CONTINUUM MECHANICS for ENGINEERS

FIGURE 9.8
Applied stress histories: (a) stepped stress; (b) arbitrary stress.
9.5 Superposition Principle, Hereditary Integrals
For linear viscoelasticity, the principle of superposition is valid just as in elasticity.
In the context of stress/strain relationships under discussion here, the
principle asserts that the total strain (stress) resulting from the application
of a sequence of stresses (strains) is equal to the sum of the strains (stresses)
caused by the individual stresses (strains). Thus, for the stepped stress history
in simple shear displayed in Figure 9.8a
when applied to a material
having a creep function J(
t)
, the resulting strain will be
( )
2
γ ()= t ∆σ J()+ t ∆σ J( t−t )+ ∆σ J( t−t )= ∆σ
J t−t 12 o 1 1 2 2 ∑ i i
i=
0
(9.5-1)
and by an obvious generalization to the arbitrary stress loading considered
as an infinity of infinitesimal step loadings, Figure 9.8b,
the strain is given by
γ
12
t
⎡dσ12
t′
⎤
()= t J( t− t′
) ⎢ dt
⎣ dt′
⎥ ′ ∫
⎦
−∞
(9.5-2)
which is called a hereditary integral since it expresses the strain at time t as
a function of the entire stress history from time t = – ∞.
If there is an initial
discontinuity in the stress at time t = 0, and if the stress is zero up until that
time (Figure 9.9), the strain becomes
( )