The forces on an element of a solid are of two types: body forces and surface forces. Body forces act throughout the volume of the solid. The magnitude of the body force on an element is thus directly proportional to its volume or mass. An example is the downward force of gravity, that is, the weight of an element, which is the product of its mass and the acceleration of gravity. Since density ρ is mass per unit volume, the gravitational body force on an element is also the product of ρg and the element’s volume. Thus the downward gravitational body force is g per unit mass and ρg per unit volume. The densities of some common rocks are listed in Appendix 2, Section E. The densities of rocks depend on the pressure; the values given are zero-pressure densities. Under the high pressures encountered deep in the mantle, rocks are as much as 50% denser than the zero-pressure values. The variation of density with depth in the Earth is discussed in Chapter 4. Typical mantle rocks have zero-pressure densities of 3250 kg m−3. Basalt and gabbro, which are the principal constituents of the oceanic crust, have densities near 2950 kg m−3. Continental igneous rocks such as granite and diorite are significantly lighter with densities of 2650 to 2800 kg m−3. Sedimentaryrocks are generally the lightest and have the largest variations in density, in large part because of variations in porosity and water content in the rocks. Surface forces act on the surface area bounding an element of volume. They arise from interatomic forces exerted by material on one side of the surface onto material on the opposite side. The magnitude of the surface force is directly proportional to the area of the surface on which it acts.
It also depends on the orientation of the surface. As an example, consider the force that must act at the base of the column of rock at a depth y beneath the surface to support the weight of the column, as illustrated in Figure 2–1. The weight of the column of cross-sectional area δA, is ρgyδA. This weight must be balanced by an upward surface force σyyδA distributed on the horizontal surface of area δA at depth y. We are assuming that no vertical forces are acting on the lateral surfaces of the column and that the density ρ is constant; σyy is thus the surface force per unit area acting perpendicular to a horizontal surface, that is, stress.