Creep equations

An attempt to predict the buckling time was made using the isotropic creep equation with Mises stress potential as outlined above for the plastic liner in the warpage problem. 

The Eyring reduced time model provides the framework for the derivation of the creep equation of polymer fibres [10]. The creep shear strain of a domain... 

Probably the most plausible explanation for the scatter of the creep parameters is based on a single mechanism involving GBS with a threshold stress (Oq).7,10,13 When a threshold stress is introduced into the creep equation (16.1), all the creep parameters in YTZP become n = 2, p= 2 and Q= 460 kJ/mol whatever the stress or temperature of the test. The value of this ct0 was found experimentally 13... 

For well-oriented fibres with a single orientation angle 0 the creep equation reduces to... 

To more clearly understand the transient changes in constituent stress and strain rate that occurs between the limits of Eqns. (7) and (8), the transient creep and changes in fiber and matrix stress are plotted in Fig. 5.4 as a function of log(cr) and log(e). For convenience in presenting the interaction of the constituents, a normalized creep equation is used for each constituent ... 

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. 

Colley and Peel -S proposed the following master creep equation to describe the compression of paper ... 

Because Formula (2.45) is identical to the experimentally obtained Formula (2.41), it is obvious that the structural diagram (Figure 2.44) adequately describes the behavior of rubber concretes at compression and can be added to the basis of the creep equation of these materials. (The similar diagram corresponds to a number of plastic [rigid PVC, polyethylene] and furan polymeric concrete.)... 

By comparison with Equation 3.56 the creep equation under a constant shear stress To may he written as... 

Viscoelastic models to simulate soil creep. Equations to the right of model govern dashport behavior, (a) Murayama and Shibata (1964) (b) Schifhnan (1959) (c) Christensen and Wu (1964) (d) Singh (1966). (From Hirst, T.J., The influence of compositional factors on the stress-slrain-time behavior of soils, PhD Thesis, University of Cahfomia,... 

Table 7.1 Creep equation exponents and diffusion paths for various creep mechanisms...

DIFFUSION- TERMS IN CREEP EQUATIONS Because of the depend-... 

TABLE 17.6 Creep Equation Exponents and Diffusion Paths for Various Creep Mechanisms ... 

Creep equations can be appropriately modified as models of intermediate and final stages of sintering. For simplicity, the matter transport during creep of a dense solid is considered first. For a pure single crystal solid with cubic structure, which is a rod with a cross section of length L. Normal stresses pa nre applied to the rod on the sides, as shown in Fig. 5.23a. It is assumed that self-diffusion within the crystal... 

As shown in Fig. 5.23c, creep in ceramics can also be caused by the diffusion along the grain boundaries, with the creep equation being given by [55, 56] ... 

Hot pressing equations can be derived from the creep equations with certain modifications. In practices, sintering data are usually present as density versus time to obtain densification rate. Because sintering compacts are porous before they are fuUy densified, the creep equations developed for dense solids should be modified to be suitable for hot pressing equations. First, the creep rate that is a linear strain rate should be related to the densification rate which is a volumetric strain rate. Second, the porosity should be compensated. In hot pressing, the mass M of the powder and the cross-sectional area A of the die, are constants, while the density... 

Provided that the strains are sufficiently small, the Boltzmann superposition principle can be applied to this problem. We can treat the creep as the resultant of two responses one due to a stress a applied over the year (3.15 X 10 s) and the other due to a stress - cr applied over the final 8 months (2.10 x 10 s). Noting the restriction to small stresses and strains, we shall neglect the term in exp(0.9o ) in the creep equation, and use the simplified equation... 

The mechanical behaviour of asphalts is determined by resistance to tensile forces, known as resistance to fatigue cracking, and resistance to compressive forces, known as resistance to permanent deformation. The first is expressed by the relationship between tensile strain and number of loadings, known as the fatigue equation. The second is expressed by the relationship between compressive strain and number of loadings, known as the creep equation. 

From the Wakai s creep equation, the predicted n-values differ from 1, but this may lead to severe physical disagreement with the experimental evidence, as it yields an apparent stress exponent given by ... 

When using the creep model for duplex microstructure, as developed by French et al. [98], it is possible to fabricate a composite with a defined creep resistance that will control the grain size, the width of the layers, and the compression axis. In the case of isostrain, where the strain and strain rate are the same for each phase, and assuming the creep equation in a compact form as, e = Ajo the stress for this configuration, o, is given as ... 

In Fig. 6.38, the strain-rate dependence on flow stress is shown for the indicated temperatures. The stress exponents at each temperature are also shown on the plots. A steady-state creep equation was used to define the dependence of strain rate on the flow stress, as given in Eq. (6.4), resulting in an activation energy of 628 24 kJ/mol for diffusion-controlled deformation. 

Most creep analyses for long-term applications involve the determination of a steady-state or minimum creep rate and its examination as a function of applied stress and temperature. The steady-state or minimum, compressive creep rate (dc/ dtmin) may be related to the applied stress and temperature by an empirical Arrhenius power law or the familiar Norton-Badey creep equation . Often in creep testing, a minimal 1 % (of the expected lifetime) criterion is used. However, a 10 % criterion is preferable in order to obtain a meaningtid prediction of the usable lifetime for creep applications. In critical applications, such as for turbine components, an 25 % criterion is recommended, even preferred, so as to avoid failure by creep rupture. 

J. Rosier, E. Arzt A new model-based creep equation for dispersion strengthened materials, Acta Met. Mat. 38(4), 671 (1990)... 

Models for hot pressing by diffusional mass transport under the driving forces of surface curvature and applied stress were formulated by Coble (39,40). In one approach, the analytical models for sintering were adapted to include the effects of an applied sdess, while in a second approach, hot pressing is viewed in a manner analogous to that of creep in dense solids and the creep equations... 

Coble did not specifically adapt the intermediate and final stage sintering models to account for hot pressing, but as described below, he used a different approach based on a modification of creep equations. However, as we shall show later, if the differential equations for matter transport are formulated with the proper dependence of the chemical potential on surface curvature and applied... 

Creep in a polycrystalline solid can also occur by diffusion along the grain boundaries (Fig. 8.25c), commonly referred to as Coble creep, and for this mechanism, the creep equation is (43)... 

In the hot pressing of powders, the data are normally acquired in the form of density versus time from which the densification rate can be determined. Furtha--more, considerable porosity is present over a large part of the process. The modification of the creep equations derived for dense solids therefore attempt to incorporate these two factors (1) relating the creep rate (a linear strain rate) to the densification rate (a volumetric strain rate) and (2) compensating for the presence of porosity. 

Table 8.6 Hot Pressing Equations Obtained by Modification of the Creep Equations...

General Isothermal Creep Equation for Porous Solids... 

The procedure used in the derivation of Eq. (8.116) can also be used to analyze diffusional creep of porous solids under an applied uniaxial stress p. In this case, the mean stress on the grain boundary is 4>p, and the general creep equation has the form... 

The maximum strain developed in the cylinder, after 1.05 x 10 s (4 months) under a stress of 2.27 MPa, is 0.84 The non-linear term in the creep equation contributes an... 

---