1
Chapter 2 Problems
2.1 Consider a Kelvin-Voigt solid and a linear spring in series:
Figure-Problem 2.1
Show that the constitutive equation implied by this mechanical analog has the same form
as that for a Maxwell element and spring in parallel. In other words show that
!
p1˙
“ + p0“=q1˙
# + q0#
in which the constants satisfy the relation
!
q1
p1
>q0
p0
.
Using a physical argument based on the response of the springs and viscous damper in
the mechanical analog to a step change in force or elongation, deduce the initial condition
!
p1“(0+)=q1#(0+)
.
SOLUTION
(a) Geometry
“=“K+“S
!
2
“ “
E + µD
( )
F
S+“
E FK=“
E “ “
E + µD
( )
#
This has the form
po“+p1˙
“ = qo#+q1˙
#
!
Physical argument for the initial condition
“0+
( )
=“K0+
( )
+“S0+
( )
=“S0+
( )
because
“K0+
( )
=0
!
!
3
4
(c2)
F2=E2+ µ2D
( )
“2
E1+ µ2D
( )
E3+ µ3D
( )
LEN+ µND
( )
M
5
This implies
J(t) =“(t)
F=1
Eo
+1
E1
1#e#t$1
( )
+L+1
EN
1#e#t$N
( )
!
!
!
complicated.