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Young's modulus
Mechanical property that measures stiffness of a solid material

Young's modulus is a mechanical property that measures the tensile or compressive stiffness of solid materials when force is applied lengthwise. It represents the modulus of elasticity for axial tension or compression, defined as the ratio of stress to resulting axial strain within the linear elastic region. Although named after 19th-century scientist Thomas Young, the concept originated with Leonhard Euler in 1727, and was experimentally developed by Giordano Riccati in 1782. The term modulus comes from the Latin root modus, meaning measure.

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Definition

Young's modulus, E {\displaystyle E} , quantifies the relationship between tensile or compressive stress σ {\displaystyle \sigma } (force per unit area) and axial strain ε {\displaystyle \varepsilon } (proportional deformation) in the linear elastic region of a material:2 E = σ ε {\displaystyle E={\frac {\sigma }{\varepsilon }}}

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

Examples:

  • Rubber (increasing pressure: large length increase, meaning low E {\displaystyle E} )
  • Aluminium (increasing pressure: small length increase, meaning high E {\displaystyle E} )

Linear elasticity

Main article: Linear elasticity

A solid material undergoes elastic 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 is linear, and the relationship between stress and strain is described by Hooke'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 idealized rigid 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

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, an I-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

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 G {\displaystyle G} , bulk modulus K {\displaystyle K} , and Poisson's ratio ν {\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 materials simple relations exist between elastic constants 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 ).}

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.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 include wood and reinforced concrete. Engineers can use this directional phenomenon to their advantage in creating structures.

Calculation

Young's modulus is 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 {\displaystyle E} is the Young's modulus (modulus of elasticity);
  • F {\displaystyle F} is the force exerted on an object under tension;
  • A {\displaystyle A} is the actual cross-sectional area, which equals the area of the cross-section perpendicular to the applied force;
  • Δ L {\displaystyle \Delta L} is the amount by which the length of the object changes ( Δ L {\displaystyle \Delta L} is positive if the material is stretched, and negative when the material is compressed);
  • L 0 {\displaystyle L_{0}} is the original length of the object.

Force exerted by stretched or contracted material

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

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 {\frac {EA}{L_{0}}}\,} and x ≡ Δ L . {\displaystyle x\equiv \Delta L.}

Note that the elasticity of coiled springs comes from shear 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.

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 explicating 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 (that is, per unit volume) is given by:

U e A L 0 = E Δ L 2 2 L 0 2 = 1 2 × E Δ L L 0 × Δ L L 0 = 1 2 × 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: u e ( ε ) = ∫ E ε d ε = 1 2 E ε 2 {\textstyle u_{e}(\varepsilon )=\int {E\,\varepsilon }\,d\varepsilon ={\frac {1}{2}}E{\varepsilon }^{2}} , since the strain is defined ε ≡ Δ L L 0 {\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 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}}

Examples

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
MaterialYoung's modulus (GPa)Megapound per square inch (Mpsi)5Ref.
Aluminium (13Al)689.8667891011
Amino-acid molecular crystals21–443.05–6.3812
Aramid (for example, Kevlar)70.5–112.410.2–16.313
Aromatic peptide-nanospheres230–27533.4–39.914
Aromatic peptide-nanotubes19–272.76–3.921516
Bacteriophage capsids1–30.145–0.43517
Beryllium (4Be)28741.618
Bone, human cortical142.0319
Brass10615.420
Bronze11216.221
Carbon nitride (CN2)82211922
Carbon-fiber-reinforced plastic (CFRP), 50/50 fibre/matrix, biaxial fabric30–504.35–7.2523
Carbon-fiber-reinforced plastic (CFRP), 70/30 fibre/matrix, unidirectional, along fibre18126.324
Cobalt-chrome (CoCr)23033.425
Copper (Cu), annealed1101626
Diamond (C), synthetic1050–1210152–17527
Diatom frustules, largely silicic acid0.35–2.770.051–0.05828
Flax fiber588.4129
Float glass47.7–83.66.92–12.130
Glass-reinforced polyester (GRP)17.22.4931
Gold77.211.232
Graphene105015233
Hemp fiber355.0834
High-density polyethylene (HDPE)0.97–1.380.141–0.235
High-strength concrete304.3536
Lead (82Pb), chemical131.8937
Low-density polyethylene (LDPE), molded0.2280.033138
Magnesium alloy45.26.5639
Medium-density fiberboard (MDF)40.5840
Molybdenum (Mo), annealed33047.9414243444546
Monel18026.147
Mother-of-pearl (largely calcium carbonate)7010.248
Nickel (28Ni), commercial2002949
Nylon 662.930.42550
Osmium (76Os)525–56276.1–81.551
Osmium nitride (OsN2)194.99–396.4428.3–57.552
Polycarbonate (PC)2.20.31953
Polyethylene terephthalate (PET), unreinforced3.140.45554
Polypropylene (PP), molded1.680.24455
Polystyrene, crystal2.5–3.50.363–0.50856
Polystyrene, foam0.0025–0.0070.000363–0.0010257
Polytetrafluoroethylene (PTFE), molded0.5640.081858
Rubber, small strain0.01–0.10.00145–0.014559
Silicon, single crystal, different directions130–18518.9–26.860
Silicon carbide (SiC)90–13713.1–19.961
Single-walled carbon nanotube > {\displaystyle >} 1000 > {\displaystyle >} 1406263
Steel, A362002964
Stinging nettle fiber8712.665
Titanium (22Ti)11616.866676869707172
Titanium alloy, Grade 511416.573
Tooth enamel, largely calcium phosphate831274
Tungsten carbide (WC)600–68687–99.575
Wood, American beech9.5–11.91.38–1.7376
Wood, black cherry9–10.31.31–1.4977
Wood, red maple9.6–11.31.39–1.6478
Wrought iron1932879
Yttrium iron garnet (YIG), polycrystalline1932880
Yttrium iron garnet (YIG), single-crystal2002981
Zinc (30Zn)10815.782
Zirconium (40Zr), commercial9513.883

See also

Further reading

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 formulae K = {\displaystyle K=\,} E = {\displaystyle E=\,} λ = {\displaystyle \lambda =\,} G = {\displaystyle G=\,} ν = {\displaystyle \nu =\,} M = {\displaystyle M=\,} Notes
( K , E ) {\displaystyle (K,\,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 )} 9 K ( K − λ ) 3 K − λ {\displaystyle {\tfrac {9K(K-\lambda )}{3K-\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)} 9 K G 3 K + G {\displaystyle {\tfrac {9KG}{3K+G}}} K − 2 G 3 {\displaystyle K-{\tfrac {2G}{3}}} 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 )} 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 )}}} 3 K ( 1 − ν ) 1 + ν {\displaystyle {\tfrac {3K(1-\nu )}{1+\nu }}}
( K , M ) {\displaystyle (K,\,M)} 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}}}
( E , λ ) {\displaystyle (E,\,\lambda )} E + 3 λ + R 6 {\displaystyle {\tfrac {E+3\lambda +R}{6}}} 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)}}} G ( E − 2 G ) 3 G − E {\displaystyle {\tfrac {G(E-2G)}{3G-E}}} 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 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 )}}} 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}}} 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}}} 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 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 ( 1 − 2 ν ) 2 ν {\displaystyle {\tfrac {\lambda (1-2\nu )}{2\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 }}} M − λ 2 {\displaystyle {\tfrac {M-\lambda }{2}}} λ M + λ {\displaystyle {\tfrac {\lambda }{M+\lambda }}}
( 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 }}} 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\,} M − 2 G 2 M − 2 G {\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 − 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 )}}}
2D formulae K 2 D = {\displaystyle K_{\mathrm {2D} }=\,} E 2 D = {\displaystyle E_{\mathrm {2D} }=\,} λ 2 D = {\displaystyle \lambda _{\mathrm {2D} }=\,} G 2 D = {\displaystyle G_{\mathrm {2D} }=\,} ν 2 D = {\displaystyle \nu _{\mathrm {2D} }=\,} M 2 D = {\displaystyle M_{\mathrm {2D} }=\,} Notes
( K 2 D , E 2 D ) {\displaystyle (K_{\mathrm {2D} },\,E_{\mathrm {2D} })} 2 K 2 D ( 2 K 2 D − E 2 D ) 4 K 2 D − E 2 D {\displaystyle {\tfrac {2K_{\mathrm {2D} }(2K_{\mathrm {2D} }-E_{\mathrm {2D} })}{4K_{\mathrm {2D} }-E_{\mathrm {2D} }}}} K 2 D E 2 D 4 K 2 D − E 2 D {\displaystyle {\tfrac {K_{\mathrm {2D} }E_{\mathrm {2D} }}{4K_{\mathrm {2D} }-E_{\mathrm {2D} }}}} 2 K 2 D − E 2 D 2 K 2 D {\displaystyle {\tfrac {2K_{\mathrm {2D} }-E_{\mathrm {2D} }}{2K_{\mathrm {2D} }}}} 4 K 2 D 2 4 K 2 D − E 2 D {\displaystyle {\tfrac {4K_{\mathrm {2D} }^{2}}{4K_{\mathrm {2D} }-E_{\mathrm {2D} }}}}
( K 2 D , λ 2 D ) {\displaystyle (K_{\mathrm {2D} },\,\lambda _{\mathrm {2D} })} 4 K 2 D ( K 2 D − λ 2 D ) 2 K 2 D − λ 2 D {\displaystyle {\tfrac {4K_{\mathrm {2D} }(K_{\mathrm {2D} }-\lambda _{\mathrm {2D} })}{2K_{\mathrm {2D} }-\lambda _{\mathrm {2D} }}}} K 2 D − λ 2 D {\displaystyle K_{\mathrm {2D} }-\lambda _{\mathrm {2D} }} λ 2 D 2 K 2 D − λ 2 D {\displaystyle {\tfrac {\lambda _{\mathrm {2D} }}{2K_{\mathrm {2D} }-\lambda _{\mathrm {2D} }}}} 2 K 2 D − λ 2 D {\displaystyle 2K_{\mathrm {2D} }-\lambda _{\mathrm {2D} }}
( K 2 D , G 2 D ) {\displaystyle (K_{\mathrm {2D} },\,G_{\mathrm {2D} })} 4 K 2 D G 2 D K 2 D + G 2 D {\displaystyle {\tfrac {4K_{\mathrm {2D} }G_{\mathrm {2D} }}{K_{\mathrm {2D} }+G_{\mathrm {2D} }}}} K 2 D − G 2 D {\displaystyle K_{\mathrm {2D} }-G_{\mathrm {2D} }} K 2 D − G 2 D K 2 D + G 2 D {\displaystyle {\tfrac {K_{\mathrm {2D} }-G_{\mathrm {2D} }}{K_{\mathrm {2D} }+G_{\mathrm {2D} }}}} K 2 D + G 2 D {\displaystyle K_{\mathrm {2D} }+G_{\mathrm {2D} }}
( K 2 D , ν 2 D ) {\displaystyle (K_{\mathrm {2D} },\,\nu _{\mathrm {2D} })} 2 K 2 D ( 1 − ν 2 D ) {\displaystyle 2K_{\mathrm {2D} }(1-\nu _{\mathrm {2D} })\,} 2 K 2 D ν 2 D 1 + ν 2 D {\displaystyle {\tfrac {2K_{\mathrm {2D} }\nu _{\mathrm {2D} }}{1+\nu _{\mathrm {2D} }}}} K 2 D ( 1 − ν 2 D ) 1 + ν 2 D {\displaystyle {\tfrac {K_{\mathrm {2D} }(1-\nu _{\mathrm {2D} })}{1+\nu _{\mathrm {2D} }}}} 2 K 2 D 1 + ν 2 D {\displaystyle {\tfrac {2K_{\mathrm {2D} }}{1+\nu _{\mathrm {2D} }}}}
( E 2 D , G 2 D ) {\displaystyle (E_{\mathrm {2D} },\,G_{\mathrm {2D} })} E 2 D G 2 D 4 G 2 D − E 2 D {\displaystyle {\tfrac {E_{\mathrm {2D} }G_{\mathrm {2D} }}{4G_{\mathrm {2D} }-E_{\mathrm {2D} }}}} 2 G 2 D ( E 2 D − 2 G 2 D ) 4 G 2 D − E 2 D {\displaystyle {\tfrac {2G_{\mathrm {2D} }(E_{\mathrm {2D} }-2G_{\mathrm {2D} })}{4G_{\mathrm {2D} }-E_{\mathrm {2D} }}}} E 2 D 2 G 2 D − 1 {\displaystyle {\tfrac {E_{\mathrm {2D} }}{2G_{\mathrm {2D} }}}-1} 4 G 2 D 2 4 G 2 D − E 2 D {\displaystyle {\tfrac {4G_{\mathrm {2D} }^{2}}{4G_{\mathrm {2D} }-E_{\mathrm {2D} }}}}
( E 2 D , ν 2 D ) {\displaystyle (E_{\mathrm {2D} },\,\nu _{\mathrm {2D} })} E 2 D 2 ( 1 − ν 2 D ) {\displaystyle {\tfrac {E_{\mathrm {2D} }}{2(1-\nu _{\mathrm {2D} })}}} E 2 D ν 2 D ( 1 + ν 2 D ) ( 1 − ν 2 D ) {\displaystyle {\tfrac {E_{\mathrm {2D} }\nu _{\mathrm {2D} }}{(1+\nu _{\mathrm {2D} })(1-\nu _{\mathrm {2D} })}}} E 2 D 2 ( 1 + ν 2 D ) {\displaystyle {\tfrac {E_{\mathrm {2D} }}{2(1+\nu _{\mathrm {2D} })}}} E 2 D ( 1 + ν 2 D ) ( 1 − ν 2 D ) {\displaystyle {\tfrac {E_{\mathrm {2D} }}{(1+\nu _{\mathrm {2D} })(1-\nu _{\mathrm {2D} })}}}
( λ 2 D , G 2 D ) {\displaystyle (\lambda _{\mathrm {2D} },\,G_{\mathrm {2D} })} λ 2 D + G 2 D {\displaystyle \lambda _{\mathrm {2D} }+G_{\mathrm {2D} }} 4 G 2 D ( λ 2 D + G 2 D ) λ 2 D + 2 G 2 D {\displaystyle {\tfrac {4G_{\mathrm {2D} }(\lambda _{\mathrm {2D} }+G_{\mathrm {2D} })}{\lambda _{\mathrm {2D} }+2G_{\mathrm {2D} }}}} λ 2 D λ 2 D + 2 G 2 D {\displaystyle {\tfrac {\lambda _{\mathrm {2D} }}{\lambda _{\mathrm {2D} }+2G_{\mathrm {2D} }}}} λ 2 D + 2 G 2 D {\displaystyle \lambda _{\mathrm {2D} }+2G_{\mathrm {2D} }\,}
( λ 2 D , ν 2 D ) {\displaystyle (\lambda _{\mathrm {2D} },\,\nu _{\mathrm {2D} })} λ 2 D ( 1 + ν 2 D ) 2 ν 2 D {\displaystyle {\tfrac {\lambda _{\mathrm {2D} }(1+\nu _{\mathrm {2D} })}{2\nu _{\mathrm {2D} }}}} λ 2 D ( 1 + ν 2 D ) ( 1 − ν 2 D ) ν 2 D {\displaystyle {\tfrac {\lambda _{\mathrm {2D} }(1+\nu _{\mathrm {2D} })(1-\nu _{\mathrm {2D} })}{\nu _{\mathrm {2D} }}}} λ 2 D ( 1 − ν 2 D ) 2 ν 2 D {\displaystyle {\tfrac {\lambda _{\mathrm {2D} }(1-\nu _{\mathrm {2D} })}{2\nu _{\mathrm {2D} }}}} λ 2 D ν 2 D {\displaystyle {\tfrac {\lambda _{\mathrm {2D} }}{\nu _{\mathrm {2D} }}}} Cannot be used when ν 2 D = 0 ⇔ λ 2 D = 0 {\displaystyle \nu _{\mathrm {2D} }=0\Leftrightarrow \lambda _{\mathrm {2D} }=0}
( G 2 D , ν 2 D ) {\displaystyle (G_{\mathrm {2D} },\,\nu _{\mathrm {2D} })} G 2 D ( 1 + ν 2 D ) 1 − ν 2 D {\displaystyle {\tfrac {G_{\mathrm {2D} }(1+\nu _{\mathrm {2D} })}{1-\nu _{\mathrm {2D} }}}} 2 G 2 D ( 1 + ν 2 D ) {\displaystyle 2G_{\mathrm {2D} }(1+\nu _{\mathrm {2D} })\,} 2 G 2 D ν 2 D 1 − ν 2 D {\displaystyle {\tfrac {2G_{\mathrm {2D} }\nu _{\mathrm {2D} }}{1-\nu _{\mathrm {2D} }}}} 2 G 2 D 1 − ν 2 D {\displaystyle {\tfrac {2G_{\mathrm {2D} }}{1-\nu _{\mathrm {2D} }}}}
( G 2 D , M 2 D ) {\displaystyle (G_{\mathrm {2D} },\,M_{\mathrm {2D} })} M 2 D − G 2 D {\displaystyle M_{\mathrm {2D} }-G_{\mathrm {2D} }} 4 G 2 D ( M 2 D − G 2 D ) M 2 D {\displaystyle {\tfrac {4G_{\mathrm {2D} }(M_{\mathrm {2D} }-G_{\mathrm {2D} })}{M_{\mathrm {2D} }}}} M 2 D − 2 G 2 D {\displaystyle M_{\mathrm {2D} }-2G_{\mathrm {2D} }\,} M 2 D − 2 G 2 D M 2 D {\displaystyle {\tfrac {M_{\mathrm {2D} }-2G_{\mathrm {2D} }}{M_{\mathrm {2D} }}}}

References

  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. /wiki/John_Wiley_%26_Sons,_Inc ↩

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