Young's modulus

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Young's modulus is the slope of the linear part of the stress-strain curve for a material under tension or compression. Stress strain ductile.svg
Young's modulus is the slope of the linear part of the stress–strain curve for a material under tension or compression.

Young's modulus (or Young modulus) is a mechanical property of solid materials that measures the tensile or compressive stiffness when the force is applied lengthwise. It is the modulus of elasticity for tension or axial compression. Young's modulus is defined as the ratio of the stress (force per unit area) applied to the object and the resulting axial strain (displacement or deformation) in the linear elastic region of the material.

Contents

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, , quantifies the relationship between tensile or compressive stress (force per unit area) and axial strain (proportional deformation) in the linear elastic region of a material: [2]

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:

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.

Material stiffness is a distinct property from the following:

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 , bulk modulus , and Poisson's ratio . 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:

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.

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 [5] 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 where the electron work function varies with the temperature as and is a calculable material property which is dependent on the crystal structure (for example, BCC, FCC). is the electron work function at T=0 and is constant throughout the change.

Calculation

Young's modulus is calculated by dividing the tensile stress, , by the engineering extensional strain, , in the elastic (initial, linear) portion of the physical stress–strain curve:

where

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.

where is the force exerted by the material when contracted or stretched by .

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

where it comes in saturation

and

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

Elastic potential energy

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

now by explicating the intensive variables:

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

or, in simple notation, for a linear elastic material: , since the strain is defined .

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:

Examples

Influences of selected glass component additions on Young's modulus of a specific base glass SpiderGraph YoungMod.gif
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
MaterialYoung's modulus (GPa)Megapound per square inch (M psi) [6] Ref.
Aluminium (13Al)689.86 [7] [8] [9] [10] [11] [12]
Amino-acid molecular crystals 21–443.05–6.38 [13]
Aramid (for example, Kevlar)70.5–112.410.2–16.3 [14]
Aromatic peptide-nanospheres230–27533.4–39.9 [15]
Aromatic peptide-nanotubes19–272.76–3.92 [16] [17]
Bacteriophage capsids 1–30.145–0.435 [18]
Beryllium (4Be)28741.6 [19]
Bone, human cortical142.03 [20]
Brass 10615.4 [21]
Bronze 11216.2 [22]
Carbon nitride (CN2)822119 [23]
Carbon-fiber-reinforced plastic (CFRP), 50/50 fibre/matrix, biaxial fabric30–504.35–7.25 [24]
Carbon-fiber-reinforced plastic (CFRP), 70/30 fibre/matrix, unidirectional, along fibre18126.3 [25]
Cobalt-chrome (CoCr)23033.4 [26]
Copper (Cu), annealed11016 [27]
Diamond (C), synthetic1050–1210152–175 [28]
Diatom frustules, largely silicic acid 0.35–2.770.051–0.058 [29]
Flax fiber588.41 [30]
Float glass 47.7–83.66.92–12.1 [31]
Glass-reinforced polyester (GRP)17.22.49 [32]
Gold 77.211.2 [33]
Graphene 1050152 [34]
Hemp fiber355.08 [35]
High-density polyethylene (HDPE)0.97–1.380.141–0.2 [36]
High-strength concrete 304.35 [37]
Lead (82Pb), chemical131.89 [12]
Low-density polyethylene (LDPE), molded0.2280.0331 [38]
Magnesium alloy 45.26.56 [39]
Medium-density fiberboard (MDF)40.58 [40]
Molybdenum (Mo), annealed33047.9 [41] [8] [9] [10] [11] [12]
Monel 18026.1 [12]
Mother-of-pearl (largely calcium carbonate)7010.2 [42]
Nickel (28Ni), commercial20029 [12]
Nylon 66 2.930.425 [43]
Osmium (76Os)525–56276.1–81.5 [44]
Osmium nitride (OsN2)194.99–396.4428.3–57.5 [45]
Polycarbonate (PC)2.20.319 [46]
Polyethylene terephthalate (PET), unreinforced3.140.455 [47]
Polypropylene (PP), molded1.680.244 [48]
Polystyrene, crystal2.5–3.50.363–0.508 [49]
Polystyrene, foam0.0025–0.0070.000363–0.00102 [50]
Polytetrafluoroethylene (PTFE), molded0.5640.0818 [51]
Rubber, small strain0.01–0.10.00145–0.0145 [13]
Silicon, single crystal, different directions130–18518.9–26.8 [52]
Silicon carbide (SiC)90–13713.1–19.9 [53]
Single-walled carbon nanotube 1000140 [54] [55]
Steel, A36 20029 [56]
Stinging nettle fiber8712.6 [30]
Titanium (22Ti)11616.8 [57] [58] [8] [10] [9] [12] [11]
Titanium alloy, Grade 511416.5 [59]
Tooth enamel, largely calcium phosphate 8312 [60]
Tungsten carbide (WC)600–68687–99.5 [61]
Wood, American beech 9.5–11.91.38–1.73 [62]
Wood, black cherry 9–10.31.31–1.49 [62]
Wood, red maple 9.6–11.31.39–1.64 [62]
Wrought iron 19328 [63]
Yttrium iron garnet (YIG), polycrystalline19328 [64]
Yttrium iron garnet (YIG), single-crystal20029 [65]
Zinc (30Zn)10815.7 [66]
Zirconium (40Zr), commercial9513.8 [12]

See also

Related Research Articles

<span class="mw-page-title-main">Hooke's law</span> Physical law: force needed to deform a spring scales linearly with distance

In physics, Hooke's law is an empirical law which states that the force needed to extend or compress a spring by some distance scales linearly with respect to that distance—that is, Fs = kx, where k is a constant factor characteristic of the spring, and x is small compared to the total possible deformation of the spring. The law is named after 17th-century British physicist Robert Hooke. He first stated the law in 1676 as a Latin anagram. He published the solution of his anagram in 1678 as: ut tensio, sic vis. Hooke states in the 1678 work that he was aware of the law since 1660.

In engineering, deformation may be elastic or plastic. If the deformation is negligible, the object is said to be rigid.

<span class="mw-page-title-main">Stress–strain curve</span> Curve representing a materials response to applied forces

In engineering and materials science, a stress–strain curve for a material gives the relationship between stress and strain. It is obtained by gradually applying load to a test coupon and measuring the deformation, from which the stress and strain can be determined. These curves reveal many of the properties of a material, such as the Young's modulus, the yield strength and the ultimate tensile strength.

<span class="mw-page-title-main">Poisson's ratio</span> Measure of material deformation perpendicular to loading

In materials science and solid mechanics, Poisson's ratioν (nu) is a measure of the Poisson effect, the deformation of a material in directions perpendicular to the specific direction of loading. The value of Poisson's ratio is the negative of the ratio of transverse strain to axial strain. For small values of these changes, ν is the amount of transversal elongation divided by the amount of axial compression. Most materials have Poisson's ratio values ranging between 0.0 and 0.5. For soft materials, such as rubber, where the bulk modulus is much higher than the shear modulus, Poisson's ratio is near 0.5. For open-cell polymer foams, Poisson's ratio is near zero, since the cells tend to collapse in compression. Many typical solids have Poisson's ratios in the range of 0.2 to 0.3. The ratio is named after the French mathematician and physicist Siméon Poisson.

Dynamic mechanical analysis is a technique used to study and characterize materials. It is most useful for studying the viscoelastic behavior of polymers. A sinusoidal stress is applied and the strain in the material is measured, allowing one to determine the complex modulus. The temperature of the sample or the frequency of the stress are often varied, leading to variations in the complex modulus; this approach can be used to locate the glass transition temperature of the material, as well as to identify transitions corresponding to other molecular motions.

In physics and materials science, elasticity is the ability of a body to resist a distorting influence and to return to its original size and shape when that influence or force is removed. Solid objects will deform when adequate loads are applied to them; if the material is elastic, the object will return to its initial shape and size after removal. This is in contrast to plasticity, in which the object fails to do so and instead remains in its deformed state.

Linear elasticity is a mathematical model of how solid objects deform and become internally stressed by prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mechanics.

An elastic modulus is the unit of measurement of an object's or substance's resistance to being deformed elastically when a stress is applied to it.

<span class="mw-page-title-main">Stiffness</span> Resistance to deformation in response to force

Stiffness is the extent to which an object resists deformation in response to an applied force.

A Maxwell material is the most simple model viscoelastic material showing properties of a typical liquid. It shows viscous flow on the long timescale, but additional elastic resistance to fast deformations. It is named for James Clerk Maxwell who proposed the model in 1867. It is also known as a Maxwell fluid. A generalization of the scalar relation to a tensor equation lacks motivation from more microscopic models and does not comply with the concept of material objectivity. However, these criteria are fulfilled by the Upper-convected Maxwell model.

In materials science and continuum mechanics, viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials, like water, resist both shear flow and strain linearly with time when a stress is applied. Elastic materials strain when stretched and immediately return to their original state once the stress is removed.

In physics and engineering, a constitutive equation or constitutive relation is a relation between two or more physical quantities that is specific to a material or substance or field, and approximates its response to external stimuli, usually as applied fields or forces. They are combined with other equations governing physical laws to solve physical problems; for example in fluid mechanics the flow of a fluid in a pipe, in solid state physics the response of a crystal to an electric field, or in structural analysis, the connection between applied stresses or loads to strains or deformations.

A Kelvin–Voigt material, also called a Voigt material, is the most simple model viscoelastic material showing typical rubbery properties. It is purely elastic on long timescales, but shows additional resistance to fast deformation. The model was developed independently by the British physicist Lord Kelvin in 1865 and by the German physicist Woldemar Voigt in 1890.

Dynamic modulus is the ratio of stress to strain under vibratory conditions. It is a property of viscoelastic materials.

Elastic energy is the mechanical potential energy stored in the configuration of a material or physical system as it is subjected to elastic deformation by work performed upon it. Elastic energy occurs when objects are impermanently compressed, stretched or generally deformed in any manner. Elasticity theory primarily develops formalisms for the mechanics of solid bodies and materials. The elastic potential energy equation is used in calculations of positions of mechanical equilibrium. The energy is potential as it will be converted into other forms of energy, such as kinetic energy and sound energy, when the object is allowed to return to its original shape (reformation) by its elasticity.

In continuum mechanics, Lamé parameters are two material-dependent quantities denoted by λ and μ that arise in strain-stress relationships. In general, λ and μ are individually referred to as Lamé's first parameter and Lamé's second parameter, respectively. Other names are sometimes employed for one or both parameters, depending on context. For example, the parameter μ is referred to in fluid dynamics as the dynamic viscosity of a fluid ; whereas in the context of elasticity, μ is called the shear modulus, and is sometimes denoted by G instead of μ. Typically the notation G is seen paired with the use of Young's modulus E, and the notation μ is paired with the use of λ.

<span class="mw-page-title-main">Flexural modulus</span> Intensive property in mechanics

In mechanics, the flexural modulus or bending modulus is an intensive property that is computed as the ratio of stress to strain in flexural deformation, or the tendency for a material to resist bending. It is determined from the slope of a stress-strain curve produced by a flexural test, and uses units of force per area. The flexural modulus defined using the 2-point (cantilever) and 3-point bend tests assumes a linear stress strain response.

<span class="mw-page-title-main">Rock mass plasticity</span>

Plasticity theory for rocks is concerned with the response of rocks to loads beyond the elastic limit. Historically, conventional wisdom has it that rock is brittle and fails by fracture while plasticity is identified with ductile materials. In field scale rock masses, structural discontinuities exist in the rock indicating that failure has taken place. Since the rock has not fallen apart, contrary to expectation of brittle behavior, clearly elasticity theory is not the last word.

In continuum mechanics, a hypoelastic material is an elastic material that has a constitutive model independent of finite strain measures except in the linearized case. Hypoelastic material models are distinct from hyperelastic material models in that, except under special circumstances, they cannot be derived from a strain energy density function.

<span class="mw-page-title-main">Resilience (materials science)</span> Material ability to absorb energy when deformed elastically

In material science, resilience is the ability of a material to absorb energy when it is deformed elastically, and release that energy upon unloading. Proof resilience is defined as the maximum energy that can be absorbed up to the elastic limit, without creating a permanent distortion. The modulus of resilience is defined as the maximum energy that can be absorbed per unit volume without creating a permanent distortion. It can be calculated by integrating the stress–strain curve from zero to the elastic limit. In uniaxial tension, under the assumptions of linear elasticity,

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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 formulaeNotes

There are two valid solutions.
The plus sign leads to .

The minus sign leads to .
Cannot be used when
2D formulaeNotes
Cannot be used when