2. Benchmark solution#

2.1. Calculation method#

2D case

Since Poisson’s ratio \(\nu\) is zero, the solution is written independently in the \(x\) direction and the \(y\) direction.

Neglecting gravity, the equation is written (in total constraints):

\(\text{Div}(\sigma )=0\)

Being in the elastic case, we have \(\sigma =Eϵ\), which is finally \(\text{Div}(ϵ)=0\).

According to \(x\), \(\frac{\partial {ϵ}_{\mathit{xx}}}{\partial x}=0\) from where:

  • movements above the interface are spelled \({u}_{x}(x,y)=\frac{-0.01\ast x}{\mathit{LX}}\ast (\mathit{LY}-y)\)

  • the movements below the interface are spelled \({u}_{x}(x,y)={f}_{x}(x,y)\ast \frac{x}{\mathit{LX}}\)

According to \(y\), travel is imposed everywhere so:

  • movements above the interface are spelled \({u}_{y}(x,y)={f}_{y}(x,y)\)

  • the movements below the interface are spelled \({u}_{y}(x,y)={f}_{y}(x,y)\)

3D case

Since Poisson’s ratio \(\nu\) is zero, the solution is written independently in the direction \(x\), the direction \(y\) and the direction \(z\).

Neglecting gravity, the equation is written (in total constraints):

\(\text{Div}(\sigma )=0\)

Being in the elastic case, we have \(\sigma =Eϵ\), which is finally \(\text{Div}(ϵ)=0\).

According to \(x\), \(\frac{\partial {ϵ}_{\mathit{xx}}}{\partial x}=0\) from where:

  • movements above the interface are spelled \({u}_{x}(x,y,z)={f}_{x}(x,y,z)\ast \frac{x}{\mathit{LX}}\)

  • the movements below the interface are spelled \({u}_{x}(x,y,z)={f}_{x}(x,y,z)\ast \frac{x}{\mathit{LX}}\)

According to \(z\), travel is imposed everywhere so:

  • movements above the interface are spelled \({u}_{z}(x,y,z)={f}_{z}(x,y,z)\)

  • the movements below the interface are spelled \({u}_{z}(x,y,z)={f}_{z}(x,y,z)\)

According to \(y\), travel sucks everywhere.

2.2. Reference quantities and results#

We test the movements above and below the interface.

2.2.1. In 2D#

In modeling A (linear) \({f}_{x}(x,y)=\{\begin{array}{c}0.01\ast y\mathit{si}Y<{L}_{d}\\ -0.01\ast (\mathit{LY}-y)\mathit{si}Y>{L}_{d}\end{array}\)

In B modeling (quadratic) \({f}_{x}(x,y)=\{\begin{array}{c}0.01\ast {y}^{2}\mathit{si}Y<{L}_{d}\\ -0.01\ast {(\mathit{LY}-y)}^{2}\mathit{si}Y>{L}_{d}\end{array}\)

with \(\mathit{LX}=\mathrm{1m},\mathit{LY}=\mathrm{5m}\) and \({f}_{y}(x,y)=\{\begin{array}{c}-0.01\ast y\mathit{si}Y<{L}_{d}\\ 0.01\ast (\mathit{LY}-y)\mathit{si}Y>{L}_{d}\end{array}\)

According to \(x\):

  • movements above the interface are spelled \({u}_{x}(x,y)=\frac{-0.01\ast x}{\mathit{LX}}\ast (\mathit{LY}-y)\)

  • the movements below the interface are spelled \({u}_{x}(x,y)={f}_{x}(x,y)\ast \frac{x}{\mathit{LX}}\)

According to \(y\), travel is imposed everywhere so:

  • movements above the interface are spelled \({u}_{y}(x,y)={f}_{y}(x,y)\)

  • the movements below the interface are spelled \({u}_{y}(x,y)={f}_{y}(x,y)\)

For A-modelling, \(Y={L}_{d}=\frac{13\ast \mathit{LY}}{25}\).

The displacement along \(y\) of the two nodes of the interface respectively on the lower and upper lip of the crack is tested.

Quantities tested

Reference type

Reference value

DY (below)

“ANALYTIQUE”

-2.6E-02

DY (above)

“ANALYTIQUE”

2.4E-02

The displacement along \(x\) of the two nodes of the interface respectively on the lower and upper lip of the crack in \(x=\mathrm{1m}\) is also tested.

Quantities tested

Reference type

Reference value

DX (below)

“ANALYTIQUE”

2.6E-02

DX (above)

“ANALYTIQUE”

-2.4E-02

For B modelling, \(Y={L}_{d}=\frac{13\ast \mathit{LY}}{25}\).

The displacement along \(y\) of the two nodes of the interface respectively on the lower and upper lip of the crack is tested.

Quantities tested

Reference type

Reference value

DY (below)

“ANALYTIQUE”

-2.6E-02

DY (above)

“ANALYTIQUE”

2.4E-02

The displacement along \(x\) of the two nodes of the interface respectively on the lower and upper lip of the crack in \(x=\mathrm{1m}\) is also tested.

Quantities tested

Reference type

Reference value

DX (below)

“ANALYTIQUE”

6.76E-02

DX (above)

“ANALYTIQUE”

-5.76E-02

2.2.2. In 3D#

In C modeling (linear) \({f}_{x}(x,y,z)=\{\begin{array}{c}0.01\ast z\mathit{si}Z<{L}_{d}\\ -0.01\ast (\mathit{LZ}-z)\mathit{si}Z>{L}_{d}\end{array}\)

In D modeling (quadratic) \({f}_{x}(x,y,z)=\{\begin{array}{c}0.01\ast {z}^{2}\mathit{si}Z<{L}_{d}\\ -0.01\ast {(\mathit{LZ}-z)}^{2}\mathit{si}Z>{L}_{d}\end{array}\)

with \(\mathit{LX}=\mathrm{1m},\mathit{LY}=\mathrm{1m},\mathit{LZ}=\mathrm{5m}\) and \({f}_{z}(x,y,z)=\{\begin{array}{c}-0.01\ast z\mathit{si}Z<{L}_{d}\\ 0.01\ast (\mathit{LZ}-z)\mathit{si}Z>{L}_{d}\end{array}\).

According to \(x\):

  • movements above the interface are spelled \({u}_{x}(x,y,z)={f}_{x}(x,y,z)\ast \frac{x}{\mathit{LX}}\)

  • the movements below the interface are spelled \({u}_{x}(x,y,z)={f}_{x}(x,y,z)\ast \frac{x}{\mathit{LX}}\)

According to \(z\), travel is imposed everywhere so:

  • movements above the interface are spelled \({u}_{z}(x,y,z)={f}_{z}(x,y,z)\)

  • the movements below the interface are spelled \({u}_{z}(x,y,z)={f}_{z}(x,y,z)\)

According to \(y\), travel sucks everywhere.

For C modelling, \(Z={L}_{d}=\frac{2\ast \mathit{LZ}}{5}\).

The displacement along \(z\) of the two nodes of the interface respectively on the lower and upper lip of the crack is tested.

Quantities tested

Reference type

Reference value

DZ (below)

“ANALYTIQUE”

-2.0E-02

DZ (above)

“ANALYTIQUE”

3.0E-02

The displacement along \(x\) of the two nodes of the interface respectively on the lower and upper lip of the crack in \(x=\mathrm{1m}\) is also tested.

Quantities tested

Reference type

Reference value

DX (below)

“ANALYTIQUE”

2.0E-02

DX (above)

“ANALYTIQUE”

-3.0E-02

For D modelling, \(Z={L}_{d}=\frac{\mathit{LZ}}{2}\).

The displacement along \(z\) of the two nodes of the interface respectively on the lower and upper lip of the crack is tested.

Quantities tested

Reference type

Reference value

DZ (below)

“ANALYTIQUE”

-2.5E-02

DZ (above)

“ANALYTIQUE”

2.5E-02

The displacement along \(x\) of the two nodes of the interface respectively on the lower and upper lip of the crack in \(x=\mathrm{1m}\) is also tested.

Quantities tested

Reference type

Reference value

DX (below)

“ANALYTIQUE”

6.25E-02

DX (above)

“ANALYTIQUE”

-6.25E-02

2.3. Uncertainty about the solution#

None, the values tested are analytical.