When studying the regular polygonal inclusion in 1997, Nozaki and Taya discovered numerically some remarkable properties of Eshelby tensor： Eshelby tensor at the center and the averaged Eshelby tensor over the inclus...When studying the regular polygonal inclusion in 1997, Nozaki and Taya discovered numerically some remarkable properties of Eshelby tensor： Eshelby tensor at the center and the averaged Eshelby tensor over the inclusion domain are equal to that of a circular inclusion and independent of the orientation of the inclusion. Then Kawashita and Nozaki justified the properties mathematically. In the present paper, some other properties of a regular polygonal inclusion are discovered. We find that for an N-fold regular polygonal inclusion except for a square, the arithmetic mean of Eshelby tensors at N rotational symmetrical points in the inclusion is also equal to the Eshelby tensor for a circular inclusion and independent of the orientation of the inclusion. Furthermore, in two corollaries, we point out that Eshelby tensor at the center, the averaged Eshelby tensor over the inclusion domain, and the line integral average of Eshelby tensors along any concentric circle of the inclusion are all identical with the arithmetic mean.展开更多
Based on the deformation characteristic of regular polygonal box stamped parts and the superfluous triangle material wrinkle model,the criterion of regular polygonal box stamped parts without wrinkle was deduced and u...Based on the deformation characteristic of regular polygonal box stamped parts and the superfluous triangle material wrinkle model,the criterion of regular polygonal box stamped parts without wrinkle was deduced and used to predict and control the wrinkle limit.According to the fracture model,the criterion of regular polygonal box stamped parts without fracture was deduced and used to predict and control the fracture limit.Combining the criterion for stamping without wrinkle with that without fracture,the stamping criterion of regular polygonal box stamped parts was obtained to predict and control the stamping limit.Taken the stainless steel0Cr18Ni9(SUS304)sheet and the square box stamped part as examples,the limit diagram was given to predict and control the wrinkle,fracture and stamping limits.It is suitable for the deep drawing without flange,the deep drawing and stretching combined forming with flange and the rigid punch stretching of plane blank.The limit deep-drawing coefficient and the minimum deep-drawing coefficient can be determined,and the appropriate BHF(blank holder force)and the deep-drawing force can be chosen.These provide a reference for the technology planning,the die and mold design and the equipment determination,and a new criterion evaluating sheet stamping formability,which predicts and controls the stamping process,can be applied to the deep drawing under constant or variable BHF conditions.展开更多
Load behavior is one of the most critical factors affecting mills＇ energy consumption and grinding efficiency, and is greatly affected by the liner profiles. Generally, as liner profiles vary, the ball mill performan...Load behavior is one of the most critical factors affecting mills＇ energy consumption and grinding efficiency, and is greatly affected by the liner profiles. Generally, as liner profiles vary, the ball mill performances are extremely different. In order to study the performance of the ball mill with regular polygon angle-spiral liners（RPASLs）, experimental and numerical studies on three types of RPASLs, including regular quadrilateral, pentagonal and hexagonal, are carried out. For the fine product of desired size, two critical parameters are analyzed： the energy input to the mill per unit mass of the fine product, E＊, and the rate of production of the fine product, F＊. Results show that the optimal structure of RPASLs is Quadrilateral ASL with an assembled angle of 50°. Under this condition, the specific energy consumption E＊ has the minimum value of 303 J per fine product and the production rate F＊ has the maximum value of 0.323. The production rate F＊ in the experimental result is consistent with the specific collision energy intensity to total collision energy intensity ratio Es/Et in the simulation. The relations between the production rate F＊ and the specific energy consumption E＊ with collision energy intensity Es and Et are obtained. The simulation result reveals the essential reason for the experimental phenomenon and correlates the mill performance parameter to the collision energy between balls, which could guide the practical application for Quadrilateral ASL.展开更多
In this paper we study the necessary conditions for the masses of the nested regular polygon solutions of the planar 2N-body problem.We prove that the masses at the vertices of each regular polygon must be equal to ea...In this paper we study the necessary conditions for the masses of the nested regular polygon solutions of the planar 2N-body problem.We prove that the masses at the vertices of each regular polygon must be equal to each other.展开更多
In this study,a large eddy simulation(LES)for fully-developed turbulent flows through a duct of regular-polygon cross-section using the immersed boundary(IB)method is performed.In case of the turbulent flow through th...In this study,a large eddy simulation(LES)for fully-developed turbulent flows through a duct of regular-polygon cross-section using the immersed boundary(IB)method is performed.In case of the turbulent flow through the square duct,though there are some disagreements of the mean quantities related with the streamwise velocity among the present LES,the previous direct numerical simulation(DNS)and the LES without the IB method,and the present LES can reproduce the secondary flow of the DNS and LES.The LES result for ten types of regular-polygon duct shows that the secondary-flow speed decreases as the number of sides of the regular polygon n increases and that the secondary flow in case of the regular icosagon duct disappears like the turbulent pipe flow.In case of low n,the behavior of the turbulent structures near the side center is different from that near the vertex.展开更多
A problem of the plane elasticity theory is addressed for a doubly connected body with an external boundary of the regular hexagon shape and with a 6-fold symmetric hole at the center. It is assumed that all the six s...A problem of the plane elasticity theory is addressed for a doubly connected body with an external boundary of the regular hexagon shape and with a 6-fold symmetric hole at the center. It is assumed that all the six sides of the hexagon are subjected to uniform normal displacements via smooth rigid stamps, while the uniformly distributed normal stress is applied to the internal hole boundary. Using the methods of complex analysis, the analytical image of Kolosov-Muskhelishvili＇s complex potentials and the shape of the hole contour are determined from the condition that the circumferential normal stress is constant along the hole contour. Numerical results are given and shown in relevant graphs.展开更多
After having laid down the Axiom of Algebra, bringing the creation of the square root of -1 by Euler to the entire circle and thus authorizing a simple notation of the nth roots of unity, the author uses it to organiz...After having laid down the Axiom of Algebra, bringing the creation of the square root of -1 by Euler to the entire circle and thus authorizing a simple notation of the nth roots of unity, the author uses it to organize homogeneous divisions of the limited development of the exponential function, that is opening the way to the use of a whole bunch of new primary functions in Differential Calculus. He then shows how new supercomplex products in dimension 3 make it possible to calculate fractals whose connexity depends on the product considered. We recall the geometry of convex polygons and regular polygons.展开更多
基金the National Natural Science Foundation of China(10172003 and 10372003)
文摘When studying the regular polygonal inclusion in 1997, Nozaki and Taya discovered numerically some remarkable properties of Eshelby tensor： Eshelby tensor at the center and the averaged Eshelby tensor over the inclusion domain are equal to that of a circular inclusion and independent of the orientation of the inclusion. Then Kawashita and Nozaki justified the properties mathematically. In the present paper, some other properties of a regular polygonal inclusion are discovered. We find that for an N-fold regular polygonal inclusion except for a square, the arithmetic mean of Eshelby tensors at N rotational symmetrical points in the inclusion is also equal to the Eshelby tensor for a circular inclusion and independent of the orientation of the inclusion. Furthermore, in two corollaries, we point out that Eshelby tensor at the center, the averaged Eshelby tensor over the inclusion domain, and the line integral average of Eshelby tensors along any concentric circle of the inclusion are all identical with the arithmetic mean.
文摘Based on the deformation characteristic of regular polygonal box stamped parts and the superfluous triangle material wrinkle model,the criterion of regular polygonal box stamped parts without wrinkle was deduced and used to predict and control the wrinkle limit.According to the fracture model,the criterion of regular polygonal box stamped parts without fracture was deduced and used to predict and control the fracture limit.Combining the criterion for stamping without wrinkle with that without fracture,the stamping criterion of regular polygonal box stamped parts was obtained to predict and control the stamping limit.Taken the stainless steel0Cr18Ni9(SUS304)sheet and the square box stamped part as examples,the limit diagram was given to predict and control the wrinkle,fracture and stamping limits.It is suitable for the deep drawing without flange,the deep drawing and stretching combined forming with flange and the rigid punch stretching of plane blank.The limit deep-drawing coefficient and the minimum deep-drawing coefficient can be determined,and the appropriate BHF(blank holder force)and the deep-drawing force can be chosen.These provide a reference for the technology planning,the die and mold design and the equipment determination,and a new criterion evaluating sheet stamping formability,which predicts and controls the stamping process,can be applied to the deep drawing under constant or variable BHF conditions.
基金Supported by National Natural Science Foundation of China(Grant Nos.51675484,51275474,51505424)Zhejiang Provincial Natural Science Foundation of China(Grant Nos.LZ12E05002,LY15E050019)
文摘Load behavior is one of the most critical factors affecting mills＇ energy consumption and grinding efficiency, and is greatly affected by the liner profiles. Generally, as liner profiles vary, the ball mill performances are extremely different. In order to study the performance of the ball mill with regular polygon angle-spiral liners（RPASLs）, experimental and numerical studies on three types of RPASLs, including regular quadrilateral, pentagonal and hexagonal, are carried out. For the fine product of desired size, two critical parameters are analyzed： the energy input to the mill per unit mass of the fine product, E＊, and the rate of production of the fine product, F＊. Results show that the optimal structure of RPASLs is Quadrilateral ASL with an assembled angle of 50°. Under this condition, the specific energy consumption E＊ has the minimum value of 303 J per fine product and the production rate F＊ has the maximum value of 0.323. The production rate F＊ in the experimental result is consistent with the specific collision energy intensity to total collision energy intensity ratio Es/Et in the simulation. The relations between the production rate F＊ and the specific energy consumption E＊ with collision energy intensity Es and Et are obtained. The simulation result reveals the essential reason for the experimental phenomenon and correlates the mill performance parameter to the collision energy between balls, which could guide the practical application for Quadrilateral ASL.
基金This work was supported by the National Natural Science Foundation of China (Grant No. 19871096), QSSTE and MOST.
文摘In this paper we study the necessary conditions for the masses of the nested regular polygon solutions of the planar 2N-body problem.We prove that the masses at the vertices of each regular polygon must be equal to each other.
文摘In this study,a large eddy simulation(LES)for fully-developed turbulent flows through a duct of regular-polygon cross-section using the immersed boundary(IB)method is performed.In case of the turbulent flow through the square duct,though there are some disagreements of the mean quantities related with the streamwise velocity among the present LES,the previous direct numerical simulation(DNS)and the LES without the IB method,and the present LES can reproduce the secondary flow of the DNS and LES.The LES result for ten types of regular-polygon duct shows that the secondary-flow speed decreases as the number of sides of the regular polygon n increases and that the secondary flow in case of the regular icosagon duct disappears like the turbulent pipe flow.In case of low n,the behavior of the turbulent structures near the side center is different from that near the vertex.
文摘A problem of the plane elasticity theory is addressed for a doubly connected body with an external boundary of the regular hexagon shape and with a 6-fold symmetric hole at the center. It is assumed that all the six sides of the hexagon are subjected to uniform normal displacements via smooth rigid stamps, while the uniformly distributed normal stress is applied to the internal hole boundary. Using the methods of complex analysis, the analytical image of Kolosov-Muskhelishvili＇s complex potentials and the shape of the hole contour are determined from the condition that the circumferential normal stress is constant along the hole contour. Numerical results are given and shown in relevant graphs.
文摘After having laid down the Axiom of Algebra, bringing the creation of the square root of -1 by Euler to the entire circle and thus authorizing a simple notation of the nth roots of unity, the author uses it to organize homogeneous divisions of the limited development of the exponential function, that is opening the way to the use of a whole bunch of new primary functions in Differential Calculus. He then shows how new supercomplex products in dimension 3 make it possible to calculate fractals whose connexity depends on the product considered. We recall the geometry of convex polygons and regular polygons.
