Creep law

We consider a boundary value problem for equations describing an equilibrium of a plate being under the creep law (1.31)-(1.32). The plate is assumed to have a vertical crack. As before, the main peculiarity of the problem is determined by the presence of an inequality imposed on a solution which represents a mutual nonpenetration condition of the crack faces... 

It was found, [13], that the concept of the distributed damage, addressed by L.M. Kachanov [lA], allows us to account for this discrepancy Introducing the damage parameter Y varying from 1 (no damage) to 0 (total damage), the creep law can be presented, [15], as... 

Consider a continuous fiber-reinforced ceramic as a multiphase system where the individual phases are parallel to one another and to the uniaxial loading direction. The fibers (or fiber bundles), matrix, and interface zone are treated as individual phases. In general, each phase undergoes elastic-plastic (creep) deformation. In the present analysis, the creep rate of each phase, e is assumed to obey a general creep law of the following form... 

Creep laws for materials with long intact fibers are relevant to cases where the fibers are unbroken at the outset, and never fracture during life. As a model, it also applies to cases where some but not all of the fibers are broken so that some fibers remain intact during service. Obviously these situations would occur only when the manufacturing procedure can produce composites with many or all of the fibers intact. 

In the problem of the creep of materials with intact unidirectional fibers, as shown in Fig. 9.1, most of the insights arise from the compatibility of the strain rates in the fibers and in the matrix. When a stress composite parallel to the fibers, the strains and strain rates of the fibers and the matrix in the z-direction must be all the same.5 This gives rise to a creep law of the form... 

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 [ ]. 

Table 15.1 Creep laws as predicted by Wakai s model [81], depending on the steps density and the controlling mechanism.

Equation (6.5) may also be expressed in logarithmic terms and many transient regimes of creep curves may be fitted to a logarithmic law when n = 1. In the extreme case, when n = 1, which is often observed experimentally, one obtains the logarithmic creep law as ... 

Considering Eq. (6.4), the experimental value of the stress exponent, n, obtained for the Al204Mg single crystal is 3.9 with an activation energy of Q — 5.3 eV. The creep range of this single crystal is 0.65 T -0.71 T and it follows a dislocation mechanism of the creep law. More specifically, the values of... 

Table 6.4 Creep law parameters [86] (with kind permission of Elsevier)...

Finally, we can see the relationship between the response given by the hereditary integral form and that given by conventional creep laws such as the logarithmic form... 

Therefore the logarithmic creep law employs an averaged relaxation spectrum for all elapsed time (i.e., only one time-dependent mechanism is assumed) as shown in Fig. 2.16, which may cause difficulty under real, complex situations such as the long term behavior of rock. On the other hand the power law (2.292) gives... 

Sketch a spring-and-dashpot model suitable to describe creep deformation Consider a material with Young s modulus E and the creep law e = Act . Calculate the time-dependence of the strain in a retardation experiment. Due to its low melting temperature, lead creeps already at ambient temperatures. A thin-walled lead tube fixed at its ends bends under its own weight in the course of time. Estimate by how much the centre of the tube is displaced within one year ... 

Creep TL-15-3 follows a stress e tonential creep law Properties at 430 C (800 F). Product aged 8 h at 495 °C (925 °F) fits the following equation ... 

In an early attempt by Pao and Marin [6] at a three-dimensional extension of their uniaxial creep law Equation (11.1), strains in the principal directions are given by terms such as... 

Here, T is a temperature-dependent constant, and p is the creep law exponent having values between 1 and 8. For p > 1, we have power law creep. There are three mechanisms—diffusion, viscous, and dislocation creeps. 

We understand that the traditional mechanical models (law of linear elasticity described by an elastic modulus whose value depends on the temperature associated with a creep law) do not allow the satisfactory description of the behavioral laws of such materials by taking into account the damage and viscoplastic effects. To take these effects into account, it is necessary to use behavior models using... 

There are two important special cases of Eqs. (26) and (30). For a solid which obeys Norton s creep law, i.e., creep rate is proportional to stress to the power n,... 

The aim of creep characterization in this case is to determine defoimation behavior at high temperature. It is feasible to describe deformation of complete laminas using anisotropic creep laws which will be discussed in this paragraph. As a supplement more information about the interaction between fibers and matrix can be gained by assigning isotropic creep laws to matrix and fibers on a microscopic scale (next section). 

Figure 10. Resulting strain rates of the unit cell simulations at a compression stress of 50 MPa at 1473 K. For both fiber and matrix a Norton creep law was used (parameters Table III).

Quite likely the sliding process is mainly responsible for the high creep rates of the 30 , 45 or 60° unit cells. Yet, a Norton aeep law was assigned to both fibers and matrix. Assuming that the matrix dominates creep in the 90 samples, its creep behavior can be better described by a primary creep law with regard to the experimental results (90 curve. Figure 4 left) ... 

A Norton creep law seems still appropriate for die fibers. After a short transient regime, experiments with fiber-dominated creep (0 curve Figure 4, left) owed an almost constant aeep rate. Therefore new simulations with different creep parameters for fiber and matrix (Table IV) were conducted. Figure 13 shows the resulting creep curves for the diosen parameters. 

Table IV Simulation creep constants for equation (7) Fibers are assigned to a strain-independent creep law, die matrix to a primaiy creep law.

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