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Prediction of tensile yield strength of rigid inorganic particulate filled thermoplastic composites
Journal of Materials Processing Technology 83 (1998) 127 – 130
Prediction of tensile yield strength of rigid inorganic particulate filled thermoplastic composites J.Z. Liang *, R.K.Y. Li Department of Physics and Materials Science, City Uni6ersity of Hong Kong, Tat Chee A6enue, Kowloon, Hong Kong, Hong Kong Received 30 August 1997
Abstract Advances in the study of the tensile yield strength of rigid inorganic particulate filled thermoplastic composites have been reviewed and brief comments on the existing expressions for predicting it have been made in the present paper. The main factors affecting the yield strength of such composites are: (i) the interfacial adhesion between the fillers and the matrix; (ii) the shape, size and its distribution of the fillers; (iii) the dispersion of the particles in the matrix. For factor (i), this depends on a large extent upon the properties of the matrix materials and the surface pre-treatment (or modification) of the inclusions. There have been relatively few theoretical descriptions of the tensile yield-strength behaviour for rigid particulate-filled composites with interfacial adhesion lying between poor and good. © 1998 Elsevier Science S.A. All rights reserved. Keywords: Yield strength; Rigid particles; Filled composites; Thermoplastics
1. Introduction When thermoplastics are filled with particles, their morphological structure and properties will suffer change, leaving the question of how the mechanical properties of the particulate-filled thermoplastic composites are changed. This is one of the aspects in which people have been most interested. Over the last 20 years, mechanical performances such as the modulus and the fracture behaviour of composites have been widely investigated. However, as pointed out by Nielsen [1], the theory for the strength of particulate-filled systems is less developed than that for moduli. This is because there is an increased number of more complex factors to be considered, such as the state of the interface between the particles and the matrix and the distribution, size and shape of the filler particles, which affect the macroscopic behaviour of the particulatefilled composites. However, these have not been satisfactorily modeled theoretically [2]. In 1990, Ahmed and Jones [2] presented an extensive concerted survey of the theories for predicting the strength and modulus of particulate-filled polymeric composites. * Corresponding author. Tel: + 852 27844026; 27887830; e-mail: [email protected]
fax:
+ 852
0924-0136/98/$19.00 © 1998 Elsevier Science S.A. All rights reserved. PII S0924-0136(98)00051-X
Among the mechanical properties, the yield stress of the composites is of primary importantce, giving information on the maximum allowable load without considerable plastic deformation. Thus far, two (i.e. no adhesion and strong adhesion) intensive studies have been carried out on only various forms of empirical and semi-empirical expressions, such as power law, linear or other forms having been proposed. However, there have been relatively few studies on the yield strength for composites with general interfacial adhesion between the inclusion and matrix. In the present paper, the main factors affecting the yield strength of rigid inorganic particulate-filled thermoplastics are discussed and analyzed and existing expressions which best predict the yield strength from experimental data are reviewed.
2. Modeling description After polymeric materials filled with rigid inorganic particles, an interfacial layer will be formed between the inclusions and the matrix. The degree of interfacial bonding is usually divided into poor adhesion, good adhesion and some adhesion and depends, to a great
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extent, upon the properties of the matrix materials (e.g. ductile or brittle) and the shape and surface modification of the fillers.
2.1. Poor adhesion In the case of poor bonding between the matrix and the filler, one can assume that the strength of a particulate composite is determined by the effective available area of load-bearing matrix in the absence of the filler [3 – 5], i.e. the filler cannot transfer stress. Thus, the yield strength of a composites depends on the effective load bearing cross-section area fraction (1− c), i.e.: syc = sym(1− c)
(1)
If it is assumed that c is a power law function of the volume fraction of the filler, ff, then: syc = sym(1− af ) b f
(3)
Nielsen [5] has introduced a stress concentration factor, K with a suggested value of 0.5. For cubic embedded in a cubic matrix Eq. (2) can be written as: syc = sym(1− ff2/3)K
(4)
Juncar et al.[6] believed that the stress concentration depends upon the concentration of the particles, with the reduction of the effective matrix cross section being the principal factor and presented a modified form of Eq. (3) syc = sym(1− 1.21ff2/3)S
(5)
The strength reduction factor S can be determined by finite-element analysis and in general varies between 1.0 and 0.2, for low-and high filler volume fractions, respectively. Considering the effect of the particulate size, Landon et al. [7] proposed an empirical linear relationship as follows: syc = sym(1− ff)−k(ff)d
(6)
where k is the slope of the plot of tensile strength against mean particle diameter, d. Leidner and Woodhams [8] noted the contribution of particle–matrix friction and residual compressive stress, sth, to the strength of the filled composite and derived a similar equation: syc = 0.83sthlff + ksym(1 − ff)
1−c=
1−ff 1+ Aff
(8)
where A is a constant related to the packing characteristic and shape of particles. The value of A is 2.318– 2.427. If A= 2.5 is taken, then Eq. (1) can be expressed approximately as syc =
1− ff s 1+ 2.5ff ym
(9)
(2)
where syc and sym are the yield strength of the composite and the matrix, respectively and a and b are constants depending on the assumed particle shape and arrangement in the model composite. For spherical particles with no adhesion and failing by random fracture, Eq. (2) becomes [3,4]: syc = sym(1− 1.21ff2/3)
where l is the coefficient of friction. It can be seen by comparing Eqs. (1) and (3) that the matrix cross-section is zero at a filler volume fraction of less than unity. In fact, the cross-section of the matrix can be zero only at ff = 1. Considering the packing phenomenon of particles in the matrix, Turcsanyi et al. [9] chose a simple hyperbolic function to represent (1− c):
(7)
2.2. Good adhesion In the case of well-bonded particles the stress is transferred through shear: Therefore, Eq. (7) can be rewritten as [8] syc = (sa + 0.83tm )+ Ksa (1−ff)
(10)
where sa is the strength of the interfacial bond and tm is the shear strength of the matrix. Piggott and Leidner [10] argued that the uniform filler arrangement assumed in most models was unlikely in practice and proposed an empirical relationship introducing a coefficient of particle-matrix adhesion, a, as follows: syc = Ksym − aff
(11)
Some results show that syc is an increasing function of ff for the filled composites with very strong interfacial bonding. Turcsanyi, et al. [9] introduced a parameter proportional to the load carried by the dispersed component, Bsy, and presented an empirical equation syc = sym
1− ff exp (Bsyff) 1+ 2.5ff
(12)
2.3. Some adhesion In the case of some interfacial adhesion between the matrix and the particles (i.e. the interfacial adhesion is between poor and good), where the interfacial layer can transfer a small part of the stress when the deformation of the matrix is very small., then debonding between the matrix and the particles will be produced with increase in the deformation (or stress), i.e. the yield strength should be the contribution of both the matrix and the particles. Therefore, the value of a in Eq. (2) becomes smaller than 1.21. Several researchers have attempted to modify Eq. (2) for better predicting the
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yield strength of filled composites. For example, Bigg [11] proposed an empirical expression: syc = sym(1− af bf + cf df )
(13)
where c and d are the constants related to the interfacial adhesion.
3. Comparison of experimental data with theory The relative tensile yield strength (sYR) of calcium carbonate (CaCO3) filled polypropylene (PP) composites with ‘zero’ adhesion, at 23 and 80°C are shown in Fig. 1. The curve is calculated using Eq. (5). The failure mechanism is primarily by micro-cavitation and crazing throughout the whole specimen. The slightly greater values of experimental sYR when compared with the prediction probably being caused by the polydispersity, size distribution and irregular shape of the CaCO3 particles. Surface treatment with stearic acid prevents particle agglomeration; however, it reduces the effect of adhesion, giving a weak interface. The ductility of the matrix increases with increased temperature, resulting in a decreased stress-concentration effect and providing increasing yield strength. Fig. 2 shows the relationship between sYR and ff of PP filled respectively, by three types of glass beads with three different diameters: 4 (6000), 35 (3000) and 219 (1922) mm. It can be seen that the values of sYR measured from the experiments are greater than those calculated using Eq. (3), which suggests that this composite has some interfacial bonding between the matrix and the glass beads. In addition, the syc values of the system filled with smaller diameter glass beads are greater than those of the system with large diameter beads at the same composition.
Fig. 1. sYR versus ff of PP with CaCO3 [6].
Fig. 2. sYR versus ff of glass-bead-filled PP.
4. Discussion It can be see from Figs. 1 and 2 that the prediction of the values of syc by employing Eq. (3) are lower than those of the data measured in the experiments. This is because in no case is there complete lack of adhesion between the polymer matrix and the particles. However, the packing phenomenon will occur easily and stress concentration will be enhanced be increasing the content of particles. In addition, the composites will tend to brittle from ductile when the composition exceeds the critical value. Eq. (11) includes an adhesion parameter, but its value is very difficulty to measure under general conditions. Similarly, k in Eq. (6), l and sth in Eq. (7) and sa and tm in Eq. (10) are also not easy to estimate. As stated above however, the yield strength of a composite depends in a very complex way on the microstructure, including the interfacial structure, the matrix characterization and the size, shape and dispersion of the fillers in the matrix. It is not surprising that theoretical analysis of simple models gives no satisfactory description of the yield strength as a function of the composition. Fig. 3 displays the correlation between sYR calculated employing Eq. (12) and ff. When Bsy =0, this means the case of no adhesion; when Bsy is greater than 3, interfacial bonding enhances with increasing ff. Fig. 4 shows the results of the predicted and the measured values of sYR of the filled composites from the experiments. Good agreement is shown between them, although B has no direct physical meaning. It can be seen by comparing lines 1 and 2 (glass-bead-filled ABS systems) that the interfacial bond is improved when the bead surface has been pretreated, thus the relative strength of the filled composites is correspondingly increased. Chiang, et al. [12] measured the relative
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Fig. 3. sYR as a function of ff and B [9].
tensile strength of PP-g-Aa/EPDM/ mica composites, finding that good agreement between the values measured in experiments and those calculated using Eq. (12).
Fig. 4. sYR versus ff for different polymer filler systems [9]: (1) ABS/glass bead; (2) ABS/glass bead, surface treated; (3) PP/wollastonite; (4) PP/talc; (5) LDPE/CaCO3. The full lines were calculated with the following values of B: 0.246, 1.059, 2537, 3.010 and 4.512.
particulate-filled composites with interfacial adhesion lying between poor and good.
5. Conclusions The yield failure of rigid inorganic particulate-filled thermoplastic composites is a complex procedure in tension. The main factors affecting the yield strength of the composites are: (i) the interfacial adhesion between the fillers and the matrix; (ii) the shape, size and distribution of the fillers; and (iii) the dispersion of the particles in the matrix. Factor (i) depends to a great extent upon the properties of the matrix materials (e.g. ductile or brittle) and the surface pre-treatment (or modification) of the inclusions. Existing expressions predicting the tensile yield strength of particle-filled polymer composites have been usually established for the case of either no adhesion or good adhesion. There have been relatively few theoretical works on the tensile yield strength behaviour of rigid
.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
L.E. Nielsen, J. Compos. Mater. 1 (1967) 100. S. Ahmed, F.R. Jones, J. Mater. Sci. 25 (1990) 4933. L. Nicolais, M. Narkis, Polym. Eng. Sci. 11 (1971) 194. N. Nicolais, L. Nicodemo, Int. J. Polym. Mater. 3 (3) (1974) 229. L.E. Nielsen, J. Appl. Polym. Sci. 10 (1966) 97. J. Jancar, A. Dianselmo, A.T. Dibenedetto, Polym. Eng. Sci. 32 (1992) 1394. G. Landon, G. Lewis, G. Boden, J. Mater. Sci. 12 (1977) 1619. J. Leidner, R.T. Woodhams, J. Appl. Polym. Sci. 18 (1974) 1639. B. Turcsanyi, B. Pukanszky, F. Tudos, J. Mater. Sci. Lett. 7 (1988) 160. M.R. Piggott, J. Leidner, J. Mater. Sci. Lett. 18 (1974) 1619. D.M. Bigg, Polym. Compos. 8 (1987) 115. W.-Y. Chiang, W.-D. Yeng, B. Pukanszky, Polym. Eng. Sci. 32 (1992) 641.
Prediction of tensile yield strength of rigid inorganic particulate filled thermoplastic composites J.Z. Liang *, R.K.Y. Li Department of Physics and Materials Science, City Uni6ersity of Hong Kong, Tat Chee A6enue, Kowloon, Hong Kong, Hong Kong Received 30 August 1997
Abstract Advances in the study of the tensile yield strength of rigid inorganic particulate filled thermoplastic composites have been reviewed and brief comments on the existing expressions for predicting it have been made in the present paper. The main factors affecting the yield strength of such composites are: (i) the interfacial adhesion between the fillers and the matrix; (ii) the shape, size and its distribution of the fillers; (iii) the dispersion of the particles in the matrix. For factor (i), this depends on a large extent upon the properties of the matrix materials and the surface pre-treatment (or modification) of the inclusions. There have been relatively few theoretical descriptions of the tensile yield-strength behaviour for rigid particulate-filled composites with interfacial adhesion lying between poor and good. © 1998 Elsevier Science S.A. All rights reserved. Keywords: Yield strength; Rigid particles; Filled composites; Thermoplastics
1. Introduction When thermoplastics are filled with particles, their morphological structure and properties will suffer change, leaving the question of how the mechanical properties of the particulate-filled thermoplastic composites are changed. This is one of the aspects in which people have been most interested. Over the last 20 years, mechanical performances such as the modulus and the fracture behaviour of composites have been widely investigated. However, as pointed out by Nielsen [1], the theory for the strength of particulate-filled systems is less developed than that for moduli. This is because there is an increased number of more complex factors to be considered, such as the state of the interface between the particles and the matrix and the distribution, size and shape of the filler particles, which affect the macroscopic behaviour of the particulatefilled composites. However, these have not been satisfactorily modeled theoretically [2]. In 1990, Ahmed and Jones [2] presented an extensive concerted survey of the theories for predicting the strength and modulus of particulate-filled polymeric composites. * Corresponding author. Tel: + 852 27844026; 27887830; e-mail: [email protected]
fax:
+ 852
0924-0136/98/$19.00 © 1998 Elsevier Science S.A. All rights reserved. PII S0924-0136(98)00051-X
Among the mechanical properties, the yield stress of the composites is of primary importantce, giving information on the maximum allowable load without considerable plastic deformation. Thus far, two (i.e. no adhesion and strong adhesion) intensive studies have been carried out on only various forms of empirical and semi-empirical expressions, such as power law, linear or other forms having been proposed. However, there have been relatively few studies on the yield strength for composites with general interfacial adhesion between the inclusion and matrix. In the present paper, the main factors affecting the yield strength of rigid inorganic particulate-filled thermoplastics are discussed and analyzed and existing expressions which best predict the yield strength from experimental data are reviewed.
2. Modeling description After polymeric materials filled with rigid inorganic particles, an interfacial layer will be formed between the inclusions and the matrix. The degree of interfacial bonding is usually divided into poor adhesion, good adhesion and some adhesion and depends, to a great
J.Z. Liang, R.K.Y. Li / Journal of Materials Processing Technology 83 (1998) 127–130
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extent, upon the properties of the matrix materials (e.g. ductile or brittle) and the shape and surface modification of the fillers.
2.1. Poor adhesion In the case of poor bonding between the matrix and the filler, one can assume that the strength of a particulate composite is determined by the effective available area of load-bearing matrix in the absence of the filler [3 – 5], i.e. the filler cannot transfer stress. Thus, the yield strength of a composites depends on the effective load bearing cross-section area fraction (1− c), i.e.: syc = sym(1− c)
(1)
If it is assumed that c is a power law function of the volume fraction of the filler, ff, then: syc = sym(1− af ) b f
(3)
Nielsen [5] has introduced a stress concentration factor, K with a suggested value of 0.5. For cubic embedded in a cubic matrix Eq. (2) can be written as: syc = sym(1− ff2/3)K
(4)
Juncar et al.[6] believed that the stress concentration depends upon the concentration of the particles, with the reduction of the effective matrix cross section being the principal factor and presented a modified form of Eq. (3) syc = sym(1− 1.21ff2/3)S
(5)
The strength reduction factor S can be determined by finite-element analysis and in general varies between 1.0 and 0.2, for low-and high filler volume fractions, respectively. Considering the effect of the particulate size, Landon et al. [7] proposed an empirical linear relationship as follows: syc = sym(1− ff)−k(ff)d
(6)
where k is the slope of the plot of tensile strength against mean particle diameter, d. Leidner and Woodhams [8] noted the contribution of particle–matrix friction and residual compressive stress, sth, to the strength of the filled composite and derived a similar equation: syc = 0.83sthlff + ksym(1 − ff)
1−c=
1−ff 1+ Aff
(8)
where A is a constant related to the packing characteristic and shape of particles. The value of A is 2.318– 2.427. If A= 2.5 is taken, then Eq. (1) can be expressed approximately as syc =
1− ff s 1+ 2.5ff ym
(9)
(2)
where syc and sym are the yield strength of the composite and the matrix, respectively and a and b are constants depending on the assumed particle shape and arrangement in the model composite. For spherical particles with no adhesion and failing by random fracture, Eq. (2) becomes [3,4]: syc = sym(1− 1.21ff2/3)
where l is the coefficient of friction. It can be seen by comparing Eqs. (1) and (3) that the matrix cross-section is zero at a filler volume fraction of less than unity. In fact, the cross-section of the matrix can be zero only at ff = 1. Considering the packing phenomenon of particles in the matrix, Turcsanyi et al. [9] chose a simple hyperbolic function to represent (1− c):
(7)
2.2. Good adhesion In the case of well-bonded particles the stress is transferred through shear: Therefore, Eq. (7) can be rewritten as [8] syc = (sa + 0.83tm )+ Ksa (1−ff)
(10)
where sa is the strength of the interfacial bond and tm is the shear strength of the matrix. Piggott and Leidner [10] argued that the uniform filler arrangement assumed in most models was unlikely in practice and proposed an empirical relationship introducing a coefficient of particle-matrix adhesion, a, as follows: syc = Ksym − aff
(11)
Some results show that syc is an increasing function of ff for the filled composites with very strong interfacial bonding. Turcsanyi, et al. [9] introduced a parameter proportional to the load carried by the dispersed component, Bsy, and presented an empirical equation syc = sym
1− ff exp (Bsyff) 1+ 2.5ff
(12)
2.3. Some adhesion In the case of some interfacial adhesion between the matrix and the particles (i.e. the interfacial adhesion is between poor and good), where the interfacial layer can transfer a small part of the stress when the deformation of the matrix is very small., then debonding between the matrix and the particles will be produced with increase in the deformation (or stress), i.e. the yield strength should be the contribution of both the matrix and the particles. Therefore, the value of a in Eq. (2) becomes smaller than 1.21. Several researchers have attempted to modify Eq. (2) for better predicting the
J.Z. Liang, R.K.Y. Li / Journal of Materials Processing Technology 83 (1998) 127–130
129
yield strength of filled composites. For example, Bigg [11] proposed an empirical expression: syc = sym(1− af bf + cf df )
(13)
where c and d are the constants related to the interfacial adhesion.
3. Comparison of experimental data with theory The relative tensile yield strength (sYR) of calcium carbonate (CaCO3) filled polypropylene (PP) composites with ‘zero’ adhesion, at 23 and 80°C are shown in Fig. 1. The curve is calculated using Eq. (5). The failure mechanism is primarily by micro-cavitation and crazing throughout the whole specimen. The slightly greater values of experimental sYR when compared with the prediction probably being caused by the polydispersity, size distribution and irregular shape of the CaCO3 particles. Surface treatment with stearic acid prevents particle agglomeration; however, it reduces the effect of adhesion, giving a weak interface. The ductility of the matrix increases with increased temperature, resulting in a decreased stress-concentration effect and providing increasing yield strength. Fig. 2 shows the relationship between sYR and ff of PP filled respectively, by three types of glass beads with three different diameters: 4 (6000), 35 (3000) and 219 (1922) mm. It can be seen that the values of sYR measured from the experiments are greater than those calculated using Eq. (3), which suggests that this composite has some interfacial bonding between the matrix and the glass beads. In addition, the syc values of the system filled with smaller diameter glass beads are greater than those of the system with large diameter beads at the same composition.
Fig. 1. sYR versus ff of PP with CaCO3 [6].
Fig. 2. sYR versus ff of glass-bead-filled PP.
4. Discussion It can be see from Figs. 1 and 2 that the prediction of the values of syc by employing Eq. (3) are lower than those of the data measured in the experiments. This is because in no case is there complete lack of adhesion between the polymer matrix and the particles. However, the packing phenomenon will occur easily and stress concentration will be enhanced be increasing the content of particles. In addition, the composites will tend to brittle from ductile when the composition exceeds the critical value. Eq. (11) includes an adhesion parameter, but its value is very difficulty to measure under general conditions. Similarly, k in Eq. (6), l and sth in Eq. (7) and sa and tm in Eq. (10) are also not easy to estimate. As stated above however, the yield strength of a composite depends in a very complex way on the microstructure, including the interfacial structure, the matrix characterization and the size, shape and dispersion of the fillers in the matrix. It is not surprising that theoretical analysis of simple models gives no satisfactory description of the yield strength as a function of the composition. Fig. 3 displays the correlation between sYR calculated employing Eq. (12) and ff. When Bsy =0, this means the case of no adhesion; when Bsy is greater than 3, interfacial bonding enhances with increasing ff. Fig. 4 shows the results of the predicted and the measured values of sYR of the filled composites from the experiments. Good agreement is shown between them, although B has no direct physical meaning. It can be seen by comparing lines 1 and 2 (glass-bead-filled ABS systems) that the interfacial bond is improved when the bead surface has been pretreated, thus the relative strength of the filled composites is correspondingly increased. Chiang, et al. [12] measured the relative
J.Z. Liang, R.K.Y. Li / Journal of Materials Processing Technology 83 (1998) 127–130
130
Fig. 3. sYR as a function of ff and B [9].
tensile strength of PP-g-Aa/EPDM/ mica composites, finding that good agreement between the values measured in experiments and those calculated using Eq. (12).
Fig. 4. sYR versus ff for different polymer filler systems [9]: (1) ABS/glass bead; (2) ABS/glass bead, surface treated; (3) PP/wollastonite; (4) PP/talc; (5) LDPE/CaCO3. The full lines were calculated with the following values of B: 0.246, 1.059, 2537, 3.010 and 4.512.
particulate-filled composites with interfacial adhesion lying between poor and good.
5. Conclusions The yield failure of rigid inorganic particulate-filled thermoplastic composites is a complex procedure in tension. The main factors affecting the yield strength of the composites are: (i) the interfacial adhesion between the fillers and the matrix; (ii) the shape, size and distribution of the fillers; and (iii) the dispersion of the particles in the matrix. Factor (i) depends to a great extent upon the properties of the matrix materials (e.g. ductile or brittle) and the surface pre-treatment (or modification) of the inclusions. Existing expressions predicting the tensile yield strength of particle-filled polymer composites have been usually established for the case of either no adhesion or good adhesion. There have been relatively few theoretical works on the tensile yield strength behaviour of rigid
.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
L.E. Nielsen, J. Compos. Mater. 1 (1967) 100. S. Ahmed, F.R. Jones, J. Mater. Sci. 25 (1990) 4933. L. Nicolais, M. Narkis, Polym. Eng. Sci. 11 (1971) 194. N. Nicolais, L. Nicodemo, Int. J. Polym. Mater. 3 (3) (1974) 229. L.E. Nielsen, J. Appl. Polym. Sci. 10 (1966) 97. J. Jancar, A. Dianselmo, A.T. Dibenedetto, Polym. Eng. Sci. 32 (1992) 1394. G. Landon, G. Lewis, G. Boden, J. Mater. Sci. 12 (1977) 1619. J. Leidner, R.T. Woodhams, J. Appl. Polym. Sci. 18 (1974) 1639. B. Turcsanyi, B. Pukanszky, F. Tudos, J. Mater. Sci. Lett. 7 (1988) 160. M.R. Piggott, J. Leidner, J. Mater. Sci. Lett. 18 (1974) 1619. D.M. Bigg, Polym. Compos. 8 (1987) 115. W.-Y. Chiang, W.-D. Yeng, B. Pukanszky, Polym. Eng. Sci. 32 (1992) 641.