2. Benchmark solution

2.1. Calculation method

It is an analytical solution. The normal cohesive stress \({t}_{c,n}\) as a function of the normal displacement jump \([{u}_{n}]\) for the law “CZM_LIN_MIX” is shown in Figure. For each of the 8 moments of calculation, the position in which one is located is represented in this same Figure. The critical displacement jump that corresponds to the disappearance of cohesion efforts is \({\mathrm{\delta }}_{c}=\frac{2{G}_{c}}{{\mathrm{\sigma }}_{c}}=\mathrm{0,0016363636}m\)

Figure 2.1-a: Normal cohesive stress as a function of the normal displacement jump for the law “CZM_LIN_MIX”

By neglecting gravity, the global equilibrium equation is written (in total constraints):

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

In the case of coupled modeling, the total stress tensor is written as:

\(\sigma =\sigma \text{'}-{p}_{1}1\)

\(\sigma \text{'}\) is the stress tensor in the skeleton and \({p}_{1}\) is the pore pressure in the massif. Since the Poisson module \(\nu\) is zero, and being elastic in the case, we have \(\sigma \text{'}=Eϵ\).

But \(\forall x,{p}_{1}(x)=0\) so finally \(\text{Div}(ϵ)=0\)

Given the imposed displacements and the zero Poisson’s ratio \(\mathrm{\nu }\), the problem is unidirectional in the \(z\) direction. In the solid matrix, the following displacement field \(z\) verifies:

\(\frac{d\mathrm{²}{u}_{z}}{\mathit{dz}\mathrm{²}}=0\)

Instant1

The column is in compression, the lips of the cohesive interface are in contact. So the \([{u}_{z}]\) movement jump is zero. So \({\mathrm{ϵ}}_{\mathit{zz}}=g(1)/\mathit{LZ}\) and \({\mathrm{\sigma }}_{\mathit{zz}}=E{\mathrm{ϵ}}_{\mathit{zz}}=E\ast g(1)/\mathit{LZ}\). The normal cohesive stress \({t}_{c,n}\) is equal to the constraint \({\mathrm{\sigma }}_{\mathit{zz}}\).

Instant 2

The column is in traction. The hypothesis is made that the cohesive interface is in a situation of adhesion. This hypothesis is true if \({t}_{c,n}\le {\mathrm{\sigma }}_{c}\). If the cohesive interface is adherent, then the lips of the cohesive interface are in contact. So the \([{u}_{z}]\) movement jump is zero. So \({\mathrm{ϵ}}_{\mathit{zz}}=g(2)/\mathit{LZ}\) and \({\mathrm{\sigma }}_{\mathit{zz}}=E{\mathrm{ϵ}}_{\mathit{zz}}=E\ast g(2)/\mathit{LZ}\). The normal cohesive stress \({t}_{c,n}\) is then equal to the constraint \({\mathrm{\sigma }}_{\mathit{zz}}=\mathrm{0,116}\mathit{MPa}\le {\mathrm{\sigma }}_{c}\). The hypothesis made initially is therefore validated.

Instant 3

The column is in traction. The hypothesis is made that the cohesive interface is in a situation of adhesion. This hypothesis is true if \({t}_{c,n}\le {\mathrm{\sigma }}_{c}\). If the cohesive interface is adherent, then the lips of the cohesive interface are in contact. So the \([{u}_{z}]\) movement jump is zero. So \({\mathrm{ϵ}}_{\mathit{zz}}=g(3)/\mathit{LZ}\) and \({\mathrm{\sigma }}_{\mathit{zz}}=E{\mathrm{ϵ}}_{\mathit{zz}}=E\ast g(3)/\mathit{LZ}\). The normal cohesive stress \({t}_{c,n}\) is then equal to the constraint \({\mathrm{\sigma }}_{\mathit{zz}}=\mathrm{1,16}\mathit{MPa}>{\mathrm{\sigma }}_{c}\). The hypothesis made initially is therefore false.

It is therefore hypothesized that the cohesive interface is in a state of damage. This hypothesis is true if \(0<[{u}_{n}]\le {\mathrm{\delta }}_{c}\). The cohesive constraint \({t}_{c,n}\) is then linked to the normal displacement jump \([{u}_{z}]\) by the relationship \({t}_{c,n}={\mathrm{\sigma }}_{c}\ast (1-\frac{[{u}_{z}]}{{\mathrm{\delta }}_{c}})\). Moreover, the total elongation of the column is \(g(3)=[{u}_{z}]+\mathit{LZ}\ast {\mathrm{ϵ}}_{\mathit{zz}}\). Finally, the normal cohesive stress is equal to the vertical stress \({t}_{c,n}=E\ast {\mathrm{ϵ}}_{\mathit{zz}}\). Finally:

\({t}_{c,n}=\frac{E\ast {\mathrm{\sigma }}_{c}\ast (1-\frac{g(3)}{{\mathrm{\delta }}_{c}})}{E-\frac{\mathit{LZ}\ast {\mathrm{\sigma }}_{c}}{{\mathrm{\delta }}_{c}}}\)

\([{u}_{z}]=\frac{-\mathit{LZ}\ast {\mathrm{\sigma }}_{c}+E\ast g(3)}{E-\frac{\mathit{LZ}\ast {\mathrm{\sigma }}_{c}}{{\mathrm{\delta }}_{c}}}\)

Digitally, we find, \([{u}_{z}]=\mathrm{0,0001230068}m<{\mathrm{\delta }}_{c}\). The hypothesis made initially is validated.

Instant 4

The column is still in traction, but less traction than at the previous moment. We are therefore in a situation of elastic return in the cohesive zone (discharge during the damage process). The normal cohesive traction is then given by \({t}_{c,n}=\frac{{t}_{c,n}(3)\ast [{u}_{z}]}{[{u}_{z}(3)]}\) and the total elongation of the column is \(g(4)=[{u}_{z}]+\mathit{LZ}\ast {\mathrm{ϵ}}_{\mathit{zz}}\). Finally, the normal cohesive stress is equal to the vertical stress \({t}_{c,n}=E\ast {\mathrm{ϵ}}_{\mathit{zz}}\). Finally:

\([{u}_{z}]=\frac{E\ast g(4)\ast [:ref:`{u}_{z}(3) <{u}_{z}(3)>\)]} {mathit {LZ}ast {t} _ {c, n} (3) +East [{u}_{z}(3)]}} `

\({t}_{c,n}=\frac{{t}_{c,n}(3)\ast E\ast g(4)}{\mathit{LZ}\ast {t}_{c,n}(3)+E\ast [:ref:`{u}_{z}(3) <{u}_{z}(3)>\)]} `

Instant 5

The column is back in traction, at a level not yet reached. It is therefore hypothesized that the cohesive interface is in a state of damage. This hypothesis is true if \(0<[{u}_{n}]\le {\mathrm{\delta }}_{c}\). The cohesive constraint \({t}_{c,n}\) is then linked to the normal displacement jump \([{u}_{z}]\) by the relationship \({t}_{c,n}={\mathrm{\sigma }}_{c}\ast (1-\frac{[{u}_{z}]}{{\mathrm{\delta }}_{c}})\). Moreover, the total elongation of the column is \(g(5)=[{u}_{z}]+\mathit{LZ}\ast {\mathrm{ϵ}}_{\mathit{zz}}\). Finally, the normal cohesive stress is equal to the vertical stress \({t}_{c,n}=E\ast {\mathrm{ϵ}}_{\mathit{zz}}\). Finally:

\({t}_{c,n}=\frac{E\ast {\mathrm{\sigma }}_{c}\ast (1-\frac{g(5)}{{\mathrm{\delta }}_{c}})}{E-\frac{\mathit{LZ}{\mathrm{\sigma }}_{c}}{{\mathrm{\delta }}_{c}}}\)

\([{u}_{z}]=\frac{-\mathit{LZ}{\mathrm{\sigma }}_{c}+E\ast g(5)}{E-\frac{\mathit{LZ}{\mathrm{\sigma }}_{c}}{{\mathrm{\delta }}_{c}}}\)

Digitally, we find, \([{u}_{z}]=\mathrm{0,000059863325}m<{\mathrm{\delta }}_{c}\). The hypothesis made initially is validated.

Instant 6

The column is still in traction, at a level not yet reached. It is therefore hypothesized that the cohesive interface is in a state of damage. This hypothesis is true if \(0<[{u}_{n}]\le {\mathrm{\delta }}_{c}\). The cohesive constraint \({t}_{c,n}\) is then linked to the normal displacement jump \([{u}_{z}]\) by the relationship \({t}_{c,n}={\mathrm{\sigma }}_{c}\ast (1-\frac{[{u}_{z}]}{{\mathrm{\delta }}_{c}})\). Moreover, the total elongation of the column is \(g(6)=[{u}_{z}]+\mathit{LZ}\ast {\mathrm{ϵ}}_{\mathit{zz}}\). Finally, the normal cohesive stress is equal to the vertical stress \({t}_{c,n}=E\ast {\mathrm{ϵ}}_{\mathit{zz}}\). Finally:

\({t}_{c,n}=\frac{E\ast {\mathrm{\sigma }}_{c}\ast (1-\frac{g(6)}{{\mathrm{\delta }}_{c}})}{E-\frac{\mathit{LZ}{\mathrm{\sigma }}_{c}}{{\mathrm{\delta }}_{c}}}\)

\([{u}_{z}]=\frac{-\mathit{LZ}{\mathrm{\sigma }}_{c}+E\ast g(6)}{E-\frac{\mathit{LZ}{\mathrm{\sigma }}_{c}}{{\mathrm{\delta }}_{c}}}\)

Digitally, we find, \([{u}_{z}]=\mathrm{0,001787699}m>{\mathrm{\delta }}_{c}\). The assumption made at the beginning is wrong. The cohesive interface is no longer in the damage regime, it is broken. So the cohesive forces are zero \({t}_{c,n}=0\) and the displacement jump is \([{u}_{z}]=g(6)\).

Instant 7

During this time step, the vertical elongation of the column remains fixed. On the other hand, lateral displacement is applied to the upper face of the column in order to verify that the cohesive forces remain zero in the event of shear in the broken cohesive interface. We thus have:

\([{u}_{z}]=g(7)\)

\([{u}_{x}]=f(7)\)

\({t}_{c,n}=0\)

\({t}_{c,s}=0\)

Instant 8

Finally, the column is compressed to verify that the contact on the lips of the cohesive interface actually applies even when the cohesive zone has been broken. The load is the same as at instant 1. We have \({\mathrm{ϵ}}_{\mathit{zz}}=g(8)/\mathit{LZ}\) and \({\mathrm{\sigma }}_{\mathit{zz}}=E{\mathrm{ϵ}}_{\mathit{zz}}=E\ast g(8)/\mathit{LZ}\). The normal cohesive stress \({t}_{c,n}\) is equal to the constraint \({\mathrm{\sigma }}_{\mathit{zz}}\).

2.2. Reference quantities and results

The value of normal cohesive traction at the level of the cohesive interface and of tangential cohesive traction (next \(x\)) is tested at each moment. To test all the nodes of the cohesive interface at the same time we test MIN and MAX.

	Normal cohesive pull (MPa)	Tangential cohesive pull (MPa)
Instant 1	-0.116	0
Instant 2	0.116	0
Instant 3	1.0173120729	0
Instant 4	0.50865603645	0
Instant 5	0.69758542141	0
Instant 6	0	0
Instant 7	0	0
Instant 8	-0.116	0

2.3. Uncertainties about the solution

None the solution is analytical.

2.4. Bibliographical references

1. Reference documentation R7.02.18 (Hydromechanical elements coupled with the extended finite element method).