Solid Mechanics

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.