4.18 A linear viscoelastic test specimen is subjected to a strain history
!
“1(t)
. An identical
specimen is subjected to a strain history
!
“2(t)
which differs from
!
“1(t)
by a disturbance of
finite duration
!
t2“t1
, that is,
!
“2(t) =“1(t) +ˆ
“ (t)
!
ˆ
“ (t) =
0, t <t1
arbitraty,t1#t#t2
0, t2<t
$
%
&
‘
&
Consider the effect of the disturbance on the difference in stress histories,
!
“2(t) # “1(t)
,
as follows.
(a) Set up an expression for
!
“2(t) # “1(t)
for any time
!
t>t2
. Use a general
stress relaxation function.
(b) Let
!
G(t) =G“+Go#G“
[ ]
e#t$R
. Show that
!
“2(t) # “1(t)
for times
!
t>t2
can be expressed in terms of
!
“2(t2)# “1(t2)
.
(c) Using the relation in part (b), estimate how long it takes for the effect of the
disturbance to become negligible.
SOLUTION
(a)
“1(t) =G(0)#1(t) +˙
G (t $s)
0
t
%#1(s)ds
“2(t) =G(0)#2(t) +˙
G (t $s)
0
t
%#2(s)ds
!
!
!
!
!
!
“2(t) # “1(t) =˙
G (t #s)
t1
t2
$ˆ
% (s)ds
(b)
G(t) =G“+Go#G“
[ ]
e#t$R
!
(c) The stress at time t will be 0.01 of the stress at time
t=t2
when
e“t“t2
( )
#R=0.01
or
“t“t2
( )
#R=ln0.01 =“4.61
!
4.19 A linear viscoelastic solid has a creep function which becomes constant
after a finite time
!
T*
. The solid is to be subjected to a specified stress history
!
“1(t)
.
However, the specified stress history is altered by a disturbance
!
ˆ
“ (t)
of finite duration
!
To
which occurs at time
!
T
1
.
!
ˆ
“ (t) =
0, t <T1
arbitrary,
0, T1+To<t
#
$
%
&
%
T1‘t‘T1+To
.
Let
!
“1(t)
be the strain history corresponding to stress history
!
“1(t)
. Discuss the effect of
the disturbance on the deviation from strain history
!
“1(t)
, for times
!
t>T
1+To
.
SOLUTION
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
%
!
Subtract these two relations in order to express the difference in strain in terms of the
difference in stress
!
“2(t) # “1(t) =$2(t) # $1(t)
[ ]
J(0) +$2(s) # $1(s)
[ ]
˙
J (t #s)ds
0
t
%
!
“2(t) # “1(t) =ˆ
$ (s)˙
J (t #s)ds
T1
T1+To
%
.
!
!
!
!
!
in the interval
!
0“s“T
1+To
and the integrand is therefore zero. The difference in
strains becomes zero for times
!
t“T
1+To+T*
.