3. Move jump#
The joints are intended to represent two facing faces, they only involve interpolation functions and the integration points of the corresponding surfacic (in 3D) or lineic (in 2D) elements (in 2D):
In 2D: for joint QUAD4 (or joint HYME QUAD8), the line element is SEG2
In 3D: for joint PENTA6 (or joint HYME PENTA15) the surface element is TRIA3
for joint HEXA8 (or joint HYME HEXA20) the surface element is QUAD4.
We call \({N}_{n}\) the shape function of the node \(n\) of the surface element [1] _ . \({U}^{\text{+n}}\) and \({U}^{\text{-n}}\) respectively designate the nodal movements of the segments \({\Gamma }^{\text{+}}\) and \({\Gamma }^{\text{-}}\) in 2D or the faces \({S}^{\text{+}}\) and \({S}^{\text{-}}\) in 3D.
In the local coordinate system, the displacement jump \(\delta\) is discretized based on the form functions \({N}_{n}\). At Gauss point \(g\), it is expressed as the difference in the displacements of the + and - faces (or segments):
\({\delta }_{g}=\sum _{n=1}^{\mathrm{Nb}}R({U}^{+n}-{U}^{-n}){N}_{n}^{g}\)
where \(\mathrm{Nb}\) is the number of nodes of the surface element and where \(R\) transition matrix in 2D, in 3D, which allows the nodal movements to be expressed in the local coordinate system. We can summarize the previous expression in a matrix \({M}_{g}^{U}\) which acts on the vector of the nodal displacements of the element: \(U\), to build the displacement jump in the local coordinate system:
\({\delta }_{g}={M}_{g}^{U}U\)
The \({M}_{g}^{U}\) matrix is of dimension \(\mathrm{ndim}\times {\mathrm{Nddl}}_{U}\), with \({\mathrm{Nddl}}_{U}\) number of mechanic degrees of freedom:
\({\mathrm{Nddl}}_{U}=8\) for the 2D joint,
\({\mathrm{Nddl}}_{U}=24\) for the 3D joint HEXA
\({\mathrm{Nddl}}_{U}=18\) for the 3D joint PENTA
\({\mathrm{Nddl}}_{U}=12\) for the HYME 2D joint
\({\mathrm{Nddl}}_{U}=48\) for the joint HYME 3D HEXA
\({\mathrm{Nddl}}_{U}=36\) for the joint HYME 3D PENTA