Mori-Tanaka

Recently, some models (e.g., Halpin-Tsai, Mori- Tanaka, lattice spring model, and FEM) have been applied to estimate the thermo-mechanical properties [247, 248], Young s modulus[249], and reinforcement efficiency [247] of PNCs and the dependence of the materials modulus on the individual filler parameters (e.g., aspect ratio, shape, orientation, clustering) and on the modulus ratio of filler to polymer matrix. 

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. 

The elasto plastic behavior of a compositionally graded metal-ceramic structure is investigated. The deformation under uniaxial loading is predicted using both an incremental Mori-Tanaka method and periodic as well as random microstructure extended unit cell approaches. The latter are able to give an accurate description of the local microfields within the phases. Furthermore, the random microstructure unit cell model can represent the interwoven structure at volume fractions close to 50%. Due to the high computational costs, such unit cell analyses are restricted to two-dimensional considerations. 

Computationally less demanding mean-field methods provide a tool to account for the out-of-plane constraints, but have the disadvantage of using phase averaged stress and stain fields. In the present work, an incremental Mori Tanaka approach is employed, which is implemented as a constitutive material law in a finite element code. Both two-dimensional and three-dimensional investigations are performed and the results are compared to the predictions of the extended unit cell approaches. 

Figure 2. Predicted axial strain response for the extended unit cell models and the corresponding incremental Mori-Tanaka approach (IMT) the matrix of the center sublayer being nickel (Ni) or alumina (A1203)...

The deformation behavior of a compositionally graded metal-ceramic structure has been investigated by numerical and (semi)analytical simulations. Random microstructure models are able to predict the response of an FGM-structure in a more accurate way than the other approaches. The interwoven structure in the middle of the FGM can be accounted for using this modeling strategy. For the extended periodic unit cell models the predicted stress strain response depends strongly on the micro-arrangement of the inclusions. Detailed information on the microfields of the stresses and strains can only be obtained by the extended unit cell models. The incremental Mori-Tanaka method... 

Figure 4. Predicted axial strain response for plane stress (PST) calculations with internally full and reduced constraint and predicted axial and bending strain response for interally fully constraint generalized plane strain (GPE) calculations by the incremental Mori-Tanaka approach...

This chapter describes the method of the stiffness homogenisation for textile composites based on Eshelby solution of the elastic problem for an ellipsoidal inclusion and Mori-Tanaka homogenisation scheme. The approach was proposed by Huysmans et al. [91—94] and is successfully applied to very different textile composites, woven [95], braided [96] and knitted [91,92]. In short, in the following discussion the approach will be called method of inclusions (Mol). 

Apply the Mori-Tanaka/Eshelhy theory to calculate the equivalent stiffness of the RVE... 

Method and the Mori-Tanaka Method (Mori and Tanaka, 1973). In the former technique, the composite is viewed as a sequence of dilute suspensions and, thus, one can use the exact solutions for these cases to determine the effective composite properties. For example, the solution by Eshelby (1957) for ellipsoidal inclusions can be used. The increments of added inclusions are taken to an infinitesimal limit and one obtains differential forms for the bulk and shear moduli, which are then solved. The Mori-Tanaka method involves complex manipulations of the field variables. This approach also builds on the dilute suspension solutions (low F,) and then forces the correct solution as F, -> 1. 

ABSTRACT It is very important to determine the thermal and mechanical parameters of mortar and concrete in mesoscopic simulation. In this paper, on the basis of the Mori-Tanaka formula of mesoscopic mechanics and the concrete is treated as a two-phase composite material constituted by aggregates and mortar, the inversion of coefficient of thermal expansion, autogenous shrinkage, elastic modulus and creep were studied. This paper proposed some inversion formulas regarding these four mechanical parameters of mortar in concrete. The accuracy of these formulas was verified by FEM numerical test and demonstrated by some examples. 

Until now, much research work has been done on the prediction of composite material coefficient of thermal expansion and elastic modulus by forefathers, and many prediction methods have been developed such as the sparse method (Guanhn Shen, et al. 2006), the Self-Consistent Method (Hill R.A. 1965), the Mori-Tanaka method (Mori T, Tanaka K. 1973) and so on. However, none of these formulas take into account the parameters variation with concrete age, and there is little research on the autogenous shrinkage and creep. In the mesoscopic simulation of concrete, thermal and mechanical parameters of mortar and aggregate (coefficient of thermal expansion, autogenous shrinkage, elasticity modulus, creep, strength) are important input parameters. In fact, there is abundant of test data on concrete, but much less data on mortar while it is one of the important components. Also parameter inversion is an essential method to obtain the data, but there are few studies on this so far. 

Ignoring the interface s effect, concrete can be considered as a two-phase composite material constituted by aggregate and mortar. If the bulk modulus and shear modulus of mortar and aggregates are given, equivalent bulk modulus K and shear modulus G of concrete can be estimated according to Mori-Tanaka method. 

Conversely, if both of the elasticity modulus of aggregate and concrete are given, the mortar elasticity modulus can be obtained by the inverse function of Mori-Tanaka formula. Because it is difficult to obtain the explicit expression of inverse function for mortar bulk modulus and shear modulus, iterative computation becomes a good choice. Eqs. 8 and 9, which are the iterative formulas to obtain mortar bulk modulus and shear modulus respectively, are transformed by the simplification and deduction of Eqs. 1-4 ... 

Concrete is treated as a two-phase composite material constituted by aggregate and mortar. This paper provides the inversion formulas of several vital thermal and mechanical parameters of mortar by Mori-Tanaka theory in meso-mechanics. With these formulas, the mortar coefficient of thermal expansion, autogenous shrinkage curve, elasticity modulus curve and creep curve can be determined conveniently. However, this paper takes no consideration of the influence of interface between aggregate and mortar. Thus further studies are needed to be done to show the effect of this factor. 

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-... 

Luo, et al. [80] have used multi-scale homogenization (MH) and FEM for wavy and straight SWCNTs, they have compare their results with Mori-Tanaka, Cox, and Halpin-Tsai, Fu, et al. [81] and Lauke [82], Trespass, et al. [83] used 3D elastic beam for C-C bond, 3D space frame for CNT, and progressive fracture model for prediction of elastic modulus, they used rule of mixture for compression of their results. Their assumption was embedded a single SWCNT in polymer with perfect bonding. The multi-scale modeling, MC, FEM, and using equivalent continuirm method was used by Spanos and Kontsos [84] and compared with Zhu, et al. [85] and Paiva, et al. [86] results. 

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). 

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]. 

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