Fiber Composite Model

Fig. 4.4. Axi-symmelric single fiber composite model employed by Rosen (1964).

Fig. 4.6. Schematic drawing of a partially debonded single fiber composite model subject to external stress, (Ta, in the fiber fragmentation test.

In contrast, the single fiber composite model predicts that the IFSS concentration becomes higher at the embedded end than at the loaded end if fiber Kf is greater than a critical value, suggesting the possibility of debond initiation at the embedded fiber... 

Following on Katz s [1976,1980] adaptation oftheHashin-RosenhoUow fiber composite model [1964], Gottesman and Hashin [1979] presented a viscoelastic calculation using the same major assumptions. 

Figure 9.17 Parameter x in Equation (9.22) as a function of the taper draw ratio t to the 3/2 power. (Reproduced from Barham, P.j. and Arridge, R.C.C. (1977) A fiber composite model of highly oriented polyethylene. J. Polym. Sci. Polym. Phys., 15, 1177. Copyright (1977).)...

Mechanical Properties. Although wool has a compHcated hierarchical stmcture (see Fig. 1), the mechanical properties of the fiber are largely understood in terms of a two-phase composite model (27—29). In these models, water-impenetrable crystalline regions (generally associated with the intermediate filaments) oriented parallel to the fiber axis are embedded in a water-sensitive matrix to form a semicrystalline biopolymer. The parallel arrangement of these filaments produces a fiber that is highly anisotropic. Whereas the longitudinal modulus of the fiber decreases by a factor of 3 from dry to wet, the torsional modulus, a measure of the matrix stiffness, decreases by a factor of 10 (30). 

The composites with the conducting fibers may also be considered as the structurized systems in their way. The fiber with diameter d and length 1 may be imagined as a chain of contacting spheres with diameter d and chain length 1. Thus, comparing the composites with dispersed and fiber fillers, we may say that N = 1/d particles of the dispersed filler are as if combined in a chain. From this qualitative analysis it follows that the lower the percolation threshold for the fiber composites the larger must be the value of 1/d. This conclusion is confirmed both by the calculations for model systems [27] and by the experimental data [8, 15]. So, for 1/d 103 the value of the threshold concentration can be reduced to between 0.1 and 0.3 per cent of the volume. 

The Three-Term Unfolding Model for Fiber Composites.176... 

A three-layer model for fiber composites may be developed, based on the theory of self-consistent models and adapting this theory to a three-layered cylinder, delineating the representative volume element for the fiber composite. 

A better approach for the Rosen-Hashin models is to adopt models, whose representative volume element consists of three phases, which are either concentric spheres for the particulates, or co-axial cylinders for the fiber-composites, with each phase maintaining its constant volume fraction 4). 

The three-layer model, as previously mentioned, as well as the multi-layer model, were previously applied to study the behaviour, especially of fiber composites 3A). The three-layer model, based on the self-consistency of phases, gave relationships between stresses and displacements between phases, which, when solved, may give... 

However, since measurements of Tg s and the thermal expansion coefficients are not very sensitive and accurate, the results derived from such model present some scattering and their reliability needs further proof for its validity. Therefore, in the following we shall concentrate to the unfolding models for fiber composites, as they have been extended from the respective models for particulates, which present significant stability and unquestionned reliability. 

For the three-term unfolding model in fiber composites the E (r)-modulus of the interphase is again expressed by relation (22). 

Fig. 15. The variation of the adhesion coefficient A = (ri, — t 2) for the three-term unfolding model and the exponent 2r for the two-term model of a series of E-glass-epoxy fiber composites, versus the fiber-volume content uf...

A series of models were introduced in this study, which take care of the existence of this boundary layer. The first model, the so-called three-layer, or N-layer model, introduces the mesophase layer as an extra pseudophase, and calculates the thickness of this layer in particulates and fiber composites by applying the self-consistent technique and the boundary- and equilibrium-conditions between phases, when the respective representative volume element of the composite is submitted to a thermal potential, concretized by an increase AT of the temperature of the model. 

A modified shear-lag model has been proposed by Rosen (1964, 1965) based on a multiple fiber composite. Fig. 4.4 shows the composite model Rosen considered wherein a fiber is embedded in a matrix which in turn is surrounded by an average composite material. The FAS and IFSS are given in the same form as those of Eqs. (4.1) and (4.2) given earlier by Cox (1952) ... 

Fig. 4.31. Distributions of (a) fiber axial stress and (b) interface shear stress along the axial direction obtained from micromechanics analysis for different fiber volume fractions, Vf = 0.03, 0.3 and 0.6 (—) single fiber composite (--------) three cylinder composite model. After Kim et al. (1994b).

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