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This book is centred about the Principle of virtual work and the related method for mechanical modelling. It aims at showing and enhancing the polyvalence and versatility of the virtual work approach in the mechanical modelling process. The virtual work statement is set as the principle at the root of a force modelling method that can be implemented on any geometrical description. After experimentally induced hypotheses have been made on the geometrical parameters that describe the concerned system and subsystems, the method provides a unifying framework for building up consistently associated force models where external and internal forces are introduced through their virtual rates of work. Systems described as three-dimensional, curvilinear or planar continua are considered: force models are established with the corresponding equations of motion; the validation process points out that enlarging the domain of relevance of the model for practical applications calls for an enrichment of the geometrical description that takes into account the underlying microstructure.
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Virtual Work Approach to Mechanical Modeling In memory of Marcel Save (1930–2016) Virtual Work Approach to Mechanical Modeling Jean Salençon First published 2018 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd 27-37 St George’s Road London SW19 4EU UK John Wiley & Sons, Inc. 111 River Street Hoboken, NJ 07030 USA www.iste.co.uk www.wiley.com © ISTE Ltd 2018 The rights of Jean Salençon to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Control Number: 2017959464 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-78630-295-3 Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Chapter 1. The Emergence of the Principle of Virtual Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1. In brief . . . . . . . . . . . . . . . . . . . . 1.2. Setting the principle as a cornerstone . . . 1.3. The “simple machines” . . . . . . . . . . . 1.4. Leonardo, Stevin, Galileo . . . . . . . . . . 1.5. Descartes and Bernoulli . . . . . . . . . . . 1.5.1. René Descartes (1596–1650) . . . . . 1.5.2. Johann Bernoulli (1667–1748). . . . . 1.6. Lagrange (1736–1813) . . . . . . . . . . . 1.6.1. Lagrange’s statement of the principle . 1.6.2. Lagrange’s proof of the principle . . . 1.6.3. Lagrange’s multipliers . . . . . . . . . . . . . . . . . . . . 1 1 2 5 9 9 11 12 12 13 15 Chapter 2. Dualization of Newton’s Laws . . . . . . . . . . . . . . . . . . . . . . 19 2.1. In brief . . . . . . . . . . . . . 2.2. Newton’s statements. . . . . . 2.2.1. First law . . . . . . . . . . 2.2.2. Second law . . . . . . . . 2.2.3. Third law . . . . . . . . . 2.2.4. Material points . . . . . . 2.3. System of material points . . . 2.3.1. System of material points 2.3.2. Subsystem . . . . . . . . . 2.3.3. Law of mutual actions . . 2.3.4. Summing up… . . . . . . . . . . . . . . . .