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Young’s
modulus, shear modulus, bulk modulus, Poisson’s
ratio
Drilling
down: the origins of moduli.
Definition and
measurement. Figure 1
shows a typical tensile stress-strain curve. The initial part, up to the yield
strength
or elastic limit
, defined under Yield strength (elastic limit), is
linear (Hooke’s law), and it is elastic, meaning that the strain is recoverable
– the material returns to its original shape when the stress is removed. Stresses above the elastic limit cause
permanent deformation or fracture (see notes for Yield
strength (elastic limit) and Fracture
toughness).
Within the linear elastic regime, strain is proportional to stress, but
stress can be applied in more than one way (Figure 2). The tensile stress
produces a
proportional tensile strain
:
and the same is true in compression. The constant of proportionality, E, is called Young’s modulus. Similarly, a shear stress
causes a proportional shear strain
and a pressure p results in a
proportional fractional volume change (or “dilatation”)
:
where G is the shear modulus and K the bulk modulus. All three of these moduli have the same dimensions as stress, that of force per unit area (N/m2 or Pa). It is convenient to use a larger unit, that of 109 Pa, Giga-Pascals, or GPa.
Young’s modulus, the shear modulus, and the bulk modulus are related, but
to relate them we need one more quantity, Poisson’s ratio. When stretched in one direction, a
material generally contracts in the other two directions. Poisson’s ratio,
, is the negative of the ratio of the lateral or transverse
strain,
, to the axial strain,
, in tensile loading:
You might think that the way to measure the elastic modulus of a material would be to apply a small stress (to be sure to remain in the linear-elastic region of the stress-strain curve), measure the strain, and divide one by the other. In reality, moduli measured as slopes of stress-strain curves are inaccurate, often by a factor of 2 or more, because of contributions to the strain from material creep or deflection of the test machine. Accurate moduli are measured dynamically: by exciting the natural vibrations of a beam or wire, or by measuring the velocity of longitudinal or shear sound waves in the material.
Drilling down:
the origins of moduli. Atoms bond together, some weakly, some strongly. If they bind strongly enough they form
solids; the stronger the bond, the higher is the melting point of the
solid. Think of the bonds as little
springs (Figure 3). The atoms
have an equilibrium spacing
; a force
pulls them apart
a little, to
, but when it is released they jump back to their original
spacing. The same happens in
compression because the energy of the bond increases no matter in which
direction the force is applied, as the lower part of the figure suggests. The bond energy is a minimum at the
equilibrium spacing. A spring that
stretches by
under a force F has a stiffness, S,
defined by
and this is the same in compression as in
tension.
Table 1 lists the stiffnesses of the different bond types; these
stiffnesses largely determine the value of the modulus,
. The covalent
bond is particularly stiff (S =
20–200 N/m); diamond, for instance, has a very high modulus because the carbon
atom is small (giving a high bond density) and its atoms are linked by the
stiffest springs (S = 200 N/m). The metallic bond is a little less stiff
(S =15–100 N/m) and metal atoms are
often close-packed, giving metals high moduli too, though not as high as that of
diamond. Ionic bonds, found in many
ceramics, have stiffnesses comparable with those of metals, giving them, also,
high moduli. Polymers contain both
strong diamond-like covalent bonds along the polymer chain and weak hydrogen or
Van-der-Waals bonds (S = 0.5–2 N/m)
between the chains; it is the weak bonds that stretch when the polymer is
deformed, giving them low moduli.
When a force
is applied to a
pair of atoms, they stretch apart by
. A force
applied to an
atom corresponds to a stress
where
is the atom
spacing. A stretch
between two
atoms separated by
corresponds to a
strain
. Substituting
these into the last equation gives
Table 1 Bond stiffnesses,
S
Bond
type
|
Examples
|
Bond Stiffness
S |
Young’s Modulus
E |
Covalent
Metallic
Ionic
Hydrogen bond
Van der Walls
|
Carbon-carbon bond
All metals
Alumina,
Al203
Polyethylene
Waxes
|
50 –
180
15 –
75
8 –
24
6 –
3
0.5 -
1
|
200 –
1000
60 –
300
32 –
96
2 –
12
1 -
4
|
Comparing this with the definition of Young’s modulus reveals that E is roughly
The largest atoms
(
= 4 x 10-10 m) bonded with the weakest bonds (S = 0.5 N/m) will have a modulus of
roughly
This is the lower limit for true solids and many
polymers do have moduli of about this value; metals and ceramics have values
50–1000 times larger because, as Table 1 shows, their bonds are
stiffer.
One class of
materials – elastomers (rubber) – have moduli that are much less than 1
GPa. An elastomer is a tangle of
long-chain molecules with occasional cross-links, as in Figure 4 (a), as
explained in Density
and atom packing. The
bonds between the molecules, apart from the cross-links, are weak – so weak
that, at room temperature, they have melted. We describe this by saying that the
glass temperature
of the elastomer
– the temperature at which the bonds first start to melt – is below room
temperature. Segments are free to
slide over each other, and were it not for the cross-links, the material would
have no stiffness at all.
Temperature favors randomness. That is why crystals melt into disordered fluids at their melting
point. The tangle of Figure 4
(a) has high randomness, or
expressed in the terms of thermodynamics, its entropy is high. Stretching it, as at (b), aligns the
molecules – some parts of it now begin to resemble the crystallites shown in the
notes on Density
and atom packing. Crystals are ordered, the opposite of randomness; their entropy is
low. The effect of temperature is
to try to restore disorder, making the material try to revert to a random
tangle, and the cross-links give it a “memory” of the disordered shape it had to
start with. So there is a
resistance to stretching – a stiffness – that has nothing to do with
bond-stretching, but with strain-induced ordering. A full theory is complicated – it
involves the statistical mechanics of long-chain tangles – so it is not easy to
calculate the value of the modulus. The main thing to know is that the moduli of elastomers are low because
they have this strange origin and that they increase with temperature (because
of the increasing tendency to randomness), whereas those of true solids decrease
(because of thermal expansion).
| Author | Title | Chapter |
| Ashby et al | Materials: Engineering, Science, Processing and Design | 4, 5 |
| Ashby & Jones | Engineering Materials Vol 1& 2 | Vol. 1, Chap. 3, 6, 7 |
| Askeland | The Science and Engineering of Materials | 2, 6 |
| Budinski | Engineering Materials: Properties and Selection | 2 |
| Callister | Materials Science and Engineering: An Introduction | 6 |
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