2. Force and Stress - Supplements
Contents
2S.1.1 Terminology and Symbols of Force and Stress
2S 1.1 Units of Stress and their Conversions
2S.2.1 Mohr Construction Activity
2S.2.3 Stress Trajectories and Stress Fields
2S.1 SUPPORTING INFORMATION
2S.1.1 Terminology and Symbols of Force and Stress
Force Mass times acceleration (F = m ⋅ a; Newton’s second law); symbol F.
Stress Force per unit area (F/A); symbol σ.
Anisotropic stress At least one principal stress has a magnitude unequal to the other principal stresses (describes an ellipsoid).
Deviatoric stress Component of the stress that remains after the mean stress is removed; this component of the stress contains the six shear stresses; symbol σdev. Note this is not the same as differential stress.
Differential stress The difference between two principal stresses (e.g., σ1 – σ3), which by definition is ≥0; symbol σd.
Homogeneous stressStress at each point in a body has the same magnitude and orientation
Hydrostatic stressIsotropic component of the stress; strictly, the pressure at the base of a water column; symbol P (or Ph).
Inhomogeneous stress Stress at each point in a body has different magnitude and/or orientation.
Isotropic stress All three principal stresses have equal magnitude (describes a sphere).
Lithostatic stress Isotropic pressure at depth in the Earth arising from the overlying rock column (density × gravity × depth, ρ ⋅ g ⋅ h); symbol Pl .
Mean stress (σ1+σ2+σ3)/3; symbol σm (or σmean).
Normal stress Stress component oriented perpendicular to a given plane; symbol σn .
Principal plane Plane of zero shear stress; three principal planes exist.
Principal stress The normal stress on a plane with zero shear stress; three principal stresses exist, with the (geologic) convention σ1 ≥ σ2 ≥ σ3, where compressive stresses are positive.
Shear stress Stress parallel to a given plane; symbol σs (sometimes the symbol τ is used).
Stress ellipsoid Geometric representation of stress; the axes of the stress ellipsoid are the principal stresses.
Stress field The orientation and magnitudes of stresses in a body.
Stress tensor Mathematical description of stress (stress is a second-order tensor).
Stress trajectory Principal stress directions in a body.
2S.1.2 Units of Stress and their Conversions
2S.2 MORE EXPLORATION
2S.2.1 Mohr Construction Activity
Figure S2.1a shows six planes in a stressed body at different angles with σ3. Using the graph in Figure S2.1b, draw the Mohr circle and estimate the normal and shear stresses for these six planes. You can check your estimates by using Equations 2.8 and 2.11.
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FIGURE S2.1. Adventures with the Mohr circle. To estimate the normal and shear stresses on the six planes in (a) apply the Mohr construction in (b). The principal stresses and angles θ are given in (a). You can check your estimates from the construction in σn – σs space by using Equations 3.7 and 3.10. |
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2S.2.2 The Stress Tensor
The stress ellipsoid is a convenient way to visualize the state of stress, but it is cumbersome for calculations. For example, it is difficult to determine the stresses acting on a randomly chosen plane in a three-dimensional body, or the corresponding stresses when we change the reference system (e.g., by a rotation). In contrast, the stress tensor, which mathematically describes the stress state in terms of three orthogonal stress axes, makes such determinations relatively easy. So let us take a look at the stress tensor in a little more detail.
A vector is a physical quantity that has magnitude and direction; it is visualized as an arrow with length and orientation at a point in space. A vector is represented by three coordinates in a Cartesian reference frame that we describe by a matrix consisting of three components. Figure 2.5 showed that stress at a point is a physical quantity that is defined by nine components, which is called a second-rank tensor. This is represented by an ellipsoid with orientation, size, and shape at a point in space. The rank of a tensor reflects the number of matrix components and is determined by raising the number 3 to the power of a tensor’s rank; for the stress tensor this means, 32 = 9 components. It follows that a vector is a first-rank tensor (31 = 3 components) and a scalar is a zero-rank tensor (30 = 1 component). Geologic examples of zero-rank tensors are pressure, temperature, and time; whereas force, velocity, and acceleration are examples of first-rank tensors.
Consider the transformation of a point P in three-dimensional space defined by coordinates P(x, y, z) to point P′(x′, y′, z′). The transformed condition is identified by adding the prime symbol (′). We can describe the transformation of the three coordinates of P as a function of P′ by
x′ = ax + by + cz
y′ = dx + ey + fz
z′ = gx + hy + iz
The tensor that describes the transformation from P to P′ is the matrix
a b c
d e f
g h i
In matrix notation, the nine components of a stress tensor are
σ11 σ12 σ13
σ21 σ22 σ23
σ31 σ32 σ33
with σ11 oriented parallel to the 1-axis and acting on a plane perpendicular to the 1-axis, σ12 oriented parallel to the 1-axis and acting on a plane perpendicular to the 2-axis, and so on. The systematics of these nine components make for an unnecessarily long notation, so in shorthand we write
[σij]
where i refers to the row (component parallel to the i-axis) and j refers to the column (component acting on the plane perpendicular to the j-axis).
You will notice the similarity between our approach to the stress tensor and our earlier approach to the description of stress at a point, consisting of one normal stress (i = j) and two shear stresses (i ≠ j) for each of three orthogonal planes. The stress tensor is simply the mathematical representation of this condition. Now we use this notation for decomposing the total stress into the mean stress and deviatoric stress
Decomposing the stress state in this manner demonstrates the property that shear stresses (i ≠ j) are restricted to the deviatoric component of the stress, whereas the mean stress contains only normal stresses. Because σij = σji, both the mean stress and the deviatoric stress are symmetric tensors.
Once you have determined the stress tensor, it is relatively easy to change the reference system. In this context, you are reminded that the values of the nine stress components are a function of the reference frame. Thus, when changing the reference frame, say by a rotation, the components of the stress tensor are changed. These transformations are greatly simplified by using mathematics for stress analysis, but we’d need another few pages explaining vectors and matrix transformations before we could show some examples. If you would like to see a more in-depth treatment of this topic, several useful references are given in the reading list.
2S.2.3 Stress Trajectories and Stress Fields
By connecting the orientation of principal stress directions at several points in a body, we obtain trajectories that show the variation in orientation of that direction within the body, which are called stress trajectories. Generally, stress trajectories for the maximum and minimum principal stresses are drawn, and a change in trend means a change in orientation of these principal stresses. Collectively, principal stress trajectories represent the orientation of the stress field in a body. In some cases the magnitude of a particular stress is represented by varying the spacing between the trajectories. An example of the stress field in a block that is pushed on one side is shown in Figure S2.2.
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FIGURE S2.2. (a) Theoretical stress trajectories of σ1 (full lines) and σ3 (dashed lines) in a crustal block that is pushed from the left resisted by frictional forces at its base and a free surface at the top. Using the predicted angle between maximum principal stress (σ1) and fault surface of ~30° (Coulomb failure criterion; Chapter 3) we can predict the orientation of faults (red dashes), as schematically shown in (b). Note that these stress trajectories from interaction between push, frictional resistance, loading and a free top surface provide an explanation for the common occurrence of curved (or listric) faults. |
If the stress at each point in the field is the same in magnitude and orientation, the stress field is homogeneous; otherwise it is heterogeneous. Homogeneity and heterogeneity of the stress field should not be confused with isotropic and anisotropic stress. Isotropic means that the principal stresses are equal (describing a sphere), whereas homogeneous stress implies that the orientation and shape of the stress ellipsoids are equal throughout the body. In a homogeneous stress field, all principal stresses have the same orientation and magnitude. The orientation of stress trajectories under natural conditions typically varies, arising from the presence of discontinuities in rocks, the complex interplay of more than one stress field (like gravity and friction; S2.2), or contrasts in rheology (explored in a later chapter).
2S.2.3 READING
Anderson, D. L., 1989. Theory of the Earth. Blackwell Scientific: Oxford.
Angelier, J., 1994. Fault slip analysis and paleostress reconstruction. In Hancock, P. L., ed., Continental deformation. Pergamon, pp. 53–100.
Engelder, T., 1993. Stress regimes in the lithosphere. Princeton University Press.
Jaeger, J. C., and Cook, N. G. W., 1976. Fundamentals of rock mechanics. Chapman and Hall: London.
Means,W. D., 1976. Stress and strain—basic concepts of continuum mechanics for geologists. Springer- Verlag: New York.
Nye, J. F., 1985. Physical properties of crystals, their representation by tensors and matrices (2nd edition). Oxford University Press: Oxford.
Turcotte, D. L., and Schubert, G., 1982. Geodynamics—applications of continuum physics to geological problems. J. Wiley & Sons: New York.
Zoback, M. L., 1992. First and second order patterns of stress in the lithosphere: the World Stress Map project. J. Geophysical Research, 97, 11703–11728.
v1.1. Last update 8-may-20