Inphysics,elasticity(fromGreek "ductible") is the tendency of solid materials to return to their original shape after being deformed. Solid objects willdeformwhenforcesare applied on them. If the material is elastic, the object will return to its initial shape and size when these forces are removed.The physical reasons for elastic behavior can be quite different for different materials. In metals, the atomic lattice changes size and shape when forces are applied (energy is added to the system). When forces are removed, the lattice goes back to the original lower energy state. Forrubbersand other polymers, elasticity is caused by the stretching of polymer chains when forces are applied.Perfect elasticity is an approximation of the real world, and few materials remain purely elastic even after very small deformations. In engineering, the amount of elasticity of a material is determined by two types of material parameter. The first type of material parameter is called amodulus, which measures the amount of force per unit area (stress) needed to achieve a given amount of deformation. The units of modulus are pascals (Pa) or pounds of force per square inch (psi, also lbf/in2). A higher modulus typically indicates that the material is harder to deform. The second type of parameter measures theelastic limit. The limit can be a stress beyond which the material no longer behaves elastic and deformation of the material will take place. If the stress is released, the material will elastically return to a permanent deformed shape instead of the original shape.When describing the relative elasticities of two materials, both the modulus and the elastic limit have to be considered. Rubbers typically have a low modulus and tend to stretch a lot (that is, they have a high elastic limit) and so appear more elastic than metals (high modulus and low elastic limit) in everyday experience. Of two rubber materials with the same elastic limit, the one with a lower modulus will appear to be more elastic.Contents[hide] 1Overview 2Linear elasticity 3Finite elasticity 3.1Cauchy elastic materials 3.2Hypoelastic materials 3.3Hyperelastic materials 4Applications 5Factors affecting elasticity 6See also 7ReferencesOverview[edit]When an elastic material is deformed due to an external force, it experiences internal forces that oppose the deformation and restore it to its original state if the external force is no longer applied. There are variouselastic moduli, such asYoung's modulus, theshear modulus, and thebulk modulus, all of which are measures of the inherent stiffness of a material as a resistance to deformation under an applied load. The various moduli apply to different kinds of deformation. For instance, Young's modulus applies to uniform extension, whereas the shear modulus applies toshearing.The elasticity of materials is described by astress-strain curve, which shows the relation betweenstress(the average restorative internalforceper unitarea) andstrain(the relative deformation).[1]For most metals or crystalline materials, the curve is linear for small deformations, and so the stress-strain relationship can adequately be described byHooke's law, and higher-order terms can be ignored. However, for larger stresses beyond theelastic limit, the relation is no longer linear. For even higher stresses, materials exhibitplastic behavior, that is, they deform irreversibly and do not return to their original shape after stress is no longer applied.[2]Forrubber-like materials such aselastomers, thegradientof the stress-strain curve increases with stress, meaning that rubbers progressively become more difficult to stretch, while for mostmetals, the gradient decreases at very high stresses, meaning that they progressively become easier to stretch.[3]Elasticity is not exhibited only by solids;non-Newtonian fluids, such asviscoelastic fluids, will also exhibit elasticity in certain conditions. In response to a small, rapidly applied and removed strain, these fluids may deform and then return to their original shape. Under larger strains, or strains applied for longer periods of time, these fluids may start to flow like aviscousliquid.Because the elasticity of a material is described in terms of a stress-strain relation, it is essential that the termsstressandstrainbe defined without ambiguity. Typically, two types of relation are considered. The first type deals with materials that are elastic only for small strains. The second deals with materials that are not limited to small strains. Clearly, the second type of relation is more general.For small strains, the measure of stress that is used is theCauchy stresswhile the measure of strain that is used is theinfinitesimal strain tensor. The stress and strain measures are related by a linear relation known asHooke's law.Linear elasticitydescribes the behavior of such materials.Cauchy elastic materialsandHypoelastic materialsare models that extend Hooke's law to allow for the possibility of large rotations.For more general situations, any of a number ofstress measurescan be used provided they arework conjugateto an appropriatefinite strainmeasure, i.e., the product of the stress measure and the strain measure should be equal to the internal energy (which does not depend on how the stress or strain are measured).Hyperelasticityis the preferred approach for dealing with finite strains and several material models analogous to Hooke's law are in use.Linear elasticity[edit]Main article:Linear elasticityAs noted above, for small deformations, most elastic materials such asspringsexhibit linear elasticity and can be described by a linear relation between the stress and strain. This relationship is known asHooke's law. A geometry-dependent version of the idea[4]was first formulated byRobert Hookein 1675 as a Latinanagram, "ceiiinosssttuv". He published the answer in 1678: "Ut tensio, sic vis" meaning "As the extension, so the force",[5][6][7]a linear relationship commonly referred to asHooke's law. This law can be stated as a relationship betweenforceFanddisplacementx,

wherekis a constant known as therateorspring constant. It can also be stated as a relationship betweenstressandstrain:

whereEIs known as theelastic modulusorYoung's modulus.Although the general proportionality constant between stress and strain in three dimensions is a 4th ordertensor, systems that exhibitsymmetry, such as a one-dimensional rod, can often be reduced to applications of Hooke's law.Finite elasticity[edit]The elastic behavior of objects that undergo finite deformations have been described using a number of models, such asCauchy elastic materialmodels,Hypoelastic materialmodels, andHyperelastic materialmodels. The primary measure that is used to quantity finite strains is thedeformation gradient(F). More convenient strain measures can be derived from this primary quantity.Cauchy elastic materials[edit]Main article:Cauchy elastic materialA material is said to be Cauchy-elastic if theCauchy stress tensoris a function of thestrain tensor(deformation gradient)Falone:

Even though the stress in a Cauchy-elastic material depends only on the state of deformation, the work done by stresses may depend on the path of deformation. Therefore a Cauchy elastic material has a non-conservative structure, and the stress cannot be derived from a scalar "elastic potential" function.Hypoelastic materials[edit]Main article:Hypoelastic materialHypoelastic materials are described by a relation of the form

This model is an extension of linear elasticity and suffers from the same form of non-conservative behaviour as Cauchy elastic materials.Hyperelastic materials[edit]Main article:Hyperelastic materialHyperelastic materials (also called Green elastic materials) are conservative models that are derived from astrain energy density function(W). The stress-strain relation for such materials takes the form

Applications[edit]Linear elasticity is used widely in the design and analysis of structures such asbeams,plates and shells, andsandwich composites. This theory is also the basis of much offracture mechanics.Hyperelasticity is primarily used to determine the response ofelastomer-based objects such asgasketsand of biological materials such assoft tissuesandcell membranes.Factors affecting elasticity[edit]