粘弹性力学中的辛方法
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摘要
哈密顿体系方法是一种直接的求解方法。由于基本控制方程得到了降阶，克服了传统方法如半逆法等求解高阶微分方程的困难，近年来这套方法受到越来越多的关注，并在弹性力学中得到了成功的应用。然而由于存在能量耗散，粘弹性力学问题属于非保守系统，哈密顿体系方法不能直接应用。本文的工作是在研究粘弹性力学问题特点的基础上，引入哈密顿体系方法，在辛体系下讨论和研究基本问题。
根据辛体系的性质和积分变换，本文以平面粘弹性问题为突破口，将问题归结为零本征值本征解即圣维南问题的解和反映局部效应的非零本征值本征解问题，并将本征解之间的辛正交归一关系从相空间推广到时域，使得问题可以在时域中的辛本征值本征解空间直接讨论，克服了反复使用Laplace反变换带来的不必要的麻烦。同时，根据辛本征解展开方法给出了一套求解非齐次方程和边界条件问题的具体方法。利用这种技术，在零本征值本征解空间就圣维南问题讨论了粘弹性材料整体的蠕变和松弛特征，并在整个辛本征值本征解空间对边界的局部效应问题进行了深入分析，给出了几种典型问题的应力应变分布场，展示了由于边界的约束条件带来的应力集中现象。
众所周知，温度对材料性能特别是对粘弹性材料性能的影响是非常重要的。在热粘弹性问题中，采用变量代换等技术可将温度效应与侧边条件都归结为非齐次的对偶方程问题。因此热粘弹性问题的关键是求解非齐次方程的特解。基于粘弹性问题的研究成果，论文将辛体系应用于热粘弹性问题。数值结果揭示了温度条件在拉伸、弯曲等问题中对粘弹性材料整体性质的影响，并讨论了由于温度的不均匀分布和边界约束条件共同作用下产生的局部效应问题。
在平面问题的基础上，将哈密顿体系方法推广到空间柱体问题。通过研究问题的正则方程，根据辛本征解之间的辛正交归一关系和本征解展开等技术，给出了一种求解空间问题的具体方法。作为特殊情况，详细和深入地研究和讨论了轴对称问题，给出了问题的解析形式的解，并将辛体系下的侧边条件和端部条件亦转化为非齐次方程问题。算例给出了几种端部条件和侧边条件问题以及一些典型的温度条件带来的非齐次问题的数值结果。这些结果描述了粘弹性柱体问题的蠕变和应力松弛特征，以及温度对材料性能的影响。
论文又将哈密顿体系进一步引进到粘弹性力学厚壁筒问题中。在辛体系下描述了该问题的对偶方程和相应的边界条件。为了求解该系统的对偶方程，提出构造下一层辛子体系的方法，把偏微分方程问题转化为常微分方程问题，从而得到了拉伸、扭转和弯曲等问题以及边界局部效应问题的解，揭示了粘弹性厚壁筒问题的蠕变和应力松弛的现象和特征。
The Hamiltonian system is a direct method by which the order of differential governing equations can be reduced. Since the difficulty of solving high-order differential equations in the traditional methods, such as the semi-inverse method, is overcome, the Hamiltonian system gained much attention in recent years and has been applied successfully into elasticity. However this method can not be applied directly into viscoelasticity for the energy non-conservation. Based on the investigation of the character of viscoelastic material, the Hamiltonian system is introduced into viscoelasticity in this dissertation and the fundamental problems are discussed in the symplectic space.
With the aid of the symplectic character and the integral transformation, the plane problem is transformed into problem of solving zero-eigenvalue eigensolutions and non zero-eigenvalue eigensolutions, which are solutions of Saint-Venant problems and local effect problems respectively. Meanwhile, the adjoint relationships of the symplectic orthogonality in the Laplace domain are generalized to the time domain. Therefore the problem can be discussed directly in the eigensolution space of the time domain and the iterative application of Laplace transformation is not needed. In addition, an effective method of solving non-homogeneous equations and boundary conditions is given with the help of the expansion of the symplectic eigenvalue eigensolutions. Based on this method, the Saint-Venant problems are studied in the zero-eigenvalue solution space and the whole character of creep and relaxation for viscoelasticity is revealed. On the other hand, the local effect near the boundary is thoroughly discussed in the whole symplectic eigenvalue solution space and the stress and strain fields of some typical problems are obtained, in which the stress concentrations caused by the boundary condition restraints are well exhibited.
As everyone knows, temperature has an important effect on materials, especially on viscoelastic material. The variable substitution method is applied in this dissertation to transform the non-homogeneous lateral boundary condition and temperature effect into homogeneous problem. Thus the key point of thermo-viscoelasticity problem is to find a special solution of the non-homogeneous equations. Based on the research of viscoelasticity, the symplectic system is applied into thermo-viscoelasticity. Numerical results show the thermal effect on the whole character of viscoelasticity in the simple tensional problems and bending problems, and exhibit the local effects caused by the boundary condition restraints and uneven distribution of thermal condition.
Based on the research of the plain problem, the Hamiltonian system is generalized to three-dimensional cylinder problem. By studying the canonical equations of the original problem and employing the adjoint relationships of the symplectic orthogonality and the symplectic eigensolution expansion technology, an effective method of solving three-dimensional problems is given. As a particular case, the axisymmetric problem is thoroughly discussed. Since the lateral boundary condition and end condition can also be transformed into problem of non-homogenous equation, the analytical solution of the axisymmetric problem can be given. Moreover the non-homogeneous problems caused by the boundary conditions and typical temperature conditions are discussed in the numerical results, in which the creep and relaxation character and thermal effect of viscoelasticity are well described.
The thick walled cylinder problem is also investigated in this dissertation under Hamiltonian system. The dual equations and corresponding boundary conditions are described in the symplectic space. Besides the sub-symplectic system is constructed so that the partial differential equations can be transformed into the ordinary differential equations. Based on this method, solutions of tension, torsion, bending and local effect problems are obtained. These solutions exhibit the characters of the creep and relaxation of viscoelasticity in the thick walled cylinder problems.
根据辛体系的性质和积分变换，本文以平面粘弹性问题为突破口，将问题归结为零本征值本征解即圣维南问题的解和反映局部效应的非零本征值本征解问题，并将本征解之间的辛正交归一关系从相空间推广到时域，使得问题可以在时域中的辛本征值本征解空间直接讨论，克服了反复使用Laplace反变换带来的不必要的麻烦。同时，根据辛本征解展开方法给出了一套求解非齐次方程和边界条件问题的具体方法。利用这种技术，在零本征值本征解空间就圣维南问题讨论了粘弹性材料整体的蠕变和松弛特征，并在整个辛本征值本征解空间对边界的局部效应问题进行了深入分析，给出了几种典型问题的应力应变分布场，展示了由于边界的约束条件带来的应力集中现象。
众所周知，温度对材料性能特别是对粘弹性材料性能的影响是非常重要的。在热粘弹性问题中，采用变量代换等技术可将温度效应与侧边条件都归结为非齐次的对偶方程问题。因此热粘弹性问题的关键是求解非齐次方程的特解。基于粘弹性问题的研究成果，论文将辛体系应用于热粘弹性问题。数值结果揭示了温度条件在拉伸、弯曲等问题中对粘弹性材料整体性质的影响，并讨论了由于温度的不均匀分布和边界约束条件共同作用下产生的局部效应问题。
在平面问题的基础上，将哈密顿体系方法推广到空间柱体问题。通过研究问题的正则方程，根据辛本征解之间的辛正交归一关系和本征解展开等技术，给出了一种求解空间问题的具体方法。作为特殊情况，详细和深入地研究和讨论了轴对称问题，给出了问题的解析形式的解，并将辛体系下的侧边条件和端部条件亦转化为非齐次方程问题。算例给出了几种端部条件和侧边条件问题以及一些典型的温度条件带来的非齐次问题的数值结果。这些结果描述了粘弹性柱体问题的蠕变和应力松弛特征，以及温度对材料性能的影响。
论文又将哈密顿体系进一步引进到粘弹性力学厚壁筒问题中。在辛体系下描述了该问题的对偶方程和相应的边界条件。为了求解该系统的对偶方程，提出构造下一层辛子体系的方法，把偏微分方程问题转化为常微分方程问题，从而得到了拉伸、扭转和弯曲等问题以及边界局部效应问题的解，揭示了粘弹性厚壁筒问题的蠕变和应力松弛的现象和特征。
The Hamiltonian system is a direct method by which the order of differential governing equations can be reduced. Since the difficulty of solving high-order differential equations in the traditional methods, such as the semi-inverse method, is overcome, the Hamiltonian system gained much attention in recent years and has been applied successfully into elasticity. However this method can not be applied directly into viscoelasticity for the energy non-conservation. Based on the investigation of the character of viscoelastic material, the Hamiltonian system is introduced into viscoelasticity in this dissertation and the fundamental problems are discussed in the symplectic space.
With the aid of the symplectic character and the integral transformation, the plane problem is transformed into problem of solving zero-eigenvalue eigensolutions and non zero-eigenvalue eigensolutions, which are solutions of Saint-Venant problems and local effect problems respectively. Meanwhile, the adjoint relationships of the symplectic orthogonality in the Laplace domain are generalized to the time domain. Therefore the problem can be discussed directly in the eigensolution space of the time domain and the iterative application of Laplace transformation is not needed. In addition, an effective method of solving non-homogeneous equations and boundary conditions is given with the help of the expansion of the symplectic eigenvalue eigensolutions. Based on this method, the Saint-Venant problems are studied in the zero-eigenvalue solution space and the whole character of creep and relaxation for viscoelasticity is revealed. On the other hand, the local effect near the boundary is thoroughly discussed in the whole symplectic eigenvalue solution space and the stress and strain fields of some typical problems are obtained, in which the stress concentrations caused by the boundary condition restraints are well exhibited.
As everyone knows, temperature has an important effect on materials, especially on viscoelastic material. The variable substitution method is applied in this dissertation to transform the non-homogeneous lateral boundary condition and temperature effect into homogeneous problem. Thus the key point of thermo-viscoelasticity problem is to find a special solution of the non-homogeneous equations. Based on the research of viscoelasticity, the symplectic system is applied into thermo-viscoelasticity. Numerical results show the thermal effect on the whole character of viscoelasticity in the simple tensional problems and bending problems, and exhibit the local effects caused by the boundary condition restraints and uneven distribution of thermal condition.
Based on the research of the plain problem, the Hamiltonian system is generalized to three-dimensional cylinder problem. By studying the canonical equations of the original problem and employing the adjoint relationships of the symplectic orthogonality and the symplectic eigensolution expansion technology, an effective method of solving three-dimensional problems is given. As a particular case, the axisymmetric problem is thoroughly discussed. Since the lateral boundary condition and end condition can also be transformed into problem of non-homogenous equation, the analytical solution of the axisymmetric problem can be given. Moreover the non-homogeneous problems caused by the boundary conditions and typical temperature conditions are discussed in the numerical results, in which the creep and relaxation character and thermal effect of viscoelasticity are well described.
The thick walled cylinder problem is also investigated in this dissertation under Hamiltonian system. The dual equations and corresponding boundary conditions are described in the symplectic space. Besides the sub-symplectic system is constructed so that the partial differential equations can be transformed into the ordinary differential equations. Based on this method, solutions of tension, torsion, bending and local effect problems are obtained. These solutions exhibit the characters of the creep and relaxation of viscoelasticity in the thick walled cylinder problems.
引文
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