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Spatial and temporal scales of anisotropic effects in ice-sheet flow
- Throstur Thorsteinsson, Edwin D. Waddington, Raymond C. Fletcher
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- Annals of Glaciology / Volume 37 / 2003
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- 14 September 2017, pp. 40-48
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Along most particle paths in polar ice sheets, ice experiences a slowly changing local deviatoric-stress pattern, and develops a fabric characteristic of its current stress state, in which there is generally a correspondence between non-zero strain-rate components and non-zero deviatoric-stress components. Where the stress pattern changes rapidly in special transition zones, fabric may evolve more slowly than the local stress field, and unusual or unexpected deformation patterns can result. The degree to which fabric tracks the local stress is determined by the relative characteristic times for changes in stress, given by the transition-zone width and ice velocity, and for changes in crystal orientation, given, in the absence of recrystallization, by the inverse of the local strain rate due to the principal stress. Recrystallization can significantly reduce the time-scale of fabric adjustment. We examine transition zones where ice (a) enters ice-stream margins, (b) is overrun by a migrating divide, and (c) flows through a strong saddle. Stress and fabric tend to be significantly misaligned in ice-stream margins and in flow through a saddle. When stresses that are markedly different from in situ stresses are applied to ice specimens during creep tests, deformation may be difficult to interpret.
Index
- David D. Pollard, Stanford University, California, Raymond C. Fletcher, Pennsylvania State University
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- Fundamentals of Structural Geology
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- 01 September 2005, pp 497-500
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Chapter 1 - Motivations and opportunities
- David D. Pollard, Stanford University, California, Raymond C. Fletcher, Pennsylvania State University
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- Fundamentals of Structural Geology
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- 01 September 2005, pp 1-24
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Mt. Hillers, southern Henry Mountains, UT. The mountain is cored by igneous rock and surrounded by upturned beds of sandstone and shale. G. K. Gilbert coined the term “laccolite” for these structures in the late 1870s and proposed models for this process of mountain building based on mechanical principles. Inset: Frontispiece from G. K. Gilbert's Report on the Geology of the Henry Mountains (Gilbert, 1877). To the rear of this illustration the sedimentary strata form the structural dome of Mt. Ellsworth, and to the front the eroded remnant of the dome represents the current topography of this mountain. Photograph by D.D. Pollard.
The sciences do not try to explain, they hardly even try to interpret, they mainly make models. By a model is meant a mathematical construct which, with the addition of certain verbal interpretations, describes observed phenomena. The justification of such a mathematical construct is solely and precisely that it is expected to work (quote from John von Neumann; Gleick, 1987, p. 273).
In this chapter we motivate the study of structural geology by introducing selected topics that illustrate the extraordinary breadth of interesting problems and important practical applications of this discipline. For example, we use the Imperial Valley earthquake of 1979 along the San Andreas Fault zone to describe techniques for geological hazard analysis. In a second example the lineaments visible in radar images of Venus provide the data for investigating tectonic processes on a planet other than our own.
Chapter 6 - Force, traction, and stress
- David D. Pollard, Stanford University, California, Raymond C. Fletcher, Pennsylvania State University
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- Fundamentals of Structural Geology
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- 01 September 2005, pp 194-242
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Photoelastic image of maximum shear stress contours in grains of model rock. Stress is concentrated at grain contacts. Inset: photoelastic image of three circular disks with point contact loads. Reprinted from Gallagher et al. (1974) with permission from Elsevier.
The concept of stress is the heart of our subject. It is the unique way continuum mechanics has for specifying the interaction between one part of a material body and another (Fung, 1969, p. 41).
In this chapter we define the relationships among forces, tractions, and stresses. One of the first concepts encountered in a physics class is that of the resultant force, F, acting on a particle with mass, m, and the associated linear acceleration, a, of that particle in the direction that the force acts (Fig. 6.1a). For a rigid body (Fig. 6.1b) one considers, for example, the resultant torque, τ, about the axis z, due to the force, f, acting at position, r, and the associated angular acceleration. For a deformable body the traction vector, t(n), is a measure of force per unit area acting on the surface of a body (Fig. 6.1c), where the surface has an orientation specified by the outward unit normal vector, n. This surface can be the exterior boundary of a rock mass or an imagined surface within the rock mass. The traction vector is defined at a point on such a surface in a limiting process as the area of a small element of this surface shrinks toward zero about the point.
Chapter 3 - Characterizing structures using differential geometry
- David D. Pollard, Stanford University, California, Raymond C. Fletcher, Pennsylvania State University
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- Fundamentals of Structural Geology
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- 01 September 2005, pp 75-119
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Two aerial views of the southeastern margin of the San Rafael Swell, UT. Mesozoic clasitic sedimentary rocks are upturned in the Waterpocket monocline. Photographs by D.D. Pollard.
The strange combination of mathematics and physics is a Greek invention, pioneered by Archimedes. Modern science is a mythical monster: half-goat, half-bird. The student of physics is led simultaneously to the laboratory, to face the phenomena of physical reality; and to the math course, to forget about the phenomena and to contemplate pure abstractions. That this hybrid existence is at all fertile is amazing: we use it, because we have discovered its effectiveness through experience.
The structure of the application of mathematics to physics by Archimedes, then, is this: by making explicit, clear assumptions, one draws the logical implications of the assumptions, which then have to hold for the world – as long as the assumptions themselves do.
Mathematics may have little to say, directly, about the physical world, but it is the only way to say anything at all with any certainty. The bet of modern science – following on Archimedes – is that we are willing to say very little, as long as what we say is well argued. Good arguments are good starting points for truly productive discussion, and so it is not surprising that the mathematical route has been so productive in modern science (Netz, 2000).
In the previous chapter we illustrated examples of geological surfaces, such as the top of the Triassic Chinle Formation throughout the San Rafael Swell in southern Utah, using structure contours (Fig. 2.27).
Chapter 2 - Structural mapping techniques and tools
- David D. Pollard, Stanford University, California, Raymond C. Fletcher, Pennsylvania State University
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- Fundamentals of Structural Geology
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- 01 September 2005, pp 25-74
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Structural geologists use a GPS receiver to determine the UTM coordinates of the point under their feet on this outcrop of Aztec sandstone in the Valley of Fire, NV. Photograph by D.D. Pollard.
‘You think these raindrops are random?’ his uncle had asked. And Leaphorn had been surprised. He'd said of course they were random. Didn't his uncle think they were random?
‘The stars,’ Haskie Jim said. ‘We have a legend about how First Man and First Woman, over by Huerfano Mesa, had the stars in their blanket and were placing them carefully in the sky. And then Coyote grabbed the blanket and whirled it around and flung them into the darkness and that is how the Milky Way was formed. Thus order in the sky became chaos. Random. But even then … Even then, what Coyote did was evil, but was there not a pattern, too, in the evil deed?’
That had not been the time in Leaphorn's life when he had patience for the old metaphysics. He remembered telling Haskie Jim about modern astronomy and the cosmic mechanics of gravity and velocity. Leaphorn had said something like ‘Even so, you couldn't expect to find anything except randomness in the way the rain fell.’ And Haskie Jim had watched the rain awhile, silently. And then he had said, and Joe Leaphorn still remembered not just the words but the old man's face when he said them: ‘I think from where we stand the rain seems random.
Chapter 9 - Brittle behavior
- David D. Pollard, Stanford University, California, Raymond C. Fletcher, Pennsylvania State University
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- Fundamentals of Structural Geology
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- 01 September 2005, pp 333-383
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Jointed limestone bed at Lilstock Beach on the southern coast of the Bristol Channel, England. Photograph by D.D. Pollard.
An admirer of Nature may be excused becoming enraptured when he takes a view from any of these noble terraces [in County Clare, Ireland]. Looking north, or south, his eyes are riveted on vast surfaces of gray limestone rocks, split up to an extent, and with a regularity of direction, truly wonderful … The observer becomes so absorbed with the scene that he unconsciously begins to feel as if the rocks under and around him were in process of being illimitably cleft from north to south–as if the earth's crust were in course of splitting up from one pole to the other; and he only rids himself of the feeling to become bewildered with the question, as to what mysterious agent produced the singular phenomenon he is contemplating (King, 1875).
In the preceding chapter we learned that the mechanical behavior of rock under certain conditions can be approximated with a linear elastic material, a mathematical construct formulated using Hooke's Law to relate stress and infinitesimal strain. The elastic material is useful for describing both ancient and modern deformation in the Earth at a variety of length and time scales. However, the limestone described by King as “illimitably cleft from north to south” provides an evocative example of fracturing which is inelastic, non-recoverable deformation. Even if the fracture surfaces were pushed back together they would not heal.
Contents
- David D. Pollard, Stanford University, California, Raymond C. Fletcher, Pennsylvania State University
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- Fundamentals of Structural Geology
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- 01 September 2005, pp v-viii
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Chapter 8 - Elastic deformation
- David D. Pollard, Stanford University, California, Raymond C. Fletcher, Pennsylvania State University
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- Fundamentals of Structural Geology
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- 01 September 2005, pp 287-332
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Comparison of displacement field components near an edge dislocation from phase images of experiments (left column) and from anisotropic elastic theory (right column). (a) and (b) displacement component ux parallel to bottom edge of image. (c) and (d) displacement component uy parallel to left edge of image. Photograph reproduced from (Hytch et al., 2003) with kind permission of Martin J. Hytch.
The conceptual success of the [infinitesimal theory of elasticity and the linear theory of viscosity] is perhaps the broadest we know in science: in terms of them we face, “explain”, and in varying amount control, our daily environment: winds and tides, earthquakes and sounds, structures and mechanisms, sailing and flying, heat and light (Truesdell and Noll, 1965).
In this chapter we describe how the elastic properties of rock are measured in the laboratory and provide tables of numbers representing the range of values for different rock types. However, the need to understand and measure the resistance to deformation of rocks goes well beyond the simple accumulation of numbers in handbooks of rock properties. To paraphrase Truesdell and Noll (1965), the aim of structural geology is to construct mathematical models that enable us, from use of knowledge gathered in a few observations, to predict by logical processes the outcomes in many other circumstances. To analyze a geologic structure one must choose the appropriate boundary or initial value problem to serve as a mechanical model. To formulate such a problem one must postulate a particular mechanical behavior.
Chapter 12 - Model development and methodology
- David D. Pollard, Stanford University, California, Raymond C. Fletcher, Pennsylvania State University
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- Fundamentals of Structural Geology
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- 01 September 2005, pp 456-477
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Oblique aerial photograph of the eastern slope of Mt. Hillers in the Henry Mountains, Utah. The Black Mesa laccolith and Maiden Creek sill form the black, cliff-forming outcrops.
Three main approaches are needed to unscramble a complicated system. (1) One can take it apart and characterize all the isolated bits – what they are made of and how they work. (2) Then one can find exactly where each part is located in the system in relation to all the other parts and how they interact with each other. These two approaches are unlikely, by themselves, to reveal exactly how the system works. (3) To do this one must also study the behavior of the system and its components while interfering very delicately with its various parts, to see what effect such alterations have on behavior at all levels (Crick, 1988).
Nature, or what we might call natural reality, can appear to our senses as a very complicated system when we view geological structures in outcrop. Francis Crick (1988) also faced a complicated system when viewing the constituents of living cells. He suggests in his book What Mad Pursuit that one should first characterize all the parts of the system and then understand their geometric relationships. This is what we attempt to do as structural geologists when mapping structures in the field.
Preface
- David D. Pollard, Stanford University, California, Raymond C. Fletcher, Pennsylvania State University
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- Fundamentals of Structural Geology
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- 01 September 2005, pp ix-x
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Fundamentals of Structural Geology is a textbook that emphasizes modern techniques of field data acquisition and analysis, the principles of continuum mechanics, and the mathematical and computational skills necessary to describe, model, and explain quantitatively the deformation of rock in Earth's lithosphere.
With precise location data now available from the Global Positioning System (GPS) and powerful computer systems now transportable in a backpack, the quantity of reproducible field data has increased dramatically. These new data sets demand better methods for describing the geometry of structures, and we address this demand by introducing the basic concepts of differential geometry, which provide unambiguous descriptions of curved lineations and surfaces in three dimensions. Data sets from a variety of field areas are provided via the textbook website to promote the practice of opening field “notebooks” to the entire community of researchers, and as input for student exercises (see below).
Textbooks in structural geology provide elements of continuum mechanics (e.g. separate chapters on stress and strain), but rarely are these concepts tied together with constitutive laws or formulated into equations of motion or equilibrium to solve boundary or initial value problems. These textbooks largely beg the questions: what methodology should one adopt to solve the problems of structural geology; and what are the fundamental constructs that must be acknowledged and honored? These constructs are the conservation laws of mass, momentum, and energy, combined with the constitutive laws for material behavior and the kinematic relationships for strain and rate of deformation.
Acknowledgments
- David D. Pollard, Stanford University, California, Raymond C. Fletcher, Pennsylvania State University
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- Fundamentals of Structural Geology
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- 01 September 2005, pp xi-xii
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Chapter 4 - Physical quantities, fields, dimensions, and scaling
- David D. Pollard, Stanford University, California, Raymond C. Fletcher, Pennsylvania State University
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- Fundamentals of Structural Geology
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- 01 September 2005, pp 120-151
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Earth's crust under the state of Texas being lifted by a crane. Scaling laws demonstrate that the good state of Texas is utterly incapable of self-support. Reprinted from Hubbert (1945) by permission of the AAPG whose permission is required for further use.
In physical science a first essential step in the direction of learning any subject is to find principles of numerical reckoning and practical methods for measuring some quality connected with it. I often say that when you can measure what you are speaking about, and express it in numbers you know something about it; but when you cannot measure it, when you cannot express it in numbers your knowledge is of a meager and unsatisfactory kind: it may be the beginning of knowledge, but you have scarcely, in your thoughts, advanced to the stage of science, whatever the matter may be (Thomson, 1891).
In an insightful article about research in geology in the early twentieth century M. King Hubbert refers to Sir William Thomson (Lord Kelvin) as the “Patron Saint” of geologists, including himself, who espouse a quantitative agenda, and he cites this quotation from Thomson as their “guiding credo” (Hubbert, 1974). Thomson is not advocating numeration purely for the sake of collecting numbers; rather this is a call to measure relevant physical quantities and express them as numbers. Thomson was a physicist, not a geologist, but Hubbert recognized the importance of quantification in the geological sciences and was a leader among geologists of his generation in this regard (Hubbert, 1972).
Chapter 11 - Rheological behavior
- David D. Pollard, Stanford University, California, Raymond C. Fletcher, Pennsylvania State University
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- Fundamentals of Structural Geology
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- 01 September 2005, pp 421-455
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Pressure solution in sandstone from the Olympic accretionary wedge, Olympic Mountains, NW Washington State, USA. The horizontal dimension is ~ 10 mm. Photograph by J. M. Rahl.
No mathematical theory can completely describe the complex world around us. Every theory is aimed at a certain class of phenomena, formulates their essential features, and disregards what is of minor importance. The theory meets its limits of applicability where a disregarded influence becomes important (Flugge, 1967).
Departures from linear viscous flow
Laboratory determinations of steady-state rock creep and field observations indicate that ductile rocks may not be well-approximated by homogeneous, isotropic, incompressible, and linear viscous fluids of uniform viscosity. In this chapter, we consider other constitutive relations that broaden the range in rock behavior described and provide a basis for understanding some of the differences arising. Because the linear Newtonian viscous fluid is the simplest material that undergoes large permanent deformation, the formulation and analysis of models using it and their application to interpret a set of field observations is always a useful first step (see Chapter 10). The results obtained establish a benchmark from which to understand differences in behavior associated with other constitutive relations. Whether the new model results in a large or subtle contrast in behavior relative to an already well-understood viscous model, we will achieve a better understanding of the reasons for the differences and a greater confidence in data interpretation.
Chapter 5 - Deformation and flow
- David D. Pollard, Stanford University, California, Raymond C. Fletcher, Pennsylvania State University
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- Fundamentals of Structural Geology
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- 01 September 2005, pp 152-193
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Nearly undeformed (upper left) and deformed (lower right) oöids with spherulitic cores viewed in thin section. Matrix is calcite with some mud and average ratios of long to short axes are 1.16 and 1.56 respectively. Samples from near Harrisonburg, VA. Reprinted from Cloos (1971, Plates 9 and 11) with permission of The John Hopkins University Press.
A clear separation between geometrical and dynamic considerations was maintained by Becker, the American geologist, at a very early date, and he referred to the English physicist Thomson (Lord Kelvin), who says very clearly: ‘We can see, therefore, that there are many attributes of movement, displacement, and deformation which can be considered independently of force, mass, chemical composition, elasticity, heat, magnetism and electricity; and it is of greatest use to science for such properties to be considered as a first step’ (Sander, 1970, p. 12).
Most structural geologists think about deformation and flow in an inverse problem mode: from the final state back toward the initial state of the deformed body of rock. For example, the lower right photograph in the frontispiece for this chapter shows deformed oöids with elliptical shapes: ratios of long to short axes are about 1.56 (Cloos, 1947, 1971). The upper left photograph shows nearly undeformed oöids with approximately circular shapes: ratios of long to short axes are about 1.16.
References
- David D. Pollard, Stanford University, California, Raymond C. Fletcher, Pennsylvania State University
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- Fundamentals of Structural Geology
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- 01 September 2005, pp 478-496
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Chapter 7 - Conservation of mass and momentum
- David D. Pollard, Stanford University, California, Raymond C. Fletcher, Pennsylvania State University
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- Fundamentals of Structural Geology
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- 01 September 2005, pp 243-286
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Structural block diagram of a part of the Penninic Alps, Switzerland (Argand, 1911).
At the heart of all this calculation lies the deeply held conviction that natural phenomena are, in essence, the consequence of just a small number of physical laws, and that these laws are best expressed in the language of mathematics. The goal is to construct a working model of the universe out of commonplace notions: ideas of number and order and measures of time and distance. With such a working model, we can leap ahead in time and predict what the otherwise opaque future has in store for us (Peterson, 1993).
In the context of structural geology we can construct a working model of mountain building from those small number of physical laws and then leap backward in time and understand the development of geological structures such as those depicted in the structural block diagram (Chapter 7, frontispiece) of a part of the Penninic Alps of Switzerland constructed by Emile Argand and published in 1911 (Argand, 1911). This is one of the earliest published block diagrams in the literature of structural geology (McIntyre and Weiss, 1956; Howarth, 1999) and it illustrates what was known in the early part of the twentieth century about one of the most interesting and complex regions of folding and faulting in that mountain chain.
Among the small number of physical laws that can be employed to understand tectonic processes and their structural products are those of mass, momentum, and energy conservation.
Frontmatter
- David D. Pollard, Stanford University, California, Raymond C. Fletcher, Pennsylvania State University
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- Fundamentals of Structural Geology
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- 01 September 2005, pp i-iv
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Chapter 10 - Viscous flow
- David D. Pollard, Stanford University, California, Raymond C. Fletcher, Pennsylvania State University
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- Fundamentals of Structural Geology
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- 01 September 2005, pp 384-420
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Straight-limbed chevron-like folds in Cretaceous strata in the footwall of the Lewis Thrust, Canadian Rockies, from the Kananaskis Highway between Banff and Blairmore, looking south. Photograph by D. Wiltschko.
This bulletin is the second contribution to the general investigation of the physical constants of rocks, the experiments concerning which follow a general plan devised by Mr. Clarence King. Questions bearing directly on the viscosity of rock masses make up so large a part of Mr. King's geological observations, that the duty of enquiring into the physics of this enormously complicated subject devolved seriously upon me. Above all things some form of reliable working hypothesis was to be discovered; and this is what the present bulletin endeavors to accomplish. I believe the physical hypothesis has been found and that the data afford substantial corroboration of Maxwell's theory of the viscosity of solids (Barus, 1891).
Rock deformation by viscous flow
As the above quotation indicates, over one hundred years ago, the first director of the United States Geological Survey, Clarence King, hired a scientist to determine the viscosity of rock. Searching through King's account of the survey of the fortieth parallel, we find no mention of “Questions bearing directly on the viscosity of rock masses …” King did not write much, and so it appears his thoughts on this subject and the observations that motivated them may have been lost.