Solid Mechanics: Hooke's Law
Finding Young's Modulus and Poisson's Ratio

Youngs Modulus from Uniaxial Tension
When a specimen made from an isotropic material is subjected to uniaxial tension, say in the x direction, sxx is the only non-zero stress. The strains in the specimen are obtained by,

The modulus of elasticity in tension, also known as Young's modulus E, is the ratio of stress to strain on the loading plane along the loading direction,

Common sense (and the 2nd Law of Thermodynamics) indicates that a material under uniaxial tension must elongate in length. Therefore the Young's modulus E is required to be non-negative for all materials,

E > 0
Poisson's Ratio from Uniaxial Tension
A rod-like specimen subjected to uniaxial tension will exhibit some shrinkage in the lateral direction for most materials. The ratio of lateral strain and axial strain is defined as Poisson's ratio n,

The Poisson ratio for most metals falls between 0.25 to 0.35. Rubber has a Poisson ratio close to 0.5 and is therefore almost incompressible. Theoretical materials with a Poisson ratio of exactly 0.5 are truly incompressible, since the sum of all their strains leads to a zero volume change. Cork, on the other hand, has a Poisson ratio close to zero. This makes cork function well as a bottle stopper, since an axially-loaded cork will not swell laterally to resist bottle insertion.

The Poisson's ratio is bounded by two theoretical limits: it must be greater than -1, and less than or equal to 0.5,

The proof for this stems from the fact that E, G, and K are all positive and mutually dependent. However, it is rare to encounter engineering materials with negative Poisson ratios. Most materials will fall in the range,




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