Benchmark solution ===================== Calculation method ----------------- *2D case* Since Poisson's ratio :math:`\nu` is zero, the solution is written independently in the :math:`x` direction and the :math:`y` direction. Neglecting gravity, the equation is written (in total constraints): :math:`\text{Div}(\sigma )=0` Being in the elastic case, we have :math:`\sigma =Eϵ`, which is finally :math:`\text{Div}(ϵ)=0`. According to :math:`x`, :math:`\frac{\partial {ϵ}_{\mathit{xx}}}{\partial x}=0` from where: * movements above the interface are spelled :math:`{u}_{x}(x,y)=\frac{-0.01\ast x}{\mathit{LX}}\ast (\mathit{LY}-y)` * the movements below the interface are spelled :math:`{u}_{x}(x,y)={f}_{x}(x,y)\ast \frac{x}{\mathit{LX}}` According to :math:`y`, travel is imposed everywhere so: * movements above the interface are spelled :math:`{u}_{y}(x,y)={f}_{y}(x,y)` * the movements below the interface are spelled :math:`{u}_{y}(x,y)={f}_{y}(x,y)` *3D case* Since Poisson's ratio :math:`\nu` is zero, the solution is written independently in the direction :math:`x`, the direction :math:`y` and the direction :math:`z`. Neglecting gravity, the equation is written (in total constraints): :math:`\text{Div}(\sigma )=0` Being in the elastic case, we have :math:`\sigma =Eϵ`, which is finally :math:`\text{Div}(ϵ)=0`. According to :math:`x`, :math:`\frac{\partial {ϵ}_{\mathit{xx}}}{\partial x}=0` from where: * movements above the interface are spelled :math:`{u}_{x}(x,y,z)={f}_{x}(x,y,z)\ast \frac{x}{\mathit{LX}}` * the movements below the interface are spelled :math:`{u}_{x}(x,y,z)={f}_{x}(x,y,z)\ast \frac{x}{\mathit{LX}}` According to :math:`z`, travel is imposed everywhere so: * movements above the interface are spelled :math:`{u}_{z}(x,y,z)={f}_{z}(x,y,z)` * the movements below the interface are spelled :math:`{u}_{z}(x,y,z)={f}_{z}(x,y,z)` According to :math:`y`, travel sucks everywhere. Reference quantities and results ----------------------------------- We test the movements above and below the interface. In 2D ~~~~~ In modeling A (linear) :math:`{f}_{x}(x,y)=\{\begin{array}{c}0.01\ast y\mathit{si}Y<{L}_{d}\\ -0.01\ast (\mathit{LY}-y)\mathit{si}Y>{L}_{d}\end{array}` In B modeling (quadratic) :math:`{f}_{x}(x,y)=\{\begin{array}{c}0.01\ast {y}^{2}\mathit{si}Y<{L}_{d}\\ -0.01\ast {(\mathit{LY}-y)}^{2}\mathit{si}Y>{L}_{d}\end{array}` with :math:`\mathit{LX}=\mathrm{1m},\mathit{LY}=\mathrm{5m}` and :math:`{f}_{y}(x,y)=\{\begin{array}{c}-0.01\ast y\mathit{si}Y<{L}_{d}\\ 0.01\ast (\mathit{LY}-y)\mathit{si}Y>{L}_{d}\end{array}` According to :math:`x`: * movements above the interface are spelled :math:`{u}_{x}(x,y)=\frac{-0.01\ast x}{\mathit{LX}}\ast (\mathit{LY}-y)` * the movements below the interface are spelled :math:`{u}_{x}(x,y)={f}_{x}(x,y)\ast \frac{x}{\mathit{LX}}` According to :math:`y`, travel is imposed everywhere so: * movements above the interface are spelled :math:`{u}_{y}(x,y)={f}_{y}(x,y)` * the movements below the interface are spelled :math:`{u}_{y}(x,y)={f}_{y}(x,y)` **For A-modelling**, :math:`Y={L}_{d}=\frac{13\ast \mathit{LY}}{25}`. The displacement along :math:`y` of the two nodes of the interface respectively on the lower and upper lip of the crack is tested. .. csv-table:: "Quantities tested", "Reference type", "Reference value" "DY (below)", "'ANALYTIQUE'", "-2.6E-02" "DY (above)", "'ANALYTIQUE'", "2.4E-02" The displacement along :math:`x` of the two nodes of the interface respectively on the lower and upper lip of the crack in :math:`x=\mathrm{1m}` is also tested. .. csv-table:: "Quantities tested", "Reference type", "Reference value" "DX (below)", "'ANALYTIQUE'", "2.6E-02" "DX (above)", "'ANALYTIQUE'", "-2.4E-02" **For B modelling**, :math:`Y={L}_{d}=\frac{13\ast \mathit{LY}}{25}`. The displacement along :math:`y` of the two nodes of the interface respectively on the lower and upper lip of the crack is tested. .. csv-table:: "Quantities tested", "Reference type", "Reference value" "DY (below)", "'ANALYTIQUE'", "-2.6E-02" "DY (above)", "'ANALYTIQUE'", "2.4E-02" The displacement along :math:`x` of the two nodes of the interface respectively on the lower and upper lip of the crack in :math:`x=\mathrm{1m}` is also tested. .. csv-table:: "Quantities tested", "Reference type", "Reference value" "DX (below)", "'ANALYTIQUE'", "6.76E-02" "DX (above)", "'ANALYTIQUE'", "-5.76E-02" In 3D ~~~~~ In C modeling (linear) :math:`{f}_{x}(x,y,z)=\{\begin{array}{c}0.01\ast z\mathit{si}Z<{L}_{d}\\ -0.01\ast (\mathit{LZ}-z)\mathit{si}Z>{L}_{d}\end{array}` In D modeling (quadratic) :math:`{f}_{x}(x,y,z)=\{\begin{array}{c}0.01\ast {z}^{2}\mathit{si}Z<{L}_{d}\\ -0.01\ast {(\mathit{LZ}-z)}^{2}\mathit{si}Z>{L}_{d}\end{array}` with :math:`\mathit{LX}=\mathrm{1m},\mathit{LY}=\mathrm{1m},\mathit{LZ}=\mathrm{5m}` and :math:`{f}_{z}(x,y,z)=\{\begin{array}{c}-0.01\ast z\mathit{si}Z<{L}_{d}\\ 0.01\ast (\mathit{LZ}-z)\mathit{si}Z>{L}_{d}\end{array}`. According to :math:`x`: * movements above the interface are spelled :math:`{u}_{x}(x,y,z)={f}_{x}(x,y,z)\ast \frac{x}{\mathit{LX}}` * the movements below the interface are spelled :math:`{u}_{x}(x,y,z)={f}_{x}(x,y,z)\ast \frac{x}{\mathit{LX}}` According to :math:`z`, travel is imposed everywhere so: * movements above the interface are spelled :math:`{u}_{z}(x,y,z)={f}_{z}(x,y,z)` * the movements below the interface are spelled :math:`{u}_{z}(x,y,z)={f}_{z}(x,y,z)` According to :math:`y`, travel sucks everywhere. **For C modelling**, :math:`Z={L}_{d}=\frac{2\ast \mathit{LZ}}{5}`. The displacement along :math:`z` of the two nodes of the interface respectively on the lower and upper lip of the crack is tested. .. csv-table:: "Quantities tested", "Reference type", "Reference value" "DZ (below)", "'ANALYTIQUE'", "-2.0E-02" "DZ (above)", "'ANALYTIQUE'", "3.0E-02" The displacement along :math:`x` of the two nodes of the interface respectively on the lower and upper lip of the crack in :math:`x=\mathrm{1m}` is also tested. .. csv-table:: "Quantities tested", "Reference type", "Reference value" "DX (below)", "'ANALYTIQUE'", "2.0E-02" "DX (above)", "'ANALYTIQUE'", "-3.0E-02" **For D modelling**, :math:`Z={L}_{d}=\frac{\mathit{LZ}}{2}`. The displacement along :math:`z` of the two nodes of the interface respectively on the lower and upper lip of the crack is tested. .. csv-table:: "Quantities tested", "Reference type", "Reference value" "DZ (below)", "'ANALYTIQUE'", "-2.5E-02" "DZ (above)", "'ANALYTIQUE'", "2.5E-02" The displacement along :math:`x` of the two nodes of the interface respectively on the lower and upper lip of the crack in :math:`x=\mathrm{1m}` is also tested. .. csv-table:: "Quantities tested", "Reference type", "Reference value" "DX (below)", "'ANALYTIQUE'", "6.25E-02" "DX (above)", "'ANALYTIQUE'", "-6.25E-02" Uncertainty about the solution ---------------------------- None, the values tested are analytical.