Page 28 - Microtectonics

2.5 · Homogeneous and Inhomogeneous Flow and Deformation 15
as complex as may be supposed from Fig. B.2.1 since, con-
trary to cars, the velocities of neighbouring particles in
an experiment or deforming rock are not independent.
Flow in nature is generally inhomogeneous and diffi-
cult to describe in numbers or simple phrases. However,
if considered at specific scales (Fig. 2.4), flow may be ap-
proximately homogeneous with an identical flow pattern
throughout a volume of material, wherever we choose the
origin of the reference frame (Fig. 2.4a). The result after
some time is homogeneous deformation.
Characteristic for homogeneous deformation is that
straight and parallel marker grid lines remain straight and
parallel, and that any circle is deformed into an ellipse. Ho-
mogeneous flow or deformation can (in two dimensions)
be completely defined by just four numbers; they are ten-

Box 2.3 Tensors
All physical properties can be expressed in numbers, but dif-
ferent classes of such properties can be distinguished. Tem-
perature and viscosity are independent of reference frame and
can be described by a single number and a unit, e.g. 25 °C and
5
10 Pas. These are scalars. Stress and homogeneous finite
strain, incremental strain, finite deformation, incremental de-
formation and flow at a point need at least four mutually in-
dependent numbers to be described completely in two dimen-
sions (nine numbers in three dimensions). These are tensors.
For example, the curves for the flow type illustrated in Fig. 2.6a
need at least four numbers for a complete description, e.g.
amplitude (the same in both curves), elevation of the Ö-curve,
elevation of the ω-curve, and orientation of one of the maxima
or minima of one of the curves in space. We might choose
another reference frame to describe the flow, but in all cases
four numbers will be needed for a full description.
Homogeneous deformation can be expressed by two equa-
tions:
x'= ax + by

y'= cx + dy
where (x',y') is the position of a particle in the deformed state,
(x,y) in the undeformed state and a, b, c, d are four parameters
describing the deformation tensor. Homogeneous flow can be
described by similar equations that give the velocity compo-
nents v an v in x and y direction for a particle at point x, y:
y
x
v x = px + qy
v = rx + ty
y
p, q, r, t are four parameters describing the flow tensor. Both
Fig. 2.4. Illustration of the concepts of homogeneous and inhomo-
geneous deformation. a For homogeneous deformation, straight and tensors can be abbreviated by describing just their param-
eters in a matrix as follows:
parallel lines remain straight and parallel, and a circle deforms into
an ellipse, the axes of which are finite strain axes. Inhomogeneous
and homogenous deformation occur on different scales. b Five scales
of observation in a rock. From top to bottom Layering and foliation
on a km scale – approximately homogeneous deformation; layering
Multiplication of these matrices with the coordinates of a
and foliation on a metre scale – inhomogeneous deformation; folia-
tion on a cm scale – approximately homogeneous deformation; thin particle or a point in space gives the complete equations. Ma-
section scale – inhomogeneous deformation; crystal scale – approxi- trices are used instead of the full equations because they are
mately homogeneous deformation easier to use in calculations.