# Deformation

It goes almost without saying that any mathematical study of continua (rocks, butter, water) must address the issue of "What do we mean by deformation?" This goes back to the more basic mathematical issue of how we assign coordinates and how we specify regions, especially the region occupied by our continuum.

One usually begins their study of coordinates by graphing various functions (straight lines, parabolas and so on). Here the inputs are taken to run along the horizontal (or x) axis and the outputs are taken to run along the vertical (or y) axis. This set up for graphing is such a basic part of our mathematical training that we forget that it involves many assumptions. Indeed there is no requirement for the axes to be perpendicular, for the x-axis to represent the inputs, or even for positive values to increase to the right. However many of these assumptions are good ones and have stood the test of time.

The assumption that we will accept nearly without reservation is the one of perpendicular axes. From linear algebra we know this means we are choosing the two vectors (1,0) and (0,1) as the basis vectors which we subsequently use to build up any other vectors, for example (3,-2)=3(1,0)-2(0,1). The orthogonality of the axes will come in handy later. For now however we return to the specification of the region occupied by our continuum.

Clearly just about any continuum will be more than just the one dimensional graph we are used to drawing. Whether a particular shape can have its region specified as a formula is determined byits complexity. Indeed for many of the shapes from classical geometry we have formulae involving inequalities, such as:

(x-5)2​ + (y-3)2​ < 52

for a disk of radius 5 centered at (5,3). For more realistic shapes we may have to make do with an approximation, but in practice most regions in two dimensions can be handled.

Of course once we have a region, we can imagine transforming the region in various geometric ways and showing how the formula changes. Thus for example a reflection across the x axis consists of mapping positive y to negative y, and thus the new disk is given by

(x-5)2​ + (y+3)2​ < 52

However notice that if we only wish to preserve the shape of the disk we have TWO choices for what to do:

Translate every point in the disk down by exactly 6 units, so that, for example (5,6) is mapped to (5,0) and (5,0) is mapped to (5,-6), as in the following figure:

Actually reflect each point across the x-axis, so that (5,6) is mapped to (5,-6) while (5,0) remains unchanged, as in the following figure:

Even though the two transformations used above to make the new disk are different, it seems fairly intuitive that neither actually deforms the disk. A fairly intuitive example of a deformation that does deform a disk would consist of starting with the unit disk centered at the origin:

x2​ + y2​ < 1

and applying the stretching map given by 2(x,y). It is easy to confirm for yourself that the new region is a disk that is still centered at the origin, but that now has a radius of 2. How do we know that there has been some deformation? Intuitively, one idea is to say that since the AREA has increased (quadrupled, in fact) there must be some deformation. Another way to characterize deformation would be to look at the distance between two nearby points before and after the transformation. Say we look at (0,0) and (0.01,0) which are mapped to (0,0) and (0.02,0) after the transformation. Initially the distance between the two points is 0.01 and after the transformation it has doubled. Thus the deformation which has increased the area has also increased the local distance between neighbours.

This brings up an interesting point. We have taken the initial configuration as the one with which we are comparing, but there is no reason for doing this (habit may be). Indeed if we take the transformed disk as the reference state then the distance between neighbours DECREASES due to the transformation.

Indeed for a complex region undergoing a complex transformation (pushing your finger into modeling clay) it may well be that in some regions the material is "squished" and in some regions it is "stretched" as evidence by the red (stretched) and orange (squished) squares in the following figure.

The mathematical complication comes in from the fact that we would like information about the stretching or squishing at every point actually occupied by the material. To analogize this with simple graphs of functions of one variable we would say that a one dimensional function that is differentiable has a linear approximation at every point and this linear approximation characterizes the local rate of increase and decrease.

This analogy suggests that a good way to build an understanding of deformation is to consider a one dimensional example (think of a thin rod that can be stretched or compressed). We divide the rod in its undeformed state into regions of equal length,

and label these regions. After deformation some of these regions will have increased in length while others decreases,

and the task is to develop a theory to describe these changes in the limit of many vanishingly small regions.