# Stresses and strain : change of volume

I was reading about stresses and strains and I came across the concept of volumetric strain and am now having a slight conceptual difficulty. I used to believe that despite changes in linear and lateral dimensions due to stresses, volume remains constant. Could someone please clarify if the body's volume really does increasing and how does this physically occur ?

• Have you heard of Poisson's ratio? – JMac Mar 12 '17 at 14:32
• " I used to believe that despite changes in linear and lateral dimensions due to stresses, volume remains constant" That doesn't make sense, unless you believe that all solids and liquids are incompressible. – alephzero Mar 12 '17 at 14:33
• Well....ok. But how does this happen on a molecular level? – SaitamaSensei Mar 12 '17 at 15:30
• Very slight changes in molecular spacing due to the stress that add up. They are stressed, why would they keep the exact same f0rmation? It's also not really appropriate to consider this as a system of molecules when dealing with stress and strain. They deal with continuum mechanics, which deals with non-discrete masses. Stress and strain aren't really properties that work on a molecular level. – JMac Mar 12 '17 at 15:50
• I have read about poisson's ratio and I guess that was the basis of my intuition since we generally see objects that show some form of positive poisson's ratio. Are objects with zero poisson's ratio relatively rare? Since we usually ignore the factoring poisson effect (Atleast they do not make it clear that they are ignoring poisson effect when expositing the definitions ) when defining tensile strain I was wondering if this is an abstraction or what.. – SaitamaSensei Mar 12 '17 at 21:34

## 1 Answer

You can break down the elastic properties to two effects. $G$ the shear modulus and $K$ the bulk modulus. There is a great conversion table at wikipedia about it.

Given the elastic modulus $E$ and Poisson's ratio $\nu$ the above properties are \begin{align} G & = \frac{E}{2(1+\nu)} & K & = \frac{E}{3(1-2\nu)} \end{align}

Now the shear modulus $G$ is the resistance to shear deformation (which preserves volume) and bulk modulus $K$ is the resistance to volume changes due to hydrfostatic pressures (all stress components are equal). To find a material with no volumetric changes under any loading, you have to find $K=\infty$ or $\nu=0.5$. Any material with $\nu<0.5$ will have a positive bult modulus meaning its volume will actually reduce on comrpessive stresses.

The definion of bulk modulus is

$$K = -V \frac{{\rm d} P}{{\rm d} V}$$ where $P$ is the hydrostatic pressure and $V>0$ the volume.

A positive $K$ means ${\rm d}V = -\frac{V}{K} {\rm d}P$ or the volume descreases with pressure (like being underwater).