Practical Handbook of Physical Properties of Rocks and Minerals (1988)

Robert S. Carmichael

• Publisher: CRC Press
• Year: 2017
• ISBN:
  9781351359184
• Edition: 1
• Topic: Physical Sciences
• Subtopic: Physics
• Index: Physics

CRC Practical Handbooks are a series of single-volume bench manuals that feature a synthesis of the most frequently used, basic reference information. These highly abridged versions of existing CRC multi-volume Handbooks contain largely tabular and graphic data. They provide extensive coverage in a scientific discipline and enable quick, convenient access to the most practical reference information...on the spot! Leading professionals in their respective fields collaborated to provide individuals and institutions with an economical and easy-to-use source of classic reference information.The CRC Practical Handbook of PHYSICAL PROPERTIES of ROCKS and MINERALS, prepared by leaders in their specialties, has been constructed to serve as a convenient, compact, yet comprehensive source of basic information. The technical data have been compiled and selectively edited to provide an organized and definitive presentation of the physical properties of rocks and their constituent minerals. The format is primarily tabular and graphical, for easy reference and comparisons. There is also instructive textual material to present, explain, and clarify the data.
This edited and abridged version of the CRC Handbook of Physical Properties of Rocks, published in three volumes in 1982 - 1984, will serve as an easy-to-use source of current and useful reference information.

Section VI
Seismic Velocities

By
Nikolas I. Christensen

INTRODUCTION

Seismology has provided a wealth of evidence relating to the physical nature of the interior of the Earth. Most seismological studies using earthquakes or reflection and refraction techniques from artificially generated waves present layered models in which velocities and layer thicknesses are tabulated. Many significant results have emerged from these studies including:

1. The broad subdivision of the Earth into crust, mantle, and core
2. The recognition of seismic discontinuities within the core and mantle that are probably related to phase changes
3. The marked difference in overall structure of the oceanic and continental crust

Several intracrustal discontinuities have, in turn, been recognized which differ from one region to another, and as increasing data become available it is apparent that many regions of the Earth’s interior are anisotropic and heterogeneous.

The information desired from seismic studies is not ultimately velocity-depth functions, but knowledge of the nature and distribution of materials with depth so we may understand the origin and evolution of the Earth. The velocities of elastic waves in materials for the interpretation of seismic data must be obtained through carefully controlled laboratory experiments which realistically simulate the physical conditions that exist within the Earth’s interior.

In addition to being of interest to Earth scientists, velocities are of considerable significance to materials scientists since they yield information important in understanding forces between atoms and ions. Also, since velocities are related to the elastic properties of solids, they are important in describing the mechanical behavior of materials.

For homogeneous isotropic elastic materials, compressional (V_p) and shear (V_s) wave velocities, density (ϱ), and the elastic moduli are related by the following equations:

Bulk Modulus	K=ϱ(Vp2−4/3s2)
Shear Modulus	μ=ϱVs2
Poisson’s Ratio	σ=(r2−2)2(r2−1),r=Vp/VS
Young’s Modulus	E = 2μ(1 + σ)
Compressional Wave Velocity	Vp = √[K + (4/3)μ]/ϱ
Shear Wave Velocity	Vs = √μ/ϱ

Laboratory studies of velocities in materials generally fall into three categories: (1) measurements of velocities in naturally occurring materials such as rocks, (2) studies of hot-pressed polycrystalline aggregates, and (3) velocity measurements in single crystals. The velocities in rocks and hot-pressed aggregates are commonly affected by porosity. Values useful for the interpretation of field measurements, except in near-surface studies, are obtained only after porosity has been reduced by application of a few kilobars pressure. Measurements of velocities in single crystals are useful in the interpretation of seismic anisotropy resulting from preferred mineral orientation. In addition, if the elastic constants have been completely determined it is possible to estimate the velocities of quasi-isotropic aggregates of single crystals.

The prediction of velocities of a quasi-isotropic rock containing a large number of randomly oriented, highly anisotropic crystals, from single-crystal data is complicated in many aspects. In theory, it is difficult to compromise between assumptions of uniform local strain and uniform local stress. Voigt¹ assumed that strain is uniform throughout the rock and averaged over solid angles the elastic constants (C_ij), whereas Reuss² assumed that uniform local stress was operative and averaged the elastic compliances (S_ij) over all directions. The appropriate relationships for the bulk moduli and shear moduli according to the two theories are as follows:

Voigt’s Moduli

9K_v - (C₁₁ + C₂₂ + C₃₃) + 2(C₁₂ + C₂₃ + C₃₁)

15μ_v = (C₁₁ + C₂₂ + C₃₃) - (C₁₂ + C₂₃ + C₃₁) + 3(C₄₄ + C₅₅ + C₆₆)

Reuss’s Moduli

1/K_r = (S₁₁ + S₂₂ + S₃₃) + 2(S₁₂ + S₂₃ + S₃₁)

15/μ_r = 4(S₁₁ + S₂₂ + S₃₃) - 4(S₁₂ + S₂₃ + S₃,) + 3(S₄₄ + S₅₅ + S₆₆)

Calculated compressional and shear wave velocities for quasi-isotropic monomineralic rocks are obtained from the relationships ϱVp2=K+4μ/3 and ϱVs2=μ where ϱ is the density of the mineral.

Voigt’s and Reuss’s velocity averages frequently show considerable variance especially for the silicate minerals of low symmetry. Hill³ has shown theoretically that the true values lie between the Voigt and Reuss Moduli and the Hill average is commonly taken as the mean of the Voigt and Reuss values.

The accuracies of seismic structure within the Earth depend to a large extent on the combination of field and analytical techniques used to identify the velocities and probably vary between 3 and 10% for most models. The accuracies of laboratory velocities in materials also depend on the specific technique employed, varying from 0.5% to 3% for the pulse-transmission method commonly used for rocks to approximately 0.01 % with interferometric methods. The laboratory techniques typically use frequencies much higher than the field studies. However, several studies have demonstrated that dispersion in the frequency range of 10^_1 to 10⁷ Hz is negligible, thus allowing direct use of the laboratory data in the interpretation of field measurements.

The ...

Table of contents

• Cover Page
• Title Page
• Copyright
• Preface
• The Author
• Contributors
• Table of Contents
• SECTION I Mineral Composition of Rocks
• SECTION II Densities of Rocks and Minerals
• SECTION III Inelastic Properties of Rocks and Minerals: Strength and Rheology
• SECTION IV Magnetic Properties of Minerals and Rocks
• SECTION V Electrical Properties
• SECTION VI Seismic Velocities
• SECTION VII Seismic Attenuation
• SECTION VIII Radioactivity Properties of Minerals and Rocks
• SECTION IX Spectroscopic Properties of Rocks and Minerals
• SECTION X Engineering Properties of Rock
• Index