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5.12 The purpose of this problem is to compare the 'stress - strain' plots obtained
under constant strain and constant stress rate conditions. With this in mind, consider
plotting the following graphs on the same set of axes:
For a test with constant strain rate
!
"
, plot
!
" #G(0)
vs.
!
" #
For a test with constant stress rate
!
"
, plot
!
" #
vs.
!
" # J(0)
(a) Show that time is the common dimension of each of these quantities
(b) Show that
!
"
#G(0) =G(x)
G(0)0
$ #
%dx
!
"
#J(0) =J(x)
J(0)0
$ #
%dx
(c) In Problem 4.1, it was shown that
!
G(t) =100 1+9e"t 2
( )
!
J(t) =1
1000 10 "9e"t 20
( )
are a stress relaxation function and its corresponding creep compliance. Use these to plot
graphs of the relations in part (b). Indicate which plot is for constant strain rate conditions
and which is for constant stress rate conditions.
SOLUTION
Since
"
is a strain rate,
dimension(")=dim(")=1
T
Since
"
is a stress rate,
dim(")=F
1
T
!
"(t) =#G(x)dx
0
t
$
!
"(t) =#t
Divide the integral by
"
, replace time using
"(t) =#t
and divide by
G(0)
to get
!
!
!
(c)
G(t) =100 1+9e"t 2
( )
G(0) =1000
G(t)
G(0) =1
10 +9
10 e"t 2
#
$
% &
'
(
!
&
)
The normalized stress -strain graph at constant stress rate lies above the normalized stress -strain
5.13 A linear viscoelastic material, solid or fluid, is subjected to the stress
history shown in Figure-Problem 5.13, in which the stress increases at a constant rate to
the value
!
"o
at time
!
T*
, and then remains at the value
!
"o
.
Let
!
ˆ
z
denote a time measured from the time
!
T*
when the stress becomes constant, as
shown. Determine an expression for the strain in terms of the time
!
ˆ
z
.
Determine the limiting value for the strain as the rise time
!
T*
goes to zero.
Figure-Problem 5.13
SOLUTION
The stress history can be written as
"(s) ="o
T*s
0"s"T*
="o
T*"s
!
!
="o
0
T*
$
T*"t
Consider the limit as
!
T*"0
.
!
ˆ
x "ˆ
z
and the strain becomes
!
"(ˆ
z ) =#oJ(ˆ
z )
