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Dynamic behaviour of two-phase viscoelastic materials
Composites Science and Technology 25 (1986) 301-309
Dynamic Behaviour of Two-phase Viscoelastic Materials
J. Milios and G. Spathis Department of Theoretical and Applied Mechanics, The National Technical University of Athens, 5, Heroes of Polytechnion Avenue, 157 73 Athens (Greece)
SUMMARY Cylindrical two-phase epoxy resin specimens have been tested under oscillating loading over a wide range of temperatures and frequencies, taking into consideration the role of the interfaces between the main constituent elements of the composites. New bounds for the loss moduli, which were about 10 % higher than those predicted by the law of mixtures, have been established for the ease of specimens of good interfacial adhesion, a result which supports the assumption of the importance of the role of the interface region.
INTRODUCTION Theoretical and experimental investigation of fibre-reinforced composite materials is always accompanied by the problem of approaching or predicting the effective macromechanical properties of the composite from the mechanical properties and volume fractions of the constituent elements. The first classical approaches to the mechanical behaviour of composite materials composed of a viscoelastic matrix containing elastic fibres 1-3 were extended by investigations seeking expressions for--and, accordingly, bounds to--the elastic modulus of the composite.4 - 6 On the other hand, other investigations have applied the correspondence principle to obtain bounds for the viscoelastic moduli of the composite. 7,a 301 Composites Science and Technology 0266-3538/86/$03.50 © Elsevier Applied Science Publishers Ltd, England, 1986. Printed in Great Britain
302
J. Milios, G. Spathis
The notion of bounds has been introduced because of the substantial differences between experimental results and predicted values for a great number of composites, and there are cases where the moduli of the composite do not even fall within the bounds. 4 Apart from the above-mentioned direction in approaching the problem of the fibre-reinforced composite, by investigating the stress-strain distribution, another approach has been established lately which takes into account the effects of the interfacial interactions between the constituent elements of the composite. 9 This approach introduces the concept of a separate intermediate region, known as the interphase, between the two main phases of the composite and which possesses mechanical properties distinct from those of the two main phases. 10 - 1 2 This intermediate region was assumed to concentrate the stress gradients and singularities, as well as possible physicochemical irregularities introduced from fibre-matrix contact. Retardation spectra analysis 13 led to important conclusions on the nature of these disturbances at the surface of contact and led to the idea that a composite cannot in reality be any longer treated as an elastostatic problem and that other mechanisms of purely viscoelastic origin are responsible for its dynamic behaviour. These conclusions were supported by the results of dynamic crack propagation experiments performed in two-phase plates, 14-16 which have shown that the interphase region alone is responsible for a phenomenon of temporary crack-arrest, caused by a rapid redistribution of the stress field concentrated at the tip of the propagating crack as it reaches the interface, in all cases of crack propagation normal to a bimaterial interface. To investigate, thus, the dynamic behaviour of a fibre-reinforced composite, a suitable model has to be chosen, for which the interphase effects would be decisive. For this purpose we chose in the present investigation a model of a cylindrical composite, to ensure large interfacial areas, consisting entirely of viscoelastic materials--interesting in themselves--where a measurable interaction effect was expected to occur. E X P E R I M E N T A L P R O C E D U R E A N D RESULTS Two types of two-phase specimens were prepared from epoxy resin (Fig. 1) for dynamic measurements. The inner cylindrical phase was made
Dynamic behaviour of two-phase viscoelastic materials 20"1. plost epoxy
oxy
ztrix
303
O008m
0.00/,2 iZ
I~l.sillcor
,=,""
Good Adhesion Sod Adhesion Pure Specimen Specimen Specimen
Fig. 1. Types of specimens used for the experiments.
from 8 wt ~o amine (TETA)-cured epoxy resin and the outer one was made from 8 w t ~ TETA-cured epoxy resin with 2 0 w t ~ of Thiokol LP3 polysulphide as plasticizer.
Good adhesion specimens A cylindrical mould was used to cast the hard component of the twophase system, and 24 h later the 20 ~ plasticized part was cast around the hard kernel. The final composite was taken out of the mould after a further 24 h and it was postcured for 24 h at 130 °C. We shall refer to these composites as 'good adhesion specimens'.
Bad adhesion specimens The same procedure as before was followed in constructing the second series of specimens, the only difference being that before casting the second phase (20 ~ plasticized) the area of contact was coated with silicone oil. These composites are referred to as 'bad adhesion specimens', since the adhesion between the two phases is expected to be poor compared with that of the other type of composites. For comparison purposes the same moulds were used to prepare single component specimens purely of 8 ~ TETA-cured epoxy resin and purely of 8 ~ TETA-cured epoxy resin with 20Wt~o Thiokol LP3. The curing procedure was the same as that used for the composites of both levels of adhesion.
J. Milios, G. Spathis
304
All specimens had the same dimensions, being standard cylindrical tensile specimens convenient for dynamic testing in Dynastat and Dynalizer equipment. The dimensions of the specimens are indicated in Fig. 1. After identical curing procedures, all specimens were subjected to sinusoidal loading of prescribed and constant maximum amplitude of 1 N, over a range of temperature from 50 ° to 130 °C, applied by Dynastat and Dynalizer machines. The specimens were mounted between a long upper rod of high elastic modulus, which was connected in series with a load cell, and a shorter lower rod, coupled together with the displacement transducer. By passing a servo-controlled current through the coil of the transducer, each specimen was subjected to a sinusoidal load of prescribed frequency. The frequency range selected in our tests varied between O)min 0" l Hz and COmax 100 HZ for all specimens. All specimens were dynamically tested from the rubbery plateau, through the transition zone, and into the glassy region, the temperature ~---
=
6.5
r 0=, ee
•
O.O
xX x X X X x
x X X ° o o o XX
xx o ° ° X
."
7.5
/ 7,0 -1
..... ,-." -
X X X X
O
• •
[og~
,~
Fig. 2. Curves for the loss modulus, E", versus frequency, plotted on a logarithmic scale, reduced to To= I10°C. O, Specimen with 0 ~ plasticizer; x , specimen with 20~o plasticizer; ©, good-adhesion specimen; - - , curve predicted by the law of mixtures.
Dynamic behaviour of two-phase viscoelastic materials
305
being raised in step-wise fashion in order to accurately obtain the shift factors. By application of the time-temperature superposition principle, 17 the data were reduced to a reference temperature of 110°C. The equations used for the reduction are
Ep¢ = E'(ToPo/TP)
(1) (2)
tt
Ep = E"(ToPo/Tp)
where p and Po are the densities of the specimens at the absolute temperature of measurement, T, and the reference temperature, To, and subscript 'p' denotes the respective reduced moduli. The temperature shift factor, a T, which is required to superpose the reduced data, was the same for both loss and storage moduli for each system. In Fig. 2 the loss modulus of the 'good adhesion specimen' is plotted as a function of frequency, together with the loss moduli of the 0 % and 20 % plasticizer specimens. It is obvious from Fig. 2 that the two peaks of the 11.5
T g, 04
8.o
X X x
° o
x
X
XX
x x X X
X
7,5
X? ° X
•
Xx
7.0 -1
log~
Fig. 3. Curves of the loss modulus, E", versus frequency, plotted on a logarithmic scale, reduced to To= l l0°C. O, Specimen with 0% plasticizer; x , specimen with 20% plasticizer; (3, bad-adhesion specimen; - - , curve predicted by the law of mixtures.
306
J. Milios, G. Spathis
1.0
eeee
T
xXX x x x
xX X
x
9
O.S
X
X
X X X X
x
•
x •°
X X
X
x
x
x
X
X
X x X x
X° l
•
-1
Fig. 4.
Logw
Curves for tan 6 versus frequency (logarithmic scale) reduced to To = 110°C. O, Specimen with 0% plasticizer; x, specimen with 20% plasticizer.
c o m p o s i t e are clearly higher t h a n the predicted values f r o m the law o f mixtures (solid curve): ?t _ _
~
E c -rlE
¢!
1 + vzE~
where v I = v 2 = 0.5. In Fig. 3 the loss m o d u l u s o f the 'bad a d h e s i o n specimen' and the loss m o d u l i o f the 0 ~o and 20 % plasticizer specimens are p l o t t e d as f u n c t i o n s 1,0
T x 0.5
!
?
IK
eX Xe oX ~X
.x eX
.x 'e eoxX
, x -1
Fig. 5.
X
eeeee
.x. '~xx
x ~'x"
[ogw
Curves for tan 6 versus frequency (logarithmic scale) reduced to To = 110 °C. O, Good-adhesion specimen; x, bad-adhesion specimen.
Dynamic behaviour of two-phase viscoelastic materials
307
of frequency. Here it is clear that the two peaks of the unbonded composite are close to the predicted values from the law of mixtures. In Figs 4 and 5 the loss tangents of each type of composite specimen and the loss tangents of the 0 % and 20 % plasticizer specimens are plotted with respect to frequency. F r o m these figures it is obvious that the second peak of the more viscous material is highly depressed, a result which was expected because the high modulus of the stiffer material in the denominator of the expression determines the value of the loss-tangent of a viscoelastic specimen.
DISCUSSION Our experimental investigation has revealed unpredicted dynamic behaviour of the cylindrical bi-phase specimens. Specimens with good interfacial adhesion form master curves with peaks clearly higher than the values predicted from the law of mixtures. By contrast, the dynamic behaviour of the bad adhesion specimens is in accordance with the law of mixtures. A first explanation of this 'peculiarity' could possibly be connected to the fact that the law of mixtures in its simple form ignores the stress redistribution due to the Poisson ratio effect in the iso-strain, goodadhesion cylindrical specimens. On the other hand, in specimens of bad interfacial adhesion we can ignore the interfacial stress redistributions due to the difference of the Poisson ratios of the main phases, thereby assuming that relative motion between the phases is possible. The law of mixtures in its complete form, that is when one takes into account the Poisson ratio effect, is given by the following relations: 17
E'c= [2(v I
-
v2)2E'l(1
vl)vx]{[(1 - Vl)L2E~2] + [ L l v 1 + (1 + v2)]E'IE'2} {(1 - v ~ ) L z E ' z + [L~v~ + (1 + vz)E'~]} z
+ E'2 + E'lvl - E'zVl
E~ =
[2(v 1 - v2)ZE'l(1 - v l ) v l ] [ L l v 1 + (1 + v2)]E'IE ~ {(1 - V l ) L z E ' 2 + [ L l v I + (1 + v2)E'l]} z
E2vx + E [
where L a = 1 -- vz - 2v 2, L z = 1 - v 1 - 2v 2, v being the Poisson ratios, v the volume fractions and the indices 1 and 2 referring to the first (0 % plasticizer) and the second (20 % plasticizer) phase of the composite. The values o f E ' c and E~, for this case of applying the law of mixtures in
308
J. Milios, G. Spathis
its complete form, are indeed higher than the values given by the simple law of mixtures, since a new term is now added to the terms provided by the simple law of mixtures. Nevertheless, this preliminary improvement of our theoretical considerations has proved to be inadequate to explain our experimental data, since the new factor is calculated, for the materials used, to add only about 2 ~ to the values of E'c and E~ predicted by the simple law of mixtures. These considerations indicate that the law of mixtures (even in its complete form) cannot explain the experimental data for the master curves of the good-adhesion bi-phase specimens, since our experimental values for the two peaks of the composite lie more than 10 ~o higher than the values predicted by the law of mixtures. Our theoretical and experimental considerations therefore imply the existence, in the interphase region, of forces of purely viscoelastic origin which cause a stress redistribution in the main phases. These viscoelastic forces are a result of the forced tendency to equalize the relaxation processes of each separate phase at their c o m m o n contact area. From a stress field point of view, this means that the interphase constitutes a locus of viscoelastic stresses. Our results show that a difficulty exists in applying the correspondence principle--given that one has solved the elastic problem--in multiphase materials, since new stress fields arise when the specimens are tested in such a range of temperatures and frequencies where the above-described viscoelastic phenomena take place. Furthermore, the results of our investigation hold, not only for the case examined in this paper, where both material phases were viscoelastic, but also in the typical case of a viscoelastic matrix reinforced with elastic fibres, since the mechanism creating the viscoelastic stresses is connected with the different time response to the applied strain of each constituent element of the composite.
REFERENCES 1. 2. 3. 4. 5. 6. 7.
S. W. Tsai, NASA CR-71, July 1964. J. C. Halpin and S. W. Tsai, A F M L - T R 67-423, June 1969. J. C. Halpin and R. L. Thomas, J. Comp. Mat., 2 (1968) p. 488. B. Paul, Trans. Metallurgical Society AIME, 218 (1960) p. 36. R. Hill, J. Mech. Phys. Solids, 11 (1963) p. 357. R. Hill, J. Mech.:Phys. Solids, 12 (1964) p. 199. Z. Hashin, Int. J. Solids Structures, 6 (1970) p. 539.
Dynamic behaviour of two-phase viscoelastic materials
309
8. Z. Hashin, Int. J. Solids Structures, 6 (1970) p. 797. 9. R. F. Gibson, J. Comp. Mat., 14 (1980) p. 155. 10. G.C. Papanicolaou, S. A. Paipetis and P. S. Theocaris, Colloid and Polymer Sci., 256(7) (1978) p. 625. 11. G. C. Papanicolaou and P. S. Theocaris, Colloid and Polymer Sci., 257 (1979) p. 239. 12. G.C. Papanicolaou, P. S. Theocaris and G. D. Spathis, ColloidandPolymer Sci., 258 (1980) p. 1231. 13. P.S. Theocaris, V. Kefalas and G. Spathis, Applied Polymer Sci., 28 (1983) p. 3641. 14. P. S. Theocaris and J. Milios, Int. J. Fracture, 16 (1980) p. 31. 15. P. S. Theocaris and J. Milios, Engng. Fracture Mech., 13 (1979) p. 599. 16. P. S. Theocaris and J. Milios, Int. J. Solids Structures, 17 (1981) p. 217. 17. P. S. Theocaris, G. Spathis and E. Sideridis, Fibre Science and Technology, 17 (1982) p. 169.
Dynamic Behaviour of Two-phase Viscoelastic Materials
J. Milios and G. Spathis Department of Theoretical and Applied Mechanics, The National Technical University of Athens, 5, Heroes of Polytechnion Avenue, 157 73 Athens (Greece)
SUMMARY Cylindrical two-phase epoxy resin specimens have been tested under oscillating loading over a wide range of temperatures and frequencies, taking into consideration the role of the interfaces between the main constituent elements of the composites. New bounds for the loss moduli, which were about 10 % higher than those predicted by the law of mixtures, have been established for the ease of specimens of good interfacial adhesion, a result which supports the assumption of the importance of the role of the interface region.
INTRODUCTION Theoretical and experimental investigation of fibre-reinforced composite materials is always accompanied by the problem of approaching or predicting the effective macromechanical properties of the composite from the mechanical properties and volume fractions of the constituent elements. The first classical approaches to the mechanical behaviour of composite materials composed of a viscoelastic matrix containing elastic fibres 1-3 were extended by investigations seeking expressions for--and, accordingly, bounds to--the elastic modulus of the composite.4 - 6 On the other hand, other investigations have applied the correspondence principle to obtain bounds for the viscoelastic moduli of the composite. 7,a 301 Composites Science and Technology 0266-3538/86/$03.50 © Elsevier Applied Science Publishers Ltd, England, 1986. Printed in Great Britain
302
J. Milios, G. Spathis
The notion of bounds has been introduced because of the substantial differences between experimental results and predicted values for a great number of composites, and there are cases where the moduli of the composite do not even fall within the bounds. 4 Apart from the above-mentioned direction in approaching the problem of the fibre-reinforced composite, by investigating the stress-strain distribution, another approach has been established lately which takes into account the effects of the interfacial interactions between the constituent elements of the composite. 9 This approach introduces the concept of a separate intermediate region, known as the interphase, between the two main phases of the composite and which possesses mechanical properties distinct from those of the two main phases. 10 - 1 2 This intermediate region was assumed to concentrate the stress gradients and singularities, as well as possible physicochemical irregularities introduced from fibre-matrix contact. Retardation spectra analysis 13 led to important conclusions on the nature of these disturbances at the surface of contact and led to the idea that a composite cannot in reality be any longer treated as an elastostatic problem and that other mechanisms of purely viscoelastic origin are responsible for its dynamic behaviour. These conclusions were supported by the results of dynamic crack propagation experiments performed in two-phase plates, 14-16 which have shown that the interphase region alone is responsible for a phenomenon of temporary crack-arrest, caused by a rapid redistribution of the stress field concentrated at the tip of the propagating crack as it reaches the interface, in all cases of crack propagation normal to a bimaterial interface. To investigate, thus, the dynamic behaviour of a fibre-reinforced composite, a suitable model has to be chosen, for which the interphase effects would be decisive. For this purpose we chose in the present investigation a model of a cylindrical composite, to ensure large interfacial areas, consisting entirely of viscoelastic materials--interesting in themselves--where a measurable interaction effect was expected to occur. E X P E R I M E N T A L P R O C E D U R E A N D RESULTS Two types of two-phase specimens were prepared from epoxy resin (Fig. 1) for dynamic measurements. The inner cylindrical phase was made
Dynamic behaviour of two-phase viscoelastic materials 20"1. plost epoxy
oxy
ztrix
303
O008m
0.00/,2 iZ
I~l.sillcor
,=,""
Good Adhesion Sod Adhesion Pure Specimen Specimen Specimen
Fig. 1. Types of specimens used for the experiments.
from 8 wt ~o amine (TETA)-cured epoxy resin and the outer one was made from 8 w t ~ TETA-cured epoxy resin with 2 0 w t ~ of Thiokol LP3 polysulphide as plasticizer.
Good adhesion specimens A cylindrical mould was used to cast the hard component of the twophase system, and 24 h later the 20 ~ plasticized part was cast around the hard kernel. The final composite was taken out of the mould after a further 24 h and it was postcured for 24 h at 130 °C. We shall refer to these composites as 'good adhesion specimens'.
Bad adhesion specimens The same procedure as before was followed in constructing the second series of specimens, the only difference being that before casting the second phase (20 ~ plasticized) the area of contact was coated with silicone oil. These composites are referred to as 'bad adhesion specimens', since the adhesion between the two phases is expected to be poor compared with that of the other type of composites. For comparison purposes the same moulds were used to prepare single component specimens purely of 8 ~ TETA-cured epoxy resin and purely of 8 ~ TETA-cured epoxy resin with 20Wt~o Thiokol LP3. The curing procedure was the same as that used for the composites of both levels of adhesion.
J. Milios, G. Spathis
304
All specimens had the same dimensions, being standard cylindrical tensile specimens convenient for dynamic testing in Dynastat and Dynalizer equipment. The dimensions of the specimens are indicated in Fig. 1. After identical curing procedures, all specimens were subjected to sinusoidal loading of prescribed and constant maximum amplitude of 1 N, over a range of temperature from 50 ° to 130 °C, applied by Dynastat and Dynalizer machines. The specimens were mounted between a long upper rod of high elastic modulus, which was connected in series with a load cell, and a shorter lower rod, coupled together with the displacement transducer. By passing a servo-controlled current through the coil of the transducer, each specimen was subjected to a sinusoidal load of prescribed frequency. The frequency range selected in our tests varied between O)min 0" l Hz and COmax 100 HZ for all specimens. All specimens were dynamically tested from the rubbery plateau, through the transition zone, and into the glassy region, the temperature ~---
=
6.5
r 0=, ee
•
O.O
xX x X X X x
x X X ° o o o XX
xx o ° ° X
."
7.5
/ 7,0 -1
..... ,-." -
X X X X
O
• •
[og~
,~
Fig. 2. Curves for the loss modulus, E", versus frequency, plotted on a logarithmic scale, reduced to To= I10°C. O, Specimen with 0 ~ plasticizer; x , specimen with 20~o plasticizer; ©, good-adhesion specimen; - - , curve predicted by the law of mixtures.
Dynamic behaviour of two-phase viscoelastic materials
305
being raised in step-wise fashion in order to accurately obtain the shift factors. By application of the time-temperature superposition principle, 17 the data were reduced to a reference temperature of 110°C. The equations used for the reduction are
Ep¢ = E'(ToPo/TP)
(1) (2)
tt
Ep = E"(ToPo/Tp)
where p and Po are the densities of the specimens at the absolute temperature of measurement, T, and the reference temperature, To, and subscript 'p' denotes the respective reduced moduli. The temperature shift factor, a T, which is required to superpose the reduced data, was the same for both loss and storage moduli for each system. In Fig. 2 the loss modulus of the 'good adhesion specimen' is plotted as a function of frequency, together with the loss moduli of the 0 % and 20 % plasticizer specimens. It is obvious from Fig. 2 that the two peaks of the 11.5
T g, 04
8.o
X X x
° o
x
X
XX
x x X X
X
7,5
X? ° X
•
Xx
7.0 -1
log~
Fig. 3. Curves of the loss modulus, E", versus frequency, plotted on a logarithmic scale, reduced to To= l l0°C. O, Specimen with 0% plasticizer; x , specimen with 20% plasticizer; (3, bad-adhesion specimen; - - , curve predicted by the law of mixtures.
306
J. Milios, G. Spathis
1.0
eeee
T
xXX x x x
xX X
x
9
O.S
X
X
X X X X
x
•
x •°
X X
X
x
x
x
X
X
X x X x
X° l
•
-1
Fig. 4.
Logw
Curves for tan 6 versus frequency (logarithmic scale) reduced to To = 110°C. O, Specimen with 0% plasticizer; x, specimen with 20% plasticizer.
c o m p o s i t e are clearly higher t h a n the predicted values f r o m the law o f mixtures (solid curve): ?t _ _
~
E c -rlE
¢!
1 + vzE~
where v I = v 2 = 0.5. In Fig. 3 the loss m o d u l u s o f the 'bad a d h e s i o n specimen' and the loss m o d u l i o f the 0 ~o and 20 % plasticizer specimens are p l o t t e d as f u n c t i o n s 1,0
T x 0.5
!
?
IK
eX Xe oX ~X
.x eX
.x 'e eoxX
, x -1
Fig. 5.
X
eeeee
.x. '~xx
x ~'x"
[ogw
Curves for tan 6 versus frequency (logarithmic scale) reduced to To = 110 °C. O, Good-adhesion specimen; x, bad-adhesion specimen.
Dynamic behaviour of two-phase viscoelastic materials
307
of frequency. Here it is clear that the two peaks of the unbonded composite are close to the predicted values from the law of mixtures. In Figs 4 and 5 the loss tangents of each type of composite specimen and the loss tangents of the 0 % and 20 % plasticizer specimens are plotted with respect to frequency. F r o m these figures it is obvious that the second peak of the more viscous material is highly depressed, a result which was expected because the high modulus of the stiffer material in the denominator of the expression determines the value of the loss-tangent of a viscoelastic specimen.
DISCUSSION Our experimental investigation has revealed unpredicted dynamic behaviour of the cylindrical bi-phase specimens. Specimens with good interfacial adhesion form master curves with peaks clearly higher than the values predicted from the law of mixtures. By contrast, the dynamic behaviour of the bad adhesion specimens is in accordance with the law of mixtures. A first explanation of this 'peculiarity' could possibly be connected to the fact that the law of mixtures in its simple form ignores the stress redistribution due to the Poisson ratio effect in the iso-strain, goodadhesion cylindrical specimens. On the other hand, in specimens of bad interfacial adhesion we can ignore the interfacial stress redistributions due to the difference of the Poisson ratios of the main phases, thereby assuming that relative motion between the phases is possible. The law of mixtures in its complete form, that is when one takes into account the Poisson ratio effect, is given by the following relations: 17
E'c= [2(v I
-
v2)2E'l(1
vl)vx]{[(1 - Vl)L2E~2] + [ L l v 1 + (1 + v2)]E'IE'2} {(1 - v ~ ) L z E ' z + [L~v~ + (1 + vz)E'~]} z
+ E'2 + E'lvl - E'zVl
E~ =
[2(v 1 - v2)ZE'l(1 - v l ) v l ] [ L l v 1 + (1 + v2)]E'IE ~ {(1 - V l ) L z E ' 2 + [ L l v I + (1 + v2)E'l]} z
E2vx + E [
where L a = 1 -- vz - 2v 2, L z = 1 - v 1 - 2v 2, v being the Poisson ratios, v the volume fractions and the indices 1 and 2 referring to the first (0 % plasticizer) and the second (20 % plasticizer) phase of the composite. The values o f E ' c and E~, for this case of applying the law of mixtures in
308
J. Milios, G. Spathis
its complete form, are indeed higher than the values given by the simple law of mixtures, since a new term is now added to the terms provided by the simple law of mixtures. Nevertheless, this preliminary improvement of our theoretical considerations has proved to be inadequate to explain our experimental data, since the new factor is calculated, for the materials used, to add only about 2 ~ to the values of E'c and E~ predicted by the simple law of mixtures. These considerations indicate that the law of mixtures (even in its complete form) cannot explain the experimental data for the master curves of the good-adhesion bi-phase specimens, since our experimental values for the two peaks of the composite lie more than 10 ~o higher than the values predicted by the law of mixtures. Our theoretical and experimental considerations therefore imply the existence, in the interphase region, of forces of purely viscoelastic origin which cause a stress redistribution in the main phases. These viscoelastic forces are a result of the forced tendency to equalize the relaxation processes of each separate phase at their c o m m o n contact area. From a stress field point of view, this means that the interphase constitutes a locus of viscoelastic stresses. Our results show that a difficulty exists in applying the correspondence principle--given that one has solved the elastic problem--in multiphase materials, since new stress fields arise when the specimens are tested in such a range of temperatures and frequencies where the above-described viscoelastic phenomena take place. Furthermore, the results of our investigation hold, not only for the case examined in this paper, where both material phases were viscoelastic, but also in the typical case of a viscoelastic matrix reinforced with elastic fibres, since the mechanism creating the viscoelastic stresses is connected with the different time response to the applied strain of each constituent element of the composite.
REFERENCES 1. 2. 3. 4. 5. 6. 7.
S. W. Tsai, NASA CR-71, July 1964. J. C. Halpin and S. W. Tsai, A F M L - T R 67-423, June 1969. J. C. Halpin and R. L. Thomas, J. Comp. Mat., 2 (1968) p. 488. B. Paul, Trans. Metallurgical Society AIME, 218 (1960) p. 36. R. Hill, J. Mech. Phys. Solids, 11 (1963) p. 357. R. Hill, J. Mech.:Phys. Solids, 12 (1964) p. 199. Z. Hashin, Int. J. Solids Structures, 6 (1970) p. 539.
Dynamic behaviour of two-phase viscoelastic materials
309
8. Z. Hashin, Int. J. Solids Structures, 6 (1970) p. 797. 9. R. F. Gibson, J. Comp. Mat., 14 (1980) p. 155. 10. G.C. Papanicolaou, S. A. Paipetis and P. S. Theocaris, Colloid and Polymer Sci., 256(7) (1978) p. 625. 11. G. C. Papanicolaou and P. S. Theocaris, Colloid and Polymer Sci., 257 (1979) p. 239. 12. G.C. Papanicolaou, P. S. Theocaris and G. D. Spathis, ColloidandPolymer Sci., 258 (1980) p. 1231. 13. P.S. Theocaris, V. Kefalas and G. Spathis, Applied Polymer Sci., 28 (1983) p. 3641. 14. P. S. Theocaris and J. Milios, Int. J. Fracture, 16 (1980) p. 31. 15. P. S. Theocaris and J. Milios, Engng. Fracture Mech., 13 (1979) p. 599. 16. P. S. Theocaris and J. Milios, Int. J. Solids Structures, 17 (1981) p. 217. 17. P. S. Theocaris, G. Spathis and E. Sideridis, Fibre Science and Technology, 17 (1982) p. 169.