# Bulk modulus -- solids

#### Prerequisites

From our discussions of phenomenological macroscopic forces like normal forces, tension, and friction, we expect that solids will deform a little if we exert a force on it. (We also expect this from our understanding of interatomic forces.) Let's consider how we might describe how a solid would respond to being squeezed.

Conceptually, the simplest situation for seeing how a solid might deform in response to forces is to consider a rectangular block of some solid material and push on it on all sides. We can do this by immersing it in a fluid — air or water — and increasing the pressure in the fluid. This will push on every surface of the object with a normal force.

How should we think about such forces? If the sides are not the same size the forces on each side could be different. How do we expect the block to respond? From our understanding of the structure of solids, we expect it to squeeze down. How much?

We are interested in extracting a parameter that tells how a solid object responds to forces that is a property of the material in general — not that of the specific object. Sort of like the way density tells us about the mass of a material generically — where the mass (density times volume) tells us about a specific object. How can we extract the object independent property of the material?

A more natural way to think about what might be an appropriate way to exert forces on a solid it to think of a few molecules in the surface of the solid.

If the pressure pushing on all of the surfaces of a block increases, the block will compress and take up less volume as in the figure. This change in volume might be very small if the atoms or molecules making up the block push against each other and resist the change. We describe the change in volume in response to a uniform increase in pressure as the *bulk modulus*, $B$ in a way analogous to our stress/strain relation in considering Young's modulus: change in shape (stress) is proportional to the force per unit area (strain). Here, this becomes

$$\Delta P = B \frac{\Delta V}{V}$$

or

$$B = \frac{\Delta P}{{\Delta V} / V}$$

Here, $ΔP$ is the change in pressure necessary to cause a fractional change in volume $ΔV/V$, where $ΔV$ is the volume change, and $V$ is the original volume.

The more resistant a solid is to compression, the more pressure it takes to change the volume, and the larger the bulk modulus is. Since $ΔV/V$ has units of volume/volume, the denominator is unitless. Therefore, $B$ has units of pressure, e.g. Pa = N/m^{2}. Since the pressure is pushing on each of the surfaces, the block will shrink along all three dimensions.

The bulk modulus of solid materials can be quite large. For steel, B= 1.60x10^{11} Pa. This means that to get a volume change of 0.1% requires a pressure increase of 1.6x10^{8} Pa which is 1580 times atmospheric pressure (1 atm ~ 10^{5} Pa). Obviously, not much happens under typical atmospheric conditions.

The bulk modulus is important for another reason, as it tells how easy it is for a pressure wave to move through the solid. Since sound is essentially a pressure wave (alternating higher and lower pressure) $B$ will be important in how well sound is transmitted through different materials.

The bulk modulus for biological materials is generally relatively high, though not as high as engineering materials such as steel (160 x 10^{9} Pa). For bone, the bulk modulus is 15 x 10^{9}Pa. The bulk modulus will determine how much a body changes volume in response to a change in pressure. Because the size of a body is mostly determined by the size of its bones, we can consider whether bones change size in response to different pressures.

Organisms do encounter a range of pressures on earth. For example, air pressure changes with altitude. In Denver (the Mile high city) air pressures is 17% less than at sea level. Air pressure is 49% lower on the top of Denali in Alaska, and 65% on the top of Mt Everest. Animals that fly very high will also encounter lower pressures. These pressure decreases could result in a small increase in the volume of bones in the body. While these are unlikely to be large, they are not insignificant.

Even larger pressure changes are encountered when animals swim to depth in the ocean. For example, sperm whales dive to 400 m depths where the absolute pressure is 40 times greater than at the surface. Elephant seals may dive up to a mile down (1600 m) where the pressure is 160 times that at the surface. It is possible that bone volume shrinks a bit in response to this increase in pressure.

Karen Carleton and Joe Redish 10/27/11

#### Follow-on

Last Modified: July 11, 2021