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.

1 Thermodynamics
2 Mechanics
3 Continuum Systems
4 Potential energy

[edit] 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

$d U = d W = - P d V$

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

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

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.

[edit] 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:

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

where $C i j k l$ is a 4th rank tensor of the elastic constants and $u i j$ is the strain tensor (we use Einstein summation notation). The values of $C i j k l$ depend upon the crystal structure of the material. For an isotropic material, $C i j k l = λδ i j δ k l + μ(δ i k δ j l + δ i l δ j k)$ , where $λ$ and $μ$ are the Lamé constants, and $δ i j$ is the Kronecker delta.

[edit] Potential energy

The elastic potential energy is defined as a work (force x distance) needed to compress or expand an elastic body. The potential energy of a string or spring of length l that has modulus of elasticity λ under an extension of x is then

$E = \frac{\lambda x^2}{2l}$