4.2 Elasticity: stress and strain (2d)  (Page 7/14)

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.

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.

Section summary

• Hooke’s law is given by
  $F=k\text{Δ}L,$

  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

  $\mathrm{\Delta }L=\frac{1}{Y}\frac{F}{A}{L}_0,$

  where $Y$ is Young’s modulus , which depends on the substance, $A$ is the cross-sectional area, and $L_0$ is the original length.

• The ratio of force to area, $\frac{F}{A}$ , is defined as stress , measured in N/m ² .
• The ratio of the change in length to length, $\frac{\mathrm{\Delta }L}{L_0}$ , is defined as strain (a unitless quantity). In other words,
  $\text{stress}=Y\times \text{strain}.$
• The expression for shear deformation is
  $\mathrm{\Delta }x=\frac{1}{S}\frac{F}{A}{L}_0,$

  where $S$ is the shear modulus and $F$ is the force applied perpendicular to $L_{\text{0}}$ and parallel to the cross-sectional area $A$ .

• The relationship of the change in volume to other physical quantities is given by
  $\mathrm{\Delta }V=\frac{1}{B}\frac{F}{A}{V}_0,$

  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.