YOUNG’S MODULUS

Young's modulus, also known as the tensile modulus, is a measure of the stiffness of an elastic material and is a quantity used to characterize materials. It is defined as the ratio of the uniaxial stress over the uniaxial strain in the range of stress in which Hooke's Law holds. In solid mechanics, the slope of the stress-strain curve at any point is called the tangent modulus. The tangent modulus of the initial, linear portion of a stress-strain curve is called Young's 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. In anisotropic materials, Young's modulus may have different values depending on the direction of the applied force with respect to the material's structure.

It is also commonly called the elastic modulus or modulus of elasticity, because Young's modulus is the most common elastic modulus used, but there are other elastic module measured, too, such as the bulk modulus and the shear modulus.

Young's modulus is named after Thomas Young, the 19th century British scientist. 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, predating Young's work by 25 years

Units

Young's modulus is the ratio of stress, which has units of pressure, to strain, which is dimensionless; therefore, Young's modulus has units of pressure.

The SI unit of modulus of elasticity (E, or less commonly Y) is the pascal (Pa or N/m² or m−1·kg·s−2). The practical units used are megapascals (MPa or N/mm²) or gigapascals (GPa or kN/mm²). In United States customary units, it is expressed as pounds (force) per square inch (psi). The abbreviation ksi refers to thousands of psi.

Usage

The Young's modulus calculates 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. Young's modulus is used in order to predict the deflection that will occur in a statically determinatebeam when a load is applied at a point in between the beam's supports. Some calculations also require the use of other material properties, such as the shear modulus, density, or Poisson's ratio.

Linear versus non-linear

For many materials, Young's modulus is essentially constant over a range of strains. Such materials are called linear, and are said to obey Hooke's law. Examples of linear materials are steel, carbon fiber and glass. Non-linear materials include rubber and soils, except under very small strains.

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 by the tensile strain in the elastic (initial, linear) portion of the stress-strain curve:

where

E is the Young's modulus (modulus of elasticity)F is the force exerted on an object under tension;A0 is the original 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 compressed material

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

where F is the force exerted by the material when compressed or stretched by ΔL.

Hooke's law can be derived from this formula, which describes the stiffness of an ideal spring:

where

Elastic potential energy

The elastic potential energy stored is given by the integral of this expression with respect to L:

where Ue is the elastic potential energy.

The elastic potential energy per unit volume is given by:

, where is the strain in the material.

This formula can also be expressed as the integral of Hooke's law:

Relation among elastic constants

For homogeneous isotropic materials simple relations exist between elastic constants (Young's modulus E, shear modulusG, bulk modulusK, and Poisson's ratioν) that allow calculating them all as long as two are known:

Approximate values

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 lbf/in² (psi)Rubber (small strain) 0.01–0.1

PTFE (Teflon) 0.5 75,000Low density polyethylene 0.238 34,000HDPE 0.8Polypropylene 1.5-2Bacteriophage capsids 1–3 150,000–435,000Polyethylene terephthalate (PET) 2-2.7Polystyrene 3-3.5Nylon 2–4 290,000–580,000Diatomfrustules (largely silicic acid) 0.35–

2.7750,000–400,000

Medium-density fiberboard 4 580,000Pine wood (along grain) 9 1,300,000Oak wood (along grain) 11

Human Cortical Bone 14 2,030,000High-strength concrete 30Hemp fiber [9] 35Magnesiummetal (Mg) 45Flax fiber 58Aluminium 69 10,000,000Stinging nettle fiber 87Glass (see chart) 50–90Aramid 70.5–

112.4Mother-of-pearl (nacre, largely calcium carbonate) 70 10,000,000Tooth enamel (largely calcium phosphate) 83 12,000,000Brass 100–125Bronze 96-120Titanium (Ti) 16,000,000[3]

Titanium alloys 105–120 15,000,000–17,500,000

Copper (Cu) 117 17,000,000Glass-reinforced plastic (70/30 by weight fibre/matrix, unidirectional, along grain)

40–45 5,800,000–6,500,000

Glass-reinforced polyester matrix 17.2 2,500,000Carbon fiber reinforced plastic 150[

Carbon fiber reinforced plastic (70/30 fibre/matrix, unidirectional, along grain)

181 26,300,000

Silicon single crystal, different directions 130-185Wrought iron 190–210Steel (ASTM-A36) 200 29,000,000polycrystalline Yttrium iron garnet (YIG) 193 28,000,000

single-crystal Yttrium iron garnet (YIG) 200 30,000,000Beryllium (Be) 287 42,000,000Molybdenum (Mo) 329Tungsten (W) 400–410Sapphire (Al2O3) along C-axis 435 63,000,000Silicon carbide (SiC) 450Osmium (Os) 550 79,800,000Tungsten carbide (WC) 450–650Single-walled carbon nanotube 1,000+ 145,000,000+Graphene 1000Diamond (C) 1220 150,000,000–

175,000,000

SHEAR MODULUS

In materials science, shear modulus or modulus of rigidity, denoted by G, or sometimes S or μ, is defined as the ratio of shear stress to the shear strain:

where

= shear stress;F is the force which actsA is the area on which the force acts

in engineering, = shear strain. Elsewhere, γxy = θΔx is the transverse displacementl is the initial length

Shear modulus is usually expressed in gigapascals (GPa) or in thousands of pounds per square inch (kpsi).

The shear modulus is always positive

Explanation

Shear modulusSI symbol: GSI unit: gigapascalDerivations from other quantities: G = τ / γ

The shear modulus is one of several quantities for measuring the stiffness of materials. All of them arise in the generalized Hooke's law:

Young's modulus describes the material's response to linear strain (like pulling on the ends of a wire),

the bulk modulus describes the material's response to uniform pressure, and the shear modulus describes the material's response to shearing strains.

The shear modulus is concerned with the deformation of a solid when it experiences a force parallel to one of its surfaces while its opposite face experiences an opposing force (such as friction). In the case of an object that's shaped like a rectangular prism, it will deform into a parallelepiped. Anisotropic materials such as wood, paper and also essentially all single crystals exhibit differing material response to stress or strain when tested in different directions. In this case one may need to use the full tensor-expression of the elastic constants, rather than a single scalar value.

The shear modulus is one of several quantities for measuring the stiffness of materials. All of them arise in the generalized Hooke's law:

Waves

In homogeneous and isotropic solids, there are two kinds of waves, pressure waves and shear waves. The velocity of a shear wave, (vs) is controlled by the shear modulus,

where

G is the shear modulusρ is the solid's density.

Shear modulus of metals

Material Typical values forshear modulus (GPa)

(at room temperature)

Diamond 478.Steel 79.3Copper 44.7Titanium 41.4Glass 26.2Aluminium 25.5Polyethylene 0.117Rubber 0.0006

shear modulus of metals is usually observed to decrease with increasing temperature. At high pressures, the shear modulus also appears to increase with the applied pressure. Correlations between the melting temperature, vacancy formation energy, and the shear modulus have been observed in many metals.

Several models exist that attempt to predict the shear modulus of metals (and possibly that of alloys). Shear modulus models that have been used in plastic flow computations include:

1. The MTS shear modulus model developed by and used in conjunction with the Mechanical Threshold Stress (MTS) plastic flow stress modelthe Steinberg-Cochran-Guinan (SCG) shear modulus model developed by and used in conjunction with the Steinberg-Cochran-Guinan-Lund (SCGL) flow stress model.

2. The Nadal and LePoac (NP) shear modulus model that uses Lindemann theory to determine the temperature dependence and the SCG model for pressure dependence of the shear modulus.

MTS shear modulus model

The MTS shear modulus model has the form:

where µ0 is the shear modulus at 0 K, and D and T0 are material constants.

SCG shear modulus model

The Steinberg-Cochran-Guinan (SCG) shear modulus model is pressure dependent and has the form

where, µ0 is the shear modulus at the reference state (T = 300 K, p = 0, η = 1), p is the pressure, and T is the temperature.

NP shear modulus model

The Nadal-Le Poac (NP) shear modulus model is a modified version of the SCG model. The empirical temperature dependence of the shear modulus in the SCG model is replaced with an equation based on Lindemann melting theory. The NP shear modulus model has the form:

where

and µ0 is the shear modulus at 0 K and ambient pressure, ζ is a material parameter, kb is the Boltzmann constant, m is the atomic mass, and f is the Lindemann constant.

Bulk modulusThe bulk modulus (K) of a substance measures the substance's resistance to uniform compression. It is defined as the ratio of the infinitesimalpressure increase to the resulting relative decrease of the volume. Its base unit is the pascal.

Definition

The bulk modulus K>0 can be formally defined by the equation:

where P is pressure, V is volume, and ∂P/∂V denotes the partial derivative of pressure with respect to volume. The inverse of the bulk modulus gives a substance's compressibility.

Other moduli describe the material's response (strain) to other kinds of stress: the shear modulus describes the response to shear, and Young's modulus describes the response to linear stress. For a fluid, only the bulk modulus is meaningful. For an anisotropic solid such as wood or paper, these three moduli do not contain enough information to describe its behaviour, and one must use the full generalized Hooke's law.

Thermodynamic relation

Strictly speaking, the bulk modulus is a thermodynamic quantity, and it is necessary to specify how the temperature varies in order to specify a bulk modulus: constant-temperature (isothermal

KT), constant-entropy (adiabaticKS), and other variations are possible. In practice, such distinctions are usually only relevant for gases.

For a gas, the adiabatic bulk modulus KS is approximately given by

and the isothermal bulk modulus KT is approximately given by

where

γ is the adiabatic index, sometimes called κ.

P is the pressure.

In a fluid, the bulk modulus K and the densityρ determine the speed of soundc (pressure waves), according to the Newton-Laplace formula

Solids can also sustain transverse waves: for these materials one additional elastic modulus, for example the shear modulus, is needed to determine wave speeds.

Measurement

It is possible to measure the bulk modulus using powder diffraction under applied pressure.

Approximate bulk modulus (K) for common materialsMaterial Bulk modulus in

PaBulk modulus in ksi

Glass (see also diagram below table) 3.5×1010 to 5.5×1010 5.8×103

Steel 1.6×1011 23×103

Diamond 4.42×1011 64×103

Material with bulk modulus value of 35GPa needs external pressure of 0.35 GPa (~3500Bar) to reduce the volume by one percent.

Approximate bulk modulus (K) for other substancesWater 2.2×109 Pa (value increases at higher pressures)

Air 1.42×105 Pa (adiabatic bulk modulus)

Air 1.01×105 Pa (constant temperature bulk modulus)

Solid helium

5×107 Pa (approximate)

TABLE OF

YOUNG’S MODULUS

SHEAR MODULUS

BULK MODULUS

YOUNG’S MODULUS

Aluminum 70.GPa Holmium 65.GPa Roentgenium N/A

Americium N/A Hydrogen N/A Rubidium 2.4GPaAntimony 55.GPa Indium 11.GPa Ruthenium 447.GPaArgon N/A Iodine N/A Rutherfordium N/AArsenic 8.GPa Iridium 528.GPa Samarium 50.GPaAstatine N/A Iron 211.GPa Scandium 74.GPaBarium 13.GPa Krypton N/A Seaborgium N/ABerkelium N/A Lanthanum 37.GPa Selenium 10.GPa

Material Modulus of Elasticity, E(lb/in2 x 106)

Shear Modulusof Elasticity, G 

(lb/in2 x 106)

Poisson's Ratiou

Weight Density(lb/in3)

Aluminum Alloys 10.2 3.9 0.33 0.098Beryllium

Copper18.0 7.0 0.29 0.30

Carbon Steel 29.0 11.5 0.29 0.28Cast Iron 14.5 6.0 0.21 0.26Inconel 31.0 11.5 0.29 0.31

Magnesium 6.5 2.4 0.35 0.07Molybdenum 48.0 17.1 0.31 0.37Monel Metal 26.0 9.5 0.32 0.32Nickel Silver 18.5 7.0 0.32 0.32Nickel Steel 29.0 11.0 0.29 0.28

Nylon 1.5 0.6 - 0.04Phosphor Bronze 16.1 6.0 0.35 0.30

Stainless Steel 27.6 10.6 0.31 0.28Titanium 16.5 6.5 - 0.16

Beryllium 287.GPa Lawrencium N/A Silicon 47.GPaBismuth 32.GPa Lead 16.GPa Silver 83.GPaBohrium N/A Lithium 4.9GPa Sodium 10.GPaBoron N/A Lutetium 69.GPa Strontium N/ABromine N/A Magnesium 45.GPa Sulfur N/ACadmium 50.GPa Manganese 198.GPa Tantalum 186.GPaCalcium 20.GPa Meitnerium N/A Technetium N/ACalifornium N/A Mendelevium N/A Tellurium 43.GPaCarbon N/A Mercury N/A Terbium 56.GPaCerium 34.GPa Molybdenum 329.GPa Thallium 8.GPaCesium 1.7GPa Neodymium 41.GPa Thorium 79.GPaChlorine N/A Neon N/A Thulium 74.GPaChromium 279.GPa Neptunium N/A Tin 50.GPaCobalt 209.GPa Nickel 200.GPa Titanium 116.GPaCopper 130.GPa Niobium 105.GPa Tungsten 411.GPaCurium N/A Nitrogen N/A Ununbium N/ADarmstadtium N/A Nobelium N/A Ununhexium N/ADubnium N/A Osmium N/A Ununoctium N/ADysprosium 61.GPa Oxygen N/A Ununpentium N/AEinsteinium N/A Palladium 121.GPa Ununquadium N/AErbium 70.GPa Phosphorus N/A Ununseptium N/AEuropium 18.GPa Platinum 168.GPa Ununtrium N/AFermium N/A Plutonium 96.GPa Uranium 208.GPaFluorine N/A Polonium N/A Vanadium 128.GPaFrancium N/A Potassium N/A Xenon N/AGadolinium 55.GPa Praseodymium 37.GPa Ytterbium 24.GPaGallium N/A Promethium 46.GPa Yttrium 64.GPaGermanium N/A Protactinium N/A Zinc 108.GPaGold 78.GPa Radium N/A Zirconium 68.GPaHafnium 78.GPa Radon N/AHassium N/A Rhenium 463.GPa

SHEAR MODULUS

Actinium N/A Helium N/A Rhodium 150.GPaAluminum 26.GPa Holmium 26.GPa Roentgenium N/AAmericium N/A Hydrogen N/A Rubidium N/A

Antimony 20.GPa Indium N/A Ruthenium 173.GPaArgon N/A Iodine N/A Rutherfordium N/A

Arsenic N/A Iridium 210.GPa Samarium 20.GPaAstatine N/A Iron 82.GPa Scandium 29.GPaBarium 4.9GPa Krypton N/A Seaborgium N/A

Berkelium N/A Lanthanum 14.GPa Selenium 3.7GPaBeryllium 132.GPa Lawrencium N/A Silicon N/A

Bismuth 12.GPa Lead 5.6GPa Silver 30.GPaBohrium N/A Lithium 4.2GPa Sodium 3.3GPa

Boron N/A Lutetium 27.GPa Strontium 6.1GPaBromine N/A Magnesium 17.GPa Sulfur N/A

Cadmium 19.GPa Manganese N/A Tantalum 69.GPaCalcium 7.4GPa Meitnerium N/A Technetium N/A

Californium N/A Mendelevium N/A Tellurium 16.GPaCarbon N/A Mercury N/A Terbium 22.GPaCerium 14.GPa Molybdenum 20.GPa Thallium 2.8GPaCesium N/A Neodymium 16.GPa Thorium 31.GPa

Chlorine N/A Neon N/A Thulium 31.GPaChromium 115.GPa Neptunium N/A Tin 18.GPa

Cobalt 75.GPa Nickel 76.GPa Titanium 44.GPaCopper 48.GPa Niobium 38.GPa Tungsten 161.GPaCurium N/A Nitrogen N/A Ununbium N/A

Darmstadtium N/A Nobelium N/A Ununhexium N/ADubnium N/A Osmium 222.GPa Ununoctium N/A

Dysprosium 25.GPa Oxygen N/A Ununpentium N/AEinsteinium N/A Palladium 44.GPa Ununquadium N/A

Erbium 28.GPa Phosphorus N/A Ununseptium N/AEuropium 7.9GPa Platinum 61.GPa Ununtrium N/AFermium N/A Plutonium 43.GPa Uranium 111.GPaFluorine N/A Polonium N/A Vanadium 47.GPa

Francium N/A Potassium 1.3GPa Xenon N/AGadolinium 22.GPa Praseodymium 15.GPa Ytterbium 9.9GPa

Gallium N/A Promethium 18.GPa Yttrium 26.GPaGermanium N/A Protactinium N/A Zinc 43.GPa

Gold 27.GPa Radium N/A Zirconium 33.GPaHafnium 30.GPa Radon N/AHassium N/A Rhenium 178.GPa

Actinium N/A Helium N/A Rhodium 380 GPaAluminum 76 GPa Holmium 40 GPa Roentgenium N/AAmericium N/A Hydrogen N/A Rubidium 2.5 GPa

Antimony 42 GPa Indium N/A Ruthenium 220 GPaArgon N/A Iodine 7.7 GPa Rutherfordium N/A

Arsenic 22 GPa Iridium 320 GPa Samarium 38 GPaAstatine N/A Iron 170 GPa Scandium 57 GPaBarium 9.6 GPa Krypton N/A Seaborgium N/A

Berkelium N/A Lanthanum 28 GPa Selenium 8.3 GPa

Beryllium 130 GPa Lawrencium N/A Silicon 100 GPaBismuth 31 GPa Lead 46 GPa Silver 100 GPaBohrium N/A Lithium 11 GPa Sodium 6.3 GPa

Boron 320 GPa Lutetium 48 GPa Strontium N/ABromine 1.9 GPa Magnesium 45 GPa Sulfur 7.7 GPa

Cadmium 42 GPa Manganese 120 GPa Tantalum 200 GPaCalcium 17 GPa Meitnerium N/A Technetium N/A

Californium N/A Mendelevium N/A Tellurium 65 GPaCarbon 33 GPa Mercury 25 GPa Terbium 38.7 GPaCerium 22 GPa Molybdenum 230 GPa Thallium 43 GPaCesium 1.6 GPa Neodymium 32 GPa Thorium 54 GPa

Chlorine 1.1 GPa Neon N/A Thulium 45 GPaChromium 160 GPa Neptunium N/A Tin 58 GPa

Cobalt 180 GPa Nickel 180 GPa Titanium 110 GPaCopper 140 GPa Niobium 170 GPa Tungsten 310 GPaCurium N/A Nitrogen N/A Ununbium N/A

Darmstadtium N/A Nobelium N/A Ununhexium N/ADubnium N/A Osmium N/A Ununoctium N/A

Dysprosium 41 GPa Oxygen N/A Ununpentium N/AEinsteinium N/A Palladium 180 GPa Ununquadium N/A

Erbium 44 GPa Phosphorus 11 GPa Ununseptium N/AEuropium 8.3 GPa Platinum 230 GPa Ununtrium N/AFermium N/A Plutonium N/A Uranium 100 GPaFluorine N/A Polonium N/A Vanadium 160 GPa

Francium N/A Potassium 3.1 GPa Xenon N/AGadolinium 38 GPa Praseodymium 29 GPa Ytterbium 31 GPa

Gallium N/A Promethium 33 GPa Yttrium 41 GPaGermanium N/A Protactinium N/A Zinc 70 GPa

Gold 220 GPa Radium N/A Zirconium N/AHafnium 110 GPa Radon N/AHassium N/A Rhenium 370 GPa

BULK MODULUS

ASSIGNMENT

IN

PHYSICS

SUB BY: ALI ISMAEILI

SECTION I-A

SUB TO: Engr. ARNI BULSECO

INSTRUCTOR