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Abstract: The overall properties of a binary elastic periodic fiber-reinforced composite, with transversely isotropic constituents in an anti-plane-strain deformation state, are studied here for a cell periodicity of square type. This analysis considers four different orientations of the axis of transverse isotropy of constituents with respect to the direction of fibers. Each case is characterized by very simple closed-form expressions for the effective coefficients, which were obtained using the asymptotic homogenization method. Local problems defined on a periodic square unit cell are solved using Weierstrassian and Natanzon’s functions and perturbation theory relative to small anisotropy. In the isotropic limit, comparison with rigorous bounds and some well-known mixing rules are made. Also, comparisons with finite element calculations show that the derived closed-form formulae provide excellent results even for large anisotropy. [Copyright &y& Elsevier]

Abstract: A binary composite is studied here, where the electroelastic properties of the constituent materials belong to the crystal class 622. A square arrangement of long continuous circular cylinders, the fiber phase, embedded in a homogeneous medium is consider here. The composite is in a state of antiplane shear piezoelectricity, that is, a coupled state of out-of-plane mechanical displacement and in-plane electric field, which is characterized by three electroelastic parameters: longitudinal shear modulus, shear stress piezoelectric coefficient and transverse dielectric constant. Our interest here lies in the determination of its effective properties. They are derived by means of the method of two spatial scales. Closed-form expressions are obtained for them. Only one of the four local (or canonical) problems that arise is needed. Two properties are thus found. The Milgrom–Shtrikman compatibility relation is used to fix the remaining one. The local problem is solved using potential methods of a complex variable. The solution involves doubly periodic Weierstrass elliptic and related functions. The final formulae for the overall properties show explicitly the dependence on (i) the properties of the phases, (ii) the radius of the cylindrical fiber and (iii) the lattice sums associated with the square array. The shear modulus is shown to depend explicitly not only on the rigidity of the phases but also on their piezoelectric and dielectric coefficients. Some natural organic substances have the symmetry 622 like collagen. Recently Silva et al. measured its electroelastic properties. Their data is used to show some numerical results of the derived formulae as a function of the fiber volumetric fraction. [Copyright &y& Elsevier]

• An easy-to-use tool to find all the roots of Rachford-Rice equation is presented. • A simple explanation to why most of the numerical methods fail is given. • An interval of location for all real roots is given. • This tool eliminates the need of robust methods and parallel computing. • This tool is stable under perturbations even singular perturbation. Theoretical results about Rachford-Rice equation for an isothermal two-phase flash of multicomponent systems are presented. The alternative method finds all the roots of the equation by means of an Associated Polynomial in the vapor fraction. All real roots are located and the negative flash condition is applied to the study cases. Associated polynomial significantly reduces calculations by eliminating asymptotes, which prevent the convergence of traditional numerical methods. Some numerical examples are shown in order to assess the method, principally when Rachford-Rice equation fails. Besides, proposed methodology is compared with other methods and variants of Rachford-Rice equations obtaining same results with no computational effort and its implementation is straightforward. [ABSTRACT FROM AUTHOR]