Material Constitution of Rocks

Homogeneity and continuity of materials

A material is homogeneous if the properties of one piece are the same as the properties of any other piece

A material that does not fit this definition is said to be inhomogeneous

A material is continuous if every subvolume within it is occupied by the material and if its properties vary smoothly from one point to another

A material that does not have these characteristics, in which properties such as density and strength vary abruptly across internal surfaces, is said to be discontinuous

The concepts of perfect homogeneity and perfect continuity are nevertheless useful because they define the properties of the idealized materials dealt with in much of the mathematical theory of mechanical behavior.

These concepts also permit us to classify material as approximately homo- geneous or approximately continuous if we specify permissible departures from the ideal.

This is illustrated for homogeneity in Figure 1.1.

Notice that to call a region of real material homogeneous in this looser, or "statistical," sense, we have to specify a scale on which the region is sampled and also the particular properties with respect to which we are examining it

Granitic dike rock showing biotite grains and joints. The rock is approximately homogeneous on a scale of 1 cm with respect to biotite content (i.e., 1 cm 3 anywhere in the specimen contains approximately the same number of biotite grains). The rock is not homogeneous on this scale with respect to joint frequency.

Material constituents and structure of rocks

Figure 1.2 shows rock bodies on various scales.

In each field of view there are approximately homogeneous regions of more than one kind.

These types of homogeneous regions are the material constituents of rock bodies.

In Figure 1.2e, for example, the material constituents are quartz grains, mica films, and kinked biotites; in Figure 1.2c they are folded sediments and igneous rock. Whatever scale we look on, we tend to see regions occupied by different material constituents.

Such regions are separated by bound- aries, and it is these boundaries-in particular their geometrical configuration in space-that define the structure of a rock body.

If it takes a microscope to see the structure, we call it the microstructure.

Figure 1.2 Rock bodies on various scales, showing material constituents and structure. (a) The whole earth, made up of core, mantle, and crust. (b) A

continental margin, showing crust and mantle. (c) Part of a mountain belt, showing folded sediments and igneous rock. (d) A fold hinge, show- ing sandstone and slate. (e) Part of a thin section, showing quartz grains, mica films, and kinked

biotite. (f) An intra- crystalline region, showing normal biotite and biotite with abundant cleavage cracks, with a kink boundary between them. (g) A region

within the macroscopic kink boundary of (f), showing, schematically, regions of normal crystal structure separated by a subgrain boundary region o f very

abnormal structure. In each picture the material constitu- ents are the labeled entities and the structure is the con- figuration of the boundaries between them.)

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Continuum mechanics and discontinuous rocks (1)

The branch of mechanics that treats materials as if they were continuous is called continuum mechanics.

We ask: Why study this subject in geology if rocks are full of grain boundaries, bedding planes, and other structures across which there is an abrupt and essentially discontinuous change in material properties?

There are two main parts to the answer:

1. Experience shows that continuum mechanics often makes approximately correct predictions, even when applied to discontinuous materials.

2. The second reason for learning continuum mechanics is that it is much simpler mathematically than the theory of discontinue. This latter theory is based on continuum concepts anyway, so for reasons of simplicity and priority, students need to learn continuum concepts first.

Continuum mechanics and discontinuous rocks (2)

Because we restrict attention in most of the rest of this course to the continuum view of rocks, we should emphasize before leaving them that the discontinuities in rocks are all important in controlling actual material behavior.

These are the features that have to be studied ultimately to learn the origin of deformational features in rocks.

Faults are initiated at cracks, and mineral grains flow because tiny regions of abnormal structure become activated and start migrating through the crystals.

In each case, deformation occurs at and propagates through regions that are not typical of the main mass of the material. Deformational processes of most types are thus highly localized phenomena.

Continuum mechanics and discontinuous rocks (3)

Mechanical State

The instantaneous condition of a rock system is referred to here as the mechanical state of the system.

We distinguish carefully between quantities describing the instantaneous state and other quantities comparing two or more states.

The mechanical state at an instant is charac- terized by features like the position of each part of the system, the velocity with which each part is changing position, and the forces acting on and between parts.

To describe any of these features quantitatively, we need a reference frame and a coordinate system.

In Figure 2.1 the reference frame is fixed to the footwall block of a microfault in sandstone, and two coordinate systems are shown.

Any quartz grain has different coordinates in the two systems, but these are always related by the equations

because the two coordinate systems are fixed in this manner relative to one another in the same reference frame. Equations 2.1 are examples of coordinate transformation formulae.

Mechanical State

Reference frames are usually defined (as above)by arbitrarily designating some convenient part ofa system as motionless.

In Figure 2.1 the footwall block of the fault is taken as motionless, and this means in effect that points in space that maintain fixed distances from particles of the footwall block are assigned fixed coordinates.

Choice of a reference frame therefore has to do with assigning coordinates to points in space.

The particular numbers assigned each point depend on the choice of coordinate system.

Notice the distinction made above between particles (which are very small bits of matter) and points (which are locations in space).

PositionThe positions of parts of a rock system can be specified by giving the coordinates of the point occupied by each particle.

Returning to Figure 2.1, and taking individual quartz grains as "particles," we can describe the positions by listing the coordinates of each quartz grain.

A second way to describe positions is to specify a position vector for each quartz grain.

A position vector is represented by an arrow drawn with itstail at the origin of the coordinate system and its tip at the particle represented (e.g., vectors P and Q in Figure 2.1).

The positions of all particles of a rock body are thus represented by an infinite number of position vectors.

Notice that the components of a position vector, parallel to the coordinate directions, are exactly equal in magnitude to the coordinates of the particle.

Configuration

Once the positions of all particles are specified,we automatically also know the configuration ofthe system-that is, the length and relative orientations of lines connecting all possible pairs of particles.

For each pair of particles the vector difference be- tween their position vectors is a third vector (Figure 2.2) that has magnitude equal to the length of the connecting line between the particles and direction parallel to the connecting line.

Velocities

The instantaneous velocities in a system can be pictured as an array of small arrows, one for each particle, pointing in the instantaneous direction of particle movement and having lengths proportional to the speed of the particle.

Each arrow represents a vector quantity (having magnitude and direction) which is the velocity of each particle.

The array of arrows represents the velocity field. The velocity field for Figure 2.1 is shown in

Figure 2.3. Notice that this is the velocity field for one instant only, the instant referred to as time t in Figure 2.1.

Forces (1)At each point on the boundary of a quartz grain A there will be some force exerted, either by a neighboring grain B or by neighboring pore fluid Figure 2.4a). An equal and opposite force will be exerted on B, or pore fluid, by A.

The magnitude of these forces will be very small since the point in question is very small, and for this reason it is difficult to think about forces at points in rocks or other materials.

If, however, we think instead about the ratio of force to area across bits of the bound- aries of the quartz grains, this ratio will generally have a finite value even for very small areas.

Forces (2) This useful ratio is called stress, and at any given instant there will be a definite stress across all parts of all the quartz grain boundaries.

The stress across each small area is a vector quantity, and, as for velocity, we can picture the distribution of grain boundary stresses by an array of stress vectors, part of which is drawn in Figure 2.4b.

Notice that we cannot draw a stress vector for a whole quartz grain, or for any other particle, because stress vectors by definition refer to particular planes and not to three-dimensional regions, however small.

Mechanical Homogeneity

In Chapter 1 we explained what is meant by a homogeneous material. We can use similar definitions to classify the mechanical condition of a rock system as homogeneous or inhomogeneous.

A velocity field is homogeneous if all the velocity vectors have the same magnitude and orientation. A stress field is homogeneous if the stress vector associated with each orientation of plane has the same magnitude and orientation, regardless of the position of the plane in the body.

Thus, for example, all vertical, North-South planes in a rock body must be subject to the same stress vector if the body is homogeneously stressed.

If a vertical, North-South plane at one point in a body is asso- ciated with a different stress vector from a vertical, North-South plane at some other point, the body is inhomogeneously stressed.

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