Stochastic calculus

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 Thomas Young, the 19th century British scientist. 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:

$E\equiv {\frac {\mbox{tensile stress}}{\mbox{extensional strain}}}={\frac {\sigma }{\varepsilon }}={\frac {F/A_{0}}{\Delta L/L_{0}}}={\frac {FL_{0}}{A_{0}\Delta L}}$

where

E is the Young's modulus (modulus of elasticity)

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.

$F={\frac {EA_{0}\Delta L}{L_{0}}}$

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:

$F=\left({\frac {EA_{0}}{L_{0}}}\right)\Delta L=kx\,$

where it comes in saturation

$k={\begin{matrix}{\frac {EA_{0}}{L_{0}}}\end{matrix}}\,$ and $x=\Delta L.\,$

Elastic potential energy

The elastic potential energy stored is given by the integral of this expression with respect to L:

$U_{e}=\int {\frac {EA_{0}\Delta L}{L_{0}}}\,d\Delta L={\frac {EA_{0}}{L_{0}}}\int {\Delta L}\,d\Delta L={\frac {EA_{0}{\Delta L}^{2}}{2L_{0}}}$

where U_e is the elastic potential energy.

The elastic potential energy per unit volume is given by:

${\frac {U_{e}}{A_{0}L_{0}}}={\frac {E{\Delta L}^{2}}{2L_{0}^{2}}}={\frac {1}{2}}E{\varepsilon }^{2}$, where $\varepsilon ={\frac {\Delta L}{L_{0}}}$ is the strain in the material.M

This formula can also be expressed as the integral of Hooke's law:

$U_{e}=\int {kx}\,dx={\frac {1}{2}}kx^{2}.$

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:

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

Approximate values

Influences of selected glass component additions on Young's modulus of a specific base glass

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.

Approximate Young's modulus for various materials

Material	GPa	lbf/in² (psi)
Rubber (small strain)	0.01–0.1	1,450–14,503
PTFE (Teflon)	0.5	75,000
Low density polyethylene	0.11–0.45	16,000–65,000
HDPE	0.8	116,000
Polypropylene	1.5–2	218,000–290,000
Bacteriophage capsids	1–3	150,000–435,000
Polyethylene terephthalate (PET)	2–2.7	290,000–390,000
Polystyrene	3–3.5	440,000–510,000
Nylon	2–4	290,000–580,000
Diatom frustules (largely silicic acid)	0.35–2.77	50,000–400,000
Medium-density fiberboard (MDF)	4	580,000
Oak wood (along grain)	11	Template:Val
Human Cortical Bone	14	Template:Val
Aromatic peptide nanotubes	19–27	Template:Val–Template:Val
High-strength concrete	30	Template:Val
Hemp fiber	35	Template:Val
Magnesium metal (Mg)	45	Template:Val
Flax fiber	58	Template:Val
Aluminum	69	Template:Val
Stinging nettle fiber	87	Template:Val
Glass (see chart)	50–90	Template:Val – Template:Val
Aramid	70.5–112.4	Template:Val – Template:Val
Mother-of-pearl (nacre, largely calcium carbonate)	70	Template:Val
Tooth enamel (largely calcium phosphate)	83	Template:Val
Brass	100–125	Template:Val – Template:Val
Bronze	96–120	Template:Val – Template:Val
Titanium (Ti)	110.3	Template:Val
Titanium alloys	105–120	Template:Val – Template:Val
Copper (Cu)	117	Template:Val
Glass-reinforced polyester matrix	17.2	Template:Val
Carbon fiber reinforced plastic (50/50 fibre/matrix, biaxial fabric)	30–50	Template:Val – Template:Val
Carbon fiber reinforced plastic (70/30 fibre/matrix, unidirectional, along grain)	181	Template:Val
Silicon Single crystal, different directions	130–185	Template:Val – Template:Val
Wrought iron	190–210	Template:Val – Template:Val
Steel (ASTM-A36)	200	Template:Val
polycrystalline Yttrium iron garnet (YIG)	193	Template:Val
single-crystal Yttrium iron garnet (YIG)	200	Template:Val
Aromatic peptide nanospheres	230–275	Template:Val – Template:Val
Beryllium (Be)Template:Cn	287	Template:Val
Molybdenum (Mo)	329	Template:Val
Tungsten (W)	400–410	Template:Val – Template:Val
Silicon carbide (SiC)	450	Template:Val
Osmium (Os)Template:Cn	550	Template:Val
Tungsten carbide (WC)	450–650	Template:Val – Template:Val
Single-walled carbon nanotube	1,000+	Template:Val+
GrapheneTemplate:Cn	1,000	Template:Val
Diamond (C)	1,220	Template:Val – Template:Val
Carbyne (C)	32,700	Template:Val – Template:Val

See also

• Deflection
• Deformation
• Hardness
• Hooke's law
• Shear modulus
• Bulk Modulus
• Bending stiffness
• Impulse excitation technique
• Toughness
• Yield (engineering)
• List of materials properties

References

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Further reading

• ASTM E 111, "Standard Test Method for Young's Modulus, Tangent Modulus, and Chord Modulus," [1]
• The ASM Handbook (various volumes) contains Young's Modulus for various materials and information on calculations. Online version Template:Subscription required

External links

• Matweb: free database of engineering properties for over 63,000 materials
• Young's Modulus for groups of materials, and their cost

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