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Stress Regimes in the Lithosphere
By Terry Engelder PRINCETON UNIVERSITY PRESS
Copyright © 1993 Princeton University Press
All rights reserved.
ISBN: 978-0-691-08555-5
CHAPTER 1
Basic Concepts
One of the most interesting and unexpected results yielded by rock (stress) measurements is the demonstration of large horizontal stresses in the earth's crust.... In the virgin rock and at depths of about 400 m measurements have revealed horizontal [stresses] as great as 1 5–3.5 times the dead weight of the overlying strata. —Nils Hast (1958) during his survey of earth stress within the Scandinavian Peninsula
The dynamic nature of our earth is apparent in the deformation of its lithosphere as manifest by mountain belts, continental rifts, rapidly subsiding basins, high plateaus, and deep oceanic trenches. Evidence from magnetic anomalies, earthquake distribution, and continental shape suggests that large-scale deformation is a consequence of plate tectonics. On a more esoteric level, a geoscientist might explain that large-scale deformation occurs in response to stress within the lithosphere. But, what is meant by the expression, stress within the lithosphere? Although the word, stress, commonly appears in the geoscience literature, doubt about its meaning arises because its definition, simply stated as force per unit area, is too vague for the context of the discussion. In the use of the word, stress, there are often subtle nuances which are either missed by the reader or never clarified by the author. The purpose of this book is to clarify some of the subtle nuances associated with the use of the term, stress, through a discussion of in situ stress measurements and their interpretation.
Starting in the early part of the twentieth century, geoscientists began making some insightful inferences about the state of stress in the lithosphere. For example, Anderson (1905) deduced that SH > Sh > Sv2 is necessary for thrust faulting during mountain building. Later, more certain knowledge about stress within the lithosphere was gained through in situ measurements. Fifty years after Anderson's prediction, systematic in situ stress measurements began when Hast (1958) found horizontal stresses (maximum, SH, and minimum, Sh) exceeding the stress developed under the weight of the overburden (Sv) in the shallow continental crust of Scandinavia. With his seminal work Hast (1958) was the first to confirm, using in situ measurements over a broad region, that the relative magnitudes of the principal stresses in the upper crust met conditions specified by Anderson (1905) for thrust faulting. Because Hast's (1958) in situ measurements preceded the theory of plate tectonics, interpretation of his stress data in the context of large-scale deformation was difficult.
Prior to the formulation of plate tectonic theory, interpretation of stress data was based on a few rather simple models for the response of the earth (i.e., the lithosphere) to stress. For example, Seager (1964) was skeptical that Hast's in situ stress data had anything to do with tectonics and chose, instead, to use a simple elastic-plastic model to interpret the data. In Seager's model, rocks are subject to Poisson's effect which is a tendency for an elastic rock to expand laterally under its own weight. According to Seager's assumption, horizontal stresses build because adjacent rock serves as a rigid boundary and prevents lateral strain (i.e., εH = εh = 0). Because rocks generally expand less than they shorten from overburden weight, horizontal stress generated through Poisson's effect remains a fraction of overburden stress. In Seager's (1964) model, differential stress, the difference between Sv and Sh, increases as more overburden is added until the rock deforms plastically. Even though tectonic stress is left out of Seager's model, such models invariably lead to the conclusion that rock strength places definite limits on differential stress in the lithosphere, a concept developed in detail in chapters 2, 3, and 4. Although the theory that horizontal stresses arise solely from Poisson's effect does not explain upper crustal stresses, a state of stress represented by this model is convenient as one of three reference states.
In introducing some basic concepts concerning stress in the lithosphere, the first step is to discuss three convenient reference states of stress. These reference states are based on boundary conditions expected within planetary bodies with globally continuous lithospheres (i.e., one-plate planetary bodies such as Mercury and the Moon). From the equations of linear elasticity we will derive a reference state of stress based on a state of uniaxial strain (Price, 1959). Another reference state of stress arises from a constant stress boundary condition and is independent of elasticity (Artyushkov, 1973; McGarr, 1988). A third reference state of stress assumes all principal stresses are equal. A second step in the introduction of basic concepts is to add tectonic stress to the lithospheric reference states of stress. The third step is to define several terms which are used in the literature as expressions for state of stress. Commonly, the word, "stress," appears in the literature when the author could have been more precise. While there are precise definitions for total stress, effective stress, deviatoric stress, and differential stress, there is no consensus for a definition of tectonic stress and it is often used in a context in which the terms, deviatoric or differential stress, are more appropriate. Because there is more than one lithospheric reference state of stress there is more than one definition for tectonic stress.
Elasticity and Lithospheric Stress
Discussion of three reference states is preceded by an introduction to elementary elasticity. Such an introduction is appropriate because many in situ stress measurement techniques capitalize on the elastic behavior of rocks. Furthermore, many notions concerning state of stress in the lithosphere arise from the assumption that the upper crust behaves as a linear elastic body.
On scales ranging from granitoid plutons and salt domes (≈ 1–5 km) to the thickness of lithospheric plates (≈100 km), the earth is approximately isotropic with elastic properties independent of direction. If we assume that the lithosphere is subject to small strains, as is the case for elastic behavior, principal stress axes must coincide with the principal strain axes. Then, elastic behavior of the lithosphere is represented by the equations of linear elasticity which define the principal stresses of the three-dimensional stress tensor as linear functions of the principal strains (Jaeger and Cook, 1969, section 5.2):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1–1)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1–2)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1–3)
where λ and ζ are elastic properties of the rock, known as the Lame's constants. ζ is commonly known as the modulus of rigidity, the ratio of shear stress to shear strain, λ + 2ζ relates stress and strain in the same direction, and λrelates stress with strain in two perpendicular directions.
If volumetric strain is defined as
Δε = ε1 + ε2 + ε3, (1–4)
then we may combine equations 1–1 to 1–3 as
σ1 = λΔε + 2ζε1. (1–5)
The uniaxial stress state is of immediate interest in developing an understanding of the elastic constants, Young's modulus (E) and Poisson's ratio (υ), both of which will appear several times during discussions of lithospheric stress in this book. Uniaxial stress, a stress state for which only one component of principal stress is not zero, is rare in the earth's crust except in the pillars of underground mines. For uniaxial stress, σ1 ≠ σ2 = σ3 = 0, and the three equations of elasticity are written as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1–6)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1–7)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1–8)
Using equations 1–7 and 1–8, strain parallel to the applied stress, ε1, is related to strain in the directions of zero stress, ε2 and ε3,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1–9)
To derive the ratio between stress and strain for uniaxial stress, values for ε2 and ε3 are substituted back into equation 1–6. This ratio is Young's modulus,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1–10)
Equation 1–10 is a simplified form of Hooke's Law,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1–11)
where Cijkl is the stiffiness tensor. Under uniaxial stress, an elastic rock will shorten under a compressive stress in one direction while expanding in orthogonal directions. The ratio of the lateral expansion to the longitudinal shortening is Poisson's ratio,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1–12)
v for rocks is typically in the range of .15 to .30.
Three Reference States of Stress
While developing an understanding of lithospheric stress, it is convenient to start with reference states which may occur in a planet with just one lithospheric plate and thus devoid of plate tectonics. The next step is to describe the difference between these reference states and the actual state of stress generated by plate tectonic processes. In the earth, geologically appropriate reference states are those expected in "young" rocks shortly after lithification. Two general types of "young" rocks are intrusive igneous rocks, shortly after solidification within large plutons deep in the crust, and sedimentary rocks, shortly after the onset of burial and diagenesis within large basins.
Lithostatic Reference State
The simplest reference state is that of lithostatic stress found in a magma which has no shear strength and, therefore, behaves like a fluid with
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1–13a)
where Pm is the pressure within the magma. At the time enough crystals have solidified from the magma to form a rigid skeleton and support earth stress, the "young" rock, an igneous intrusion, is subject to a lithostatic state of stress
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1–13b)
where Pc is confining pressure. This, presumably, is the state of stress at the start of polyaxial strength experiments in the laboratory. Strictly speaking, there are no principal stresses in this case, because the lithostatic state of stress is isotropic. Equation 1–4 applies to calculate volumetric elastic strain accompanying the complete erosion of an igneous intrusion assuming no temperature change. Adding equations 1–1 to 1–3 gives an expression for the response of rocks, including laboratory test specimens, to changes in confining pressure,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1–14)
where κ is the bulk modulus and its reciprocal, β, is the compressibility.
Lithostatic stress will develop if a rock has no long-term shear strength. Although some rocks such as weak shales and halite have very little shear strength, experiments suggest that all rocks support at least a small differential stress for very long periods (Kirby, 1983). Indeed, heat will cause rocks to relax, but they never reach a lithostatic stress state. Although metamorphic rocks retain a shear strength during deformation, the metamorphic petrologist commonly uses the term pressure when actually referring to a state of stress which may approximate lithostatic stress (e.g., Philpotts, 1990). A structural geologist reserves the term pressure for describing confined pore fluid or other material with no shear strength. The terms signifying states of stress, of which lithostatic stress is one, apply to rock and other materials that can support a shear stress. Although the lithostatic state of stress is rare in the lithosphere, it is a convenient reference state.
Uniaxial-Strain Reference State
A second reference state is based on the postulated boundary condition that strain is constrained at zero across all fixed vertical planes (Terzaghi and Richart, 1952; Price, 1966. 1974; Savage et al., 1985). Such a boundary condition leads to a stress state which approximates newly deposited sediments in a sedimentary basin: the state of stress arising from uniaxial strain (fig. 1–1). Prior to complete diagenesis and lithificaiton "young" sediments compact by a process called uniaxial consolidation (Karig and Hou, 1992). During uniaxial consolidation horizontal stress, SH = Sh, increases as function of depth of burial but at a rate less than the vertical stress, Sv. Diagenesis under uniaxial strain conditions leads to a stress ratio,
k0 = Sh/Sv< 1, (1–15)
which develops independently of the elastic properties of the rock.
Uniaxial strain is also used to constrain changes in horizontal stress as a function of changing overburden load, assuming that rocks develop fixed elastic properties at some point after deposition. Under elastic conditions, σ1 is still the vertical stress, Sv, arising from the weight of overburden. If rocks were unconfined in the horizontal direction the response to an addition of overburden weight would be a horizontal expansion. Because rocks are confined at depth in the crust, Price (1966) suggests that horizontal expansion is restricted by adjacent rock so that in the ideal case, ε2 = ε3 = 0. For the case of uniaxial strain, ε1 ≠ 0, ε2 = ε = 0, the equations of elasticity (1–1 to 1–3) are written (Jaeger and Cook, 1969, sec. 5.3):
σ1 = (λ + 2ζ)ε1 (1–16a)
σ1 = σ3 λε1 (1–16b)
From equations 1–16a and b, the relationship between vertical (Sv = pgz = σ1) and horizontal stresses (SH = Sh = σ2 = σ3) are given in terms of the Poisson's ratio
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1–17)
where ρ is the integrated density of the overburden, g is the gravitational acceleration, and z is the depth within the earth. Assuming that the rock in question is fully lithified, one application of equation 1–17 is the calculation of changes in horizontal stress during sedimentation and erosion. During erosion and removal of overburden weight, lack of contraction in the horizontal direction by uniaxial-strain behavior also leads to large changes in horizontal stresses. Major deviations from the functional relationship between Sh and Sv predicted by equation 1–17 signal that elastic behavior using the uniaxial-strain model is not a complete representation for state of stress in the lithosphere.
(Continues...)
Excerpted from Stress Regimes in the Lithosphere by Terry Engelder. Copyright © 1993 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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