CONTINUUM MECHANICS for ENGINEERS

Development of details relative to the stress relaxation test follows closely
that of the creep test. With an imposed strain at time t = 0
γ = γ ()
(9.4-15)
the resulting stress associated with Kelvin behavior is given directly by
inserting into Eq 9.3-6 resulting in
˙ γ = γ δ()
t
12
o
(9.4-16)
The delta function in this equation indicates that it would require an infinite
stress at time t = 0 to produce the instantaneous strain γ o. For Maxwell
behavior, when the instantaneous strain, Eq 9.4-15, is substituted into
Eq 9.3-7, the stress relaxation is the solution to the differential equation
which upon integration using Eq 9.4-12 yields
(9.4-17)
(9.4-18)
The initial time-rate of decay of this stress is seen to be γ oG/τ, which if it
were to continue would reduce the stress to zero at time t = τ. Thus, τ is
called the relaxation time for the Maxwell model.
Analogous to the creep function J(t) associated with the creep test we
define the stress relaxation function, G(t), for any material by expressing
σ 12(t) in its most general form
From Eq 9.4-18, the stress relaxation function for the Maxwell model is
and for the generalized Maxwell model it is
12
o Ut
σ t γ GU t η δ t
12
σ˙
()= ()+ ()
o
[ ]
1
+ σ = Gγ o δ()
t
τ
12 12
− t/
τ
σ t γ Ge U t
12
()= ()
o
σ t G t γ U t
12
()= () ()
Gt Ge t − /τ ()=
()=
∑
Gt G e
i=
1
(9.4-19)
(9.4-20)
(9.4-21)
Creep compliance and relaxation modulus for several simple viscoelastic
models are given in Table 9.4-1. The differential equations governing the
various models are also given in this table.
N
i
o
−t/τ
i