1. Reference problem#

The reference problem is inspired by an article by F. Auricchio*et al*. [1] who offers an analytical solution.

1.1. Geometry#

We consider a square with dimension \(2m\times 2m\).

The geometry can be visualized in the figure, with the mesh that will be used for both models A and B.

_images/10000000000005DB000003214C69DF1A38759B8B.svg

Figure 1.1-1: Geometry and meshing

1.2. Material properties#

The material is elastic and almost incompressible, that is to say that its Poisson’s ratio tends to 0.5:

  • Modulus of elasticity: \(E=120\mathit{Pa}\)

  • Poisson’s ratio: \(\mathrm{\nu }=0.499999\)

These values may seem surprising but have no physical meaning since the test is purely mathematical.

1.3. Boundary conditions and loads#

The four edges of the square (\(\mathit{DX}=\mathit{DY}=0\)) are embedded, and a volume force is applied to the entire square. This force is variable in space and has the following components:

\(\mathit{FX}=\mathrm{\mu }y(\frac{-3}{2}{x}^{4}+6{x}^{2}-3{x}^{2}{y}^{2}+{y}^{2}-\frac{5}{2})-15{x}^{2}(y-1)\)

\(\mathit{FY}=\mathrm{\mu }x(\frac{3}{2}{y}^{4}-6{y}^{2}+3{y}^{2}{x}^{2}-{x}^{2}+\frac{5}{2})-3{y}^{2}-5{x}^{3}\)