2. Benchmark solution#
2.1. General case#
In this part, we detail the analytical solution in pure \(I\) mode in its \(\mathrm{3D}\) form. For \(\mathrm{2D}\) plan calculations, the solution is identical, the jump component and the stress vector following \(\tau\) are not involved, and it is enough to replace the area \(S\) by the length \(L\) in the solution.
For loads in shear mode, the elastic element does not intervene. Only a test is carried out on the law of tangential behavior. For cohesive laws, a displacement jump is imposed and the cohesive stress obtained is verified.
2.2. In pure \(\text{I}\) mode#
The analytical solution of the global response of the system written in the local coordinate system \((n,t,\tau )\) is presented. We apply a loading collinear to the normal: \(U=Un\), the cohesive element opens in pure \(I\) mode and the tangential constraints and tangential jumps remain zero. So we are back to a scalar problem. We note \(\sigma =n\mathrm{.}\sigma \mathrm{.}n\) the only non-zero component of the stress tensor of the elastic element in the local coordinate system. The global response solution for cohesive laws is presented:
CZM_EXP_REG, CZM_EXP_MIX
The cohesive behavior relationship is given by (see doc [R7.02.11]):
\(\overrightarrow{\sigma }=(\begin{array}{}{\sigma }_{n}\\ {\sigma }_{t}\\ {\sigma }_{\tau }\end{array})=(\begin{array}{}{\sigma }_{c}\cdot {e}^{-\frac{{\sigma }_{c}}{{G}_{c}}\cdot {\delta }_{n}}\\ 0\\ 0\end{array})\)
with \({\delta }_{n}\) the normal move jump. The elastic law of the volume element gives:
\(\sigma =E\varepsilon =F/S\)
where \(\varepsilon\) is the elastic deformation and \(F\) is the force corresponding to the displacement imposed on the surface \(S\). In the case where the threshold stress in the cohesive element is not reached, the solution is elastic, the overall response is linear, it is expressed in the following way:
When the breaking threshold is reached, the jump in the cohesive element is no longer zero, the response is no longer linear. The balance of the system is given by:
Moreover, in this simple case of loading, the imposed displacement is equal to the sum of the displacement jump and the displacement associated with the deformation \(\varepsilon\) of the elastic element:
\(U={\delta }_{n}+L\varepsilon\)
From this we deduce the relationship between force and imposed displacement:
\(U(F)=-\frac{{G}_{c}}{{\sigma }_{c}}\mathrm{log}(\frac{F}{S{\sigma }_{c}})+\frac{\mathrm{FL}}{\mathrm{SE}}\)
Note: -depending on the material data, we may not have a step back in the global response that we capture when controlling the load.
the control of the interface elements is based on the percentage of energy dissipated.
CZM_LIN_REG, CZM_OUV_MIX, CZM_TAC_MIX, CZM_FAT_MIX
The cohesive behavioral relationship is given by:
\(\vec{\sigma }=\left(\begin{array}{c}{\sigma }_{n}\\ {\sigma }_{t}\\ {\sigma }_{\tau }\end{array}\right)=\left(\begin{array}{c}{\sigma }_{c}\left(1-{\delta }_{n}\frac{{\sigma }_{c}}{2{G}_{c}}\right)\\ 0\\ 0\end{array}\right)\)
We adopt the same reasoning as with the exponential law, the analytical solution of the global response is expressed in the following way:
\(U(F)=\frac{F}{S}(\frac{L}{E}-\frac{2{G}_{c}}{{\sigma }_{c}^{2}})+2\frac{{G}_{c}}{{\sigma }_{c}}\)
Note: for the law CZM_FAT_MIXla the previous global answer is only valid if the loading is monotonic. In most cases this one is cyclical since this law is intended for fatigue. It is suggested to refer to the documentation [R7.02.11] of cohesive laws for more information.
CZM_LIN_MIX
Since the deformation is uniform in the block, in the same way as before, we can relate the displacement imposed \(U\) on the face to the jump in displacement through the cohesive element by \(U=⟦u⟧+Lϵ\), the deformation \(ϵ\) being given by \(ϵ=\frac{\sigma }{E}\). Since the constraint is uniform in the block, it is given by the cohesive law as \(\sigma ={\sigma }_{c}\left(1-⟦u⟧\frac{{\sigma }_{c}}{2{G}_{c}}\right)\). By combining these expressions, we get:
\(U=⟦u⟧+L\frac{{\sigma }_{c}}{E}\left(1-⟦u⟧\frac{{\sigma }_{c}}{2{G}_{c}}\right)\)
In this test, we validate the correct implementation of load management in this test, in addition to validating the good implementation of the cohesive law. In general, when load control is used, the displacement increment \(\Delta u\) is linked to the time step \(\Delta t\) by the relationship (see documentation [R5.03.80]):
\(f(\Delta u)=\frac{\Delta t}{\text{COEF\_MULT}}\)
where:
\(f\) is the control function, which depends on the quantity you want to control,
COEF_MULT is the value entered under the keyword of the same name, under the PILOTAGE factor keyword of the STAT_NON_LINE command.
More particularly for the cohesive law CZM_LIN_MIX, the steering function is:
\(f(\Delta u)=\frac{⟦u⟧}{{w}_{c}}\) where \({w}_{c}=\frac{2{G}_{c}}{{\sigma }_{c}}\) is the critical displacement jump.
To recap, the movement jump increment \(⟦\Delta u⟧\) applied during a \(\Delta t\) time step is written as:
\(⟦\Delta u⟧=\frac{\Delta t}{\text{COEF\_MULT}}\frac{2{G}_{c}}{{\sigma }_{c}}\)
For this test \(\text{COEF\_MULT}=10\). The reference load to be controlled is defined by an imposed displacement \({U}_{0}=2.5\). The parameter ETA_PILO giving the intensity of the load will therefore be given by \(\text{ETA\_PILO}=\frac{U}{{U}_{0}}\).
2.3. In pure \(\text{II}\) and \(\text{III}\) mode#
We only test the law of behavior (see doc [R7.02.11] and [R7.02.13]):
CZM_EXP_REG
\(T\) designating \(t\) in mode \(\mathrm{II}\) and \(\tau\) in mode \(\mathrm{III}\) respectively
CZM_LIN_REG, CZM_OUV_MIX, CZM_TAC_MIX
\({\sigma }_{T}={\sigma }_{c}(1-{\delta }_{T}\frac{{\sigma }_{c}}{2{G}_{c}})\), \(T\) designating respectively \(t\) in mode \(\mathrm{II}\) and \(\tau\) in mode \(\mathrm{III}\)