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Harmonic Motion of A Maxwell Model: Trigonometric Notation

This document derives the stress response of a Maxwell model to a sinusoidal strain input. It first uses trigonometric notation to show that the stress is a sinusoid with the same frequency but a phase shift. It then uses complex notation to express the stress and strain as complex quantities, allowing the stress to be written in terms of the storage and loss moduli. The storage modulus represents energy storage during periodic deformation, while the loss modulus represents energy dissipation.

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AmirAmiri
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92 views4 pages

Harmonic Motion of A Maxwell Model: Trigonometric Notation

This document derives the stress response of a Maxwell model to a sinusoidal strain input. It first uses trigonometric notation to show that the stress is a sinusoid with the same frequency but a phase shift. It then uses complex notation to express the stress and strain as complex quantities, allowing the stress to be written in terms of the storage and loss moduli. The storage modulus represents energy storage during periodic deformation, while the loss modulus represents energy dissipation.

Uploaded by

AmirAmiri
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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A8

Harmonic Motion
of a Maxwell Model
Trigonometric Notation
Starting with a sinusoidal input of strain in a Maxwell element (see Chapter 3), we derive the resulting
sinusoidal stress. First we let the strain e be a function of a maximum or peak strain e
0
and time t with a
frequency u:
e Z3
0
sin ut A8:1
For the Maxwell element:
de
dt
Z
1
E
ds
dt
C
s
lE
A8:2
Differentiating Equation A8.1:
de
dt
Zu3
0
cos ut A8:3
Rearranging Equation A8.2:
ds
dt
C
s
l
Zue
0
E cos ut A8:4
This is a simple linear differential equation of the form
dy
dx
CPy ZQ
The general solution for such an equation, when P and Q are functions of x only, is
y expj Z

expjQdx CC; j Z

Pdx A8:5
For Equation A8.4, the analogy is
j Z
t
l
A8:6
A8-1
q 2006 by Taylor & Francis Group, LLC
s exp
t
l

Zue
0
E

exp
t
l

cos ut dt CC A8:7
s exp
t
l

Z
ue
0
El
1 Cu
2
l
2
cos ut Cul sin utexp
t
l

CC A8:8
or
s Z
ul
1 Cu
2
l
2
e
0
Ecos ut Cul sin ut CC exp
Kt
l

A8:9
The second term on the right is a transient one which drops out in the desired steady-state solution
for t/lOO1.
Let us now dene an angel d by
tan d Z
1
ul
Z
sin d
cos d
and sin d Z
1
1 Cu
2
l
2

1=2
A8:10
Then, making use of trigonometric identities:
cos ut Cul sin ut Z
cos utsin d
sin d
C
sin utcos d
sin d
A8:11
Z
sinut Cd
sin d
A8:12
Z1 Cu
2
l
2

1=2
sinut Cd A8:13
Finally, combining Equation. A8.13 and Equation A8.9 with the transient term dropped, one arrives at
s Z
ul
1 Cu
2
l
2

1=2
e
0
E sinut Cd A8:14
Complex Notation
Starting with a complex strain, the real part of which is the actual strain:
e

Ze
0
expiut A8:15
The motion of the Maxwell element, in terms of a complex stress and strain, is
de

dt
Z
1
E
ds

dt
C
s

lE
A8:16
Differentiating Equation A8.15:
de

dt
Ziue
0
expiut A8:17
A8-2 Plastics Technology Handbook
q 2006 by Taylor & Francis Group, LLC
Rearranging Equation A8.16 and Equation A8.17:
ds

dt
C
s

l
ZE
de

dt
Ziue
0
E expiut ZQ A8:18
As in Equation A8.5, the general solution is
s

exp
t
l

Z

exp
t
l

Qdt CC A8:19

exp
t
l
Qdt Ziu3
0
E

expiut C
t
l
dt
Z
iu3
0
E expiut Ct=l
iuC1=l
A8:20
Substitution and rearrangement yields
s

Z
iu3
0
lE exp iut
iul C1
CC exp K
t
l

A8:21
Once again, the second term on the right-hand side is a transient term that drops out at t/lOO1.
Multiplying both numerator and denominator by 1Kiul and substituting e* for its equivalent, e
0
exp(iut) gives
s

Z
u
2
l
2
e

E Ciule

E
1 Cu
2
l
2
A3:22
Rearranging gives
s

Z
Eu
2
l
2
1 Cu
2
l
2
C
iEul
1 Cu
2
l
2
A3:23
The denition of complex E* is
E

ZE
0
CiE
00
Z
s

A3:24
Comparing Equation A8.23 and Equation A8.24 one concludes that
E
0
Z
Eu
2
l
2
1 Cu
2
l
2
and E
00
Eul
1 Cu
2
l
2
A3:25
The dynamic modulus E
0
, which is the real component of E*, is associated with energy storage and release
in the periodic deformation and is therefore called the storage modulus. The imaginary part of the
modulus, E
00
, is associated with viscous energy dissipation and is called the loss modulus (see Chapter 3
for more details).
Harmonic Motion of a Maxwell Model A8-3
q 2006 by Taylor & Francis Group, LLC

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