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4.15 Consider a linear viscoelastic material for which
!
G(t)
reaches
!
G"
at a
finite time
!
t*
, as shown in Figure-Problem 4.15a. Suppose that two strain histories,
!
"1(t)
and
!
"2(t)
, coincide after some time
!
t1
, as shown in Figure-Problem 4.15b. Show that
after some finite time, the difference in their stress histories,
!
"1(t) # "2(t)
, becomes
zero. Estimate this time.
Figure-Problem 4.15a Figure-Problem 4.15b
SOLUTION
Since the strain histories are prescribed, it is convenient to express the stress in terms of
the strain in the form
"(t) =G(0)#(t) +#(s) ˙
G (t $s)ds
0
t
%
For
!
t>t1
,
"(1)(t) # "(2) (t) =0
!
The graphs of the factors in the integrand are shown below
As t increases,
˙
G (t "s)
slides to the right. The values of factor
˙
G (t "s)
in the integrand of
!
!
!
!
!
4.16 Consider a linear viscoelastic solid whose creep function becomes constant
after a finite time
!
T*
.
(a) Sketch a graph of
!
˙
J (t)
(b) Consider stress histories
!
"1(t)
and
!
"2(t)
which coincide after time
!
To
.
When will the corresponding strain histories coincide?
SOLUTION
(a)
(b)
Since the stress is prescribed, the most convenient form of the constitutive equation is
"(t) =#(t)J(0) +#(s)˙
J (t $s)ds
0
t
%
!
For times
t>To
, the stresses coincide and the above reduces to
!
!
!
!
4.17 In a step strain test, the stress relaxes during a finite time interval
!
T
1
. Now,
let the strain history be arbitrary during a time interval
!
T2
, after which the strain is zero.
Using the constitutive equation in the form
!
"(t) =G(0)#(t) +˙
G
0
t
$(s)#(t %s)ds
,
determine the time it takes for the stress to reduce to zero.
SOLUTION
Graphs for
!
G(s) =G(")
and
!
˙
G (s) =0
for
!
t"T
1
are shown in the figure.
The expression for the stress becomes
!
"(t) =G(0)#(t) +˙
G
T1
$(s)#(t %s)ds
(1)
!
!
!
!
obtained by reversing graph of
!
"(s)
for
!
0"s"t
.
Since
"(s) =0
for
T2"s"t
, then
"(t #s) =0
for
T2"t#s"t
. It follows that
t"s#t
!
!
!
t"T2#T
1
, or when
!
t"T
1+T2
. Since
!
"(t) =0
for
!
t"T2
, the stress given by (1) is zero
when
!
t"T
1+T2
.
