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


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**1.1 Geometry**

As shown in Figure 1, the structure under investigation is a 2D plate containing a centrally placed crack. The crack is subjected to tensile loading conditions in the presence of initial stress fields.


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*Figure 1: Geometry description.*

The geometry parameters of this rectangular plate are a width, 2w = 50 mm, a height, 2h = 140 mm, and a crack length with the ratio a/w = 0.4.

Figure 1 also describes the shows the analysis sequence, which is split into three steps:


* Step 1: Uncracked plate is stretched beyond yielding until an equivalent plastic strain of 55.9% is achieved before the load is released. This generates an initial stress field.


* Step 2: A crack is introduced into the middle of the plate, and a dummy step run in order to produce an auto-equilibriated stress state in the plate.


* Step 3: The cracked plate is loaded again to 0.2% proof stress, thereby achieving a load lever Lr = 1.

**1.2 Material Properties**

For code_aster model to compute initial stress, the material is elastoplastic of Von Mises with a bi-linear strain-stress relationship


* Young modulus E = 500 MPa


* Plastic modulus Ep = (-y) /p


* Elastic limit y = 1 MPa


* E/Ep = 500


* Fish's ratio = 0.3

**1.3 Boundary conditions and loading**

Thanks to symmetry, a quarter scale symmetric model of the geometry is considered, ABCDE (see Figure 1-ii). The boundary conditions applied to the different sides of the geometry during each step are shown below:


*
    * Step 1:


        *
            * AE: UX = 0


            * FROM: UY = 0


            * CD: UY=0


            * AB: P1 = -1.8 MPa


    * Step 2:


        *
            * AE: UX = 0


            * CD: UY=0


    * Step 3:


        *
            * AE: UX = 0


            * CD: UY=0


            * AB: P2 = -0.7 MPa


            * The stress obtained at the end of step 2 is introduced into step 3 as the initial stress.