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

ChapterPage Content （内容） 11 1.Introduction （概述） 2.Mathematical Preliminaries （数学基础） 3.Stress and Equilibrium （应力与平衡） 4.Displacements and Strains （位移与应变） 5.Material Behavior- Linear Elastic Solids （弹性应力应变关系 ) 6.Formulation and Solution Strategies （弹性力学问题求解） 7.Two-Dimensional Formulation （平面问题基本理论） 8.Two-Dimensional Solution （平面问题的直角坐标求解） 9.Two-Dimensional Solution （平面问题的极坐标求解） 10.Three-Dimensional Problems （三维空间问题） 11.Bending of Thin Plates （薄板弯曲） 12.Plastic deformation – Introduction （塑性力学基础） 13.Introduction to Finite Element Mechod （有限元方法介绍）

Chapter Page 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 ) 7 2

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

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

7.1 Plane Strain （平面应变） Chapter Page all cross-sections have identical displacements （横截面位移相同） 3-D2-D (1)A prismatic body whose length is much larger than any in-plane dimension,. (2)In-plane loads are independent of the out-of- plane coordinate z. (3)Absence of normal strain, in a direction perpendicular to the plane. 7 ５ Example: long cylindrical body ( 长圆柱体 )

Chapter Page 7.1 Plane Strain （平面应变） Plain Strain Examples 7 ６

7.1 Plane Strain （平面应变） Chapter Page Hooke’s lawstrain-displacement equations the equilibrium equations 7 7 Field equations( 基本方程 )

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

7.1 Plane Stress and Plane Strain Chapter Page Plain Stress 平面应力问题 Plain Strain 平 面应变问题 非平面问题 Not Plain Problem 7 9 Problems ：

Chapter Page 7.2 Displacement Formulation ( 位移法 ) Displacements Formulation （ Navier equations for plane stress ） (B.C.)

Chapter Page Displacements Formulation （ Navier equations for plane strain ） 7.2 Displacement Formulation ( 位移法 ) (B.C.)

Chapter Page 7.3 Stress Formulation ( 应力法 ) Stress Formulation （ for plane stress ） + or (B.C.)

7.3 Stress Formulation ( 应力法 ) Chapter Page Stress Formulation （ for plane strain ） + or (B.C.)

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

7.3 Stress Formulation ( 应力法 ) Chapter Page The difference in Physical Equation between Plain Stress and Plain Strain 7 15 Plain StressPlain Strain

7.3 Stress Formulation ( 应力法 ) Chapter Page Plain Stress Plain Strain Plain Strain Plain Stress 7 16

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

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

Chapter Page 7.3 Airy Stress Function （应力函数） 由微分方程理论，必存在一函 数 A(x,y) ，使得 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 齐次方程的通解 7 19

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

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

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

7.4 Photoelastic stress Measurement ( 光弹应力测试 ) Chapter Page Photoelastic experiment （光弹性实验） 7 23 光程差模型厚度主应力差值

7.4 Photoelastic stress Measurement ( 光弹应力测试 ) Chapter Page 7 24 Example:

7.4 Photoelastic stress Measurement ( 光弹应力测试 ) Chapter Page 7 25

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

7.4 Photoelastic stress Measurement ( 光弹应力测试 ) Chapter Page 7 27 Example:

7.4 Photoelastic stress Measurement ( 光弹性测试 ) Chapter Page 7 28 Example: FEM

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

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

Vocabulary( 词汇 ) Chapter Page 6 31 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

Homework Chapter Page 7 32 思考题： 6 － 1 6 － 5