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 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.
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, predating Young's work by 25 years.
The SI unit of modulus of elasticity (E, or less commonly Y) is 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.
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
For many materials, the ratio of stress over strain is essentially constant over a range of strains. Such materials are called linear, and are said to obey Hooke's law. Examples of linear materials are steel, carbon fiber and glass. Young's modulus does not apply to Non-linear materials, including rubber and soils, except under very small strains.
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.
- 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 compressed 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 compressed 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 Ue 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:
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.
|Rubber (small strain)||0.01–0.1|
|PTFE (Teflon)||0.5||75,000|
|Low density polyethylene||0.11 - 0.45||16,000-65,000|
|Polyethylene terephthalate (PET)||2-2.7|
|Diatom frustules (largely silicic acid)||0.35–2.77||50,000–400,000|
|Medium-density fiberboard (MDF)||4||580,000|
|Pine wood (along grain)||9||1,300,000|
|Oak wood (along grain)||11|
|Human Cortical Bone||14||2,030,000|
|Aromatic peptide nanotubes,||19-27|
|Hemp fiber ||35|
|Magnesium metal (Mg)||45|
|Flax fiber ||58|
|Stinging nettle fiber ||87|
|Glass (see chart)||50–90|
|Mother-of-pearl (nacre, largely calcium carbonate) ||70||10,000,000|
|Tooth enamel (largely calcium phosphate)||83||12,000,000|
|Glass-reinforced plastic (70/30 by weight fibre/matrix, unidirectional, along grain)||40–45||5,800,000–6,500,000|
|Glass-reinforced polyester matrix ||17.2||2,500,000|
|Carbon fiber reinforced plastic (50/50 fibre/matrix, biaxial fabric)||30-50|
|Carbon fiber reinforced plastic (70/30 fibre/matrix, unidirectional, along grain)||181||26,300,000|
|Silicon Single crystal, different directions ||130-185|
|polycrystalline Yttrium iron garnet (YIG)||193||28,000,000|
|single-crystal Yttrium iron garnet (YIG)||200||30,000,000|
|Aromatic peptide nanospheres ||230-275|
|Sapphire (Al2O3) along C-axis||435||63,000,000|
|Silicon carbide (SiC)||450|
|Tungsten carbide (WC)||450–650|
|Single-walled carbon nanotube||1,000+||145,000,000+|
- Hooke's law
- Shear modulus
- Bending stiffness
- Impulse excitation technique
- Yield (engineering)
- List of materials properties
- IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "modulus of elasticity (Young's modulus), E".
- The Rational Mechanics of Flexible or Elastic Bodies, 1638–1788: Introduction to Leonhardi Euleri Opera Omnia, vol. X and XI, Seriei Secundae. Orell Fussli.
- "Elastic Properties and Young Modulus for some Materials". The Engineering ToolBox. Retrieved 2012-01-06.
- "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)
- Rho, JY (1993). "Young's modulus of trabecular and cortical bone material: ultrasonic and microtensile measurements". Journal of Biomechanics 26 (2): 111–119.
- Kol, N. et al. (June 8, 2005). "Self-Assembled Peptide Nanotubes Are Uniquely Rigid Bioinspired Supramolecular Structures". Nano Letters 5 (7): 1343–1346. doi:10.1021/nl0505896.
- Niu, L. et al. (June 6, 2007). "Using the Bending Beam Model to Estimate the Elasticity of Diphenylalanine Nanotubes". Langmuir 23 (14): 7443–7446. doi:10.1021/la7010106.
- Nabi Saheb, D.; Jog, JP. (1999). "Natural fibre polymer composites: a review". Advances in Polymer Technology 18 (4): 351–363. doi:10.1002/(SICI)1098-2329(199924)18:4<351::AID-ADV6>3.0.CO;2-X.
- Bodros, E. (2002). "Analysis of the flax fibres tensile behaviour and analysis of the tensile stiffness increase". Composite Part A 33 (7): 939–948. doi:10.1016/S1359-835X(02)00040-4.
- Bodros, E.; Baley, C. (15 May 2008). "Study of the tensile properties of stinging nettle fibres (Urtica dioica)". Materials Letters 62 (14): 2143–2145. doi:10.1016/j.matlet.2007.11.034.
- DuPont (2001). Kevlar Technical Guide. p. 9.
- A. P. Jackson,J. F. V. Vincent and R. M. Turner (1988). "The Mechanical Design of Nacre". Proc. R. Soc. Lond. B 234 (1277): 415–440. Bibcode:1988RSPSB.234..415J. doi:10.1098/rspb.1988.0056.
- 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] (2008-05-17). Retrieved on 2011-03-30.
- E-G-nu.htm "Composites Design and Manufacture (BEng) - MATS 324".
- 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.
- E.J. Boyd et al. (February, 2012). "Measurement of the Anisotropy of Young's Modulus in Single-Crystal 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 non-destructive methods". Journal of Materials Science Letters 7 (11): 1217–1220. doi:10.1007/BF00722341.
- YIG properties
- Adler-Abramovich, L. et al. (December 17, 2010). "Self-Assembled Organic Nanostructures with Metallic-Like 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 single-walled carbon nanotube measured by atomic force microscopy". Applied Physics Letters 98 (4): 041901. Bibcode:2011ApPhL..98d1901Y. doi:10.1063/1.3546170.
- Spear and Dismukes (1994). Synthetic Diamond – Emerging CVD Science and Technology. Wiley, NY. ISBN 978-0-471-53589-8.
- ASTM E 111, "Standard Test Method for Young's Modulus, Tangent Modulus, and Chord Modulus," 
- The ASM Handbook (various volumes) contains Young's Modulus for various materials and information on calculations. Online version (subscription required)
- Matweb: free database of engineering properties for over 63,000 materials
- Young's Modulus for groups of materials, and their cost
|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.|