50.5: Volume Stress

The other types of stress described so far (longitudinal and transverse) deform a solid body, but do not change its volume. A third type of stress, the volume stress (or hydrostatic pressure), involves a change in volume. It typically occurs with the compression or expansion of a gas.

For a gas, the volume stress is just the gas pressure $P$. Pressure in SI units is measured in Pascals ( $\mathrm{Pa}$ ), named for the French mathematician and physicist Blaise Pascal. One pascal is equal to $1 \mathrm{~N} / \mathrm{m}^{2}$.

Strain

For volume stress, the strain is the fractional change in volume:

$$\varepsilon=-\Delta V / V_{0},$$

where $\Delta V=V-V_{0}$ is the change in volume, $V_{0}$ is the original (unstressed) volume and $V$ is the stressed volume. If the gas is compressed, then $\Delta V$ is negative and the strain $\varepsilon$ is positive; if the gas expands, then $\Delta V$ is positive and the strain $\varepsilon$ is negative.

Bulk Modulus

In the case of volume stress, the appropriate elastic modulus is the bulk modulus B. Since the elastic modulus is the ratio of the stress to the strain, we have

$$B=-\frac{P}{\Delta V / V_{0}}$$