Mechanics of deformable solids Definition

Bulk modulus

Definition

Consider a substance whose equilibrium pressure $p$ is a differentiable function $p(V)$ of its volume $V > 0$ along a given process, for example at constant temperature. The bulk modulus of the substance at volume $V$ is $K = -V \frac{dp}{dV},$ where $dp/dV$ is the derivative of $p$ . For an isotropic linear elastic solid with Lamé parameters $\lambda$ and $\mu$ , a hydrostatic stress $\sigma = -p I$ produces the volume strain $\tr \varepsilon = -p / K$ with $K = \lambda + \tfrac23 \mu.$

Remarks

Example: an ideal gas at constant temperature has $p V = c$ for a constant $c > 0$ , so $dp/dV = -c/V^2$ and $K = c/V = p$ .

Non-example: for an incompressible material the volume does not change under any pressure. So $p$ is not a function of $V$ , and the definition does not apply; one says informally that $K = \infty$ .

For the elastic solid, $\tr \sigma = (3\lambda + 2\mu) \tr \varepsilon$ , and $\tr \sigma = -3p$ gives $\tr \varepsilon = -3p/(3\lambda + 2\mu) = -p/K$ . In terms of Young's modulus $E$ and Poisson's ratio $\nu$ , $K = E / (3(1 - 2\nu))$ , which tends to infinity as $\nu \to 1/2$ . The isothermal and adiabatic bulk moduli of a gas differ; for an ideal gas the adiabatic one is $\gamma p$ , where $\gamma$ is the heat capacity ratio.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Derivative concept
Lamé parameters concept
Bulk modulus definition

Tag	`WTDGF7AZ35`
URI	`m2://WTDGF7AZ35`
Kind	definition
Writing	W0 stub
MSC	74B05, 74A20
Defines	Bulk modulus
Uses	Lamé parameters, Derivative
Changed	3 November 2026

Bulk modulus

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here