I noticed that the bulk modulus often has a negative sign appended to it in order to cancel with a negative change in an object's volume; this allows the bulk modulus to be positive.
However, I noticed that the compression modulus, or Young's Modulus but with respect to compressive forces, does not have a negative sign attached (at least according to what I have seen). Since the change in length of an object is negative when compressive stress occurs, I would assume that, without a negative sign attached to the equation to cancel the negative change in length, the modulus would result in a negative value.
Is there a physical reason for this, if any? What would it be?
Best Answer
An increase of the hydrostatic pressure $p$ requires a volume reduction $\Delta V/V_0$,
$ p = K \Delta V/V_0 $.
Therefore, in this definition, $ K$ must be negative. From a more general perspective, the pressure is the negative trace of the stress matrix $\sigma$ (over 3 in three dimensions), but the volume change is the positive trace of the strain matrix $\varepsilon$ (over 3 in three dimensions). Thus, we have
$trace(\sigma)=-K\ trace(\varepsilon)$.
We take $C:=-K$ as the new definition of the compression modulus, where $C$ is positive.
$C$ is an eigenvalue of the elasticity matrix, and these have to be positive. Otherwise, there may be wave propagation with complex propagation velocities, and the material is unstable and would decompose into phases. It is a quite complicated topic, you need to study elasticity. But maybe this thought experiment is sufficient:
Imagine that a volume decrease causes a negative pressure in the material. Hence, the surrounding would compress the sample even more, reducing the volume, reducing the pressure etc...