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Elastic energy is the mechanical potential energy stored in the configuration of a material or physical system as it is subjected to elastic deformation by work performed upon it. 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. It is an example of entropic elasticity.) The elastic potential energy equation is used in calculations of positions of mechanical equilibrium. The energy is potential as it will be converted into other forms of energy, such as kinetic energy and sound energy, when the object is allowed to return to its original shape (reformation) by its elasticity.
The essence of elasticity is reversibility. Forces applied to an elastic material transfer energy into the material which, upon yielding that energy to its surroundings, can recover its original shape. 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. Beyond the elastic limit, a material is no longer storing all of the energy from mechanical work performed on it in the form of elastic energy.
Elastic energy of or within a substance is static energy of configuration. It corresponds to energy stored principally by changing the interatomic distances between nuclei. Thermal energy is the randomized distribution of kinetic energy within the material, resulting in statistical fluctuations of the material about the equilibrium configuration. There is some interaction, however. 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. [2] : 107 et seq.
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 simple thermodynamic formula: where dU is an infinitesimal change in recoverable internal energy U, P is the uniform pressure (a force per unit area) applied to the material sample of interest, and dV is the infinitesimal change in volume that corresponds to the change in internal energy. 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. In other words, the system loses stored internal energy when doing work on its surroundings. Pressure is stress and volumetric change corresponds to changing the relative spacing of points within the material. The stress-strain-internal energy relationship of the foregoing formula is repeated in formulations for elastic energy of solid materials with complicated crystalline structure.
Components of mechanical systems store elastic potential energy if they are deformed when forces are applied to the system. Energy is transferred to an object by work when an external force displaces or deforms the object. 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. While some of the energy transferred can end up stored as the kinetic energy of acquired velocity, the deformation of component objects results in stored elastic energy.
A prototypical elastic component is a coiled spring. The linear elastic performance of a spring is parametrized by a constant of proportionality, called the spring constant. 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. Within a certain range of deformation, k remains constant and is defined as the negative ratio of displacement to the magnitude of the restoring force produced by the spring at that displacement.
The deformed length, L, can be larger or smaller than Lo, the undeformed length, so to keep k positive, Fr must be given as a vector component of the restoring force whose sign is negative for L>Lo and positive for L< Lo. If the displacement is abbreviated as then Hooke's Law can be written in the usual form
Energy absorbed and held in the spring can be derived using Hooke's Law to compute the restoring force as a measure of the applied force. This requires the assumption, sufficiently correct in most circumstances, that at a given moment, the magnitude of applied force.
For each infinitesimal displacement dx, the applied force is simply k x and the product of these is the infinitesimal transfer of energy into the spring dU. The total elastic energy placed into the spring from zero displacement to final length L is thus the integral
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, : where Ue is the elastic potential energy.
The elastic potential energy per unit volume is given by: where is the strain in the material.
In the general case, elastic energy is given by the free energy per unit of volume f as a function of the strain tensor components εij where λ and μ are the Lamé elastic coefficients and we use Einstein summation convention. Noting the thermodynamic connection between stress tensor components and strain tensor components, [1] 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: where is a 4th rank tensor, called the elastic tensor or stiffness tensor [3] which is a generalization of the elastic moduli of mechanical systems, and is the strain tensor (Einstein summation notation has been used to imply summation over repeated indices). The values of depend upon the crystal structure of the material: in the general case, due to symmetric nature of and , the elastic tensor consists of 21 independent elastic coefficients. [4] This number can be further reduced by the symmetry of the material: 9 for an orthorhombic crystal, 5 for an hexagonal structure, and 3 for a cubic symmetry. [5] Finally, for an isotropic material, there are only two independent parameters, with , where and are the Lamé constants, and is the Kronecker delta.
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: where is the displacement at a point in the -th direction and is the partial derivative in the -th direction. Note that: where no summation is intended. 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. Thus beware (as here) that in some contexts a repeated index does not imply a sum overvalues of that index ( in this case), but merely a single component of a tensor.
In continuum mechanics, the infinitesimal strain theory is a mathematical approach to the description of the deformation of a solid body in which the displacements of the material particles are assumed to be much smaller than any relevant dimension of the body; so that its geometry and the constitutive properties of the material at each point of space can be assumed to be unchanged by the deformation.
Young's modulus is a mechanical property of solid materials that measures the tensile or compressive stiffness when the force is applied lengthwise. It is the modulus of elasticity for tension or axial compression. Young's modulus is defined as the ratio of the stress applied to the object and the resulting axial strain in the linear elastic region of the material.
In physics, Hooke's law is an empirical law which states that the force needed to extend or compress a spring by some distance scales linearly with respect to that distance—that is, Fs = kx, where k is a constant factor characteristic of the spring, and x is small compared to the total possible deformation of the spring. The law is named after 17th-century British physicist Robert Hooke. He first stated the law in 1676 as a Latin anagram. He published the solution of his anagram in 1678 as: ut tensio, sic vis. Hooke states in the 1678 work that he was aware of the law since 1660.
In engineering, deformation may be elastic or plastic. If the deformation is negligible, the object is said to be rigid.
In thermodynamics, the Helmholtz free energy is a thermodynamic potential that measures the useful work obtainable from a closed thermodynamic system at a constant temperature (isothermal). The change in the Helmholtz energy during a process is equal to the maximum amount of work that the system can perform in a thermodynamic process in which temperature is held constant. At constant temperature, the Helmholtz free energy is minimized at equilibrium.
Linear elasticity is a mathematical model of how solid objects deform and become internally stressed by prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mechanics.
In physics and engineering, a constitutive equation or constitutive relation is a relation between two or more physical quantities that is specific to a material or substance or field, and approximates its response to external stimuli, usually as applied fields or forces. They are combined with other equations governing physical laws to solve physical problems; for example in fluid mechanics the flow of a fluid in a pipe, in solid state physics the response of a crystal to an electric field, or in structural analysis, the connection between applied stresses or loads to strains or deformations.
In mechanics, virtual work arises in the application of the principle of least action to the study of forces and movement of a mechanical system. The work of a force acting on a particle as it moves along a displacement is different for different displacements. Among all the possible displacements that a particle may follow, called virtual displacements, one will minimize the action. This displacement is therefore the displacement followed by the particle according to the principle of least action.
The work of a force on a particle along a virtual displacement is known as the virtual work.
In continuum mechanics, the finite strain theory—also called large strain theory, or large deformation theory—deals with deformations in which strains and/or rotations are large enough to invalidate assumptions inherent in infinitesimal strain theory. In this case, the undeformed and deformed configurations of the continuum are significantly different, requiring a clear distinction between them. This is commonly the case with elastomers, plastically deforming materials and other fluids and biological soft tissue.
In thermodynamics, the internal energy of a system is expressed in terms of pairs of conjugate variables such as temperature and entropy, pressure and volume, or chemical potential and particle number. In fact, all thermodynamic potentials are expressed in terms of conjugate pairs. The product of two quantities that are conjugate has units of energy or sometimes power.
Betti's theorem, also known as Maxwell–Betti reciprocal work theorem, discovered by Enrico Betti in 1872, states that for a linear elastic structure subject to two sets of forces {Pi} i=1,...,n and {Qj}, j=1,2,...,n, the work done by the set P through the displacements produced by the set Q is equal to the work done by the set Q through the displacements produced by the set P. This theorem has applications in structural engineering where it is used to define influence lines and derive the boundary element method.
The elasticity tensor is a fourth-rank tensor describing the stress-strain relation in a linear elastic material. Other names are elastic modulus tensor and stiffness tensor. Common symbols include and .
In the mathematical theory of elasticity, Saint-Venant's compatibility condition defines the relationship between the strain and a displacement field by
In continuum mechanics, a compatible deformation tensor field in a body is that unique tensor field that is obtained when the body is subjected to a continuous, single-valued, displacement field. Compatibility is the study of the conditions under which such a displacement field can be guaranteed. Compatibility conditions are particular cases of integrability conditions and were first derived for linear elasticity by Barré de Saint-Venant in 1864 and proved rigorously by Beltrami in 1886.
The viscous stress tensor is a tensor used in continuum mechanics to model the part of the stress at a point within some material that can be attributed to the strain rate, the rate at which it is deforming around that point.
The acoustoelastic effect is how the sound velocities of an elastic material change if subjected to an initial static stress field. This is a non-linear effect of the constitutive relation between mechanical stress and finite strain in a material of continuous mass. In classical linear elasticity theory small deformations of most elastic materials can be described by a linear relation between the applied stress and the resulting strain. This relationship is commonly known as the generalised Hooke's law. The linear elastic theory involves second order elastic constants and yields constant longitudinal and shear sound velocities in an elastic material, not affected by an applied stress. The acoustoelastic effect on the other hand include higher order expansion of the constitutive relation between the applied stress and resulting strain, which yields longitudinal and shear sound velocities dependent of the stress state of the material. In the limit of an unstressed material the sound velocities of the linear elastic theory are reproduced.
Plasticity theory for rocks is concerned with the response of rocks to loads beyond the elastic limit. Historically, conventional wisdom has it that rock is brittle and fails by fracture while plasticity is identified with ductile materials. In field scale rock masses, structural discontinuities exist in the rock indicating that failure has taken place. Since the rock has not fallen apart, contrary to expectation of brittle behavior, clearly elasticity theory is not the last word.
In continuum mechanics, objective stress rates are time derivatives of stress that do not depend on the frame of reference. Many constitutive equations are designed in the form of a relation between a stress-rate and a strain-rate. The mechanical response of a material should not depend on the frame of reference. In other words, material constitutive equations should be frame-indifferent (objective). If the stress and strain measures are material quantities then objectivity is automatically satisfied. However, if the quantities are spatial, then the objectivity of the stress-rate is not guaranteed even if the strain-rate is objective.
In continuum mechanics, a hypoelastic material is an elastic material that has a constitutive model independent of finite strain measures except in the linearized case. Hypoelastic material models are distinct from hyperelastic material models in that, except under special circumstances, they cannot be derived from a strain energy density function.
The Fradkin tensor, or Jauch-Hill-Fradkin tensor, named after Josef-Maria Jauch and Edward Lee Hill and David M. Fradkin, is a conservation law used in the treatment of the isotropic multidimensional harmonic oscillator in classical mechanics. For the treatment of the quantum harmonic oscillator in quantum mechanics, it is replaced by the tensor-valued Fradkin operator.