bulk-modulus-turns-volume-strain-into-pressure - Elasticity and Nonlinear Dynamics Step by Step

The bulk modulus K ties a pressure change to the volume strain it produces, Delta-P = -K * Delta-V / V, so squeezing a material raises its internal pressure. The arithmetic is exact only for the stated bulk modulus, pressure change, and volume strain in the ideal linear elastic model. Real materials add nonlinear response, porosity, temperature/loading history, geometry tolerance, calibration, and measurement/source uncertainty.

Bulk response uses positive pressure and volume-change magnitudes. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Bulk response starts from volume and volume change

The book uses positive compression magnitudes. Volume and volume change are checked before pressure is accepted.

V=6\ \text{m}^{3}\qquad \Delta V=1\ \text{m}^{3}

Pressure follows the volume-change ratio

Hold the starting volume and bulk modulus fixed. The pressure column follows the checked volume change.

\begin{array}{c|c|c|c}V&\Delta V&K&\Delta P\\6\ \text{m}^{3}&\tfrac{1}{2}\ \text{m}^{3}&12\ \text{Pa}&1\ \text{Pa}\\6\ \text{m}^{3}&1\ \text{m}^{3}&12\ \text{Pa}&2\ \text{Pa}\\6\ \text{m}^{3}&2\ \text{m}^{3}&12\ \text{Pa}&4\ \text{Pa}\\\end{array}

Bulk modulus turns volume strain into pressure change

This checked ledger uses positive compression magnitudes; sign conventions are outside this training surface.

\Delta P=K\frac{\Delta V}{V}=12\ \text{Pa}\cdot\frac{1\ \text{m}^{3}}{6\ \text{m}^{3}}=2\ \text{Pa}