IINNTTEERRNNAATTIIOONNAALL CCEENNTTRREE FFOORR MMEECCHHAANNIICCAALL SSCCIIEENNCCEESS 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 ISBN 978-3-7091-2746-9 (eBook) DOI 10.1007/978-3-7091-2746-9 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 and B be two different configurations of the same 0 continuous body B, ~ and~ the positions, in B0 and B, of the same particle of B ln a fixed frame of reference. The mapping B0 ~ B is a deformation of the body. A deformation is regular if: 2 T. Manacorda 1) the correspondence X~ ! is one-to-one; 2) if we put on ! = x ( 1.1) -1 the functiona X and ~ are continuous up to their third de- rivatives ( 1) • ' 3) the determinant of the defoPmation gPadient h ax h h F = Grad ! = Grad X = llx 'H 11 X 'H = axH ( 1. 2) h,H = 1,2,3 is strictly positive: J = det F > O. Let d! be a linear element of B, and d~ the corresponding one in the reference configuration B0 ; obviously 1 dx = F dX dX = F- dx ( 1. 3) In these lectures we shall only need the continuity of the second derivatives, but the third derivatives must be con sidered for the so called congruence conditions. ~et dx~ be a triad of linear ele~er.ts issuing from x and non c~planar; the volume v of the thetraedron d~l, d~2 ,d~3 is given by v = d~1 • d~2x d~3 , and therefore by v = [dr_1 • [dr_2x !:_dr_3 = J dEl · d~2x dr_3, as is well knovr,. Therefore, the condition J > O is eouivalent to the condi tion that the two triads dxh and d!B be equally oriented. Topics in Elastodynamics 3 so that the knowledge of the gradient of deformation implies the knowledge of the deformation of a first order neigh·bour hood of X. For instance, from (1.3) it is easily obtained ( 1. 4) with = T C F F· ( 1. 5) Q is a symmetric tensor defined on B called the right Cauchy 0 , Green deformation te~sor. The ueformation tensor F. is given by = C 1 + 2 E ( 1. 6) where 1 is the unit tensor, whose cartesians components 0 nk are 0 if h ¥ k, 1 if h = k. The proper values AH of c are strictly positive, as it can easily proved using (1.4). Let u be a symmetric, positive de- finite tensor whose proper values coincide with hH, H = 1,2,3; the polar decomposition od F = F R U ( 1. 7) = = where R is a proper rotation, ~ ~T ~' det R 1, is easily proved. In fact, put = -1 y F U ( 1. 8) we obtain successively ·yT y = U(l )T.ET!:. .!:!.-1 = u-1 c u- 1 = -u1 u2 u- 1 = 1 ' det. y = det.F det. (Q.-1) = J I det.Q. = J I laet. II = 1 4 T. Manacorda = ir J det.f > 0. The uniqueness of ~ can be proved by assum ing th& existence of a differente rotation, say~', and proving that this assumption is false. The symmetric, positive definite tensor B = F FT = V2 ( l. 9) is called. the left Cauchy-Green deformation tensor. B is an Eulerian (or loca~) tensor. The following identities hold: = B -R -C -RT ( 1.10) £ is a Lagrangian (or molecular) tensor; his proper direc tions are called the principal directions of deformation. The rotation R brings the proper directions of C into coinci- 3 - dence with the proper directions of B ( ) f determines also the correspondence between oriented sur- face elements. Let dO be a surface element of B, whose unit normal is ~' and let dE be the corresponding element of B and N the unit 0 normal to dE, whose orientation is choosen so that ~·f! > 0 (in general, f! is neither normal nor tangent to dcr). Then dcr ~ = dE J (fT ) -1 !· ( 1.11) Let the reference configuration be changed in B~, and let the gradient of deformation B~ + B be f'; obviously, (3) Occasionally, it will be convenient to introduce the ten- sor £ = (fT ) -1 f -1 = ~ -1 = 1 + 2e £ possesses the same proper directions as ~' but his proper numbers are the reciprocal of the proper numbers of 3. Topics in Elastodynamics 5 F = F' P' F' = F P (1.12) where and (1.13) Bb. is the gradient of deformation from B0 to For instance, we obtain = C' F'T F' (1.14) In particular, let us consider the case P'= g, with g or thogonal; we obtain (1.15) and C and C' have the same principal invariants. If P' re- = duces to a uniform dilatation, P' d l C I : d -Z C (1.16) and C and~· have the same principal directions. More gener- ally, for a conformal transformation P' = d R (B. RT = 1 ' det . R 1) (1.17) C' = d -2 R C RT (1.18) Till now we have reviewed some geometrical concepts con- nected with the deformation of B; we now discuss the motion of B. The motion of B is an one-to-one mapping of B on one-para meter family of configurations Bt• where the parameter is the