Solid Mechanics: Hooke's Law 
Shear Modulus from Pure Shear 
When a specimen made from an isotropic
material is subjected to pure shear, for instance, a cylindrical bar
under torsion in the xy sense, s_{xy} is the only nonzero
stress. The strains in the specimen are obtained by,
The shear modulus G, is defined as the ratio of shear stress to engineering shear strain on the loading plane,
where . The shear modulus G is also known as the rigidity modulus, and is equivalent to the 2nd Lamé constant m mentioned in books on continuum theory. Common sense and the 2nd Law of Thermodynamics require that a positive shear stress leads to a positive shear strain. Therefore, the shear modulus G is required to be nonnegative for all materials,
G > 0
Since both G and the elastic modulus E are required to be positive, the quantity in the denominator of G must also be positive. This requirement places a lower bound restriction on the range for Poisson's ratio,
n > 1 
Bulk Modulus from Hydrostatic Pressure 
When an isotropic
material specimen is subjected to hydrostatic pressure s, all shear stress will be zero and the normal
stress will be uniform, . The strains in the specimen are given by,
In response to the hydrostatic load, the specimen will change its volume. Its resistance to do so is quantified as the bulk modulus K, also known as the modulus of compression. Technically, K is defined as the ratio of hydrostatic pressure to the relative volume change (which is related to the direct strains),
Common sense and the 2nd Law of Thermodynamics require that a positive hydrostatic load leads to a positive volume change. Therefore, the bulk modulus K is required to be nonnegative for all materials,
K > 0
Since both K and the elastic modulus E are required to be positive, the following requirement is placed on the upper bound of Poisson's ratio by the denominator of K,
n < 1/2 
Relation Between Relative Volume Change and Strain 
For simplicity, consider a rectangular block of
material with dimensions a_{0}, b_{0},
and c_{0}. Its volume V_{0} is given
by,
When the block is loaded by stress, its volume will change since each dimension now includes a direct strain measure. To calculate the volume when loaded V_{f}, we multiply the new dimensions of the block,
Products of strain measures will be much smaller than individual strain measures when the overall strain in the block is small (i.e. linear strain theory). Therefore, we were able to drop the strain products in the equation above. The relative change in volume is found by dividing the volume difference by the initial volume,
Hence, the relative volume change (for small strains) is equal to the sum of the 3 direct strains. 
Copyright © efunda.com
