Elasticities and Strengths

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You are here: Chapter:2 General physics

Section: 2.2 Mechanical properties of materials

SubSection: 2.2.2 Elasticities and strengths

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2.2.2 Elasticities and strengths

Elastic properties isotropic materials

Listed below are the elastic constants in common use, any two of which are sufficient to define the elastic properties of a

homogeneous isotropic solid. The two fundamental constants are those which relate change of volume and change of shape to

applied stress. They are respectively, the bulk modulus K(as inp= K. V/V) and the shear modulus G.

For many practical purposes, the following constants are commonly used:

Youngs Modulus, or longitudinal elasticity, E.

Poissons ratio, = lateral contraction per unit breadth divided by the longitudinal extension per unit length under an applied

longitudinal stress.

Compressibility, = 1/K.

Longitudinal modulus, M, which is the longitudinal modulus for zero lateral strain and determines the velocity of ultrasonic stress

pulses in solids.

For a homogeneous isotropic solid, the following relations exist between the constants.

(a) G=E

2(1 + )

(b) K=E

3 (1 2)

(c) K=1 EG

3 3(3GE)

(d) M= K+4

G3

The value of Poissons ratio is usually positive and lies between 0 and , but in some cases it may be negative.

Elasticities of metals and alloys

Material

20 C

E

GPa

G

GPa

K

GPa

Aluminium . . . . . 70.3 26.1 0.345 75.5

Bismuth . . . . . . 31.9 12.0 0.330 31.3

Cadmium . . . . . 49.9 19.2 0.300 41.6

Chromium . . . . . 279.1 115.4 0.210 160.1

Copper . . . . . . 129.8 48.3 0.343 137.8

Gold . . . . . . . 78.0 27.0 0.44 217.0

Iron (soft) . . . . . 211.4 81.6 0.293 169.8

Iron (cast) . . . . . 152.3 60.0 0.27 109.5

Lead . . . . . . 16.1 5.59 0.44 45.8

Magnesium . . . . . 44.7 17.3 0.291 35.6

Nickel (unmag., soft) . . 199.5 76.0 0.312 177.3

,, ,, hard) . . 219.2 83.9 0.306 187.6

Niobium . . . . . 104.9 37.5 0.397 170.3

Platinum . . . . . 168.0 61.0 0.377 228.0

Silver . . . . . . 82.7 30.3 0.367 103.6

Tantalum . . . . . 185.7 69.2 0.342 196.3

Tin . . . . . . . 49.9 18.4 0.357 58.2

Titanium . . . . . 115.7 43.8 0.321 107.7

Tungsten . . . . . 411.0 160.6 0.280 311.0

Vanadium . . . . . 127.6 46.7 0.365 158.0

Zinc . . . . . . . 108.4 43.4 0.249 72.0

Brass (70 Zn, 30 Cu) . 100.6 37.3 0.350 111.8

Constantan . . . . . 162.4 61.2 0.327 156.4

ticities and strengths 2.2.2 http://www.kayelaby.npl.co.uk/general_physics/2_2/2_2_2.html

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Hidurax Special . . . 144.5 54.4 0.333 144.1

Invar (36 Ni, 63.8 Fe, 0.2 C) 144.0 57.2 0.259 99.4

Nickel Silver . . . . 132.5 49.7 0.333 132.0

Steel (Mild) . . . . 211.9 82.2 0.291 169.2

,, ( C) . . . . 210.0 81.1 0.293 168.7

,, ( C hardened) . 201.4 77.8 0.296 165.0

,, Tool||. . . . . 211.6 82.2 0.287 165.3

,, Tool (hardened)|| . 203.2 78.5 0.295 165.2

,, Stainless . . . 215.3 83.9 0.293 166.0

Tungsten Carbide . . 534.4 219.0 0.22 319.0

Approx. value or values for materials of variable composition.

Cu-Ni alloy with Al, Fe and Mn additions.

Approx. %composition: Cu 55, Ni 8, Zn 27.

||Oil hardening non-deforming tool steel of approx. %composition: C 0.98, Mn 1.03, Cr 0.65, W 1.01, V 0.1, remainder Fe.

Approx. %composition: C 0.02, Si 0.5, Mn 0.7, Ni 2, Cr 18, remainder Fe.

Elasticities of glasses

Material

20 C

E

GPa

G

GPa

K

GPa

Glass (Heavy Flint) . . . 80.1 31.5 0.27 57.6

Glass (Crown) . . . . 71.3 29.2 0.22 41.2Quartz (fused) . . . . 73.1 31.2 0.17 36.9

Several values in these tables are taken from Bradfield (1964).

Bulk moduli of elements

Element K

GPa

Element K

GPa

Element K

GPa

Element K

GPa

Aluminium . . 75.5 Chlorine Molybdenum 231.0 Selenium . . 8.3

Antimony . . 42.0 (liq) . . . 1.1 Nickel Silicon . . . 100.0

Arsenic . . 22.0 Chromium . 160.1 (soft) . . 177.3 Silver . . . 103.6

Bismuth . . 31.3 Copper . . 137.8 (hard) . . 187.6 Sodium . . 6.3

Bromine . . 1.9 Gold . . . 217.0 Palladium . 182.0 Sulphur . . 7.7

Cadmium . . 41.6 Iodine . . . 7.7 Phosphorus . Thallium . . 43.0

Caesium . . 1.6 Iron . . . 169.8 (red) . 10.9 Tin . . . . 58.2

Calcium . . 17.2 Lead . . . 45.8 Phosphorus Zinc . . . 72.0

Carbon . . Lithium . . 11.1 (white) . 4.9

(diamond) . 542.0 Magnesium . 44.7 Platinum . 228.0

Carbon Manganese . 118.0 Potassium . 3.1

(graphite) . 33.0 Mercury . . 25.0 Rubidium . 2.5

Bradfield (1964).

Markham (1968).

Bulk moduli of liquids

As the pressure increases, Kincreases. In general a rise in temperature decreases the bulk modulus of a liquid; water, however,

shows a maximum value of Kat about 50 C (see J. H. Poynting and J. J. Thomson (1920) Properties of Matter, London, Charles

Griffin; Bridgman (1949)).

Liquid Temp.

C

K

GPa

Liquid Temp.

C

K

GPa

Acetic acid, 116 atm 20 1.45 Mercury:

Amyl alcohol, 8 atm . 17.7 1.12 837 atm . . . 20 26.2

Benzene, 8 atm . . 17.9 1.10 100200 atm . . 15 30.0

Butyl alcohol, 8 atm . 17.4 1.13 Methyl acetate, 837 atm 14.3 1.04

Butyl alcohol, iso-, 8 atm 17.9 1.03 Methyl alcohol, 37 atm . 14.7 0.97

Carbon bisulphide, 837 atm 15.6 1.16 Olive oil . . . . 20.5 1.60

Carbon tetrachloride . . 20 1.12 Paraffin oil . . . 14.8 1.62

Chloroform, 100-200 atm . 20 1.1 Pentane . . . . 20 0.318

Ether: Petroleum . . . . 16.5 1.46

150 atm . . . 0 0.689 Propyl alcohol, 8 atm . 17.7 1.04

9001000 atm . . 0 1.56 Propyl alcohol, iso-, 8 atm 17.8 0.983

9001000 atm . . 198 0.703 Turpentine . . . 19.7 1.280


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Ethyl acetate, 837 atm . 13.3 0.974 Water:

Ethyl alcohol: 125 atm . . . 15 2.05

1500 atm . . 0 1.32 9001000 atm . . 15 2.75

150200 atm . . 310 0.024 9001000 atm . . 198 1.81

Ethyl bromide, 837 atm . 99.3 0.343 25003000 atm . . 14.2 3.88

Ethyl chloride, 837 atm . 15.2 0.662 Water (sea) . . . 2.32

Glycerine . . . . 20.5 4.03

Elasticities of plastics

All plastics are visco-elastic and consequently the elasticity varies considerably with temperature and strain rate. The table below

gives approximate values at 20 C for slow rates of strain.

Material E

GPa

Material E

GPa

ABS . . . . . . 1.43.1 Polyethylene (high density) . . 0.41.3

Epoxy . . . . . ~3.2 Polyimide . . . . . . . ~3.1

Nylon 6 (cast) . . . 2.43.1 Polymethylmethacrylate (PMMA) 2.43.4

Nylon 6 (moulded) . . 0.83.1 Polypropylene . . . . . . 1.11.6

Nylon 66 . . . . 1.22.9 Polystyrene . . . . . . 2.74.2

Polybenzoxazole . . . ~3.5 Polytetrafluoroethylene (PTFE) 0.4

Polycarbonate . . . 2.4 Polyvinylchloride (PVC) . .

(unplasticised)

2.44.1

Temperature coefficient of elastic constants for a range of materials

Temperature coefficient in

Et= E15{1 - (t15)}

Gt= G15{1 - '(t15)}

At 15C

104for E

'104

for G

Aluminium . . . . 4.8 5.2

Brass . . . . . 3.7 4.6

Copper . . . . . 3.0 3.1

German silver . . . 6.5

Gold . . . . . 4.8 3.3

Iron . . . . . . 2.3 2.8

Phosphor-bronze . . 3.0

Platinum . . . . . 0.98 1.0

Quartz fibre . . . . 1.5 1.1

Silver . . . . . . 7.5 4.5

Steel . . . . . . 2.4 2.6Tin . . . . . . 5.9

Elastic properties anisotropic materials

Anisotropic materials can be either naturally occurring (e.g. wood) or manufactured (e.g. fibre reinforced composites). In general

they are characterised by twenty-one independent constants, but this is reduced to nine for orthotropic materials and five for

transversely isotropic materials. They are frequently planar in form.

The main engineering constants in use for orthotropic composites are:

longitudinal modulus of elasticity, E11

transverse modulus of elasticity, E22

through-thickness modulus of elasticity, E33

longitudinal in-plane shear modulus, G12

longitudinal through-thickness shear modulus, G13

transverse through-thickness shear modulus, G23

major Poissons ratio, 12

minor Poissons ratio, 13


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transverse Poissons ratio, 23

For unidirectionally reinforced composites, 1 = fibre direction in-plane, 2 = transverse to fibre in-plane and 3 = transverse

through-thickness (i.e. perpendicular to plane). For other materials, the directions would be defined by other features, such as the

production length-wise direction. Poissons ratio can be greater than 0.5 for angle-ply or multidirectionally reinforced materials.

Composites with fully unidirectional reinforcement are approximately transversely isotropic materials (i.e. 2 and 3 directions are

equal). The following relations exist in this case:

E33= E22, G13= G12, 13= 12,

and E22= 2(1 + 23)G23

For orthotropic symmetry the following relations exist:

12=

21,

23=

32,

13=

31

E11 E22 E22 E33 E11 E33

Elasticities of woods

All woods are elastically anisotropic and in general there are nine independent elastic constants. The values in the table below are

for some common woods and give the three principal values of Youngs modulus measured along the grain EL, in a radial direction

ERand tangential direction ET(Hearmon, 1948).

WoodRelative

density

EL

GPa

ER

GPa

ET

GPa

Ash . . . . . . . . 0.7 16 1.6 0.9

Balsa . . . . . . . 0.2 6 0.3 0.1

Beech . . . . . . . 0.7 14 2.2 1.1

Birch . . . . . . . 0.6 16 1.1 0.6

Mahogany . . . . . . 0.5 12 1.1 0.6

Oak . . . . . . . . 0.7 11

Walnut . . . . . . . 0.6 11 1.2 0.6

Teak . . . . . . . . 0.6 13

Douglas Fir . . . . . . 0.5 16 1.1 0.8

Scots Pine . . . . . . 0.5 16 1.1 0.6Spruce . . . . . . . 0.40.5 1016 0.40.9 0.40.6

Elasticities of fibre-reinforced plastics full set

Material E11

GPa

E22

GPa

E33

GPa

G12

GPa

G13

GPa

G23

GPa

v12 v21 v23

High Modulus Carbon Fibre/Epoxy

unidirectionally reinforced

specimen 287 7.80 7.75 6.7 6.7 2.5 0.30 0.01 0.55

High Strength Carbon Fibre/Epoxy

unidirectionally reinforced

specimen 172 11.6 11.6 7.8 7.8 3.9 0.36 0.02 0.48

Elastic Constants measured at NPL by the Ultrasonic Technique (Read and Dean, 1978).

Elasticities of fibre-reinforced plastics in-plane properties

Material23 C

E11

GPa

E22

GPa

v12 v21

Injection moulded, discontinuous (long) fibre thermoplastic: glass-fibre/nylon

(30%fibre by volume) 10.6 7.9 0.34 0.22


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Hot compression moulded, sheet moulding (thermoset) compond (SMC): glass

fibre strands/filler/polyester resin (62%fibre + filler by volume) 10.0 9.8 0.30 0.31

Thermoformed (press) moulded, mat + unidirectional fibres/thermoplastic

glass-fibre/polypropylene (18%fibre by volume) 9.2 4.4 0.41 0.22

Autoclaved, unidirectional continuous fibre/thermoset resin: glass-fibre/epoxy

(59%fibre by volume) 47.0 16.4 0.28 0.08

Autoclaved, unidirectional continuous fibre/thermoset resin: carbo-fibre/epoxy

(61%

fibre by volume) 146 9.9 0.30 0.02

Typical values measured at NPL using mechanical test methods (Sims et al., 1993) actual values depend on fibre type,

orientation and distribution, also on resin properties and process route.

Strength properties isotropic materials

The strength properties of many materials are dependent on the rate of loading and the test temperature. This particularly applies

to plastics and glass-fibre reinforced plastics. Generally materials will reach their elastic l imit prior to failure.

Substance Tensile

strength

MPa

Metals

Aluminium (cast) 90100

(rolled) 90150

Brass (66% Cu, 34% Zn) (cast) 150190

" (rolled) 230270

Calcium 4260

Cobalt 260750

Copper (cast) 120170

,, (rolled) 200400

Gun metal (90% Cu, 10% Sn) 190260

Iron (cast) 100230

,, (wrought) 290450

Lead (cast) 1217

Magnesium (cast) 6080

,, (extruded) 170190

Phosphor-bronze (cast) 180280

Steel (castings). 400600

Steel (mild) (0.2% C) 430490

High-carbon spring steel:

(annealed 700770

(tempered) 9301080

(nickel) (5% Ni) 8001000

(nickel-chromium) 10001500

Soft solder 5575

Tin (cast) 2035

Zinc (rolled) 110150

Plastics

Nylon 6 7697

Nylon 66 6283

Polyacetal ~69

Polybenzoxazole 82117

Polycarbonate 5565

Polyethylene 2135

Polyimide 69-104

Polymethylmethacrylate 50-76

Polypropylene 30-40

Polystyrene 34-52

Miscellaneous


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Catgut 420

Glass 3090

Hemp rope 60100

Leather belt 3050

Silk fibre 260

Spider thread 180

Woods:

Ash, beech, oak, teak, mahogany 60110

Fir, pitch-pine 4080 Red or white deal 3070

White or yellow pine 2050

Quartz fibre (fused) ~1000

Wires

Aluminium 200-450

Brass 350-550

Copper (hard-drawn) 400460

,, (annealed) 280-310

Duralumin 400-550 German Silver 460

Gold 200250

Iron (charcoal, hard-drawn 540-620

,, (annealed) 460

Molybdenum 11003000

Nickel 500-900

Palladium 350450

Phosphor-bronze (hard-drawn) 6901080

Platinum 330370

Pt + 10% Rh 630

Silver 290

Steel (ordinary) ~1100

,, (tempered) 1550 ,, (pianoforte, hard-drawn) . 18602330

Tantalum 8001100

Tungsten 15003500

Zirconium (annealed) 260390

,, (hard-drawn) 1000

Along the grain

Strength properties anisotropic materials

The strength properties of anisotropic materials measured in different directions may differ considerably. Differences in strengthscan be higher than those in elastic properties.

Ultimate tensile strength properties of fibre-reinforced plastics in-plane properties. (Sims et al., 1993). Typical values;

actual values depend on fibre type, orientation and distribution; resin properties and process route (NB. 11 = longitudinal

ultimate tensile strength and 22= transverse ultimate tensile strength).

Material

23 C11

MPa

22

MPa

Injection moulded discontinuous (long) glass-fibre/nylon (30%fibre by volume) 148 113

Hot comperssion moulded, sheet moulding material (SMC) glass fibre strands/filler/polyester

resin (62%fibre + filler by volume)

60

59

Thermoformed (press) moulded, mat + unidirectional/thermoplastic glass-fibre/

polypropylene (18%fibre by volume )

143 38

Autoclaved, unidirectional glass-fibre/epoxy (59%fibre by volume) 1139 63


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Autoclaved, unidirectional carbon-fibre/epoxy (61%fibre by volume) 2386 76

References

G. Bradfield (1964) Notes on Applied Science No. 30, Use in Industry of Elasticity with the Help of Mechanical Vibrations, HMSO.

P. W. Bridgman (1949) The Physics of High Pressure, Bell.

J. A. Ewing (1899) Strength of Material, Cambridge University Press.

R. F. S. Hearmon (1948). See also R. F. S. Hearmon, Elasticity of Wood and Plywood, Forest Products Research Special Report No.

7, HMSO.

M. F. Markham (1968) Measurements made at the NPL.

B. E. Read and G. D. Dean (1978) The determination of dynamic properties of polymers and composites, Adam Hilger.

G. D. Sims, W. Nimmo and W. R. Broughton (1993) Data measured at the NPL.

Others sources of data include,

W. Bolton (1989) Engineering Materials Pocket Book, Newnes.

Handbook of Industrial Materials(1992) Elsevier Adv. Tech.

N. A. Waterman and M. F. Ashby (1992) Elsevier Materials Selector, Elsevier Applied Science.

G.Sims

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