The micromechanical models

The aim of this chapter is to describe the micro-mechanical processes that occur close to an interface during adhesive or cohesive failure of polymers. Emphasis will be placed on both the nature of the processes that occur and the micromechanical models that have been proposed to describe these processes. The main concern will be processes that occur at size scales ranging from nanometres (molecular dimensions) to a few micrometres. Failure is most commonly controlled by mechanical process that occur within this size range as it is these small scale processes that apply stress on the chain and cause the chain scission or pull-out that is often the basic process of fracture. The situation for elastomeric adhesives on substrates such as skin, glassy polymers or steel is different and will not be considered here but is described in a chapter on tack . Multiphase materials, such as rubber-toughened or semi-crystalline polymers, will not be considered much here as they show a whole range of different micro-mechanical processes initiated by the modulus mismatch between the phases. 

One of the major differences between the results obtained from the micromechanics and FE analyses is the relative magnitude of the stress concentrations. In particular, the maximum IFSS values at the loaded and embedded fiber ends tend to be higher for the micromechanics analysis than for the FEA for a large Vf. This gives a slightly lower critical Vf required for the transition of debond initiation in the micromechanics model than in the FE model of single fiber composites. All these... 

Let us consider that the model is based on the approach, which is principally different from the micromechanical models it is assumed that polymer composites properties are defined by their matrix structural state only and that the role of the filler consists in modification and fixation of the matrix polymer structure. 

The micromechanical models used for the comparison was Halpin-Tsai (H-T) [89] and Tandon-Weng (T-W) [90] model and the comparison was performed for 5 wt% CNT/PP. It was noted that the H-T model results to lower modulus compared to FEA because H-T equation does not account for maximum packing fraction and the arrangement of the reinforcement in the composite. A modified H-T model that account for this has been proposed in the literature [91], The effect of maximum packing fraction and the arrangement of the reinforcement within the composite become less significant at higher aspect ratios [92],... 

For CNTs not well bonded to polymers, Jiang et al. [137] established a cohesive law for CNT/polymer interfaces. The cohesive law and its properties (e g. cohesive strength and cohesive energy) are obtained directly fiom the Lennard-Jones potential from the vdW interactions. Such a cohesive law is incorporated in the micromechanics model to study the mechanical behavior of CNT-reinforced composite materials. CNTs indeed improves the mechanical behavior of composite at the small strain. However, such improvement disappears at relatively large strain because the completely debonded nanotubes behave like voids in the matrix and may even weaken the composite. The increase of interface adhesion between CNTs and polymer matrix may significantly improve the composite behavior at the large strain [138]. 

It should be mentioned that in general, hard phase clusters can be non-spherical, as discussed in various earlier papers. In this case, the modulus increase could strongly depend on the aspect ratio the effect of the aspect ratio can be modeled through the micromechanical models of Halpin and Tsai [55] or Mori and Tanaka [56]. However, as we already commented above, below the spherical-to-cylindrical transition, most of the hard phase nano-domains have an aspect ratio close to 1. Above the spherical-to-cylindiical transition that is in our model associated with percolation threshold, most of the cylinders participate in the formation of the percolated hard phase, while the soft phase primarily contains hard phase islands with smaller aspect ratios. Therefore, in our analysis we assume that all the fillers dispersed within the soft phase are spherical (or have aspect ratios close to one). 

In what follows, the micromechanical model is used to evaluate the effective viscoelastic response of two polyamide-6-based nanocomposite systems. Several inputs related to the structure of nanocomposites and the mechanical properties of constituents are required by the model. The thickness of a single silicate layer is known to be about 1 nm. The structural parameters of the... 

A micromechanics-based model recently proposed by Anoukou et al. [7,8] was adopted in the present investigation to develop a pertinent model for describing the viscoelastic response of polyamide-6-based nanocomposite systems. Comparisons between the results from the micromechanical model and experimental data were considered for nanocomposites reinforced with modified and unmodified montmorillonite clay. Reasonable agreement between theoretical predictions and experimental data was noticed, the discrepancies being attributed to both uncertainties in the input data and a possible effect of reduced chain segment mobility in the vicinity of clay nanoplatelets. 

In the micromechanical model presented, like most classical micromechanics models for two component composites (Christensen 1979), we have assumed perfect interfacial adhesion and absence of physical or chemical interactions of constituents at the interface. Additionally, we assumed perfect oriented fibers in the matrix. The validity of these assumptions in our model will be one of the tasks to be undertaken for future studies on this subject. For example, we have not taken into account the fact that transesterification reaction might occur during the processing of TP/LCP blends, as well as non-perfect fiber orientation. According to the literature, transesterification reaction between the LCP and TP molecules could lead to enhanced compatibility between the polymer blend constituents. 

Figure 3.18e shows the effective force law (force versus displacement) between two parallel dimers with aspect ratio = 1.3 undergoing compression for the micromechanical model in which (1) lobe interactions are multiply counted or (2) the interaction potential is given by Equation 3.15. The two force laws are the same as long as overlaps between lobes have not merged, or < 0.021 for the configuration in Figure 3.18a. Beyond Sm, the two force laws differ. The force law based on the total area of overlap converges to linear behavior f 5 more quickly than the one that multiply counts lobe interactions, for example, it is not sensitive to the formation of the fourth lobe contact at 8/a = 84/a = 0.075. In future studies, these results can be compared to finite element analyses of linear elastic particles with complex shapes.

The predicted results are in a good agreement with their experiments and show that the micromechanical model can be used an indirect characterization technique to quantify the exfoliation/aggregation degree in the plasticized starch/clay nanocomposites. 

The prediction of macromechanical strength properties of a unidirectional fiber-reinforced composite lamina using micromechanics models has met with less success than the elastic moduli predictions of the earlier sections. The structural designer will most likely rely primarily on results from mechanical tests that measure the macromechanical strength properties of the composite lamina directly. Nevertheless, it is instructive to look at the micromechanics model for tensile strength in the fiber direction of a lamina to gain a better understanding of how the composite functions. 

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