Elastic Properties of SolidsTopics Discussed in Kittel, Ch. 3, pages 73-85

Hooke's “Law”• A property of an ideal spring of spring constant k is that

it takes twice as much force to stretch the spring twice as far. That is, if it is stretched a distance x, the restoring force is given by F = - kx. The spring is then said to obey

Hooke's “Law”. • An elastic medium is one in which a disturbance can be

analyzed in terms of Hooke’s “Law” forces. Consider the propagation of a mechanical wave

(disturbance) in a solid. • We are interested in the case of very long wavelengths, when the

wavelength is much, much larger than the interatomic spacing:

>> a so that the solid can be treated as a continuous elastic

medium & the fact that there are atoms on a lattice is irrelevant to the wave propagation.

A Mass-Spring Systemin which a mass m isattached to an idealspring of springconstant k. That is, the

Simple HarmonicOscillator

(SHO)

A Prototype Hooke’s Law System

Simple Harmonic Oscillator Stretch the spring a distance A & release it:

• In the absence of friction, the oscillations go on forever.• The Newton’s 2nd Law equation of motion is:

F = ma = m(d2x/dt2) = -kxDefine: (ω0)2 k/m (d2x/dt2) + (ω0)2 x = 0

A standard 2nd order time dependent differential equation!

Fig. 1 Fig. 2

Fig. 3

Simple Harmonic Oscillator Hooke’s “Law” for a vertical spring (take + x as down):

Static Equilibrium:

∑Fx = 0 = mg - kx0

or x0 = (mg/k)

Newton’s 2nd Law

Equation of Motion: This is the same asbefore, but the equilibrium position isx0 instead of x = 0

• An Elastic Medium is defined to be one in which a disturbance from equilibrium obeys Hooke’s “Law” so that a local deformation is proportional to an applied force.

• If the applied force gets too large, Hooke’s “Law” no longer holds. If that happens the medium is no longer elastic. This is called the Elastic Limit.

• The Elastic Limit is the point at which permanent deformation occurs, that is, if the force is taken off the medium, it will not return to its original size and shape.

Sound Waves• Sound waves are mechanical waves

which propagate through a material medium (solid, liquid, or gas) at a speed which depends on the elastic & inertial properties of the medium. There are 2 types of wave motion for sound waves:

Longitudinaland

Transverse

Sound Waves• Because we are considering only long wavelength

mechanical waves ( >> a) the presence of atoms is irrelevant & the medium may be treated as continuous.

Longitudinal Waves

Sound Waves• Because we are considering only long wavelength

mechanical waves ( >> a) the presence of atoms is irrelevant & the medium may be treated as continuous.

Longitudinal Waves

Transverse Waves

• Sound waves propagate through solids. This tells us that wavelike lattice vibrations of wavelength long compared to the interatomic spacing are possible. The detailed atomic structure is unimportant for these waves & their propagation is governed by the macroscopic elastic properties of the crystal.

• So, the reason for discussing sound waves is thatthey correspond to the low frequency, longwavelength limit of the more general latticevibrations we have been considering up to now.

• At a given frequency and in a given direction in a crystal it is possible to transmit 3 different kinds of sound waves, differing in their direction of polarization and in general also in their velocity.

Elastic Waves• So, consider sound waves propagating in a solid,

when their wavelength is very long, so that the solid may be treated as a continous medium. Such waves are referred to as elastic waves.

• At the point x the elastic displacement (or change in length) is U(x) & the strain e is defined as the change in length per unit length.

dUe

dx

x x+dx

Consider Longitudinal Elastic WavePropagation in a Solid Bar

• In general, a Stress S at a point in space is defined as the force per unit area at that point.

C Young’s Modulus

dUe

dx

• Hooke’s “Law” tells us that, at point x & time t in the bar, the stress S produced by an elastic wave propagation is proportional to the strain e. That is:

x x+dx

• To analyze the dynamics of the bar, choose an arbitrary segment of length dx as shown above. Use Newton’s 2nd Law to write for the motion of this segment,

2

2( ) ( ) ( )

uAdx S x dx S x A

t

C Young’s Modulus

Mass Acceleration = Net Force resulting from stress

dUe

dx

x x+dx

.S C e

due

dx

2

2

.

.

uS C

x

S uC

x x

Equation of Motion

2

2( ) ( ) ( )

uAdx S x dx S x A

t

( ) ( )S

S x dx S x dxx

2 2

2 2( )

u uAdx C Adx

t x

2 2

2 2

u uC

t x

( )i kx tu Ae /

s

s

v k

v C

Cancelling common terms in Adx gives:This is the wave equation a planewave solution which gives thesound velocity vs:

Plane wave solution:

So, this becomes:

k = wave number = (2π/λ), ω = frequency, A = amplitude

sv k

• Unlike the case for the discrete lattice, the dispersion relation ω(k) in this long wavelength limit is the simple equation:

• At small λ (k → ∞), scattering from discrete atoms occurs.

• At long λ (k → 0), (continuum) no scattering occurs.

• When k increases the sound velocity decreases.

• As k increases further, the scattering becomes greater since the strength of scattering increases as the wavelength decreases, and the velocity decreases even further.

Speed of Sound• The speed VL with which a longitudinal elastic wave

moves through a medium of density ρ is given by:

L C

V

C Bulk Modulusρ Mass Density

• The velocity of sound is in general a function of the direction of propagation in crystalline materials.

• Solids will sustain the propagation of transverse waves, which travel more slowly than longitudinal waves.

• The larger the elastic modulus & the smaller the density, the larger the sound speed is.

Speed of Sound for Several Common Solids

Most calculated VL values are in reasonable agreement with measurements. Sound speeds are of the order of 5000 m/s in typical metallic, covalent & ionic solids :