CONTINUUM MECHANICS for ENGINEERS

FIGURE 9.9
Stress history with an initial discontinuity.
t
⎡dσ12
t′
⎤
γ 12()=
t σoJ()+
t J( t− t′
) ⎢ dt
⎣ dt′
⎥ ′ ∫
⎦
0
(9.5-3)
Upon integrating the integral in this equation by parts and inserting the
assigned limits of integration, the alternative form
()= ()+ ( ′ )
γ t J σ t σ t
12 o 12 12
0
t
∫
(9.5-4)
is obtained where J o = J(0).
In a completely analogous way, we may develop hereditary integrals
expressing stress as the result of arbitrary strains. The basic forms are as
follows:
t
(9.5-5a)
(9.5-5b)
(9.5-5c)
where G o = G(0).
The hereditary integral Eq 9.5-2, derived on the basis of simple shear, is a
special case of the general viscoelastic constitutive equations in hereditary
integral form, as given by the pair below expressed in terms of the distortional
and dilatational responses
( )
( )
⎡dJ
t− t′
⎤
⎢ ⎥ dt′
⎣ dt ( − t′
) ⎦
( )
⎡dγ12
t′
⎤
σ 12()=
t G( t− t′
) ⎢ dt
⎣ dt′
⎥ ′ ∫
⎦
−∞
t
⎡dγ12
t′
⎤
σ12()= t γ oG()+
t G( t− t′
) ⎢ dt
⎣ dt′
⎥ ′ ∫
⎦
12 12 o 12
0
0
t
( )
( )
⎡dG
t− t′
⎤
σ ()= t γ () t G + γ ( t′
) ⎢ ⎥ dt′
∫
⎣ dt ( − t′
) ⎦