Elasticities and strengths 2.2.2



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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
(c) K=	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

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

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

GPa

Liquid

Temp.

°C

GPa

Acetic acid, 1–16 atm

1.45

Mercury:

Amyl alcohol, 8 atm .

17.7

1.12

8–37 atm . . .

26.2

Benzene, 8 atm . .

17.9

1.10

100–200 atm . .

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 . .

1.12

Paraffin oil . . .

14.8

1.62

Chloroform, 100-200 atm .

1.1

Pentane . . . .

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 . . .

2.05

1–500 atm . .

1.32

900–1000 atm . .

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

GPa

Material

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
E_t = E₁₅{1 - α(t − 15)}
G_t = G₁₅{1 - α'(t − 15)}

At 15°C

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 E_L, in a radial direction E_R and tangential direction E_T (Hearmon, 1948).

Wood

Relative
density

E_L

GPa

E_R

GPa

E_T

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