Elasticity

The subject of Elasticity can be approached from several points of view, - pending on whether the practitioner is principally interested in the mat- matical structure of the subject or in its use in engineering applications and, in the latter case, whether essentially numerical or analytical methods are envisaged as the solution method. My ?rst introduction to the subject was in response to a need for information about a speci?c problem in Tribology. As a practising Engineer with a background only in elementary Mechanics of - terials, I approached that problem initially using the concepts of concentrated forces and superposition. Today, with a rather more extensive knowledge of analytical techniques in Elasticity, I still ?nd it helpful to go back to these roots in the elementary theory and think through a problem physically as well as mathematically, whenever some new and unexpected feature presents di?culties in research. This way of thinking will be found to permeate this book. My engineering background will also reveal itself in a tendency to work examples through to ?nal expressions for stresses and displacements, rather than leave the derivation at a point where the remaining manipulations would be mathematically routine. The ?rst edition of this book, published in 1992, was based on a one semester graduate course on Linear Elasticity that I have taught at the U- versity of Michigan since 1983.

Expansion of topics to include more on complex variable methods, variational methods and three-dimensional plate and beam solutions Additional end-of-chapter problems Detailed improvements elsewhere, some suggested by users of the earlier editions Modern treatment of the subject Clarity of presentation Supplementary electronic files in Mathematica and Maple available for download Includes supplementary material: sn.pub/extras

Autorentext
Jim Barber graduated in Mechanical Sciences from Cambridge University in 1963 and joined British Rail, who later sponsored his research at Cambridge between 1965 and 1968 on the subject of thermal effects in braking systems. In 1969 he became Lecturer and later Reader in Solid Mechanics at the University of Newcastle upon Tyne. In 1981 he moved to the University of Michigan, where he is presently Professor of Mechanical Engineering and Applied Mechanics. His current research interests are in solid mechanics with particular reference to thermoelasticity, contact mechanics and tribology. He is a chartered engineer in the U.K., Fellow of the Institution of Mechanical Engineers and has engaged extensively in consulting work in the field of stress analysis for engineering design.

Klappentext

This is a first year graduate textbook in Linear Elasticity. It is written with the practical engineering reader in mind, dependence on previous knowledge of solid mechanics, continuum mechanics or mathematics being minimized. Emphasis is placed on engineering applications of elasticity and examples are generally worked through to final expressions for the stress and displacement fields in order to explore the engineering consequences of the results.

The topics covered are chosen with a view to modern research applications in fracture mechanics, composite materials, tribology and numerical methods. Thus, significant attention is given to crack and contact problems, problems involving interfaces between dissimilar media, thermoelasticity, singular asymptotic stress fields and three-dimensional problems.

This third edition includes new chapters on complex variable methods, variational methods and three-dimensional solutions for the prismatic bar. Other detailed changes have been made throughout the work, many suggested by users of earlier editions.

The new edition includes over 300 end-of-chapter problems, expressed wherever possible in the form they would arise in engineering - i.e. as a body of a given geometry subjected to prescribed loading - instead of inviting the student to 'verify' that a given candidate stress function is appropriate to the problem. Solution of these problems is considerably facilitated by the use of modern symbolic mathematical languages such as Maple and Mathematica. Electronic files and hints on this method of solution, as well as further supplementary software are available for download via the webpage for this volume on www.springer.com.

Inhalt

Part I GENERAL CONSIDERATIONS ; 1 Introduction; 1.1 Notation for stress and displacement ; 1.1.1 Stress; 1.1.2 Index and vector notation and the summationconvention; 1.1.3 Vector operators in index notation; 1.1.4 Vectors, tensors and transformation rules; 1.1.5 Principal stresses and Von Mises stress ; ; 1.1.6 Displacement; ; 1.2 Strains and their relation to displacements; ; 1.2.1 Tensile strain; ; 1.2.2 Rotation and shear strain; ; 1.2.3 Transformation of co¨ordinates; ; 1.2.4 Definition of shear strain; ; 1.3 Stressstrain relations; ; 1.3.1 Lam´e's content; 1.3.2 Dilatation and bulk modulus ; PROBLEMS; 2 Equilibrium and compatibility; 2.1 Equilibrium equations ; 2.2 Compatibility equations; 2.2.1 The significance of the compatibility equations ; 2.3 Equilibrium equations in terms of displacements ; PROBLEMS; Part II TWODIMENSIONAL PROBLEMS ; 3 Plane strain and plane stress ; 3.1 Plane strain ; 3.1.1 The corrective solution; 3.1.2 SaintVenant's principle ; 3.2 Plane stress; 3.2.2 Relationship between plane stress and plane strain; PROBLEMS; 4 Stress function formulation ; 4.1 The concept of a scalar stress function ; 4.2 Choice of a suitable form ; 4.3 The Airy stress function; 4.3.1 Transformation of co¨ordinates; 4.3.2 Nonzero body forces ; 4.4 The governing equation ; 4.4.1 The compatibility condition ; 4.4.2 Method of solution ; 4.4.3 Reduced dependence on elastic constants ; PROBLEMS; 5 Problems in rectangular co¨ordinates ; 5.1 Biharmonic polynomial functions ; 5.1.1 Second and third degree polynomials; 5.2 Rectangular beam problems ; 5.2.1 Bending of a beam by an end load; 5.2.2 Higher order polynomials a general strategy; 5.2.3 Manual solutions symmetry considerations ; 5.3 Fourier series and transform solutions ; 5.3.1 Choice of form ; 5.3.2 Fourier transforms; PROBLEMS; 6 End effects ; 6.1 Decaying solutions; 6.2 The corrective solution; 6.2.1 Separatedvariable solutions ; 6.2.2 The eigenvalue problem ; 6.3 Other SaintVenant problems; 6.4 Mathieu's solution; PROBLEMS; 7 Body forces ; 7.1 Stress function formulation ; 7.1.1 Conservative vector fields; 7.1.2 The compatibility condition ; 7.2 Particular cases ; 7.2.1 Gravitational loading ; 7.2.2 Inertia forces ; 7.2.3 Quasistatic problems; 7.2.4 Rigidbody kinematics ; 7.3 Solution for the stress function ; 7.3.1 The rotating rectangular beam; 7.3.2 Solution of the governing equation;7.4 Rotational acceleration; 7.4.1 The circular disk ; 7.4.2 The rectangular bar; 7.4.3 Weak boundary conditions and the equation of motion; PROBLEMS; 8 Problems in polar co¨ordinates ; 8.1 Expressions for stress components; 8.2 Strain components; 8.3 Fourier series expansion; ; 8.3.1 Satisfaction of boundary conditions; 8.3.3 Degenerate cases ; 8.4 The Michell solution ; 8.4.1 Hole in a tensile field ; PROBLEMS; 9 Calculation of displacements ; 9.1 The cantilever with an end load ; 9.1.1 Rigidbody displacements and end conditions ; 9.1.2 Deflection of the free end; 9.2 The circular hole ; 9.3 Displacements for the Michell solution ; 9.3.1 Equilibrium considerations ; 9.3.2 The cylindrical pressure vessel ; PROBLEMS; 10 Curved beam problems ; 10.1 Loading at the ends; 10.1.1 Pure bending ;;10.1.2 Force transmission; 10.2 Eigenvalues and eigenfunctions ; 10.3 The inhomogeneous problem; 10.3.1 Beam with sinusoidal loading ; 10.3.2 The nearsingular problem; 10.4 Some general considerations ; 10.4.1 Conclusions; PROBLEMS; 11 Wedge problems ; 11.1 Power law tractions; 11.1.1 Uniform tractions ; 11.1.2 The rectangular body revisited; 11.1.3 More general uniform loading ; 11.1.4 Eigenvalue…