Microstructure, mechanical properties and deformation characteristics of Al-Mg-Si alloys processed by a continuous expansion extrusion approach

In the present work,a double-pass continuous expansion extrusion forming(CEEF) process was proposed for an Al-Mg-Si alloy,in which the diameter of rods was gradually expanded.The microstructural evolution,mechanical properties and deformation characteristics were investigated by utilizing microstructural observations,mechanical testing and a finite element method coupled with a cellular automata model.The results showed that the strength and ductility of the double-pass CEEF processed Al-Mg-Si alloys were improved synchronously,especially in artificially aged alloys.The grain size of the processed Al-Mg-Si alloy rods was refined remarkably by continuous dynamic recrystallization(CDRX)and geometric dynamic recrystallization(GDRX),and the homogeneity of microstructure was gradually improved with increasing number of processing passes.The artificially aged alloy processed with double-pass CEEF and water quenching contained fine(sub)grains and high-density dislocations,which resulted in more needle-shaped β" precipitates and a larger precipitate aspect ratio than the as-received and air-cooled CEEF alloys owing to the different precipitation kinetics.The severe cumulate strain and microshear bands were found to accelerate CDRX and GDRX for grain refinement between adjacent positions of the parabolic metal flow due to the special temperature characteristics and la rge shear straining during the CEEF process.

Two Dual Expansions for Generalized Bivariate Thiele-Type Matrix Valued Interpolating Continued Fractions

A new method for the construction of bivariate matrix valued rational interpolants on a rectangulargrid is introduced. The rational interpolants are expressed in the continued fraction form with scalardenominator. Tile matrix quotients are based oil the generalized inverse for a matrix, Which is found to beeffective in continued fraction interpolation. In this paper, tWo dual expansions for bivariate matrix valuedThiele-type interpolating continued fractions are presented, then, tWo dual rational interpolants are definedout of them.

On the Family of Thue Equation |x^3+mx^2y-(m+3)xy^2+y^3|=k

The family of cubic Thue equation which depend on two parameters ｜ x^3 ＋ mx^2 y-（m＋3） xy^2＋y^3｜=k is studied. Using rational approximation, we give a smaller upper bound of the solution of the equation, that is quite better than the present result. Moreover, we study two inequalities ｜ x^3 ＋ mx^2y-（m ＋ 3） xy^2＋y^3 ｜ =k≤2m＋3 and ｜x^3 ＋mx^2y- （m＋3）xy^2 ＋ y^3｜ = k≤ （2m＋3）^2 separately. Our result of upper bound make it easy to solve those inequalities by simple method of continuous fraction expansion.

A unified direct approach for discretizing fractional-order differentiator in delta-domain

Fractional-order differentiator is a principal component of the fractional-order controller.Discretization of fractional-order differentiator is essential to implement the fractionalorder controller digitally.Discretization methods generally include indirect approach and direct approach to find the discrete-time approximation of fractional-order differentiator in the Z-domain as evident from the existing literature.In this paper,a direct approach is proposed for discretization of fractional-order differentiator in delta-domain instead of the conventional Z-domain as the delta operator unifies both analog system and digital system together at a high sampling frequency.The discretization of fractional-order differentiator is accomplished in two stages.In the first stage,the generating function is framed by reformulating delta operator using trapezoidal rule or Tustin approximation and in the next stage,the fractional-order differentiator has been approximated by expanding the generating function using continued fraction expansion method.The proposed method has been compared with two well-known direct discretization methods taken from the existing literature.Two examples are presented in this context to show the efficacy of the proposed discretization method using simulation results obtained from MATLAB.