2. Local coordinate system and transition matrix

It is necessary to build a coordinate system local to the element to define the displacement jump \(\delta\) (input data for the laws of behavior: see [R7.02.11] and [R7.01.25]). Moreover, the transition matrix \(R\) is defined from the global coordinate system to the local coordinate system. This part is valid for joint models in pure mechanics and for coupled hydromechanical models.

2.1. 2D case

Let \((X,Y)\) be the global benchmark. The direction given by the long sides [12] and [34] of the 2D joint element makes it possible to define a local coordinate system \((n,t)\) to the joint element (see figure 1):

\(t=\frac{\overrightarrow{12}}{\parallel \overrightarrow{12}\parallel }\), \(n=t\wedge (X\wedge Y)\)

The transition matrix from the global coordinate system to the local coordinate system is expressed:

\(R=\left[\begin{array}{cc}{n}_{x}& {n}_{y}\\ {t}_{x}& {t}_{y}\end{array}\right]\)

2.2. 3D case

The global benchmark is \((X,Y,Z)\). To build the coordinate system local to the joint element, the covariant base of the corresponding surface element is used. If we note \(s({\xi }^{\mathrm{1,}}{\xi }^{2})\) the parameterized position of a point on the surface element:

\(s({\xi }^{\mathrm{1,}}{\xi }^{2})=\sum _{n=1}^{\mathrm{Nb}}{N}_{n}({\xi }^{1},{\xi }^{2}){s}^{n}\)

where \({N}_{n}\) and \({s}^{n}\) respectively designate the shape function and the geometric position of the node \(n\), and \(\mathrm{Nb}\) the number of nodes of the surface element. The local covariant base \(({a}_{\mathrm{1,}}{a}_{2})\) is defined as follows:

\({a}_{1}=\frac{\partial s}{\partial {\xi }_{1}}=\sum _{n=1}^{\mathrm{Nb}}\frac{\partial {N}_{n}}{\partial {\xi }^{1}}{s}^{n}\) \({a}_{2}=\frac{\partial s}{\partial {\xi }_{2}}=\sum _{n=1}^{\mathrm{Nb}}\frac{\partial {N}_{n}}{\partial {\xi }^{2}}{s}^{n}\)

These two vectors are in fact vectors tangent to the element at a given point. The local direct orthonormal base \((n,t,\tau )\) is then constructed in the following manner:

\(t=\frac{{a}_{1}}{\parallel {a}_{1}\parallel }\) \(n=\frac{t\wedge {a}_{2}}{\parallel {a}_{2}\parallel }\) \(\tau =n\wedge t\)

The transition matrix from the global coordinate system to the local coordinate system is given by:

\(R=\left[\begin{array}{ccc}{n}_{x}& {n}_{y}& {n}_{z}\\ {t}_{x}& {t}_{y}& {t}_{z}\\ {\tau }_{x}& {\tau }_{y}& {\tau }_{z}\end{array}\right]\)