Differential Equations of Linear Elasticity of Homogeneous Media

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

Differential Equations of Linear Elasticity of Homogeneous Media