![Stress strain ductile](https://upload.wikimedia.org/wikipedia/commons/c/c1/Stress_strain_ductile.svg)
Young's modulus
, the Young modulus, or the
modulus of elasticity
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 ...
in
tension
Tension may refer to:
Science
* Psychological stress
* Tension (physics), a force related to the stretching of an object (the opposite of compression)
* Tension (geology), a stress which stretches rocks in two opposite directions
* Voltage or el ...
or
compression
Compression may refer to:
Physical science
*Compression (physics), size reduction due to forces
*Compression member, a structural element such as a column
*Compressibility, susceptibility to compression
* Gas compression
*Compression ratio, of a ...
(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 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 b ...
of a
solid
Solid is one of the State of matter#Four fundamental states, four fundamental states of matter (the others being liquid, gas, and Plasma (physics), plasma). The molecules in a solid are closely packed together and contain the least amount o ...
material when the force is applied lengthwise. It quantifies the relationship between tensile/compressive
stress
Stress may refer to:
Science and medicine
* Stress (biology), an organism's response to a stressor such as an environmental condition
* Stress (linguistics), relative emphasis or prominence given to a syllable in a word, or to a word in a phrase ...
(force per unit area) and axial
strain
Strain may refer to:
Science and technology
* Strain (biology), variants of plants, viruses or bacteria; or an inbred animal used for experimental purposes
* Strain (chemistry), a chemical stress of a molecule
* Strain (injury), an injury to a mu ...
(proportional deformation) in the
linear elastic region of a material and is determined using the formula:
Young's moduli are typically so large that they are expressed not in
pascals
The pascal (symbol: Pa) is the unit of pressure in the International System of Units (SI), and is also used to quantify internal pressure, stress, Young's modulus, and ultimate tensile strength. The unit, named after Blaise Pascal, is defined ...
but in
gigapascals (GPa).
Example:
*
Silly Putty
Silly Putty is a toy based on silicone polymers that have unusual physical properties. It bounces, but it breaks when given a sharp blow, and it can also flow like a liquid. It contains a viscoelastic liquid silicone, a type of non-Newtonian f ...
(increasing pressure:
length increases quickly, meaning tiny )
*
Aluminum
Aluminium (aluminum in American and Canadian English) is a chemical element with the symbol Al and atomic number 13. Aluminium has a density lower than those of other common metals, at approximately one third that of steel. It has ...
(increasing pressure:
length increases slowly, meaning high )
Higher Young's modulus corresponds to greater (lengthwise) stiffness.
Although Young's modulus is named after the 19th-century British scientist
Thomas Young, the concept was developed in 1727 by
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
. The first experiments that used the concept of Young's modulus in its current form were performed by the Italian scientist
Giordano Riccati in 1782, pre-dating Young's work by 25 years. The term modulus is derived from the
Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...
root term ''
modus'' which means ''measure''.
Definition
Linear elasticity
A solid material will undergo
elastic deformation
In engineering, deformation refers to the change in size or shape of an object. ''Displacements'' are the ''absolute'' change in position of a point on the object. Deflection is the relative change in external displacements on an object. Strain ...
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'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
, and the relationship between stress and strain is described 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 ...
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, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external 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
Strength may refer to:
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 c ...
: 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 language, Polish, Bulgarian language, Bulgarian, Spanish language, Spanish ...
has a higher bending stiffness than a rod of the same material for a given mass per length;
*
Hardness
In materials science, hardness (antonym: softness) is a measure of the resistance to localized plastic deformation induced by either mechanical indentation or abrasion. In general, different materials differ in their hardness; for example hard ...
: relative resistance of the material's surface to penetration by a harder body;
*
Toughness
In materials science and metallurgy, toughness is the ability of a material to absorb energy and plastically deform without fracturing.[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 describe ...]
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 determinate 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 grou ...
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, 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 \ \stackre ...
,
bulk modulus
The bulk modulus (K or B) of a substance is a measure of how resistant to compression the substance is. It is defined as the ratio of the infinitesimal pressure increase to the resulting ''relative'' decrease of the volume.
Other moduli describ ...
, and
Poisson's ratio
In materials science and solid mechanics, Poisson's ratio \nu ( nu) is a measure of the Poisson effect, the deformation (expansion or contraction) of a material in directions perpendicular to the specific direction of loading. The value of Po ...
. 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:
:
Linear versus non-linear
Young's modulus represents the factor of proportionality in
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 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 made up of iron with added carbon to improve its strength and fracture resistance compared to other forms of iron. Many other elements may be present or added. Stainless steels that are corrosion- and oxidation-resistant ty ...
,
carbon fiber
Carbon fiber-reinforced polymers (American English), carbon-fibre-reinforced polymers (Commonwealth English), carbon-fiber-reinforced plastics, carbon-fiber reinforced-thermoplastic (CFRP, CRP, CFRTP), also known as carbon fiber, carbon compo ...
and
glass
Glass is a non-crystalline, often 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 rapid cooling (quenching) of ...
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 polymers of the organic compound isoprene, with minor impurities of other organic compounds. Thailand, Malaysia, and ...
and
soils
Soil, also commonly referred to as earth or dirt, is a mixture of organic matter, minerals, gases, liquids, and organisms that together support life. Some scientific definitions distinguish ''dirt'' from ''soil'' by restricting the former term ...
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 under the action of forces, temperature changes, phase changes, and ot ...
, the slope of the
stress–strain curve at any point is called the
tangent modulus {{Use dmy dates, date=November 2017
In solid mechanics, the tangent modulus is the slope of the stress– strain curve at any specified stress or strain. Below the proportional limit (the limit of the linear elastic regime) the tangent modulus is eq ...
. It can be experimentally determined from the
slope
In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is use ...
of a stress–strain curve created during
tensile test
Tensile testing, also known as tension testing, is a fundamental materials science and engineering test in which a sample is subjected to a controlled tension until failure. Properties that are directly measured via a tensile test are ultimat ...
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 . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe ...
, 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. It can be defined as a difference, when measured along different axes, in a material's physic ...
, 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 polymers (American English), carbon-fibre-reinforced polymers (Commonwealth English), carbon-fiber-reinforced plastics, carbon-fiber reinforced-thermoplastic (CFRP, CRP, CFRTP), also known as carbon fiber, carbon compo ...
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 trees and other woody plants. It is an organic materiala natural composite of cellulose fibers that are strong in tension and embedded in a matrix of lignin th ...
and
reinforced concrete
Reinforced concrete (RC), also called reinforced cement concrete (RCC) and ferroconcrete, is a composite material in which concrete's relatively low tensile strength and ductility are compensated for by the inclusion of reinforcement having hig ...
. 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
where the electron work function varies with the temperature as
and
is a calculable material property which is dependent on the crystal structure (for example, BCC, FCC).
is the electron work function at T=0 and
is constant throughout the change.
Calculation
Young's modulus ''E'', can be calculated by dividing the
tensile stress
In continuum mechanics, stress is a physical quantity. It is a quantity that describes the magnitude of forces that cause deformation. Stress is defined as ''force per unit area''. When an object is pulled apart by a force it will cause elonga ...
,
, by the
engineering extensional strain,
, in the elastic (initial, linear) portion of the physical
stress–strain curve:
where
*
is the Young's modulus (modulus of elasticity)
*
is the force exerted on an object under tension;
*
is the actual cross-sectional area, which equals the area of the cross-section perpendicular to the applied force;
*
is the amount by which the length of the object changes (
is positive if the material is stretched, and negative when the material is compressed);
*
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.
:
where
is the force exerted by the material when contracted or stretched 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 ...
for a stretched wire can be derived from this formula:
:
where it comes in saturation
:
and
But note that the elasticity of coiled springs comes from
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 \ \stackre ...
, not Young's modulus.
Elastic potential energy
The
elastic potential energy
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, ...
stored in a linear elastic material is given by the integral of the Hooke's law:
:
now by explicating the intensive variables:
:
This means that the elastic potential energy density (that is, per unit volume) is given by:
:
or, in simple notation, for a linear elastic material:
, since the strain is defined
.
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
In mathematics, a quadratic polynomial is a polynomial of degree two in one or more variables. A quadratic function is the polynomial function defined by a quadratic polynomial. Before 20th century, the distinction was unclear between a polynomial ...
of the strain:
:
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
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 an ...
. The values here are approximate and only meant for relative comparison.
See also
*
Bending stiffness The bending stiffness (K) is the resistance of a member against bending deformation. It is a function of the Young's modulus E, the second moment of area I of the beam cross-section about the axis of interest, length of the beam and beam boundary c ...
*
Deflection
Deflection or deflexion may refer to:
Board games
* Deflection (chess), a tactic that forces an opposing chess piece to leave a square
* Khet (game), formerly ''Deflexion'', an Egyptian-themed chess-like game using lasers
Mechanics
* Deflection ...
*
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 (physics), such changes considered and analyzed as displacements of continuum bodies.
* Defor ...
*
Flexural modulus In mechanics, the flexural modulus or bending modulus is an intensive property that is computed as the ratio of stress to strain in flexural deformation, or the tendency for a material to resist bending. It is determined from the slope of a stres ...
*
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 ...
*
Impulse excitation technique
The 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 resonant frequencies in order to calculate the You ...
*
List of materials properties
A materials property is an intensive property of a material, i.e., a physical property that does not depend on the amount of the material. These quantitative properties may be used as a metric by which the benefits of one material versus another c ...
*
Yield (engineering)
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 wi ...
References
Further reading
*
ASTM E 111
"Standard Test Method for Young's Modulus, Tangent Modulus, and Chord Modulus"
* The ''
ASM Handbook
ASM may refer to:
Codes
* American Samoa, ISO 3166-1 alpha-3 country code
* Asmara International Airport, IATA airport code
* Assamese language, ISO 639 language code ''asm''
Organizations
*Aare Seeland mobil, a Swiss bus and train operator
* Al ...
'' (various volumes) contains Young's Modulus for various materials and information on calculations
Online version
External links
Matweb: free database of engineering properties for over 115,000 materials
{{Authority control
Elasticity (physics)
Physical quantities
Structural analysis