Behavioral relationship of mechanisms
=======================================

The behavior of mechanisms 1 and 2 is similar. It is non-linear between a rigid initial tangent behavior and an asymptotic limiting behavior.

It is described by two essential parameters: the nonlinearity parameter and the boundary surface parameter.

The stop (mechanism 1) or the ruin (mechanism 2) are described by an associated kinematic criterion.




Unidirectional behavior
----------------------------

We told [:ref:`§2 <§2>`] that unidirectional behaviors during normal effort and around time are similar [:ref:`Figure 2-b <Figure 2-b>`].

They can be described by the same relationship if the dimensionless quantities are used:


1. reduced forces: :math:`n=\frac{{N}_{x}}{\overline{N}}\text{et}m=\frac{{M}_{y}}{\overline{M}}`


2. reduced trips: :math:`{U}_{r}=\frac{U}{\overline{U}}\text{et}{\theta }_{r}=\frac{\theta }{\overline{\theta }}`

The [:ref:`Figure 3.1-a <Figure 3.1-a>`] represents unidirectional behavior in dimensionless form. Analytically, it can be written (it is a choice):

:math:`\begin{array}{c}\begin{array}{}{U}_{r}=h(n)\text{ou}{\mathrm{\theta }}_{r}=h(m)\\ \text{avec}h(x)=\frac{1}{\overline{d}}\frac{{x}^{a+1}}{1-{x}^{a}}\\ \overline{d}=\frac{{\overline{n}}^{a+1}}{1-{\overline{n}}^{a}}\end{array}\end{array}`

:math:`a` is the scalar nonlinearity parameter. :math:`\overline{n}` and :math:`a` are identified on unidirectional tests. :math:`\overline{n}` that takes into account the variability of the trials generally takes the value :math:`0.95`.


.. image:: images/100015EA00002DFE000014C6890AEB74205214A0.svg
    :width: 593
    :height: 268


.. _RefImage_100015EA00002DFE000014C6890AEB74205214A0.svg:

**Figure 3.1-a: assembly behavior relationship**

Note that :math:`h(\overline{n})=1` or :math:`h(\overline{m})=1`, that is to say: :math:`{U}_{r}=1` or :math:`{\mathrm{\theta }}_{r}=1`, or again: :math:`U=\overline{U}` or :math:`\mathrm{\theta }=\overline{\mathrm{\theta }}`.

The unidirectional kinematic criterion is therefore verified for :math:`n=\overline{n}` or :math:`m=\overline{m}`.




Incremental two-dimensional behavior
------------------------

The boundary coupling is defined by the boundary surface:

:math:`{(\frac{{N}_{x}}{\overline{N}})}^{2}+{(\frac{{M}_{y}}{\overline{M}})}^{2}=1`

Unidirectional behavior in reduced variables is described by the [:ref:`§3.1 <§3.1>`] relationship:

:math:`\begin{array}{c}\underline{d}\mathrm{=}h(\underline{f})\end{array}`


.. csv-table::

    "where", ":math:`\underline{d}` is the reduced displacement vector :math:`(\begin{array}{}{U}_{r}\\ {\mathrm{\theta }}_{r}\end{array})`"
    "", ":math:`\underline{f}` is the reduced forces vector :math:`(\begin{array}{}n\\ m\end{array})`"


In two-dimensional behavior, isotropy is translated by a model with an internal variable **scalar** :math:`p` such as:

:math:`\begin{array}{c}p=h(\text{feq})\text{en chargement}\end{array}`

where :math:`\text{feq}` is the equivalent reduced force (scalar).

:math:`\text{feq}` is defined as:

:math:`\underline{F}=\text{feq}\ast \underline{\overline{F}}\ast`


.. csv-table::

    "where", ":math:`\underline{F}` is the current loading point :math:`(\begin{array}{}{N}_{x}\\ {M}_{y}\end{array})`"
    "", ":math:`\begin{array}{}\underline{\overline{F}}\ast \\ \end{array}` is the load limit associated with :math:`\underline{F}` :math:`(\begin{array}{}{\overline{N}}_{x}^{\text{*}}\\ {\overline{M}}_{y}^{\text{*}}\end{array})`


The expression for :math:`\text{feq}` is derived from the expression for the boundary surface. :math:`\begin{array}{}\underline{\overline{F}}\ast \\ \end{array}`'s membership in the boundary surface is written as:

:math:`{(\frac{{\overline{N}}_{x}^{\text{*}}}{\overline{N}})}^{2}+{(\frac{{\overline{M}}_{y}^{\text{*}}}{\overline{M}})}^{2}=1`

By the definition of :math:`\text{feq}`, we can write:

:math:`{(\frac{{N}_{x}}{\text{feq}\overline{N}})}^{2}+{(\frac{{M}_{y}}{\text{feq}\overline{M}})}^{2}=1`

that is to say according to the reduced forces :math:`n` and :math:`m`:

:math:`{(\frac{n}{\text{feq}})}^{2}+{(\frac{m}{\text{feq}})}^{2}=1`

Hence :math:`\begin{array}{c}\text{feq}=\sqrt{{n}^{2}+{m}^{2}}\end{array}`

The loading surface :math:`F`, which is homothetic to the boundary surface, is then defined by:

:math:`\begin{array}{c}\begin{array}{cc}F:& \text{feq}-R(p)=0\\ & \text{où}R(p)={h}^{-1}(p)\end{array}\end{array}`

For a formalism similar to that of isotropic work hardening plasticity [:ref:`bib2 <bib2>`], we obtain the continuous behavior relationship expressed in reduced quantities:

:math:`\begin{array}{c}\begin{array}{}\underline{\stackrel{\text{.}}{d}}=\stackrel{\text{.}}{p}\frac{\partial F}{\partial \underline{f}}=\stackrel{\text{.}}{p}\frac{\underline{f}}{\text{feq}}\\ \stackrel{\text{.}}{p}=0\text{si}\text{feq}-R(p)<0\\ \stackrel{\text{.}}{p}=h\text{'}(\text{feq})\stackrel{\text{.}}{f}\text{eq}\text{si}\text{feq}-R(p)=0\end{array}\end{array}`

The rigid - plastic behavior relationship without elasticity is finally written:

:math:`\begin{array}{}\underline{\stackrel{\text{.}}{D}}=\frac{\stackrel{\text{.}}{p}}{\text{feq}}\left[\overline{D}\right]{\left[\overline{F}\right]}^{\text{-}1}\underline{F}\\ \text{où}\underline{D}=(\begin{array}{}U\\ \theta \end{array})\mathrm{et}\underline{F}=(\begin{array}{}{N}_{x}\\ {M}_{y}\end{array})\\ \left[\overline{D}\right]=\left[\begin{array}{cc}\overline{U}& 0\\ 0& \overline{\theta }\end{array}\right]\mathrm{et}\left[\overline{F}\right]=\left[\begin{array}{cc}\overline{N}& 0\\ 0& \overline{M}\end{array}\right]\end{array}`

The incremental behavior relationship in reduced quantities is obtained by integrating the continuous relationship between :math:`t` (variables -) and :math:`\text{t+ dt}` (variables +).

While loading, :math:`\mathrm{\Delta \; p}` checks :math:`F` = 0 to :math:`t` + :math:`\text{dt}`:


.. _RefEquation 2.2-1:

:math:`\begin{array}{c}{\text{feq}}^{\text{+}}=R({p}^{\text{-}}+\mathrm{\Delta \; p})\end{array}` eq 2.2-1

By introducing the behavioral relationship,


.. _RefEquation 2.2-2:

:math:`\begin{array}{c}\mathrm{\Delta }\underline{d}=\mathrm{\Delta \; p}\frac{{\underline{f}}^{\text{+}}}{{\text{feq}}^{\text{+}}}\end{array}` eq 2.2-2

we deduct the value of :math:`\mathrm{\Delta \; p}`,


.. _OLE_LINK1:

:math:`\begin{array}{c}\mathrm{\Delta \; p}=\parallel \mathrm{\Delta }\underline{d}\text{.}\mathrm{\Delta }\underline{d}\parallel =\sqrt{{\mathrm{\Delta \; U}}_{r}^{2}+{\mathrm{\Delta \; \theta }}_{r}^{2}}\end{array}`

and we calculate the value of :math:`{\text{feq}}^{\text{+}}` by [:ref:`éq 2.2-1 <éq 2.2-1>`]. The behavioral relationship [:ref:`éq 2.2-2 <éq 2.2-2>`] results in reduced efforts:

:math:`\begin{array}{c}\begin{array}{}{n}^{\text{+}}=\frac{{\mathrm{\Delta \; U}}_{r}}{\mathrm{\Delta \; p}}R({p}^{\text{-}}+\mathrm{\Delta \; p})\\ {m}^{\text{+}}=\frac{{\mathrm{\Delta \; \theta }}_{r}}{\mathrm{\Delta \; p}}R({p}^{\text{-}}+\mathrm{\Delta \; p})\end{array}\end{array}`

When unloading, :math:`\mathrm{\Delta \; p}` = 0 and we have by [:ref:`éq 2.2-2 <éq 2.2-2>`]:

:math:`\mathrm{\Delta }\underline{d}=0`