Prove G = E/2(1+V) – Homework+2+Answer+Key

Relationship Between Moduli of Elasticity

For all isotropic polymeric materials where Hook”s law is valid, simple relationships exist between the elastic constants such as Young”s modulus E, shear modulus G, bulk modulus B, and Poisson”s ratio ν) As long as two of them are known, all others can be predicted. There are many different elastic constants from which we can choose. Often the relation between two moduli is expressed as a function of the Poisson ratio:

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From these relations it follows that −1 1/5 1 For isotropic weakly compressible materials such as liquids and rubbers, the Poisson”s ratio approaches the upper bound ν = 1/2. In that case the elastic tensile modulus is three time the shear modulus and the bulk modulus is much larger than the Young”s modulus,

E ≈ 3 G and B »E

The Poisson”s ratio of rubber is never exact 0.5 because this means the elastic bulk modulus would go to infinity which is not possible. However, values very close to 1/2 have been reported (0.4999)1. In the case of quasi-isotropic polycrystalline polymers and other rather stiff isotropic polymers like polystyrene (polymers with bulky side groups), the Poisson”s ratio reaches a value of about ν ≈ 1/3.

E ≈ 8/3 G and EB

For most engineering plastics, ν is in the range 0.35 table of Poisson”s ratio”s).

For a (hypothetical) material with no lateral contraction, ν = 0, the elastic tensile modulus is about two times the shear modulus and three times the bulk modulus:

E ≈ 2 G = 3 B

However, most common plastics have much larger Poisson”s ratios, ν > 0.3.

References

P. H. Mott and C. M. Roland, Physical Review B 80, 132104 (2009).

M.D. Lechner, K. Gehrke, E.H. Nordmeier, Makromolekulare Chemie, 1993

L.H. Sperling, Introduction to Physical Polymer Science, New York 1992

Summary

For isotropic materials and for deformations where Hook”s law applies, the elastic constants and the Poisson”s ratio are related. If two elastic constants are known, the others can be calculated from simple relationships.For isotropic (nearly) incompressible materials such as liquids and elastomers, the Poisson”s ratio approaches the value ν = 0.5. For these materials, the elastic tensile modulus is about three times the shear modulus.For materials with no lateral contraction (ν = 0) the elastic tensile modulus is about two times the shear modulus and three times the bulk modulus.

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