Slip Forces for Bodies in Contact With Large Spin and Small Slip Velocities

1974 ◽  
Vol 41 (1) ◽  
pp. 296-298
Author(s):  
E. F. Kurtz
Keyword(s):  
Contact Surface ◽  
Elliptic Integrals ◽  
Elastic Bodies ◽  
Complete Elliptic Integrals ◽  
Large Spin

Two elastic bodies are treated, which are in contact over an elliptical surface, are subjected to a large spin about an axis normal to the contact surface, and have small slip velocities tangential to that contact surface. It is shown that these slip velocities are proportional to the tangential components of force acting at the contact surface, and that the coefficients of proportionality can be written as simple expressions in terms of complete elliptic integrals of the first and second kind.

Compliance of Elastic Bodies in Contact

1949 ◽  
Vol 16 (3) ◽  
pp. 259-268
Author(s):  
R. D. Mindlin
Keyword(s):  
Contact Area ◽  
Contact Surface ◽  
Poisson’S Ratio ◽  
Tangential Force ◽  
Normal Component ◽  
Poisson's Ratio ◽  
Elastic Bodies ◽  
Complete Elliptic Integrals ◽  
Elliptic Contact ◽  
Circular Contact

Abstract A small tangential force and a small torsional couple are applied across the elliptic contact surface of a pair of elastic bodies which have been pressed together. If there is no slip at the contact surface, considerations of symmetry and continuity lead to the conclusion that there is no change in the normal component of traction across the surface and, aside from warping of the surface, there is no relative displacement of points on the contact surface. The problem is thus reduced to a “problem of the plane” in which the tangential displacements and normal component of traction are given over part of the boundary and the three components of traction are given over the remainder. In the case of the tangential force it is observed that, when Poisson’s ratio is zero, the problem is a simple one, in potential theory, which is then generalized by means of a special device. An expression for tangential compliance is found as a linear combination of complete elliptic integrals. In general, the compliance is greater in the direction of the major axis of the elliptic contact surface than in the direction of the minor axis. Both components of tangential compliance increase as Poisson’s ratio decreases and become equal when Poisson’s ratio is zero. Over the practical range of Poisson’s ratio, the tangential compliance is greater than the normal compliance, but never more than twice as great as long as there is no slip. The tangential traction on the contact surface is everywhere parallel to the applied force. Contours of constant traction are ellipses homothetic with the elliptic boundary. The magnitude of the traction rises from one half the average at the center of the contact surface to infinity at the edge. Due to this infinity, there will be slip, the effect of which is studied for the circular contact surface. In the case of the torsional couple, the solution is obtained by generalizing a solution by H. Neuber pertaining to a hyperbolic groove in a twisted shaft. The torsional compliance is expressed in terms of complete elliptic integrals and, for the circular contact area, reduces to that found by E. Reissner and H. F. Sagoci. The resultant traction at a point rises from zero at the center to infinity at the edge of the contact surface, but is constant along and parallel to homothetic ellipses only in the case of the circular contact area.

scholarly journals Master integrals for bipartite cuts of three-loop photon self energy

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
R. N. Lee ◽  
A. I. Onishchenko
Keyword(s):  
Spectral Density ◽  
High Energy ◽  
Elliptic Integrals ◽  
Iterated Integrals ◽  
Self Energy ◽  
Complete Elliptic Integrals ◽  
The One

Abstract We calculate the master integrals for bipartite cuts of the three-loop propagator QED diagrams. These master integrals determine the spectral density of the photon self energy. Our results are expressed in terms of the iterated integrals, which, apart from the 4m cut (the cut of 4 massive lines), reduce to Goncharov’s polylogarithms. The master integrals for 4m cut have been calculated in our previous paper in terms of the one-fold integrals of harmonic polylogarithms and complete elliptic integrals. We provide the threshold and high-energy asymptotics of the master integrals found, including those for 4m cut.

Some Applications of Elliptic Integrals of First Kind

2013 ◽  
Vol 65 (1) ◽  
Author(s):  
Yasamin Barakat ◽  
Nor Haniza Sarmin
Keyword(s):  
Algebraic Function ◽  
High Accuracy ◽  
Elliptic Integrals ◽  
Definite Integrals ◽  
Engineering Mathematics ◽  
Optimal Values ◽  
Complete Elliptic Integrals

One of the most important applications of elliptic integrals in engineering mathematics is their usage to solve integrals of the form  (Eq. 1), where  is a rational algebraic function and  is a polynomial of degree  with no repeated roots. Nowadays, incomplete and complete elliptic integrals of first kind are estimated with high accuracy using advanced calculators.  In this paper, several techniques are discussed to show how definite integrals of the form (Eq. 1) can be converted to elliptic integrals of the first kind, and hence be estimated for optimal values. Indeed, related examples are provided in each step to help clarification.

Remarks on generalized elliptic integrals

2009 ◽  
Vol 139 (2) ◽  
pp. 417-426 ◽  
Author(s):  
Xiaohui Zhang ◽  
Gendi Wang ◽  
Yuming Chu
Keyword(s):  
Elliptic Integrals ◽  
Complete Elliptic Integrals

We study the monotonicity for certain combinations of generalized elliptic integrals, thus generalizing analogous well-known results for classical complete elliptic integrals, and prove a conjecture put forward by Heikkala, Vamanamurthy and Vuorinen.

scholarly journals Closed Analytic Formulas for the Approximation of the Legendre Complete Elliptic Integrals of the First and Second Kinds

2021 ◽  
Vol 8 ◽  
pp. 23-28
Author(s):  
Richard Selescu
Keyword(s):  
Normal Form ◽  
Asymptotic Expansions ◽  
Elliptic Integrals ◽  
Analytic Study ◽  
Regression Functions ◽  
Complete Elliptic Integrals ◽  
The Right ◽  
Approximate Formulas

The author proposes two sets of closedanalytic functions for the approximate calculus of thecomplete elliptic integrals of the first and secondkinds in the normal form due to Legendre, therespective expressions having a remarkablesimplicity and accuracy. The special usefulness of theproposed formulas consists in that they allowperforming the analytic study of variation of thefunctions in which they appear, by using thederivatives. Comparative tables including theapproximate values obtained by applying the two setsof formulas and the exact values, reproduced fromspecial functions tables are given (all versus therespective elliptic integrals modulus, k = sin ). It is tobe noticed that both sets of approximate formulas aregiven neither by spline nor by regression functions,but by asymptotic expansions, the identity with theexact functions being accomplished for the left end k= 0 ( = 0) of the domain. As one can see, the secondset of functions, although something more intricate,gives more accurate values than the first one andextends itself more closely to the right end k = 1 ( =90) of the domain. For reasons of accuracy, it isrecommended to use the first set until  = 70.5 only,and if it is necessary a better accuracy or a greaterupper limit of the validity domain, to use the secondset, but on no account beyond  = 88.2.

scholarly journals INEQUALITIES OF THE COMPLETE ELLIPTIC INTEGRALS

1998 ◽  
Vol 29 (3) ◽  
pp. 165-169
Author(s):  
FENG QI ◽  
ZHENG HUANG
Keyword(s):  
Integral Inequality ◽  
Elliptic Integrals ◽  
Complete Elliptic Integrals

In this article, using Tchebycheff's integral inequality, the authors establish some estimates and inequalities for three kinds of the complete elliptic integrals.

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