Fiber matrix adhesion models

Some researchers have used approximate microscopic descriptions to develop more rigorous macroscopic constitutive laws. A microstructural model of AC [5] linked the directionality of mechanical stiffness of cartilage to the orientation of its microstructure. The biphasic composite model of [6] uses an isotropic fiber network described by a simple linear-elastic equation. A homogenization method based on a unit cell containing a single fiber and a surrounding matrix was used to predict the variations in AC properties with fiber orientation and fiber-matrix adhesion. A recent model of heart valve mechanics [8] accounts for fiber orientation and predicts a wide range of behavior but does not account for fiber-fiber interactions. 

In polymer matrix composites, there appears to be the optimum level of fiber-matrix adhesion which provides the best mechanical properties. Several models which relate the structure and properties of composites to fiber-matrix interfacial behavior have been proposed based either on mechanical principles with some assumptions made about the level of fiber-matrix adhesion in the composite or have taken a surface chemistry approach in which the fiber-matrix interphase was assumed to be the only factor of importance in controlling the final properties of the composite. Neither effort has had much success. 

The remaining part of this chapter will review the three most common direct methods for measuring fiber-matrix adhesion, focusing on the sample preparation and fabrication, the experimental protocols and the underlying theoretical analyses upon which evaluation of these methods are based. In addition, finite-element nonlinear analyses and photoelastic analyses will be used to identify differences in the state of stress that is induced in each specimen model of the three different techniques. In order to provide an objective comparison between the three different techniques to measure the interfacial shear strength for the prospective user, data and a carbon fiber-epoxy resin system will be used as a baseline system throughout this chapter, However, these methods and procedures can be applied for adhesion measurements to any fiber-matrix combination. 

However, it must be stated that this model assmnes perfect fiber-matrix adhesion. It was foimd fi om vahdation studies that large discrepancies existed between actual mechanical response and theoretical results calculated fiom the model. This has been documented elsewhere where poor fiber-matrix adhesion occurred for the prepared composite catheters [1]. 

Assuming the work of adhesion to be measurable, one must next ask if it can be related to practical adhesion. If so, it may be a useful predictor of adhesion. The prospect at first looks bleak. The perfect disjoining of phases contemplated by Eq. 1 almost never occurs, and it takes no account of the existence of an interphase , as discussed earlier. Nonetheless, modeling the complex real interphase as a true mathematical interface has led to quantitative relationships between mechanical quantities and the work of adhesion. For example, Cox [22] suggested a linear relationship between Wa and the interfacial shear strength, r, in a fiber-matrix composite as follows ... 

The distribution of stress around discontinuous fibers in composites has been studied by a number of researchers. Theoretical analyses have been performed by Cox [82] and Rosen [83]. In these models only fiber axial stress distribution and the fiber-matrix interfacial shear stress distribution are determined. Amirbayat and Hearle [84] studied the effect of different levels of adhesion on the stress distribution, that is, no bond, no adhesion, perfect bond, and the intermediate case of limited friction. They also considered the inhibition of slippage by frictional forces resulting from interfacial pressure due to Poisson s lateral contractions of the matrix but did not consider the shrinkage of the matrix during curing. 

Another three-dimensional axisymmetric stress distribution for the stress around fiber breaks was obtained by Naim [93] using variational mechanics. In this study, breaks interaction was also included and it was assumed that both fiber and matrix were linearly elastic and a perfect adhesion at the fiber-matrix interface. To account for the stress singularity at the matrix crack tip of the fiber break, the matrix plastic-model was also included. Imperfect adhesion to mimic a failed fiber-matrix interface was added to this model to study the mechanism of interfacial failure, that is, the stress conditions that cause the extent of interfacial failure or its increase. It was suggested that due to the complexity of the multi-axial stress state, a simple maximum stress failure criterion was unrealistic and an energy release rate analysis was necessary to calculate the total energy release rate associated with the growth of interfacial damage. 

The above equations correspond to the intuitive understanding that in the former case, the mechanical response of the laminar composite is essentially dominated by the performance of the fibers, thus giving the upper-boimd reinforcing effect, while in the latter case, the matrix (and the fibers-to-matrix adhesion) is playing the key role and dictates the lower-bound reinforcement. As we will see, some micro-mechanical models for short fibers composites clearly emphasize such upper and lower limits, and it follows that what can really be achieved when manufacturing a short fiber-reinforced polymer object is indeed between those two extremes. 

The exponentials Ki and K2 in the model account for the degree of heterogeneity of the interphase layer. The ratio of these exponentials is representative, as suggested in [121], of the load transmission between the fiber and the matrix (the adhesion factor a). 

The interaction of two substrates, the bond strength of adhesives are frequently measured by the peel test [76]. The results can often be related to the reversible work of adhesion. Due to its physical nature such a measurement is impossible to carry out for particulate filled polymers. Even interfacial shear strength widely applied for the characterization of matrix/fiber adhesion cannot be used in particulate filled polymers. Interfacial adhesion of the components is usually deduced indirectly from the mechanical properties of composites with the help of models describing composition dependence. Such models must also take into account interfacial interactions. 

In optimized commercial materials, the interface functions as an efficient transmitter of forces between fiber and matrix. As such, as long as the interface is intact, composite material behavior can be adequately described by models which assume ideal adhesion between fiber and matrix and consider the interface to be a two-dimensional boundary. 

The compressive strengths of the composites obtained increased and their temperature dependencies decreased with increasing fiber length, fiber-volume fraction, and density of the matrix foam. More specifically, the compressive strengtii of the composite was found to be proportional to that of the matrix and increased linearly with increased fiber-volume fraction in the experimental range employed (below 2% by volume). This result could be explained by Swift s sinusoidal model, assuming that the adhesion between fiber and matrix foam is perfect. 

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