Shtrikman

Zvi Hashin and S. Shtrikman, A Variational Approach to the Theory of the Elastic Behaviour of Multiphase Materials, Journal of the Mechanics and Physics of Solids, March-April 1963, pp. 127-140. 

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. 

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. 

Frei, E.H., Shtrikman, S. and Treves, D. (1957) Critical size and nucleation field of ideal ferromagnetic particles. Physical Review, 106 (3), 446-454. 

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. 

J. Gores, D. Goldhaber-Gordon, S. Heemeyer, M.A. Kastner, H. Shtrikman, D. Mahalu, U. Meirav, Fano resonances in electronic transport through a single-electron transistor, Phys. Rev. B 62 (2000) 2188. 

Watt IP (1979) Hashin-Shtrikman boimds on the effective elastic modrrh of polycrystals with orthorhombic... 

Mertes KM, Suzuki Y, Sarachik MP, Myasoedov Y, Shtrikman H, Zeldov E, Rum-berger EM, Hendrickson DN, Christou G (2003) Solid State Commun 127 131... 

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. 

Q— Random Hollow-Sphere 0 FCC Hollow-Sphere —Octet Truss — — Hashin-Shtrikman Bound... 

The relative strength of hollow-sphere foams lies between the theoretical performance of open- and closed-cell foams. The performance of optimized truss structures is similar to that of closed-cell foams and, for the Kagome truss, approaches the behavior of a Hashin-Shtrikman porous material. Honeycombs are the most efficient structures when loaded purely out-of-plane. However, plastic buckling can decrease its performance at low relative densities. Further, since honeycomb is highly anisotropic, any inplane loading results in severely reduced performance. Although the theoretical performance of closed-cell foams far exceeds that of open-cell foams, processing defects result in commercially available material that behaves similar to an open-cell material at low relative densities. Commercially available samples of other types of low-density metallic structures behave nearly as predicted. [17]... 

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