基于Reddy简化高阶剪切理论的复合材料对称角铺设矩形板横向弯曲一般解析解
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摘要
采用复数级数法求解基于Reddy简化高阶剪切理论的复合材料对称角铺设矩形板横向弯曲问题。将待定位移函数展开为复数级数,代入该弯曲问题控制偏微分方程组,确定特征根和挠度待定常数与其他位移函数待定常数之间关系式。首次给出了该弯曲问题实数形式的一般解析解。将该一般解析解代入矩形板弯曲边界条件和角点条件,根据正弦级数的正交性建立关于挠度函数待定常数的线性代数方程组,求解此线性代数方程组可确定挠度函数待定常数。建立了该问题解析求解模式。将Reddy高阶剪切理论解析解与经典理论、一阶剪切理论解析解进行对比计算,验证了一般解析解,并给出数值算例。
The complex series method(CSM)is applied to slove the transverse bending problem of symmetrical angle-ply laminated rectangular plate based on Reddy simplified high order shear deformation theory.The underdetermined displacement functions can be expanded into the complex series and these complex series are substituted into the system of partial differential equation which governs this bending problem.The characteristic root and the relationship between the undersolved coefficients of deflection and those of other displacements can be determined.The general analytical solution in real form for this problem is presented for the first time.The general analytical solution can be substituted into the boundary conditions and the corner conditions in the rectangular laminates in transverse bending.The system of linear algebraic equations for the undersolved coefficients of deflection is conducted by the orthogonality of sine series and the undersolved coefficients of deflection can be determined by solving this system of linear algebraic equations.The analytical method for this bending problem is conducted.The computations using the general analytical solutions of this bending problem based on Reddy simplified high order shear deformation theory are carried out to compare between those of classical laminated plate theory and the first-order shear deformation theory.The general analytical solutions based on Reddy simplified high order shear deformation theory are tested and verified.Some numerical results are presented.
The complex series method(CSM)is applied to slove the transverse bending problem of symmetrical angle-ply laminated rectangular plate based on Reddy simplified high order shear deformation theory.The underdetermined displacement functions can be expanded into the complex series and these complex series are substituted into the system of partial differential equation which governs this bending problem.The characteristic root and the relationship between the undersolved coefficients of deflection and those of other displacements can be determined.The general analytical solution in real form for this problem is presented for the first time.The general analytical solution can be substituted into the boundary conditions and the corner conditions in the rectangular laminates in transverse bending.The system of linear algebraic equations for the undersolved coefficients of deflection is conducted by the orthogonality of sine series and the undersolved coefficients of deflection can be determined by solving this system of linear algebraic equations.The analytical method for this bending problem is conducted.The computations using the general analytical solutions of this bending problem based on Reddy simplified high order shear deformation theory are carried out to compare between those of classical laminated plate theory and the first-order shear deformation theory.The general analytical solutions based on Reddy simplified high order shear deformation theory are tested and verified.Some numerical results are presented.
引文
[1]王震鸣.复合材料力学和复合材料结构力学[M].北京:机械工业出版社,1991.
[2]Reddy J N,Khdeir A A,Librescu L.Lévy type solutions for symmetrically laminated rectangular plates using first-order shear deformation theory[J].Journal of Applied Mechanics,1987,54(3):740-742.
[3]韩建强.Reissner厚板的一般解析解及其应用[J].工程力学,1990,7(3):17-27.
[4]王春玲,高典,刘俊卿.横观各向同性弹性半空间地基上四边自由各向异性矩形薄板弯曲解析解[J].力学季刊,2015,36(1):95-104
[5]钟万勰.弹性力学求解新体系[M].大连:大连理工大学出版社,1995.
[6]姚伟岸,钟万勰.辛弹性力学[M].北京:高等教育出版社,2002.
[7]胡文锋,刘一华.含固支边矩形叠层厚板的状态空间新解法[J].力学学报,2015,47(5):762-771.
[8]卿光辉,张小欣.基于状态方程矩形层合板多种边界条件下的解析解[J].应用数学和力学,2015,36(11):1167-1177.
[9]张承宗,杨光松.各向异性板结构横向弯曲一般解析解[J].力学学报,1996,28(4):429-440.
[10]孙卫明,杨光松,张承宗.双参数地基上弹性厚板弯曲的一般解析解[J].工程力学,1999,16(2):71-78.
[11]张承宗.复合材料板壳结构力学求解新体系[D].武汉:海军工程学院,1999.
[12]Zhang C Z.General analytical solution for the bending of anisotropic skew plate with first-order shear deformation theory[J].Mechanics of Advanced Materials and Structures,2003,10(3):161-181.
[13]黄炎,雷勇军,申慧君.各向异性矩形板自由振动的一般解析解[J].应用数学和力学,2006,27(4):411-416.
[14]杨端生,袁端才,黄炎.受压对称迭层矩形板的自由振动分析[J].工程力学,2007,27(4):41-45.
[15]张承宗.复合材料板壳结构力学解析理论[M].北京:国防工业出版社,2009.
[16]张承宗.A new type complex series method for composite structure mechanics and mathematical physics[M].北京:国防工业出版社,2009.
[17]杨端生,黄炎,李广利.各向异性矩形板的剪切屈曲分析[J].应用力学,2012,29(2):220-222.
[18]张承宗.数学物理方法与复数特殊函数[M].北京:中国宇航出版社,2014.
[19]张承宗.Anisotropic mathematical physics and complex special functions[M].北京:中国宇航出版社,2014.
[20]Reddy J N.A simple higher-order theory for laminated composite plates[J].Journal of Applied Mechanics,1984,51(4):745-752.
[21]罗祖道,王震鸣.复合材料力学进展[M].北京:北京大学出版社,1992.
[22]贺丹,杨万里,陈万吉.基于各向异性修正偶应力理论的Reddy型层合板的自由振动[J].固体力学学报,2016,37(2):161-171.
[23]Zhang J W,Hai J W,Feng H.Elasticity(second edition)[M].北京:北京理工大学出版社,2017.
[2]Reddy J N,Khdeir A A,Librescu L.Lévy type solutions for symmetrically laminated rectangular plates using first-order shear deformation theory[J].Journal of Applied Mechanics,1987,54(3):740-742.
[3]韩建强.Reissner厚板的一般解析解及其应用[J].工程力学,1990,7(3):17-27.
[4]王春玲,高典,刘俊卿.横观各向同性弹性半空间地基上四边自由各向异性矩形薄板弯曲解析解[J].力学季刊,2015,36(1):95-104
[5]钟万勰.弹性力学求解新体系[M].大连:大连理工大学出版社,1995.
[6]姚伟岸,钟万勰.辛弹性力学[M].北京:高等教育出版社,2002.
[7]胡文锋,刘一华.含固支边矩形叠层厚板的状态空间新解法[J].力学学报,2015,47(5):762-771.
[8]卿光辉,张小欣.基于状态方程矩形层合板多种边界条件下的解析解[J].应用数学和力学,2015,36(11):1167-1177.
[9]张承宗,杨光松.各向异性板结构横向弯曲一般解析解[J].力学学报,1996,28(4):429-440.
[10]孙卫明,杨光松,张承宗.双参数地基上弹性厚板弯曲的一般解析解[J].工程力学,1999,16(2):71-78.
[11]张承宗.复合材料板壳结构力学求解新体系[D].武汉:海军工程学院,1999.
[12]Zhang C Z.General analytical solution for the bending of anisotropic skew plate with first-order shear deformation theory[J].Mechanics of Advanced Materials and Structures,2003,10(3):161-181.
[13]黄炎,雷勇军,申慧君.各向异性矩形板自由振动的一般解析解[J].应用数学和力学,2006,27(4):411-416.
[14]杨端生,袁端才,黄炎.受压对称迭层矩形板的自由振动分析[J].工程力学,2007,27(4):41-45.
[15]张承宗.复合材料板壳结构力学解析理论[M].北京:国防工业出版社,2009.
[16]张承宗.A new type complex series method for composite structure mechanics and mathematical physics[M].北京:国防工业出版社,2009.
[17]杨端生,黄炎,李广利.各向异性矩形板的剪切屈曲分析[J].应用力学,2012,29(2):220-222.
[18]张承宗.数学物理方法与复数特殊函数[M].北京:中国宇航出版社,2014.
[19]张承宗.Anisotropic mathematical physics and complex special functions[M].北京:中国宇航出版社,2014.
[20]Reddy J N.A simple higher-order theory for laminated composite plates[J].Journal of Applied Mechanics,1984,51(4):745-752.
[21]罗祖道,王震鸣.复合材料力学进展[M].北京:北京大学出版社,1992.
[22]贺丹,杨万里,陈万吉.基于各向异性修正偶应力理论的Reddy型层合板的自由振动[J].固体力学学报,2016,37(2):161-171.
[23]Zhang J W,Hai J W,Feng H.Elasticity(second edition)[M].北京:北京理工大学出版社,2017.