Assessing the complete solution set of the planar frictional wedging problem A. Pinto da Costa ⇑ Departamento de Engenharia Civil e Arquitectura and ICIST, Instituto Superior Técnico, Universidade Técnica de Lisboa, Avenida Rovisco Pais, 1049-001 Lisboa, Portugal article info Article history: Received 23 February 2010 Received in revised form 29 November 2010 Accepted 21 December 2010 Available online 30 December 2010 Keywords: Wedged equilibrium state Unilateral contact Coulomb friction Finite elements Complementarity abstract This paper examines the computation of all the (infinitely many) solutions of the frictional wedging prob- lem in the non-coercive context by an algorithm applied to its complementarity formulation. Our analysis offers insights regarding the mechanical and the geometrical meaning of the solution set. We find that the solution structure depends much on the value of the coefficient of friction. The evidence indicates that non-coercivity implies much longer computation times. Coefficients of friction at the onset of wedg- ing are computed and related. Ó 2010 Elsevier B.V. All rights reserved. 1. Introduction Jellet [23], Painlevé [31,32], Sparre [37], Lecornu [27], Delassus [8,9] and Béghin [4,5] are names associated with the discovery of and first discussions on possible ‘‘pathological behaviours’’ of the equations of static equilibrium and the equations governing the motion of mechanical systems in the presence of frictional obsta- cles. Since the discovery of ‘‘paradoxes’’ in the theoretical determi- nation of the equilibrium states or dynamic motions in systems composed of rigid bodies, numerous examples of indeterminacy (multiplicity of solution) and impossibility (absence of solution in the static case or absence of a continuous solution in the dynam- ical case) have been exhibited. Among those examples, one may cite (i) the negative effective stiffness along the direction tangent to the obstacle implying that, for the quasi-static sliding progres- sion of a particle, the applied tangent force must be opposite to the motion [25], (ii) the velocity discontinuities with or without collision occurring during the motion of a rigid body in contact with a rigid obstacle (our experience says that, if a sufficiently long piece of chalk is dragged on a blackboard with a certain inclination and sense, it is not possible to produce a continuous line on the blackboard, instead a pointed line is registered) [31,8,30,14], (iii) the intense noises coming from the self-excited vibrations gener- ated by the frictional contact of a pin on a disc [22] for certain ranges of parameters or (iv) the acoustic discomfort of the train passengers and the inhabitants of the regions near railways caused by the wheel-tail contact in close curves [12,15]. Klarbring and co-workers [24,1,25] invented frictional contact systems with a small number of degrees of freedom exhibiting solution non-uniqueness for the quasi-static incremental or rate problems, for sufficiently large coefficients of friction. Several stud- ies were also dedicated to the computation of multiple solutions to the static friction problem, generally with external loading: in [16,17,19–21] Hassani et al. and Hild use eigenvalue analyses to compute analytically and numerically branches with infinite solu- tions involving the slip of all the contact nodes. This article is devoted to a particular kind of solution non- uniqueness that the static equilibrium conditions of a linear elastic solid in the presence of frictional obstacles may exhibit: wedging. A wedged equilibrium state of a solid may be defined by a static equilibrium state involving a non-vanishing system of self- equilibrated (there are no external forces) frictional contact reac- tions from a given set of rigid obstacles. The wedging problem is connected with the static frictional contact problem (with external loading) exhibiting solution multiplicity: as concluded in [3], ‘‘the existence of distinct multiple solutions [of the static problem] involving slip also implies the existence of non-trivial solutions to the corresponding homogeneous (unloaded) problem’’. Early authors [37,27,31,4] referred to wedging as ‘‘arc-boutement statique’’ by opposition to ‘‘arc-boutement dynamique’’, that was reserved to dynamic solutions exhibiting percussions without impact which, in a very short time interval, manage to stop the rel- ative sliding velocity; for a modern discussion see, for example, [30,14]. Due to its practical importance, the subject of wedging has been receiving some attention. In order to compute the coefficient friction at the onset of wedging Barber and Hild [3] formulated a 0045-7825/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.cma.2010.12.019 ⇑ Fax: +351 21 849 7650. E-mail address: apcosta@civil.ist.utl.pt Comput. Methods Appl. Mech. Engrg. 205–208 (2012) 139–148 Contents lists available at ScienceDirect Comput. Methods Appl. Mech. Engrg. journal homepage: www.elsevier.com/locate/cma