We are surrounded by natural and man-made structures that deform when subjected to loadings. These structures span a wide spectrum of length scales, from suspension bridges and aircrafts all the way down to spider webs, human hair, micro-electro-mechanical systems, and cell membranes. In this course, we will focus on slender bodies, which by virtue of their elongated aspect can be modeled as curvilinear media. This simplified geometry permits the introduction of the fundamental concepts of the mechanics of deformable solids without recourse to the heavy mathematical formalism that is inherent to the description of their three-dimensional counterparts. It will thus allow us to solve problems and comprehend phenomena (such as the buckling of elastic beams) involving geometric or behavioral nonlinearities that, in three dimensions, do not lend themselves to analytical treatment.
We will cover the following topics:
- Geometry, deformation, and kinematics of curvilinear media
- External and internal forces and couples, equilibrium equations
- Constitutive relations, including rigid bars, extensible strings, and elastic rods
- Boundary value problems associated with various models: elastic strings, beams, and arcs
- Euler's elastica (and, time permitting, its boundary layer)
- Linearized elasticity of slender bodies and its applications
- Stability of conservative systems (first discrete systems, later, via the calculus of variations, continuous systems)
and, time permitting,
- Dynamics: wave propagation in elastic beams, forced and free vibrations of elastic rods
Prerequisites: Some knowledge of ordinary differential equations would be helpful. As would be PHY 101, 102, 105, 201, and 206. Otherwise, the course will be mostly self-contained.
- Teaching coordinator: Jabbour Michel