# modulus of elasticity

Physical size
Surname Modulus of elasticity
Formula symbol E.
Size and
unit system
unit dimension
SI Pa  = N / m 2 = kg · m -1 · s -2 M · L −1 · T −2
cgs Ba = dyn / cm 2 = cm −1 g s −2
See also: tension (mechanics) pressure p${\ displaystyle \ sigma}$

The modulus of elasticity , also E-modulus , tensile modulus , coefficient of elasticity , elongation modulus , or Young's modulus , is a material parameter from materials engineering that describes the proportional relationship between stress and elongation when a solid body is deformed in the case of linear-elastic behavior . The modulus of elasticity - in another notation directly proportional to the spring constant - is the proportionality constant in Hooke's law .

The type of magnitude of the modulus of elasticity is mechanical stress . As a formula symbol is common. ${\ displaystyle E}$

The modulus of elasticity increases with the resistance that a material opposes to its elastic deformation. A component made from a material with a high modulus of elasticity such as steel is thus stiffer than the same component made from a material with a low modulus of elasticity such as rubber.

In the case of anisotropic , especially crystalline materials, the modulus of elasticity is direction-dependent and must be described by the elasticity tensor , the components of which are represented in a simplified manner by the elastic constants . These are material constants that vary within real solids, since real solids are neither perfectly homogeneous (especially concrete ) nor have constant physical properties (e.g. temperature).

## definition

Schematic stress-strain diagram: for small strains linear, Hooke's straight line with slope E.

The modulus of elasticity is defined as the slope of the graph in the stress-strain diagram with uniaxial loading with infinitesimal change in distortion with no stress. Most materials have a (at least small) linear area, this is also known as Hooke's area .

${\ displaystyle E = {\ frac {\ sigma} {\ varepsilon}} = {\ text {const.}}}$

The mechanical stress denotes the force per area ( normal stress , not shear stress ) and the elongation. The elongation is the ratio of the change in length to the original length . The unit of the modulus of elasticity is that of a stress: ${\ displaystyle \ sigma = {\ frac {F} {A}}}$${\ displaystyle \ varepsilon = {\ frac {\ Delta \ ell} {\ ell _ {0}}}}$${\ displaystyle \ Delta \ ell = \ ell - \ ell _ {0}}$${\ displaystyle \ ell _ {0}}$

${\ displaystyle \ left [E \ right] = 1 \, {\ frac {\ mathrm {N}} {\ mathrm {mm} ^ {2}}} = 1 \, \ mathrm {MPa}}$In SI units : .${\ displaystyle \ left [E \ right] = 1 \, {\ frac {\ mathrm {N}} {\ mathrm {m} ^ {2}}} = 1 \, \ mathrm {Pa}}$

The modulus of elasticity is referred to as the material constant, since it and the Poisson's ratio are used to establish the law of elasticity . However, the modulus of elasticity is not constant with regard to all physical quantities. It depends on various environmental conditions such as B. temperature or humidity .

### application

With an ideal linear- elastic material law (proportional range in the stress-strain diagram), the spring constant c of a straight bar results from its cross-sectional area  A , its length and its modulus of elasticity  E : ${\ displaystyle L_ {0}}$

${\ displaystyle c = {\ frac {F} {\ Delta L}} = {\ frac {E \ cdot A} {L_ {0}}}}$ .

With the expressions for the stress and for the elongation, Hooke's law for the uniaxial stress state is obtained from the above equation ${\ displaystyle \ sigma = {\ frac {F} {A}}}$${\ displaystyle \ varepsilon = {\ frac {\ Delta L} {L_ {0}}}}$

${\ displaystyle \ sigma = E \ cdot \ varepsilon}$

and from this the modulus of elasticity

${\ displaystyle E = {\ frac {\ sigma} {\ varepsilon}}}$

### Typical numerical values

material E-module in GPa material E-module in GPa
Metallic materials at 20 ° C Non-metallic materials at 20 ° C
beryllium 303 PVC 1.0 ... 3.5
Structural steel 210 Glass 40 ... 90
V2A steel 180 concrete 20 ... 40
cast iron 90… 145 Ceramics 160 ... 440
Brass 78… 123 Wood 10… 15
copper 100 ... 130 Polypropylene 1.3 ... 1.8
titanium 110 rubber up to 0.05
gold 78 Ice (−4  ° C ) 10
nickel 195… 205 Hard rubber 5

## Relations of elastic constants

There are different relationships between elastic moduli , according to which one elastic modulus can be calculated from two others for isotropic materials.

In addition to the modulus of elasticity (E-module) is the shear modulus , and shear modulus or G-called used which is measured at shear and, depending on Poisson's ratio for non- auxetic materials is that 0.33 to 0.5 times the modulus of elasticity. With rigid materials, the elastic modulus is usually measured in soft ( gels , polymer - melt ) the shear modulus as the elastic modulus can usually no longer be measured well in such systems because the sample under its own weight deformed (the so-called sagging.) .

For a linear-elastic, isotropic material, the following relationship applies between the shear modulus G , the compression modulus K and the Poisson's number : ${\ displaystyle \ nu}$

${\ displaystyle E = 2 (1+ \ nu) \ cdot G = 3 (1-2 \ nu) \ cdot K = {\ frac {9KG} {3K + G}}}$

## Relation to other properties of metallic materials

The modulus of elasticity is not strictly related to the hardness , yield point and tensile strength of metallic materials (e.g. simple structural steel and high-strength special steel). The modulus of elasticity of a metal increases with its melting temperature . In addition, body-centered cubic metals have a higher modulus of elasticity than face-centered cubic metals at a comparable melting temperature . The relationship at the atomic level results from the bond strength of the atoms in the crystal lattice. ${\ displaystyle R_ {e}}$ ${\ displaystyle R_ {m}}$

## Stresses and strains in statically (un) determinate systems

In statically determined systems, the mechanical stresses in the linear- elastic range result from the load (acting forces) and the geometry, while the elongations depend on the E-module of the materials. If the material deforms plastically , stresses are limited.

In cases of static uncertainty (e.g. continuous beams, obstructed thermal expansion, ship's hull in swell or in tidal range ), the forces and induced stresses are dependent on the rigidity of the static system. In such cases, components made from more flexible materials with a lower modulus of elasticity can reduce stresses. The components adapt more flexibly to the circumstances. Stiffer materials, on the other hand, resist elastic deformation to a greater extent, as a result of which greater stresses build up.

## E-modulus versus stiffness

The term stiffness in the sense of technical mechanics generally describes the resistance of bodies or assemblies to elastic deformation caused by mechanical forces or moments . Their value does not result solely from the elastic properties of the materials used, but is also determined by the respective body geometry or construction (e.g. machine rigidity). In the case of the tensile test , the tensile or elongation stiffness of the sample is the product of its (effective) modulus of elasticity and the smallest orthogonally loaded cross-sectional area : ${\ displaystyle E}$${\ displaystyle A}$

${\ displaystyle S _ {\ mathrm {t}} = E \ cdot A}$.

The physical unit corresponds to that of a force .

The term stiffness in the sense of a material property refers to the deformation behavior of the material in the elastic range. There is no geometry dependency, which is why only the elastic material parameters, e.g. B. E-modulus and shear modulus can be used for characterization.

## Hooke's law in scalar and general form

The relationship in scalar notation only applies to materials without transverse strain or to the uniaxial stress state (e.g. uniaxial tension ). In the multiaxial stress state, Hooke's law must be applied in its general form, depending on the degree of elastic anisotropy . For example, for the lateral deformation of thin isotropic plates ( plane stress state ) ${\ displaystyle \ sigma = E \ cdot \ varepsilon}$

${\ displaystyle \ left ({\ begin {array} {c} \ sigma _ {xx} \\\ sigma _ {yy} \\\ sigma _ {xy} \ end {array}} \ right) = {\ frac {E} {1- \ nu ^ {2}}} \ left ({\ begin {array} {ccc} 1 & \ nu & 0 \\\ nu & 1 & 0 \\ 0 & 0 & {\ frac {1- \ nu} {2} } \ end {array}} \ right) \ left ({\ begin {array} {c} \ varepsilon _ {xx} \\\ varepsilon _ {yy} \\ 2 \ varepsilon _ {xy} \ end {array} } \ right)}$,

where denotes the Poisson's number . The elongation in the direction of thickness results in ${\ displaystyle \ nu}$

${\ displaystyle \ varepsilon _ {zz} = - {\ frac {\ nu} {E}} (\ sigma _ {xx} + \ sigma _ {yy})}$.

## Component stiffening through biaxial stress states

In the transition from the uniaxial (uniaxial) to the biaxial (biaxial) stress state, two simple special cases can be distinguished for components and layers made of homogeneous isotropic material. Because of the influence on the transverse contraction for non- auxetic materials with a Poisson's ratio that is truly greater than zero, a higher modulus is always measured in the loading direction.

As a result of a prevented transverse contractionyy = 0) this results in

${\ displaystyle E_ {x} ^ {*} = {\ frac {E} {1- \ nu ^ {2}}}}$ .

If there is an additional load in the transverse or y direction at the level σ yy = σ xx , then the "biaxial modulus of elasticity" is

${\ displaystyle E_ {x} ^ {**} = {\ frac {E} {1- \ nu}}}$ .

The latter has z. B. Significance for the lateral stiffness of adhesive layers, for example in the case of differences in the thermal expansion behavior between layer and substrate. The former is used in thick-walled components or very wide beams . However, the two derived quantities are not material constants in the original sense.

## Conversion between the elastic constants

The module ... ... results from:
${\ displaystyle (K, \, E)}$ ${\ displaystyle (K, \, \ lambda)}$ ${\ displaystyle (K, \, G)}$ ${\ displaystyle (K, \, \ nu)}$ ${\ displaystyle (E, \, \ lambda)}$ ${\ displaystyle (E, \, G)}$ ${\ displaystyle (E, \, \ nu)}$ ${\ displaystyle (\ lambda, \, G)}$ ${\ displaystyle (\ lambda, \, \ nu)}$ ${\ displaystyle (G, \, \ nu)}$ ${\ displaystyle (G, \, M)}$
Compression module ${\ displaystyle K \,}$ ${\ displaystyle K}$ ${\ displaystyle K}$ ${\ displaystyle K}$ ${\ displaystyle K}$ ${\ displaystyle (E + 3 \ lambda) +}$${\ displaystyle {\ tfrac {\ sqrt {(E + 3 \ lambda) ^ {2} -4 \ lambda E}} {6}}}$ ${\ displaystyle {\ tfrac {EG} {3 (3G-E)}}}$ ${\ displaystyle {\ tfrac {E} {3 (1-2 \ nu)}}}$ ${\ displaystyle \ lambda +}$${\ displaystyle {\ tfrac {2G} {3}}}$ ${\ displaystyle {\ tfrac {\ lambda (1+ \ nu)} {3 \ nu}}}$ ${\ displaystyle {\ tfrac {2G (1+ \ nu)} {3 (1-2 \ nu)}}}$ ${\ displaystyle M-}$${\ displaystyle {\ tfrac {4G} {3}}}$
modulus of elasticity ${\ displaystyle E \,}$ ${\ displaystyle E}$ ${\ displaystyle {\ tfrac {9K (K- \ lambda)} {3K- \ lambda}}}$ ${\ displaystyle {\ tfrac {9KG} {3K + G}}}$ ${\ displaystyle 3K (1-2 \ nu) \,}$ ${\ displaystyle E}$ ${\ displaystyle E}$ ${\ displaystyle E}$ ${\ displaystyle {\ tfrac {G (3 \ lambda + 2G)} {\ lambda + G}}}$ ${\ displaystyle {\ tfrac {\ lambda (1+ \ nu) (1-2 \ nu)} {\ nu}}}$ ${\ displaystyle 2G (1+ \ nu) \,}$ ${\ displaystyle {\ tfrac {G (3M-4G)} {MG}}}$
1. Lamé constant ${\ displaystyle \ lambda \,}$ ${\ displaystyle {\ tfrac {3K (3K-E)} {9K-E}}}$ ${\ displaystyle \ lambda}$ ${\ displaystyle K-}$${\ displaystyle {\ tfrac {2G} {3}}}$ ${\ displaystyle {\ tfrac {3K \ nu} {1+ \ nu}}}$ ${\ displaystyle \ lambda}$ ${\ displaystyle {\ tfrac {G (E-2G)} {3G-E}}}$ ${\ displaystyle {\ tfrac {E \ nu} {(1+ \ nu) (1-2 \ nu)}}}$ ${\ displaystyle \ lambda}$ ${\ displaystyle \ lambda}$ ${\ displaystyle {\ tfrac {2G \ nu} {1-2 \ nu}}}$ ${\ displaystyle M-2G \,}$
Shear modulus or (2nd Lamé constant) ${\ displaystyle G}$${\ displaystyle \ mu}$
${\ displaystyle {\ tfrac {3KE} {9K-E}}}$ ${\ displaystyle {\ tfrac {3 (K- \ lambda)} {2}}}$ ${\ displaystyle G}$ ${\ displaystyle {\ tfrac {3K (1-2 \ nu)} {2 (1+ \ nu)}}}$ ${\ displaystyle (E-3 \ lambda) +}$${\ displaystyle {\ tfrac {\ sqrt {(E-3 \ lambda) ^ {2} +8 \ lambda E}} {4}}}$ ${\ displaystyle G}$ ${\ displaystyle {\ tfrac {E} {2 (1+ \ nu)}}}$ ${\ displaystyle G}$ ${\ displaystyle {\ tfrac {\ lambda (1-2 \ nu)} {2 \ nu}}}$ ${\ displaystyle G}$ ${\ displaystyle G}$
Poisson's number ${\ displaystyle \ nu \,}$ ${\ displaystyle {\ tfrac {3K-E} {6K}}}$ ${\ displaystyle {\ tfrac {\ lambda} {3K- \ lambda}}}$ ${\ displaystyle {\ tfrac {3K-2G} {2 (3K + G)}}}$ ${\ displaystyle \ nu}$ ${\ displaystyle - (E + \ lambda) +}$${\ displaystyle {\ tfrac {\ sqrt {(E + \ lambda) ^ {2} +8 \ lambda ^ {2}}} {4 \ lambda}}}$ ${\ displaystyle {\ tfrac {E} {2G}}}$${\ displaystyle -1}$ ${\ displaystyle \ nu}$ ${\ displaystyle {\ tfrac {\ lambda} {2 (\ lambda + G)}}}$ ${\ displaystyle \ nu}$ ${\ displaystyle \ nu}$ ${\ displaystyle {\ tfrac {M-2G} {2M-2G}}}$
Longitudinal module ${\ displaystyle M \,}$ ${\ displaystyle {\ tfrac {3K (3K + E)} {9K-E}}}$ ${\ displaystyle 3K-2 \ lambda \,}$ ${\ displaystyle K +}$${\ displaystyle {\ tfrac {4G} {3}}}$ ${\ displaystyle {\ tfrac {3K (1- \ nu)} {1+ \ nu}}}$ ${\ displaystyle {\ tfrac {E- \ lambda + {\ sqrt {E ^ {2} +9 \ lambda ^ {2} + 2E \ lambda}}} {2}}}$ ${\ displaystyle {\ tfrac {G (4G-E)} {3G-E}}}$ ${\ displaystyle {\ tfrac {E (1- \ nu)} {(1+ \ nu) (1-2 \ nu)}}}$ ${\ displaystyle \ lambda + 2G \,}$ ${\ displaystyle {\ tfrac {\ lambda (1- \ nu)} {\ nu}}}$ ${\ displaystyle {\ tfrac {2G (1- \ nu)} {1-2 \ nu}}}$ ${\ displaystyle M}$

Wiktionary: Modulus of elasticity  - explanations of meanings, word origins, synonyms, translations

## Individual evidence

1. Spring constant # calculation
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11. ↑ Shear modulus #Connection with other material constants
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