Equation Mori-Tanaka

An interpolation procedure applied by Halpin and Tsai [17,18] has led to general expressions for the moduli of composites, as given by Eqs. (2.18) and (2.19). Note that for = 0, Eq. (2.18) reduces to that for the lower hmit, Eq. (2.8), and for = infinity, it becomes equal to the upper limit for continuous composites, Eq. (2.7). By empirical curve fitting, the value of = 2(l/d) has been shown to predict the tensile modulus of aligned short-fiber composites in the direction of the fibers, and the value of = 0.5 can be used for the transverse modulus. Other mathematical relationships for modulus calculations of composites with discontinuous fillers include the Takaya-nagi and the Mori-Tanaka equations [20]. 

Here we will consider the expanded expression of the Mori-Tanaka equation, which is useful for practical calculations. It is easy to show (Mori and Wakashima 1989) that Eqs. 6.20 and 6.21 can be rewritten as ... 

The Mori-Tanaka Average Stress Theory The Mori-Tanaka equations were derived for calculating the elastic stress field in and around an ellipsoidal particle in an infinite matrix (16,17). The shape of the ellipsoid can be altered to appear more fiber-like, disk-like, or spheroidal, thus allowing for a continuous range of shapes to be considered. 

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... 

Figure 13.9 (16,17) summarizes the results of the Halpin-Tsai equations, and also those of the Mori-Tanaka theory, to be discussed below. Note that for fibers, the Halpin-Tsai equations predict equal moduli for the 2 and 3 directions, but for platelets, the moduli are equal for the 1 and 2 directions. Also, note the symbolism where n and x represent the composite modulus parallel and perpendicular to the major axis of the filler. 

The mechanical properties of an rPET/OMMT nanocomposite are shown in Figure 3 from [45]. By increasing the OMMT content up to 1 wt%, the yield strength and ultimate strength of the nanocomposite improved by 17 and 27%, respectively. Additionally, the incorporation of OMMT into the rPET resulted in a significant increase of the modulus, which was well described by Mori-Tanaka, and solution of Eshleby at low MMT content. Also, the prepared nanocomposite exhibited reduced creep, which could be predicted by the power law equation for long-term creep. 

The Halpin-Tsai equations give excellent predictions to a good approximation in most circumstances. However, the Mori-Tanaka approach is more accurate for high aspect ratio inclusions. The reader is referred to a recent paper by Tucker and Liang [11] for a comprehensive review. 

Note that for this case the HS upper bounds are identical to the Mori-Tanaka predictions for random materials of matrix-inclusion type with spherical inclusions [Mori Tanaka 1973, Weng 1984, Zimmerman 1994] and that Equation (91) corresponds to a relation known in geophysical context under the name Kuster-Toksbz relation [Kuster Toksoz 1974, Zimmerman 1991b]. It is shown below that in the alumina-zirconia system the Kuster-Toksoz relation. Equation (91), is an excellent approximation to the HS upper boimd for the tensile modulus (error <0.1 %) and for the shear modulus (error < 2.6 %) but not for the bulk modulus (error < 14.3 %). For the purpose of later reference we note that Equation (91) can be approximated by the following second-order polynomial ... 

Note that the HS upper bound has been drawn as one single curve in Fig. 4, corresponding to Equation (91), which is identical to the Mori-Tanaka prediction and the Kuster-Toksbz relation (112). Indeed, a comparison of the values obtained by this simple... 

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 ... 

Tucker et al. [50] prepared an application review of different classes of micromechanical models. The authors remarked that Halpin-Tsai equations [46] are the most widely used, but the Mori-Tanaka type models [45] give the best results for large aspect ratio fillers. 

Eshelby used an orthorhombic crystal as an example to describe the spatial relationships of S. He also pointed out that they do not couple (51,1,2,2 52,2,i,i). Mori-Tanaka and Tandon-Weng refer to S as Eshelby s tensors or Eshelby s transformation tensors. Eshelby utilized direction cosines from an observation point to a volume element to evaluate the elastic fields in the dispersed phase. 5i,1,1,1, 52,2,2,2> 52,2,3,3, 52,2,i,i, and 5i,1,2,2 are needed to calculate K, K, and K2 for Equation (5.9). Their values for disk-shaped dispersed phases are provided in the Tandon-Weng and Eshelby publications. These values are solely a function of aspect ratio and Poisson s ratio. Knowing the proper S values, one can return to Equation (5.11) to calculate the strain in the dispersed phase as a function of the strain in the composite. 

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] ... 

Figure 7.11 The dependences of the reinforcement degree EJE on the filling degree W for nanocomposites filled by Na -montmorillonite. 1-5 - the theoretical dependences corresponding to Halpin-Tsai (1, 2) and Mori-Tanaka (3, 4) equations at L/d i = 100 (1, 3) and 50 (2,4) and to Equation 7.7 (5). 6-13 -the experimental data for nanocomposites on the basis of epoxy polymer at T < (6), polyamide-6 (7), poly(butylenes terephthalate) (8), polycarbonate...

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. 

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)... 

Tandon and Weng have derived a complete set of explicit relationships for moduli of a composite model in which randomly distributed ellipsoidal particles (i.e., short-fiber like) are unidirectionally aligned. The Mori-Tanaka s average strain concept is also used as the transformation tensor described by Eshelby when he solved the problem of an ellipsoidal inclusion in an elastic field. The Eshelby s transformation tensor is a 4 order tensor whose components depend only the aspect ratio of the inclusion and the elastic moduli of the matrix. Some of the equations obtained by Tandon and Weng must however be solved iteratively and handling them implies calculating at first quite a impressive number of "constants," in fact various... 

In the particular case of polymer films containing "liquid fillers," Gao and Tsou have extended the Tandon and Weng s work to derive easy-to-handle expressions for the moduli and the Poisson s ratio. Through a comparison with finite element calculations on idealized arrays of fibers. Tucker and Liang have evaluated models derived from the Mori-Tanaka s average field concept, the (easier-to-handle) Halpin-Tsai equations and other models. They found that the Halpin-Tsai equations give reasonable estimation for stiffness but the best predictions (of finite element calculations) were obtained with the Mori-Tanaka model. 

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