Young's modulus

Young's modulus (or Young modulus) is a mechanical property of

region of the material.

Although Young's modulus is named after the 19th-century British scientist Thomas Young, the concept was developed in 1727 by Leonhard Euler. The first experiments that used the concept of Young's modulus in its modern form were performed by the Italian scientist Giordano Riccati in 1782, pre-dating Young's work by 25 years.^[1] The term modulus is derived from the Latin root term modus which means measure.

Definition

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

Young's modulus is commonly measured in the

(GPa).

Examples:

• Rubber
  (increasing pressure: length increases quickly, meaning low ${\displaystyle E}$)
• Aluminium (increasing pressure: length increases slowly, meaning high ${\displaystyle E}$)

Linear elasticity

A solid material undergoes

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

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.

Usage

Young's modulus enables the calculation of the change in the dimension of a bar made of an

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 ${\displaystyle G}$, bulk modulus ${\displaystyle K}$, and Poisson's ratio ${\displaystyle \nu }$. Any two of these parameters are sufficient to fully describe elasticity in an isotropic material. For homogeneous isotropic materials simple relations exist between elastic constants that allow calculating them all as long as two are known:

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

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.

, 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

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

. Engineers can use this directional phenomenon to their advantage in creating structures.

Temperature dependence

The Young's modulus of metals varies with the temperature and can be realized through the change in the interatomic bonding of the atoms, and hence its change is found to be dependent on the change in the work function of the metal. Although classically, this change is predicted through fitting and without a clear underlying mechanism (for example, the Watchman's formula), the Rahemi-Li model[4] demonstrates how the change in the electron work function leads to change in the Young's modulus of metals and predicts this variation with calculable parameters, using the generalization of the Lennard-Jones potential to solids. In general, as the temperature increases, the Young's modulus decreases via ${\displaystyle E(T)=\beta (\varphi (T))^{6}}$ where the electron work function varies with the temperature as ${\displaystyle \varphi (T)=\varphi _{0}-\gamma {\frac {(k_{B}T)^{2}}{\varphi _{0}}}}$ and ${\displaystyle \gamma }$ is a calculable material property which is dependent on the crystal structure (for example, BCC, FCC). ${\displaystyle \varphi _{0}}$ is the electron work function at T=0 and ${\displaystyle \beta }$ is constant throughout the change.

Calculation

Young's modulus E, is calculated by dividing the

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

where

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

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

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

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

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

where it comes in saturation

${\displaystyle k\equiv {\frac {EA}{L_{0}}}\,}$ and ${\displaystyle x\equiv \Delta L.}$

But note that the elasticity of coiled springs comes from shear modulus, not Young's modulus. ^{[citation needed]}

Elastic potential energy

The

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

${\displaystyle U_{e}=\int {kx}\,dx={\frac {1}{2}}kx^{2}.}$

now by explicating the intensive variables:

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

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

${\displaystyle {\frac {U_{e}}{AL_{0}}}={\frac {E\,{\Delta L}^{2}}{2L_{0}^{2}}}}$

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

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:

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

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

Material	Young's modulus (GPa)	Megapound per square inch ( psi)[5]	Ref.
Aluminium (13Al)	68	9.86	[6][7][8][9][10][11]
Amino-acid molecular crystals	21–44	3.05–6.38	[12]
Aramid (for example, Kevlar)	70.5–112.4	10.2–16.3	[13]
Aromatic peptide-nanospheres	230–275	33.4–39.9	[14]
Aromatic peptide-nanotubes	19–27	2.76–3.92	[15][16]
Bacteriophage capsids	1–3	0.145–0.435	[17]
Beryllium (4Be)	287	41.6	[18]
Bone, human cortical	14	2.03	[19]
Brass	106	15.4	[20]
Bronze	112	16.2	[21]
Carbon nitride (CN2)	822	119	[22]
Carbon-fiber-reinforced plastic (CFRP), 50/50 fibre/matrix, biaxial fabric	30–50	4.35–7.25	[23]
Carbon-fiber-reinforced plastic (CFRP), 70/30 fibre/matrix, unidirectional, along fibre	181	26.3	[24]
Cobalt-chrome (CoCr)	230	33.4	[25]
Copper (Cu), annealed	110	16	[26]
Diamond (C), synthetic	1050–1210	152–175	[27]
Diatom frustules, largely silicic acid	0.35–2.77	0.051–0.058	[28]
Flax fiber	58	8.41	[29]
Float glass	47.7–83.6	6.92–12.1	[30]
Glass-reinforced polyester (GRP)	17.2	2.49	[31]
Gold	77.2	11.2	[32]
Graphene	1050	152	[33]
Hemp fiber	35	5.08	[34]
High-density polyethylene (HDPE)	0.97–1.38	0.141–0.2	[35]
High-strength concrete	30	4.35	[36]
Lead (82Pb), chemical	13	1.89	[11]
Low-density polyethylene (LDPE), molded	0.228	0.0331	[37]
Magnesium alloy	45.2	6.56	[38]
Medium-density fiberboard (MDF)	4	0.58	[39]
Molybdenum (Mo), annealed	330	47.9	[40][7][8][9][10][11]
Monel	180	26.1	[11]
Mother-of-pearl (largely calcium carbonate)	70	10.2	[41]
Nickel (28Ni), commercial	200	29	[11]
Nylon 66	2.93	0.425	[42]
Osmium (76Os)	525–562	76.1–81.5	[43]
Osmium nitride (OsN2)	194.99–396.44	28.3–57.5	[44]
Polycarbonate (PC)	2.2	0.319	[45]
Polyethylene terephthalate (PET), unreinforced	3.14	0.455	[46]
Polypropylene (PP), molded	1.68	0.244	[47]
Polystyrene, crystal	2.5–3.5	0.363–0.508	[48]
Polystyrene, foam	0.0025–0.007	0.000363–0.00102	[49]
Polytetrafluoroethylene (PTFE), molded	0.564	0.0818	[50]
Rubber, small strain	0.01–0.1	0.00145–0.0145	[12]
Silicon, single crystal, different directions	130–185	18.9–26.8	[51]
Silicon carbide (SiC)	90–137	13.1–19.9	[52]
Single-walled carbon nanotube	${\displaystyle >}$1000	${\displaystyle >}$140	[53][54]
Steel, A36	200	29	[55]
Stinging nettle fiber	87	12.6	[29]
Titanium (22Ti)	116	16.8	[56][57][7][9][8][11][10]
Titanium alloy , Grade 5	114	16.5	[58]
Tooth enamel, largely calcium phosphate	83	12	[59]
Tungsten carbide (WC)	600–686	87–99.5	[60]
Wood, American beech	9.5–11.9	1.38–1.73	[61]
Wood, black cherry	9–10.3	1.31–1.49	[61]
Wood, red maple	9.6–11.3	1.39–1.64	[61]
Wrought iron	193	28	[62]
Yttrium iron garnet (YIG), polycrystalline	193	28	[63]
Yttrium iron garnet (YIG), single-crystal	200	29	[64]
Zinc (30Zn)	108	15.7	[65]
Zirconium (40Zr), commercial	95	13.8	[11]

See also

• Bending stiffness
• Deflection
• Deformation
• Flexural modulus
• Impulse excitation technique
• List of materials properties
• Yield (engineering)

References

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 175,000 materials
• Young's Modulus for groups of materials, and their cost

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