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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.

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-static (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.

Elastic Potential Energy in Mechanical Systems

In the case of a spring of natural length l and modulus of elasticity λ under an extension of x, elastic potential energy can be calculated using the formula:

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

This formula is obtained from the integral of Hooke's Law:

$U_e = -\int\vec{F}\cdot d\vec{x}=-\int {-k x}\, dx = \frac {1} {2} k x^2.$

In the general case, elastic energy is given by the Helmholtz potential per unit of volume f as a function of the strain tensor components ε_ij:

$f(\epsilon_{ij}) = \lambda \left ( \sum_{i=1}^{3} \epsilon_{ii}\right)^2+2\mu \sum_{i=1}^{3} \sum_{j=1}^{3} \epsilon_{ij}^2$

where λ and μ are the Lamé elastical coefficients. The connection between stress tensor components and strain tensor components is:

$\sigma_{ij} = \left ( \frac{\partial f}{\partial \epsilon_{ij}} \right)_S.$

For a material of Young's modulus, Y (same as modulus of elasticity λ), cross sectional area, A₀, initial length, l₀, which is stretched by a length, $Δ l$ :

$U_e = \int {\frac{Y A_0 \Delta l} {l_0}}\, dl = \frac {Y A_0 {\Delta l}^2} {2 l_0}$

where U_e is the elastic potential energy.

The elastic potential energy per unit volume is given by:

$\frac{U_e} {A_0 l_0} = \frac {Y {\Delta l}^2} {2 l_0^2} = \frac {1} {2} Y {\varepsilon}^2$

where $\varepsilon = \frac {\Delta l} {l_0}$ is the strain in the material.

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:

$E = \frac{1}{2}C_{jikl}u_{ji}u_{lk}$ ,

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

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