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where $B$ is the bulk modulus (see [link] ), ${V}_{0}$ is the original volume, and $\frac{F}{A}$ is the force per unit area applied uniformly inward on all surfaces. Note that no bulk moduli are given for gases.
What are some examples of bulk compression of solids and liquids? One practical example is the manufacture of industrial-grade diamonds by compressing carbon with an extremely large force per unit area. The carbon atoms rearrange their crystalline structure into the more tightly packed pattern of diamonds. In nature, a similar process occurs deep underground, where extremely large forces result from the weight of overlying material. Another natural source of large compressive forces is the pressure created by the weight of water, especially in deep parts of the oceans. Water exerts an inward force on all surfaces of a submerged object, and even on the water itself. At great depths, water is measurably compressed, as the following example illustrates.
Calculate the fractional decrease in volume ( $\frac{\mathrm{\Delta}V}{{V}_{0}}$ ) for seawater at 5.00 km depth, where the force per unit area is $5\text{.}\text{00}\times {\text{10}}^{7}\phantom{\rule{0.25em}{0ex}}\mathrm{N}/{\mathrm{m}}^{2}$ .
Strategy
Equation $\mathrm{\Delta}V=\frac{1}{B}\frac{F}{A}{V}_{0}$ is the correct physical relationship. All quantities in the equation except $\frac{\mathrm{\Delta}V}{{V}_{0}}$ are known.
Solution
Solving for the unknown $\frac{\mathrm{\Delta}V}{{V}_{0}}$ gives
Substituting known values with the value for the bulk modulus $B$ from [link] ,
Discussion
Although measurable, this is not a significant decrease in volume considering that the force per unit area is about 500 atmospheres (1 million pounds per square foot). Liquids and solids are extraordinarily difficult to compress.
Conversely, very large forces are created by liquids and solids when they try to expand but are constrained from doing so—which is equivalent to compressing them to less than their normal volume. This often occurs when a contained material warms up, since most materials expand when their temperature increases. If the materials are tightly constrained, they deform or break their container. Another very common example occurs when water freezes. Water, unlike most materials, expands when it freezes, and it can easily fracture a boulder, rupture a biological cell, or crack an engine block that gets in its way.
Other types of deformations, such as torsion or twisting, behave analogously to the tension, shear, and bulk deformations considered here.
where $\mathrm{\Delta}L$ is the amount of deformation (the change in length), $F$ is the applied force, and $k$ is a proportionality constant that depends on the shape and composition of the object and the direction of the force. The relationship between the deformation and the applied force can also be written as
where $Y\phantom{\rule{0.25em}{0ex}}$ is Young’s modulus , which depends on the substance, $A$ is the cross-sectional area, and ${L}_{0}$ is the original length.
where $S$ is the shear modulus and $F$ is the force applied perpendicular to ${L}_{\text{0}}$ and parallel to the cross-sectional area $A$ .
where $B$ is the bulk modulus, ${V}_{\text{0}}$ is the original volume, and $\frac{F}{A}$ is the force per unit area applied uniformly inward on all surfaces.
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