Reference Work Entry

Structural Geology and Tectonics

Part of the series Encyclopedia of Earth Science pp 750-751

Stress ellipsoid

  • Stephen Marshak
The stress ellipsoid is a useful mental tool for geometrically representing the state of three-dimensional stress at a point. Stress is a force per unit area, and the stress applied to a specific plane can be represented by a vector (a quantity that has both magnitude and direction). The stress vector associated with a plane need not be perpendicular to the plane; if it is not, the stress vector can be resolved into a normal component (perpendicular to the plane) and a shear component (parallel to the plane). At any point in a body of rock, one can imagine that there are an infinite number of planes, each with a different orientation, and if that body of rock is subjected to a compressive stress, a pair of inward-pointing stress vectors will be associated with each plane. The members of each pair must be equal in magnitude and opposite in orientation to their partner; otherwise, the rock body could not remain at rest. If all the stress vectors associated with the infinite number of imaginary planes at a point were drawn so that their heads touch, one can imagine a three-dimensional surface defined by the tails of all the vectors. This three-dimensional surface is the stress ellipsoid (Means, 1976), and it represents the three-dimensional state of stress at the point located at the center of the ellisoid. A two-dimensional section of this ellipsoid is an ellipse (Fig. 1). If the stresses were tensional, then the vector orientations would be reversed and the ellipsoid defined by a surface composed of the heads of all the vectors.
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FIGURE 1

An elliptical section through the stress ellipsoid showing the inward-pointing pairs of stress vectors.

All ellipsoids are uniquely specified by the lengths of three mutually orthogonal axes called the principal axes (Fig. 2). In the stress ellipsoid, only those stress vectors that are parallel to the principal axes of the elipsoid are normal to the planes on which they act. All the other stress vectors have shear components. Thus, the distance from the center of the ellipsoid to its surface is not representative of the normal stress in that orientation, unless the direction of measurement happens to be on one of the principal axes. The stress vectors that are parallel to the principal axes are called the principal stresses. The maximum principal stress is usually called σ1, the intermediate principal stress is usually called σ2, and the minimum principal stress is usually called σ3. The three mutually perpendicular planes, each of which contains two of the principal stresses and is perpendicular to the third, are called the principal planes of stress.
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FIGURE 2

A stress ellipsoid showing the mutually orthogonal axes that are parallel to the three principal stresses. The ellipsoid is oriented to show σ1 > σ2 > σ3.

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).

Further reading

Germano G., Kavanagh P.B., Waechter P., et al., 2000. A new algorithm for the quantitation of myocardial perfusion SPECT. I: Technical principles and reproducibility. J Nucl Med 41 (4): 712–719.

Stapel G., Moeys R., Biermann C., 1996. Neogene evolution of the Sorbas basin (SE Spain) determined by paleostress analysis. Tectonophysics 255 (3–4): 291–305.

Rohr J., Latz M.I., Fallon S., et al., 1998. Experimental approaches towards interpreting dolphin–stimulated bioluminescence. J Exp Biol 201 (9): 1447–1460.

Florea V.G., Mareyev V.Y., Samko A.N., et al., 1999. Left ventricular remodelling: common process in patients with different primary myocardial disorders. Int J Cardiol 68 (3): 281–287.

Mazeron P., Muller S., ElAzouzi H., 1997. On intensity reinforcements in small–angle light scattering patterns of erythrocytes under shear. Eur Biophys J Biophy 26 (3): 247–252.

Nemat–Nasser S., 1999. Averaging theorems in finite deformation plasticity. Mech Mater 31 (8): 493–523.

Shima S., Kotera H., Ujie Y., 1995. A study of constitutive behaviour of powder assembly by particulate modeling. Mater Sci Res Int 1 (3): 163–168.

Nagano K., Sato K., Niitsuma H., 1996. Polarization of crack waves along an artificial subsurface fracture. Geophys Res Lett 23 (16): 2017–2020.

Luo J., Stevens R., 1996. Micromechanics of randomly oriented ellipsoidal inclusion composites .1. Stress, strain and thermal expansion. J Appl Phys 79 (12): 9047–9056.

Baer G., Beyth M., Reches Z., 1994. Dikes emplaced into fractured basement, Timna Igneous Complex, Israel. J Geophys Res–Sol Ea 99 (B12): 24039–24050.

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