Mathematical description of deformation

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 back to its undeformed state.
The deformation matrix D is very useful if one wants to model deformation using a computer. Once the deformation matrix is defined, any aspect of the deformation itself can be found. Once again, it tells us nothing about the deformation history, nor does it reveal how a given deforming medium responds to such a deformation. For more information about matrix algebra, see below.

MATRIX ALGEBRA

Matrices contain coefficients of systems of equations that represent linear transformations. In two dimensions this means that the system of equations shown in Equation 2.1 can be expressed by the matrix of Equation 2.2. A linear transformation implies a homogeneous deformation. The matrix describes the shape and orientation of the strain ellipse or ellipsoid, and the transformation is a change from a unit circle, or a unit sphere in three dimensions. 
Matrices are simpler to handle than sets of equations, particularly when applied in computer programs. The most important matrix operations in structural geology are multiplications and finding eigenvectors and eigenvalues: 
Matrix multiplied by a vector:

The determinant describes the area or volume change: If det D=1 then there is no area or volume change involved for the transformation (deformation) represented by 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 two eigenvectors for two dimensions and three for three dimensions. The eigenvectors describe the orientation of the ellipsoid (ellipse), and the eigenvalues describe its shape (length of its principal axes). Eigenvalues and eigenvectors are easily found by means of a spreadsheet or computer program such as MatLabTM.

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