Creep theory

Before we enter upon the discussion of the lifetime of a polymer fibre, a brief presentation is given of the creep theory of polymer fibres according to the continuous chain model [7-10]. The total fibre strain is given by the sum of the... 

For strong type-II superconductors a difference may exist between B 2 obtained from ac-susceptibility data and the thermodynamically relevant upper critical field, 5c2, for which (2.13) and (2.14) were derived originally. Prom extensive work on high-Tc cuprates it is known that B 2 is related to the so-called irreversibility line [196]. The basic idea of a semiquantitative flux-creep theory [197] is that pinned vortices can be activated thermally over an energy barrier Uq resulting in a reduced critical current of the form [196]... 

The temperature dependence of a was obtained in experiments on copper polycrystals [ ]. From Cu creep curves, values of a were determined in the temperature range 1.4 to 4.2 K for a constant r of 24 MPa. Figure 2 shows a plotted against temperature. From the logarithmic creep theory, based on the thermally activated creep assumption [ one expects the linear- and temperature-proportional behavior of the a factor to be... 

Certain important features of the a(T) curve are (1) there is a temperature dependence of a, a(T although weak and (2) the a(T) curve tends toward saturation in the very low temperature range and, if extrapolated to T = 0 K, ends in a finite nonzero value of ao- Contrary to the thermally activated creep theories, this suggests that there is a creep at 0 K. On the one hand, the temperature dependence of a is apparent on the other hand, its behavior differs from the predictions of the classical thermally activated creep laws. This leads one to assume that at 4 K, creep depends on two mechanisms, and as absolute zero is approached, the athermic component becomes more important [ ]. 

The second set of measurements studied the dependence of the creep rate ratio 82/81 on AT, obtained as a result of a temperature change of AT = T2—T1. It is readily seen that the classical thermally activated creep theory leads to the following relation between 82/81 and AT. 

The peculiarities of f.c.c. and h.c.p. metal creep appeared to be explainable in terms of the low-temperature creep theory developed by Natsik et al, [ ], which allows for a quantum fluctuation effect upon crystal lattice barriers surmounted by dislocations moving through the crystal. 

The rednction of ronghness on the surface is given by, in analogy to conventional creep theory. 

This chapter discusses the theory of thermally activated plastic flow including creep theory, components of the stress strain curve vs. strain rate and temperature and, isothermal and thermomechanical fatigue. Finally, the physical basis for modeling reliability of interconnects under thermal cycling is discussed. 

Second, it is easier to relax the scale stresses of the nanocrystalline coatings. According to the diffusional-creep theory, the deformation rate, e, of a polycrystalline material at a temperature at which grain-boundary diffusion predominates can be given as... 

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. 

Master curves can be used to predict creep resistance, embrittlement, and other property changes over time at a given temperature, or the time it takes for the modulus or some other parameter to reach a critical value. For example, a mbber hose may burst or crack if its modulus exceeds a certain level, or an elastomeric mount may fail if creep is excessive. The time it takes to reach the critical value at a given temperature can be deduced from the master curve. Frequency-based master curves can be used to predict impact behavior or the damping abiUty of materials being considered for sound or vibration deadening. The theory, constmction, and use of master curves have been discussed (145,242,271,277,278,299,300). 

Not only are the creep compliance and the stress relaxation shear modulus related but in turn the shear modulus is related to the tensile modulus which itself is related to the stress relaxation time 0. It is therefore in theory possible to predict creep-temperature relationships from WLF data although in practice these are still best determined by experiment. 

The theory relating stress, strain, time and temperature of viscoelastic materials is complex. For many practical purposes it is often better to use an ad hoc system known as the pseudo-elastic design approach. This approach uses classical elastic analysis but employs time- and temperature-dependent data obtained from creep curves and their derivatives. In outline the procedure consists of the following steps ... 

It is shown that solute atoms differing in size from those of the solvent (carbon, in fact) can relieve hydrostatic stresses in a crystal and will thus migrate to the regions where they can relieve the most stress. As a result they will cluster round dislocations forming atmospheres similar to the ionic atmospheres of the Debye- Huckel theory ofeleeti oly tes. The conditions of formation and properties of these atmospheres are examined and the theory is applied to problems of precipitation, creep and the yield point."... 

By 1969, when a major survey (Thompson 1969) was published, the behaviour of point defeets and also of dislocations in crystals subject to collisions with neutrons and to the eonsequential collision cascades had become a major field of researeh. Another decade later, the subjeet had developed a good deal further and a highly quantitative body of theory, as well as of phenomenological knowledge, had been assembled. Gittus (1978) published an all-embracing text that eovered a number of new topics chapter headings include Bubbles , Voids and Irradi-ation(-enhanced) Creep . 

Viscoelastic and rate theory To aid the designer the viscoelastic and rate theories can be used to predict long-term mechanical behavior from short-term creep and relaxation data. Plastic properties are generally affected by relatively small temperature changes or changes in the rate of loading application. 

Linear viscoelasticity Linear viscoelastic theory and its application to static stress analysis is now developed. According to this theory, material is linearly viscoelastic if, when it is stressed below some limiting stress (about half the short-time yield stress), small strains are at any time almost linearly proportional to the imposed stresses. Portions of the creep data typify such behavior and furnish the basis for fairly accurate predictions concerning the deformation of plastics when subjected to loads over long periods of time. It should be noted that linear behavior, as defined, does not always persist throughout the time span over which the data are acquired i.e., the theory is not valid in nonlinear regions and other prediction methods must be used in such cases. 

Note that the term y in Eqs. 2-15 and 2-16 has a different significance than that in Eq. 2-14. In the first equation it is based on a concept of relaxation and in the others on the basis of creep. In the literature, these terms are respectively referred to as a relaxation time and a retardation time, leading for infinite elements in the deformation models to complex quantities known as relaxation and retardation functions. One of the principal accomplishments of viscoelastic theory is the correlation of these quantities analytically so that creep deformation can be predicted from relaxation data and relaxation data from creep deformation data. 

Rate theory An alternate method available involves the manipulation of the rate theory based on the Arrhenius equation. This procedure requires considerable test data but the indications are that considerably more latitude is obtained and more materials obey the rate theory. The method can also be used to predict stress-rupture of plastics as well as the creep characteristics of a material, which is a strong plus for the method. 

In computing ordinary short-term characteristics of plastics, the standard stress analysis formulas may be used. For predicting creep and stress-rupture behavior, the method will vary according to circumstances. In viscoelastic materials, relaxation data can be used in Eqs. 2-16 to 2-20 to predict creep deformations. In other cases the rate theory may be used. 

In many cases, a product fails when the material begins to yield plastically. In a few cases, one may tolerate a small dimensional change and permit a static load that exceeds the yield strength. Actual fracture at the ultimate strength of the material would then constitute failure. The criterion for failure may be based on normal or shear stress in either case. Impact, creep and fatigue failures are the most common mode of failures. Other modes of failure include excessive elastic deflection or buckling. The actual failure mechanism may be quite complicated each failure theory is only an attempt to explain the failure mechanism for a given class of materials. In each case a safety factor is employed to eliminate failure. 

Rhee, H.-K., and Amundson, N. R., Equilibrium theory of creeping profiles in fixed-bed catalytic reactors. Ind. Eng. Chem. Funda. 13, 1-4 (1974). 

Creep tests give extremely important practical information and at the same time give useful data on those interested in the theory of the mechanical properties of materials. As illustrated in Figure 1, in creep tests one mea-... 

Creep and stress-relaxation tests measure the dimensional stability of a material, and because the tests can be of long duration, such tests are of great practical importance. Creep measurements, especially, are of interest to engineers in any application where the polymer must sustain loads for long periods. Creep and stress relaxation are also of major importance to anyone interested in the theory of or molecular origins of Viscoelasticity. 

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