CONTINUUM MECHANICS for ENGINEERS

(9.5-6a)
(9.5-6b)
where J S is the shear compliance and J ν the volumetric compliance. Likewise,
Eq 9.5-5a is the simple shear form of the general equations
(9.5-7a)
(9.5-7b)
where G S is the relaxation modulus in shear and G V the relaxation modulus
in dilatation. Because we have assumed elastic behavior in dilatation, J V =
1/G V = K and both Eq 9.5-6b and Eq 9.5-7b reduce to the form σ kk = 3K ε kk in
keeping with Eq 9.2-7b.
9.6 Harmonic Loadings, Complex Modulus, and
Complex Compliance
The behavior of viscoelastic bodies when subjected to harmonic stress or
strain is another important part of the theory of viscoelasticity. To investigate
this aspect of the theory, we consider the response of the material cube shown
in Figure 9.1 under an applied harmonic shear strain of frequency ω as
expressed by
or by
t ⎡∂
Sij t′
⎤
()= t J ( t− t′
) ⎢ ⎥dt′
∫ ⎣⎢
∂ t′
⎦⎥
2ηij s
−∞
( )
( )
t ⎡∂
σ kk t′
⎤
3ε t J kk()= v(
t− t′
) ⎢ dt
⎣ ∂ t′
⎥ ′
∫−∞
⎦
( )
t ⎡∂
ij t′
⎤
Sij()= t GS( t− t′
) ⎢ ⎥ dt′
∫−∞ ⎣⎢
∂ t′
⎦⎥
2
η
σ
⎡∂
εkk
t′
⎤
()= t G ( t− t′
) ⎢ ⎥ dt′
⎣ ∂ t′
⎦
t
kk ∫ V
−∞
3
γ ()= t γ sinωt
12
γ ()= t γ cosωt
12
(9.6-1a)
(9.6-1b)
Mathematically, it is advantageous to combine these two by assuming the
strain in the complex form
o
o
( )