具有高阶剪切效应的弹性和粘弹性板的理论及其静、动力学行为分析
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摘要
本文在有限变形条件下，建立了考虑高阶横向剪切效应的粘弹性板非线性分析的数学模型，建立了相关的微分求积方法；并针对该模型及其退化情形的非线性静、动力学行为展开了比较系统的研究。主要工作如下：
1．基于Reddy高阶横向剪切理论，采用Boltzmann遗传积分本构关系，综合利用Laplace变换及其反变换技术，Boltzmann算子及卷积等手段，建立了有限变形条件下，考虑高阶横向剪切效应粘弹性板非线性分析的数学模型。其控制方程是关于5个广义位移的耦合的非线性积分-偏微分方程组。将微分求积方法推广以便对上述非线性问题进行数值分析。同时，推广Wang和Bert处理边界条件的方法(简称为DQWB途径)来处理这类问题中所包括的高阶矩的边界条件。给出了推广的微分求积算法的一般格式。
2．基于本文建立的考虑高阶横向剪切效应粘弹性板非线性分析的数学模型，给出了正交各向异性层合板非线性静、动力学行为分析的数学模型；利用本文推广的微分求积格式得到了相应的非线性控制方程的DQ形式；推广运用特殊矩阵乘积及解耦技巧，对相应的耦合数值逼近方程进行解耦，在静态情况下，得到了以挠度W为基本变向量的微分求积迭代格式，避免了病态矩阵，极大的简化了非线性计算。在动态情况下，获得了简化的微分求积谐波平衡算法，并对具有不同节点分布及参数的正交各向异性层合板及其退化情形的静态弯曲、自由振动和强迫振动进行了数值模拟。对数值结果进行了收敛性和比较性研究，考察了不同的节点分布对收敛速度的影响。与传统方法相比，本文的算法在适用性、精度、效率等方面都具有相当的优势。同时考察了几何参数、材料参数及横向剪切效应和转动惯性对正交各向异性层合板非线性力学行为的影响。
3．在小变形情况下，应用反逆法，结合通常的笛卡尔内积和卷积双线性形式，构造了一种以卷积形式表示的Gurtin型泛函，建立了具有任意边界条件考虑高阶横向剪切效应粘弹性板的卷积型变分原理。同时，对所建立的高阶剪切效应粘弹性板力学行为分析的数学模型，在空域上采用本文推广的微分求积算法进行简化，时域上进一步将其化为常微分方程，分析了简化系统在阶跃载荷作用下的准静态力学行为，得出了问题的DQ近似解析解；对简化的动力系统进行研究，得到了
上海大学博士学位论文
粘弹性简支板的动态响应。研究了DQ解的收敛性和精确性;表明该方法具有收
敛性好，存贮量小，运算快，易于实现等优点，从而避免了变换法的局限性。并
考察了几何参数、材料参数及横向剪切效应对粘弹性板静、动力学行为的影响。
4.在有限变形条件下，对考虑高阶横向剪切效应粘弹性板的非线性稳定性问
题进行了分析。通过使用Galerkin平均化法，并引进新的变量，将积分一非线性偏
微分控制方程组，转化为高维非线性自治常微分方程组。综合使用非线性动力学
中数值分析方法，对动力系统的响应进行了长时间力学行为的数值模拟，研究了
系统的分岔，混沌等现象;揭示了粘弹性矩形板在周期激励下的丰富的动力学行
为。比较了1一阶和2一阶Galerkin截断系统的动力学行为，发现它们定性上是基本
相同的，但定量性质略有偏差。考察了载荷和材料等参数对粘弹性板动力稳定性
的影响，发现系统的运动关于分岔参数具有很强的敏感性。
5.建立了有限变形条件下，粘弹性基础上考虑高阶横向剪切效应粘弹性板的
动力学行为的数学模型;对模型在空域上应用前述的推广的微分求积格式、特殊
矩阵乘积及简化技巧得到了简洁的矩阵形式的非线性数值逼近公式，时域上进一
步技巧地引进新的变量，并综合利用非线性动力学中的数值分析方法，揭示了粘
弹性基础上粘弹性矩形板的丰富的动力学行为，如:不动点、极限环、拟周期、
混沌等。系统研究了板和基础的材料、载荷和几何等参数以及横向剪切效应对非
线性粘弹性板的动力学特性的影响。揭示了系统随参数的变化进入混沌的途径是
阵发性的或是拟周期分岔。
In this dissertation, nonlinear mathematical models of visco-elastic plates with finite deformations and the effect of higher-order shear deformations are established. At the same time, an extended differential quadrature method is formulated. The static and dynamical problems of visco-elastic plates and the corresponding degenerate models are systematically studied in theoretical and in numerical. The main results contain as follows:
1.Based on the Boltzmann superposition principles and the Reddy's theory of plates with the of effect of higher-order shear deformations, the nonlinear governing equations are established for static and dynamic analyses of visco-elastic plates with finite deformations and taking account of shear effects by the Laplace transformation and its inverse transformation. The governing equations are a set of coupled nonlinear integro-partial-differential equations. The differential quadrature method (DQM) is extended to analyze the nonlinear behaviors of such systems. The differential quadrature approach suggested by Wang and Bert is extended to handle the high-order moment boundary conditions of plates, too.
2.Based on the above nonlinear model of visco-elastic plates with the effect of higher-order shear deformations, the coupled nonlinear governing equations for laminated plates with finite deformations are presented. The differential quadrature method presented in this dissertation is applied to discretize the nonlinear model and the corresponding DQ form is obtained. By extending the special matrix product and decoupled technique, the coupled nonlinear equations are decoupled and the simplified iterative formulae of DQ are presented for the static problem, and hence it is able to avoid the ill-conditioning matrix and greatly reduce nonlinear computation. For the dynamic problem, the harmonic balance method of DQ is presented. The static bending, free and force vibrations of the system with different parameters and grid spacing are simulated numerically. The numerical convergence and comparison studies are carried out to validate the validity of the present method. The influence of grid spacing on the converg
ence rate is discussed. The results show that the presented differential quadrature method is more accurate and efficient than traditional one. Influences of geometric and material parameters, transverse shear deformations and rotation inertia as well as vibration amplitudes on nonlinear characteristics of laminated plates are studied in detail.
3.In the case of small deflections, the convolution-type functional and the corresponding Gurtin-type variational principle are all presented for the static- and dynamic analysis of visco-elastic plates with higher-order shear deformations and arbitrary boundary. By the extended differential quadrature method in spatial domain, the original integro-partial-differential equations are transformed into a set of integro-ordinary equations. The latter may reduced to a set of ordinary differential equations in time domain. The approximate analytical solutions are proposed for the quasi-static bending of visco-elastic plates subjected to step loads. The dynamic response of the visco-elastic plate with simply supported is obtained. The convergence and comparison of solutions are studied. The numerical results proves that the DQ method presented in this paper is very a reliable and available computational method with higher precision. Moreover, the influences of geometric and material parameters as well as the transv
erse shear deformations on dynamic behaviors are studied.
4. The nonlinear dynamic stability of visco-elastic plates with finite deformations and taking
account of higher-order shear deformation effects is discussed. By the Galerkin method, a system of integro-partial differential equations with infinite dimension is transformed into a set of nonlinear autonomous ordinary differential equations with higher-dimension via introducing new variables. The numerical methods in nonlinear dynamics, such as, time history, phase plane portrait
1．基于Reddy高阶横向剪切理论，采用Boltzmann遗传积分本构关系，综合利用Laplace变换及其反变换技术，Boltzmann算子及卷积等手段，建立了有限变形条件下，考虑高阶横向剪切效应粘弹性板非线性分析的数学模型。其控制方程是关于5个广义位移的耦合的非线性积分-偏微分方程组。将微分求积方法推广以便对上述非线性问题进行数值分析。同时，推广Wang和Bert处理边界条件的方法(简称为DQWB途径)来处理这类问题中所包括的高阶矩的边界条件。给出了推广的微分求积算法的一般格式。
2．基于本文建立的考虑高阶横向剪切效应粘弹性板非线性分析的数学模型，给出了正交各向异性层合板非线性静、动力学行为分析的数学模型；利用本文推广的微分求积格式得到了相应的非线性控制方程的DQ形式；推广运用特殊矩阵乘积及解耦技巧，对相应的耦合数值逼近方程进行解耦，在静态情况下，得到了以挠度W为基本变向量的微分求积迭代格式，避免了病态矩阵，极大的简化了非线性计算。在动态情况下，获得了简化的微分求积谐波平衡算法，并对具有不同节点分布及参数的正交各向异性层合板及其退化情形的静态弯曲、自由振动和强迫振动进行了数值模拟。对数值结果进行了收敛性和比较性研究，考察了不同的节点分布对收敛速度的影响。与传统方法相比，本文的算法在适用性、精度、效率等方面都具有相当的优势。同时考察了几何参数、材料参数及横向剪切效应和转动惯性对正交各向异性层合板非线性力学行为的影响。
3．在小变形情况下，应用反逆法，结合通常的笛卡尔内积和卷积双线性形式，构造了一种以卷积形式表示的Gurtin型泛函，建立了具有任意边界条件考虑高阶横向剪切效应粘弹性板的卷积型变分原理。同时，对所建立的高阶剪切效应粘弹性板力学行为分析的数学模型，在空域上采用本文推广的微分求积算法进行简化，时域上进一步将其化为常微分方程，分析了简化系统在阶跃载荷作用下的准静态力学行为，得出了问题的DQ近似解析解；对简化的动力系统进行研究，得到了
上海大学博士学位论文
粘弹性简支板的动态响应。研究了DQ解的收敛性和精确性;表明该方法具有收
敛性好，存贮量小，运算快，易于实现等优点，从而避免了变换法的局限性。并
考察了几何参数、材料参数及横向剪切效应对粘弹性板静、动力学行为的影响。
4.在有限变形条件下，对考虑高阶横向剪切效应粘弹性板的非线性稳定性问
题进行了分析。通过使用Galerkin平均化法，并引进新的变量，将积分一非线性偏
微分控制方程组，转化为高维非线性自治常微分方程组。综合使用非线性动力学
中数值分析方法，对动力系统的响应进行了长时间力学行为的数值模拟，研究了
系统的分岔，混沌等现象;揭示了粘弹性矩形板在周期激励下的丰富的动力学行
为。比较了1一阶和2一阶Galerkin截断系统的动力学行为，发现它们定性上是基本
相同的，但定量性质略有偏差。考察了载荷和材料等参数对粘弹性板动力稳定性
的影响，发现系统的运动关于分岔参数具有很强的敏感性。
5.建立了有限变形条件下，粘弹性基础上考虑高阶横向剪切效应粘弹性板的
动力学行为的数学模型;对模型在空域上应用前述的推广的微分求积格式、特殊
矩阵乘积及简化技巧得到了简洁的矩阵形式的非线性数值逼近公式，时域上进一
步技巧地引进新的变量，并综合利用非线性动力学中的数值分析方法，揭示了粘
弹性基础上粘弹性矩形板的丰富的动力学行为，如:不动点、极限环、拟周期、
混沌等。系统研究了板和基础的材料、载荷和几何等参数以及横向剪切效应对非
线性粘弹性板的动力学特性的影响。揭示了系统随参数的变化进入混沌的途径是
阵发性的或是拟周期分岔。
In this dissertation, nonlinear mathematical models of visco-elastic plates with finite deformations and the effect of higher-order shear deformations are established. At the same time, an extended differential quadrature method is formulated. The static and dynamical problems of visco-elastic plates and the corresponding degenerate models are systematically studied in theoretical and in numerical. The main results contain as follows:
1.Based on the Boltzmann superposition principles and the Reddy's theory of plates with the of effect of higher-order shear deformations, the nonlinear governing equations are established for static and dynamic analyses of visco-elastic plates with finite deformations and taking account of shear effects by the Laplace transformation and its inverse transformation. The governing equations are a set of coupled nonlinear integro-partial-differential equations. The differential quadrature method (DQM) is extended to analyze the nonlinear behaviors of such systems. The differential quadrature approach suggested by Wang and Bert is extended to handle the high-order moment boundary conditions of plates, too.
2.Based on the above nonlinear model of visco-elastic plates with the effect of higher-order shear deformations, the coupled nonlinear governing equations for laminated plates with finite deformations are presented. The differential quadrature method presented in this dissertation is applied to discretize the nonlinear model and the corresponding DQ form is obtained. By extending the special matrix product and decoupled technique, the coupled nonlinear equations are decoupled and the simplified iterative formulae of DQ are presented for the static problem, and hence it is able to avoid the ill-conditioning matrix and greatly reduce nonlinear computation. For the dynamic problem, the harmonic balance method of DQ is presented. The static bending, free and force vibrations of the system with different parameters and grid spacing are simulated numerically. The numerical convergence and comparison studies are carried out to validate the validity of the present method. The influence of grid spacing on the converg
ence rate is discussed. The results show that the presented differential quadrature method is more accurate and efficient than traditional one. Influences of geometric and material parameters, transverse shear deformations and rotation inertia as well as vibration amplitudes on nonlinear characteristics of laminated plates are studied in detail.
3.In the case of small deflections, the convolution-type functional and the corresponding Gurtin-type variational principle are all presented for the static- and dynamic analysis of visco-elastic plates with higher-order shear deformations and arbitrary boundary. By the extended differential quadrature method in spatial domain, the original integro-partial-differential equations are transformed into a set of integro-ordinary equations. The latter may reduced to a set of ordinary differential equations in time domain. The approximate analytical solutions are proposed for the quasi-static bending of visco-elastic plates subjected to step loads. The dynamic response of the visco-elastic plate with simply supported is obtained. The convergence and comparison of solutions are studied. The numerical results proves that the DQ method presented in this paper is very a reliable and available computational method with higher precision. Moreover, the influences of geometric and material parameters as well as the transv
erse shear deformations on dynamic behaviors are studied.
4. The nonlinear dynamic stability of visco-elastic plates with finite deformations and taking
account of higher-order shear deformation effects is discussed. By the Galerkin method, a system of integro-partial differential equations with infinite dimension is transformed into a set of nonlinear autonomous ordinary differential equations with higher-dimension via introducing new variables. The numerical methods in nonlinear dynamics, such as, time history, phase plane portrait
引文
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