An introduction to the basic concepts, equations, and methods of the mechanics of solids, including solutions of representative problems in linear elasticity. Through lectures and reading exercises, discover the concepts of stress and strain, and discuss conservation laws and constitutive equations. Derive the Navier equations of linear elasticity, and use them and the Airy stress-function method to understand the mechanics of fracture and of plastic deformation. Solve problems to illustrate the behavior of cracks, dislocations, and force-induced singularities in applications in materials science, structural engineering, geophysics, and other disciplines.
(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).
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.
General Continuum Mechanics by T. J. Chung (2007) Cambridge University Press
Scaling by G. I. Barenblatt (2003)