横向约束下FGM梁板的热过屈曲研究
|
摘要
功能梯度材料(FGM)由于能够消除传统复合材料中的界面效应和缓解热应力,在现代工程结构中,特别是在高温环境服役的结构中具有重要的应用。因此,FGM结构在热载荷作用下的力学行为研究已成为固体力学新的研究方向。约束屈曲也称为限制失稳,是指构件的屈曲变形由于受到某种限制性约束的作用而不能自由发展的屈曲,这类问题在实际工程中也有大量应用,也是固体力学中的一个经典问题。本论文主要研究FGM梁板在横向约束下的热过屈曲响应,内容包括以下几个方面：
1.FGM圆板在横向点间隙约束下的热过屈曲
考虑周边不可移夹紧的FGM圆板,其圆心处的横向两侧各有一刚性点约束,这些点约束与圆板间存在着微小间隙。假设圆板的材料性质和升温均只沿厚度方向变化。由升温引起的径向压力超过临界值后,板将产生热过屈曲变形。本文重点研究板与点间隙约束接触前后的热过屈曲响应的变化情况。首先,基于von Karman薄板理论,建立FGM圆板受到点间隙约束作用前后的轴对称热过屈曲控制方程。该方程组为非线性常微分方程组,是以中面位移为基本未知量,且包括温度载荷参数,相应的边界条件在接触前后发生改变。然后,采用打靶法数值求解控制方程,得到了FGM圆板与点约束接触前后的变形和内力的变化情况,分析了材料的非均匀性和温度的非均匀性对热过屈曲响应的影响,给出了有关的平衡构形和平衡路径,并将采用混合律模型和Mori-Tanaka模型计算的结果进行了比较。作为特例,考虑了中心固联有刚性圆盘的圆／环板的情况,分析了外边缘分别为不可移简支和夹紧边界条件下刚性圆盘尺寸对热过屈曲响应的影响。
2.非线性弹性地基上FGM Timoshenko梁的热过屈曲
研究了置于非线性弹性地基上功能梯度Timoshenko梁的热过屈曲问题。基于精确的几何非线性理论,推导了梁在非均匀升温下的热过屈曲控制方程,分别考虑两端不可移夹紧和简支两种边界条件。从而将研究问题归结为包含七个基本未知函数的非线性一阶常微分方程组的两点边值问题。采用打靶法获得了该边值问题的数值解。临界屈曲温度与地基刚度参数有关,对应的临界屈曲模态会随着地基刚度参数的增加发生跃迁,文中系统分析了临界屈曲模态跃迁问题,给出了临界屈曲模态跃迁点对应的地基刚度跃迁值及其对应的临界温度载荷；给出了热过屈曲的平衡路径和平衡构形,分析了材料和升温的非均匀性、地基刚度参数、边界条件、长细比和剪切变形等对热过屈曲响应的影响。将退化后的均质梁的模态跃迁结果与已有文献进行了比较,结果吻合良好。
3.FGM梁在横向点间隙约束下的热过屈曲
在前述研究的基础上,基于一维稳态温度场,推导了横向有刚性点间隙约束的功能梯度Euler-Bernoulli梁的热过屈曲控制方程。考虑边界条件为两端不可移夹紧,假设点约束在梁中点的横向两侧并与梁间有微小的间隙,材料性质沿厚度方向按照幂函数连续变化。推导出的控制方程为包含七个基本未知函数的非线性一阶常微分方程组,但梁与点约束接触后相应的边界条件由于点约束反力的出现而与接触前有所不同。采用打靶法数值进行求解,得到了FGM梁与点约束接触前、后的热过屈曲响应,给出了点约束反力随着温度载荷的变化曲线,讨论了材料的梯度变化、梁上下表面升温比、长细比和点间隙位置等对梁的构形、内力和点约束反力的影响。
Functionally graded materials (FGM) are widely applied in modern engineering structures, especially in high temperature environment, because they can eliminate the interfaces exiting in composite materials and reduce the thermal stress concentration in the high temterature gradient enviromemt. The researches on the mechanical behaviors of FGM structures under thermal loadings have been a new research area in solid mechanics. Constrained buckling, which is also called confined buckling, means that the bucking configurations of structures can not evolve freely due to the existence of the constraint. This type of problems also has many applications in the practical engineering and is a classic problem of solid mechanics, In this thesis, thermal post-buckling response of functionally graded beams and plates subjected to transverse constraints is studied including the following parts:
1. Thermal post-buckling of FGM circular plates subjected to transverse point-constraint.
The FGM circular plates with immovably clamped edges are bilaterally constrained by rigid point-constraints at the center with clearance. It is assumed that material properties and non-uniform temperature rise change only through the thickness direction. The plates will go into thermal post-buckling when the radial pressure caused by temperature rise exceeds the critical value. The focus of this thesis is the changes of thermal post-bucking responses during the contact. Firstly, based on von Karman's nonlinear thin plate theory, governing equations of axisymmetric thermal post-buckling of constrained FGM circular plates are derived, which including parameter of thermal loading are nonlinear ordinary differential equations in terms of the displacements of the middle plane, with boundary conditions changing when the plates contact the point-constraints. Then, by using the shooting method, thermal post-buckling responses of the FGM circular plates are obtained before and after it contact the point-space constraint. The changes in the characteristics of the deformation and the internal forces of FGM plates are discussed. The effects of gradients of material properties and non-uniform temperature rise parameters on the thermal post-buckling behaviors of the plates are also examined. The equilibrium configurations and equilibrium paths are given. The results according to the linear mixture rule and Mori-Tanaka model are compared. As an example, the behaviors of the circular/annular plates with a centric rigid mass are analyzed. The effect of the size of the centric rigid circular plate on the constrained post-buckling response is studied for the plates with both immovably clamped and simply supported edges.
2. Thermal post-buckling of FGM Timoshenko beams on nonlinear elastic foundations.
Based on the accurate geometrically nonlinear theory for Timoshenko beams, thermal post-buckling of FGM Timoshenko beams on nonlinear elastic foundations is studied. Governing equations of this problem consist of a two-point boundary value problem of nonlinear ordinary differential equations including seven basic unknown functions with two types of boundary conditions, namely; immovably clamped and simply supported ends. The numerical solutions are arrived at by the shooting method. The critical buckling temperatures vary with the foundation stiffness and the corresponding critical buckling modes transit with the increasing foundation stiffness. Stability boundaries, or critical load curves, are presented for different values of the power exponent and boundary conditions. Post-buckling equilibrium configurations and equilibrium paths are demonstrated for different values of nonlinear elastic foundation parameters, the power exponent, boundary conditions, non-uniform temperature rise, and slenderness ratio. The results of Timoshenko beams are compared with Euler-Bernoulli beams to examine the effect of the shearing deformation. The results of uniform beams are compared with those in the literature, which show a good agreement.
3. Thermal post-buckling of FGM beams subjected to transverse point-constraint.
On the basis of the aforementioned study, governing equations for large post-buckling deformation of FGM beams with point-constraint are formulated. One dimensional steady state temperature field is considered. It is assumed that the material properties of the beam vary continuously as a power function of the thickness coordinate and the point-constraint is at the side of the middle point of the beam and the space value is in the range of thermal post-buckling deformation. The derived governing equations are nonlinear ordinary differential equations including seven basic unknown functions, with boundary conditions changing due to the appearance of point constraint force. By using shooting method to solve the nonlinear bounadary value promlem numerically, thermal post-buckling responses of the FGM beam are obtained when it contacts the point constraint. Curves of the point constraint force changing with the thermal load are plotted. The effects of parameters of the gradients of material properties, non-uniform temperature rise, slenderness ratio and the space value on the thermal post-buckling behaviors of FGM beams are also examined.
1.FGM圆板在横向点间隙约束下的热过屈曲
考虑周边不可移夹紧的FGM圆板,其圆心处的横向两侧各有一刚性点约束,这些点约束与圆板间存在着微小间隙。假设圆板的材料性质和升温均只沿厚度方向变化。由升温引起的径向压力超过临界值后,板将产生热过屈曲变形。本文重点研究板与点间隙约束接触前后的热过屈曲响应的变化情况。首先,基于von Karman薄板理论,建立FGM圆板受到点间隙约束作用前后的轴对称热过屈曲控制方程。该方程组为非线性常微分方程组,是以中面位移为基本未知量,且包括温度载荷参数,相应的边界条件在接触前后发生改变。然后,采用打靶法数值求解控制方程,得到了FGM圆板与点约束接触前后的变形和内力的变化情况,分析了材料的非均匀性和温度的非均匀性对热过屈曲响应的影响,给出了有关的平衡构形和平衡路径,并将采用混合律模型和Mori-Tanaka模型计算的结果进行了比较。作为特例,考虑了中心固联有刚性圆盘的圆／环板的情况,分析了外边缘分别为不可移简支和夹紧边界条件下刚性圆盘尺寸对热过屈曲响应的影响。
2.非线性弹性地基上FGM Timoshenko梁的热过屈曲
研究了置于非线性弹性地基上功能梯度Timoshenko梁的热过屈曲问题。基于精确的几何非线性理论,推导了梁在非均匀升温下的热过屈曲控制方程,分别考虑两端不可移夹紧和简支两种边界条件。从而将研究问题归结为包含七个基本未知函数的非线性一阶常微分方程组的两点边值问题。采用打靶法获得了该边值问题的数值解。临界屈曲温度与地基刚度参数有关,对应的临界屈曲模态会随着地基刚度参数的增加发生跃迁,文中系统分析了临界屈曲模态跃迁问题,给出了临界屈曲模态跃迁点对应的地基刚度跃迁值及其对应的临界温度载荷；给出了热过屈曲的平衡路径和平衡构形,分析了材料和升温的非均匀性、地基刚度参数、边界条件、长细比和剪切变形等对热过屈曲响应的影响。将退化后的均质梁的模态跃迁结果与已有文献进行了比较,结果吻合良好。
3.FGM梁在横向点间隙约束下的热过屈曲
在前述研究的基础上,基于一维稳态温度场,推导了横向有刚性点间隙约束的功能梯度Euler-Bernoulli梁的热过屈曲控制方程。考虑边界条件为两端不可移夹紧,假设点约束在梁中点的横向两侧并与梁间有微小的间隙,材料性质沿厚度方向按照幂函数连续变化。推导出的控制方程为包含七个基本未知函数的非线性一阶常微分方程组,但梁与点约束接触后相应的边界条件由于点约束反力的出现而与接触前有所不同。采用打靶法数值进行求解,得到了FGM梁与点约束接触前、后的热过屈曲响应,给出了点约束反力随着温度载荷的变化曲线,讨论了材料的梯度变化、梁上下表面升温比、长细比和点间隙位置等对梁的构形、内力和点约束反力的影响。
Functionally graded materials (FGM) are widely applied in modern engineering structures, especially in high temperature environment, because they can eliminate the interfaces exiting in composite materials and reduce the thermal stress concentration in the high temterature gradient enviromemt. The researches on the mechanical behaviors of FGM structures under thermal loadings have been a new research area in solid mechanics. Constrained buckling, which is also called confined buckling, means that the bucking configurations of structures can not evolve freely due to the existence of the constraint. This type of problems also has many applications in the practical engineering and is a classic problem of solid mechanics, In this thesis, thermal post-buckling response of functionally graded beams and plates subjected to transverse constraints is studied including the following parts:
1. Thermal post-buckling of FGM circular plates subjected to transverse point-constraint.
The FGM circular plates with immovably clamped edges are bilaterally constrained by rigid point-constraints at the center with clearance. It is assumed that material properties and non-uniform temperature rise change only through the thickness direction. The plates will go into thermal post-buckling when the radial pressure caused by temperature rise exceeds the critical value. The focus of this thesis is the changes of thermal post-bucking responses during the contact. Firstly, based on von Karman's nonlinear thin plate theory, governing equations of axisymmetric thermal post-buckling of constrained FGM circular plates are derived, which including parameter of thermal loading are nonlinear ordinary differential equations in terms of the displacements of the middle plane, with boundary conditions changing when the plates contact the point-constraints. Then, by using the shooting method, thermal post-buckling responses of the FGM circular plates are obtained before and after it contact the point-space constraint. The changes in the characteristics of the deformation and the internal forces of FGM plates are discussed. The effects of gradients of material properties and non-uniform temperature rise parameters on the thermal post-buckling behaviors of the plates are also examined. The equilibrium configurations and equilibrium paths are given. The results according to the linear mixture rule and Mori-Tanaka model are compared. As an example, the behaviors of the circular/annular plates with a centric rigid mass are analyzed. The effect of the size of the centric rigid circular plate on the constrained post-buckling response is studied for the plates with both immovably clamped and simply supported edges.
2. Thermal post-buckling of FGM Timoshenko beams on nonlinear elastic foundations.
Based on the accurate geometrically nonlinear theory for Timoshenko beams, thermal post-buckling of FGM Timoshenko beams on nonlinear elastic foundations is studied. Governing equations of this problem consist of a two-point boundary value problem of nonlinear ordinary differential equations including seven basic unknown functions with two types of boundary conditions, namely; immovably clamped and simply supported ends. The numerical solutions are arrived at by the shooting method. The critical buckling temperatures vary with the foundation stiffness and the corresponding critical buckling modes transit with the increasing foundation stiffness. Stability boundaries, or critical load curves, are presented for different values of the power exponent and boundary conditions. Post-buckling equilibrium configurations and equilibrium paths are demonstrated for different values of nonlinear elastic foundation parameters, the power exponent, boundary conditions, non-uniform temperature rise, and slenderness ratio. The results of Timoshenko beams are compared with Euler-Bernoulli beams to examine the effect of the shearing deformation. The results of uniform beams are compared with those in the literature, which show a good agreement.
3. Thermal post-buckling of FGM beams subjected to transverse point-constraint.
On the basis of the aforementioned study, governing equations for large post-buckling deformation of FGM beams with point-constraint are formulated. One dimensional steady state temperature field is considered. It is assumed that the material properties of the beam vary continuously as a power function of the thickness coordinate and the point-constraint is at the side of the middle point of the beam and the space value is in the range of thermal post-buckling deformation. The derived governing equations are nonlinear ordinary differential equations including seven basic unknown functions, with boundary conditions changing due to the appearance of point constraint force. By using shooting method to solve the nonlinear bounadary value promlem numerically, thermal post-buckling responses of the FGM beam are obtained when it contacts the point constraint. Curves of the point constraint force changing with the thermal load are plotted. The effects of parameters of the gradients of material properties, non-uniform temperature rise, slenderness ratio and the space value on the thermal post-buckling behaviors of FGM beams are also examined.
引文
[1]王保林,杜善义,韩杰才.功能梯度材料的热/机械耦合分析研究进展[J].力学进展,1999,29(4)：528-548.
[2]沈惠申.功能梯度复合材料板壳结构的弯曲、屈曲和振动[J].力学进展,2004,34(1)：53-60.
[3]Birman V, Byrd L W. Modeling and analysis of functionally graded materials and structures[J]. Applied Mechanics Reviews,2007,60(5):195-216.
[4]仲政,吴林志,陈伟球.功能梯度材料与结构的若干力学问题研究进展[J].力学进展,2010,40(5)：528-541.
[5]李华东,朱锡,梅志远,等.功能梯度板壳的力学研究进展[J].材料导报,2012,26(1)：110-118.
[6]郭英涛,任文敏.关于限制失稳的研究进展[J].力学进展,2004,34(1)：41-52.
[7]李世荣,程昌钧.加热弹性杆的热过屈曲分析[J].应用数学和力学,2000,21(2)：12-18.
[8]Vaz M A, Solano R F. Postbuckling analysis of slender elastic rods subjected to uniform thermal loads[J]. Journal of Thermal Stresses,2003,26(9):847-860.
[9]Vaz M A, Solano R F. Thermal postbuckling of slender elastic rods with hinged ends constrained by a linear spring[J]. Journal of Thermal Stresses,2004,27(4):367-380.
[10]Coffin D W, Bloom F. Elastica solution for the hygrothermal buckling of a beam[J]. International Journal of Non-Linear Mechanics,1999,34(5):935-947.
[11]马连生,顾春龙.经典梁热过屈曲问题的解析解[J].应用力学学报,2011,28(4)：372-375.
[12]Rao G V, Raju K K. Thermal postbuckling of uniform columns:a simple intuitive method[J]. AIAA journal,2002,40(10):2138-2140.
[13]Li S R, Zhou Y H, Zheng X J. Thermal post-buckling of a heated elastic rod with pinned-fixed ends[J]. Journal of Thermal Stresses,2002,25(1):45-56.
[14]Li S R, Cheng C J, Zhou Y H. Thermal post-buckling of an elastic beams subjected to a transversely non-uniform temperature rising[J]. Applied Mathematics and Mechanics,2003,24(5):514-520.
[15]李世荣,周凤玺.弹性曲梁在机械和热载荷共同作用下的几何非线性模型及其数值解[J].计算力学学报,2008,25(1)：25-28.
[16]马连生,顾春龙.剪切可变形梁热过屈曲解析解[J].工程力学,2012,2：28.
[17]Li S R, Zhou Y H. Geometrically nonlinear analysis of Timoshenko beams under thermomechanical loadings[J]. Journal of Thermal Stresses,2003,26(9):861-872.
[18]Kiani Y, Eslami M R. Thermal buckling analysis of functionally graded material beams[J]. International Journal of Mechanics and Materials in Design,2010,6(3): 229-238.
[19]Anandrao K S, Gupta R K, Ramchandran P, et al. Thermal post-buckling analysis of uniform slender functionally graded material beams [J]. Structural Engineering and Mechanics,2010,36(5):545-560.
[20]Zhao F Q, Wang Z M, Liu H Z. Thermal post-buckling analyses of functionally graded material rod[J]. Applied Mathematics and Mechanics,2007,28(1):59-67.
[21]马连生,牛牧华.基于物理中面FGM经典梁的非线性力学行为[J].兰州理工大学学报,2010,36(5)：163-167.
[22]Kiani Y, Taheri S, Eslami M R. Thermal buckling of piezoelectric functionally graded material beams[J]. Journal of Thermal Stresses,2011,34(8):835-850.
[23]Kiani Y, Eslami M R. Thermomechanical Buckling of Temperature-dependent FGM Beams[J]. Latin American Journal of Solids and Structures,2013,10(2):223-245.
[24]李世荣,张靖华,赵永刚.功能梯度材料Timoshenko梁的热过屈曲分析[J].应用数学和力学,2006,27(6)：709-715.
[25]Ma L S, Lee D W. Exact solutions for nonlinear static responses of a shear deformable FGM beam under an in-plane thermal loading [J]. European Journal of Mechanics-A/Solids,2012,31(1):13-20.
[26]牛牧华,马连生.基于物理中面FGM梁的非线性力学行为[J].工程力学,2011,28(6)：219-225.
[27]Ma L S, Lee D W. A further discussion of nonlinear mechanical behavior for FGM beams under in-plane thermal loading[J]. Composite Structures,2011,93(2): 831-842.
[28]Fu Y M, Chen Y, Zhang P. Thermal buckling analysis of functionally graded beam with longitudinal crack[J]. Meccanica,2013,48(5):1227-1237.
[29]Kiani Y, Rezaei M, Taheri S, et al. Thermo-electrical buckling of piezoelectric functionally graded material Timoshenko beams [J]. International Journal of Mechanics and Materials in Design,2011,7(3):185-197.
[30]Kargani A, Kiani Y, Eslami M R. Exact solution for nonlinear stability of piezoelectric FGM Timoshenko beams under thermo-electrical loads[J]. Journal of Thermal Stresses,2013,36(10):1056-1076.
[31]苏厚德,李世荣,高颖.粘贴压电层功能梯度材料Timoshenko梁的热过屈曲分析[J].计算力学学报,2010,27(6)：1067-1072.
[32]马连生,徐刚年.FGM梁稳定性分析的DQ法[J].兰州理工大学学报,2011,37(2)：164-167.
[33]Wattanasakulpong N, Gangadhara Prusty B, Kelly D W. Thermal buckling and elastic vibration of third-order shear deformable functionally graded beams[J]. International Journal of Mechanical Sciences,2011,53(9):734-743.
[34]Tauchert T R. Thermally induced flexure, buckling, and vibration of plates [J]. Applied Mechanics Reviews,1991,44(8):347-360.
[35]Thornton E A. Thermal buckling of plates and shells[J]. Applied Mechanics Reviews,1993,46(10):485-506.
[36]郝际平,吴子燕.圆板的热后屈曲分析[J].西北工业大学学报,1996,14(3)：391-396.
[37]Raju K K, Rao G V. Thermal post-buckling of circular plates[J]. Computers & structures,1984,18(6):1179-1182.
[38]李世荣.弹性圆板的热过屈曲[J].固体力学学报,1997,18(2)：179-182.
[39]Li S R, Zhou Y H. Shooting method for non-linear vibration and thermal buckling of heated orthotropic circular plates[J]. Journal of Sound and Vibration,2001,248(2): 379-386.
[40]沈惠申,朱湘赓.中厚板热后屈曲分析[J].应用数学和力学,1995,16(5)：443-450.
[41]Kanaka Raju K, Venkateswara Rao G. Thermal post-buckling of thin simply supported orthotropic square plates[J]. Composite structures,1989,12(2):149-154.
[42]Jha D K, Kant T, Singh R K. A critical review of recent research on functionally graded plates[J].2013,96:833-849.
[43]Najafizadeh M M, Eslami M R. First-order-theory-based thermoelastic stability of functionally graded material circular plates[J]. AIAA journal,2002,40(7): 1444-1450.
[44]Ma L S, Wang T J. Nonlinear bending and post-buckling of a functionally graded circular plate under mechanical and thermal loadings[J]. International Journal of Solids and Structures,2003,40(13):3311-3330.
[45]Li S R, Zhang J H, Zhao Y G. Nonlinear thermomechanical post-buckling of circular FGM plate with geometric imperfection[J]. Thin-Walled Structures,2007,45(5): 528-536.
[46]Najafizadeh M M, Hedayati B. Refined theory for thermoelastic stability of functionally graded circular plates[J]. Journal of thermal stresses,2004,27(9): 857-880.
[47]王铁军,马连生,石朝锋.功能梯度中厚圆/环板轴对称弯曲问题的解析解[J].力学学报,2004,36(3)：348-353.
[48]Sepahi O, Forouzan M R, Malekzadeh P. Thermal buckling and postbuckling analysis of functionally graded annular plates with temperature-dependent material properties[J]. Materials & Design,2011,32(7):4030-4041.
[49]Jalali S K, Naei M H, Poorsolhjouy A. Thermal stability analysis of circular functionally graded sandwich plates of variable thickness using pseudo-spectral method[J]. Materials & Design,2010,31(10):4755-4763.
[50]Prakash T, Ganapathi M. Asymmetric flexural vibration and thermoelastic stability of FGM circular plates using finite element method[J]. Composites Part B: Engineering,2006,37(7):642-649.
[51]Najafizadeh M M, Heydari H R. Thermal buckling of functionally graded circular plates based on higher order shear deformation plate theory[J]. European Journal of Mechanics-A/Solids,2004,23(6):1085-1100.
[52]武兰河,王立彬,刘淑红.四边简支功能梯度矩形板的热屈曲分析[J].工程力学,2004,21(2)：152-156.
[53]Javaheri R, Eslami M R. Thermal buckling of functionally graded plates[J].2002, 40(1):162-169.
[54]Samsam Shariat B A, Eslami M R. Thermal buckling of imperfect functionally graded plates[J]. International Journal of Solids and Structures,2006,43(14-15): 4082-4096.
[55]Morimoto T, Tanigawa Y, Kawamura R. Thermal buckling analysis of inhomogeneous rectangular plate due to uniform heat supply[J]. Journal of thermal stresses,2003,26(11-12):1151-1170.
[56]Ghannadpour S, Ovesy H R, Nassirnia M. Buckling analysis of functionally graded plates under thermal loadings using the finite strip method[J]. Computers & Structures,2012,108-109:93-99.
[57]Lanhe W. Thermal buckling of a simply supported moderately thick rectangular FGM plate[J]. Composite Structures,2004,64(2):211-218.
[58]Bouazza M, Tounsi A, Adda-Bedia E A, et al. Thermoelastic stability analysis of functionally graded plates:An analytical approach[J]. Computational Materials Science,2010,49(4):865-870.
[59]Zhao X, Lee Y Y, Liew K M. Mechanical and thermal buckling analysis of functionally graded plates[J]. Composite Structures,2009,90(2):161-171.
[60]Lee Y Y, Zhao X, Reddy J N. Postbuckling analysis of functionally graded plates subject to compressive and thermal loads[J]. Computer Methods in Applied Mechanics and Engineering,2010,199(25-28):1645-1653.
[61]Javaheri R, Eslami M R. Thermal buckling of functionally graded plates based on higher order theory [J]. Journal of Thermal Stresses,2002,25(7):603-625.
[62]Shariat B A, Eslami M R. Buckling of thick functionally graded plates under mechanical and thermal loads[J]. Composite Structures,2007,78(3):433-439.
[63]Shen H S, Li S R. Postbuckling of sandwich plates with FGM face sheets and temperature-dependent properties[J]. Composites Part B:Engineering,2008,39(2): 332-344.
[64]Mansouri M H, Shariyat M. Thermal buckling predictions of three types of high-order theories for the heterogeneous orthotropic plates, using the new version of DQM[J]. Composite Structures,(0).
[65]Zenkour A M, Sobhy M. Thermal buckling of various types of FGM sandwich plates[J]. Composite Structures,2010,93(1):93-102.
[66]Chen J S, Ro W C. Deformations and stability of an elastica subjected to an off-axis point constraint[J]. Journal of Applied Mechanics,2010,77(3):31006.
[67]郭锐,李世荣,张靖华.点间隙约束下轴向受压弹性梁的过屈曲[J].甘肃科学学报,2010,22(4)：96-99.
[68]Chen J S, Hung S Y. Snapping of an elastica under various loading mechanisms [J]. European Journal of Mechanics-A/Solids,2011,30(4):525-531.
[69]李世荣,郭锐.点间隙约束下弹性梁的湿热后屈曲问题[J].固体力学学报,2011,32(2)：197-202.
[70]Domokos G A B, Holmes P, Royce B. Constrained Euler buckling[J]. Journal of Nonlinear Science,1997,7(3):281-314.
[71]Holmes P, Domokos G, Schmitt J, et al. Constrained Euler buckling:an interplay of computation and analysis[J]. Computer Methods in Applied Mechanics and Engineering,1999,170(3):175-207.
[72]Adan N, Altus E, Sheinman Ⅰ. Post-buckling behavior of beams under contact constraints [J]. Journal of Applied Mechanics,1994,61(4):764-772.
[73]Chai H. The post-buckling response of a bi-laterally constrained column[J]. Journal of the Mechanics and Physics of Solids,1998,46(7):1155-1181.
[74]Plaut R H, Suherman S, Dillard D A, et al. Deflections and buckling of a bent elastica in contact with a flat surface[J]. International Journal of Solids and Structures,1999,36(8):1209-1229.
[75]Ro W C, Chen J S, Hong S Y. Vibration and stability of a constrained elastica with variable length[J]. International Journal of Solids and Structures,2010,47(16): 2143-2154.
[76]Chen J S, Wu H H. Deformation and stability of an elastica under a point force and constrained by a flat surface[J]. International Journal of Mechanical Sciences,2011, 53(1):42-50.
[77]Chen J S. On the contact behavior of a buckled Timoshenko beam constrained laterally by a plane wall[J]. Acta Mechanica,2011,222(3-4):225-232.
[78]武秀根,郑百林,贺鹏飞.限制失稳杆的后屈曲分析[J].同济大学学报(自然科学版),2009,37(1)：26-30.
[79]武秀根,郑百林,贺鹏飞.限制失稳模态方程及约束载荷计算[J].应用力学学报,2009,26(2)：375-378.
[80]尤国萍,吴京.屈曲约束支撑线接触长度的简化计算[J].东南大学学报(自然科学版),2009,39：222-226.
[81]吴京,梁仁杰,王春林,等.屈曲约束支撑核心单元的多波屈曲过程研究[J].工程力学,2012,29(8)：136-142.
[82]Chen J S, Li C W. Planar elastica inside a curved tube with clearance[J]. International Journal of Solids and Structures,2007,44(18-19):6173-6186.
[83]Lu Z H, Chen J S. Deformations of a clamped-clamped elastica inside a circular channel with clearance[J]. International Journal of Solids and Structures,2008,45(9): 2470-2492.
[84]Kuru E, Qiu W, Martinez A, et al. The buckling behavior of pipes and its influence on the axial force transfer in directional wells[J]. Journal of energy resources technology,2000,122(3):129-135.
[85]Liu C, Chen J. Effect of Coulomb friction on the deformation of an elastica constrained in a straight channel with clearance[J]. European Journal of Mechanics-A/Solids,2013,39:50-59.
[86]Liu C, Chen J. Effect of friction on the planar elastica constrained inside a circular channel with clearance[J]. International Journal of Solids and Structures,2013,50(1): 270-278.
[87]Chai H. Contact buckling and postbuckling of thin rectangular plates[J]. Journal of the Mechanics and Physics of Solids,2001,49(2):209-230.
[88]Chai H. On the post-buckling behavior of bilaterally constrained plates[J], International Journal of Solids and Structures,2002,39(11):2911-2926.
[89]Roman B, Pocheau A. Postbuckling of bilaterally constrained rectangular thin plates[J]. Journal of the Mechanics and Physics of Solids,2002,50(11):2379-2401.
[90]申波,邓长根.柔性套管约束下轴心受压杆件的屈曲分析[J].力学与实践,2006,28(5)：43-46.
[91]申波,邓长根.双钢管构件由点接触到线接触的连续过渡[J].工程力学,2007,24(2)：154-160.
[92]申波,邓长根.套管构件中轴压内核与柔性套筒线接触的屈曲[J].工程力学,2007,24(11)：63-69.
[93]杨骁,程昌钧.弹性基础上环形板的屈曲和过屈曲[J].应用数学和力学,1991,12(8)：737-747.
[94]Gupta U S, Lal R, Jain S K. Buckling and vibrations of polar orthotropic circular plates of linearly varying thickness resting on an elastic foundation[J]. Journal of sound and vibration,1991,147(3):423-434.
[95]Dumir P C. Non-linear vibration and postbuckling of isotrophic thin circular plates on elastic foundations[J]. Journal of sound and vibration,1986,107(2):253-263.
[96]李世荣,刘平.弹性地基上Euler-Bernoulli梁的热屈曲模态跃迁特性[J].应用力学学报,2011,28(1)：90-94.
[97]Vaz M A, Nascimento M S, Solano R F. Initial postbuckling of elastic rods subjected to thermal loads and resting on an elastic foundation[J]. Journal of Thermal Stresses, 2007,30(4):381-393.
[98]Fallah A, Aghdam M M. Thermo-mechanical buckling and nonlinear free vibration analysis of functionally graded beams on nonlinear elastic foundation [J]. Composites Part B:Engineering,2012,43(3):1523-1530.
[99]Fallah A, Aghdam M M. Nonlinear free vibration and post-buckling analysis of functionally graded beams on nonlinear elastic foundation [J]. European Journal of Mechanics-A/Solids,2011,30(4):571-583.
[100]Rao G V, Raju K K. Thermal postbuckling of uniform columns on elastic foundation-intuitive solution[J]. Journal of engineering mechanics,2003,129(11): 1351-1354.
[101]Song X, Li S R. Thermal buckling and post-buckling of pinned-fixed Euler-Bernoulli beams on an elastic foundation[J]. Mechanics Research Communications,2007, 34(2):164-171.
[102]Li S R, Batra R C. Thermal buckling and postbuckling of Euler-Bernoulli beams supported on nonlinear elastic foundations[J]. AIAA Journal,2007,45(3):712-720.
[103]Ghiasian S E, Kiani Y, Eslami M R. Dynamic buckling of suddenly heated or compressed FGM beams resting on nonlinear elastic foundation [J]. Composite Structures,2013,106:225-234.
[104]Esfahani S E, Kiani Y, Eslami M R. Non-linear thermal stability analysis of temperature dependent FGM beams supported on non-linear hardening elastic foundations [J]. International Journal of Mechanical Sciences,2013,69:10-20.
[105]李世荣,郁汶山.弹性地基上加热弹性圆板的热过屈曲及临界屈曲模态跃迁[J].工程力学,2007,24(5)：63-66.
[106]Tomota Y, Kuroki K, Mori T, et al. Tensile deformation of two-ductile-phase alloys: Flow curves of a-y Fe-Cr-Ni alloys[J]. Materials Science and Engineering,1976, 24(1):85-94.
[107]Williamson R L, Rabin B H, Drake J T. Finite element analysis of thermal residual stresses at graded ceramic-metal interfaces. Part I. Model description and geometrical effects[J]. Journal of Applied Physics,1993,74(2):1310-1320.
[108]Drake J T, Williamson R L, Rabin B H. Finite element analysis of thermal residual stresses at graded ceramic-metal interfaces. Part II. Interface optimization for residual stress reduction[J]. Journal of Applied Physics,1993,74(2):1321-1326.
[109]Kapuria S, Bhattacharyya M, Kumar A N. Theoretical modeling and experimental validation of thermal response of metal-ceramic functionally graded beams [J]. Journal of Thermal Stresses,2008,31(8):759-787.
[110]Bhattacharyya M, Kapuria S, Kumar A N. On the stress to strain transfer ratio and elastic deflection behavior for Al/SiC functionally graded material[J]. Mechanics of Advanced Materials and Structures,2007,14(4):295-302.
[111]Ferreira A J M, Batra R C, Roque C M C, et al. Static analysis of functionally graded plates using third-order shear deformation theory and a meshless method[J]. Composite Structures,2005,69(4):449-457.
[112]Shen H S. Nonlinear vibration of shear deformable FGM cylindrical shells surrounded by an elastic medium[J]. Composite Structures,2012,94(3):1144-1154.
[113]Lu C F, Chen W Q, Lim C W. Elastic mechanical behavior of nano-scaled FGM films incorporating surface energies[J]. Composites Science and Technology,2009, 69(7):1124-1130.
[114]Hill R. A self-consistent mechanics of composite materials[J]. Journal of the Mechanics and Physics of Solids,1965,13(4):213-222.
[115]Reiter T, Dvorak G J, Tvergaard V. Micromechanical models for graded composite materials[J]. Journal of the Mechanics and Physics of Solids,1997,45(8): 1281-1302.
[116]Gibson L J, Ashby M F, Karam G N, et al. The mechanical properties of natural materials. Ⅱ. Microstructures for mechanical efficiency[J]. Proceedings of the Royal Society of London. Series A:Mathematical and Physical Sciences,1995,450(1938): 141-162.
[117]Mori T, Tanaka K. Average stress in matrix and average elastic energy of materials with misfitting inclusions[J]. Acta metallurgica,1973,21(5):571-574.
[118]何光渝,高永利.Visual Fortran常用数值算法集[M].北京：科学出版社,2002.
[119]Kanaka Raju K, Venkateswara Rao G. Thermal postbuckling of uniform columns on elastic foundation[J]. Journal of engineering mechanics,1993,119(3):626-629.
[120]李世荣.非线性柔韧梁板结构的热过屈曲和振动：[兰州大学博士学位论文][D].兰州：兰州大学,2003.
[121]Wu B S, Zhong H X. Postbuckling and imperfection sensitivity of fixed-end and free-end struts on elastic foundation[J]. Archive of Applied Mechanics,1999,69(7): 491-498.
[122]Fu Y, Wang J, Mao Y. Nonlinear analysis of buckling, free vibration and dynamic stability for the piezoelectric functionally graded beams in thermal environment[J]. Applied Mathematical Modelling,2012,36(9):4324-4340.
[2]沈惠申.功能梯度复合材料板壳结构的弯曲、屈曲和振动[J].力学进展,2004,34(1)：53-60.
[3]Birman V, Byrd L W. Modeling and analysis of functionally graded materials and structures[J]. Applied Mechanics Reviews,2007,60(5):195-216.
[4]仲政,吴林志,陈伟球.功能梯度材料与结构的若干力学问题研究进展[J].力学进展,2010,40(5)：528-541.
[5]李华东,朱锡,梅志远,等.功能梯度板壳的力学研究进展[J].材料导报,2012,26(1)：110-118.
[6]郭英涛,任文敏.关于限制失稳的研究进展[J].力学进展,2004,34(1)：41-52.
[7]李世荣,程昌钧.加热弹性杆的热过屈曲分析[J].应用数学和力学,2000,21(2)：12-18.
[8]Vaz M A, Solano R F. Postbuckling analysis of slender elastic rods subjected to uniform thermal loads[J]. Journal of Thermal Stresses,2003,26(9):847-860.
[9]Vaz M A, Solano R F. Thermal postbuckling of slender elastic rods with hinged ends constrained by a linear spring[J]. Journal of Thermal Stresses,2004,27(4):367-380.
[10]Coffin D W, Bloom F. Elastica solution for the hygrothermal buckling of a beam[J]. International Journal of Non-Linear Mechanics,1999,34(5):935-947.
[11]马连生,顾春龙.经典梁热过屈曲问题的解析解[J].应用力学学报,2011,28(4)：372-375.
[12]Rao G V, Raju K K. Thermal postbuckling of uniform columns:a simple intuitive method[J]. AIAA journal,2002,40(10):2138-2140.
[13]Li S R, Zhou Y H, Zheng X J. Thermal post-buckling of a heated elastic rod with pinned-fixed ends[J]. Journal of Thermal Stresses,2002,25(1):45-56.
[14]Li S R, Cheng C J, Zhou Y H. Thermal post-buckling of an elastic beams subjected to a transversely non-uniform temperature rising[J]. Applied Mathematics and Mechanics,2003,24(5):514-520.
[15]李世荣,周凤玺.弹性曲梁在机械和热载荷共同作用下的几何非线性模型及其数值解[J].计算力学学报,2008,25(1)：25-28.
[16]马连生,顾春龙.剪切可变形梁热过屈曲解析解[J].工程力学,2012,2：28.
[17]Li S R, Zhou Y H. Geometrically nonlinear analysis of Timoshenko beams under thermomechanical loadings[J]. Journal of Thermal Stresses,2003,26(9):861-872.
[18]Kiani Y, Eslami M R. Thermal buckling analysis of functionally graded material beams[J]. International Journal of Mechanics and Materials in Design,2010,6(3): 229-238.
[19]Anandrao K S, Gupta R K, Ramchandran P, et al. Thermal post-buckling analysis of uniform slender functionally graded material beams [J]. Structural Engineering and Mechanics,2010,36(5):545-560.
[20]Zhao F Q, Wang Z M, Liu H Z. Thermal post-buckling analyses of functionally graded material rod[J]. Applied Mathematics and Mechanics,2007,28(1):59-67.
[21]马连生,牛牧华.基于物理中面FGM经典梁的非线性力学行为[J].兰州理工大学学报,2010,36(5)：163-167.
[22]Kiani Y, Taheri S, Eslami M R. Thermal buckling of piezoelectric functionally graded material beams[J]. Journal of Thermal Stresses,2011,34(8):835-850.
[23]Kiani Y, Eslami M R. Thermomechanical Buckling of Temperature-dependent FGM Beams[J]. Latin American Journal of Solids and Structures,2013,10(2):223-245.
[24]李世荣,张靖华,赵永刚.功能梯度材料Timoshenko梁的热过屈曲分析[J].应用数学和力学,2006,27(6)：709-715.
[25]Ma L S, Lee D W. Exact solutions for nonlinear static responses of a shear deformable FGM beam under an in-plane thermal loading [J]. European Journal of Mechanics-A/Solids,2012,31(1):13-20.
[26]牛牧华,马连生.基于物理中面FGM梁的非线性力学行为[J].工程力学,2011,28(6)：219-225.
[27]Ma L S, Lee D W. A further discussion of nonlinear mechanical behavior for FGM beams under in-plane thermal loading[J]. Composite Structures,2011,93(2): 831-842.
[28]Fu Y M, Chen Y, Zhang P. Thermal buckling analysis of functionally graded beam with longitudinal crack[J]. Meccanica,2013,48(5):1227-1237.
[29]Kiani Y, Rezaei M, Taheri S, et al. Thermo-electrical buckling of piezoelectric functionally graded material Timoshenko beams [J]. International Journal of Mechanics and Materials in Design,2011,7(3):185-197.
[30]Kargani A, Kiani Y, Eslami M R. Exact solution for nonlinear stability of piezoelectric FGM Timoshenko beams under thermo-electrical loads[J]. Journal of Thermal Stresses,2013,36(10):1056-1076.
[31]苏厚德,李世荣,高颖.粘贴压电层功能梯度材料Timoshenko梁的热过屈曲分析[J].计算力学学报,2010,27(6)：1067-1072.
[32]马连生,徐刚年.FGM梁稳定性分析的DQ法[J].兰州理工大学学报,2011,37(2)：164-167.
[33]Wattanasakulpong N, Gangadhara Prusty B, Kelly D W. Thermal buckling and elastic vibration of third-order shear deformable functionally graded beams[J]. International Journal of Mechanical Sciences,2011,53(9):734-743.
[34]Tauchert T R. Thermally induced flexure, buckling, and vibration of plates [J]. Applied Mechanics Reviews,1991,44(8):347-360.
[35]Thornton E A. Thermal buckling of plates and shells[J]. Applied Mechanics Reviews,1993,46(10):485-506.
[36]郝际平,吴子燕.圆板的热后屈曲分析[J].西北工业大学学报,1996,14(3)：391-396.
[37]Raju K K, Rao G V. Thermal post-buckling of circular plates[J]. Computers & structures,1984,18(6):1179-1182.
[38]李世荣.弹性圆板的热过屈曲[J].固体力学学报,1997,18(2)：179-182.
[39]Li S R, Zhou Y H. Shooting method for non-linear vibration and thermal buckling of heated orthotropic circular plates[J]. Journal of Sound and Vibration,2001,248(2): 379-386.
[40]沈惠申,朱湘赓.中厚板热后屈曲分析[J].应用数学和力学,1995,16(5)：443-450.
[41]Kanaka Raju K, Venkateswara Rao G. Thermal post-buckling of thin simply supported orthotropic square plates[J]. Composite structures,1989,12(2):149-154.
[42]Jha D K, Kant T, Singh R K. A critical review of recent research on functionally graded plates[J].2013,96:833-849.
[43]Najafizadeh M M, Eslami M R. First-order-theory-based thermoelastic stability of functionally graded material circular plates[J]. AIAA journal,2002,40(7): 1444-1450.
[44]Ma L S, Wang T J. Nonlinear bending and post-buckling of a functionally graded circular plate under mechanical and thermal loadings[J]. International Journal of Solids and Structures,2003,40(13):3311-3330.
[45]Li S R, Zhang J H, Zhao Y G. Nonlinear thermomechanical post-buckling of circular FGM plate with geometric imperfection[J]. Thin-Walled Structures,2007,45(5): 528-536.
[46]Najafizadeh M M, Hedayati B. Refined theory for thermoelastic stability of functionally graded circular plates[J]. Journal of thermal stresses,2004,27(9): 857-880.
[47]王铁军,马连生,石朝锋.功能梯度中厚圆/环板轴对称弯曲问题的解析解[J].力学学报,2004,36(3)：348-353.
[48]Sepahi O, Forouzan M R, Malekzadeh P. Thermal buckling and postbuckling analysis of functionally graded annular plates with temperature-dependent material properties[J]. Materials & Design,2011,32(7):4030-4041.
[49]Jalali S K, Naei M H, Poorsolhjouy A. Thermal stability analysis of circular functionally graded sandwich plates of variable thickness using pseudo-spectral method[J]. Materials & Design,2010,31(10):4755-4763.
[50]Prakash T, Ganapathi M. Asymmetric flexural vibration and thermoelastic stability of FGM circular plates using finite element method[J]. Composites Part B: Engineering,2006,37(7):642-649.
[51]Najafizadeh M M, Heydari H R. Thermal buckling of functionally graded circular plates based on higher order shear deformation plate theory[J]. European Journal of Mechanics-A/Solids,2004,23(6):1085-1100.
[52]武兰河,王立彬,刘淑红.四边简支功能梯度矩形板的热屈曲分析[J].工程力学,2004,21(2)：152-156.
[53]Javaheri R, Eslami M R. Thermal buckling of functionally graded plates[J].2002, 40(1):162-169.
[54]Samsam Shariat B A, Eslami M R. Thermal buckling of imperfect functionally graded plates[J]. International Journal of Solids and Structures,2006,43(14-15): 4082-4096.
[55]Morimoto T, Tanigawa Y, Kawamura R. Thermal buckling analysis of inhomogeneous rectangular plate due to uniform heat supply[J]. Journal of thermal stresses,2003,26(11-12):1151-1170.
[56]Ghannadpour S, Ovesy H R, Nassirnia M. Buckling analysis of functionally graded plates under thermal loadings using the finite strip method[J]. Computers & Structures,2012,108-109:93-99.
[57]Lanhe W. Thermal buckling of a simply supported moderately thick rectangular FGM plate[J]. Composite Structures,2004,64(2):211-218.
[58]Bouazza M, Tounsi A, Adda-Bedia E A, et al. Thermoelastic stability analysis of functionally graded plates:An analytical approach[J]. Computational Materials Science,2010,49(4):865-870.
[59]Zhao X, Lee Y Y, Liew K M. Mechanical and thermal buckling analysis of functionally graded plates[J]. Composite Structures,2009,90(2):161-171.
[60]Lee Y Y, Zhao X, Reddy J N. Postbuckling analysis of functionally graded plates subject to compressive and thermal loads[J]. Computer Methods in Applied Mechanics and Engineering,2010,199(25-28):1645-1653.
[61]Javaheri R, Eslami M R. Thermal buckling of functionally graded plates based on higher order theory [J]. Journal of Thermal Stresses,2002,25(7):603-625.
[62]Shariat B A, Eslami M R. Buckling of thick functionally graded plates under mechanical and thermal loads[J]. Composite Structures,2007,78(3):433-439.
[63]Shen H S, Li S R. Postbuckling of sandwich plates with FGM face sheets and temperature-dependent properties[J]. Composites Part B:Engineering,2008,39(2): 332-344.
[64]Mansouri M H, Shariyat M. Thermal buckling predictions of three types of high-order theories for the heterogeneous orthotropic plates, using the new version of DQM[J]. Composite Structures,(0).
[65]Zenkour A M, Sobhy M. Thermal buckling of various types of FGM sandwich plates[J]. Composite Structures,2010,93(1):93-102.
[66]Chen J S, Ro W C. Deformations and stability of an elastica subjected to an off-axis point constraint[J]. Journal of Applied Mechanics,2010,77(3):31006.
[67]郭锐,李世荣,张靖华.点间隙约束下轴向受压弹性梁的过屈曲[J].甘肃科学学报,2010,22(4)：96-99.
[68]Chen J S, Hung S Y. Snapping of an elastica under various loading mechanisms [J]. European Journal of Mechanics-A/Solids,2011,30(4):525-531.
[69]李世荣,郭锐.点间隙约束下弹性梁的湿热后屈曲问题[J].固体力学学报,2011,32(2)：197-202.
[70]Domokos G A B, Holmes P, Royce B. Constrained Euler buckling[J]. Journal of Nonlinear Science,1997,7(3):281-314.
[71]Holmes P, Domokos G, Schmitt J, et al. Constrained Euler buckling:an interplay of computation and analysis[J]. Computer Methods in Applied Mechanics and Engineering,1999,170(3):175-207.
[72]Adan N, Altus E, Sheinman Ⅰ. Post-buckling behavior of beams under contact constraints [J]. Journal of Applied Mechanics,1994,61(4):764-772.
[73]Chai H. The post-buckling response of a bi-laterally constrained column[J]. Journal of the Mechanics and Physics of Solids,1998,46(7):1155-1181.
[74]Plaut R H, Suherman S, Dillard D A, et al. Deflections and buckling of a bent elastica in contact with a flat surface[J]. International Journal of Solids and Structures,1999,36(8):1209-1229.
[75]Ro W C, Chen J S, Hong S Y. Vibration and stability of a constrained elastica with variable length[J]. International Journal of Solids and Structures,2010,47(16): 2143-2154.
[76]Chen J S, Wu H H. Deformation and stability of an elastica under a point force and constrained by a flat surface[J]. International Journal of Mechanical Sciences,2011, 53(1):42-50.
[77]Chen J S. On the contact behavior of a buckled Timoshenko beam constrained laterally by a plane wall[J]. Acta Mechanica,2011,222(3-4):225-232.
[78]武秀根,郑百林,贺鹏飞.限制失稳杆的后屈曲分析[J].同济大学学报(自然科学版),2009,37(1)：26-30.
[79]武秀根,郑百林,贺鹏飞.限制失稳模态方程及约束载荷计算[J].应用力学学报,2009,26(2)：375-378.
[80]尤国萍,吴京.屈曲约束支撑线接触长度的简化计算[J].东南大学学报(自然科学版),2009,39：222-226.
[81]吴京,梁仁杰,王春林,等.屈曲约束支撑核心单元的多波屈曲过程研究[J].工程力学,2012,29(8)：136-142.
[82]Chen J S, Li C W. Planar elastica inside a curved tube with clearance[J]. International Journal of Solids and Structures,2007,44(18-19):6173-6186.
[83]Lu Z H, Chen J S. Deformations of a clamped-clamped elastica inside a circular channel with clearance[J]. International Journal of Solids and Structures,2008,45(9): 2470-2492.
[84]Kuru E, Qiu W, Martinez A, et al. The buckling behavior of pipes and its influence on the axial force transfer in directional wells[J]. Journal of energy resources technology,2000,122(3):129-135.
[85]Liu C, Chen J. Effect of Coulomb friction on the deformation of an elastica constrained in a straight channel with clearance[J]. European Journal of Mechanics-A/Solids,2013,39:50-59.
[86]Liu C, Chen J. Effect of friction on the planar elastica constrained inside a circular channel with clearance[J]. International Journal of Solids and Structures,2013,50(1): 270-278.
[87]Chai H. Contact buckling and postbuckling of thin rectangular plates[J]. Journal of the Mechanics and Physics of Solids,2001,49(2):209-230.
[88]Chai H. On the post-buckling behavior of bilaterally constrained plates[J], International Journal of Solids and Structures,2002,39(11):2911-2926.
[89]Roman B, Pocheau A. Postbuckling of bilaterally constrained rectangular thin plates[J]. Journal of the Mechanics and Physics of Solids,2002,50(11):2379-2401.
[90]申波,邓长根.柔性套管约束下轴心受压杆件的屈曲分析[J].力学与实践,2006,28(5)：43-46.
[91]申波,邓长根.双钢管构件由点接触到线接触的连续过渡[J].工程力学,2007,24(2)：154-160.
[92]申波,邓长根.套管构件中轴压内核与柔性套筒线接触的屈曲[J].工程力学,2007,24(11)：63-69.
[93]杨骁,程昌钧.弹性基础上环形板的屈曲和过屈曲[J].应用数学和力学,1991,12(8)：737-747.
[94]Gupta U S, Lal R, Jain S K. Buckling and vibrations of polar orthotropic circular plates of linearly varying thickness resting on an elastic foundation[J]. Journal of sound and vibration,1991,147(3):423-434.
[95]Dumir P C. Non-linear vibration and postbuckling of isotrophic thin circular plates on elastic foundations[J]. Journal of sound and vibration,1986,107(2):253-263.
[96]李世荣,刘平.弹性地基上Euler-Bernoulli梁的热屈曲模态跃迁特性[J].应用力学学报,2011,28(1)：90-94.
[97]Vaz M A, Nascimento M S, Solano R F. Initial postbuckling of elastic rods subjected to thermal loads and resting on an elastic foundation[J]. Journal of Thermal Stresses, 2007,30(4):381-393.
[98]Fallah A, Aghdam M M. Thermo-mechanical buckling and nonlinear free vibration analysis of functionally graded beams on nonlinear elastic foundation [J]. Composites Part B:Engineering,2012,43(3):1523-1530.
[99]Fallah A, Aghdam M M. Nonlinear free vibration and post-buckling analysis of functionally graded beams on nonlinear elastic foundation [J]. European Journal of Mechanics-A/Solids,2011,30(4):571-583.
[100]Rao G V, Raju K K. Thermal postbuckling of uniform columns on elastic foundation-intuitive solution[J]. Journal of engineering mechanics,2003,129(11): 1351-1354.
[101]Song X, Li S R. Thermal buckling and post-buckling of pinned-fixed Euler-Bernoulli beams on an elastic foundation[J]. Mechanics Research Communications,2007, 34(2):164-171.
[102]Li S R, Batra R C. Thermal buckling and postbuckling of Euler-Bernoulli beams supported on nonlinear elastic foundations[J]. AIAA Journal,2007,45(3):712-720.
[103]Ghiasian S E, Kiani Y, Eslami M R. Dynamic buckling of suddenly heated or compressed FGM beams resting on nonlinear elastic foundation [J]. Composite Structures,2013,106:225-234.
[104]Esfahani S E, Kiani Y, Eslami M R. Non-linear thermal stability analysis of temperature dependent FGM beams supported on non-linear hardening elastic foundations [J]. International Journal of Mechanical Sciences,2013,69:10-20.
[105]李世荣,郁汶山.弹性地基上加热弹性圆板的热过屈曲及临界屈曲模态跃迁[J].工程力学,2007,24(5)：63-66.
[106]Tomota Y, Kuroki K, Mori T, et al. Tensile deformation of two-ductile-phase alloys: Flow curves of a-y Fe-Cr-Ni alloys[J]. Materials Science and Engineering,1976, 24(1):85-94.
[107]Williamson R L, Rabin B H, Drake J T. Finite element analysis of thermal residual stresses at graded ceramic-metal interfaces. Part I. Model description and geometrical effects[J]. Journal of Applied Physics,1993,74(2):1310-1320.
[108]Drake J T, Williamson R L, Rabin B H. Finite element analysis of thermal residual stresses at graded ceramic-metal interfaces. Part II. Interface optimization for residual stress reduction[J]. Journal of Applied Physics,1993,74(2):1321-1326.
[109]Kapuria S, Bhattacharyya M, Kumar A N. Theoretical modeling and experimental validation of thermal response of metal-ceramic functionally graded beams [J]. Journal of Thermal Stresses,2008,31(8):759-787.
[110]Bhattacharyya M, Kapuria S, Kumar A N. On the stress to strain transfer ratio and elastic deflection behavior for Al/SiC functionally graded material[J]. Mechanics of Advanced Materials and Structures,2007,14(4):295-302.
[111]Ferreira A J M, Batra R C, Roque C M C, et al. Static analysis of functionally graded plates using third-order shear deformation theory and a meshless method[J]. Composite Structures,2005,69(4):449-457.
[112]Shen H S. Nonlinear vibration of shear deformable FGM cylindrical shells surrounded by an elastic medium[J]. Composite Structures,2012,94(3):1144-1154.
[113]Lu C F, Chen W Q, Lim C W. Elastic mechanical behavior of nano-scaled FGM films incorporating surface energies[J]. Composites Science and Technology,2009, 69(7):1124-1130.
[114]Hill R. A self-consistent mechanics of composite materials[J]. Journal of the Mechanics and Physics of Solids,1965,13(4):213-222.
[115]Reiter T, Dvorak G J, Tvergaard V. Micromechanical models for graded composite materials[J]. Journal of the Mechanics and Physics of Solids,1997,45(8): 1281-1302.
[116]Gibson L J, Ashby M F, Karam G N, et al. The mechanical properties of natural materials. Ⅱ. Microstructures for mechanical efficiency[J]. Proceedings of the Royal Society of London. Series A:Mathematical and Physical Sciences,1995,450(1938): 141-162.
[117]Mori T, Tanaka K. Average stress in matrix and average elastic energy of materials with misfitting inclusions[J]. Acta metallurgica,1973,21(5):571-574.
[118]何光渝,高永利.Visual Fortran常用数值算法集[M].北京：科学出版社,2002.
[119]Kanaka Raju K, Venkateswara Rao G. Thermal postbuckling of uniform columns on elastic foundation[J]. Journal of engineering mechanics,1993,119(3):626-629.
[120]李世荣.非线性柔韧梁板结构的热过屈曲和振动：[兰州大学博士学位论文][D].兰州：兰州大学,2003.
[121]Wu B S, Zhong H X. Postbuckling and imperfection sensitivity of fixed-end and free-end struts on elastic foundation[J]. Archive of Applied Mechanics,1999,69(7): 491-498.
[122]Fu Y, Wang J, Mao Y. Nonlinear analysis of buckling, free vibration and dynamic stability for the piezoelectric functionally graded beams in thermal environment[J]. Applied Mathematical Modelling,2012,36(9):4324-4340.
© 2004-2018 中国地质图书馆版权所有 京ICP备05064691号 京公网安备11010802017129号 地址：北京市海淀区学院路29号 邮编：100083 电话：办公室：(+86 10)66554848；文献借阅、咨询服务、科技查新：66554700 |