2. Benchmark solution#
2.1. Calculation method used for the reference solution#
This problem admits an analytical solution that is explained in this part.
The load is such that the solution is homogeneous. The constraint that is installed in the cube is equal to \(\sigma ={\sigma }_{0}\text{}{e}_{x}\otimes {e}_{x}\). And the stress vector on the interface has normal and tangential components \({\sigma }_{n}={\sigma }_{0}/2\) and \({\sigma }_{t}=-{\sigma }_{0}/2\), where the local coordinate system \((n,t,{e}_{z})\) is direct.
The result is a deformation in the volume equal to:
\(\epsilon =\frac{{\sigma }_{0}}{E}({e}_{x}\otimes {e}_{x}-\nu {e}_{y}\otimes {e}_{y}-\nu {e}_{z}\otimes {e}_{z})\)
By noting \(\delta\) the jump in movement through the interface, we deduce that the displacement at a point \(x\) of the cube is equal to:
\(u(x)=\epsilon \cdot x+\delta H(x)\)
where \(H\) is the Heaviside function that is 0 on the left side of the interface and 1 on the right side of the interface.
Finally, the jump in movement through the interface depends on the choice of grip conditions. In the table below, the values of its normal and tangential components are given according to the combination of adhesion conditions selected. The vertical component \({\delta }_{z}\) is zero in all cases.
Elastic normal grip |
Single-sided normal grip |
Perfect normal grip |
|
Elastic tangential grip |
\({\delta }_{n}={\sigma }_{n}/{k}_{n}\) \({\delta }_{t}={\sigma }_{t}/{k}_{t}\) |
\({\delta }_{t}={\sigma }_{t}/{k}_{t}\) |
\({\delta }_{t}={\sigma }_{t}/{k}_{t}\) |
Perfect tangential grip |
\({\delta }_{n}={\sigma }_{n}/{k}_{n}\) \({\delta }_{t}=0\) |
\({\delta }_{t}=0\) |
\({\delta }_{t}=0\) |
2.2. Benchmark results#
The displacement is evaluated at point \(x=y=z=1000\) in 3D and \(x=y=1000\) in 2D, according to the analytical solution above.
We will ensure that the calculation returns the expected displacement for all combinations of adhesion conditions and for both values (tension and compression) of the stress.
2.3. Uncertainty about the solution#
Nil.
2.4. Bibliographical references#
Lorentz E. (2021). ADELAHYD II - Recommendations relating to the numerical simulation of cohesive zones using finite interface elements. Internal note EDF R&D 6125-1724-2021-01404.