Shock and friction modeling DIS_CONTACT#

Behavior DIS_CONTACT reflects contact with shock and friction between two structures, via two types of relationships:

  • the unilateral contact relationship which expresses the non-inter-penetrability between solid bodies,

  • the friction relationship that governs the variation of tangential forces in contact. For the present developments, a simple relationship will be retained: Coulomb’s law of friction.

lookout

This behavior is integrated with a direct implicit time schema and is written as a classical law of behavior. The stiffness and damping parameters that are involved in this behavior are therefore « physical ».

Recommendation

In the case where the stiffness is no longer « physical » but rather « penalizing », it is desirable to use behavior DIS_CHOC, which has the same material parameters and whose relationship is integrated explicitly by treating contact and friction by penalization.

Unilateral contact relationship#

_images/10000201000004EF000002762EC438E1833E9A02.png

Defining distances for relationship DIS_CHOC.#

Definition of contact distance#

We define the contact distance \(d_c\). In the case of a discrete with 2 knots of length \(l_d\)

\[\DeclareMathOperator{\dist}{dist} \DeclareMathOperator{\sgn}{sgn} d_c = {l}_d–\dist_1–\dist_2\]

In the case of a discrete with 1 node:

\[d_c = \text{jeu}-\dist_1\]

The contact distance is positive in the direction of the local axis \({X}_{\text{loc}}\). To establish initial contact, it is therefore necessary to choose the values of the parameters wisely. \(\dist_1\), \(\dist_2\), and \(\text{jeu}\)

Formulation of behavior#

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

  • Friction is represented by a threshold describing nonlinear behavior.

  • The elastic stiffness along the local axes \({K}_{x},{K}_{y},{K}_{z}\).

Under these conditions:

()#\[f = \sqrt{ ({F}_{cy}^{2} + {F}_{cz}^{2})} - {\mu} \, \vert {F}_{x} \vert\]

Where:

  • \({F}_{cy}\) and \({F}_{cz}\): Coulomb’s tangent efforts..

  • \({F}_{x}\): normal effort.

  • \({\mu}\): Coulomb coefficient of friction.

The behavior is described by the following equations:

()#\[\begin{split}\left \{ \begin{array}{ll} {\dot{F}}_{cy}={K}_{T}({\dot{U}}_{y}-{\dot{U}}_{y}^{p})+{C}_{T}{\gamma}_{y} \\ {\dot{F}}_{cz}={K}_{T}({\dot{U}}_{z}-{\dot{U}}_{z}^{p})+{C}_{T}{\gamma}_{z} \end{array} \right.\end{split}\]

The efforts in the discreet are:

\[\begin{split}\left \{ \begin{array}{ll} {\dot{F}}_{x}={K}_{N}{\dot{U}}_{x}+{C}_{N}{\gamma}_{x} \\ {\dot{F}}_{y}={K}_{y}{\dot{U}}_{y}+{\dot{F}}_{cy} \\ {\dot{F}}_{z}={K}_{z}{\dot{U}}_{z}+{\dot{F}}_{cz} \end{array} \right.\end{split}\]

If \(f\le 0\):

\[\dot{\lambda}=0\]

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

\[\begin{split}\left \{ \begin{array}{ll} {\dot{U}}_{y}^{p}=\dot{{\lambda}}\frac{\partial f}{\partial {F}_{y}} \\ {\dot{U}}_{z}^{p}=\dot{{\lambda}}\frac{\partial f}{\partial {F}_{z}} \end{array} \right.\end{split}\]

with

\[F_t = \sqrt{({F}_{cy}^{2}+{F}_{cz}^{2})}\]

and

\[\dot{\lambda} = \frac{{F}_{cy}}{{F}_{t}}\,{\dot{U}}_{y} + \frac{{F}_{cz}}{{F}_{t}}\,{\dot{U}}_{z} - \frac{{K}_{N}}{{K}_{T}} \, \mu {\dot{U}}_{x} \, \sgn({F}_{x})\]

Coulomb friction is governed by behavior equations () and (). An example of a loading path is shown in Figure r5.03.17-fig-EvolutionFrottementCoulomb.

_images/frottement_coulomb.png

Evolution of Coulomb friction.#

The loading path is as follows:

  1. loading with stiffness RIGI_TAN up to the Coulomb threshold.

  2. sliding on the threshold.

  3. unloading with stiffness RIGI_TAN up to the threshold.

  4. sliding on the threshold.

  5. charging up to the threshold.

  6. sliding on the threshold.

Local integration of behavior#

The scheme used is that of Runge-Kutta of order 5 with error control. Figure RK5 is a combination of the 4th order diagram and the 6th order diagram [bib9], for which the coefficients are determined so as to minimize the error between order 4 and order 6.

The local integration of this model uses the 5th order RUNGE - KUTTA schema.

The error between orders 4 and 5 is known, so it is possible to trigger local subdivisions of the time step if the convergence criterion is not met.

Defining contact parameters#

Here are the keywords used to define the parameters of contact, damping and friction in DEFI_MATERIAU [U4.43.01]:

  • The RIGI_NOR operand allows you to give the value of the normal shock stiffness \({K}_{N}\).

  • The AMOR_NOR operand allows you to give the value of the normal shock absorption \({C}_{N}\).

  • The RIGI_TAN operand allows you to give the value of the tangential stiffness \({K}_{T}\).

  • The AMOR_TAN operand allows you to give the value of the tangential shock damping \({C}_{T}\).

  • The COULOMB operand is used to give the value of the Coulomb coefficient.

  • The DIST_1 operand defines the characteristic distance of matter surrounding the first shock node

  • The DIST_2 operand defines the characteristic distance of material surrounding the second shock node (shock between two mobile structures).

  • The operand JEU defines the distance between the shock node and an unmodelled obstacle (case of a shock between a mobile structure and an undeformable and immobile obstacle).

  • Operand EVOLJEU which is a function of the multiplicative time of JEU. This function makes it possible to progressively establish the contact distance, as defined above.

Internal variables#

There are nine of them:

  • \(V1\): component of the tangent force along the local axis \(y\): \({F}_{cy}\).

  • \(V2\): component of the tangent force along the local axis \(z\): \({F}_{cz}\).

  • \(V3\): displacement due to sliding in the tangential plane along the local \(y\) axis.

  • \(V4\): displacement due to sliding in the tangential plane along the local \(z\) axis.

  • \(V5\): speed along the local \(x\) axis.

  • \(V6\): speed along the local \(y\) axis.

  • \(V7\): speed along the local \(z\) axis.

  • \(V8\): management of the initial interpenetration of the discrete.

  • \(V9\): initial contact management.