Hashin-Shtrikman model

Most polyphase materials have elastic moduli which lie between these two bounds, and more sophisticated models have been developed to narrow the gap. The Hashin-Shtrikman model, in particular [27], has given better predictions for the elastic modulus of a composite, while, more recently, Aboudi has used a cell model [28] and finite element analysis has also been used to derive the relationship between the volume fraction and effective modulus [29-32], These methods also estimate the displacement and stress field in the composite. 

Fig. 10. Sequential IPNs of poly(butyl acrylate) and polystyrene as a function of composition. A Upper-Lower Bound Model B Hashin-Shtrikman Model C Davies Model D Budiansky Model and E Coran—Patel Model, dyne/cm = 0.1 Pa.

Gray and McCrum735 used the Hashin-Shtrikman theory to explain the origin of the y relaxation in PE and PTFE, Maeda et al.745 have given exact analyses of several two phase models for semi-crystalline polymers and Buckley755 represented a biaxially oriented sheet of linear polyethylene by a two phase composite model. 

The foregoing summary of applications of composites theory to polymers does not claim to be complete. There are many instances in the literature of the use of bounds, either the Voigt and Reuss or the Hashin-Shtrikman, of simplified schemes such as the Halpin-Tsai formulation84, of simple models such as the shear lag or the two phase block and of the well-known Takayanagi models. The points we wish to emphasize are as follows. 

Figure 11.25 Shear viscosity of the composite, G, normalized by the bulk viscosity of the matrix, K, as a function of the volume fraction of inclusions (0.6 times the number fraction). Calculations by finite element method (A), self-consistent model (solid line), and Hashin-Shtrikman lower bound (dashed line) are shown. (From Ref. 39.)...

The best known bounds modelling, without information about the geometrical combination are given by Hashin-Shtrikman [5] relation as ... 

Young s modulus data is plotted below with the following models also in evidence Voigt (Rule of Mixtures), Ress, and Hashin-Shtrikman. The Voigt bound assumes proportional stiffness... 

FIGURE 6.26 Voigt, Reuss, Voigt-Reuss-Hill average, and Hashin-Shtrikman bounds for com-pressional modulus as function of porosity. Input parameters are quartz = 11 GPa, )Tma = 44GPa, and water 4fl=2.2GPa. [An Excel worksheet you find in the folder Elastic-mechanical, bound models on the website (see also Section 11.2) http /A>ooksite.elsevier.com/ 9780081004043/.]... 

Figure 6.26 shows Voigt, Reuss, and Hashin-Shtrikman bounds for com-pressional modulus as a function of porosity. The lower Hashin-Shtrikman bound is equal the Reuss bound in case of a porous medium, where mie constituent is a fluid (shear modulus zero). Gommensen et al. (2007) therefore implemented a modified upper Hashin-Shtrikman bound this model crosses the Reuss bound at critical porosity. 

For the same two-component model, the Hashin-Shtrikman upper and lower bounds are calculated. 

While the Takayanagi models have proved useful because of their simplicity, the effects of changes in mechanical behavior with composition and phase structure may also be profitably explored using several analytical relations, which include equations derived by Kerner (1956b), Hashin and Shtrikman (1963), and Halpin and Tsai (Ashton et al, 1969, Chapter 5). The most widely applied of these is the Kerner equation, which presents the... 

More quantitative information depends on the use of models. The Takayanagi models were already mentioned in connection with Figure 6.29. More analytical models have been evolved by Kerner/ Hashin and Shtrikman/ and Davies.Briefly, the first two theories assume spherical particles dispersed in an isotropic matrix. From the modulus of each phase, the composite modulus is calculated. An upper or lower bound modulus is arrived at by assuming the higher or lower modulus phase to be the matrix, respectively. The theory is reviewed elsewhere. 

Hashin and Shtrikman ako provided an upper bound model for the thermal conductivity of spherical particles dispersed randomly in a continuous matrix... 

The mathematical representation of the elastic behavior of oriented heterogeneous solids can be somewhat improved through a more appropriate choice of the boundary conditions such as proposed by Hashin and Shtrikman [66] and Stern-stein and Lederle [86]. In the case of lamellar polymers the formalisms developed for reinforced materials are quite useful [87—88]. An extensive review on the experimental characterization of the anisotropic and non-linear viscoelastic behavior of solid polymers and of their model interpretation had been given by Hadley and Ward [89]. New descriptions of polymer structure and deformation derive from the concept of paracrystalline domains particularly proposed by Hosemann [9,90] and Bonart [90], from a thermodynamic treatment of defect concentrations in bundles of chains according to the kink and meander model of Pechhold [10—11], and from the continuum mechanical analysis developed by Anthony and Kroner [14g, 99]. 

Bound computational models describe the upper and lower limits of elastic parameters of a composite medium. Voigt (1910) gives the upper and Reuss (1929) gives the lower bound. More narrow bounds are derived by Hashin and Shtrikman (1962a,b, 1963). 

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