Continuum Mechanics

Behavior of materials using an generalization from discrete particles to a continuous distribution of mass. Collected of three courses: Solids, Fluids, and Incompressible Flow.

One of my favorite realizations is that at the most infinitesimal scales, most fields of study are use discrete units of some material. Fluids studies the movement of particles. Solids study the movement of dislocations. Electronics is study the movement of electrons in a medium and their effect on the electric field. Most of chemistry is studying the sharing of electrons in orbitals and the resultant molecules. Radiation is the movement of neutrons in and out of the nucleus and how much power that can produce.

Matter which makes up all this stuff is are just discrete atoms. Even waves are made of individual repeating wave numbers. It is quite strange how school first teaches a discrete, then distributed, and then back to discrete: Classical mechanics show forces acting through the center of mass assumes rigid bodies, this continuum method to show distributed loading, and then quantum mechanics to show interactions.

But quantum mechanics only function at the smallest scales: the meso and macro scales observations are orders of magnitude larger than those discrete units. For example, electrical current SI unit is the Ampere which is a rate of transfer of 1 Coulomb per second. For reference, an electron carries $1.60217662 \times 10^{-19}$ Coulomb.

That is $\bold{19}$ orders of magnitude fewer than the fundamental unit. Even working in milliamps, there are still 16 decimal points of precision to work with. Functionally, that is an arbitrary level of precision to work with.

Without the knowledge of fundamentally discrete units, our mathematical models have arbitrary precision as well and that is where we start: from the math which starts our understanding to build towards that more accurate discrete representation.

Rundown

Once again, going back to etymology, a continuum is a smooth contiguous body and mechanics being the study of forces.

Comment: what an awful intro to use etymology

Material Derivative

$$\frac{D}{Dt} = \frac{\partial}{\partial t} + {\bf v}\cdot{\nabla}$$

Links

Internet fluids book

RPI study on teaching

Nikander compressible flow helps shows the flexibility of the plaintext file hosting, vs a more traditional method of course hostng.