Creep rate equation

Atom Movements by Jean Philibert, Les Editions de Physique, Les Ulis Cedex A Prance, 1991. As noted already in chap. 7, this is my favorite book on diffusion. Philibert s chap. 10 is on The Study of Some Diffusion-Controlled Processes and has a thoughtful description of the approximate models used to obtain the type of creep rate equations developed in the present chapter. 

We consider the material to be in a visco-elastic state. A transient stress distribution will therefore occur after each change of the applied stress and/or temperature profile. Only very small local deformations and, thus, short times are necessary to adjust local stresses to the general continuity condition. After the transition, the whole specimen will creep in tension under the action of a radial distribution of axial stresses o(r) which assures, respecting the creep rate equation, an equal creep rate for the whole specimen. From the viewpoint of continuum mechanics, a chemically homogeneous specimen with a radial temperature gradient is indeed a "graded material" inasmuch as each coaxial shell offers a different resistance to the applied stress and has a different time constant for relaxation. We may speak of a "thermally graded material". 

Nabarro-Herring and Coble creep can take place in parallel so that actual creep rates will involve both components and both diffusion coefficients. In ceramics we also have a situation in which both anions and cations are diffusing adding further complications to the creep rate equations. If there is a large difference in the diffusion rates then the creep rate is controlled by the slower diffusing species along the faster diffusing path. 

In this section an experimentally tractable creep rate equation, long used to described deformation in metals, is suitably modified to permit empirical description of the densification kinetics of ceramics and ceramic-like materials. This particular approach is predicated upon a few simple assumptions ... 

In this section it is demonstrated that the densification of particulate materials during hot pressing may be adequately and conveniently described by a modified form of the creep rate equation developed by Zener and Hollomon ( ) ... 

Here o is the stress, A and n are creep constants and Q is the activation energy for creep. Most engineering design against creep is based on this equation. Finally, the creep rate accelerates again into tertiary creep and fracture. 

Creep rate (c) during this period can be predicted from the following equations ... 

This relationship holds for both polymers, only the constants being different. One quantity of particular interest is the rate of deflection at a given time, which is the same as the creep rate at that time for the stressed sample. By differentiating the above equation with respect to time ... 

This differential equation states that the creep rate during irradiation is directly related not only to the deflection of the unstressed sample which has occurred up to that time (D2, which may be related to the increase in polymer free volume) but also to the rate of increase of D2 with time. The reason for the dependence of the creep rate on dDo/dt is not apparent but may be related to the fact that gas is being generated at an increasing rate as time progresses (as in the case of PVC). This relationship emphasizes the strong dependence of the ac-... 

Although the flux equations for grain boundary and volume transport are of the same type, the creep kinetics are different because the boundary conditions of the transport differ for the two models (Fig. 14-3). Finally, we observe that creep in compound crystals requires the simultaneous motion of all components [R.L. Coble (1963)] so that the slow ones necessarily determine the creep rate. 

If Vt 1240 meters/sec in the matrix and branching will occur in the rubber at 29 meters/sec, we calculate A/Co = 0.047. Thus, branching can occur after a matrix crack acceleration distance of only 2 to 5/x (assuming a Griffith crack length of 50-100fi) hence, ample room for the development of fast cracks or fast crazes exists in the ABS structure. Note that the expressions for craze instability, acceleration, and speed (Equations 1, 6, 7) show that the macro strain rate of the specimen is irrelevant— fast cracks and crazes propagate in specimens strained even at slow creep rates. 

According to the equation, the diffusional creep rate of a polycrystal may be enhance by reducing the crystal size cl, and by increasing the boundary diffusivity I. Nanoceramics are therefore expected to exhibit enhanced diffusional creep for two reasons first, the reduction of the crystal size from about 10 pm to -10 nm enhances the creep rate by a factor of 109, and second, the enhanced boundary diffusivity may increase the creep rate by 103, so that the total enhancement is 1012. 

Fig. 4.20 Monkman-Grant curves for two commercial grades of silicon nitride. Some grades give curves that are temperature-independent (a) AY6, SiC -reinforced others give a series of curves depending on temperature (b) NT154. The temperature independent curves have creep rate exponents, m, for the Monkman-Grant equation, tf = ce L, that are approximately 1, whereas the creep rate exponent for the temperature-dependent curves are greater than 1 e.g., 1.7 for NT154.

Suppose that one conducts a series of experiments to determine the stress and temperature dependence of creep behavior for the fibers and matrix these experiments would provide curves such as those shown schematically in Fig. 5.6a and b. Conducting these experiments over a range of temperatures and stresses would provide a family of curves that could be combined to provide a relationship between strain rate, stress, and temperature. Such a temperature and stress dependence of constituent intrinsic creep rates, together with the intrinsic creep mismatch ratio, is schematically illustrated in Fig. 5.6c. In this plot, the creep equations for the two constituents at a given temperature and stress are represented by planes in (1 IT, logo-, logs) space, with different slopes, described by <2/> Qm and ny, nm. The intersection of the two planes represents the condition where CMR = 1, which separates temperature and stress into two regimes CMR< 1 and CMR> 1. 

In both polymers, creep of compression-molded specimens is caused mainly by crazing, with shear processes accounting for less than 20% of the total time-dependent deformation. Crazing is associated with an increasing creep rate and a substantial drop in modulus. The effects of stress upon creep rates are described by the Eyring equation, which also offers an explanation for the effects of rubber content upon creep kinetics. Hot-drawing reduces creep rates parallel to the draw direction and increases the relative importance of shear mechanisms. 

Creep and yielding are stress- and temperature-activated processes, which in many mateiMs, including pdymers, follow the Eyring rate equation ... 

Exact relationships between creep rate and diffusion coefficients have been proposed, such as the Bird-Dorn-Mukherjee equation ... 

At a shear stress close to the breaking point, or 7 kg force per cm2, -j,g creep rate was about 10 " cm mln i at V -10°C. If the boundary region is taken to be of about the thickness of the vapor adsorbed film, Xq = lOA, Equation 9 yields an effective boundary region viscosity of 6x10 poise. This value, while... 

One final note In Chap. 7 it was stated that it was of no consequence whether the flux of atoms or defects were considered. To illustrate this important notion once again, it is worthwhile to derive an expression for the creep rate based on the flux of defects. Substituting Eq. (12.11) in the appropriate flux equation for the diffusion of vacancies, i.e.. 

The creep strength of AljNb is comparatively low - a stress of 10 MN/m produces 1 % strain in only 500 h and fracture in 2300 h - whereas the yield stress compares favorably with the superalloys. This illustrates the fact that the difference between the yield stress and the creep strength is much more pronounced for intermetallics than for conventional alloys. Creep of Al3Nb is controlled by dislocation climb which is accompanied by subgrain formation. The observed creep behavior corresponds to that of conventional disordered alloys and the creep rates are described by the known constitutive equations. This will be discussed in more detail with respect to NiAl (Sec. 4.3). The secondary creep rate follows the power law, i.e. Dorn equation for dislocation creep... 

The creep behavior of the ternary B2 phase (Ni,Fe)Al was studied in detail as a function of stress, temperature, composition, and grain size (Rudy and Sauthoff, 1985 Rudy, 1986 Jung etal., 1987). At high temperatures, e.g. 60% of the melting temperature or higher, the secondary creep at rates between about 10" s" and 10" s exhibits power law behavior, i.e. the observed secondary creep rates are described by the familiar Dorn equation for dislocation creep [Eq. (2)] (Mukherjee etal., 1969). 

Similar effects have been observed in other intermetallic NiAl-base alloys with less regular distributions of hexagonal C14 Laves phases (Machon, 1992 Sauthoff, 1993 a), and have been discussed by Sauthoff (1991b). In those alloys with coarse phase distributions the observed secondary creep rates follow a rule of mixtures at a first approximation, and additional strengthening effects are only observed for alloys with fine phase distributions. From this it is concluded that particulate and nonparticulate intermetallic alloys creep in similar ways and can be described by the same constitutive equations as conventional multiphase alloys. 

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