向钱伟长先生致敬 细环壳钱伟长方程的精确解 引言 今天是钱伟长先生的诞辰日,钱先生于1912年10月9日出生,其后半生最重要的学术贡献是关于环形壳体解析理论的系统研究,证明了级数解的收敛性问题并做了大量计算,对该科学问题做出了非常重要的贡献,同时也给后人留下了一些待解问题,如细环壳钱伟长方程的特殊函数表示问题,作者在此给出了这个问题的结果。继承发展是对前辈最好的纪念,在先生诞辰之日作者愿以此文向钱先生致敬。 环壳 环壳是中国现代力学的二位主要奠基人钱伟长和张维都做过系统研究的唯一的一种壳体。圆环壳形如救生圈或汽车轮胎,它是一种形状比较复杂的旋转壳,环壳是壳体理论中难度比较大的问题之一,钱伟长曾说“环壳理论有两个特点:方程复杂和求解不易”。 环壳问题难于求解的数学原因是环壳的基本方程是变系数的高阶偏微分方程,其系数是分数型的且分母在其二个几何顶点有“零点”即具有奇异性,由于在顶点两边的高斯曲率变号而使方程变性,是壳体理论中最复杂的问题之一。 德国的Hans Reissner非常有名,有关黑洞的4个解中其中一个(角动量J=0, 质量M和电荷Q不为零)是他1916求得的,见论文 Reissner, H. (1916). über die Eigengravitation des electrischen Feldes nach der Einsteinschen Theorie. Annalen der Physik (in German) 50 : 106。 另外, 他的儿子Eric Reissner是大力学家,弹性力学的2变量变分原理就是由Eric提出的,后来中国的胡海昌和日本的Washizu推广到3变量变分原理。 有关环壳的最早研究是从Hans Reissner开始的,他指导博士生Gustav Weihs在1911年完成其有关环壳的博士论文(我们没有看到这篇论文)。Weihs在1911年分析了薄轮胎在承受旋转对称和非旋转对称载荷时的应力状态时,第一次用级数解对环壳的应力状态进行了分析。瑞士苏黎世联邦高等工业大学的Hans Wissler 在1916年利用Reissner-Meissner的旋转对称壳体微分方程系统地以级数的形式给出了在旋转对称载荷作用下环壳的弯曲应力状态解。他得到的级数解对于细环壳收敛较快,但对于粗环壳收敛极慢,无法在工程中应用。求解对于粗环壳全域一致收敛的解就变成了一个壳体理论难题,1944年张维在国际上第一次求得了粗环壳的一致收敛的渐进解。 由于环壳非常复杂,所以世界上研究它一般都使用复变量方程。本文作者首先导出环壳的位移型方程,系统研究了细环壳的弯曲、振动和屈曲,在世界上第一次得到这个问题的封闭解, 作为封闭解的应用修改了其中一个60年来国际工程届一直使用的膨胀接头的经典设计公式。 环壳方程和细环壳的解 钱伟长,应用数学与力学文集,江苏科技出版社,1979。 孙博华,环壳百年忆张维,力学与实践,2013, 35:94-97. Wissler, H., Festigkeiberechung von Ringsflachen, Promotionarbeit, Zurich (1916). Zhang Wei, Dissertation, TH Berlin, published partly in Science Report of Tsinghua University, Ser.A, Vol5.,289-349,1949. Bohua Sun, Closed Form Solution of Axisymmetric Slender Elastic Toroidal Shells, J. Engrg. Mech., Volume 136, Issue 10, pp. 1281-1288 (2010). Novozhilov, V.V., The Theory of Thin Shells, Noordhoff, Groningen, 1959. Zhang, R.J. and Zhang, W., Toroidal shells under nonsymmetrical loading, Int. J. Solids and Structures, Vol.1, No.19,1994. 孙博华,李群对称方法和应用,DOI: 10.13140/RG.2.1.2500.1685, http://www.researchgate.net/publication/282613969_Lie_Group_Symmetry_Methods_and_Applications Bohua Sun, Exact solution of some shells, The first int. conference on shells, plates and beams, Bologna, Italy, 2015.
2005年,93岁高龄的钱伟长校长在出席其生平最后一次上海大学本科生毕业典礼时,动情地向台下的学生说“今天你们毕业了,我有几句话要告诉你们,这就是:先天下之忧而忧,后天下之乐而乐!天下就是老百姓,百姓之忧、民族之忧,你们是否放在心上?先天下之忧而忧,忧过没有?后天下之乐而乐,乐过没有?”这就是著名的“钱伟长之问”。 h ttp://www.longquanzs.org/articledetail.php?id=13578 一直知道钱学森之问,不知有“钱伟长之问”,真是孤陋寡闻!
钱老因为划为右派耽误了做学问最好的近三十年光景,再次出来做学问的时候已经70高龄。在美国数学学会的数据库中找到钱老发表的34篇文章,可能不是全部,其中有著名的发表于1944年的《 The intrinsic theory of thin shells and plates 》,尽管该理论早已写进教科书,但是至今仍被引用52次,另外还有一篇文章 《 Torsion with variable twist 》 是钱老在加州理工做博后期间和冯卡门的写的。在学术宗谱网站上也找到了 钱老和他的导师John Lighton Synge的师生关系 ,无奈钱老虽然也是桃李满天下,但是学术后代的数据没有添加进去。 v\:* {behavior:url(#default#VML);} o\:* {behavior:url(#default#VML);} w\:* {behavior:url(#default#VML);} .shape {behavior:url(#default#VML);} Normal 0 false false false MicrosoftInternetExplorer4 /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-parent:""; mso-padding-alt:0in 5.4pt 0in 5.4pt; mso-para-margin:0in; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman"; mso-ansi-language:#0400; mso-fareast-language:#0400; mso-bidi-language:#0400;} Publications results for "Items authored by Chien, Wei Zang " MR1449801 Xu, Kai Yu ; Zhou, Zhe Wei ; Qian, Wei Chang The subharmonic bifurcations of a laminated anisotropic circular cylindrical shell. (Chinese) J. Shanghai Univ. Nat. Sci. 3 (1997), no. 1, 88--92. 73H05 (73K15) PDF Doc Del Clipboard Journal Article MR1343065 (96e:73057) Chien, Wei Zang The second order approximation theory of three-dimensional elastic plates and its boundary conditions without using Kirchhoff-Love assumptions. Appl. Math. Mech. (English Ed.) 16 (1995), no. 5, 405--427; translated from Appl. Math. Mech. 16 (1995), no. 5, 381--402(Chinese) (Reviewer: L. P. Lebedev) 73K10 (73B05) PDF Doc Del Clipboard Journal Article MR1334386 (96b:73061) Chien, Wei Zang Approximation theory of three-dimensional elastic plates and its boundary conditions without using Kirchhoff-Love assumptions. Appl. Math. Mech. (English Ed.) 16 (1995), no. 3, 203--224; translated from Appl. Math. Mech. 16 (1995), no. 3, 189--209(Chinese) (Reviewer: L. P. Lebedev) 73K10 (73C99) PDF Doc Del Clipboard Journal Article MR1144280 (92m:42005) Chien, Weizang Summation of trigonometric series by Fourier transforms. Advances in applied mathematics and mechanics in China, Vol. 3, 1--14, Internat. Acad. Publ., Beijing, 1991. 42A16 PDF Doc Del Clipboard Journal Article MR1009230 (90h:42014) Chien, Wei-zang Summation of trigonometric series by Fourier transforms. Appl. Math. Mech. (English Ed.) 10 (1989), no. 5, 385--397; translated from Appl. Math. Mech. 10 (1989), no. 5, 371--383(Chinese) (Reviewer: Rafat N. Siddiqi) 42A24 PDF Doc Del Clipboard Journal Article MR0943882 (89f:73021) Chien Wei-zang Variational principles and generalized variational principles for nonlinear elasticity with finite displacement. Appl. Math. Mech. (English Ed.) 9 (1988), no. 1, 1--12; translated from Appl. Math. Mech. 9 (1988), no. 1, 1--11(Chinese) 73C60 (49A29 73C50) PDF Doc Del Clipboard Journal Article MR0901374 (88f:73018) Chien, Wei Zang Variational principles in elasticity with nonlinear stress-strain relation. Appl. Math. Mech. (English Ed.) 8 (1987), no. 7, 589--601; translated from Appl. Math. Mech. 8 (1987), no. 7, 567--577(Chinese) 73C60 PDF Doc Del Clipboard Journal Article MR0795444 (86m:73043) Chien, Wei Zang Involutory transformations and variational principles with multivariables in thin plate bending problems. Appl. Math. Mech. (English Ed.) 6 (1985), no. 1, 25--40; translated from Appl. Math. Mech. 6 (1985), no. 1, 25--39(Chinese) (Reviewer: L. Jentsch) 73K10 PDF Doc Del Clipboard Journal Article MR0889290 (88d:73038) Chien, Wei-Zang Incompatible elements and generalized variational principles. Adv. in Appl. Mech. 24 (1984), 93--153. 73K25 (49H05) PDF Doc Del Clipboard Journal Article MR0794384 (86k:73017) Chien, Wei Zang Classification of variational principles in elasticity. Appl. Math. Mech. (English Ed.) 5 (1984), no. 6, 1737--1743; translated from Appl. Math. Mech. 5 (1984), no. 6, 765--770(Chinese) (Reviewer: G. Aubert) 73C60 PDF Doc Del Clipboard Journal Article MR0768444 (85j:76025) Chien, Wei Zang Variational principles and generalized variational principles in hydrodynamics of viscous fluids. Appl. Math. Mech. (English Ed.) 5 (1984), no. 3, 1281--1296; translated from Appl. Math. Mech. 5 (1984), no. 3, 305--322(Chinese) 76N10 (49H05 76D99) PDF Doc Del Clipboard Journal Article MR0735866 (85b:73008) Chien, Wei Zang Method of high-order Lagrange multiplier and generalized variational principles of elasticity with more general forms of functionals. Appl. Math. Mech. (English Ed.) 4 (1983), no. 2, 143--157. 73C60 (49H05) PDF Doc Del Clipboard Journal Article MR0718349 (85d:73019) Chien, Wei Zang Incompatible plate elements based upon the generalized variational principles. Hybrid and mixed finite element methods (Atlanta, Ga., 1981), 381--404, Wiley Ser. Numer. Methods Engrg., Wiley, Chichester, 1983. 73K25 (65N30) PDF Doc Del Clipboard Journal Article MR0624371 (83f:65180) Qian, Wei Chang Recent developments in the finite element method. (Chinese) Lixue yu Shijian 2 (1980), no. 4, 4--11, 16. 65N30 PDF Doc Del Clipboard Journal Article MR0615120 (83e:65169) Chien, Wei Zang The explicit forms of field functions in tetrahedral finite elements with $16$ and $20$ degrees of freedom. (Chinese) Appl. Math. Mech. 1 (1980), no. 2, 153--158. 65N30 PDF Doc Del Clipboard Journal Article MR0615262 (82k:73055) Chien, Wei Zang Finite element analysis of axisymmetric elastic body problems. (Chinese) Appl. Math. Mech. 1 (1980), no. 1, 25--35. 73K25 PDF Doc Del Clipboard Journal Article MR0081080 (18,350c) Chien, Wei-zang Problem of large deflection of circular plate. Arch. Mech. Stos. 8 (1956), 3--12. (Reviewer: B. R. Seth) 73.2X PDF Doc Del Clipboard Journal Article MR0069719 (16,1070e) Chien, Wei-Zang ; Yeh, Kai-Yuan On the large deflection of circular plate. Sci. Sinica 3, (1954). 405--436. (Reviewer: E. Reissner) 73.2X PDF Doc Del Clipboard Journal Article MR0066882 (16,644f) Chien, Wei-Zang Assumptions in Saint-Venant's solution for the torsion of an elastic cylinder. Acta Sci. Sinica 3, (1954). 165--170. (Reviewer: J. L. Ericksen) 73.2X PDF Doc Del Clipboard Journal Article MR0073427 (17,432g) Chien, Wei-Zang ; Chen, Chih-Ta Theory of rolling. (Chinese) Chinese J. Phys. 9 (1953), 57--92. 73.2X PDF Doc Del Clipboard Journal Article MR0073410 (17,430b) Chien, Wei-Zang Continuous beams with non-uniform stiffness. (Chinese) Chinese J. Phys. 9 (1953), 170--182. 73.2X PDF Doc Del Clipboard Journal Article MR0071986 (17,211b) Chien, Wei-Zang Assumptions in Saint-Venant's solution for the torsion of an elastic cylinder. (Chinese) Chinese J. Phys. 9 (1953), 215--220. 73.2X PDF Doc Del Clipboard Journal Article MR0040890 (12,763d) Chien, Wei-Zang Hydrodynamic theory of lubrication for plane sliders of finite width. Chinese J. Phys. 7, (1949). 278--299. (Reviewer: G. F. Carrier) 76.1X PDF Doc Del Clipboard Journal Article MR0042303 (13,90d) Chien, Wei-Zang ; Ho, Shui-Tsing Asymptotic method on the problems of thin elastic ring shell with rotational symmetrical load. Eng. Rep. Nat. Tsing Hua Univ. 3, (1948). no. 2, 71--86. (Reviewer: R. A. Clark) 73.2X PDF Doc Del Clipboard Journal Article MR0041606 (12,871a) Chien, Wei-Zang The true leaving angle for diaphragm and bucket wheel with curved guides at the discharge end. Eng. Rep. Nat. Tsing Hua Univ. 4, (1948). no. 1, 78--102. (Reviewer: C. Saltzer) 76.1X PDF Doc Del Clipboard Journal Article MR0034206 (11,558j) Chien, Wei Zang Derivation of the equations of equilibrium of an elastic shell from the general theory of elasticity. Sci. Rep. Nat. Tsing Hua Univ. Ser. A. 5, (1948). 240--251. (Reviewer: E. Reissner) 73.2X PDF Doc Del Clipboard Journal Article MR0026893 (10,218a) Chien, Wei Zang Asymptotic behavior of a thin clamped circular plate under uniform normal pressure at very large deflection. Sci. Rep. Nat. Tsing Hua Univ. 5, (1948). 71--94. (Reviewer: D. L. Holl) 73.2X PDF Doc Del Clipboard Journal Article MR0042902 (13,180d) Chien, Wei-Zang Symmetrical conical flow at supersonic speed by perturbation method. Eng. Rep. Nat. Tsing Hua Univ. 3, (1947). no. 1, 1--14. (Reviewer: Sears. W. R.) 76.1X PDF Doc Del Clipboard Journal Article MR0024321 (9,481b) Chien, Wei-Zang Large deflection of a circular clamped plate under uniform pressure. Chinese J. Phys. 7, (1947). 102--113. (Reviewer: D. L. Holl) 73.2X PDF Doc Del Clipboard Journal Article MR0017173 (8,119b) von Kármán, Theodore ; Chien, Wei-Zang Torsion with variable twist. J. Aeronaut. Sci. 13, (1946). 503--510. (Reviewer: D. L. Holl) 73.2X PDF Doc Del Clipboard Journal Article MR0010867 (6,81i) Chien, Wei-Zang The intrinsic theory of thin shells and plates. III. Application to thin shells. Quart. Appl. Math. 2, (1944). 120--135. (Reviewer: G. E. Hay) 73.2X PDF Doc Del Clipboard Journal Article MR0010102 (5,250f) Chien, Wei-Zang The intrinsic theory of thin shells and plates. II. Application to thin plates. Quart. Appl. Math. 2, (1944). 43--59. (Reviewer: G. E. Hay) 73.2X PDF Doc Del Clipboard Journal Article MR0009744 (5,195c) Chien, Wei-Zang The intrinsic theory of thin shells and plates. I. General theory. Quart. Appl. Math. 1, (1944). 297--327. (Reviewer: G. E. Hay) 73.2X PDF Doc Del Clipboard Journal Article MR0009079 (5,98a) Weinstein, Alexander ; Chien, Wei Zang On the vibrations of a clamped plate under tension. Quart. Appl. Math. 1, (1943). 61--68. (Reviewer: R. Courant) 36.0X PDF Doc Del Clipboard Journal Article
非常意外三钱中仅存的 钱伟长 先生与世长辞。不久前,还听与钱老有交往的人透露,他已经进入无喜无忧的境界,将益寿延年。因此,我在大会发言中说, Aged 98, he is a legendary figure belonging to the realm beyond the wildest imagine of common people. 重新通读钱老的《教育和教学问题的思考》。记得这本书是十来年前钱老召集学校的教授开会,会后每人赠送了这本书,上面还有钱老的签名 ( 章 ) 。 自己并非教育方面的专家,只是从事具体教学工作的人,更倾向于从比较朴素的观点 ( 也就是比较低的层面上 ) 理解钱老的教诲。对于钱老关于教师如何搞科研、研究生如何选题、学生如何自学等非常有共鸣。至于办学方针、学制等,其实也超出了我们普通教师关注的范围。 如果一定要概括,就我的理解,钱老教育思想的核心是强调创新和贯通。无论学生还是教师,都需要有创新意识,要全面发展,能与时俱进地适应甚至引领时代的需求。 在负责申报国家精品时,按要求需要列出 200 字以内不超过三项课程的主要特色及创新点。我觉得这应该是申报的点睛之笔。其中前两点都与钱老的教育思想有关。 贯彻钱伟长院士培养全面的人和打破四堵墙的办学理念。 结合本校三制办学模式,