Creep models fibers

This review is intended to focus on ceramic matrix composite materials. However, the creep models which exist and which will be discussed are generic in the sense that they can apply to materials with polymer, metal or ceramic matrices. Only a case-by-case distinction between linear and nonlinear behavior separates the materials into classes of response. The temperature-dependent issue of whether the fibers creep or do not creep permits further classification. Therefore, in the review of the models, it is more attractive to use a classification scheme which accords with the nature of the material response rather than one which identifies the materials per se. Thus, this review could apply to polymer, metal or ceramic matrix materials equally well. 

Boltzmann s constant, and T is tempeiatuie in kelvin. In general, the creep resistance of metal is improved by the incorporation of ceramic reinforcements. The steady-state creep rate as a function of appHed stress for silver matrix and tungsten fiber—silver matrix composites at 600°C is an example (Fig. 18) (52). The modeling of creep behavior of MMCs is compHcated because in the temperature regime where the metal matrix may be creeping, the ceramic reinforcement is likely to be deforming elastically. 

DiCarlo, J. A., Creep Stress Relaxation Modeling of Polycrystalline Ceramic Fibers, NASA, 1994. 

The 1-D concentric cylinder models described above have been extended to fiber-reinforced ceramics by Kervadec and Chermant,28,29 Adami,30 and Wu and Holmes 31 these analyses are similar in basic concept to the previous modeling efforts for metal matrix composites, but they incorporate the time-dependent nature of both fiber and matrix creep and, in some cases, interface creep. Further extension of the 1-D model to multiaxial stress states was made by Meyer et a/.,32-34 Wang et al.,35 and Wang and Chou.36 In the work by Meyer et al., 1-D fiber-composites under off-axis loading (with the loading direction at an angle to fiber axis) were analyzed with the... 

To gain a better understanding of the creep behavior of fiber-reinforced ceramics, a simple 1-D analytical approach will be used to examine the effects of constituent behavior on composite creep deformation and changes in internal stress. Since the derivation of the model provides valuable insight into the parameters that influence composite creep behavior, the derivation of the 1-D concentric cylinder model will be outlined first. 

Fig. 5.2 Comparison of creep behavior and time-dependent change in fiber and matrix stress predicted using a 1-D concentric cylinder model (ROM model) (solid lines) and a 2-D finite element analysis (dashed lines). In both approaches it was assumed that a unidirectional creep specimen was instantaneously loaded parallel to the fibers to a constant creep stress. The analyses, which assumed a creep temperature of 1200°C, were conducted assuming 40 vol.% SCS-6 SiC fibers in a hot-pressed SijN4 matrix. The constituents were assumed to undergo steady-state creep only, with perfect interfacial bonding. For the FEM analysis, Poisson s ratio was 0.17 for the fibers and 0.27 for the matrix, (a) Total composite strain (axial), (b) composite creep rate, and (c) transient redistribution in axial stress in the fibers and matrix (the initial loading transient has been ignored). Although the fibers and matrix were assumed to exhibit only steady-state creep behavior, the transient redistribution in stress gives rise to the transient creep response shown in parts (a) and (b). After Wu et al 1...

Various models will be used for the interface between the fiber and the matrix. For bonded interfaces, complete continuity of all components of the velocity will be invoked. The simplest model for a weak interface is that a shear drag equal to r opposes the relative shear velocity jump across the interface. The direction of the shear drag is determined by the direction of the relative velocity. However, the magnitude of r is independent of the velocities. This model is assumed to represent friction occurring mainly because of roughness of the surfaces or due to a superposed large normal pressure on the interface. Creep can, of course, relax the superposed normal stress over time, but on a short time scale the parameter r can be assumed to be relatively invariant. No attempt will be made to account for Coulomb friction associated with local normal pressures on the interface. 

On the other hand, a model for the viscous flow of creeping material along a fiber surface is exploited in some of the cases covered. This model is thought to represent the movement of material in steady state along a rough fiber surface and is given by... 

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. 

The exact laws, based on continuum analysis of the fibers and the matrix, would be very complicated. The analysis would involve equilibrium of stresses around, and in, the fibers and compatibility of matrix deformation with the fiber strains. Furthermore, end and edge effects near the free surfaces of the composite material would introduce complications. However, a simplified model can be developed for the interior of the composite material based on the notion that the fibers and the matrix interact only by having to experience the same longitudinal strain. Otherwise, the phases behave as two uniaxially stressed materials. McLean5 introduced such a model for materials with elastic fibers and he notes that McDanels et al.6 developed the model for the case where both the fibrous phase and the matrix phase are creeping. In both cases, the longitudinal parameters are the same, namely... 

For materials with a strong bond between the matrix and the fiber, models for steady transverse creep are available. The case of a linear matrix is represented exactly by the effect of rigid fibers in an incompressible linear elastic matrix and is covered in texts on elastic materials.7,11,12 For example, the transverse shear modulus, and therefore the shear viscosity, of a material containing up to about 60% rigid fibers in a square array is approximated well... 

Consider a specimen of length Ls containing a very large number of wholly intact fibers. A stress a is suddenly applied to the specimen parallel to the fibers. The temperature has already been raised to the creep level and is now held fixed. Upon first application of the load, some of the fibers will break. The sudden application of the load means that the initial response is elastic. This elastic behavior has been modeled by Curtin,16 among others, but details will not be given here. If the applied stress exceeds the ultimate strength of the composite in this elastic mode of response, then the composite will fail and long-term creep is obviously not an issue. However, it will be assumed that... 

When the applied stress a is less than Su, creep of the matrix will commence after application of the load. During this creep, the matrix will relax and the stress on the fibers will increase. Therefore, further fiber failure will occur. In addition, the process of matrix creep will depend on the extent of prior fiber failure and, as mentioned previously, on the amount of matrix cracking. The details will be rather complicated. However, the question of whether steady-state creep or, perhaps, rupture will occur, or whether sufficient fibers will survive to provide an intact elastic specimen, can be answered by consideration of the stress in the fibers after the matrix has been assumed to relax completely. Clearly, when the matrix carries no stress, the fibers will at least fail to the extent that they do in a dry bundle. It is possible that a greater degree of fiber failure will be caused by the transient stresses during creep relaxation, but this effect has not yet been modeled. Instead, the dry bundle behavior will be used to provide an initial estimate of fiber failure in these circumstances. 

First, consider a composite with a volume fraction/of fibers, all of which are broken. There are two possible models for the steady-state creep behavior of such a material. In one, favored by Mileiko18 and Lilholt,19 among others, the matrix serves simply to transmit shear stress from one fiber to another and the longitudinal stress in the matrix is negligible. The kinematics of this model requires void space to increase in volume at the ends of the fibers. However,... 

In a version of the Mileiko18 model in which it is assumed that each of six neighboring fibers has a break somewhere within the span of the length of a given fiber, but that the location of those breaks is random within the span, the relationship between the steady-state creep rate and the composite stress is... 

For the Mileiko18 model of composite creep leading to the steady-state creep rate for fixed fiber length given in Eqn. (63), a rudimentary fiber fragment length model gives... 

It should be noted that the model leading to Eqn. (69) is incomplete since the stress required to cause the enlargement of void space at the fiber breaks is omitted from consideration. At high strain rates this contribution to stress can be expected to dominate other contributions. Therefore, at high stress or strain, the creep behavior will diverge from Eqn. (69) and perhaps exhibit the nth power dependence on stress as controlled by the matrix. The creep rate at these high stresses can be expected to exceed the creep rate of the matrix at the same applied stress since the void space at the fiber ends is a form of damage. 

It seems unlikely that long-fiber ceramic matrix composites with strong bonds will find application because of their low temperature brittleness. However, for completeness, a model which applies to the creep of such materials can be stated. It is that due to Kelly and Street.21 It is possible also that the model applies to aligned whisker-reinforced composites since they may have strong bonds. In addition, the model has a wide currency since it is believed to apply to weakly bonded composites as well. However, the Mileiko18 model predicts a lower creep strength for weakly bonded or unbonded composites and therefore is considered to apply in that case. 

As previously noted, this chapter has been concerned mainly with those models for the creep of ceramic matrix composite materials which feature some novelty that cannot be represented simply by taking models for the linear elastic properties of a composite and, through transformation, turning the model into a linear viscoelastic one. If this were done, the coverage of models would be much more comprehensive since elastic models for composites abound. Instead, it was decided to concentrate mainly on phenomena which cannot be treated in this manner. However, it was necessary to introduce a few models for materials with linear matrices which could have been developed by the transformation route. Otherwise, the discussion of some novel aspects such as fiber brittle failure or the comparison of non-linear materials with linear ones would have been incomprehensible. To summarize those models which could have been introduced by the transformation route, it can be stated that the inverse of the composite linear elastic modulus can be used to represent a linear steady-state creep coefficient when the kinematics are switched from strain to strain rate in the relevant model. 

In structural ceramic composites, the principal effect considered was one of crack-face closure tractions, or cohesive forces, brought about, for instance, by bridging fibers. A rigorous evaluation of the crack tip fields where the crack faces are not traction free has not yet been attempted. However, an approximate approach for the small-scale creep case is to assume that the crack tip fields are not functionally altered by crack-face tractions, with the effect of the traction being only to introduce a zone of crack tip shielding. This allows for the development of preliminary models for creep crack growth which is inclusive of the role of crack bridging. These preliminary models predict that,... 

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