Steady creep

Equations 9.6 and 9.8 were derived by Stokes and are known as Stokes s equations for steady creeping flow round a sphere. 

Stokes s solution (S9) for steady creeping flow past a rigid sphere may be obtained directly from the results of the previous section with co. The same results are obtained by solving Eq. (3-1) with Eqs. (3-4) to (3-6) replaced by the single condition that Uq O a.tr = a. The corresponding streamlines are shown in Figs. 3.3a and 3.4a. As for fluid spheres, the particle causes significant... 

By analogy with the solution for steady creeping flow, we assume that the stream function relative to the particle takes the form... 

For CO 0, Eq. (11-7) reduces to the stream function for steady creeping flow past a rigid sphere, i.e., Eq. (3-7) with k = co. The parameter 3 may be regarded as a characteristic length scale for diffusion of vorticity generated at the particle surface into the surrounding fluid. When co is very large, 3 is small, and the flow can be considered irrotational except in the immediate vicinity of the particle. In the limit co go, Eq. (11-7) reduces to Eq. (1-29), the result for potential flow past a stationary sphere. 

The results of Experiment II show that when the sample is first irradiated under no stress and then the stress is applied, the creep rate has an initially high value but decays to a steady rate after a few minutes. In contrast to this behavior, the same steady creep rate is achieved in a much shorter time—e.g., less than 30 seconds in Experiment III for the same stress and radiation conditions in this latter case, the beam... 

The previous paragraph has made it clear that if there are elastic fibers and a constant macroscopic stress is applied, the longitudinal creep rate will eventually fall to zero. With constant transverse stresses applied as well, the process of transient creep will be much more complicated than that associated with Eqns. (27) and (28). However, it can be deduced that the longitudinal creep rate will still fall to zero eventually. Furthermore, any transverse steady creep rate must occur in a plane strain mode. During such steady creep, the fiber does not deform further because the stress in the fiber is constant. In addition, any debonding which might tend to occur would have achieved a steady level because the stresses are fixed. 

In practical terms this means that, near the melting point, a stress of 50 bars causes steady creep at a rate near i per cent per minute, this rate decreasing about a factor of 10 for a 10 degC fall in temperature. 

The differential equation system (DES) for viscous, incompressible media under conditions of steady, creeping flow with no allowance for gravity reads [4], [11] ... 

Ji3) has units of 1/G, although they are not equivalent for most cases. With this change of variables. Fig. 14 may be drawn. For steady creeping regime, i.e., when the slope of the curve becomes constant. 

Rigid sphere-. When Re —> 0, the nonlinear inertia terms in the Navier-Stokes equations vanish and an analytical solution exists. The steady creeping flow past a rigid sphere was first determined by... 

Fluid sphere Studies by Hadamard [4] and Rybczynski [5] have addressed the problems of steady creeping flow past a fluid sphere analytically. The stream functions representing the motion are given by... 

For example, for Sn-3.8Ag-0.7Cu, the creep mechanism contour line that gives equal contribution from the two creep rates ( 50% GBC, 50% MC line) is close to the 75 ""C (167 °F) creep rate line, at least up to about 25 to 30 MPa (3.63 to 4.35 ksi). This is the same temperature that Ref 51 identified as the dividing point for the analysis of steady-creep rates, fitting SAC creep data with separate models in the two temperature ranges -25 to 75 T ( -13 to 167 °F), and 75 to... 

As indicated in Figure 3.3.2c, a steady state creep compliance Jf is defined by extrapolation of the limiting slope to r = 0. The slope is the inverse of the viscosity at low shear rate, ri . Thus, in the steady creeping regime, we have... 

Time response of different rheological systems to applied forces. The Maxwell model gives steady creep with some post stress recovery, representative of a polymer with no cross-linking. The Kelvin-Voigt model gives a retarded viscoelastic behavior expected from a cross-linked polymer. 

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