非线性非保守系统弹性力学拟变分原理研究
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摘要
非线性力学是一门研究物体的几何非线性和物理非线性的科学，它广泛地存在于自然世界中。对非线性问题，尤其是大位移问题，体系常常是非保守的。非保守系统是指载荷在使物体发生位移和变形的过程中，其输入功与路径有关的系统。有一类典型的非保守系统，作用于系统的非保守力随物体的变形而变化，这种系统称为“伴生力”系统，本文研究的非保守系统特指“伴生力”系统。
本文从弹性力学的基本方程出发，将变积方法推广应用到物理非线性和几何非线性弹性力学问题中，研究了非线性非保守弹性力学系统的拟变分原理。
首先，研究了非线性非保守弹性静力学系统的拟变分原理。推导了非线性非保守弹性静力学系统的虚功原理、拟势能原理，余虚功原理、拟余能原理。论述了虚功原理和余虚功原理的宽广适用性，并给出拟势能原理和拟余能原理的其它表示形式。建立了第一类两类变量、第二类两类变量的广义拟势能原理和广义拟余能原理；一对有先决条件的两类变量广义拟势能原理和广义拟余能原理。建立了三类变量的广义拟势能原理和广义拟余能原理。提出了非线性弹性力学问题的拟驻值条件的概念，通过推导拟变分原理的拟驻值条件对拟变分原理进行了检验。应用非线性非保守弹性静力学的拟势能原理建立了非线性Leipholz杆的平衡方程，并研究了其屈曲特性；建立了承受伴生力作用的大挠度矩形薄板的第一类两类变量的广义拟势能原理，并推导了它的拟驻值条件。
其二，研究了非线性非保守弹性动力学系统时域边值问题的拟变分原理。建立了非线性非保守弹性动力学系统时域边值问题的拟Hamilton原理、拟余Hamilton原理，以及它们的其它表示形式。建立了第一类两类变量、第二类两类变量的广义拟Hamilton原理和广义拟余Hamilton原理。建立了三类变量的广义拟Hamilton原理和广义拟余Hamilton原理。应用非线性非保守系统弹性动力学的拟Hamilton原理推导了非线性Leipholz杆的运动方程，并研究了它的动力响应；建立了大挠度非保守矩形薄板的三类变量的广义拟Hamilton原理，并通过推导其拟驻值条件，得到该问题的全部基本方程。
其三，研究了非线性非保守弹性动力学系统时间初值问题的卷积型拟变分原理。借助于限制变分思想，并综合运用变积方法、加零变换法和卷积知识，建立了非线性非保守系统弹性动力学卷积型的拟势能原理和拟余能原理。建立了第一类两类变量、第二类两类变量的卷积型广义拟势能原理和广义拟余能原理。建立了三类变量的卷积型广义拟势能原理和广义拟余能原理。
其四，以非线性非保守弹性动力学系统时域边值问题为例，研究了应用Lagrange乘子不参加变分的程式推导经典拟变分原理的拟驻值条件的方法，以及应用Lagrange乘子参加变分的程式建立广义拟变分原理的方法。
其五，以弹性静力学为例，研究了建立适用于有限元计算的非线性非保守弹性力学系统拟变分原理、广义拟变分原理以及修正的拟变分原理、广义拟变分原理的方法。
此外，说明了本文所建立的非线性非保守系统弹性力学各级拟变分原理的含义非常广泛，可以将其退化到非线性保守弹性力学系统、线性非保守弹性力学系统、线性保守弹性力学系统中，从而得到相应的各级变分原理或拟变分原理。
Nonlinear mechanics is a subject to research geometric and physical nonlinearity, which exists in nature widely. The systems are usually non-conservative for nonlinear problems, especially for large displacement problems. Non-conservative system is that the work input is dependent on the loading path during the process of displacement and deformation-making when loads are imposed. There is a typical non-conservative system called accompanying force system, in which the force changes with the deformation of the object. In present paper non-conservative system is specified as accompanying force system. .
Started from the nonlinear basic equations, the variational integral method was generalized to nonlinear non-conservative elasticity, and the quasi-variational principles in nonlinear non-conservative elasticity were studied.
First of all, the quasi-variational principles in nonlinear non-conservative elastostatics were studied. The virtual work principle, quasi-potential energy principle, complementary virtual work principle and quasi-complementary energy principle were deduced. The broad applicability of virtual work and complementary virtual work principle was discussed, and the other forms of quasi-potential energy and quasi-complementary energy principle were introduced. The first and second types generalized quasi-potential energy principles and quasi-complementary energy principles with two kinds of variables were established, and so were a couple of quasi-variational principles with two kinds of variables which had precedent conditions. The generalized quasi-potential energy principles and quasi-complementary energy principles with three kinds of variables were established. The concept of quasi-stationary value condition in nonlinear non-conservative elasticity was proposed, and the quasi-variational principles were examined by deriving their quasi-stationary value conditions. The equilibrium equation of nonlinear Leipholz bar was gained by quasi-potential energy principle, and its buckling property was researched. The first type generalized quasi-potential energy principle with two kinds of variables for large deflection rectangular sheet which bore accompanying force was established, and its quasi-stationary value conditions were obtained.
Secondly, the quasi-variational principles of time boundary value problem in nonlinear non-conservative elastodynamics were studied. The quasi-Hamilton principle, quasi-complementary Hamilton principle and their other forms were gained. The first and second types generalized quasi-Hamilton principles and quasi-complementary Hamilton principles with two kinds of variables, and generalized quasi-Hamilton principle and quasi-complementary Hamilton principle with three kinds of variables were established. The motion equation of nonlinear Leipholz bar was gained by quasi-Hamilton principle, and its dynamical property was researched. The generalized quasi-Hamilton principle with three kinds of variables for large deflection rectangular sheet which bore accompanying force was established, and all of its basic equations were gained by deducing its quasi-stationary value conditions.
Thirdly, the quasi-variational principles of time initial value problem in nonlinear non-conservative elastodynamics were studied. Using restricted variation, variational integral method, transformation of naught addition and convolution knowledge, the convolutional quasi-potential energy principle, quasi-complementary energy principle, the first and second types generalized quasi-potential energy principles and quasi-complementary energy principles with two kinds of variables, and the generalized quasi-potential energy principles and quasi-complementary energy principles with three kinds of variables were established.
Fourthly, taking the time boundary value problem in nonlinear non-conservative elastodynamics for example, it was studied the method of deducing quasi-stationary value condition of classical quasi-variational principles by the operation of Lagrange Multiplier not joining variational, and establishing generalized quasi-variational principles by the operation of Multiplier joining variational.
Fifthly, taking nonlinear non-conservative elastostatics for example, it was worked out the quasi-variational principles and generalized quasi-variational principles which were applicable for the calculation of finite element method.
Furthermore, it was concluded that the meaning of the quasi-variational principles in nonlinear non-conservative elasticity established in the paper was very abundant. It could be degenerated to nonlinear conservative elasticity, linear non-conservative elasticity and linear conservative elasticity, so the corresponding variational principles or quasi-variational principles were derived.
本文从弹性力学的基本方程出发，将变积方法推广应用到物理非线性和几何非线性弹性力学问题中，研究了非线性非保守弹性力学系统的拟变分原理。
首先，研究了非线性非保守弹性静力学系统的拟变分原理。推导了非线性非保守弹性静力学系统的虚功原理、拟势能原理，余虚功原理、拟余能原理。论述了虚功原理和余虚功原理的宽广适用性，并给出拟势能原理和拟余能原理的其它表示形式。建立了第一类两类变量、第二类两类变量的广义拟势能原理和广义拟余能原理；一对有先决条件的两类变量广义拟势能原理和广义拟余能原理。建立了三类变量的广义拟势能原理和广义拟余能原理。提出了非线性弹性力学问题的拟驻值条件的概念，通过推导拟变分原理的拟驻值条件对拟变分原理进行了检验。应用非线性非保守弹性静力学的拟势能原理建立了非线性Leipholz杆的平衡方程，并研究了其屈曲特性；建立了承受伴生力作用的大挠度矩形薄板的第一类两类变量的广义拟势能原理，并推导了它的拟驻值条件。
其二，研究了非线性非保守弹性动力学系统时域边值问题的拟变分原理。建立了非线性非保守弹性动力学系统时域边值问题的拟Hamilton原理、拟余Hamilton原理，以及它们的其它表示形式。建立了第一类两类变量、第二类两类变量的广义拟Hamilton原理和广义拟余Hamilton原理。建立了三类变量的广义拟Hamilton原理和广义拟余Hamilton原理。应用非线性非保守系统弹性动力学的拟Hamilton原理推导了非线性Leipholz杆的运动方程，并研究了它的动力响应；建立了大挠度非保守矩形薄板的三类变量的广义拟Hamilton原理，并通过推导其拟驻值条件，得到该问题的全部基本方程。
其三，研究了非线性非保守弹性动力学系统时间初值问题的卷积型拟变分原理。借助于限制变分思想，并综合运用变积方法、加零变换法和卷积知识，建立了非线性非保守系统弹性动力学卷积型的拟势能原理和拟余能原理。建立了第一类两类变量、第二类两类变量的卷积型广义拟势能原理和广义拟余能原理。建立了三类变量的卷积型广义拟势能原理和广义拟余能原理。
其四，以非线性非保守弹性动力学系统时域边值问题为例，研究了应用Lagrange乘子不参加变分的程式推导经典拟变分原理的拟驻值条件的方法，以及应用Lagrange乘子参加变分的程式建立广义拟变分原理的方法。
其五，以弹性静力学为例，研究了建立适用于有限元计算的非线性非保守弹性力学系统拟变分原理、广义拟变分原理以及修正的拟变分原理、广义拟变分原理的方法。
此外，说明了本文所建立的非线性非保守系统弹性力学各级拟变分原理的含义非常广泛，可以将其退化到非线性保守弹性力学系统、线性非保守弹性力学系统、线性保守弹性力学系统中，从而得到相应的各级变分原理或拟变分原理。
Nonlinear mechanics is a subject to research geometric and physical nonlinearity, which exists in nature widely. The systems are usually non-conservative for nonlinear problems, especially for large displacement problems. Non-conservative system is that the work input is dependent on the loading path during the process of displacement and deformation-making when loads are imposed. There is a typical non-conservative system called accompanying force system, in which the force changes with the deformation of the object. In present paper non-conservative system is specified as accompanying force system. .
Started from the nonlinear basic equations, the variational integral method was generalized to nonlinear non-conservative elasticity, and the quasi-variational principles in nonlinear non-conservative elasticity were studied.
First of all, the quasi-variational principles in nonlinear non-conservative elastostatics were studied. The virtual work principle, quasi-potential energy principle, complementary virtual work principle and quasi-complementary energy principle were deduced. The broad applicability of virtual work and complementary virtual work principle was discussed, and the other forms of quasi-potential energy and quasi-complementary energy principle were introduced. The first and second types generalized quasi-potential energy principles and quasi-complementary energy principles with two kinds of variables were established, and so were a couple of quasi-variational principles with two kinds of variables which had precedent conditions. The generalized quasi-potential energy principles and quasi-complementary energy principles with three kinds of variables were established. The concept of quasi-stationary value condition in nonlinear non-conservative elasticity was proposed, and the quasi-variational principles were examined by deriving their quasi-stationary value conditions. The equilibrium equation of nonlinear Leipholz bar was gained by quasi-potential energy principle, and its buckling property was researched. The first type generalized quasi-potential energy principle with two kinds of variables for large deflection rectangular sheet which bore accompanying force was established, and its quasi-stationary value conditions were obtained.
Secondly, the quasi-variational principles of time boundary value problem in nonlinear non-conservative elastodynamics were studied. The quasi-Hamilton principle, quasi-complementary Hamilton principle and their other forms were gained. The first and second types generalized quasi-Hamilton principles and quasi-complementary Hamilton principles with two kinds of variables, and generalized quasi-Hamilton principle and quasi-complementary Hamilton principle with three kinds of variables were established. The motion equation of nonlinear Leipholz bar was gained by quasi-Hamilton principle, and its dynamical property was researched. The generalized quasi-Hamilton principle with three kinds of variables for large deflection rectangular sheet which bore accompanying force was established, and all of its basic equations were gained by deducing its quasi-stationary value conditions.
Thirdly, the quasi-variational principles of time initial value problem in nonlinear non-conservative elastodynamics were studied. Using restricted variation, variational integral method, transformation of naught addition and convolution knowledge, the convolutional quasi-potential energy principle, quasi-complementary energy principle, the first and second types generalized quasi-potential energy principles and quasi-complementary energy principles with two kinds of variables, and the generalized quasi-potential energy principles and quasi-complementary energy principles with three kinds of variables were established.
Fourthly, taking the time boundary value problem in nonlinear non-conservative elastodynamics for example, it was studied the method of deducing quasi-stationary value condition of classical quasi-variational principles by the operation of Lagrange Multiplier not joining variational, and establishing generalized quasi-variational principles by the operation of Multiplier joining variational.
Fifthly, taking nonlinear non-conservative elastostatics for example, it was worked out the quasi-variational principles and generalized quasi-variational principles which were applicable for the calculation of finite element method.
Furthermore, it was concluded that the meaning of the quasi-variational principles in nonlinear non-conservative elasticity established in the paper was very abundant. It could be degenerated to nonlinear conservative elasticity, linear non-conservative elasticity and linear conservative elasticity, so the corresponding variational principles or quasi-variational principles were derived.
引文
[1] Turner M J, Clough R J, Martin C H, Topp L J. Stiffness and deflection analysis of complete structures. J.A.S.23. 1956(9): 805—823P
[2] K.Washizu. Variational methods in elasticity and plasticity. 3rd ed. New York: Pergamon Press, 1982: 325—332P
[3] 钱伟长．应用数学与力学论文集．江苏科学技术出版社，1980：138—143页
[4] 钱伟长．变分法及有限元，北京：科学出版社，1980：220—254页
[5] 梁立孚，石志飞．粘性流体力学的变分原理及其广义变分原理．应用力学学报．1993，10(1)：119—123页
[6] 梁立孚，石志飞．关于变分学中逆问题的研究．应用数学和力学．1994，15(9)：775—788页
[7] 梁立孚．应用Lagrange乘子法推导一般力学中的广义变分原理．中国科学(A辑)．1999，29(12)：1102—1108页
[8] 梁立孚．胡海昌．一般力学中的三类变量的广义变分原理．中国科学(A辑)．2000，30(12)：1130—1135页
[9] 赵学仁，赵旭生，程兆雄．固体力学．北京：中国科学技术出版社，1993：144—147页
[10] 刘正兴，孙雁，王国庆．计算固体力学．上海交通大学出版社，2000：35—48页，230—236页
[11] 胡海昌．弹性力学的变分原理及其应用．北京：科学出版社，1981：382—388页，12—13页
[12] Reissner E. On a variational theorem in elasticity. Journal of Mathematics and Physics. 1950, 29(2): 90—98P
[13] 胡海昌．论弹性力学与受范体力学中的一般变分原理．物理学报．1954， 10(3)：359—363页
[14] K.Washizu. On the variational principles of elasticity. Massachusetts Institute of technology Technical Report 25—18, March 1955: 386—389P
[15] 钱令希．论固体力学中的极限分析并建立一个一般变分原理．力学学报．1963，6(4)：287—302页
[16] 张汝清．固体力学变分原理及其应用．重庆大学出版社，1991：12—28页
[17] 梁立孚著．变分原理及应用．哈尔滨工程大学等五联社，2005：262—272页，37页，80—85页，8—11页
[18] 梁立孚，刘殿魁，宋海燕．非保守系统的两类变量的广义拟变分原理．中国科学(G辑)，2005，35(2)：201—212页
[19] 陈惠发，A．F．萨里普著．余天庆，王勋文译．土木工程材料的本构方程(第一卷)．武汉：华中科技大学出版社，2001：133—148，94—99页
[20] 李卓球，董文堂．非线性弹性理论基础．北京：科学出版社，2004：73—75页
[21] 殷有泉著．固体力学非线性有限元引论．北京：北京大学出版社，清华大学出版社，1987：141—146页
[22] 何君毅，林祥都．工程结构非线性问题的数值解法．国防工业出版社，1994：102页
[23] Buffer H. Generalized variational principles with relaxed continuity requirements for certain nonlinear problems with an application to nonlinear elasticity. Computer Methods in Applied Mechanics and Engineering. 1979, 19(2): 235—255P
[24] Ogden R W. A note on variational theorems in nonlinear elastostatics. Math. Proc. Camb. Phil. Sot., 1975(77): 609-615P
[25] 钱伟长．大位移非线性弹性理论的变分原理和广义变分原理．应用数学和力学．1988，9(1)：1—11页
[26] 郭仲衡．非线性弹性理论变分原理的统一理论．应用数学和力学．1980， 1(1)：11—29页
[27] 陈至达．钱氏定理在有限变形极矩弹性力学广义变分原理的应用．应用数学和力学．1981，2(2)：191—196页
[28] 戴天民．论非线性弹性理论的各种变分原理．应用数学和力学．1982，3(5)：585—595页
[29] 梁立孚，章梓茂．推导弹性力学变分原理的一种凑合法(续)．哈尔滨船舶工程学院学报．1985，6(4)：1—12页
[30] 罗恩，潘小强，张贺忻．变形体动力学的变分原理及其应用．工程力学(增刊)．1988：48—54页
[31] 邢京堂．非线性弹性理论余能原理的注记．北京航空航天大学学报．1996，22(5)：555—562页
[32] 赵玉祥，顾祥珍，李欢秋．大变形对称弹性理论的广义变分原理．应用数学和力学．1994，15(9)：767—774页
[33] 沈孝明．大变形非线性弹性动力学的分区变分原理．南京理工大学学报．1994，73(1)：12—16页
[34] 沈孝明．非线性弹性体大变形问题的新广义变分原理．上海力学．1988，9(4)：66—79页
[35] 沈孝明．大变形非线性弹性理论中的新的能量原理．上海第二工业大学学报．1993，2：59—63页
[36] 赵玉祥，顾祥珍，宋熙太．用变分方法求解大变形对称弹性力学问题．应用数学和力学．1993，14(8)：679—685页
[37] 赵玉祥，顾祥珍．大变形对称弹性理论的广义变分原理．国际非线性力学会议论文集(1985)
[38] 薛大为．大变形非线性弹性力学的广义变分原理．应用数学和力学．1991，12(3)：209—216页
[39] 金伏生．非线性问题Gurtin型的变分原理和边界积分方程．自然科学进展—国家重点实验室通讯．1992，(6)：533—543页
[40] 张汝清．非线性弹性体的弹性动力学变分原理．应用数学和力学．1992，13(1)：1—9页
[41] 罗恩，罗志国．有限变形压电弹性动力学的非传统Hamilton型变分原理．技术科学思想与力学．北京：国防工业出版社，2001：337—342页
[42] 罗恩，邝君尚，黄伟江等．非线性耦合热弹性动力学的非传统Hamilton型变分原理．中国科学(A辑)．2002，32(4)：337—347页
[43] 牛庠均．蠕变理论的有限变形广义变分原理．北京工业大学学报．1992，18(1)：1—10页
[44] 宋彦琦，陈至达．有限变形非对称弹性理论变分原理．应用数学和力学．1999，20(11)：1115—1120页
[45] 黄泊．变分原理在有限变形压电热弹性体中的应用．北京理工大学学报．2004，24(3)：189—196页
[46] 高行山，张汝清．有限变形弹性理论随机变量变分原理及有限元法．应用数学和力学．1994，15(10)：855—862页
[47] Leipholz H H E. On conservative elastic system of the first and second kind. Ing. Arch. 1974, 43: 255—271P
[48] Leipholz H H E. On a generalization of the self-adjointness and of Rayleigh's quotient, Mech. Res. Commun 1974, 1: 67—72P
[49] Leipholz H H E. Stability of elastic rods via Liapunov's second method. Ing. Arch., 1975, 44:21—26P
[50] Leipholz H H E. On the buckling of thin plates subjected to nonconservative follower forces. Trans. Can. Soc. Mech. Eng., 1975, 3: 25—34P
[51] Leipholz H H E, Waddington D. On the mode of instability of a simply supported rectangular plate subjected to follwer forces. Mech. Res. Comm., 1981, 8: 223—229P
[52] Leipholz H H E. Direct variational methods and eigenvalue problems in engineering. Noordhoff Int. Publ., Leyden, 1977
[53] Leipholz H H E. On some developments in direct methods of the calculus of variations. Appl Mech Rev, 1987, 40(10): 1379—1392P
[54] Leipholz H H E, et al. On the solution of the stability problem of elastic rods subjected to uniformly distributed tangential forces. Ing. Arch., 1975: 347—357P
[55] Leipholz H H E. Variational principles for nonconservative problems-a foundation for a finite element approach. Comp. Maths. Appl. Mech. Eng.. 1979, 17/18, Part Ⅲ
[56] 刘殿魁，张其浩．弹性理论中非保守问题的一般拟变分原理．力学学报．1981，(6)：562—570页
[57] 郑泉水．非线性弹性理论的泛变分原理．应用数学和力学．1984，5(2)：205—216页
[58] 黄玉盈，王武久．弹性非保守系统的拟固有频率变分原理及其应用．固体力学学报．1987，(2)：127—136页
[59] Glabisz W. Stability of one-degree-of-freedom system under velocity and acceleration dependent nonconservative forces. Computers and Structures. 2001, 79: 757—768P
[60] Glabisz W. Stability of simple discrete systems under nongcongservative loading with dynamic follower parameter. Computer and Structures. 1996, 60(4): 653—663P
[61] Detinko F M. On the elastic stability of uniform beams and circular arches under nonconservative loading. Int. J. Solids and Structures. 2000, 37:550—551P
[62] Glabisz W. Vibration and stability of a beam with elastic supports and concentrated masses under conservative and nonconservative forces. Computers and Structures. 1999, 70:305—31P
[63] Si-Ung Ryu, Yoshiko Sugiyama. Computational dynamics approach to the effect of damping stability of a cantilevered column subjected to a follower force. Computers and Structures. 2003, 81: 265 —271P
[64] Renato V, Vitaliani et al. Finite element solution of the stability problem for nonlinear undamped and damped systems under nonconservative loading. Int. J. Solids Structures. 1997, 34(19): 2497-2516P
[65] Kounadis A N. et al. An improved energy criterion for dynamic buckling of imperfection sensitive nonconservative systems. Int. J. Solids Structures. 2001,38:7487-7500P
[66] Zuo Q H, Schreyer H.L. Flutter and divergence instability of nogconservative beams and plates. Int. J. Solids Structures. 1996, 33(9):1353 —1367P
[67] Adali S. Stability of a rectangular plate under nonconservative and conservative forces. Int. J. Solids Structures. 1982, 18(12): 1043 —1052P
[68] Reissner E, Wan F Y M. A follower load buckling problem for rectangular plates. J. Appl. Mech.. 1992, 59: 674-676P
[69] Kounadis A N. Some new instability aspects for nonconservative systems under follower loads. Int. J. Mech. Sci.. 1991, 33: 297-331P
[70] Anantha S et al. Stability analysis of stochastically parametered nonconservative columns. Int. J. Solids Structures. 1992,29: 2973—2988P
[71] Plaut R.H, Infante E F. The effect of external damping on the stability of Beck's column. Int. J. Solids Structures. 1970, 6: 491 —496P
[72] Inman D J, Olsen C L. Dynamics of symmetrizable nonconservative systems. J. Appl. Meth.. 1988, 55: 206—212P
[73] Kounadis A N. The existence of region of divergence instability for nonconservative systems under follower forces. Int. J. Solids Structures Sci.. 1983,19: 725—733P
[74] 张其洁，单文秀．非保守稳定向题的有限元方法．上海力学．1981，2(3)：40—46页
[75] 宋志远，万虹，梅占馨．非保守力作用下矩形薄板的稳定问题．固体力学学报．1991，12(4)：359—363页
[76] 王忠民，计伊周．矩形薄板在随从力作用下的动力稳定性分析．振动工程学报．1992，5(1)：78—83页
[77] 王忠民，赵凤群．弹性非保守简支矩形薄板的后屈曲性态．西安理工大学学报．1997，13(3)：238—242页
[78] 莫宵依，计伊周，王忠民．矩形薄板在非保守力作用下的动力稳定性．西安理工大学学报．2000，16(4)：370—375页
[79] 王忠民，高教伯，李会侠．非保守圆薄板的轴对称振动和稳定性．固体力学学报．2003，24(2)：155—162页
[80] 吴莹，李世荣，黄达文．非保守力作用下可伸长简支梁的过屈曲．甘肃工业大学学报．2002，28(2)：123—125页
[81] 揭敏．非保守力作用下杆的塑性动态稳定性．应用数学和力学．1997，18(4)：379—379页
[82] 钱伟长．广义变分原理．上海：知识出版社，1985
[83] 鹫津久一郎．弹性和塑性力学中变分法．北京：科学出版社，1984
[84] C L Dym, I H Shames著．袁祖贻，姚金山，应达之译．固体力学变分法．中国铁道出版社，1984：370—373页
[85] 金问鲁．变分原理和非线性力学．计算结构力学及其应用．1984，4(3)：21—28页
[86] 熊跃熙．非保守系统的有限变形弹性变分原理．力学学报．1983(1)：86—90页
[87] 付宝连．弹性力学中的能量原理及其应用．北京：科学出版社，2004：601—602页，409页
[88] 宋海燕，梁立孚，周轶雷．一般力学广义变分原理的对偶形式．哈尔滨 工程大学学报．2004，25(6)：740-743，760页
[89] 宋海燕．非保守系统的拟变分原理及其应用研究．哈尔滨工程大学博士学位论文．2005：39页
[90] 郭仲衡．非线性弹性理论．科学出版社，1980：8，18，218页
[91] 钱令希．余能原理．中国科学．1950，1(2-4)：449—456页
[92] 王新志，赵永刚，叶开沅，黄达文．正交各向异性板的非对称大变形问题．应用数学和力学．2002，23(9)：881—888页
[93] 王林祥，王新志，邱平．圆薄板非对称大变形弯曲问题．应用数学和力学．1992，13(12)：1103—1116页
[94] 钟万勰．弹性力学求解新体系．大连理工大学出版社，1995
[95] 钱伟长，卢文达，王蜀．动力学分区变分原理及其广义变分原理．力学学报．1989，21(3)：300—305页
[96] 罗恩．几何非线性弹性动力学中广义Hamilton型拟变分原理．中山大学学报(自然科学版)．1990，29(2)：15—19页．
[97] 罗恩，潘小强，张贺忻等．一种基于Hamilton型拟变分原理的时间子域法．工程力学．2000，17(4)：13—20页
[98] 邢京堂．包含终值条件为驻值条件的哈氏型弹性动力学变分原理．上海力学．1990，11(3)：24—34页
[99] 邢京堂；郑兆昌．非线性弹性动力学两时端边值问题的变分原理．上海力学．1992，13(3)：45—51页
[100] 邢京堂，郑兆昌．线弹性动力学的某些一般定理及广义与广义分区变分原理．应用数学和力学．1992，13(9)：795—810页
[101] Gurtin M E. Variationnal principles for elastodynamics. Arch. Rat. Mech.. 1964, 16L: 34—50P
[102] Tonti E. On the variational formulation of linear initial value problems. Anal Mat Pura Appl. 1973.95: 18—28P
[103] 罗恩．关于线性动力学中各种Gurtin型变分原理．中国科学(A辑)．1987， (9)：936—948页
[104] 罗恩，邝尚君．压电热弹性动力学的一些基本原理．中国科学(A辑)．1999，29(9)：851—860页
[105] 罗恩，黄伟江，张贺沂．相空间非传统Hamilton型变分原理与辛算法．中国科学(A辑)．2002，32(12)：1120—1126页
[106] 罗恩，邝尚君．微孔压电热弹性动力学的能量原理．力学学报．2001，33(2)：195—204页
[107] Oden J T, Reddy J N. Variational method in theoretical mechanics. New York: Springer-Verlag, 1983: 137—177P
[108] 罗恩，朱慧坚．有限变形弹性动力学的Gurtin型变分原理．固体力学学报．2003，24(1)：1—7页
[109] 罗恩，梁立孚，李纬华．分析力学的非传统Hamilton型变分原理．中国科学(G辑)．2006，36(6)：633-643页
[110] Finlayson B A, Scriven L. On the search for variational principles. Int J HeatMassTransfer, 1967, 10: 799P
[111] 梁立孚，罗恩，冯晓九．分析力学初值问题的一种变分原理形式．力学学报．2007，39(1)：106—111页
[112] Levinson M. The complementary energy theorem in finite elasticity. J Appl Mech. 1965, 32: 826—828P
[113] Gao D Y, Strang G. Geometric nonlinearity: potential energy, complementary energy and the gap function. Quartl Appl Math, 1989, 47(3): 487-504P
[114] Gao D Y. Pure complementary energy principle and triality theory in finite elasticity. Mechanics Research Communications. 1999, 26(1): 31—37P
[115] 钱伟长．弹性理论中广义变分原理的研究及其在有限元计算中的应用．力学与实践．1979，1(1)：16—24页
[116] 钱伟长．弹性理论中广义变分原理的研究及其在有限元计算中的应用 (续)．力学与实践．1979，1(2)：18—26页
[117] 钱伟长．高阶拉氏乘子法和弹性理论中更一般的广义变分原理．应用数学和力学．1983，4(2)：137—149页
[118] 梁立孚，章梓茂．应用线性Lagrange乘子法推导广义变分原理的泛函．哈尔滨船舶工程学院学报．1985(3)：1—8页
[119] 钱伟长．论拉氏乘子法及其唯一性问题．力学学报．1988，20(4)：313—324页
[2] K.Washizu. Variational methods in elasticity and plasticity. 3rd ed. New York: Pergamon Press, 1982: 325—332P
[3] 钱伟长．应用数学与力学论文集．江苏科学技术出版社，1980：138—143页
[4] 钱伟长．变分法及有限元，北京：科学出版社，1980：220—254页
[5] 梁立孚，石志飞．粘性流体力学的变分原理及其广义变分原理．应用力学学报．1993，10(1)：119—123页
[6] 梁立孚，石志飞．关于变分学中逆问题的研究．应用数学和力学．1994，15(9)：775—788页
[7] 梁立孚．应用Lagrange乘子法推导一般力学中的广义变分原理．中国科学(A辑)．1999，29(12)：1102—1108页
[8] 梁立孚．胡海昌．一般力学中的三类变量的广义变分原理．中国科学(A辑)．2000，30(12)：1130—1135页
[9] 赵学仁，赵旭生，程兆雄．固体力学．北京：中国科学技术出版社，1993：144—147页
[10] 刘正兴，孙雁，王国庆．计算固体力学．上海交通大学出版社，2000：35—48页，230—236页
[11] 胡海昌．弹性力学的变分原理及其应用．北京：科学出版社，1981：382—388页，12—13页
[12] Reissner E. On a variational theorem in elasticity. Journal of Mathematics and Physics. 1950, 29(2): 90—98P
[13] 胡海昌．论弹性力学与受范体力学中的一般变分原理．物理学报．1954， 10(3)：359—363页
[14] K.Washizu. On the variational principles of elasticity. Massachusetts Institute of technology Technical Report 25—18, March 1955: 386—389P
[15] 钱令希．论固体力学中的极限分析并建立一个一般变分原理．力学学报．1963，6(4)：287—302页
[16] 张汝清．固体力学变分原理及其应用．重庆大学出版社，1991：12—28页
[17] 梁立孚著．变分原理及应用．哈尔滨工程大学等五联社，2005：262—272页，37页，80—85页，8—11页
[18] 梁立孚，刘殿魁，宋海燕．非保守系统的两类变量的广义拟变分原理．中国科学(G辑)，2005，35(2)：201—212页
[19] 陈惠发，A．F．萨里普著．余天庆，王勋文译．土木工程材料的本构方程(第一卷)．武汉：华中科技大学出版社，2001：133—148，94—99页
[20] 李卓球，董文堂．非线性弹性理论基础．北京：科学出版社，2004：73—75页
[21] 殷有泉著．固体力学非线性有限元引论．北京：北京大学出版社，清华大学出版社，1987：141—146页
[22] 何君毅，林祥都．工程结构非线性问题的数值解法．国防工业出版社，1994：102页
[23] Buffer H. Generalized variational principles with relaxed continuity requirements for certain nonlinear problems with an application to nonlinear elasticity. Computer Methods in Applied Mechanics and Engineering. 1979, 19(2): 235—255P
[24] Ogden R W. A note on variational theorems in nonlinear elastostatics. Math. Proc. Camb. Phil. Sot., 1975(77): 609-615P
[25] 钱伟长．大位移非线性弹性理论的变分原理和广义变分原理．应用数学和力学．1988，9(1)：1—11页
[26] 郭仲衡．非线性弹性理论变分原理的统一理论．应用数学和力学．1980， 1(1)：11—29页
[27] 陈至达．钱氏定理在有限变形极矩弹性力学广义变分原理的应用．应用数学和力学．1981，2(2)：191—196页
[28] 戴天民．论非线性弹性理论的各种变分原理．应用数学和力学．1982，3(5)：585—595页
[29] 梁立孚，章梓茂．推导弹性力学变分原理的一种凑合法(续)．哈尔滨船舶工程学院学报．1985，6(4)：1—12页
[30] 罗恩，潘小强，张贺忻．变形体动力学的变分原理及其应用．工程力学(增刊)．1988：48—54页
[31] 邢京堂．非线性弹性理论余能原理的注记．北京航空航天大学学报．1996，22(5)：555—562页
[32] 赵玉祥，顾祥珍，李欢秋．大变形对称弹性理论的广义变分原理．应用数学和力学．1994，15(9)：767—774页
[33] 沈孝明．大变形非线性弹性动力学的分区变分原理．南京理工大学学报．1994，73(1)：12—16页
[34] 沈孝明．非线性弹性体大变形问题的新广义变分原理．上海力学．1988，9(4)：66—79页
[35] 沈孝明．大变形非线性弹性理论中的新的能量原理．上海第二工业大学学报．1993，2：59—63页
[36] 赵玉祥，顾祥珍，宋熙太．用变分方法求解大变形对称弹性力学问题．应用数学和力学．1993，14(8)：679—685页
[37] 赵玉祥，顾祥珍．大变形对称弹性理论的广义变分原理．国际非线性力学会议论文集(1985)
[38] 薛大为．大变形非线性弹性力学的广义变分原理．应用数学和力学．1991，12(3)：209—216页
[39] 金伏生．非线性问题Gurtin型的变分原理和边界积分方程．自然科学进展—国家重点实验室通讯．1992，(6)：533—543页
[40] 张汝清．非线性弹性体的弹性动力学变分原理．应用数学和力学．1992，13(1)：1—9页
[41] 罗恩，罗志国．有限变形压电弹性动力学的非传统Hamilton型变分原理．技术科学思想与力学．北京：国防工业出版社，2001：337—342页
[42] 罗恩，邝君尚，黄伟江等．非线性耦合热弹性动力学的非传统Hamilton型变分原理．中国科学(A辑)．2002，32(4)：337—347页
[43] 牛庠均．蠕变理论的有限变形广义变分原理．北京工业大学学报．1992，18(1)：1—10页
[44] 宋彦琦，陈至达．有限变形非对称弹性理论变分原理．应用数学和力学．1999，20(11)：1115—1120页
[45] 黄泊．变分原理在有限变形压电热弹性体中的应用．北京理工大学学报．2004，24(3)：189—196页
[46] 高行山，张汝清．有限变形弹性理论随机变量变分原理及有限元法．应用数学和力学．1994，15(10)：855—862页
[47] Leipholz H H E. On conservative elastic system of the first and second kind. Ing. Arch. 1974, 43: 255—271P
[48] Leipholz H H E. On a generalization of the self-adjointness and of Rayleigh's quotient, Mech. Res. Commun 1974, 1: 67—72P
[49] Leipholz H H E. Stability of elastic rods via Liapunov's second method. Ing. Arch., 1975, 44:21—26P
[50] Leipholz H H E. On the buckling of thin plates subjected to nonconservative follower forces. Trans. Can. Soc. Mech. Eng., 1975, 3: 25—34P
[51] Leipholz H H E, Waddington D. On the mode of instability of a simply supported rectangular plate subjected to follwer forces. Mech. Res. Comm., 1981, 8: 223—229P
[52] Leipholz H H E. Direct variational methods and eigenvalue problems in engineering. Noordhoff Int. Publ., Leyden, 1977
[53] Leipholz H H E. On some developments in direct methods of the calculus of variations. Appl Mech Rev, 1987, 40(10): 1379—1392P
[54] Leipholz H H E, et al. On the solution of the stability problem of elastic rods subjected to uniformly distributed tangential forces. Ing. Arch., 1975: 347—357P
[55] Leipholz H H E. Variational principles for nonconservative problems-a foundation for a finite element approach. Comp. Maths. Appl. Mech. Eng.. 1979, 17/18, Part Ⅲ
[56] 刘殿魁，张其浩．弹性理论中非保守问题的一般拟变分原理．力学学报．1981，(6)：562—570页
[57] 郑泉水．非线性弹性理论的泛变分原理．应用数学和力学．1984，5(2)：205—216页
[58] 黄玉盈，王武久．弹性非保守系统的拟固有频率变分原理及其应用．固体力学学报．1987，(2)：127—136页
[59] Glabisz W. Stability of one-degree-of-freedom system under velocity and acceleration dependent nonconservative forces. Computers and Structures. 2001, 79: 757—768P
[60] Glabisz W. Stability of simple discrete systems under nongcongservative loading with dynamic follower parameter. Computer and Structures. 1996, 60(4): 653—663P
[61] Detinko F M. On the elastic stability of uniform beams and circular arches under nonconservative loading. Int. J. Solids and Structures. 2000, 37:550—551P
[62] Glabisz W. Vibration and stability of a beam with elastic supports and concentrated masses under conservative and nonconservative forces. Computers and Structures. 1999, 70:305—31P
[63] Si-Ung Ryu, Yoshiko Sugiyama. Computational dynamics approach to the effect of damping stability of a cantilevered column subjected to a follower force. Computers and Structures. 2003, 81: 265 —271P
[64] Renato V, Vitaliani et al. Finite element solution of the stability problem for nonlinear undamped and damped systems under nonconservative loading. Int. J. Solids Structures. 1997, 34(19): 2497-2516P
[65] Kounadis A N. et al. An improved energy criterion for dynamic buckling of imperfection sensitive nonconservative systems. Int. J. Solids Structures. 2001,38:7487-7500P
[66] Zuo Q H, Schreyer H.L. Flutter and divergence instability of nogconservative beams and plates. Int. J. Solids Structures. 1996, 33(9):1353 —1367P
[67] Adali S. Stability of a rectangular plate under nonconservative and conservative forces. Int. J. Solids Structures. 1982, 18(12): 1043 —1052P
[68] Reissner E, Wan F Y M. A follower load buckling problem for rectangular plates. J. Appl. Mech.. 1992, 59: 674-676P
[69] Kounadis A N. Some new instability aspects for nonconservative systems under follower loads. Int. J. Mech. Sci.. 1991, 33: 297-331P
[70] Anantha S et al. Stability analysis of stochastically parametered nonconservative columns. Int. J. Solids Structures. 1992,29: 2973—2988P
[71] Plaut R.H, Infante E F. The effect of external damping on the stability of Beck's column. Int. J. Solids Structures. 1970, 6: 491 —496P
[72] Inman D J, Olsen C L. Dynamics of symmetrizable nonconservative systems. J. Appl. Meth.. 1988, 55: 206—212P
[73] Kounadis A N. The existence of region of divergence instability for nonconservative systems under follower forces. Int. J. Solids Structures Sci.. 1983,19: 725—733P
[74] 张其洁，单文秀．非保守稳定向题的有限元方法．上海力学．1981，2(3)：40—46页
[75] 宋志远，万虹，梅占馨．非保守力作用下矩形薄板的稳定问题．固体力学学报．1991，12(4)：359—363页
[76] 王忠民，计伊周．矩形薄板在随从力作用下的动力稳定性分析．振动工程学报．1992，5(1)：78—83页
[77] 王忠民，赵凤群．弹性非保守简支矩形薄板的后屈曲性态．西安理工大学学报．1997，13(3)：238—242页
[78] 莫宵依，计伊周，王忠民．矩形薄板在非保守力作用下的动力稳定性．西安理工大学学报．2000，16(4)：370—375页
[79] 王忠民，高教伯，李会侠．非保守圆薄板的轴对称振动和稳定性．固体力学学报．2003，24(2)：155—162页
[80] 吴莹，李世荣，黄达文．非保守力作用下可伸长简支梁的过屈曲．甘肃工业大学学报．2002，28(2)：123—125页
[81] 揭敏．非保守力作用下杆的塑性动态稳定性．应用数学和力学．1997，18(4)：379—379页
[82] 钱伟长．广义变分原理．上海：知识出版社，1985
[83] 鹫津久一郎．弹性和塑性力学中变分法．北京：科学出版社，1984
[84] C L Dym, I H Shames著．袁祖贻，姚金山，应达之译．固体力学变分法．中国铁道出版社，1984：370—373页
[85] 金问鲁．变分原理和非线性力学．计算结构力学及其应用．1984，4(3)：21—28页
[86] 熊跃熙．非保守系统的有限变形弹性变分原理．力学学报．1983(1)：86—90页
[87] 付宝连．弹性力学中的能量原理及其应用．北京：科学出版社，2004：601—602页，409页
[88] 宋海燕，梁立孚，周轶雷．一般力学广义变分原理的对偶形式．哈尔滨 工程大学学报．2004，25(6)：740-743，760页
[89] 宋海燕．非保守系统的拟变分原理及其应用研究．哈尔滨工程大学博士学位论文．2005：39页
[90] 郭仲衡．非线性弹性理论．科学出版社，1980：8，18，218页
[91] 钱令希．余能原理．中国科学．1950，1(2-4)：449—456页
[92] 王新志，赵永刚，叶开沅，黄达文．正交各向异性板的非对称大变形问题．应用数学和力学．2002，23(9)：881—888页
[93] 王林祥，王新志，邱平．圆薄板非对称大变形弯曲问题．应用数学和力学．1992，13(12)：1103—1116页
[94] 钟万勰．弹性力学求解新体系．大连理工大学出版社，1995
[95] 钱伟长，卢文达，王蜀．动力学分区变分原理及其广义变分原理．力学学报．1989，21(3)：300—305页
[96] 罗恩．几何非线性弹性动力学中广义Hamilton型拟变分原理．中山大学学报(自然科学版)．1990，29(2)：15—19页．
[97] 罗恩，潘小强，张贺忻等．一种基于Hamilton型拟变分原理的时间子域法．工程力学．2000，17(4)：13—20页
[98] 邢京堂．包含终值条件为驻值条件的哈氏型弹性动力学变分原理．上海力学．1990，11(3)：24—34页
[99] 邢京堂；郑兆昌．非线性弹性动力学两时端边值问题的变分原理．上海力学．1992，13(3)：45—51页
[100] 邢京堂，郑兆昌．线弹性动力学的某些一般定理及广义与广义分区变分原理．应用数学和力学．1992，13(9)：795—810页
[101] Gurtin M E. Variationnal principles for elastodynamics. Arch. Rat. Mech.. 1964, 16L: 34—50P
[102] Tonti E. On the variational formulation of linear initial value problems. Anal Mat Pura Appl. 1973.95: 18—28P
[103] 罗恩．关于线性动力学中各种Gurtin型变分原理．中国科学(A辑)．1987， (9)：936—948页
[104] 罗恩，邝尚君．压电热弹性动力学的一些基本原理．中国科学(A辑)．1999，29(9)：851—860页
[105] 罗恩，黄伟江，张贺沂．相空间非传统Hamilton型变分原理与辛算法．中国科学(A辑)．2002，32(12)：1120—1126页
[106] 罗恩，邝尚君．微孔压电热弹性动力学的能量原理．力学学报．2001，33(2)：195—204页
[107] Oden J T, Reddy J N. Variational method in theoretical mechanics. New York: Springer-Verlag, 1983: 137—177P
[108] 罗恩，朱慧坚．有限变形弹性动力学的Gurtin型变分原理．固体力学学报．2003，24(1)：1—7页
[109] 罗恩，梁立孚，李纬华．分析力学的非传统Hamilton型变分原理．中国科学(G辑)．2006，36(6)：633-643页
[110] Finlayson B A, Scriven L. On the search for variational principles. Int J HeatMassTransfer, 1967, 10: 799P
[111] 梁立孚，罗恩，冯晓九．分析力学初值问题的一种变分原理形式．力学学报．2007，39(1)：106—111页
[112] Levinson M. The complementary energy theorem in finite elasticity. J Appl Mech. 1965, 32: 826—828P
[113] Gao D Y, Strang G. Geometric nonlinearity: potential energy, complementary energy and the gap function. Quartl Appl Math, 1989, 47(3): 487-504P
[114] Gao D Y. Pure complementary energy principle and triality theory in finite elasticity. Mechanics Research Communications. 1999, 26(1): 31—37P
[115] 钱伟长．弹性理论中广义变分原理的研究及其在有限元计算中的应用．力学与实践．1979，1(1)：16—24页
[116] 钱伟长．弹性理论中广义变分原理的研究及其在有限元计算中的应用 (续)．力学与实践．1979，1(2)：18—26页
[117] 钱伟长．高阶拉氏乘子法和弹性理论中更一般的广义变分原理．应用数学和力学．1983，4(2)：137—149页
[118] 梁立孚，章梓茂．应用线性Lagrange乘子法推导广义变分原理的泛函．哈尔滨船舶工程学院学报．1985(3)：1—8页
[119] 钱伟长．论拉氏乘子法及其唯一性问题．力学学报．1988，20(4)：313—324页