2. Benchmark solution#
2.1. Calculation method used for the reference solution#
It is a flat problem. We can look for the solution in the form of a rigid rotation and an elongation by a factor
In the direction
.
The transformation gradient and the Green-Lagrange deformation are then:
The behavioral relationship then leads to a diagonal Lagrangian stress tensor:
The boundary condition of the equilibrium equation then allows us to determine the value of the elongation
:
The Cauchy constraint is given by:
Finally, the force exerted on the faces:
[2,4]:
[4,3]:
[1,2,3,4]:
where
represent the initial surfaces of the faces.
2.2. Benchmark results#
Displacements, the Cauchy constraint and the force exerted on the [2,4] and [4,3] faces are adopted as reference results.
At time \(t=\mathrm{2 }s\) :
We’re looking for
such as lengthening
be \(T=\mathrm{31 096}.154\mathrm{MPa}\).
The Cauchy constraint is then:
The forces exerted are:
At time \(t=\mathrm{3 }s\) :
The bar is back in its original state:
2.3. Uncertainty about the solution#
Analytical solution.
2.4. Bibliographical references#
Eric LORENTZ « A non-linear hyperelastic behavior relationship » Internal Note EDF/DER HI‑74/95/011/0