Mori-Tanaka Method

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. 

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... 

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. 

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. 

Since the sequential stacking procedure could be shown to produce results of equivalent quality, see Section 5.5, with a little more flexibility and less numerical expenditure in comparison to the Mori-Tanaka method, it will be applied for the example calculations. Following the discussion of Section 5.4.6, the factor C3 intended to map the disturbance of electrostatic fields close to the electrodes will be omitted ... 

The calculation procedure is based on the effective field method (EFM), i.e., the Mori -Tanaka method [26] generalized for heterogeneous piezoelectric media [16, 23], The EFM based on Eshelby s concept of spheroidal inclusions [27, 28] is a variant of the self-consistent scheme for the calculation of effective constants of the piezoactive composites. Following the EFM, we take into account the electromechanical interaction between the piezo-active inclusions in the matrix and related coupled effects. The effective electromechanical properties of the 0-3 composite are represented in the matrix form as... 

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. 

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). 

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. 

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 ... 

A hybrid atomistie/eontinuum mechanics method is established in the Feng et al. [70] study the deformation and fracture behaviors of CNTs in composites. The unit eell eontaining a CNT embedded in a matrix is divided in three regions, whieh are simulated by the atomic-potential method, the continumn method based on the modified Cauchy-Bom rule, and the classical continuum mechanics, respectively. The effect of CNT interaction is taken into account via the Mori-Tanaka effective field method of micromechanics. This method not only can predict the formation of Stone-Wales (5-7-7-5) defects, but also simulate the subsequent deformation and fracture process of CNTs. It is found that the critical strain of defect nucleation in a CNT is sensitive to its chiral angle but not to its diameter. The critical strain of Stone-Wales defect formation of zigzag CNTs is nearly twice that of armchair CNTs. Due to the constraint effect of matrix, the CNTs embedded in a composite are easier to fracture in comparison with those not embedded. With the increase in the Young s modulus of the matrix, the critical breaking strain of CNTs decreases. 

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

Clay is very anisotropic and small with a large surface area and modulus. To account for these characteristics of clay in a polymer, Fornes and Paul evaluated two analytical methods, those of Halpin-Tsai [6,7,8] and Mori-Tanaka [9], that were developed to calculate the Young s moduli for the types of morphology that can be associated with clay particles. Assuming that the polymer and the reinforcing dispersed phase are the only components in the polymer composite, the volume... 

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