Continuum Mechanics / A.J.M
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Classical Mechanics
Classical Mechanics Hyoungsoon Choi Spring, 2014 Contents 1 Introduction4 1.1 Kinematics and Kinetics . .5 1.2 Kinematics: Watching Wallace and Gromit ............6 1.3 Inertia and Inertial Frame . .8 2 Newton's Laws of Motion 10 2.1 The First Law: The Law of Inertia . 10 2.2 The Second Law: The Equation of Motion . 11 2.3 The Third Law: The Law of Action and Reaction . 12 3 Laws of Conservation 14 3.1 Conservation of Momentum . 14 3.2 Conservation of Angular Momentum . 15 3.3 Conservation of Energy . 17 3.3.1 Kinetic energy . 17 3.3.2 Potential energy . 18 3.3.3 Mechanical energy conservation . 19 4 Solving Equation of Motions 20 4.1 Force-Free Motion . 21 4.2 Constant Force Motion . 22 4.2.1 Constant force motion in one dimension . 22 4.2.2 Constant force motion in two dimensions . 23 4.3 Varying Force Motion . 25 4.3.1 Drag force . 25 4.3.2 Harmonic oscillator . 29 5 Lagrangian Mechanics 30 5.1 Configuration Space . 30 5.2 Lagrangian Equations of Motion . 32 5.3 Generalized Coordinates . 34 5.4 Lagrangian Mechanics . 36 5.5 D'Alembert's Principle . 37 5.6 Conjugate Variables . 39 1 CONTENTS 2 6 Hamiltonian Mechanics 40 6.1 Legendre Transformation: From Lagrangian to Hamiltonian . 40 6.2 Hamilton's Equations . 41 6.3 Configuration Space and Phase Space . 43 6.4 Hamiltonian and Energy . 45 7 Central Force Motion 47 7.1 Conservation Laws in Central Force Field . 47 7.2 The Path Equation . -
The Development of the Action Principle a Didactic History from Euler-Lagrange to Schwinger Series: Springerbriefs in Physics
springer.com Walter Dittrich The Development of the Action Principle A Didactic History from Euler-Lagrange to Schwinger Series: SpringerBriefs in Physics Provides a philosophical interpretation of the development of physics Contains many worked examples Shows the path of the action principle - from Euler's first contributions to present topics This book describes the historical development of the principle of stationary action from the 17th to the 20th centuries. Reference is made to the most important contributors to this topic, in particular Bernoullis, Leibniz, Euler, Lagrange and Laplace. The leading theme is how the action principle is applied to problems in classical physics such as hydrodynamics, electrodynamics and gravity, extending also to the modern formulation of quantum mechanics 1st ed. 2021, XV, 135 p. 23 illus., 1 illus. and quantum field theory, especially quantum electrodynamics. A critical analysis of operator in color. versus c-number field theory is given. The book contains many worked examples. In particular, the term "vacuum" is scrutinized. The book is aimed primarily at actively working researchers, Printed book graduate students and historians interested in the philosophical interpretation and evolution of Softcover physics; in particular, in understanding the action principle and its application to a wide range 54,99 € | £49.99 | $69.99 of natural phenomena. [1]58,84 € (D) | 60,49 € (A) | CHF 65,00 eBook 46,00 € | £39.99 | $54.99 [2]46,00 € (D) | 46,00 € (A) | CHF 52,00 Available from your library or springer.com/shop MyCopy [3] Printed eBook for just € | $ 24.99 springer.com/mycopy Order online at springer.com / or for the Americas call (toll free) 1-800-SPRINGER / or email us at: [email protected]. -
On the Geometric Character of Stress in Continuum Mechanics
Z. angew. Math. Phys. 58 (2007) 1–14 0044-2275/07/050001-14 DOI 10.1007/s00033-007-6141-8 Zeitschrift f¨ur angewandte c 2007 Birkh¨auser Verlag, Basel Mathematik und Physik ZAMP On the geometric character of stress in continuum mechanics Eva Kanso, Marino Arroyo, Yiying Tong, Arash Yavari, Jerrold E. Marsden1 and Mathieu Desbrun Abstract. This paper shows that the stress field in the classical theory of continuum mechanics may be taken to be a covector-valued differential two-form. The balance laws and other funda- mental laws of continuum mechanics may be neatly rewritten in terms of this geometric stress. A geometrically attractive and covariant derivation of the balance laws from the principle of energy balance in terms of this stress is presented. Mathematics Subject Classification (2000). Keywords. Continuum mechanics, elasticity, stress tensor, differential forms. 1. Motivation This paper proposes a reformulation of classical continuum mechanics in terms of bundle-valued exterior forms. Our motivation is to provide a geometric description of force in continuum mechanics, which leads to an elegant geometric theory and, at the same time, may enable the development of space-time integration algorithms that respect the underlying geometric structure at the discrete level. In classical mechanics the traditional approach is to define all the kinematic and kinetic quantities using vector and tensor fields. For example, velocity and traction are both viewed as vector fields and power is defined as their inner product, which is induced from an appropriately defined Riemannian metric. On the other hand, it has long been appreciated in geometric mechanics that force should not be viewed as a vector, but rather a one-form. -
Some Insights from Total Collapse
Some insights from total collapse S´ergio B. Volchan Abstract We discuss the Sundman-Weierstrass theorem of total collapse in its histor- ical context. This remarkable and relatively simple result, a type of stability criterion, is at the crossroads of some interesting developments in the gravi- tational Newtonian N-body problem. We use it as motivation to explore the connections to such important concepts as integrability, singularities and typ- icality in order to gain some insight on the transition from a predominantly quantitative to a novel qualitative approach to dynamical problems that took place at the end of the 19th century. Keywords: Celestial Mechanics; N-body problem; total collapse, singularities. I Introduction Celestial mechanics is one of the treasures of physics. With a rich and fascinating history, 1 it had a pivotal role in the very creation of modern science. After all, it was Hooke’s question on the two-body problem and Halley’s encouragement (and financial resources) that eventually led Newton to publish the Principia, a water- shed. 2 Conversely, celestial mechanics was the testing ground par excellence for the new mechanics, and its triumphs in explaining a wealth of phenomena were decisive in the acceptance of the Newtonian synthesis and the “clockwork universe”. arXiv:0803.2258v1 [physics.hist-ph] 14 Mar 2008 From the beginning, special attention was paid to the N-body problem, the study of the motion of N massive bodies under mutual gravitational forces, taken as a reliable model of the the solar system. The list of scientists that worked on it, starting with Newton himself, makes a veritable hall of fame of mathematics and physics; also, a host of concepts and methods were created which are now part of the vast heritage of modern mathematical-physics: the calculus, complex variables, the theory of errors and statistics, differential equations, perturbation theory, potential theory, numerical methods, analytical mechanics, just to cite a few. -
Leonhard Euler - Wikipedia, the Free Encyclopedia Page 1 of 14
Leonhard Euler - Wikipedia, the free encyclopedia Page 1 of 14 Leonhard Euler From Wikipedia, the free encyclopedia Leonhard Euler ( German pronunciation: [l]; English Leonhard Euler approximation, "Oiler" [1] 15 April 1707 – 18 September 1783) was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function.[2] He is also renowned for his work in mechanics, fluid dynamics, optics, and astronomy. Euler spent most of his adult life in St. Petersburg, Russia, and in Berlin, Prussia. He is considered to be the preeminent mathematician of the 18th century, and one of the greatest of all time. He is also one of the most prolific mathematicians ever; his collected works fill 60–80 quarto volumes. [3] A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is our teacher in all things," which has also been translated as "Read Portrait by Emanuel Handmann 1756(?) Euler, read Euler, he is the master of us all." [4] Born 15 April 1707 Euler was featured on the sixth series of the Swiss 10- Basel, Switzerland franc banknote and on numerous Swiss, German, and Died Russian postage stamps. The asteroid 2002 Euler was 18 September 1783 (aged 76) named in his honor. He is also commemorated by the [OS: 7 September 1783] Lutheran Church on their Calendar of Saints on 24 St. Petersburg, Russia May – he was a devout Christian (and believer in Residence Prussia, Russia biblical inerrancy) who wrote apologetics and argued Switzerland [5] forcefully against the prominent atheists of his time. -
The Phase Space Elementary Cell in Classical and Generalized Statistics
Entropy 2013, 15, 4319-4333; doi:10.3390/e15104319 OPEN ACCESS entropy ISSN 1099-4300 www.mdpi.com/journal/entropy Article The Phase Space Elementary Cell in Classical and Generalized Statistics Piero Quarati 1;2;* and Marcello Lissia 2 1 DISAT, Politecnico di Torino, C.so Duca degli Abruzzi 24, Torino I-10129, Italy 2 Istituto Nazionale di Fisica Nucleare, Sezione di Cagliari, Monserrato I-09042, Italy; E-Mail: [email protected] * Author to whom correspondence should be addressed; E-Mail: [email protected]; Tel.: +39-0706754899; Fax: +39-070510212. Received: 05 September 2013; in revised form: 25 September 2013/ Accepted: 27 September 2013 / Published: 15 October 2013 Abstract: In the past, the phase-space elementary cell of a non-quantized system was set equal to the third power of the Planck constant; in fact, it is not a necessary assumption. We discuss how the phase space volume, the number of states and the elementary-cell volume of a system of non-interacting N particles, changes when an interaction is switched on and the system becomes or evolves to a system of correlated non-Boltzmann particles and derives the appropriate expressions. Even if we assume that nowadays the volume of the elementary cell is equal to the cube of the Planck constant, h3, at least for quantum systems, we show that there is a correspondence between different values of h in the past, with important and, in principle, measurable cosmological and astrophysical consequences, and systems with an effective smaller (or even larger) phase-space volume described by non-extensive generalized statistics. -
Fundamental Governing Equations of Motion in Consistent Continuum Mechanics
Fundamental governing equations of motion in consistent continuum mechanics Ali R. Hadjesfandiari, Gary F. Dargush Department of Mechanical and Aerospace Engineering University at Buffalo, The State University of New York, Buffalo, NY 14260 USA [email protected], [email protected] October 1, 2018 Abstract We investigate the consistency of the fundamental governing equations of motion in continuum mechanics. In the first step, we examine the governing equations for a system of particles, which can be considered as the discrete analog of the continuum. Based on Newton’s third law of action and reaction, there are two vectorial governing equations of motion for a system of particles, the force and moment equations. As is well known, these equations provide the governing equations of motion for infinitesimal elements of matter at each point, consisting of three force equations for translation, and three moment equations for rotation. We also examine the character of other first and second moment equations, which result in non-physical governing equations violating Newton’s third law of action and reaction. Finally, we derive the consistent governing equations of motion in continuum mechanics within the framework of couple stress theory. For completeness, the original couple stress theory and its evolution toward consistent couple stress theory are presented in true tensorial forms. Keywords: Governing equations of motion, Higher moment equations, Couple stress theory, Third order tensors, Newton’s third law of action and reaction 1 1. Introduction The governing equations of motion in continuum mechanics are based on the governing equations for systems of particles, in which the effect of internal forces are cancelled based on Newton’s third law of action and reaction. -
A Re-Interpretation of the Concept of Mass and of the Relativistic Mass-Energy Relation
A re-interpretation of the concept of mass and of the relativistic mass-energy relation 1 Stefano Re Fiorentin Summary . For over a century the definitions of mass and derivations of its relation with energy continue to be elaborated, demonstrating that the concept of mass is still not satisfactorily understood. The aim of this study is to show that, starting from the properties of Minkowski spacetime and from the principle of least action, energy expresses the property of inertia of a body. This implies that inertial mass can only be the object of a definition – the so called mass-energy relation - aimed at measuring energy in different units, more suitable to describe the huge amount of it enclosed in what we call the “rest-energy” of a body. Likewise, the concept of gravitational mass becomes unnecessary, being replaceable by energy, thus making the weak equivalence principle intrinsically verified. In dealing with mass, a new unit of measurement is foretold for it, which relies on the de Broglie frequency of atoms, the value of which can today be measured with an accuracy of a few parts in 10 9. Keywords Classical Force Fields; Special Relativity; Classical General Relativity; Units and Standards 1. Introduction Albert Einstein in the conclusions of his 1905 paper “Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?” (“Does the inertia of a body depend upon its energy content? ”) [1] says “The mass of a body is a measure of its energy-content ”. Despite of some claims that the reasoning that brought Einstein to the mass-energy relation was somehow approximated or even defective (starting from Max Planck in 1907 [2], to Herbert E. -
CONTINUUM MECHANICS (Lecture Notes)
CONTINUUM MECHANICS (Lecture Notes) Zdenekˇ Martinec Department of Geophysics Faculty of Mathematics and Physics Charles University in Prague V Holeˇsoviˇck´ach 2, 180 00 Prague 8 Czech Republic e-mail: [email protected]ff.cuni.cz Original: May 9, 2003 Updated: January 11, 2011 Preface This text is suitable for a two-semester course on Continuum Mechanics. It is based on notes from undergraduate courses that I have taught over the last decade. The material is intended for use by undergraduate students of physics with a year or more of college calculus behind them. I would like to thank Erik Grafarend, Ctirad Matyska, Detlef Wolf and Jiˇr´ıZahradn´ık,whose interest encouraged me to write this text. I would also like to thank my oldest son Zdenˇekwho plotted most of figures embedded in the text. I am grateful to many students for helping me to reveal typing misprints. I would like to acknowledge my indebtedness to Kevin Fleming, whose through proofreading of the entire text is very much appreciated. Readers of this text are encouraged to contact me with their comments, suggestions, and questions. I would be very happy to hear what you think I did well and I could do better. My e-mail address is [email protected]ff.cuni.cz and a full mailing address is found on the title page. ZdenˇekMartinec ii Contents (page numbering not completed yet) Preface Notation 1. GEOMETRY OF DEFORMATION 1.1 Body, configurations, and motion 1.2 Description of motion 1.3 Lagrangian and Eulerian coordinates 1.4 Lagrangian and Eulerian variables 1.5 Deformation gradient 1.6 Polar decomposition of the deformation gradient 1.7 Measures of deformation 1.8 Length and angle changes 1.9 Surface and volume changes 1.10 Strain invariants, principal strains 1.11 Displacement vector 1.12 Geometrical linearization 1.12.1 Linearized analysis of deformation 1.12.2 Length and angle changes 1.12.3 Surface and volume changes 2. -
Non-Equilibrium Continuum Physics
Non-Equilibrium Continuum Physics Lecture notes by Eran Bouchbinder (Dated: June 10, 2019) Abstract This course is intended to introduce graduate students to the essentials of modern continuum physics, with a focus on non-equilibrium solid mechanics and within a thermodynamic perspective. Emphasis will be given to general concepts and principles, such as conservation laws, symmetries, material frame-indifference, thermodynamics, dissipation inequalities and non-equilibrium behaviors, stability criteria and configurational forces. Examples will cover a wide range of physical phenomena and applications in diverse disciplines. The power of field theory as a mathematical structure that does not make direct reference to microscopic length scales well below those of the phenomenon of interest will be highlighted. Some basic mathematical tools and methods will be introduced. The course will be self-contained and will highlight essential ideas and basic physical intuition. Together with courses on fluid mechanics and soft condensed matter, a broad background and understanding of continuum physics will be established. The course will be given within a framework of 12-13 two-hour lectures and 12-13 two-hour tutorial sessions with a focus on problem-solving. No prior knowledge of the subject is assumed. Basic knowledge of statistical thermodynamics, vector calculus, differential equations and complex analysis is required. These extended lecture notes are self-contained and in principle no textbook will be followed. Some general references (mainly for the first half of the course), can be found below. 1 Contents I. Introduction: Background and motivation 4 II. Mathematical preliminaries: Tensor Analysis 7 III. Motion, deformation and stress 13 A. -
Rethinking 'Classical Physics'
This is a repository copy of Rethinking 'Classical Physics'. White Rose Research Online URL for this paper: http://eprints.whiterose.ac.uk/95198/ Version: Accepted Version Book Section: Gooday, G and Mitchell, D (2013) Rethinking 'Classical Physics'. In: Buchwald, JZ and Fox, R, (eds.) Oxford Handbook of the History of Physics. Oxford University Press , Oxford, UK; New York, NY, USA , pp. 721-764. ISBN 9780199696253 © Oxford University Press 2013. This is an author produced version of a chapter published in The Oxford Handbook of the History of Physics. Reproduced by permission of Oxford University Press. Reuse Unless indicated otherwise, fulltext items are protected by copyright with all rights reserved. The copyright exception in section 29 of the Copyright, Designs and Patents Act 1988 allows the making of a single copy solely for the purpose of non-commercial research or private study within the limits of fair dealing. The publisher or other rights-holder may allow further reproduction and re-use of this version - refer to the White Rose Research Online record for this item. Where records identify the publisher as the copyright holder, users can verify any specific terms of use on the publisher’s website. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing [email protected] including the URL of the record and the reason for the withdrawal request. [email protected] https://eprints.whiterose.ac.uk/ 1 Rethinking ‘Classical Physics’ Graeme Gooday (Leeds) & Daniel Mitchell (Hong Kong) Chapter for Robert Fox & Jed Buchwald, editors Oxford Handbook of the History of Physics (Oxford University Press, in preparation) What is ‘classical physics’? Physicists have typically treated it as a useful and unproblematic category to characterize their discipline from Newton until the advent of ‘modern physics’ in the early twentieth century. -
Ch.9. Constitutive Equations in Fluids
CH.9. CONSTITUTIVE EQUATIONS IN FLUIDS Multimedia Course on Continuum Mechanics Overview Introduction Fluid Mechanics Lecture 1 What is a Fluid? Pressure and Pascal´s Law Lecture 3 Constitutive Equations in Fluids Lecture 2 Fluid Models Newtonian Fluids Constitutive Equations of Newtonian Fluids Lecture 4 Relationship between Thermodynamic and Mean Pressures Components of the Constitutive Equation Lecture 5 Stress, Dissipative and Recoverable Power Dissipative and Recoverable Powers Lecture 6 Thermodynamic Considerations Limitations in the Viscosity Values 2 9.1 Introduction Ch.9. Constitutive Equations in Fluids 3 What is a fluid? Fluids can be classified into: Ideal (inviscid) fluids: Also named perfect fluid. Only resists normal, compressive stresses (pressure). No resistance is encountered as the fluid moves. Real (viscous) fluids: Viscous in nature and can be subjected to low levels of shear stress. Certain amount of resistance is always offered by these fluids as they move. 5 9.2 Pressure and Pascal’s Law Ch.9. Constitutive Equations in Fluids 6 Pascal´s Law Pascal’s Law: In a confined fluid at rest, pressure acts equally in all directions at a given point. 7 Consequences of Pascal´s Law In fluid at rest: there are no shear stresses only normal forces due to pressure are present. The stress in a fluid at rest is isotropic and must be of the form: σ = − p01 σδij =−∈p0 ij ij,{} 1, 2, 3 Where p 0 is the hydrostatic pressure. 8 Pressure Concepts Hydrostatic pressure, p 0 : normal compressive stress exerted on a fluid in equilibrium. Mean pressure, p : minus the mean stress.