(Difference between revisions)
 Revision as of 11:59, 2 May 2008 (view source) (New page: == General description of the condition == This conditions represents the main part of the algorithm to solve contact problems. The algorithm is based on a surface to surface penalty formu...) Revision as of 12:19, 2 May 2008 (view source) (→Formulation of the two body contact problem)Newer edit → Line 8: Line 8: [[Image:Structural Appl Contact Figure 2.gif|thumb|right|300px|Slave $\Gamma_1$ and master $\Gamma_2$ contact surface, and gap $g=\nu\cdot\left(\boldsymbol{y}_2-\boldsymbol{y}_1\right)$ between the surfaces]] [[Image:Structural Appl Contact Figure 2.gif|thumb|right|300px|Slave $\Gamma_1$ and master $\Gamma_2$ contact surface, and gap $g=\nu\cdot\left(\boldsymbol{y}_2-\boldsymbol{y}_1\right)$ between the surfaces]] [[Image:Structural Appl Contact Figure 3.gif|thumb|right|300px|Three dimensional view on the contact surfaces and the gap between the two surfaces]] [[Image:Structural Appl Contact Figure 3.gif|thumb|right|300px|Three dimensional view on the contact surfaces and the gap between the two surfaces]] + The contact of two (or more) bodies in three dimension is a constrained continuum mechanical problem. The surface $\partial\Omega =\Gamma$ of the contacting continua  $\Omega_1$ and $\Omega_2$ are constructed by the Neumann boundary $\Gamma^{\sigma}$, the Dirichlet + boundary $\Gamma^{u}$ and the contact boundary $\Gamma^c$. For the treated area $\Omega$ the following assumptions hold: + + *$\Omega = \Omega_1 \cup \Omega_2$ and $\Omega_1 \cap \Omega_2=0$, + + *having the boundaries $\partial\Omega=\Gamma^{\sigma}_i\cup\Gamma^{u}_i\cup\Gamma^{c}_i$ and $\Gamma^{\sigma}_i\cap \Gamma^{u}_i=0$, $\Gamma^{\sigma}_i\cap \Gamma^{c}_i=0$, $\Gamma^{u}_i\cap \Gamma^{c}_i=0$ + + The problem is constrained by the prescription of a penetration of the two bodies, e.g. of the penetration of the master surface $\Gamma^2$ by the slave surface $\Gamma^1$. This constrained is expressed by the gap function defined as the scalar projection of the distance vector between the point on the slave surface $\mathbf{x}\in \Gamma^1$ and its closest point projection onto the slave surface $\mathbf{y}(\mathbf{x})=\mbox{arg}\;\mbox{min}_{\mathbf{y}\epsilon\Gamma_2^c}\| \mathbf{x}-\mathbf{y}\left(\mathbf{x}\right)\|$  and the normal vector $\nu$ on the master surface: + *$g\left(\mathbf{x}\right)=-\mathbf{\nu}\left(\mathbf{x}-\mathbf{y}\right)$ + :::$=-\mathbf\nu\left(\left[\mathbf{X}+\mathbf{u}_1\right]-\left[\mathbf{Y}-\mathbf{u}_2\right]\right)$ + :::$=g_0-\mathbf\nu\left(\mathbf{u}_1-\mathbf{u}_2\right)$ + The constrained that defines the contact problem is now stated as: + *$g(x)\leq 0$ + Within the finite element formulation the problem is now solved by use of an penalty treatment of the contact constrained. To solve the problem the Energie of the system must be minimised by fulfilling the penetration constraint. After having introduced a contact stress $t_N$ that acts on the surfaces if they are in contact the constraint can be rewritten in form of Kuhn-Tucker constraints: + *$g\leq 0\;\;\;\;\;\;t_N\geq 0\;\;\;\;\;\;g t_N=0\;\;\;\forall\;\mathbf{x}\epsilon\Gamma^c$ + The penalty method now penalizes violation of the constraint, leading to the following potential: + *$\Pi^c=\sum_{i=1}^2\int_{\Gamma_c^{i}}<\epsilon_Ng>^2d\Gamma^i$ + The minimization of the potential is now been done by its variation: + *$\delta W^c=\sum_{i=1}^2\int_{\Gamma_c^{i}}\delta g\epsilon_Nd\Gamma^i$ + Together with the minimisation of the system energy + *$\delta W^{int, ext}=\sum_{i=1}^2\delta W_i^{int, ext}$ + the minimisation of the system energie with fulfillment of the penetration constraint can be written as + *$\delta W=\delta W^{int, ext}+\delta W^c=0$ + This equation states a initial boundary value problem and can be solved by normal discretization methods in time and space. + ==References== ==References== * T.A. Laursen ''Computational Contact and Impact Mechanics'', Springer, 2002 * T.A. Laursen ''Computational Contact and Impact Mechanics'', Springer, 2002

## General description of the condition

This conditions represents the main part of the algorithm to solve contact problems. The algorithm is based on a surface to surface penalty formulation using a master and a slave contact surface. It is formulated to solve frictional contact problems in structural mechanics, the algorithm is formulated in a finite deformation context.

This condition and the underlying algorithm for contact problems has not been tested by means of a broader range of benchmark examples so far. It seem tio work quite fine for a number of different examples, but I do not give a guarantee that it is free of errors in its formulation or its implementation.

## Formulation of the two body contact problem

Two bodies Ωi staying in contact over a contact surface $\Gamma^c_i$
Slave Γ1 and master Γ2 contact surface, and gap $g=\nu\cdot\left(\boldsymbol{y}_2-\boldsymbol{y}_1\right)$ between the surfaces
Three dimensional view on the contact surfaces and the gap between the two surfaces

The contact of two (or more) bodies in three dimension is a constrained continuum mechanical problem. The surface $\partial\Omega =\Gamma$ of the contacting continua Ω1 and Ω2 are constructed by the Neumann boundary Γσ, the Dirichlet boundary Γu and the contact boundary Γc. For the treated area Ω the following assumptions hold:

• $\Omega = \Omega_1 \cup \Omega_2$ and $\Omega_1 \cap \Omega_2=0$,
• having the boundaries $\partial\Omega=\Gamma^{\sigma}_i\cup\Gamma^{u}_i\cup\Gamma^{c}_i$ and $\Gamma^{\sigma}_i\cap \Gamma^{u}_i=0$, $\Gamma^{\sigma}_i\cap \Gamma^{c}_i=0$, $\Gamma^{u}_i\cap \Gamma^{c}_i=0$

The problem is constrained by the prescription of a penetration of the two bodies, e.g. of the penetration of the master surface Γ2 by the slave surface Γ1. This constrained is expressed by the gap function defined as the scalar projection of the distance vector between the point on the slave surface $\mathbf{x}\in \Gamma^1$ and its closest point projection onto the slave surface $\mathbf{y}(\mathbf{x})=\mbox{arg}\;\mbox{min}_{\mathbf{y}\epsilon\Gamma_2^c}\| \mathbf{x}-\mathbf{y}\left(\mathbf{x}\right)\|$ and the normal vector ν on the master surface:

• $g\left(\mathbf{x}\right)=-\mathbf{\nu}\left(\mathbf{x}-\mathbf{y}\right)$
$=-\mathbf\nu\left(\left[\mathbf{X}+\mathbf{u}_1\right]-\left[\mathbf{Y}-\mathbf{u}_2\right]\right)$
$=g_0-\mathbf\nu\left(\mathbf{u}_1-\mathbf{u}_2\right)$

The constrained that defines the contact problem is now stated as:

• $g(x)\leq 0$

Within the finite element formulation the problem is now solved by use of an penalty treatment of the contact constrained. To solve the problem the Energie of the system must be minimised by fulfilling the penetration constraint. After having introduced a contact stress tN that acts on the surfaces if they are in contact the constraint can be rewritten in form of Kuhn-Tucker constraints:

• $g\leq 0\;\;\;\;\;\;t_N\geq 0\;\;\;\;\;\;g t_N=0\;\;\;\forall\;\mathbf{x}\epsilon\Gamma^c$

The penalty method now penalizes violation of the constraint, leading to the following potential:

• $\Pi^c=\sum_{i=1}^2\int_{\Gamma_c^{i}}<\epsilon_Ng>^2d\Gamma^i$

The minimization of the potential is now been done by its variation:

• $\delta W^c=\sum_{i=1}^2\int_{\Gamma_c^{i}}\delta g\epsilon_Nd\Gamma^i$

Together with the minimisation of the system energy

• $\delta W^{int, ext}=\sum_{i=1}^2\delta W_i^{int, ext}$

the minimisation of the system energie with fulfillment of the penetration constraint can be written as

• δW = δWint,ext + δWc = 0

This equation states a initial boundary value problem and can be solved by normal discretization methods in time and space.

## References

• T.A. Laursen Computational Contact and Impact Mechanics, Springer, 2002