Winter 2012 Math 272A – Foundations of Continuum Mechanics


Instructor: Prof. Marcus Roper,

Meeting times: M: 3-5pm, W: 3-4pm, Math Sci. 5217

Office hours: Tu:4-5pm, W:1.30-2.30pm, Th.:11am-noon, Math Sci. 7619B

Many of the most important ideas and developments in applied math, and particularly the set of numerical and analytical tools that we use generally to solve and understand PDEs originated in the 19th and 20th century study of continuum mechanics. Continuum mechanics is still a vital subject in the 21st century: in addition to its traditional role in shaping the design and study of structures (from MEMS to bridges), an increasing amount of data points to the importance of mechanical stresses in biology: organisms musy deal with physically challenging environments to disperse, move and grow. Mechanics is not just inconvenient to organisms: ontogeny, how organisms complexify from small undifferentiated clumps of cells into complex differentiated forms, is now known to be shaped and guided by the physical forces across and between cells.

This class will focus on developing the basic ideas, equations and solution techniques of Solid Mechanics (a corresponding introductory course in Fluid Mechanics will be offered in 2012-13). We will assume that this is your first mechanics class. Our emphasis will be on classical problem solving techniques, many of which have been around since the 1930s (but that remain as useful and important today). However, we will try to use the time available to us to also provide an introduction to the Finite Element Method, which is at the heart of most modern computational solid mechanics.

Holidays: No class on Mon Jan 16th and Mon Feb 20th.

Conflicts: I am away from UCLA on Mon Feb 27th, we will have to reschedule those two hours of class.

Course outline:

Principles of linear elasticity: Superposition, Work, Maxwell and Betti-Rayleigh reciprocity, Strain Energy, Principle of Virtual Work

Tensor analysis

Stress: Symmetry of the stress tensor, Stress Transformation, Invariants

Strain: Measuring deformation (Green and Cauchy strain tensors), Infinitesimal strain and interpretation of strain tensor components

Physics: Conservation laws, constitutive modeling, Navier’s equation

Applications of linear elasticity I: Simple beam bending, pressurized sphere. Linear elastodynamics (P-, SH-, SV- waves).

Potential methods: St. Venant’s theory of torsion of a pipe. Lame’s strain potential, Papkovich-Neubar potentials, Galerkin vector, Airy stress functions

Applications of linear elasticity II: Boussinesq’s problem, and Hertzian contact

Introduction to large deformation: Ogden’s method

Simplifying geometry: Elastic rods and plates. Membranes.

Computational solid mechanics: Foundations of the Finite Element Model

Assessment: You will be assessed by three homeworks (50%) and a take home final (50%).

References: Although we don’t follow any specific text closely, you may find the following references helpful (list compiled by J. Vlassak):

Fung, Foundations of solid mechanics

Timoskenko and Goodier, Theory of elasticity

Malvern, Introduction to the mechanics of a continuous medium

Atkin and Fox, An introduction to the theory of elasticity

Landau and Lifschitz, Theory of elasticity

Fung’s book is probably the most directly relevant reference for the first three weeks of the class