Frictionless Contact

Contact mechanics, in the classical sense, describes the behavior of solids in contact under the action of an external load. The first studies in the area of contact mechanics date back to the seminal publication "On the contact of elastic solids of Heinrich Hertz in 1882 [ 1 ]. The original Hertz theory was applied to frictionless non-adhering surfaces of perfectly elastic solids. Lee and Radok [2], Graham [3], and Yang [4] developed the theories of contact mechanics of viscoelastic solids. None of these treatments, however, accounted for the role of interfacial adhesive interactions. 

During a collision, the colliding solids undergo both elastic and inelastic (or plastic) deformations. These deformations are caused by the changes of stresses and strains, which depend on the material properties of the solids and the applied external forces. Theories on the elastic deformations of two elastic bodies in contact are introduced in the literature utilizing Hertzian theory for frictionless contact and Mindlin s approach for frictional contact. As for inelastic deformations, few theories have been developed and the available ones are usually based on elastic contact theories. Hence, an introduction to the theories on elastic contact of solids is essential. 

In this section, an introduction of the general relations of stresses in equilibrium in an infinitely large solid medium is presented, followed by a special application where a concentrated force is acting on a point inside the solid. Also presented is the case of forces on the boundary of a semiinfinite solid medium, which is of importance to the contact of two solid objects. As consequences of the boundary compression, displacements due to the changes of stresses and strains in the region of contact can be linked to the contact force by the Hertzian theory for frictionless contacts and by Mindlin s theory for frictional contacts. For more details on the Hertzian theory for contact, interested readers may refer to books on elasticity [Goldsmith, 1960 Timoshenko and Goodier, 1970 Landau and Lifshitz, 1970]. 

When two elastic and frictionless spheres are brought into contact under compressional forces or pressures, deformation occurs. The maximum displacement and contact area depend not only on the compressional force but also on the elastic material properties and radii of the spheres. The contact between two elastic and frictionless spherical bodies under compression was first investigated by Hertz (1881) and is known as the Hertzian contact. 

Consider two frictionless spheres in contact as shown in Fig. 2.9. M is a point on the surface of sphere 1 with a distance r from the Z -axis. N is the opposite point on the surface of sphere 2 with the same distance r from the Z2-axis. O is the contact point and also serves as the origin of both Zi-axis and Z2-axis. N, M, and O are in the same plane. The distances from M or N to the tangential plane which is normal to the Z i -axis and Z2-axis are denoted by i and Z2, respectively. In Fig. 2.9, the triangle O2AO is similar to the triangle NBO. Hence, the theorem of similarity of triangles gives... 

Figure 2.10. Hertz pressure distribution for frictionless contact.

Collisions between particles with smooth surfaces may be reasonably approximated as elastic impact of frictionless spheres. Assume that the deformation process during a collision is quasi-static so that the Hertzian contact theory can be applied to establish the relations among impact velocities, material properties, impact duration, elastic deformation, and impact force. 

When the particle concentration is high, the shear motion of particles leads to interparticle collisions. The transfer of momentum between particles can be described in terms of a pseudoshear stress and the viscosity of particle-particle interactions. Let us first examine the transfer of momentum in an elastic collision between two particles, as shown in Fig. 5.8(a). Particle 1 is fixed in space while particle 2 collides with particle 1 with an initial momentum in the x-direction. Assume that the contact surface is frictionless so that the rebound of the particle is in a form of specular reflection in the r-x-plane. The rate of change of the x-component of the momentum between the two particles is given by... 

To model the elastic properties of dispersions of soft particles, we consider a dispersion of N spheres in a periodic box, as shown in Fig. 6. The particles are either monodisperse with radius R or polydisperscd with a Gaussian distribution around a mean radius R. The concentration of particles is above the random close-packed volume fraction of 0c = 0.64 so that the particles are jammed together and form facets at contact. The contacts are assumed to be purely repulsive and frictionless and hence exert only a normal repulsive force at contact. The total elastic energy stored in the structure is the summation of the pairwise contact energies. Even at the highest volume fraction at near-equilibrium conditions, i.e., without flow, deformation of a particle is no more than 10% of its radius. Thus, the particle deformation is small compared to the size of the undeformed sphere and the contacts obey the Hertzian contact potential given by (1). 

Another technique widely used in the estimation of the properties of cells and their components is atomic force microscopy, where the sample is probed by a sharp tip located at the end of a cantilever of a prescribed stiffness, and the corresponding indentation is tracked with a laser. The force/indentation relationship is a characterization of the cell (cellular component) properties. A traditional interpretation of this experiment is based on Hertz theory of a frictionless contact of a rigid tip with an elastic isotropic half-space [Radmacher et al, 1996]. The finite thickness of the cell can be taken into account by considering an elastic layer adhered to a substrate. More details of cell geometry and rheology can be considered by using the finite element method. 

For this example, we will initially assume that the tip of the manipulator is already in motion relative to the contact surface (slipping). The coefficient of friction is finite. In this case, we may assume that the contact forces applied in the directions of motion are already laige enough to overcome static friction. We will also examine the same ccxitact when the coefficient of friction is negligible (frictionless surface), and when the manipulatcx tip is not slipping on the surface. 

If the coefficient of friction, //, is negligible (frictionless surface), then this contact becomes a Class I contact That is, if... 

In a recent paper. Hall and Savage (1987) point out that every E.H.L. problem has an associated dry frictionless contact problem with identical surface displacement v(x) and pressure P(x) related by equation [5]. They provide a detailed derivation of the classical Poritsky Solution [2] for dry, frictionless contacts which constitutes an exact solution of the elasticity problem given by equation (5). Writing... 

If the clearances and pressures in a highly deformed soft contact, with a constant entrainment velocity, are examined, it may be seen that over the majority of the contact the pressure distribution lies close to that found under dry, frictionless conditions. Similarly, the clearance under the contact, see Fig. 1, is nearly uniform and well outside the contact is only changed slightly from the dry profile. The non-dimensional clearance, H, used in that figure is that of ref. [2]. The only significant departures from the dry contact... 

Solutions for simplified cases in which it is assumed that plane sections of the coating remain plane after indentation-induced compression, can be found in the book by Johnson [75]. Johnson s approach was subsequently extended by Jaffar [76]. An early work [24] assumed frictionless contact and used elastic continuum mechanics to estimate the stress distribution in layered materials. In this analysis, force and displacement continuity at the interfaces was satisfied exactly, while the surface displacement condition (profile of the indenter), was only approximated. These authors analyzed indentation stresses created by circular flat-ended and parabolic indenters numerically on both single and multilayer systems, with good results. 

The common assumption of contact mechanics is that the two surfaces in contact are frictionless, so that shear stresses cannot be sustained at the interface. 

This frictionless assumption is often appropriate for very stiff materials where adhesive forces are relatively unimportant, but it is often not the case for softer materials such as elastomers, where adhesive forces play a very important role. In these cases, a full-friction boundary condition, where sliding of the two surfaces is not allowed, is often more appropriate, In many important cases (contact of a very thick, incompressible elastic layer, for example) there is little or no practical difference for these two boundary conditions. Nevertheless, in the discussion that follows, we are careful to indicate that boundary condition (frictionless or full-friction) that formally applies in each case. In all cases we assume that the contacting materials are isotropic and homogeneous, each being characterized by two independent elastic constants. 

Our notation is to refer to these non-adhesive values of the load and displacement as P and (5, respectively. Actual values of the load and displacement will generally differ from these values when the contacting surfaces adhere to one another. These differences are quantified in subsequent sections. Barquins has given values of P for a variety of other frictionless, axisymmetric half spaces, including rounded and flat-ended cones, etc. [4]. The reader is referred to this reference for details. 

The compliance based treatment of contact mechanics is useful because it relies only on directly measurable quantities, these being the load, displacement and contact radius. It does not rely on information relating to the detailed stress distribution within the material. Often, however, it is useful to have more information about this stress distribution. Analytic theories of non-adhesive contact generally assume that the contact is frictionless, so that the interface is not able to support shear stresses. The Hertzian form of the radial distribution of the normal stress, or contact pressure, for frictionless contact is given by the following expression [1 ] ... 

In the adhesive case tensile loads can be supported, so that the actual load P is generally less than P. An additional stress field arises that is proportional to the difference, P — P. For frictionless contact with a/h = 0, this additional normal stress, referred to here as is given by the following expression [3] ... 

The expressions given above are valid only for frictionless contacts in the limit where a/h is very small. Additional information about the stress distribution... 

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