 |
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.
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
| |
|
|
|
|
|
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
|
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 |
|
|
Liquid |
|
|
| |
|
|
|
|
|
|
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 |
0.689 |
Propyl alcohol, 8
atm . |
17.7 |
1.04 |
|
900–1000
atm . . |
0 |
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 . . |
0 |
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.
|
|
|
|
|
|
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)} |
|
|
|
|
|
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).
|
|
|
|
|
|
|
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
|
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
|
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).
|
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
|
|
