Structural Geology
Lecture 12
Mohr's Circle and Earth Stress
(The Elastic Earth)

In the equations that we derived for Mohr’s circle, we measured the angle, q, as the angle between s1 and the normal to the fault plane.  On the Mohr diagram, q is represented by measuring 2q counter clockwise from the maximum normal stress, s1.  In general, q will be approximately 60° so that the cos 2q will be a negative number.  The significance of this is that the normal stress on active fault planes is less than the mean stress (i.e., on the diagram below A < C).  The components of Mohr's Circle are the following:

A:   

B:  

C:    The center of the Mohr's Circle  -  

D:    The radius of the Mohr's Circle  -  

Elasticity and reference states for lithospheric stress

Discussion of two reference states is preceded by an introduction to elementary elasticity.  Such an introduction to elasticity 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 (≈ 12-5 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): 

s1 =  (l  +  2z)e1 +  le2  +  le3	(12-1)

s2 =  le1 +  (l  +  2z)e2  +  le3	(12-2)

s3 =  le1 +  le2  +  (l  +  2z)e3	(12-3)

where l and z are elastic properties of the rock, known as the Lame's constants.  z is commonly known as the modulus of rigidity, the ratio of shear stress to simple shear strain.  l  +  2z relates stress and strain in the same direction, and l relates stress with strain in two perpendicular directions.  
If volumetric strain is defined as

De   =  e1  +   e2  +   e3,	(12-4)

then then we may combine equations 12-1 to 12-3 as

si  =   lDe  +  2zei. 	(12-5)

The uniaxial stress state is of immediate interest in developing an understanding of the elastic constants, Young's modulus (E) and Poisson's ratio (u), 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, s1  ≠  0,  s2  =  s3  =  0, and the three equations of elasticity are written as

s1 =  (l  +  2z)e1 +  le2  +  le3,	(12-6)

0 =  le1 +  (l  +  2z)e2  +  le3,	(12-7)

0 =  le1 +  le2  +  (l  +  2z)e3.	(12-8)

Using equations 12-7 and 12-8, strain parallel to the applied stress, e1, is related to strain in the directions of zero stress, e2 and e3,

.	(12-9)

To derive the ratio between stress and strain for uniaxial stress, values for e2  and  e3 are substituted back into equation 12-6.  This ratio is Young's modulus,

.	(12-10)

Equation 12-10 is a simplified form of Hooke's Law, 

 	(12-11)

where Cijkl is the stiffness 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,

.	(12-12)

u for rocks is typically in the range of .15 to .30.  

Two reference states of stress

While developing an understanding of lithospheric stress, it is convenient to start with reference states which occur in a planet devoid of plate tectonics and, then describe the difference between these reference states and the actual state of stress.   Stress generated by plate tectonic processes make the difference.  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

s1 = s2 = s3     Þ     SH = Sh = Sv    Þ    Pm 	(12-13a)

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

s1 = s2 = s3     Þ     SH = Sh = Sv     Þ      Pc 	(12-13b)

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.  Looking back at the Mohr diagram, the student will see that Mohr’s circle will plot as a point.  Equation 12-4 applies to calculate volumetric elastic strain accompanying the complete erosion of an igneous intrusion assuming no temperature change.   Adding equations 12-1 to 12-3 gives an expression for the response of rocks, including laboratory test specimens, to changes in confining pressure,

	(12-14)

where k is the bulk modulus and its reciprocal, b, 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 state of stress, of which lithostatic stress in 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 (Figure 12-1).  Uniaxial strain is also used as the model for the stress state describing the effect of overburden load on rocks at depth assuming these rocks develop fixed elastic properties at some point after deposition.  s1 is the vertical stress 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, e2 = e3 = 0.  In practice, it is understood that horizontal stress arising from a uniaxial strain state in a sedimentary basin is modified by changes in elastic properties during diagenesis, creep relaxation, and slumping along listric normal faults.  Each of these processes bring the state of stress in sedimentary basins closer to lithostatic.  Mandl (1988) refers to any stresses arising from such inelastic deformation mechanisms as prestress or a stress which is not explained by simple incremental elastic deformation.  As discussed in Chapter 10, remnant stress is a version of prestress.  Upon initiation of tectonic processes, horizontal stresses can vary widely from those calculated using the uniaxial reference state.  During erosion and removal of overburden weight, lack of contraction in the horizontal direction by uniaxial strain behavior also leads to a large changes in horizontal stresses.  
For the case of uniaxial strain, e1 ≠ 0, e2 = e3 = 0, the equations of elasticity (12-1 to 12-3) are written (Jaeger and Cook, 1969, section 5.3):

s1  =  (l  +  2z)e1	(12-15)

s2  =  s3  =  le1.  	(12-16)

From equations 12-15 and 12-16, the relationship between vertical (Sv = rgz = s1) and horizontal stresses (SH = Sh  =  s2  =  s3) are given in terms of the Poisson's ratio

	(12-17)

where r is the integrated density of the overburden, g is the gravitational acceleration, and z is the depth within the earth. Major deviations from this reference state may signal that the uniaxial-strain model is not a particularly effective model for state of stress in the lithosphere.  Assuming a u = 0.2, the Sh = 0.25 Sv.  This stress state is illustrated by the Mohr’s circle on page 62. 

Tectonic stress 

Stresses can vary from the reference state as a consequence of either natural or man-made processes.  Those components of the in-situ stress field which are a deviation from a reference state as a consequence of natural processes include tectonic stresses, residual stresses, and near-surface thermal stresses induced by diurnal and annual heating.  Large deviations from a reference state of stress arising from mining, drilling, excavation, and other societal activities are man-made and, thus, unrelated to stress in the lithosphere.  Traditionally, tectonic stresses are associated with stresses arising from the largest-scale natural sources such as plate-boundary tractions (e.g., Zoback et al., 1989,  Hickman, 1991) and local stresses arising from topographic loading, thermoelastic loading, and unloading due to erosion are considered nontectonic.  The term, contemporary tectonic stress, grew from Sbar and Sykes' (1973) paper to mean present stress fields which are reasonably homogeneous on a regional or plate-wide scale.  However, the distinction between plate-wide and local stresses breaks down in certain geological contexts.  For example, the orientation of some late-formed joint sets is controlled by plate-wide stress fields and, yet, they propagate in response to stresses developed during local unloading due to erosion (e.g., Hancock and Engelder, 1989).  Thus, the definition of tectonic stress is simplified if most constraints concerning scale and source are removed from the definition.   Tectonic stresses are usually horizontal components of the in-situ stress field which are a deviation from a reference state as a consequence of natural processes on all scales from plate-wide to local.  By this definition local stresses developed under topographic loading are considered tectonic.  Unfortunately, this definition is still so loose that geoscientists will continue to quibble about what qualifies as a tectonic stress.