  Elastic Impact with Friction  by J.Jaeger, Blattwiesenstr. 7, D-76227 Karlsruhe, Germany Thesis, Delft University, Dep. Maths. & Infs., Delft, The Netherlands, 1992 1 Summary The  classical impact theories consider either frictionless surfaces or gross slip of the whole contact area. A more realistic theory must consider the partition of the contact area into stick and slip areas. Three coefficients of  restitution determine the rotations and velocities of the two bodies after the impact. These coefficients will be calculated for some examples. This work is based on the fundamental equations of the three dimensional theory of linear elasticity. H. Hertz (1882) solved these equations for the normal impact of two bodies. He used the half-space solutions for single forces of Boussinesq (1885) and Cerruti (1882) to determine the stress field in  the vicinity of the contact area. This holds for compact bodies, where the dimensions of the contact area are small compared to the global curvatures and dimensions of the bodies. We generalize the Hertz solution on the basis of  this simplification. This work is exclusively confined to the half-space approximation and elastic materials. Chapter 3 is the introduction to this work. The contact processes during impact require the analysis of  some basic load-histories. Chapter 4 summarizes the basic formulae for half-space problems. Some well-known load-histories for similar material are discussed in chapter 5. Chapter 6 presents the theory of Mindlin and Deresiewicz for varying normal and oblique forces under a new point of view and generalizes their result. We introduce the so-called Cattaneo-Mindlin functions and we will arrive at a formula for the forces, which depends only on so-called  points of instantaneous adhesion, where the entire contact area is stuck for a moment, and on the current displacements, but not on the specific form of the previous load history. In Chapter 7 we generalize this theory for elliptical contact areas under oblique forces with varying directions. This theory is based on the simplifying assumption that the stress direction in the entire slip area is constant between two points of instantaneous adhesion.  In con sequence of this theory different load histories with the same stress distribution are possible. Elliptical contact areas under varying torques can be approximated by a simple formula, which we compare with some numerical calculations in chapter 7.6. The basic equations for the impact of two bodies are stated in chapter 8. The three different contact regimes of full adhesion, partial slip and complete sliding are introduced and the equations of  motion are deduced. Some analytical solutions are presented in chapter 9. Finally we discuss the torsional impact in chapter 10. The resulting torque is much smaller than the moments produced by the tangential forces, because the  dimensions of the contact area should be less than 10% of the the dimensions of the body, in the frame of the Hertz theory. Consequently the lever-arm of the torsional stress is almost negligible compared with the lever-arm of the tangential stress, and the torsional torque and the moments inside of the contact area can be neglected in a first order impact theory. Chapter 11 is dedicated to the numerical solutions by J.J. Kalker, because our numerical procedures are based on his methods. The application of the Gauss-Seidel procedure, explained in chapter 12, improves the calculation time and the storage requirements considerably. In chapter 13 we prove convergence of our modified Gauss-Seidel method for a few special load histories. The convergence of the linearized frictional law is not investigated, because convergence is certain for infinitely small increments of a load history with Hertzian surfaces, where the stress directions hardly change. In this case the adhesive area also varies in a few points only, and the stick area loop converges very well. The problem of a flat punch pressing on an elastic half space is  more difficult, because at the beginning of contact the stick area changes very much. Furthermore the stress directions can only be calculated, if the first estimation of the stick area is not too wrong. We introduced a number of control parameters, which control the convergence of the program, like the maximal number of iterations for each loop, the precision and the size of the increments. Empirical experience shows, that a correct parameter setting yields convergence of the program, even for large contact areas of 1000 poins. The velocity of convergence is also very important for large contact areas, because calculation times of one or two days on an 80386/33MHz computer are normal for 1000 points. The calculation time increases with n2 approximately, while the storage requirements increase with n, where n denotes the number of points in the area of integration. In chapter 14 we compare the simplified solutions of chapter 7 with some numerical results. It turns out that in contrast to our theory the stress directions in the slip area vary considerably. The correspondence between numerics and theory for the size of  the stick area and the absolute value of the tangential stress is much better. A new conclusion of the numerical results is that for elliptical contact areas with n1=n2=0 the stresses szx, szy are constant on ellipses, as long as torsion is absent. Even if the Poisson numbers are different from zero, the stresses are still almost constant on ellipses as long as the materials are similar. Small frictional coefficients with different materials yield the most unpredictable stress distributions. An example for the superposition of torsion and tangential shift is also presented in section 14.4. In section 14.5 we present some results for non-Hertzian contact areas in form of flat punches. The tangential coefficient of restitution is the ratio of the tangential velocity in the contact point before impact to the velocity after impact. Chapter 15 shows that the correspondence between the Cattaneo-Mindlin theory and the numerical result is very good. Furthermore the coefficients of restitution for dissimilar materials, superposition of torsion and ellipsoid bodies are presented. Additional information  about the above mentioned topics can be found in the introduction.