Differential Equations of Linear Elasticity of Homogeneous Media Theory of Linear Elasticity

The transmission of forces from without to within solid medium comprises a mathematical challenge of utmost complexity. The sources of difficulties are as follows: 1. Surface indeterminate conditions 2. Medium indeterminate relationships 3- Spatial indeterminate continuity 4. Fixing and loading indeterminate conditions 5. Inertial rotational indeterminate equilibrium STATICS OF STRESS Navier's Partial differential equations of stress Surface conditions for projection of stress Cauchy's quadratic or surface of normal stresses Spherical stress tensor Stress deviator tensor Vanishing deviator of the first invariant of the GEOMETRY OF STRAIN Cauchy's equations for displacement, elongation, shear, and rotational strains General strain tensor Deviator and spherical strain tensors and invariants Cubic deviations of the third invariant of the relative strain tensor VOLUMETRIC HOOKE'S LAW The three components of Hooke's law Elastic properties of material Relationships between Young's modulus, Poisson's ratio, and Lamé's coefficients Elastic potential energy LAMÉ'S EQUATIONS OF CONTINUITY ELASTIC VIBRATION Vibration of unbound surfaces Longitudinal vibration Transverse vibration Harmonic longitudinal vibrations Vibration of bound surfaces TORSION, BENDING, AND SUSPENSION OF A BAR Pure shear stress Torsion of a circular bar Pure bending stress Suspension of a bar PLANE ELASTICITY PROBLEMS Plane strain approximations Modified Hooke's law for planar strains Planar stress approximations Hooke's law for planar stress Interpretation of Maurice Lévy's equation Polynomial stress function Pure bending of cantilever Forced bending of cantilever Uniformly loaded beam supported at both ends Vertically loaded triangular dam Separation of variables or geometrical polynomials Beam with infinite span Cylindrical tube with infinite length Cylindrical polar radial Levy's stress function Lame's circular cylindrical tube Bending a circular ring Finite force applied on half plane Flamant Boussinesg BIHARMONIC EQUATION BiHarmonic equation of plane stress in polar cylindrical coordinates Variable separation constant TORSION OF PRISMATICAL BARS Prismatical Circular Cylindrical Bar Torsion of prismatical bars Ludwig Prandtl's shear stress function Fx, y Prismatical Elliptic Cylindrical Bar Complex stress and torsion functions Torsional angle or angle of twist Deformed crosssection contour Triangular Prismatical Bar Complex function representation of triangular geometry Prismatical bar with rectangular crosssection Membrane surface tension with Ludwig Prandtl's stress function GENERAL SOLUTION OF ELASTICITY PROBLEMS Beltrami Michell Equations Maxwell's stress functions Morera's stress functions Plane stress in cylindrical coordinates Harmonic equation Concentrated load on half space medium Distributed load on half space medium Filon's solution of plain stress problem by complex variables Airy stress function with complex harmonic function Elastic vibrational waves THIN SLAB SOLUTION BY PLANE APPROXIMATION Bending of rod versus bending of thin slab Sophie Germain's equation for bending and torsion of thin slab Elliptic plate Circular plate Rectangular plate Navier's method Levy's method VARIATIONAL METHOD OF SOLUTION IN PLANAR ELASTICITY Clapeyron's Theorem in Linear Elasticity Lagrange's geometrical variation Vibrational perturbation of displacements and strains Elastic body energy Virtual work done Plane crosssection approximations in thick media Lagrange's equation for threedimensional arbitrary body Castigliano's static variation Torsion of prismatical rod Castigliano's variation equation for torsion of rod Laplace's form of Castigliano's variation equation for torsion of rod Practical approximate solution of elasticity by method of variation of elastic energy Lame's problem of rectangular prism