Creep case

In structural ceramic composites, the principal effect considered was one of crack-face closure tractions, or cohesive forces, brought about, for instance, by bridging fibers. A rigorous evaluation of the crack tip fields where the crack faces are not traction free has not yet been attempted. However, an approximate approach for the small-scale creep case is to assume that the crack tip fields are not functionally altered by crack-face tractions, with the effect of the traction being only to introduce a zone of crack tip shielding. This allows for the development of preliminary models for creep crack growth which is inclusive of the role of crack bridging. These preliminary models predict that,... 

At 4 years and a stress level of 2924 psi the apparent modulus is 620,000 psi for glass-filled polypropylene according to the curve in Fig. 5-9. The new solution for the creep case first approximation is ... 

Thermal event start System delay time Temperature difference between set value and actual value Heating power Creep case Oscillating case Aperiodic limit... 

The analytical solution of Eq. (31) can be obtained using the diagonalization technique or modal projection (Kreyszig, 2006), This is fully detailed in (Fafard et al., 2001). The specific solution for the uniaxial creep case ( =cr = constant) of a cylindrical sample is... 

Finally, for the uniaxial creep case (constant loading) of a cylindrical sample, Eq. (49) can be rewritten as... 

Fig. 7. Normal stresses at the center of each component of the model (creep case).

The governing equations used in this case are identical to Equations (4.1) and (4.4) describing the creeping flow of an incompressible generalized Newtonian fluid. In the air-filled sections if the pressure exceeds a given threshold the equations should be switched to the following set describing a compressible flow... 

In this section we consider a different experimental situation the case of creep. In a creep experiment a is maintained at a constant value and the time dependence of the strain is measured. Thus it is the exact inverse of the relaxation... 

As we did in the case of relaxation, we now compare the behavior predicted by the Voigt model—and, for that matter, the Maxwell model—with the behavior of actual polymer samples in a creep experiment. Figure 3.12 shows plots of such experiments for two polymers. The graph is on log-log coordinates and should therefore be compared with Fig. 3.11b. The polymers are polystyrene of molecular weight 6.0 X 10 at a reduced temperature of 100°C and cis-poly-isoprene of molecular weight 6.2 X 10 at a reduced temperature of -30°C. 

The procedure we followed in adapting the relaxation model to experimental findings immediately suggests how to handle the discrepancies between the model and experiment in the case of creep ... 

The purpose of these comparisons is simply to point out how complete the parallel is between the Rouse molecular model and the mechanical models we discussed earlier. While the summations in the stress relaxation and creep expressions were included to give better agreement with experiment, the summations in the Rouse theory arise naturally from a consideration of different modes of vibration. It should be noted that all of these modes are overtones of the same fundamental and do not arise from considering different relaxation processes. As we have noted before, different types of encumbrance have different effects on the displacement of the molecules. The mechanical models correct for this in a way the simple Rouse model does not. Allowing for more than one value of f, along the lines of Example 3.7, is one of the ways the Rouse theory has been modified to generate two sets of Tp values. The results of this development are comparable to summing multiple effects in the mechanical models. In all cases the more elaborate expressions describe experimental results better. 

We continue the investigation of the contact problem for a plate under creep conditions. In this section the case of both normal and tangential displacements of the plate is considered. 

We shall consider an equilibrium problem with a constitutive law corresponding to a creep, in particular, the strain and integrated stress tensor components (IT ), ay(lT ) will depend on = (lT, w ), where (lT, w ) are connected with (IT, w) by (3.1). In this case, the equilibrium equations will be nonlocal with respect to t. 

Another aspect of plasticity is the time dependent progressive deformation under constant load, known as creep. This process occurs when a fiber is loaded above the yield value and continues over several logarithmic decades of time. The extension under fixed load, or creep, is analogous to the relaxation of stress under fixed extension. Stress relaxation is the process whereby the stress that is generated as a result of a deformation is dissipated as a function of time. Both of these time dependent processes are reflections of plastic flow resulting from various molecular motions in the fiber. As a direct consequence of creep and stress relaxation, the shape of a stress—strain curve is in many cases strongly dependent on the rate of deformation, as is illustrated in Figure 6. 

If S and are two-thinds the yield stress of the materials at room and operating temperatures (which is the case below the creep range), then... 

Water Treatment. Water and steam chemistry must be rigorously controlled to prevent deposition of impurities and corrosion of the steam cycle. Deposition on boiler tubing walls reduces heat transfer and can lead to overheating, creep, and eventual failure. Additionally, corrosion can develop under the deposits and lead to failure. If steam is used for chemical processes or as a heat-transfer medium for food and pharmaceutical preparation there are limitations on the additives that may be used. Steam purity requirements set the allowable impurity concentrations for the rest of most cycles. Once contaminants enter the steam, there is no practical way to remove them. Thus all purification must be carried out in the boiler or preboiler part of the cycle. The principal exception is in the case of nuclear steam generators, which require very pure water. These tend to provide steam that is considerably lower in most impurities than the turbine requires. A variety of water treatments are summarized in Table 5. Although the subtieties of water treatment in steam systems are beyond the scope of this article, uses of various additives maybe summarized as follows ... 

When a fiber is stressed, the instantaneous elongation that occurs is defined as instantaneous elastic deformation. The subsequent delayed additional elongation that occurs with increasing time is creep deformation. Upon stress removal, the instantaneous recovery that occurs is called instantaneous elastic recovery and is approximately equal to the instantaneous elastic deformation. If the subsequent creep recovery is 100%, ie, equal to the creep deformation, the specimen exhibits primary creep only and is thus completely elastic. In such a case, the specimen has probably not been extended beyond its yield point. If after loading and load removal, the specimen fails to recover to its original length, the portion of creep deformation that is recoverable is still called primary creep the portion that is nonrecoverable is called secondary creep. This nonrecoverable elongation is typically called permanent set. 

In the constant-strain method, the specimen is stretched or bent to a fixed position at the start of the test. The most common shape of the specimens used for constant-strain testing is the U-beud, hairpin, or horseshoe type. A bolt is placed through holes in the legs of the specimen, and it is loaded by tightening a nut on the bolt. In some cases, the stress may be reduced during the test as a result of creep. In the constant-load test the specimen is supported horizontally at each end... 

An account of the mechanism for creep in solids placed under a compressive hydrostatic suess which involves atom-vacancy diffusion only is considered in Nabano and Hemirg s (1950) volume diffusion model. The counter-movement of atoms and vacancies tends to relieve the effects of applied pressure, causing extension normal to the applied sU ess, and sluinkage in the direction of the applied sU ess, as might be anticipated from Le Chatelier s principle. The opposite movement occurs in the case of a tensile sU ess. The analysis yields the relationship... 

At constant displacement, creep causes stresses to relax with time. Bolts in hot turbine casings must be regularly tightened. Plastic paper-clips are not, in the long term, as good as steel ones because, even at room temperature, they slowly lose their grip. 

Metallic alloys are usually designed to resist power-law creep diffusional flow is only rarely considered. One major exception is the range of directionally solidified ( DS ) alloys described in the Case Study of Chapter 20 here special techniques are used to obtain very large grains. 

The turbine blade - a case study in creep-limited design... 

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