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 Z 3

0

sin ut

ðA8

:1Þ

For the Maxwell element:

de
dt

Z

1

E

ds

dt

C

s

l

E

ðA8

:2Þ

Differentiating Equation A8.1:

de
dt

Z u3

0

cos ut

ðA8

:3Þ

Rearranging Equation A8.2:

ds

dt

C

s

l

Z ue

0

E cos ut

ðA8

:4Þ

This is a simple linear differential equation of the form

dy
dx

C

Py Z Q

The general solution for such an equation, when P and Q are functions of x only, is

y expðjÞ Z

ð

expðjÞQdx C C

; j Z

ð

Pdx

ðA8

:5Þ

For Equation A8.4, the analogy is

j Z

t

l

ðA8

:6Þ

A8-1

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2006 by Taylor & Francis Group, LLC

s

exp

t

l

Z ue

0

E

ð

exp

t

l

cos ut dt C C

ðA8

:7Þ

s

exp

t

l

Z

ue

0

El

1 C u

2

l

2

ðcos ut C ul sin utÞexp

t

l

C

C

ðA8

:8Þ

or

s Z

ul

1 C u

2

l

2

e

0

Eðcos ut C ul sin utÞ C C exp

K

t

l

ðA8

:9Þ

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 define an angel d by

tan d Z

1

ul

Z

sin d

cos d

and sin d Z

1

ð1 C u

2

l

2

Þ

1

=2

ðA8

:10Þ

Then, making use of trigonometric identities:

cos ut C ul sin ut Z

cos utðsin dÞ

sin d

C

sin utðcos dÞ

sin d

ðA8

:11Þ

Z

sinðut C dÞ

sin d

ðA8

:12Þ

Z ð1 C u

2

l

2

Þ

1

=2

sinðut C dÞ

ðA8

:13Þ

Finally, combining Equation. A8.13 and Equation A8.9 with the transient term dropped, one arrives at

s Z

ul

ð1 C u

2

l

2

Þ

1

=2

e

0

E sinðut C dÞ

ðA8

:14Þ

Complex Notation

Starting with a complex strain, the real part of which is the actual strain:

e

Z e

0

expðiutÞ

ðA8

:15Þ

The motion of the Maxwell element, in terms of a complex stress and strain, is

de

dt

Z

1

E

ds

dt

C

s

l

E

ðA8

:16Þ

Differentiating Equation A8.15:

de

dt

Z iue

0

expðiutÞ

ðA8

:17Þ

A8-2

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2006 by Taylor & Francis Group, LLC

Rearranging Equation A8.16 and Equation A8.17:

ds

dt

C

s

l

Z E

de

dt

Z iue

0

E expðiutÞ Z Q

ðA8

:18Þ

As in Equation A8.5, the general solution is

s

exp

t

l

Z

ð

exp

t

l

Qdt C C

ðA8

:19Þ

Ð

expð

t

l

ÞQdt Z iu3

0

E

ð

expðiut C

t

l

Þdt

Z

iu3

0

E expðiut C t

=lÞ

iu C 1

=l

ðA8

:20Þ

Substitution and rearrangement yields

s

Z

iu3

0

l

E exp ðiutÞ

iul C 1

C

C exp K

t

l

ðA8

:21Þ

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 C iule

E

1 C u

2

l

2

ðA3

:22Þ

Rearranging gives

s

e

Z

Eu

2

l

2

1 C u

2

l

2

C

iðEulÞ

1 C u

2

l

2

ðA3

:23Þ

The definition of complex E* is

E

Z E

0

C

iE

00

Z

s

e

ðA3

:24Þ

Comparing Equation A8.23 and Equation A8.24 one concludes that

E

0

Z

Eu

2

l

2

1 C u

2

l

2

and E

00

Eul

1 C u

2

l

2

ðA3

:25Þ

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

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2006 by Taylor & Francis Group, LLC