Linear Rule of Mixtures

Young s modulus of the block copolymer fibers compared favorably with that of the physical blends as shown in Table 6, and in general they follow the linear rule of mixtures. The modulus data suggested that one does not need very large PBZT molecules to have the reinforcing efficiency. From the tensile data, one clear trend is that the tensile strength of the block copolymer system is much... 

DENSITY. The specific gravity of all formulations was measured by a displacement method in accordance with ASTM D792. The density composition data for Groups 1,2,and 3 are shown in Figure 1. Straight lines were drawn for each system based on a linear rule of mixtures. ... 

Observed IPN densities, higher than those predicted by a linear rule of mixtures assumption, have been interpreted to indicate the degree of molecular mixing (7) and hence the extent of system interpenetration. Kim et al (8) have explained the increased density of IPNs qualitatively by means of chain entanglements at the domain boundaries. 

Figure 3 illustrates the relationship between Vickers hardness and composition in Zr02-Ni system. The hardness decreases remarkably with the rise of Ni content and conforms with linear rule of mixture approximately. However, the slope of the hardness curve changes at 40 vol% Ni, which seems to reflect that the matrix phase alters from PSZ to Ni at about that composition. 

Deviations of equations (12.48), (12.50), and (12.51) from a linear rule of mixtures are shown in Figure 12.31 for a typical case considered by Nielsen (1967a), who used the following values of properties for the calculations ... 

The early work on properties of molecular composites was done by Hwang et al. (1983a, b). They performed experimental studies of solution processing of films based on poly(p-phenylene benzobisthiazole) with both poly(2,5 (6 ) benzimidazole (ABPBI) and poly(2,5 (6 ) benzothiazole) (ABPBT). The films were shown to possess very high modulus and strength values, which improve upon heat treatment. The uniaxial modulus of highly oriented molecular composites was shown to follow the linear rule of mixtures. 

The common fillers used in plastics are minerals (densities from 2.4-2.S g/cm ), which give a composite of higher density than that of the unfilled polymer (densities of 0.8-1.9 g/cm ). The density of a composite of known composition can be calculated according to the linear rule of mixtures (Equation 8.1), where p,., pf and pp are the densities of the composite, filler and matrix, respectively, and mf is the mass fraction of filler. 

The stiffness and strength should be predictable by the rule of mixtures, i.e. the volume fraction of filler will be the ultimate determining factor. In practice this ideal state is very difficult to obtain, due to various processing problems, especially affecting dispersion and particle alignment. (Note, the linear rule of mixtures also only strictly applies for two components that have the same Poisson s ratios and have interfacial coherence [31]). 

Fig. 2.6. The coefficient of linear thermal expansion a at room temperature of Ni-Fe alloys containing 0.4% Mn and 0.1% C. The dotted line denotes the value of a predicted by a linear rule of mixtures. (After Eskin and Pritze, 1940.)...

Then, applying the linear rule of mixtures for the alloy, it follows that EsiGe 125 GPa and i siGe = 0.276. The mismatch stress is then... 

The Fox equation relationship (Rg. 7.10) defines or describes the composition dependence of the Tg of ideal co-polymer or ideal polymer blends (or miscible blends of two polymers) in which no strong interactions are involved. The Fox equation typically is expected to lead to a lower value of Tg than would be given by a simple linear Rule of Mixtures (Blend Inherent Viscosity vs. % Component in the Solution Blends) and reflects the effective higher free volume or randomness due to the presence of two components in the mixture ... 

The stress-strain curves for cortical bones at various strain rates are shown in Figure 5.130. The mechanical behavior is as expected from a composite of linear elastic ceramic reinforcement (HA) and a compliant, ductile polymer matrix (collagen). In fact, the tensile modulus values for bone can be modeled to within a factor of two by a rule-of-mixtures calculation on the basis of a 0.5 volume fraction HA-reinforced... 

Although the linear coefficient of thermal expansion varies with temperature, it can be considered constant within typical design and processing conditions. It is especially high for polyolefins, where it ranges from 1.5 x 10 4K-1 to 2 x 10 4K 1 however, fibers and other fillers significantly reduce thermal expansion. A rule of mixtures is sufficient to calculate the thermal expansion coefficient of polymers that are filled with powdery or small particles as well as with short fibers. In this case, the rule of mixtures is written as... 

Equation 3.126 represents the rule of mixture for stresses. It is valid only for the linear elastic region of the stress-strain curve (see Figure 3.2). Since [Pg.335]

Linear thermal expansion testing helps to determine if failure by thermal stress may occur in products and materials. Precise knowledge of the CTE can be utilized to estimate the thermal stresses. This aspect makes CTE to an important property of the used fiber for composite materials. A rule of mixtures is sufficient for calculating the CTE of polymers filled with powder or short fibers. In case of long libers, the rule of mixtures is valid perpendicular to the reinforcing fibers. Molecular orientation affects the thermal expansion of polymers. Processing also affects CTE, for semicrystalline polymers this fact is very important. For that reason, CTE measurements are often used to predict shrinkage in injection moulded parts. 

Density was measured with a bulk mass/volume method using the same flexure specimens that were used in elastic modulus measurements. A total of five specimens were used for each of alumina contents. Figure 4 depicts density as a function of alumina content for both particulate and platelet composites [3, 4]. Density decreased linearly with increasing alumina content, yielding good agreement with the prediction based on the rule of mixture. The difference in density between particulate and platelet composites was negligible. 

Figure 12.35. Comparison of experimental values of linear coefficient of expansion of a Ti02-filled epoxy resin with values predicted by equation (12.54). The dashed curve is for the rule of mixture, the solid curve for the predictions of equation (12.54). Temperature range -50 C to 100°C df = 1 x 10" °C Ef = 7.25 x lO psi Vf = 0.23 dip = 55 X lO" C = 7.25 x 10 psi = 0.3. (Wang and Kwei,...

The linear additivity of moduli expressed by eqn 6.14 is another example of the rule of mixtures applying (see Section 6.1). The second term in the equation, however, makes only a small contribution, since Ex-... 

This view of traditional composite micromechanics, underlies the widely accepted rule-of-mixtures approach to modeling fiber reinforced composite materials. It states that the modulus of the composite is a linear combination of the moduli of the materials from which it is composed, and weights each modulus with the volume fraction of that component. Its basis lies in continuity of parallel strain between the fibers and matrix provided a linearly elastic response of the composite occurs for small strains. 

Rule of Mixtures Properties P depend on respective contributions of fibers Pf and matrix P, fiber volume fraction V/, geometric effectiveness G, rule of mixtures assumes that contributions of components can be summed linearly in proportion to respective fractions,... 

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