Behavioral relationship of threaded assemblies#

Model DIS_GOUJ2E_PLAS equations#

They are deduced from 3D behavior VMIS_ISOT_TRAC [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 \(y\) direction.

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

The nonlinear behavior only applies to the local \(y\) direction.

Noting \(\Delta \varepsilon =\Delta {u}_{y}^{1}-\Delta {u}_{y}^{2}\) and \(\sigma ={F}_{y}^{1}={F}_{y}^{2}\).

Relationships are written in the same form as Von Mises 1D relationships [R5.03.09]:

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

In these expressions, p represents a « cumulative plastic displacement », and the isotropic work hardening function \(R(p)\) 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 [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 \(E\) which is the slope of this linear part (\(E\) is independent of temperature). The function \(R(p)\) 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]: \(F=f(u-v)\).

Relationship integration DIS_GOUJ2E_PLAS#

By direct implicit discretization of behavioral relationships, analogous to 1D integration [R5.03.09], we obtain:

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

There are two cases:

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

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

The resolution algorithm is deduced from this:

Let’s say \({\sigma }^{\text{e}}={\sigma }^{\text{-}}+E\Delta \varepsilon\)

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

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

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

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

so taking the absolute value:

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

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

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

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

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

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

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

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

  • otherwise \(\frac{\delta \sigma }{\delta \epsilon }=E\)

Internal variables#

Behavioral relationship DIS_GOUJ2E_PLAS produces two internal variables:

  • \(V1\): the « cumulative plastic displacement » \(p\), and

  • \(V2\): an indicator equal to \(1\) if the increase in plastic deformation is non-zero and \(0\) otherwise.