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Rate of change of volume and poisson's ratio

  1. Oct 26, 2009 #1

    NDO

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    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. jcsd
  3. Oct 26, 2009 #2

    lanedance

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    Homework Helper

    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: Oct 26, 2009
  4. Oct 26, 2009 #3

    NDO

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    I am still unsure as to how i can relate this to Young's modulus E, shear modulus G
     
  5. Oct 26, 2009 #4

    lanedance

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    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
     
  6. Oct 26, 2009 #5

    lanedance

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    cleaned up original post for clarity
     
  7. Oct 27, 2009 #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.
     
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