Why poisson ratio of imcompressible substance is 0.5?
Why poisson ratio of imcompressible substance is 0.5?
An easy way of explaining this is to use the relationships between the Young's modulus EE, shear modulus GG, bulk modulus KK, and Poisson's Ratio νν.
E=2G(1+ν)=3K(1−2ν)E=2G(1+ν)=3K(1−2ν)
Now under conditions of simple tension and simple shear, all materials known experience displacements in the direction of force and increase in volume in case of hydrostatic loading. So E,GE,G and KK cannot be negative. So,
1+ν≥0⟹ν≥−11+ν≥0⟹ν≥−1
and
1−2ν≥0⟹ν≤121−2ν≥0⟹ν≤12
Combining the above,
−1≤ν≤12
OR
If you remember the relationship E=3K(1-2n), where E is young's modulus of elasticity, K is bulk modulus of elasticity and n is poission's ratio.
If value of poission's ratio is more than half, implies that value of young's modulus is negative which is not possible.
So value of poission's ratio is always less than 0.5
EE, shear modulus GG, bulk modulus KK, and Poisson's Ratio νν.
E=2G(1+ν)=3K(1−2ν)E=2G(1+ν)=3K(1−2ν)
Now under conditions of simple tension and simple shear, all materials known experience displacements in the direction of force and increase in volume in case of hydrostatic loading. So E,GE,G and KK cannot be negative. So,
1+ν≥0⟹ν≥−11+ν≥0⟹ν≥−1
and
1−2ν≥0⟹ν≤121−2ν≥0⟹ν≤12
Combining the above,
−1≤ν≤12
OR
If you remember the relationship E=3K(1-2n), where E is young's modulus of elasticity, K is bulk modulus of elasticity and n is poission's ratio.
If value of poission's ratio is more than half, implies that value of young's modulus is negative which is not possible.
So value of poission's ratio is always less than 0.5
