1. Reference problem

1.1. Geometry

Shell features:

• thickness \(h=0.05\mathrm{mm}\),

• radius of curvature \(R=1\mathrm{mm}\),

• width \(L=\mathrm{AB}=\mathrm{CD}=0.1\mathrm{mm}\),

• position of the first center of curvature:

,

the angle :math:`alpha` is chosen so that the**upper* surface of the shell at point \(X\) is at \((y=0)\), that is to say aligned with \(A\) and \(C\),

• position of the second center of curvature:

.

1.2. Material properties

\(E=\mathrm{2 000}\mathrm{MPa}\)

\(\nu =0.3\)

An elasto-plastic behavior law with the Von Misès criterion for linear isotropic work hardening is used: \({\sigma }_{y}=100\mathrm{MPa}\) \({E}_{T}:200\mathrm{MPa}\).

1.3. Boundary conditions and loads

• on \(\mathrm{AB}\): embedding: \(\mathrm{DX}=\mathrm{DY}=\mathrm{DZ}=\mathrm{DRX}=\mathrm{DRY}=\mathrm{DRZ}=0\),

• all over the shell: plane deformation according to \(\mathrm{Oz}\) or \(\mathrm{DZ}=\mathrm{DRX}=\mathrm{DRY}=0\),

• on \(\mathrm{CD}\): linear force (per unit length \(\mathrm{Oz}\)) according to \(\mathrm{Ox}\) given by: \({f}_{x}=50N/\mathrm{mm}\) It is equivalent to a pressure of \({p}_{X}={f}_{x}/h=100\mathrm{MPa}\) exerted on the side \(\mathrm{CD}\),

• the load is gradually applied to the structure. The loading path is divided into 10 equal increments.