Contact relationships between two structures
==========================================

Two relationships govern the contact between two structures:


* The one-sided contact relationship that expresses the non-interpenetrability between solid bodies,


* the friction relationship that governs the variation of tangential forces in contact. For the present developments, a simple relationship will be retained: Coulomb's law of friction.




Unilateral contact relationship
------------------------------

Let's say two structures :math:`{\mathrm{\Omega }}_{1}` and :math:`{\mathrm{\Omega }}_{2}`. We note :math:`{d}_{N}^{1/2}` the normal distance between structures and :math:`{F}_{N}^{1/2}` the normal reaction force of :math:`{\mathrm{\Omega }}_{1}` out of :math:`{\mathrm{\Omega }}_{2}` (see figure).


.. image:: images/100009340000175C000012805317E9CFA5F97175.svg
    :width: 301
    :height: 238


.. _RefImage_100009340000175C000012805317E9CFA5F97175.svg:

Figure 2-1: Normal distance and normal reaction

The law of action and reaction requires:

.. math::
 :label: eq-1

    {F} _ {N} ^ {1/2} =- {F} _ {N} ^ {2/1}

The conditions of unilateral contact, also called Signorin conditions (see [:ref:`1 <1>`]), are expressed as follows:

.. math::
 :label: eq-2

    \ {\ begin {array} {c} {d} _ {N} _ {N} ^ {1/2}\ ge 0\\ {F} _ {N} ^ {1/2}\ ge 0\\ {d} _ {N} _ {N} _ {N}} ^ {1/2} _ {1/2} _ {N} {D} _ {D} _ {N} {d} _ {N} _ {D} _ {N} _ {N} _ {N} ^ {1/2} _ {N} _ {N} ^ {1/2} _ {N} ^ {1/2} _ {N} ^ {1/2} _ {N


.. image:: images/100003FE00001AC50000159A8FEBFA504DAA0327.svg
    :width: 301
    :height: 238


.. _RefImage_100003FE00001AC50000159A8FEBFA504DAA0327.svg:

Figure 2-2: Unilateral contact relationship graph

The graphic representation of the law of unilateral contact in the figure reflects a force-displacement relationship that is not differentiable. It is therefore not usable in a simple manner in a dynamic calculation algorithm.

If we restrict the study to the case of a tubular structure in the presence of an undeformable support, we note :math:`{d}_{n}\left({d}_{n}={d}_{N}^{1/2}\right)` the normal distance to the support, and :math:`{F}_{n}` the reaction of the latter (:math:`{F}_{n}={F}_{N}^{2/1}=-{F}_{N}^{1/2}` see figure).


.. image:: images/10002ED800002DAF00000F816C0D28EB8C40FB75.svg
    :width: 301
    :height: 238


.. _RefImage_10002ED800002DAF00000F816C0D28EB8C40FB75.svg:

Figure 2-3: Normal distance and normal reaction between a structure and a support

The expression of the conditions of normal contact, expressing the limitation of movements due to the support, is equivalent to:

.. math::
 :label: eq-3

    \ {\ begin {array} {c} {d} _ {n} _ {n}\ ge 0\\ {F} _ {n}\ ge 0\\ {d} _ {n} {F} _ {f} _ {n} _ {n} = 0\ end {array}




Coulomb's law of friction
----------------------------

Coulomb's law expresses a limitation of the tangential force :math:`{F}_{T}^{1/2}` of tangential reaction of :math:`{O}_{1}` on :math:`{O}_{2}`. Let :math:`{\dot{u}}_{T}^{1/2}` be the relative speed of :math:`{\Omega }_{1}` with respect to :math:`{O}_{2}` at a point of contact and let :math:`\mu` be the Coulomb coefficient of friction, we have (see [1]):

.. math::
: label: eq-4

    \ {\ begin {array} {c} s=\ Green {F} s=\ Green {F} _ {F} _ {N} ^ {1/2}\ le 0\\\ exists\ mathrm {\ lambda}\ _ {1/2}\ le 0\\\ exists\ mathrm {\ lambda}\ {\ lambda}\ text {\ lambda}\ text {\ lambda}\ text {such as} {\ dot {u}}} _ {T} ^ {1/2} =\ mathrm {\ lambda} F} _ {T} ^ {1/2}\\\ mathrm {\ lambda}\ le 0\\\ mathrm {\ lambda}\ mathrm {.} s=0\ end {array}

and the law of action and reaction:

.. math::
: label: eq-5

    {F} _ {T} ^ {1/2} =- {F} _ {T} ^ {2/1}

The graphical representation of Coulomb's law on the figure also reflects the non-differentiable nature of the law and is therefore not simple to use in a dynamic algorithm.


.. image:: images/10000000000001C1000001647CD67B3488EB0697.png
    :width: 257
    :height: 208


.. _RefImage_10000000000001C1000001647CD67B3488EB0697.png:

Figure 2-4: Graph of Coulomb's law of friction

If the study is restricted to the case of a tubular structure in the presence of an undeformable support, only the tangential force :math:`{F}_{T}^{2/1}={F}_{T}` is used, the law of friction is expressed in the following way:

.. math::
 :label: eq-6

    \ {\ begin {array} {c} s=\ Green {F} s=\ Green {F} _ {F} s=\ Green -\ mathrm {\ mu} _ {n}\\ exists\ mathrm {\ lambda}\\ s=\ mathrm {\ lambda}\\ s=\ mathrm {\ lambda} {\ exists\ mathrm {\ lambda}\\ s=\\ mathrm {\ lambda}\ s=\\\ mathrm {\ lambda}\ le 0\\\ mathrm {\ lambda} s=0\ end {array}

A common extension of Coulomb's law, resulting from experience, consists in having two coefficients of friction: one for adhesion, noted :math:`{\mu }_{s}`, the other for sliding, noted :math:`{\mu }_{d}`, with :math:`{\mu }_{s}>{\mu }_{d}`. We then have an adhesion phase :math:`\Vert {F}_{T}\Vert \le {\mathrm{\mu }}_{s}{F}_{n}` and a sliding phase :math:`\Vert {F}_{T}\Vert ={\mathrm{\mu }}_{d}{F}_{n}`.


.. _refheading__2298_864480060: