Research On A Systematic Methodology For Theory Of Elasticity

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...