Generalized Measures for Large Deformations
In engineering materials, even in the largest purely elastic deformations, the strain is small even though the displacement gradients and rotations may be large. In such cases there is very little gained in altering classical linear strain-stress tensor c
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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 strict
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