Discussion of: Tangential loading of general 3D contacts, Ciavarella, M. by J. Jäger, IBAMA, Goerdelerstr. 8, 91058 Erlangen, Germany, J. Appl. Mech., 1999, Vol. 66, 10481049, Copyright © 19962000 ASME International This page comprises an identical preprint of the discussion, and Dr. Ciavarella´s closure is appended at the end. Since it is rewritten in HTMLquality, the reader may consult the originials in printing quality for scientific purposes. email: j_jaeger@tonline.de Recently, Dr. Ciavarella (of University of Southampton ) published a paper in the J. Appl. Mech., where he presented a generalization of the well known CattaneoMindlin solution for equal halfspaces with arbitrary contact areas and Poisson's ratio n=0. We will shortly explain the background of this generalization, before we discuss some remarks. For vanishing Poisson's ratio, the normal and tangential stressdisplacement relations are uncoupled and identical, with exception of a constant factor. Then, each normal solution is also a tangential solution. The two boundary conditions for unidirectional tangential shift are a constant displacement of the stick area and Coulomb's law q=f p in the slip area, where q denotes the tangential traction, p the pressure and f the coefficient of friction. When the normal pressure p_{1} between two bodies in contact on the area C_{1} is increased to a new value p_{2}, the additional pressureDp=p_{2}p_{1} causes a constant displacement of the old contact area C_{1}. The increment Dp satisfies the first boundary condition for the tangential problem, when we assume that the old contact area is identical with the stick area and Coulomb's law can be satisfied by setting q=fp. Thus, q=fDp is the solution for the tangential problem. The side conditions of Coulomb's law are also satisfied, as discussed in the article. In the next paragraphs, we will discuss some points, which can mislead the reader. Some suggestions on future research are presented at the end. 1. The results for axisymmetric contact, summarized by Ciavarella in formulas (29)(33), have already been published by Jäger [1995a], and presented at the ASME summer meeting (Jäger, 1995b). Table 1 shows a comparison of identical equations by Ciavarella [1998a] and Jäger [1995a] Table 1 Identical equations by Ciavarella [1998a] and Jäger [1995a]
Ciavarella (29) (30) (31) (32) (33)
Jäger (32)&(15a) (33)&(14) (34) (32) (33)
Equation (31) by Ciavarella was written in the form Q*=fP(a*) by Jäger, [1995a], with c=a* in Ciavarella's notation. The general solution for plane contact was also published by Jäger [1997a], and some examples have been presented in Jäger [1997b]. Dr. Ciavarella requested and received Jäger's papers in April 30, 1998. Nevertheless, in a new publication from July 17th, 1998 (Barber & Ciavarella, 1999), Dr. Ciavarella wrote: "A significant generalization of CattaneoMindlin was discovered by Ciavarella ..... for any plane contact problem ...". Although this statement was corrected later, Ciavarella did not relate his axisymmetric solution to Jäger [1995a]. 2. "Flat regions are either entirely in full stick or are in full slip." This statement is only correct when all flat areas have the same height. Otherwise, only the lowest area is in initial contact. With increasing normal force, higher flat areas enter in contact, and the new contact areas change their form with the compression. When a tangential force is applied, the part of that area slides first, which was coming in contact latest, and the rest of that area sticks. Thus, partial slip of a flat area is possible. 3. For the direction of the slip traction, Ciavarella published the following formula (eq. 18 in Ciavarella, 1998a) (1) In the case of rigid bodies, the displacement disappears, i. e. u_{t}=0 and the direction of the slip traction can not be determined. To set this right, the slip traction must be opposite to the slip velocity ds/dt, when the slip s denotes the difference between the rigid body shift d_{t} and the elastic displacement u_{t}_{} (2) When the slip velocity is unidirectional, which was the case investigated by Ciavarella, the derivation is unnecessary q_{t}/ q_{t}+ s / s = 0. The slip disappears in the stick region, thus: u_{t}=d_{t} in the stick region. 4. Formula (23) seems not correct. The tangential force Q_{x} consists of two components: Q_{x}=f(PP*). The first component f P is the resultant of the full slip traction in the actual contact area. The other component:  f P*, is the resultant of the full slip traction in the stick area. With the distance d* between f P and f P*, the right formula for the distance d between Q_{x} and f P must be: (3) In contrast to Ciavarella's result, the distance for vanishing Q_{x} is d = d*/2. It is not necessary to calculate the point of application of the tangential load. It is sufficient to restrict the rotation around the contact normal. In the experiment, the support mechanism must be constructed appropriately and the numerical solution requires the displacement increments anyway (Jäger, 1992). For most contacts, the contact area is very small compared with the dimensions of the body, and the local moments in the contact area can be neglected. In this context, it should be noted that the generalization of the CattaneoMindlin solution requires nonrotating bodies, because the stick condition is always a constant displacement, when the bodies do not roll. Therefore, it is consequent to suppress all rotations, especially the rotation around the tangential axes, which appeared in Ciavarella's equation (5) and (6). It is worth remarking that a special case of transient rolling exists, where the CattaneoMindlin solution can also be applied. General problems with transient rolling can only be solved numerically. 5. At the beginning of Section 3, Ciavarella classified the decreasing stick zone as a 'receding' contact problem. This is in contrast to Dundurs' definition, who used the term for the case when the contact area decreases with increasing forces. In normal direction, the type of contact is 'advancing', because the contact area increases with the force. The tangential traction is the superposition of two advancing solutions. The slip zone increases with the tangential force, while the stick area and the corresponding component Q_{x}* decrease. It seems that the Cattaneo Mindlin generalization can not be used for receding contact problems. In fact, for a receding contact problem, the generalized CattaneoMindlin theory would suggest that the slip area decreases with increasing tangential force, which contradicts gross slip. 6. There are also contradictions in Ciavarella's other publications. Formula (A9) in Ciavarella [1998b] for the pressure of surfaces x^{k} has the wrong dimension and contains several typing errors. Furthermore, the hypergeometric function is singular for odd k. Jäger's formula (18), [1997b] should be used instead. A possible field for future research are more general structures. The necessary condition for the generalization of the CattaneoMindlin solution is that the stress displacement equations are equal, with the exception of a constant factor. This is the case for thin compressible layers in plane contact, and approximately for thick layers too (Jäger, 1999c). For equal bodies with thin layers of equal material and geometry, the normal and tangential coupling disappears. When the coupling between the tangential components is also neglected, the Winkler model is obtained as limiting case for very thin layers. This model allows solutions for very complex contact surfaces (Jäger, 1999b). This generalization can also be used for impact calculations, where only the rigid friction model has been solved to date. For very thin, incompressible layers, the tangential displacement is much larger than the normal displacement and Literature Ciavarella, M., 1998a, "Tangential loading of general threedimensional contacts", J. Appl. Mech., Vol. 65, 9981003. Ciavarella, M., 1998b, "The generalized Cattaneo partial slip plane contact problem, IIExamples", Int. J. Solids Structures, Vol. 35, 23632378. Jäger, J., 1992, "Elastic impact with friction", Thesis, Inst. Inf. and Maths., Delft, The Netherlands. Jäger, J., 1995a, "Axisymmetric bodies of equal material in contact under torsion or shift", Arch. Appl. Mech., Vol. 65, 478487. Jäger, J., 1995b, "Contact with friction of elastically similar bodies", Impact, Waves and Fracture, eds.: R.C. Batra, A.K. Mal, G. P. MacSithigh, AMDVol. 205, ASME, New York, ASME summer meeting, 129152. Jäger, J., 1997a, "Ein neues Prinzip der Kontakmechanik", Z. angew. Math. Mech., Vol. 7, S143S144, presented at the GAMMmeeting in April 1996. Jäger, J., 1997b, "Halfplanes without coupling under contact loading", Arch. Appl. Mech., Vol. 67, 247259. Jäger, J., 1999a, "Uniaxial deformation of a random packing of particles", Arch. Appl.Mech., Vol. 69, 181203. Jäger, J., 1999b, "Conditions for the generalization of CattaneoMindlin, presented at the GAMM meeting in April 1999, to be published in ZAMM 2000. Jäger, J., 1999c, "Equal layers in contact with friction", Int. Conf. Contact Mechanics 99, Stuttgart, August 1999, in: Contact Mechanics IV, eds.: L. Gaul, C.A. Brebbia, WITPress, Southampton, 99108. The author´s closure follows Author´s closure^{3}^{} JAM, Vol. 66, Dec. 1999, 10491050, Copyright © 19962000 ASME International First of all, I thank Dr. Jäger for his interest in my paper “Tangential loading of general Threedimensional contacts”. I confess I don´t understand very well the scope of Dr. Jäger´s “discussion”. At first I thought he was questioning the originality of my paper, but this is not the case as there is no previous paper where an extension to the Cattaneo´s solution for isotropic materials but general threedimensional geometry is given. Secondly, I thought Dr. Jäger was trying to improve the clarity of the presentation of my paper, by discussing what he calls the “background” of my generalization. He writes literally “For vanishing Poisson´s ratio, the normal and the tangential stressdisplacement relations are uncoupled and identical.... each normal solution is also a tangential solution. The two boundary conditions for unidirectional tangential shift are a constant displacement of the stick area and Coulomb´s law q=fp in the slip area”. The point is that I do not agree with this oversimplification of the contact problem, for the reason that the two problems of normal contact and of tangential contact are governed also by inequalities, and there is no reason a priori to think that the inequalities in the normal contact (basically the condition for no overlap outside the contact area and positive pressure in the contact area) are connected at all with the inequalities of Coulomb´s law, which Dr. Jäger seems to forget assuming a personal, reduced version of Coulomb´s law “q=fp in the slip area”. Therefore, also this scope is not reached, and I suggest the reader to refer to the original paper where the problem is approached with rigor. Having said this, he then proceeds to “discuss some points, which can mislead the reader”, to which I respond in order. 1 It may he true that some equations of his papers on axisymmetric contact and my paper where the axisymmetric case was just an example are similar, it remains the question on whether Dr. Jäger had proved them at all, given my remarks above about inequalities. As regarding my reading and referencing of his papers that I confirm I received from him in the course of a review paper written with Prof. J. R. Barber of University of Michigan, I invite Dr. Jäger to wait for when the review paper will be out in the press, not accusing me of omissions based on preliminary versions of the paper that I may have sent him in the course of this work, and which is largely different from the final version soon in print.^{4}^{} 2 For flat regions, I also assumed the same height. Other definitions of “flat” could be clearly used, and in the limit, every point is a flat region. I doubt this point was unclear in my text. 3 I agree his notation can be more general, but for the scopes of my paper I was happy with my Eq. (18) 4 Dr. Jäger here is correct in pointing to a misprint, as in my Eq. (23) a denominator (2Qx/fP) is missing as it can be obtained from elementary vector analysis. The results and the comments are not significantly altered. I don´t follow Dr. Jäger when he says “a special case of transient rolling exist, where the CM solution can also be applied”, without further reference. But this is not important for the scope of this discussion. 5 I refer to Dundurs´ reference obviously not in the section about receding contact areas, but receding stick areas in frictional problems. 6 This point is raised about my other publications and I will reply on the appropriate journals. Finally, Dr. Jäger proposes his “suggestion for further research” which look to me more likely to be his further research than mine, and concludes with publicity on his work on other generalizations of the “CM” theory, on which I don´t have much to say, apart that I hope the proofs are rigorous, i.e. the inequalities are included. In conclusion, it looks to me the whole of the “Discussion” comes down to the correction of the mispring in eq. (23). ^{3} M. Ciavarella, Primo Ricercatore, CNRIRIS, str. Crocefisso 2B, 70126 Bari, Italy. email: m.ciavarella@bari.ba.cnr.it ^{4} The paper is now in print. Please see Barber and Ciavarella (2000) in which Ciavarella wrote “A significant generalization of CattaneoMindlin was discovered by Ciavarella .... for any plane contact problem”. Although this statement was corrected later, Ciavarella did not relate his axisymmetric solution to Jäger (1995a).
I would like to remark here that Coulomb´s inequalities are proved in: “J. Jäger, A new principle in contact mechanics, J. Tribology, Vol. 120, 677685”, and others. The proof is a priori clear for a superposition of flat punches.
