2. Benchmark solution#

2.1. Calculation method#

Analytical solution for damage variable \(D\):

\(D(t)\mathrm{=}1\mathrm{-}{(1\mathrm{-}(1+k){(\frac{{\sigma }_{0}}{A})}^{R}t)}^{\frac{1}{1+k}}\)

_images/Object_2.png

Analytical solution for the viscoplastic isotropic work hardening variable, \(r\), in the case of a zero \({\sigma }_{Y}\) threshold:

\(r(t)\mathrm{=}{\left[\frac{(M+N)}{M(1+k\mathrm{-}N)}{(\frac{{\sigma }_{0}}{A})}^{\mathrm{-}R}{(\frac{{\sigma }_{0}}{K})}^{N}(1\mathrm{-}{(1\mathrm{-}(1+k){(\frac{{\sigma }_{0}}{A})}^{R}t)}^{\frac{1+k\mathrm{-}N}{1+k}})\right]}^{\frac{M}{M+N}}\)

_images/Object_4.png

In the previous expressions, \(D\) is the damage variable corresponding to the internal variable \(\mathit{V9}\) and \(r\) is the multiplicative viscoplastic work hardening variable corresponding to the internal variable \(\mathit{V8}\).

We also have the following correspondence, in relation to the parameters of the VENDOCHAB keyword:

\(N\mathrm{=}{N}_{\mathit{VP}}\)

\(M\mathrm{=}{M}_{\mathit{VP}}\)

\(K\mathrm{=}{K}_{\mathit{VP}}\)

\(A\mathrm{=}{A}_{D}\)

\(R\mathrm{=}{R}_{D}\)

\(k\mathrm{=}{K}_{D}\)

2.2. Reference quantities and results#

Evolution of the damage variable, \(D\), as a function of time. This value is tested at various times:

Instant

Reference

520000

1.52596E-02

1000000

3.30676E-02

2000000

9.9465369E-02

2250000

1.37520763E-01

2500000

2.66018229E-01

Evolution of the viscoplastic isotropic work hardening variable, \(r\), as a function of time. This value is tested at various times:

Instant

Reference

520000

2.300147E-03

1000000

3.179469E-03

2000000

4.95103E-03

2250000

5.592847E-03

2500000

6.99749E-03

The difference observed on \(D\) for \(t=2.5{10}^{6}s\) is due to the very high non-linearity of the evolution of the damage variable.

2.3. Uncertainties about the solution#

Code accuracy