Formulation of Constitutive Equations - Continuum Mechanics of Anisotropic Materials - page 116

not. The type of material symmetry, that is to say either isotropy or the type of
anisotropy, characterizing a constitutive relation is reflected in the form of the
material coefficient tensors ( H , C or G

), for example the forms listed in Tables 4.3,
4.4 and 4.5. The tensor H may have any of the forms in Table 4.3 and the tensor C
may have any forms in Tables 4.4 and 4.5. Material symmetry, that is to say the
isotropy or type of anisotropy, is the property of a constitutive relation at a particle X ,
while inhomogeneity or homogeneity of materials relates to how the material
properties change from particle to particle. Thus a constitutive relation may be
either anisotropic and homogeneous or anisotropic and inhomogeneous. The most
mathematically simplifying assumptions are those of an isotropic symmetry and
homogeneous material.
The Newtonian law of viscosity, ( 5.7N ), is characterized by these most
simplifying assumptions, homogeneity, and isotropy. These assumptions are easily
justified when one thinks about the structure of, say, distilled water. Absent gravity,
there is no preferred direction in distilled water, and distilled water has the same
mechanical and thermal properties at all locations in the volume and in all volumes
of distilled water. One can then generalize this thought process to see that fluids are
isotropic.
The isotropic form of the Newtonian law of viscosity, ( 5.7N ), is obtained by
using the representation for the isotropic form of N obtained from Table 4.5, thus

ð s Þ

2 3 2 3 2 3

T 1 þ N 11 N 12 N 12 0 0 0

N 12 N 11 N 12 0 0 0
N 12 N 12 N 11 0 0 0
000 N 11 N 12 0 0
000 0 N 11 N 12 0
000 0 0 N 11 N 12

D 1
D 2
D 3
D 4
D 5
D 6

p

4 5 4 5 4 5

T 2 þ

p

T 3 þ

p

¼ ;

T 4
T 5
T 6

(5.8N)

This six-dimensional representation is converted to the three-dimensional repre-
sentation by employing the relations (A163) and introducing the following new
notation for the two distinct elements of the 6 by 6 matrix in ( 5.8N ),

N 11 ¼ l þ

m; N 12 ¼ l

2 (5.9N)

then

2 3 2 3 2 3

T 11 þ p l þ 2 ml l 000 D 11

D 22
D 33

4 5 4 5 4 5

T 22 þ p l l þ 2 ml 000

T 33 þ p l l l þ 2 m 000

p T 23 p D 23

¼ (5.10N)

0 0 0 2 m 00

p T 13 p D 13 0 0 0 0 2 m 0

p T 12 p D 12

0 0 0 0 0 2 m