Tensors for Physics

308 16 Constitutive Relations

In addition to the bulk and shear moduliBandG, which already occur for isotropic
systems, a third modulusGc, specifically associated with the cubic symmetry, is
needed here. The fourth rank tensor, cf. Sect.9.5.1,

Hμνλκ(^4 ) ≡

∑^3

i= 1

eμ(i)e(νi)e(λi)e(κi) =

∑^3

i= 1

eμ(i)eν(i)eλ(i)e(κi)−

1

5

(δμνδλκ+δμλδνκ+δμκδνλ),

reflects the full cubic symmetry. The unit vectorse(i)are identified with the unit
vectorsex,eyandezparallel to the coordinate axes. Due toHμνλκ(^4 ) δμνδλκ =0,

Hμνλκ(^4 ) Δμν,λκ =0, andHμνλκ(^4 ) Hμνλκ(^4 ) =^65 , multiplication of (16.23) by the cubic

tensorHμνλκ(^4 ) yields

Gc=

5

12

Hμνλκ(^4 ) Gμν,λκ. (16.24)

For the cubic symmetry, the Voigt coefficients are

c 11 =B+

4

3

G+

4

5

Gc, c 12 =B−

2

3

G−

2

5

Gc, c 44 =G−

2

5

Gc. (16.25)

By symmetry, one hasc 11 =c 22 =c 33 ,c 12 =c 23 =c 31 , andc 44 =c 55 =c 66. Other
coefficients, likec 14 orc 45 are equal to zero. The coefficientc 66 =c 44 is the shear
modulus for a displacementuin thex-direction with its gradient in they-direction.
The shear modulus for a deformation in thexy-plane rotated by an angle of 45◦from
these directions is

c ̃ 66 = ̃c 44 =G+

3

5

Gc. (16.26)

In contradistinction toBandG, the cubic coefficient may have either sign. In fact, it
is negative for body centered (bcc) and face centered (fcc) cubic crystals, while it is
positive for simple cubic (sc) crystals. Mechanical stability requires that both shear
modulic 44 andc ̃ 44 be positive. This sets lower and upper bounds onGc:

−

5

3

G<Gc<

5

2

G. (16.27)

In terms of thec-coefficients, the bulk and shear moduli are given by

B=

1

3

(c 11 + 2 c 12 ), G=

1

5

(c 11 −c 12 + 3 c 44 ), Gc=

1

2

(c 11 −c 12 − 2 c 44 ).

(16.28)