# Rate of change of volume and poisson's ratio

1. Oct 26, 2009

### NDO

1. The problem statement, all variables and given/known data

Consider a rectangular block of isotropic material of dimensions a, b and c, with c >> a
or b. It is characterised by its elastic constants: Young's modulus E, shear modulus G
and Poisson's ratio .
The block of material is subjected to axial deformation along the c dimension.

1. Derive an expression for the relative change in volume, change in V/
V , in term of Poisson's ratio.
2. Make a plot of the relative change in volume, change inV/ V , as a function of Poisson's
ratio varying from 0 to 0.5.

2. Relevant equations

Poisson's ratio = - Transverse strain / Axial strain

E = dl/L

3. The attempt at a solution

can the following formula be used G = E/(2(1+v)) i dont know whether v is poisson's ratio or what it is?

assuming the axial load is acting through c

the cross sectional area would be a*b

any help would be great especially if u can help me link poisson's ratio with G and E or explain why i would be required to use change in volume instead of length

cheers NDO

2. Oct 26, 2009

### lanedance

so say, where K is some constant

$$V(x,y,z) = Kxyz$$
where x,y,z, represent the linear dimensions of the object

independent small changesdenoted by dx, dy, dz gives (using partial differntiation)

$$dV = Kyz(dx) + Kxz(dy) + Kxy(dz)$$

now try dividing through by the volume to get dV/V... and what is dx/x?

Last edited: Oct 26, 2009
3. Oct 26, 2009

### NDO

I am still unsure as to how i can relate this to Young's modulus E, shear modulus G

4. Oct 26, 2009

### lanedance

I don't think the question asks for that...

though if you follow the steps given previously it should be possible anyway

the v in that equation does represent poisson's ratio, have a look at the following

http://www.efunda.com/formulae/solid_mechanics/mat_mechanics/elastic_constants_G_K.cfm

5. Oct 26, 2009

### lanedance

cleaned up original post for clarity

6. Oct 27, 2009

### mapril

for isotropic material,

the deformation of a material in one direction will produce a deformation of material along the other axis in 3 dimensions.
so,

strain in x direction = $$\frac{1}{E}$$[stressX - Vpoisson(stressY+stressZ)]

and the similar for the other 2 directions

not sure this could be use in ur question.