颗粒振动高密度堆积的数值模拟研究
|
摘要
在本文中采用离散元方法,数值模拟了单一尺寸球在三维振动条件下的堆积。其目的是实现高堆积密度,表征其形成的致密结构,并最终确定硬球结晶机理。
在三维间歇振动和批量加料条件下,使用DEM模型研究了2500个直径为1厘米玻璃球的堆积。通过适当选择振动参数如振动振幅A和振动频率ω,可以实现堆积密度0.728的高密度堆积,它是比最大随机紧密堆积更致密的堆积。通过宏观特征如堆积密度ρ、微观特征如配位数(CN, Coordination Number)、径向分布函数(RDF, Radial Distribution Function)、角分布函数(ADF, Angle Distribution Function)及Voronoi/Delaunay孔尺寸分布以及伴随力场和速度场的变化发现：
(1)可以数值实现超越最大无规密排(p≈0.64)的有序结构,堆积密度可达0.728。
(2)CN和Voronoi/Delaunay分布表明,模拟产生的结构与随机堆积结构有很大不同,通过对比发现不论是致密堆积还是松散堆积,其宏观特征与微观特征是相互对应的。
(3)RDF和ADF的进一步分析证明所获得的结构并不是纯随机的,RDF和ADF曲线分布证明了堆积结构无论是在较大距离上还是在特定角度上都存在相关性,这是有序结构所具有的特点。从它们的分布上还发现一些局部的无序结构,即所谓的缺陷。
(4)孔的尺寸分布呈现出高且窄的主峰,表明孔分布小且均匀。Voronoi曲线有一个子峰,表明堆积结构中有较大孔存在。
(5)对致密和松散堆积进行了Voronoi多面体的特征分析和对比。这些特征包括：每个Voronoi多面体顶点分布,周长分布,面积分布,面数分布,体积分布；每个Voronoi多面体面的边分布,周长分布,面积分布。它们与松散结构相比表现很大差距。对于高密度堆积,它们的分布更趋于均匀,这是有序结构的一个典型特点。
(6)对致密和松散堆积进行了Delaunay四面体的特征分析和对比。这些特征包括：每个Delaunay四面体的面积、体积、直径、球形度。它们与松散结构相比差别很大,堆积密度越高其分布越均匀。
(7)通过静态和动态分析发现获得的堆积结构是FCC晶体,但内有少量的缺陷。结晶机理可以归因于小岛(核)的形成及生长,在长大过程中,一个晶粒被另一个晶粒吞并形成一个大的晶粒。
In this paper, numerical simulation of monosize particle packing under three dimensional (3D) vibration was carried out by using Discrete Element Method (DEM). The aim is to realize high packing density, followed by the characterization of the formed dense structure, and finally the identification of the hard sphere crystallization mechanism.
Here mainly focus on the packing of 2500 monosize glass beads with 1 cm in diameter under 3D interval vibration and batch-wised feeding with the aid of DEM modeling. By properly choosing the vibration parameters such as amplitude A and frequencyω, highly dense packing with the packing density of 0.728 can be realized, which is much denser than the maximum packing density of random close packing. Through the analyse on both the macro-property, e.g. packing density, and the micro-properties, e.g. Coordination Number (CN), Radial Distribution Function (RDF), Angle Distribution Function (ADF), Voronoi/Delaunay pore size distribution, and the variation of accompanied force field and velocity field, the following conclusions are obtained.
(1) Ordered structure beyond the maximum random packing density (p≈0.64) can be numerically achieved under special conditions, and the packing density can reach 0.728.
(2) The CN and Voronoi/Delaunay distributions show that the created structure of simulation is quite different from that of random packing, and through comparison it can be found that the macroscopic property is in good agreement with microscopic ones at both the dense and loose packings.
(3) The analyse on RDF and ADF further proves that ordered structure is formed, which can be identified by their correlation either at long distance or at specific angle, which is the characteristic of ordered structure. From their distributions some local disorder structures are also obtained, and they are the so called defects.
(4) The pore size distribution shows a high and narrow main peak, which indicates that the pore distribution is narrow and uniform. The Voronoi curve corresponds to a sub-peak structure, and this implies that there exist large pores in the packing structure.
(5) The properties of Voronoi polyhedron have been analyzed and compared with those of loose packing as well. These properties include:vertex distribution, perimeter distribution, surface area distribution, face number distribution, volume distribution of each Voronoi polyhedron; and edge distribution, perimeter distribution, face area distribution of each face on Voronoi polyhedron. They all showed large difference from those of loose structure. For highly dense packing, these distributions tend to be more uniform, which is a typical characteristic of ordered structure.
(6) The properties of Delaunay tessellation have been analyzed and compared with those of loose packing as well. These properties include:area, volume, diameter, and sphericity of each tetrahedron. They all showed large difference from those of loose structure. For highly dense packing, these distributions tend to be more uniform.
(7) Through the static and dynamic analysis, it is found that the obtained packing structure is FCC crystal, but with a small amount of defects. The crystallization mechanism can be ascribed to the formation of small ordered islands (core) and then their growth. During the growing, one grain is devoured by another to form a large one.
在三维间歇振动和批量加料条件下,使用DEM模型研究了2500个直径为1厘米玻璃球的堆积。通过适当选择振动参数如振动振幅A和振动频率ω,可以实现堆积密度0.728的高密度堆积,它是比最大随机紧密堆积更致密的堆积。通过宏观特征如堆积密度ρ、微观特征如配位数(CN, Coordination Number)、径向分布函数(RDF, Radial Distribution Function)、角分布函数(ADF, Angle Distribution Function)及Voronoi/Delaunay孔尺寸分布以及伴随力场和速度场的变化发现：
(1)可以数值实现超越最大无规密排(p≈0.64)的有序结构,堆积密度可达0.728。
(2)CN和Voronoi/Delaunay分布表明,模拟产生的结构与随机堆积结构有很大不同,通过对比发现不论是致密堆积还是松散堆积,其宏观特征与微观特征是相互对应的。
(3)RDF和ADF的进一步分析证明所获得的结构并不是纯随机的,RDF和ADF曲线分布证明了堆积结构无论是在较大距离上还是在特定角度上都存在相关性,这是有序结构所具有的特点。从它们的分布上还发现一些局部的无序结构,即所谓的缺陷。
(4)孔的尺寸分布呈现出高且窄的主峰,表明孔分布小且均匀。Voronoi曲线有一个子峰,表明堆积结构中有较大孔存在。
(5)对致密和松散堆积进行了Voronoi多面体的特征分析和对比。这些特征包括：每个Voronoi多面体顶点分布,周长分布,面积分布,面数分布,体积分布；每个Voronoi多面体面的边分布,周长分布,面积分布。它们与松散结构相比表现很大差距。对于高密度堆积,它们的分布更趋于均匀,这是有序结构的一个典型特点。
(6)对致密和松散堆积进行了Delaunay四面体的特征分析和对比。这些特征包括：每个Delaunay四面体的面积、体积、直径、球形度。它们与松散结构相比差别很大,堆积密度越高其分布越均匀。
(7)通过静态和动态分析发现获得的堆积结构是FCC晶体,但内有少量的缺陷。结晶机理可以归因于小岛(核)的形成及生长,在长大过程中,一个晶粒被另一个晶粒吞并形成一个大的晶粒。
In this paper, numerical simulation of monosize particle packing under three dimensional (3D) vibration was carried out by using Discrete Element Method (DEM). The aim is to realize high packing density, followed by the characterization of the formed dense structure, and finally the identification of the hard sphere crystallization mechanism.
Here mainly focus on the packing of 2500 monosize glass beads with 1 cm in diameter under 3D interval vibration and batch-wised feeding with the aid of DEM modeling. By properly choosing the vibration parameters such as amplitude A and frequencyω, highly dense packing with the packing density of 0.728 can be realized, which is much denser than the maximum packing density of random close packing. Through the analyse on both the macro-property, e.g. packing density, and the micro-properties, e.g. Coordination Number (CN), Radial Distribution Function (RDF), Angle Distribution Function (ADF), Voronoi/Delaunay pore size distribution, and the variation of accompanied force field and velocity field, the following conclusions are obtained.
(1) Ordered structure beyond the maximum random packing density (p≈0.64) can be numerically achieved under special conditions, and the packing density can reach 0.728.
(2) The CN and Voronoi/Delaunay distributions show that the created structure of simulation is quite different from that of random packing, and through comparison it can be found that the macroscopic property is in good agreement with microscopic ones at both the dense and loose packings.
(3) The analyse on RDF and ADF further proves that ordered structure is formed, which can be identified by their correlation either at long distance or at specific angle, which is the characteristic of ordered structure. From their distributions some local disorder structures are also obtained, and they are the so called defects.
(4) The pore size distribution shows a high and narrow main peak, which indicates that the pore distribution is narrow and uniform. The Voronoi curve corresponds to a sub-peak structure, and this implies that there exist large pores in the packing structure.
(5) The properties of Voronoi polyhedron have been analyzed and compared with those of loose packing as well. These properties include:vertex distribution, perimeter distribution, surface area distribution, face number distribution, volume distribution of each Voronoi polyhedron; and edge distribution, perimeter distribution, face area distribution of each face on Voronoi polyhedron. They all showed large difference from those of loose structure. For highly dense packing, these distributions tend to be more uniform, which is a typical characteristic of ordered structure.
(6) The properties of Delaunay tessellation have been analyzed and compared with those of loose packing as well. These properties include:area, volume, diameter, and sphericity of each tetrahedron. They all showed large difference from those of loose structure. For highly dense packing, these distributions tend to be more uniform.
(7) Through the static and dynamic analysis, it is found that the obtained packing structure is FCC crystal, but with a small amount of defects. The crystallization mechanism can be ascribed to the formation of small ordered islands (core) and then their growth. During the growing, one grain is devoured by another to form a large one.
引文
1. Yang R Y, Zou R P, Yu A B. Computer simulation of the packing of fine particles [J], Phys. Rev. E,2000,62(3):3900-3908.
2. Per Bak, Chao Tang, Kurt Wisesenfeld. Formation of saturated solutions in a nonlinear dispersive system with instability and dissipation [J], Phys. Rev. Lett,1983,51(5): 381-383.
3. 张林.二维颗粒系统中力分布的理论研究[D],南京师范大学,2005
4. 蒋红英.颗粒介质传力特性及其在岩土工程中的若干应用问题[D],兰州大学,2006
5. 陶珍东,郑少华.粉体工程与设备[M],北京：化学工业出版社,2003.
6. 李艳洁.堆积问题的离散元模拟-实验研究[D],中国农业大学,2005
7. Cundall P A. A computer model for simulating progressive large scale movements in blocky system [J], Proceedings of the Symposium of the International Society for Rock Mechanics. Rotterdam:BalaamA A,1971,1:8-12.
8. Cundall P A, Strack O D L. A discrete numerical model for granular assembles [J], Geotechnique,1979,29(1):47-65.
9. Walton O R. Particle dynamics modeling of geological materials. Lawrence Livermore National Lab. Report UCRL-52915,1980.
10. Campbell C S, Brennen C E. Computer simulation of granular shear flows [J], J Fluid Mech,1985,151:167-188.
11. Matuttis H G, Luding S, Hermann H J. Discrete element simulations of dense packing and heaps made of spherical and non-spherical particles [J], Powder Technol, 2000,109:278-292.
12. Robinson D A, Friedman S P. Observations of the effects of particle shape and particle size distribution on avalanching of granular media [J], Physica A,2002,311(1-2): 97-110.
13. Liffman K, Nguyen M, et al. Forces in piles of granular material:an analytic and 3D DEM study [J], Granular Matter,2001,3(11):165-176.
14. Thornton C. A direct approach to micromechanically based continuum models for granular material [M], 1n Satake M, eds. Mechanics of Granular Material, Japanese SMFE society,1989:145-150.
15. Owen D R J, Feng Y T. Parallelized finite-discrete element simulation of multi-fracture solids and discrete systems [J], Engineering Computation,2001, 18(3-4):557-576.
16. John R. Williams, Eric Perkins et al. A contact algorithm for partitioning N arbitrary sized objects [J], Engineering Computation,2004,21(2-4):235-248.
17. Oda M. Mechanics of granular materials, an introduction [J], Rotterdam:Balkema A A,1999,147-223.
18. Yang R Y, Zou R P, and Yu A B. Voronoi tessellation of the packing of fine uniform spheres [J], PHYSICAL REVIEW E,2002.
19.邢继波,王泳嘉.离散元法的改进及其在颗粒介质研究中的应用[J],岩土工程学报,1990,12(5)：51-57.
20.李世海,高波,燕琳.三峡永久船闸高边坡开挖三维离散元数值模拟[J],岩土力学,2002,23(3)：272-277.
21.王海兵,刘咏,黄伯云等.粉末颗粒线性堆积密度模型的改进[J],粉末冶金技术,2001,19(4)：208-211.
22.李伟,朱德惫,胡选利等.不连续散粒体的离散单元法[J],南京航空航天大学学报,1999,31(1)：85-91.
23.俞良群,邢纪波.筒仓装卸料时力场及流场的离散单元法模拟[J],农业工程学报,2000,16(4)：15-19.
24.徐泳,孙其诚等.颗粒离散元法研究进展[J],力学进展,2003,33(2)：251-260.
25.程远方,果世驹,赖和怡.球形颗粒随机排列过程的计算机模拟[J],北京科技大学学报,1999,21(4).
26. Scott G D. Packing of equal spheres [J], Nature,1960,188:908.
27. Yu A B, An X Z, Zou R P, et al. Self assembly of particles for densest packing by mechanical vibration [J], Phys. Rev. Lett,2006.
28. Hopkins M A. Constitutive relations for rapidly sheared granular flows:a Monte Carlo form based on the kinetic theory of dense gases, Ph. D. thesis, Clarkon University, Postdam,N.Y,1987.
29. Beddow J K, Nasta M D, Kostelnik M C. The use of a Monte Carlo technique to simulate the tap densification of spheres [J], Powder Technology,1974,9:221.
30. Alder B J, Wanwright T E. Phase transition for a hard sphere system [J], The Journal of Chemical Physics,1957,27:1208-1209.
31.周耐根.薄膜晶体缺陷形成与控制的分子动力学模拟研究[D],南昌大学,2005
32. Ghaboussi J, Barbosa R. Three-dimensional discrete element method for granular materials [J], Int J Numer Analytic Meth Geomech,1990,14:451-472.
33.范建华.面粉筒仓的侧压力测试和分析[J],郑州工程学院学报,1991,1：75-81.
34.肖国先.料仓内散体流动的数值模拟研究[D],南京工业大学,2004
35. Campbell C S. The stress tensor for simple shear flows of a granular material [J], J. Fluid Mech,1989,203:449-473.
36. Walton C S. Particle-dynamics calculation of shear flow, in Mechanics of Granular Materials:New Models and Constitutive Relations, ed. by Jenkins J T and Satake M, Amsterdam:Elsevier,1983,327-338.
37. Cundall P A, Strack O D L. Modeling of microscopic mechanisms in granular material, In Mechanics of Granular Materials:New Models and Constitutive Relations, ed. by Jenkins J T and Satake M, Amsterdam:Elsvier,1983,137-150.
38. Werner B T, Haff P K. A simulation study of low energy ejects resulting from single impacts in eolian saltation, In Advancements in Aerodynamics, Fluid Mechanics and Hydraulics, ed. Aradt R E A et al, New York, ASCE,.1986,337-345.
39. Zhou Y C, Wright B D, et al. Rolling friction in the dynamic simulation of sand pile formation [J], Physica A 1999,269(2-4):536-553.
40. Allen M P, Tildesley D J. Computer simulation of liquids [M], Oxford:Oxford University Press,1987.
41. Johnson K L. Contact Mechanics [M], Cambridge:Cambridge University Press,1985.
42. Schwager T, Poschel T. Coecient of normal restitution of viscous particles and cooling rate of granular gases [J], Phys. Rev. E,1998,57:650.
43. Langston P A, Tuzun U, Heyes D M. Discrete element simulation of granular flow in 2D and 3D hoppers:Dependence of discharge rate and wall stress on particle interactions [J], Chemical Engineering Science,1995,50(6):967.
44. Mehta A, Barker G C. Vibrated powders:A microscopic approach [J], Phys. Rev. Lett, 1991,67(3):394.
45. An X Z, Yang R Y, Zou R P, et al. Effect of vibration conditions and material friction on dense packing of vibrated equal spheres [J], Phys. Rev. E,2008(To be published).
46. Thiessen A H, Precipitation averages for large areas [J], Monthly Weather Review, 1911,39:1082-1084.
47. Delaunay B. Sur la sphere vide [J], Bulletin of the Academy of Sciences of the USSR, Classe des Sciences Mathematiques et Naturelles,1934,8:793-800.
48.周培德.计算几何—算法设计与分析[M],北京：清华大学出版社,2005,147-151.
49.崔凌国.约束Delaunay四面体剖分及其相关算法的研究[D],西安西北工业大学,2006
50. Finney J L. Random packing and the structure of simple liquids 1.The geometry of random close packing [J], Proc. Roy. Soc. Lond. A,1970,319:479-493.
51. Yang R Y, Zou R P, Yu A B. Effect of material properties on the packing of fine particles [J], J. Appl. Phys,2003,94(5):3025-3034.
52. Scott G D. Radial distribution of the random close packing of equal spheres [J], Nature,1962,194:956-957.
53. Bernal J D, The structure of liquids [J], Proc. Roy. Soc. A,1964,280,299-322.
2. Per Bak, Chao Tang, Kurt Wisesenfeld. Formation of saturated solutions in a nonlinear dispersive system with instability and dissipation [J], Phys. Rev. Lett,1983,51(5): 381-383.
3. 张林.二维颗粒系统中力分布的理论研究[D],南京师范大学,2005
4. 蒋红英.颗粒介质传力特性及其在岩土工程中的若干应用问题[D],兰州大学,2006
5. 陶珍东,郑少华.粉体工程与设备[M],北京：化学工业出版社,2003.
6. 李艳洁.堆积问题的离散元模拟-实验研究[D],中国农业大学,2005
7. Cundall P A. A computer model for simulating progressive large scale movements in blocky system [J], Proceedings of the Symposium of the International Society for Rock Mechanics. Rotterdam:BalaamA A,1971,1:8-12.
8. Cundall P A, Strack O D L. A discrete numerical model for granular assembles [J], Geotechnique,1979,29(1):47-65.
9. Walton O R. Particle dynamics modeling of geological materials. Lawrence Livermore National Lab. Report UCRL-52915,1980.
10. Campbell C S, Brennen C E. Computer simulation of granular shear flows [J], J Fluid Mech,1985,151:167-188.
11. Matuttis H G, Luding S, Hermann H J. Discrete element simulations of dense packing and heaps made of spherical and non-spherical particles [J], Powder Technol, 2000,109:278-292.
12. Robinson D A, Friedman S P. Observations of the effects of particle shape and particle size distribution on avalanching of granular media [J], Physica A,2002,311(1-2): 97-110.
13. Liffman K, Nguyen M, et al. Forces in piles of granular material:an analytic and 3D DEM study [J], Granular Matter,2001,3(11):165-176.
14. Thornton C. A direct approach to micromechanically based continuum models for granular material [M], 1n Satake M, eds. Mechanics of Granular Material, Japanese SMFE society,1989:145-150.
15. Owen D R J, Feng Y T. Parallelized finite-discrete element simulation of multi-fracture solids and discrete systems [J], Engineering Computation,2001, 18(3-4):557-576.
16. John R. Williams, Eric Perkins et al. A contact algorithm for partitioning N arbitrary sized objects [J], Engineering Computation,2004,21(2-4):235-248.
17. Oda M. Mechanics of granular materials, an introduction [J], Rotterdam:Balkema A A,1999,147-223.
18. Yang R Y, Zou R P, and Yu A B. Voronoi tessellation of the packing of fine uniform spheres [J], PHYSICAL REVIEW E,2002.
19.邢继波,王泳嘉.离散元法的改进及其在颗粒介质研究中的应用[J],岩土工程学报,1990,12(5)：51-57.
20.李世海,高波,燕琳.三峡永久船闸高边坡开挖三维离散元数值模拟[J],岩土力学,2002,23(3)：272-277.
21.王海兵,刘咏,黄伯云等.粉末颗粒线性堆积密度模型的改进[J],粉末冶金技术,2001,19(4)：208-211.
22.李伟,朱德惫,胡选利等.不连续散粒体的离散单元法[J],南京航空航天大学学报,1999,31(1)：85-91.
23.俞良群,邢纪波.筒仓装卸料时力场及流场的离散单元法模拟[J],农业工程学报,2000,16(4)：15-19.
24.徐泳,孙其诚等.颗粒离散元法研究进展[J],力学进展,2003,33(2)：251-260.
25.程远方,果世驹,赖和怡.球形颗粒随机排列过程的计算机模拟[J],北京科技大学学报,1999,21(4).
26. Scott G D. Packing of equal spheres [J], Nature,1960,188:908.
27. Yu A B, An X Z, Zou R P, et al. Self assembly of particles for densest packing by mechanical vibration [J], Phys. Rev. Lett,2006.
28. Hopkins M A. Constitutive relations for rapidly sheared granular flows:a Monte Carlo form based on the kinetic theory of dense gases, Ph. D. thesis, Clarkon University, Postdam,N.Y,1987.
29. Beddow J K, Nasta M D, Kostelnik M C. The use of a Monte Carlo technique to simulate the tap densification of spheres [J], Powder Technology,1974,9:221.
30. Alder B J, Wanwright T E. Phase transition for a hard sphere system [J], The Journal of Chemical Physics,1957,27:1208-1209.
31.周耐根.薄膜晶体缺陷形成与控制的分子动力学模拟研究[D],南昌大学,2005
32. Ghaboussi J, Barbosa R. Three-dimensional discrete element method for granular materials [J], Int J Numer Analytic Meth Geomech,1990,14:451-472.
33.范建华.面粉筒仓的侧压力测试和分析[J],郑州工程学院学报,1991,1：75-81.
34.肖国先.料仓内散体流动的数值模拟研究[D],南京工业大学,2004
35. Campbell C S. The stress tensor for simple shear flows of a granular material [J], J. Fluid Mech,1989,203:449-473.
36. Walton C S. Particle-dynamics calculation of shear flow, in Mechanics of Granular Materials:New Models and Constitutive Relations, ed. by Jenkins J T and Satake M, Amsterdam:Elsevier,1983,327-338.
37. Cundall P A, Strack O D L. Modeling of microscopic mechanisms in granular material, In Mechanics of Granular Materials:New Models and Constitutive Relations, ed. by Jenkins J T and Satake M, Amsterdam:Elsvier,1983,137-150.
38. Werner B T, Haff P K. A simulation study of low energy ejects resulting from single impacts in eolian saltation, In Advancements in Aerodynamics, Fluid Mechanics and Hydraulics, ed. Aradt R E A et al, New York, ASCE,.1986,337-345.
39. Zhou Y C, Wright B D, et al. Rolling friction in the dynamic simulation of sand pile formation [J], Physica A 1999,269(2-4):536-553.
40. Allen M P, Tildesley D J. Computer simulation of liquids [M], Oxford:Oxford University Press,1987.
41. Johnson K L. Contact Mechanics [M], Cambridge:Cambridge University Press,1985.
42. Schwager T, Poschel T. Coecient of normal restitution of viscous particles and cooling rate of granular gases [J], Phys. Rev. E,1998,57:650.
43. Langston P A, Tuzun U, Heyes D M. Discrete element simulation of granular flow in 2D and 3D hoppers:Dependence of discharge rate and wall stress on particle interactions [J], Chemical Engineering Science,1995,50(6):967.
44. Mehta A, Barker G C. Vibrated powders:A microscopic approach [J], Phys. Rev. Lett, 1991,67(3):394.
45. An X Z, Yang R Y, Zou R P, et al. Effect of vibration conditions and material friction on dense packing of vibrated equal spheres [J], Phys. Rev. E,2008(To be published).
46. Thiessen A H, Precipitation averages for large areas [J], Monthly Weather Review, 1911,39:1082-1084.
47. Delaunay B. Sur la sphere vide [J], Bulletin of the Academy of Sciences of the USSR, Classe des Sciences Mathematiques et Naturelles,1934,8:793-800.
48.周培德.计算几何—算法设计与分析[M],北京：清华大学出版社,2005,147-151.
49.崔凌国.约束Delaunay四面体剖分及其相关算法的研究[D],西安西北工业大学,2006
50. Finney J L. Random packing and the structure of simple liquids 1.The geometry of random close packing [J], Proc. Roy. Soc. Lond. A,1970,319:479-493.
51. Yang R Y, Zou R P, Yu A B. Effect of material properties on the packing of fine particles [J], J. Appl. Phys,2003,94(5):3025-3034.
52. Scott G D. Radial distribution of the random close packing of equal spheres [J], Nature,1962,194:956-957.
53. Bernal J D, The structure of liquids [J], Proc. Roy. Soc. A,1964,280,299-322.