Young 's modulus - wikipedia Young 's modulus Rubber, a material with an extremely low Young 's modulus Young 's modulus, also known as the elastic modulus, is a measure of the stiffness of a solid material. It is a mechanical property of linear elastic solid materials. It defines the relationship between stress ( force per unit area ) and strain ( proportional deformation ) in a material. Young 's modulus 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. The term modulus is the diminutive of the Latin term modus which means measure. A solid material will deform when a load is applied to it. If it returns to its original shape after the load is removed, this is called elastic deformation. In the range where the ratio between load and deformation remains constant, the stress - strain curve is linear. Not many materials are linear and elastic beyond a small amount of deformation. A stiff material needs more force to deform compared to a soft material, and an infinite force would be needed to deform a perfectly rigid material, implying that it would have an infinite Young 's modulus. Although such a material can not exist, a material with a very high Young 's modulus can be approximated as rigid. Material stiffness should not be confused with : Strength : the strength of material is the amount of force it can withstand and still recover its original shape ; Geometric stiffness : the geometric stiffness depends on shape, e.g. the bending stiffness of an I beam is much higher than that of a rod made of the same steel, thus having the same rigidity, and same mass of material per length ; Hardness : the hardness of a material defines the relative resistance that its surface imposes against the penetration of a harder body ; Toughness : toughness is the amount of energy that a material can absorb before fracture. Contents 1 Technical 1.1 Units 1.1. 1 United States 2 Usage 2.1 Linear versus non-linear 2.2 Directional materials 3 Calculation 3.1 Force exerted by stretched or contracted material 3.2 Elastic potential energy 3.3 Relation among elastic constants 4 Approximate values 5 See also 6 References 7 Further reading 8 External links Technical The technical definition is : 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. 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 or kg m s ). The practical units used are megapascals ( MPa or N / mm ) or gigapascals ( GPa or kN / mm ). United States In United States customary units, it is expressed as pounds ( force ) per square inch ( psi ). The abbreviation ksi refers to kips per square inch. A `` kip '' is an Imperial unit of force and is equal to 1000 pounds - force, making a ksi equal to 1000 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 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 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 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 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 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. 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 a 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, σ ( ε ) ( displaystyle sigma ( \varepsilon ) ), by the engineering extensional strain, ε ( displaystyle \varepsilon ), in the elastic ( initial, linear ) portion of the physical stress -- strain curve : E ≡ σ ( ε ) ε = F / A Δ L / L 0 = F L 0 A Δ L ( displaystyle E equiv ( \frac ( sigma ( \varepsilon ) ) ( \varepsilon ) ) = ( \frac ( F / A ) ( Delta L / L_ ( 0 ) ) ) = ( \frac ( FL_ ( 0 ) ) ( A Delta L ) ) ) where E is the Young 's modulus ( modulus of elasticity ) F is the force exerted on an object under tension ; A is the actual cross-sectional area, which equals the area of the cross-section perpendicular to the applied force ; ΔL is the amount by which the length of the object changes ( ΔL is positive if the material is stretched, and negative when the material is compressed ) ; L 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. F = E A Δ L L 0 ( displaystyle F = ( \frac ( EA Delta L ) ( L_ ( 0 ) ) ) ) where F is the force exerted by the material when contracted or stretched by ΔL. Hooke 's law for a stretched wire can be derived from this formula : F = ( E A L 0 ) Δ L = k x ( displaystyle F = left ( ( \frac ( EA ) ( L_ ( 0 ) ) ) \right ) Delta L = kx , ) where it comes in saturation k ≡ E A L 0 ( displaystyle k equiv ( \begin ( matrix ) ( \frac ( EA ) ( L_ ( 0 ) ) ) end ( matrix ) ) , ) and x ≡ Δ L. ( displaystyle x equiv Delta L. , ) But note that the elasticity of coiled springs comes from shear modulus, not Young 's modulus. Elastic potential energy The elastic potential energy stored in a linear elastic material is given by the integral of the Hooke 's law : U e = ∫ k x d x = 1 2 k x 2. ( displaystyle U_ ( e ) = int ( kx ) , dx = ( \frac ( 1 ) ( 2 ) ) kx ^ ( 2 ). ) now by expliciting the intensive variables : U e = ∫ E A Δ L L 0 d Δ L = E A L 0 ∫ Δ L d Δ L = E A Δ L 2 2 L 0 ( 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 ( i.e., per unit volume ) is given by : U e A L 0 = E Δ L 2 2 L 0 2 ( displaystyle ( \frac ( U_ ( e ) ) ( AL_ ( 0 ) ) ) = ( \frac ( E ( Delta L ) ^ ( 2 ) ) ( 2L_ ( 0 ) ^ ( 2 ) ) ) ) or, in simple notation, for a linear elastic material : u e ( ε ) = ∫ E ε d ε = 1 2 E ε 2 ( displaystyle u_ ( e ) ( \varepsilon ) = int ( E , \varepsilon ) , d \varepsilon = ( \frac ( 1 ) ( 2 ) ) E ( \varepsilon ) ^ ( 2 ) ), since the strain is defined ε ≡ Δ L L 0 ( displaystyle \varepsilon equiv ( \frac ( Delta L ) ( L_ ( 0 ) ) ) ). In a nonlinear elastic material the Young modulus is a function of the strain, so the second equivalence no longer holds and the elastic energy is not a quadratic function of the strain : u e ( ε ) = ∫ E ( ε ) ε d ε ≠ 1 2 E ε 2 ( displaystyle u_ ( e ) ( \varepsilon ) = int ( E ( \varepsilon ) , \varepsilon ) , d \varepsilon \neq ( \frac ( 1 ) ( 2 ) ) E ( \varepsilon ) ^ ( 2 ) ) 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 : E = 2 G ( 1 + ν ) = 3 K ( 1 − 2 ν ). ( displaystyle E = 2G ( 1 + \nu ) = 3K ( 1 - 2 \nu ). , ) Approximate values 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 Material GPa M psi Rubber ( small strain ) 0.01 -- 0.1 1.45 -- 6998145000000000000 ♠ 14.5 × 10 Low - density polyethylene 0.11 -- 0.86 1.6 -- 6998650000000000000 ♠ 6.5 × 10 Diatom frustules ( largely silicic acid ) 0.35 -- 2.77 0.05 -- 0.4 PTFE ( Teflon ) 0.5 0.075 HDPE 0.8 0.116 Bacteriophage capsids 1 -- 3 0.15 -- 0.435 Polypropylene 1.5 -- 2 0.22 -- 0.29 Polyethylene terephthalate ( PET ) 2 -- 2.7 0.29 -- 0.39 Nylon 2 -- 4 0.29 -- 0.58 Polystyrene, solid 3 -- 3.5 0.44 -- 0.51 Polystyrene, foam 0.0025 -- 0.007 0.00036 -- 0.00102 Medium - density fiberboard ( MDF ) 0.58 Wood ( along grain ) 11 1.60 Human Cortical Bone 14 2.03 Glass - reinforced polyester matrix 17.2 2.49 Aromatic peptide nanotubes 19 -- 27 2.76 -- 3.92 High - strength concrete 30 4.35 Carbon fiber reinforced plastic ( 50 / 50 fibre / matrix, biaxial fabric ) 30 -- 50 4.35 -- 7.25 Hemp fiber 35 5.08 Magnesium metal ( Mg ) 45 6.53 Glass ( see chart ) 50 -- 90 7.25 -- 13.1 Flax fiber 58 8.41 Aluminum 69 10 Mother - of - pearl ( nacre, largely calcium carbonate ) 70 10.2 Aramid 70.5 -- 112.4 10.2 -- 16.3 Tooth enamel ( largely calcium phosphate ) 83 12 Stinging nettle fiber 87 12.6 Bronze 96 -- 120 13.9 -- 17.4 Brass 100 -- 125 14.5 -- 18.1 Titanium ( Ti ) 110.3 16 Titanium alloys 105 -- 120 15 -- 17.5 Copper ( Cu ) 117 17 Carbon fiber reinforced plastic ( 70 / 30 fibre / matrix, unidirectional, along fibre ) 181 26.3 Silicon Single crystal, different directions 130 -- 185 18.9 -- 26.8 Wrought iron 190 -- 210 27.6 -- 30.5 Steel ( ASTM - A36 ) 209 29 polycrystalline Yttrium iron garnet ( YIG ) 193 28 single - crystal Yttrium iron garnet ( YIG ) 200 29 Cobalt - chrome ( CoCr ) 220 -- 258 29 Aromatic peptide nanospheres 230 -- 275 33.4 -- 40 Beryllium ( Be ) 287 41.6 Molybdenum ( Mo ) 329 -- 330 47.7 -- 47.9 Tungsten ( W ) 400 -- 410 58 -- 59 Silicon carbide ( SiC ) 450 65 Tungsten carbide ( WC ) 450 -- 650 65 -- 94 Osmium ( Os ) 525 -- 562 76.1 -- 81.5 Single - walled carbon nanotube 1,000 + 150 + Graphene ( C ) 1050 152 Diamond ( C ) 1050 -- 1210 152 -- 175 Carbyne ( C ) 32100 7007321000000000000 ♠ 4,660 See also Bending stiffness Deflection Deformation Flexural modulus Impulse excitation technique List of materials properties Yield ( engineering ) References Jump up ^ The Rational Mechanics of Flexible or Elastic Bodies, 1638 -- 1788 : Introduction to Leonhardi Euleri Opera Omnia, vol. X and XI, Seriei Secundae. Orell Fussli. Jump up ^ In computer graphics visual effects, the terms linear elastic material and rigid material are used interchangeably if the Young 's modulus is high. Jump up ^ IUPAC, Compendium of Chemical Terminology, 2nd ed. ( the `` Gold Book '' ) ( 1997 ). Online corrected version : ( 2006 -- ) `` modulus of elasticity ( Young 's modulus ), E ''. ^ Jump up to : `` Elastic Properties and Young Modulus for some Materials ''. The Engineering ToolBox. Retrieved 2012 - 01 - 06. Jump up ^ `` Overview of materials for Low Density Polyethylene ( LDPE ), Molded ''. Matweb. Archived from the original on January 1, 2011. Retrieved February 7, 2013. Jump up ^ 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. PMID 15762160. doi : 10.1166 / jnn. 2005.006. Jump up ^ Ivanovska IL, de Pablo PJ, Sgalari G, MacKintosh FC, Carrascosa JL, Schmidt CF, Wuite GJ ( 2004 ). `` Bacteriophage capsids : Tough nanoshells with complex elastic properties ''. Proc Natl Acad Sci USA. 101 ( 20 ) : 7600 -- 5. Bibcode : 2004PNAS... 101.7600 I. PMC 419652. PMID 15133147. doi : 10.1073 / pnas. 0308198101. Jump up ^ `` Styrodur Technical Data '' ( PDF ). BASF. Retrieved 2016 - 03 - 15. Jump up ^ `` Medium Density Fiberboard ( MDF ) Material Properties : : MakeItFrom.com ''. Retrieved February 4, 2016. Jump up ^ Rho, JY ( 1993 ). `` Young 's modulus of trabecular and cortical bone material : ultrasonic and microtensile measurements ''. Journal of Biomechanics. 26 ( 2 ) : 111 -- 119. PMID 8429054. doi : 10.1016 / 0021 - 9290 ( 93 ) 90042 - d. Jump up ^ `` Polyester Matrix Composite reinforced by glass fibers ( Fiberglass ) ''. ( SubsTech ) ( 2008 - 05 - 17 ). Retrieved on 2011 - 03 - 30. Jump up ^ 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... 5.1343 K. doi : 10.1021 / nl0505896. Jump up ^ 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. Jump up ^ `` Composites Design and Manufacture ( BEng ) -- MATS 324 ''. Jump up ^ 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. Jump up ^ 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. Jump up ^ 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. Jump up ^ DuPont ( 2001 ). `` Kevlar Technical Guide '' : 9. Jump up ^ M. Staines, W.H. Robinson and J.A.A. Hood ( 1981 ). `` Spherical indentation of tooth enamel ''. Journal of Materials Science. 16 ( 9 ) : 2551 -- 2556. Bibcode : 1981JMatS... 16.2551 S. doi : 10.1007 / bf01113595. Jump up ^ 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. Jump up ^ Epoxy Matrix Composite reinforced by 70 % carbon fibers ( SubsTech ). Substech.com ( 2006 - 11 - 06 ). Retrieved on 2011 - 03 - 30. Jump up ^ `` Physical properties of Silicon ( Si ) ''. Ioffe Institute Database. Retrieved on 2011 - 05 - 27. Jump up ^ 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. Jump up ^ 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. Jump up ^ YIG properties Jump up ^ `` Properties of cobalt - chrome alloys -- Heraeus Kulzer cara ''. Archived from the original on 1 July 2015. Retrieved February 4, 2016. Jump up ^ Adler - Abramovich, L. ; et al. ( December 17, 2010 ). `` Self - Assembled Organic Nanostructures with Metallic - Like Stiffness ''. Angewandte Chemie International Edition. 49 ( 51 ) : 9939 -- 9942. PMID 20878815. doi : 10.1002 / anie. 201002037. Jump up ^ Foley, James C. ; et al. ( 2010 ). `` An Overview of Current Research and Industrial Practices of Be Powder Metallurgy ''. In Marquis, Fernand D.S. Powder Materials : Current Research and Industrial Practices III. Hoboken, NJ, USA : John Wiley & Sons, Inc. p. 263. doi : 10.1002 / 9781118984239. ch32. Jump up ^ `` Molybdenum : physical properties ''. webelements. Retrieved January 27, 2015. Jump up ^ `` Molybdenum, Mo '' ( PDF ). Glemco. Retrieved January 27, 2014. Jump up ^ D.K. Pandey ; Singh, D. ; Yadawa, P.K. ; et al. ( 2009 ). `` Ultrasonic Study of Osmium and Ruthenium '' ( PDF ). Platinum Metals Rev. 53 ( 4 ) : 91 -- 97. doi : 10.1595 / 147106709X430927. Retrieved November 4, 2014. Jump up ^ L. Forro ; et al. `` Electronic and mechanical properties of carbon nanotubes '' ( PDF ). Jump up ^ Y.H. Yang ; Li, W.Z. ; et al. ( 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. Jump up ^ Fang Liu ; Pingbing Ming & Ju Li. `` Ab initio calculation of ideal strength and phonon instability of graphene under tension '' ( PDF ). Jump up ^ Spear and Dismukes ( 1994 ). Synthetic Diamond -- Emerging CVD Science and Technology. Wiley, N.Y. p. 315. ISBN 978 - 0 - 471 - 53589 - 8. Jump up ^ Owano, Nancy ( Aug 20, 2013 ). `` Carbyne is stronger than any known material ''. phys.org. Jump up ^ Mingjie Liu, Vasilii I. Artyukhov, Hoonkyung Lee, Fangbo Xu and Boris I. Yakobson ( Dec 2, 2013 ). `` Carbyne From First Principles : Chain of C Atoms, a Nanorod or a Nanorope ? '' ( PDF ). arxiv.org. CS1 maint : Multiple names : authors list ( link ) Further reading 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 ) External links Matweb : free database of engineering properties for over 115,000 materials Young 's Modulus for groups of materials, and their cost Branches of physics Divisions Applied Experimental Theoretical Energy Motion Thermodynamics Mechanics Classical Ballistics Lagrangian Hamiltonian Continuum Celestial Statistical Solid Fluid Quantum Waves Fields Gravitation Electromagnetism Optics Geometrical Physical Nonlinear Quantum Quantum field theory Relativity Special General By speciality Accelerator Acoustics Astrophysics Nuclear Stellar Heliophysics Solar Space Astroparticle Atomic -- molecular -- optical ( AMO ) Communication Computational Condensed matter Mesoscopic Solid - state Soft Digital Engineering Material Mathematical Molecular Nuclear Particle Phenomenology Plasma Polymer Statistical Physics in life science Biophysics Virophysics Biomechanics Medical physics Cardiophysics Health physics Laser medicine Medical imaging ‎ Nuclear medicine Neurophysics Psychophysics Physics with other sciences Agrophysics Soil Atmospheric Cloud Chemical Econophysics Geophysics Elastic moduli for homogeneous isotropic materials Bulk modulus ( K ( displaystyle K ) ) Young 's modulus ( E ( displaystyle E ) ) Lamé 's first parameter ( λ ( displaystyle lambda ) ) Shear modulus ( G, μ ( displaystyle G, mu ) ) Poisson 's ratio ( ν ( displaystyle \nu ) ) P - wave modulus ( M ( displaystyle M ) ) 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 = ( displaystyle K = , ) E = ( displaystyle E = , ) λ = ( displaystyle lambda = , ) G = ( displaystyle G = , ) ν = ( displaystyle \nu = , ) M = ( displaystyle M = , ) Notes ( K, E ) ( displaystyle ( K, , E ) ) K ( displaystyle K ) E ( displaystyle E ) 3 K ( 3 K − E ) 9 K − E ( displaystyle ( \tfrac ( 3K ( 3K - E ) ) ( 9K - E ) ) ) 3 K E 9 K − E ( displaystyle ( \tfrac ( 3KE ) ( 9K - E ) ) ) 3 K − E 6 K ( displaystyle ( \tfrac ( 3K - E ) ( 6K ) ) ) 3 K ( 3 K + E ) 9 K − E ( displaystyle ( \tfrac ( 3K ( 3K + E ) ) ( 9K - E ) ) ) ( K, λ ) ( displaystyle ( K, , lambda ) ) K ( displaystyle K ) 9 K ( K − λ ) 3 K − λ ( displaystyle ( \tfrac ( 9K ( K - lambda ) ) ( 3K - lambda ) ) ) λ ( displaystyle lambda ) 3 ( K − λ ) 2 ( displaystyle ( \tfrac ( 3 ( K - lambda ) ) ( 2 ) ) ) λ 3 K − λ ( displaystyle ( \tfrac ( lambda ) ( 3K - lambda ) ) ) 3 K − 2 λ ( displaystyle 3K - 2 lambda , ) ( K, G ) ( displaystyle ( K, , G ) ) K ( displaystyle K ) 9 K G 3 K + G ( displaystyle ( \tfrac ( 9KG ) ( 3K + G ) ) ) K − 2 G 3 ( displaystyle K - ( \tfrac ( 2G ) ( 3 ) ) ) G ( displaystyle G ) 3 K − 2 G 2 ( 3 K + G ) ( displaystyle ( \tfrac ( 3K - 2G ) ( 2 ( 3K + G ) ) ) ) K + 4 G 3 ( displaystyle K+ ( \tfrac ( 4G ) ( 3 ) ) ) ( K, ν ) ( displaystyle ( K, , \nu ) ) K ( displaystyle K ) 3 K ( 1 − 2 ν ) ( displaystyle 3K ( 1 - 2 \nu ) , ) 3 K ν 1 + ν ( displaystyle ( \tfrac ( 3K \nu ) ( 1 + \nu ) ) ) 3 K ( 1 − 2 ν ) 2 ( 1 + ν ) ( displaystyle ( \tfrac ( 3K ( 1 - 2 \nu ) ) ( 2 ( 1 + \nu ) ) ) ) ν ( displaystyle \nu ) 3 K ( 1 − ν ) 1 + ν ( displaystyle ( \tfrac ( 3K ( 1 - \nu ) ) ( 1 + \nu ) ) ) ( K, M ) ( displaystyle ( K, , M ) ) K ( displaystyle K ) 9 K ( M − K ) 3 K + M ( displaystyle ( \tfrac ( 9K ( M-K ) ) ( 3K + M ) ) ) 3 K − M 2 ( displaystyle ( \tfrac ( 3K - M ) ( 2 ) ) ) 3 ( M − K ) 4 ( displaystyle ( \tfrac ( 3 ( M-K ) ) ( 4 ) ) ) 3 K − M 3 K + M ( displaystyle ( \tfrac ( 3K - M ) ( 3K + M ) ) ) M ( displaystyle M ) ( E, λ ) ( displaystyle ( E, , lambda ) ) E + 3 λ + R 6 ( displaystyle ( \tfrac ( E + 3 lambda + R ) ( 6 ) ) ) E ( displaystyle E ) λ ( displaystyle lambda ) E − 3 λ + R 4 ( displaystyle ( \tfrac ( E-3 lambda + R ) ( 4 ) ) ) 2 λ E + λ + R ( displaystyle ( \tfrac ( 2 lambda ) ( E+ lambda + R ) ) ) E − λ + R 2 ( displaystyle ( \tfrac ( E - lambda + R ) ( 2 ) ) ) R = E 2 + 9 λ 2 + 2 E λ ( displaystyle R = ( sqrt ( E ^ ( 2 ) + 9 lambda ^ ( 2 ) + 2E lambda ) ) ) ( E, G ) ( displaystyle ( E, , G ) ) E G 3 ( 3 G − E ) ( displaystyle ( \tfrac ( EG ) ( 3 ( 3G - E ) ) ) ) E ( displaystyle E ) G ( E − 2 G ) 3 G − E ( displaystyle ( \tfrac ( G ( E-2G ) ) ( 3G - E ) ) ) G ( displaystyle G ) E 2 G − 1 ( displaystyle ( \tfrac ( E ) ( 2G ) ) - 1 ) G ( 4 G − E ) 3 G − E ( displaystyle ( \tfrac ( G ( 4G - E ) ) ( 3G - E ) ) ) ( E, ν ) ( displaystyle ( E, , \nu ) ) E 3 ( 1 − 2 ν ) ( displaystyle ( \tfrac ( E ) ( 3 ( 1 - 2 \nu ) ) ) ) E ( displaystyle E ) E ν ( 1 + ν ) ( 1 − 2 ν ) ( displaystyle ( \tfrac ( E \nu ) ( ( 1 + \nu ) ( 1 - 2 \nu ) ) ) ) E 2 ( 1 + ν ) ( displaystyle ( \tfrac ( E ) ( 2 ( 1 + \nu ) ) ) ) ν ( displaystyle \nu ) E ( 1 − ν ) ( 1 + ν ) ( 1 − 2 ν ) ( displaystyle ( \tfrac ( E ( 1 - \nu ) ) ( ( 1 + \nu ) ( 1 - 2 \nu ) ) ) ) ( E, M ) ( displaystyle ( E, , M ) ) 3 M − E + S 6 ( displaystyle ( \tfrac ( 3M - E + S ) ( 6 ) ) ) E ( displaystyle E ) M − E + S 4 ( displaystyle ( \tfrac ( M - E + S ) ( 4 ) ) ) 3 M + E − S 8 ( displaystyle ( \tfrac ( 3M + E-S ) ( 8 ) ) ) E − M + S 4 M ( displaystyle ( \tfrac ( E-M + S ) ( 4M ) ) ) M ( displaystyle M ) S = ± E 2 + 9 M 2 − 10 E M ( displaystyle S = pm ( sqrt ( E ^ ( 2 ) + 9M ^ ( 2 ) - 10EM ) ) ) There are two valid solutions. The plus sign leads to ν ≥ 0 ( displaystyle \nu geq 0 ). The minus sign leads to ν ≤ 0 ( displaystyle \nu leq 0 ). ( λ, G ) ( displaystyle ( lambda, , G ) ) λ + 2 G 3 ( displaystyle lambda + ( \tfrac ( 2G ) ( 3 ) ) ) G ( 3 λ + 2 G ) λ + G ( displaystyle ( \tfrac ( G ( 3 lambda + 2G ) ) ( lambda + G ) ) ) λ ( displaystyle lambda ) G ( displaystyle G ) λ 2 ( λ + G ) ( displaystyle ( \tfrac ( lambda ) ( 2 ( lambda + G ) ) ) ) λ + 2 G ( displaystyle lambda + 2G , ) ( λ, ν ) ( displaystyle ( lambda, , \nu ) ) λ ( 1 + ν ) 3 ν ( displaystyle ( \tfrac ( lambda ( 1 + \nu ) ) ( 3 \nu ) ) ) λ ( 1 + ν ) ( 1 − 2 ν ) ν ( displaystyle ( \tfrac ( lambda ( 1 + \nu ) ( 1 - 2 \nu ) ) ( \nu ) ) ) λ ( displaystyle lambda ) λ ( 1 − 2 ν ) 2 ν ( displaystyle ( \tfrac ( lambda ( 1 - 2 \nu ) ) ( 2 \nu ) ) ) ν ( displaystyle \nu ) λ ( 1 − ν ) ν ( displaystyle ( \tfrac ( lambda ( 1 - \nu ) ) ( \nu ) ) ) Cannot be used when ν = 0 ⇔ λ = 0 ( displaystyle \nu = 0 Leftrightarrow lambda = 0 ) ( λ, M ) ( displaystyle ( lambda, , M ) ) M + 2 λ 3 ( displaystyle ( \tfrac ( M + 2 lambda ) ( 3 ) ) ) ( M − λ ) ( M + 2 λ ) M + λ ( displaystyle ( \tfrac ( ( M - lambda ) ( M + 2 lambda ) ) ( M+ lambda ) ) ) λ ( displaystyle lambda ) M − λ 2 ( displaystyle ( \tfrac ( M - lambda ) ( 2 ) ) ) λ M + λ ( displaystyle ( \tfrac ( lambda ) ( M+ lambda ) ) ) M ( displaystyle M ) ( G, ν ) ( displaystyle ( G, , \nu ) ) 2 G ( 1 + ν ) 3 ( 1 − 2 ν ) ( displaystyle ( \tfrac ( 2G ( 1 + \nu ) ) ( 3 ( 1 - 2 \nu ) ) ) ) 2 G ( 1 + ν ) ( displaystyle 2G ( 1 + \nu ) , ) 2 G ν 1 − 2 ν ( displaystyle ( \tfrac ( 2G \nu ) ( 1 - 2 \nu ) ) ) G ( displaystyle G ) ν ( displaystyle \nu ) 2 G ( 1 − ν ) 1 − 2 ν ( displaystyle ( \tfrac ( 2G ( 1 - \nu ) ) ( 1 - 2 \nu ) ) ) ( G, M ) ( displaystyle ( G, , M ) ) M − 4 G 3 ( displaystyle M - ( \tfrac ( 4G ) ( 3 ) ) ) G ( 3 M − 4 G ) M − G ( displaystyle ( \tfrac ( G ( 3M - 4G ) ) ( M-G ) ) ) M − 2 G ( displaystyle M - 2G , ) G ( displaystyle G ) M − 2 G 2 M − 2 G ( displaystyle ( \tfrac ( M - 2G ) ( 2M - 2G ) ) ) M ( displaystyle M ) ( ν, M ) ( displaystyle ( \nu, , M ) ) M ( 1 + ν ) 3 ( 1 − ν ) ( displaystyle ( \tfrac ( M ( 1 + \nu ) ) ( 3 ( 1 - \nu ) ) ) ) M ( 1 + ν ) ( 1 − 2 ν ) 1 − ν ( displaystyle ( \tfrac ( M ( 1 + \nu ) ( 1 - 2 \nu ) ) ( 1 - \nu ) ) ) M ν 1 − ν ( displaystyle ( \tfrac ( M \nu ) ( 1 - \nu ) ) ) M ( 1 − 2 ν ) 2 ( 1 − ν ) ( displaystyle ( \tfrac ( M ( 1 - 2 \nu ) ) ( 2 ( 1 - \nu ) ) ) ) ν ( displaystyle \nu ) M ( displaystyle M ) Retrieved from `` https://en.wikipedia.org/w/index.php?title=Young%27s_modulus&oldid=807855856 '' Categories : Elasticity ( physics ) Physical quantities Hidden categories : CS1 maint : Multiple names : authors list Use mdy dates from May 2017 Articles needing more detailed references Pages containing links to subscription - only content Talk Contents About Wikipedia Беларуская Boarisch Català Čeština Deutsch Español فارسی Français Gaeilge 한국어 Հայերեն हिन्दी Hrvatski Bahasa Indonesia Қазақша 日本 語 Polski Русский Scots සිංහල Simple English Slovenčina தமிழ் Türkçe Українська Tiếng Việt 中文 Edit links This page was last edited on 30 October 2017, at 14 : 08. 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Question: what are the units of stress strain and young’s modulus
Answer: megapascals (MPa or N/mm2)