2. Benchmark solution

2.1. Calculation method used for the reference solution for modeling A

It is a flat problem. We can look for the solution in the form of a rigid rotation followed by an expansion by a factor \(a\) in one direction and \(b\) in the other:

The transformation gradient and the Green-Lagrange deformation are then:

The behavioral relationship leads to a diagonal Lagrangian stress tensor (with \(\lambda\) and \(\mu\) the Lamé coefficients):

We deduce the Cauchy stress tensor, which is also diagonal:

Finally, the boundary conditions are written as:

It is also possible to calculate the forces exerted on the faces:

where

represent the initial surfaces of the faces.

2.2. Model B reference

The reference results for this modeling are provided by the same calculation to which the rotation step was not applied.

2.3. Benchmark results

As reference results, we adopt the displacements, the Grenn-Lagrange deformations, the Cauchy stresses and the forces exerted on faces \([\mathrm{1,}3]\), \([\mathrm{3,}4]\) and \([\mathrm{1,}\mathrm{2,}\mathrm{3,}4]\) at the end of loading (\(t=2s\)).

We’re looking for

such as dilation

be \(p=–26610.3\mathrm{MPa}\).

Dilation \(b\) and displacements are then:

The Cauchy constraints are as follows:

Finally, the forces exerted are:

2.4. Uncertainty about the solution

Analytical solution.

2.5. Bibliographical references

1. Eric LORENTZ « A non-linear hyperelastic behavior relationship » Internal Note EDF/DER HI‑74/95/011/0