.. _use-case-chaboche: The Chaboche mechanical model ============================= Deterministic model ------------------- The Chaboche mechanical law predicts the stress depending on the strain: .. math:: \sigma = G(\epsilon,R,C,\gamma) = R + \frac{C}{\gamma} (1-\exp(-\gamma\epsilon)) where: - :math:\epsilon is the strain, - :math:\sigma is the stress (Pa), - :math:R, :math:C, :math:\gamma are the parameters. The variables have the following distributions and are supposed to be independent. ================ =========================================================== Random var. Distribution ================ =========================================================== :math:R Lognormal (:math:\mu = 750 MPa, :math:\sigma = 11 MPa) :math:C Normal (:math:\mu = 2750 MPa, :math:\sigma = 250 MPa) :math:\gamma Normal (:math:\mu = 10, :math:\sigma = 2) :math:\epsilon Uniform(a=0, b=0.07). ================ =========================================================== Parameters to calibrate ----------------------- The vector of parameters to calibrate is: .. math:: \theta = (R,C,\gamma). We set: - :math:R = 750\times 10^6, - :math:C = 2750\times 10^6, - :math:\gamma = 10. Observations ------------ In order to create a calibration problem, we make the hypothesis that the strain has the following distribution: .. math:: \epsilon \sim Uniform(0,0.07). Moreover, we consider a gaussian noise on the observed constraint: .. math:: \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: .. math:: n = 100. We generate a Monte-Carlo samplg with size :math:n: .. math:: \sigma_i = G(\epsilon_i,R,C,\gamma) + (\epsilon_\sigma)_i, for :math:i = 1,..., n. The observations are the pairs :math:\{(\epsilon_i,\sigma_i)\}_{i=1,...,n}, i.e. each observation is a couple made of the strain and the corresponding stress. Thanks to --------- - Antoine Dumas, Phimeca References ---------- - J. Lemaitre and J. L. Chaboche (2002) "Mechanics of solid materials" Cambridge University Press. Load the use case ----------------- We can load this classical model from the use cases module as follows : .. code-block:: python >>> from openturns.usecases import chaboche_model as chaboche_model >>> # Load the Chaboche use case >>> cm = chaboche_model.ChabocheModel() API documentation ----------------- See :class:~openturns.usecases.chaboche_model.ChabocheModel. Examples based on this use case ------------------------------- .. raw:: html
.. only:: html .. figure:: /auto_calibration/least_squares_and_gaussian_calibration/images/thumb/sphx_glr_plot_calibration_chaboche_thumb.png :alt: :ref:sphx_glr_auto_calibration_least_squares_and_gaussian_calibration_plot_calibration_chaboche.py .. raw:: html
.. toctree:: :hidden: /auto_calibration/least_squares_and_gaussian_calibration/plot_calibration_chaboche