Hooke's Law

The concepts of stress and strain are introduced to provide scalability between geometries for laboratory specimens and real-world applications. The generalized Hooke's law for relating stress and strain is given by  σij = Eijkl εkl where the 81 components of the fourth-order tensor Eijkl are known as the elastic constants.  For materials described by anisotropic, linear-elastic behavior, it can be shown that this tensor can be reduced to 21 distinct components. For orthotropic materials this number is reduced to nine components. For isotropic materials this number is further reduced to two components. Details

from Engineering Mechanics of Solids by E.P. Popov, Prentice Hall, 1991, p. 7

Assumptions

Linear - straight line relationship between load and deformation (or stress and strain)

Elastic - deformation is fully recoverable when load is removed

Homogeneous - material composition is the same throughout

Isotropic - material properties are the same in all directions

Elastic Constants

- the factors of proportionality (K, E, ν, and G) that relate elastic displacement of a material to applied forces

Modulus of Elasticity, E The measure of rigidity or stiffness of a metal; the ratio of stress, below the proportional limit, to the corresponding strain. Also known as Young's modulus.

Poisson's Ratio, ν The absolute value of the ratio of transverse (lateral) strain to the corresponding axial strain resulting from uniformly distributed axial stress below the proportional limit of the material.

Shear Modulus, G The ratio of shear stress to the corresponding shear strain for shear stresses below the proportional limit of the material. Also known as the modulus of rigidity.

Bulk Modulus, K The measure of resistance to change in volume. Also known as compression modulus, hydrostatic modulus, and volumetric modulus of elasticity.