Model Mori-Tanaka

Hbaieb et al. [260] recently suggested that Mori-Tanaka and 2D FEM models do not predict accurately the elastic modulus of real clay/PNCs. The Mori-Tanaka model underestimates the stiffness at higher volume fractions (>5%) and overestimates the stiffness of exfoliated clay/PNCs. 

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. 

The Mori-Tanaka model is derived based on the principles of Eshelby s inclusion model for predicting an elastic stress field in and around elUpsoidal filler in an infinite matrix. The complete analytical solufions for longitudinal SI and transverse elastic moduh of an isotropic matrix filled with aligned spherical inclusion are [45,... 

Step 4—overall corrstitutive properties of the dilute and unidirectiortal SWCNT/ polymer composite are determined with Mori-Tanaka model with the mechanical properties of the effective fiber and the bttlk polymer. The layer of polymer molecules that are near the polymer/nanotube interface (Figure 2) is included in the effective fiber, and it is assttmed that the matrix polymer surrotmding the effective fiber has me-... 

The Mori-Tanaka model is uses for prediction an elastic stress field for in and around an ellipsoidal reinforcement in an infinite matrix. This method is based on Eshebly s model. Longitudinal and transverse elastic modulus, Ejj and for isotropic matrix and directed spherical reinforcement are ... 

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). 

Equation 6.20 is the required equation for the effective stiffness tensor Cyia-Since Cp, Cp and q/ are all known, one only needs to find the strain-eoneen-tration tensor Ayu- Different expressions of Ayi i represent different models. Many models have been reviewed by Tucker and Liang (1999). They recommend the Mori-Tanaka model as the best choice for injection molded composites. The model was proposed by Mori and Tanaka (1973) and has later been described by Benveniste (1987) and Christensen (1990) in a simpler direct way. The Mori-Tanaka strain-concentration tensor is given by... 

Fig. 6.2 Numerical results based on the Mori-Tanaka model for the elastic moduli En, E22 and Gi2 as a function of volume fraction of fibers...

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. 

Mori-Tanaka model Kalpin-Tsai model Lattice-spring model Finite element method Equivalent continuum approach Seif-similar approach... 

The Halpin-Tsai equations and the Mori-Tanaka model are the most used to predict mechanical properties of composites. The Halpin-Tsai equations predict stiffness of the unidirectional composites as a function of aspect ratio. In this model, the longitudinal stiffness and transverse engineering moduli are expressed in the following general form ... 

The Mori-Tanaka model is derived from a model developed by Eshelby [15]. The terms in Eshelby s arguments are predicated in the field of metallurgy, hence, they may seem foreign to investigators in polymer... 

With these values, a comparison was made between the predicted values from the Halpin-Tsai and Mori-Tanaka models and the measured Young s modulus as a function of montmorillonite concentration in the polymer. When the experimentally determined number-average aspect ratio, 57, was employed, Halpin-Tsai predicted higher values and Mori-Tanaka predicted lower values than the experimental results. When the aspect ratio for the theoretical perfect exfoliation of montmorillonite was utilized in the Mori-Tanaka model, 97, the values were virtually identical to the experimental values. When the Halpin-Tsai model was altered to accommodate dispersed phases that have more than... 

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. 

The Mori-Tanaka model for an aspect ratio of 100 of perfectly exfoliated montmorillonite aligned in the direction of stress predicts a slope of 10.8 for the same copolymer. When the r/min of the extruder is increased to 200 and 380, the slope increased to about 8.4 for the copolymer-montmorillonite composite with 25 wt.% acrylonitrile content. 

This is the basic equation for implementing a Mori-Tanaka model. Using Eq. 10 in Eq.9, the composite effective elastic constant could be derived. Based on Mori-Tanaka model, several attempts have been made to develop and apply expressions for the effective moduli of imidirectional nanoeomposites with dispersed and parallel flake-like inclusions (Hui and Shia, 1998 Shia et al., 1998). However, these models assume complete exfoliation of clay layers, full dispersion and imiform orientation. As a consequence, these idealized models are not in agreement with experimental results. 

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