B14
Theoretical and Applied Solid Mechanics
Course Coordinator: 
Gustavo Gioia
Description: 

Students are introduced to the concepts of stress and strain, and discuss conservation laws and constitutive equations. We derive the Navier equations of linear elasticity, introduce the Airy stress-function method, and solve problems to illustrate the behavior of cracks, dislocations, and force-induced singularities in applications relating to materials science, structural engineering, geophysics and other disciplines.

Aim: 
To introduce basic concepts, equations, and methods of the mechanics of solids, including solutions of representative problems in linear elasticity.
Detailed Syllabus: 

(1) Mathematical Preliminaries:

  • Summation convention, Cartesian, spherical, and cylindrical coordinates.
  • Vectors, tensors, linear operators, functionals.
  • Eigenvalues and eigenvectors of second-order symmetric tensors, eigenvalues as extrema of the quadratic form.
  • Fields, vector and tensor calculus.

(2) Stress, Strain, Energy, and Constitutive Relations:

  • Cauchy stress tensor, traction, small strain tensor, compatibility.
  • Strain energy, strain energy function, symmetries, elastic modulii.

(3) Elasticity and the Mechanics of Plastic Deformation:

  • Navier equations, problems with spherical symmetry and problems with cylindrical symmetry (tunnels, cavities, centers of dilatation).
  • Anti-plane shear. Plane stress, plane strain.
  • The Airy stress-function method in polar and Cartesian coordinates.
  • Superposition and Green's functions.
  • Problems without a characteristic lengthscale.
  • Flamant's problem, Cerruti's problem, Hertz's problem.
  • Load-induced versus geometry-induced singularities (unbounded versus bounded energies).
  • Problems with an axis of symmetry.
  • Disclinations, dislocations, Burgers vector, energetics; relation to plastic deformation in crystalline solids.

(4) Fracture Mechanics:

  • The Williams expansion, crack-tip fields and opening displacements via the Airy stress-function method (modes I, II) and via the Navier equations (mode III), crack-tip-field exponents as eigenvalues, stress intensity factors.
  • Energy principles in fracture mechanics, load control and displacement control.
  • Energy release rate and its relation to the stress intensity factors, specific fracture energy, size effect, stability. The Griffith crack and the Zener-Stroh crack. Anticracks.

(5) Possible Additional Topics (if time allows):

  • Elasticity and variational calculus, nonconvex potentials, two-phase strain fields, frustration, microstructures.
  • Stress waves in solids, P, S, and R waves, waveguides, dispersion relations, geophysical applications.
  • Dislocation-based fracture mechanics, the Bilby-Cotterell-Swindon solution, small- and large-scale yielding, T-stress effects, crack-tip dislocation emission, the elastic enclave model.
  • Deterministic versus statistical size effects in quasibrittle materials.
  • Vlasov beam theory, coupled bending-torsional instabilities.
  • Dynamic forms of instability, nonconservative forces, fluttering (Hopf bifurcation).
Course Type: 
Elective
Credits: 
2
Text Book: 
No textbook is set. Students are expected to take good notes in class. The Professor will from time to time distribute essential readings, as needed.
Reference Book: 
General Continuum Mechanics by T. J. Chung (2007) Cambridge University Press
Scaling by G. I. Barenblatt (2003)