Composite Mori-Tanaka

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... 

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). 

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. 

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. 

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-... 

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]. 

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... 

Baltussen MGHM, Hulsen MA, Peters GWM (2010) Numerical simulation of the fountain flow instability in injection molding. J Non-Newtonian Fluid Mech 165 631-640 Batoz JL, Lardeur P (1989) A discrete shear triangular nine d.o.f element for the analysis of thick to very thin plates. Int J Numer Meth Eng 28 533-560 Bay RS (1991) Orientation in injection molded composites A comparison of theory and experiment. Ph.D Thesis, University of Illinois, Uibana, II Benveniste Y (1987) A new approach to the application of Mori-Tanaka s theory in composite materials. Mech Mater 6 147-157... 

Benveniste Y. (1987) A new approach to the application of Mori-Tanaka s theory in composite materials. Mechanics of Materials, 6(2), 147-157. 

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. 

Almost a standard procedure, due to its relative simplicity while allowing for dependable results and documented by a thorough theoretical discussion in the literature, the Mori-Tanaka approach was initiated by Mori and Tanaka [127]. Its major assumption may be formulated for electromechanically coupled composites as follows ... 

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. 

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. 

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. 

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