Solid Mechanics: Hooke's Law 
Youngs Modulus from Uniaxial Tension 
When a specimen made from an isotropic
material is subjected to uniaxial tension, say in the x
direction, s_{xx} is the
only nonzero 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 nonnegative for all materials,
E > 0 
Poisson's Ratio from Uniaxial Tension 
A rodlike 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 axiallyloaded 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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