Young's modulus
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.^{1} 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 the 19thcentury 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, predating 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 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.
Contents
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 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 nonlinear
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 nonlinear.
Steel, carbon fiber and glass among others are usually considered linear materials, while other materials such as rubber and soils are nonlinear. However, this is not an absolute classification : if very small stresses or strains are applied to a nonlinear 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:
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 crosssectional 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
 and
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:
 , 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 (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 
PTFE (Teflon)  0.5 ^{3}  75,000 
Low density polyethylene^{4}  0.11–0.45  16,000–65,000 
HDPE  0.8  116,000 
Polypropylene  1.5–2^{3}  218,000–290,000 
Bacteriophage capsids^{5}  1–3  150,000–435,000 
Polyethylene terephthalate (PET)  2–2.7^{3}  290,000–390,000 
Polystyrene  3–3.5^{3}  440,000–510,000 
Nylon  2–4  290,000–580,000 
Diatom frustules (largely silicic acid)^{6}  0.35–2.77  50,000–400,000 
Mediumdensity fiberboard (MDF)^{7}  4  580,000 
Oak wood (along grain)  11^{3}  1.60×10^{6} 
Human Cortical Bone^{8}  14  2.03×10^{6} 
Aromatic peptide nanotubes ^{9}^{10}  19–27  2.76×10^{6}–3.92×10^{6} 
Highstrength concrete  30^{3}  4.35×10^{6} 
Hemp fiber ^{11}  35  5.08×10^{6} 
Magnesium metal (Mg)  45^{3}  6.53×10^{6} 
Flax fiber ^{12}  58  8.41×10^{6} 
Aluminum  69^{3}  10.0×10^{6} 
Stinging nettle fiber ^{13}  87  12.6×10^{6} 
Glass (see chart)  50–90^{3}  7.25×10^{6} – 13.1×10^{6} 
Aramid^{14}  70.5–112.4  10.2×10^{6} – 16.3×10^{6} 
Motherofpearl (nacre, largely calcium carbonate) ^{15}  70  10.2×10^{6} 
Tooth enamel (largely calcium phosphate)^{16}  83  12.0×10^{6} 
Brass  100–125^{3}  14.5×10^{6} – 18.1×10^{6} 
Bronze  96–120^{3}  13.9×10^{6} – 17.4×10^{6} 
Titanium (Ti)  110.3  16.0×10^{6}^{3} 
Titanium alloys  105–120^{3}  15.0×10^{6} – 17.5×10^{6} 
Copper (Cu)  117  17.0×10^{6} 
Glassreinforced polyester matrix ^{17}  17.2  2.49×10^{6} 
Carbon fiber reinforced plastic (50/50 fibre/matrix, biaxial fabric)  30–50^{18}  4.35×10^{6} – 7.25×10^{6} 
Carbon fiber reinforced plastic (70/30 fibre/matrix, unidirectional, along grain)^{19}  181  26.3×10^{6} 
Silicon Single crystal, different directions ^{20}^{21}  130–185  18.9×10^{6} – 26.8×10^{6} 
Wrought iron  190–210^{3}  27.6×10^{6} – 30.5×10^{6} 
Steel (ASTMA36)  200^{3}  29.0×10^{6} 
polycrystalline Yttrium iron garnet (YIG)^{22}  193  28.0×10^{6} 
singlecrystal Yttrium iron garnet (YIG)^{23}  200  29.0×10^{6} 
Aromatic peptide nanospheres ^{24}  230–275  33.4×10^{6} – 39.9×10^{6} 
Beryllium (Be)^{citation needed}  287  41.6×10^{6} 
Molybdenum (Mo)^{citation needed}  329  47.7×10^{6} 
Tungsten (W)  400–410^{3}  58.0×10^{6} – 59.5×10^{6} 
Silicon carbide (SiC)  450^{3}  65.3×10^{6} 
Osmium (Os)^{citation needed}  550  79.8×10^{6} 
Tungsten carbide (WC)  450–650^{3}  65.3×10^{6} – 94.3×10^{6} 
Singlewalled carbon nanotube^{25}^{26}  1,000+  145×10^{6}+ 
Graphene  1,050^{27}  145×10^{6} 
Diamond (C)^{28}  1,220  150×10^{6} – 175×10^{6} 
Carbyne (C)^{29}  32,700  5388×10^{6} – 5402×10^{6} 
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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 ^ The Rational Mechanics of Flexible or Elastic Bodies, 1638–1788: Introduction to Leonhardi Euleri Opera Omnia, vol. X and XI, Seriei Secundae. Orell Fussli.
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} ^{f} ^{g} ^{h} ^{i} ^{j} ^{k} ^{l} ^{m} ^{n} ^{o} ^{p} ^{q} ^{r} ^{s} "Elastic Properties and Young Modulus for some Materials". The Engineering ToolBox. Retrieved 20120106.
 ^ "Overview of materials for Low Density Polyethylene (LDPE), Molded". Matweb. Retrieved Feb 7, 2013.
 ^ Ivanovska IL, de Pablo PJ, Sgalari G, MacKintosh FC, Carrascosa JL, Schmidt CF, Wuite GJL (2004). "Bacteriophage capsids: Tough nanoshells with complex elastic properties". Proc Nat Acad Sci USA. 101 (20): 7600–5. Bibcode:2004PNAS..101.7600I. doi:10.1073/pnas.0308198101. PMC 419652. PMID 15133147.
 ^ Subhash G, Yao S, Bellinger B, Gretz MR. (2005). "Investigation of mechanical properties of diatom frustules using nanoindentation". J Nanosci Nanotechnol. 5 (1): 50–6. doi:10.1166/jnn.2005.006. PMID 15762160.
 ^ Material Properties Data: Medium Density Fiberboard (MDF)
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 ^ M. Staines, W. H. Robinson and J. A. A. Hood (1981). "Spherical indentation of tooth enamel". Journal of Materials Science.
 ^ Polyester Matrix Composite reinforced by glass fibers (Fiberglass). [SubsTech] (20080517). Retrieved on 20110330.
 ^ EGnu.htm "Composites Design and Manufacture (BEng) – MATS 324".
 ^ Epoxy Matrix Composite reinforced by 70% carbon fibers [SubsTech]. Substech.com (20061106). Retrieved on 20110330.
 ^ Physical properties of Silicon (Si). Ioffe Institute Database. Retrieved on 20110527.
 ^ E.J. Boyd et al. (February 2012). "Measurement of the Anisotropy of Young's Modulus in SingleCrystal Silicon". Journal of Microelectromechanical Systems 21 (1): 243–249. doi:10.1109/JMEMS.2011.2174415.
 ^ Chou, H. M.; Case, E. D. (November 1988). "Characterization of some mechanical properties of polycrystalline yttrium iron garnet (YIG) by nondestructive methods". Journal of Materials Science Letters 7 (11): 1217–1220. doi:10.1007/BF00722341.
 ^ YIG properties
 ^ AdlerAbramovich, L. et al. (December 17, 2010). "SelfAssembled Organic Nanostructures with MetallicLike Stiffness". Angewandte Chemie International Edition 49 (51): 9939–9942. doi:10.1002/anie.201002037.
 ^ L. Forro et al. "Electronic and mechanical properties of carbon nanotubes".
 ^ Y.H.Yang et al.; Li, W. Z. (2011). "Radial elasticity of singlewalled carbon nanotube measured by atomic force microscopy". Applied Physics Letters 98 (4): 041901. Bibcode:2011ApPhL..98d1901Y. doi:10.1063/1.3546170.
 ^ http://li.mit.edu/A/Papers/07/Liu07.pdf. Missing or empty
title=
(help)  ^ Spear and Dismukes (1994). Synthetic Diamond – Emerging CVD Science and Technology. Wiley, NY. ISBN 9780471535898.
 ^ Owano, Nancy (Aug 20, 2013). "Carbyne is stronger than any known material". phys.org.
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 (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

Conversion formulas  

Homogeneous isotropic linear elastic materials have their elastic properties uniquely determined by any two moduli among these, thus given any two, any other of the elastic moduli can be calculated according to these formulas.  