On a Uniform Treatment of Darboux’s Method

Let F(z) be an analytic function in |z| < 1. If F(z) has only a finite number of algebraic singularities on the unit circle |z| = 1, then Darbouxfq9lyn6y2kv28fn/xxlarge8217.gif" alt="rsquo" align="BASELINE" BORDER="0">s method can be used to give an asymptotic expansion for the coefficient of zⁿ in the Maclaurin expansion of F(z). However, the validity of this expansion ceases to hold, when the singularities are allowed to approach each other. A special case of this confluence was studied by Fields in 1968. His results have been considered by others to be too complicated, and desires have been expressed to investigate whether any simplification is feasible. In this paper, we shall show that simplification is indeed possible. In the case of two coalescing algebraic singularities, our expansion involves only two Bessel functions of the first kind.     

Zr-based bulk metallic glass with super-plasticity under uniaxial compression at room temperature

Commonly, bulk metallic glasses exhibit very limited plastic deformation (<2%) at room temperature. In this letter, through appropriate composition choices, rods of Zr_62.55Cu_17.55Ni_9.9Al₁₀ and Zr_64.80Cu_14.85Ni_10.35Al₁₀ bulk metallic glasses (BMGs) were prepared by using copper-mold suction-casting. X-ray diffraction and differential scanning calorimetry were utilized to determine their structures and thermal stabilities, and uniaxial compression tests were adopted to study their plastic deformation behaviors (PDBs) at room temperature. The results showed that T_g and T_x of the former are 651.5 K and 748 K, respectively, while those of the latter are 646 K and 750 K, respectively. Micro-hardness values are 693 ± 7 Hv and 595 ± 5 Hv for T1 and T2, respectively. During the compression tests, the former underwent about 2.5% engineering strain, i.e., 2.8% true strain, then fractured. While the latter was compressed into a flake-like form, of which, engineering strain values are larger than 70%, i.e., true strain exceeds 120%, showing super-plasticity at room temperature. It can be concluded that micro-hardness and PDBs of BMGs are very sensitive to composition.     

An equation of state for anisotropic solids under shock loading

An anisotropic equation of state is proposed for accurate extrapolation of high-pressure shock Hugoniot states to other thermodynamics states for shocked single crystals and polycrystalline alloys. The proposed equation of state represents mathematical and physical generalization of the Mie-Grüneisen equation of state for isotropic material and reduces to this equation in the limit of isotropy. Using an anisotropic nonlinear continuum framework and generalized decomposition of a stress tensor [Int. J. Plasticity 24, 140 (2008)], the shock waves propagation along arbitrary directions in anisotropic solids of any symmetry can be examined. The non-associated strength model includes the distortion effect of the yield surface which can be used to describe the anisotropic strength differential effect. A numerical calculation showed that the general pulse shape, Hugoniot Elastic Limits (HELs), and Hugoniot stress levels for aluminum alloy 7010-T6 agree with the experimental data. The results are presented and discussed, and future studies are outlined.