.. _R5.03.17-ARME:

Behavior ARME
====

Definition
----
Behavior ARME is used to model the behavior of airline armament. The arm of each broken phase weapon, represented by a discrete element, has a non-linear force-displacement behavior.

The law of non-linear behavior only applies in the local direction y and is defined by the following parameters:

*   :math:`{d}_{e}` limiting displacement of the elastic domain,
*   :math:`{d}_{l}` limiting displacement of the plastic domain,
*   :math:`{K}_{\mathrm{el}}` slope of the elastic domain,
*   :math:`{K}_{\mathrm{pl}}` slope of the plastic domain,
*   :math:`{K}_{G}` ultimate slope,

It relates the differential displacement :math:`{U}_{y}` (local coordinate system) between the two nodes of the discrete element and the force to the nodes :math:`{F}_{y}` (local coordinate system).


.. figure:: images/1000104E00001F510000112816547A497DB92680.svg
    :width: 80%
    :align: center

    Behavior of the law ARME


For the three response areas identified, the law of behavior is expressed by:

:math:`\dot{{F}_{y}}=k\cdot \dot{{U}_{y}}`

Where stiffness :math:`k` varies according to the phase:

*   In the elastic phase and the discharge phase:

    :math:`k\mathrm{=}{K}_{\mathit{el}}` if :math:`{U}_{y}<{d}_{e}` or :math:`{U}_{y}<{d}_{l}\text{et}\mathrm{\Delta }{U}_{y}/{U}_{y}<0`

*   In the plastic phase:

    :math:`k\mathrm{=}{K}_{\mathit{pl}}` if :math:`{U}_{y}<{d}_{l}\text{et}\mathrm{\Delta }{U}_{y}/{U}_{y}>0`

*   In the limit phase or discharge on the limit curve:

    :math:`k\mathrm{=}{K}_{G}` if :math:`{U}_{y}>{d}_{l}` or :math:`{V}_{1}\ge ({d}_{l}-{d}_{e})\text{et}\mathrm{\Delta }{U}_{y}/{U}_{y}<0`

Where :math:`{V}_{1}` is the internal variable corresponding to the maximum difference between the displacement and the limiting displacement :math:`{V}_{1}\mathrm{=}∣{U}_{y}∣\mathrm{-}{d}_{e}`.


**Behavior integration**

The integration of the law of behavior respects the following equations:

*   If we are in charge :math:`\Delta {U}_{y}\mathrm{/}{U}_{y}>0`, we determine if we are on the elastic, plastic or limit curve, according to the value of the internal variable :math:`{V}_{1}^{\text{-}}` at the previous step.

    *   if :math:`{V}_{1}^{\text{-}}<({d}_{l}-{d}_{e})`, we test the new position

    *   if :math:`∣{F}^{\text{-}}∣+{K}_{\mathit{el}}∣\Delta {U}_{y}∣\mathrm{\le }{K}_{\mathit{el}}\mathrm{\cdot }{d}_{e}`, we are on the elastic curve and therefore:

        :math:`\mathrm{\{}\begin{array}{c}{k}_{t}\mathrm{=}{K}_{\mathit{el}}\\ {V}_{1}^{\text{+}}\mathrm{=}{V}_{1}^{\text{-}}\end{array}`

    *   if :math:`{F}^{\text{-}}+{K}_{\mathit{el}}\Delta {U}_{y}\mathrm{\ge }{K}_{\mathit{el}}\mathrm{\cdot }{d}_{e}` and :math:`{U}_{y}\mathrm{\le }{d}_{l}`, we are on the plastic curve and therefore:

        :math: `\ {\ begin {array} {c} {c} {k} _ {k} _ {t} =\ frac {{K} _ {el}}\ cdot {d} _ {e} + {K} _ {K} _ {K} _ {e} _ {e} _ {e} _ {e}) -\ left| {e} _ {e}) -\ left| {e}) -\ left| {e}) -\ left| {e}) -\ left| {e}) -\ left| {f}} ^ {\ text {-}}\ right|} {\ left|} {\ left|\ mathrm {\ Delta} {U} {U}\ right|}\\ {V} _ {1} ^ {\ text {+}}} =\ left| {\ left| {U} _ {U} _ {y}\ right|- {d} _ {e}\ hfill\ end {array}}} =\ left| {U} _ {U} _ {y}\ right|- {d} _ {e}\ hfill\ end {array}}}

    *   if :math:`{F}^{\text{-}}+{K}_{\mathit{el}}\Delta {U}_{y}\ge {K}_{\mathit{el}}\cdot {d}_{e}` and :math:`{U}_{y}>{d}_{p}`, we are on the limit curve:

        :math:`\mathrm{\{}\begin{array}{c}{k}_{t}\mathrm{=}\frac{{K}_{\mathit{el}}\mathrm{\cdot }{d}_{e}+{K}_{\mathit{pl}}\mathrm{\cdot }{d}_{l}+{K}_{G}\mathrm{\cdot }(∣{U}_{y}∣\mathrm{-}{d}_{l})\mathrm{-}∣{F}^{\text{-}}∣}{∣\Delta {U}_{y}∣}\\ {V}_{1}^{\text{+}}\mathrm{=}∣{U}_{y}∣\mathrm{-}{d}_{e}\end{array}`

    *   if :math:`{V}_{1}^{\text{-}}\mathrm{\ge }({d}_{l}\mathrm{-}{d}_{e})`, we are on the limit curve:

        :math:`\mathrm{\{}\begin{array}{c}{k}_{t}\mathrm{=}{K}_{G}\\ {V}_{1}^{\text{+}}\mathrm{=}{d}_{l}\mathrm{-}{d}_{e}\end{array}`

*   If you are in unloads :math:`\Delta {U}_{y}\mathrm{/}{U}_{y}<0`, you determine if you are on the elastic or limit curve, according to the value of the internal variable :math:`{V}_{1}^{\text{-}}` at the previous step.

    *   if :math:`{V}_{1}^{\text{-}}<({d}_{l}\mathrm{-}{d}_{e})`, we unload according to the elastic curve i.e. :math:`{k}_{t}\mathrm{=}{K}_{\mathit{el}}` and :math:`{V}_{1}^{\text{+}}\mathrm{=}{V}_{1}^{\mathrm{-}\text{}}`.

    *   if :math:`{V}_{1}^{\text{-}}\mathrm{\ge }({d}_{l}\mathrm{-}{d}_{e})`, we discharge according to the limit curve i.e. :math:`{k}_{t}\mathrm{=}{K}_{G}` and :math:`{V}_{1}^{\text{+}}\mathrm{=}{d}_{l}\mathrm{-}{d}_{e}`.

The strength in the element is then expressed by :math:`{F}^{\text{+}}\mathrm{=}{F}^{\text{-}}+{k}_{t}\Delta {U}_{y}`.


Internal variables
----
There is an internal variable:

*   :math:`V1`: Maximum value of the local displacement-limit displacement difference.