Young's modulus
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Young's modulus, also known as the tensile modulus or elastic modulus, is a measure of the stiffness of an elastic material and is a quantity used to characterize materials. It is defined as the ratio of the stress (force per unit area) along an axis to the strain (ratio of deformation over initial length) along that axis in the range of stress in which Hooke's law holds.^{[1]}
Young's modulus is the most common elastic modulus, sometimes called the modulus of elasticity, but there are other elastic moduli 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.^{[2]}
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 to its length. The rigidity characterises the materials (it is an intensive property) while the stiffness regards products and constructions (it is an extensive property): 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.
- rigidity and toughness: toughness is the amount of energy that a material can absorb before fracturing when subjected to strain.
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/m^{2} or m^{−1}·kg·s^{−2}). The practical units used are megapascals (MPa or N/mm^{2}) or gigapascals (GPa or kN/mm^{2}). In United States customary units, it is expressed as pounds (force) per square inch (psi). 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. The Young's modulus directly applies to cases uniaxial stress, that is tensile or compressive stress in one direction and no stress in the other directions. Young's modulus is also 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. Other elastic calculations usually require the use of one additional elastic property, such as the shear modulus, bulk modulus or Poisson's ratio. Any two of these parameters are sufficient to fully describe elasticity in an isotropic material.
Linear versus non-linear
The Young's modulus represents the factor of proportionality in Hooke's law, which relates the stress and the strain. However, Hooke's law is only valid under the assumption of an elastic and 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 solid 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. Otherwise (if the typical stress one would apply is outside the linear range) 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 in such a case of catastrophic failure.
In solid mechanics, the slope of the stress–strain curve at any point is called the tangent 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. The tangent modulus of the initial, linear portion of a stress–strain curve is called Young's modulus.
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
- E is the Young's modulus (modulus of elasticity)
- F is the force exerted on an object under tension;
- A_{0} is the original cross-sectional area through which the force is applied;
- ΔL is the amount by which the length of the object changes;
- L_{0} 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
Elastic potential energy
The elastic potential energy stored is given by the integral of this expression with respect to L:
where U_{e} 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.
Material | GPa | lbf/in² (psi) |
---|---|---|
Rubber (small strain) | 0.01–0.1^{[3]} | 1,450–14,503 |
Low density polyethylene^{[4]} | 0.11–0.45 | 16,000–65,000 |
Diatom frustules (largely silicic acid)^{[5]} | 0.35–2.77 | 50,000–400,000 |
PTFE (Teflon) | 0.5 ^{[3]} | 75,000 |
HDPE | 0.8 | 116,000 |
Bacteriophage capsids^{[6]} | 1–3 | 150,000–435,000 |
Polypropylene | 1.5–2^{[3]} | 218,000–290,000 |
Polyethylene terephthalate (PET) | 2–2.7^{[3]} | 290,000–390,000 |
Nylon | 2–4 | 290,000–580,000 |
Polystyrene | 3–3.5^{[3]} | 440,000–510,000 |
Medium-density fiberboard (MDF)^{[7]} | 4 | 580,000 |
Oak wood (along grain) | 11^{[3]} | Template:Val |
Human Cortical Bone^{[8]} | 14 | Template:Val |
Glass-reinforced polyester matrix ^{[9]} | 17.2 | Template:Val |
Aromatic peptide nanotubes ^{[10]}^{[11]} | 19–27 | Template:Val–Template:Val |
High-strength concrete | 30^{[3]} | Template:Val |
Carbon fiber reinforced plastic (50/50 fibre/matrix, biaxial fabric) | 30–50^{[12]} | Template:Val – Template:Val |
Hemp fiber ^{[13]} | 35 | Template:Val |
Magnesium metal (Mg) | 45^{[3]} | Template:Val |
Glass (see chart) | 50–90^{[3]} | Template:Val – Template:Val |
Flax fiber ^{[14]} | 58 | Template:Val |
Aluminum | 69^{[3]} | Template:Val |
Mother-of-pearl (nacre, largely calcium carbonate) ^{[15]} | 70 | Template:Val |
Aramid^{[16]} | 70.5–112.4 | Template:Val – Template:Val |
Tooth enamel (largely calcium phosphate)^{[17]} | 83 | Template:Val |
Stinging nettle fiber ^{[18]} | 87 | Template:Val |
Bronze | 96–120^{[3]} | Template:Val – Template:Val |
Brass | 100–125^{[3]} | Template:Val – Template:Val |
Titanium (Ti) | 110.3 | Template:Val^{[3]} |
Titanium alloys | 105–120^{[3]} | Template:Val – Template:Val |
Copper (Cu) | 117 | Template:Val |
Carbon fiber reinforced plastic (70/30 fibre/matrix, unidirectional, along grain)^{[19]} | 181 | Template:Val |
Silicon Single crystal, different directions ^{[20]}^{[21]} | 130–185 | Template:Val – Template:Val |
Wrought iron | 190–210^{[3]} | Template:Val – Template:Val |
Steel (ASTM-A36) | 200^{[3]} | Template:Val |
polycrystalline Yttrium iron garnet (YIG)^{[22]} | 193 | Template:Val |
single-crystal Yttrium iron garnet (YIG)^{[23]} | 200 | Template:Val |
Aromatic peptide nanospheres ^{[24]} | 230–275 | Template:Val – Template:Val |
Beryllium (Be)Template:Cn | 287 | Template:Val |
Molybdenum (Mo) | 329 | Template:Val |
Tungsten (W) | 400–410^{[3]} | Template:Val – Template:Val |
Silicon carbide (SiC) | 450^{[3]} | Template:Val |
Tungsten carbide (WC) | 450–650^{[3]} | Template:Val – Template:Val |
Osmium (Os)^{[25]} | 560 | Template:Val |
Single-walled carbon nanotube^{[26]}^{[27]} | 1,000+ | Template:Val+ |
Graphene | 1,050^{[28]} | Template:Val |
Diamond (C)^{[29]} | 1,220 | Template:Val – Template:Val |
Carbyne (C)^{[30]} | ~2,000 | 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
- ↑ Template:GoldBookRef
- ↑ The Rational Mechanics of Flexible or Elastic Bodies, 1638–1788: Introduction to Leonhardi Euleri Opera Omnia, vol. X and XI, Seriei Secundae. Orell Fussli.
- ↑ ^{3.00} ^{3.01} ^{3.02} ^{3.03} ^{3.04} ^{3.05} ^{3.06} ^{3.07} ^{3.08} ^{3.09} ^{3.10} ^{3.11} ^{3.12} ^{3.13} ^{3.14} ^{3.15} ^{3.16} ^{3.17} ^{3.18} Template:Cite web
- ↑ Template:Cite web
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- ↑ Material Properties Data: Medium Density Fiberboard (MDF)
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- ↑ Polyester Matrix Composite reinforced by glass fibers (Fiberglass). [SubsTech] (2008-05-17). Retrieved on 2011-03-30.
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- ↑ Epoxy Matrix Composite reinforced by 70% carbon fibers [SubsTech]. Substech.com (2006-11-06). Retrieved on 2011-03-30.
- ↑ Physical properties of Silicon (Si). Ioffe Institute Database. Retrieved on 2011-05-27.
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- ↑ YIG properties
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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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