1. Reference problem#

1.1. Geometry#

We consider a three-dimensional bar whose dimensions are:

  • height: \(\mathit{LZ}\) = 4 mm,

  • side: \(\mathit{LX}\) = \(\mathit{LY}\) = 1 mm.

This bar has a plane, semi-elliptical crack. The crack is located in plane \(\mathit{Oxy}\). The characteristics of the cracks are as follows:

  • semi-major axis: \(a\) = 119 µm

  • semi-minor axis: \(b\) = 100 µm.

_images/Cadre11.gif _images/Cadre11-1.gif _images/Cadre11-2.gif _images/Cadre11-3.gif

● A

● B

● C Figure 1.1-1: Geometry of the initial crack

1.2. Material properties#

The material is isotropic elastic whose properties are:

\(E\mathrm{=}200000\mathit{MPa}\)

\(\nu \mathrm{=}\mathrm{0,3}\)

1.3. Boundary conditions and loads#

1.3.1. Cyclic loading for fatigue study#

The structure is subject to a loading of fatigue under constant amplitude: traction \({\sigma }_{\mathit{max}}=220\mathit{MPa}\) and a ratio \(R\mathrm{=}\mathrm{0,1}\). The temperature is room temperature. The charging frequency is \(40\mathit{Hz}\). A load of 4000 cycles is applied.

The traction force is applied to the upper and lower faces.

The blocking of rigid modes is carried out in the following way:

  • point \(A\) is stuck in the \(\mathit{Oy}\) and \(\mathit{Oz}\) directions,

  • point \(B\) is stuck in the \(\mathit{Oy}\) and \(\mathit{Oz}\) directions,

  • point \(C\) is stuck in the \(\mathit{Ox}\) and \(\mathit{Oz}\) directions.

1.3.2. Modeling with cohesive zones: monotonic loading#

For this modeling, the load is monotonic instead of cyclical: the structure is subjected to \({\sigma }_{\mathit{max}}\mathrm{=}220\mathit{MPa}\) traction. The traction force is applied to the upper and lower faces.

The blocking of rigid modes is carried out in the following way:

  • point \(A\) is stuck in the \(\mathit{Oy}\) and \(\mathit{Oz}\) directions,

  • point \(B\) is stuck in the \(\mathit{Oy}\) and \(\mathit{Oz}\) directions,

  • point \(C\) is stuck in the \(\mathit{Ox}\) and \(\mathit{Oz}\) directions.