Stress state, particles

It is important to note that the state determined by this analysis refers only to the pressure (or normal stress) and particle velocity. The material on either side of the point at which the shock waves collide reach the same pressure-particle velocity state, but other variables may be different from one side to the other. The material on the left-hand side experienced a different loading history than that on the right-hand side. In this example the material on the left-hand side would have a lower final temperature, because the first shock wave was smaller. Such a discontinuity of a variable, other than P or u that arises from a shock wave interaction within a material, is called a contact discontinuity. Contact discontinuities are frequently encountered in the context of inelastic behavior, which will be discussed in Chapter 5. 

Einstein coefficient b, in (5) for viscosity 2.5 by a value dependent on the ratio between the lengths of the axes of ellipsoids. However, for the flows of different geometry (for example, uniaxial extension) the situation is rather complicated. Due to different orientation of ellipsoids upon shear and other geometrical schemes of flow, the correspondence between the viscosity changed at shear and behavior of dispersions at stressed states of other types is completely lost. Indeed, due to anisotropy of dispersion properties of anisodiametrical particles, the viscosity ceases to be a scalar property of the material and must be treated as a tensor quantity. 

In an elastic material medium a deformation (strain) caused by an external stress induces reactive forces that tend to recall the system to its initial state. When the medium is perturbed at a given time and place the perturbation propagates at a constant speed (or celerity) c that is characteristic of the medium. This propagating strain is called an elastic (or acoustic or mechanical) wave and corresponds to energy transport without matter transport. Under a periodic stress the particles of matter undergo a periodic motion around their equilibrium position and may be considered as harmonic oscillators. 

A threshold level of interfacial adhesion is also necessary to produce a triaxial tensile state around rubber particles as the result of the cure process. When the two-phase material is cooled from the cure temperature to room temperature, internal stresses around particles are generated due to the difference of thermal expansion coefficients of both phases. If particles cannot debond from the matrix, this stress field magnifies the effect produced upon mechanical loading. 

In order to start the multiscale modeling, internal state variables were adopted to reflect void/crack nucleation, void growth, and void coalescence from the casting microstructural features (porosity and particles) under different temperatures, strain rates, and deformation paths [115, 116, 221, 283]. Furthermore, internal state variables were used to reflect the dislocation density evolution that affects the work hardening rate and, thus, stress state under different temperatures and strain rates [25, 283-285]. In order to determine the pertinent effects of the microstructural features to be admitted into the internal state variable theory, several different length scale analyses were performed. Once the pertinent microstructural features were determined and included in the macroscale internal state variable model, notch tests [216, 286] and control arm tests were performed to validate the model s precision. After the validation process, optimization studies were performed to reduce the weight of the control arm [287-289]. 

M.D. Dighe et al Effect of loading condition and stress state on damage evolution of silicon particles in an Al-Si-Mg-Base cast alloy. Metall. Matls. Trans. A 33, 555-565 (2002)... 

ABS and HIPS. The yield stress vs. W/t curves of ABS and HIPS are very similar. They are somewhat surprising because the yield stresses reach their respective maximum values near the W/t (or W/b) where plane strain predominates. This behavior is not predicted by either the von Mises-type or the Tresca-type yield criteria. This also appears to be typical of grafted-rubber reinforced polymer systems. A plausible explanation is that the rubber particles have created stress concentrations and constraints in such a way that even in very narrow specimens plane strain (or some stress state approaching it) already exists around these particles. Consequently, when plane strain is imposed on the specimen as a whole, these local stress state are not significantly affected. This may account for the similarity in the appearance of fracture surface electron micrographs (Figures 13a, 13b, 14a, and 14b), but the yield stress variation is still unexplained. 

Silica particles do not induce any modification of the stress state in the material, and so no extended plasticity in the matrix. Naiiosized silica particles can be considered as a modifier of polymer chain displacement, and not as a reinforcement filler. There is adsorption of PP on silica surface and consequent reduction of molecular mobility with a large increase of elastic modulus. We do not observe any process zone, and mechanisms as particle/matrix decohesion as well as crack pinning or blunting are not effective. 

For a confined powder which is composed of the particles the response to the application of stress follows the steps discussed above. That is, before the transmission of stress to an individual particle in the body of the powder is the same as if it were directly stressed, the particles have to rearrange to allow multiple points of inter-particulate contact and approach the Tully dense" state (relative to the individual particles). This results in a lag in the stress-strain prorile however the yield point and energy of deformation should remain close to that of the individual particle property once corrected for the number of particles in the powder. A schematic stress-strain prorile for a particle and for a powder made up of the particles is illustrated in Figure 2A. Figure 2B shows the same plots normalized. 

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... 

The aim of this work is to study the influence of particle size, interparticle distance, particle volume content, and local stress state on the toughening mechanism in several dispersed systems. The systems consist of a matrix of an amorphous or semicrystalline thermoplastic (see Figure 1). It is necessary to determine whether the particle diameter or the interparticle distance is of primary importance. But it is difficult to check the influence of both parameters because there is an interrelation between D, the average minimum value of A, and the particle volume content, t>P ... 

The same mechanism can appear in ABS polymers. Besides the formation of the fibrillated crazes, and depending on the matrix and local stress state, a homogeneous plastic deformation between particles, comparable to the appearance of homogeneous crazes in SAN (12, 13), is also possible (Figure 6). The homogeneous deformation in ABS is associated with cavitation inside the rubber particles. In general, this mechanism precedes the formation of the fibrillated crazes. 

The initiation point of crazes is directly at the surface of the rubber particles. Along with the formation of crazes, there is the creation of microvoids, which increase the stress concentration at the craze tip. By this stress concentration, the polymeric material at the craze tip is transformed into the craze. Thus, with craze propagation the stress state necessary for craze initiation is reproduced continuously at the craze tip in the matrix (Figure 10). 

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. 

Figure 18. Effect of interparticle distance, A, on plastic deformation of matrix strands between particles (a) definitions of the size parameters D = particle diameter, vP = particle volume content, aQ = applied stress, and aK = stress concentration (b) with a small interparticle distance, a uniaxial stress state is dominant between the particles and microvoids after cracking of the particles, and plastic yielding can be obtained and (c) with a large interparticle distance, thick matrix strands favor a triaxial stress state between the particles and microvoids, and plastic yielding is hindered.

The effect necessary for stress transformation from a triaxial stress state into a more uniaxial stress state requires relatively small interparticle distances because the stress state must be changed in the whole polymeric material between particles. This fact may explain the experimental results for critical in-... 

Case a stress-induced formation of fibrillated crazes. The weak rubber particles act as stress concentrators. Crazes are formed starting from the particle-matrix interface around the equatorial region of particles. The voids inside the crazes initiate a stress concentration at the craze tip, which propagates together with the propagating craze therefore, the crazes reproduce the stress state necessary for their propagation. Cavitation inside the rubber particles is not necessary, but it enables a higher stress concentration and easier deformation of the 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. 

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