The Chaboche mechanical model

Deterministic model

The Chaboche mechanical law predicts the stress depending on the strain:

\sigma = G(\epsilon,R,C,\gamma) = R + \frac{C}{\gamma} (1-\exp(-\gamma\epsilon))

where:

  • \epsilon is the strain,

  • \sigma is the stress (Pa),

  • R, C, \gamma are the parameters.

The variables have the following distributions and are supposed to be independent.

Random var.

Distribution

R

Lognormal (\mu = 750 MPa, \sigma = 11 MPa)

C

Normal (\mu = 2750 MPa, \sigma = 250 MPa)

\gamma

Normal (\mu = 10, \sigma = 2)

\epsilon

Uniform(a=0, b=0.07).

Parameters to calibrate

The vector of parameters to calibrate is:

\theta = (R,C,\gamma).

We set:

  • R = 750\times 10^6,

  • C = 2750\times 10^6,

  • \gamma = 10.

Observations

In order to create a calibration problem, we make the hypothesis that the strain has the following distribution:

\epsilon \sim Uniform(0,0.07).

Moreover, we consider a gaussian noise on the observed constraint:

\epsilon_\sigma \sim \mathcal{N} \left(0,10\times 10^6\right)

and we make the hypothesis that the observation errors are independent. We set the number of observations to:

n = 100.

We generate a Monte-Carlo samplg with size n:

\sigma_i = G(\epsilon_i,R,C,\gamma) + (\epsilon_\sigma)_i,

for i = 1,..., n. The observations are the pairs \{(\epsilon_i,\sigma_i)\}_{i=1,...,n}, i.e. each observation is a couple made of the strain and the corresponding stress.

Variables

In the particular situation where we want to calibrate this model, the following list presents which variables are observed input variables, input calibrated variables and observed output variables.

  • \epsilon: Input. Observed.

  • R, C, \gamma : Inputs. Calibrated.

  • \sigma: Output. Observed.

Thanks to

  • Antoine Dumas, Phimeca

References

    1. Lemaitre and J. L. Chaboche (2002) “Mechanics of solid materials” Cambridge University Press.

Load the use case

We can load this model from the use cases module as follows :

>>> from openturns.usecases import chaboche_model
>>> # Load the Chaboche use case
>>> cm = chaboche_model.ChabocheModel()

API documentation

See ChabocheModel.

Examples based on this use case

Calibration of the Chaboche mechanical model