7R29. Three-Dimensional Contact Problems. Solid Mechanics and its Applications, Vol 93. - VM Alexandrov (Dept of Mech and Math, Moscow State Univ, Moscow, Russia) and DA Pozharskii (Mech and Appl Math Inst, Rostov-on-Don State Univ, Rostov-on-Don, Russia). Kluwer Acad Publ, Dordrecht, Netherlands. 2001. 406 pp. ISBN 0-7923-7165-8. \$131.00.

Reviewed by JT Tielking (PO Box 1009, Daleville VA 24083-1009).

The calculation of stress and displacement in elastic bodies loaded by contact with one another or against a rigid surface is a difficult problem in applied mechanics. Even for linear material and infinitesimal displacements, the problem is nonlinear when the boundary of the contact region is unknown a priori. This book presents the derivation of analytic solutions for a variety of problems involving isotropic linearly-elastic bodies in frictionless, static contact. Separate chapters are devoted to contact problems for cylinders, wedges, cones, and spheres. Numerical results for spherical problems are compared with Hertz solutions to show the ranges of validity of the Hertz analysis.

The exact calculation of pressure in the contact interface requires solution of an integral equation, a system of integral equations in the case of 3D problems. For certain punch problems, and others of engineering interest, the contact boundary is specified so that the governing equations are linear. Various transformations are used to extract solutions in terms of special functions or in asymptotic expansions. The important problem of determining the contact boundary, as well as the normal contact pressure, is formulated as a nonlinear boundary integral equation of the Hammerstein type. This formulation is due to B A Galanov who also developed an algorithm based on Newton’s method for solving this type of nonlinear integral equation. A Fortran listing of Galanov’s computer program is given in the Appendix. Numerical results from the program are compared in the text with results from analytic solutions for two test problems.

The large collection of analytic solutions presented in this book will be useful in evaluating test results calculated by boundary element and finite element methods. The book is well-written by the authors (a translator is not listed). Good drawings are included to define problem geometry and coordinate systems. The analyses are presented in considerable detail, and the work of others is carefully referenced, often including the page numbers where relevant equations will be found. The collected reference list covers eight pages. Many are by Russian authors, some not yet translated. There is a good index. Three-Dimensional Contact Problems is recommended for purchase by engineering libraries and for individuals interested in the mathematical analysis of contact problems.