In

^{−1}⋅M⋅T^{−2}.
For most commonly used engineering materials, the elastic modulus is on the scale of gigapascals (GPa, 10^{9} Pa).

Hooke's law
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, where is a constant factor characteristic of ...

. A geometry-dependent version of the idea was first formulated by Hooke's law
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, where is a constant factor characteristic of ...

. This law can be stated as a relationship between tensile

The Feynman Lectures on Physics Vol. II Ch. 38: Elasticity

{{Authority control

physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which rela ...

and materials science, elasticity is the ability of a body
Body may refer to:
In science
* Physical body, an object in physics that represents a large amount, has mass or takes up space
* Body (biology), the physical material of an organism
* Body plan, the physical features shared by a group of anim ...

to resist a distorting influence and to return to its original size and shape when that influence or force
In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as ...

is removed. Solid objects will deform when adequate 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'', 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
A metal (from Greek μέταλλον ''métallon'', "mine, quarry, metal") is a material that, when freshly prepared, polished, or fractured, shows a lustrous appearance, and conducts electricity and heat relatively well. Metals are typical ...

s, 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. For rubbers and other polymer
A polymer (; Greek '' poly-'', "many" + '' -mer'', "part")
is a substance or material consisting of very large molecules called macromolecules, composed of many repeating subunits. Due to their broad spectrum of properties, both synthetic a ...

s, elasticity is caused by the stretching of polymer chains when forces are applied.
Hooke's law
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, where is a constant factor characteristic of ...

states that the force required to deform elastic objects should be 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 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
Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...

, the elasticity of a material is quantified by the elastic modulus
An elastic modulus (also known as modulus of elasticity) is the unit of measurement of an object's or substance's resistance to being deformed elastically (i.e., non-permanently) when a stress is applied to it. The elastic modulus of an object is ...

such as the Young's modulus
Young's modulus E, the Young modulus, or the modulus of elasticity in tension or compression (i.e., negative tension), is a mechanical property that measures the tensile or compressive stiffness of a solid material when the force is applied le ...

, bulk modulus or shear modulus
In materials science, shear modulus or modulus of rigidity, denoted by ''G'', or sometimes ''S'' or ''μ'', is a measure of the elastic shear stiffness of a material and is defined as the ratio of shear stress to the shear strain:
:G \ \stack ...

which measure the amount of stress needed to achieve a unit of strain; a higher modulus indicates that the material is harder to deform. The SI unit
The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. E ...

of this modulus is the pascal (Pa). The material's ''elastic limit'' or yield strength
In materials science and engineering, the yield point is the point on a stress-strain curve that indicates the limit of elastic behavior and the beginning of plastic behavior. Below the yield point, a material will deform elastically and w ...

is the maximum 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 moduli, such asYoung's modulus
Young's modulus E, the Young modulus, or the modulus of elasticity in tension or compression (i.e., negative tension), is a mechanical property that measures the tensile or compressive stiffness of a solid material when the force is applied le ...

, the shear modulus
In materials science, shear modulus or modulus of rigidity, denoted by ''G'', or sometimes ''S'' or ''μ'', is a measure of the elastic shear stiffness of a material and is defined as the ratio of shear stress to the shear strain:
:G \ \stack ...

, 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 shear. 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
In engineering and materials science, a stress–strain curve for a material gives the relationship between stress and strain. It is obtained by gradually applying load to a test coupon and measuring the deformation, from which the stress a ...

, which shows the relation between stress (the average restorative internal force
In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as ...

per unit area) and strain (the relative deformation). The curve is generally nonlinear, but it can (by use of a Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor seri ...

) be approximated as linear for sufficiently small deformations (in which higher-order terms are negligible). If the material is isotropic
Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence '' anisotropy''. ''Anisotropy'' is also used to describ ...

, the linearized stress–strain relationship is called Hooke's law
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, where is a constant factor characteristic of ...

, 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 plastic behavior, that is, they deform irreversibly and do not return to their original shape after stress is no longer applied. For rubber-like materials such as elastomer
An elastomer is a polymer with viscoelasticity (i.e. both viscosity and elasticity) and with weak intermolecular forces, generally low Young's modulus and high failure strain compared with other materials. The term, a portmanteau of ''elasti ...

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. Elasticity is not exhibited only by solids; non-Newtonian fluid
A non-Newtonian fluid is a fluid that does not follow Newton's law of viscosity, i.e., constant viscosity independent of stress. In non-Newtonian fluids, viscosity can change when under force to either more liquid or more solid. Ketchup, for exa ...

s, such as viscoelastic fluids, will also exhibit elasticity in certain conditions quantified by the Deborah number The Deborah number (De) is a dimensionless number, often used in rheology to characterize the fluidity of materials under specific flow conditions. It quantifies the observation that given enough time even a solid-like material might flow, or a flu ...

. 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 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 while the measure of strain that is used is the infinitesimal strain tensor; the resulting (predicted) material behavior is termed linear elasticity
Linear elasticity is a mathematical model of how solid objects deform and become internally stressed due to prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mec ...

, which (for isotropic
Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence '' anisotropy''. ''Anisotropy'' is also used to describ ...

media) is called the generalized Hooke's law
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, where is a constant factor characteristic of ...

. Cauchy elastic materials and hypoelastic materials are models that extend Hooke's law to allow for the possibility of large rotations, large distortions, and intrinsic or induced anisotropy
Anisotropy () is the property of a material which allows it to change or assume different properties in different directions, as opposed to isotropy. It can be defined as a difference, when measured along different axes, in a material's phys ...

.
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 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
In thermodynamics, an adiabatic process (Greek: ''adiábatos'', "impassable") is a type of thermodynamic process that occurs without transferring heat or mass between the thermodynamic system and its environment. Unlike an isothermal process, ...

that remains below the elastic limit.
Units

International System

The SI unit for elasticity and the elastic modulus is the pascal (Pa). This unit is defined as force per unit area, generally a measurement ofpressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and ...

, which in mechanics corresponds to stress. The pascal and therefore elasticity have the dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordi ...

LLinear elasticity

As noted above, for small deformations, most elastic materials such as springs exhibit linear elasticity and can be described by a linear relation between the stress and strain. This relationship is known asRobert Hooke
Robert Hooke FRS (; 18 July 16353 March 1703) was an English polymath active as a scientist, natural philosopher and architect, who is credited to be one of two scientists to discover microorganisms in 1665 using a compound microscope th ...

in 1675 as a Latin anagram
An anagram is a word or phrase formed by rearranging the letters of a different word or phrase, typically using all the original letters exactly once. For example, the word ''anagram'' itself can be rearranged into ''nag a ram'', also the word ...

, "ceiiinosssttuv". He published the answer in 1678: "''Ut tensio, sic vis''" meaning "''As the extension, so the force''", a linear relationship commonly referred to as force
In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as ...

and corresponding extension displacement ,
:$F=k\; x,$
where is a constant known as the ''rate'' or ''spring constant''. It can also be stated as a relationship between stress and strain $\backslash varepsilon$:
:$\backslash sigma\; =\; E\backslash varepsilon,$
where is known as the elastic modulus
An elastic modulus (also known as modulus of elasticity) is the unit of measurement of an object's or substance's resistance to being deformed elastically (i.e., non-permanently) when a stress is applied to it. The elastic modulus of an object is ...

or Young's modulus
Young's modulus E, the Young modulus, or the modulus of elasticity in tension or compression (i.e., negative tension), is a mechanical property that measures the tensile or compressive stiffness of a solid material when the force is applied le ...

.
Although the general proportionality constant between stress and strain in three dimensions is a 4th-order tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tens ...

called stiffness
Stiffness is the extent to which an object resists deformation in response to an applied force.
The complementary concept is flexibility or pliability: the more flexible an object is, the less stiff it is.
Calculations
The stiffness, k, of a bo ...

, systems that exhibit symmetry
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...

, 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 infinite strain theory
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 stra ...

.
Cauchy elastic materials

A material is said to be Cauchy-elastic if theCauchy stress tensor
In continuum mechanics, the Cauchy stress tensor \boldsymbol\sigma, true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components \sigma_ that completely ...

''σ'' is a function of the deformation gradient ''F'' alone:
:$\backslash \; \backslash boldsymbol\; =\; \backslash mathcal(\backslash boldsymbol)$
It is generally incorrect to state that Cauchy stress is a function of merely a strain 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 (indeed, infinitesimall ...

, 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

A hypoelastic material can be rigorously defined as one that is modeled using aconstitutive equation
In physics and engineering, a constitutive equation or constitutive relation is a relation between two physical quantities (especially kinetic quantities as related to kinematic quantities) that is specific to a material or substance, and ap ...

satisfying the following two criteria:
# The Cauchy stress $\backslash boldsymbol$ at time $t$ 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 $G$ such that $\backslash dot\; =\; G(\backslash boldsymbol,\backslash boldsymbol)\; \backslash ,,$ in which $\backslash dot$ is the material rate of the Cauchy stress tensor, and $\backslash boldsymbol$ 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 $G$ ''exists''. As detailed in the main hypoelastic material article, specific formulations of hypoelastic models typically employ so-called objective rates so that the $G$ 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

Hyperelastic materials (also called Green elastic materials) are conservative models that are derived from astrain energy density function
A strain energy density function or stored energy density function is a scalar-valued function that relates the strain energy density of a material to the deformation gradient.
:
W = \hat(\boldsymbol) = \hat(\boldsymbol^T\cdot\boldsymbol) = ...

(''W''). A model is hyperelastic if and only if it is possible to express the Cauchy stress tensor
In continuum mechanics, the Cauchy stress tensor \boldsymbol\sigma, true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components \sigma_ that completely ...

as a function of the deformation gradient via a relationship of the form
:$\backslash boldsymbol\; =\; \backslash cfrac~\; \backslash cfrac\backslash boldsymbol^\backslash textsf\; \backslash quad\; \backslash text\; \backslash quad\; J\; :=\; \backslash det\backslash boldsymbol\; \backslash ,.$
This formulation takes the energy potential (''W'') as a function of the deformation gradient ($\backslash boldsymbol$). By also requiring satisfaction of material objectivity, the energy potential may be alternatively regarded as a function of the Cauchy-Green deformation tensor ($\backslash boldsymbol\; :=\; \backslash boldsymbol^\backslash textsf\backslash boldsymbol$), in which case the hyperelastic model may be written alternatively as
:$\backslash boldsymbol\; =\; \backslash cfrac~\; \backslash boldsymbol\backslash cfrac\backslash boldsymbol^\backslash textsf\; \backslash quad\; \backslash text\; \backslash quad\; J\; :=\; \backslash det\backslash boldsymbol\; \backslash ,.$
Applications

Linear elasticity is used widely in the design and analysis of structures such as beams, plates and shells, and sandwich composites. This theory is also the basis of much of fracture mechanics. Hyperelasticity is primarily used to determine the response ofelastomer
An elastomer is a polymer with viscoelasticity (i.e. both viscosity and elasticity) and with weak intermolecular forces, generally low Young's modulus and high failure strain compared with other materials. The term, a portmanteau of ''elasti ...

-based objects such as gasket
Some seals and gaskets
A gasket is a mechanical seal which fills the space between two or more mating surfaces, generally to prevent leakage from or into the joined objects while under compression. It is a deformable material that is used to ...

s and of biological materials such as soft tissue
Soft tissue is all the tissue in the body that is not hardened by the processes of ossification or calcification such as bones and teeth. Soft tissue connects, surrounds or supports internal organs and bones, and includes muscle, tendons, ligam ...

s and cell membranes
The cell membrane (also known as the plasma membrane (PM) or cytoplasmic membrane, and historically referred to as the plasmalemma) is a biological membrane that separates and protects the interior of all cells from the outside environment ( ...

.
Factors affecting elasticity

Forisotropic material
Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence '' anisotropy''. ''Anisotropy'' is also used to describ ...

s, 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 density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematica ...

increases, indicating that the presence of cracks makes bodies brittler. Microscopic
The microscopic scale () is the scale of objects and events smaller than those that can easily be seen by the naked eye, requiring a lens or microscope to see them clearly. In physics, the microscopic scale is sometimes regarded as the scale b ...

ally, the stress–strain relationship of materials is in general governed by the Helmholtz free energy
In thermodynamics, the Helmholtz free energy (or Helmholtz 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 ener ...

, a thermodynamic quantity. Molecule
A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioch ...

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
Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodyna ...

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 equilibrium distance between molecules, can affect the elasticity of materials: for instance, in inorganic
In chemistry, an inorganic compound is typically a chemical compound that lacks carbon–hydrogen bonds, that is, a compound that is not an organic compound. The study of inorganic compounds is a subfield of chemistry known as ''inorganic chemis ...

materials, as the equilibrium distance between molecules at 0 K increases, the bulk modulus decreases. 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 lattice, its behavior under expansion, as well as the vibrations of the molecules, all of which are dependent on temperature.
See also

References

External links

The Feynman Lectures on Physics Vol. II Ch. 38: Elasticity

{{Authority control