Reference problem
=====================


Geometry
---------

We consider a one-dimensional bar of length :math:`\mathrm{2L}=199\mathit{mm}`, at the center of which a discontinuity governed by cohesive behavior is arranged.


.. image:: images/100002010000053800000128C82A270D778CF653.png
    :width: 4.478in
    :height: 1.0634in


.. _RefImage_100002010000053800000128C82A270D778CF653.png:

**Figure** 1.1-a **: One-dimensional bar under tension**

We denote the space variable :math:`\widehat{x}`, and we mark the center of the bar by :math:`\widehat{x}=0`.


Material properties
----------------------

The behavior of the massif is linear elastic:


*Young's modulus: :math:`E=30000\mathit{MPa}`*


* Poisson's ratio: :math:`\nu =0`

Cohesive behavior is described by an affine law in a softening regime (CZM_TAC_MIX for the interface elements used in modeling A, and CZM_LIN_REG for the joint elements used in modeling B), whose parameters are as follows:


* Critical constraint: :math:`{\sigma }_{c}=3\mathit{MPa}`


* Breakthrough energy: :math:`{G}_{c}=0.1N/\mathit{mm}`

The values for the additional numerical parameters are as follows:


* :math:`\text{PENA\_LAGR}=45000` for modeling A (CZM_TAC_MIX)


* :math:`\text{PENA\_ADHERENCE}={10}^{-4}` for B modeling (CZM_LIN_REG)

*Note:*

When only half of the bar is modelled (as is the case for modeling B) by imposing a symmetry condition, the value of the rupture energy is divided by 2.


.. _RefNumPara__11653_1489988600:


Boundary conditions and loads
-------------------------------------

The problem is one dimensional. The bar is called upon in traction, by imposing the same displacement of intensity :math:`U` at each of its ends.

We increase the imposed displacement according to a ramp ranging from :math:`U=0` to :math:`{U}_{\mathit{test}}=2L\frac{{\sigma }_{c}}{E}=0.0199\mathit{mm}`