离心场中广义弹性体的动力学建模与数值分析
|
摘要
在旋转机械、航空航天、车辆工程、柔性机械臂以及现代微机电系统(MEMS)等领域中含有各种结构形式的变形体,由这些构件组成的单体或多体系统在历经大范围刚体运动的同时发生自身的变形,运动与变形耦合产生的附加的惯性力将对结构动力学特性和响应产生重要影响,为此大范围运动下变形体刚柔耦合动力学研究对促进这些领域结构动力分析、系统设计与控制技术的发展等都至关重要。运动与变形耦合理论涉及到刚体动力学与变形体力学的统一,要求所建立的耦合动力学模型既能反映刚柔耦合效应,又能在无刚体运动的时候退化为变形体力学,而在不变形的时候退化为刚体动力学。目前对刚柔耦合动力学的研究集中在力学建模、数值求解、多体系统下的接触与碰撞以及多物理场下的刚柔耦合效应等领域,然而由于刚柔耦合动力学涉及刚体运动与弹性变形的统一描述,且具有运动非线性和几何非线性效应,这些问题的解决至今尚不圆满,且随着广义弹性理论的发展,在基于广义弹性理论的基础上研究变形体的刚体运动与变形耦合问题尚较少涉及。
单体运动与变形耦合动力学研究是柔性多体动力学研究的基础,在分析国内外对柔性多体动力学以及广义弹性力学研究现状的基础上,本文集中关注离心环境下结构的运动与变形耦合理论建模和数值算法开发,引入了含有三个材料参数的广义弹性体模型,采用广义弹性体模型分析了不同尺度下结构的静力和动力特性,建立了离心环境下广义弹性体的运动与变形耦合动力学模型并开发了相应的数值算法,考察了离心环境下典型悬臂梁结构的动力特性。论文的主要工作和结论如下:
①考虑含旋转变形的广义弹性体,在偶应力弹性理论和微极弹性理论的研究基础上,确定动量和动量矩守恒方程形式和广义弹性体的运动学表达。应用虚功原理和各向同性张量函数定理,改进偶应力和曲率张量的本构关系。建立广义弹性体的动力学方程和含力偶的初边值条件。
②采用约束变分原理,考虑位移和转角为独立变量,采用罚方法引入约束条件,建立广义弹性体的有限元控制方程。构造了4结点12自由度的平面等参元和8结点48自由度的六面体等参元。对平面单剪问题和悬臂梁静力和动力分析表明广义弹性体模型能够把经典弹性理论的应用范围扩展到微观领域,且在力学分析时给出了更加丰富的信息,有利于结构分析,能够计及尺寸效应的影响。
③建立了简化的双自由度弹簧质量系统的耦合动力学分析模型。利用简化的双自由度弹簧质量系统,引入描述质点刚体旋转运动和变形运动的惯性坐标系和浮动坐标系,应用正交张量和欧拉罗德里格斯旋转公式,给出质点在惯性坐标系和浮动坐标系的速度和加速度表达,建立定轴旋转的离心环境下的动力学模型,对典型算列的计算结果表明离心力效应、科式力效应和偏心效应对转动系统的动力特性有重要影响。
④建立了定轴变速离心环境下的广义弹性体的运动与变形耦合动力学方程。解析浮动标架下的离心力、科氏力、切向惯性力,基于连续介质力学分析方法,在浮动坐标系下建立广义弹性体动量和动量矩守恒方程。应用广义弹性体关于应力和偶应力的本构关系,给出浮动坐标系下含力偶的初边值条件,建立离心场中广义弹性体运动与变形耦合动力学方程,并采用初应力法解析了柔性结构的动力刚化效应。
⑤开发离心环境下广义弹性体运动与变形耦合动力学问题的数值算法。利用虚功原理建立广义弹性体运动与变形耦合的有限元方程,采用8结点48自由度的六面体等参元对广义弹性体进行离散。解析柔性结构由于几何非线性带来的动力刚化效应,考察广义惯性力形式,建立广义弹性体运动与变形耦合动力分析的有限元控制方程。
⑥将梁视为广义弹性体,针对尺寸和几何特征不同的梁型结构,应用所建立的广义弹性体有限元控制方程,数值求解梁型结构在恒速和变速的离心环境下的动频及动力响应特性,结合挠度和强度两方面考察了离心场下旋转梁的临界转速。研究结果表明科式力和离心力效应取决与转速、结构形态以及旋转姿态,旋转姿态影响动频与动力响应、高转速下柔性结构动力刚化效应明显,微观结构尺度下的尺寸效应提高刚度并严重影响动力特性。
Flexible bodies with different structure form are widely used in many industrial and technological systems, such as rotary machine, aerospace, vehicle engineering, mechanical arm and micro electro mechanical systems(MEMS). Construction members in these systems undergo large scale rigidity motion as well as elastic deformation. Inertial forces caused by the coupling between the rigid motion and elastic deformation play a key role to the dynamics characteristics for structures. Since that researches to the dynamics on the rigidity-flexible coupling are very important to dynamic analysis, system design and control of flexible multibody system(FMS). Theories on the motion and deformation coupling is a unification between the rigid-body dynamics and the deformable body mechanics. The dynamic formulations developed in this field must take account of the coupling effects between the rigid displacement and the deformation in deformable body, moreover, when ignoring the elastic deformation, the formulation can degenerate to the rigid-body dynamics, and when ignoring the rigid motion, the formulation can degenerate to the deformable body mechanics. There are several research topics which are of current interest, among these topics are the construction for dynamic formulation, computational strategies, impact and contact problem for the FMS and the fluid-structure interaction. Many problems are still not solved completely as a result of the nonlinearity in geometry, motion, Physics and material. Otherwise, the generalized elasticity has been developed deeply in recent years, few research has been given to the generalized elastic body undergoing large scale displacement.
The dynamic problem on a single flexible body should firstly be solved, when to formulate the kinetic equation for the FMS. On the base of the research at home and abroad on both the flexible multibody system dynamics(FMD) and generalized elasticity, In this dissertation, we focus on the dynamic formulation for the generalized elastic body undergoing rigid rotating and its computational strategies. The generalized elasticity contains three material parameter is put forward, and the structure with different size scale were analyzed. The dynamic models are developed for the generalized elastic body in the centrifugal field with fixed rotational axis and various rotational speed, also the finite element formulation was derived. The classic rotating cantilever beam is simulated in centrifugal field with both the constant rotational speed and various rotational speed. The main works and conclusions in this dissertation are as follows:
①The equations of linear and angular momentum conservation are derived for the generalized elasticity. basing on the couple stress elasticity theory and the micropolar theory. Combining the principle of virtual work with the theorem for the isotropic tensor, the modified relationship was put forward for the couple stress and the curvature tensor, moreover the kinetic equation and the boundary conditions were given for the generalized elastic body.
②The finite element equation was formulated for the generalized elastic body, on the base of the method of weighted residuals and the constrained variational principle,with both the displacement and rotational angle considered as independent variables. The 4-nodes and 12-Dofs plane isoparametric element was constructed as well as the 8-nodes and 48-Dofs hexahedron isoparametric element for generalized elastic body. The analysis results to the simple shear problem and the cantilever with different size scale conclude that the generalized elasticity expands the classic elasticity to the micro size scale, and provides much more information than the classic elasticity and the beam theory in the structure analysis.
③The dynamic model was developed for the double freedom vibration system, so as to investigate the effects of the centrifugal force, coriolis force and force caused by the eccentricity. For a two-Dofs springmass model, both the float frame reference and the fixed frame reference were introduced to described the motion of the mass point, applying the orthogonal tensor and the Rodriguez rotating formula, the velocity and the acceleration are given, and the dynamics formulation are derived for the fixed axis rotational springmass. The simulation results discovered that the centrifugal force and Coriolis force are play a key role to the dynamic characteristic as well as the eccentricity effect.
④The dynamic formulation was constructed for a generalized elastic body rotating along a fixed axis, for the various speed conditions. In the float frame reference, the centrifugal force, Coriolis force and tangential inertial force was investigated. The equations of linear and angular momentum conservation are depicted based on the continuum mechanics analysis method as well as the boundary conditions.
⑤The principle of virtual work was adopted to derived the finite element equation for the generalized elastic body in the centrifugal field. The finite element formulation for the classic elastic body undergoing large rigid rotation was also been deduced by the lagrange methods, and the dynamic stiffening effects because of the geometrical non-linearity was discussed in the initial stress methods. The 8-nodes and 48-Dofs hexahedron isoparametric element was extended to the finite element formulation in the motion-deformation coupling simulation for the generalized elastic body.
⑥Simulations has been given to a typical cantilever beam rotating around a fixed axis with both the constant speed status and various speed status. Dynamic frequencies and responses for cantilever with different rotating configuration and size scale were evaluated. The results indicate that the dynamic frequencies and responses are different for the different rotating configuration, the dynamic stiffening effect become obviously for the flexible structure in the higher rotating speed conditions, and the size effects can’t be ignored when for the micro size structure.
单体运动与变形耦合动力学研究是柔性多体动力学研究的基础,在分析国内外对柔性多体动力学以及广义弹性力学研究现状的基础上,本文集中关注离心环境下结构的运动与变形耦合理论建模和数值算法开发,引入了含有三个材料参数的广义弹性体模型,采用广义弹性体模型分析了不同尺度下结构的静力和动力特性,建立了离心环境下广义弹性体的运动与变形耦合动力学模型并开发了相应的数值算法,考察了离心环境下典型悬臂梁结构的动力特性。论文的主要工作和结论如下:
①考虑含旋转变形的广义弹性体,在偶应力弹性理论和微极弹性理论的研究基础上,确定动量和动量矩守恒方程形式和广义弹性体的运动学表达。应用虚功原理和各向同性张量函数定理,改进偶应力和曲率张量的本构关系。建立广义弹性体的动力学方程和含力偶的初边值条件。
②采用约束变分原理,考虑位移和转角为独立变量,采用罚方法引入约束条件,建立广义弹性体的有限元控制方程。构造了4结点12自由度的平面等参元和8结点48自由度的六面体等参元。对平面单剪问题和悬臂梁静力和动力分析表明广义弹性体模型能够把经典弹性理论的应用范围扩展到微观领域,且在力学分析时给出了更加丰富的信息,有利于结构分析,能够计及尺寸效应的影响。
③建立了简化的双自由度弹簧质量系统的耦合动力学分析模型。利用简化的双自由度弹簧质量系统,引入描述质点刚体旋转运动和变形运动的惯性坐标系和浮动坐标系,应用正交张量和欧拉罗德里格斯旋转公式,给出质点在惯性坐标系和浮动坐标系的速度和加速度表达,建立定轴旋转的离心环境下的动力学模型,对典型算列的计算结果表明离心力效应、科式力效应和偏心效应对转动系统的动力特性有重要影响。
④建立了定轴变速离心环境下的广义弹性体的运动与变形耦合动力学方程。解析浮动标架下的离心力、科氏力、切向惯性力,基于连续介质力学分析方法,在浮动坐标系下建立广义弹性体动量和动量矩守恒方程。应用广义弹性体关于应力和偶应力的本构关系,给出浮动坐标系下含力偶的初边值条件,建立离心场中广义弹性体运动与变形耦合动力学方程,并采用初应力法解析了柔性结构的动力刚化效应。
⑤开发离心环境下广义弹性体运动与变形耦合动力学问题的数值算法。利用虚功原理建立广义弹性体运动与变形耦合的有限元方程,采用8结点48自由度的六面体等参元对广义弹性体进行离散。解析柔性结构由于几何非线性带来的动力刚化效应,考察广义惯性力形式,建立广义弹性体运动与变形耦合动力分析的有限元控制方程。
⑥将梁视为广义弹性体,针对尺寸和几何特征不同的梁型结构,应用所建立的广义弹性体有限元控制方程,数值求解梁型结构在恒速和变速的离心环境下的动频及动力响应特性,结合挠度和强度两方面考察了离心场下旋转梁的临界转速。研究结果表明科式力和离心力效应取决与转速、结构形态以及旋转姿态,旋转姿态影响动频与动力响应、高转速下柔性结构动力刚化效应明显,微观结构尺度下的尺寸效应提高刚度并严重影响动力特性。
Flexible bodies with different structure form are widely used in many industrial and technological systems, such as rotary machine, aerospace, vehicle engineering, mechanical arm and micro electro mechanical systems(MEMS). Construction members in these systems undergo large scale rigidity motion as well as elastic deformation. Inertial forces caused by the coupling between the rigid motion and elastic deformation play a key role to the dynamics characteristics for structures. Since that researches to the dynamics on the rigidity-flexible coupling are very important to dynamic analysis, system design and control of flexible multibody system(FMS). Theories on the motion and deformation coupling is a unification between the rigid-body dynamics and the deformable body mechanics. The dynamic formulations developed in this field must take account of the coupling effects between the rigid displacement and the deformation in deformable body, moreover, when ignoring the elastic deformation, the formulation can degenerate to the rigid-body dynamics, and when ignoring the rigid motion, the formulation can degenerate to the deformable body mechanics. There are several research topics which are of current interest, among these topics are the construction for dynamic formulation, computational strategies, impact and contact problem for the FMS and the fluid-structure interaction. Many problems are still not solved completely as a result of the nonlinearity in geometry, motion, Physics and material. Otherwise, the generalized elasticity has been developed deeply in recent years, few research has been given to the generalized elastic body undergoing large scale displacement.
The dynamic problem on a single flexible body should firstly be solved, when to formulate the kinetic equation for the FMS. On the base of the research at home and abroad on both the flexible multibody system dynamics(FMD) and generalized elasticity, In this dissertation, we focus on the dynamic formulation for the generalized elastic body undergoing rigid rotating and its computational strategies. The generalized elasticity contains three material parameter is put forward, and the structure with different size scale were analyzed. The dynamic models are developed for the generalized elastic body in the centrifugal field with fixed rotational axis and various rotational speed, also the finite element formulation was derived. The classic rotating cantilever beam is simulated in centrifugal field with both the constant rotational speed and various rotational speed. The main works and conclusions in this dissertation are as follows:
①The equations of linear and angular momentum conservation are derived for the generalized elasticity. basing on the couple stress elasticity theory and the micropolar theory. Combining the principle of virtual work with the theorem for the isotropic tensor, the modified relationship was put forward for the couple stress and the curvature tensor, moreover the kinetic equation and the boundary conditions were given for the generalized elastic body.
②The finite element equation was formulated for the generalized elastic body, on the base of the method of weighted residuals and the constrained variational principle,with both the displacement and rotational angle considered as independent variables. The 4-nodes and 12-Dofs plane isoparametric element was constructed as well as the 8-nodes and 48-Dofs hexahedron isoparametric element for generalized elastic body. The analysis results to the simple shear problem and the cantilever with different size scale conclude that the generalized elasticity expands the classic elasticity to the micro size scale, and provides much more information than the classic elasticity and the beam theory in the structure analysis.
③The dynamic model was developed for the double freedom vibration system, so as to investigate the effects of the centrifugal force, coriolis force and force caused by the eccentricity. For a two-Dofs springmass model, both the float frame reference and the fixed frame reference were introduced to described the motion of the mass point, applying the orthogonal tensor and the Rodriguez rotating formula, the velocity and the acceleration are given, and the dynamics formulation are derived for the fixed axis rotational springmass. The simulation results discovered that the centrifugal force and Coriolis force are play a key role to the dynamic characteristic as well as the eccentricity effect.
④The dynamic formulation was constructed for a generalized elastic body rotating along a fixed axis, for the various speed conditions. In the float frame reference, the centrifugal force, Coriolis force and tangential inertial force was investigated. The equations of linear and angular momentum conservation are depicted based on the continuum mechanics analysis method as well as the boundary conditions.
⑤The principle of virtual work was adopted to derived the finite element equation for the generalized elastic body in the centrifugal field. The finite element formulation for the classic elastic body undergoing large rigid rotation was also been deduced by the lagrange methods, and the dynamic stiffening effects because of the geometrical non-linearity was discussed in the initial stress methods. The 8-nodes and 48-Dofs hexahedron isoparametric element was extended to the finite element formulation in the motion-deformation coupling simulation for the generalized elastic body.
⑥Simulations has been given to a typical cantilever beam rotating around a fixed axis with both the constant speed status and various speed status. Dynamic frequencies and responses for cantilever with different rotating configuration and size scale were evaluated. The results indicate that the dynamic frequencies and responses are different for the different rotating configuration, the dynamic stiffening effect become obviously for the flexible structure in the higher rotating speed conditions, and the size effects can’t be ignored when for the micro size structure.
引文
[1] Kane T R, Ryan R R, and Banerjee A K. Dynamics of a cantilever beam attached to a moving base[J]. J of Guidance, 1987, 10(2): 139–151.
[2] Schilhans, M J. Bending frequency of a rotating cantilever beam[J]. Trans. ASME, J. Appl. Mechanics, 1958, 25: 28–30.
[3] Shabana A A. Flexible multibody dynamics: review of past and recent developments[J]. Multibody System Dynamics, 1997, 1(2):189–222.
[4] Wasfy T M, Noor A K. Computational strategies for ?exible multibody systems. Applied Mechanics Reviews, 2003, 56(6):553–613.
[5]洪嘉振,刘铸永.刚柔耦合动力学的建模方法[J].上海交通大学学报, 2008, 42(11): 1922-1927
[6] Likins P W. Modal method for the analysis of free rotations of spacecraft[J], AIAA J., 1967, 5(7): 1304–1308.
[7] Winfry R C. Elastic Link Mechanism Dynamics[J], ASME Journal of Engineering for Industry,1971,93:268-272
[8] Winfry R C. Dynamics Analysis of Elastic Link Mechanisms by Reduction of Coordinates[J], ASME Journal of Engineering for Industry,1972,94:577-582
[9] Agrawal O P , Shabana A A: Dynamic analysis of multibody systems using component modes[J]. Comput. Struct. 21(6), 1985, 1303–1312
[10] Canavin J R, Likins P W: Floating reference frames for flexible spacecraft, J. Spacecr. Rockets 14(12), 1977, 724–732
[11] Argyris J H, Kelsey S, and Kaneel H. Matrix Methods for Structural Analysis[M]: A Precis of Recent Developments, MacMillan, New York. 1964.
[12] Belytschko T, Hsieh B J. Non-linear transient finite element analysis with convected co-ordinates[J]. Int. J. Numer. Methods Eng, 1973, 7(3): 255–271.
[13] Takahashi Y and Shimizu N. Study on derivation and application of mean axis for deformable beam by means of the absolute nodal coordinate multibody dynamics formulation[R]. DETC20001/ VIB-21340, Proc of the ASME DETC, 2001
[14] Wu S C, Chang C W, and Housner J M. Finite element approach for transient analysis of multibody systems[J]. J. Guid. Control Dyn. 1992, 15(4), 847–854.
[15] Hughes T J R. The Finite Element Method[M]. Prentice Hall, Englewood Cliffs NJ, 1987
[16] Belytschko T, Schwer L, Klein M J. Large displacement, transient analysis of space frames[J].Int. J. for Numer. Methods in English , 1977, 11(1): 65–84.
[17] Belytschko T and Tsay C S. A stabilization procedure for the quadrilateral plate element with one-point quadrature[J]. Int. J. Numer. Methods Eng, 1983, 19(3): 405–419.
[18] Belytschko T, Lin J I, and Tsay C S. Explicit algorithms for the nonlinear dynamics of shells[J]. Comput. Methods Appl. Mech. Eng. 1984, 42(2): 225–251.
[19] Simo J C and Vu-Quoc L. On the dynamics of flexible beams under large overall motions-The plane case: Part I, Part II[J]. ASME J. Appl. Mech. 1986,53(4): 849–863
[20] Shabana A A, Dynamics of Multibody Systems[M], 2nd Edition, Cambridge Univ Press. 1998.
[21] Shabana A A. An absolute nodal coordinate formulation for the large rotation and deformation analysis of ?exible bodies[R]. Tech Report # MBS96-1-UIC, Dept of Mechanical Engineering, Univ of Illinois at Chicago, 1996.
[22] Liu Jinyang , Hong Jiazhen. Dynamic modeling and modal truncation approach for a high speed rotating elastic beam[J]. Archive of Applied Mechanics, 2002, 72(8) : 554-563.
[23] Yang Hui, Hong Jiazhen, Yu Zhengyue. Dynamics modeling of a flexible hub-beam system with a tip mass[J]. Journal of Sound and Vibration, 2003, 266(4): 759-774.
[24]蒋丽忠,洪嘉振.作大范围运动弹性梁的动力刚化研究[J].计算力学学报, 1998, 15(4): 407-413.
[25]刘锦阳,洪嘉振.作大范围运动矩形薄板的建模理论和有限元离散方法[J].振动工程学报,2003, 16(2): 175-179.
[26]胡振东,洪嘉振.刚柔耦合系统动力学建模及分析[J].应用数学和力学, 1999, 20(10): 1087-1093.
[27]刘锦阳,李彬等.做大范围运动的柔性梁的刚-柔耦合动力学[J].力学学报,2006,38(2):276-282.
[28]肖世富,陈滨.一类刚-柔耦合系统的建模与稳定性分析[J].力学学报,1997, 29(4): 439-447.
[29]肖世富,陈滨.离心场中纵向悬臂梁的大范围分岔分析[J].力学学报,2000, 32(5): 439-447.
[30]和兴锁,邓岩峰,王睿.具有大范围运动和非线性变形的空间柔性梁的精确动力学建模[J].物理学报, 2010, 59(3): 1428-1437.
[31] Cardona A and Geradin M. A beam finite element non-linear theory with finite rotations[J]. Int. J. Numer. Methods Eng, 1988, 26(11): 2403–2438
[32] Nygard M K and Bergan P G. Advances in treating large rotations for nonlinear problems[C]. States of the Arts Surveys on Computational Mechanics, AK Noor and JT Oden (eds), ASME, New York, 1989, 305–333.
[33] Jelenic G and Crisfield M A. Geometrically exact 3D beam theory: implementation of a strain-invariant finite element for static and dynamics[J]. Comput. Methods Appl. Mech. Eng,1999,171(2): 141–171.
[34] Campanelli M, Berzeri M, and Shabana A A. Performance of the incremental and non- incremental finite element formulations in flexible multibody problems[J]. ASME J. Mech. Des. 2000, 122(4): 498–507.
[35] Shabana A A. Finite element incremental approach and exact rigid body inertia[J]. ASME J. Mech. Des. 1996, 118(2): 171–178.
[36] Shabana A A. Computer implementation of the absolute nodal coordinate formulation for flexible multibody dynamics[J], Nonlinear Dynamics, 1998, 16(3): 293–306.
[37] Ambrosio J A C and Ravn P. Elastodynamics of multibody systems using generalized inertial coordinates and structural damping[J]. Mech. Struct. Mach. 1997, 25(2): 201–219.
[38] Ryu J, Kim S S. A criterion on inclusion of stress stiffening effects in flexible multibody dynamic system simulation[J]. Comput. Struct. 1997, 62(6): 1035–1048
[39] Crisfield M A, Galvanetto U, and Jelenic G. Dynamics of 3-D co-rotational beams[J]. Computational Mechanics, 1997, 20(6): 507–519
[40] Simo J C and Vu-Quoc L. On the dynamics of flexible beams under large overall motions—The plane case: Part I, Part II[J]. J. Appl. Mech, 1986, 53(4): 849–863.
[41] W S Shum, L Zhang. Dynamic motion of whirling rods with Coriolis effect[J]. Applied Mathematical Modeling, 2010, 34(5): 1203-1216.
[42] CL Huang, WY Lin, KM Hsiao. Free vibration analysis of rotating Euler beams at high angular velocity[J]. Computers and Structures , 2010 , 88(17-18): 991–1001.
[43] O Turhan, G Bulut. On nonlinear vibrations of a rotating beam[J]. J. Sound and Vibration, 2009, 322(1-2): 314–335.
[44] S C Lin, K M Hsiao. Vibration analysis of a rotating Timoshenko beam[J]. J. Sound and Vibration , 2001, 240(2): 303-322.
[45] W S Shum, R D Entwistle. Longitudinal vibration frequencies of steadily whirling rods[J], J. Acoust. Soc. Amer., 2006, 119(8): 909–916.
[46] J B Yang, L J Jiang, D C Chen. Dynamic modeling and control of a rotating Euler–Bernoulli beam[J]. J. Sound and Vibration, 2004, 274(3-5): 863–875.
[47] G P Cai, J Z Hong, S X Yang. Dynamic analysis of a flexible hub-beam system with tip mass[J]. Mechanics Research Communications, 2005, 32(2): 173–190.
[48] W Kim, J Chung. Free non-linear vibration of a rotating thin ring with the in-plane and out-of-plane motions[J]. J. Sound and Vibration, 2002, 258(1): 167–178.
[49] J S Chen. On the linearization of the equations of motion of a rotating disk[J]. Applied Mathematical Modelling, 2011, 35(1): 392–397.
[50] N Baddour, J W Zu. A revisit of spinning disk models. Part I: Derivation of equations of motion[J]. Applied Mathematical Modelling, 2001, 25(7): 541–559.
[51] N Baddour, J W Zu. A revisit of spinning disk models. Part II: Linear transverse vibrations[J]. Applied Mathematical Modeling, 2001, 25(7): 561–578.
[52] S H Hashemi, S Farhadi, S Carra. Free vibration analysis of rotating thick plates[J]. J. of Sound and Vibration, 2009, 323(1-2): 366–384.
[53] R Maretic. Transverse vibration and stability of an eccentric rotating circular plate[J]. J. of Sound and Vibration 280(3-5): 467–478, 2005
[54] Kane T R, Ryan R R, Banerjee A K. Dynamics of a Cantilever Beam Attached to a Moving Base[J]. Journal of Guidance, Control and Dynamics, 1987, 10(2):139-151
[55] Banerjee A K, Kane T R. Dynamics of a Plate in Large Overall Motion[J]. Journal of Applied Mechanics, 1989, 56: 887-892
[56] Banerjee A K, Dickens J M. Dynamics of an Arbitrary Flexible Body in Large Rotation and Translation[J]. Journal of Guidance, Control and Dynamics, 1990, 13(2): 221-227
[57] Banerjee A K, Lemak M K. Multi-Flexible Body Dynamics Capturing Motion-Induced Stiffness[J]. Journal of Applied Mechanics, 1991, 58: 766-775
[58] Ryu J, Kim S S. A General Approach to Stress Stiffening Effects on Flexible Multibody Dynamic Systems[J]. Mech.Strut & Mach., 1994, 22(2): 157-180
[59] Wallrapp O. Linearized Flexible Multibody Dynamics Including Geometric Stiffening Effects[J]. Mech. Struct. and Mach.,1991, 19(3): 385-409
[60] Zhang D J, Huston R L. On dynamic stiffening of flexible bodies having high angular velocity[J]. Mech Struct and Mach , 1996 , 24 (3) : 313-329.
[61] Mayo J, Dominguez J. Geometrically nonlinear for simulation of flexible multibody systems in terms of beam elements: Geometric stiffness[J]. Computers & Structures, 1996, 59: 1039-1050.
[62] Simo J C, Quoc L V. The Role of Non-linear Theories in Transient Dynamics Analysis of Flexible Structures[J]. Journal of Sound and Vibration, 1987, 119(3): 487-508.
[63] Sugiyamaa H, Gerstmayrb J, Shabana A A. Deformation modes in the finite element absolute nodal coordinate formulation[J]. Jounal of Sound and Vibration, 2006, 298(4-5): 1129–1149.
[64] Garcia-Vallejo D, Sugiyama H, Shabana A A. Finite element analysis of the geometric stiffening effect. Part 1:a correction in the floating frame of reference of formulation[J]. J. Multi-body Dynamics, 2005, 219(K): 187-202.
[65] Garcia-Vallejo D, Sugiyama H, Shabana A A. Finite element analysis of the geometric stiffening effect. Part 2: Nonlinear elasticity[J]. J. Multibody Dynamics,2005, 219(K): 203-211.
[66] Langlois R G, Anderson R J. Multibody dynamics of very flexible damped systems[J].Multibody Syst. Dyn. 1999, 3(2): 109–136.
[67] Iura M, Atluri S N. Dynamic analysis of planar flexible beams with finite rotations by using inertial and rotating frames[J], Comput. Struct., 1995, 55(3): 453–462.
[68] Ibrahimbegovic A. On FE implementation of geometrically nonlinear Reissner beam theory: Three-dimensional curved beam finite elements[J]. Comput. Methods Appl. Mech. Eng. 1995, 122: 10–26.
[69] Ibrahimbegovic A, Frey F, and Kozar I. Computational aspects of vector-like parameterization of three-dimensional finite rotations[J]. Int. J. Numer. Methods Eng. 1995, 38: 3653–3673.
[70] Shabana A A. Definition of the slopes and the finite element absolute nodal coordinate formulation[J], Multibody Syst. Dyn. 1997, 1(3): 339–348.
[71] Gontier C, Vollmer C. A large displacement analysis of a beam using a CAD geometric definition[J]. Comput. Struct. 1995, 57(6): 981–989.
[72] Cardona A, Huespe A. Nonlinear path following with turning and bifurcation points in multibody systems analysis[C]. Comput Methods in Appl Sci 96, 3rd ECCOMAS Comput Fluid Dyn Conf and the 2nd ECCOMAS Conf on Numer Methods in Eng, 1996,440–446.
[73] Rao D V, Sheikh A H, Mukopadhyay M. Finite element large displacement analysis of stiffened plates[J]. Comput. Struct, 1993, 47(6): 987–993.
[74] Simo J C, Fox D D . On a stress resultant geometrically exact shell model. Part I: Formulation and optimal parameterization[J]. Comput. Methods Appl. Mech. Eng. 1989,72: 267–304.
[75] Simo J C, Fox D D, Rifai M S. Geometrically exact stress resultant shell models: Formulation and computational aspects of thenonlinear theory, Analytical and Comput Models of Shells[C]. ASME Winter Annual Meeting, San Fransisco CA, 1989, 161–190.
[76] Oden T D. Finite Elements of Nonlinear Continua[M]. McGrawHill, New York, 1972.
[77] Bathe K J, Ramm E, and Wilson E L. Finite element formulations for large deformation dynamic analysis[J]. Int. J. Numer. Methods Eng. 1975, 9(2): 353–386.
[78] Wasfy T M. Modeling ?exible multibody systems including impact and thermal effects using the finite element method and element convected frames[M]. PhD Dissertation, Columbia Uni., 1994.
[79] Sunada W, Dubowsky S. The application of the finite element methods to the dynamic analysis of ?exible spatial and co-planar linkage systems[J]. Journal of Mechanical Design ,1981,103(3): 643–651.
[80] Sunada W, Dubowsky S. On the dynamic analysis and behavior of industrial robotic manipulator with elastic members[J]. Journal of Mechanisms, Transmissions, and Automation in Design, 1983, 105(1): 42–51.
[81] Agrawal O P , Shabana A A. Dynamic analysis of multibody systems using component modes[J]. Computer and Structures, 1985, 21(6): 1301–1312.
[82] Ashley, H. Observations on the dynamic behavior of large ?exible bodies in orbit[J], AIAA Journal,1967, 5(3): 460–469.
[83] Cavin R K, Dusto A R. Hamilton’s principle: Finite element methods and ?exible body dynamics[J]. AIAA Journal,1977, 15(2): 1684–1690.
[84] Veubeke B F. The dynamics of ?exible bodies[J]. International Journal for Engineering Science, 1976, 14: 895–913.
[85] Gofron M. Driving elastic forces in ?exible multibody systems[M]. Ph.D. Thesis, University of Illinois at Chicago, 1995
[86] Koppens W P. The dynamics of systems of deformable bodies[M]. Ph.D. Thesis, Technical University of Eindhoven, 1989
[87] Erdman A G,Sandor G N. Kineto-elastodynamics–A review of the state of the art and trends[J]. Mechanism and Machine Theory, 1972,7(1): 19–33.
[88] Lowen G G, Chassapis C. The elastic behavior of linkages: An update[J]. Mechanism and Machine Theory, 1986, 21(1), 33–42.
[89] Lowen G G, Jandrasits W G. Survey of investigations into the dynamic behavior of mechanisms containing links with distributed mass and elasticity[J]. Mechanism and Machine Theory, 1972, 7(2): 13–17.
[90] Winfrey R C. Elastic link mechanism dynamics[J]. ASME Journal of Engineering for Industry, 1971, 93: 268–272.
[91] Winfrey R C. Dynamic analysis of elastic link mechanisms by reduction of coordinates[J]. ASME Journal of Engineering for Industry ,1972, 94: 577–582.
[92] Turcic D A, Midha A , Dynamic analysis of elastic mechanism systems, Part I: Applications[J]. ASME J. Dyn. Syst., Meas., Control, 1984, 106(4): 249–254.
[93] Turcic D A , Midha A, and Bosnik J R.Dynamic analysis of elastic mechanism systems, Part II: Experimental results[J]. J. Dyn. Syst., Meas., Control,1984,106(4), 255–260
[94] Viscomi B V, Ayre R S. Nonlinear dynamic response of elastic slider-crank mechanism[J]. ASME J. Eng. Ind.,1971, 93(1): 251–262.
[95] Chu S C, Pan K C. Dynamic response of a high-speed slider-crank mechanism with an elastic connecting rod[J]. ASME J. Eng.Ind.1975, 97(2): 542–550.
[96] Sadler J P, Sandor G N . Nonlinear vibration analysis of elastic four-bar linkages[J]. ASME J. Eng. Ind. May, 1974, 96(2): 411–419.
[97] Sadler J P. On the analytical lumped-mass model of an elastic four-bar mechanism[J]. ASME J.Eng.Ind. May,1975, 97(2): 561–565
[98] Cavin R K, Dusto A R. Hamilton’s principle: Finite element methods and ?exible body dynamics[J]. AIAA J. 1977, 15: 1684–1690.
[99] Shabana A A. Resonance conditions and deformable body coordinate systems[J]. J. Sound Vib,1996, 192(1): 389–398.
[100] Likins P W. Geometric stiffness characteristics of a rotating elastic appendage[J]. Int. J. Solids Struct. 1974, 10(2): 161–167.
[101] Wright A, Smith C, and Thresher R. Vibration modes of centrifugally stiffened beams[J]. ASME J. Appl. Mech. 1982,49(1): 197–202.
[102] Likins P W. Hybrid-coordinate spacecraft dynamics using large deformation modal coordinates s[J]. Astronautical Acta , 1973,18(5): 331–348.
[103] Shabana A A, Wehage R A. Coordinate reduction technique for transient analysis of spatial substructures with large angular rotations[J]. Journal of Structural Mechanics, 1983, 11(3): 401–431.
[104] Winfrey R C. Dynamic analysis of elastic link mechanisms by reduction of coordinates[J],. ASME J. Eng. Ind. 1972, 94(2): 577–582.
[105] Hablani H B. Constrained and unconstrained modes: Some modeling aspects of ?exible spacecraft[J]. J. Guid. Control Dyn. 1982, 5(2): 164–173.
[106] Song J O and Haug E J. Dynamic analysis of planar ?exible mechanisms[J]. Comput. Methods Appl. Mech. Eng, 1980, 24(2): 359–381.
[107] Imam I, Sandor G N, and Kramer S N. De?ection and stress analysis in high speed planar mechanisms with elastic links[J]. ASME J. Eng. Ind. 1973, 95(2): 541–548.
[108] Laurenson R M. Modal analysis of rotating ?exible structures[J]. AIAA J, 1976,14(10), 1444–1450.
[109] Hoa S V. Vibration of a rotating beam with tip mass[J]. J. Sound Vib. 1979, 63(3): 369–381.
[110]陈志达.刚体动力学和变形体力学的统一理论[C].固体力学及其工程应用,张维教授八十寿辰纪念文集(黄克智,徐秉业主编),北京,清华大学出版社,1993.
[111] Toupin R A. Elastic Materials with Couple-stresses[J]. Arch. Rational Mech. Anal, 1962, 11(1): 415-448.
[112] Mindlin R D, Tiersten H F. Effects of Couple-stresses in Linear Elasticity[J]. Arch. Rational Mech. Anal, 1962, 11(1): 415-448.
[113] Koiter W T. Couple-stresses in the theory of elasticity, I and II[J]. Proc. Ned. Akad. Wet. (B). 1964, 67: 17–44.
[114] Mindlin R D. Micro-structure in linear elasticity[J]. Arch. Rat. Mech. Anal. 1964, 16: 51–78.
[115] Mindlin R D. Influence of couple-stresses on stress concentrations[J]. Exp. Mech., 1963, 3: 1-7.
[116] Leroy Y M, Molinari A. Spatial patterns and size effects in shear zones: A hyperelastic model with higer-order gradients[J]. J. Mech. Phys. Solids, 1993, 41(4):631-663.
[117] Fleck N A, Hutchinson J W. Strain Gradient Plasticity. Advance in Applied Mechanics[J]. 1997, 33: 295-362
[118] Fleck N A, Hutchinson J W. A reformulation of strain gradient plasticity [J]. Journal of Mechanics and Physics of Solids , 2001, 49(10): 2245–2271.
[119] Fleck N A , Muller G M , Ashby M F . Strain gradient plasticity: theory and experiment . A ctaM etallM ater, 1994, 42(2): 475-487
[120] Stolken J S, Evans A G. A microbend test method fo r measuring the plasticity length scale[R]. Harvard University Report Mech 320, Division of Engineering and App lied Sciences, Harvard U niversity, Cambridge, 1997
[121] Lloyd D J. Particle reinforced aluminum and magnesium matrix compsites[J]. Int Mater Reviews, 1994,39:1-23
[122] Yang F, Chong A C M, Lam D C C. Couple Stress Based Strain Gradient Theory for Elasticity[J]. Int. J. Solids and Structure, 2002, 39(10): 2731-2743.
[123] Lam D C C, Yang F, Chong, A C M. Experiments and theory in strain gradient elasticity[J]. Journal of Mechanics and Physics of Solids, 2003, 51: 1477-1508.
[124] Eringen A C. Linear theory of micropolar elasticity[J]. J. Math. Mech. 1966, 15: 909-924
[125] Eringen A C. Theory of micropolar elasticity[C]. In: Liebowitz H. (Ed.), Fracture. An Advanced Treatise, Vol. II, Mathematical Fundamentals, Academic Press, New York and London, 1968, 621–729
[126] Aifantis E C. Update on a class of gradient theories[J]. Mechanics of Materials 2003, 35: 259–280.
[127] Soh A K, Chen W. Finite element formulations of strain gradient theory for micro structures and the C0-1 patch test[J]. International Journal for Numerical Methods in Engineering , 2004 ,61(3) :433-454.
[128] Xia Z C, Hutchnson J W. Crack tip fields in strain gradient plasticity[J]. J. Mech .Phys . Solids, 1996, 44:1621-1648.
[129] Hermann L R. Mixed Finite Elements of Couple-Stresses Analysis[C]. In: S N Atluri et al. eds. Hybrid and Mixed Finite Element Methods. New York: Wiley,1983,1-17.
[130] Wood R D. Finite element analysis of plane couple-stress Problems using first order stress functions[J]. Int J Num Mech Engng, 1988, 26:489-509.
[131]肖其林,凌中,吴永礼.偶应力问题的杂交/混合元分析[J].计算力学学报, 2003, 20(4): 427-434.
[132]黄若煜,吴长春,钟万勰.基于平面偶应力Reissner/Mindlin板比拟的偶应力有限元[J].力学学报, 2004, 36 (03): 176-180.
[133]钟万勰,姚伟岸,郑长良. Reissner板弯曲与平面偶应力模拟[J].大连理工大学学报, 2002 , 42 (05): 519-521.
[134]郑长良,任明法,张志峰,宋和平.应用离散偶应力单元分析弹性Cosserat介质[J].计算物理,2004, 21(3): 377-382
[135] Tang Z, Shen S, Atluri S N. Analysis of materials with strain-gradient effects: a mesh free local Petrov-Galerkin(MPLG) approach, with nodal displacements only[J]. Comput. Model Eng. Sci., 2003, 4(1): 177-196.
[136] Askes H, Aifantis E C. Numerical modeling of size effects with gradient elasticity-Formulation, meshless discretization and example[J]. International Journal of Fracture, 2002, 117(4): 347-358.
[137] Yeon J H, Youn S K. Variational multiscale analysis of solids with strain gradient plastic effects using meshfree approximation[J]. International Journal of Computational Methods, 2005, 2(4): 601-626.
[138]张敦福,朱维中,李术才.基于偶应力理论的无网格Galerkin方法及尺度效应数值模拟研究[J].机械强度, 2008, 30(4): 625-632.
[139]聂志峰,周慎杰,王凯,孔胜利,C1自然邻近迦辽金法在偶应力理论中的应用[J].山东大学学报,2008, 38(6): 112-118.
[140]王卫东,张敦福,赵国群,程刚,基于偶应力理论的自然单元法研究[J].机械强度,2009 , 31(4) : 634-637.
[141] Mindlin R D. Influence of couple stresses on stress concentration [ J ]. Experiment Mechanics, 1962, 3 (1) : 1～7.
[142] Zhang H W, Wang H, Liu G Z. Quadrilateral isoparametric finite elements for plane elastic Cosserat bodies[J]. Acta Mechanica Sinica, 2005, 21(4): 388-394
[143]陈万吉.细观尺度C0和C1理论及有限元分片检验函数[C].中国科学G辑:物理学力学天文学,2009,39(10):1480-1486.
[144] Atkinson A, Leppington F G. The effect of couple stress on the tip of a crack[J]. International Journal of Solids and Structures, 1977, 13:1103-1122.
[145] Xia Z C, Hutchinson J W. Crack tip fields in strain gradient plasticity[J]. Journal of the Mechanics and Physics of Solids, 1996, 44(10): 1621-1648.
[146] Hwang K C, Huang Y. Mechanism-based strain gradient(MSG) plasticity and the associatedasymptotic crack-tip fields[J]. Fracture and Strength of Solids, Pts 1 and 2, 2000, 183(1): 9-18
[147] Jiang H, Huang Y, Zhuang Z, et al. Fracture in mechanism-based strain gradient plasticity[J]. Journal of the Mechanics and Physics of Solids, 2001, 49(5): 979-993.
[148] Radi E. Effects of characteristic material lengths on modeⅢcrack propagation in couple stress elastic-plastic materials[J]. International Journal of Plasticity, 2007, 23(8): 1439-1456.
[149] Chen S H, Wang T C. Finite element solutions for plane strain modeⅠcrack with strain gradient effects[J]. International Journal of Solids and Structures, 2002, 39: 1241-1257.
[150] Muhlhaus H B. Application of Cosserat theory in numerical solutions of limit load problems[J]. Ingenieur-Archiv,1989,59:124-137.
[151] Muhlhaus H B. Stress and couple stress in a layered half plane with surface loading[J]. International Journal for Numerical and Analytical Methods in Geomechanics, 1990, 14(9): 545-563.
[152] de Borst R, Muhlhaus H B. Gradient-dependent plasticity: formulation and algorithmic aspects[J]. International Journal for Numerical Methods in Engineering, 1992, 35(3): 521-539.
[153] Adhikary D P, Dyskin A V, Jewell R J. Numerical modeling of the flexural deformation of foliated rock slopes[J]. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, 1996, 33(6): 595-606.
[154] Voyiadjis G Z, Alsaleh M I, Alshibli K A. Evolving internal length scales in plastic strain localization for granular materials[J]. International Journal of Plasticity, 2005, 21(10): 2000-2024.
[155]陈胜宏,王鸿儒,熊文林.节理岩体的偶应力影响研究[J].水利学报,1990,l(l): 1-8.
[156]佘成学,熊文林,陈胜宏.具有弯曲效应的层状结构岩体变形的cosserat介质分析方法[J].岩土力学,1994,15(4):12-19.
[157]佘成学,熊文林,陈胜宏.层状岩体的弹粘塑性Cosserat介质理论及其工程应用[J].水利学报,1996,4:10-17.
[158] Shu J Y, Barlow C Y. Strain gradient effects on microscopic strain fields in a metal matrix composite[J]. International Journal of Plasticity, 2000, 16(5): 563-591.
[159] Liu X N, Hu G K. A continuum micromechanical theory of overall plasticity for particulate composites including particle size effects[J]. International Journal of Plasticity, 2005, 21(4): 777-799.
[160] Mori T ,Tanaka K. Average stress in matrix and average elastic energy of materials with misfitting inclusions[J]. Acta Metallurgica et Materialia, 1973, 21(5): 571-574.
[161] Qu S, Siegmund T, Huang Y, et al. A study of particle size effect and interface fracture inaluminum alloy composite via an extended conventional theory of mechanism-based strain-gradient plasticity[J]. Composites Science and Technology, 2005, 65(7-8): 1244-1253.
[162] Stolken J S, Evans A G. A microbend test method for measuring the plasticity length scale[R]. Harvard University Report Mech 320, Division of Engineering and Applied Sciences, Harvard University, Cambridge, 1997.
[163] Aifantis E C. Strain gradient interpretation of size effects[J]. International Journal of Fracture, 1999, 95(l): 299- 314.
[164] Wang W, Huang Y, Hsia K J. A study of micro-bend test by strain gradient Plasticity[J]. International Journal of plasticity, 2003, 19(3): 365-382.
[165] Anthoine A. Effeet of couple stresses on the elastic bending of beams[J]. International Journal of Solids and structures, 2000, 37(7): 1003-1018.
[166] Park S K, X-L Gao. Bernoulli-Euler Beam Model Based on a Modified Couple Stress Theory[J]. Journal of Micromechanics and Microengineering. 2006, 16(11): 2355-2359.
[167] Park S K, X-L Gao. Variational Formulation of a Modified Couple Stress Theory and Its Application to a Simple Shear Problem[J]. Z.angew Math. Phys, 2008, 59(5): 904-917.
[168] Ma H M, X-L Gao, Reddy J N. A Microstructure-Dependent Timoshenko Beam Model Based on a Modified Couple Stress Theory[J]. Journal of the Mechanics and Physics of Solids, 2008, 56(12): 3379-3391.
[169] Chen S H, Wang T C. Strain gradient theory with couple stress for Crystalline solids[J]. European Journal of Mechanics A/Solids, 2001, 24(5): 1497-1505.
[170] Hwang K C, Jiang H, Huang Y. A finite deformation theory of strain gradient Plasticity[J]. Journal of the Mechanics and physics of Solids, 2002, 50(l):81-99.
[171] Hwang K C, Jiang H, Huang Y. Finite deformation analysis of mechanism-based strain gradient Plasticity: torsion and crack tip field[J]. International Journal of Plasticity, 2003 19(2): 235-251.
[172] Wei Y G, Wang X Z, Zhao M H. Size effect and geometrical effects of solids in micro-indentation test[J]. Acta Mechanica Sinica, 2003, 19(1):59-70.
[173] Qiu X, Huang Y, Wei Y. The flow theory of mechanism-based strain gradient plasticity Mechanics of Materials, 2003, 35(3-6): 245-258.
[174]章定国,吴胜宝,康新.考虑尺度效应的微梁刚柔耦合动力学分析[J].固体力学学报, 2010, 31(1): 32-39.
[175]徐芝纶.弹性力学(第4版)[M].北京:高等教育出版社,2006.
[176] Ottosen NS, Ristinmaa M, Ljung C. Rayleigh waves obtained by the indeterminate couple-stress theory[J]. Eur. J. Mech. A/Solids, 2000, 19 (6): 929–947.
[177] Truesdell C, Noll W. The Non-linear Field Theories of Mechanics[M]. Handbuch der physic,(S Flugge ed) III/3. Berlin: Springer-verlag, 1965.
[178] J H Park, H Y Park, S Y Jeong. Linear vibration analysis of rotating wind-turbine blade. Current Applied Physics, 2010, 10(2): 332–334.
[179] B Palais, R Palais. Euler’s fixed point theorem: the axis of a rotation[J]. J. Fixed Point Theory and Applications, 2007, 2: 215-220.
[180] Oskar W, Richard S. Representation of geometric stiffening in multibody system simulation[J]. International Journal for Numerical Methods in Engineering, 1991, 32(8):1833-1850.
[2] Schilhans, M J. Bending frequency of a rotating cantilever beam[J]. Trans. ASME, J. Appl. Mechanics, 1958, 25: 28–30.
[3] Shabana A A. Flexible multibody dynamics: review of past and recent developments[J]. Multibody System Dynamics, 1997, 1(2):189–222.
[4] Wasfy T M, Noor A K. Computational strategies for ?exible multibody systems. Applied Mechanics Reviews, 2003, 56(6):553–613.
[5]洪嘉振,刘铸永.刚柔耦合动力学的建模方法[J].上海交通大学学报, 2008, 42(11): 1922-1927
[6] Likins P W. Modal method for the analysis of free rotations of spacecraft[J], AIAA J., 1967, 5(7): 1304–1308.
[7] Winfry R C. Elastic Link Mechanism Dynamics[J], ASME Journal of Engineering for Industry,1971,93:268-272
[8] Winfry R C. Dynamics Analysis of Elastic Link Mechanisms by Reduction of Coordinates[J], ASME Journal of Engineering for Industry,1972,94:577-582
[9] Agrawal O P , Shabana A A: Dynamic analysis of multibody systems using component modes[J]. Comput. Struct. 21(6), 1985, 1303–1312
[10] Canavin J R, Likins P W: Floating reference frames for flexible spacecraft, J. Spacecr. Rockets 14(12), 1977, 724–732
[11] Argyris J H, Kelsey S, and Kaneel H. Matrix Methods for Structural Analysis[M]: A Precis of Recent Developments, MacMillan, New York. 1964.
[12] Belytschko T, Hsieh B J. Non-linear transient finite element analysis with convected co-ordinates[J]. Int. J. Numer. Methods Eng, 1973, 7(3): 255–271.
[13] Takahashi Y and Shimizu N. Study on derivation and application of mean axis for deformable beam by means of the absolute nodal coordinate multibody dynamics formulation[R]. DETC20001/ VIB-21340, Proc of the ASME DETC, 2001
[14] Wu S C, Chang C W, and Housner J M. Finite element approach for transient analysis of multibody systems[J]. J. Guid. Control Dyn. 1992, 15(4), 847–854.
[15] Hughes T J R. The Finite Element Method[M]. Prentice Hall, Englewood Cliffs NJ, 1987
[16] Belytschko T, Schwer L, Klein M J. Large displacement, transient analysis of space frames[J].Int. J. for Numer. Methods in English , 1977, 11(1): 65–84.
[17] Belytschko T and Tsay C S. A stabilization procedure for the quadrilateral plate element with one-point quadrature[J]. Int. J. Numer. Methods Eng, 1983, 19(3): 405–419.
[18] Belytschko T, Lin J I, and Tsay C S. Explicit algorithms for the nonlinear dynamics of shells[J]. Comput. Methods Appl. Mech. Eng. 1984, 42(2): 225–251.
[19] Simo J C and Vu-Quoc L. On the dynamics of flexible beams under large overall motions-The plane case: Part I, Part II[J]. ASME J. Appl. Mech. 1986,53(4): 849–863
[20] Shabana A A, Dynamics of Multibody Systems[M], 2nd Edition, Cambridge Univ Press. 1998.
[21] Shabana A A. An absolute nodal coordinate formulation for the large rotation and deformation analysis of ?exible bodies[R]. Tech Report # MBS96-1-UIC, Dept of Mechanical Engineering, Univ of Illinois at Chicago, 1996.
[22] Liu Jinyang , Hong Jiazhen. Dynamic modeling and modal truncation approach for a high speed rotating elastic beam[J]. Archive of Applied Mechanics, 2002, 72(8) : 554-563.
[23] Yang Hui, Hong Jiazhen, Yu Zhengyue. Dynamics modeling of a flexible hub-beam system with a tip mass[J]. Journal of Sound and Vibration, 2003, 266(4): 759-774.
[24]蒋丽忠,洪嘉振.作大范围运动弹性梁的动力刚化研究[J].计算力学学报, 1998, 15(4): 407-413.
[25]刘锦阳,洪嘉振.作大范围运动矩形薄板的建模理论和有限元离散方法[J].振动工程学报,2003, 16(2): 175-179.
[26]胡振东,洪嘉振.刚柔耦合系统动力学建模及分析[J].应用数学和力学, 1999, 20(10): 1087-1093.
[27]刘锦阳,李彬等.做大范围运动的柔性梁的刚-柔耦合动力学[J].力学学报,2006,38(2):276-282.
[28]肖世富,陈滨.一类刚-柔耦合系统的建模与稳定性分析[J].力学学报,1997, 29(4): 439-447.
[29]肖世富,陈滨.离心场中纵向悬臂梁的大范围分岔分析[J].力学学报,2000, 32(5): 439-447.
[30]和兴锁,邓岩峰,王睿.具有大范围运动和非线性变形的空间柔性梁的精确动力学建模[J].物理学报, 2010, 59(3): 1428-1437.
[31] Cardona A and Geradin M. A beam finite element non-linear theory with finite rotations[J]. Int. J. Numer. Methods Eng, 1988, 26(11): 2403–2438
[32] Nygard M K and Bergan P G. Advances in treating large rotations for nonlinear problems[C]. States of the Arts Surveys on Computational Mechanics, AK Noor and JT Oden (eds), ASME, New York, 1989, 305–333.
[33] Jelenic G and Crisfield M A. Geometrically exact 3D beam theory: implementation of a strain-invariant finite element for static and dynamics[J]. Comput. Methods Appl. Mech. Eng,1999,171(2): 141–171.
[34] Campanelli M, Berzeri M, and Shabana A A. Performance of the incremental and non- incremental finite element formulations in flexible multibody problems[J]. ASME J. Mech. Des. 2000, 122(4): 498–507.
[35] Shabana A A. Finite element incremental approach and exact rigid body inertia[J]. ASME J. Mech. Des. 1996, 118(2): 171–178.
[36] Shabana A A. Computer implementation of the absolute nodal coordinate formulation for flexible multibody dynamics[J], Nonlinear Dynamics, 1998, 16(3): 293–306.
[37] Ambrosio J A C and Ravn P. Elastodynamics of multibody systems using generalized inertial coordinates and structural damping[J]. Mech. Struct. Mach. 1997, 25(2): 201–219.
[38] Ryu J, Kim S S. A criterion on inclusion of stress stiffening effects in flexible multibody dynamic system simulation[J]. Comput. Struct. 1997, 62(6): 1035–1048
[39] Crisfield M A, Galvanetto U, and Jelenic G. Dynamics of 3-D co-rotational beams[J]. Computational Mechanics, 1997, 20(6): 507–519
[40] Simo J C and Vu-Quoc L. On the dynamics of flexible beams under large overall motions—The plane case: Part I, Part II[J]. J. Appl. Mech, 1986, 53(4): 849–863.
[41] W S Shum, L Zhang. Dynamic motion of whirling rods with Coriolis effect[J]. Applied Mathematical Modeling, 2010, 34(5): 1203-1216.
[42] CL Huang, WY Lin, KM Hsiao. Free vibration analysis of rotating Euler beams at high angular velocity[J]. Computers and Structures , 2010 , 88(17-18): 991–1001.
[43] O Turhan, G Bulut. On nonlinear vibrations of a rotating beam[J]. J. Sound and Vibration, 2009, 322(1-2): 314–335.
[44] S C Lin, K M Hsiao. Vibration analysis of a rotating Timoshenko beam[J]. J. Sound and Vibration , 2001, 240(2): 303-322.
[45] W S Shum, R D Entwistle. Longitudinal vibration frequencies of steadily whirling rods[J], J. Acoust. Soc. Amer., 2006, 119(8): 909–916.
[46] J B Yang, L J Jiang, D C Chen. Dynamic modeling and control of a rotating Euler–Bernoulli beam[J]. J. Sound and Vibration, 2004, 274(3-5): 863–875.
[47] G P Cai, J Z Hong, S X Yang. Dynamic analysis of a flexible hub-beam system with tip mass[J]. Mechanics Research Communications, 2005, 32(2): 173–190.
[48] W Kim, J Chung. Free non-linear vibration of a rotating thin ring with the in-plane and out-of-plane motions[J]. J. Sound and Vibration, 2002, 258(1): 167–178.
[49] J S Chen. On the linearization of the equations of motion of a rotating disk[J]. Applied Mathematical Modelling, 2011, 35(1): 392–397.
[50] N Baddour, J W Zu. A revisit of spinning disk models. Part I: Derivation of equations of motion[J]. Applied Mathematical Modelling, 2001, 25(7): 541–559.
[51] N Baddour, J W Zu. A revisit of spinning disk models. Part II: Linear transverse vibrations[J]. Applied Mathematical Modeling, 2001, 25(7): 561–578.
[52] S H Hashemi, S Farhadi, S Carra. Free vibration analysis of rotating thick plates[J]. J. of Sound and Vibration, 2009, 323(1-2): 366–384.
[53] R Maretic. Transverse vibration and stability of an eccentric rotating circular plate[J]. J. of Sound and Vibration 280(3-5): 467–478, 2005
[54] Kane T R, Ryan R R, Banerjee A K. Dynamics of a Cantilever Beam Attached to a Moving Base[J]. Journal of Guidance, Control and Dynamics, 1987, 10(2):139-151
[55] Banerjee A K, Kane T R. Dynamics of a Plate in Large Overall Motion[J]. Journal of Applied Mechanics, 1989, 56: 887-892
[56] Banerjee A K, Dickens J M. Dynamics of an Arbitrary Flexible Body in Large Rotation and Translation[J]. Journal of Guidance, Control and Dynamics, 1990, 13(2): 221-227
[57] Banerjee A K, Lemak M K. Multi-Flexible Body Dynamics Capturing Motion-Induced Stiffness[J]. Journal of Applied Mechanics, 1991, 58: 766-775
[58] Ryu J, Kim S S. A General Approach to Stress Stiffening Effects on Flexible Multibody Dynamic Systems[J]. Mech.Strut & Mach., 1994, 22(2): 157-180
[59] Wallrapp O. Linearized Flexible Multibody Dynamics Including Geometric Stiffening Effects[J]. Mech. Struct. and Mach.,1991, 19(3): 385-409
[60] Zhang D J, Huston R L. On dynamic stiffening of flexible bodies having high angular velocity[J]. Mech Struct and Mach , 1996 , 24 (3) : 313-329.
[61] Mayo J, Dominguez J. Geometrically nonlinear for simulation of flexible multibody systems in terms of beam elements: Geometric stiffness[J]. Computers & Structures, 1996, 59: 1039-1050.
[62] Simo J C, Quoc L V. The Role of Non-linear Theories in Transient Dynamics Analysis of Flexible Structures[J]. Journal of Sound and Vibration, 1987, 119(3): 487-508.
[63] Sugiyamaa H, Gerstmayrb J, Shabana A A. Deformation modes in the finite element absolute nodal coordinate formulation[J]. Jounal of Sound and Vibration, 2006, 298(4-5): 1129–1149.
[64] Garcia-Vallejo D, Sugiyama H, Shabana A A. Finite element analysis of the geometric stiffening effect. Part 1:a correction in the floating frame of reference of formulation[J]. J. Multi-body Dynamics, 2005, 219(K): 187-202.
[65] Garcia-Vallejo D, Sugiyama H, Shabana A A. Finite element analysis of the geometric stiffening effect. Part 2: Nonlinear elasticity[J]. J. Multibody Dynamics,2005, 219(K): 203-211.
[66] Langlois R G, Anderson R J. Multibody dynamics of very flexible damped systems[J].Multibody Syst. Dyn. 1999, 3(2): 109–136.
[67] Iura M, Atluri S N. Dynamic analysis of planar flexible beams with finite rotations by using inertial and rotating frames[J], Comput. Struct., 1995, 55(3): 453–462.
[68] Ibrahimbegovic A. On FE implementation of geometrically nonlinear Reissner beam theory: Three-dimensional curved beam finite elements[J]. Comput. Methods Appl. Mech. Eng. 1995, 122: 10–26.
[69] Ibrahimbegovic A, Frey F, and Kozar I. Computational aspects of vector-like parameterization of three-dimensional finite rotations[J]. Int. J. Numer. Methods Eng. 1995, 38: 3653–3673.
[70] Shabana A A. Definition of the slopes and the finite element absolute nodal coordinate formulation[J], Multibody Syst. Dyn. 1997, 1(3): 339–348.
[71] Gontier C, Vollmer C. A large displacement analysis of a beam using a CAD geometric definition[J]. Comput. Struct. 1995, 57(6): 981–989.
[72] Cardona A, Huespe A. Nonlinear path following with turning and bifurcation points in multibody systems analysis[C]. Comput Methods in Appl Sci 96, 3rd ECCOMAS Comput Fluid Dyn Conf and the 2nd ECCOMAS Conf on Numer Methods in Eng, 1996,440–446.
[73] Rao D V, Sheikh A H, Mukopadhyay M. Finite element large displacement analysis of stiffened plates[J]. Comput. Struct, 1993, 47(6): 987–993.
[74] Simo J C, Fox D D . On a stress resultant geometrically exact shell model. Part I: Formulation and optimal parameterization[J]. Comput. Methods Appl. Mech. Eng. 1989,72: 267–304.
[75] Simo J C, Fox D D, Rifai M S. Geometrically exact stress resultant shell models: Formulation and computational aspects of thenonlinear theory, Analytical and Comput Models of Shells[C]. ASME Winter Annual Meeting, San Fransisco CA, 1989, 161–190.
[76] Oden T D. Finite Elements of Nonlinear Continua[M]. McGrawHill, New York, 1972.
[77] Bathe K J, Ramm E, and Wilson E L. Finite element formulations for large deformation dynamic analysis[J]. Int. J. Numer. Methods Eng. 1975, 9(2): 353–386.
[78] Wasfy T M. Modeling ?exible multibody systems including impact and thermal effects using the finite element method and element convected frames[M]. PhD Dissertation, Columbia Uni., 1994.
[79] Sunada W, Dubowsky S. The application of the finite element methods to the dynamic analysis of ?exible spatial and co-planar linkage systems[J]. Journal of Mechanical Design ,1981,103(3): 643–651.
[80] Sunada W, Dubowsky S. On the dynamic analysis and behavior of industrial robotic manipulator with elastic members[J]. Journal of Mechanisms, Transmissions, and Automation in Design, 1983, 105(1): 42–51.
[81] Agrawal O P , Shabana A A. Dynamic analysis of multibody systems using component modes[J]. Computer and Structures, 1985, 21(6): 1301–1312.
[82] Ashley, H. Observations on the dynamic behavior of large ?exible bodies in orbit[J], AIAA Journal,1967, 5(3): 460–469.
[83] Cavin R K, Dusto A R. Hamilton’s principle: Finite element methods and ?exible body dynamics[J]. AIAA Journal,1977, 15(2): 1684–1690.
[84] Veubeke B F. The dynamics of ?exible bodies[J]. International Journal for Engineering Science, 1976, 14: 895–913.
[85] Gofron M. Driving elastic forces in ?exible multibody systems[M]. Ph.D. Thesis, University of Illinois at Chicago, 1995
[86] Koppens W P. The dynamics of systems of deformable bodies[M]. Ph.D. Thesis, Technical University of Eindhoven, 1989
[87] Erdman A G,Sandor G N. Kineto-elastodynamics–A review of the state of the art and trends[J]. Mechanism and Machine Theory, 1972,7(1): 19–33.
[88] Lowen G G, Chassapis C. The elastic behavior of linkages: An update[J]. Mechanism and Machine Theory, 1986, 21(1), 33–42.
[89] Lowen G G, Jandrasits W G. Survey of investigations into the dynamic behavior of mechanisms containing links with distributed mass and elasticity[J]. Mechanism and Machine Theory, 1972, 7(2): 13–17.
[90] Winfrey R C. Elastic link mechanism dynamics[J]. ASME Journal of Engineering for Industry, 1971, 93: 268–272.
[91] Winfrey R C. Dynamic analysis of elastic link mechanisms by reduction of coordinates[J]. ASME Journal of Engineering for Industry ,1972, 94: 577–582.
[92] Turcic D A, Midha A , Dynamic analysis of elastic mechanism systems, Part I: Applications[J]. ASME J. Dyn. Syst., Meas., Control, 1984, 106(4): 249–254.
[93] Turcic D A , Midha A, and Bosnik J R.Dynamic analysis of elastic mechanism systems, Part II: Experimental results[J]. J. Dyn. Syst., Meas., Control,1984,106(4), 255–260
[94] Viscomi B V, Ayre R S. Nonlinear dynamic response of elastic slider-crank mechanism[J]. ASME J. Eng. Ind.,1971, 93(1): 251–262.
[95] Chu S C, Pan K C. Dynamic response of a high-speed slider-crank mechanism with an elastic connecting rod[J]. ASME J. Eng.Ind.1975, 97(2): 542–550.
[96] Sadler J P, Sandor G N . Nonlinear vibration analysis of elastic four-bar linkages[J]. ASME J. Eng. Ind. May, 1974, 96(2): 411–419.
[97] Sadler J P. On the analytical lumped-mass model of an elastic four-bar mechanism[J]. ASME J.Eng.Ind. May,1975, 97(2): 561–565
[98] Cavin R K, Dusto A R. Hamilton’s principle: Finite element methods and ?exible body dynamics[J]. AIAA J. 1977, 15: 1684–1690.
[99] Shabana A A. Resonance conditions and deformable body coordinate systems[J]. J. Sound Vib,1996, 192(1): 389–398.
[100] Likins P W. Geometric stiffness characteristics of a rotating elastic appendage[J]. Int. J. Solids Struct. 1974, 10(2): 161–167.
[101] Wright A, Smith C, and Thresher R. Vibration modes of centrifugally stiffened beams[J]. ASME J. Appl. Mech. 1982,49(1): 197–202.
[102] Likins P W. Hybrid-coordinate spacecraft dynamics using large deformation modal coordinates s[J]. Astronautical Acta , 1973,18(5): 331–348.
[103] Shabana A A, Wehage R A. Coordinate reduction technique for transient analysis of spatial substructures with large angular rotations[J]. Journal of Structural Mechanics, 1983, 11(3): 401–431.
[104] Winfrey R C. Dynamic analysis of elastic link mechanisms by reduction of coordinates[J],. ASME J. Eng. Ind. 1972, 94(2): 577–582.
[105] Hablani H B. Constrained and unconstrained modes: Some modeling aspects of ?exible spacecraft[J]. J. Guid. Control Dyn. 1982, 5(2): 164–173.
[106] Song J O and Haug E J. Dynamic analysis of planar ?exible mechanisms[J]. Comput. Methods Appl. Mech. Eng, 1980, 24(2): 359–381.
[107] Imam I, Sandor G N, and Kramer S N. De?ection and stress analysis in high speed planar mechanisms with elastic links[J]. ASME J. Eng. Ind. 1973, 95(2): 541–548.
[108] Laurenson R M. Modal analysis of rotating ?exible structures[J]. AIAA J, 1976,14(10), 1444–1450.
[109] Hoa S V. Vibration of a rotating beam with tip mass[J]. J. Sound Vib. 1979, 63(3): 369–381.
[110]陈志达.刚体动力学和变形体力学的统一理论[C].固体力学及其工程应用,张维教授八十寿辰纪念文集(黄克智,徐秉业主编),北京,清华大学出版社,1993.
[111] Toupin R A. Elastic Materials with Couple-stresses[J]. Arch. Rational Mech. Anal, 1962, 11(1): 415-448.
[112] Mindlin R D, Tiersten H F. Effects of Couple-stresses in Linear Elasticity[J]. Arch. Rational Mech. Anal, 1962, 11(1): 415-448.
[113] Koiter W T. Couple-stresses in the theory of elasticity, I and II[J]. Proc. Ned. Akad. Wet. (B). 1964, 67: 17–44.
[114] Mindlin R D. Micro-structure in linear elasticity[J]. Arch. Rat. Mech. Anal. 1964, 16: 51–78.
[115] Mindlin R D. Influence of couple-stresses on stress concentrations[J]. Exp. Mech., 1963, 3: 1-7.
[116] Leroy Y M, Molinari A. Spatial patterns and size effects in shear zones: A hyperelastic model with higer-order gradients[J]. J. Mech. Phys. Solids, 1993, 41(4):631-663.
[117] Fleck N A, Hutchinson J W. Strain Gradient Plasticity. Advance in Applied Mechanics[J]. 1997, 33: 295-362
[118] Fleck N A, Hutchinson J W. A reformulation of strain gradient plasticity [J]. Journal of Mechanics and Physics of Solids , 2001, 49(10): 2245–2271.
[119] Fleck N A , Muller G M , Ashby M F . Strain gradient plasticity: theory and experiment . A ctaM etallM ater, 1994, 42(2): 475-487
[120] Stolken J S, Evans A G. A microbend test method fo r measuring the plasticity length scale[R]. Harvard University Report Mech 320, Division of Engineering and App lied Sciences, Harvard U niversity, Cambridge, 1997
[121] Lloyd D J. Particle reinforced aluminum and magnesium matrix compsites[J]. Int Mater Reviews, 1994,39:1-23
[122] Yang F, Chong A C M, Lam D C C. Couple Stress Based Strain Gradient Theory for Elasticity[J]. Int. J. Solids and Structure, 2002, 39(10): 2731-2743.
[123] Lam D C C, Yang F, Chong, A C M. Experiments and theory in strain gradient elasticity[J]. Journal of Mechanics and Physics of Solids, 2003, 51: 1477-1508.
[124] Eringen A C. Linear theory of micropolar elasticity[J]. J. Math. Mech. 1966, 15: 909-924
[125] Eringen A C. Theory of micropolar elasticity[C]. In: Liebowitz H. (Ed.), Fracture. An Advanced Treatise, Vol. II, Mathematical Fundamentals, Academic Press, New York and London, 1968, 621–729
[126] Aifantis E C. Update on a class of gradient theories[J]. Mechanics of Materials 2003, 35: 259–280.
[127] Soh A K, Chen W. Finite element formulations of strain gradient theory for micro structures and the C0-1 patch test[J]. International Journal for Numerical Methods in Engineering , 2004 ,61(3) :433-454.
[128] Xia Z C, Hutchnson J W. Crack tip fields in strain gradient plasticity[J]. J. Mech .Phys . Solids, 1996, 44:1621-1648.
[129] Hermann L R. Mixed Finite Elements of Couple-Stresses Analysis[C]. In: S N Atluri et al. eds. Hybrid and Mixed Finite Element Methods. New York: Wiley,1983,1-17.
[130] Wood R D. Finite element analysis of plane couple-stress Problems using first order stress functions[J]. Int J Num Mech Engng, 1988, 26:489-509.
[131]肖其林,凌中,吴永礼.偶应力问题的杂交/混合元分析[J].计算力学学报, 2003, 20(4): 427-434.
[132]黄若煜,吴长春,钟万勰.基于平面偶应力Reissner/Mindlin板比拟的偶应力有限元[J].力学学报, 2004, 36 (03): 176-180.
[133]钟万勰,姚伟岸,郑长良. Reissner板弯曲与平面偶应力模拟[J].大连理工大学学报, 2002 , 42 (05): 519-521.
[134]郑长良,任明法,张志峰,宋和平.应用离散偶应力单元分析弹性Cosserat介质[J].计算物理,2004, 21(3): 377-382
[135] Tang Z, Shen S, Atluri S N. Analysis of materials with strain-gradient effects: a mesh free local Petrov-Galerkin(MPLG) approach, with nodal displacements only[J]. Comput. Model Eng. Sci., 2003, 4(1): 177-196.
[136] Askes H, Aifantis E C. Numerical modeling of size effects with gradient elasticity-Formulation, meshless discretization and example[J]. International Journal of Fracture, 2002, 117(4): 347-358.
[137] Yeon J H, Youn S K. Variational multiscale analysis of solids with strain gradient plastic effects using meshfree approximation[J]. International Journal of Computational Methods, 2005, 2(4): 601-626.
[138]张敦福,朱维中,李术才.基于偶应力理论的无网格Galerkin方法及尺度效应数值模拟研究[J].机械强度, 2008, 30(4): 625-632.
[139]聂志峰,周慎杰,王凯,孔胜利,C1自然邻近迦辽金法在偶应力理论中的应用[J].山东大学学报,2008, 38(6): 112-118.
[140]王卫东,张敦福,赵国群,程刚,基于偶应力理论的自然单元法研究[J].机械强度,2009 , 31(4) : 634-637.
[141] Mindlin R D. Influence of couple stresses on stress concentration [ J ]. Experiment Mechanics, 1962, 3 (1) : 1～7.
[142] Zhang H W, Wang H, Liu G Z. Quadrilateral isoparametric finite elements for plane elastic Cosserat bodies[J]. Acta Mechanica Sinica, 2005, 21(4): 388-394
[143]陈万吉.细观尺度C0和C1理论及有限元分片检验函数[C].中国科学G辑:物理学力学天文学,2009,39(10):1480-1486.
[144] Atkinson A, Leppington F G. The effect of couple stress on the tip of a crack[J]. International Journal of Solids and Structures, 1977, 13:1103-1122.
[145] Xia Z C, Hutchinson J W. Crack tip fields in strain gradient plasticity[J]. Journal of the Mechanics and Physics of Solids, 1996, 44(10): 1621-1648.
[146] Hwang K C, Huang Y. Mechanism-based strain gradient(MSG) plasticity and the associatedasymptotic crack-tip fields[J]. Fracture and Strength of Solids, Pts 1 and 2, 2000, 183(1): 9-18
[147] Jiang H, Huang Y, Zhuang Z, et al. Fracture in mechanism-based strain gradient plasticity[J]. Journal of the Mechanics and Physics of Solids, 2001, 49(5): 979-993.
[148] Radi E. Effects of characteristic material lengths on modeⅢcrack propagation in couple stress elastic-plastic materials[J]. International Journal of Plasticity, 2007, 23(8): 1439-1456.
[149] Chen S H, Wang T C. Finite element solutions for plane strain modeⅠcrack with strain gradient effects[J]. International Journal of Solids and Structures, 2002, 39: 1241-1257.
[150] Muhlhaus H B. Application of Cosserat theory in numerical solutions of limit load problems[J]. Ingenieur-Archiv,1989,59:124-137.
[151] Muhlhaus H B. Stress and couple stress in a layered half plane with surface loading[J]. International Journal for Numerical and Analytical Methods in Geomechanics, 1990, 14(9): 545-563.
[152] de Borst R, Muhlhaus H B. Gradient-dependent plasticity: formulation and algorithmic aspects[J]. International Journal for Numerical Methods in Engineering, 1992, 35(3): 521-539.
[153] Adhikary D P, Dyskin A V, Jewell R J. Numerical modeling of the flexural deformation of foliated rock slopes[J]. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, 1996, 33(6): 595-606.
[154] Voyiadjis G Z, Alsaleh M I, Alshibli K A. Evolving internal length scales in plastic strain localization for granular materials[J]. International Journal of Plasticity, 2005, 21(10): 2000-2024.
[155]陈胜宏,王鸿儒,熊文林.节理岩体的偶应力影响研究[J].水利学报,1990,l(l): 1-8.
[156]佘成学,熊文林,陈胜宏.具有弯曲效应的层状结构岩体变形的cosserat介质分析方法[J].岩土力学,1994,15(4):12-19.
[157]佘成学,熊文林,陈胜宏.层状岩体的弹粘塑性Cosserat介质理论及其工程应用[J].水利学报,1996,4:10-17.
[158] Shu J Y, Barlow C Y. Strain gradient effects on microscopic strain fields in a metal matrix composite[J]. International Journal of Plasticity, 2000, 16(5): 563-591.
[159] Liu X N, Hu G K. A continuum micromechanical theory of overall plasticity for particulate composites including particle size effects[J]. International Journal of Plasticity, 2005, 21(4): 777-799.
[160] Mori T ,Tanaka K. Average stress in matrix and average elastic energy of materials with misfitting inclusions[J]. Acta Metallurgica et Materialia, 1973, 21(5): 571-574.
[161] Qu S, Siegmund T, Huang Y, et al. A study of particle size effect and interface fracture inaluminum alloy composite via an extended conventional theory of mechanism-based strain-gradient plasticity[J]. Composites Science and Technology, 2005, 65(7-8): 1244-1253.
[162] Stolken J S, Evans A G. A microbend test method for measuring the plasticity length scale[R]. Harvard University Report Mech 320, Division of Engineering and Applied Sciences, Harvard University, Cambridge, 1997.
[163] Aifantis E C. Strain gradient interpretation of size effects[J]. International Journal of Fracture, 1999, 95(l): 299- 314.
[164] Wang W, Huang Y, Hsia K J. A study of micro-bend test by strain gradient Plasticity[J]. International Journal of plasticity, 2003, 19(3): 365-382.
[165] Anthoine A. Effeet of couple stresses on the elastic bending of beams[J]. International Journal of Solids and structures, 2000, 37(7): 1003-1018.
[166] Park S K, X-L Gao. Bernoulli-Euler Beam Model Based on a Modified Couple Stress Theory[J]. Journal of Micromechanics and Microengineering. 2006, 16(11): 2355-2359.
[167] Park S K, X-L Gao. Variational Formulation of a Modified Couple Stress Theory and Its Application to a Simple Shear Problem[J]. Z.angew Math. Phys, 2008, 59(5): 904-917.
[168] Ma H M, X-L Gao, Reddy J N. A Microstructure-Dependent Timoshenko Beam Model Based on a Modified Couple Stress Theory[J]. Journal of the Mechanics and Physics of Solids, 2008, 56(12): 3379-3391.
[169] Chen S H, Wang T C. Strain gradient theory with couple stress for Crystalline solids[J]. European Journal of Mechanics A/Solids, 2001, 24(5): 1497-1505.
[170] Hwang K C, Jiang H, Huang Y. A finite deformation theory of strain gradient Plasticity[J]. Journal of the Mechanics and physics of Solids, 2002, 50(l):81-99.
[171] Hwang K C, Jiang H, Huang Y. Finite deformation analysis of mechanism-based strain gradient Plasticity: torsion and crack tip field[J]. International Journal of Plasticity, 2003 19(2): 235-251.
[172] Wei Y G, Wang X Z, Zhao M H. Size effect and geometrical effects of solids in micro-indentation test[J]. Acta Mechanica Sinica, 2003, 19(1):59-70.
[173] Qiu X, Huang Y, Wei Y. The flow theory of mechanism-based strain gradient plasticity Mechanics of Materials, 2003, 35(3-6): 245-258.
[174]章定国,吴胜宝,康新.考虑尺度效应的微梁刚柔耦合动力学分析[J].固体力学学报, 2010, 31(1): 32-39.
[175]徐芝纶.弹性力学(第4版)[M].北京:高等教育出版社,2006.
[176] Ottosen NS, Ristinmaa M, Ljung C. Rayleigh waves obtained by the indeterminate couple-stress theory[J]. Eur. J. Mech. A/Solids, 2000, 19 (6): 929–947.
[177] Truesdell C, Noll W. The Non-linear Field Theories of Mechanics[M]. Handbuch der physic,(S Flugge ed) III/3. Berlin: Springer-verlag, 1965.
[178] J H Park, H Y Park, S Y Jeong. Linear vibration analysis of rotating wind-turbine blade. Current Applied Physics, 2010, 10(2): 332–334.
[179] B Palais, R Palais. Euler’s fixed point theorem: the axis of a rotation[J]. J. Fixed Point Theory and Applications, 2007, 2: 215-220.
[180] Oskar W, Richard S. Representation of geometric stiffening in multibody system simulation[J]. International Journal for Numerical Methods in Engineering, 1991, 32(8):1833-1850.