Four Linear Continuum Theories - Continuum Mechanics of Anisotropic Materials - page 163

s¼s 1 < s¼s 2

D G ij ð D G ij ð s Þ s Þ

for all i

; j ¼ ; ; ... ;

and for s 1 >

s 2 > :

1 2 6 0 (6.48)

Ds Ds

The isotropic form of the viscoelastic stress-strain relations (5.36V) is obtained by
using the representation for the isotropic form of G

ð s Þ

obtained from Table 5.4, thus

2
4

3
5

G 11 ðsÞ G 12 ðsÞ G 12 ðsÞ

0 0 0

G 12 ðsÞ G 11 ðsÞ G 12 ðsÞ

0 0 0

G 12 ðsÞ G 12 ðsÞ G 11 ðsÞ

0 0 0

G

ð s Þ¼

G 11 ðsÞ G 12 ðsÞ

0 0 0 0 0

G 11 ðsÞ G 12 ðsÞ

0 0 0 0 0

G 11 ðsÞ G 12 ðsÞ

0 0 0 0 0

This six-dimensional representation is converted to the three-dimensional repre-
sentation by employing the relations in Table 6.1 and introducing the following new
notation for the two distinct elements of this 6-by-6 matrix, thus

k tr ð s Þþ 2 G dev ð s Þ k tr ð s Þ G dev ð s Þ

G 11 ð

; G 12 ð s Þ¼ s Þ¼ : (6.49)

3 3

The isotropic form of the viscoelastic stress-strain relations (5.36V) may then be
rewritten in three dimensions as

Z s¼t

D
Ds f

tr T ð x ; t Þ¼ k tr ð t s Þ tr E ð x ; s Þg d s (6.50)

s¼1

and

Z s¼t

D
Ds f

dev T

ð x ; t Þ¼ G dev ð t s Þ

dev E

ð x ; s Þg

d s

;

(6.51)

s ¼1

where k tr ( s ) and G dev ( s ) represent independent relaxation functions. In a similar set
of arguments it may be shown that the isotropic form of the viscoelastic
strain-stress relations ( 6.41 ) may be expressed in terms of two isotropic creep
functions, j tr ( s ) and J dev ( s ), thus

Z s¼t

D
Ds f

tr E ð x ; t Þ¼ j tr ð t s Þ tr T ð x ; s Þg d s (6.52)

s¼1

and

Z s¼t

D
Ds f

dev E ð x ; t Þ¼ J dev ð t s Þ dev T ð x ; s Þg d s : (6.53)

s ¼1

Viscoelastic materials have properties characteristic of both fluids and solids and
it is sometimes important to distinguish between viscoelastic fluids and viscoelastic