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.

Contents

1 Units 2 Other units 3 Usage 3.1 Linear vs Non-linear 3.2 Directional Materials 4 Calculation 4.1 Tension 4.2 Elastic potential energy 5 Approximate values 6 See also

Units

The SI unit of modulus of elasticity is the Pascal. However, given the large values typical of many common materials, figures are often quoted in Mega- or Giga- Pascals for convenience.

Other units

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 uniform - their mechanical properties are the same in all directions.

However, this 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, they 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.

<math> \lambda = \frac{stress}{strain} = \frac{F/A}{x/l} = \frac{F l} {A x} <math>

where

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.

<math>T = \frac{\lambda A x}{l}<math>

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:

<math>E = \frac{\lambda A x^2}{2 l}<math>

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.

Approximate Young's Moduli of Various Solids

Material	Young's modulus (E) in GPa	Young's modulus (E) in PSI
Rubber (small strain)	0.01-0.1	1,500-15,000
polystyrene	3-3.5
nylon	2-4
Oak Wood (along grain)	11	1,600,000
High-Strength Concrete (under compression)	30	4,350,000
Magnesium metal	45	6,500,000
glass	50-90	7,250,000-13,000,000
Aluminium alloys	69	10,000,000
Brasses and bronzes	103-124	17,000,000
Titanium (Ti)	105-120
Carbon Fiber Reinforced Plastic (unidirectional, along grain)	150
Wrought iron and steel	190-210	30,000,000
Tungsten	400-410
Silicon carbide (SiC)	450
Tungsten carbide (WC)	450-650
Diamond	1,050-1,200	150,000,000-175,000,000

See also

• Deformation
• Stress
• Strain
• Tensile_strength
• Yield_strength
• Toughness
• Hardness

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