CONTINUUM MECHANICS for ENGINEERS

where the constant J, the reciprocal of the shear modulus G, is called the
shear compliance. As a general rule, the creep function of any viscoelastic
model is the sum of the creep functions of its series-connected units. Thus,
for the standard linear solid of Figure 9.5a,
−t/τ
Jt ()= J( −e)+
J
(9.4-7)
1 1
2
and for the generalized Kelvin model of Figure 9.6a
(9.4-8)
With respect to creep loading for the Maxwell model, Eq 9.4-2 is substituted
into the constitutive equation, Eq 9.3-7, resulting in the differential
equation
(9.4-9)
where δ(t) is the delta function, the time derivative of the unit step function.
In general,
and is defined by the equations
from which it may be shown that
N
Jt ()= ∑ Ji −e
1
i=
1
(9.4-10)
(9.4-11a)
(9.4-11b)
(9.4-12)
for any continuous function, f(t). Accordingly, Eq 9.4-9 integrates to yield the
Maxwell creep response as
from which the Maxwell creep function is
−t/τ
i ( )
γ σ δ
12 σ
η
=
() t Ut
o + o
G
dU t− t
δ( t−t1)= dt
()
( )
δ t t t t
− ( )= 0, ≠
+
1 1
t1
δ( t−t ) dt=
∫ 1 1
−
t
1
t
f( t′ ) δ( t′−t) dt′ = f( t ) U( t − t ) t > t
∫ 1 1 1 for
1
−∞
⎛ t ⎞
γ 12()=
t σoJ
1+
Ut
⎝ τ ⎠
⎛ t ⎞
Jt ()= J 1+ ⎝ τ
⎠
1
()
(9.4-13)
(9.4-14)