CONTINUUM MECHANICS for ENGINEERS

FIGURE 9.7
Graphic representation of the unit step function (often called the Heaviside step function).
which is shown by the diagram in Figure 9.7. If the creep loading is applied
at time t = 0, the stress is written as
σ = σ ()
(9.4-2)
By inserting this stress into the constitutive equation for a Kelvin material
model, Eq 9.3-6, the resulting differential equation
may be integrated to yield the creep response
12
o Ut
σ Ut ()= Gγ
+ η γ˙
o
γ ()= t σ 1−e
12
12 12
− t/
τ
o(
)
Ut
G
(9.4-3)
(9.4-4)
where e is the base of the natural logarithm system. It is interesting to note
that as t → ∞, the strain approaches a terminal value of σo/G. Also, when t =
0, the strain rate equals σo/η and if the creep were to continue at this rate
it would reach its terminal value at time t = τ. For this reason, τ is called the
retardation time. Eq 9.4-4, as well as any creep response, may always be
written in the general form
˙ γ 12
γ t J t σ U t
12
()= () ()
(9.4-5)
in which J(t) is called the creep function. Thus, for the Kelvin solid, the creep
function is seen to be
e
Jt ()=
G
− 1
−t/
τ
o
= J 1−e
()
−t/
τ
( )
(9.4-6)