Four Linear Continuum Theories - Continuum Mechanics of Anisotropic Materials - page 165

vibrations and pulse and harmonic wave propagation techniques have the ranges of
effectiveness indicated in Fig. 6.8 . One of the easiest probing tests for a viscoelastic
material is to subject the material to forced steady state oscillations. This testing
method is effective over a wide range of frequencies, Fig. 6.8 .
In the special case of forced steady state oscillations, special forms of the
stress-strain relations emerge. As an example we consider the case of the deviatoric
part of the isotropic stress-strain relations ( 6.51 ). It is assumed that the material is
subjected to a forced deviatoric strain specified as a harmonic function of time,

e i o

t

dev E

ð t Þ¼f

dev E o g

¼f

dev E o gf

o t þ o t g:

cos isin (6.57)

Upon substitution of the strain ( 6.57 ) into the viscoelastic stress-strain relation
( 6.51 ), along with the decomposition of G dev ( t ) given by ( 6.55 ), it follows that

dev E o Z s¼t
s

o

G dev dev E o e its G dev ð

e i

s d s

dev T

ð x ; t Þ¼ þ i o t s Þ :

(6.58)

¼1

Now, making a change of variable, t s ¼ ,

dev E o e i

Z 1 Z 1

o

dev Tðx; tÞ¼ G dev þo sin oZG dev ðZÞ d Zþio cos oZ G dev ðZÞ d Z

t

;

0 0

(6.59)

or

o

G dev ð dev E o e i

t

dev T ð x ; t Þ¼ i oÞ ; (6.60)

where

G 0 dev ðoÞþ G dev ð iG 00 dev ðoÞ; i oÞ¼ (6.61)

and

Z 1 Z 1

G 0 dev ðoÞ¼G dev þ o sin oZG dev ðZÞ d Z; G 0 dev ðoÞ¼o cos oZG dev ðZÞ d Z:

0 0

(6.62)

The functions G 0 dev ðoÞ and G 0 dev ðoÞ are called the storage and loss moduli,

respectively. The formulas ( 6.62 ) show that the real and complex parts of the
complex modulus G dev ð

, the only material function in the specialized steady-
state oscillatory viscoelastic stress-strain relation ( 6.60 ), are determined by the
relaxation function, G dev ( t ). The stress-strain relations ( 6.60 ) may also be written as

i oÞ

dev E o e ið o

tþ j dev Þ ;

G dev ð

dev T

ð x ; t Þ¼j i oÞj

(6.63)