Theory of Elasticity 弹性力学

1. Theory of Elasticity弹性力学 Chapter 7 Two-Dimensional Formulation 平面问题基本理论

2. Theory of Elasticity Chapter Page Content（内容） • Introduction（概述） • Mathematical Preliminaries （数学基础） • Stress and Equilibrium（应力与平衡） • Displacements and Strains （位移与应变） • Material Behavior- Linear Elastic Solids（弹性应力应变关系) • Formulation and Solution Strategies（弹性力学问题求解） • Two-Dimensional Formulation （平面问题基本理论） • Two-Dimensional Solution （平面问题的直角坐标求解） • Two-Dimensional Solution （平面问题的极坐标求解） • Three-Dimensional Problems（三维空间问题） • Bending of Thin Plates （薄板弯曲） • Plastic deformation – Introduction（塑性力学基础） • Introduction to Finite Element Mechod（有限元方法介绍） 1 1

3. Theory of Elasticity Page Chapter Two-Dimensional Formulation • 7.1 Plane Stress and Plane Strain (平面应力和平面应变) • 7.2 Displacement Formulation (位移求解) • 7.3 Stress Formulation and Airy Stress Function (应力求解与应力函数) • 7.4 Photoelastic stress measurement (光弹应力测试) 2 7

4. Theory of Elasticity Page Chapter 7.1Plane Stress （平面应力） Example: thin elastic plate(弹性薄板) h, is small in comparison to other dimensions z =±h, are stress free Not only on the surface, but also throughout the entire domain.(整个实体) 3 7

5. Theory of Elasticity Page Chapter 7.1Plane Stress （平面应力） Field equations(基本方程) strain-displacement equations Hooke’s law The equilibrium equations （平衡方程） 4 7

6. Theory of Elasticity • A prismatic body whose length is much larger than any in-plane dimension, . • In-plane loads are independent of the out-of-plane coordinate z. • Absence of normal strain , in a direction perpendicular to the plane. 3-D 2-D Page Chapter 7.1 Plane Strain （平面应变） Example: long cylindrical body (长圆柱体) all cross-sections have identical displacements（横截面位移相同） ５ 7

7. Theory of Elasticity Page Chapter 7.1 Plane Strain （平面应变） Plain Strain Examples ６ 7

8. Theory of Elasticity Page Chapter 7.1 Plane Strain（平面应变） Field equations(基本方程) strain-displacement equations Hooke’s law the equilibrium equations 7 7

9. Theory of Elasticity Page Chapter 7.1 Plane Stress and Plane Strain Difference xyxy xy xy Plane “Stress” 6 component , 3 are zero Plane “Strain” 6 component , 3 are zero 8 7

10. Theory of Elasticity Page Chapter 7.1 Plane Stress and Plane Strain Problems： Plain Stress 平面应力问题 Plain Strain平面应变问题 非平面问题 Not Plain Problem 9 7

11. Theory of Elasticity Page Chapter 7.2Displacement Formulation (位移法) Displacements Formulation（Navier equations for plane stress） (B.C.) + 10 7

12. Theory of Elasticity Page Chapter 7.2Displacement Formulation (位移法) Displacements Formulation（ Navier equationsfor plane strain） + (B.C.) 11 7

13. Theory of Elasticity Page Chapter 7.3Stress Formulation (应力法) StressFormulation（for plane stress） + or (B.C.) + 12 7

14. Theory of Elasticity Page Chapter 7.3Stress Formulation(应力法) StressFormulation（for plane strain） + or (B.C.) + 13 7

15. Theory of Elasticity Plain Strain Page Chapter 7.3Stress Formulation (应力法) Difference in solution the equilibrium equations(平衡方程) Compatibility Equations　　（相容方程） Plain Stress Which factor causes the difference? 14 7

16. Plain Strain Plain Stress Theory of Elasticity Page Chapter 7.3Stress Formulation (应力法) The difference in Physical Equation between Plain Stress and Plain Strain 15 7

17. Plain Stress Plain Strain Plain StrainPlain Stress Theory of Elasticity Page Chapter 7.3Stress Formulation (应力法) 16 7

18. Theory of Elasticity Page Chapter 7.3 Airy Stress Function（应力函数） Solution of plain problems(平面问题的应力求解) Plain Strain Plain Stress Single Connected (单连通域) 3 unknowns Solution is not easy employs theAiry stress function Single unknown 17 7

19. Theory of Elasticity Page Chapter 7.3 Airy Stress Function （按应力求解） 方程的解 齐次方程通解 非齐次方程的特解 全解 = 齐次方程通解＋ +非齐次方程的特解。 18 7

20. Theory of Elasticity Page Chapter 7.3 Airy Stress Function （应力函数） 也必存在一函数 B(x,y)，使得 由微分方程理论，必存在一函数 φ(x,y)，使得 由微分方程理论，必存在一函数 A(x,y)，使得 齐次方程的通解 19 7

21. 通解 Theory of Elasticity 特解 Page Chapter 7.3 Airy Stress Function （应力函数） 满足相容方程 biharmonic equation +边界条件＋单值条件 20 7

22. 3 D 15 unknowns including 3 displacements, 6 strains, and 6 stresses. Theory of Elasticity 2 D 1 unknowns Page Chapter 7.3 Airy Stress Function （应力函数） 21 7

23. Theory of Elasticity Page Chapter 7.4Photoelastic stress Measurement (光弹应力测试) Solution of plain problems（平面问题的应力求解） Stress distribution doesn’t depend on material constants Photoelastic stress measurement (光弹应力测试) Single Connected (单连通域) 22 7

24. Theory of Elasticity 光程差 模型厚度 主应力差值 Page Chapter 7.4Photoelastic stress Measurement (光弹应力测试) Photoelastic experiment（光弹性实验） 23 7

25. Theory of Elasticity Page Chapter 7.4Photoelastic stress Measurement(光弹应力测试) Example: 24 7

26. Theory of Elasticity Page Chapter 7.4 Photoelastic stress Measurement (光弹应力测试) 25 7

27. Theory of Elasticity Page Chapter 7.4Photoelastic stress Measurement(光弹应力测试) Example: indirect tension test (ASTM D-4123 1987) bituminous and other brittle materials such as concrete, asphalt, rock, and ceramics. 26 7

28. Theory of Elasticity Page Chapter 7.4 Photoelastic stress Measurement (光弹应力测试) Example: 27 7

29. Theory of Elasticity Page Chapter 7.4Photoelastic stress Measurement(光弹性测试) Example: FEM 28 7

30. Theory of Elasticity Page Chapter 7.4Photoelastic stress Measurement(光弹性测试) Example: granular（颗粒状）materials 29 7

31. Theory of Elasticity Page Chapter 7.4Photoelastic stress Measurement(光弹性测试) Example: Photoelastic studies of the stress distribution around the tip of a crack 30 7

32. Theory of Elasticity Page Chapter Vocabulary(词汇) Plane stress Plane strain Photoelastic stress measurement Airy Stress Function biharmonic equation 平面应力 平面应变 光弹应力测试 艾里应力函数 双调和方程 31 6

33. Theory of Elasticity Page Chapter Homework 思考题： 6－1 6－5 32 7