Young's modulus (or the Young modulus) is a mechanical property of solid materials 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 ...

when the force is applied lengthwise. It is the modulus of elasticity for tension or axial compression. Young's modulus is defined as the ratio of the stress (force per unit area) applied to the object and the resulting axial strain (displacement or deformation) in the linear elastic region of the material. 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 polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...

. The first experiments that used the concept of Young's modulus in its modern 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 languages, Italic branch of the Indo-European languages. Latin was originally spoken by the Latins (Italic tribe), Latins in Latium (now known as Lazio), the lower Tiber area aroun ...

root term '' modus'', which means ''measure''.

Definition

Young's modulus,

E

, quantifies the relationship between tensile or compressive stress

\sigma

(force per unit area) and axial strain

\varepsilon

(proportional deformation) in the linear elastic region of a material:

E = \frac

Young's modulus is commonly measured in the

International System of Units The International System of Units, internationally known by the abbreviation SI (from French ), is the modern form of the metric system and the world's most widely used system of measurement. It is the only system of measurement with official s ...

(SI) in multiples of the pascal (Pa) and common values are in the range of gigapascals (GPa). Examples: *

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. Types of polyisoprene ...

(increasing pressure: ''large length increase, meaning low

E

'') *

Aluminium Aluminium (or aluminum in North American English) is a chemical element; it has chemical symbol, symbol Al and atomic number 13. It has a density lower than that of other common metals, about one-third that of steel. Aluminium has ...

(increasing pressure: ''small length increase, meaning high

E

'')

Linear elasticity

A solid material undergoes elastic deformation 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, and the relationship between stress and strain is described by Hooke's law 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 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.

Related but distinct properties

Material stiffness is a distinct property from the following: * Strength: 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 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 plastic deformation, such as an indentation (over an area) or a scratch (linear), induced mechanically either by Pressing (metalworking), pressing or abrasion ...

: 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 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 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

G

, bulk modulus

K

, and Poisson's ratio

\nu

. Any two of these parameters are sufficient to fully describe elasticity in an isotropic material. For example, calculating physical properties of cancerous skin tissue, has been measured and found to be a Poisson’s ratio of 0.43±0.12 and an average Young’s modulus of 52 KPa. Defining the elastic properties of skin may become the first step in turning elasticity into a clinical tool. For homogeneous isotropic materials simple relations exist between elastic constants that allow calculating them all as long as two are known: :

E = 2G(1+\nu) = 3K(1-2\nu).

Linear versus non-linear

Young's modulus represents the factor of proportionality in Hooke's law, 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 of iron and carbon that demonstrates improved mechanical properties compared to the pure form of iron. Due to steel's high Young's modulus, elastic modulus, Yield (engineering), yield strength, Fracture, fracture strength a ...

, carbon fiber and

glass Glass is an amorphous (non-crystalline solid, non-crystalline) solid. Because it is often transparency and translucency, transparent and chemically inert, glass has found widespread practical, technological, and decorative use in window pane ...

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. Types of polyisoprene ...

and soils 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, the slope of the stress–strain curve at any point is called the tangent modulus. It can be experimentally determined from the slope of a stress–strain curve created during tensile tests 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, 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 structural property of non-uniformity in different directions, as opposed to isotropy. An anisotropic object or pattern has properties that differ according to direction of measurement. For example, many materials exhibit ver ...

, 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 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 and reinforced concrete. Engineers can use this directional phenomenon to their advantage in creating structures.

Calculation

Young's modulus is calculated by dividing the tensile stress,

\sigma(\varepsilon)

, by the engineering extensional strain,

\varepsilon

, in the elastic (initial, linear) portion of the physical stress–strain curve:

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

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 for a stretched wire can be derived from this formula: :

F = \left( \frac  \right) \, \Delta L = k x

where it comes in saturation :

k \equiv \frac   \,

and

x \equiv \Delta L.

Note that the elasticity of coiled springs comes from shear modulus, not Young's modulus. When a spring is stretched, its wire's length doesn't change, but its shape does. This is why only the shear modulus of elasticity is involved in the stretching of a spring.

Elastic potential energy

The elastic potential energy 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   =\frac \times \frac  \times \frac  = \frac  \times \sigma(\varepsilon) \times \varepsilon

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(\varepsilon) = \int E(\varepsilon) \, \varepsilon \, d\varepsilon \ne \frac   E \varepsilon^2

Examples

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

polymer A polymer () is a chemical substance, substance or material that consists of very large molecules, or macromolecules, that are constituted by many repeat unit, repeating subunits derived from one or more species of monomers. Due to their br ...

s. The values here are approximate and only meant for relative comparison.

References

External links

Matweb: free database of engineering properties for over 175,000 materials

{{Authority control Elasticity (physics) Physical quantities Structural analysis