Four Linear Continuum Theories - Continuum Mechanics of Anisotropic Materials - page 166

Fig. 6.9 The complex modulus G *( i o ). The vertical scale represents both the real and imaginary

parts of G *( i ) which have the dimensions of one upon stress. The horizontal scale is the log of the

frequency, log

o

. The monotonically increasing curve represents G 0 ( o o ), the real part of G *( i o ),

and the curve with a peak represents G 00 ( o ), the imaginary part of G *( io ).

where the phase angle ' dev ðoÞ is given by

tan 1 G 0 dev ðoÞ
G 0 dev ðoÞ

' dev ðoÞ¼ : (6.64)

is called the loss tangent. The steady-state harmonic
strain lags behind the stress by the phase angle

The quantity tan ' dev ðoÞ

' dev ðoÞ . Typical plots of the storage

and loss moduli, G 0 dev ðoÞ and G 0 dev ðoÞ are shown in

Fig. 6.9 . However these curves for real material seldom look exactly like these
examples.

, respectively as a function of o

Example 6.5.1
An isotropic viscoelastic material is subjected to a step loading in shear strain E 12 .
The magnitude of the step loading is E o . The unit step function h ( t ) is used to
represent the step loading, E 12 ¼

E o h ( t ). Recall that h ( t ) is a function that is defined

as 0 for t < 0 and as 1 for t > 0. The derivative of the unit step function is the delta

d

( t ), ( d /d t )( h ( t ))

¼ d

( t ), where the delta function has the property that function

Z 1
1 dð

t Þ f ð t Þ ¼ t f ð t Þ d t :

Determine the stress response to the strain loading E 12 ¼ E o h ( t ).

Solution : Substitution of E o h ( t ) into the appropriate

stress-strain relation ( 6.51 ) yields the following simple formula,

strain loading E 12 ¼

Z s¼t

T 12 ¼ G dev ð t s Þ E o dð s Þ d s ¼ G dev ð t Þ E o :

s¼1