Stress concentration between particles

Figure 11. Stress concentration between particles (Top) smallest average interparticle distance, A, and stress concentration, adB> at point P1 and between the particles (point P2), as a function of rubber-particle volume content, Vp. (Bottom) stress concentration between particles for two different rubber volume contents.

Figure 17. Schematic representation of the three-stage mechanism of toughening in PA6 blends (a) elastic stress concentration, between particles and the for-...

Stress State at Particles. If the modifier particles consist of rubber-like material, they act as stress concentrators as in HIPS and ABS. Whereas in HIPS and related polymers the maximum stress component, aee, at the equatorial regions around the particles is responsible for initiating crazes, in polymers with a tendency to shear deformation the maximum shear stress at the particles, yielding the formation of shear bands, must be considered (see Figure 17a). But in contrast to crazes, which have a stress-concentrating ability, shear bands to not increase the stress between particles as effectively. Therefore, the formation of microvoids inside the particles is necessary as an additional mechanism to increase the stress at and between particles. To make the polymeric material between particles yield, not only is the stress concentration at particles necessary (as in craze formation), but so is the stress field between particles. 

Case c stress-induced formation of shear deformation. The stress concentration of the modifier particles, usually too small, is increased by the formation of cavities inside the particles. By these cavities, and if A is small enough, the originally triaxial stress state between particles is transformed into a more uniaxial stress state. Then the matrix strands between particles can be plastically deformed where the necessary volume increase, which arises from the cavitated particles, appears. 

Choosing a modifier with a solubility parameter close to that of epoxy produces a less segregated structure after cure and leads to more effective toughening. A modifier with a solubility parameter considerably dififeient from that of epoi causes phase separation of large particles which do not toughen epoxy effectively. An effective modifier has a particle surface which is compatible with epoxy one which forms an intermixed boundary reduces stress concentration between the two phases and therefore toughens effectively. 

Void formation. Owing to stress concentrations higher hydrostatic stresses are built up inside the particles, causing particles to crack and microvoids to form inside (cf. Figure 6), yielding a higher local stress concentration between the particles. 

A primary effect of the first two mechanisms is the stress concentration at softer rubber particles. Depending on the size of the shear modulus ratio between particie and matrix, the maximum stress concentration at spherical particles (and at voids) is 2.045 (i.e., the locally increased stress is 2.045 times larger than the applied stress). With increasing particle modulus (and decrease of the Poisson ratio), the stress concentration decreases. The stress concentration at particles/ voids decrease very rapidly with increasing distance r from the particle/matrix interface (with about 1/r see Fig. 5.6). If the distance r is larger than the particle diameter D, no stress concentration occurs [6]. 

The stress acting on particles is due to a relative velocity between the particles and the fluid. If their mean velocities also differ, contact between the particles or between a particle and the tank wall or the impeller elements leads to impact stress. However, this impact stress is negligible if the density differences and the particle concentrations are low. 

The mechanism by which sulphur has these observed effects is as follows. Immersion of native magnesium oxychloride cement in water brings about a slow dissolution which creates pores. When those pores are filled with sulphur, sites of possible stress concentration at points of contact between particles are modified. Similar effects occur when sulphur is used to impregnate hydraulic cements based on Portland cement and silica (Beaudoin, Ramachandran Feldman, 1977). 

Concentrated particle suspensions may also show a yield point which must be exceeded before flow will occur. This may result from interaction between irregularly shaped particles, or the presence of water bridges at the interface between particles which effectively bind them together. Physical and chemical attractive forces between suspended particles can also promote flocculation and development of particle network structures, which can be broken down by an applied shear stress [2]. 

Figure 13.9 Sequence of events in a croid formation, (a) Initial state at the crack tip. (b) Cavitation ofthe rubber particles dueto loading head of the crack tip. (c) Cavitation of rubber particles near the already cavitated particles due to stress-concentration effect. The croid is forming, (d) Croids are propagating ahead ofthe crack and inside the craze-like damaged zone many shear bands develop between cavitated rubber particles. (Sue, 1992 with kind permission from Kluwer Academic Publisher.)...

When the particle concentration is high, the shear motion of particles leads to interparticle collisions. The transfer of momentum between particles can be described in terms of a pseudoshear stress and the viscosity of particle-particle interactions. Let us first examine the transfer of momentum in an elastic collision between two particles, as shown in Fig. 5.8(a). Particle 1 is fixed in space while particle 2 collides with particle 1 with an initial momentum in the x-direction. Assume that the contact surface is frictionless so that the rebound of the particle is in a form of specular reflection in the r-x-plane. The rate of change of the x-component of the momentum between the two particles is given by... 

Many industrial processes are affected by the influence of particulate materials on the flow properties of material. Flow properties of materials can be adjusted by fillers to meet the requirements. Flow properties can also be adversely affected by numerous phenomena related to the presence of filler in formulations.One common example is related to the flow of industrial slurries which contain concentrated suspensions of small particles. Such suspensions are usually non-Newtonian fluids with a yield stress which is formed through strong interactions between particles. During flow, these interactions are continuously broken and rebuilt. A solid deposit formed on the slopes and walls is an adverse effect of this property. 

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. 

Fig. 8.2. Representative data associated with four of the key strengthening mechanisms within a material, (a) Increase in flow stress as a function of concentration of substitutional impurities, (b) Dependence of flow stress on mean particle size for material in which there are second-phase particles, (c) Dependence of yield stress on mean grain size of material, (d) Relation between yield stress and mean dislocation density. (Adapted from (a) Neuhauser and Schwink (1993), (b) Reppich (1993), (c) Hansen (1985), (d) Basinski and Basinski (1979).)...

Local stress concentration in the matrix at and between particles (depending on particle shape, particle content, Youngs modulus, and Poisson s ratio of matrix polymer and particles)... 

Electron Microscopic Results. The fundamental deformation step is the formation of crazes at the rubber particles (Figure 4). The crazes start directly at the interface between rubber particles and matrix in the equatorial zones around the particles, that is, in the zones of highest stress concentration. The structure of the amorphous material is transformed by local plastic defor-... 

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