepsilon }_{\text{yy}})\) at points \(A\), \(B\), \(C\) and \(D\)
2.3. Uncertainty about the solution#
Analytical solution
2.4. Bibliographical references#
CHAVANT: THHM models. Overview and algorithms, document R7.01.10
CHAVANT, B. CIREE: Drücker-Prager double criterion behavior law for concrete cracking and compression, document R7.01.03