Generalized Creep Expression

The steady-state creep can be expressed by a general formula [2] given by Equation 15.42. 

In this equation, G is the shear modulus, r is the grain size exponent, and p is the stress exponent as indicated in Equation 15.42. Based on nation 15.42, the creep behavior of ceramics can be divided into two regimes  

A low-stress, small-grain-size regime where the creep rate is a function of grain size and the stress exponent is unity. Rearranging Equation 15.42, we get  

Normalized creep rate versus normalized stress for alumina. The dotted Line is the predicted line according to Nabarro-Herring creep. (From W. R. Cannon and T. G. Langdon, /. Mater. Sci., 18,1-50,1983.) 

When the stress is high, the creep rate becomes independent of grain size. That is, r becomes equal to zero. Now, Equation 15.43 becomes  

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The solder alloys are often subjected to steady-state creep regime under typical thermomechanical loading conditions. Steady-state creep deformation can generally be expressed by the following relationship (Ref 8) for low- and medium-stress values ... 

The limiting value of the creep is equal to 00 = a0/E = Voigt-Kelvin element is only able to describe qualitatively the creep behaviour of rubberlike materials with a limited creep and not the creep of an elastic liquid. In general the creep compliance may be expressed as... 

Consider the tensile experiment of Fig. 11 -12a as a creep study in which a steady stress To is suddenly applied to the polymer specimen. In general, the resulting strain c(t) will be a function of time starting from the imposition of the load. The results of creep experiments are often expressed in terms of compliances rather than moduli. Tlie tensile creep compliance D(t) is... 

Usually creep test results are expressed in terms of the compliance J° = Axlt)/Ay ylt). For an ideally elastic interface J° =1/G°. However, for visco elastic films J°(t) l/G lt). General relations exist by which J°(t) can be converted into G (t) and vice versa, but for an accurate conversion J°(t) has to be known over quite a long range of time. In dilation, creep tests are hard to perform except for elastic interfaces, because in such a test a constant Ay should be applied to the interface and this is experimentally very difficult. 

Now, we have expressed the general streamfunction, (7-149), and the disturbance flow contribution in (7-150) and (7-151), in terms of spherical coordinates. However, we have not yet specified a body shape. Thus the linear decrease of the disturbance flow with distance from the body must clearly represent a property of creeping-flows that has nothing to do with specific coordinate systems. Indeed, this is the case, and the velocity field (7-151) plays a very special and fundamental role in creeping-flow theory. It is commonly known as the Stokeslet velocity field and represents the motion induced in a fluid at Re = 0 by a point force at the origin (expressed here in spherical coordinates).17 We shall see later that the Stokeslet solution plays an important role in many aspects of creeping-flow theory. 

We shall see that the stokeslet solution plays a fundamental role in creeping flow theory. We have already seen in Section E of Chap. 7 that it describes the disturbance velocity far away from a body of any shape that exerts a nonzero force on an unbounded fluid. Indeed, when nondimensionalized and expressed in spherical coordinates, it is identical to the velocity field, (7 151). In the next section we use the stokeslet solution to derive a general integral representation for solutions of the creeping-flow equations. 

In general, the steady-state creep of ceramics may be expressed in the form " ... 

The general case of this expression cannot be given in closed form, but it is nevertheless possible to obtain the value of da/de for the cases for which M has been calculated as a function of e Y (= e2 Y/R) by graphical methods, and then plot d as a function of e. This has been done in Fig. 21 for the model with = 6. The curves obtained show, as expected from the basic assumptions for the model, a decreasing sensitivity to rate as the rate of extension increases. At very low rates, the curve appears to become almost parallel to the strain-axis this is only an illusion because the gradient of the stress-strain curve at such slow rates is determined by the relative values of m and /, and the lengths p and N. The particular case selected has a creep rate only slightly smaller than the strain rate at this level of stress. 

From the previous sections you can see that there are a large number of creep mechanisms. These can be expressed by one general equation ... 

Nevertheless, in the case of drying, the moisture content only decreases and some simplifications apply. Here, only the most common way to express creep and mechano-sorptive effect will be presented. The general formulation... 

Trantina (3) analyzed the creep behavior of talc-filled polypropylene composites under constant uniaxial stress loads. He proposed a general expression for the strain ec attributed to creep alone as a superposition of time (f), stress (a), and temperature (T) dependence. The model equation is written as a product of three separate functions for each parameter ... 

To improve the performance of SiC and to increase its resistance against creep failure, generally various constituents are added to monolithic SiC ceramics. Additives in various shapes and sizes are usually added to SiC to achieve a better material for structural use and to extend its service lifetime. An evaluation of creep failure, commonly referred to as creep rupture or stress rupture , is a critical step in evaluating the suitability of a certain ceramic for use in the desired application. The stress rupture and creep properties of a SiC matrix reinforced with SiC fiber (i.e., a SiC/SiC composite) has been evaluated by tests conducted in order to assess the propensity of SiC/SiC for high-temperature appfications over an extended lifetime. In Fig. 6.105, plots of stress versus time-to-rupture are shown for several temperatures. As commonly done, these plots are on a log-log scale. Each curve can be fitted by means of an empirical relation, similar to the earlier exponential equation expressing the time-to-rupture, h, to a stress exponent for stress rupture as ... 

In a general way the ratio of strain to stress at a certain time is expressed as creep compliance D(t) as shown in Eq. (3). 

Due to the complicated geometry, the solution of the flow equation for more than one particle at the interface is prohibitively complex and can only be obtained for the case of two particles by applying elaborate numerical techniques. In general, the solution of the creeping flow (Stokes) equation can be expressed in the integral form [106]... 

These equations express the close connection between the creep and relaxation functions, either of which, it will be recalled, can be determined in principle from knowledge of the other. This point is discussed further, in general terms, in Sect. 1.5. How it may be done in specific cases is discussed in Sect. 1.6. 

This relationship determines h P) and therefore the creep function in terms of the relaxation function. A similar equation obtained by interchanging the creep and relaxation parameters expresses g a) in terms of A (a). This relationship, in somewhat different, and slightly more general, form, was given by Gross (1953). Ferry (1970) discusses theoretically based forms of the function J a). In many cases, (1.6.45) does not give analytic expressions and so it may be that approximate methods are more useful, for example than that of Rapp and Romas ko (1972) who assume that (1.2.Ip) is an equality. 

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