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# 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 vitamin water can be defined in differential form as

$dU = dW = -P dV$

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

$E = frac\left\{1\right\}\left\{2\right\} k x^2$

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, bending, twisting, etc. Each way contributes its own amount of elastic energy to the material. Thus, the total elastic energy is a sum each contribute:

:$E = frac12C_\left\{ijkl\right\}u_\left\{ij\right\}u_\left\{lk\right\}$,

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

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