Elastic Energy - Continuum Systems

Continuum Systems

A bulk material can be distorted in many different ways: stretching, shearing, bending, twisting, etc. Each kind of distortion contributes to the elastic energy of a deformed material. In orthogonal coordinates, the elastic energy per unit volume due to strain is thus a sum of contributions:

,

where is a 4th rank tensor, called the elastic, or sometimes stiffness, tensor which is a generalization of the elastic moduli of mechanical systems, and is the strain tensor (Einstein summation notation has been used to imply summation over repeated indices). The values of depend upon the crystal structure of the material. For an isotropic material, where and are the Lamé constants, and is the Kronecker delta.

The strain tensor itself can be defined to reflect distortion in any way that results in invariance under total rotation, but the most common definition which regard to which elastic tensors are usually expressed defines strain as the symmetric part of the gradient of displacement with all nonlinear terms suppressed:

where is the displacement at a point in the direction and is the partial derivative in the direction. Note that:

where no summation is intended. Although full Einstein notation sums over raised and lowered pairs of indices, the values of elastic and strain tensor components are usually expressed with all indices lowered. Thus beware (as here) that in some contexts a repeated index does not imply a sum over values of that index ( in this case), but merely a single component of a tensor.