Reference problem
=====================


Geometry
---------


.. image:: images/100000000000012C00000168470D7C91D38568BB.png
    :width: 3.1228in
    :height: 3.748in


.. _RefImage_100000000000012C00000168470D7C91D38568BB.png:


.. csv-table::

    "Radius of sphere", ":math:`R=500\mathrm{mm}`"
    "Imposed displacement", ":math:`100\mathrm{mm}`"



Material properties
----------------------

Two different models to represent the rigid sphere:

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

Stiffening by kinematic conditions

Block: Steel, perfect elasto‑plastic behavior law.


.. csv-table::

    "Module Young", ":math:`E=210000\mathrm{MPa}`"
    "Poisson's Ratio", ":math:`\nu =\mathrm{0,3}`"
    "Work hardening module", ":math:`\mathrm{Et}=0`"
    "Elastic limit", ":math:`{\sigma }_{y}=50\mathrm{MPa}`"



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 :math:`–100\mathrm{mm}` on the upper part of the sphere in models A and D


* Loading from 0 to :math:`–100\mathrm{mm}` on the contact surface of the sphere in models B and C