Rate of change of volume and poisson's ratio

In summary, Poisson's ratio affects the change in volume as a function of stress in the x, y and z directions.
  • #1
NDO
8
0

Homework Statement



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.


Homework Equations



Poisson's ratio = - Transverse strain / Axial strain

E = dl/L

The Attempt at a Solution



can the following formula be used G = E/(2(1+v)) i don't 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
 
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  • #2
so say, where K is some constant

[tex] V(x,y,z) = Kxyz [/tex]
where x,y,z, represent the linear dimensions of the object

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

[tex] dV = Kyz(dx) + Kxz(dy) + Kxy(dz)[/tex]

now try dividing through by the volume to get dV/V... and what is dx/x?
 
Last edited:
  • #3
I am still unsure as to how i can relate this to Young's modulus E, shear modulus G
 
  • #4
NDO said:

Homework Statement



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.

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

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

NDO said:
can the following formula be used G = E/(2(1+v)) i don't know whether v is poisson's ratio or what it is?

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
cleaned up original post for clarity
 
  • #6
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 = [tex]\frac{1}{E}[/tex][stressX - Vpoisson(stressY+stressZ)]

and the similar for the other 2 directions

not sure this could be use in ur question.
 

1. What is the rate of change of volume?

The rate of change of volume is a measure of how quickly the volume of an object changes in relation to a change in its dimensions. It is represented by the symbol ∇V/∇t, where ∇V is the change in volume and ∇t is the change in time.

2. How is the rate of change of volume calculated?

The rate of change of volume can be calculated by dividing the change in volume by the change in time. This can be represented by the formula ∇V/∇t = (V2 - V1)/(t2 - t1), where V2 and V1 are the final and initial volumes respectively, and t2 and t1 are the final and initial times respectively.

3. What is Poisson's ratio?

Poisson's ratio is a measure of the ratio of lateral strain to axial strain in a material. It is represented by the symbol ν and is typically between -1 and 0.5. It describes how a material will deform under stress, specifically how it will change in width compared to its length.

4. How is Poisson's ratio related to the rate of change of volume?

Poisson's ratio is related to the rate of change of volume through the formula ν = -∇W/∇L, where ∇W is the change in width and ∇L is the change in length. This means that a material's Poisson's ratio is equal to the negative of its rate of change of volume divided by its initial volume.

5. What are the practical applications of understanding rate of change of volume and Poisson's ratio?

Understanding the rate of change of volume and Poisson's ratio is crucial in engineering and material science. It can help in predicting how a material will behave under different stress conditions and inform design decisions for structures and products. It is also important in fields such as geology and seismology, as it can help in understanding the behavior of rocks and other natural materials under pressure and stress.

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