Behaviors CHOC_ENDO and CHOC_ENDO_PENA#

These behaviors are dedicated to discrete elements like K_T_D_L. They make it possible to model shocks taking into account:

  • a threshold function, which limits the shock force during loading depending on the displacement,

  • damage to shock stiffness during loading,

  • the evolution of the « gap », due to repeated shocks,

  • the variable depreciation during the calculation.

The material that gives the characteristics is DIS_CHOC_ENDO [U4.43.01].

Attention:

Behavior CHOC_ENDO is implicit and is written in a manner similar to a law of plastic behavior. The CHOC_ENDO_PENA behavior is explicitly integrated and treats contact as a penalty.

Behavior Formulation CHOC_ENDO#

The formulation of the behavior is very similar to that found in perfect plasticity.

The shock force is limited by a threshold depending on the relative displacement of the discrete nodes.

So we have: :math: `f=left|left|stackrel {~} {{F}} _ {x}}right|-R (p) `

Where:

  • \(\stackrel{~}{{F}_{x}}\): is the effort in the discreet, with or without damping.

  • \(R(p)\): the threshold function limiting the effort of the discrete.

The behavior is described by the following equation:

With depreciation taken into account in the threshold

\(\stackrel{~}{{F}_{x}}={K}_{\mathit{nor}}(p).\left({U}_{x}–{U}_{x}^{p}\right)+{A}_{n}(p).{\dot{U}}_{x}\)

Without taking depreciation into account in the threshold

\(\stackrel{~}{{F}_{x}}={K}_{\mathit{nor}}(p).\left({U}_{x}–{U}_{x}^{p}\right)\)

Where:

  • \({K}_{\mathit{nor}}(p)\) represents the function of shock stiffness.

  • \({U}_{x}\), \({\dot{U}}_{x}\) represent respectively the displacement and the relative speed of the discrete nodes.

  • \({U}_{x}^{p}\) represents the anelastic displacement corresponding to the evolution of the game.

  • \({A}_{n}(p)\) represents the function of depreciation.

The formulation is part of the thermodynamics of irreversible processes. In this framework, the equations are written as follows:

if \(f\le 0\):

\({\dot{F}}_{x}={K}_{\mathit{nor}}(p).\left({\dot{U}}_{x}–{\dot{U}}_{x}^{p}\right)+{A}_{n}(p).{\ddot{U}}_{x}\) and \(\dot{\mathrm{p}}=0\).

if not:

\(f=0\) and \(\dot{f}=0\)

The plastic multiplier:

Taking into account depreciation in the threshold

Without damping in the threshold

\(\dot{p}=\frac{{K}_{\mathit{nor}}.{\dot{U}}_{x}+{A}_{n}.{\ddot{U}}_{x}}{R\text{'}+{K}_{\mathit{nor}}+k{\text{'}}_{\mathit{nor}}.({U}_{x}-{U}_{x}^{P})+A{\text{'}}_{n}.\dot{{U}_{x}}}\)

\(\dot{p}=\frac{{K}_{\mathit{nor}}.{\dot{U}}_{x}}{R\text{'}+{K}_{\mathit{nor}}+k{\text{'}}_{\mathit{nor}}.({U}_{x}-{U}_{x}^{P})}\)

In all cases the anelastic displacement is: \({\dot{U}}_{x}^{p}=\dot{p}\)

The increment of effort in the discreet:

\({\dot{F}}_{x}={K}_{\mathit{nor}}(p).({\dot{U}}_{x}-{\dot{U}}_{x}^{p})+K{\text{'}}_{\mathit{nor}}(p).\dot{p}.({U}_{x}-{U}_{x}^{P})+{A}_{n}(p).{\ddot{U}}_{x}+A{\text{'}}_{n}(p).\dot{p}.{\dot{U}}_{x}\)

Internal variables#

The four internal variables are:

  • \(V1\): cumulative plasticity \(p\).

  • \(V2\): anelastic displacement \({U}_{x}^{p}\).

  • \(V3\): the relative speed of the knots of the discrete \({\dot{U}}_{x}\).

  • \(V4\): the evolution of the game.

Behavior Formulation CHOC_ENDO_PENA#

The behavior is expressed explicitly and the shock is treated by penalization:

  • Shock force is a compression force, so it is negative when there is contact.

  • The indentations are negative.

  • Depreciation may be included or excluded when assessing the threshold.

    • if included: \({A}_{\text{in}}={A}_{n}\), \({A}_{\text{out}}=0\)

    • if excluded: \({A}_{\text{in}}=0\), \({A}_{\mathit{out}}={A}_{n}\)

A few notations:

  • \(\mathit{Dx}\): sinking.

  • \(\mathit{Dxm}\): maximum indentation \(\text{Sup}(\mathit{Dx})\).

  • \(\mathit{Dxr}\): residual or anelastic depression.

  • \({K}_{\mathit{nor}}\), \({A}_{n}\), \(\mathit{Seuil}\): are pressing functions.

In the initial state, the maximum depression and the residual depression are zero. They are internal variables in the law of behavior.

In elastic prediction, the effort is given by:

\({F}_{x}={K}_{\mathit{nor}}(\mathit{Dxm}).(\mathit{Dx}-\mathit{Dxr})\)

If \({F}_{X}>\mathit{Seuil}(\mathit{Dxm})\) then

\(\mathit{Dxr}=\mathit{Dx}+\mathit{Seuil}(\mathit{Dxm})/{K}_{\mathit{nor}}(\mathit{Dxm})\)

Consideration of depreciation when included in the threshold check.

\({F}_{x}={K}_{\mathit{nor}}(\mathit{Dxm}).(\mathit{Dx}-\mathit{Dxr})+{A}_{\text{in}}.\dot{D}x\)

If \({F}_{X}>\mathit{Seuil}(\mathit{Dx})\) then

\({F}_{X}=\mathit{Seuil}(\mathit{Dx})+{A}_{\mathit{out}}.\dot{D}x\)

If not

\({F}_{x}={K}_{\mathit{nor}}(\mathit{Dxm}).(\mathit{Dx}-\mathit{Dxr})+{A}_{n}.\dot{D}x\)

If \({F}_{x}>0\) then \({F}_{x}=0\). The effort can only be negative.

Internal variables#

The three internal variables are:

  • \(V1\): maximum indentation \(\mathit{Dxm}\).

  • \(V2\): residual indentation \(\mathit{Dxr}\).

  • \(V3\): load indicator (0 contact steps, 1 elastic contact, 2 on the threshold)

Definition of material parameters#

Here are the keywords used to define the parameters of material DIS_CHOC_ENDO [U4.43.01]:

  • FX makes it possible to give the threshold function of the shock force, which depends on the relative movements of the discrete nodes.

  • RIGI_NOR allows you to give the function of the normal shock stiffness, which depends on the relative displacements of the discrete nodes.

  • AMOR_NOR allows you to give the damping function, which depends on the relative displacements of the discrete nodes.

  • DIST_1 allows you to define the characteristic distance of material surrounding the first shock node.

  • DIST_2 allows you to define the characteristic distance of material surrounding the second shock node.

  • CRIT_AMOR allows amortization to be included or excluded in the threshold assessment.