Stiffening particles

Determine the expression for the modulus of a composite material stiffened by particles of any cross section but prismatic along the direction in which the modulus is desired as in Figure 3-18. 

Use the bounding techniques of elasticity to determine upper and lower bounds on the shear modulus, G, of a dispersion-stiffened composite materietl. Express the results In terms of the shear moduli of the constituents (G for the matrix and G for the dispersed particles) and their respective volume fractions (V and V,j). The representative volume element of the composite material should be subjected to a macroscopically uniform shear stress t which results in a macroscopically uniform shear strain y. 

Determine the bounds on E for a dispersion-stiffened composite material of more than two constituents, i.e., more than one type of particle is dispersed in a matrix material. 

Particle-in-cell simulation, 154 Phonon stiffening, 36 Phonon-magnon coupled mode, 39 Photo-absorption cross section, 156 Photo-induced phase transitions, 42 Photo-nuclear activation, 173 PIC, 135... 

With the introduction of particles in a polymer the stiffness increases, but the tensile strength and the impact strength are hardly increased. It is, therefore, better, to use the term stiffening additives . 

So far the micro-mechanical origin of the Mullins effect is not totally understood [26, 36, 61]. Beside the action of the entropy elastic polymer network that is quite well understood on a molecular-statistical basis [24, 62], the impact of filler particles on stress-strain properties is of high importance. On the one hand the addition of hard filler particles leads to a stiffening of the rubber matrix that can be described by a hydrodynamic strain amplification factor [22, 63-65]. On the other, the constraints introduced into the system by filler-polymer bonds result in a decreased network entropy. Accordingly, the free energy that equals the negative entropy times the temperature increases linear with the effective number of network junctions [64-67]. A further effect is obtained from the formation of filler clusters or a... 

For structuring, the IL has to be immobilised. This can be done using i.e. zeolitic structures or molecular sieves. It is obvious that with increasing surface area of the solid phase, the motion of the liquid and the proton transport will be hindered. From polymerisation experiments it is known that the stiffening of polymers by cross-linking can be compared with the polymer-surface interaction. Electrode surfaces and solids such as silica, carbon black or cathode powder also stiffen the polymer [52]. This can be explained by different transport properties at the interfaces. As a consequence it must be expected that at the surface of the added particles the ionic liquid will behave in a different way than in the immobilised liquid phase. 

One other mechanism appears to operate when fractioning paper and boxboard. The application of water to the surface will raise the coefficient of friction by lifting fibers out of the surface of the paper. If only water is applied, these fibers tend to go back into the surface if another box or paper bag is passed over its surface. However, if these fibers are coated with colloidal silica particles from the application, they are stiffened and strengthened and tend to remain upright creating a rougher surface. 

It is often convenient to stiffen or harden a material, commonly a polymer, by the incorporation of particulate inclusions. The shape ofthe particles is important [see Christensen, 1979]. In isotropic systems, stiff platelet (or flake) inclusions are the most effective in creating a stiff composite, followed by fibers and the least effective geometry for stiff inclusions is the spherical particle, as shown in Figure 41.3. A dilute concentration of spherical particulate inclusions of stiffness , and volume fraction Vj, in a matrix (with Poisson s ratio assumed to be 0.5) denoted by the subscript m, gives rise to a composite with a stiffness E ... 

The stiffness of such a composite is close to the Hashin-Shtrikman lower bound for isotropic composites. Even if the spherical particles are perfectly rigid compared with the matrix, their stiffening effect at low concentrations is modest. Conversely, when the inclusions are more compliant than the matrix, spherical ones reduce the stiffness the least and platelet ones reduce it the most. Indeed, soft platelets are suggestive of crack-like defects. Soft platelets, therefore result not only in a compliant composite, but also a weak one. Soft spherical inclusions are used intentionally as crack stoppers to enhance the toughness of polymers such as polystyrene (high impact polystyrene), with a small sacrifice in stiffness. 

In other situations, the form and average particle size of a stiffening filler like talc can be specifically tailored to fit an application s mechanical property needs. 

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