Mining in Manitoba
Rock Mechanics
Mine Development
In elastic deformation, there are various constants that relate the magnitude of the strain response to the applied stress. These elastic constants include the following:
(1) Young's modulus (E) is the ratio of the applied stress to the fractional extension (or shortening) of the sample length parallel to the tension (or compression). The strain is the linear change in dimension divided by the original length.
(2) Shear modulus () is the ratio of the applied stress to the distortion (rotation) of a plane originally perpendicular to the applied shear stress; it is also termed the modulus of rigidity.
(3) Bulk modulus (k) is the ratio of the confining pressure to the fractional reduction of volume in response to the applied hydrostatic pressure. The volume strain is the change in volume of the sample divided by the original volume. Bulk modulus is also termed the modulus of incompressibility.
(4) Poisson's ratio (p) is the ratio of lateral strain (perpendicular to an applied stress) to the longitudinal strain (parallel to applied stress).
For elastic and isotropic materials, the elastic constants are interrelated. For example,
and
The following are the common units of stress:
Thus 10 kilobars = 1 gigapascal (i.e., 109 Pa).
Rock mechanics
The study of deformation resulting from the strain of rocks in response to stresses is called rock mechanics. When the scale of the deformation is extended to large geologic structures in the crust of the Earth, the field of study is known as geotectonics.
The mechanisms and character of the deformation of rocks and Earth materials can be investigated through laboratory experiments, development of theoretical models based on the properties of materials, and study of deformed rocks and structures in the field. In the laboratory, one can simulate--either directly or by appropriate scaling of experimental parameters--several conditions. Two types of pressure may be simulated: confining (hydrostatic), due to burial under rock overburden, and internal (pore), due to pressure exerted by pore fluids contained in void space in the rock. Directed applied stress, such as compression, tension, and shear, is studied, as are the effects of increased temperature introduced with depth in the Earth's crust. The effects of the duration of time and the rate of applying stress (i.e., loading) as a function of time are examined. Also, the role of fluids, particularly if they are chemically active, is investigated.
Some simple apparatuses for deforming rocks are designed for biaxial stress application: a directed (uniaxial) compression is applied while a confining pressure is exerted (by pressurized fluid) around the cylindrical specimen. This simulates deformation at depth within the Earth. An independent internal pore-fluid pressure also can be exerted. The rock specimen can be jacketed with a thin, impermeable sleeve (e.g., rubber or copper) to separate the external pressure medium from the internal pore fluids (if any). The specimen is typically a few centimetres in dimension.
Another apparatus for exerting high pressure on a sample was designed in 1968 by Akira Sawaoka, Naoto Kawai, and Robert Carmichael to give hydrostatic confining pressures up to 12 kilobars (1.2 gigapascal), additional directed stress, and temperatures up to a few hundred degrees Celsius. The specimen is positioned on the baseplate; the pressure is applied by driving in pistons with a hydraulic press. The end caps can be locked down to hold the pressure for time experiments and to make the device portable.
Apparatuses have been developed, typically using multianvil designs, which extend the range of static experimental conditions--at least for small specimens and limited times--to pressures as high as about 1,700 kilobars and temperatures of about 2,000º C. Such work has been pioneered by researchers such as Peter M. Bell and Ho-Kwang Mao, who conducted studies at the Geophysical Laboratory of the Carnegie Institution in Washington, D.C. Using dynamic techniques (i.e., shock from explosive impact generated by gun-type designs), even higher pressures up to 7,000 kilobars (700 gigapascal)--which is nearly twice the pressure at the centre of the Earth and seven million times greater than the atmospheric pressure at the Earth's surface--can be produced for very short times. A leading figure in such ultrapressure work is A. Sawaoka at the Tokyo Institute of Technology.
In the upper crust of the Earth, hydrostatic pressure increases at the rate of about 320 bars per kilometre, and temperature increases at a typical rate of 20º-40º C per kilometre, depending on recent crustal geologic history. Additional directed stress, as can be generated by large-scale crustal deformation (tectonism), can range up to 1 to 2 kilobars. This is approximately equal to the ultimate strength (before fracture) of solid crystalline rock at surface temperature and pressure (see below). The stress released in a single major earthquake--a shift on a fault plane--is about 50-150 bars.
In studying the deformation of rocks one can start with the assumption of ideal behaviour: elastic strain and homogeneous and isotropic stress and strain. In reality, on a microscopic scale there are grains and pores in sediments and a fabric of crystals in igneous and metamorphic rocks. On a large scale, rock bodies exhibit physical and chemical variations and structural features. Furthermore, conditions such as extended length of time, confining pressure, and subsurface fluids affect the rates of change of deformation. Figure 7 shows the generalized transition from brittle fracture through faulting to plastic-flow deformation in response to applied compressional stress and the progressive increase of confining pressure.
Young's modulus
Metal bar under tension increases in length and decreases in cross section
numerical constant, named for the 18th-century English physician and physicist Thomas Young, that describes the elastic properties of a solid undergoing tension or compression in only one direction, as in the case of a metal rod that after being stretched or compressed lengthwise returns to its original length. Young's modulus is a measure of the ability of a material to withstand changes in length when under lengthwise tension or compression. Sometimes referred to as the modulus of elasticity, Young's modulus is equal to the longitudinal stress divided by the strain. Stress and strain may be described as follows in the case of a metal bar under tension.
If a metal bar of cross-sectional area A is pulled by a force F at each end, the bar stretches from its original length Lo to a new length Ln. (Simultaneously the cross section decreases.) The stress is the quotient of the tensile force divided by the cross-sectional area, or F/A. (Its units in the English system are pounds per square inch, usually abbreviated psi.) The strain or relative deformation is the change in length, Ln - Lo, divided by the original length, or (Ln - Lo)/Lo. (Strain is dimensionless.) Thus Young's modulus may be expressed mathematically as
Young's modulus =strainstress = (Ln - Lo)/LoF/A
This is a specific form of Hooke's law of elasticity. The units of Young's modulus in the English system are pounds per square inch, and in the metric system newtons per square metre (N/m2). The value of Young's modulus for aluminum is about 1.0 107 psi, or 7.0 1010 N/m2. The value for steel is about three times greater, which means that it takes three times as much force to stretch a steel bar the same amount as a similarly shaped aluminum bar.
Young's modulus is meaningful only in the range in which the stress is proportional to the strain, and the material returns to its original dimensions when the external force is removed. As stresses increase, Young's modulus may no longer remain constant but decrease, or the material may either flow, undergoing permanent deformation, or finally break.
When a metal bar under tension is elongated, its width is slightly diminished. This lateral shrinkage constitutes a transverse strain that is equal to the change in the width divided by the original width. The ratio of the transverse strain to the longitudinal strain is called Poisson's ratio. The average value of Poisson's ratio for steels is 0.28, and for aluminum alloys, 0.33. The volume of materials that have Poisson's ratios less than 0.50 increase under longitudinal tension and decrease under longitudinal compression.
