APPENDIX PHYSICAL PRINCIPLES MENTIONED IN THE TEXT Bernoulli's Theorem In a stream ofliquid, the sum of the elevation head, the pressure head, and the velocity head (1) remains constant along any line of flow provided that no work is done by or upon the liquid in the course of its flow and (2) decreases in proportion to the energy lost in viscous flow. Hooke's Law The stress within an elastic solid up to the elastic limit is prop0l1ional to the strain responsible for it. Level of Neutral Buoyancy In the lithosphere, this is the level at which a rising column of magma no longer can rise because below that level the country rock is denser than the magma, and above that level the country rock is less dense than the magma (see Walker, 1989). This level also has been called the level of gravitational stability. Navier-Coulomb Maximum Shear-Stress Theory We do not treat this concept in detail or with mathematics. Those interested in details are referred to Jaeger (1962, p. 75-80). The Navier-Coulomb maximum shear-stress theory states that failure (by shear) of a material occurs when the maximum shear stress equals the shear strength of the material. Compressive shear strength is always greater than tensile shear strength. Because of this, shear caused by compressive stress must be at an angle of less than 45° to the direction of that stress. In any three-dimensional system, there are three principal stresses. If this three-dimensional system is the Eatih's lithosphere, one of the three principal stresses is essentially vertical. Thus three cases----three types offaulting----arise according to whether the vertical stress direction is the greatest, intermediate, or the least (Jaeger, 1962) Case 1, Thrust faults----As shown on Figure Al a (from Meyerhoff et aI., I 992b) the vertical principal stress is the least in magnitude and the other two principal stresses are compressive. The planes of fracture pass through the direction of the intermediate principal stress and make angles of less than 45° with the direction of the greatest compressive (horizontal) stress. This case is illustrated in nature by Figures 2.36,3.1 and 3.18, which show the lithosphere Benioff zones dip at an angle less than 45° to the Em1h's sUIface (see Scheidegger and Wilson, 1950). APPENDIX 259 "okCl2 S OJ --7 f-- OJ ~ OJ (a) (b) THRUST FAULT C B ~2ll t~1 (- a-~ 0 v3 3 ~3 A +"1 D (e) (d) STRIKE-SLIP FAULT ~3t& t V3 "i (e) (f)-r- NORMAL FAULT From Jaeger (1962) Fig. A.I. lbree examples of Navier-Coulomb maximum shear-stress theory: in each case, fracture takes place at an angle >45 0 with the direction of maximum principal stress. s[ in each diagram is the direction of least principal stress. S, is the direction of intennediate principal stress. '" is the direction of maximum principal stress. (a) and (b) describe thrustfauhing; (c) and (d) describe strike-slip faulting; (e) and (I) describe nonnal faulting. From Jaeger (1962, p. 75- 80). Case 2, Streamline-strike-slip faults----As shown on Figure A.l c (from Meyerhoff et al.,1992b), the vertical principal stress is the intermediate stress. Of the other two principal stresses, one will be a compression and the other will be small and may even be a tension. In this case, failure can take place on either oftwo vertical planes (AOB, COD) that are equally inclined at angles less then 45° to it. Case 3, Normal faults----As shown on Figure A.le (from Meyerhoff et aI., 1992b), the vertical principal stress is the greatest in magnitude. This state can be expected mainly at considerable depths (Jaeger, 1962, p. 80). Failure will take place at an angle of less then 45° to the vertical, that is, at an angle greater than 45° to the Earth's surface. Figures 2.36, 3.1 and 3.18 illustrate natural examples in which the strictosphere Benioff zone dips at an angle greater than 45° to the Earth's surface. 260 APPENDIX Newton's Laws All laws, theorems, and principles that involve motion are derived from Newton's three laws of motion and the law of gravity. First Law of Motion----A body at rest remains at rest and a body in motion remains in uniform motion in a straight line unless acted upon by an external force. Second Law of Motion----The rate of change of the velocity of a body, that is, its acceleration, is equal to the resultant of all external forces acting on the body divided by the mass of the body, and is in the direction ofthe straight line in which the force acts (that is, in the same direction as the resultant force). Third Law of Motion----To every action there is always opposed an equal reaction; that is, the mutual actions of two bodies upon each other are always equal and in the opposite direction; or, for every force, there is an equal and opposite force or reaction. Law of Gravitation----Every pilliic1e of matter in the universe attracts every other particle with a force that is directly propOliional to the produce of the masses of the particles and inversely propOliional to the square of the distance between them. Pascal's Law, Theorem, or Principle Pressm-e applied to a confined liquid at any point is transmitted undiminished through the fluid in all directions and acts upon every pati of the coni ning vessel at right angles to its interior surfaces and equally upon equal areas. Peach-Kohler Climb Force Although the following works best in mafic lavas because of their low viscosities, the statements apply to more silicic lavas as well. However, the relatively lower viscosity of mafic lavas makes it easier for them to move through the lithosphere and seek a position of gravitational stability (Walker, 1989). In the case of a veliical crack filled with magma, the upward force exerted by the crack is a result of the lower density of the magma filling the crack (Corry, 1988), Weeliman (1971, p. 1177) explained the process. "The force of (p - p') gV [where p is the density of the country rock at a given depth; p' is the magma density; g is the acceleration due to gravity; and V is the magma volume per unit of crack width] experienced by the crack is identical to the Archimedian buoyancy that would act on the crack if it were a solid of density p' embedded in a liquid of density p. There is a fundamental difference, however, between these two forces. For the tme Archimedian force the body force p'gY differs fi-om the net force pgY exerted on the embedded solid by the hydrostatic pressm-e. In the case of the liquid-filled crack, the body force p'g Yexelied on the liquid within the crack must be, and is, exactly balanced by hydrostatic forces exerted against the crack walls. "There is no net force exerted on the liquid within the crack." The Peach-Kohler climb force is produced by a gradient in elastic strain energy within the solid (rock body). The origin of the APPENDIX 261 true and the pseudo-Archimedian forces is the same, namely, the gravity field. "The crack will stop propagating, and thus the magma rise will stop, when the magma reaches its level of neutral buoyance, which is the level below which the country rock is denser than the magma, and above which the country rock is less dense than the magma" (Walker, 1989). Poiseuille's Law The velocity of flow of a liquid through a capillary tube varies directly as the pressure and the fomih power of the diameter of the tube, and inversely as the length of the tube and the coefficient of viscosity. Stokes's Law and Linear Geologic Structures Stokes's Law, one of many expressions of Newton's Second Law of Motion, is little used by geologists, despite its fundamental impOliance. Because the law usually is thought of as a description of the flow of solids through air and water, it almost never is regarded as describing the interactions between any flowing medium and a solid substance (e.g., the flow of lava through lava tubes). One critic of an earlier version of this manuscript wrote that our use of Stokes's Law" ... only confuses a discussion of the well-known streamline pattems of viscous flow." This critic apparently did not know that streamline pattems of viscous flow (in glaciers, for example) are commonly cited in physics texts as examples of Stokes's Law (e.g., Blatt, 1983)! Nor did he realize that such pattems of streamline flow are what led in the first place to the 1845 formulation ofthis law by Sir George G. Stokes (e.g., Sears et a!., 1974). Stokes's Law may be stated most simply as follows: "The force required to move a sphere through a given viscous fluid at a low unifom1 velocity is directly proportional to the velocity and radius of the sphere" (Webster's Third New international Dictionary, 1971, p. 2248). Under the heading of Stokes's Law, Sears et a!. (1974, p. 225-226) wrote that "When an ideal fluid of zero viscosity flows past a sphere, or when a sphere moves through a stationary fluid, the streamlines form a peIfectly symmetrical pattem around the sphere .... " If the fluid is viscous, however, there will be a viscous drag on the sphere. Moreover, ifthe smface ofthe sphere consists of altemate parallel bands of rough and smooth texture, differential (viscous) drag will take place tween the fluid that flows over the rough-textured suIfaces and that which flows over the smooth-textured suIfaces.