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Young's modulus $E$, the Young modulus, or the
modulus of elasticity An elastic modulus (also known as modulus of elasticity) is a quantity that measures an object 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 defi ...
in tension or compression (i.e., negative tension), is a mechanical property that measures the tensile or compressive
stiffness Stiffness is the extent to which an object resists Deformation (mechanics), 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 ...
of a
solid Solid is one of the four fundamental states of matter (the others being liquid A liquid is a nearly incompressible fluid In physics, a fluid is a substance that continually Deformation (mechanics), deforms (flows) under an applied ...

material when the force is applied lengthwise. It quantifies the relationship between tensile/compressive stress $\sigma$ (force per unit area) and axial strain $\varepsilon$ (proportional deformation) in the linear elastic region of a material and is determined using the formula: $E = \frac$ Young's moduli are typically so large that they are expressed not in pascals but in gigapascals (GPa). Although Young's modulus is named after the 19th-century British scientist , the concept was developed in 1727 by
Leonhard Euler Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ) ...

. The first experiments that used the concept of Young's modulus in its current form were performed by the Italian scientist
Giordano RiccatiImage:Federici, Domenico Maria - Commentario sopra la vita e gli studi del conte Giordano Riccati, 1790 - BEIC 1516076.jpg, Domenico Maria Federici, ''Commentario sopra la vita e gli studi del conte Giordano Riccati'', 1790 Giordano Riccati or Jorda ...
in 1782, pre-dating Young's work by 25 years. The term modulus is derived from the Latin root term ''modus'' which means ''measure''.

# Definition

## Linear elasticity

A solid material will undergo
elastic deformation 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 ran ...
when a small load is applied to it in compression or extension. Elastic deformation is reversible, meaning that the material returns to its original shape after the load is removed. At near-zero stress and strain, the stress–strain curve is
linear Linearity is the property of a mathematical relationship (''function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out se ...

, and the relationship between stress and strain is described by
Hooke's law The balance wheel at the core of many mechanical clocks and watches depends on Hooke's law. Since the torque generated by the coiled spring is proportional to the angle turned by the wheel, its oscillations have a nearly constant period. Hooke ...
that states stress is proportional to strain. The coefficient of proportionality is Young's modulus. The higher the modulus, the more stress is needed to create the same amount of strain; an idealized
rigid body In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. " ...
would have an infinite Young's modulus. Conversely, a very soft material (such as a fluid) would deform without force, and would have zero Young's modulus. Not many materials are linear and elastic beyond a small amount of deformation.

# Note

Material stiffness should not be confused with these properties: *
Strength Physical strength *Physical strength, as in people or animals *Hysterical strength, extreme strength occurring when people are in life-and-death situations *Superhuman strength, great physical strength far above human capability *A common attrib ...
: maximum amount of stress that material can withstand while staying in the elastic (reversible) deformation regime; * Geometric stiffness: a global characteristic of the body that depends on its shape, and not only on the local properties of the material; for instance, an
I-beam An I-beam, also known as H-beam (for universal column, UC), w-beam (for "wide flange"), universal beam (UB), rolled steel joist (RSJ), or double-T (especially in Polish Polish may refer to: * Anything from or related to Poland Polan ...

has a higher bending stiffness than a rod of the same material for a given mass per length; *
Hardness Hardness (antonym: softness) is a measure of the resistance to localized induced by either mechanical or . In general, different materials differ in their hardness; for example hard metals such as and are harder than soft metals such as and ...
: relative resistance of the material's surface to penetration by a harder body; *
Toughness In materials science The Interdisciplinarity, interdisciplinary field of materials science, also commonly termed materials science and engineering, covers the design and discovery of new materials, particularly solids. The intellectual origin ...
: amount of energy that a material can absorb before fracture.

# Usage

Young's modulus enables the calculation of the change in the dimension of a bar made of an
isotropic Isotropy is uniformity in all orientations; it is derived from the Greek ''isos'' (ἴσος, "equal") and ''tropos'' (τρόπος, "way"). Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by t ...
elastic material under tensile or compressive loads. For instance, it predicts how much a material sample extends under tension or shortens under compression. The Young's modulus directly applies to cases of uniaxial stress; that is, tensile or compressive stress in one direction and no stress in the other directions. Young's modulus is also used in order to predict the deflection that will occur in a
statically determinateIn statics Statics is the branch of mechanics Mechanics (Ancient Greek, Greek: ) is the area of physics concerned with the motions of physical objects, more specifically the relationships among force, matter, and motion. Forces applied to obj ...
beam Beam may refer to: Streams of particles or energy *Light beam, or beam of light, a directional projection of light energy **Laser beam *Particle beam, a stream of charged or neutral particles **Charged particle beam, a spatially localized group ...
when a load is applied at a point in between the beam's supports. Other elastic calculations usually require the use of one additional elastic property, such as the
shear modulus In materials science The interdisciplinary field of materials science, also commonly termed materials science and engineering, covers the design and discovery of new materials, particularly solids. The intellectual origins of materials scienc ...
$G$,
bulk modulus The bulk modulus (K or B) of a substance is a measure of how resistant to compression that substance is. It is defined as the ratio of the infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to ze ...
$K$, and
Poisson's ratio In materials science The Interdisciplinarity, interdisciplinary field of materials science, also commonly termed materials science and engineering, covers the design and discovery of new materials, particularly solids. The intellectual origins o ...
$\nu$. Any two of these parameters are sufficient to fully describe elasticity in an isotropic material. For homogeneous isotropic materials simple relations exist between elastic constants that allow calculating them all as long as two are known: :$E = 2G\left(1+\nu\right) = 3K\left(1-2\nu\right).$

## Linear versus non-linear

Young's modulus represents the factor of proportionality in
Hooke's law The balance wheel at the core of many mechanical clocks and watches depends on Hooke's law. Since the torque generated by the coiled spring is proportional to the angle turned by the wheel, its oscillations have a nearly constant period. Hooke ...
, which relates the stress and the strain. However, Hooke's law is only valid under the assumption of an ''elastic'' and ''linear'' response. Any real material will eventually fail and break when stretched over a very large distance or with a very large force; however all solid materials exhibit nearly Hookean behavior for small enough strains or stresses. If the range over which Hooke's law is valid is large enough compared to the typical stress that one expects to apply to the material, the material is said to be linear. Otherwise (if the typical stress one would apply is outside the linear range) the material is said to be non-linear.
Steel Steel is an alloy An alloy is an admixture of metal A metal (from Ancient Greek, Greek μέταλλον ''métallon'', "mine, quarry, metal") is a material that, when freshly prepared, polished, or fractured, shows a lustrous appe ...

,
carbon fiber Carbon fiber reinforced polymer (American English American English (AmE, AE, AmEng, USEng, en-US), sometimes called United States English or U.S. English, is the set of varieties of the English language native to the United States. Cur ...
and
glass Glass is a non- crystalline, often transparency and translucency, transparent amorphous solid, that has widespread practical, technological, and decorative use in, for example, window panes, tableware, and optics. Glass is most often formed by ...

among others are usually considered linear materials, while other materials such as
rubber Rubber, also called India rubber, latex, Amazonian rubber, ''caucho'', or ''caoutchouc'', as initially produced, consists of polymer A polymer (; Greek ''poly- Poly, from the Greek :wikt:πολύς, πολύς meaning "many" or "much" ...

and
soils Soil is a of , , es, s, and s that together support . 's body of soil, called the , has four important : * as a medium for plant growth * as a means of , supply and purification * as a modifier of * as a habitat for organisms All of these ...
are non-linear. However, this is not an absolute classification: if very small stresses or strains are applied to a non-linear material, the response will be linear, but if very high stress or strain is applied to a linear material, the linear theory will not be enough. For example, as the linear theory implies reversibility, it would be absurd to use the linear theory to describe the failure of a steel bridge under a high load; although steel is a linear material for most applications, it is not in such a case of catastrophic failure. In
solid mechanics Solid mechanics, also known as mechanics of solids, is the branch of continuum mechanics that studies the behavior of solid materials, especially their motion and deformation (mechanics), deformation under the action of forces, temperature change ...
, the slope of the
stress–strain curve In engineering Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompa ...
at any point is called the tangent modulus. It can be experimentally determined from the
slope In mathematics, the slope or gradient of a line Line, lines, The Line, or LINE may refer to: Arts, entertainment, and media Films * ''Lines'' (film), a 2016 Greek film * ''The Line'' (2017 film) * ''The Line'' (2009 film) * ''The Line'', ...

of a stress–strain curve created during
tensile test Tensile testing, also known as tension testing, is a fundamental materials science The interdisciplinary field of materials science, also commonly termed materials science and engineering, covers the design and discovery of new materials, par ...
s conducted on a sample of the material.

## Directional materials

Young's modulus is not always the same in all orientations of a material. Most metals and ceramics, along with many other materials, are
isotropic Isotropy is uniformity in all orientations; it is derived from the Greek ''isos'' (ἴσος, "equal") and ''tropos'' (τρόπος, "way"). Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by t ...
, and their mechanical properties are the same in all orientations. However, metals and ceramics can be treated with certain impurities, and metals can be mechanically worked to make their grain structures directional. These materials then become
anisotropic Anisotropy () is the property of a material which allows it to change or assume different properties in different directions as opposed to isotropy Isotropy is uniformity in all orientations; it is derived from the Greek ''isos'' (ἴσος, ...
, and Young's modulus will change depending on the direction of the force vector. Anisotropy can be seen in many composites as well. For example,
carbon fiber Carbon fiber reinforced polymer (American English American English (AmE, AE, AmEng, USEng, en-US), sometimes called United States English or U.S. English, is the set of varieties of the English language native to the United States. Cur ...
has a much higher Young's modulus (is much stiffer) when force is loaded parallel to the fibers (along the grain). Other such materials include
wood Wood is a porous and fibrous structural tissue found in the stems and roots of tree In botany, a tree is a perennial plant with an elongated Plant stem, stem, or trunk (botany), trunk, supporting branches and leaves in most species. ...

and
reinforced concrete Reinforced concrete (RC), also called reinforced cement concrete (RCC), is a composite material A composite material (also called a composition material or shortened to composite, which is the common name) is a material which is produced from ...
. Engineers can use this directional phenomenon to their advantage in creating structures.

## Temperature dependence

The Young's modulus of metals varies with the temperature and can be realized through the change in the interatomic bonding of the atoms and hence its change is found to be dependent on the change in the work function of the metal. Although classically, this change is predicted through fitting and without a clear underlying mechanism (for example, the Watchman's formula), the Rahemi-Li model demonstrates how the change in the electron work function leads to change in the Young's modulus of metals and predicts this variation with calculable parameters, using the generalization of the Lennard-Jones potential to solids. In general, as the temperature increases, the Young's modulus decreases via $E\left(T\right) = \beta\left(\varphi\left(T\right)\right)^6$ where the electron work function varies with the temperature as $\varphi\left(T\right)=\varphi_0-\gamma\frac$ and $\gamma$ is a calculable material property which is dependent on the crystal structure (for example, BCC, FCC). $\varphi_0$ is the electron work function at T=0 and $\beta$ is constant throughout the change.

# Calculation

Young's modulus ''E'', can be calculated by dividing the
tensile stress In continuum mechanics Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as point particle, discrete particles. The French mathematician Augustin-Louis ...
, $\sigma\left(\varepsilon\right)$, by the engineering extensional strain, $\varepsilon$, in the elastic (initial, linear) portion of the physical
stress–strain curve In engineering Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompa ...
: $E \equiv \frac= \frac = \frac$ where *$E$ is the Young's modulus (modulus of elasticity) *$F$ is the force exerted on an object under tension; *$A$ is the actual cross-sectional area, which equals the area of the cross-section perpendicular to the applied force; *$\Delta L$ is the amount by which the length of the object changes ($\Delta L$ is positive if the material is stretched, and negative when the material is compressed); *$L_0$ is the original length of the object.

## Force exerted by stretched or contracted material

The Young's modulus of a material can be used to calculate the force it exerts under specific strain. :$F = \frac$ where $F$ is the force exerted by the material when contracted or stretched by $\Delta L$.
Hooke's law The balance wheel at the core of many mechanical clocks and watches depends on Hooke's law. Since the torque generated by the coiled spring is proportional to the angle turned by the wheel, its oscillations have a nearly constant period. Hooke ...
for a stretched wire can be derived from this formula: :$F = \left\left( \frac \right\right) \, \Delta L = k x$ where it comes in saturation :$k \equiv \frac \,$ and $x \equiv \Delta L.$ But note that the elasticity of coiled springs comes from
shear modulus In materials science The interdisciplinary field of materials science, also commonly termed materials science and engineering, covers the design and discovery of new materials, particularly solids. The intellectual origins of materials scienc ...
, not Young's modulus.

## Elastic potential energy

The
elastic potential energy Elastic energy is the mechanical potential energy In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of p ...
stored in a linear elastic material is given by the integral of the Hooke's law: :$U_e = \int \, dx = \frac k x^2.$ now by explicating the intensive variables: :$U_e = \int \frac \, d\Delta L = \frac \int \Delta L \, d\Delta L = \frac$ This means that the elastic potential energy density (that is, per unit volume) is given by: :$\frac = \frac$ or, in simple notation, for a linear elastic material: $u_e(\varepsilon) = \int \, d\varepsilon = \frac E ^2$, since the strain is defined $\varepsilon \equiv \frac$. In a nonlinear elastic material the Young's modulus is a function of the strain, so the second equivalence no longer holds and the elastic energy is not a quadratic function of the strain: : $u_e\left(\varepsilon\right) = \int E\left(\varepsilon\right) \, \varepsilon \, d\varepsilon \ne \frac E \varepsilon^2$

# Approximate values

Young's modulus can vary somewhat due to differences in sample composition and test method. The rate of deformation has the greatest impact on the data collected, especially in polymers. The values here are approximate and only meant for relative comparison.

*
Bending stiffnessThe bending stiffness Stiffness is the extent to which an object resists deformation in response to an applied force In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, fro ...
*
Deflection Deflection or deflexion may refer to: * Deflection (ballistics), a technique of shooting ahead of a moving target so that the target and projectile will collide * Deflection (chess), a tactic that forces an opposing chess piece to leave a square ...
*
Deformation Deformation can refer to: * Deformation (engineering), changes in an object's shape or form due to the application of a force or forces. ** Deformation (mechanics), such changes considered and analyzed as displacements of continuum bodies. * Defo ...
*
Flexural modulusIn mechanics Mechanics (Ancient Greek, Greek: ) is the area of physics concerned with the motions of physical objects, more specifically the relationships among force, matter, and motion. Forces applied to objects result in Displacement (vector) ...
*
Hooke's law The balance wheel at the core of many mechanical clocks and watches depends on Hooke's law. Since the torque generated by the coiled spring is proportional to the angle turned by the wheel, its oscillations have a nearly constant period. Hooke ...
*
Impulse excitation techniqueThe impulse excitation technique (IET) is a non-destructive material characterization technique to determine the elastic properties and internal friction of a material of interest. It measures the Fundamental frequency, resonant frequencies in order ...
*
List of materials properties A material's property (or material property) is an Intensive properties, intensive property of some chemical change, material, i.e. a physical property that does not depend on the amount of the material. These quantitative properties may be used as ...
*
Yield (engineering) In materials science The Interdisciplinarity, interdisciplinary field of materials science, also commonly termed materials science and engineering, covers the design and discovery of new materials, particularly solids. The intellectual origin ...

# References

*
ASTM ASTM International, formerly known as American Society for Testing and Materials, is an international standards organization A standards organization, standards body, standards developing organization (SDO), or standards setting organization ( ...
E 111
"Standard Test Method for Young's Modulus, Tangent Modulus, and Chord Modulus"
* The '' ASM Handbook'' (various volumes) contains Young's Modulus for various materials and information on calculations
Online version