Elastic energy
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The elastic energy is the energy which causes or is released by the elastic distortion of a solid or a fluid.
[edit] Thermodynamics
The elastic energy is internal energy (U) that can be converted into mechanical energy (work) under adiabatic conditions by virtue of spontaneous changes in volume or shape of a body. Elastic distortion is characterised by reversibility. As the process is adiabatic, heat=0 and using the first law of thermodynamics, for a gas 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.
[edit] Mechanics
For a spring the elastic energy is
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.
[edit] 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:
,
where Cijkl is a 4th rank tensor of the elastic constants and uij is the strain tensor (we use Einstein summation notation). The values of Cijkl depend upon the crystal structure of the material. For an isotropic material, Cijkl = λδijδkl + μ(δikδjl + δilδjk), where λ and μ are the Lamé constants, and δij is the Kronecker delta.
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