CONTINUUM MECHANICS for ENGINEERS

the absolute compliance, in which γ o is the peak value of the strain as given by
(9.6-18)
Based upon the definitions of G * and J * , it is clear that these complex
quantities are reciprocals of one another. Thus
(9.6-19)
A simple procedure for calculating G * (or J * ) of a specific model or material
is to replace the partial differential operator ∂ t in the material’s constitutive
equation by iω and solve the resulting algebraic equation for the ratio σ 12/γ 12 =
G * , (or γ 12/σ 12 = J * if that is the quantity required). Accordingly, from Eq 9.3-6
for the Kelvin solid, the constitutive equation σ 12 = {G + η ∂ t }γ 12 becomes
σ 12 = (G + iηω)γ 12 which yields σ 12/γ 12 = G(1 + iτω) = G * . Likewise, from
Eq 9.3-7 for the Maxwell fluid, {∂ t + 1/τ}σ 12 = {G∂ t}γ 12 becomes (iω + 1/τ) σ 12 =
(Giω)γ 12 from which σ 12/γ 12 = G(τ 2 ω 2 + iτω)/(1 + τ 2 ω 2 ) = G * .
In developing the formulas for the complex modulus and the complex
compliance we have used the differential operator form of the fundamental
viscoelastic constitutive equations. Equivalent expressions for these complex
quantities may also be derived using the hereditary integral form of constitutive
equations. To this end we substitute Eq 9.6-2 into Eq 9.5-5a. However,
before making this substitution it is necessary to decompose the stress relaxation
function G(t) into two parts as follows,
G(t) = G o [1 – φ(t)] (9.6-20)
where G 0 = G(0), the value of G(t) at time t = 0. Following this decomposition
and the indicated substitution, Eq 9.5-5a becomes
(9.6-21)
In this equation, let t – t′ = ξ so that dt′ = –dξ and such that when t′ = t, ξ =
0, and when t′ = –∞, ξ = ∞. Now
which reduces to
γ J˜
ω σ
= ( )
o o
( ) ( )=
* ∗
G iω J iω
iωt′ σ ()= t iωγ G e G 1−φ
t− t′ dt
12
o o
∫
t
−∞
ω
σ12 t iωtG ω
e
i t ⎧
o⎨
⎩ i
∫0
o
1
[ ( ) ] ′
iωξ iωt ⎫
e e φξ dξ⎬
⎭
()= − ( )
∞ ⎧
⎫
σ12()= t γ o⎨Go − Go ( iωcosωξ+ ωsinωξ) φ( ξ) dξ⎬e ⎩ ∫0 ⎭
∞
iωt (9.6-22)
(9.6-23)