.. _R5.03.17-DIS_GOUJ2E:

Behavioral relationship of threaded assemblies
====

Model DIS_GOUJ2E_PLAS equations
----
They are deduced from 3D behavior VMIS_ISOT_TRAC [:ref:`R5.03.02<r5.03.02>`]: we represent a behavioral relationship of the elastoplastic type with isotropic work hardening, linking the forces in the discrete element to the difference in displacement of the two nodes in the local :math:`y` direction.

In the local :math:`x` direction, the behavior is elastic, linear, and the coefficient relating the force :math:`\mathrm{Fx}` to the displacement :math:`\mathrm{Dx}` is the stiffness :math:`\mathrm{Kx}` provided via AFFE_CARA_ELEM.

The nonlinear behavior only applies to the local :math:`y` direction.

Noting :math:`\Delta \varepsilon =\Delta {u}_{y}^{1}-\Delta {u}_{y}^{2}` and :math:`\sigma ={F}_{y}^{1}={F}_{y}^{2}`.

Relationships are written in the same form as Von Mises 1D relationships [:ref:`R5.03.09<r5.03.09>`]:

:math:`\begin{array}{l}{\dot{e}}^{p}=\dot{p}\frac{s}{\mid s\mid }\\ s=E(e-{e}^{p})\\ {s}_{\text{eq}}-R(p)=\mid s\mid -R(p)\le 0\\ {s}_{\text{eq}}-R(p)<0\Rightarrow \dot{p}=0\\ {s}_{\text{eq}}-R(p)=0\Rightarrow \dot{p}\mathrm{³0}\end{array}`

In these expressions, p represents a "cumulative plastic displacement", and the isotropic work hardening function :math:`R(p)` is refined piecewise, given in the form of a force-displacement curve defined point by point, provided under the keyword factor TRACTION of the operator DEFI_MATERIAU [:ref:`U4.43.01<u4.43.01>`], provided under the keyword factor of the operator [].

The first point corresponds to the end of the linear domain, and is therefore used to define both the linearity limit (analogous to the elastic limit), and :math:`E` which is the slope of this linear part (:math:`E` is independent of temperature). The function :math:`R(p)` is deduced from a characteristic curve of the assembly (modeling of a few threads) expressing the force on the stud as a function of the difference in average displacement between the stud and the flange [bib1]_: :math:`F=f(u-v)`.


Relationship integration DIS_GOUJ2E_PLAS
----
By direct implicit discretization of behavioral relationships, analogous to 1D integration [:ref:`R5.03.09<r5.03.09>`], we obtain:

:math:`\begin{array}{l}E\Delta \varepsilon -\Delta \sigma =E\Delta p\frac{{\sigma }^{\text{-}}+\Delta \sigma }{\mid {\sigma }^{\text{-}}+\Delta \sigma \mid }\\ \mid {\sigma }^{\text{-}}+\Delta \sigma \mid -R({p}^{\text{-}}+\Delta p)\le 0\\ \mid {\sigma }^{\text{-}}+\Delta \sigma \mid -R({p}^{\text{-}}+\Delta p)<0\Rightarrow \Delta p=0\\ \mid {\sigma }^{\text{-}}+\Delta \sigma \mid -R({p}^{\text{-}}+\Delta p)=0\Rightarrow \Delta p\ge 0\end{array}`

There are two cases:

*   :math:`\mid {\sigma }^{\text{-}}+\Delta \sigma \mid <R({p}^{\text{-}}+\Delta p)` then :math:`\Delta p=0` is :math:`\Delta \sigma =E\Delta \varepsilon` so :math:`\mid {\sigma }^{\text{-}}+E\Delta \varepsilon \mid <R({p}^{\text{-}})`

*   :math:`\mid {\sigma }^{\text{-}}+\Delta \sigma \mid =R({p}^{\text{-}}+\Delta p)` so :math:`\Delta p\ge 0` so :math:`\mid {\sigma }^{\text{-}}+E\Delta \varepsilon \mid \ge R({p}^{\text{-}})`

The resolution algorithm is deduced from this:

Let's say :math:`{\sigma }^{\text{e}}={\sigma }^{\text{-}}+E\Delta \varepsilon`

If :math:`\mid {\sigma }^{\text{e}}\mid \le R({p}^{\text{-}})` then :math:`\Delta p=0` and :math:`\Delta \sigma =E\Delta \varepsilon`

If :math:`\mid {\sigma }^{\text{e}}\mid >R({p}^{\text{-}})` then you have to solve:

:math:`{\sigma }^{\text{e}}={\sigma }^{\text{-}}+\Delta \sigma +E\Delta p\frac{{\sigma }^{\text{-}}+\Delta \sigma }{\mid {\sigma }^{\text{-}}+\Delta \sigma \mid }`

:math:`{\sigma }^{\text{e}}=({\sigma }^{\text{-}}+\Delta \sigma )(1+\frac{E\Delta p}{\mid {\sigma }^{\text{-}}+\Delta \sigma \mid })`

so taking the absolute value:

:math:`\mid {\sigma }^{\text{e}}\mid =\mid {\sigma }^{\text{-}}+\Delta \sigma \mid (1+\frac{E\Delta p}{\mid {\sigma }^{\text{-}}+\Delta \sigma \mid })`

Or, using :math:`\mid {\sigma }^{\text{-}}+\Delta \sigma \mid =R({p}^{\text{-}}+\Delta p)`

:math:`\mid {\sigma }^{\text{e}}\mid =R({p}^{\text{-}}+\Delta p)+E\Delta p`

Taking into account the piecewise linearity of :math:`R(p)`, we can explicitly solve this equation to find :math:`\Delta p`

We deduce: :math:`\frac{{\sigma }^{\text{e}}}{\mid {\sigma }^{\text{e}}\mid }=\frac{\sigma }{R({p}^{\text{-}}+\Delta p)}`

So: :math:`\sigma ={\sigma }^{\text{-}}+\Delta \sigma =\frac{{\sigma }^{\text{e}}}{\mid {\sigma }^{\text{e}}\mid }R(p)=\frac{{\sigma }^{\text{e}}}{1+\frac{E\Delta p}{R(p)}}`

In addition, option FULL_MECA allows you to calculate the tangent matrix :math:`{K}_{i}^{n}` at each iteration. The tangent operator used to construct it is calculated directly on the previous discretized system. We obtain directly:

*   If :math:`\mid {\sigma }^{\text{e}}\mid >R({p}^{\text{-}})` then :math:`\frac{\delta \sigma }{\delta \epsilon }={E}_{t}=\frac{ER\text{'}(p)}{E+R\text{'}(p)}`

*   otherwise :math:`\frac{\delta \sigma }{\delta \epsilon }=E`


Internal variables
----
Behavioral relationship DIS_GOUJ2E_PLAS produces two internal variables:

*   :math:`V1`: the "cumulative plastic displacement" :math:`p`, and
*   :math:`V2`: an indicator equal to :math:`1` if the increase in plastic deformation is non-zero and :math:`0` otherwise.