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.^[1] 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 cannot exist, a material with a very high Young's modulus can be approximated as rigid.^[2]

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 stiffness of an I beam is much higher than that of a round tube 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 fracturing.

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.^[3]

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/m² or kg·m⁻¹·s⁻²). The practical units used are megapascals (MPa or N/mm²) or gigapascals (GPa or kN/mm²).

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 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, $\sigma (\varepsilon )$ , by the extensional strain, $\varepsilon$ , in the elastic (initial, linear) portion of the physical stress–strain curve:

E\equiv {\frac {\sigma (\varepsilon )}{\varepsilon }}={\frac {F/A_{0}}{\Delta L/L_{0}}}={\frac {FL_{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 actual 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

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

F={\frac {EA_{0}\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=\left({\frac {EA_{0}}{L_{0}}}\right)\Delta L=kx\,

where it comes in saturation

k\equiv {\begin{matrix}{\frac {EA_{0}}{L_{0}}}\end{matrix}}\,

and

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}=\int {kx}\,dx={\frac {1}{2}}kx^{2}.

now by expliciting the intensive variables:

U_{e}=\int {\frac {EA_{0}\Delta L}{L_{0}}}\,d\Delta L={\frac {EA_{0}}{L_{0}}}\int {\Delta L}\,d\Delta L={\frac {EA_{0}{\Delta L}^{2}}{2L_{0}}}

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

{\frac {U_{e}}{A_{0}L_{0}}}={\frac {E{\Delta L}^{2}}{2L_{0}^{2}}}

or, in simple notation, for a linear elastic material: $u_{e}(\varepsilon )=\int {E\,\varepsilon }\,d\varepsilon ={\frac {1}{2}}E{\varepsilon }^{2}$ , since the strain is defined $\varepsilon \equiv {\frac {\Delta L}{L_{0}}}$ .

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:

$u_{e}(\varepsilon )=\int {E(\varepsilon )\,\varepsilon }\,d\varepsilon \neq {\frac {1}{2}}E{\varepsilon }^{2}$

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=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^[4]	1.45–6998145000000000000♠14.5×10⁻³
Low-density polyethylene^[5]	0.11–0.86	1.6-6998650000000000000♠6.5×10⁻²
Diatom frustules (largely silicic acid)^[6]	0.35–2.77	0.05–0.4
PTFE (Teflon)	0.5 ^[4]	0.075
HDPE	0.8	0.116
Bacteriophage capsids^[7]	1–3	0.15–0.435
Polypropylene	1.5–2^[4]	0.22–0.29
Polyethylene terephthalate (PET)	2–2.7^[4]	0.29–0.39
Nylon	2–4	0.29–0.58
Polystyrene, solid	3–3.5^[4]	0.44–0.51
Polystyrene, foam^[8]	0.0025-0.007	0.00036-0.00102
Medium-density fiberboard (MDF)^[9]	4	0.58
wood (along grain)	11^[4]	1.60
Human Cortical Bone^[10]	14	2.03
Glass-reinforced polyester matrix ^[11]	17.2	2.49
Aromatic peptide nanotubes ^[12]^[13]	19–27	2.76-3.92
High-strength concrete	30^[4]	4.35
Carbon fiber reinforced plastic (50/50 fibre/matrix, biaxial fabric)	30–50^[14]	4.35 – 7.25
Hemp fiber ^[15]	35	5.08
Magnesium metal (Mg)	45^[4]	6.53
Glass (see chart)	50–90^[4]	7.25 – 13.1
Flax fiber ^[16]	58	8.41
Aluminum	69^[4]	10
Mother-of-pearl (nacre, largely calcium carbonate) ^[17]	70	10.2
Aramid^[18]	70.5–112.4	10.2 – 16.3
Tooth enamel (largely calcium phosphate)^[19]	83	12
Stinging nettle fiber ^[20]	87	12.6
Bronze	96–120^[4]	13.9 – 17.4
Brass	100–125^[4]	14.5 – 18.1
Titanium (Ti)	110.3	16^[4]
Titanium alloys	105–120^[4]	15 – 17.5
Copper (Cu)	117	17
Carbon fiber reinforced plastic (70/30 fibre/matrix, unidirectional, along fibre)^[21]	181	26.3
Silicon Single crystal, different directions ^[22]^[23]	130–185	18.9 – 26.8
Wrought iron	190–210^[4]	27.6 – 30.5
Steel (ASTM-A36)	200^[4]	29
polycrystalline Yttrium iron garnet (YIG)^[24]	193	28
single-crystal Yttrium iron garnet (YIG)^[25]	200	29
Cobalt-chrome (CoCr)^[26]	220-258	29
Aromatic peptide nanospheres ^[27]	230–275	33.4 – 40
Beryllium (Be)^[28]	287	41.6
Molybdenum (Mo)	329 - 330 ^[4] ^[29] ^[30]	47.7 – 47.9
Tungsten (W)	400 – 410 ^[4]	58 – 59
Silicon carbide (SiC)	450 ^[4]	65
Tungsten carbide (WC)	450 – 650 ^[4]	65 – 94
Osmium (Os)	525 - 562 ^[31]	76.1 – 81.5
Single-walled carbon nanotube	1,000 + ^[32]^[33]	150 +
Graphene	1050^[34]	152
Diamond (C)	1050 - 1210^[35]	152 - 175
Carbyne (C)^[36]	32100^[37]	7007321000000000000♠4,660

References

↑ The Rational Mechanics of Flexible or Elastic Bodies, 1638–1788: Introduction to Leonhardi Euleri Opera Omnia, vol. X and XI, Seriei Secundae. Orell Fussli.
↑ In computer graphics visual effects, the terms linear elastic material and rigid material are used interchangeably if the Young's modulus is high.
↑ IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "modulus of elasticity (Young's modulus), E".
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 "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.
↑ 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.
↑ 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..101.7600I. doi:10.1073/pnas.0308198101. PMC 419652. PMID 15133147.
↑ "Styrodur Technical Data" (PDF). BASF. Retrieved 2016-03-15.
↑ "Medium Density Fiberboard (MDF) Material Properties :: MakeItFrom.com". Retrieved 4 February 2016.
↑ Rho, JY (1993). "Young's modulus of trabecular and cortical bone material: ultrasonic and microtensile measurements". Journal of Biomechanics. 26 (2): 111–119. doi:10.1016/0021-9290(93)90042-d.
↑ Polyester Matrix Composite reinforced by glass fibers (Fiberglass). [SubsTech] (2008-05-17). Retrieved on 2011-03-30.
↑ 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.
↑ 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.
↑ "Composites Design and Manufacture (BEng) – MATS 324".
↑ 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.
↑ 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.
↑ DuPont (2001). "Kevlar Technical Guide": 9.
↑ 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..16.2551S. doi:10.1007/bf01113595.
↑ 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.
↑ 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
↑ "Properties of cobalt-chrome alloys – Heraeus Kulzer cara". Retrieved 4 February 2016.
↑ 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.
↑ 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.
↑ . webelements http://www.webelements.com/molybdenum/physics.html. Retrieved Jan 27, 2015. Missing or empty |title= (help)
↑ (PDF). Glemco http://www2.glemco.com/pdf/NEW_MARTERIAL_LIST/Molybdenum.pdf. Retrieved Jan 27, 2014. Missing or empty |title= (help)
↑ 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.
↑ L. Forro; et al. "Electronic and mechanical properties of carbon nanotubes" (PDF).
↑ 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.
↑ Fang Liu; Pingbing Ming & Ju Li. "Ab initio calculation of ideal strength and phonon instability of graphene under tension" (PDF).
↑ Spear and Dismukes (1994). Synthetic Diamond – Emerging CVD Science and Technology. Wiley, NY. p. 315. ISBN 978-0-471-53589-8.
↑ Owano, Nancy (Aug 20, 2013). "Carbyne is stronger than any known material". phys.org.
↑ 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.

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Elastic moduli for homogeneous isotropic materials

Bulk modulus ( $K$ ) Young's modulus ( $E$ ) Lamé's first parameter ( $\lambda$ ) Shear modulus ( $G,\mu$ ) Poisson's ratio ( $\nu$ ) P-wave modulus ( $M$ )

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.
	$K=\,$	$E=\,$	$\lambda =\,$	$G=\,$	$\nu =\,$	$M=\,$	Notes
$(K,\,E)$	$K$	$E$	${\tfrac {3K(3K-E)}{9K-E}}$	${\tfrac {3KE}{9K-E}}$	${\tfrac {3K-E}{6K}}$	${\tfrac {3K(3K+E)}{9K-E}}$
$(K,\,\lambda )$	$K$	${\tfrac {9K(K-\lambda )}{3K-\lambda }}$	$\lambda$	${\tfrac {3(K-\lambda )}{2}}$	${\tfrac {\lambda }{3K-\lambda }}$	$3K-2\lambda \,$
$(K,\,G)$	$K$	${\tfrac {9KG}{3K+G}}$	$K-{\tfrac {2G}{3}}$	$G$	${\tfrac {3K-2G}{2(3K+G)}}$	$K+{\tfrac {4G}{3}}$
$(K,\,\nu )$	$K$	$3K(1-2\nu )\,$	${\tfrac {3K\nu }{1+\nu }}$	${\tfrac {3K(1-2\nu )}{2(1+\nu )}}$	$\nu$	${\tfrac {3K(1-\nu )}{1+\nu }}$
$(K,\,M)$	$K$	${\tfrac {9K(M-K)}{3K+M}}$	${\tfrac {3K-M}{2}}$	${\tfrac {3(M-K)}{4}}$	${\tfrac {3K-M}{3K+M}}$	$M$
$(E,\,\lambda )$	${\tfrac {E+3\lambda +R}{6}}$	$E$	$\lambda$	${\tfrac {E-3\lambda +R}{4}}$	${\tfrac {2\lambda }{E+\lambda +R}}$	${\tfrac {E-\lambda +R}{2}}$	$R={\sqrt {E^{2}+9\lambda ^{2}+2E\lambda }}$
$(E,\,G)$	${\tfrac {EG}{3(3G-E)}}$	$E$	${\tfrac {G(E-2G)}{3G-E}}$	$G$	${\tfrac {E}{2G}}-1$	${\tfrac {G(4G-E)}{3G-E}}$
$(E,\,\nu )$	${\tfrac {E}{3(1-2\nu )}}$	$E$	${\tfrac {E\nu }{(1+\nu )(1-2\nu )}}$	${\tfrac {E}{2(1+\nu )}}$	$\nu$	${\tfrac {E(1-\nu )}{(1+\nu )(1-2\nu )}}$
$(E,\,M)$	${\tfrac {3M-E+S}{6}}$	$E$	${\tfrac {M-E+S}{4}}$	${\tfrac {3M+E-S}{8}}$	${\tfrac {E-M+S}{4M}}$	$M$	$S=\pm {\sqrt {E^{2}+9M^{2}-10EM}}$ There are two valid solutions. The plus sign leads to $\nu \geq 0$ . The minus sign leads to $\nu \leq 0$ .
$(\lambda ,\,G)$	$\lambda +{\tfrac {2G}{3}}$	${\tfrac {G(3\lambda +2G)}{\lambda +G}}$	$\lambda$	$G$	${\tfrac {\lambda }{2(\lambda +G)}}$	$\lambda +2G\,$
$(\lambda ,\,\nu )$	${\tfrac {\lambda (1+\nu )}{3\nu }}$	${\tfrac {\lambda (1+\nu )(1-2\nu )}{\nu }}$	$\lambda$	${\tfrac {\lambda (1-2\nu )}{2\nu }}$	$\nu$	${\tfrac {\lambda (1-\nu )}{\nu }}$	Cannot be used when $\nu =0\Leftrightarrow \lambda =0$
$(\lambda ,\,M)$	${\tfrac {M+2\lambda }{3}}$	${\tfrac {(M-\lambda )(M+2\lambda )}{M+\lambda }}$	$\lambda$	${\tfrac {M-\lambda }{2}}$	${\tfrac {\lambda }{M+\lambda }}$	$M$
$(G,\,\nu )$	${\tfrac {2G(1+\nu )}{3(1-2\nu )}}$	$2G(1+\nu )\,$	${\tfrac {2G\nu }{1-2\nu }}$	$G$	$\nu$	${\tfrac {2G(1-\nu )}{1-2\nu }}$
$(G,\,M)$	$M-{\tfrac {4G}{3}}$	${\tfrac {G(3M-4G)}{M-G}}$	$M-2G\,$	$G$	${\tfrac {M-2G}{2M-2G}}$	$M$
$(\nu ,\,M)$	${\tfrac {M(1+\nu )}{3(1-\nu )}}$	${\tfrac {M(1+\nu )(1-2\nu )}{1-\nu }}$	${\tfrac {M\nu }{1-\nu }}$	${\tfrac {M(1-2\nu )}{2(1-\nu )}}$	$\nu$	$M$

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