Alternating shear stress

Cooke BJ, Matheson AJ (1976) Dynamic viscosity of dilute polymer solutions at high frequencies of alternating shear stress. J Chem Soc Faraday Trans II 72(3) 679-685 Curtiss CF, Bird RB (1981a) A kinetic theory for polymer melts. I The equation for the single-link orientational distribution function. J Chem Phys 74 2016—2025 Curtiss CF, Bird RB (1981b) A kinetic theory for polymer melts. II The stress tensor and the rheological equation of state. J Chem Phys 74(3) 2026—2033 Daoud M, de Gennes PG (1979) Some remarks on the dynamics of polymer melts. J Polym Sci Polym Phys Ed 17 1971-1981... 

The dynamic viscosity r](cai) is conveniently defined under alternating shear-stress conditions. Assuming the fluid velocity along x and the velocity gradient along z, the solvent velocity at the position of the hth atom in the absence of the chain is... 

As the number of cycles is increased, the amount of alternating shear stress that the adhesive layer can withstand is progressively decreased until the adhesive attains its service life resistance at around 20 million cycles, i.e. no further reduction in strength is observed after this point. 

Similar principles to the above apply when a liquid is subjected to an alternating shear stress. However, the measure of viscosity under shear is the rate of strain and not the strain itself, i.e. whereas Hooke s law said O = Ey, Newton s law says that CT = rj(dyfdt). Now, the phase of a differential of a periodic function leads the phase of the function itself (Figure 10.3). 

Fig. 9.14 a Fatigue S-N curves for various static mean stress normal to the planes of maximum alternating shear stress, b Relationship between alternating shear stress on the maximum shear planes and static mean stress for SAE 1045 steel (BHN 456) (mcxlified from [26])... 

B. J. Cooke and A. J. Matheson. Dynamic viscosity of dilute polymer solutions at high frequencies of alternating shear stress. J. Chem. Soc. Faraday Trans. 2, 72 (1975), 679-685. 

Two alternate core structures of the ordinary 1/2[110] dislocation, shown schematically in 1 gs. 2a amd b, respectively, were obtained using different starting configurations. The core shown in Fig. 2a is planar, spread into the (111) plame, while the core shown in Fig. 2b is non-plamar, spread concomitcmtly into the (111) amd (111) plames amd thus sessile. The sessile core is energetically favored since when a shear stress parallel to the [110] direction was applied in the (111) plane the planar core transformed into the non-plamar one. However, in a similar study emplo3dng EAM type potentials (Rao, et al. 1991) it was found that the plamar core configuration is favored (Simmons, et al. 1993 Rao, et al. 1995). 

In order to overcome the shortcomings of the power-law model, several alternative forms of equation between shear rate and shear stress have been proposed. These are all more complex involving three or more parameters. Reference should be made to specialist works on non-Newtonian flow 14-171 for details of these Constitutive Equations. 

Equation 11.12 does not fit velocity profiles measured in a turbulent boundary layer and an alternative approach must be used. In the simplified treatment of the flow conditions within the turbulent boundary layer the existence of the buffer layer, shown in Figure 11.1, is neglected and it is assumed that the boundary layer consists of a laminar sub-layer, in which momentum transfer is by molecular motion alone, outside which there is a turbulent region in which transfer is effected entirely by eddy motion (Figure 11.7). The approach is based on the assumption that the shear stress at a plane surface can be calculated from the simple power law developed by Blasius, already referred to in Chapter 3. 

Alternatives to compounding in the melt are solution mixing or powder blending of solid particles. Mixing with the aid of solvents can be performed at lower temperatures with minimal shear. However, difficulties in removal of the solvent results in plasticization of tJie polymer matrix and altered erosion/drug release performance in addition to residual solvent toxicity concerns. Powder blending at room temperature minimizes thermal/shear stresses, but achieving intimate mixtures is difficult. 

Though this new algorithm still requires some time step refinement for computations with highly inelastic particles, it turns out that most computations can be carried out with acceptable time steps of 10 5 s or larger. An alternative numerical method that is also based on the compressibility of the dispersed particulate phase is presented by Laux (1998). In this so-called compressible disperse-phase method the shear stresses in the momentum equations are implicitly taken into account, which further enhances the stability of the code in the quasi-static state near minimum fluidization, especially when frictional shear is taken into account. In theory, the stability of the numerical solution method can be further enhanced by fully implicit discretization and simultaneous solution of all governing equations. This latter is however not expected to result in faster solution of the TFM equations since the numerical efforts per time step increase. 

The standard wall function is of limited applicability, being restricted to cases of near-wall turbulence in local equilibrium. Especially the constant shear stress and the local equilibrium assumptions restrict the universality of the standard wall functions. The local equilibrium assumption states that the turbulence kinetic energy production and dissipation are equal in the wall-bounded control volumes. In cases where there is a strong pressure gradient near the wall (increased shear stress) or the flow does not satisfy the local equilibrium condition an alternate model, the nonequilibrium model, is recommended (Kim and Choudhury, 1995). In the nonequilibrium wall function the heat transfer procedure remains exactly the same, but the mean velocity is made more sensitive to pressure gradient effects. 

Another alternative is to correlate data in terms of a reduced shear stress, —-, in... 

An alternative criterion is derived by considering that the velocity of the stirrer tips, v = jrdsco, determines the shear stress on the fluid and, consequently, its micro mixing. The scale-up criterion ds(o = constant gives... 

An alternative version of the same expression describing flow-induced shear stresses for a circular pipe operating in the turbulent flow regime is described by the relation (6)... 

The constants s and c ( = 1 /s) are known as the elastic compliance constant and the elastic stiffness constant, respectively. The elastic stiffness constant is the elastic modulus, which is seen to be the ratio of stress to strain. In the case of normal stress-normal strain (Fig. 10.3a) the ratio is the Young s modulus, whereas for shear stress-shear strain the ratio is called the rigidity, or shear, modulus (Fig. 10.36). The Young s modulus and rigidity modulus are the slopes of the stress-strain curves and for nonHookean bodies they may be defined alternatively as da-/ds. They are requited to be positive quantities. Note that the higher the strain, for a given stress, the lower the modulus. 

Equation 3B.18 is known as the Weissenberg-Rabinowitsch-Mooney (WRM) equation in honor of the three rheologists who have worked on this problem. An alternate equation can be derived for fluids obeying the power law model between shear stress and the pseudo shear rate ... 

These rheological parameters have been successfully correlated to textural attributes of hardness and spreadabUity and provide information pertaining to the fat crystal network (69). The value of G is useful in assessing the solid-like stmcture of the fat crystal network. Increases in the value of G typically correspond to a stronger network and a harder fat (66). Alternatively, G" represents the fluid-like behavior of the fat system. This value can be related to the spreadability of a fat system, because increases in G" indicate more fluid-like behavior under an applied shear stress. The tan 8 is the ratio of these two values. As the value of 5 approaches 0° (stress wave in phase with stress wave), the G" value approaches zero, and therefore, the sample behaves like an ideal solid and is referred to as perfectly elastic (68). As 8 approaches 90° (stress is completely out of phase relative to the strain). 

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