Kounadis A.N., Gdoutos E.E. (Eds.) Recent Advances in Mechanics

The Fracture Toughness of a Highly Filled Polymer Composite 449

Table 2 shows the experimental data for the matrix (PMMA). No tests were per-

formed at 0

0

C and so these values are extrapolated. The modulus values are taken

from the fracture tests and the yield stresses were obtained in compression. A value

of 0.4 is assumed for the Poisson’s ratio,

ν

, in subsequent calculations [2].

The G

m

values presented some difficulties. At 22, 40 and 65

Ԩ

the data showed

little evidence of ductility and a small decrease in G

m

was observed at 65

Ԩ

. At

90

Ԩ

, however, a value of 1.10 kJ/m

2

was obtained and the very low yield stress

led to crack blunting although the results were valid. Again the standard devia-

tions are about 10%.

Table 1. Fracture Toughness Data, (S is standard deviation).

T Ԩ

0

22

40

65

90

I

G

c

±S

(kJ/m

2

)

0.35

0.54±0.08

0.54±0.07

0.69±0.06

0.68±0.06

0.77±0.14

0.37

0.48± 0.05

0.44±0.05

0.61±0.04

0.62±0.09

0.62±0.05

0.38

0.56± 0.06

0.55±0.10

0.56±0.06

0.59±0.03

0.61±0.05

0.40

0.48±0.06

0.44±0.07

0.51±0.07

0.54±0.02

0.56±0.03

0.42

0.47±0.03

0.41±0.10

0.51±0.07

0.55±0.09

0.62±0.07

0.42

0.45± 0.02

0.39±0.06

0.47±0.07

0.50±0.06

0.59±0.17

0.45

0.45±0.03

0.35±0.08

0.47±0.06

0.48±0.06

0.59±0.06

0.47

0.43±0.05

0.37±0.06

0.43±0.04

0.45±0.05

0.49±0.09

0.49

0.36±0.02

0.26±0.04

0.40±0.06

0.41±0.02

0.44±0.06

450 O.A. Stapountzi, M.N. Charalambides, and J.G. Williams

Fig. 1. Composite Fracture Toughness Data versus filler volume fraction at various tem-

peratures.

Table 2. PMMA Matrix Property Data, ( ) indicate extrapolated values.

T °C

σ

y

(MPa)

E

(GPa)

G

m

± S (kJ/m

2

)

0

(150) (3·5) (0·45)

22

125 3·4 0·45 ±·03

40

100 3·2 0·45 ±·03

65

71 2·6 0·35 ±·03

90

39 1·4 1·10 ±·07

The Fracture Toughness of a Highly Filled Polymer Composite 451

4 Toughness Model

A model to describe these high volume fraction data is based on that given in [3].

In the model the toughening is assumed to arise from a single mechanism is which

particles around the crack tip are subjected to sufficient hydrostatic stress to cause

debonding. This freeing of the particle interface gives rise to plastic work being

dissipated in a zone surrounding the particle.

The model is based on a cell of side length l containing a spherical particle of

radius

r

°

as shown in Fig. 2. The volume fraction of particles,

φ

, is given by,

3

4

3

π

φ

°

⎛⎞

=

⎜⎟

⎝⎠

r

l

(1)

(The upper limit for the particles to touch is

2

=

o

l

r

and

0.52

6

π

φ

==

).

(a) (b)

Fig. 2. Particle unit cell and fracture area.

The area fraction of matrix on a plane through the equator of the particle, AA in

Fig 2.a and shown in Fig 2.b, is

2

3

2

3

11 11.21

4

φ

ππ

φ

π

⎛⎞

=− =− =−

⎜⎟

⎝⎠

o

a

r

l

(2)

The rigid particle is assumed to be subjected to a surface radial stress of

σ

c

and

that, prior to yielding, the matrix is elastic and debonding occurs when [3],

2

4

1

σ

ν

=

+

a

c

o

EG

r

(3)

On debonding this gives rise to plastic work being performed, via a constant yield

stress

σ

y

in a zone of radius r

2

also shown in Fig 2a. r

2

is given by [3]:

452 O.A. Stapountzi, M.N. Charalambides, and J.G. Williams

3

11

ln 1

21

σ

ν

νσ

+

⎛⎞

=−

⎜⎟

−

⎝⎠

c

y

R

(4)

where

2

=

o

r

R

r

.

This gives rise to a plastic energy dissipation per unit of particle surface area,

2

4 r

π

°

of

()

2

3

2

11

3

σ

ν

°

⎛⎞

=− −−

⎜⎟

⎝⎠

y

p

r

R

Gx

Ex

(5)

where

11

21

σ

ν

σ

+

⎛⎞

=

⎜⎟

−

⎝⎠

c

y

x

For micron scale particles only those in the fracture plane are involved [3] and so

the measured G

c

is given by,

()

222 2

4

comopa

lG l r G r G G

ππ

=− + +

i.e.

()

41

φφ

⎛⎞

=+ + −

⎜⎟

⎝⎠

p

ca

aa

mmm

G

GG

GGG

(6)

where G

m

is the matrix toughness. If we now introduce a matrix plastic zone size

2

πσ

=

m

y

EG

r

then from equation (5) we have,

3

2

0

(1 )

1

3

ν

π

⎛⎞

−

=−

⎜⎟

⎝⎠

p

mm

G

rR

x

Grx

Equation (3) gives

()

2

0

1

21

ν

πν

−

⎛⎞

=

⎜⎟

+

⎝⎠

a

mm

Gr

x

Gr

and hence

1

2

0

4(1 ) 5 1

4

32(1)

νν

πν

−

⎛⎞

⎛⎞ ⎛⎞

⎛⎞

−−

+= −

⎜⎟

⎜⎟ ⎜⎟

⎜⎟

+

⎝⎠

⎝⎠ ⎝⎠

⎝⎠

x

p

c

mm m

G

Gre

x

GG r x

since

31−

=

x

R

e

from equation (4).

The Fracture Toughness of a Highly Filled Polymer Composite 453

From equation (6) we may write,

1

2

0

4(1 ) 5 1

.

13 2(1)

φ

νν

φπ ν

−

⎛⎞

−−

=−

⎜⎟

−+

⎝⎠

c

a

x

m

am

G

Gr

e

x

rx

(7)

This is the relationship explored in [3] and it was shown that for micron scale par-

ticles for

φ

<0.20 the right hand side of equation (7) did remain constant so that

c

m

G

increased with

φ

(i.e. with

2

3

11.21

φφ

−=

a

) with x approximately 3.1 for

ν

=1/3 and for G

a

being a substantial proportion of G

m

, i.e. 0.2-0.4 G

m

. The anal-

ysis assumes that there is no inter-particle interaction and so would be appropriate

for low volume fractions. Here we are considering cases where 0.3<

φ

<0.5 and

c

m

G

decreases with increasing

φ

.

To accommodate inter-particle effects it will be assumed that r

2

is constrained

by adjacent particles. This is taken as when r

2

encompasses the whole unit cell

and

1

3

33

22

43

ˆˆ

,.. 0.62

34

π

⎛⎞

===

⎜⎟

⎝⎠

rl ier l l

as illustrated in Fig 3 and

33

3

2

00

ˆ

31

ˆ

4

π

φ

⎛⎞ ⎛⎞

⎛⎞

== =

⎜⎟ ⎜⎟

⎜⎟

⎝⎠

⎝⎠ ⎝⎠

r

l

R

rr

(8)

Fig. 3. Upper limit of plastic radius

2

ˆ

r

.

454 O.A. Stapountzi, M.N. Charalambides, and J.G. Williams

Thus we now have,

()

2

0

1

3

ν

πφ

−

⎛⎞

=−

⎜⎟

⎝⎠

p

mm

G

r

x

Grx

and

()

2

0

1

21

ν

πν

−

⎛⎞

=

⎜⎟

+

⎝⎠

a

mm

Gr

x

Gr

The equivalent relationship to equation [7] becomes,

()

2

0

41

151

13 21

φ

ν

φπ φ ν

−

⎛⎞

−

⎛⎞

−

== −

⎜⎟

−+

⎝⎠

a

m

am

G

Gr

Yx

rx

(9)

The transition from constant Y occurs when

1

x

e

φ

−

=

and for

3.1, 0.12.

φ

≈≈x

The various parameters may be found by plotting Y versus

φ

−

and fitting a

best straight line,

1

Ym c

φ

−

=+

and for

ν

=0.4 for PMMA [2]:

2

00

0.25 and .089

⎛⎞ ⎛⎞

=−=

⎜⎟ ⎜⎟

⎝⎠ ⎝⎠

mm

rr

mxcx

rr

2

0

. . 2.81 , 1.42

⎛⎞

−

⎛⎞

==

⎜⎟

−

⎝⎠

m

rcm

ie x

mr c

()

and 0.46=−

a

m

G

c

G

The standard deviations of these parameters may be found from a best fit line

which determines m, c and R

2

, the correlation coefficient. The necessary formulae

are given in the Appendix.

5 Analysis of Results

Table 3 gives the results of plotting the fracture data in Table 1 according to equa-

tion [9], i.e. Y versus

1

φ

−

. The values of R

2

, the correlation coefficient, indicate

The Fracture Toughness of a Highly Filled Polymer Composite 455

that the data at 40 and 65°C are a good linear fit with R

2

>0.9 and with standard

deviations in both m and c of about 10% and 20% respectively. That for 0 and

22°C shows considerably more scatter with R≈0.7 and standard deviations of

about double the 40 and 65°C values. The 90°C labelled (1) used G

m

=1.1 kJ/m

2

which made Y small and gave a very poor correlation with R

2

≈0.5. The high val-

ue of G

m

is open to some doubt since there is some plastic blunting involved in the

bend specimens which would probably not be present in the highly constrained

layers between particles. With this in mind G

m

=0.55 kJ/m

2

was used in 90(2)

which gave R

2

≈0.8 and standard deviation similar to those at 0 and 22°C.

Table 3. Derived parameters.

456 O.A. Stapountzi, M.N. Charalambides, and J.G. Williams

Also shown in Table 3 are the derived values of x,

o

m

r

and

a

m

G

. The nature of

these derivations increases the standard deviations and so the possible variation in

these sets, other than the 40 and 65

Ԩ

data, is quite large. However x is generally

greater than three as expected from [3] and

o

m

r

is around unity.

a

m

G

is less than

unity, as expected.

6 Discussion

In [3] all the toughness data analysed increased with volume fraction and the right

hand side of equation (7) was constant. Thus to determine x and hence G

a

, r

0

had

to be known. Values were taken from particle size distributions but there was

some uncertainty about the effects of agglomeration. In this case equation (9) per-

tains and from m and c it is possible to find x, and hence

σ

c

,

o

m

r

hence r

o

and

a

m

G

and hence G

a

. G

a

comes directly from c and G

m

and has the least scatter. The

values deduced are given in Table 4 and are somewhat less than G

m

but a substan-

tial proportion as would be expected for particles treated to enhance bonding as

used here.

Table 4. Surface energy, particle radius and interfacial stress values.

T°C

G

a

± S (kJ/m

2

)

r

m

± S (μm)

r

0

± S (μm)

σ

c

± S (MPa)

0

0.09±0.07 11 14±19 250±240

22

0.32±0.12 16 15±19 476±270

40

0.33±0.06 23 28±11 330±96

65

0.32±0.04 29 51±11 220±60

90(1)

0.14±0.11 160 51±85 110±120

90(2)

0.28±0.11 80 86±68 110±70

The Fracture Toughness of a Highly Filled Polymer Composite 457

The values of

σ

c

are determined by G

a

and r

0

via equation (3). The variations

in both are substantial and arise from the derivations from equation (9). Since x

does not change greatly the changes in

σ

c

with temperature arise mostly from

changes in

σ

y

. One would expect r

0

to be independent of temperature since it is a

measure of the effective particle size. Again the variations are large but r

o

does

show some evidence of varying with temperature. The average value is 39 μm.

Particle size analysis of the filler gave a minimum of 10 μm with agglomerates up

to 100 μm. The Talysurf roughness measurements of the fracture surfaces gave an

average of 37 μm over the temperature range with no evidence of a trend. How-

ever the agreement is acceptable but the trend persists.

The quality of the data limits how far one can pursue the cause of this apparent

trend in r

o

. In [1] some effects were attributed to thermal stresses which were es-

timated at about 40 MPa. These would change

σ

c

but are not sufficient to account

for the changes to r

o

. Such stresses may be the cause of the high scatter at 0°C

and 22°C though they would not for 90°C where the low modulus greatly reduces

their value. A further possibility is that at the lower temperatures crazing occurs

rather than shear yielding [4]. This would reduce the effective yield stresses to

about 100 MPa and so increase r

m

to about 25 μm and thus increase r

o

to about 30

μm and greatly reduce the increasing trend in r

0

. The presence of crazing is an-

other possible cause of the increased scatter at 0 and 22°C.

Fig. 4 and Fig. 5 show micro graphs of the fracture surfaces for two tempera-

tures. At the higher temperature there is clear evidence of plastic deformation and

debonding of the particles as postulated in the toughness model. At the lower

temperatures this is less and may indicate the onset of crazing.

Fig. 4. SEM micrograph of fracture surface for 35% volume fraction sample fractured at 22°C.

458 O.A. Stapountzi, M.N. Charalambides, and J.G. Williams

Fig. 5. SEM micrograph of fracture surface for 35% volume fraction sample fractured

at 90°C.

7 Conclusions

It is difficult to achieve consistent fracture data with this type of material particu-

larly when sharp cracks are needed as in these tests. However the data described

do show clear trends with both volume fraction and temperature. The latter effec-

tively give composites of different matrix behaviour with large variations in

modulus and yield and/or craze stress.

The notion that the toughness arises from the debonding of the particles fol-

lowed by plastic deformation is confirmed by the description of the data by the

model and the micrographs. The values of G

a

are sensible as is the predicted par-

ticle size when compared to direct measurements such as surface roughness and

size distributions. The assumption that the plastic deformation is limited by

31

ˆ

φ

−

=R

is, to some extent, arbitrary but is justified by the fit to the data. More

evidence is needed to confirm this extension to the model.

Unlike the fatigue behaviour described in [1] there was no evidence here of re-

sidual or thermal stresses. It is possible that this is due to low cooling rates in the

moulding process.

Acknowledgments

The authors wish to thank DuPont for funding this work and supplying the sam-

ples and Dr Clyde Hutchins for helpful advice. Mr Tomas Procházka performed

the fracture tests on the composites and Mr Koucheng Zuo the tests on PMMA.

Kounadis A.N., Gdoutos E.E. (Eds.) Recent Advances in Mechanics

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