1. Reference problem#

1.1. Geometry#

_images/100000000000012C00000168470D7C91D38568BB.png

Radius of sphere

\(R=500\mathrm{mm}\)

Imposed displacement

\(100\mathrm{mm}\)

1.2. Material properties#

Two different models to represent the rigid sphere:

Material stiffness: \(E=\mathrm{2,1E9}\mathrm{Mpa}\) and \(\nu =\mathrm{0,3}\)

Stiffening by kinematic conditions

Block: Steel, perfect elasto‑plastic behavior law.

Module Young

\(E=210000\mathrm{MPa}\)

Poisson’s Ratio

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

Work hardening module

\(\mathrm{Et}=0\)

Elastic limit

\({\sigma }_{y}=50\mathrm{MPa}\)

1.3. Boundary conditions and loads#

The deformations are axisymmetric and the block forming the plane is assumed to be embedded on its base.

An imposed displacement is applied:

  • Loading from 0 to \(–100\mathrm{mm}\) on the upper part of the sphere in models A and D

  • Loading from 0 to \(–100\mathrm{mm}\) on the contact surface of the sphere in models B and C