2. Benchmark solution

2.1. Calculation method used for the reference solution

We are looking for the \(U\) displacement field in the form:

The transformation gradient, the deformation and its mechanical part are then:

with:

Note:

\({({E}^{m})}_{\text{eq}}\mathrm{=}\mathrm{\mid }a\mathrm{-}b\mathrm{\mid }\mathrm{=}a\mathrm{-}b\) (it is assumed that \(a>b\))

The behavioral relationship is written as:

with:

To determine

taking into account linear work hardening, the following are introduced:

• the shear modulus:

  

• the work hardening module:

  

,

The « internal pseudo-variable »

It is then worth:

Finally,

is written:

Taking into account boundary conditions:

The system to be solved is written as:

It is also written:

A

fixed, so it is a non-linear system in

and

, since

Is quadratic in

and

quadratic in

.

However, one can choose to fix

(so

) and solve a linear system by

and

(from which we deduct

and

):

• 
• 
• 
• 

It then remains to express Cauchy’s constraint:

Or here:

As for the force exerted on the face [3,4], due to the hypothesis of dead charges, it is simply written:

2.2. Benchmark results

As reference results, we will adopt displacements, the Cauchy constraint and the force exerted on face \(\mathrm{[}\mathrm{3,4}\mathrm{]}\) (in \(\mathrm{3D}\) only):

At time \(t\mathrm{=}\mathrm{2 }s\) (\(\Delta T\mathrm{=}100°C\), traction

)

In fact, we’re looking for

such as elongation:

• \(K\mathrm{=}166\mathrm{666 }\mathit{MPa}\) \(\mu \mathrm{=}\mathrm{76 923 }\mathit{MPa}\) \(R\text{'}\mathrm{=}\mathrm{2 020 }\mathit{MPa}\)

• \(a\mathrm{=}0.095\)

• 
• 
• 
• 

At time \(t\mathrm{=}\mathrm{3 }s\)

The bar is back in its original state:

2.3. Uncertainty about the solution

The solution is analytical. Except for rounding errors, it can be considered accurate.

2.4. Bibliographical references

Reference may be made to:

1. 5. LORENTZ: A nonlinear hyperelastic behavior relationship - Internal note EDF DER HI-74/95/011/0