【作者】 罗建辉；

【导师】 刘光栋； 尚守平；

【作者基本信息】 湖南大学 ， 结构工程， 2002， 博士

【摘要】 以微分形式与积分形式的等价和联系为基础，研究弹性力学的求解体系。研究侧重于揭示微分形式与积分形式的相互联系和对应关系。在本文的框架下研究与对偶向量体系有关的对偶向量、正交关系、板状弹性体等问题。本文共有9章。第一章对弹性力学求解研究进行综述。第二章研究3维弹性力学求解体系。第三章研究3维弹性理论分区变分原理。第四章研究薄板弹性弯曲理论的求解体系。第五章总结工程弹性力学的统一求解体系。第六章对涉及多个坐标方向的对偶向量的构造、对偶方程组及相应的变分原理的建立进行研究。第七章提出一种新的对偶向量形式和建立一种新的对偶微分矩阵，并研究正交关系。第八章研究1维力学和2维弹性力学问题的特征函数展开解法。第九章讨论板状物体3维弹性力学问题的求解。最后给出研究结论和展望。本文完成了下列有特色的工作： 1．比较系统地研究了弹性力学求解体系。证明了微分形式与积分形式的等价关系。在统一的构架下，导出了各种变分原理。提出了一种建立变分原理的新方法。使得建立变分原理的工作“定理化”、“规范化” 2．给出了工程弹性力学求解体系微分形式的统一描述。证明了工程弹性力学求解体系微分形式与积分形式的等价关系。提出了一个对于工程弹性力学普遍成立的积分恒等式。导出了工程弹性力学普遍适用的变分原理的统一表达式。 3．提出了应力张量与位移梯度张量的对偶关系。避免用时间坐标对一个坐标方向的模拟，将弹性力学求解新体系提出的对偶向量推广到多坐标方向。 4．提出了新的对偶向量形式和新的对偶微分矩阵。对于z方向材料正交的各向异性材料，发现了对偶向量体系的正交关系可以分解为2个独立的、对称的子正交关系，新的正交关系包含对偶向量体系的正交关系。对偶向量体系的正交关系是一个普遍成立的广义关系。但对偶向量体系的正交关系可以在z方向材料正交的各向异性的条件下以狭义的强形式出现。 5．与对偶向量体系采用零特征的解对应圣维南解不同，本文采用微分算子解法，将2维弹性力学的解进行分解。将圣维南解以及特解用以z为变量n 博士学位论文的常微分方程表示。揭示了梁的弯曲理论与弹性弯曲理论圣维南解的关系。 6．利用新的正交条件，对于条状区域2维弹性力学求解中的可对角化处理的边界条件，求解衰减解积分常数的方程组可化为互不耦合的2元1次方程组，实现了系数矩阵的对角化，求得了问题的解析解。对于不可对角化的边界条件，求解衰减解积分常数的方程组不能解耦，但可以实现系数矩阵的对称性。 7 采用偏微分方程的算子解法，对板状区域的弹性力学求解进行了研究。构造了2个方向展开的对偶向量和对偶微分方程组。在板弹性弯曲理论中，分离出了与零特征解对应的圣维南解关于X、y方向的偏微分方程，分析了薄板弯曲理论与弹性弯曲理论圣维南解的关系。利用功互等定理，导出了板状区域的正交关系。更多还原

【Abstract】 A systematic methodology for theory of elasticity is presented which founds on the equivalence and relation between differential form and integral form. The research emphasizes particularly on that the relation between differential form and integral form is uncovered. In the framework of this dissertation the problems of the dual vectors, orthogonality relationship and a elastic body with plate domain involved in the system of dual vectors are researched. This dissertation consists of 9 chapters. In the first chapter a comprehensive study for theory of elasticity is presented. In chapter 2 a systematic methodology for 3-dimensional theory of elasticity is presented. In chapter 3 partitioned variational principles of 3-dimensional theory of elasticity are researched. In chapter 4 a systematic methodology for bend theory of elastic thin plates are researched. In chapter 5 a uniform systematic methodology for theory of engineering elasticity are summarized. In chapter 6 constructing dual vectors involved in multi-directions of coordinates, establishing dual equations and corresponding variational principles are researched. In chapter 7 a new form of dual vectors is presented, a new dual differential matrix is founded and the orthogonality relationship is researched. In chapter 8 solutions by eigenfunction expansion to 1-dimensional problems of mechanics and 2-dimensional problems of theory of elasticity are researched. In chapter 9 a solution to 3-dimensional problems, of theory of elasticity is discussed in a body with plate domain. Lastly, in this dissertation the conclusion and prospects of researches are presented. In this dissertation the following characteristic work is completed:1. A systematic methodology for theory of elasticity is systematically established. The equivalence between differential form and integral form is proved. All kinds of variational principles are educed under a uniform framework. A new method to establish variational principles is presented. Work to establish variational principles is theorized and standardized.2. A uniform description of differential form of a systematic methodology for theory of engineering elasticity is given. The equivalence between differential form and integral form of a systematic methodology for theory of engineering elasticity is proved. An integral identical equation is presented which hold generally true for theory of engineering elasticity. A uniform expression of variational principles is educed which hold generally true for theory of engineering elasticity.3. A dual relationship between stress tensor and displacement grads tensor is presented. It is avoided that a direction of coordinate is simulated as a time coordinate. The dual vectors presented by a new systematic methodology for theory of elasticity are generalized to multi-directions of coordinates.4. A new form of dual vectors and a new dual differential matrix is presented. For anisotropic materials in which this direction of coordinate z is an orthogonal direction of materials it is discovered that the orthogonality relationship of the system of dual vectors may be decomposed into two independent and symmetrical orthogonality subrelationships. The new orthogonality relationship includes the orthogonality relationship in the system of dual vectors .The research of this dissertation indicates that the orthogonality relationship of the system of dual vectors may be appeared in a strong form with narrow sense in the condition that this direction of coordinate z of a anisotropic materials is a orthogonal direction of materials.5. The solution of this dissertation is different from one of the system of dual vectors in which the solution of zero eigenvalue is corresponding withSaint-Venant’s solution. The solutions of 2-dimensional theory of elasticity are decomposed by a solution with differential operators. Its Saint-Venant’s solution and special solution are expressed by ordinary different equations containing one variable z. The relationship is uncovered between the bend the更多还原