Hashin-Shtrikman equation

Analysis of the relationships between the moduli and bond strength between particles [222] has shown that for Vf = 0.1 — 0.15 the concentration dependence of the modulus corresponds to the lower curve in the Hashin-Shtrikman equation [223] (hard inclusion in elastic matrix), and for Vf — 0.34 to the upper boundary (elastic inclusion in a hard matrix). The 0.1 to 0.34 range is the phase inversion region. 

An approximation to Gc can be obtained from the Hashin-Shtrikman equation (35), which, for a viscous matrix, is given by... 

A variety of lattice-type materials have been studied in the recent literature including lattice block material [20,21], the octet-truss lattice [22], and most recently, the Kagome-truss lattice [23], The octet-truss lattice was studied in depth by [24] and is shown in Figure 1 because it is nearly an isotropic material. The equations for the behavior of the octet-truss lattice were obtained from [24], The Hashin-Shtrikman upper bound for an isotropic porous material [24,25] is also plotted for comparison. 

A chart of density and Young s modulus as a function of volume percent SiC is provided in Figure 2. The density follows the typical additive rule of noixtures. Over this range of SiC volume percent, the Young s modulus also appears to follow a linear relationship. When plotted in relation to the Hashin-Shtrikman bounds, however, a slight parabolic shape is observed, as shown in Figure 3. These boimds were calculated based on the equations provided by Hashin et al. [16] for Young s modulus. 

HASAWA. Health and Safety at Work, etc Act, 1975. See health and safety. Hashin s Equations. Expressions for the bulk and shear moduli, and hence Young s modulus, of composite materials, including porous materials. (A.G.Hashin and S. Shtrikman, J. Mech. Phy. Solids 11, 1963, pl27-140)... 

More narrow boundaries than Voigt and Reuss can be calculated as Hashin-Shtrikman bounds (Hashin and Shtrikman, 1962a,b, 1963). Mavko et al. (1998) present the equations in a comfortable form for a two-component medium ... 

This equation is identical to the Maxwell [236,237] solution originally derived for electrical conductivity in a dilute suspension of spheres. Hashin and Shtrikman [149] using variational theory showed that Maxwell s equation is in fact an upper bound for the relative diffusion coefficients in isotropic medium for any concentration of suspended spheres and even for cases where the solid portions of the medium are not spheres. However, they also noted that a reduced upper bound may be obtained if one includes additional statistical descriptions of the medium other than the void fraction. Weissberg [419] demonstrated that this was indeed true when additional geometrical parameters are included in the calculations. Batchelor and O Brien [34] further extended the Maxwell approach. 

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... 

Due to the negligible electrical conductivity of the polymer matrix, the electrical conductivity of the composite holds the lower bound equation found by Hashin and Shtrikman assuming a high condueting phase (metal) surrounded by an insulator (polymer). 

Hashin and Shtrikman produced the following lower bound equation to describe the effect of spherical filler particles on the thermal conductivity of a randomly dispersed, particle-in-matrix, two phase syston [16] ... 

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