vectors

We all live in a three-dimensional space, in which objects can be located by their distances from an arbitrary point in the space - call it the "origin." To facilitate this, we imagine a rectilinear, orthogonal coordinate system. "Rectilinear," because the space can be defined with straight lines (rather than curves, as, say, on the surface of a sphere), and "orthogonal" because a point in the space can be uniquely determined by specifying its location on three mutually perpendicular directions.

In the drawing, we have marked off the three perpendicular axes, labelled x, y and z. The location of the tip of the arrow (or vector) in the drawing can be represented in standard notation by (a,b,c), where a is the distance from the origin along the x axis, and similarly for b and c along the y and z axes. We say that the vector x = (a,b,c).

In the previous blog I showed a drawing with a point x on the surface of an arbitrary volume within a solid body. Now I can say more precisely, that x is the vector from the origin of the coordinate system to the point on the surface, but to keep the drawing simpler I omitted the coordinate system. Anyhow, I wanted to explain this so you'd know why x is a vector.

Now, to continue that earlier discussion: We have the unit vector n perpendicular to the arbitrary surface drawn through the point located by the vector x. It's easier to talk about n if we choose a coordinate system so that the surface through x is perpendicular to one of the axes. For example, if this surface is perpendicular to the x axis, then n = (1,0,0). This notation means that n has a value of 1 in the x direction, and zero in the y and z directions. (Remember, this value 1 has whatever unit of length you choose, as long as you are consistent, and use the same unit for every length).

Now we can talk about the stress vector, t. What makes it so interesting to me is that t depends not only on the location in the solid defined by x, but also on the orientation of the surface we've chosen to examine, that contains the point defined by x.

Before we go any further, let's define stress: stress results from the force of a solid material acting on itself.

Imagine that we cut a rubber band, so that we have a single piece of material, and we pull on it from each end. We are then exerting a force on the ends (it has to be the same force on each end, for the material to hold still), but what about in the interior of the body?

Imagine a very small cube, somewhere inside the rubber, with its surfaces lined up so one pair is perpendicular to the direction we are pulling, and the others are therefore parallel to that direction. The forces acting on those surfaces are exactly balanced by forces in the the opposite directions, since we are holding the rubber band still - i.e., the elastic body is stationary. The material is actually pulling on itself.

From a microscopic point of view, we are stretching the bonds between molecules of rubber (or even stretching the long chain molecules themselves), and the magnitude of the stress depends on the strength of the molecular bonds. These molecular forces act only within short distances, so the force on each molecule is due only to other molecules nearby. Electrical shielding by the nearby molecules prevents long range effects, except in special cases - piezoelectric materials, for example - when long range forces are important.

But here we are confining ourselves to a continuum model, in which the stress and the properties of the material can be described by mathematically continuous functions in space. That's why we just say that the material "acts on itself." Note also that this means for any region of the elastic body we want to consider, the deformation of that region is entirely determined by the forces acting on its surface. Also, the sum of these forces, taking their directions into account, must add up to zero - otherwise the body would move.

The forces on the perpendicular surfaces mentioned above are perpendicular, or "normal" to the surfaces (they are equal and in opposite directions), whereas the forces on the parallel surfaces are parallel to them, or "shear," (and they are also equal and in opposite directions). Since we are pulling on the band, the normal forces are called "tensile." If we had a material we could squeeze, like a sponge, the normal forces would be "compressive."

I'll add a drawing of this cube next.