strain transformation

Let E be a Euclidean planeMathworldPlanetmath. Fix a line in E and a real number r0. Take any point pE. Drop a line mp from p perpendicularMathworldPlanetmathPlanetmathPlanetmath to . Denote d(p,) the distance from p to . Then there is a unique point p on mp such that


The function sr:EE such that sr(p)=p is called a strain transformation, or simply a strain.

One can visualize a strain stretches a geometric figure if |r|>1 and compresses it if |r|<1. If r=1, then sr is the identity function, the only time when a strain is a rigid motionMathworldPlanetmath. For example, let be the x-axis and C be a circle in the upper half plane of the x-y plane. Then the following diagrams show how a strain transforms C:

Again, if is the x-axis, then sr is the function that sends (x,y) to (x,ry). Representing the ordered pairs as column vectorsMathworldPlanetmath and sr as a matrix , we have


Nevertheless, a strain, as a (non-singular) linear transformation, takes lines to lines, and parallel linesMathworldPlanetmath to parallel lines.

In general, given any finite dimensional vector spaceMathworldPlanetmath V over a field k, a strain sr is a non-singular diagonalizable linear transformation on V such that sr leaves a subspacePlanetmathPlanetmathPlanetmath W of codimension 1 fixed. 0rk is called the strain coefficient.

Remark. By choosing an appropriate base for V of dimension n, sr can be represented as a diagonal matrixMathworldPlanetmath whose diagonalsMathworldPlanetmath are 1 in at least n-1 cells and r in at most one cell.

It is easy to see that every non-singular diagonalizable linear transformation on V can be written as a productPlanetmathPlanetmath of n strains, where n=dim(V).

Title strain transformation
Canonical name StrainTransformation
Date of creation 2013-03-22 17:25:45
Last modified on 2013-03-22 17:25:45
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 5
Author CWoo (3771)
Entry type Definition
Classification msc 15A04
Synonym strain
Defines strain coefficient