# Earthquake and Volcano Deformation

PAUL SEGALL
Edition: STU - Student edition
Pages: 464
https://www.jstor.org/stable/j.ctt7sg19

1. Front Matter
(pp. i-vi)
(pp. vii-x)
3. Preface
(pp. xi-xiv)
4. Acknowledgments
(pp. xv-xvi)
5. Origins
(pp. xvii-xxiv)
6. 1 Deformation, Stress, and Conservation Laws
(pp. 1-31)

In this chapter, we will develop a mathematical description of deformation. Our focus is on relating deformation to quantities that can be measured in the field, such as the change in distance between two points, the change in orientation of a line, or the change in volume of a borehole strain sensor. We will also review the Cauchy stress tensor and the conservation laws that generalize conservation of mass and momentum to continuous media. Last, we will consider constitutive equations that relate the stresses acting on a material element to the resultant strains and/or rates of strain. This necessarily abbreviated...

7. 2 Dislocation Models of Strike-Slip Faults
(pp. 32-50)

In this chapter, we will begin by considering a simple two-dimensional model of a very long strike-slip fault (figure 2.1). We consider a homogeneous, linear elastic half-space. The fault is taken to be infinitely long in the${x_3}$direction, and the deformation is antiplane strain. This means that the only nonzero displacement is parallel to the fault in the${x_3}$direction, and it varies only in the plane perpendicular to the fault. That is,${u_3}$varies only with${x_1}$and${x_2}$,${\bf{u}}({\bf{x}}) = {u_3}({x_1},\;{x_2}){{\bf{\hat e}}_3}$. As a starting point, we will consider the slip to be uniform with depth along the fault. Later,...

8. 3 Dip-Slip Faults and Dislocations in Three Dimensions
(pp. 51-91)

In chapter 2, we developed methods for analyzing very long strike-slip faults. This chapter begins by deriving expressions for infinitely long thrust and normal faults, as shown in figure 3.1, and eventually presents methods for computing displacements for general three-dimensional dislocations.

The fault shown in figure 3.1 is understood to extend indefinitely in and out of the page. In this case, the displacements are restricted to the${x_1}$,${x_2}$plane—that is,${u_3} = 0$. Furthermore, the remaining displacement components do not vary in the${x_3}$direction:$\partial {u_1}/\partial {x_3} = \partial {u_2}/\partial {x_3} = 0$. The resulting state of deformation is one ofplane strain.

Plane-strain problems are significantly...

9. 4 Crack Models of Faults
(pp. 92-117)

With the dislocation models discussed in the previous two chapters, the displacement discontinuity, or slip, was prescribed as a boundary condition. As discussed previously, such models cannot hope to explainwhythe fault slip occurs. We have also seen that constant slip on the fault surface leads to nonphysical stress singularities at the fault tip.

In the earth, shear stress accumulates on a fault until the stress exceeds the fault’s strength and the fault slips. During the earthquake, the shear stress acting on the fault decreases by some amount$\Delta \tau = {\tau _i} - {\tau _f},$, where${\tau _i}$is the initial shear stress acting on the...

10. 5 Elastic Heterogeneity
(pp. 118-165)

The earth is not a homogeneous isotropic elastic half-space.We know from both geologic and geophysical observations that the crust is composed of different rock types, with different elastic wave speeds, densities, and elastic moduli. In this chapter, we will explore three different methods for computing dislocation solutions in elastically heterogeneous media. Image methods are useful when there are different material properties separated by planar boundaries. Images are relatively straightforward for antiplane problems but require more experience when solving plane strain and three-dimensional problems. Propagator matrices, on the other hand, are quite general and are widely used for horizontally stratified media...

11. 6 Postseismic Relaxation
(pp. 166-199)

Up to this point, we have considered only time-independent processes, except for simple models of interseismic strain accumulation in which the material behavior is elastic and the only parameter that varies with time is the amount of slip on the fault (chapter 2). There is considerable evidence that such descriptions are inadequate. First of all, we expect a priori that with increasing depth, and hence temperature, rock will behave in a ductile fashion. Ductile flow in response to stress changes induced by earthquakes, among other forcings, can lead to transient deformation at the earth’s surface. We also recognize that rock...

12. 7 Volcano Deformation
(pp. 200-254)

Measurements of deformation are one of the most important means for studying magmatic processes and monitoring active volcanoes. Indeed, deformation along with seismicmonitoring is one of the principal means of assessing the potential for future eruptive activity. The reason for this is straightforward. As magma migrates toward the earth’s surface, it forces aside the surrounding crust. This inevitably causes deformation that can be detected by a variety of modern techniques. Because the shallow crust is brittle, the deformation usually results in earthquakes that are also easily detected. In some cases, there is evidence to suggest that the onset of measurable...

13. 8 Topography and Earth Curvature
(pp. 255-266)

Up to this point, we have treated the earth as a half-space, ignoring both earth curvature and topography. The earth is, of course, an oblate spheroid with mountains, valleys, and volcanic edifices. Except in the most extreme cases, however, the slope of the topography at true scale is modest, and approximatemethods accurate for small slope to be presented here are adequate for treating topographic effects. For extreme topography, onewould need to resort to boundary element or other numerical procedures.

Semianalytical methods also exist for dislocation sources in a spherical earth (Ben-Menahem et al. 1969; Smylie and Mansinha 1971; Pollitz 1996;...

14. 9 Gravitational Effects
(pp. 267-296)

It might strike the reader as strange that we have been able to put off a discussion of the effects of gravity for so long. After all, gravity is a dominant force for problems on the scale of the earth. Yet, up to this point, we have ignored all body forces, including gravity, in writing the equilibrium equations. As you will see, the reason we have been able to get away with this is that the preexisting stress state, prior to fault slip or magma chamber inflation, equilibrates the gravitational body forces. Fault slip or magmatic intrusion perturbs this equilibrium...

15. 10 Poroelastic Effects
(pp. 297-331)

Up to this point, we have considered the earth’s crust to be a solid nonporous elastic, or viscoelastic, medium. In actuality, the crust is porous and, except for the very shallow near-surface region, at least partially liquid saturated. That rock is a multiphase composite consisting of solid and liquid filled pores adds a significant richness to its mechanical behavior. The liquid phase flows under gradients in pore-fluid pressure. Changes in fluid pressure induce strains,andconversely, changes in stress or strain induce changes in pore pressure.

To see this at an intuitive level, consider a fluid-saturated sponge. Rapidly squeezing the...

16. 11 Fault Friction
(pp. 332-371)

We began our study with dislocation models of faulting, in which the slip on a fault is prescribed. In chapter 4, we explored models in which the shear stress in the slipping zone is specified, leading to mixed stress-displacement boundary conditions. In neither of these classes of models does the slip on the fault arise naturally due to the physical properties of the fault zone and its interactions with its elastic surroundings. In order to develop such models, we must consider the constitutive properties of faults—the topic of the present chapter. This also leads to a discussion of fault...

17. 12 Interseismic Deformation and Plate Boundary Cycle Models
(pp. 372-414)

Strain accumulates in the crust adjacent to major faults during the interseismic period between large earthquakes. Measurement of strain accumulation is a first-order predictor of future seismic hazard since elastic strain is ultimately released in earthquakes. Higher rates of strain accumulation should generally be associated with higher fault slip rates and thus either larger or more frequent earthquakes. To make these ideas quantitative requires mathematical models, the predictions of which can be compared to observations. Chapter 2 introduced the elastic screw-dislocation model of interseismic deformation for very long strike-slip faults as well as the backslip concept. We will begin this...

18. APPENDIX A: Integral Transforms
(pp. 415-419)
19. APPENDIX B: A Solution of the Diffusion Equation
(pp. 420-422)
20. APPENDIX C: Displacements Due to Crack Model of Strike-Slip Fault by Contour Integration
(pp. 423-424)
21. Author Index
(pp. 425-432)