Chapter 6

Mechanical Properties of Solids

We now turn to discuss the properties of solids, especially of monatomic solids, alloys, and simple compounds. Many materials of technological interest fall in these categories, but wood for instance does not. Rocks in the earth's crust often consist of aggregates of very small crystals that are themselves fairly simple; commonly, they have variable composition. The study of solids goes under various names: crystallography, solid state physics, and materials science. There are also the related fields of solid state chemistry, mineralogy, surface science, and others. These names reflect the different backgrounds and interests of workers in this vast area. Human technology has relied heavily on solids from the stone age, to the iron age, to the transistor age.

We can broadly divide the subject into two parts: mechanical properties and transport properties (electrical and thermal conduction). We start with mechanical properties such as such as stiffness, strength, and toughness, which we will define as we go along. For specific applications, we would like to engineer or design materials with the appropriate combination of these qualities.

6.1 Stiffness

How "stiff" or "flexible" is a material? It depends on whether we pull on it, twist it, bend it, or simply compress it. We shall see that in the simplest case the material is characterized by two independent "stiffness constants" and that different combinations of these constants determine the response to a pull, twist, bend, or pressure. This simplest behavior occurs when the material deforms elastically, or more precisely within the bounds of linear elasticity, and further it is effectively homogeneous and elastically isotropic. A material behaves elastically if it returns to its original shape when the stress is removed; linear elastic behavior is an idealization that is quite accurate for small deformations (typically less than 1%). A piece of an amorphous solid or a glass of constant composition is homogeneous and isotropic in all its properties, on a scale larger than atomic size. On the other hand, a piece of wood is not isotropic in many of its properties, including elasticity, and is not homogeneous on a small enough scale ( it has a "grain"), but is effectively homogeneous for deformations on a scale larger than the grain. Single crystals are not elastically isotropic (and are anisotropic in other ways too: for instance, they show preferred faceting). Rocks and metals often consist of many microcrystals and are elastically isotropic for deformations on a scale larger than the microcrystals. This is true despite the fact that the microcrystals themselves are elastically anisotropic, provided they are randomly oriented. Since the microcrystal size can be as small as 0.1mm, it is hard to distinguish a finely microcrystalline material from an amorphous one.

If a material is anisotropic, we must repeat the arguments given below for pulls (or bends) in different directions, and the question will be how many independent such pulls must be considered, or how many independent elastic constants characterize the material. If a material is inhomogeneous, we can think of it as consisting of homogeneous little pieces, infinitely many of them if the composition varies continuously.

6.1.1 Stress and strain

Pull on a bar of length l with a force F : it will elongate by an amount Dl (see figure 6.1). The stress which is applied at the ends of the bar is F/A, where A is the cross-sectional area of the bar. Stress has the same dimensions as pressure (its SI units are pascals, with 1 psi = 6891 Pa), but conventionally a positive stress (pulling out) corresponds to a negative pressure.

Fig. 6.1 Deformation of a pulled bar. In equilibrium, opposite forces are acting on opposite faces, whether we are pulling on a bar that is fixed at one end (top, reaction force shown dashed) or we are physically pulling on both ends (bottom).

The deformation of the bar is described by dimensionless ratios called strains. An obvious strain is the fractional elongation Dl/l, which is positive for a bar in tension, negative for a bar in compression.
Hooke's law, valid for small strains, tells us that the fractional elongation is proportional to the stress:

E Dl/l = F/A

where E is the Young modulus (also denoted by Y). The dimensions of E are the same as those of stress and pressure.

The bar also gets thinner in the directions perpendicular to the force: as shown in Fig. 5.2 a transverse dimension w is reduced to w+Dw (with Dw < 0). The fractional shrinkage Dw/w is also a strain; if small, it is proportional to the stress F/A, and thus proportional to Dl/l. The Poisson ratio n (also called s) is defined by

Dw/w = -n Dl/l

and can have any value between 0 and ½. The reason for the upper limit on n is that the volume change DV must be positive, and one can show that DV/V = (1 - 2n)Dl/l.

6.2 Elastic constants

In an isotropic elastic medium, the two elastic constants we have introduced, Young's modulus E and Poisson's ratio n, are sufficient to determine the strains given the stresses, or vice versa, in any possible case. However, some combinations of E and n are often encountered and are given special names:

When considering the effects of hydrostatic pressure, it is convenient to introduce the bulk modulus B, defined by DV/V = -Bp, as it was for a fluid. It is left as a problem to show that E = 3B(1 -2n )
Shear stresses are related to shear strains by the shear modulus m, as discussed in detail by Feynman, who also shows that E = 2m (1 +n).

The general stress-strain relation is conveniently written as

$\begin{displaymath} S_{ij}=B\Delta \,\delta _{ij}+2\mu \left( \mathcal{E}_{ij}-\frac{\Delta }{3}\,\delta _{ij}\right) \end{displaymath}$

where $\Delta =\mathcal{E}_{11}+\mathcal{E}_{22}+\mathcal{E}_{33}$ and, as in Ashcroft - Mermin,

$\begin{displaymath} \mathcal{E}_{ij}=\frac{1}{2}\left( \frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}\right) \end{displaymath}$

6.3 Speeds of sound

Consider sound waves propagating down a bar or beam that is much longer than the wavelength. The bar's cross-section is constant, but it is not necessarily a simple geometrical shape, such as a square or a circle. To be definite, orient the bar in the z direction, as shown in the figure for a rectangular bar of transverse dimensions L_x, L_y and for a circular cylinder of radius R.

We consider first the simplest "mode" of propagation, in which all points in a cross-section of the bar have the same displacement. In general, the bar is the mechanical analogue of an electromagnetic wave guide of constant cross-section; however, the bar can transmit both longitudinal (or compression) waves and transverse (shear) waves, while in the electromagnetic case only transverse waves are present. The following limiting cases give an idea of the variety of possible behaviors:

6.3.1 Compression waves

Bulk longitudinal waves.
If the transverse dimensions (such as L_x, and L_y, or R) are much larger than the wavelength $\lambda$ , the speed of sound is that of longitudinal waves in an infinite medium and is called c_l. Naively we may expect that $\rho c_{l}^{2}=$ B, but the correctresult is instead

$\begin{displaymath} \rho c_{l}^{2}=3B\,\frac{1-\nu }{1+\nu }=E\frac{1-\nu }{(1+\nu )(1-2\nu )} \end{displaymath}$

Bar compression waves.
If the transverse dimensions are much smaller than the wavelength $\lambda$ , the speed of sound, c_bar, is given by

$\begin{displaymath} \rho c_{bar}^{2}=E=3B(1-2\nu ) \end{displaymath}$

6.3.2 Transverse waves

Bulk shear waves.
If all transverse dimensions are much larger than $\lambda ,$ the speed of sound is that of transverse waves (shear waves) in an infinite medium and is called c_t. As one might expect,

$\begin{displaymath} \rho c_{t}^{2}=\mu =\frac{E}{2(1+\nu )}\,. \end{displaymath}$

Torsion waves.
If the transverse dimensions are much smaller than $\lambda$ , the result depends on the cross-section. For a circular cross-section there are simple torsional waves with speed given again by c_t, independent of the radius R. This is true both for a rod (solid cylinder) and for a tube. Pure torsional waves also exist if the cross-section is an ellipse or an equilateral triangle, but in general torsion couples to flexion (see next).

Flexion waves.
These are waves in a thin beam that vibrates from side to side, when $\lambda$ is larger than the lateral dimensions. For a vibration in the y direction, the equation of motion is

$\begin{displaymath} EI\,\frac{\partial ^{4}u_{y}}{\partial x^{4}}=\rho A\,\frac{\partial^{2}u_{y}}{\partial t^{2}} \end{displaymath}$ (1)

where I is the bending moment of inertia (see Feynman, 38-4). Hence, the dispersion relation is

$\begin{displaymath} \omega =\frac{EI}{\rho A}\,k^{2}\end{displaymath}$

where as usual $k=2\pi /\lambda$ . The phase velocity is

$\begin{displaymath}c_{flex}(k)=2k\sqrt{\frac{EI}{\rho A}} \end{displaymath}$

and the group velocity is twice this. Long waves move very slowly.

Waves in a string under tension.
These are also transverse waves, and for vibrations in the y direction obey the standard wave equation

$\begin{displaymath} \rho A\,\frac{\partial ^{2}u_{y}}{\partial t^{2}}=T\,\,\frac{\partial ^{2}u_{y}}{\partial x^{2}}\,, \end{displaymath}$ (2)

where T is the tension in the string. Hence, $\omega =c_{str}k$ , with

$\begin{displaymath} \rho Ac_{str}^{2}=T \end{displaymath}$

and c_str is the wave velocity. Note that $\rho A$ is the mass per unit length of the string.

Beam or bar under tension
A beam under tension is kept straight by it. and also resists bending because of its rigidity (and a rather thick, short string does too). The two restoring forces add and give the equation

$\begin{displaymath} \rho A\,\frac{\partial ^{2}u_{y}}{\partial t^{2}}=T\,\,\frac... ...partial x^{2}}-EI\,\frac{\partial ^{4}u_{y}}{\partial x^{4}} \end{displaymath}$ (3)

which manifestly reduces to ( 1) or to ( 2) when one of the restoring forces can be neglected. The dispersion relation is

$\begin{displaymath} \rho A\omega ^{2}=Tk^{2}+EIk^{4} \end{displaymath}$

At small k, large $\lambda$ , the T term is dominant, and the contrary is true at small $\lambda$ .

Beam under compression: buckling

If T is negative, it describes a compression. Eq. (3) is still valid, but for small k it gives a negative value of $\omega ^{2}$ . That means that $\omega$ is imaginary and the amplitude grows exponentially, instead of oscillating. A beam of length L subject to a compression T=-EIk² will buckle into the shape described by $\sin kx$ , with $k=\pi /L$ . The buckling stress, then, has magnitude $EI\pi ^{2}/L^{2.}$ , in agreement with the result given by a purely static analysis (see Feynman, 38-5).

References

1: R. P. Feynman et al., The Feynman Lectures on Physics (Redwood City, Calif.: Addison-Wesley, 1989), Vol. II, Chap. 30, 31, 38, and 39.
2: E. R. Cohen, The Physics Quick Reference Guide (American Institute of Physics, 1991).
3: Karl F. Graff, Wave Motion in Elastic Solids (Dover Publications, 1991).