Stress Dependency

For some materials the linear constitutive relation of Newtonian fluids is not accurate. Either stress depends on strain in a more complex way, or variables other than the instantaneous rate of strain must be taken into account. Such fluids are known collectively as non-Newtonian. Many different types of behavior have been observed, ranging from fluids for which the viscosity in the Navier-Stokes equation is a simple function of the shear rate to the so-called viscoelastic fluids, for which the constitutive equation is so different that the normal stresses can cause the fluid to flow in a manner opposite to that predicted for a Newtonian fluid. 

Many industrially important fluids cannot be described in simple terms. Viscoelastic fluids are prominent offenders. These fluids exhibit memory, flowing when subjected to a stress, but recovering part of their deformation when the stress is removed. Polymer melts and flour dough are typical examples. Both the shear stresses and the normal stresses depend on the history of the fluid. Even the simplest constitutive equations are complex, as exemplified by the Oldroyd expression for shear stress at low shear rates ... 

Fluids without any sohdlike elastic behavior do not undergo any reverse deformation when shear stress is removed, and are called purely viscous fluids. The shear stress depends only on the rate of deformation, and not on the extent of derormation (strain). Those which exhibit both viscous and elastic properties are called viscoelastic fluids. 

Purely viscous fluids are further classified into time-independent and time-dependent fluids. For time-independent fluids, the shear stress depends only on the instantaneous shear rate. The shear stress for time-dependent fluids depends on the past history of the rate of deformation, as a result of structure or orientation buildup or breakdown during deformation. 

Y.M. Gupta, Stress Dependence of Elastic-Wave Attenuation in LiF, J. Appl. Phys. 46, 3395-3401 (1975). 

There is considerable literature on material imperfections and their relation to the failure process. Typically, these theories are material dependent flaws are idealized as penny-shaped cracks, spherical pores, or other regular geometries, and their distribution in size, orientation, and spatial extent is specified. The tensile stress at which fracture initiates at a flaw depends on material properties and geometry of the flaw, and scales with the size of the flaw (Carroll and Holt, 1972a, b Curran et al., 1977 Davison et al., 1977). In thermally activated fracture processes, one or more specific mechanisms are considered, and the fracture activation rate at a specified tensile-stress level follows from the stress dependence of the Boltzmann factor (Zlatin and Ioffe, 1973). 

Figure 8.11. Fragment size and fracture stress dependence on tensile loading strain rate for oil shale.

The theoretieal fraeture parameters in (8.22) and (8.23), based on a model assuming an inherent power law fracture flaw distribution and a constant fracture growth velocity, can be determined with the strain rate dependent fracture data in Fig. 8.11 (Grady and Kipp, 1980). Using the fracture data for oil shale provides a value of m = 8 and a fracture stress dependence on strain... 

So much for the stress dependence of P. But what of its volume dependence We have already seen that the probability of one sample surviving a stress <7 is Ps(Vq). The probability that a batch of n such samples all survive the stress is just P fl/g) . If these n samples were stuck together to give a single sample of volume 1/ = nVo then its survival probability would still be (P fl/o) . So... 

TA2 mode ([IlO] polarization) visibly depends on the influence of the electrons on the lattice vibrations (see Fig. 7). Without V2, the stiffness of the TAi mode is enlarged. Therefore, we can conclude that this TA2 mode is very sensitive to changes in the electronic structure and measures the stability of the B2 phase. This may be a hint of the temperature or stress dependence of the Cu-Zn system in the martensitic region (Zn < 42 at%) close to the stoichiometric concentration CuZn... 

Figure 7 Intensity lines from (a) neutron (temperature dependent, courtesy Zheludev et al. ) and (b) x-ray elastic scattering experiments (stress dependent, courtesy Martynov et al. ) showing the existence of a satellite at 1/6 [110] corresponding with the modulation wavelength in the ISO s visible in figure 6.

S.M. Shapiro, Svensson E.C., Vettier C. and Hennion B., Uniaxial-stress dependence of the phonon... 

Does yield stress depend on temperature Probably, not, and flow curves constructed at different temperatures look as is shown in Fig. 7, where the arrow indicates the direction of temperature increase. 

The situation becomes most complicated in multicomponent systems, for example, if we speak about filling of plasticized polymers and solutions. The viscosity of a dispersion medium may vary here due to different reasons, namely a change in the nature of the solvent, concentration of the solution, molecular weight of the polymer. Naturally, here the interaction between the liquid and the filler changes, for one, a distinct adsorption layer, which modifies the surface and hence the activity (net-formation ability) of the filler, arises. Therefore in such multicomponent systems in the general case we can hardly expect universal values of yield stress, depending only on the concentration of the filler. Experimental data also confirm this conclusion [13],... 

How does yield stress depend on a filler concentration It is shown in Fig. 9 that appreciable values of Y appear beginning from a certain critical concentration cp and then increase rather sharply. Though the existence of cp seems to be quite obvious from the view point of the possibility of contacts of the filler, i.e. the beginning of a netformation in the system, practically the problem turns on the accuracy of measuring small stresses in high-viscosity media. It is quite possible to represent the Y(cp) dependence by exponential law, as follows from Fig. 10, for example, leaving aside the problem of the behavior of this function at very low concentrations of the filler, all the more the small values of are measured with a significant part of uncertainty. 

How does yield stress depend on the size of particles We have mentioned above that increasing the specific surface, i.e. decreasing an average size of particles of one type, causes an increase in yield stress. This fact was observed in many works (for example [14-16]). Clear model experiments the purpose of which was to reveal the role of a particle s size were carried out in work [8], By an example of suspensions of spherical particles in polystyrene melt it was shown that yield stress of equiconcentrated dispersions may change by a hundred of times according to the diameter d of non-... 

It is well known that mechanical loads on a structure induce stresses within the material such as those shown in Fig. 2-4. It is also well known that the magnitudes of these static and dynamic stresses depends on many factors, including forces, angle of loads, rate and point... 

The response of a plastic to an applied stress depends on the temperature and the time at that temperature to a much greater extent than does that of a metal or ceramic. The variation of an amorphous TP over an extended temperature range can be exemplified by the behavior of its elastic modulus as a function of temperature. 

In semi-cristalline polymers, rate-enhancement under stress has been frequently observed, e.g. in UV-photooxidation of Kapron, natural silk [80], polycaprolactam and polyethylene terephthalate [81]. Quantitative interpretation is, however, difficult in these systems although the overall rate is determined by the level of applied stress, other stress-dependent factors like the rate of oxygen diffusion or change in polymer morphology could occur concurrently and supersede the elementary molecular steps [82, 83], Similar experiments in the fluid state showed unequivocally that flow-induced stresses can accelerate several types of reactions, the best studied being the hydrolysis of DNA [84] and of polyacrylamide [85]. In these examples, hydrolysis involves breaking of the ester O —PO and the amide N —CO bonds. The tensile stress stretches the chain, and therefore, facilitates the... 

Fig. 16 a. Stress dependence of the molar fraction of the P form (x ) and b) bulk stress-strain curve measured for uniaxially oriented polybutyleneterephtalate [83]... 

Maximum aiiowabie stress depends sharpiy on temperature. 

The special flow conditions in circular (capillaries, tubes) or rectangular channels cause very different stresses depending on the position of the particles in the flow cross section. With laminar flow, for example the following applies to velocity gradient (see e.g. [37]) ... 

The stress depends on the extent of reaction, p(tf), which progresses with time. However, it is not enough to enter the instantaneous value of p(t ). Needed is some integral over the crosslinking history. The solution of the mutation problem would require a constitutive model for the fading memory functional Gf Zflt, t p(t") which is not yet available. This restricts the applicability of dynamic mechanical experiments to slowly crosslinking systems. 

A stress-dependent viscosity model, which has the same general characteristics as the Carreau model, is the Meter model (Meter, 1964) ... 

Many other data in the literature show a strong dependence of creep compliance on the applied load, although in some cases the authors did not discuss this aspect of creep. Stress dependence is found with all kinds of plastics. For example, the creep of polyethylene has been studied by... 

It can be shown that for o>2 GPa a constant transition density function I=I0 yields almost the same stress dependence of the creep rate as the linear function. Therefore, in order to keep the calculations tractable we derive the lifetime of a fibre by applying the same density transition function as was used in the calculation of the dependence of the strength on the load rate, viz. I(U)=IQ on the interval [U0, Um and I(U)=0 elsewhere. This results for the shear strain of a domain in... 

In Chapter 8, Stavola and Pearton discuss the local vibrational modes of complexes in Si that contain hydrogen or deuterium. They also show how one can use applied stress and polarized light to determine the symmetry of the defects. In the case of the B-H complex, the bond-center location of H is confirmed by vibrational and other measurements, although there are some remaining questions on the stress dependence of the Raman spectrum. The motion of H in different acceptor-H complexes is discussed for the Be-H complex, the H can tunnel between bond-center sites, while for B-H the H must overcome a 0.2 eV barrier to move between equivalent sites about the B. In the case of the H-donor complexes, instead of bonding directly to the donor, H is in the antibonding site beyond the Si atom nearest to the donor. The main experimental evidence for this is that nearly the same vibrational frequency is obtained for the different donor atoms. There is also a discussion of the vibrational modes of H tied to crystal defects such as those introduced by implantation. The relationship of the experimental results to recent theoretical studies is discussed throughout. 

The uniaxial stress dependence of the IR absorption and Raman bands due to the B—H complex have been studied by Bergman et al. (1988b) and by Stutzmann and Herrero (1988a,b) and Herrero and Stutzmann (1988, 1989). The observed stress splittings and their interpretation for the two studies do not agree. Here, the two experiments and their interpretation will be summarized. While the differences between the experiments will be discussed, they will not be resolved. 

It was proposed (Herrero and Stutzmann, 1988a,b) that the H is not on the trigonal axis of the B—H complex but that it is displaced toward the C site to explain the splitting observed for a [100] stress. The stress dependence of the relative intensities of the stress-split components of the stretching band was proposed to be due to stress-induced alignment of the... 

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