Chapter 9 Rheology, Stress in the Crust, and Shear Zones Introduction Why is it that some rocks break whereas other rocks appear to flow seamless- ly? Sometimes, one can observe these contrasting types of behavior in different minerals in the same rock? Rheology is the study of flow of rocks and to delve into this topic, and understand the question raised in the first sentence, requires us to understand understand the relations between stress and strain (or strain rate), the effects of environmental factors on deformation, and how materials actually de- form at the scale of the crystal lattice. Relationship Between Stress and Strain At its simplest, deformation can be either non-permanent or permanent. In the former case, the deformation exists only while the stress is applied and the material returns to its undeformed state upon removal of the stress. Permanent de- formation, on the other hand, is forever. CHAPTER 9 SHEAR ZONES & STRESS IN THE CRUST Elasticity Elastic deformation is, by definition, non-permanent and instantaneous. The material suffers distortion only while stress is applied and quickly returns to normal when he stress is removed. Many processes in geophysics and geology are mostly or completely elastic: the propagation of seismic waves, the earthquake cycle, or flex- ure of the lithosphere beneath a load such as a mountain belt or a sedimentary basin. The GPS data that you analyzed in Chapter 7 is largely non-permanent de- formation that occurs when the earth on one side of a fault “snaps back” during an earthquake, something known as the elastic rebound theory. What distinguishes all of these deformations is that they are very small even though the stresses are larger. From your physics courses and experiments with springs, you probably re- member Hooke’s Law, in which there is a linear relationship between force and displacement. However, you also know that both stress and infinitesimal strain are second order tensors and, therefore, the relationship between them should be a fourth order tensor: "σ ij = −Cijklε kl (9.1) where, Cijkl is the stiffness tensor. Although Equation (9.1) looks nasty with 81 terms, they re not all independent. In fact because of symmetry, there are at most 36 in- dependent parameters and for all practical purposes, we only refer to a few elastic moduli. They are: • Young’s modulus, E, for axial strain (elongations or shortenings), where σ = Eε. A material with a high Young’s modulus is very rigid. • The shear modulus or modulus of rigidity, G, is appropriate for simple shear deformations. • The bulk modulus, or incompressibility, K, is likewise the one to use for simple contractions or dilations (i.e., volume strains). These moduli can be related to one another if we know and independent parame- ter, known as Poissons Ratio, ν, which describes to what extent a shortening in one direction is balanced out by a lengthening in an orthogonal direction. Poissons ratio is the ratio of the transverse to the longitudinal extension: MODERN STRUCTURAL PRACTICE "174 R. W. ALLMENDINGER © 2015-16 CHAPTER 9 SHEAR ZONES & STRESS IN THE CRUST wi Figure 9.1 — The blue rectangle is elastically deformed into the red rectangle (strains are highly exaggerated). Poissons ratio is defined as the ratio of the transverse extensions to the f i longitudinal extensions, calculated from the w’s and ’s, respectively as shown in equation (9.2). wf ⎛ w f − wi ⎞ e ⎝⎜ w ⎠⎟ "ν = − t = − i (9.2) eℓ ⎛ ℓ f − ℓi ⎞ ⎜ ⎟ ⎝ ℓi ⎠ For volume constant deformation, ν = 0.5 but for most rocks, 0.25 ≤ ν ≤ 0.33. All of these parameters are related by the following equation: E 3K (1− 2ν ) " G = = (9.3) 2(1+ν ) 2(1+ν ) Rocks only experience a very small amount of elastic strain before perma- nent deformation ensues. That permanent deformation can be in the form of a fracture or fault, something we saw in Chapter 6. When a fracture cuts across the material, there is a loss of cohesion and the sample falls apart. However, the mater- ial can also deform permanently without losing cohesion, a type of deformation we call… Plasticity Plastic deformation results when a critical threshold stress, known as the yield stress (σy), is exceeded and the ratio of the change in differential stress to the change strain decreases drastically. Three different behaviors are possible (Fig. 9.2): MODERN STRUCTURAL PRACTICE "175 R. W. ALLMENDINGER © 2015-16 CHAPTER 9 SHEAR ZONES & STRESS IN THE CRUST ing arden in h tra s Figure 9.2 — Idealized differential perfect σy stress(Δσ)-strain(ε) curves exhibiting dif- strain s plastic ferent types of plastic deformation once oftenin g the yield stress, σy, is surpassed. Strain Δσ hardening occurs at lower temperatures where as strain softening at higher temper- atures. If the stress is removed (dashed stress line), the initial elastic deformation is re- removed couped but the deformation beyond the yield stress is not. ε (a) the slope of the stress strain curve decreases but remains positive, which is known as strain hardening, (b) strain increases continually without any further in- crease in stress (perfect plastic behavior), or (c) the strain increases with decreasing differential stress, known as strain softening. Temperature largely controls which of these behaviors will occur. Strain Rate and Viscosity So far, we haven’t said anything about time except that elastic deformation is instantaneous. Consider the deformation shown in Figure 9.3a. The same material deforms continuously for a constantly applied differential stress, but the rate of de- formation tensor3, ε! , increases with increasing stress. So, we can make a new curve (Fig. 9.3b) where the differential stress is plotted against strain rate; the ratio between stress and strain rate is known as the viscosity, η, which is a measure of a fluid’s resistance to flow. The material shown in Figure 9.3b exhibits a constant vis- cosity and thus is known as a Newtonian fluid. Over a long period of time, even rocks within the earth can exhibit fluid-like behavior, but in contrast to Newtonian fluids, they tend to exhibit non-linear, power law viscosity as we shall see, below. 3 The rate of deformation tensor is commonly confused with the time derivative of the strain tensor. For infinitesimal strain, the two are equivalent but in finite strain, the former is defined with respect to the spatial coordinates whereas the latter is defined with respect to the material coordinates (Malvern, 1969). MODERN STRUCTURAL PRACTICE "176 R. W. ALLMENDINGER © 2015-16 CHAPTER 9 SHEAR ZONES & STRESS IN THE CRUST (a) (b) Δσ c Δσ ε Δσ Δσb c Δσb strain, Δσa Δσa differential stress, differential time, t strain rate, ε Figure 9.3 — (a) A material that accrues strain over time at constant stress. Δσa ≤ Δσb ≤ Δσc and thus the strain rate varies differential stress. (b) Same data plotted with dif- ferential stress against strain rate. The slope of the line is known as the viscosity and the simple material shown, with constant viscosity, is known as a Newtonian fluid. Viscous and elastic idealized models are combined in various ways. For ex- ample, viscoelastic deformation is non-permanent but develops over time and is recovered over time as well. Likewise, there are viscoplastic models that combine elements of viscosity and plasticity. There are many additional hybrid mechanisms. Environmental Factors The type of deformation that a rock experiences is due primarily to its com- position and the environmental conditions under which the deformation occurred. You have undoubtedly reviewed the environmental factors in the lecture part of your course. The most important are: • Confining Pressure — This is the uniform pressure surrounding the rock at the time of deformation. It commonly corresponds to the vertical stress or lithostatic load, that is the weight of the overlying rocks. An increase in confining pressure makes rocks stronger (i.e., the yield stress increases), as reflected by the slope of the Coulomb part of the failure envelop. Because confining pres- sure increases with depth, rocks should get stronger deeper in the earth. The formula for lithostatic load is: MODERN STRUCTURAL PRACTICE "177 R. W. ALLMENDINGER © 2015-16 CHAPTER 9 SHEAR ZONES & STRESS IN THE CRUST z " P = ρgdz ≈ ρgz (9.4) lith ∫ 0 The right side of this equation is a common approximation for the case where the density, ρ, does not vary with depth, z, and the change in gravitational acceleration, g, is small with changes in depth corresponding to crustal conditions. The confining pressure effect is relatively insensitive to rock composition, except of course to the extent that composition determines density. • Temperature — With increasing temperature, mechanisms of crystal plasticity described below, which depend on composition, begin to kick in and reduce the yield stress. The increase in con- ductive temperature with depth varies with tectonic setting, being about 15-20°C/km with depth (a heat flow of about 60 mW/m2) in stable continental interiors, ~30°C/km in rift provinces (~90 mW/m2), and >40°C/km in active volcanic provinces (~120 mW/ m2). Because temperature increases with depth, its effect is opposite to that of confining pressure. Thus these are the two great compet- ing factors, but they are modified by other factors as well, includ- ing… • Fluids — There are three distinct ways in which fluids act to weaken rocks: (a) increasing the pore fluid pressure counteracts confining pressure by reducing the effective normal stress; (b) pres- sure solution dissolves soluble minerals, especially at high stress grain-to-grain contacts, redepositing the material locally in low stress “pressure shadows” or flushing the dissolved material com- pletely out of the rock; and (c) hydrolytic weakening where wa- ter in the crystal structure weakens the bonds of the crystal.