Bending, Ringing, Breaking and Flowing

Elastic Solids and Complex Fluids in 10 chapters — from “Strain and stress in a deformed solid” to “Bubbles, drops and cavitation”.

Chapters

  1. Strain and stress in a deformed solid
    • Index notation and tensor rotation
    • The displacement field and its gradient
    • The strain tensor and the local rotation
    • Dilatation and principal strains
    • The equilibrium equations inside a body
    • Cauchy's equation of motion
    • Principal stresses and Mohr's circle
    • Deviatoric stress and maximum shear stress
    • Problems
  2. Hooke's law for an isotropic solid
    • Isotropic Hooke's law from symmetry
    • Young's modulus and Poisson's ratio in three dimensions
    • Bulk modulus and the incompressible limit
    • Strain energy density and its bounds
    • The Navier equation of elastic equilibrium
    • Plane strain and plane stress
    • The thick-walled pressure vessel
    • Thermal stress in constrained bodies
    • Saint-Venant's principle and elastic anisotropy
    • Problems
  3. Torsion and bending of rods and beams
    • Torsion of a circular shaft
    • Torsional rigidity of tubes and fibres
    • Pure bending and the neutral axis
    • Second moment of area and the moment-curvature relation
    • The Euler-Bernoulli beam equation
    • Cantilever deflection from diving boards to AFM probes
    • Bending energy and the slender-beam approximation
    • Problems
  4. Buckling, plates and elastic foundations
    • The Euler buckling load
    • Effective length and end conditions
    • An energy criterion for stability
    • Column slenderness ratio and yield
    • Self-weight buckling and the height of trees
    • Plate flexural rigidity and the plate equation
    • Flexure on an elastic foundation
    • Bending versus stretching in thin sheets
    • Problems
  5. Elastic waves in a solid and at its surface
    • The elastodynamic Navier equation
    • Helmholtz decomposition into two wave types
    • Bar wave speed and uniaxial stress
    • Elastic impedance and normal reflection
    • Mode conversion at a free surface
    • The Rayleigh wave
    • Locating an earthquake from its S-P time
    • Ultrasonic pulse-echo measurement of elastic constants
    • Problems
  6. Vibrations of rods, beams and plates
    • Longitudinal and torsional modes of a rod
    • Flexural wave dispersion in a beam
    • Cantilever modes and the tuning fork
    • Free-free bar modes and tuned percussion
    • Rayleigh's method for frequencies
    • Plate flexural waves and coincidence
    • Loss factor and the ringing time
    • Limits of thin-beam theory
    • Problems
  7. Dislocations and the onset of plastic flow
    • Theoretical shear strength and measured yield
    • The Volterra dislocation and its Burgers vector
    • The screw dislocation field
    • The edge dislocation field
    • Dislocation line energy and line tension
    • The Peach-Koehler force
    • The Frank-Read source and the yield stress
    • The Orowan relation for plastic strain rate
    • Yield criterion of Tresca and von Mises
    • The fully plastic moment of a beam
    • Problems
  8. Fracture: stress concentration, cracks and toughness
    • Flaws and Griffith's estimate of brittle strength
    • Stress concentration at a hole
    • The energy release rate of a crack
    • The stress intensity factor at a crack tip
    • Fracture toughness and critical flaw size
    • The crack-tip plastic zone
    • Crack stability under load and displacement control
    • Fatigue crack growth and service life
    • The ductile-to-brittle transition in steel
    • Problems
  9. Non-Newtonian and viscoelastic fluids
    • Rubber elasticity and a putty that flows
    • The generalised Newtonian fluid
    • Power-law fluid in a pipe
    • The Bingham plastic and plug flow
    • The Maxwell model and stress relaxation
    • The Kelvin-Voigt model and creep
    • The complex shear modulus in oscillatory shear
    • The Deborah number from putty to mantle
    • Normal stress differences and rod climbing
    • Problems
  10. Bubbles, drops and cavitation
    • The Rayleigh-Plesset equation
    • Rayleigh collapse time of an empty cavity
    • Minnaert resonance of a gas bubble
    • Nucleation and the Blake threshold
    • The cavitation number of a flow
    • Cavitation damage by collapse jets
    • Bubble rise velocity from micrometres to centimetres
    • Drop oscillation and breakup
    • Problems