by J. Jäger, Lauterbach Verfahrenstechnik, Postfach 711 117, D-76338 Eggenstein, Germany email: j_jaeger@t-online.de Copyright © 1999-2001 Elsevier Science. Preprint for Int. J. Solids & Structures, Feb. 2001, vol. 38, issue 14, pages: 2453-2457. Version: May, 2000
In the present note, it is shown that Ciavarella's calculation of the tangential tractions, forces and displacements is not necessary. Furthermore, the example of wedge and conical indenters can be written as a superposition of a Hertzian and a flat profile with rounded corners. The solution for flat rounded profiles was published by Schubert (1942). In the first section of this note, Jäger's theory is shortly summarized. Some comments on Ciavarella's theory for axisymmetric contact follow in Section 2 and the last section treats Ciavarella's theory for plane contact. In order to keep the present note short, we discuss only the surface traction and displacements for a single contact area. Ciavarella's examples for multiple plane contact and the sub-surface stress for axisymmetric contact have not been examined. More remarks can be found in another discussion by Jäger (1999c).
Historically, the first generalization of Cattaneo-Mindlin was published by Jäger (1995) for axisymmetric contact. From that publication follows the generalization for plane elasticity with multiple contact areas, which was presented at the GAMM-meeting in April 1996 (Jäger, 1997a). It was also shown by Jäger (1999a,b) that this theory can be used for thin and thick layers and some conditions for the generalization of Cattaneo-Mindlin have been presented at the GAMM-meeting (1999d). In order to simplify Ciavarella's solution, we will summarize the theory by Jäger shortly. For two bodies in contact, the combined displacements are the sum of the displacements of each body. The contact condition for a normal pressure increment is a constant combined displacement of the initial contact area. The stick condition in tangential direction is also a constant rigid body displacement in the stick area. In the slip area, Coulomb's law requires the tangential traction to be proportional to the normal pressure. When the normal and tangential stress-displacement equations are identical, with the exception of a constant factor, each normal solution is also a tangential solution. In this case, the tangential traction
In the case of an axisymmetric contact area with the radius
Ciavarella (1999a) summarized his theory in equations (36) for the corrective shear force and (38) for the tangential displacement, which are identical with equations (34) and (33) by Jäger (1995). For the special case of a flat punch with rounded corners, the gap
where, b, the radius of the flat area and a, the contact radius. Ciavarella obtained the solution for the normal force P, the normal displacement a_{1}(a) and the pressure _{n1}(a)p . We shortly summarize these formulas in a modified form, because they are useful below _{1}(a)
where the variables n G denote Poisson's ratio and the shear modulus of body _{k}k, respectively. It seems that the integral with the term arccos(b/s) in Eq. (5) cannot be expressed with tabulated functions, but it is possible to evaluate the derivative dp(r)/dr. Using integration by parts, the square-root singularity under the integral must first be eliminated. The resulting integral can easily be differentiated, which gives some elliptic integrals listed in Gradstein and Ryshik (1981)
In section 3.2, Ciavarella discussed the example of a conical punch with rounded tip
The profile (8) can be written as the difference of a Hertzian profile and a flat punch (2)
In linear elasticity, solutions can linearly be superposed, and it is not necessary to perform any calculations for the solution of this case. The Hertzian result is the special case
where the index 2 characterizes values for the punch with rounded tip and index 1 values from Eqs. (3)-(6). Equations (10)-(13) are identical with Ciavarella's equations (19)-(21). For In section 4.1, Ciavarella (1999) discussed the tangential solution for a flat punch. Again, there is no necesity for calculation and the results are given by Equation (1). For completeness, we summarize the results shortly
Equation (17) holds for axisymmetric contact, only. Formulas (14)-(17) can also be used for the conical punch with rounded tip, after substitution of index 2 for 1. Equations (14)-(17) cover all cases of equations (39)-(47) by Ciavarella (1999a).
The theory for plane contact with multiples contact areas was reported by Jäger (1997a) and proved in Jäger (1998). Ciavarella (1998a) summarized his theory in equations (18) and (19), which are better expressed with equation (9) Table of identical formulas by Ciavarella and Jäger
Eqs. (22), (23) by Ciavarella (1998b), for the flat punch with rounded corners, have been written in a simpler form by Schubert (1942)
As in the case of axisymmetric contact, the slope of the pressure is infinite at r=b. The solution for a wedge with rounded apex is the difference of a Hertzian profile and a flat punch (18)-(20)
with H(x) given by (13) and P It should be noted that Ciavarella’s equation (A9) has the wrong dimension and becomes singular for odd k. It should be replaced with equation (18) by Jäger [1997b]
with the hypergeometric function F(a,b;c;x). Ciavarella [1998a-I] distinguishes on page 2355 among known or unknown contact areas, but only the profile is known, a priori. When the profile has edges with vertical or concave tangents, such edges are a border of the contact area. In this case, a rigid body displacement in form of a flat punch solution with sharp edges can be superposed. When the slope of the profile is finite and convex, the pressure falls to zero at the border of the contact area. The remarks on page 2356 by Ciavarella [1998a-I] should be replaced with the following - The corrective shear traction
*q**is always the full slip stress*fp**of the stick area. - The stick condition is a constant rigid body displacement, and therefore, identical only with the contact condition, when the bodies do not rotate. Consequently, the bodies must not rotate during tangential loading.
- For contact profiles with flat areas of different height, higher areas enter in contact with increasing normal forces and the new contact areas change their form with the compression. When a tangential force is applied, a slip area propagates at the highest flat area and a partial slip solution exists. Only when all flat areas have the same height, all contact areas stick completely for forces below Coulomb's limit.
Finally, it can be concluded from the pressure (7), (18) for flat rounded punches that a discontinuous second derivative of the displacement produces an infinite slope of the pressure.
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