Rubber particles modulus

At the present time it is generally accepted that the toughening effect is associated with the crazing behaviour.Because of the presence of the low-modulus rubber particles most of the loading caused when a polyblend is subject to mechanical stress is taken up by the rigid phase (at least up to the moment of... 

Figure 3.9. Rubber particle straddling craze perpendicular to stress is subjected to triaxial stresses and because of its high bulk modulus becomes load bearing. (After Bucknall )...

Compo- sition number Resin type/parts per 100 parts of mbber (phr) Sulfur (phr) Y Method of preparation Cross- link density, r/2 (moles X 10 per ml of mbber) Rubber particle size (pm) Young s modulus (MPa) Stress at 100% strain (MPa) Tens. str. (MPa) UlL elong. (%) Tens. set (%)... 

The extremely high energy dissipation in the craze layer, 5 X 108 ergs/gram, would lead to an adiabatic temperature rise of about 24°C during crack/craze propagation, which is insufficient to cause most matrix polymers to pass through their T0 at usual environmental temperatures. Introduction of rubber particles can only lower this temperature rise since rubber secant modulus is very low and rubber deformation does not exceed the 300% or so of the deformed matrix. 

The actual stress distribution around a rubber particle has been calculated for the case of a rubber particle embedded in a rigid matrix, following Goodier (14). Assuming the shear modulus of the matrix, / i, is much greater than the shear modulus of the rubber, /a2, we find ... 

Profusion of branching should be proportional to number of rubber particles greater in size than the minimum discussed above. At a given rubber content, the number of rubber particles varies as the reciprocal of the third power of particle diameter. Thus, number of particles drops rapidly as particle size climbs above the effective minimum. Laboratory tests show that stiffness properties depend on total rubber content irrespective of particle size (provided the specimen dimensions are large compared with particle dimensions) hence, narrow particle size distribution is essential if maximum toughness is to be combined with minimum loss in stiffness properties (modulus, creep). 

Since these rubber particles are highly filled with a homopolymer or a copolymer, the rubber is already reinforced with a resin to give a higher modulus particle than the grafted rubber latex. On the basis of the uniqueness of these rubber particles, this process is also more appropriate in manufacturing high-strength medium-impact ABS polymer (31), or rubber-reinforced styrene-methyl methacrylate copolymer (32). The... 

An additional problem arises in attenqiting to predict the moduli of rubber-modified dastics the rubber particles are themselves composite in structure. A typfeal hi inqiact polystyrene (HIPS) m t contain only 6 v(d.% of pcdybutadi-ene, whereas the volume fraction of rubber particles is between 20 and 30%. The additional vcdume comes from sniall sub-inclusions of polystyrene (PS) embedded in the rubber. The problem of relating modulus to structure in rubber-toughened plastics is discussed in a recent review ). 

Goodier (42) showed that for a particle which possesses a considerably lower shear modulus than the matrix, the maximum stress concentration occurs at the equator of the particle. Rubbers are commonly found to undergo cavitation quite readily under the action of a triaxial tensile stress field. Thus, the microvoids are produced by cavitation around the rubbery particles during fatigue crack propagation, together with localized plastic deformation due to interaction between the stress field ahead of the crack and the rubber particles. 

The major, and most important, advantage of being able to use high values of the Poisson ratio is that this implies that the value of K of the rubber particle is high. Indeed, input values for of 1 MPa and v of 0.49992 imply a value of K of about 2 GPa, which is of the order expected for a rubbery polymer (19). Thus, the values of the bulk modulus, K, and of the shear modulus,... 

Figure 6. Comparison of experimental and predicted values of Youngs modulus of epoxy resin toughened with rubber particles. Finite-element predicted values were calculated using two values of the Poisson ratio of the rubber phase and the predicted bounds using the Ishai and Cohen model (26).

The stress distributions for the different properties of the rubber sphere, for this pure hydrostatic applied stress, have been found to be unique functions of the bulk modulus, K, of the rubber (27). In other words, for a given volume fraction, the values of maximum stress for the different rubber properties fall on single curves when plotted as functions of the bulk modulus of the rubber. The relationships are shown for a 20% volume fraction of rubber in Figure 8 the values plotted are the hydrostatic stress in the rubber particle and the maximum von Mises stress in the epoxy, occurring at the interface. The results shown in Figure 8 demonstrate that the hydrostatic stress in the rubber sphere increases steadily with increasing values of K of the rubber, although the rate of increase is lower as the value of K rises. When the value of K of the rubber equals that of the epoxy annulus (i.e., 3.333 GPa), the model responds as an isotropic sphere and the stress state is pure hydrostatic tension. The maximum von Mises stress in the epoxy annulus decreases relatively... 

Figure 8. Variation of different types of maximum stresses with the bulk modulus, K, of the rubber particle. The applied pure hydrostatic tension is 100 MPa, and the volume fraction of rubber is 20%.

Figure 9 shows the variation of hydrostatic stress in the rubber particle as a function of the volume fraction of the rubber phase, for two values of K, 2.083 and 0.0167 GPa, of the rubber particle. These values of K are equivalent, for example, to the value of E of 1 MPa for both cases, and values of v of 0.49992 and 0.490 for the higher and lower values of K, respectively. The hydrostatic stresses for the higher value of K for the rubber particle are about 40 times greater than those for the lower value of K. For both values of K, the hydrostatic stress in the rubber increases slightly as the volume fraction of rubber phase increases. Thus, the hydrostatic stress is greater in the rubber particle when its bulk modulus is higher, as expected. 

Stress Bate at Particles. The stress component, avv, acting parallel to the boundary between rubber particles and matrix is important for the initiation of crazes. It reaches a maximum value (which can be about twice the outer stress, a0) at the equatorial regions of the particles. Besides depending on the shape of the particles and Poisson s ratio, the elastic-stress concentration at the rubber particles depends mainly on the ratio x = Gp/GM, where GP and GM are the Youngs modulus of the particles and the matrix, respectively. This ratio has been calculated by Michler (14) on the basis of the solution obtained by Goodier for an isolated particle embedded in a matrix and subjected to uniaxial tension (15) (see Figure 9). 

The stress concentration at rubber particles starts to increase by the superposition effect if the interparticle distance, A, is smaller than about 0.5D, that is, if the rubber volume content is higher than 15 vol% (Figure 11). There is an increasing tendency of craze formation with increasing stress concentration, which means with decreasing particle modulus and increasing rubber volume content. 

Figure 9. Dependence of the size of the stress component, cree> at rubber particles on the ratio Gp/GM, where GP and GM are the Youngs modulus of the particles and the matrix, respectively.

Figure 12. Dependence of the modulus of rubber particles on particle diameter in HIPS (a) increase of the length-to-width ratio of particles with increasing diameter (b) decrease of the modulus, G, of particles with increasing diameter. The modulus is shown as the ratio between G and G3fJLW the value of G of large particles.

The result of the diameter-dependent modulus is an increased stress concentration and increased tendency to craze initiation with increasing particle diameter. This result is demonstrated in the micrograph of an ABS polymer in Figure 7, which shows the preferred craze formation at the largest rubber particles. Therefore, toughened materials with a broad particle-diameter distribution or a bimodal diameter distribution often show preferred craze initiation at the largest particles, which has the disadvantage of reduced effectiveness (23). The maximum formation of crazes appears in a material with rubber particles of optimum diameter, Dopt and a small diameter distribution. 

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