# Relation between Youngs modulus $Y$ and Bulks modulus $B$ for a cube

lets suppose a cube of side $$l$$, Young's modulus $$Y$$, bulk's modulus $$K$$ under a force F across all sides. so $$Y=\frac{F*l}{\Delta l*l^2}$$ now $$\Delta v=l^3-(l-\Delta l)^3$$ now ignoring powers of $$\Delta l$$ as they are too small $$\Delta v=3l^2\Delta l$$ so $$\frac{\Delta v}{v}=3\frac{\Delta l}{l}$$ so bulks modulus is $$K=\frac{F*l}{3\Delta l*6l^2}$$ that is $$18K=Y$$ does this hold true for a cube? also i dont know what tag does this question come under,can someone please refer which tag should i use for this?

i am a high schooler just playing with what i know so can someone please tell me where i am going wrong??

• You might like a note I’m working on that derives the elastic moduli (and the relations between them). Commented Jan 29, 2022 at 16:37

The first is the rogue factor of 6 that appears in your penultimate line. Although a cube has 6 faces, $$F/A$$ is the same for all of them, so the bulk stress has the same value as the longitudinal stress.
The other thing is something that you probably haven't been taught: the Poisson effect! When a cuboid of metal (say) is stretched it contracts in the transverse directions and when it is compressed longitudinally it expands in the transverse directions. The Poisson ratio, $$\nu$$ is defined by $$\nu=-\frac{\text{transverse strain}}{\text{longitudinal strain}}$$ A value of $$\nu=\frac 14$$ would be typical for a metal. There is a pretty simple relationship between $$Y, K$$ and $$\nu$$. You might like to see if you can derive it!
• (a) I'd stick with a cube. Go for a cuboid if you'd be more comfortable. That's surely general enough. You can consider any shape of solid to be built from small cubes. (b) Start by calculating the volume change for stretching along one axis (𝑥, say), taking account of the increase of length along that axis and the decrease in the two transverse directions. Work to first order in $\Delta 𝐿/L$ as you have been doing. Then repeat for stretches along the 𝑦 and 𝑧 axes remembering that the effects on the volume strain will be additive. Good luck! Commented Jan 29, 2022 at 15:52