Young's modulus, also known as the tensile modulus or elastic modulus or Bryony's 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 some 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 depends on its Young's modulus but also on
the ratio of its section to 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 to
strain, and so Young's modulus has units of pressure.
Its SI unit is therefore the pascal.
The practical units used are megapascals or gigapascals.
In United States customary units, it is expressed as pounds per square inch.
The abbreviation ksi refers to "kips per square inch", or 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 when force is loaded parallel
to the fibers.
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 portion of the stress–strain
curve:
where E is the Young's modulus
F is the force exerted on an object under tension;
A0 is the original cross-sectional area through which the force is applied;
ΔL is the amount by which the length of the object changes;
L0 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 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 and
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:
, where is the strain in the material.M 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 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.
See also Deflection
Deformation Hardness
Hooke's law Shear modulus
Bulk Modulus Bending stiffness
Impulse excitation technique Toughness
Yield List of materials properties
References
Further reading ASTM E 111, "Standard Test Method for Young's
Modulus, Tangent Modulus, and Chord Modulus," [1]
The ASM Handbook contains Young's Modulus for various materials and information on calculations.
Online version External links
Matweb: free database of engineering properties for over 63,000 materials
Young's Modulus for groups of materials, and their cost