Elastic energy

The elastic energy is the energy which causes or is released by the elastic distortion of a solid or a fluid.

Thermodynamics

Elastic energy is internal energy (U) that can be converted into mechanical energy (work) under adiabatic conditions.

The elastic energy can be defined in differential form as

$dU=dW=-PdV$

where P is the external pressure, equal to the internal pressure as the process is quasi-estatic (reversible), and V is the volume of the gas. The minus sign appears as the external pressure exerts a force contrary to the expansion. In Thermodynamics the work that is carried out by a gas (in general by a system) is negative, whilst the work exerted over a system is positive.

Mechanics

For a spring the elastic energy is

$E={\frac {1}{2}}kx^{2}$

where k is the elastic constant of the spring (see Hooke's law) and x is the elongation of the spring. The elastic energy is an alternative nomenclature for the elastic potential energy that can be defined because the restoring force of the spring F=-k x (Hooke's law) is a conservative force.

Continuum Systems

A bulk material can be distorted in many different ways: stretching, shearing, beading, twisting, etc. Each way contributes its own amount of elastic energy to the material. Thus, the total elastic energy is a sum each contribute:

$E={\frac {1}{2}}C_{ijkl}u_{ij}u_{lk}$,

where $C_{ijkl}$ is a 4th rank tensor of the elastic constants and $u_{ij}$ is the strain tensor (we use Einstein summation notation). The values of $C_{ijkl}$ depend upon the crystal structure of the material. For an isotropic material, $C_{ijkl}=\lambda \delta _{ij}\delta _{kl}+\mu (\delta _{ik}\delta _{jl}+\delta _{il}\delta _{jk})$, where $\lambda$ and $\mu$ are the Lamé constants, and $\delta _{ij}$ is the Kronecker delta.