Young's modulus

From Wikipedia, the free encyclopedia - View original article

 
Jump to: navigation, search

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 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:

Units[edit]

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/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.

Usage[edit]

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[edit]

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 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 for this one.

Directional materials[edit]

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[edit]

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:

 E \equiv \frac{\mbox {tensile stress}}{\mbox {extensional strain}} = \frac{\sigma}{\varepsilon}= \frac{F/A_0}{\Delta L/L_0} = \frac{F L_0} {A_0 \Delta L}

where

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 contracted material[edit]

The Young's modulus of a material can be used to calculate the force it exerts under specific strain.

F = \frac{E A_0 \Delta L} {L_0}

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:

F = \left( \frac{E A_0} {L_0} \right) \Delta L = k x \,

where it comes in saturation

k = \begin{matrix} \frac {E A_0} {L_0} \end{matrix} \, and x = \Delta L. \,

Elastic potential energy[edit]

The elastic potential energy stored is given by the integral of this expression with respect to L:

U_e = \int {\frac{E A_0 \Delta L} {L_0}}\, d\Delta L = \frac {E A_0} {L_0} \int { \Delta L }\, d\Delta L = \frac {E A_0 {\Delta L}^2} {2 L_0}

where Ue is the elastic potential energy.

The elastic potential energy per unit volume is given by:

\frac{U_e} {A_0 L_0} = \frac {E {\Delta L}^2} {2 L_0^2} = \frac {1} {2} E {\varepsilon}^2, where \varepsilon = \frac {\Delta L} {L_0} is the strain in the material.M

This formula can also be expressed as the integral of Hooke's law:

U_e = \int {k x}\, dx = \frac {1} {2} k x^2.

Relation among elastic constants[edit]

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:

E = 2G(1+\nu) = 3K(1-2\nu).\,

Approximate values[edit]

Influences of selected glass component additions on Young's modulus of a specific base glass

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.

Approximate Young's modulus for various materials
MaterialGPalbf/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]0.11–0.4516,000–65,000
HDPE0.8116,000
Polypropylene1.5–2[3]218,000–290,000
Bacteriophage capsids[5]1–3150,000–435,000
Polyethylene terephthalate (PET)2–2.7[3]290,000–390,000
Polystyrene3–3.5[3]440,000–510,000
Nylon2–4290,000–580,000
Diatom frustules (largely silicic acid)[6]0.35–2.7750,000–400,000
Medium-density fiberboard (MDF)[7]4580,000
Oak wood (along grain)11[3]1.60×106
Human Cortical Bone[8]142.03×106
Aromatic peptide nanotubes [9][10]19–272.76×1063.92×106
High-strength concrete30[3]4.35×106
Hemp fiber [11]355.08×106
Magnesium metal (Mg)45[3]6.53×106
Flax fiber [12]588.41×106
Aluminum69[3]10.0×106
Stinging nettle fiber [13]8712.6×106
Glass (see chart)50–90[3]7.25×10613.1×106
Aramid[14]70.5–112.410.2×10616.3×106
Mother-of-pearl (nacre, largely calcium carbonate) [15]7010.2×106
Tooth enamel (largely calcium phosphate)[16]8312.0×106
Brass100–125[3]14.5×10618.1×106
Bronze96–120[3]13.9×10617.4×106
Titanium (Ti)110.316.0×106[3]
Titanium alloys105–120[3]15.0×10617.5×106
Copper (Cu)11717.0×106
Glass-reinforced polyester matrix [17]17.22.49×106
Carbon fiber reinforced plastic (50/50 fibre/matrix, biaxial fabric)30–50[18]4.35×1067.25×106
Carbon fiber reinforced plastic (70/30 fibre/matrix, unidirectional, along grain)[19]18126.3×106
Silicon Single crystal, different directions [20][21]130–18518.9×10626.8×106
Wrought iron190–210[3]27.6×10630.5×106
Steel (ASTM-A36)200[3]29.0×106
polycrystalline Yttrium iron garnet (YIG)[22]19328.0×106
single-crystal Yttrium iron garnet (YIG)[23]20029.0×106
Aromatic peptide nanospheres [24]230–27533.4×10639.9×106
Beryllium (Be)[citation needed]28741.6×106
Molybdenum (Mo)[citation needed]32947.7×106
Tungsten (W)400–410[3]58.0×10659.5×106
Silicon carbide (SiC)450[3]65.3×106
Osmium (Os)[citation needed]55079.8×106
Tungsten carbide (WC)450–650[3]65.3×10694.3×106
Single-walled carbon nanotube[25][26]1,000+145×106+
Graphene1,050[27]145×106
Diamond (C)[28]1,220150×106175×106
Carbyne (C)[29]32,7005388×1065402×106

See also[edit]

References[edit]

  1. ^ IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version:  (2006–) "modulus of elasticity (Young's modulus), E".
  2. ^ 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. ^ 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 2012-01-06. 
  4. ^ "Overview of materials for Low Density Polyethylene (LDPE), Molded". Matweb. Retrieved Feb 7, 2013. 
  5. ^ 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. 
  6. ^ 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. 
  7. ^ Material Properties Data: Medium Density Fiberboard (MDF)
  8. ^ Rho, JY (1993). "Young's modulus of trabecular and cortical bone material: ultrasonic and microtensile measurements". Journal of Biomechanics 26 (2): 111–119. 
  9. ^ Kol, N. et al. (June 8, 2005). "Self-Assembled Peptide Nanotubes Are Uniquely Rigid Bioinspired Supramolecular Structures". Nano Letters 5 (7): 1343–1346. Bibcode:2005NanoL...5.1343K. doi:10.1021/nl0505896. 
  10. ^ 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. 
  11. ^ 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. 
  12. ^ 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. 
  13. ^ 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. 
  14. ^ DuPont (2001). Kevlar Technical Guide. p. 9. 
  15. ^ A. P. Jackson,J. F. V. Vincent and R. M. Turner (1988). "The Mechanical Design of Nacre". Proceedings of the Royal Society B 234 (1277): 415–440. Bibcode:1988RSPSB.234..415J. doi:10.1098/rspb.1988.0056. 
  16. ^ M. Staines, W. H. Robinson and J. A. A. Hood (1981). "Spherical indentation of tooth enamel". Journal of Materials Science. 
  17. ^ Polyester Matrix Composite reinforced by glass fibers (Fiberglass). [SubsTech] (2008-05-17). Retrieved on 2011-03-30.
  18. ^ E-G-nu.htm "Composites Design and Manufacture (BEng) – MATS 324". 
  19. ^ Epoxy Matrix Composite reinforced by 70% carbon fibers [SubsTech]. Substech.com (2006-11-06). Retrieved on 2011-03-30.
  20. ^ Physical properties of Silicon (Si). Ioffe Institute Database. Retrieved on 2011-05-27.
  21. ^ 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. 
  22. ^ 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. 
  23. ^ YIG properties
  24. ^ 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. 
  25. ^ L. Forro et al. "Electronic and mechanical properties of carbon nanotubes". 
  26. ^ 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. 
  27. ^ http://li.mit.edu/A/Papers/07/Liu07.pdf.  Missing or empty |title= (help)
  28. ^ Spear and Dismukes (1994). Synthetic Diamond – Emerging CVD Science and Technology. Wiley, NY. ISBN 978-0-471-53589-8. 
  29. ^ Owano, Nancy (Aug 20, 2013). "Carbyne is stronger than any known material". phys.org. 

Further reading[edit]

External links[edit]

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.
(K,\,E)(K,\,\lambda)(K,\,G)(K,\, \nu)(E,\,G)(E,\,\nu)(\lambda,\,G)(\lambda,\,\nu)(G,\,\nu)(G,\,M)
K=\,KKKK\tfrac{EG}{3(3G-E)}\tfrac{E}{3(1-2\nu)}\lambda+ \tfrac{2G}{3}\tfrac{\lambda(1+\nu)}{3\nu}\tfrac{2G(1+\nu)}{3(1-2\nu)}M - \tfrac{4G}{3}
E=\, E\tfrac{9K(K-\lambda)}{3K-\lambda}\tfrac{9KG}{3K+G}3K(1-2\nu)\,EE\tfrac{G(3\lambda + 2G)}{\lambda + G}\tfrac{\lambda(1+\nu)(1-2\nu)}{\nu}2G(1+\nu)\,\tfrac{G(3M-4G)}{M-G}
\lambda=\,\tfrac{3K(3K-E)}{9K-E}\lambdaK-\tfrac{2G}{3}\tfrac{3K\nu}{1+\nu}\tfrac{G(E-2G)}{3G-E}\tfrac{E\nu}{(1+\nu)(1-2\nu)}\lambda\lambda\tfrac{2 G \nu}{1-2\nu}M - 2G\,
G=\, \tfrac{3KE}{9K-E}\tfrac{3(K-\lambda)}{2}G\tfrac{3K(1-2\nu)}{2(1+\nu)}G\tfrac{E}{2(1+\nu)}G\tfrac{\lambda(1-2\nu)}{2\nu}GG
\nu=\,\tfrac{3K-E}{6K}\tfrac{\lambda}{3K-\lambda}\tfrac{3K-2G}{2(3K+G)}\nu\tfrac{E}{2G}-1\nu\tfrac{\lambda}{2(\lambda + G)}\nu\nu\tfrac{M - 2G}{2M - 2G}
M=\,\tfrac{3K(3K+E)}{9K-E}3K-2\lambda\,K+\tfrac{4G}{3}\tfrac{3K(1-\nu)}{1+\nu}\tfrac{G(4G-E)}{3G-E}\tfrac{E(1-\nu)}{(1+\nu)(1-2\nu)}\lambda+2G\,\tfrac{\lambda(1-\nu)}{\nu}\tfrac{2G(1-\nu)}{1-2\nu} M