第二讲 固体力学基础 张俊乾 jqzhang2@shu

1. 近代力学基础 第二讲 固体力学基础张俊乾jqzhang2@shu.edu.cn

2. Contents • Stress and Kinetics • Strain and Kinematics • Constitutive Models for Materials • Material Failure • Boundary Value Problems

3. Boundary Value Problems:Basic equations 15 unknown mechanical variables

4. Boundary Value Problems:Basic equations Equations of equilibrium Boundary conditions Strain-displacement relations Constitutive relations Thermoelastic: 15 field equations Plastic:

5. Boundary Value Problems:Boundary conditions h x h a q y Boundary conditions Examples: • The displacement boundary condition and traction boundary condition are mutually exclusive. Either displacement or traction is specified on the boundary. They cannot be specified simultaneously. • A boundary may be subjected to a combination of displacement and traction (“mixed”) boundary conditions, in other words, displacement boundary conditions in some directions may be given whereas the traction boundary conditions in remaining directions are specified. • If you are solving a static problem with only tractions prescribed on the boundary, you must ensure that the total external force acting on the solid sums to zero (otherwise a static equilibrium solution cannot exist).

6. Boundary Value Problems:Boundary conditions P P P P/2 P/2 Saint-Venant Principle 若把物体的一小部分边界上的面力，变换为分布不同但静力等效的面力，则近处的应力分布将有显著改变，而远处所受的影响可忽略不计。

7. Boundary Value Problems:Interfacial conditions Two materials jointed together Perfect interface: Interface crack (debonding): Spring-like interface:

8. Boundary Value Problems:in terms of displacements Navier’s equations 3 field equations

9. Boundary Value Problems:in terms of displacements Papkovich–Neuber’s solution(without body force) 4 harmonic functions

10. Boundary Value Problems:in terms of displacements P x y z Boussinesq problem Boundary conditions:

11. Boundary Value Problems:in terms of displacements P x y z Cerruti’s problem Boundary conditions:

12. Boundary Value Problems:in terms of displacements Flat punch indenting a half-space Boundary conditions: distributed pressure: Governing eqaution: Solution:

13. Boundary Value Problems:2-dimensional x z b t y y a Plane stress Plane strain

14. Boundary Value Problems:2-dimensional Plane stress Plane strain Constitutive equations for isotropic elasticity

15. Boundary Value Problems:2-dimensional Airy Function Polar coordinates Rectangular coordinates

16. Boundary Value Problems:2-dimensional Airy Function: Polynomials Polynomial of degree 2 Chapter 5.4 24

17. Boundary Value Problems:2-dimensional Airy Function: Polynomials Polynomial of degree 3 Pure bending Chapter 5.4 24

18. Boundary Value Problems:2-dimensional Lateral Bending of a Slender Rectangle BCs : Chapter 5.4 26

19. Boundary Value Problems:2-dimensional BCs : A Hole Under Remote Shear Chapter 5.5 48

20. A Hole Under Remote Shear Boundary Value Problems:2-dimensional Stresses Along the rim of the hole The maximum hoop stress Chapter 5.5 50

21. Boundary Value Problems:2-dimensional BCs : A Circular Hole Under Tension Chapter 5.5 54

22. Boundary Value Problems:2-dimensional Pure Bending of Curved Beams Weak form Boundary conditions: 3

23. Boundary Value Problems:2-dimensional boundary conditions A curved beam loaded by a transverse force 6

24. Boundary Value Problems:2-dimensional Stresses: BCs:

25. Boundary Value Problems:2-dimensional or Chapter 6.3 22