1. Reference problem#

1.1. Geometry#

It is a rectangular plate whose lower left corner corresponds to the center of the coordinate system and the left side coincides with the \(\vec{\mathit{oy}}\) axis.

L

X

H

Y

O

Plate height: \(H=200\mathit{mm}\)

Width of the plate: \(l=100\mathit{mm}\)

This plate constitutes a semi-structure, to be extended symmetrically with respect to axis \(\vec{\mathit{ox}}\).

The crack, in red in the figure above, is located on this axis of symmetry, on the half-width of the structure.

The 3D structure is obtained by rotating this plate around the \(\vec{\mathit{oy}}\) axis.

1.2. Material properties#

Elastic properties:

Young’s module:

\(E=\mathrm{2,0}\times {10}^{5}\mathit{MPa}\)

Poisson’s ratio:

\(\nu =0.3\)

Thermal properties:

Thermal expansion coefficient

\(\mathrm{\alpha }=\mathrm{1,0}\times {10}^{-5}/°C\)

Properties in the plastic field:

B, C, and G models

SE, F and H to L modeling

Elasticity limit:

\(\mathit{SY}=50\mathit{MPa}\)

\(\mathit{SY}=30\mathit{MPa}\)

Tangent module:

\(D\mathit{SIGM}\mathit{EPSI}=E/15\)

1.3. Boundary conditions and loads#

  • Travel for \(y=0\) and \(y=H\): \(v=0.\)

  • Travel for \(x=l\): \(u=0.\) for A, B, C and G models

  • Pre-deformations: \({\varepsilon }_{\mathrm{xx}}={\varepsilon }_{\mathrm{yy}}={\varepsilon }_{\mathrm{zz}}=f(x)\) for A to L models.

  • Initial deformations imposed using the command variable EPSA (anelastic deformation) for M and N models.

For each modeling, two calculations are carried out:

  • a thermomechanical calculation

  • a calculation with imposed pre-deformations for models A to L.

  • a calculation with initial deformations imposed using the command variable EPSA (anelastic deformation) for M and N models.

The thermal load is defined by a linear function equal to -50 in \(x=0\) and 50 in \(x=l\).

Pre-deformations are defined by a linear function equal to \(-{5.10}^{-4}\) in \(x=0\) and \({5.10}^{-4}\) in \(x=l\).

The control variable EPSA is determined by extracting the thermal deformations from the thermomechanical calculation.

Note that the deformation field generated by the thermal loading is identical to the imposed pre-deformation field.