1. Reference problem#
1.1. Geometry#
The global frame of reference is frame \((A,X,Y,Z)\). In this coordinate system the coordinates of the nodes are:
\(A(0.,0.,0.)\)
\(B(3.,1.,0.)\)
\(C(2.,3.,0.)\)
\(D(3.,1.,-1)\)
We will study the behavior of the tetrahedron \(\mathrm{ABCD}\) whose material properties are defined in a local coordinate system \((A,x,y,z)\) obtained by rotating the global coordinate system according to nautical angles \((\alpha =30°,\beta =20°,\gamma =10°)\).
1.2. Material properties#
The materials used are orthotropic and transverse isotropic. In order to validate orthotropic deformations of thermal origin, a thermomechanical calculation is also carried out.
We adopt the terminology convention used in Code_Aster. Let the suffixes \(L\), \(T\) and \(N\) mean Longitudinal, Transverse, and Normal.
The units are not specified.
(We know that
,
either
For transverse isotropy, we keep the same values knowing that:
It should be noted that these coefficients are defined in a local coordinate system \((A,L,T,N)\) rotated with the nautical angles \((30°,20°,10°)\) with respect to the global coordinate system.
1.3. Boundary conditions and loads#
The boundary conditions are of the Dirichlet type. A linear displacement field is assumed in \(x\) and \(y\) so that the deformation field is constant.
Dirichlet conditions |
\(\mathrm{dX}=\mathrm{2x}+\mathrm{3y}+\mathrm{4z}\) |
\(\mathrm{dY}=\mathrm{3x}+\mathrm{5y}+\mathrm{6z}\) |
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\(\mathrm{dZ}=\mathrm{4x}+\mathrm{6y}+\mathrm{7z}\) |
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Thermal conditions |
Temperature imposed on the entire structure of 100 |
We will therefore impose:
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