Elasticity is concerned with the determination of displacement, strain and stress distributions in elastic solids. Solutions are typically developed based on a system of PDEs. This course emphasizes the seamless integration of mathematical theories and mechanical principles.
——课程团队
课程概述
Welcome to this open online course on the theory of elasticity. In this course, we will help you to get familiar with the major contents of elasticity theory. As you may already know, elasticity is one of the most important courses in many engineering subjects such as mechanics, civil engineering, mechanical engineering, transportation engineering and materials science and engineering. This course will require some reasonable background knowledge in advanced mathematics, linear algebra, statics and mechanics of materials. The major contents of this course involve the fundamentals of tensor, three and two-dimensional theory of elasticity, governing equations, boundary conditions, typical solution strategy and solution method, as well as many illustrating examples. This course particularly emphasizes on the integration of mathematical knowledge and mechanical principles. Throughout the process, we will go through the origin, development and maturation of the basic equations of elasticity, such that you can get a deep understanding about the logic flow of this theory. If you wish to lay a solid mathematical foundation for the further learning of mechanics courses, taking this course is a wise choice. Upon completion of this course, your abilities of calculating, analyzing and self-learning mechanical knowledge will be enhanced. You will also able to propose some new mechanical models and make some routine analysis by yourself. This course will also prepare you better for more advanced courses in mechanics and engineering, both in terms of theory and engineering practice.
授课目标
Upon the completion of this course, students are expected to: (1) understand the assumptions, research objects and major topics of elasticity theory; (2) familiarize with the governing equations, boundary conditions, formulations and conventional analytical solution methods; (3) know how to solve simple two and three-dimensional elastic problems. Through these rigorous trainings, students’ capabilities on analyzing, calculating and self-learning mechanics problems can be significantly solidified. As a result, students should be more prepared for mechanics courses at a more advanced level and more capable of conducting research activities on solid mechanics.
课程大纲
Introduction to elasticity
Get familiar with the history, solution methods, applications and the fundamentals of elasticity theory.
1.1 A brief history of elasticity
1.2 Solution tools, applications and some concepts
1.3 Research objects, assumptions and course contents
Mathematical preliminaries
Master the mathematical preliminaries that are required for the development of governing equations of elasticity theory.
2.1 Fundamentals of tensor analysis
2.2 Tensor algebra
2.3 Tensor calculus
2.4 Tensor calculus in curvilinear coordinates
Displacement and strain
Understand the derivation of strain-displacement relation and the concepts of strain compatibility and domain connectivity.
3.1 Strain-displacement relation
3.2 Strain transformation
3.3 Strain compatibility and domain connectivity
Stress and equilibrium
Comprehend the physical interpretations of Cauchy stress and stresses on oblique planes and boundaries. Command the derivation of equilibrium equations from the principles of momentum balances.
4.1 Cauchy stress and Cauchy’s relation
4.2 Stress transformation
4.3 Momentum balance and equilibrium
Constitutive relations
Grasp the stress-strain relationship of isotropic solids and comprehend the physical meanings of elastic constants.
5.1 Constitutive relations of anisotropic solids
5.2 Reductions due to symmetry
5.3 Hooke’s law for isotropic solids
Formulations and solution strategy
Familiarize with the stress and displacement formulations of governing equations. Understand the general principles and solution strategies of elastic problems.
6.1 Governing equations and boundary conditions
6.2 Stress and displacement formulations
6.3 General principles and solution strategy
Formulations of plane elasticity
Command the reduction of three-dimensional elastic problems to plane strain and plane stress ones. Understand the development of Airy stress function method.
7.1 Plane strain
7.2 Plane stress
7.3Connections between plane strain and plane stress
7.4 Method of Airy stress function
Plane problems in rectangular coordinates
Understand the methods of power series and Fourier series for solving plane elastic problems in Cartesian coordinates.
8.1 Method of power series
8.2Transverse bending of beams
8.2 Method of Fourier series
Plane problems in polar coordinates
Command the solution methods for solving both the axially symmetric and general plane problems in polar coordinates.
9.1 Axially symmetric solutions
9.2 Examples of axially symmetric problems
9.3 General solution of Airy stress function in polar coordinates
9.4 Hole in an infinite plane under uniaxial remote tension
9.5 Stress concentrations around holes
9.6 Curved beams subjected to end loads
9.7 Wedge problems
9.8 Half-plane problems
9.9 Flamant solution
Torsion of prismatic shafts
Learn how to solve the torsional stresses in non-circular shafts with a few representative cross-sectional shapes.
10.1 Elastic cylinders subjected to end loadings
10.2 Torsion formulation
10.3 Membrane analogy
10.4 Solutions derived from boundary equation
10.5 Torsion solutions using Fourier methods
10.6 Torsion of cylinders with hollow sections
Three-dimensional problems
Familiarize with the typical displacement potentials that are used to solve three-dimensional elastic problems. Also understand the solution procedures for a few representative three-dimensional problems.
11.1 A brief review on displacement formulation
11.2 Three-dimensional problems with symmetries
11.3 Helmholtz representation and Lame potential
11.4 Galerkin and Papkovich-Neuber displacement potentials
11.5 Concentrated force in infinite and semi-infinite domain
11.6 Distributed pressure on half-space boundary
11.7 Hertzian Contact
Bending of thin-plates
Understand the derivation and physical meanings of governing equations for the bending of thin-plates. Also command the conventional methods for solving the deflection, bending stresses and internal forces in thin-plates subjected to transverse loads.
12.1 Assumptions
12.2 Derivation of governing equation
12.3 Internal forces and moments
12.4 Boundary conditions
12.5 Sample problems
Thermoelasticity
This chapter extends the theory of elasticity from isothermal condition to thermoelasticity. You'll learn how to solve two-dimensional thermoelastic problems in both rectangular and polar coordinates.
13.1 Governing equations of thermoelasticity
13.2 Stress formulation for two-dimensional thermoelasticity
13.3 Displacement formulation for two-dimensional thermoelasticity
13.4 Examples in rectangular coordinates
13.5 Two-dimensional thermoelasticity in polar coordinates
13.6 Axisymmetric problems in polar coordinates
Energy method and variational principle
This chapter introduces the evaluation of strain energy as well as complementary strain energy. Base on the energy method, some fundamentals and applications of virtual displacements and virtual stresses follow.
14.1 Strain energy due to normal stresses
14.2 Strain energy due to generalized stresses
14.3 Principle of virtual work
14.4 Principle of minimum total potential energy
14.5 Applications of Ritz and Galerkin method
14.6 Principle of complementary virtual work
14.7 Approximate solutions of virtual stresses
预备知识
Background knowledge in advanced mathematics, linear algebra, statics and mechanics of materials is desired
参考资料
M.H. Sadd, Elasticity: Theory, Applications, and Numerics, 2005, Elsevier. (A modern textbook, using tensor notation, that is relatively easier to follow.)
Y.C. Fung, Foundations of Solid Mechanics, Prentice Hall, 1965. (A deep explanation on the deformation and motion of solids. Explained and proved many fundamental principles on the subject matter.)
S.P. Timoshenko & J.N. Goodier, 1970, Theory of Elasticity, 3rd ed., McGraw-Hill. (A classic textbook, using older notation, emphasizing practical solution of engineering problems.)
I.S. Sokolnikoff, 1956, Mathematical Theory of Elasticity, McGraw-Hill. (This is also a classic textbook, using extensively tensor notation, more emphasis on mathematical methods.)
R.W. Little, 1973, Elasticity, Prentice Hall. (A good textbook emphasizing series solutions, and 3-D problems, uses modern tensor notation. It hosts many interesting problems.)
A.P. Boresi and K.P. Chong, 1987, Elasticity in Engineering Mechanics, Elsevier. (Solid textbook, good discussion of 2-D and 3-D problems.)
A.H. England, 1971, Complex Variable Methods in Elasticity, Wiley-Interscience. (Good textbook for the application of complex variable methods to the solution of 2-D problems.)
N.I. Muskhelishvili, 1975, Some Basic Problems of the Mathematical Theory of Elasticity, Noordhoff International Publishing. (A huge book, very detailed, the standard reference on the application of complex variable methods to elasticity problems.)
A.K. Mal and S.J. Singh, 1991, Deformation of Elastic Solids, Prentice Hall. (Good, modern textbook.)
A.S. Saada, 1993, Elasticity: Theory & Applications, 2nd ed., Krieger. (Good, modern textbook, includes problems at end of chapter.)
B.A. Boley and J.H. Weiner, 1960, Theory of Thermal Stresses, John Wiley. (The standard reference on thermal stresses in elastic and plastic solids.)
G.L.M. Gladwell, 1980, Contact Problems in the Classical Theory of Elasticity, Sijthoff and Nordhoff. (Emphasis on contact of elastic solids.)
A.E. Green and W. Zerna, 1968, Theoretical Elasticity, Oxford University Press (also in Dover edition, 1992). (If you can master the notation, the book is very well written. Heavy emphasis on mathematical approach.)
L.D. Landau and E.M. Lifshitz, 1986, Theory of Elasticity, 3rd ed., Pergamon Press.
A.E.H. Love, 1944, The Mathematical Theory of Elasticity, Dover Publications. (Older notation, contains a wealth of solved difficult problems. A standard reference.)
T. Mura, 1987, Micromechanics of Defects in Solids, 2nd ed. Martinus Nijhoff. (Exclusive emphasis on the theory of defects in elastic solids, and on various applications of the Eshelby transforming inclusion problem.)
J.F. Nye, 1957, Physical Properties of Crystals, Oxford University Press. (Superb discussion of crystalline anisotropy effects for many material properties, including compliance and stiffness.)
S.G. Lekhnitskii, 1963, Theory of Elasticity of an Anisotropic Elastic Body, Holden Day.
I. Sneddon, 1951, Fourier Transforms, McGraw-Hill (Recently in Dover edition). (Discusses in detail the application of Fourier and Hankel transforms to elasticity solutions for indentation of half-planes and half-spaces.)
S.P. Timoshenko, 1953, A History of Strength of Materials, McGraw-Hill (Recently in Dover edition). (For historical details.)
A.E.H. Love, 1934, A Treatise of the Mathematical Theory of Elasticity, 4th ed., Cambridge University Press. (For historical details.)
Todhunter and Pearson, 1893, History of the Theory of Elasticity, University Press. (For historical details.)
