2. Benchmark solution#
2.1. Calculation method#
The calculation is analytical.
We used the computer calculation program Mathematica to do it.
We know that the field of movement is:
\(\mathrm{dX}=\mathrm{2x}+\mathrm{3y}+\mathrm{4z}\)
\(\mathrm{dY}=\mathrm{3x}+\mathrm{5y}+\mathrm{6z}\)
\(\mathrm{dZ}=\mathrm{4x}+\mathrm{6y}+\mathrm{7z}\)
The deformation field \({\varepsilon }_{G}\) in the global coordinate system is therefore constant and equal to:
Let \(P\) be the transition matrix allowing a vector to pass from the global coordinate system \((A,X,Y,Z)\) to the local coordinate system \((A,L,N,T)\).
Either
the deformation tensor in the local coordinate system. We have: \({\varepsilon }_{L}\mathrm{=}P\mathrm{\cdot }{\varepsilon }_{G}\mathrm{\cdot }{P}^{T}\)
The Hooke tensor
is known in the local coordinate system, or
the stress tensor in this coordinate system. We have:
We get the tensor
constraints in the global frame of reference by:
In the case where a temperature field is applied, the equations above are modified as follows:
The field of deformations
in the global frame of reference is always the same:
Either
the deformation tensor in the local coordinate system. We have:
The tensor of mechanical deformations in the local coordinate system is therefore equal to:
with
, the other components being zero
The Hooke tensor
is known in the local coordinate system. Either
the stress tensor in this coordinate system. We have:
We get the tensor
constraints in the global frame of reference by:
2.2. Benchmark results#
They are obtained by performing the operations described above with Mathematica.
2.3. Uncertainties about the solution#
The uncertainty is zero because the solution is analytical.
2.4. Bibliographical references#
For the description of Hooke matrices for transverse isotropic and orthotropic materials for 3D models, plane stresses and plane deformations, the reference chosen was:
“Hooke matrix for orthotropic materials “. Internal report applications in Mechanics No. 79‑018 by Jean-Claude Masson CISI.