Editing Elasticity (physics)

{{short description|Physical property when materials or objects return to original shape after deformation}}
{{redirect|Elasticity theory|the economics measurement|Elasticity (economics)|the computing system resource term|Elasticity (system resource)}}
{{Use dmy dates|date=November 2017}}
{{more citations needed|date=February 2017}}
{{Continuum mechanics|cTopic=[[Solid mechanics]]}}

In [[physics]] and [[materials science]], '''elasticity''' is the ability of a [[Physical object|body]] to resist a distorting influence and to return to its original size and shape when that influence or [[force]] is removed. Solid objects will [[Deformation (engineering)|deform]] when adequate [[Structural load|loads]] are applied to them; if the material is elastic, the object will return to its initial shape and size after removal. This is in contrast to [[Plasticity (physics)|''plasticity'']], in which the object fails to do so and instead remains in its deformed state.

The physical reasons for elastic behavior can be quite different for different materials. In [[metal]]s, the [[Crystal structure|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. For [[rubber elasticity|rubbers]] and other [[polymer]]s, elasticity is caused by the stretching of polymer chains when forces are applied.

[[Hooke's law]] states that the force required to deform elastic objects should be [[Proportionality (mathematics)|directly proportional]] to the distance of deformation, regardless of how large that distance becomes. This is known as ''perfect elasticity'', in which a given object will return to its original shape no matter how strongly it is deformed. This is an [[Idealization (science philosophy)|ideal concept]] only; most materials which possess elasticity in practice remain purely elastic only up to very small deformations, after which plastic (permanent) deformation occurs.

In [[engineering]], the elasticity of a material is quantified by the [[elastic modulus]] such as the [[Young's modulus]], [[bulk modulus]] or [[shear modulus]] which measure the amount of [[Stress (mechanics)|stress]] needed to achieve a unit of [[deformation (engineering)|strain]]; a higher modulus indicates that the material is harder to deform. The [[International System of Units|SI unit]] of this modulus is the [[Pascal (unit)|pascal]] (Pa). The material's ''elastic limit'' or [[yield strength]] is the maximum [[Stress (mechanics)|stress]] that can arise before the onset of plastic deformation. Its SI unit is also the pascal (Pa).

==Overview==
When an elastic material is deformed due to an external force, it experiences internal resistance to the deformation and restores it to its original state if the external force is no longer applied. There are various [[Elastic modulus|elastic moduli]], such as [[Young's modulus]], the [[shear modulus]], and the [[bulk modulus]], all of which are measures of the inherent elastic properties 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 extension/compression of a body, whereas the shear modulus applies to its [[Shearing (physics)|shear]].<ref>Landau LD, Lipshitz EM. Theory of Elasticity, 3rd Edition, 1970: 1–172.</ref> Young's modulus and shear modulus are only for solids, whereas the [[bulk modulus]] is for solids, liquids, and gases.

The elasticity of materials is described by a [[stress–strain curve]], which shows the relation between [[stress (mechanics)|stress]] (the average restorative internal [[force]] per unit area) and [[Strain (engineering)|strain]] (the relative deformation).<ref>{{cite book|last=Treloar|first=L. R. G.|title=The Physics of Rubber Elasticity|year=1975|publisher=Clarendon Press|location=Oxford|isbn=978-0-1985-1355-1}}</ref> The curve is generally nonlinear, but it can (by use of a [[Taylor series]]) be approximated as linear for sufficiently small deformations (in which higher-order terms are negligible). If the material is [[isotropic]], the linearized stress–strain relationship is called [[Hooke's law]], which is often presumed to apply up to the elastic limit for most metals or crystalline materials whereas [[nonlinear elasticity]] is generally required to model large deformations of rubbery materials even in the elastic range. For even higher stresses, materials exhibit [[Plasticity (physics)|plastic behavior]], that is, they deform irreversibly and do not return to their original shape after stress is no longer applied.<ref>{{cite book|last=Sadd|first=Martin H.|title=Elasticity: Theory, Applications, and Numerics|year=2005|publisher=Elsevier|location=Oxford|isbn=978-0-1237-4446-3}}</ref> For rubber-like materials such as [[elastomer]]s, the slope of the stress–strain curve increases with stress, meaning that rubbers progressively become more difficult to stretch, while for most metals, the gradient decreases at very high stresses, meaning that they progressively become easier to stretch.<ref>{{cite book|last=de With|first=Gijsbertus|title=Structure, Deformation, and Integrity of Materials, Volume I: Fundamentals and Elasticity|year=2006|publisher=Wiley VCH|location=Weinheim|isbn=978-3-527-31426-3|page=32}}</ref> Elasticity is not exhibited only by solids; [[non-Newtonian fluid]]s, such as [[Viscoelasticity|viscoelastic fluids]], will also exhibit elasticity in certain conditions quantified by the [[Deborah number]]. 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 a [[viscosity|viscous]] liquid.

Because the elasticity of a material is described in terms of a stress–strain relation, it is essential that the terms ''stress'' and ''strain'' be 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 in the sense that it must include the first type as a special case.

For small strains, the measure of stress that is used is the [[Cauchy stress tensor|Cauchy stress]] while the measure of strain that is used is the [[infinitesimal strain theory|infinitesimal strain tensor]]; the resulting (predicted) material behavior is termed [[linear elasticity]], which (for [[isotropic]] media) is called the generalized [[Hooke's law]]. [[Cauchy elastic material]]s and [[hypoelastic material]]s are models that extend Hooke's law to allow for the possibility of large rotations, large distortions, and intrinsic or induced [[anisotropy]].

For more general situations, any of a number of [[stress measures]] can be used, and it is generally desired (but not required) that the elastic stress–strain relation be phrased in terms of a [[finite strain theory|finite strain]] measure that is [[work conjugate]] to the selected stress measure, i.e., the time integral of the inner product of the stress measure with the rate of the strain measure should be equal to the change in internal energy for any [[adiabatic process]] that remains below the elastic limit.

== Units ==

=== International System ===
The SI unit for elasticity and the elastic modulus is the [[Pascal (unit)|pascal]] (Pa). This unit is defined as force per unit area, generally a measurement of [[pressure]], which in mechanics corresponds to [[Stress (mechanics)|stress]]. The pascal and therefore elasticity have the [[Dimensional analysis|dimension]] L<sup>−1</sup>⋅M⋅T<sup>−2</sup>.

For most commonly used engineering materials, the elastic modulus is on the scale of gigapascals (GPa, 10<sup>9</sup> Pa).

== Linear elasticity ==
{{Main|Linear elasticity}}
As noted above, for small deformations, most elastic materials such as [[Spring (device)|spring]]s exhibit linear elasticity and can be described by a linear relation between the stress and strain. This relationship is known as [[Hooke's law]]. A geometry-dependent version of the idea{{efn|Descriptions of material behavior should be independent of the geometry and shape of the object made of the material under consideration.  The original version of Hooke's law involves a stiffness constant that depends on the initial size and shape of the object.  The stiffness constant is therefore not strictly a material property.{{cn|date=October 2023}}}} was first formulated by [[Robert Hooke]] in 1675 as a Latin [[anagram]], "ceiiinosssttuv". He published the answer in 1678: "''Ut tensio, sic vis''" meaning "''As the extension, so the force''",<ref>{{cite book|last1=Atanackovic|first1=Teodor M.|first2= Ardéshir |last2=Guran |title=Theory of elasticity for scientists and engineers|year=2000|publisher=Birkhäuser|location=Boston, Mass.|isbn=978-0-8176-4072-9|chapter=Hooke's law}}</ref><ref>{{cite web |url=http://www.lindahall.org/events_exhib/exhibit/exhibits/civil/design.shtml |title=Strength and Design |website=Centuries of Civil Engineering: A Rare Book Exhibition Celebrating the Heritage of Civil Engineering |publisher=Linda Hall Library of Science, Engineering & Technology |url-status=dead |archive-url=https://web.archive.org/web/20101113210736/http://www.lindahall.org/events_exhib/exhibit/exhibits/civil/design.shtml |archive-date=13 November 2010 }}{{page needed|date=November 2012}}</ref> a linear relationship commonly referred to as [[Hooke's law]]. This law can be stated as a relationship between tensile [[force]] {{math|''F''}} and corresponding extension [[displacement (vector)|displacement]] <math>x</math>,
:<math>F=k x,</math>
where {{math|''k''}} is a constant known as the ''rate'' or ''spring constant''. It can also be stated as a relationship between [[tensile stress|stress]] <math>\sigma</math> and [[strain (materials science)|strain]] <math>\varepsilon</math>:
:<math>\sigma = E\varepsilon,</math>
where {{math|''E''}} is known as the [[Young's modulus]].<ref>{{cite book |last1=Ibrahimbegovic |first1=Adnan |title=Nonlinear Solid Mechanics: Theoretical Formulations and Finite Element Solution Methods |date=2 June 2009 |publisher=Springer Science & Business Media |isbn=978-90-481-2330-8 |pages=20–26 |url=https://books.google.com/books?id=z8oI0hCMt40C&dq=%22hooke%27&pg=PA20 |language=en |access-date=9 July 2023 |archive-date=28 May 2024 |archive-url=https://web.archive.org/web/20240528030534/https://books.google.com/books?id=z8oI0hCMt40C&dq=%22hooke%27&pg=PA20#v=onepage&q=%22hooke'&f=false |url-status=live }}</ref>

Although the general proportionality constant between stress and strain in three dimensions is a 4th-order [[tensor]] called [[stiffness]], systems that exhibit [[symmetry]], such as a one-dimensional rod, can often be reduced to applications of Hooke's law.

== Finite elasticity ==
The elastic behavior of objects that undergo finite deformations has been described using a number of models, such as [[Cauchy elastic material]] models, [[Hypoelastic material]] models, and [[Hyperelastic material]] models. The [[deformation gradient]] ('''''F''''') is the primary deformation measure used in [[finite strain theory]].

=== Cauchy elastic materials ===
{{main|Cauchy elastic material}}
A material is said to be Cauchy-elastic if the [[Cauchy stress tensor]] '''''σ''''' is a function of the [[deformation gradient]]  '''''F''''' alone:
:<math>\ \boldsymbol{\sigma} = \mathcal{G}(\boldsymbol{F}) </math>
It is generally incorrect to state that Cauchy stress is a function of merely a [[strain tensor]], as such a model lacks crucial information about material rotation needed to produce correct results for an anisotropic medium subjected to vertical extension in comparison to the same extension applied horizontally and then subjected to a 90-degree rotation; both these deformations have the same spatial strain tensors yet must produce different values of the Cauchy stress tensor.

Even though the stress in a Cauchy-elastic material depends only on the state of deformation, the work done by stresses might depend on the path of deformation. Therefore, Cauchy elasticity includes non-conservative "non-hyperelastic" models (in which work of deformation is path dependent) as well as conservative "[[hyperelastic material]]" models (for which stress can be derived from a scalar "elastic potential" function).

=== Hypoelastic materials ===
{{main|Hypoelastic material}}
A hypoelastic material can be rigorously defined as one that is modeled using a [[constitutive equation]] satisfying the following two criteria:<ref>{{cite book|last1=Truesdell|first1=Clifford|last2=Noll|first2=Walter|title=The Non-linear Field Theories of Mechanics|year=2004|publisher=Springer-Verlag|location=Berlin Heidelberg New York|edition=3rd|isbn=978-3-540-02779-9|page=401}}</ref>

# The Cauchy stress <math>\boldsymbol{\sigma}</math> at time <math>t</math> depends only on the order in which the body has occupied its past configurations, but not on the time rate at which these past configurations were traversed. As a special case, this criterion includes a [[Cauchy elastic material]], for which the current stress depends only on the current configuration rather than the history of past configurations.
# There is a tensor-valued function <math>G</math> such that <math>
   \dot{\boldsymbol{\sigma}} = G(\boldsymbol{\sigma},\boldsymbol{L}) \,,
</math> in which <math>\dot{\boldsymbol{\sigma}}</math> is the material rate of the Cauchy stress tensor, and <math>\boldsymbol{L}</math> is the spatial [[velocity gradient]] tensor.

If only these two original criteria are used to define hypoelasticity, then [[hyperelasticity]] would be included as a special case, which prompts some constitutive modelers to append a third criterion that specifically requires a hypoelastic model to ''not'' be hyperelastic (i.e., hypoelasticity implies that stress is not derivable from an energy potential). If this third criterion is adopted, it follows that a hypoelastic material might admit nonconservative adiabatic loading paths that start and end with the same [[deformation gradient]] but do ''not'' start and end at the same internal energy.

Note that the second criterion requires only that the function <math>G</math> ''exists''. As detailed in the main [[hypoelastic material]] article, specific formulations of hypoelastic models typically employ so-called objective rates so that the <math>G</math> function exists only implicitly and is typically needed explicitly only for numerical stress updates performed via direct integration of the actual (not objective) stress rate.

=== Hyperelastic materials ===
{{main|Hyperelastic material}}
Hyperelastic materials (also called Green elastic materials) are conservative models that are derived from a [[strain energy density function]] (''W''). A model is hyperelastic if and only if it is possible to express the [[Cauchy stress tensor]] as a function of the [[deformation gradient]] via a relationship of the form
:<math>
   \boldsymbol{\sigma} = \cfrac{1}{J}~ \cfrac{\partial W}{\partial \boldsymbol{F}}\boldsymbol{F}^\textsf{T} \quad \text{where} \quad J := \det\boldsymbol{F} \,.
 </math>

This formulation takes the energy potential (''W'') as a function of the [[deformation gradient]] (<math>\boldsymbol{F}</math>). By also requiring satisfaction of [[material objectivity]], the energy potential may be alternatively regarded as a function of the [[Cauchy-Green deformation tensor]] (<math>\boldsymbol{C} := \boldsymbol{F}^\textsf{T}\boldsymbol{F}</math>), in which case the hyperelastic model may be written alternatively as
:<math>
   \boldsymbol{\sigma} = \cfrac{2}{J}~ \boldsymbol{F}\cfrac{\partial W}{\partial \boldsymbol{C}}\boldsymbol{F}^\textsf{T} \quad \text{where} \quad J := \det\boldsymbol{F} \,.
 </math>

== Applications ==
Linear elasticity is used widely in the design and analysis of structures such as [[beam bending|beams]], [[plate theory|plates and shells]], and [[sandwich theory|sandwich composites]]. This theory is also the basis of much of [[fracture mechanics]].

Hyperelasticity is primarily used to determine the response of [[elastomer]]-based objects such as [[gasket]]s and of biological materials such as [[soft tissue]]s and [[elasticity of cell membranes|cell membranes]].

==Factors affecting elasticity==
In a given [[isotropic material | isotropic solid]], with known theoretical elasticity for the bulk material in terms of Young's modulus,the effective elasticity will be governed by [[porosity]]. Generally a more porous material will exhibit lower stiffness. More specifically, the fraction of pores, their distribution at different sizes and the nature of the fluid with which they are filled give rise to different elastic behaviours in solids.<ref>{{Cite journal |last=Liu |first=Mingchao |last2=Wu |first2=Jian |last3=Gan |first3=Yixiang |last4=Hanaor |first4=Dorian AH |last5=Chen |first5=C.Q. |date=1 May 2019 |title=Multiscale modeling of the effective elastic properties of fluid-filled porous materials |url=https://unsworks.unsw.edu.au/bitstreams/1dd2eacb-33d5-41d6-9ff2-401e255cee3a/download |journal=International Journal of Solids and Structures |language=en |volume=162 |pages=36–44 |doi=10.1016/j.ijsolstr.2018.11.028}}</ref> 

For [[isotropic material]]s containing cracks, the presence of fractures affects the Young and the shear moduli perpendicular to the planes of the cracks, which decrease (Young's modulus faster than the shear modulus) as the fracture [[number density|density]] increases,<ref>{{cite book|last=Sadd|first=Martin H.|title=Elasticity: Theory, Applications, and Numerics|year=2005|publisher=Elsevier|location=Oxford|isbn=978-0-1237-4446-3}}</ref> indicating that the presence of cracks makes bodies brittler. [[Microscopic]]ally, the stress–strain relationship of materials is in general governed by the [[Helmholtz free energy]], a [[Functions of state|thermodynamic quantity]]. [[Molecule]]s settle in the configuration which minimizes the free energy, subject to constraints derived from their structure, and, depending on whether the energy or the [[entropy]] term dominates the free energy, materials can broadly be classified as ''energy-elastic'' and ''entropy-elastic''. As such, microscopic factors affecting the free energy, such as the [[Mechanical equilibrium|equilibrium]] distance between molecules, can affect the elasticity of materials: for instance, in [[inorganic]] materials, as the equilibrium distance between molecules at [[Absolute zero|0 K]] increases, the [[bulk modulus]] decreases.<ref>{{cite book|last=Sadd|first=Martin H.|title=Elasticity: Theory, Applications, and Numerics|year=2005|publisher=Elsevier|location=Oxford|isbn=978-0-1237-4446-3}}</ref> The effect of temperature on elasticity is difficult to isolate, because there are numerous factors affecting it. For instance, the bulk modulus of a material is dependent on the form of its [[Crystal structure|lattice]], its behavior under [[Thermal expansion|expansion]], as well as the [[vibrations]] of the molecules, all of which are dependent on temperature.<ref>{{cite book|last=Sadd|first=Martin H.|title=Elasticity: Theory, Applications, and Numerics|year=2005|publisher=Elsevier|location=Oxford|isbn=978-0-1237-4446-3}}</ref>

== See also ==
{{Columns-list|colwidth=20em|
* [[Elasticity tensor]]
* [[Elastography]]
* [[Tactile imaging]]
* [[Elastic modulus]]
* [[Linear elasticity]]
* [[Pseudoelasticity]]
* [[Resilience (materials science)|Resilience]]
* [[Rubber elasticity]]
* [[Stiffness]]
* [[Ductility]]
* [[Physical crystallography before X-rays#Elastic properties|Physical crystallography before X-rays]]
}}

== Notes ==
{{notelist}}

== References ==
{{reflist}}

== External links ==
*[https://feynmanlectures.caltech.edu/II_38.html The Feynman Lectures on Physics Vol. II Ch. 38: Elasticity]

{{Topics in continuum mechanics}}
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[[Category:Elasticity (physics)| ]]