Elastic energy occurs when objects are impermanently compressed, stretched or generally deformed in any manner.
Elasticity theory primarily develops formalisms for the mechanics of solid bodies and materials.
[1] (Note however, the work done by a stretched rubber band is not an example of elastic energy.
The elastic potential energy equation is used in calculations of positions of mechanical equilibrium.
However, all materials have limits to the degree of distortion they can endure without breaking or irreversibly altering their internal structure.
Hence, the characterizations of solid materials include specification, usually in terms of strains, of its elastic limits.
It corresponds to energy stored principally by changing the interatomic distances between nuclei.
For example, for some solid objects, twisting, bending, and other distortions may generate thermal energy, causing the material's temperature to rise.
Thermal energy in solids is often carried by internal elastic waves, called phonons.
Elastic waves that are large on the scale of an isolated object usually produce macroscopic vibrations .
Although elasticity is most commonly associated with the mechanics of solid bodies or materials, even the early literature on classical thermodynamics defines and uses "elasticity of a fluid" in ways compatible with the broad definition provided in the Introduction above.
Solids include complex crystalline materials with sometimes complicated behavior.
By contrast, the behavior of compressible fluids, and especially gases, demonstrates the essence of elastic energy with negligible complication.
The minus sign appears because dV is negative under compression by a positive applied pressure which also increases the internal energy.
Upon reversal, the work that is done by a system is the negative of the change in its internal energy corresponding to the positive dV of an increasing volume.
The system loses stored internal energy when doing work on its surroundings.
The quantity of energy transferred is the vector dot product of the force and the displacement of the object.
As forces are applied to the system they are distributed internally to its component parts.
This constant is usually denoted as k (see also Hooke's Law) and depends on the geometry, cross-sectional area, undeformed length and nature of the material from which the coil is fashioned.
This requires the assumption, sufficiently correct in most circumstances, that at a given moment, the magnitude of applied force.
The total elastic energy placed into the spring from zero displacement to final length L is thus the integral
where λ and μ are the Lamé elastic coefficients and we use Einstein summation convention.
where the subscript T denotes that temperature is held constant, then we find that if Hooke's law is valid, we can write the elastic energy density as
Matter in bulk can be distorted in many different ways: stretching, shearing, bending, twisting, etc.
Each kind of distortion contributes to the elastic energy of a deformed material.
In orthogonal coordinates, the elastic energy per unit volume due to strain is thus a sum of contributions:
depend upon the crystal structure of the material: in the general case, due to symmetric nature of
{\displaystyle C_{ijkl}=\lambda \delta _{ij}\delta _{kl}+\mu \left(\delta _{ik}\delta _{jl}+\delta _{il}\delta _{jk}\right)}
The strain tensor itself can be defined to reflect distortion in any way that results in invariance under total rotation, but the most common definition with regard to which elastic tensors are usually expressed defines strain as the symmetric part of the gradient of displacement with all nonlinear terms suppressed:
Although full Einstein notation sums over raised and lowered pairs of indices, the values of elastic and strain tensor components are usually expressed with all indices lowered.