Mori-Tanaka model nanocomposites

Since the assumption of uniformity in continuum mechanics may not hold at the microscale level, micromechanics methods are used to express the continuum quantities associated with an infinitesimal material element in terms of structure and properties of the micro constituents. Thus, a central theme of micromechanics models is the development of a representative volume element (RVE) to statistically represent the local continuum properties. The RVE is constracted to ensure that the length scale is consistent with the smallest constituent that has a first-order effect on the macroscopic behavior. The RVE is then used in a repeating or periodic nature in the full-scale model. The micromechanics method can account for interfaces between constituents, discontinuities, and coupled mechanical and non-mechanical properties. Their purpose is to review the micromechanics methods used for polymer nanocomposites. Thus, we only discuss here some important concepts of micromechanics as well as the Halpin-Tsai model and Mori-Tanaka model. 

Shah et al. 2007 Spencer et al. 2010). They have also modified the Mori-Tanaka model using a two-population approach to predict modulus of ternary systems such as polymer blend nanocomposites and polymer composites containing two different fillers. More details can be found in papers describing the extensive modeling work by Paul and coworkers (Spencer and Paul 2011 Yoo et al. 2011 Tiwari et al. 2012). Other composite models such as the Christensen model has been also used to predict modulus of PS/PP/PP-g-MA/MMT nanocomposites where PP particles form the dispersed phase (Istrate et al. 2012). 

Micromechanical models have been widely used to estimate the mechanical and transport properties of composite materials. For nanocomposites, such analytical models are still preferred due to their predictive power, low computational cost, and reasonable accuracy for some simplified stmctures. Recenfly, these analytical models have been extended to estimate the mechanical and physical properties of nanocomposites. Among them, the rule of mixtures is the simplest and most intuitive approach to estimate approximately the properties of composite materials. The Halpin-Tsai model is a well-known analytical model for predicting the stiffness of unidirectional composites as a function of filler aspect ratio. The Mori-Tanaka model is based on the principles of the Eshelby s inclusion model for predicting the elastic stress field in and around the eflipsoidal filler in an infinite matrix. 

This examination of experimentally determined Young s modulus values for montmorillonite-nylon 6 nanocomposites compared with the Halpin-Tsai and Mori-Tanaka models seems to indicate that the extremely effective reinforcing efficiency of montmorillonite in polymer can be explained by its high modulus and large aspect ratio when fully exfoliated. 

Hbaieb et al. [24] compared the utility of modeling polymer-montmorillonite nanocomposites by finite element analysis (TEA) in relation to the Mori-Tanaka model. The three-dimensional finite element model (FEM) was found to be superior to the two-dimensional one. For the calculations, the aspect ratio (A) was chosen to be 50, YfjYp = 100, the Poisson ratio for the polymer (Pp) was assumed to be 0.35, and the Poisson ratio for the montmorillonite was assumed to be Pf = 0.2. The FEA was performed using the commercial package, ABAQUS. The morphology of the montmorillonite was assumed to be disk-shaped. The limitations of this assumption are exposed in the discussion above by Lee and Paul. 

Chivrac et al. (2008) developed micromechanical models based on Mori-Tanaka model to predict the effective elastic properties of starch-based nanocomposites. Except the exfoliation ratio, they considered the effect of plasticizers and storage conditions on effective mechanical properties of the nanocomposites (Equations 11 and 12, respectively)... 

Models of increasing sophistication have been developed to predict the elastic properties of composite materials from the properties of their constituent parts. These range from the simple rule-of-mixtures approach to the Halpin-Tsai and Mori-Tanaka analyses, where the geometry - essentially, the aspect ratio - of the reinforcing particles can be taken into account. This has the potential to model the effects of extreme aspect ratios that are seen in nanocomposites. Direct finite element simulation of the microstructure is an option that is becoming increasingly feasible at both the micro and nano levels. 

By virtue of the strong anisotropy of the shape of particles of Na -montmorillonite mentioned above, for theoretical estimation of the degree of reinforcement of nanocomposites filled by them the models of Halpin-Tsai and Mori-Tanaka are used [19]. For the case of isotropic (spherical) filler particles EJE estimation can be carried out according to the equation, obtained in paper [35] ... 

---