Figure 2.7 Hypothetical vertical column extending from the surface to depths in the lithosphere. If each column were free to expand laterally, d would move to d' and d" and c would move to c' and c". However, lateral stresses develop and increase with depth so that the tendency of every column is constrained by the tendency if adjacent columns are to expand laterally so that horizontal strain is preserved.
of the body. If we are dealing with hot rocks or minerals at considerable depth, they will behave as a fluid, so that the horizontal stress will equal the vertical stress (i.e. the stresses are 'hydrostatic').
In the upper 40-60 km of oceanic lithosphere, much of the rock mass away from the spreading-ridge will behave, to a very close approximation, as an elastic body, even when the stresses act for very long periods (i.e. 1.5*108 years). Consequently, it can be shown (Price, 1966) that when the horizontal strain (eh) is zero, the horizontal stress (.S'h) is given by:
where m is Poisson's number, the reciprocal of Poisson's ratio.
The value of m depends upon rock type. Moreover, it changes, in real rock, with the magnitude of ambient stresses (Price, 1966). However, for convenience, we shall deal only with linear-elastic theory, in which m does not change with ambient stress. Also, we shall assume that m=4.0 (a reasonable approximation for many basic rock types). Hence, when the conditions of zero horizontal strain are met, then, from Equation 2.2, it follows that the horizontal stress, at any specific depth, is about one third the vertical pressure.
It is usual to refer to the vertical direction as the z axis and the two orthogonal horizontal axes as the x and y directions, where, commonly, the x direction is attributed to the greatest horizontal stress, and the y direction indicates the direction of action of the least horizontal stress. These stresses may, or may not, be principal stresses.
The stresses which result from a shortening of a plate parallel to the direction of movement are given by a combination of the horizontal stresses induced by gravity, plus the increment of compressive stress caused by the compressive strain, so that (Price, 1966) the maximum horizontal stress Sx is given by
The compressive stress Sc causes strains to develop in the directions z and x. In the vertical direction (z), the surface can move upward as the result of this strain. However, as elastic strains are usually very small, the increase in vertical stress can conveniently be ignored.
The increase in stress in the v direction, assuming that strain ey=0, is equal to .S'6/m. so that
If the plate is being pulled, there will be extension in the direction of plate movement so that Eq. 2.3a and b, respectively, will become:
These equations1 confirm our simple model, i.e. that if the plates are being pushed, then at or near the surface the axis of maximum horizontal stress should be aligned parallel to the direction of absolute plate motion. However, if the plates are being pulled, it is the axis of least horizontal stress that should be aligned parallel with the direction of absolute plate motion.
If a hypothetical plate had a rectangular shape and uniform boundary stresses, these equations would hold throughout much of the plate. However, since real plates are irregular in plan, boundary conditions vary from place to place. These equations indicate the type of stress conditions which are likely to exist in the central region of the plate which includes the axis of the resultant force.
Let us now ascertain to what extent evidence of stress orientations and their relative magnitudes in real plates may be inferred from studies using various methods.
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