Solid Mechanics: An Introduction(15 Solid Mechanics and Its Applications)

This book is a concise and readable introductory text on solid mechanics suitable for engineers, scientists and applied mathematicians. It presents the foundations of stress, strain and elasticity theory and consistently employs the use of vectors and (particularly) Cartesian tensor notation. The first chapter introduces vectors with particular emphasis being paid to applications which arise in later chapters. Chapter 2 introduces Cartesian tensors and describes some of their important applications. In particular, finite and infinitessimal rotations are examined as are isotropic tensors and second order symmetric tensors. The last topic of this chapter includes a full discussion on eigenvalues and eigenvectors. There are separate introductions, in Chapters 3 and 4, to stress and strain and to their practical measurement using, respectively, photoelastic methods and strain gauges. In Chapter 5 the concepts of stress and strain are brought together and, in conjunction with Newton's equilibrium equations, used to deduce the basic equations of linear elasticity theory. These fundamental equations are then examined and analyzed by obtaining simple exact solutions, including solutions which describe twisting, bending and stretching of beams. Chapter 6 introduces the fundamental concept of strain energy and uses this concept to derive the Kirchoff uniqueness theorem, Rayleigh's reciprocal theorem and the important Castigliano relations. The chapter concludes with a thorough treatment of the theorem of minimum potential energy and examines some of its applications. The final three chapters examine the application of the fundamental equations to the theory of torsion, to structural analysis and to the treatment of two dimensional elastostatics by analytical and approximate (finite element) methods.

Table of Contents:
1 Vectors.- 1.1 Introduction to vector algebra.- 1.2 The scalar product.- 1.3 The vector product.- 1.4 Applications of vectors to forces.- 1.5 Triple products.- 1.6 The index notation for vectors.- 1.7 Vector differential calculus.- 1.8 Vector integral calculus.- 2 Cartesian Tensors.- 2.1 Introduction.- 2.2 Rotation of Cartesian coordinates.- 2.3 Cartesian tensors.- 2.4 Properties of tensors.- 2.5 The rotation tensor.- 2.6 Isotropic tensors.- 2.7 Second order symmetric tensors.- 3 The Analysis of Stress.- 3.1 Introduction.- 3.2 The stress tensor.- 3.3 Principal axes.- 3.4 Maximum normal and shear stresses.- 3.5 Plane stress.- 3.6 Photoelastic measurement of principal stresses.- 4 The Analysis of Strain.- 4.1 The strain tensor.- 4.2 Physical interpretation of the strain tensor.- 4.3 Principal axes, principal strains.- 4.4 Principal strains and the strain rosette.- 4.5 The compatibility equations for strain.- 5 Linear Elasticity.- 5.1 Hooke’s law and the simple tension experiment.- 5.2 The governing equations of linear eleasticity.- 5.3 Simple solutions.- 5.4 The Navier equation in linear elasticity.- 6 Energy.- 6.1 Strain energy and work.- 6.2 Kirchoff’s uniqueness theorem.- 6.3 The reciprocal theorem.- 6.4 The Castigliano theorem.- 6.5 Potential energy.- 7 The General Torsion Problem.- 7.1 Introduction.- 7.2 The torsion function.- 7.3 Shearing stress in the torsion problem.- 7.4 Simple exact solutions in the torsion problem.- 7.5 Approximate formulae in the torsion problem.- 8 The Matrix Analysis of Structures.- 8.1 Introduction.- 8.2 Pin-jointed elements.- 8.3 Two and three dimensional pin-jointed structures.- 8.4 Beam elements.- 8.5 Equivalent nodal forces.- 9 Two Dimensional Elastostatics.- 9.1 Plane strain, plane stress and generalised plane stress.- 9.2 Exactsolutions to problems in plane strain.- 9.3 Approximations in two dimensional elastostatics.- Appendix 1 The Variational Calculus.- A1.1 The fundamental lemma.- A1.2 Functionals and the variational calculus.- A1.3 Construction of functionals.- A1.4 One dimensional fourth-order problems.- A1.5 Variational formulation of fourth-order problems.- References.