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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 in p = − K . ΔV/V) and the shear modulus G.

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

Young’s Modulus, or longitudinal elasticity, E.
Poisson’s ratio, ν = lateral contraction per unit breadth divided by the longitudinal extension per unit length under an applied longitudinal stress.
Compressibility, κ = 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 + ν)

  (b)  

K = 

E

 

3 (1 − 2ν)

(c)  K=  

1

 

EG

     

3

 

3(3G − E)

  (d)  

M = K + 

4

 G

3


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


Elasticities of metals and alloys

Material
20 ºC

E

GPa

G

GPa

 ν

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

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

 ν

K

GPa

Glass (Heavy Flint)    .    .    .

80.1

31.5

0.27

57.6

Glass (Crown)      .    .    .    .

71.3

29.2

0.22

41.2

Quartz (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, K increases. In general a rise in temperature decreases the bulk modulus of a liquid; water, however, shows a maximum value of K at 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, 1–16 atm

    20  

  1.45

 Mercury:

 

 

Amyl alcohol, 8 atm      .

    17.7

  1.12

      8–37 atm    .      .      .

  20

26.2

Benzene, 8 atm      .      .

    17.9

  1.10

  100–200 atm  .      .

  15

30.0

Butyl alcohol, 8 atm      .

    17.4

  1.13

 Methyl acetate, 8–37 atm 

     14.3

    1.04

Butyl alcohol, iso-, 8 atm 

    17.9

  1.03

 Methyl alcohol, 37 atm   .

     14.7

    0.97

Carbon bisulphide, 8–37 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

       1–50 atm   .         .      .

    0.689

 Propyl alcohol, 8 atm      .

     17.7

    1.04

   900–1000 atm         .      .

  1.56

 Propyl alcohol, iso-, 8 atm

     17.8

      0.983

   900–1000 atm         .      .

198    

    0.703

 Turpentine        .      .      .

     19.7

      1.280

Ethyl acetate, 8–37 atm      .

 13.3

    0.974

 Water:

 

 

Ethyl alcohol:

 

 

        1–25 atm   .      .      .

  15

    2.05

       1–500 atm           .      .

  1.32

    900–1000 atm      .      .

  15

    2.75

   150–200 atm           .      .

310   

    0.024

    900–1000 atm      .      .

198

    1.81

Ethyl bromide, 8–37 atm    .

 99.3

    0.343

  2500–3000 atm      .      .

     14.2

    3.88

Ethyl chloride, 8–37 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    .    .    .    .    .    .

1.4–3.1

  Polyethylene (high density)    .    .

0.4–1.3

Epoxy       .    .    .    .    .

~3.2

  Polyimide     .    .    .    .    .    .    .

~3.1

Nylon 6 (cast)     .    .    .

2.4–3.1

  Polymethylmethacrylate (PMMA)

2.4–3.4

Nylon 6 (moulded)   .    .

0.8–3.1

  Polypropylene  .    .    .    .    .    .

1.1–1.6

Nylon 66       .    .    .    .

1.2–2.9

  Polystyrene      .    .    .    .    .    .

2.7–4.2

Polybenzoxazole   .    .    .

~3.5

  Polytetrafluoroethylene (PTFE)

0.4

Polycarbonate     .    .    .

   2.4

  Polyvinylchloride (PVC)      .    .
      (unplasticised)

2.4–4.1

       

Temperature coefficient of elastic constants for a range of materials


Temperature coefficient α in

Et = E15{1 - α(t − 15)}

Gt = G15{1 - α'(t − 15)}

At 15°C

α

10−4

 for E

α'

10−4

 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

2.6

Tin       .    .    .    .    .    .

 

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 Poisson’s ratio, ν12
       • minor Poisson’s ratio, ν13
       • transverse Poisson’s ratio, ν23

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. Poisson’s 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,  ν13 = ν12,

and
   E22 = 2(1 + ν23)G23

For orthotropic symmetry the following relations exist:

ν12

=

ν21

,

ν23

 =

ν32

,

ν13

 =

ν31

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 Young’s modulus measured along the grain EL, in a radial direction ER and tangential direction ET (Hearmon, 1948).

Wood

Relative
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

0.6

Spruce    .    .    .    .    .    .    .

0.4–0.5

10–16

0.4–0.9

0.4–0.6

         

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


Material
23 °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

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 limit prior to failure.


Substance Tensile
  strength
  MPa
Metals  
   
Aluminium (cast)  90–100
          (rolled)  90–150
Brass (66% Cu, 34% Zn)  (cast)  150–190  
  "        (rolled) 230–270  
Calcium               42–60  
Cobalt                 260–750  
Copper (cast)   120–170  
      ,,     (rolled) 200–400  
Gun metal (90% Cu, 10% Sn) 190–260  
Iron (cast) 100–230 
   ,,  (wrought) 290–450 
Lead (cast) 12–17 
Magnesium (cast) 60–80 
         ,,       (extruded) 170–190 
Phosphor-bronze (cast) 180–280 
Steel (castings). 400–600 
  Steel (mild) (0.2% C) 430–490
  High-carbon spring steel:  
     (annealed 700–770
     (tempered)   930–1080
     (nickel) (5% Ni)   800–1000
     (nickel-chromium) 1000–1500
  Soft solder 55–75
  Tin (cast) 20–35
  Zinc (rolled) 110–150
   
   
Plastics  
   
  Nylon 6 76–97
  Nylon 66 62–83
  Polyacetal ~69
  Polybenzoxazole   82–117
  Polycarbonate 55–65
  Polyethylene 21–35
Polyimide   69-104
Polymethylmethacrylate 50-76
Polypropylene 30-40
Polystyrene 34-52
   
   
Miscellaneous  
   
   
Catgut 420
Glass 30–90
Hemp rope   60–100
Leather belt 30–50
Silk fibre 260
Spider thread 180
Woods†:  
     Ash, beech, oak, teak, mahogany   60–110
    Fir, pitch-pine 40–80
    Red or white deal 30–70
    White or yellow pine 20–50
Quartz fibre (fused) ~1000
   
   
Wires  
   
   
Aluminium  200-450
Brass  350-550
  Copper (hard-drawn) 400–460
       ,,      (annealed) 280-310
  Duralumin 400-550
  German Silver 460
  Gold 200–250
  Iron (charcoal, hard-drawn 540-620
   ,,    (annealed) 460
  Molybdenum 1100–3000
  Nickel 500-900
  Palladium 350–450
  Phosphor-bronze (hard-drawn)   690–1080
  Platinum 330–370
  Pt + 10% Rh 630
  Silver 290
  Steel (ordinary) ~1100
     ,,    (tempered) 1550
     ,,    (pianoforte, hard-drawn) . 1860–2330
  Tantalum   800–1100
   
  Tungsten 1500–3500
  Zirconium (annealed) 260–390
        ,,        (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 strengths can 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. σ11 = longitudinal ultimate tensile strength and σ22 = transverse ultimate tensile strength).


Material
23 °C    

σ11

MPa

σ22

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

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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