The modulus of elasticity (Young’s modulus) is the lower limit of the energy density

Consider a linear elastic body with density $\rho$ and Young’s modulus $E$ . The speed of sound through this body is

$v = \sqrt{\dfrac{E}{\rho}}$

Letting $c$ be the speed of light in a vacuum, the theory of Special Relativity requires that $v < c$ , or

$\dfrac{E}{\rho}<c^2$

The density of a homogeneous body is the ratio between its mass and volume: $\rho = \dfrac{M}{V}$ . In turn, the mass is $M = \dfrac{E_{en}}{c^2}$ , where $E_{en}$ is the energy corresponding to the mass $M$ .

The following inequality holds:

$E<\rho c^2 = \dfrac{M}{V}c^2 = \dfrac{E_{en}}{V c^2}c^2$

that is

$E< \dfrac{E_{en}}{V} = e$

where we have indicated with $e$ the energy density.

The previous inequality justifies the statement in the title.
It can be deduced that, within the limits of linear approximations, there is a maximum stiffness associated with energy density.

Conversely, there is a minimum energy density associated to elastic stiffness.

The modulus of elasticity (Young’s modulus) is the lower limit of the energy density

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