16.2 Elasticity 307
The Voigt coefficients are
c 11 =c 22 =c 33 =B+4
3
G, c 12 =c 23 =c 31 =B−2
3
G, c 44 =c 55 =c 66 =G,other coefficients, likec 14 orc 45 are equal to zero. In terms of thec-coefficients, the
bulk and shear moduli ate given by
B=
1
3
(c 11 + 2 c 21 ), G=1
5
(c 11 −c 21 + 3 c 44 ).Hooke’s law (15.19) can be inverted to express the deformation in terms of the stress
tensor. The relative volume change is given by
uλλ=1
3 B
σλλ.The full strain tensor obeys the relation
uμν=1
9 B
σλλδμν+1
2 G
σμν. (16.21)A simple application is a homogeneous deformation of a body, e.g. the elongation or
compression of a brick-shaped solid by a forceFzstretching or squeezing it along
thez-direction. Than one hasσzz=Fz/A=kz, whereAis the area of the face
normal to thez-direction. In this case, all non-diagonal components of the strain
tensor vanish and the diagonal ones are given by
uzz=
1
3
(
1
3 B
+
1
G
)
kz=1
E
kz, uxx=uyy=−1
3
(
1
2 G
−
1
3 B
)
kz=−σuzz.HereEis the Young elastic modulus andσis the contraction number. These material
properties are related to the bulk and shear moduli by
E=
9 BG
3 B+G
, 2 σ=3 B− 2 G
3 B+G
. (16.22)
For a practically incompressible substance, whereB Gapplies, these expressions
reduce toE= 3 Gandσ= 1 /2.
16.2.4 Cubic System.
For a system with cubic symmetry, the elastic tensor, cf. the Hooke’s law (16.14), is
Gμν,λκ=Bδμνδλκ+ 2 GΔμν,λκ+ 2 GcHμνλκ(^4 ). (16.23)