CONTINUUM MECHANICS (Lecture Notes)
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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. -
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. -
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. -
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. -
Chapter 1 Continuum Mechanics Review
Chapter 1 Continuum mechanics review We will assume some familiarity with continuum mechanics as discussed in the context of an introductory geodynamics course; a good reference for such problems is Turcotte and Schubert (2002). However, here is a short and extremely simplified review of basic contin- uum mechanics as it pertains to the remainder of the class. You may wish to refer to our math review if notation or concepts appear unfamiliar, and consult chap. 1 of Spiegelman (2004) for some clean derivations. TO BE REWRITTEN, MORE DISCUSSION ADDED. 1.1 Definitions and nomenclature • Coordinate system. x = fx, y, zg or fx1, x2, x3g define points in 3D space. We will use the regular, Cartesian coordinate system throughout the class for simplicity. Note: Earth science problems are often easier to address when inherent symmetries are taken into account and the governing equations are cast in specialized spatial coordinate systems. Examples for such systems are polar or cylindrical systems in 2-D, and spherical in 3-D. All of those coordinate systems involve a simpler de- scription of the actual coordinates (e.g. fr, q, fg for spherical radius, co-latitude, and longitude, instead of the Cartesian fx, y, zg) that do, however, lead to more compli- cated derivatives (i.e. you cannot simply replace ¶/¶y with ¶/¶q, for example). We will talk more about changes in coordinate systems during the discussion of finite elements, but good references for derivatives and different coordinate systems are Malvern (1977), Schubert et al. (2001), or Dahlen and Tromp (1998). • Field (variable). For example T(x, y, z) or T(x) – temperature field – temperature varying in space. -
Elastoplasticity Theory (Backmatter Pages)
Appendixes Appendix 1: Projection of Area Consider the projection of the area having the unit normal vector n onto the surface having the normal vector m in Fig. A.1. Now, suppose the plane (abcd in Fig. A.1) which contains the unit normal vectors m and n. Then, consider the line ef obtained by cutting the area having the unit normal vector n by this plane. Further, divide the area having the unit normal vector n to the narrow bands perpendicular to this line and their projections onto the surface having the normal vector m. The lengths of projected bands are same as the those of the original bands but the projected width db are obtained by multiplying the scalar product of the unit normal vectors, i.e. m • n to the original widths db . Eventually, the projected area da is related to the original area da as follows: da = m • nda (A.1) a e n b db f m d db= nm• db c Fig. A.1 Projection of area 422 Appendixes Appendix 2: Proof of ∂(FjA/J)/∂x j = 0 ∂ FjA 1 ∂(∂x /∂X ) ∂x ∂J J = j A J − j ∂x J2 ∂x ∂X ∂x j ⎧ j A j ⎫ ∂ ∂ ∂ ⎨ ∂ε x1 x2 x3 ⎬ ∂(∂x /∂X ) ∂ ∂ ∂ ∂x PQR ∂X ∂X ∂X = 1 j A ε x1 x2 x3 − j P Q R 2 PQR J ⎩ ∂x j ∂XP ∂XQ ∂XR ∂XA ∂x j ⎭ ∂ 2 ∂ 2 ∂ 2 ∂ ∂ ∂ = 1 x1 + x2 + x3 ε x1 x2 x3 J2 ∂X ∂x ∂X ∂x ∂X ∂x PQR ∂X ∂X ∂X A 1 A 2 A 3 P Q R ∂ ∂ 2 ∂ ∂ ∂ ∂ ∂ 2 ∂ − ε x1 x1 x2 x3 + x2 x1 x2 x3 PQR ∂X ∂X ∂x ∂X ∂X ∂X ∂X ∂X ∂x ∂X A P 1 Q R A P Q 2 R ∂ ∂ ∂ ∂ 2 + x3 x1 x2 x3 ∂X ∂X ∂X ∂X ∂x A P Q R 3 ∂ 2 ∂ ∂ ∂ ∂ ∂ 2 ∂ ∂ = 1 ε x1 x1 x2 x3 + x1 x2 x2 x3 J2 PQR ∂X ∂x ∂X ∂X ∂X ∂X ∂X ∂x ∂X ∂X A 1 P Q R P A 2 Q R ∂ ∂ ∂ 2 ∂ + x1 x2 x3 x3 ∂X ∂X ∂X ∂x ∂X P Q A 3 -
Euler, Newton, and Foundations for Mechanics
Euler, Newton, and Foundations for Mechanics Oxford Handbooks Online Euler, Newton, and Foundations for Mechanics Marius Stan The Oxford Handbook of Newton Edited by Eric Schliesser and Chris Smeenk Subject: Philosophy, History of Western Philosophy (Post-Classical) Online Publication Date: Mar 2017 DOI: 10.1093/oxfordhb/9780199930418.013.31 Abstract and Keywords This chapter looks at Euler’s relation to Newton, and at his role in the rise of Newtonian mechanics. It aims to give a sense of Newton’s complicated legacy for Enlightenment science and to point out that key “Newtonian” results are really due to Euler. The chapter begins with a historiographical overview of Euler’s complicated relation to Newton. Then it presents an instance of Euler extending broadly Newtonian notions to a field he created: rigid-body dynamics. Finally, it outlines three open questions about Euler’s mechanical foundations and their proximity to Newtonianism: Is his mechanics a monolithic account? What theory of matter is compatible with his dynamical laws? What were his dynamical laws, really? Keywords: Newton, Euler, Newtonian, Enlightenment science, rigid-body dynamics, classical mechanics, Second Law, matter theory Though the Principia was an immense breakthrough, it was also an unfinished work in many ways. Newton in Book III had outlined a number of programs that were left for posterity to perfect, correct, and complete. And, after his treatise reached the shores of Europe, it was not clear to anyone how its concepts and laws might apply beyond gravity, which Newton had treated with great success. In fact, no one was sure that they extend to all mechanical phenomena. -
From Continuum Mechanics to General Relativity
From continuum mechanics to general relativity Christian G. B¨ohmer∗ and Robert J. Downes† ∗Department of Mathematics, †Centre for Advanced Spatial Analysis, University College London, Gower Street, London, WC1E 6BT, UK 19 May 2014 This essay received a honorable mention in the 2014 essay competition of the Gravity Research Foundation. Abstract Using ideas from continuum mechanics we construct a theory of gravity. We show that this theory is equivalent to Einstein’s theory of general relativity; it is also a much faster way of reaching general relativity than the conventional route. Our approach is simple and natural: we form a very general model and then apply two physical assumptions supported by experimental evidence. This easily reduces our construction to a model equivalent to general relativity. Finally, arXiv:1405.4728v1 [gr-qc] 19 May 2014 we suggest a simple way of modifying our theory to investigate non- standard space-time symmetries. ∗[email protected] †[email protected] 1 Einstein himself, of course, arrived at the same Lagrangian but without the help of a developed field theory, and I must admit that I have no idea how he guessed the final result. We have had troubles enough arriving at the theory - but I feel as though he had done it while swimming underwater, blindfolded, and with his hands tied behind his back! Richard P. Feynman, Feynman lectures on gravitation, 1995. 1 Introduction Einstein’s theory of general relativity is one of the crowning achievements of modern science. However, the route from a special-relativistic mind game to a fully-fledged theory capable of producing physical predictions is long and tortuous. -
General Overview of Continuum Mechanics – Jose Merodio and Anthony D.Rosato
CONTINUUM MECHANICS – General Overview of Continuum Mechanics – Jose Merodio and Anthony D.Rosato GENERAL OVERVIEW OF CONTINUUM MECHANICS Jose Merodio Department of Continuum Mechanics and Structures, Universidad Politécnica de Madrid, Madrid, Spain Anthony D. Rosato Department of Mechanical Engineering, New Jersey Institute of Technology University Heights, Newark, NJ 0710, USA Keywords: Deformation, Motion, Balance Laws, Constitutive Equations, Stress, Strain, Energy, Entropy, Solids, Fluids Contents 1. Preliminary concepts: Definitions and theoretical framework 2. Kinematics of deformation 2.1. The continuum setting: Mathematical description 2.2. Displacement, deformation and motion 2.3. Stretch, shear and strain 2.4. Change of reference frame and objective tensors 3. Balance laws 3.1. Mass and conservation of mass 3.2. Force, torque and theory of stress 3.3. Euler’s laws of motion 3.4. Energy 3.5. Summary of equations and unknowns 4. Constitutive behavior 4.1. Basic concepts of constitutive theories 4.2. Isothermal Solid Mechanics 4.3. Isothermal Fluid Mechanics Glossary Bibliography Biographical Sketches SummaryUNESCO – EOLSS This chapter presents the essential concepts of continuum mechanics – starting from the basic notion ofSAMPLE deformation kinematics and CHAPTERS progressing to a sketch of constitutive behavior of simple materials. We intend neither to develop it with the mathematical sophistication used in the different topics related to this subject nor to develop every mathematical tool. On the contrary, we try to put in words the ideas behind this analysis as an initial approximation prior to further reading. This chapter is a general overview of the topic, an outline that covers the main and basic ideas. Our attempt is to develop a pedagogical approach accessible to a wide audience by introducing at every step the necessary mathematical machinery in conjunction with a physical interpretation. -
Universit`A Di Trento Dottorato Di Ricerca in Ingegneria Dei Materiali
Universit`adi Trento Dottorato di Ricerca in Ingegneria dei Materiali e delle Strutture XV ciclo BOUNDARY ELEMENT TECHNIQUES IN FINITE ELASTICITY MICHELE BRUN Relatore: Prof. Davide Bigoni Correlatore: Dr. Domenico Capuani Trento, Febbraio 2003 Universit`adegli Studi di Trento Facolt`adi Ingegneria Dottorato di Ricerca in Ingegneria dei Materiali e delle Strutture XV ciclo Esame finale: 14 febbraio 2003 Commissione esaminatrice: Prof. Giorgio Novati, Universit`adegli Studi di Trento Prof. Francesco Genna, Universit`adegli Studi di Brescia Prof. Alberto Corigliano, Politecnico di Milano Membri esperti aggiunti: Prof. John R. Willis, University of Cambridge, England Prof. Patrick J. Prendergast, Trinity College, Dublin Acknowledgements The research reported in this thesis has been carried out in the Department of Mechanical and Structural Engineering at the University of Trento during the last three years. I would like to gratefully acknowledge my supervisor, Prof. Davide Bigoni, for all the stimulating and brilliant discussions, for his non-stop encourage- ments, remarks, corrections and suggestions. I would like to thank Dr. Domenico Capuani for his accurate, precise and fruitful help. I wish to express my sincere gratitude to Prof. Giorgio Novati, and to all members of the group of mechanics at the University of Trento: Prof. Marco Rovati, Prof. Antonio Cazzani (and Lavinia), Dr. Roberta Springhetti (and child) and little scientists Arturo di Gioia, Giulia Frances- chini, Massimilano Margonari and Andrea Piccolroaz. I am indebted to Massimiliano Gei for his suggestions about the topic of my research and to Katia Bertoldi for the help in the development of numerical applications. Last but not the least, I thank my family and the unique Elisa for their love, patience and fully support. -
Introduction to Continuum Mechanics
Introduction to Continuum Mechanics David J. Raymond Physics Department New Mexico Tech Socorro, NM Copyright C David J. Raymond 1994, 1999 -2- Table of Contents Chapter1--Introduction . 3 Chapter2--TheNotionofStress . 5 Chapter 3 -- Budgets, Fluxes, and the Equations of Motion . ......... 32 Chapter 4 -- Kinematics in Continuum Mechanics . 48 Chapter5--ElasticBodies. 59 Chapter6--WavesinanElasticMedium. 70 Chapter7--StaticsofElasticMedia . 80 Chapter8--NewtonianFluids . 95 Chapter9--CreepingFlow . 113 Chapter10--HighReynoldsNumberFlow . 122 -3- Chapter 1 -- Introduction Continuum mechanics is a theory of the kinematics and dynamics of material bodies in the limit in which matter can be assumed to be infinitely subdividable. Scientists have long struggled with the question as to whether matter consisted ultimately of an aggregate of indivisible ‘‘atoms’’, or whether any small parcel of material could be subdivided indefinitely into smaller and smaller pieces. As we all now realize, ordinary matter does indeed consist of atoms. However, far from being indivisible, these atoms split into a staggering array of other particles under sufficient application of energy -- indeed, much of modern physics is the study of the structure of atoms and their constituent particles. Previous to the advent of quantum mechanics and the associated experimen- tal techniques for studying atoms, physicists tried to understand every aspect of the behavior of matter and energy in terms of continuum mechanics. For instance, attempts were made to characterize electromagnetic waves as mechani- cal vibrations in an unseen medium called the ‘‘luminiferous ether’’, just as sound waves were known to be vibrations in ordinary matter. We now know that such attempts were misguided. However, the mathematical and physical techniques that were developed over the years to deal with continuous distributions of matter have proven immensely useful in the solution of many practical problems on the macroscopic scale. -
Leonhard Euler: the First St. Petersburg Years (1727-1741)
HISTORIA MATHEMATICA 23 (1996), 121±166 ARTICLE NO. 0015 Leonhard Euler: The First St. Petersburg Years (1727±1741) RONALD CALINGER Department of History, The Catholic University of America, Washington, D.C. 20064 View metadata, citation and similar papers at core.ac.uk brought to you by CORE After reconstructing his tutorial with Johann Bernoulli, this article principally investigates provided by Elsevier - Publisher Connector the personality and work of Leonhard Euler during his ®rst St. Petersburg years. It explores the groundwork for his fecund research program in number theory, mechanics, and in®nitary analysis as well as his contributions to music theory, cartography, and naval science. This article disputes Condorcet's thesis that Euler virtually ignored practice for theory. It next probes his thorough response to Newtonian mechanics and his preliminary opposition to Newtonian optics and Leibniz±Wolf®an philosophy. Its closing section details his negotiations with Frederick II to move to Berlin. 1996 Academic Press, Inc. ApreÁs avoir reconstruit ses cours individuels avec Johann Bernoulli, cet article traite essen- tiellement du personnage et de l'oeuvre de Leonhard Euler pendant ses premieÁres anneÂes aÁ St. PeÂtersbourg. Il explore les travaux de base de son programme de recherche sur la theÂorie des nombres, l'analyse in®nie, et la meÂcanique, ainsi que les reÂsultats de la musique, la cartographie, et la science navale. Cet article attaque la theÁse de Condorcet dont Euler ignorait virtuellement la pratique en faveur de la theÂorie. Cette analyse montre ses recherches approfondies sur la meÂcanique newtonienne et son opposition preÂliminaire aÁ la theÂorie newto- nienne de l'optique et a la philosophie Leibniz±Wolf®enne.