11-6 BULK STRESS AND STRAIN
Den a submersible plunges deep into the ocean, the water exerts nearly uniform pres everywhere on its surface and squeezes the submersible to a slightly smaller volume c. ll-8b). This is a different situation from the tensile and compression stresses and we have discussed. The stress is now a uniform pressure on all sides, and the . g deformation is a volume change. We use the terms bulk stress (or volume and bulk strain (or volume strain) to describe these quantities. Another exampled is the compression of a gas under pressure, such as the air in a car’s tire. If we choose an arbitrary cross section within a fluid (liquid or gas) at rest, the force on each side of the section is always perpendicular to it If we tried to exert a force to a section, the fluid would slip sideways to counteract the effort. When a solid is immersed in a fluid and both are at rest, the forces that the fluid exerts on the surface solid are always perpendicular to the surface at each point. The force FJ. per unit on such a surface is called the pressure p in the fluid:
p=~ F: A (pressure in a fluid). (11-11) been we apply pressure to the surface of a fluid in a container, such as the cylinder piston shown in Fig. 11-14, the pressure is transmitted through the fluid and also on the surface of any body immersed in the fluid. This principle is called Pascal’s . If pressure differences due to differences in depth within the fluid can be neglected, pressure is the same at every point in the fluid and at every point on the surface of merged body. Pressure has the same units as stress commonly used units include 1 Pa (= 1 N/m2) and 1 lbftD.2 (I psi). Also in common use is the ionosphere, abbreviated atm. One atom Sphere is the approximate average pressure of the earth’s atmosphere at sea level: 1 atmosphere = 1 atm = 1.013 x lOs Pa = 14.7 lb/in.’. Pressure is a scalar quantity, not a vector quantity; it has no direction.
Pressure plays the role of stress in a volume deformation, The corresponding strain is the fractional change in volume (Fig. 11-15), that is, the ratio of the volume change L1 V to the original volume Va: Bulk (volume) strain = L1V. (11-12) Va Volume strain is the change in volume per unit volume. Like tensile or compression strain, it is a pure number, without units. When Hooke’s law is obeyed, an increase in pressure (bulk stress) produces a proportional sulk strain (fractional change in volume. The corresponding elastic modulus (ratio of stress to strain) is called the bulk modulus, denoted by B. When the pressure on a body changes by a small amount Sp, from Po to Po+Sp, and the resulting bulk strain is Hooke’s law takes the form. (bulk modulus) (11-13) We include a minus sign in this equation because an increase of pressure always causes a decrease in volume. In other words, if is positive, is negative. The bulk modulus B itself is a positive quantity. For small pressure changes in a solid or a liquid, we consider B to be constant. The bulk modulus of a gas, however, depends on the initial pressure Po’ able 11-1 includes values of the bulk modulus for several solid materials. Its units, force per unit area, are the same as those of pressure (and of tensile or compression stress). The reciprocal of the bulk modulus is called the comprehensibility and is denoted by From Eq. (comprehensibility). (11-14) Comprehensibility is the fractional decrease in volume, per unit increase in pressure. The units of comprehensibility are those of reciprocal pressure, Values of comprehensibility k for several liquids are listed in Table 11-2. From this table the comprehensibility of water is 46.4 x 1O-(i atm -I. This means that for each atmosphere increase in pressure, the volume of water decreases by 46.4 parts per Materials with small bulk modulus and large comprehensibility are easy to compress; those with larger bulk modulus and smaller comprehensibility compress less with the same pressure increase.