Created: 2003
Last updated: April 12, 2007 [added i to eq.4]
Anisotropic and Inhomogeneous Elastic Continuum Mechanics




1. Basic Definitions:

R = material coordinate, or the "mean position" of a particular ion in a crystal. [A&M page 422]

r = position in space [A&M]

r(R) = position of the ion whose equilibrium position is R. [A&M page 422]

u = change of position or "deviation from equilibrium" of an ion [A&M page 422]

r(R) = R + u(R)
[2]
[equation (22.1) in A&M]

However, (unlike me) Ashcroft and Mermin allow u to be complex valued, as in the equation:

u(R,t) = ε ei(k·R-ωt)
[4]
[ A & M (22.55) on page 439]

Ashcroft & Mermin call ε the "polarization vector".

In the formalism of this webpage, u is always real valued, which is natural since both R and r are real valued vectors. However in many solid state physics and engineering calculations involving vibrations it is convenient for u to be complex valued, with only the real part being understood to have physical significance.

2. Strain:

   The strain tensor is defined by:
εσμ = (1/2) (   ∂
------
∂xσ 		uμ +   ∂
------
∂xμ 		uσ )
[8]
[ A & M eq. (22.77) ]

Caution is needed here since there are two popular definitions of the strain tensor's components. Only the off diagonal components are affected. It is common (though not universal) to use eij to represent the other popular convention for strain. In this convention:

eij = εij if i = j
eij = 2 εij if i ≠ j
[10]
[ This is equation (22.80) in A & M ]

In "elegant" literature, more often εij is used, while in "practical" literature more often eij is used. Quite often it is not made clear by the writer which convention is in use. In either case symbol eij or εij could be used, and only a very few references (those that have to make a connection between the "elegant" and "experimental" worlds like A&M and Kittel) use both εij and eij.

3. Elasticity Theory:

The harmonic potential energy (which is basically the same thing as the total potential energy of the crystal, as discussed in A & M on page 425) is given by:

U = (1/2) ∫ d3R   Σ
i j k l 		εij cijkl εkl
[14]

[adapted from (22.78) in A & M]

cijkl are the elastic constants.

i j k and l each correspond to the Cartesian axes x, y and z. So this 3 × 3 × 3 × 3 tensor has 81 elements. In the most general case there are only 21 independent elements. cijkl has the general symmetries

cijkl = cklij
cijkl = cjikl
cijkl = cijlk
[15]

The symmetry of the material reduces the number of independent elements of cijkl:

Triclinic 18
Monoclinic 12
Orthohombic 9
Tetragonal 6
Rhombohedral 6
Hexagonal 5
Cubic 3
Isotropic 2

In many situations it is convenient to use six components of the strain defined in the style of a 6-vector.

e1 = exx = εxx
e2 = eyy = εyy
e3 = ezz = εzz
e4 = eyz = 2 εyz
e5 = ezx = 2 εzx
e6 = exy = 2 εxy

The potential energy can also be written as:

U = (1/2)  6
Σ
i=1 		 6
Σ
j=1 		∫ d3R ei Cij ej
[18]
[ equation (22.82) from A & M;  (14) on page 140 from Kittel ]

   The Cij are called the "elastic stiffness constants." Cij is a 6 × 6 matrix. It is these quantities that are almost always used in practical, experimental or engineering applications. Note that Cij = Cji.

The 6 x 6 matrix of elastic stiffness constants is related to the 3x3x3x3 elasticity tensor.
A & M have the following incorrect equations (22.83) on page 446:
Cαβ = cijkl if α and β are both among xx, yy or zz
Cαβ = (1/2) cijkl if α and β are both among yz, xz or xy
Cαβ = (1/4) cijkl otherwise
However, I believe that A & M are wrong on this point, and that the correct relationship is

Cαβ = cijkl.
[20]

i.e. what is meant by equation [20] is
C11 = c1111 C22 = c2222 C33 = c3333
C12 = c1122 C13 = c1133 C23 = c2233
C14 = c1123 C15 = c1113 C16 = c1112
etc...

   Here is why I think so: In particular, consider C45 in relation to cyzxz. Consider the contribution to the potential energy U in equation [14] and [18]:
The sum over α and β includes the following identical terms:
   C45e4e5
+ C54e4e5 .
This sum of these two terms is equal to the sum of eight identical terms in the sum over i, j, k and l:
   c2313ε23ε13
+ c3213ε32ε13
+ c2331ε23ε31
+ c3231ε32ε31
+ c1323ε13ε23
+ c3123ε31ε23
+ c1332ε13ε32
+ c3132ε31ε32
Note that ε23 = ε32 = (1/2) e4 and ε13 = ε31 = (1/2) e5. If c2313 equals C45 then the two expressions are equal, as they must be. The mistake that Ashcroft & Mermin made is to confuse the two different conventions for the strain tensor.
   The other kind of case that needs to be checked is where α≤3 but β≥4.
   C16e1e6
+ C61e6e1
which must equal the sum of the four identical terms:
   c1211ε12ε11
+ c2111ε21ε11
+ c1112ε11ε12
+ c1121ε11ε21
Note that ε12 = (1/2) e6 and ε11 = e1. So the sums work out the same as long as c1211 = C16.

The kinetic energy is

T = ρ ∫ d3R (1/2) ( ∂u
-----
∂t 		)·( ∂u
-----
∂t 		)
[22]
[ based on equation 22.84 in A & M ]

here ρ is the mass density.
The Lagrangian of the medium is L = T - U. Using Hamilton's principle (least action) we get the equation of motion:

ρ ∂2
-----
∂t2 		ui =  Σ
j k l 		cijkl     ∂2
----------
∂xj∂xk 		ul
[24]
[ Based on equation (22.86) in A & M ]

   Take note that equation [24] can handle the general case of an anisotropic material, but it is not correct for an inhomogeneous material (a material where cijkl are position dependent).

5. Comma and Summation Notation

   An alternative more compact notation for partial derivatives is to use commas followed by the variable with respect to which the derivative is taken. Also, the Einstein summation convention means that an implicit summation is made in a term in which an index is repeated. Using this notation, equation [24] becomes

ρ ui,tt = cijkl ul,jk
[26]

   Note that equation [26] applies only for a homogeneous material. Making use of the symmetry properties of the cijkl as well as the irrelevance of the order of partial derivatives, this equation can also be written equivalently as:
ρ ui,tt = cijkl uk,lj
[27]
which is more similar to the right side of equation (1) in Saviot, Murray and Marco de Lucas.
Equation [8] for the strain becomes

εij = (1/2) (uj,i + ui,j)
[28]

Equation [14] for the potential energy becomes
U = (1/8) ∫ d3R cijkl (uj,i + ui,j) (ul,k + uk,l)
[30]
and using the symmetry of cijkl this can be simplified to:
U = (1/2) ∫ d3R cijkl ui,j uk,l
[31]
Equation [22] for the kinetic energy becomes
T = (1/2) ∫ d3R ρ ui,t ui,t
[32]

6. Inhomogeneous Elasticity:

   Consider an elastic material that has position dependent elastic constants cijkl(R). R denotes the material coordinate.
   The stress tensor is σij. It is related to the strain tensor εij (defined above in equation [8]) through

σij = cijkl εkl
[6.1]
this is the same as:
σij = (1/2) cijkl (uk,l+ul,k)
[6.2]
which is which same as:
σij = cijkl uk,l
[6.3]
Consider now an infinitesimal spherical region of the material, dV. This infinitessimal region has surface S. Consider a small piece of S, denoted as dS. The outward pointing normal vector to dS is n with Cartesian components ni. Form a vector dS = dS n. The force exerted by the surrounding environment on region dV through surface dS is dF where
dF = σ dS
[6.4]
The total force on region dV is F, which is the integral over the surface S of dV:
F = ∫
S 		dF = ∫
S 		σ dS
[6.5]
It is more familiar to see Green's theorem in the context of a vector field, for example B:
∫
S 		B · dS = ∫
V 		(∇·B) dV
[6.6]
however it makes just as much sense mathematically to apply Green's theorem to the 2nd rank tensor σ rather than a vector field.
F = ∫
V 		(∇·σ) dV
[6.7]
Specifically, "∇·σ" means σij,j.
The mass of the region is m, where
m = ∫
V 		ρ dV
[6.8]
The acceleration of the infinitessimal region of material is a = ui,tt.
Next, applying Newton's second law F = m a to this infinitessimal region dV, we get
ρ ui,tt = σij,j
[6.9]
Using [6.3], and noting that both u and c are functions of position, we get the equation of motion:
ρ ui,tt = cijkl,j uk,l + cijkl uk,lj
[6.10]

References:

N. W. Ashcroft and N. D. Mermin, "Solid States Physics" 1976 Holt, Rinehart and Winston   (particularly chapter 22 "Classical Theory of the Harmonic Crystal")

C. Kittel "Introduction to Solid States Physics" fourth edition. Wiley.

Snieder, R., "General theory of elastic wave scattering" , in Scattering and Inverse Scattering in Pure and Applied Science , Eds. Pike, R. and P. Sabatier, Academic Press, San Diego, 528-542, 2002 pdf

Daniel Murray
Associate Professor
Math, Stats & Physics Unit
University of British Columbia - Okanagan
Kelowna, BC, Canada
daniel "dot" murray "at" ubc "dot" ca

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