If the three principal stresses are equal in magnitude, the ellipsoid simplifies to a sphere, and each of the infinite number of stress vectors becomes a principal stress vector. A state of stress defined by a spherical stress ellipsoid called hydrostatic stress ; under conditions, of hydrostatic stress none of the infinite number of planes at the center of the sphere feels a shear stress. A stress condition defined by a stress ellipsoid in which two of the axes are equal but the third is not is called an axial stress, and a stress condition defined by a stress ellipsoid in which all three axes are unequal is called a general, or triaxial, stress (Means, 1976).
It is not obvious from this discussion that the geometric description of the three-dimensional stress at a point should be an ellipsoid. The ellipsoidal shape can be demonstrated by solving for the balance of forces exerted on a plane (Means, 1976; Nadai, 1950).
The usefulness of a stress ellipsoid in picturing stress can best be illustrated by an example (following Means, 1976). Imagine a grain of a rock that has been buried to a distance h beneath the surface of the Earth. During burial, the rock could not expand in the horizontal plane. The grain will feel the stress in the vertical direction of magnitude ρgh, where ρ is the rock's density, g is the acceleration due to gravity, and h is the vertical distance beneath the surface of the Earth. If no horizontal stresses are applied, the stresses in all directions in the horizontal plane will be only confining stresses and will be approximately equal to v/(1 − v)ρgh, where v is Poisson's ratio. Because the stresses in the horizontal plane are equal and are less than the vertical stress, the vertical stress is σ1 and the horizontal stresses are σ2 = σ3. The stress ellipsoid representing this situation is an axial ellipsoid. If a horizontal tectonic stress is applied on one direction, the three principal axes become unequal, and the stress ellipsoid becomes a general ellipsoid. If the grain were somehow separated from the rock body and were to be suspended in a fluid-filled pore, it would feel a hydrostatic stress.
Calculations involving stress can also be handled by the use of stress tensors. The stress at a point is a second-rank tensor that is defined by nine components. Each component of the tensor is a component of stress across the face of a cube of known orientation. Components of the principal diagonal are normal components; all others are shear. If the stress tensor is specified, the normal and shear components of stress on any given plane can be calculated (Means, 1976; Nye, 1957).
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