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:

  • \({d}_{e}\) limiting displacement of the elastic domain,

  • \({d}_{l}\) limiting displacement of the plastic domain,

  • \({K}_{\mathrm{el}}\) slope of the elastic domain,

  • \({K}_{\mathrm{pl}}\) slope of the plastic domain,

  • \({K}_{G}\) ultimate slope,

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

_images/1000104E00001F510000112816547A497DB92680.svg

Behavior of the law ARME#

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

\(\dot{{F}_{y}}=k\cdot \dot{{U}_{y}}\)

Where stiffness \(k\) varies according to the phase:

  • In the elastic phase and the discharge phase:

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

  • In the plastic phase:

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

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

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

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

Behavior integration

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

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

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

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

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

    • if \({F}^{\text{-}}+{K}_{\mathit{el}}\Delta {U}_{y}\mathrm{\ge }{K}_{\mathit{el}}\mathrm{\cdot }{d}_{e}\) and \({U}_{y}\mathrm{\le }{d}_{l}\), 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}hfillend {array}}} =left| {U} _ {U} _ {y}right|- {d} _ {e}hfillend {array}}}

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

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

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

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

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

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

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

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

Internal variables#

There is an internal variable:

  • \(V1\): Maximum value of the local displacement-limit displacement difference.