1. Reference problem#

1.1. Geometry and boundary conditions#

_images/10028CBC00000A8B00000B791E48614B1DA079E2.svg

Figure 1.1-a: Geometry and boundary conditions

Edge length:

\(a=0.56m\)

Thickness:

\(0.1m\)

The loading is such that a state of homogeneous stress and of the plane stress type is obtained even if the modeling in Code_Aster was carried out in 3D. Loading is imposed in the form of trips imposed in two stages:

  • direction \((\Delta {\varepsilon }_{\mathit{xx}},\Delta {\varepsilon }_{\mathit{yy}},\Delta {\varepsilon }_{\mathit{xy}})\mathrm{=}(1,\mathrm{-}\nu ,0)\), up to maximum stress (initiation of the softening phase)

  • direction \((\Delta {\varepsilon }_{\mathrm{xx}},\Delta {\varepsilon }_{\mathrm{yy}},\Delta {\varepsilon }_{\mathrm{xy}})=(\mathrm{1 ,}1.5,1)\), up to \({\varepsilon }_{\mathrm{xx}}=0.0015\)

Where

  • \({\varepsilon }_{\mathrm{xx}}\): movement of the side delimited by the side

_images/Object_1.svg

in the direction \(\mathrm{OX}\)

  • \({\varepsilon }_{\mathrm{yy}}\): movement of the side delimited by the side

_images/Object_2.svg

in the direction \(\mathrm{OY}\)

  • \({\varepsilon }_{\mathrm{xy}}\): movement of the side delimited by the side

_images/Object_3.svg

in the direction \(\mathrm{OY}\)

Let \(\mathit{P5P6P7P8}\) be the plane of the cube in \(z=0.1\).

In practice, the following conditions are imposed:

  • while loading:

\(\mathrm{P1P4P8P5}\): \(\mathrm{dx}=0\) \(\mathrm{P1}\): \(\mathrm{dy}=\mathrm{dz}=0\) \(\mathrm{P5}\): \(\mathrm{dy}=0\)

  • during phase 1:

\(\mathrm{P2}\), \(\mathrm{P6}\): \(\mathrm{dy}=0\) \(\mathit{P2P3P7P6}\): \(\mathrm{dx}=1\) \(\mathit{P3P4P8P7}\): \(\mathrm{dy}=-0.2\)

  • during phase 2:

\(\mathit{P2P3P7P6}\): \(\mathrm{dx}=1\) \(\mathrm{P4}\), \(\mathrm{P8}\): \(\mathrm{dy}=1.5\) \(\mathrm{P3}\), \(\mathrm{P7}\): \(\mathrm{dy}=3.5\) \(\mathrm{P2}\), \(\mathrm{P6}\): \(\mathrm{dy}=2.\)

1.2. Material properties#

For the Mazars model, the following parameters were used:

Elastic behavior:

\(E\mathrm{=}32000\mathit{MPa},\nu \mathrm{=}0.2\)

Damaging behavior:

\({\varepsilon }_{\mathit{d0}}\mathrm{=}9\text{.}\text{375}\text{.}{\text{10}}^{\mathrm{-}4};\text{Ac}\mathrm{=}1\text{.}\text{15};\text{At}\mathrm{=}0\text{.}8;\text{Bc}\mathrm{=}\text{1391}\text{.}3;\text{Bt}\mathrm{=}\text{10}\text{000};k\mathrm{=}0\text{.}\text{7}\)