Example calculations composite moduli

Today, thermosets such as epoxy resins are frequently reinforced with either glass fibers or carbon fibers (also known as graphite fibers) (8,9). As an example calculation, the modulus of a unidirectional glass fiber-epoxy composite will be calculated in both the longitudinal and transverse directions. The properties of the materials are... 

However, not aU composite deck boards would satisfy even this low support span requirement. Trex deck board, for example, has flexural modulus equal to 175,000 psi (hsted on Trex technical data sheet). Using Eq. (7.45), one can calculate that at L = 12 in., I = bhVl2 = 0.895 (b = 5-1/2", h = 1-1/4"), load at the deflection of L/180 =... 

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). 

While there has been limited investigation of mechanical properties of three-component (polymer/polymer/filler) nanocomposites, it would appear that the final result is strongly dependent on the morphology of the system. For example, if two polymers are immiscible, and nanopartides are localized in one of the domains, the modulus could be evaluated in two stages (i) separately calculate the modulus of the phase containing particles and (ii) utilize composite theories to determine the overall modulus of the entire system. 

Example 3.2 PEEK is to be reinforced with 30% by volume of unidirectional carbon fibres and the properties of the individual materials are given below. Calculate the density, modulus and strength of the composite in the fibre direction. 

Example 3.5 Calculate the transverse modulus of the PEEK/carbon fibre composite referred to in Example 3.2, using both the simplified solid mechanics approach and the empirical approach. For PEEK = 0.36. 

Rapidly solidified in-situ metal matrix composites. A design project for alloys based on the Fe-Cr-Mo-Ni-B system, and produced by rapid solidification, was undertaken by Pan (1992). During processing a mixture of borides is formed inside a ductile Fe-based matrix which makes the alloys extremely hard with high moduli. These alloys provide a good example of how phase-diagram calculations were able to provide predictions which firstly helped to identify unexpected boride formation (Saunders et al. 1992) and were ultimately used in the optimisation of the modulus of a shaft material for gas turbines (Pan 1992). 

Values of Example were calculated for the constituting domains of SEBS (PS and PEB) and for the nanoclay regions in the SEBS/clay nanocomposite using (6) and are provided in Table 2. The modulus of the clay platelets was found to be 100 MPa, whereas the modulus for PS and PEB blocks was determined to be 22 and 12 MPa, respectively. These modulus values tallied with the slow strain-rate macromechanical tensile data of 26 MPa for the SEBS/clay nanocomposite (Table 2). The lower calculated modulus values of nanoclays compared to the literature might be due to adhering soft rubber on the nanoclays, which reduces the overall modulus of clay regions in the composite. 

The relationship between the structure of the disordered heterogeneous material (e.g., composite and porous media) and the effective physical properties (e.g., elastic moduli, thermal expansion coefficient, and failure characteristics) can also be addressed by the concept of the reconstructed porous/multiphase media (Torquato, 2000). For example, it is of great practical interest to understand how spatial variability in the microstructure of composites affects the failure characteristics of heterogeneous materials. The determination of the deformation under the stress of the porous material is important in porous packing of beds, mechanical properties of membranes (where the pressure applied in membrane separations is often large), mechanical properties of foams and gels, etc. Let us restrict our discussion to equilibrium mechanical properties in static deformations, e.g., effective Young s modulus and Poisson s ratio. The calculation of the impact resistance and other dynamic mechanical properties can be addressed by discrete element models (Thornton et al., 1999, 2004). 

Theory. Basic theories for the prediction of the modulus of a composite from those of the components were derived by, for example, Hashin in 1955 (I), Kerner in 1956 (2), and van der Poel in 1958 (3). Takayanagi (4, 5), and Fujino et al. (6) developed a very promising and instructive model theory which includes calculation of the loss spectra of composites, and it may easily be extended to anisotropic morphologies. Furthermore, Nielsen and coworkers may be cited for fundamental theoretical and experimental contributions (7, 8, 9,10). 

If the stresses in the composite vary only slightly, the deformations may be calculated using an effective modulus of elasticity, E(t). which may, for example, be determined from Figure 4.13. 

An example of the application of theoretical models is the calculation of Young s modulus of a two-phase composite. Both phases behave as linear elastic and homogeneous materials according to Hooke s model, but having different Young s moduli and 2. They are randomly distributed in the space with respective volume contents and 1 2, + V2 = 1. To obtain... 

Another approach is to calculate cost-to-performance ratios for diverse materials and/or compositions. For example, it may be asked how much a unit of the tensile modulus or the strength at yield will cost. Since the performance, P(wi), usually changes with composition on a logarithmic scale one can write ... 

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