Mathematical description of deformation
Deformation is conveniently and accurately described and modelled by means of elementary linear algebra. Let us use a local coordinate system, such as one attached to a shear zone, to look at some fundamental deformation types. We will think in terms of particle positions (or vectors) and see how particles change positions during deformation. If (x, y) is the original position of a particle, then the new position will be denoted (x',y'). For homogeneous deformation in two dimensions (i.e. in a section) we have that
The reciprocal or inverse deformation takes the deformed rock lower back to its undeformed kingdom.
The deformation matrix D may be very useful if one desires to model deformation the use of a pc. Once the deformation matrix is de?Ned, any component of the deformation itself can be found. Once again, it tells us not anything approximately the deformation history, nor does it display how a given deforming medium responds to the sort of deformation. For extra statistics approximately matrix algebra, see beneath.
Matrices include coef?Cients of systems of equations that represent linear changes. In two dimensions because of this the machine of equations shown in Equation 2.1 can be expressed via the matrix of Equation 2.2. A linear transformation implies a homogeneous deformation. The matrix describes the shape and orientation of the stress ellipse or ellipsoid, and the transformation is a trade from a unit circle, or a unit sphere in 3 dimensions.
Matrices are less complicated to deal with than units of equations, mainly whilst applied in pc programs. The most crucial matrix operations in structural geology are multiplications and ?Nding eigenvectors and eigenvalues:
Matrix increased with the aid of a vector:
The determinant describes the region or extent change: If det D=1 then there is no location or quantity trade concerned for the transformation (deformation) represented by means of D. Eigenvectors (x) and eigenvalues (lambda) of a matrix A are the vectors and values that fulfil
If A=DDTT, then the deformation matrix has eigenvectors for 2 dimensions and three for three dimensions. The eigenvectors describe the orientation of the ellipsoid (ellipse), and the eigenvalues describe its form (period of its principal axes). Eigenvalues and eigenvectors are easily located by using a spreadsheet or laptop program together with MatLabTM.