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Continuum Mechanics Using Mathematica® Fundamentals, Methods, and Applications Modeling and Simulation in Science, Engineering and Technology Antonio Romano Addolorata Marasco Continuum Mechanics using Mathematica® Fundamentals, Methods, and Applications Second Edition Modeling and Simulation in Science, Engineering and Technology Series Editor Nicola Bellomo Politecnico di Torino Tor ino, Italy Editorial Advisory Board K.J. Bathe P. Koumoutsakos Department of Mechanical Engineering Computational Science & Engineering Massachusetts Institute of Technology Laboratory Cambridge, MA, USA ETH Zürich Zürich, Switzerland M. Chaplain Division of Mathematics H.G. Othmer University of Dundee Department of Mathematics Dundee, Scotland, UK University of Minnesota Minneapolis, MN, USA P. Degond Department of Mathematics, K.R. Rajagopal Imperial College London, Department of Mechanical Engineering London, United Kingdom Texas A&M University College Station, TX, USA A. Deutsch Center for Information Services T.E. Tezduyar and High-Performance Computing Department of Mechanical Engineering & Technische Universität Dresden Materials Science Dresden, Germany Rice University Houston, TX, USA M.A. Herrero Departamento de Matematica Aplicada A. Tosin Universidad Complutense de Madrid Istituto per le Applicazioni del Calcolo Madrid, Spain “M. Picone” Consiglio Nazionale delle Ricerche Roma, Italy More information about this series at http://www.springer.com/series/4960 Antonio Romano • Addolorata Marasco Continuum Mechanics using MathematicaR Fundamentals, Methods, and Applications Second Edition Antonio Romano Addolorata Marasco Department of Mathematics Department of Mathematics and Applications “R. Caccioppoli” and Applications “R. Caccioppoli” University of Naples Federico II University of Naples Federico II Naples, Italy Naples, Italy Additional material to this book can be downloaded from http://extras.springer.com ISSN 2164-3679 ISSN 2164-3725 (electronic) ISBN 978-1-4939-1603-0 ISBN 978-1-4939-1604-7 (eBook) DOI 10.1007/978-1-4939-1604-7 Springer New York Heidelberg Dordrecht London Library of Congress Control Number: 2014948090 Mathematics Subject Classification: 74-00, 74-01, 74AXX, 74BXX, 74EXX, 74GXX, 74JXX © Springer Science+Business Media New York 2006, 2014 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.birkhauser-science.com) Preface The motion of any body depends both on its characteristics and on the forces acting on it. Although taking into account all possible properties makes the equations too complex to solve, sometimes it is possible to consider only the properties that have the greatest influence on the motion. Models of ideals bodies, which contain only the most relevant properties, can be studied using the tools of mathematical physics. Adding more properties into a model makes it more realistic, but it also makes the motion problem harder to solve. In order to highlight the above statements, let us first suppose that a system S of N unconstrained bodies Ci , i D 1;:::;N, is sufficiently described by the model of N material points whenever the bodies have negligible dimensions with respect to the dimensions of the region containing the trajectories. This means that all the physical properties of Ci that influence the motion are expressed by a positive number, the mass mi , whereas the position of Ci with respect to a frame I is given by the position vector ri .t/ versus time. To determine the functions ri .t/, one has to integrate the following system of Newtonian equations: mi rRi D Fi Á fi .r1;:::;rN ; rP1;:::;rPN ;t/; i D 1;:::;N, where the forces Fi , due both to the external world and to the other points of S, are assigned functions fi of the positions and velocities of all the points of S, as well as of time. Under suitable regularity assumptions about the functions fi , the previous (vector) system of second-order ordinary differential equations in the unknowns ri .t/ has one and only one solution satisfying the given initial conditions 0 0 ri .t0/ D ri ; rPi .t0/ D rPi ;iD 1;:::;N: A second model that more closely matches physical reality is represented by asystemS of constrained rigid bodies Ci , i D 1;:::;N. In this scheme, the v vi Preface extension of Ci and the presence of constraints are taken into account. The position of Ci is represented by the three-dimensional region occupied by Ci in the frame I . Owing to the supposed rigidity of both bodies Ci and constraints, the configurations of S are described by n Ä 6N parameters q1;:::;qn, which are called Lagrangian coordinates. Moreover, the mass mi of Ci is no longer sufficient for describing the physical properties of Ci since we have to know both its density and geometry. To determine the motion of S, that is, the functions q1.t/; : : : ; qn.t/, the Lagrangian expressions of the kinetic energy T.q;q/P and active forces Qh.q; q/P are necessary. Then a possible motion of S is a solution of the Lagrange equations d @T @T D Qh.q; q/;P h D 1;:::;n; dt @qPh @qh satisfying the given initial conditions 0 0 qh.t0/ D qh; qPh.t0/ DPqh;hD 1;:::;n; which once again fix the initial configuration and the velocity field of S. We face a completely different situation when, to improve the description, we adopt the model of continuum mechanics. In fact, in this model the bodies are deformable and, at the same time, the matter is supposed to be continuously distributed over the volume they occupy, so that their molecular structure is completely erased. In this book we will show that the substitution of rigidity with the deformability leads us to determine three scalar functions of three spatial variables and time, in order to find the motion of S. Consequently, the fundamental evolution laws become partial differential equations. This consequence of deformability is the root of the mathematical difficulties of continuum mechanics. This model must include other characteristics which allow us to describe the different macroscopic material behaviors. In fact, bodies undergo different deformations under the influence of the same applied loads. The mathematical description of different materials is the object of the constitutive equations. These equations, although they have to describe a wide variety of real bodies, must in any case satisfy some general principles. These principles are called constitutive axioms and they reflect general rules of behavior. These rules, although they imply severe restrictions on the form of the constitutive equations, permit us to describe different materials. The constitutive equations can be divided into classes describing the behavior of material categories: elastic bodies, fluids, etc. The choice of a particular constitutive relation cannot be done a priori but instead relies on experiments, due to the fact that the macroscopic behavior of a body is strictly related to its molecular structure. Since the continuum model erases this structure, the constitutive equation of a particular material has to be determined by experimental procedures. However, the introduction of deformability into the model does not permit us to describe all the phenomena accompanying the motion. In fact, the viscosity of S as well as the friction between S and any external bodies produce heating, which in turn causes heat exchanges among parts of S or between S and Preface vii its surroundings. Mechanics is not able to describe these phenomena, and we must resort to the thermomechanics of continuous systems. This theory combines the laws of mechanics and thermodynamics, provided that they are suitably generalized to a deformable continuum at a nonuniform temperature. The situation is much more complex when the continuum carries charges and currents. In such a case, we must take into account Maxwell’s equations, which describe the electromagnetic fields accompanying the motion, together with the thermomechanic equations.
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