Nonlinear Dynamics Of Elastic Bodies

INTERNATIONAL INTERNATIONAL CENTRE FOR FOR MECHANICAL SCIENCES SCIENCES COURSES AND LECTURES No. 227 NONLINEAR DYNAMICS OF ELASTIC BODIES EDITED BY Z. WESOLOWSKI POLISH ACADEMY OF SCIENCES SPRINGER-VERLAG WIEN GMBH This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned specifically those of translation, reprinting, re-use of illustrations, broadcasting,_ reproduction by photocopying machine or similar means, and storage in data banks. © 1978 by Springer-VerlagWien Originally published by Springer Verlag Wien New York in 1978 ISBN 978-3-211-81512-0 DOI 10.1007/978-3-7091-2746-9 ISBN 978-3-7091-2746-9 (eBook) PREFACE This volume contains the texts of five series of lectures devoted to the nonlinear dynamics of elastic bodies which were delivered at the Department of Mechanics of Solids of the International Centre for Mechanical Sciences, Udine, Italy. The contributions of the various authors are closely interrelated. The two first papers, by T. Manacorda and Cz. Wozniak, provide a basis for the analysis of the problems illustrated by the other lecturers. These include acceleration waves and progression waves in nonlinear elastic materials (Z. Wezolowski) and the stability of elastic systems (S.]. Britvec). Finally, the contribution by B.R. Seth is of a somewhat different nature. It advocates, for large deformations, the use of generalized measures and discusses the ensuing results and advantages. We hope the contributions presented will be of interest to research workers inJJolved in investigating the nonlinear response of material s under various static and dynamic conditions. Z. Wezolowski LIST OF CONTRIBUTORS Tristano Manacorda Universita di Pisa, Istituto di Matematiche Applicate, Pisa. Czeslaw Wozniak University ofWarsaw, Miedzynarodowa 58 m. 63, 03-922 Warszawa, Poland. Zbigniew Wesolowski Institute of Fundamental Technological Research, Swietokrzyska 21, Warsaw, Poland. S.J. Britvec Professor of Engineering Mechanics. University of Stuttgart and University of 7.agreb. B.R. Seth Birla Institute of Technology, Mesra, Ranchi, India. TOPICS IN ELASTODYNAMICS TRISTANO MANACORDA Universita di Pisa Istituto di Matematiche Applicate CHAPTER I INTRODUCTION. MOTION AND DEFORMATiON 1 - DEFORMATIONS, The notion of a continuous body is a primitive concept. A continuous body can be put in one-to-one correspondence with regions of the Euclidean space; more exactly, with family of such regions. Each of these regions is called a configuration of the body. Let B 0 and B be two different configurations of the continuous body B, ~ and~ same the positions, in B0 and B, of the same particle of B ln a fixed frame of reference. The mapping B0 if: ~ B is a deformation of the body. A deformation is regular T. Manacorda 2 1) the correspondence X~ ! is one-to-one; 2) if we put ! =x on the functiona X and ~ -1 are continuous up to their third de- 1) • rivatives ( 3) ( 1.1) ' the determinant of the defoPmation gPadient F = Grad ! = Grad X h,H = = llx h 'H 11 X h = 'H ax h ( 1. 2) axH 1,2,3 is strictly positive: J = det F > O. Let d! be a linear element of B, and one in the reference configuration B 0 dx = F dX 1 ; dX = F- dx d~ the corresponding