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Problems
5.1 Consider the strain history shown in Figure-Problem 5.1.
(a) Sketch the corresponding stress - time plot.
(b) Sketch the corresponding stress - strain plot. Indicate points corresponding to
times t =
!
T*, 2T*, 3T*, 4 T*
and as
!
t" #
.
You may use results from the notes, but state the appropriate page and/or equation
number.
Figure-Problem 5.1
SOLUTION
(a) Decompose this strain history into two strain histories of the form in Figure 5.8
The stress history for strain history on the left is shown in Figure 5.11. By non-aging and
scaling, the stress history for the strain history on the right is obtained by shifting the first
!
The total history is the superposition of the two histories shown above, giving
Note that
!
"3T*
( )
>"T*
( )
!
"4T*
( )
<"2T*
( )
(b) The stress-strain plot is
5.2 Figure-Problem 5.2 shows two strain histories which reach the same strain
!
"o
at
the same time
!
T
1+T2
, a 'fast- slow' history and a 'slow - fast' history.
(a) For each strain history, write an expression for the stress at time
!
T
1+T2
.
(b) Using the expressions developed in part (a), and the properties of the stress relaxation
function, determine which strain history produces the greatest stress at time
!
T
1+T2
.
(c) Consider the corresponding strain increments for the histories, as indicated in the
figure. Show that the result in part (b) can also be obtained by considering the response to
such strain increments, as used in the derivation of the constitutive equation.
Figure-Problem 5.2
SOLUTION
(a) The figure shows the notation
!
fast-slow stress at time
T
1+T2
is
"(FS)(T
1+T2)=G(T
1+T2#s)$1
0
T1
%ds +G(T
1+T2#s)$2
T1
T1+T2
%ds
!
!
Let
!
x=T
1+T2"s
in both integrals and change variables of integration
"(FS)(T
1+T2)=#1G(x)
T2
T1+T2
$dx +#2G(x)
0
T2
$dx
!
!
(b) The figures show corresponding strain increments of the FS and SF histories
strain increment strain increment
fast-slow strain history slow-fast strain history
stress increment
fast-slow strain history
stress increment
slow-fast strain history
The magnitudes of the strain increments are equal. The stress response increment
!
d"(FS)(t)
starts earlier and relaxes over a longer time interval than
!
d"(SF ) (t)
. Thus,
!
d"(SF ) (T
1+T2)>d"(FS) (T
1+T2)
By superposition, get the same results in (b)
