5.5 Consider the strain history shown in Figure-Problem 5.5. Sketch the stress
vs. time plot and the corresponding stress vs. strain plot. You may use results from the
notes, but state the appropriate page and/or equation number.
Figure-Problem 5.5
SOLUTION
The strain history is the superposition of a constant strain rate and recovery histories and
the same one shifted by
!
2T
1
, as shown below.
!
!
The superposition of these two stress histories is
The stress vs. strain plot for this history is
!
!
5.6 Consider a strain history of the form shown in Figure-Problem 5.3.
(a) determine an expression for σ(t) in terms of a general stress relaxation
function G(t).
(b) Discuss the expected plot of σ vs. t for large and small values of α. Does the
stress always increase with time.
(c) Determine a condition on α so that
!
˙
“ (0) >0
. It expressed in terms of
!
G(0), ˙
G (0)
.
(d) Show that in order that
!
˙
“ (t) >0, 0 #t
, the stress relaxation function G(t)
must satisfy
!
d
dt (e“t#oG(t)) >0, t $0.
SOLUTION
(a) The strain history is described by
“(s) =“o+#s
. The stress history is
For
!
“
sufficiently small, the stress history is
For
!
“
sufficiently large, the stress history is
“(t) =#oG(t) +$G(s)ds
0
t
%
!
!
if
!
˙
“ (0) >0
, then
!
“>#$o
˙
G (0)
G(0)
(d) Suppose
˙
“ (t) =#o˙
G (t) +$G(t) >0
Divide through by
“o
which is assumed to be positive, as shown in Figure-Problem 5.3.
Multiply through by
e“t#o
which is also positive. The inequality becomes