Mori-Tanaka theory

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. 

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. 

What makes the Halpin-Tsai and Mori-Tanaka theories more complex is primarily that both of them correct for end effects. Thus, when a fiber or other particle is of finite length in the stress direction, the particle s ends present points of lesser reinforcement, and clearly the gaps between such particles are not reinforced. 

The Halpin-Tsai and Mori-Tanaka theories based on first-principle arguments adequately model the mechanical properties provided by the reinforcement of montmorillonite as a dispersed phase in all polymers [5,20]. The significant independent variables that correlate to reinforcement are aspect ratio, modulus, and the alignment of the montmorillonite in the direction of the applied stress. It is enlightening to examine some of the general predictions of these theories as they relate to the effect of clay loading, aspect ratio, and the modulus of the pristine... 

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

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. 

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. 

In order to prediet the stiffness, the strain concentration tensor A is needed. Mori and Tanaka (1973) eonsidered a composite model where the heterogeneities are diluted in the matrix. This model takes into account an interaction between the inclusion and the surroimdings (inelusions and polymer matrix) in their original model, the inclusions were considered to have the same shape and orientation. Benveniste (1987) made a reconsideration and reformulation of the Mori-Tanaka s theory in its application to the computation of the effective properties of composite. In this model the inclusions can be considered either aligned or randomly oriented. This formulation is more suitable for the morphology of clays dispersed in a polymer. The expression of the strain concentration tensor A is written as follows ... 

Chamis C.C. and Sendeckyj G. R (1968) Critique on theories predicting thermoelastic properties of fibrous composites. Journal of Composite Materials, 2, 332-358. Mori T. and Tanaka K. (1973) Average stress in matrix and average elastic energy of materials with misfitting iclusions. Acta Metallurgica, 21, 571-574. 

There is an extensive literature on composite materials stemming from the seminal papers of Eshelby [8], who considered the elastic field in and around an elliptic inclusion in an infinite matrix. His theory assumed a single particle in an infinite matrix and therefore was valid only for low-volume fractions ( 1%). The extension to more concentrated systems was undertaken by Mori and Tanaka [9], whose method was used by Tandon and Weng [10] to derive the elastic constants of an aligned fibre composite. The composite moduli for this model are given by... 

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