Composite theories

Fig. 20.6. The modulus of concrete is very close to that given by simple composite theory (eqn. 20.11).

The modulus of the polymer is an average of the stiffnesses of its bonds. But it obviously is not an arithmetic mean even if the stiff bonds were completely rigid, the polymer would deform because the weak bonds would stretch. Instead, we calculate the modulus by summing the deformation in each type of bond using the methods of composite theory (Chapter 25). A stress d produces a strain which is the weighted sum of the strains in each sort of bond... 

According to the composite theory, tensile modulus of fiber reinforced composites can be calculated by knowing the mechanical constants of the components, their volume fraction, the fiber aspect ratio, and orientation. But in the case of in situ composites injection molded, the TLCP fibrils are developed during the processing and are still embedded in the matrix. Their modulus cannot be directly measured. To overcome this problem, a calculation procedure was developed to estimate the tensile modulus of the dispersed fibers and droplets as following. 

In many cases, reaction rates cannot be adequately represented by equation 6.1-1, but are more complex functions of temperature and composition. Theories of reaction kinetics should also explain the underlying basis for this phenomenon. 

In spite of the lack of spiritual powers, these two empirically originated bodies of earth and water became widely accepted into the list of principles as the seventeenth century progressed, finally making five the most common number for compositional theory. Earth and water, however, were often distinguished from the others by being labelled passive compared to the active nature of mercury, sulphur, and salt. That they were so accepted shows a significant move toward a materially based chemistry. 

The effect of polymer-filler interaction on solvent swelling and dynamic mechanical properties of the sol-gel-derived acrylic rubber (ACM)/silica, epoxi-dized natural rubber (ENR)/silica, and polyvinyl alcohol (PVA)/silica hybrid nanocomposites was described by Bandyopadhyay et al. [27]. Theoretical delineation of the reinforcing mechanism of polymer-layered silicate nanocomposites has been attempted by some authors while studying the micromechanics of the intercalated or exfoliated PNCs [28-31]. Wu et al. [32] verified the modulus reinforcement of rubber/clay nanocomposites using composite theories based on Guth, Halpin-Tsai, and the modified Halpin-Tsai equations. On introduction of a modulus reduction factor (MRF) for the platelet-like fillers, the predicted moduli were found to be closer to the experimental measurements. 

Composites theory, which has developed from classical elasticity, combined with modelling techniques may point the way forward to a complete theory of the behaviour of polymers. However, it is clear from the literature that many experimentalists do not appreciate the niceties of the mathematical theories of elasticity and of continuum mechanics, nor, in some cases, the inaccuracies inherent in their experimental methods, while nearly all theorists have no conception of the problems encountered by the experimentalist when dealing with real materials and samples of finite size. We have therefore attempted in this review to bring theory and experiment closer together by highlighting some of the problems both of the theoretician and of the experimentalist. 

In the field of semi-crystalline polymers several workers have used composite theories to explain their elastic properties in terms of those of those of the crystalline and amorphous... 

The foregoing summary of applications of composites theory to polymers does not claim to be complete. There are many instances in the literature of the use of bounds, either the Voigt and Reuss or the Hashin-Shtrikman, of simplified schemes such as the Halpin-Tsai formulation84, of simple models such as the shear lag or the two phase block and of the well-known Takayanagi models. The points we wish to emphasize are as follows. 

The size of the phases is an important factor. If they are too small to be considered as elastic continua then composites theory cannot be applied in its usual form. 

It is noted that attempts to apply composites theory to the materials investigated have not been entirely successful. While upper and lower bounds on, e g., moduli can be established there is little quantitative ediction of the impact strei th or fracture toughness parameters of the composites. Hence, the systems cannot be considered as optimized, for example, with regard to impact strength versus particle size, shape, or distribution or matrix-particle adhesion. The complexity is, of course, due to the statistical structure of the dispersed phase and the resultant uncertainties in the calculations of local stress fields, which in turn imply uncertainty in the local mode of yielding or rate of yielding. 

Conceptually, the problems associated with the optimization of specific mechanical properties by variations of structure and morphology are the same in rubber-filled systems, ass-bead filled systems and semicrystalline polymers. When the fracture properties are singled out, our understanding of the relationships between macroscopic failure and local failure is hampered by the limited knowledge of stress transfer in statistically nonhomogeneous structures. The increased use of composites theory and micromechanics to address these problems would appear to be appropriate. 

The interest in multicomponent materials, in the past, has led to many attempts to relate their mechanical behaviour to that of the constituent phases (Hull, 1981). Several theoretical developments have concentrated on the study of the elastic moduli of two-component systems (Arridge, 1975 Peterlin, 1973). Specifically, the application of composite theories to relationships between elastic modulus and microstructure applies for semicrystalline polymers exhibiting distinct crystalline and amorphous phases (Andrews, 1974). Furthermore, as discussed in Chapter 4, the elastic modulus has been shown to be correlated to microhardness for lamellar PE. In addition, H has been shown to be a property that describes a semicrystalline polymer as a composite material consisting of stiff (crystals) and soft, compliant elements. Application of this concept to lamellar PE involves, however, certain difficulties. This material has a microstructure that requires specific methods of analysis involving the calculation of the volume fraction of crystallized material, crystal shape and dimensions, etc. (Balta Calleja et al, 1981). 

Optimum Composition Theory predicts that the optimum current-voltage performance is obtained with 33 vol% of ionomer, as validated by experiments [122, 123]. The optimum catalyst utilization is, however, found with 50 vol% of ionomer, as can be seen in Fig. 15. This discrepancy is due to nonoptimal gas diffusion. Facilitating oxygen diffusion would bring the two optima together. As mentioned before, a result of the statistical theory is that the maximum catalyst utilization does not exceed 40%. 

Educational measurement theory defines large-scale assessment as a technical activity. Consequently, each aspect of an assessment situation is treated as a variable more or less within the control of the assessment designer or administrator. Composition theory, however, treats writing as a complex of activities and influences, most of which cannot be cleanly isolated for analysis or evaluation. (Lynne, 4)... 

Wilson then derived Eq. (39) based on the local composition theory. Eqs. (40) and (41) for the activity coefficient result from Eq. (39). 

The methods developed in this book can also provide input parameters for calculations using techniques such as mean field theory and mesoscale simulations to predict the morphologies of multiphase materials (Chapter 19), and to calculations based on composite theory to predict the thermoelastic and transport properties of such materials in terms of material properties and phase morphology (Chapter 20). Material properties calculated by the correlations presented in this book can also be used as input parameters in computationally-intensive continuum mechanical simulations (for example, by finite element analysis) for the properties of composite materials and/or of finished parts with diverse sizes, shapes and configurations. The work presented in this book therefore constitutes a "bridge" from the molecular structure and fundamental material properties to the performance of finished parts. 

Many books provide detailed and general treatments of composite theory. Most notable are the book of Nemat-Nasser and Hori [1] for its mathematical thoroughness, and the book of Christensen [2] for its emphasis on the engineering aspects. These books, as well as many other publications, emphasize the prediction of the properties by closed-form (analytical) expressions, as will be discussed in Section 20.B. Such methods are very quick and easy to use, and especially useful in predicting the thermoelastic properties and the transport properties of multiphase materials with morphologies that are often considerably idealized. 

Closed-form expressions based on composite theory are especially useful in correlating and predicting the thermoelastic properties (moduli and coefficients of linear thermal expansion) of multiphase materials [1,2]. An article by Tucker et al [3] with emphasis on the internally consistent combination of a set of judiciously chosen techniques to predict the thermoelastic properties of a wide variety of multiphase polymeric systems, and the review articles by Ahmed and Jones [4] and by Chow [5], provide concise descriptions of micromechanical models and are recommended to readers interested in relatively brief discussions of several popular models. 

The discussion on the residual stresses in composites in Section ll.C.8.d had already provided an example of how composite theory can be used to correlate and/or to predict the thermoelastic properties and the related performance characteristics of a multiphase polymeric system. A much broader discussion of the use of composite theory will now be provided. 

Closed-form expressions from composite theory are also useful in correlating and predicting the transport properties (dielectric constant, electrical conductivity, magnetic susceptibility, thermal conductivity, gas diffusivity and gas permeability) of multiphase materials. The models lor these properties often utilize mathematical treatments [54,55] which are similar to those used for the thermoelastic properties, once the appropriate mathematical analogies [56,57] are made. Such analogies and the resulting composite models have been pursued quite extensively for both particulate-reinforced and fiber-reinforced composites where the filler phase consists of discrete entities dispersed within a continuous polymeric matrix. 

The Wilson equation is based on local composition theories and accounts for inter-molecular interactions between a molecule and its immediate neighbors. The Wilson expression for the excess Gibbs energy and the resulting equations for the activity coefficients are... 

This equation is also based on local composition theories. The expression for the Gibbs free energy is... 

Crystallites may also be considered to act as reinforcing fillers. For example, the rubbery modulus of poly(vinyl chloride) was shown by lobst and Manson (1970,1972,1974) to be increased by an increase in crystallinity calculated moduli in the rubbery state agreed well with values predicted by equation (12.9). Halpin and Kardos (1972) have recently applied Tsai-Halpin composite theory to crystalline polymers with considerable success, and Kardos et al (1972) have used in situ crystallization of an organic filler to prepare and characterize a model composite system. More recently, the concept of so-called molecular composites —based on highly crystalline polymeric fibers arranged in a matrix of the same polymer—has stimulated a high level of experimental and theoretical interest (Halpin, 1975 Linden-meyer, 1975). 

The Halpin-Tsai model is a well-known composite theory to predict the stiffness of unidirectional composites as a functional of aspect ratio. In this model, the lorrgitudi-nal Ejj and transverse engineering modtrli are expressed in the following general form ... 

Fomes, T. D. and Paul, D. R. Modeling properties of nylon 6/clay nanocomposites using composite theories. Polymer, 44,4993-5013 (2003). 

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