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In contact mechanics, a unilateral contact, also called unilateral constraint, denotes a mechanical constraint which prevents penetration between two rigid/flexible bodies; see figure 1a. This kind of constraints is omnipresent in non-smooth multibody dynamics applications, such as

• granular flows;
• Legged robot;
• vehicle dynamics;
• particle damping;
• imperfect joints;
• ...

Among these applications, it is almost the most important to model and simulation of the unilateral constraints.

Modeling of the unilateral constraints

There are mainly two kinds of method to model the unilateral constraints. The first kind of method is intherited from continuum mechanics, and the second kind of method is based on the non-smooth theory.

Continuum mechanics based method

Fig. 1. Hertz contact model

In this method, normal forces generated from the unilateral constraints are obtained according to the local material properties of bodies. In detail, contact force models are derived using the continuum mechanics, and expressed as functions of gap and the impact velocity between bodies, i.e. the models have the same underlying mathematical structure for unilateral contacts.The animation of classic Hertz contact law is shown in Fig. 1, where the contact is explained by the local defomation of bodies. And more contact models can refer to the introduction in contact mechanics, and some reviews.

Non-smooth theory based method

In the non-smooth method, unilateral interactions between bodies are fundamentally modelled by the Signorini condition for the non-penetration condition, and impact laws to define the impact process. The Signorini condition can be concluded as the complementarity problem: $g\geq 0,\lambda \geq 0,\lambda \perp g$, where $g$ means the distance between two bodies and $\lambda$ denotes the contact force generated by the unilateral constraints, as shown in Fig. 2b. Also, using the proximal point, the Signorini condition can be equivalently expressed as: $\lambda ={\rm {proj}}_{\mathbb {R} ^{+}}(\lambda -\rho g)$, where $\rho >0$ denotes an auxiliary parameter, and ${\rm {proj_{\bf {C}}()}}$ represents the proximal point of a set ${\bf {C}}$ to the variable. Both two expressions above represent the dynamic behaviors of unilateral constraints: when the normal distance $g$ is above zero, the contact is open, which means that there is no contact force between bodies, $\lambda =0$; while when the normal distance $g$ is equal to zero, the contact is closed, resulting in $\lambda \geq 0$.

Figure 2: a) unilateral contact, b) the Signorini graph, c) continuum mechanics based model

During the implementation of non-smooth theory based method, velocity Signorini condition, and acceleration Signorini condition are also employed sometimes. The velocity Signorini condition is expressed as: $U^{+}\geq 0,\lambda \geq 0,U^{+}\lambda =0$, where $U^{+}$means the normal velocity after impact. This velocity Signorini condition should be understood together with the Signorini $g\geq 0,\lambda \geq 0,\lambda \perp g$. The acceleration Signorini condition is considered under the close contact ($g=0,U^{+}=0$), as: ${\ddot {g}}\geq 0,\lambda \geq 0,{\ddot {g}}\lambda =0$, where the over dot means the derivative with repect to time.

Further reading

Books and articles

• Acary V., Brogliato B. Numerical Methods for Nonsmooth Dynamical Systems. Applications in Mechanics and Electronics. Springer Verlag, LNACM 35, Heidelberg, 2008.
• Brogliato B. Nonsmooth Mechanics. Communications and Control Engineering Series Springer-Verlag, London, 1999 (2dn Ed.)
• Glocker, Ch. Dynamik von Starrkoerpersystemen mit Reibung und Stoessen, volume 18/182 of VDI Fortschrittsberichte Mechanik/Bruchmechanik. VDI Verlag, Düsseldorf, 1995
• Glocker Ch. and Studer C. Formulation and preparation for Numerical Evaluation of Linear Complementarity Systems. Multibody System Dynamics 13(4):447-463, 2005
• Jean M. The non-smooth contact dynamics method. Computer Methods in Applied mechanics and Engineering 177(3-4):235-257, 1999
• Moreau J.J. Unilateral Contact and Dry Friction in Finite Freedom Dynamics, volume 302 of Non-smooth Mechanics and Applications, CISM Courses and Lectures. Springer, Wien, 1988
• Pfeiffer F., Foerg M. and Ulbrich H. Numerical aspects of non-smooth multibody dynamics. Comput. Methods Appl. Mech. Engrg 195(50-51):6891-6908, 2006
• Potra F.A., Anitescu M., Gavrea B. and Trinkle J. A linearly implicit trapezoidal method for integrating stiff multibody dynamics with contacts, joints and friction. Int. J. Numer. Meth. Engng 66(7):1079-1124, 2006
• Stewart D.E. and Trinkle J.C. An Implicit Time-Stepping Scheme for Rigid Body Dynamics with Inelastic Collisions and Coulomb Friction. Int. J. Numer. Methods Engineering 39(15):2673-2691, 1996
• Studer C. Augmented time-stepping integration of non-smooth dynamical systems, PhD Thesis ETH Zurich, ETH E-Collection, to appear 2008
• Studer C. Numerics of Unilateral Contacts and Friction -- Modeling and Numerical Time Integration in Non-Smooth Dynamics, Lecture Notes in Applied and Computational Mechanics, Volume 47, Springer, Berlin, Heidelberg, 2009

Open-source softwares

In most cases, friction among bodies should also be considered, and the A unilateral contact is usually associated with a gap function g which measures the distance between the two bodies and a contact force. The behaviour of a unilateral contact is modeled by a force law which states a relation between the gap function and the contact force. Set-valued force laws of type Upr assume a hard contact and clearly distinguish between open contact (contact force equal to zero, gap g is positive) and closed contact (contact force is positive, gap g is zero), see figure 1b. Regularized force laws are associated to compliance models. These laws write the contact force as function of the gap, i.e. the models have the same underlying mathematical structure for closed and open contacts. Unilateral contacts are used in contact dynamics and/or contact mechanics.

See also

• contact dynamics – Motion of multibody systems
• contact mechanics – Study of the deformation of solids that touch each other
• Siconos – Open source scientific software for modeling non-smooth dynamical systems