;‘

W V
flaw
t. ‘é .e
.

 

 

4v 1
N" Kym‘ 1"“
11v W‘sr -.
:3 33.211

 

; ‘5
him" ,

asset

V: ‘
t. .
{1"1“v5‘;‘51
wm {MP

4:
Imp, .,
r" a1»!-
' “rm”. 4
€'- H 1 . ;
Btu" » 3 a
. 4(7
‘v’I' "‘3‘ n-g’fuu
3*-¢"4u’."‘r";.-\ “
w ‘ - 'h-,(;, .45.,
nun-J J"
.« rm .
. nrr' a:."" 1""?
. . ‘ .7 :9“

t

A. "“3 : ggbv,

4,
‘_ 7- 1.. -
e

(be

H' ,
m,
,..

4

3‘
‘3
u.
. 3
3‘
§

I. '4
u
- o

if! 1 1‘:

a. 23:52:

,,
,., -.,..‘,

..‘ “.1” ..

3‘:
5-H
WWI-:1." .
1;...»

554;"

l

44.31;“,

1
0;“: wt.
.. -- , .
“.1 «Inn-4y
'r _ 1‘- s1

'1': .

. .- W
.1333???"
:

LU. (
"I"
¢

 

..
.4.
“f,

m
r

V.
{‘

 

 

 

LIBRARY
Michigan State
University

 

 

 

This is to certify that the

thesis entitled

CHARACTERIZATION OF COMPOSITES BY INTERNAL FRICTION

AND ELASTIC MODULUS MEASUREMENTS
presented by

WON JAE LEE

has been accepted towards fulfillment
of the requirements for

MASTERS MATERIALS SCIENCE

 

 

degree in

%$\CM&

E. D. CASE

Major professor

 

Date Feb 29; 1788

0-7639 MS U is an Affirmative Action/Equal Opportunity Institution

 

 

 

 

MSU

LIBRARIES
-:_.

 

 

RETURNING MATERIALS:
Place in book drop to
remove this checkout from
your record. FINES will
be charged if book is
returned after the date
stamped below.

 

 

 

 

 

 

CHARACTERIZATION OF COMPOSITES BY INTERNAL
FRICTION AND ELASTIC MODULUS MEASUREMENTS

By
won.Jae Lee

A.TEESIS

Submitted to

lichigan State university

in partial fulfillnent of the require-ants

for the degree of

HASTER.0P SCIENCE

Depart-eat of’uetallurgy, Mechanics and Material Science

1988

ABSTRACT

CHARACTERIZATION OF COMPOSITES BY INTERNAL
FRICTION AND ELASTIC MODULUS MEASUREMENTS
BY
Won Jae Lee

Internal friction and Young's modulus were studied on
polymer composites and Sic whisker/alumina composites using
the sonic resonance technique. Polymer composites
reinforced with unidirectional fibers exhibit modulus and
internal friction anisotropy. Usually, elastic modulus and
internal friction showed a inverse relationship. The
internal friction increased with decreasing interfacial
shear strength. For the sic whisker reinforced alumina
composites, microcracks introduced by water quenching were
detected by internal friction and Young's modulus
measurements. As thermal shock progresses, Young's modulus
decreases and internal friction increases. A quantitative
relation between the modulus and internal friction is‘
developed. Elevated Young's modulus and internal friction
also were investigated for the Sic whisker reinfored

alumina composite.

.ACKNONEEDGEMENTS

First and formost, I would like to thank my adviser,
Professor E. D. Case for his help and support over the
course of this investigation. Special thanks are given to
D. Bray, Alcoa, for providing the Sic whisker/alumina
composites. Special thanks are also given to M. Rich and
Professor L. Drzal for providing the polymer composites.

I would like also acknowledge the support of the
Research Excellence Economic Development Fund, State of
Michigan, and of the National Science Foundation through
grant number MSM-8706915.

I would like to thank my mother and father for their
endless love and help over my entire life. Also I would

like to thank my wife Haesun for her endless love.

ii

TABLE OF CONTENTS

Page

LIST or TABLES ------------------------------------- iv

LIST or FIGURES ------------------------------------ v

1. INTRODUCTION ----------------------------------- 1

2. BACKGROUND ------------------------------------- 4
2-1. Fundamentals of Methods for Measuring

Internal Friction ----------------------------- 4

3. EXPERIMENTAL PROCEDURE ------------------------- 13

3-1. Materials ------------------------------------- 13

a) Polymer Composites -------------------------- 13

b) Whisker Reinforced Alumina Composites ------- 16

3-2. Room Temperature Elasticity Measurement ------- 16

3-3. Room Temperature Internal Friction Measurement-- 25

3-4. Elevated Temperature Elastic Moduli and

Internal Friction ----------------------------- 29
3-5. Dynamic Mechanical Tests ---------------------- 32
3-6. Thermal Shock Fatigue Tests ------------------- 32
4. RESULTS AND DISCUSSION ------------------------- 36
4-1. Polymer Composites ---------------------------- 36
4-2. SiC Whisker Reinforced Alumina Composites ----- 46

4-2-1. Room Temperature Modulus and Internal Friction 46

4-2-2. Elevated Temperature Modulus and Internal

Friction ------------------------------------ 43
4-2-3. Thermal Fatigue Test ------------------------ 55
5. CONCLUSIONS ------------------------------------ 7o
6. REFERENCES ------------------------------------- 72

Table

2.
3.

4.

10.

11.

LIST OF TABLES

Dimensions, Mass, and Mass Density of Polymer
Composites ------------------------------------

Characteristics of Polymer Composites ---------
Dimensions and Mass of Alumina/Sic Composites -

Density and Mechanical Properties of SiC Whisker
Reinforced Alumina Composites -----------------

Position of the Vibrational Modes of a Prismatic
Specimen --------------------------------------

Room Temperature Modulus and Internal Friction
of Polymer Composite --------------------------

DMA Data for Polymer Composite at 35 Degrees
CelCius.(Modulus and Tan 5) -------------------

Modulus and Internal Friction of Alumina/Sic
Composite Specimens ---------------------------

Relative Changes in Young's Modulus and Internal
Friction for the Thermal Shocked Specimen
(Sic Whisker/Alumina Composite) ---------------

Results of the Non-linear Regression Analysis
for the Sic Whisker/Alumina Composite ---------

Thermal Fatigue Constants for the 81C Whisker
Alumina Matrix Composite ----------------------

iv

Page

14
15
17

18

23

37

38

47

61

63

65

LIST OF FIGURES

Figure Caption

10.

11.

12.

13.

14.

The free deca of natural vibrations of an
anelastic sol d. ------------------------------

Lorentzian form for the resonance peak in forced
vibration. ------------------------------------

Schematic illustration of ultrasonic pulse
method. ---------------------------------------

The attenuation of an ultrasonic pulse as it
propagates through an anelastic meadium. ------

Outline of apparatus for measuring the elastic
modulus. --------------------------------------

Illustration of method of coupling accoustic
energy. ---------------------------------------

Internal friction apparatus diagram. ----------

Schematic diagram of the elevated temperature
elasticity and internal friction measurement
apparatus. ------------------------------------

Schematic diagram of the thermal shock test
apparatus. ------------------------------------

Variation of elastic moduli of a fiber composite
and a monolithic material with the angle of
reinforcement. --------------------------------

Internal friction of polymer composite as a
function of interfacial shear strength. -------

Measured internal friction as a function of
suspension position (x/l *100) for the polymer
composite specimen AS4-longitudinal. ----------

Measured internal friction as a function of
suspension position (x/l *100) for the polymer
composite specimen AS4C-longitudina1. ---------

Measured internal friction as a function of
suspension position (x/l *100) for the polymer
composite specimen AU4-longitudinal. ----------

page

10

11

19

20

26

3O

33

41

42

43

44

45

15.

16.

17.

18.

19.

20a.

20b.

21a.

21b.

22.

Measured internal friction as a function of
suspension position (x/l *100) for the .
alumina/Sic composite specimen HP158. ---------

Measured internal friction as a function of
suspension position (x/l *100) for the
alumina/Sic composite specimen HP160. ---------

Measured internal friction as a function of
suspension position (x/l *100) for the
alumina/Sic composite specimen HP171. ---------

Young’s modulus of the alumina/Sic composite
spec men HP171 as a function of temperature. --

Internal friction of the alumina/Sic composite
specimen HP171 as a function of temperature. --

Young’s modulus of the alumina/Sic composite
spec m

(AT = 310 degrees Celcius), and HP158 (AT = 380
degrees Celcius) as a function of thermal shock
cycle. ---------------------------------------

Young's modulus of the alumina/Sic composites
HP158, MPIGO, and HP171 versus thermal shock
number. ‘Data in this figure is a subset
(namely, the first twenty thermal shocks) of
the presented in figure 20a. Note that even
for n small, equation (16) fits the data

well. ----------------------------------------

Internal friction of the alumina/Sic composite

specimens HP160 (AT = 270 degrees Celcius), HP171

(AT = 310 degrees Celcius), and HP158 (AT = 380
degrees Celcius) as a function of thermal shock

cycle. ---------------------------------------

Internal fricyion of the alumina/Sic composites
HP158, HP160, and HP171 versus thermal shock
number. Data in this figure is a subset
(namely, the first twenty thermal shocks) of
the presented in figure 21a. Note that even
for n small, equation (17) fits the data

well. ----------------------------------------

Internal friction change, AQ, as a function of

crack damage parameter, c, for the SiC whisker/
alumina composite. ----------------------------

vi

ens HP160 (AT = 270 degrees Celcius), HP171

49

50

51

52

54

56

57

58

59

69

1. INTRODUCTION

Most ceramic materials are susceptible to failure
under mechanical or thermal loadings. Fiber and whisker
reinforcement of a number of ceramic matrix composite
materials have provided drastic improvements in the
strength and fracture properties, especially at elevated
temperature. Tai-il Mah et al. [1] recently reviewed
developments in fiber-reinforced ceramic composites.
Besides the strengthening and toughening of reinforced
ceramic composites, the strain-to-failure of some
reinforced ceramic composites can be increased by a factor
of ten times or more as compared to that of the
unreinforced ceramics [2,3].

In composites, there exists either chemical or
physical bonds between the fiber and the matrix. For
fiber reinforced ceramic composites, the interfacial
bonding between the fiber and the matrix can affect the
strain-to-failure curve. Bonding between the fiber and
matrix that is too strong may cause brittle fracture, but
proper bonding can cause a considerable increase in strain
to failure, by the mechanism of fiber pull out. When the
bonding between fibers and matrix is not sufficiently
strong, the fibers can debond and slip within the matrix
before the composite failure. The mechanism of fiber pull

out results in the relatively high strain [4].

An incomplete understanding of the nature of the
interfacial bonding can be attributed, at least in part,
to the lack of nondestructive techniques to characterize
the interfacial bonding. Disruption of interfacial
bonding occur due to interfacial shear stresses that
result from slippage between the matrix and the fiber.
Thus the relative slip between the fiber and the matrix is
important in assessing the nature of interfacial bonding
in a composite.

It is proposed that the fiber-matrix friction in
ceramic composites be studied nondestructively by internal
friction and elasticity techniques. The increase in
internal friction observed to accompany microcracking in
brittle materials has been explained in terms of rubbing
of opposing faces of the microcrack [5]. As the material
sonically vibrates, the opposite surfaces of the
microcracks rub against each other, resulting in a
frictional impedence (increasing in internal friction). A
similar dependence may exist for the internal friction
changes as a function of the relative slip, or rubbing, of
the matrix and embedded fibers in a composite.

Thus, internal friction and elastic modulus
measurements by sonic methods are nondestructive. In
addition, both internal friction and elastic modulus of
ceramics are thought to be very sensitive to the
microstructural changes [5-10]. For example, the elastic

moduli and internal friction are sensitive to phase

changes [10], microcracking [5,7,9], porosity [8], and
grain boundary sliding [5,6]. Elasticity and internal
friction can thus provide important information on the
interfacial bonding and the microstructural stability of
the ceramic composites.

Internal friction values for numbers of metals,
glasses, and polymers can be found in the literature
[11,12]. However, little data has been reported for the
internal friction of ceramics, especially ceramic
composites.

In the present study, internal friction and elastic
modulus of the fiber-reinforced polymer and whisker-
reinforced ceramic were measured using the flexural
resonance method. The effects of interfacial bonding and
thermal shock upon the internal friction and elastic

modulus of the materials were also investigated.

2.8ACEGROUND

2-1. Fundamentals of Methods for Measuring Internal

Friction

Damping effect within the solid is internal friction.
Anelasticity or internal friction of solids is one of the
important properties which are indirectly related to their
elastic moduli. Let us consider briefly the relationship
among the various measures of internal friction.

Generally, the method of measuring internal friction
of solids can be divided into four categories.[13]:

1. direct observation of stress-strain curve

2. free-vibration method (logarithmic decrement)

3. forced-vibration method (full width half maxima)
4. wave propagation method

The definition of mechanical damping, as the ratio
AE/E between the energy AE dissipated during one stress
cycle and the maximal elastic energy E stored in the
specimen, suggests one direct way for its measurement. If
AB is sufficiently large, a measureable rise in the
specimen temperature may result from energy dissipation.
The stored vibration energy E is determined by the maximum
applied stress. Obviously, this method is restricted to
relatively high stress levels. Another disadvantage

applies to a variation of this method, in which one

measures simultaneously the applied stress and the
resulting strain. Mechanical damping manifests itself as
a stress-strain hysterisis loop, and is proportional to
the area of this loop.

Lower values of the internal friction (order of
lO-‘or larger) are more commonly measured by a resonance
method. The free vibration method which is the oldest and
still most popular method is based on the measurement of
the decay in amplitude of vibrations during "free
vibration." The natural logarithmic of this ratio is
commonly called "logarithmic decrement." Figure 1 shows
the vibration behavior during free decay. It is defined

as follows

5 = [1n(A/B)]/tf (1)
I AE/ZE

where A and B amplitude of the first and the last
vibration considered

frequency of vibration

time interval between the two successive
vibrations

energy loss per cycle

stored vibration energy.

Mg (fl-h
ll

When the damping is small, the internal friction is
expressed by
Q- = 6/: (2)
= [ln(A/B)]/«tf.

‘-- '-'—\
‘0 WM
ENVELOPE or FREE DECAY

AMPLITUDE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

EXCITING FORCE REMOVED

Figure l. The free decay of natural vibrations of an
anelastic solid.

The product, tf, is equal to the number of cycles, N. It
may be measured directly by counting the number of cycles
during a free decay. As the magnitude of internal
friction value decreases, the number of cycles counted
during free decay increases. For values of internal
friction larger than 1x10-8, the relatively low number of
cycles, N, occuring during the free decay may induce some
experimental error. Therefore, this method is effective
with specimen having low internal friction values (order
of 10-‘or higher) and in the case of high frequencies
(hundreds Hz to a few hundred kHz) [14].

The standard resonance method can be used as a forced
vibration method (full width half maxima). The specimen
is forced to vibrate at a certain range of frequencies
with constant drive voltage. When the imposed frequency
is equal to a resonance frequency of the specimen, the
amplitude of vibration reaches a maximum value. If the
impressed frequency deviates from the resonance frequency,
the amplitude of vibration decreases. Figure 2 shows the
resonance peak in the forced vibration. The sharpness of
this peak varies inversely with the internal friction.

The relative sharpness is readily obtained by
relative sharpness = Af/fr

where Af a width of the frequency at half maximum
amplitude

fr = resonance frequency.

 

 

 

‘Figure 2. Lorentzian form for the resonance peak in
forced vibration.

Zener [15] shows that the ratio, Af/fr ,

to the specific loss in the following relation

is related

Af/fr,= 13/2: * AE/E (3)
= J3/a * 6.

Thus the internal friction is expressed as
-1
Q = 6/: = 0.5773Af/fr, (4)

The forced vibration method is most effective with
specimen having values of Af of several cycles/second.
For relatively low damping levels (say Q-1= 10") a
precision of 1 percent for the damping measurement
requires a precision of a 10-. in the frequency reading.
Since the attenuation constant of a stress wave
through a solid can be related to the internal friction,
this provides another method for measuring the internal
friction of a specimen. The wave propagation technique
consists of determining the amplitude of a stress wave
before and after traversing well-known distance. This
technique is useful at high frequencies (above several
MHz). Using the pulse technique, it is possible to
overcome this difficulty. The wave propagation method
employs a pulse or wave packet whose length is small
compared to the specimen length in the direction of
propagation. Figure 3 and figure 4 show the method

lO

TRANSDUCER SPECIMEN

F

I l 1

 

 

 

 

PULSE AT REFLECTED
TIME t PULSE AT
TIME t+At

Figure 3. Schematic illustration of ultrasonic pulse
method (After Norwick and Berry [13]).

ll

 

 

 

 

 

 

 

 

 

\
\
\
U \\ U = UOEXP(-Ax)
\ .
\\
\\\ U2
DISPLACEMENT I
,/ a [A
/
/
/
/ /
/
/ IF At "I
AT TIME 1: AT TIME t+AT

Figure 4. The attenuation of an ultrasonic pulse as it
propagates through an anelastic medium.

12

schematically. A narrow high frequency mechanical pulse
is applied to the specimen, and the internal friction is
derived from the decrease in amplitude of the wave pulse
as it propagates through the specimen. This method may be
suitable for materials having much lower values of
internal friction than 1x10-5[13]. One of the
disadvantages of this technique has been that high-
temperature effects cannot be investigated due to
transducer limitations and difficulties in bonding the
transducer to the specimen at high temperature. However,
this is being overcome with the developement of high-
temperature transducers. Lithium niobate, which can
withstand temperatures above 900 C, may provide the means

to extend the measurements to high temperatures.

3.EXPERIMENTAL PROCEDURE

3-1. Materials
a) Polymer Composites *

Samples were obtained from the Composite Center at
Michigan State University. They were processed in an
autoclave for two hours at 75 C. For reasons unknown,
they contained a large number of voids. The as-received
specimens were initially in the form of scrap plates.
They were cut in the shape of prismatic bars by using a
Beuhler Isomet cutter employing a diamond wheel. The
specimens were held in a grip provided on the diamond
cutter. The grip held the specimen by means of two
adjustable screws. The dimension of the as-cut specimens
is given in table 1. Many characteristics of the samples

used for this research are given in table 2.

 

* The polymer composites were given by M. Rich, Composite
Center.

13

14

Table 1. Dimensions, Mass, and Mass density of Polymer

 

 

Composite
Dimensions Mass Mass density

8

(cm) (9) (9/9m )
A04 (longitudinal) 9.07x2.45x0.26 8.2429 1.45
(transverse) 7.66x2.42x0.27 6.2280 1.42
A84 (longitudinal) 7.97x2.46x0.27 7.6392 1.44
(transverse) 7.66x2.42x0.27 7.1050 1.43
AS4C(1ongitudinal) 7.27x2.4lx0.34 8.4674 1.41
(transverse) 9.68x2.44x0.34 11.5600 1.42

 

15

 

 

Table 2. Characteristics of Polymer Composite

Polymer Interfacial Shear Interfacial

Composite Strength (Kpsi)* Treatment

AU4 5.4 untreated, unsized

A54 9.9 surface treated only

AS4C 11.8 surface treated and coated

with layer of epoxy

 

Matrix used: EPON 828 (DGBEA, by Shell)

Fiber used: Polyacrylonitrile based graphite fiber having

a diameter of 7.7 microns

 

 

* The measurement for this interfacial shear strength was

by the uniaxial tension test.

16

b) Whisker Reinforced Alumina Composites

Whisker reinforced alumina composites used for this
research were produced by Alcoa.* The material consists
of A-16 SG alumina (a super ground calcined alumina),
reinforced with both the Arco and Tokai carbon Sic
whiskers. Processing was carried out using pH
stabilization in a wet procedure. The material was then
dried and hot pressed. The dimension of the as-received
specimens is shown in table 3. Table 4 shows some

properties of the samples used for this research.
3-2. Room Temperature Elasticity Measurement

The sonic resonance technique described by Forster
[16] was used to determine the elastic moduli and internal
friction. Figure 5 shows schematically the experimental
arrangement used. The modulus equipment measures the
resonant frequencies of the specimen. Cotton strings were
used to suspend the prismatic specimen horizontally from a
driver and a phonograph pickup (Figure 6). An electronic
signal produced by a frequency generator (3325A
Synthesizer/Function Generator made by Hewlett-Packard)

 

* The whisker reinforced ceramic composites were obtained
from D. J. Bray, Technical Superviser, Alcoa Technical
Center, Alcoa Center, PA.

17

Table 3. Dimensions and Mass of Alumina/Sic Composites

 

 

3
Dimensions (cm ) Mass (9)
HP158 7.01x0.702x0.297 5.5893
HP160 . 7.01x0.700x0.297 5.6291

HP171 7.00X0.700x0.298 5.6039

 

18

Table 4. Density and Mechanical Properties of SiC Whisker
Reinforced Alumina Composites*

 

HP158 (1800 C) HP160 (1800.C)

A-16SG+20%TOKAMAX A-16SG+20%ARCO

Sic Whiskers SiC Whiskers

HP17l (1800. C)
A-lSSG+20%ARCO

Sic Whiskers

 

Density (g/cms)
Porosity (%)
H20 Abs. (%)
% of theor.
-density
Strength (Kpsi)
(MOR Bars)
HOODED (MPSi)

3.76
.72
.19

98.3

58.79

56.74

.07

99.3

67.39

57.11

.11
98.8

51.25

57.09

 

* The data in this table

came from Alcoa.

l9

 

 

 

 

OSCILLOSCOPE
VOLTMETER
FREQUENCY
SYNTHESIZER
FILTER
AMPLIFIER
DRIVER PICK UP

 

 

l"'""'SPECIMEN""_J

Figure 5. Outline of apparatus for measuring the elastic
modulus (Dynamic resonance method).

20

Driver Pickup
Cotton
‘///,/-Support
Thread

/ A a

 

 

Specimen

Figure 6. Illustration of method of coupling accoustic
energy.

2l

was amplified and converted into mechanical vibrations via
a high power piezoelectric driver transducer (model number
62-1 made by Astatic Corp., Conneaut, Ohio). And it was
allowed to pass through the specimen, and converted from a
mechanical to an electronic signal by the pick up
transducer. This signal was amplified, filtered (4302
Dual 24DB/Octave Filter-Amplifier made by Ithaco, Ithaca,
N.Y.), and passed through a voltmeter (8050A Digital
Multimeter made by Fluke, Everett, WA), and oscilloscope
(V-100A 100MHz Oscilloscope made by Hitachi, Japan).
Resonance frequencies for the polymer composites were
found to be the sharp. The maximum amplitude of the
vibration and the mode of vibration were determined by
probing the specimen with a needle to determine nodal
plane locations. To calculate the elasticity, one must
know the type and the mode of the vibration of the
specimen. A prismatic bar can be excited in a variety of
vibrational modes, including the fundamental flexural
frequencies. Thus if one identifies a resonant frequency,
one must determine which particular mechanical vibrational
mode is being excited before a determination of elastic
modulus can be made. The vibrational mode can be
identified by locating the position of the vibrational
nodes and antinodes along the specimen in which a
standing-wave vibration forms. The positions of the nodal
and antinodal lines on a prismatic bar-shaped specimen

vibrating at a mechanical resonance may be determined by

22

mechanically probing the bar [17].

In this study, the nodal and antinodal positions were
probed using a sewing needle, balanced at various
locations along the length of the resonating bar. When
the needle is on a nodal line, the amplitude of resonant
frequency will change very little. But if the needle is
positioned away from a nodal line, then the needle will
dampen the mechanical vibrations in the specimen, and the
vibrational amplitude, as measured by the pick-up
transducer, will be reduced. Anderson [17] indicates the
position of the nodes as a fraction of the length of the
specimen (Table 5).

The resonant frequency was taken as the average of
five or more readings. This experiment was done at room
temperature.

The resonance frequency data were converted into
moduli values by using the equations of Pickett [18] as
modified by Hasselman [19]. They were computer programmed
by E. D. Case. Shear modulus, G, was calculated from the

equation
c; = P (2L*f/N)2 *R (5)

where an integer (unity for the fundamental mode)
torsional resonance frequency

specimen length

mass density

a Shape factor.

W'UE‘HIZ
II II II II II

23

Table 5. Position of the Vibrational Modes of a Prismatic
Specimen. The Nodal Positions are Expressed as
Function of the Length of the Specimen.

 

Mode of Vibration Flexural Vibration Torsional Vibration

 

Fundamental 0.224 0.500
0.776

First Overtone 0.132 0.250
0.500 0.750

0.868

 

24

For prismatic specimens of rectangular cross section,

R can be approximated by an equation as follows;

21
II

where a a
b:

([l+(b/a)2]*[1+(0.0085N2b2/If )] / {4-2.521

(a/b) [(1-l.991)/(exp(b/a)+l) J] } "’
O.06(Nb/L)1'5 (b/a -1)2 (6)

thickness of specimen
width of specimen.

Young's modulus, E, for a rectangular specimen can be

obtained from.the equation

F!
l

UB’UUH
II II II II II

The shape

where u

2 4 2
— 0.94642PTf L /D (7)

fundamental flexural resonant frequency
length of the specimen

mass density of the material

a shape factor

cross sectional dimension in the direction of
vibration.

factor T is approximated by
1 + 6.58(1+0.0752v +0.8109 yz)(D/L) - 0.868
(D/L)‘ - [8.34(1+o.2023u+2.173u2)(D/L)‘] /

[l+6.338(l+0.148y+l.53u2)(D/L)] (8)

Poisson's ratio.

25

3-3. Room Temperature Internal Friction Measurement

Internal friction (0-1) may be determined in two ways
using Forster’s method [16].

For the polymer composite having a high damping
property, the full width half maxima method is most
effective and accurate. For the full width half maxima
method, the same equipment was used as employed for
measuring elastic modulus. The peak width, Af, can be
measured by finding the two values of frequencies at which
the amplitude of vibration is one-half the value of

o o o -1 a
resonance. The measured internal friction, Qm, is given

by
Qfil= 0.5773*Af/f (4)
where f = the resonance frequency.

The free decay method is more effective with the
specimen having smaller values of internal friction such
as the SiC/Alumina composite. For the free decay method,
two additional items of apparatus (trigger* and counter**)
are required. Figure 7 shows the "internal friction

apparatus" for the free decay method. With the specimen

 

* Constructed by the Biochemistry Electronics Shop,
Michigan State University

** Model 5314A Universal Counter made by Hewlett-Packard

26

 

 

FREQUENCY 08 CI LLOS COPE VOLTMETER UNIVERSAL
SYN THES I ZER COUNTER
TRIGGER FILTER
AMPLIFIER
DRIVER PI CK UP

 

 

 

Figure 7. Internal friction apparatus diagram.

 

27

driven at its fundamental flexural resonant frequency, the
driving signal is turned off by changing the mode from
drive to free decay and the number of cycles, N, between a
fixed amplitude, A, and a smaller fixed amplitude, B, is
counted. The measured internal friction, Qfi: is given by

-1

Qm = ln(A/B)/«N. (5)

The measured internal friction, 05: is composed of
the internal friction of the specimen, OS: and the
internal friction of the apparatus, Q5: The internal
friction of the apparatus includes the effects of the
specimen suspension position.

Wachtman and Tefft [14] have formulated an equation

relating 051 and Q51 to the measured internal friction,

-l
I

ofi’= to's‘+ mist1 (st/1U)2 whim/W)” 1 (10)

-1
where Qs = the internal friction of the specimen
k,= an empirical constant
Qa the internal friction of the apparatus
Y the vertical displacement of the suspension
point from equilibrium
Y'= the vertical displacement of the end of the
specimen.

The Y/Y' term is given by Rayleigh [20] for the transverse

fundamental mode of vibration as

 

28

Y/Y'- [1.018(cosh4.730x/l+cos4.730x/1)-(sinh4.730x/l

+sin4.730x/l)]/2.036 1(11)

where l = the length of the specimen
x = the distance to the suspension point from the
end of the specimen.
These equations were also computer programmed by E. D.
Case.

More than three pairs of suspension positions were
marked on each specimen (Figure 6 illustrates the
suspension of the specimen). The horizontal distance, x,
from the end of the specimen was measured using metric
vernier calipers. The specimen was suspended by cotton
string and internal friction was measured for the each
pair of suspension positions. The measured Q5! were put
into the program to obtain values for Q5: k and OS: The
average of three successive determinations of the internal
friction of the specimen was taken as the value of the
internal friction of the specimen. All of these
determinations of OS1 were made in air, neglecting the

damping effect of air.

 

29

3-4. Elevated Temperature Elastic Moduli and Internal

Friction

In addition to the full width half maxima method and
the free decay method described in the section 3-3, there
are other methods of measuring internal friction at room
temperature. For example, the quartz crystal composite-
oscillator method (ultrasonic pulse method) is capable of
giving very accurate measurements of small values of
internal friction. However, Forster's method is capable
of being extended to elevated temperature by the simple
adjustment of surrounding the specimen with a furnace, as
shown in Figure 8.

Elevated temperature data were obtained by suspending
the specimen horizontally by platinum wire (Ted Pella,
Inc., Tustin, CA) in the hot zone of a electric resistance
furnace. The heated Chamber of the vertical furnace was
14 inches long and 4 inches in diameter. The temperature
was controlled by a digital temperature controller (CN5000
Omega, Engineering Inc., Stamford, CT). The fundamental
flexural resonance frequency was determined over the range
from room temperature to 900 C, with the temperature being
raised by 50°C increments.

The resonance frequency ft at some temperature AT
above room temperature was converted to relative elastic

modulus Et/Eo using the relationship [17]

30

 

 

 

FREQUENCY OSCILLOSCOPE VOLTMETER UNIVERSAL
SYNTHESIZER L l COUNTER
TRIGGER ' FILTER
AMPLIFIER
DRIVER PICK UP

 

furnace

 

 

 

égéél— SPE

 

 

 

N

 

Figure 8. Schematic diagram of the elevated temperature
elasticity and internal friction measurement
apparatus.

3i

Et/E. (ti/f3 mm AT) (12)

where Et' E0 = high temperature and room temperature
elastic modulus

fo = room temperature resonance frequency

a = linear thermal expansion coefficient.

The true internal friction value of the specimen was
not measured at elevated temperature. Only one suspension
point was used for elevated temperature internal friction
measurements. Thus, these values were corrected to the

specimen room temperature corrected internal friction

value, QS1, by the equation (8)
-1 -1 -1
QSt ‘ QS ( QNt / QO ) (13)

-l
where Qst - the measured specimen corrected internal
_, friction at elevated temperature
th = the measured internal friction at elevated
_, temperature
00 = the measured internal friction with platinum
wires in air at room temperature.

The assumption inherent in this equation is that the
observed changes in internal friction as a function of

temperature are due to changes in the internal friction of

the specimen.

 

32

3-5. Dynamic Mechanical Tests

The dynamic mechanical properties of the polymer
specimen were tested in a Du Pont 982 Dynamic Mechanical
Analyzer. The specimen were heated at a rate of 5 °C/min
to 75 °C in air. The tan delta, determined in this
instrument as the ratio between the flexural loss modulus
and the flexural storage modulus, and the flexural storage

modulus were measured as a function of frequency.
3-6. Thermal Shock Fatigue Tests

For the Sic whisker reinforced alumina composite
specimens, thermal shock fatigue tests were performed
using the apparatus shown in Figure 9. The apparatus was
designed to perform one thermal shock per hour. During
one thermal shock, the specimen was suspended in the
middle of the furnace for approximately 45 minutes. After
thermal shock, the specimen remained in the water bath for
15 minutes.

The furnace used in this study was an electric
resistance furnace. The heated chamber of the furnace was
13.5 inches long and 3 inches in diameter. The power
input to the furnace was controlled by a variable
transformer (2PF136 type Powerstat, The Superior Electric
Co., Bristol, Conn). The variable transformer was

adjusted to a predetermined power level and the furnace

33

 

 

 

 

 

 

thermocoup
F

8
o \

c

b

a

*
.._.Jv--.... L___,

water

 

 

 

 

 

 

 

 

 

weight

CC!

 

 

 

 

 

‘ specimen

 

 

 

Figure 9. Schematic diagram of the thermal shock test

apparatus.

34

was allowed to approach thermal equilibrium before
operating the thermal fatigue equipment. Up to two and
one-half hours was required for thermal equilibrium to be
reached. Once thermal equilibrium was reached, the change
in thermocouple output was i 1 millivolt during the entire
thermal shock anneal, which is equivalent to i 2 degrees
Celcius uncertainty. Temperature measurement was made
with a J type thermocouple with its hot junction located
to the side of the sample near its middle.

The plastic container for the water bath was
rectangular with dimensions 24cm x 50cm x 75cm.
Approximately 8 liters of distilled water were used for
the water bath. The temperature of the water bath was
measured by a mercury-in-glass thermometer with a
temperature range of -10 degrees Celsius to 260 degrees
Celcius. The thermometer markings were at 1 degree
Celcius intervals, so temperature measurements were
accurate to within i 0.1 degree Celcius.

After attaining the preselected temperature in the
middle of the furnace, the thermal shock fatige tests were
performed. Rapid thermal quenching was done one time per
hour. The typical time interval for the specimen free
fall between the furnace and the water bath was one
second. An electric timer was used to measure the
dropping time. All specimens were removed from the water
bath after having the desired number of thermal shocks.

Specimens were dried with a paper towel and kept 10 hours

35

at room temperature before performing the elasticity and
internal friction measurements.
The thermal shock damage was examined by measuring

the changes in the elastic modulus and internal friction.

4. RESULTS AND DISCUSSION

4-1. Polymer Composite

Values of the resonant frequency, Young’s modulus,
shear modulus and internal friction at room temperature
are given in Table 6 for the unidirectional fiber
composites included in this study. In Table 7, DMA
(Dynamic Mechanical Analyzer) results are listed.

Unidirectional composites are anisotropic due to
their nature, thus their properties show a marked
differences as a function of direction. The polymer
composites in this study are unidirectional, hence a
modulus anisotropy and a internal friction anisotropy are
expected. The Young's modulus anisotropy,
E(longitudinal)/E(transverse), varied from a value of 11.6
to 12.4. The internal friction anisotropy,
Q-‘(transverse)/Q-1(longitudinal), varied from a value of
7.5 to 8.6.

Although for the present study, internal friction was
always higher in the transverse direction, the Young's
modulus was always higher in the longitudinal direction.
Ledbetter [22] observed similar directional dependences
for the elastic moduli and internal friction of fiber
reinforced polymer composites. According to Ledbetter

[22], longitudinal specimens showed higher Young's modulus

36

37

 

 

Table 6. Modulus and Internal Friction of Polymer
Composite
Flexural Torsional E G Q-1
Frequency Frequency
(Hz) (Hz) (Gpa) (Spa) (10")
AU4 (longitudinal) 2627.5 1927.5 96.73 4.33 30.7
(transverse) 1039.5 2037.5 7.81 4.11 235.8
A84 (longitudinal) 3468.6 2231.3 90.32 4.11 27.7
(transverse) 1074.3 2285.4 7.42 3.70 237.7
AS4C(longitudinal) 5115.0 2292.0 82.46 3.70 25.5
(transverse) 847.9 2215.6 7.09 3.67 190.9

 

38

Table 7. DMA Data for Polymer Composite at 35 Degrees
Celcius (Modulus and Tan 6)

 

 

 

0.1Hz 1.0Hz 10Hz resonance

AS4 (longitudinal) (42.9Hz)
E (Gpa) 42.75 43.15 44.32 49.30
Tan 6 0.00018 ---* 0.00129 0.00642

(transverse) (14.76Hz)
E (Gpa) 5.43 5.43 5.46 5.50
Tan 6 0.01500 0.00704 0.00858 0.00363
AU4 (longitudinal) (41.95Hz)
E 43.16 43.57 44.76 53.35
Tan 6 0.00173 0.00121 0.00642 0.00634

(transverse) (14.75Hz)
E 5.79 5.86 5.90 6.05
Tan 6 0.01124 0.00959 0.00840 0.00592

 

* DMA didn’t read data at this point.

39

and lower internal friction than comparable specimens with
transversely oriented fibers. '

For the fiber reinforced composites included in the
present study, the shear modulus was almost the same for
two different directions (longitudinal and transverse
fiber orientations). Figure 10 schematically illustrates
the elastic moduli of the randomly mixed composite and a
unidirectional composite as a function of fiber
orientation [21].

Ledbetter [22] observed the dynamic Young's modulus
and internal friction of several fibrous cOmposites
(including graphite-epoxy, glass cloth-epoxy, Boron-
Aluminum and Boron-epoxy) at room temperature. The data
obtained from the present study agrees well with the data
obtained by Ledbetter. First, a higher Young's modulus
corresponds to a lower internal friction, and vice versa.
Second, Ledbetter's data also showed a modulus anisotropy,
in that the modulus anisotropy varied from near 1 to 100,
but in the present study a modulus anisotropy varied from
11.6 to 12.4. Ledbetter showed an internal friction
anisotropy varied from 3 to 20. Ledbetter showed wide
modulus anisotropy variation. However, the author didn’t
observe wide modulus anisotropy variation presumably
because of the limited number of materials being measured.

An inverse relationship, between the longitudinal
internal friction of the polymer composite and the

interfacial shear strength, has been Observed [23]. This

40

relationship is shown in figure 11. This suggests that
the degree of the interfacial bonding can simply be
characterized by using the measured value of internal
friction of the polymer composite. Figure 11 shows that
internal friction of a polymer composite (longitudinal)
increases with decreasing the interfacial shear strength.
The measured room temperature internal friction,
(051), as a function of specimen's suspension position is
shown in figures 12-14. As predicted by Wachtman and
Tefft [14], the measured internal friction varied with the
position of the support strings. Figures 12-14 Show that
equation (10) fits the experimental data very well. For
several polymer composite specimens, the measured internal
friction values as a function of the specimen support
position showed relatively good agreement with the
theoretical internal friction values obtained by equation

(10).
-1 -1 -1 2 2
Qm = [Q5 + kQa (Y/Y') J/[1+k(Y/Y') ] (10)

The internal friction values measured by sonic
resonance method and DMA are different, even the author
cannot find a trend between these two measurements due to
the lack of data. The main reason for this difference may
presumably come from the difference in the frequencies

between these two method.

41

 

 

composite

 

 

 

 

 

 

 

 

 

 

 

 

E
—---- monolithic
—"" \‘/'—=
I l
0 90
.9 - -—-__
fl
2
4» G
C!
.2
in
J!
6’.
l l A l
0 90 0 90
Figure 10. Variation of elastic moduli of a fiber

 

composite and a monolithic material with the
angle of reinforcement (After Meyer and Chawla

[21])-

42

 

 

 

 

40
fi;‘
‘3
£5

30~ °
2!
g; o
|.-
33 o
0:
u.
2’ 20-
2:
0:
DJ
I—
Z

10 I I I

2 4 6 8 1'0 1'2 14
INTERFACIAL STRENGTH (Kpsi)

Figure 11. Internal friction of polymer composites as a

function of interfacial shear strength.

INTERNAL FRICTION (IE—04)

Figure 12.

43

 

 

0

Solid curve represents least—squares
best fit to equation (10).

 

U I I

I 1' l I I I 1 I

1'0 ' ' 20 30
SUSPENSION POSITION (X/LHOO)

Measured internal friction as a function of
suspension position (x/l * 100) for the
polymer composite specimen AS4-longitudinal.

 

44

 

INTERNAL FRICTION (E—O4)

Solid curve represents least-squares
best fit to equation (10).

 

 

 

10 .

Figure 13.

I U I

1'0 ' ' I 20 I 30
SUSPENSION POSITION (X/LHOO)

Measured internal friction as a function of
suspension position (x/l * 100) for the
polymer composite specimen AS4C-longitudinal.

45

 

 

 

 

 

 

90
:F‘ T
O -1
1h 70—
.1-
2:
9
F_ I
S2 50-
Q:
h.
2%
z ‘ M
0: .30-
if
g Solid curve represents least-squares
. best fit to equation (10).
10 V U U I I I I I U l I V T T l
0 10 20 30

SUSPENSION POSITION (X/Lnoo)

Figure 14. Measured internal friction as a function of
suspension position (x/l *100) for the polymer
composite AU4-longitudinal.

46

4-2. SiC Whisker Reinforced Alumina Composite
4-2-1. Room Temperature Modulus and Internal Friction

The room temperature resonant frequency, Young's
modulus and internal friction of the as-received SiC
whisker reinforced alumina composite specimens are given
in table 8.

The Young’s modulus values are 410.10 Gpa and 406.14
Gpa and the internal friction values 14.2x10-‘ and
18.6x10-‘ for the HP160 and HP171 specimens, respectively.
Comparing the Young’s modulus and internal friction
observed for HP160 and HP171, it appears that the only
difference between the two specimens is their porosity
(density). The higher Young's modulus and lower internal
friction value of HP160 may be attributed to the lower
porosity (higher density). The strength data further
supports this since the fracture strength decreases
rapidly as porosity increases. Specimen HP158 cannot
perhaps be compared directly because different whiskers
were used in this specimen. Whiskers used for the
specimen HP158 were TOKAMAX SiC whiskers, but for the
specimens HP160 and HP171, ARCO SiC whiskers were used.
However, it has not been demonstrated whether the
mechanical properties of the TOKAMAX SiC whiskers and the
ARCO SiC whiskers differ significantly. Thus, the use of

fibers from different vendors may make no significant

47

Table 8. Room Temperature Modulus and Internal Friction
of Alumina/Sic Composite Specimens

 

Resonant Frequency (Hz) E (Gpa) Q§1(10-5)

 

HP158 6387.67 404.56 13.2
HP160 6406.17 410.10 14.2
HP171 6431.00 406.14 18.6

 

48

difference in the measured elastic moduli and internal
friction. The values of mass density, porosity, water
absorption, fracture strength, and elastic modulus (Table
4) measured by Alcoa were quite similar for each of the
three Sic whisker/alumina matrix specimens. In this
study, the room temperature values of elastic modulus and
internal friction (prior to thermal shock, Table 8) were
also quite similar for each of the three specimens.

Figures 15-17 illustrate the measured internal
friction, Q-l, as a function of the position of

suspension, for the SiC whisker reinforced alumina

composites.

4-2-2. Elevated Temperature Modulus and Internal Friction

Prior to thermal shock testing, the Young's modulus
and internal friction were measured as a function of
'temperature for the as-received Sic whisker/alumina
composite specimen HP171.

Figure 18 illustrates the Young’s modulus as a
function of temperature for Sic whisker/alumina composite

HP171. The modulus decreased continuously with increasing
temperature, which is typical of polycrystalline ceramics.
A1 so, the modulus retraced the heating paths during
<3<>oling. The modulus changed linearly up to 900.C, which

‘flils the maximum test temperature for the present study.

49

 

 

   
 

 

 

90
A
10
Ol _,
EB) 70-
2: .
9.
£3 50.
E _ Solid curve represents least-squares
LL best fit to equation (10).
_] .
<( .
2:
CE 30-
DJ
(.— a
_Z_
.I ___/”A
10 ..,....I,j,rl
0 10 20 30

SUSPENSION POSITION (X/LaIz‘lOO)

Figure 15. Measured internal friction as a function of

suspension position (x/l *100) for the
alumina/Sic composite specimen HP158.

 

50

 

INTERNAL FRICTION (E-05)

   
   

Solid curve represents least-squares
best fit to equation (10).

A

 

 

Figure 16.

T I I

r j T l l I I l I r

1D 20 30
SUSPENSION POSITION (X/LHOO)

Measured internal friction as a function of
suspension position (x/l *100) for the
alumina/Sic composite specimen HP160.

 

INTERNAL FRICTION (E-OS)

51

 

 

  
   

 

90

70-

50— .
SolId curve represents least—squares
best fit to equation (10).

30-

10 ' T ' ' ' ' F . ' T ' I

0 10 20 30

SUSPENSION POSITION (X/LHOO)

Figure 17. Measured internal friction as a function of

suspension position (x/l *100) for the
alumina/Sic composite specimen HP171.

 

52

 

 

 

 

420
A .l
8. 400-
E3) .
(n I
3 380-
3 .
C3 .
O 1
2 .360-
90 .
CZD ‘ A cooling
CD) 3401 O heating
>- ' -

320 T I T I 1 T I I 1 y I U , , I U r

O 200 400 600 800
0
Temperature (C)
Figure 18. Young's modulus of the alumina/Sic composite

specimen HP171 as a function of temperature
(Measurements performed prior to thermal shock
treatment of specimen HP171).

53

The modulus versus temperature data was fit Via least-
squares to a linear function of temperature. The least-

squares procedure produced the empirical equation (in Gpa)
E = 409.5 - 0.047T (14)

The correlation coefficient for the least-square fit of
the Young's modulus versus temperature data was 0.998 and
the number of data points was 19. The linear relation
indicates a modulus decrease of z 1.1 percent/100 C which
agrees favorably with data for various other
polycrystalline ceramics [24,25]. In particular, the
value of Young's modulus versus temperature slope for
polycrystalline monolithic alumina has been determined to
be 1.2 percent/100°C [26].

Figure 19 shows the internal friction, Q-l, as a
function of temperature, which was measured along with the
modulus. The internal friction varied by approximately i
30 percent about a mean value of 2.7x10-‘. The data do
not yield a particulary smooth relation with temperature,
but scatter of this type is typical for high temperature
measurements of internal friction [8].

The room temperature internal friction value after
this elevated temperature measurements has changed from
18.6x10-5 to 11.1x10-5. The observed changes in internal
friction during the measurements on unshocked specimen

HP171 are relatively small compared to AQmax, the maximum

54

 

 

 

.)
fr?
0' 12-
uJ
v
I:
Q
Eé 8-
Q:
“- I
-J
E
4-
Q:
E3 O o o O O o o o o ‘3
2: 0 0 (3 ° °
__ 4 o O
O T I I r I T r I 1 I I I I l I I
O 200 400 600 800

TEMPERATURE (’0)

Figure 19. Internal friction of the alumina/Sic composite
specimen HP171 as a function of temperature
(Measurement performed prior to thermal shock
treatment of specimen HP171).

 

55

change in internal friction due to the thermal fatigue
testing that was performed as subsequent step of this
study. Small Changes in the internal friction could be
due to relaxation of residual stresses in the specimen, or
in part to diffusive crack healing (the anneal temperature
of 900.0 or equivalently 1173 K a 1/2 melting temperature

for the alumina matrix).
4-2-3. Thermal Fatigue Test

Figure 20 and figure 21 show the room temperature
Young's modulus and internal friction of SiC whisker
reinforced alumina plotted as a function of the number of
thermal shock cycles. As expected, the internal friction
increases while the Young's modulus decreases [5,6,7,9].
The changes in both Young's modulus and in internal
friction are presumably due to the formation of
microcracks that form due to the transient stresses the
specimens experience during thermal shock.

This study shows that for thermal shocked SiC whisker
reinforced alumina composite specimen HP160 the
microcracking increases the internal friction by a factor
of nearly 120 percent. However, the modulus values for
this specimen differ only by a factor of 0.07 percent. A
possible explanation is that specimen HP160 contains only

a few microcracks and that internal friction measurements

YOUNG'S MODULUS (Gpa)

56

 

 

 

 

 

 

 

410 -
4k
400 9* j:
390- ' Solid curves represent least-squares
best fit to equation (16)
'o
I... G . o

360- 9 J?
O HP158
o HP160
D HP171

370 ' l ' l ' I ' l ' I ' T ' l ' l ' I .

 

 

0 10 20 30 40 so 60 70 80 90 100
NUMBER OF THERMAL SHOCK

Figure 20a. Young's modulus of the alumina/Sic composite
specimens HP160 (AT=270 degrees C), HP171 ‘
(AT=310 degrees C) and HP158 (AT=380 degrees
C) as a function of thermal shock cycle.

YOUNG'S MODULUS (Gpa)

41C)-

57

 

 

  

13

SOIia curves represent least-squares
best fit to equation (16) _

 

 

 

O 3 4
380-
O HP158
‘ O HP150
D HP171
370 I
O 10 20

Figure 20b.

NUMBER OF THERMAL SHOCK

Young’s modulus of the alumina/Sic composites
HP158, HP160, and HP171 versus thermal shock
number. Data in this figure is a subset
(namely, the first twenty thermal shocks) of
the data presented in figure 20a. Note that
even for n small, equation (16) fits the data
well.

INTERNAL FRICTION (E—OS)

58

 

 

 

 

 

 

 

 

180 A
.l
160- °
. A o O 6
.40.. If"
4 O HP158
120- o HP160
‘li DHP171
100-
4 Solid curves represent least—squares
80— best fit to equation (17)
60+
J_'1_ El
40 :1 u
—o F a
20
O I T, r T I T T T T’ T I
0 20 4O 60 80 100

Figure 21a.

NUMBER OF THERMAL SHOCK

Internal friction of the alumina/Sic
composite specimens HP160 (AT=270 degrees C),
HP171 (AT=310 degrees C) and HP158

(AT=380 degrees C) as a function of thermal

shock cycle.

INTERNAL FRICTION (E-OS)

59

 

 

   
 

 

 

 

 

 

180
160-
1 c
140-
l O HP158
120+ o HP160
. D HP171
100-
- Solid curves represent least-squares
80- best fit to equation (17)
1
60-1
40-
El
20 ‘“IF
0 r
0 1'0 30
NUMBER OF THERMAL SHOCK
Figure 21b. Internal friction of the alumina/Sic

composites HP158, HP160, and HP171 versus
thermal shock number. Data in this figure is
a subset (namely, the first twenty thermal
shocks) of the data presented in figure 21a.
Note that even for n small, equation (17)
fits the data relatively well.

60

are more sensitive to the microcracking effects than are
modulus measurements [7,27]. '

Relative changes in Young’s modulus and internal
friction are listed in table 9 for each of the three
thermal shocked SiC whisker/alumina matrix composite.

The observed decrease in elastic modulus of the Sic
whisker reinforced alumina specimens with increased
thermal cycling can be explained quantitatively on the
basis of microcrack damage acculmulation with increased
thermal shock damage. The reduction in Young's modulus as
a function of.microcrack damage may be understood

quantitatively from an equation of the form [28]
E = Eo (1-f(u)Na’) (15)

where E the effective Young’s modulus

Eo = original Young’modulus
f(u) = function of Poisson’s ratio
v = Poisson's ratio
N = microcrack density (number of microcracks per
unit volume)
a = mean radius of microcrack.

The product, Mat is defined as the crack damage
parameter, which shall be denoted here as c. The
effective modulus varies linearly with the crack damage
parameter. The critical value of c at which E vanishes is
1/f(u) .

The data were fit to the empirical relations via non-

linear regression analysis in the plotIT system. The

61

 

 

Table 9. Relative Changes in Young’s Modulus and Internal
Friction for the Thermal Shocked Specimen (Sic
Whisker/Alumina Composites)

. o -1 -1
Spec1men AT ( C) AE max /E0 .1. 100 AQ max /Q o * 100
HP160 270 0.07 120
HP171 310 1.72 318
HP158 1049

380 5.83

 

62

empirical equations of the forms are for Young’s modulus,

E

E = Eo - A(l-exp(-an)) (16)

where E the effective Young's modulus in Gpa
Eo = Young's modulus for theoretically dense, non-
microcracked material
n number of the thermal shock cycle
A,a = positive constants determined by the non-linear
regression.

-1
For internal friction, Q , where the data was scaled by a
constant factor of 10-5 to facilate the least-squares

fitting procedure, the emprical equation employed was
-1 -1
Q = 00 + E(l-eXP1-finll (17)

where Q::= the effective internal friction
Qo = internal friction for the non-microcracked
material

B,p = positive constants determined by the non-linear

regression.
n = number of the thermal shock cycle.

The results of the non-linear regression analysis are
shown in table 10. For a fixed value of AT, the constant
A in equation (16) represents a "damage saturation" or
steady state level of damage for a large number of thermal
shock cycles, n. Likewise, the constant B in equation
(17) represents the large n value (or asymptotic value) of
the internal friction. The thermal fatigue damage, as

measured by both the internal friction and Young's modulus

63

Table 10. Results of the Non-linear Regression analysis
for the 81C whisker/Alumina Composites

 

 

Specimen HP160 HP171 HP158
AT (°C) 270 310 ' 380
Q51(x10-5) 14.18 11.10 13.40
Eo (Gpa) _ 404.84 406.10 404.60
A (Gpa) ' 3.00 6.21 23.17
B (x10-5) 20.38 33.65 138.31
a 0.042 0.220 2.11
p 0.061 0.150 1.11
cor. coeff. (A,a) 0.993 0.956 0.991

cor. coeff. (B,fi) 0.996 0.978 0.991

 

64

shows a saturation level, and this saturation damage level
increases with increasing AT. '

The constant a in equation (16) is a "rate of
decrement" of the Young's modulus as a function of thermal
fatigue treatment. At a number of cycles given by n =
1/a, the relative decrement in Young's modulus is (1-e-l)
a 0.632. In table 11, a fatigue constant n0e is listed,
where n0e = l/a for the given thermal shock conditions.
Similarly, n°i= l/p is the fatigue constant for the
thermal shock induced changes in internal friction. As
shown in table 11, n0e decreases rapidly as AT increases,
thus the number of thermal shock cycles required to reach
the saturation damage level (which are approximately by z
3 noeand 3 n91 respectively) is a sensitive function of
AT.

In order to understand the increase in the effective
internal friction quantitatively, a quantitative relation
between the effective internal friction and crack damage
parameter, such as equation (15), is needed. From

equation (15) and (16), one can express
EO(1-f(u)c) = Eo - A(1-exp(-an)) (18)
Solving equation (18) then yields

6 = 1/f(u) * A/Eo 4 (l-exp(-an)) (19)

 

65

Table 11. Thermal Fatigue Constants for the SiC Whisker
Alumina Matrix Composite

 

 

Specimen HP160 HP171 HP158
AT (°C) 270 310 380
(A-EO)/Eo 0.9924 0.9847 0.9427
-1 -1
(B-Qo )/Qo 0.437 2.072 9.33
noe 24.3 4.83 0.50
n . 16.3 7.25 0.89

01

 

 

66

Multipling equation (19) by f(u) * Eo/A and rearranging

gives
exp(-an) = l - Eo/A * f(u)e (20)

Taking the logarithm of each side of equation (20) and
deviding by -a then yields

n = -1/a * ln[1-EO/A*f(u)e] (21)
Substituting n from equation (21) into equation (17)
Q-1= 061+ B(1-(1-Eo/A * f(»).)fi/“) (22)

Salganik [29] developed an equation similar to the
equation (15), which is

E = Eo (1-f(u.)c) (23)

where f(u.) = 16(10-3*u.)(1-u3) / 45(2-u.) , (24)

u. = Poisson's ratio for the noncracked material.

The calculated value of Poisson's ratio, 9., was 0.22
(this study). The calculated value of f(u.) in equation
(24) was 1.775. If we use Salganik's equation (23)

instead of equation (15), then we can express internal

67

friction as a function of crack damage parameter only.

The relation is as follows
0"-’= 023‘- B(1-(1-Eo/A *f(v.)e)”/") (25)
Equation (25) can be rewritten as follows
AQ"= C1[1-(1-C2e) Ca] (26)

The entire data set of internal friction as a function of
theraml shock damage for the three SiC whisker/alumina
matrix composite specimens were fit to the empirical
relation given by equation (26) via non-linear regression
analysis. The constants C1, C2, and C3 were 13.55x10'J,
250, and 1.143, respectively.

For sufficiently small C26, the term (1-C26) C3 in
equation (26) can be written as (1-C2C3e) using the

binominal expansion theorem. Thus, equation (26) becomes
-1
AQ z C‘C’C’e = De (27)

The least-squares best fit of equation (27) gives a value
of 4.17x10-2 for constant D. From the equation (26), the
result of product C1C2C3 was 3.87x10 '2. Therefore the
equation (27) can be a relatively good approximation to
the AQ-Iversus 6 data for the thermally-shocked SiC

whisker/alumina composites in this study. The correlation

68

coefficients for equation (26) and (27) were 0.998 and
0.997, respectively. Thus while both equations fit the
data relatively well, equation (26) with three adjustable
parameters of course fits the data better than equation
(27), a simple linear equation with one adjustable
parameter.

Figure 22 shows the internal friction change, AQ-I,
as a function of crack damage parameter, c, for the Sic
whisker/alumina matrix composites. As shown in figure 22,

-1 . .
AQ increases as 6 increases.

69

 

    
 

 

 

 

160-
1 o
,-
m d
('3 120-1
Lu 1
V l
78 801
I ‘1
T . / .
O 40 .1 / SolId curve represents least-squares
4 best fit to equation (26)
Dotted curve represents least-squares
- best fit to equation (27)
0 t’ TT ”T T’ TT T T ‘TT
0 100 200 300 400

CRACK DAMAGE PARAMETER (E—04)

Figure 22. Internal friction Change, AQ-l, as a function
of crack damage parameter, c, for the SiC
whisker/alumina composites.

CONCLUSIONS

This study dealt with both polymer and ceramic
composites, although ceramic composites were the primary

focus of the study.

Polymer composites reinforced with unidirectional
fibers exhibited modulus and internal friction anisotropy.
The internal friction increased with decreasing interfacial
shear strength.

The elastic moduli of SiC whisker/Alumina decreased
linearly with temperature at a rate of 1.1 percent/100
degrees Celcius. As thermal shock progressed, Young's
modulus decreased and internal friction increased.
Empirical equations for the modulus and internal friction
as a function of thermal shock cycles were developed. The

fitting equations were

E = Eo - A(l-(l-exp(-an))
and Q-1= Q61+ B(1-(1-exp(-pn))

As AT increased, the "damage saturation" A, B, and the
"rate of decrement or increment" a, and 5 increased. Also
-1

an empirical equation for AQ as a function of crack

damage parameter, c was developed, which was

-1 ' C3
AQ "—' C1 [l-(l-Cze) ]

70

71

In order to more completely understand the thermal
shock induced damage and the empirical equation developed
in this study, further investigation is needed. Further
gains in understanding the phenomena of thermal shock of
whisker reinforced ceramic composites can be expected from
comparing the thermal shock results for composites with
thermal shock damage results from monolithic ceramic

specimens.

10.

REFERENCES

T. Mah, M. G. Mendiratta, A. P. Katz, and K. S.
Mazdiyasni, "Recent Developments in Fiber-Reinforced
High Temperature Ceramic Composites", Amer. Ceram.
Soc. Bull. 66 [2] 304-308, (1987).

P. J. Lamicq, G. A. Bernhart, M. M. Dauchin, and J. G.
Mace, "SiC/SiC Composite Ceramics", ibid., 65 [2] 336-
38 (1986).

A. J. Caputo, D. P. Stinton, R. A. Lowden, and T. M.
Besman, "Fiber-Reinforced SiC Composites with Improved
Mechanical Properties", ibid., 66 [2] 368-72 (1987).

L. J. Broutman, Composite Materials Vol. 5-Fracture
and Fatigue, Chapter 3, Academic Press, New York,
(1974).

S. L. Dale, 0. Hunter, Jr., F. W. Calderwood, and D.
J. Bray, "Microcracing of Monoclinic Hfoz", J. Amer.
Ceram. Soc., 61; 486-72 (1978)

E. Bonetti, E. Evangelista, and P. Gondi, "High
Temperature Damping in SiaN, Depending on the
Sintering Conditions", Internal Friction and
Ultrasonic Attenuation in Solids, 401-405, Pergamon
Press, New York, (1980).

S. L. Dole, 0. Hunter, Jr., and C. J. Wooge, "Elastic
Properties of Monoclinic Hafnium Oxide at Room
Temperature", J. Amer. Ceram. Soc. 60 [11-12] 488-90
(1977).

C. J. Malarky, "Effects of Microstructure on the
Elastic Properties of Selected Ta205 -Eu, q.
Compositions", M. S. Thesis, Iowa State University,
43-44 (1977).

K. Matsushita, S. Kuratani, T. Okamoto and M. Shimada,
"Young's Modulus and Internal Friction in Alumina
Subjected to Thermal Shock", J. Mat. Sci. Lett., 3
345-48 (1984).

S. L. Dole, 0. Hunter, Jr., "Elastic Properties of
Hafnium and Zirconium Oxides Stabilized with
Praseodium or Terbium Oxide", J. Amer. Ceram. Soc., 66
[3] C-47-C-49 (1983).

72

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

73

Eighth International Conference on Internal Friction
and Ultrasonic Attenuation in Solids, University of
Illinois, J. de Physique, Colloque, ISSN, C10-427-C10-
493, no.10, (1985).

Fourth European Conference on Internal Friction and

Ultrasonic Attenuation in Solids, Villeurbanne,
France, J. de Physique, Colloque, ISSN, C9-85-C9-203,
no.9, (1983).

A. S. Nowick, B. 8. Berry, Anelastic Relaxation in
Crystalline Solids, Chapter 1, ACademic Press, New
York, (1972).

J. B. Wachtman, Jr., and W. E. Tefft, "Effect of
Suspension Position on Apparant Values of Internal
Friction Determined by Forster's Method ", Rev. Sci.
Instrs., 29 [6] 517-20 (1958).

C. Zener, Elasticity and Anelasticity of Metals,
University of Chicago Press, Chicago, p.60, (1948).

F. Forster, "A New Method for the Determination of the
Modulus of Elasticity and Damping", Zeit. Metallkunde,
29 [4] 109-15 (1937). '

E. Schreiber, O. L. Anderson, and N. Soga, Elastic
Constants and Their Measurement, Chapter 4, McGraw-
Hill, New York, (1974).

G. Pickett, "Equations for Computing Elastic Constants
from Flexural and Torsional Resonant Frequencies of
Vibration of Prisms and Cylinders", ASTM Proc., 45,
846-65, (1945).

D. P. H. Hasselman, Tables for the Computation of
Shear Modulus and Young's Modulus of Elasticity from
Resonant Frequencies of Rectangular Prisms,
Camborundum Co., Niagara Falls, New York, (1961).

L. Rayleigh, The Theory of Sound, Dover Publications,
New York, Vol.1, pp. 253-83, (1945).

M. A. Meyer and K. K. Chawla, Mechanical Metallurgy -
Principles and Applications, Prentice-Hall, p.458,
(1984).

H. M. Ledbetter, "Dynamic Elastic Modulus and Internal
Friction in Fibrous Composites", Nonmetallic Materials
and Composites at Low Temperatures, Plenum, New York,
pp. 267-81, (1979).

23.

24.

25.

26.

27.

28.

29.

74

P. S. Chau, "Characterization of the Interfacial
Adhesion Using Tan Delta", Sampe Quarterly, 19 [1] 10-
15 (1987).

W. R. Manning and 0. Hunter, Jr., "Elastic Properties
of Polycrystalline Yttrium Oxide, Holumium Oxide, and
Erbium Oxide; High-Temperature Measurements", J. Amer.
Ceram. Sco., 52 [9] 492-96 (1969).

J. A. Haglund and 0. Hunter, Jr., "Elastic Properties
of Polycrystalline Monoclinic Gdgos", Ibid., 55 [6]
327-30 (1973).

E. D. Case, J. G. Smyth, and 0. Hunter, Jr., Fracture
Mechanics of Ceramics-Vol. 5, pp. 507-30, edited by R.
C. Bradt, A. G. Evans, D. P. H. Hasselman, and F. F.
Lange, Plenum Press, New York, (1983).

J. A. Coppola and R. C. Bradt, "Thermal-Shock Damage
in SiC", Ibid., 56 [4] 214-218 (1973).

B. Budiansky and R. J. O'Connell, "Elastic Moduli of a
Cracked Solid ", Int. J. Solids Structures 12 81-97
(1975).

R. L. Salganik, Mech. Solids 8 [4] 135 (1973); English
Translation.