Young's modulus? |
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In solid mechanics, Young's modulus (also known as the modulus of elasticity or elastic modulus) is a measure of the stiffness of a given material. It is defined as the limit for small strains of the rate of change of stress with strain. This can be experimentally determined from the slope of a stress-strain curve created during tensile tests conducted on a sample of the material. Young's modulus is named after Thomas Young the English physicist, physician, and Egyptologist.
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Units
The SI unit of modulus of elasticity is the pascal. Given the large values typical of many common materials, figures are often quoted in megapascals or gigapascals.
The modulus of elasticity can also be measured in other units of pressure, for example pounds per square inch (psi).
Usage
The Young's modulus allows the behavior of a material under load to be calculated. For instance, it can be used to predict the amount a wire will extend under tension, or to predict the load at which a thin column will buckle under compression. Some calculations also require the use of other material properties, such as the shear modulus, density, or Poisson's ratio.
Linear vs non-linear
For many materials, Young's modulus is a constant over a range of strains. Such materials are called linear, and are said to obey Hooke's law. Examples of linear materials include steel, carbon fiber, and glass. Rubber is a non-linear material.
Directional materials
Most metals and ceramics, along with many other materials, are isotropic - their mechanical properties are the same in all directions.
It is not always the case. Some materials, particularly those which are composites of two or more ingredients have a "grain" or similar mechanical structure. As a result, these anisotropic materials have different mechanical properties when load is applied in different directions. For example, carbon fiber is much stiffer (higher Young's Modulus) when loaded parallel to the fibers (along the grain). Other such materials include wood and reinforced concrete.
Calculation
The modulus of elasticity, λ, can be calculated by dividing the stress by the strain, i.e.
where (in SI units)
λ is the modulus of elasticity, measured in pascals
F is the force, measured in newtons
A is the cross-sectional area through which the force is applied, measured in square metres
x is the extension, measured in metres
l is the natural length, measured in metres
Tension
The modulus of elasticity of a material can be used to calculate the tension force it exerts under a specific extension.
where
T is the tension, measured in newtons
Elastic potential energy
The elastic potential energy stored is given by the integral of this expression with respect to x, i.e. energy stored E is given by:
where
E is the elastic potential energy, measured in joules
Approximate values
Note that Young's Modulus can vary considerably depending on the exact composition of the material. For example, the value for most metals can vary by 5% or more, depending on the precise composition of the alloy and any heat treatment applied during manufacture. As such, many of the values here are very approximate.
| Material | Young's modulus (E) in GPa | Young's modulus (E) in lbf/in² (psi) |
|---|---|---|
| Rubber (small strain) | 0.01-0.1 | 1,500-15,000 |
| Low density polyethylene | 0.2 | 30,000 |
| Polypropylene | 1.5-2 | 217,000-290,000 |
| Polyethylene terephthalate | 2-2.5 | 290,000-360,000 |
| Polystyrene | 3-3.5 | 435,000-505,000 |
| Nylon | 2-4 | 290,000-580,000 |
| Oak wood (along grain) | 11 | 1,600,000 |
| High-strength concrete (under compression) | 30 | 4,350,000 |
| Magnesium metal | 45 | 6,500,000 |
| Aluminium alloy | 69 | 10,000,000 |
| Glass (all types) | 72 | 10,400,000 |
| Brass and bronze | 103-124 | 17,000,000 |
| Titanium (Ti) | 105-120 | 15,000,000-17,500,000 |
| Carbon fiber reinforced plastic (unidirectional, along grain) | 150 | 21,800,000 |
| Wrought iron and steel | 190-210 | 30,000,000 |
| Tungsten (W) | 400-410 | 58,000,000-59,500,000 |
| Silicon carbide (SiC) | 450 | 65,000,000 |
| Tungsten carbide (WC) | 450-650 | 65,000,000-94,000,000 |
| Single Carbon nanotube 1 | approx. 1,000 | approx. 145,000,000 |
| Diamond | 1,050-1,200 | 150,000,000-175,000,000 |