In 1962, Wunderlich published the article “On a developable Möbius band,” in which he attempted to determine the equilibrium shape of a free standing Möbius band. In line with Sadowsky’s pioneering works on Möbius bands of infinitesimal width, Wunderlich used an energy minimization principle, which asserts that the equilibrium shape of the Möbius band has the lowest bending energy among all possible shapes of the band. By using the developability of the band, Wunderlich reduced the bending energy from a surface integral to a line integral without assuming that the width of the band is small. Although Wunderlich did not completely succeed in determining the equilibrium shape of the Möbius band, his dimensionally reduced energy integral is arguably one of the most important developments in the field. In this work, we provide a rigorous justification of the validity of the Wunderlich integral and fully formulate the energy minimization problem associated with finding the equilibrium shapes of closed bands, including both orientable and nonorientable bands with arbitrary number of twists. This includes characterizing the function space of the energy functional, dealing with the isometry and local injectivity constraints, and deriving the Euler–Lagrange equations. Special attention is given to connecting edge conditions, regularity properties of the deformed bands, determination of the parameter space needed to ensure that the deformation is surjective, reduction in isometry constraints, and deriving matching conditions and jump conditions associated with the Euler–Lagrange equations.
Europium-doped yttrium-silicon-oxide-nitride phosphors are synthesized by carbothermal reduction and nitridation method. The crystal structure of the phosphors changed gradually from oxide Y2Si2O7 to nitride YSi3N5 state with increasing dosage of Si3N4 and carbon powder. The Y2Si2O7:Eu phosphor shows a blue emission at 465 nm with 300 nm excitation and a characteristic red emission of Eu^3+ at 612 nm with 230 nm excitation. The YSi3N5:EU phosphor shows a broad emission band centred at 595nm with some sharp peaks of Eu^3+ with 325nm excitation. The absorption of the studied phosphors increases from 450 to 700hm with an increment in nitrogen content. Blue-to-orange tunable luminescence is observed with 390 nm excitation. 相似文献