Young's modulus Young's modulus, also known as the Tensile
modulus or elastic modulus, is a measure of the stiffness of an elastic isotropic material
and is a quantity used to characterize materials. It is defined as the ratio of the stress along
an axis over the strain along that axis in the range of stress in which Hooke's law holds.
In solid mechanics, the slope of the stress–strain curve at any point is called the tangent modulus.
The tangent modulus of the initial, linear portion of a stress–strain curve is called
Young's 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. In anisotropic
materials, Young's modulus may have different values depending on the direction of the applied
force with respect to the material's structure. Young's modulus is the most common elastic
modulus, sometimes called the modulus of elasticity, but there are other elastic moduli measured,
too, such as the bulk modulus and the shear modulus.
It is named after the 19th-century British scientist Thomas Young. However, the concept
was developed in 1727 by Leonhard Euler, and 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.
A material whose Young's modulus is very high is rigid. Do not confuse:
rigidity and strength: the strength of material is characterized by its yield strength and
/ or its tensile strength; rigidity and stiffness: the beam stiffness
(for example) depends on its Young's modulus but also on the ratio of its section at its
length. The rigidity characterises the materials, the stiffness regards products and constructions:
a massive mechanical plastic part can be much stiffer than a steel spring;
rigidity and hardness: the hardness of a material defines its relative resistance that its surface
opposes to the penetration of a harder body. Units
Young's modulus is the ratio of stress (which has units of pressure) to strain (which is
dimensionless), and so Young's modulus has units of pressure. Its SI unit is therefore
the pascal (Pa or N/m2 or m−1·kg·s−2). The practical units used are megapascals (MPa
or N/mm2) or gigapascals (GPa or kN/mm2). In United States customary units, it is expressed
as pounds (force) per square inch (psi). The abbreviation ksi refers to thousands of psi.
Usage The Young's modulus enables the calculation
of the change in the dimension of a bar made of an isotropic elastic material under tensile
or compressive loads. For instance, it predicts how much a material sample extends under tension
or shortens under compression. Young's modulus is 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. Some calculations also require the use of other material properties,
such as the shear modulus, density, or Poisson's ratio.
Linear versus non-linear The Young's modulus represents the factor
of proportionality in Hooke's law, relating the stress and the strain ; but this law
is only valid under the assumption of an elastic or 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 materials exhibit 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 ; if the
typical stress one would apply is outside the linear range, then the material is said
to be non-linear. Steel, carbon fiber and glass among others
are usually considered linear materials, while other materials such as rubber 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 for this one.
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, 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 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, E, can be calculated by dividing
the tensile stress by the extensional strain in the elastic (initial, linear) portion of
the stress–strain curve: where
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 F is the force exerted by the material when contracted or stretched by ΔL.
Hooke's law can be derived from this formula, which describes the stiffness of an ideal
spring: where it comes in saturation
Elastic potential energy The elastic potential energy stored is given
by the integral of this expression with respect to L:
where Ue is the elastic potential energy. The elastic potential energy per unit volume
is given by: This formula can also be expressed as the
integral of Hooke's law: Relation among elastic constants
For homogeneous isotropic materials simple relations exist between elastic constants
(Young's modulus E, shear modulus G, bulk modulus K, and Poisson's ratio ν) that allow
calculating them all as long as two are known: 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.