Movatterモバイル変換

[0]ホーム
URL:

Jump to content
WikipediaThe Free Encyclopedia
Search

Young's modulus

From Wikipedia, the free encyclopedia
Mechanical property that measures stiffness of a solid material

Young's modulus is the slope of the linear part of thestress–strain curve for a material under tension or compression.

Young's modulus (or theYoung modulus) is a mechanical property of solid materials that measures the tensile or compressivestiffness when the force is applied lengthwise. It is theelastic modulus fortension or axialcompression. Young's modulus is defined as the quotient of thestress (force per unit area) applied to the object and the resulting axialstrain (a dimensionless quantity that quantifies relative deformation) in thelinear elastic region of the material. As such, Young's modulus is similar to and proportional to the spring constant inHooke's law, but with dimensions of pressure instead of force per distance.

Although Young's modulus is named after the 19th-century British scientistThomas Young, the concept was developed in 1727 byLeonhard Euler. The first experiments that used the concept of Young's modulus in its modern form were performed by the Italian scientistGiordano Riccati in 1782, pre-dating Young's work by 25 years.[1] The term modulus is derived from theLatinroot termmodus, which meansmeasure.

Definition

[edit]

Young's modulus,E{\displaystyle E}, quantifies the relationship between tensile or compressivestressσ{\displaystyle \sigma } (force per unit area) and axialstrainε{\displaystyle \varepsilon } (proportional deformation) in thelinear elastic region of a material:[2]E=σε{\displaystyle E={\frac {\sigma }{\varepsilon }}}

Young's modulus is commonly measured in theInternational System of Units (SI) in multiples of thepascal (Pa) and common values are in the range ofgigapascals (GPa).

Examples:

Linear elasticity

[edit]
Main article:Linear elasticity

A solid material undergoeselastic deformation when a small load is applied to it in compression or extension. Elastic deformation is reversible, meaning that the material returns to its original shape after the load is removed.

At near-zero stress and strain, the stress–strain curve islinear, and the relationship between stress and strain is described byHooke's law that states stress is proportional to strain. The coefficient of proportionality is Young's modulus. The higher the modulus, the more stress is needed to create the same amount of strain; an idealizedrigid body would have an infinite Young's modulus. Conversely, a very soft material (such as a fluid) would deform without force, and would have zero Young's modulus.

Related but distinct properties

[edit]

Material stiffness is a distinct property from the following:

  • Strength: maximum amount of stress that material can withstand while staying in the elastic (reversible) deformation regime;
  • Geometric stiffness: a global characteristic of the body that depends on its shape, and not only on the local properties of the material; for instance, anI-beam has a higher bending stiffness than a rod of the same material for a given mass per length;
  • Hardness: relative resistance of the material's surface to penetration by a harder body;
  • Toughness: amount of energy that a material can absorb before fracture.
  • The point E is the elastic limit or the yield point of the material within which the stress is proportional to strain and the material regains its original shape after removal of the external force.

Usage

[edit]

Young's modulus enables the calculation of the change in the dimension of a bar made of anisotropic 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 of 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 thedeflection that will occur in astatically determinatebeam 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 theshear modulusG{\displaystyle G},bulk modulusK{\displaystyle K}, andPoisson's ratioν{\displaystyle \nu }. Any two of these parameters are sufficient to fully describe elasticity in an isotropic material. For example, calculating physical properties of cancerous skin tissue, has been measured and found to be a Poisson's ratio of 0.43±0.12 and an average Young's modulus of 52 KPa. Defining the elastic properties of skin may become the first step in turning elasticity into a clinical tool.[3] For homogeneous isotropic materialssimple relations exist between elastic constants that allow calculating them all as long as two are known:

E=2G(1+ν)=3K(1−2ν).{\displaystyle E=2G(1+\nu )=3K(1-2\nu ).}

Linear versus non-linear

[edit]

Young's modulus represents the factor of proportionality inHooke's law, which relates the stress and the strain. However, Hooke's law is only valid under the assumption of anelastic andlinear 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 nearly 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 andglass among others are usually considered linear materials, while other materials such asrubber andsoils 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 impliesreversibility, 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.

Insolid mechanics, the slope of thestress–strain curve at any point is called thetangent modulus. It can be experimentally determined from theslope of a stress–strain curve created duringtensile tests conducted on a sample of the material.

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, areisotropic, 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 becomeanisotropic, and Young's modulus will change depending on the direction of the force vector.[4] Anisotropy can be seen in many composites as well. For example,carbon fiber has a much higher Young's modulus (is much stiffer) when force is loaded parallel to the fibers (along the grain). Other such materials includewood andreinforced concrete. Engineers can use this directional phenomenon to their advantage in creating structures.

Calculation

[edit]

Young's modulus is calculated by dividing thetensile stress,σ(ε){\displaystyle \sigma (\varepsilon )}, by theengineering extensional strain,ε{\displaystyle \varepsilon }, in the elastic (initial, linear) portion of the physicalstress–strain curve:

E≡σ(ε)ε=F/AΔL/L0=FL0AΔL{\displaystyle E\equiv {\frac {\sigma (\varepsilon )}{\varepsilon }}={\frac {F/A}{\Delta L/L_{0}}}={\frac {FL_{0}}{A\,\Delta L}}}where

An equivalent definition of Young's modulus isE=L0Ad2UdL2|L0{\displaystyle E={\frac {L_{0}}{A}}\left.{\frac {d^{2}U}{dL^{2}}}\right|_{L_{0}}}whereU{\displaystyle U} is material's energy.

Force exerted by stretched or contracted material

[edit]

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

F=EAΔLL0{\displaystyle F={\frac {EA\,\Delta L}{L_{0}}}}

whereF{\displaystyle F} is the force exerted by the material when contracted or stretched byΔL{\displaystyle \Delta L}.

Hooke's law for a stretched wire can be derived from this formula:

F=(EAL0)ΔL=kx{\displaystyle F=\left({\frac {EA}{L_{0}}}\right)\,\Delta L=kx}

where it comes in saturation

k≡EAL0{\displaystyle k\equiv {\frac {EA}{L_{0}}}\,} andx≡ΔL.{\displaystyle x\equiv \Delta L.}

Note that the elasticity of coiled springs comes fromshear modulus, not Young's modulus. When a spring is stretched, its wire's length doesn't change, but its shape does. This is why only the shear modulus of elasticity is involved in the stretching of a spring.[citation needed]

Elastic potential energy

[edit]

Theelastic potential energy stored in a linear elastic material is given by the integral of the Hooke's law:

Ue=∫kxdx=12kx2.{\displaystyle U_{e}=\int {kx}\,dx={\frac {1}{2}}kx^{2}.}

now by explicating the intensive variables:

Ue=∫EAΔLL0dΔL=EAL0∫ΔLdΔL=EAΔL22L0{\displaystyle U_{e}=\int {\frac {EA\,\Delta L}{L_{0}}}\,d\Delta L={\frac {EA}{L_{0}}}\int \Delta L\,d\Delta L={\frac {EA\,{\Delta L}^{2}}{2L_{0}}}}

This means that the elastic potential energy density (that is, per unit volume) is given by:

UeAL0=EΔL22L02=12×EΔLL0×ΔLL0=12×σ(ε)×ε{\displaystyle {\frac {U_{e}}{AL_{0}}}={\frac {E\,{\Delta L}^{2}}{2L_{0}^{2}}}={\frac {1}{2}}\times {\frac {E\,{\Delta L}}{L_{0}}}\times {\frac {\Delta L}{L_{0}}}={\frac {1}{2}}\times \sigma (\varepsilon )\times \varepsilon }

or, in simple notation, for a linear elastic material:ue(ε)=∫Eεdε=12Eε2{\textstyle u_{e}(\varepsilon )=\int {E\,\varepsilon }\,d\varepsilon ={\frac {1}{2}}E{\varepsilon }^{2}}, since the strain is definedε≡ΔLL0{\textstyle \varepsilon \equiv {\frac {\Delta L}{L_{0}}}}.

In a nonlinear elastic material the Young's modulus is a function of the strain, so the second equivalence no longer holds, and the elastic energy is not aquadratic function of the strain:

ue(ε)=∫E(ε)εdε≠12Eε2{\displaystyle u_{e}(\varepsilon )=\int E(\varepsilon )\,\varepsilon \,d\varepsilon \neq {\frac {1}{2}}E\varepsilon ^{2}}

Examples

[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 inpolymers. The values here are approximate and only meant for relative comparison.

Approximate Young's modulus for various materials
MaterialYoung's modulus (GPa)Megapound per square inch (Mpsi)[5]Ref.
Aluminium (13Al)689.86[6][7][8][9][10][11]
Amino-acidmolecular crystals21–443.05–6.38[12]
Aramid (for example,Kevlar)70.5–112.410.2–16.3[13]
Aromatic peptide-nanospheres230–27533.4–39.9[14]
Aromatic peptide-nanotubes19–272.76–3.92[15][16]
Bacteriophagecapsids1–30.145–0.435[17]
Beryllium (4Be)28741.6[18]
Bone, human cortical142.03[19]
Brass10615.4[20]
Bronze11216.2[21]
Carbon nitride (CN2)822119[22]
Carbon-fiber-reinforced plastic (CFRP), 50/50 fibre/matrix, biaxial fabric30–504.35–7.25[23]
Carbon-fiber-reinforced plastic (CFRP), 70/30 fibre/matrix, unidirectional, along fibre18126.3[24]
Cobalt-chrome (CoCr)23033.4[25]
Copper (Cu), annealed11016[26]
Diamond (C), synthetic1050–1210152–175[27]
Diatomfrustules, largelysilicic acid0.35–2.770.051–0.058[28]
Flax fiber588.41[29]
Float glass47.7–83.66.92–12.1[30]
Glass-reinforced polyester (GRP)17.22.49[31]
Gold77.211.2[32]
Graphene1050152[33]
Hemp fiber355.08[34]
High-density polyethylene (HDPE)0.97–1.380.141–0.2[35]
High-strength concrete304.35[36]
Lead (82Pb), chemical131.89[11]
Low-density polyethylene (LDPE), molded0.2280.0331[37]
Magnesium alloy45.26.56[38]
Medium-density fiberboard (MDF)40.58[39]
Molybdenum (Mo), annealed33047.9[40][7][8][9][10][11]
Monel18026.1[11]
Mother-of-pearl (largelycalcium carbonate)7010.2[41]
Nickel (28Ni), commercial20029[11]
Nylon 662.930.425[42]
Osmium (76Os)525–56276.1–81.5[43]
Osmiumnitride (OsN2)194.99–396.4428.3–57.5[44]
Polycarbonate (PC)2.20.319[45]
Polyethylene terephthalate (PET), unreinforced3.140.455[46]
Polypropylene (PP), molded1.680.244[47]
Polystyrene, crystal2.5–3.50.363–0.508[48]
Polystyrene, foam0.0025–0.0070.000363–0.00102[49]
Polytetrafluoroethylene (PTFE), molded0.5640.0818[50]
Rubber, small strain0.01–0.10.00145–0.0145[12]
Silicon, single crystal, different directions130–18518.9–26.8[51]
Silicon carbide (SiC)90–13713.1–19.9[52]
Single-walledcarbon nanotube>{\displaystyle >}1000>{\displaystyle >}140[53][54]
Steel,A3620029[55]
Stinging nettle fiber8712.6[29]
Titanium (22Ti)11616.8[56][57][7][9][8][11][10]
Titanium alloy, Grade 511416.5[58]
Tooth enamel, largelycalcium phosphate8312[59]
Tungsten carbide (WC)600–68687–99.5[60]
Wood,American beech9.5–11.91.38–1.73[61]
Wood,black cherry9–10.31.31–1.49[61]
Wood,red maple9.6–11.31.39–1.64[61]
Wrought iron19328[62]
Yttrium iron garnet (YIG), polycrystalline19328[63]
Yttrium iron garnet (YIG), single-crystal20029[64]
Zinc (30Zn)10815.7[65]
Zirconium (40Zr), commercial9513.8[11]

See also

[edit]

References

[edit]
  1. ^The Rational mechanics of Flexible or Elastic Bodies, 1638–1788: Introduction to Leonhardi Euleri Opera Omnia, vol. X and XI, Seriei Secundae. Orell Fussli.
  2. ^Jastrzebski, D. (1959).Nature and Properties of Engineering Materials (Wiley International ed.).John Wiley & Sons, Inc.
  3. ^Tilleman, Tamara Raveh; Tilleman, Michael M.; Neumann, Martino H.A. (December 2004)."The Elastic Properties of Cancerous Skin: Poisson's Ratio and Young's Modulus"(PDF).Israel Medical Association Journal.6 (12):753–755.PMID 15609889.
  4. ^Gorodtsov, V.A.; Lisovenko, D.S. (2019). "Extreme values of Young's modulus and Poisson's ratio of hexagonal crystals".Mechanics of Materials.134:1–8.Bibcode:2019MechM.134....1G.doi:10.1016/j.mechmat.2019.03.017.S2CID 140493258.
  5. ^"Unit of Measure Converter".MatWeb. RetrievedMay 9, 2021.
  6. ^"Aluminum, Al".MatWeb. RetrievedMay 7, 2021.
  7. ^abcWeast, Robert C. (1981).CRC Handbook of Chemistry and Physics (62nd ed.). Boca Raton, FL:CRC Press.doi:10.1002/jctb.280500215.ISBN 978-0-84-930740-9.
  8. ^abcRoss, Robert B. (1992).Metallic Materials Specification Handbook (4th ed.). London:Chapman & Hall.doi:10.1007/978-1-4615-3482-2.ISBN 978-0-412-36940-7.
  9. ^abcNunes, Rafael; Adams, J. H.; Ammons, Mitchell; et al. (1990).Volume 2: Properties and Selection: Nonferrous Alloys and Special-Purpose Materials(PDF). ASM Handbook (10th ed.).ASM International.ISBN 978-0-87170-378-1.
  10. ^abcNayar, Alok (1997).The Metals Databook. New York, NY:McGraw-Hill.ISBN 978-0-07-462300-8.
  11. ^abcdefgLide, David R., ed. (1999). "Commercial Metals and Alloys".CRC Handbook of Chemistry and Physics (80th ed.). Boca Raton, FL:CRC Press.ISBN 978-0-84-930480-4.
  12. ^abAzuri, Ido; Meirzadeh, Elena; Ehre, David; et al. (November 9, 2015)."Unusually Large Young's Moduli of Amino Acid Molecular Crystals"(PDF).Angewandte Chemie.54 (46) (International ed.).Wiley:13566–13570.Bibcode:2015ACIE...5413566A.doi:10.1002/anie.201505813.PMID 26373817.S2CID 13717077 – viaPubMed.
  13. ^"Kevlar Aramid Fiber Technical Guide"(PDF).DuPont. 2017. RetrievedMay 8, 2021.
  14. ^Adler-Abramovich, Lihi; Kol, Nitzan; Yanai, Inbal; et al. (December 17, 2010). "Self-Assembled Organic Nanostructures with Metallic-Like Stiffness".Angewandte Chemie.49 (51) (International ed.).Wiley-VCH (published September 28, 2010):9939–9942.doi:10.1002/anie.201002037.PMID 20878815.S2CID 44873277.
  15. ^Kol, Nitzan; Adler-Abramovich, Lihi; Barlam, David; et al. (June 8, 2005)."Self-Assembled Peptide Nanotubes Are Uniquely Rigid Bioinspired Supramolecular Structures".Nano Letters.5 (7). Israel:American Chemical Society:1343–1346.Bibcode:2005NanoL...5.1343K.doi:10.1021/nl0505896.PMID 16178235 – viaACS Publications.
  16. ^Niu, Lijiang; Chen, Xinyong; Allen, Stephanie; et al. (June 6, 2007)."Using the Bending Beam Model to Estimate the Elasticity of Diphenylalanine Nanotubes".Langmuir.23 (14).American Chemical Society:7443–7446.doi:10.1021/la7010106.PMID 17550276 – viaACS Publications.
  17. ^Ivanovska, Irena L.; de Pablo, Pedro J.; Ibarra, Benjamin; et al. (May 7, 2004). Lubensky, Tom C. (ed.)."Bacteriophage capsids: Tough nanoshells with complex elastic properties".Proceedings of the National Academy of Sciences of the United States of America.101 (20).The National Academy of Sciences:7600–7605.Bibcode:2004PNAS..101.7600I.doi:10.1073/pnas.0308198101.PMC 419652.PMID 15133147.
  18. ^Foley, James C.; Abeln, Stephen P.; Stanek, Paul W.; et al. (2010). "An Overview of Current Research and Industrial Practices of be Powder Metallurgy". In Marquis, Fernand D. S. (ed.).Powder Materials: Current Research and Industrial Practices III. Hoboken, NJ:John Wiley & Sons, Inc. p. 263.doi:10.1002/9781118984239.ch32.ISBN 978-1-11-898423-9.
  19. ^Rho, Jae Young; Ashman, Richard B.; Turner, Charles H. (February 1993)."Young's modulus of trabecular and cortical bone material: Ultrasonic and microtensile measurements".Journal of Biomechanics.26 (2).Elsevier:111–119.doi:10.1016/0021-9290(93)90042-d.PMID 8429054 – viaElsevier Science Direct.
  20. ^"Overview of materials for Brass".MatWeb. RetrievedMay 7, 2021.
  21. ^"Overview of materials for Bronze".MatWeb. RetrievedMay 7, 2021.
  22. ^Chowdhury, Shafiul; Laugier, Michael T.; Rahman, Ismet Zakia (April–August 2004). "Measurement of the mechanical properties of carbon nitride thin films from the nanoindentation loading curve".Diamond and Related Materials.13 (4–8):1543–1548.Bibcode:2004DRM....13.1543C.doi:10.1016/j.diamond.2003.11.063 – viaElsevier Science Direct.
  23. ^Summerscales, John (September 11, 2019)."Composites Design and Manufacture (Plymouth University teaching support materials)".Advanced Composites Manufacturing Centre.University of Plymouth. Archived fromthe original on May 9, 2021. RetrievedMay 8, 2021.
  24. ^Kopeliovich, Dmitri (June 3, 2012)."Epoxy Matrix Composite reinforced by 70% carbon fibers".SubsTech. RetrievedMay 8, 2021.
  25. ^Bose, Susmita; Banerjee, Dishary; Bandyopadhyay, Amit (2016). "Introduction to Biomaterials and Devices for Bone Disorders". In Bose, Susmita; Bandyopadhyay, Amit (eds.).Materials for Bone Disorders.Academic Press. pp. 1–27.doi:10.1016/B978-0-12-802792-9.00001-X.ISBN 978-0-12-802792-9.
  26. ^"Copper, Cu; Annealed".MatWeb. RetrievedMay 9, 2021.
  27. ^Spear, Karl E.; Dismukes, John P., eds. (1994).Synthetic Diamond: Emerging CVD Science and Technology.Wiley. p. 315.ISBN 978-0-47-153589-8.ISSN 0275-0171.
  28. ^Subhash, Ghatu; Yao, Shuhuai; Bellinger, Brent; Gretz, Michael R. (January 2005). "Investigation of mechanical properties of diatom frustules using nanoindentation".Journal of Nanoscience and Nanotechnology.5 (1). American Scientific Publishers:50–56.doi:10.1166/jnn.2005.006.PMID 15762160 – viaIngenta Connect.
  29. ^abBodros, Edwin; Baley, Christophe (May 15, 2008). "Study of the tensile properties of stinging nettle fibres (Urtica dioica)".Materials Letters.62 (14):2143–2145.Bibcode:2008MatL...62.2143B.CiteSeerX 10.1.1.299.6908.doi:10.1016/j.matlet.2007.11.034 – viaElsevier Science Direct.
  30. ^"Float glass – Properties and Applications".AZO Materials. February 16, 2001. RetrievedMay 9, 2021.
  31. ^Kopeliovich, Dmitri (March 6, 2012)."Polyester Matrix Composite reinforced by glass fibers (Fiberglass)".SubsTech. RetrievedMay 7, 2021.
  32. ^"Gold material property data".MatWeb. RetrievedSeptember 8, 2021.
  33. ^Liu, Fang; Ming, Pingbing; Li, Ju (August 28, 2007)."Ab initio calculation of ideal strength and phonon instability of graphene under tension"(PDF).Physical Review B.76 (6) 064120.American Physical Society.Bibcode:2007PhRvB..76f4120L.doi:10.1103/PhysRevB.76.064120 – viaAPS Physics.
  34. ^Saheb, Nabi; Jog, Jyoti (October 15, 1999)."Natural fibre polymer composites: a review".Advances in Polymer Technology.18 (4).John Wiley & Sons, Inc.:351–363.doi:10.1002/(SICI)1098-2329(199924)18:4<351::AID-ADV6>3.0.CO;2-X.
  35. ^"High-Density Polyethylene (HDPE)".Polymer Database. Chemical Retrieval on the Web. RetrievedMay 9, 2021.
  36. ^Cardarelli, François (2008). "Cements, Concrete, Building Stones, and Construction Materials".Materials Handbook: A Concise Desktop Reference (2nd ed.). London:Springer-Verlag. pp. 1421–1439.doi:10.1007/978-3-319-38925-7_15.ISBN 978-3-319-38923-3.
  37. ^"Overview of materials for Low Density Polyethylene (LDPE), Molded".MatWeb. RetrievedMay 7, 2021.
  38. ^"Overview of materials for Magnesium Alloy".MatWeb. RetrievedMay 9, 2021.
  39. ^"Medium Density Fiberboard (MDF)".MakeItFrom. May 30, 2020. RetrievedMay 8, 2021.
  40. ^"Molybdenum, Mo, Annealed".MatWeb. RetrievedMay 9, 2021.
  41. ^Jackson, Andrew P.; Vincent, Julian F. V.; Turner, R. M. (September 22, 1988). "The mechanical design of nacre".Proceedings of the Royal Society B.234 (1277).Royal Society:415–440.Bibcode:1988RSPSB.234..415J.doi:10.1098/rspb.1988.0056.eISSN 2053-9193.ISSN 0080-4649.S2CID 135544277 – viaThe Royal Society Publishing.
  42. ^"Nylon® 6/6 (Polyamide)".Poly-Tech Industrial, Inc. 2011. RetrievedMay 9, 2021.
  43. ^Pandey, Dharmendra Kumar; Singh, Devraj; Yadawa, Pramod Kumar (April 2, 2009)."Ultrasonic Study of Osmium and Ruthenium"(PDF).Platinum Metals Review.53 (4).Johnson Matthey:91–97.doi:10.1595/147106709X430927. Archived fromthe original(PDF) on March 12, 2020. RetrievedMay 7, 2021 – viaIngenta Connect.
  44. ^Gaillac, Romain; Coudert, François-Xavier (July 26, 2020)."ELATE: Elastic tensor analysis".ELATE. RetrievedMay 9, 2021.
  45. ^"Polycarbonate".DesignerData. RetrievedMay 9, 2021.
  46. ^"Overview of materials for Polyethylene Terephthalate (PET), Unreinforced".MatWeb. RetrievedMay 9, 2021.
  47. ^"Overview of Materials for Polypropylene, Molded".MatWeb. RetrievedMay 9, 2021.
  48. ^"Young's Modulus: Tensile Elasticity Units, Factors & Material Table".Omnexus. SpecialChem. RetrievedMay 9, 2021.
  49. ^"Technical Data – Application Recommendations Dimensioning Aids".Stryodur.BASF. August 2019. RetrievedMay 7, 2021.
  50. ^"Overview of materials for Polytetrafluoroethylene (PTFE), Molded".MatWeb. RetrievedMay 9, 2021.
  51. ^Boyd, Euan J.; Uttamchandani, Deepak (2012). "Measurement of the Anisotropy of Young's Modulus in Single-Crystal Silicon".Journal of Microelectromechanical Systems.21 (1).Institute of Electrical and Electronics Engineers:243–249.Bibcode:2012JMemS..21..243B.doi:10.1109/JMEMS.2011.2174415.eISSN 1941-0158.ISSN 1057-7157.S2CID 39025763 – viaIEEE Xplore.
  52. ^"Silicon Carbide (SiC) Properties and Applications".AZO Materials. February 5, 2001. RetrievedMay 9, 2021.
  53. ^Forró, László; Salvetat, Jean-Paul; Bonard, Jean-Marc; et al. (January 2002). Thorpe, Michael F.;Tománek, David; Enbody, Richard J. (eds.)."Electronic and Mechanical Properties of Carbon Nanotubes".Science and Application of Nanotubes. Fundamentals Materials Research. Boston, MA:Springer:297–320.doi:10.1007/0-306-47098-5_22.ISBN 978-0-306-46372-3 – viaResearchGate.
  54. ^Yang, Yi-Hsuan; Li, Wenzhi (January 24, 2011). "Radial elasticity of single-walled carbon nanotube measured by atomic force microscopy".Applied Physics Letters.98 (4).American Institute of Physics: 041901.Bibcode:2011ApPhL..98d1901Y.doi:10.1063/1.3546170.
  55. ^"ASTM A36 Mild/Low Carbon Steel".AZO Materials. July 5, 2012. RetrievedMay 9, 2021.
  56. ^"Titanium, Ti".MatWeb. RetrievedMay 7, 2021.
  57. ^Boyer, Rodney; Welsch, Gerhard; Collings, Edward W., eds. (1994).Materials Properties Handbook: Titanium Alloys. Materials Park, OH:ASM International.ISBN 978-0-87-170481-8.
  58. ^U.S. Titanium Industry Inc. (July 30, 2002)."Titanium Alloys – Ti6Al4V Grade 5".AZO Materials. RetrievedMay 9, 2021.
  59. ^Staines, Michael; Robinson, W. H.; Hood, J. A. A. (September 1981). "Spherical indentation of tooth enamel".Journal of Materials Science.16 (9).Springer:2551–2556.Bibcode:1981JMatS..16.2551S.doi:10.1007/bf01113595.S2CID 137704231 – viaSpringer Link.
  60. ^"Tungsten Carbide – An Overview".AZO Materials. January 21, 2002. RetrievedMay 9, 2021.
  61. ^abcGreen, David W.; Winandy, Jerrold E.; Kretschmann, David E. (1999). "Mechanical Properties of Wood".Wood Handbook: Wood as an Engineering Material(PDF). Madison, WI:Forest Products Laboratory. pp. 4–8. Archived fromthe original(PDF) on July 20, 2018.
  62. ^"Wrought Iron – Properties and Applications".AZO Materials. August 13, 2013. RetrievedMay 9, 2021.
  63. ^Chou, Hung-Ming; 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.S2CID 135957639 – viaSpringerLink.
  64. ^"Yttrium Iron Garnet".Deltronic Crystal Industries, Inc. December 28, 2012. RetrievedMay 7, 2021.
  65. ^"An Introduction to Zinc".AZO Materials. July 23, 2001. RetrievedMay 9, 2021.

Further reading

[edit]

External links

[edit]
Divisions
Approaches
Classical
Modern
Interdisciplinary
Related
Elastic moduli for homogeneousisotropic materials
Conversion formulae
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, provided both for 3D materials (first part of the table) and for 2D materials (second part).
3D formulaeK={\displaystyle K=\,}E={\displaystyle E=\,}λ={\displaystyle \lambda =\,}G={\displaystyle G=\,}ν={\displaystyle \nu =\,}M={\displaystyle M=\,}Notes
(K,E){\displaystyle (K,\,E)}3K(3K−E)9K−E{\displaystyle {\tfrac {3K(3K-E)}{9K-E}}}3KE9K−E{\displaystyle {\tfrac {3KE}{9K-E}}}3K−E6K{\displaystyle {\tfrac {3K-E}{6K}}}3K(3K+E)9K−E{\displaystyle {\tfrac {3K(3K+E)}{9K-E}}}
(K,λ){\displaystyle (K,\,\lambda )}9K(K−λ)3K−λ{\displaystyle {\tfrac {9K(K-\lambda )}{3K-\lambda }}}3(K−λ)2{\displaystyle {\tfrac {3(K-\lambda )}{2}}}λ3K−λ{\displaystyle {\tfrac {\lambda }{3K-\lambda }}}3K−2λ{\displaystyle 3K-2\lambda \,}
(K,G){\displaystyle (K,\,G)}9KG3K+G{\displaystyle {\tfrac {9KG}{3K+G}}}K−2G3{\displaystyle K-{\tfrac {2G}{3}}}3K−2G2(3K+G){\displaystyle {\tfrac {3K-2G}{2(3K+G)}}}K+4G3{\displaystyle K+{\tfrac {4G}{3}}}
(K,ν){\displaystyle (K,\,\nu )}3K(1−2ν){\displaystyle 3K(1-2\nu )\,}3Kν1+ν{\displaystyle {\tfrac {3K\nu }{1+\nu }}}3K(1−2ν)2(1+ν){\displaystyle {\tfrac {3K(1-2\nu )}{2(1+\nu )}}}3K(1−ν)1+ν{\displaystyle {\tfrac {3K(1-\nu )}{1+\nu }}}
(K,M){\displaystyle (K,\,M)}9K(M−K)3K+M{\displaystyle {\tfrac {9K(M-K)}{3K+M}}}3K−M2{\displaystyle {\tfrac {3K-M}{2}}}3(M−K)4{\displaystyle {\tfrac {3(M-K)}{4}}}3K−M3K+M{\displaystyle {\tfrac {3K-M}{3K+M}}}
(E,λ){\displaystyle (E,\,\lambda )}E+3λ+R6{\displaystyle {\tfrac {E+3\lambda +R}{6}}}E−3λ+R4{\displaystyle {\tfrac {E-3\lambda +R}{4}}}2λE+λ+R{\displaystyle {\tfrac {2\lambda }{E+\lambda +R}}}E−λ+R2{\displaystyle {\tfrac {E-\lambda +R}{2}}}R=E2+9λ2+2Eλ{\displaystyle R={\sqrt {E^{2}+9\lambda ^{2}+2E\lambda }}}
(E,G){\displaystyle (E,\,G)}EG3(3G−E){\displaystyle {\tfrac {EG}{3(3G-E)}}}G(E−2G)3G−E{\displaystyle {\tfrac {G(E-2G)}{3G-E}}}E2G−1{\displaystyle {\tfrac {E}{2G}}-1}G(4G−E)3G−E{\displaystyle {\tfrac {G(4G-E)}{3G-E}}}
(E,ν){\displaystyle (E,\,\nu )}E3(1−2ν){\displaystyle {\tfrac {E}{3(1-2\nu )}}}Eν(1+ν)(1−2ν){\displaystyle {\tfrac {E\nu }{(1+\nu )(1-2\nu )}}}E2(1+ν){\displaystyle {\tfrac {E}{2(1+\nu )}}}E(1−ν)(1+ν)(1−2ν){\displaystyle {\tfrac {E(1-\nu )}{(1+\nu )(1-2\nu )}}}
(E,M){\displaystyle (E,\,M)}3M−E+S6{\displaystyle {\tfrac {3M-E+S}{6}}}M−E+S4{\displaystyle {\tfrac {M-E+S}{4}}}3M+E−S8{\displaystyle {\tfrac {3M+E-S}{8}}}E−M+S4M{\displaystyle {\tfrac {E-M+S}{4M}}}S=±E2+9M2−10EM{\displaystyle S=\pm {\sqrt {E^{2}+9M^{2}-10EM}}}

There are two valid solutions.
The plus sign leads toν≥0{\displaystyle \nu \geq 0}.

The minus sign leads toν≤0{\displaystyle \nu \leq 0}.

(λ,G){\displaystyle (\lambda ,\,G)}λ+2G3{\displaystyle \lambda +{\tfrac {2G}{3}}}G(3λ+2G)λ+G{\displaystyle {\tfrac {G(3\lambda +2G)}{\lambda +G}}}λ2(λ+G){\displaystyle {\tfrac {\lambda }{2(\lambda +G)}}}λ+2G{\displaystyle \lambda +2G\,}
(λ,ν){\displaystyle (\lambda ,\,\nu )}λ(1+ν)3ν{\displaystyle {\tfrac {\lambda (1+\nu )}{3\nu }}}λ(1+ν)(1−2ν)ν{\displaystyle {\tfrac {\lambda (1+\nu )(1-2\nu )}{\nu }}}λ(1−2ν)2ν{\displaystyle {\tfrac {\lambda (1-2\nu )}{2\nu }}}λ(1−ν)ν{\displaystyle {\tfrac {\lambda (1-\nu )}{\nu }}}Cannot be used whenν=0⇔λ=0{\displaystyle \nu =0\Leftrightarrow \lambda =0}
(λ,M){\displaystyle (\lambda ,\,M)}M+2λ3{\displaystyle {\tfrac {M+2\lambda }{3}}}(M−λ)(M+2λ)M+λ{\displaystyle {\tfrac {(M-\lambda )(M+2\lambda )}{M+\lambda }}}M−λ2{\displaystyle {\tfrac {M-\lambda }{2}}}λM+λ{\displaystyle {\tfrac {\lambda }{M+\lambda }}}
(G,ν){\displaystyle (G,\,\nu )}2G(1+ν)3(1−2ν){\displaystyle {\tfrac {2G(1+\nu )}{3(1-2\nu )}}}2G(1+ν){\displaystyle 2G(1+\nu )\,}2Gν1−2ν{\displaystyle {\tfrac {2G\nu }{1-2\nu }}}2G(1−ν)1−2ν{\displaystyle {\tfrac {2G(1-\nu )}{1-2\nu }}}
(G,M){\displaystyle (G,\,M)}M−4G3{\displaystyle M-{\tfrac {4G}{3}}}G(3M−4G)M−G{\displaystyle {\tfrac {G(3M-4G)}{M-G}}}M−2G{\displaystyle M-2G\,}M−2G2M−2G{\displaystyle {\tfrac {M-2G}{2M-2G}}}
(ν,M){\displaystyle (\nu ,\,M)}M(1+ν)3(1−ν){\displaystyle {\tfrac {M(1+\nu )}{3(1-\nu )}}}M(1+ν)(1−2ν)1−ν{\displaystyle {\tfrac {M(1+\nu )(1-2\nu )}{1-\nu }}}Mν1−ν{\displaystyle {\tfrac {M\nu }{1-\nu }}}M(1−2ν)2(1−ν){\displaystyle {\tfrac {M(1-2\nu )}{2(1-\nu )}}}
2D formulaeK2D={\displaystyle K_{\mathrm {2D} }=\,}E2D={\displaystyle E_{\mathrm {2D} }=\,}λ2D={\displaystyle \lambda _{\mathrm {2D} }=\,}G2D={\displaystyle G_{\mathrm {2D} }=\,}ν2D={\displaystyle \nu _{\mathrm {2D} }=\,}M2D={\displaystyle M_{\mathrm {2D} }=\,}Notes
(K2D,E2D){\displaystyle (K_{\mathrm {2D} },\,E_{\mathrm {2D} })}2K2D(2K2D−E2D)4K2D−E2D{\displaystyle {\tfrac {2K_{\mathrm {2D} }(2K_{\mathrm {2D} }-E_{\mathrm {2D} })}{4K_{\mathrm {2D} }-E_{\mathrm {2D} }}}}K2DE2D4K2D−E2D{\displaystyle {\tfrac {K_{\mathrm {2D} }E_{\mathrm {2D} }}{4K_{\mathrm {2D} }-E_{\mathrm {2D} }}}}2K2D−E2D2K2D{\displaystyle {\tfrac {2K_{\mathrm {2D} }-E_{\mathrm {2D} }}{2K_{\mathrm {2D} }}}}4K2D24K2D−E2D{\displaystyle {\tfrac {4K_{\mathrm {2D} }^{2}}{4K_{\mathrm {2D} }-E_{\mathrm {2D} }}}}
(K2D,λ2D){\displaystyle (K_{\mathrm {2D} },\,\lambda _{\mathrm {2D} })}4K2D(K2D−λ2D)2K2D−λ2D{\displaystyle {\tfrac {4K_{\mathrm {2D} }(K_{\mathrm {2D} }-\lambda _{\mathrm {2D} })}{2K_{\mathrm {2D} }-\lambda _{\mathrm {2D} }}}}K2D−λ2D{\displaystyle K_{\mathrm {2D} }-\lambda _{\mathrm {2D} }}λ2D2K2D−λ2D{\displaystyle {\tfrac {\lambda _{\mathrm {2D} }}{2K_{\mathrm {2D} }-\lambda _{\mathrm {2D} }}}}2K2D−λ2D{\displaystyle 2K_{\mathrm {2D} }-\lambda _{\mathrm {2D} }}
(K2D,G2D){\displaystyle (K_{\mathrm {2D} },\,G_{\mathrm {2D} })}4K2DG2DK2D+G2D{\displaystyle {\tfrac {4K_{\mathrm {2D} }G_{\mathrm {2D} }}{K_{\mathrm {2D} }+G_{\mathrm {2D} }}}}K2D−G2D{\displaystyle K_{\mathrm {2D} }-G_{\mathrm {2D} }}K2D−G2DK2D+G2D{\displaystyle {\tfrac {K_{\mathrm {2D} }-G_{\mathrm {2D} }}{K_{\mathrm {2D} }+G_{\mathrm {2D} }}}}K2D+G2D{\displaystyle K_{\mathrm {2D} }+G_{\mathrm {2D} }}
(K2D,ν2D){\displaystyle (K_{\mathrm {2D} },\,\nu _{\mathrm {2D} })}2K2D(1−ν2D){\displaystyle 2K_{\mathrm {2D} }(1-\nu _{\mathrm {2D} })\,}2K2Dν2D1+ν2D{\displaystyle {\tfrac {2K_{\mathrm {2D} }\nu _{\mathrm {2D} }}{1+\nu _{\mathrm {2D} }}}}K2D(1−ν2D)1+ν2D{\displaystyle {\tfrac {K_{\mathrm {2D} }(1-\nu _{\mathrm {2D} })}{1+\nu _{\mathrm {2D} }}}}2K2D1+ν2D{\displaystyle {\tfrac {2K_{\mathrm {2D} }}{1+\nu _{\mathrm {2D} }}}}
(E2D,G2D){\displaystyle (E_{\mathrm {2D} },\,G_{\mathrm {2D} })}E2DG2D4G2D−E2D{\displaystyle {\tfrac {E_{\mathrm {2D} }G_{\mathrm {2D} }}{4G_{\mathrm {2D} }-E_{\mathrm {2D} }}}}2G2D(E2D−2G2D)4G2D−E2D{\displaystyle {\tfrac {2G_{\mathrm {2D} }(E_{\mathrm {2D} }-2G_{\mathrm {2D} })}{4G_{\mathrm {2D} }-E_{\mathrm {2D} }}}}E2D2G2D−1{\displaystyle {\tfrac {E_{\mathrm {2D} }}{2G_{\mathrm {2D} }}}-1}4G2D24G2D−E2D{\displaystyle {\tfrac {4G_{\mathrm {2D} }^{2}}{4G_{\mathrm {2D} }-E_{\mathrm {2D} }}}}
(E2D,ν2D){\displaystyle (E_{\mathrm {2D} },\,\nu _{\mathrm {2D} })}E2D2(1−ν2D){\displaystyle {\tfrac {E_{\mathrm {2D} }}{2(1-\nu _{\mathrm {2D} })}}}E2Dν2D(1+ν2D)(1−ν2D){\displaystyle {\tfrac {E_{\mathrm {2D} }\nu _{\mathrm {2D} }}{(1+\nu _{\mathrm {2D} })(1-\nu _{\mathrm {2D} })}}}E2D2(1+ν2D){\displaystyle {\tfrac {E_{\mathrm {2D} }}{2(1+\nu _{\mathrm {2D} })}}}E2D(1+ν2D)(1−ν2D){\displaystyle {\tfrac {E_{\mathrm {2D} }}{(1+\nu _{\mathrm {2D} })(1-\nu _{\mathrm {2D} })}}}
(λ2D,G2D){\displaystyle (\lambda _{\mathrm {2D} },\,G_{\mathrm {2D} })}λ2D+G2D{\displaystyle \lambda _{\mathrm {2D} }+G_{\mathrm {2D} }}4G2D(λ2D+G2D)λ2D+2G2D{\displaystyle {\tfrac {4G_{\mathrm {2D} }(\lambda _{\mathrm {2D} }+G_{\mathrm {2D} })}{\lambda _{\mathrm {2D} }+2G_{\mathrm {2D} }}}}λ2Dλ2D+2G2D{\displaystyle {\tfrac {\lambda _{\mathrm {2D} }}{\lambda _{\mathrm {2D} }+2G_{\mathrm {2D} }}}}λ2D+2G2D{\displaystyle \lambda _{\mathrm {2D} }+2G_{\mathrm {2D} }\,}
(λ2D,ν2D){\displaystyle (\lambda _{\mathrm {2D} },\,\nu _{\mathrm {2D} })}λ2D(1+ν2D)2ν2D{\displaystyle {\tfrac {\lambda _{\mathrm {2D} }(1+\nu _{\mathrm {2D} })}{2\nu _{\mathrm {2D} }}}}λ2D(1+ν2D)(1−ν2D)ν2D{\displaystyle {\tfrac {\lambda _{\mathrm {2D} }(1+\nu _{\mathrm {2D} })(1-\nu _{\mathrm {2D} })}{\nu _{\mathrm {2D} }}}}λ2D(1−ν2D)2ν2D{\displaystyle {\tfrac {\lambda _{\mathrm {2D} }(1-\nu _{\mathrm {2D} })}{2\nu _{\mathrm {2D} }}}}λ2Dν2D{\displaystyle {\tfrac {\lambda _{\mathrm {2D} }}{\nu _{\mathrm {2D} }}}}Cannot be used whenν2D=0⇔λ2D=0{\displaystyle \nu _{\mathrm {2D} }=0\Leftrightarrow \lambda _{\mathrm {2D} }=0}
(G2D,ν2D){\displaystyle (G_{\mathrm {2D} },\,\nu _{\mathrm {2D} })}G2D(1+ν2D)1−ν2D{\displaystyle {\tfrac {G_{\mathrm {2D} }(1+\nu _{\mathrm {2D} })}{1-\nu _{\mathrm {2D} }}}}2G2D(1+ν2D){\displaystyle 2G_{\mathrm {2D} }(1+\nu _{\mathrm {2D} })\,}2G2Dν2D1−ν2D{\displaystyle {\tfrac {2G_{\mathrm {2D} }\nu _{\mathrm {2D} }}{1-\nu _{\mathrm {2D} }}}}2G2D1−ν2D{\displaystyle {\tfrac {2G_{\mathrm {2D} }}{1-\nu _{\mathrm {2D} }}}}
(G2D,M2D){\displaystyle (G_{\mathrm {2D} },\,M_{\mathrm {2D} })}M2D−G2D{\displaystyle M_{\mathrm {2D} }-G_{\mathrm {2D} }}4G2D(M2D−G2D)M2D{\displaystyle {\tfrac {4G_{\mathrm {2D} }(M_{\mathrm {2D} }-G_{\mathrm {2D} })}{M_{\mathrm {2D} }}}}M2D−2G2D{\displaystyle M_{\mathrm {2D} }-2G_{\mathrm {2D} }\,}M2D−2G2DM2D{\displaystyle {\tfrac {M_{\mathrm {2D} }-2G_{\mathrm {2D} }}{M_{\mathrm {2D} }}}}



Authority control databasesEdit this at Wikidata
Retrieved from "https://en.wikipedia.org/w/index.php?title=Young%27s_modulus&oldid=1323192779"
Categories:
Hidden categories:
[8]ページ先頭
©2009-2025 Movatter.jp