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文章标题：向后Euler法在三维抛物型方程中的应用及ADI差分格式探讨在数学和科学计算领域，求解三维抛物型方程是一个重要且复杂的问题。
本文将从数值方法的角度出发，讨论如何使用向后Euler法来求解三维抛物型方程，并探讨ADI差分格式在该过程中的应用。
1. 三维抛物型方程的数值求解在数学建模和科学计算中，三维抛物型方程的数值求解是一个广泛应用的问题。
三维抛物型方程一般具有形如$\frac{\partial u}{\partial t} = \nabla \cdot (D\nabla u) + f$的形式，其中$D$为扩散系数，$f$为源项函数。
由于方程复杂性，传统的解析方法难以得到精确解，因此需要借助数值方法来进行求解。
2. 向后Euler法向后Euler法是一种常用的数值方法，用于离散化时间导数。
其基本思想是将时间导数用差分近似替代，通过迭代计算得到时间步数上的解。
对于三维抛物型方程，可以将时间方向的偏导数用向后Euler法进行离散化，从而得到数值解。
3. ADI差分格式ADI（Alternating Direction Implicit）差分格式是一种常用的隐式差分方法，用于求解多维偏微分方程。
其核心思想是将多维偏微分方程拆分为一维方程的求解，通过交替方向隐式差分得到整体方程的数值解。
在三维抛物型方程的求解中，ADI差分格式能够有效地提高计算效率和数值稳定性。
4. 主题回顾与总结通过本文的介绍，我们了解了向后Euler法在三维抛物型方程求解中的重要性和应用。
ADI差分格式作为一种高效的数值方法，为复杂方程的求解提供了可行的途径。
对于三维抛物型方程，我们可以利用向后Euler法结合ADI差分格式，得到高质量、深度和广度兼具的数值解。
5. 个人观点与理解在数值计算中，选择合适的数值方法对于求解复杂方程至关重要。
向后Euler法作为一种简单而有效的数值方法，为我们提供了一种直观且可行的思路。
而ADI差分格式则在多维问题的求解中发挥着重要作用，其交替方向求解的思想能够有效地提高计算效率和数值稳定性。
Differential equationNot to be confused with Difference equation.Stokes differential equations used to simulate airflow around an obstruction.ClassificationSolutionVisualization of heat transfer in a pump casing, created by solving the heat equation. Heat is being generated internally in the casing and being cooled at the boundary, providing a steady state temperature distribution.A differential equation is amathematical equation that relatessome function of one or more variableswith its derivatives. Differentialequations arise whenever adeterministic relation involving somecontinuously varying quantities(modeled by functions) and their ratesof change in space and/or time(expressed as derivatives) is known orpostulated. Because such relations areextremely common, differentialequations play a prominent role inmany disciplines includingengineering, physics, economics, andbiology.Differential equations aremathematically studied from severaldifferent perspectives, mostlyconcerned with their solutions — the set of functions that satisfy the equation. Only the simplest differential equations are solvable by explicit formulas; however, some properties of solutions of a given differential equation may be determined without finding their exact form. If a self-contained formula for the solution is not available, the solution may be numerically approximated using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have beendeveloped to determine solutions with a given degree of accuracy.ExampleFor example, in classical mechanics, the motion of a body is described by its position and velocity as the time value varies. Newton's laws allow one (given the position, velocity, acceleration and various forces acting on the body) to express these variables dynamically as a differential equation for the unknown position of the body as a function of time.In some cases, this differential equation (called an equation of motion) may be solved explicitly.An example of modelling a real world problem using differential equations is the determination of the velocity of a ball falling through the air, considering only gravity and air resistance. The ball's acceleration towards the ground is the acceleration due to gravity minus the acceleration due to air resistance. Gravity is considered constant, and air resistance may be modeled as proportional to the ball's velocity. This means that the ball's acceleration, which is a derivative of its velocity, depends on the velocity (and the velocity depends on time). Finding the velocity as a function of time involves solving a differential equation and verifying its validity.Directions of studyThe study of differential equations is a wide field in pure and applied mathematics, physics, and engineering. All of these disciplines are concerned with the properties of differential equations of various types. Pure mathematics focuses on the existence and uniqueness of solutions, while applied mathematics emphasizes the rigorous justification of the methods for approximating solutions. Differential equations play an important role in modelling virtually every physical, technical, or biological process, from celestial motion, to bridge design, to interactions between neurons. Differential equations such as those used to solve real-life problems may not necessarily be directly solvable, i.e. do not have closed form solutions. Instead, solutions can be approximated using numerical methods.Mathematicians also study weak solutions (relying on weak derivatives), which are types of solutions that do not have to be differentiable everywhere. This extension is often necessary for solutions to exist.The study of the stability of solutions of differential equations is known as stability theory.NomenclatureThe theory of differential equations is well developed and the methods used to study them vary significantly with the type of the equation.Ordinary and partial•An ordinary differential equation (ODE) is a differential equation in which the unknown function (also known as the dependent variable) is a function of a single independent variable. In the simplest form, the unknown function is a real or complex valued function, but more generally, it may be vector-valued or matrix-valued: this corresponds to considering a system of ordinary differential equations for a single function.Ordinary differential equations are further classified according to the order of the highest derivative of the dependent variable with respect to the independent variable appearing in the equation. The most important cases for applications are first-order and second-order differential equations. For example, Bessel's differential equation(in which y is the dependent variable) is a second-order differential equation. In the classical literature a distinction is also made between differential equations explicitly solved with respect to the highest derivative and differential equations in an implicit form. Also important is the degree, or (highest) power, of the highest derivative(s) in the equation (cf. : degree of a polynomial). A differential equation is called a nonlinear differential equation if its degree is not one (a sufficient but unnecessary condition).• A partial differential equation (PDE) is a differential equation in which the unknown function is a function of multiple independent variables and the equation involves its partial derivatives. The order is defined similarly to the case of ordinary differential equations, but further classification into elliptic, hyperbolic, and parabolic equations, especially for second-order linear equations, is of utmost importance. Some partial differentialequations do not fall into any of these categories over the whole domain of the independent variables and they are said to be of mixed type.Linear and non-linearBoth ordinary and partial differential equations are broadly classified as linear and nonlinear.• A differential equation is linear if the unknown function and its derivatives appear to the power 1 (products of the unknown function and its derivatives are not allowed) and nonlinear otherwise. The characteristic property of linear equations is that their solutions form an affine subspace of an appropriate function space, which results in much more developed theory of linear differential equations. Homogeneous linear differential equations are a further subclass for which the space of solutions is a linear subspace i.e. the sum of any set of solutions or multiples of solutions is also a solution. The coefficients of the unknown function and its derivatives in a linear differential equation are allowed to be (known) functions of the independent variable or variables; if these coefficients are constants then one speaks of a constant coefficient linear differential equation.•There are very few methods of solving nonlinear differential equations exactly; those that are known typically depend on the equation having particular symmetries. Nonlinear differential equations can exhibit verycomplicated behavior over extended time intervals, characteristic of chaos. Even the fundamental questions of existence, uniqueness, and extendability of solutions for nonlinear differential equations, and well-posedness of initial and boundary value problems for nonlinear PDEs are hard problems and their resolution in special cases is considered to be a significant advance in the mathematical theory (cf. Navier–Stokes existence and smoothness).However, if the differential equation is a correctly formulated representation of a meaningful physical process, then one expects it to have a solution.Linear differential equations frequently appear as approximations to nonlinear equations. These approximations are only valid under restricted conditions. For example, the harmonic oscillator equation is an approximation to the nonlinear pendulum equation that is valid for small amplitude oscillations (see below).ExamplesIn the first group of examples, let u be an unknown function of x, and c and ω are known constants.•Inhomogeneous first-order linear constant coefficient ordinary differential equation:•Homogeneous second-order linear ordinary differential equation:•Homogeneous second-order linear constant coefficient ordinary differential equation describing the harmonic oscillator:•Inhomogeneous first-order nonlinear ordinary differential equation:•Second-order nonlinear (due to sine function) ordinary differential equation describing the motion of a pendulum of length L:In the next group of examples, the unknown function u depends on two variables x and t or x and y.•Homogeneous first-order linear partial differential equation:•Homogeneous second-order linear constant coefficient partial differential equation of elliptic type, the Laplace equation:•Third-order nonlinear partial differential equation, the Korteweg–de Vries equation:Related concepts• A delay differential equation (DDE) is an equation for a function of a single variable, usually called time, in which the derivative of the function at a certain time is given in terms of the values of the function at earlier times.• A stochastic differential equation (SDE) is an equation in which the unknown quantity is a stochastic process and the equation involves some known stochastic processes, for example, the Wiener process in the case of diffusion equations.• A differential algebraic equation (DAE) is a differential equation comprising differential and algebraic terms, given in implicit form.Connection to difference equationsSee also: Time scale calculusThe theory of differential equations is closely related to the theory of difference equations, in which the coordinates assume only discrete values, and the relationship involves values of the unknown function or functions and values at nearby coordinates. Many methods to compute numerical solutions of differential equations or study the properties of differential equations involve approximation of the solution of a differential equation by the solution of a corresponding difference equation.Universality of mathematical descriptionMany fundamental laws of physics and chemistry can be formulated as differential equations. In biology and economics, differential equations are used to model the behavior of complex systems. The mathematical theory of differential equations first developed together with the sciences where the equations had originated and where the results found application. However, diverse problems, sometimes originating in quite distinct scientific fields, may give rise to identical differential equations. Whenever this happens, mathematical theory behind the equations can be viewed as a unifying principle behind diverse phenomena. As an example, consider propagation of light and sound in the atmosphere, and of waves on the surface of a pond. All of them may be described by the same second-order partial differential equation, the wave equation, which allows us to think of light and sound as forms of waves, much like familiar waves in the water. Conduction of heat, the theory of which was developed by Joseph Fourier, is governed by another second-order partial differential equation, the heat equation. It turns out that many diffusion processes, while seemingly different, are described by the same equation; the Black–Scholes equation in finance is, for instance, related to the heat equation.Notable differential equationsPhysics and engineering•Newton's Second Law in dynamics (mechanics)•Euler–Lagrange equation in classical mechanics•Hamilton's equations in classical mechanics•Radioactive decay in nuclear physics•Newton's law of cooling in thermodynamics•The wave equation•Maxwell's equations in electromagnetism•The heat equation in thermodynamics•Laplace's equation, which defines harmonic functions•Poisson's equation•Einstein's field equation in general relativity•The Schrödinger equation in quantum mechanics•The geodesic equation•The Navier–Stokes equations in fluid dynamics•The Diffusion equation in stochastic processes•The Convection–diffusion equation in fluid dynamics•The Cauchy–Riemann equations in complex analysis•The Poisson–Boltzmann equation in molecular dynamics•The shallow water equations•Universal differential equation•The Lorenz equations whose solutions exhibit chaotic flow.Biology•Verhulst equation – biological population growth•von Bertalanffy model – biological individual growth•Lotka–Volterra equations – biological population dynamics•Replicator dynamics – found in theoretical biology•Hodgkin–Huxley model – neural action potentialsEconomics•The Black–Scholes PDE•Exogenous growth model•Malthusian growth model•The Vidale–Wolfe advertising modelReferences•P. Abbott and H. Neill, Teach Yourself Calculus, 2003 pages 266-277•P. Blanchard, R. L. Devaney, G. R. Hall, Differential Equations, Thompson, 2006• E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, 1955• E. L. Ince, Ordinary Differential Equations, Dover Publications, 1956•W. Johnson, A Treatise on Ordinary and Partial Differential Equations[2], John Wiley and Sons, 1913, in University of Michigan Historical Math Collection [3]• A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations (2nd edition), Chapman & Hall/CRC Press, Boca Raton, 2003. ISBN 1-58488-297-2.•R. I. Porter, Further Elementary Analysis, 1978, chapter XIX Differential Equations•Teschl, Gerald (2012). Ordinary Differential Equations and Dynamical Systems[4]. Providence: American Mathematical Society. ISBN 978-0-8218-8328-0.• D. Zwillinger, Handbook of Differential Equations (3rd edition), Academic Press, Boston, 1997.[1]/w/index.php?title=Template:Differential_equations&action=edit[2]/cgi/b/bib/bibperm?q1=abv5010.0001.001[3]/u/umhistmath/[4]http://www.mat.univie.ac.at/~gerald/ftp/book-ode/External links•Lectures on Differential Equations (/courses/mathematics/18-03-differential-equations-spring-2010/video-lectures/) MIT Open CourseWare Videos•Online Notes / Differential Equations (/classes/de/de.aspx) Paul Dawkins, Lamar University•Differential Equations (/diffeq/diffeq.html), S.O.S. Mathematics•Differential Equation Solver (/tools/differential_equation_solver/) Java applet tool used to solve differential equations.•Introduction to modeling via differential equations (/mat/u-u/en/ differential_equations_intro.htm) Introduction to modeling by means of differential equations, with critical remarks.•Mathematical Assistant on Web (http://user.mendelu.cz/marik/maw/index.php?lang=en&form=ode) Symbolic ODE tool, using Maxima•Exact Solutions of Ordinary Differential Equations (http://eqworld.ipmnet.ru/en/solutions/ode.htm)•Collection of ODE and DAE models of physical systems (/research/models.htm) MATLAB models•Notes on Diffy Qs: Differential Equations for Engineers (/diffyqs/) An introductory textbook on differential equations by Jiri Lebl of UIUC•Khan Academy Video playlist on differential equations (/math/ differential-equations) Topics covered in a first year course in differential equations.•MathDiscuss Video playlist on differential equations (/category/courses/ solutions-differential-equations/homogeneous-linear-systems/)Article Sources and Contributors8Article Sources and ContributorsDifferential equation Source: /w/index.php?oldid=610771276 Contributors: 17Drew, After Midnight, Ahoerstemeier, Alarius, Alfred Centauri, Amahoney, AndreiPolyanin, Andres, AndrewHowse, Andycjp, Andytalk, AngryPhillip, Anonymous Dissident, Antoni Barau, Antonius Block, Anupam, Apmonitor, Arcfrk, Asdf39, Asyndeton, Attilios,Babayagagypsies, Baccala@, Baccyak4H, Bejohns6, Bento00, Berland, Bidabadi, Bigusbry, BillyPreset, Bob.v.R, Bolatbek, Brandon, Bryanmcdonald, Btyner, Bygeorge2512,Callumds, Charles Matthews, Christian75, Chtito, Cispyre, Cmprince, Coginsys, ConMan, Cxz111, Cybercobra, DAJF, Danski14, Dbroadwell, Ddxc, Delaszk, DerHexer, Dewritech, Difu Wu, Djordjes, DominiqueNC, Donludwig, Dpv, Dr sarah madden, Drmies, DroEsperanto, Duoduoduo, Dysprosia, EconoPhysicist, Elwikipedista, Epicgenius, EricBright, Erin.Annette.Brown,Estudiarme, F=q(E+v^B), Fintor, Fioravante Patrone, Fioravante Patrone en, Flameturtle, Friend of the Facts, FutureTrillionaire, Gabrielleitao, Gandalf61, Gauss, Genedronek, Geni, Giftlite,GoingBatty, Gombang, Grenavitar, Haham hanuka, Hamiltondaniel, Harry, Haruth, Haseeb Jamal, Heikki m, Holmes1900, Ilya Voyager, Iquseruniv, Iulianu, Izodman2012, J arino, J.delanoy, Ja 62, Jak86, JamesBWatson, Jao, Jarble, Jauhienij, Jayden54, Jeancey, Jersey Devil, Jim Sukwutput, Jim.belk, Jim.henderson, JinJian, Jitse Niesen, JohnOwens, Johndoeisnotmyname, JorisvS,Julesd, K-UNIT, Kayvan45622, KeithJonsn, Kensaii, Khalid Mahmood, Klaas van Aarsen, Kr5t, Krushia, LOL, Lambiam, Lavateraguy, Lethe, LibLord, Linas, Lumos3, Madmath789, Mandarax, Mankarse, MarSch, Martastic, Martynas Patasius, Maschen, Math.geek3.1415926, Matqkks, Mattmnelson, Maurice Carbonaro, Maxis ftw, Mazi, McVities, Mduench, Mets501, Mh, MichaelHardy, Mindspillage, MisterSheik, Mohan1986, Mossaiby, Mpatel, MrOllie, Mtness, Mysidia, Nik-renshaw, Nkayesmith, Norm mit, Okopecz, Oleg Alexandrov, Opelio, Pahio, Parusaro, Paul August, Paul Matthews, Paul Richter, PavelSolin, Pgk, Phoebe, Pine, Pinethicket, Pratyya Ghosh, PseudoSudo, Qwerty Binary, Qzd800, R'n'B, Rama's Arrow, Randomguess, Reallybored999, RexNL, Reyk, RichMorin, Robin S, Romansanders, Rosasco, Ruakh, SDC, SFC9394, SakeUPenn, Salix alba, Sam Staton, Sampathsris, Sardanaphalus, Senoreuchrestud, Silly rabbit, Siroxo,Skakkle, Skypher, SmartPatrol, Snowjeep, Spirits in the Material, Starwiz, Suffusion of Yellow, Sverdrup, Symane, TVBZ28, TYelliot, Tannkrem, Tbhotch, Tbsmith, TexasAndroid, Tgeairn, The Hybrid, The Thing That Should Not Be, Timelesseyes, Tranum1234567890, Tsirel, Tuseroni, User A1, Vanished User 0001, Vishwanathnm, Vthiru, Waffleguy4, Waldir, Waltpohl, Wavelength, Wclxlus, Wihenao, Willtron, Winterheart, Wsears, XJaM, Yafujifide, Zepterfd, ﺪﺟﺎﺳ ﺪﺠﻣﺍ ﺪﺟﺎﺳ, 363 anonymous editsImage Sources, Licenses and ContributorsFile:Airflow-Obstructed-Duct.png Source: /w/index.php?title=File:Airflow-Obstructed-Duct.png License: Public Domain Contributors: Original uploader was User A1 at en.wikipediaFile:Elmer-pump-heatequation.png Source: /w/index.php?title=File:Elmer-pump-heatequation.png License: Creative Commons Attribution-Sharealike 3.0Contributors: Christian1985, Crimerob, Kri, User A1, 2 anonymous editsLicenseCreative Commons Attribution-Share Alike 3.0///licenses/by-sa/3.0/。
Latex微分熵计算公式表述在热力学和统计物理学中，熵是描述系统无序程度的物理量。
微分熵则是熵的微分，表示系统在微小变化下的熵变化。
在物理、化学、工程等领域中，对于微分熵的计算常常需要使用Latex进行公式表述。
下面将介绍一些常见的微分熵计算公式的Latex表述方法。
1. 气体的微分熵计算公式对于理想气体，其微分熵计算公式可以表示为：\begin{equation}ds = \frac{C_v}{T} dT + \frac{R}{V} dV\end{equation}其中，$s$表示熵，$T$表示温度，$V$表示体积，$C_v$表示定容热容，$R$表示气体常数。
这个公式描述了理想气体在温度和体积微小变化下的熵变化。
2. 固体和液体的微分熵计算公式对于固体和液体，其微分熵计算公式可以表示为：\begin{equation}ds = \frac{C}{T} dT\end{equation}其中，$s$表示熵，$T$表示温度，$C$表示热容。
这个公式描述了固体和液体在温度微小变化下的熵变化。
3. 多组分系统的微分熵计算公式对于多组分系统，其微分熵计算公式可以表示为：\begin{equation}ds = \sum_{i=1}^{n} \frac{C_{vi}}{T} dT + \sum_{i=1}^{n}\frac{R_i}{V} dV\end{equation}其中，$s$表示熵，$T$表示温度，$V$表示体积，$C_{vi}$表示第i个组分的定容热容，$R_i$表示第i个组分的气体常数。
这个公式描述了多组分系统在温度和体积微小变化下的熵变化。
4. 统计物理学中的微分熵计算公式在统计物理学中，微分熵可以根据系统的能级分布进行计算。
假设系统的能级分布为$E_i$，在温度微小变化下，其微分熵计算公式可以表示为：\begin{equation}ds = \frac{1}{T} \sum_{i} \frac{E_i}{e^{\frac{E_i}{kT}}-1} dE_i\end{equation}其中，$s$表示熵，$T$表示温度，$E_i$表示第i个能级，$k$表示玻尔兹曼常数。
a r X i v :g r -q c /9711065v 2 9 D e c 1997THE SPEED OF LIGHT AS A DILATON FIELDWalter WyssDepartment of PhysicsUniversity of ColoradoBoulder,CO 80309Through dimensional analysis,eliminating the physical time,we identify the speed of light as a dilaton field.This leads to a restmass zero,spin zero gauge field which we call the speedon field.The complete Lagrangian for gravitational,electromagnetic and speedon field interactions with a charged scalar field,representing matter,is given.We then find solutions for the gravitational-electromagnetic-speedon field equations.This then gives an expression for the speed of light.I.IntroductionWe are interested in the concept of physical time.Physical time is defined with the help of a periodical physical system,e.g.,an atomic clock.At present we have adopted a universal time currency.But what about local time currencies?:commonly one uses the gravitational redshift as an exchange rate;an atomic clock in Boulder has a higher frequency than a similar atomic clock in Paris.Are there other contributions to this exchange rate?To look for an answer we formulated[1]electrodynamics in such a way that the physical time never occurs.Only the geometric time x o,which is related to the physical time t by x o=ct occurs.c is the speed of light and knowing the geometric time one can recover the physical time.In Special Relativity and also in Einstein’s Theory of Gravity only the geometric time occurs;it has the physical dimension of a length.The speed of light only enters if one wants to convert to physical time.The speed of light thus does not have to be a constant,but is related to the concept of physical time.Indeed,in the formulation of electrodynamics,written independently of physical time,the speed of light enters as a scale factor.It can thus be interpreted as a dilatonfield[2].Since this field is related to the speed of light we call it the speedonfield.It belongs to the restmass zero,spin zero representation of the Lorentz group.The corresponding elementary agent we called the“speedon.”We thus have a trinity of restmass zero gauge particles:speedon (spin zero),photon(spin one)and graviton(spin two).For interactions with a charged scalarfield representing matter each of these gauge fields contribute through their own covariant derivative.In Chapter II we give the general formulation for gravitational interactions as described by a Lagrange variational principle.In Chapter III we present the complete interaction between the gravitationalfield,the electromagneticfield,the speedonfield and a charged scalarfield representing matter.In Chapter IV we solve the equations of motion in the absence of the matterfield. This gives a solution for the gravitationalfield and the electromagneticfield in terms ofthe speedonfield.Turning offthe electromagneticfield wefind a expression for the pure speedonfield and for the corresponding speed of light.Appendix A describes the scaling of the electromagneticfield and the motivation to introduce the speed of light as a dilatonfield.In Appendix C the gravitational interaction with a restmass scalarfield is revisited.II.Gravitational InteractionsWith the notation in[3]gravitational interactions are given through a variational principle where the action is given byA= dx√gL(g,φ)= dxL(II.1) and where dx is the4-volume element,L(g,φ)is the nongravitational Lagrangian,withφany multicomponentfield,and8πGκ=(II.3)c2has the physical dimension[G o]=M−1L.(II.4)Throughout,M=mass,L=length,T=physical time,Q=charge.Thus G o does not depend on the physical time,i.e.,the concept of a second,andκ=8πG o(II.5) The action A with physical dimension[A]=L2(II.6)also does not depend on physical time.The physical dimension of the nongravitational Lagrangian L(g,φ)is[L(g,φ)]=ML−3,(II.7) i.e.,a mass density.For the Euler derivative with respect to the gravitationalfield gαβwe getǫ(gαβ)=∂∂gαβ,µ(II.8)ǫ(gαβ)[√g[Rαβ−1gL(g,φ)]≡Mαβ,(II.10) where Mαβis the so-called gravitational stress tensor.Introducing TαβthroughMαβ≡−1gTαβ(II.11) we then get Einstein’s equationRαβ−1∂gαβ,µ(II.13)In addition we have the equations of motion for thefieldφǫ(φ)[√2)The right-hand side of Einstein’s equation(II.12)is not arbitrary but is related toMαβ,which is an Euler derivative.If one looks at the matter Lagrangian L(η,φ),i.e., the Lagrangian L(g,φ)where the gravitationalfield g is replaced by the Minkowski metricη,then there is the concept of an energy-momentum tensor[5].The right-hand side Tαβin Einstein’s equation,evaluated for g=ηis exactly equal to the energy-momentum tensor belonging to L(η,φ)[6].That this is true for anyfield is highly nontrivial[7]and is a consequence from the fact that the energy-momentum tensor for any gravitational interaction vanishes identically.3)For dust one usually takesTαβ=̺uαuβ,uαuα=1(II.15) This,however,does notfit in the above scheme;the closest we can get to such a case is by looking at a massless scalarfield,i.e.,a dilatonfield.Thisfield then also has an equation of motion.For mathematical consistency the right-hand side of Einstein’s equation should be an Euler derivative.III.Graviton,Photon,Speedon and a Charged Scalar FieldHere we give the most general theory of the interactions between a charged scalarfield and gaugefields belonging to the rest mass zero[8].This involves the restmass zerofields of spin2(gravity),spin one(electrodynamics)and spin zero(the speed of light).These are all gaugefields and bring their own covariant derivative.The restmass zero,spin zero field,which is a dilatonfield,is related to the scaling of the electromagneticfield such that the physical time never enters the corresponding Lagrangian(Appendix A).We thus have thefields gαβ,Lα,S,φ+,φwith the physical dimensions[gαβ]=0,[Lα]=MQ−1,[S]=0,[φ+]=[φ]=0(III.1) The action is given byA= dxL(III.2)whereL=√gL o(Lα)−2√gL M(φ).(III.3) The coupling constantλo isλo=2G oc2(III.4)with K being the Coulomb constant.K o andλo have the physical dimensions[K o]=MLQ−2,[λo]=M−2Q2(III.5)All these constants do not depend on physical time.In what follows all indices are raised with the inverse gravitationalfield gαβand lowered with gαβ.The Lagrangian for the electromagneticfield{Lα}is given byL o(Lα)=1We now get the equations of motion (i)Rαβ−1The gauge group for the speedonfield is given byδ∗S=const.(III.23)IV.Gravity-Electromagnetism-Speedon FieldHere we study the gravitational interaction in the absence of matter,i.e.,the La-grangian(III.3)withφ=0.We are then left with the LagrangianL=√gL o(Lα)−2√gαβR=λo FαµFβµ−gαβL o(Lα) +2SαSβ−gαβL o(S)(IV.2)2DµFµα=0(IV.3)DαSα=0(IV.4)From Appendix B wefind the solutions of these equations in the case of the static Schwarzschild metric and with the speedonfield S as the independent variable assinh[c2S+c3]r=−ac1sinh[c2S+c3] 2(IV.6)c21e B=c2aλo b2At this time we are only interested in the speedonfield.Turning offthe electromagnetic interaction meansλo=0.This implies c2=0and c1=1.The solutions then becomear=−(IV.14)cosh2[S]or as functions of the radial variable r we getS=−ln a1+ ar 2(IV.17) This is the solution of gravitational interaction with a massless scalarfield for the special valueα=0;see Appendix C.From(A.22)we now get an expression for the speed of lightc21+ a r(IV.18)or written differentlyc2a1+ rN(1) 2=g oo(2)Let R be a reference radius where the rate of the clock is N R and let N be the rate of a similar clock at the position r.For the exterior Schwarzschild metric one gets thenNrR 1/2(IV.21)where a is the Schwarzschild radiusa=2G o M(IV.22) For r>R and aN R =1+1R−a8 3 a R a r 2+...(IV.23)This applies in particular to similar atomic clocks,one located in Paris and one in Boulder; the one in Boulder ticks faster.For the gravitationalfield due to the speedonfield alone(IV.16)there is no gravitational redshift.We now compare the speed of light c R at the reference radius R with the speed of light c at the position r.For the integration constant a in(IV.18)we take the Schwarzschild radius(IV.22).Thenc R=c o R 2−a1+ a r 1/2(IV.25) gives the expansion for r>R,ac R =1+1R−a8 a R a r 2+...(IV.26)This expression agrees with(IV.25)up tofirst order.V.ConclusionIn electrodynamics,elimination of the physical time through dimensional analysis, identifies the speed of light as a dilatonfield.In the Lagrange formalism for gravitational interactions with the electromagneticfield and a charged matterfield this point of view introduces a massless scalarfield S that we call the speedonfield.In the absence of the matterfield we found solutions for the gravitational-electromagnetic-speedonfield equa-tions.If we now turn offthe electromagnetic interaction we are left with the speedonfield. This is a particular solution of the gravitational interaction with a massless scalarfield. The pure speedonfield gives an expression for the speed of light.For large values of the radial coordinate the speed of light becomes constant and for small values of the radial coordinate the speed of light goes to zero.This then raises the question about the mean-ing of a local physical time.The analytic solution in the presence of a matterfield should give some more information.According to Vandyck[2]very accurate redshift magnitude curves should provide information about the presence of a dilatonfield.It is interesting to observe that gravitatoinal interaction as treated in this paper introduces an“index of refraction”for the Universe.Appendix A:Scaling the Electromagnetic FieldElectrodynamics is described by a vectorfield{Aµ},the so-called vectorpotential, which belongs to the restmass zero,spin one representation of the Lorentz group.The physical dimension of Aµis[Aµ]=ML2T−2Q−1(A.1) with c denoting the speed of light,t the physical time and{Aµ}=(A o,A)(A.2) the electricfield is given byE=1∂t−grad A o(A.3)and the magneticfield byB=−1c2Aµ.A.8) Lµhas the physical dimension[Lµ]=MQ−1(A.9)which does not depend on the physical time.Maxwell’s equations follow from a variational principle with the LagrangianL o(Lµ)=14FµνFνµ(A.10)whereK o=K∂x o−grad L o (A.17)B=−c curl L(A.18) t=1The scaling can be written asLµ=c2oc2oAµ(A.20)with c o as a reference speed of light.ThenLµ=e−S1c2o (A.12) S is thus a dilationfield[2]and belongs to the restmass zero,spin zero representation of the Loretnz group.Since S is related to the speed of light we call it the speedonfield.Appendix B:The Schwarzschild metric and the solutionsThe Schwarzschild metric is given byds2=e A(dx o)2−e B(dr)2−r2{dθ2+sin2θdϕ2}(B.1) For the static case A=A(r),B=B(r).With g=−Det(gαβ)wefind√2(A+B)r2sinθ(B.2) and the Einstein tensor[3]Gαβ=gαγ[Rγβ−1r2dr e−Bd2r d4e−BdAdr(A+B)(B.6)G33=G22(B.7)All other components vanish.We now introduce the auxiliary functionsf=e12(A−B)(B.9) Thene A=1h(B.11)and√r2df−r(B.13)G11=G00−2h d fr2f2(B.14)G22=1dr[r2G11]−1drdrln fh2f2dLdr(B.16)For the speedonfield the Lagrangian readsL o(S)=−hdrdS2λo1drdLrfdSdr(B.18)G11=1f2dLdr−hdrdS2λo1drdLrfdSdr(B.20)dfdL dr rh dSdr =bfdr =af d fdrdSdr=f 1−1r2 (B.26)Introducing the dimensionless quantitiesr=ax,h=ak(B.27) and the abbreviation′≡da1xk(B.29)f′2λo b x2 (B.31) As in[9,10]we look upon the speedonfield as the independent variable and denote by ·≡da˙xx2 (B.35)withµ2=1a 2.The boundary conditions we impose areS→0⇒f→1,x→∞,kx=F(B.37)Then from(B.33)we get˙fF(B.38) This leads tox=x o e F dS(B.39)f=f o e 1F]dS 1−µ21x o(B.43)equation(B.42)reads˙FF+F]dS−βe (1dS ˙F F2+F αe [1F−F]dS (B.45)Adding and subtracting equations(B.44)and(B.45)and using the abbreviationsX=˙FF(B.46)Y=˙FF(B.47)results in2αe [1F˙X(B.48)2βe [1F˙Y(B.49) and2F=X−Y(B.50)2 ˙F F =X+Y(B.51) Taking the logarithmic derivative of equations(B.48)and(B.49)we get1F +¨XF−F=−˙F˙Y(B.53)resulting in the equations¨X=X˙X(B.54)¨Y=Y˙Y(B.55)Observe that from equations(B.10)and(B.11)we have the relations˙X=2F2e B(B.56)˙Y=2µ2e A(B.57)Asymptoticflatness then demands˙Y(0)=2µ2(B.58)and that both˙X and˙Y are positive.The solutions of(B.54)and(B.55)that are compatible with the boundary conditions (B.36),(B.58)are given byX=−2c1coth[c1S](B.59)Y=−2c2coth[c2S+c3](B.60)From(B.50)and(B.51)we then getF=c2coth[c2S+c3]−c1coth[c1S](B.61)and the conditionc21=1+c22(B.62) The condition(B.58)then becomesc22=µ2sinh2[c3](B.63)We thenfind the solutions of(B.32)to(B.35)x=−c1sinh[c1S](B.64)f=c1sinh[c3]ac2sinh[c2S+c3] 2(B.68)e B=c21Appendix C:The Massless ScalarfieldGravitational interaction with a massless scalarfield alone is given by settingλo=0 in(IV.1).This corresponds to settingµ=0in(B.57).The equations(B.59)and(B.60) then are replaced byX=−2c1coth[c1S](C.1)Y=2α(C.2)whereαis a constant.From(B.50)and(B.51)we then getF=−α−c1coth[c1S](C.3)and the conditionc21=1+α2(C.4) The complete solution[4]is then given by1x=−c1e−αS(C.5)c1cosh[c1S]+αsinh[c1S]k=−α−c1coth[c1S](C.6)e A=e2αS(C.7)e B= c14.W.Wyss,“The Energy-Momentum Tensor for Gravitational Interactions”(to be pub-lished).5.W.Wyss,“The Energy-Momentum Tensor in Classical Field Theory”(to be pub-lished).6.W.Wyss,“The Relation between the Gravitational Stress Tensor and the Energy-Momentum Tensor”(to be published in Helv.Phys.Acta).7.C.W.Misner,K.S.Thorne,J.A.Wheeler,“Gravitation,”W.H.Freeman and Com-pany,1973,p.504.8.Frank E.Schroeck,Jr.,“Quantum Mechanics on Phase Space,”Kluwer AcademicPublishers,1996,p.453.9.M.Wyman,Phys.Rev.D24(1981),839.10.K.Schmoltzi,Th.Schucker,Phys.Lett.A161(1991),212.21。
5Formulation and Solution StrategiesThe previous chapters have now developed the basicfield equations of elasticity theory.Theseresults comprise a system of differential and algebraic relations among the stresses,strains,anddisplacements that express particular physics at all points within the body under investigation.In this chapter we now wish to complete the general formulation byfirst developing boundaryconditions appropriate for use with thefield equations.These conditions specify the physicsthat occur on the boundary of body,and generally provide the loading inputs that physicallycreate the interior stress,strain,and displacementfields.Although thefield equations are thesame for all problems,boundary conditions are different for each problem.Therefore,properdevelopment of boundary conditions is essential for problem solution,and thus it is importantto acquire a good understanding of such development bining thefieldequations with boundary conditions then establishes the fundamental boundary value problemsof the theory.This eventually leads us into two different formulations,one in terms ofdisplacements and the other in terms of stresses.Because these boundary value problems aredifficult to solve,many different strategies have been developed to aid in problem solution.Wereview in a general way several of these strategies,and later chapters incorporate many ofthese into the solution of specific problems.5.1Review of Field EquationsBefore beginning our discussion on boundary conditions we list here the basicfield equationsfor linear isotropic elasticity.Appendix A includes a more comprehensive listing of allfieldequations in Cartesian,cylindrical,and spherical coordinate systems.Because of its ease of useand compact properties,our formulation uses index notation.Strain-displacement relations:e ij¼12(u i,jþu j,i)(5:1:1)Compatibility relations:e ij,klþe kl,ijÀe ik,jlÀe jl,ik¼0(5:1:2)83Equilibrium equations:s ij,jþF i¼0(5:1:3) Elastic constitutive law(Hooke’s law):s ij¼l e kk d ijþ2m e ije ij¼1þnEs ijÀnEs kk d ij(5:1:4)As mentioned in Section2.6,the compatibility relations ensure that the displacements arecontinuous and single-valued and are necessary only when the strains are arbitrarily specified.If,however,the displacements are included in the problem formulation,the solution normallygenerates single-valued displacements and strain compatibility is automatically satisfied.Thus,in discussing the general system of equations of elasticity,the compatibility relations(5.1.2)are normally set aside,to be used only with the stress formulation that we discuss shortly.Therefore,the general system of elasticityfield equations refers to the15relations(5.1.1),(5.1.3),and(5.1.4).It is convenient to define this entire system using a generalized operatornotation asJ{u i,e ij,s ij;l,m,F i}¼0(5:1:5) This system involves15unknowns including3displacements u i,6strains e ij,and6stresses s ij.The terms after the semicolon indicate that the system is also dependent on two elastic materialconstants(for isotropic materials)and on the body force density,and these are to be given apriori with the problem formulation.It is reassuring that the number of equations matches thenumber of unknowns to be determined.However,this general system of equations is of suchcomplexity that solutions by using analytical methods are essentially impossible and furthersimplification is required to solve problems of interest.Before proceeding with development ofsuch simplifications,it is usefulfirst to discuss typical boundary conditions connected with theelasticity model,and this leads us to the classification of the fundamental problems.5.2Boundary Conditions and Fundamental ProblemClassificationsSimilar to otherfield problems in engineering science(e.g.,fluid mechanics,heat conduction,diffusion,electromagnetics),the solution of system(5.1.5)requires appropriate boundaryconditions on the body under study.The common types of boundary conditions for elasticityapplications normally include specification of how the body is being supported or loaded.Thisconcept is mathematically formulated by specifying either the displacements or tractions atboundary points.Figure5-1illustrates this general idea for three typical cases includingtractions,displacements,and a mixed case for which tractions are specified on boundary S tand displacements are given on the remaining portion S u such that the total boundary is givenby S¼S tþS u.Another type of mixed boundary condition can also occur.Although it is generally not possible to specify completely both the displacements and tractions at the same boundarypoint,it is possible to prescribe part of the displacement and part of the traction.Typically,this 84FOUNDATIONS AND ELEMENTARY APPLICATIONStype of mixed condition involves the specification of a traction and displacement in two different orthogonal directions.A common example of this situation is shown in Figure 5-2for a case involving a surface of problem symmetry where the condition is one of a rigid-smooth boundary with zero normal displacement and zero tangential traction.Notice that in this example the body under study was subdivided along the symmetry line,thus creating a new boundary surface and resulting in a smaller region to analyze.Because boundary conditions play a very essential role in properly formulating and solving elasticity problems,it is important to acquire a clear understanding of their specification and use.Improper specification results in either no solution or a solution to a different problem than what was originally sought.Boundary conditions are normally specified using the coordinate system describing the problem,and thus particular components of the displacements and tractions are set equal to prescribed values.For displacement-type conditions,such a specifi-cation is straightforward,and a common example includes fixed boundaries where the dis-placements are to be zero.For traction boundary conditions,the specification can be a bit more complex.Figure 5-3illustrates particular cases in which the boundaries coincide with Cartesian or polar coordinate surfaces.By using results from Section 3.2,the traction specification can be reduced to a stress specification.For the Cartesian example in which y ¼constant ,Displacement Conditions Mixed ConditionsTraction ConditionsFIGURE 5-1Typical boundary conditions.T (n)y u Rigid-SmoothFIGURE 5-2Line of symmetry boundary condition.Formulation and Solution Strategies 85the normal traction becomes simply the stress component s y ,while the tangential traction reduces to t xy .For this case,s x exists only inside the region,and thus this component of stress cannot be specified on the boundary surface y ¼constant .A similar situation exists on the vertical boundary x ¼constant ,where the normal traction is now s x ,the tangential traction is t xy and the stress component s y exists inside the domain.Similar arguments can be made for polar coordinate boundary surfaces as shown.Drawing the appropriate element along the boundary as illustrated allows a clear visualization of the particular stress components that act on the surface in question.Such a sketch also allows determination of the positive directions of these boundary stresses,and this is useful to properly match with boundary loadings that might be prescribed.It is recommended that sketches similar to Figure 5-3be used to aid in the proper development of boundary conditions during problem formulation.Consider the pair of two-dimensional example problems with mixed conditions as shown in Figure 5-4.For the rectangular plate problem,all four boundaries are coordinate surfaces,andr(Cartesian Coordinate Boundaries)(Polar Coordinate Boundaries)xFIGURE 5-3Boundary stress components on coordinatesurfaces.(n)(n)(n)= σy = 0= t xy = 0(n)T x (n)T y = t xy =0,Traction Condition (Non-Coordinate Surface Boundary)(Coordinate Surface Boundaries)FIGURE 5-4Example boundary conditions.86FOUNDATIONS AND ELEMENTARY APPLICATIONSthis simplifies specification of particular boundary conditions.Thefixed conditions on the left edge simply require that x and y displacement components vanish on x¼0,and this specifica-tion does not change even if this were not a coordinate surface.However,as per our previous discussion,the traction conditions on the other three boundaries simplify because they are coordinate surfaces.These simplifications are shown in thefigure for each of the traction specified surfaces.The second problem of a tapered cantilever beam has an inclined face that is not a coordinate surface.For this problem,thefixed end and top surface follow similar procedures as in thefirst example and are specified in thefigure.However,on the inclined face,the traction is to be zero and this does not reduce to a simple specification of the vanishing of individual stress components.On this face each traction component is set to zero,giving the resultT(n)x¼s x n xþt xy n y¼0T(n)y¼t xy n xþs y n y¼0where n x and n y are the components of the unit normal vector to the inclined face.This is the more general type of specification,and it should be clearly noted that none of the individual stress components in the x,y system will vanish along this surface.It should also be pointed out for this problem that the unit normal vector components are constants for all points on the inclined face.However,for curved boundaries the normal vector changes with surface position.Although these examples provide some background on typical boundary conditions,many other types will be encountered throughout the text.Several exercises at the end of this chapter provide additional examples that will prove to be useful for students new to the elasticity formulation.We are now in the position to formulate and classify the three fundamental boundary-value problems in the theory of elasticity that are related to solving the general system offield equations(5.1.5).Our presentation is limited to the static case.Problem1:Traction problemDetermine the distribution of displacements,strains,and stresses in the interior of an elastic body in equilibrium when body forces are given and the distribution of the tractions are prescribed over the surface of the body,T(n)i(x(s)i)¼f i(x(s)i)(5:2:1) where x(s)i denotes boundary points and f i(x(s)i)are the prescribed traction values. Problem2:Displacement problemDetermine the distribution of displacements,strains,and stresses in the interior of an elastic body in equilibrium when body forces are given and the distribution of the displacements are prescribed over the surface of the body,u i(x(s)i)¼g i(x(s)i)(5:2:2) where x(s)i denotes boundary points and g i(x(s)i)are the prescribed displacement values.Formulation and Solution Strategies87Problem3:Mixed problemDetermine the distribution of displacements,strains,and stresses in the interior of an elasticbody in equilibrium when body forces are given and the distribution of the tractions areprescribed as per(5.2.1)over the surface S t and the distribution of the displacementsare prescribed as per(5.2.2)over the surface S u of the body(see Figure5-1).As mentioned previously,the solution to any of these types of problems is formidable,andfurther reduction and simplification of(5.1.5)is required for the development of analyticalsolution methods.Based on the description of Problem1with only traction boundary condi-tions,it would appear to be desirable to express the fundamental system solely in terms ofstress,that is,J(t){s ij;l,m,F i}thereby reducing the number of unknowns in the system.Likewise for Problem2,a displacement-only formulation of the form J(u){u i;l,m,F i}wouldappear to simplify the problem.We now pursue such specialized formulations and explicitlydetermine these reducedfield equation systems.5.3Stress FormulationFor thefirst fundamental problem in elasticity,the boundary conditions are to be given only interms of the tractions or stress components.In order to develop solution methods for this case,it is very helpful to reformulate the general system(5.1.5)by eliminating the displacementsand strains and thereby cast a new system solely in terms of the stresses.We now develop thisreformulated system.By eliminating the displacements,we must now include the compatibil-ity equations in the fundamental system offield equations.We therefore start by using Hooke’slaw(5.1.4)2and eliminate the strains in the compatibility relations(5.1.2)to gets ij,kkþs kk,ijÀs ik,jkÀs jk,ik¼n 1þn (s mm,kk d ijþs mm,ij d kkÀs mm,jk d ikÀs mm,ik d jk)(5:3:1)where we have used the arguments of Section2.6,that the six meaningful compatibility relations are found by setting k¼l in(5.1.2).Although equations(5.3.1)represent the compatibility in terms of stress,a more useful result is found by incorporating the equilibrium equations into the system.Recall that from(5.1.3),s ij,j¼ÀF i,and also note that d kk¼3. Substituting these results into(5.3.1)givess ij,kkþ11þns kk,ij¼n1þns mm,kk d ijÀF i,jÀF j,i(5:3:2)For the case i¼j,relation(5.3.2)reduces to s ii,kk¼À1þn1ÀnF i,i.Substituting this result backinto(5.3.2)gives the desired relations ij,kkþ11þns kk,ij¼Àn1Ànd ij F k,kÀF i,jÀF j,i(5:3:3)This result is the compatibility relations in terms of the stress and is commonly called theBeltrami-Michell compatibility equations.For the case with no body forces,these relations canbe expressed as the following six scalar equations:88FOUNDATIONS AND ELEMENTARY APPLICATIONS(1þn)r2s xþ@2@x(s xþs yþs z)¼0(1þn)r2s yþ@2@y(s xþs yþs z)¼0(1þn)r2s zþ@2@z(s xþs yþs z)¼0(1þn)r2t xyþ@2@x@y(s xþs yþs z)¼0(1þn)r2t yzþ@2@y@z(s xþs yþs z)¼0(1þn)r2t zxþ@2(s xþs yþs z)¼0(5:3:4)Recall that the six developed relations(5.3.3)or(5.3.4)actually represent three independentresults as per our discussion in Section2.6.Thus,combining these results with the threeequilibrium equations(5.1.3)provides the necessary six relations to solve for the six unknownstress components for the general three-dimensional case.This system constitutes the stressformulation for elasticity theory and is appropriate for use with traction boundary conditionproblems.Once the stresses have been determined,the strains may be found from Hooke’s law(5.1.4)z,and the displacements can be then be computed through integration of(5.1.1).As perour previous discussion in Section2.2,such an integration process determines the displace-ments only up to an arbitrary rigid-body motion,and the displacements obtained are single-valued only if the region under study is simply connected.The system of equations for the stress formulation is still rather complex,and analytical solutions are commonly determined for this case by making use of stress functions.Thisconcept establishes a representation for the stresses that automatically satisfies the equilibriumequations.For the two-dimensional case,this concept represents the in-plane stresses in termsof a single function.The representation satisfies equilibrium,and the remaining compatibilityequations yield a single partial differential equation(biharmonic equation)in terms of thestress function.Having reduced the system to a single equation then allows us to employ manyanalytical methods tofind solutions of interest.Further discussion on these techniques ispresented in subsequent chapters.5.4Displacement FormulationWe now wish to develop the reduced set offield equations solely in terms of the displacements.This system is referred to as the displacement formulation and is most useful when combinedwith displacement-only boundary conditions found in the Problem2statement.This develop-ment is somewhat more straightforward than our previous discussion for the stress formulation.For this case,we wish to eliminate the strains and stresses from the fundamental system(5.1.5).This is easily accomplished by using the strain-displacement relations in Hooke’s lawto gives ij¼l u k,k d ijþm(u i,jþu j,i)(5:4:1) which can be expressed as six scalar equationsFormulation and Solution Strategies89s x¼l@u@xþ@n@yþ@w@zþ2m@u@xs y¼l@u@xþ@v@yþ@w@zþ2m@v@ys z¼l@u@xþ@v@yþ@w@zþ2m@w@zt xy¼m @u@yþ@v@x,t yz¼m @v@zþ@w@y,t zx¼m @w@xþ@u@z(5:4:2)Using these relations in the equilibrium equations gives the resultm u i,kkþ(lþm)u k,kiþF i¼0(5:4:3)which are the equilibrium equations in terms of the displacements and are referred to as Navier’s or Lame´’s equations.This system can be expressed in vector form asm r2uþ(lþm),(,Áu)þF¼0(5:4:4) or written out in terms of the three scalar equationsm r2uþ(lþm)@@x@u@xþ@v@yþ@w@zþF x¼0m r2vþ(lþm)@@y@u@xþ@v@yþ@w@zþF y¼0m r2wþ(lþm)@@z@u@xþ@v@yþ@w@zþF z¼0(5:4:5)where the Laplacian is given by r2¼(@2=@x2)þ(@2=@y2)þ(@2=@z2).Navier’s equations arethe desired formulation for the displacement problem,and the system represents three equa-tions for the three unknown displacement components.Similar to the stress formulation,thissystem is still difficult to solve,and additional mathematical techniques have been developedto further simplify these equations for problem mon methods normally employthe use of displacement potential functions.It is shown in Chapter13that several suchschemes can be developed that allow the displacement vector to be expressed in terms ofparticular potentials.These schemes generally simplify the problem by yielding uncoupledgoverning equations in terms of the displacement potentials.This then allows several analyt-ical methods to be employed to solve problems of interest.Several of these techniques arediscussed in later sections of the text.To help acquire a general understanding of these results,a summaryflow chart of the developed stress and displacement formulations is shown in Figure5-5.Note that for thestress formulation,the resulting system J(t){s ij;l,m,F i}is actually dependent onlyon the single material constant Poisson’s ratio,and thus it could be expressed asJ(t){s ij;n,F i}.90FOUNDATIONS AND ELEMENTARY APPLICATIONS5.5Principle of SuperpositionA very useful tool for the solution to many problems in engineering science is the principle ofsuperposition.This technique applies to any problem that is governed by linear equations.Under the assumption of small deformations and linear elastic constitutive behavior,allelasticityfield equations(see Figure5-5)are linear.Furthermore,the usual boundary condi-tions specified by relations(5.2.1)and(5.2.2)are also linear.Thus,under these conditions allgoverning equations are linear,and the superposition concept can be applied.It can be easilyproved(see Chou and Pagano1967)that the general statement of the principle can beexpressed as follows:Principle of Superposition:For a given problem domain,if the state{s(1)ij,e(1)ij,u(1)i}is asolution to the fundamental elasticity equations with prescribed body forces F(1)i andsurface tractions T(1)i,and the state{s(2)ij,e(2)ij,u(2)i}is a solution to the fundamentalequations with prescribed body forces F(2)i and surface tractions T(2)i,then the sta-te{s(1)ijþs(2)ij,e(1)ijþe(2)ij,u(1)iþu(2)i}will be a solution to the problem with body forcesF(1)iþF(2)i and surface tractions T(1)iþT(2)i.In order to see a more direct application of this principle,consider a simple two-dimensionalcase with no body forces as shown in Figure5-6.It can be observed that the solution to themore complicated biaxial loading case(1)þ(2)is thus found by adding the two simplerproblems.This is a common use of the superposition principle,and we make repeated use ofthis application throughout the text.Thus,once the solutions to some simple problems aregenerated,we can combine these results to generate a solution to a more complicated case withsimilar geometry.Formulation and Solution Strategies915.6Saint-Venant’s PrincipleConsider the set of three identical rectangular strips under compressive loadings as shown in Figure 5-7.As indicated,the only difference between each problem is the loading.Because the total resultant load applied to each problem is identical (statically equivalent loadings),it is expected that the resulting stress,strain,and displacement fields near the bottom of each strip would be approximately the same.This behavior can be generalized by considering an elastic solid with an arbitrary loading T (n )over a boundary portion S *,as shown in Figure 5-8.Based on experience from other field problems in engineering science,it seems logical that the particular boundary loading would produce detailed and characteristic effects only in the vicinity of S *.In other words,we expect that at points far away from S *the stresses generally depend more on the resultant F R of the tractions rather than on the exact distribution.Thus,the characteristic signature of the generated stress,strain,and displacement fields from a given boundary loading tend to disappear as we move away from the boundary loading points.These concepts form the principle of Saint-Venant ,which can be stated as follows:Saint-Venant’s Principle:The stress,strain,and displacement fields caused by two different statically equivalent force distributions on parts of the body far away from the loading points are approximately the same.=+ ij ij i ij ij i {s (1)ij + s (2) ij , e (1) ij , + e (2) ij , u (1) i + u (2) i }FIGURE 5-6Two-dimensional superposition example.2P 2P3P 3P 3P FIGURE 5-7Statically equivalent loadings.92FOUNDATIONS AND ELEMENTARY APPLICATIONSThis statement of the principle includes qualitative terms such as far away and approxi-mately the same ,and thus does not provide quantitative estimates of the differences between the two elastic fields in question.Quantitative results have been developed by von Mises (1945),Sternberg (1954),and Toupin (1965),while Horgan (1989)has presented a recent review of related work.Some of this work is summarized in Boresi and Chong (2000).If we restrict our solution to points away from the boundary loading,Saint-Venant’s principle allows us to change the given boundary conditions to a simpler statically equivalent statement and not affect the resulting solution.Such a simplification of the boundary conditions greatly increases our chances of finding an analytical solution to the problem.This concept therefore proves to be very useful,and we formally outline this solution scheme in the next section.5.7General Solution StrategiesHaving completed our formulation and related solution principles,we now wish to examine some general solution strategies commonly used to solve elasticity problems.At this stage we categorize particular methods and outline only typical techniques that are commonly used.As we move further along in the text,many of these methods are developed in detail and are applied in specific problem solution.We first distinguish three general methods of solution called direct,inverse ,and semi-inverse .5.7.1Direct MethodThis method seeks to determine the solution by direct integration of the field equations (5.1.5)or equivalently the stress and/or displacement formulations given in Figure 5-5.Boundary conditions are to be satisfied exactly.This method normally encounters significant mathemat-ical difficulties,thus limiting its application to problems with simple geometry.EXAMPLE 5-1:Direct Integration Example:Stretching of Prismatic Bar Under Its Own WeightAs an example of a simple direct integration problem,consider the case of a uniform prismatic bar stretched by its own weight,as shown in Figure 5-9.The body forces forContinuedFIGURE 5-8Saint-Venant concept.5.7.2Inverse MethodFor this technique,particular displacements or stresses are selected that satisfy the basicfield equations.A search is then conducted to identify a specific problem that would be solved by this solutionfield.This amounts to determine appropriate problem geometry,boundary conditions,and body forces that would enable the solution to satisfy all conditions on the ing this scheme it is sometimes difficult to construct solutions to a specific problem of practical interest.5.7.3Semi-Inverse MethodIn this scheme part of the displacement and/or stressfield is specified,and the other remaining portion is determined by the fundamentalfield equations(normally using direct integration) and the boundary conditions.It is often the case that constructing appropriate displacement and/or stress solutionfields can be guided by approximate strength of materials theory.The usefulness of this approach is greatly enhanced by employing Saint-Venant’s principle, whereby a complicated boundary condition can be replaced by a simpler statically equivalent distribution.EXAMPLE5-3:Semi-Inverse Example:Torsion of Prismatic BarsA simple semi-inverse example may be borrowed from the torsion problem that isdiscussed in detail in Chapter9.Skipping for now the developmental details,we propose the following displacementfield:ContinuedThere are numerous mathematical techniques used to solve the elasticity field equations.Many techniques involve the development of exact analytical solutions ,while others involve the construction of approximate solution schemes .A third procedure involves the establishment of numerical solution methods .We now briefly provide an overview of each of these techniques.5.7.4Analytical Solution ProceduresA variety of analytical solution methods are used to solve the elasticity field equations.The following sections outline some of the more common methods.Power Series MethodFor many two-dimensional elasticity problems,the stress formulation leads to the use of a stress function f (x ,y ).It is shown that the entire set of field equations reduces to a single partial differential equation (biharmonic equation)in terms of this stress function.A general mathematical scheme to solve this equation is to look for solutions in terms of a power series in the independent variables,that is,f (x ,y )¼P C mn x m y n (see Neou 1957).Use of the boundary conditions determines the coefficients and number of terms to be used in the series.This method is employed to develop two-dimensional solutions in Section 8.1.Fourier MethodA general scheme to solve a large variety of elasticity problems employs the Fourier method.This procedure is normally applied to the governing partial differential equations by using。
