4. Implantation#

4.1. Calculation of the probability of rupture#

Tips for using this model are given in the documentation [U2.05.08] with a reminder of the sensitivity of the model to the refinement of the crack-point mesh.

Let us consider domain \({\Omega }_{c}\) of the studied structure, which can be the whole of the studied mesh, a group of elements or a mesh. Following an elastoplastic thermomechanical calculation, we know the evolution of the fields of stress, deformation and cumulative plastic deformation in this field. Using a damage criterion based on the stress field at the tip of the crack (or notch), the aim is to determine the cumulative probability of failure by cleavage of the structure. The model is used in a post-processor using the WEIBULL keyword from the POST_ELEM [U4.81.22] or POST_BEREMIN [U4.81.08] command.

Note that for the calculation with plastic deformation correction (option CORR_PLAST =” OUI “available with POST_ELEM), a prior calculation of the Green-Lagrange deformation field on the area of the structure studied (via the command CALC_CHAMP) is necessary (via the command). If not, post-processing stops.

In addition, the law of material behavior must include an internal variable corresponding to the cumulative equivalent plastic deformation \(p\). In particular, these are laws (non-exhaustive list):

VMIS_ISOT_ , VMIS_ECMI_, VMIS_ _ CHAB, ROUSS_, LEMAITRE, MONOCRISTAL. If not, post-processing stops.

The corresponding numerical integration in code_aster takes place in two stages:

  • we calculate at each Gauss point \({\mathrm{\sigma }}_{I}\) if the cumulative plastic deformation rate at this point is strictly positive,

  • by quadrature on each cell and then simple summation on the target domain \({\Omega }_{c}\), we deduce the Weibull stress as well as the associated probability of rupture. The summation is weighted by a multiplicative coefficient that takes into account any symmetries and the type of modeling used (axi, 2D, 3D, etc.). Care should be taken to define this coefficient (COEF_MULT) in accordance with the indications given in document [U4.81.22].

The first step makes it possible to introduce a variant (keyword SIGM_ELMOY instead of SIGM_ELGA) leading to substantially different results in the case of a cracked structure (presence of a gradient): in each mesh, \({\mathrm{\sigma }}_{I}\) is determined from the average weighted by the Gauss weights (for POST_ELEM and for) on this mesh of the stress field (and, possibly, of the deformation field of POST_BEREMIN Green-Lagrange). It is non-zero if the cumulative plastic deformation rate at the time in question is strictly positive at at least one Gauss point.

4.2. Material parameters#

The Beremin model requires the knowledge of three parameters: the two parameters characteristic of the material considered in Weibull’s law, \(m\) and \({\sigma }_{u}\), as well as the elementary volume of the elementary plastic zone \({V}_{0}\). Elementary volume \({V}_{0}\) must be large enough to respect hypothesis (5). In Weibull’s law, the \({V}_{0}\) and \({\mathrm{\sigma }}_{u}\) parameters are not independent. In fact, product \({V}_{0}{\mathrm{\sigma }}_{u}^{m}\) is involved. To these three parameters, it is possible to add a plastic deformation threshold making it possible to define the plastic zone on which the integration is carried out. In most cases, in order to simplify, we assume that the threshold is zero, which gives us a two-parameter law \(m\) and \({V}_{0}{\mathrm{\sigma }}_{u}^{m}\).

The identification of these parameters is not unique, but depends heavily on the calculated stress fields. code_aster has a specific operator dedicated to identifying the model from experimental data: RECA_WEIBULL [U4.82.06]. An example of using the command is available via the ssna103 test case [V6.01.103]. The test case is carried out using a tensile test on a smooth cylindrical specimen.