Tensors for Physics

306 16 Constitutive Relations

purpose, an appropriate Cartesian coordinate system is chosen and the components
ofσμνanduμνare related to the components ofσianduI,i= 1 , 2 , ..,6 according to

σ 1 =σxx,σ 2 =σyy,σ 3 =σzz,

σ 4 =σyz=σzy,σ 5 =σxz=σzx,σ 6 =σxy=σyx

u 1 =uxx, u 2 =uyy, u 3 =uzz,

u 4 =uyz+uzy, u 5 =uxz+uzx, u 6 =uxy+uyx. (16.16)

In this notation, the linear relation (16.14) between the stress and the strain tensors
reads

σi=

∑^6

j= 1

cijuj. (16.17)

The matrixcijof the Voigt elasticity coefficients is symmetric,cij=cji. The connec-
tion with the fourth rank elasticity tensor follows from (16.16), e.g.c 11 =Gxx xx,
c 12 =Gxx yy,c 44 = 2 Gyz yz.
The coordinate axes are chosen such that they match the crystallographic axes.
The choice is obvious for acubic system. In the case of ahexagonal system,thez-axis
is put parallel to the sixfold symmetry axis. There are crystallographic conventions
for the general case.

16.2.3 Isotropic Systems

The trace of the deformation tensor is essentially the relative volume changeδV/V=
uλλ. The symmetric traceless part uμν describes a squeeze or shear deformation.
In an isotropic system, the trace of the stress tensor is linked with the trace of the
strain tensor,σλλ∼uλλ. Similarly, the symmetric traceless parts of these tensors are

proportional to each other,σμν ∼uμν. Thus two material coefficients only occur
in the linear elastic relation. These are thebulk modulus Band theshear modulus G.
The ansatz for an isotropic elastic solid is

σμν=Buλλδμν+ 2 Guμν. (16.18)

In this case, the fourth rank elasticity tensor is given

Gμν,λκ=Bδμνδλκ+ 2 GΔμν,λκ. (16.19)

Mechanical stability requiresB>0 andG>0. The moduliBandGare related to
the tensor by

B=

1

9

Gμμ,λλ, G=

1

10

Δμν,λκGμν,λκ. (16.20)