B14
Course Coordinator:
Gustavo Gioia
Theoretical and Applied Solid Mechanics
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.
Course Content:

(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)
Prior Knowledge:

Prerequisite is A104 Vector and Tensor Calculus