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

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: 4

5. x Boundary Value Problems: Boundary conditions q a y h
Examples: x y a h q
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). 5

6. Boundary Value Problems: Boundary conditions
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。 P
P/2 P

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

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

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

10. Boundary Value Problems: in terms of displacementsx z P
Boussinesq problem
Boundary conditions: 10

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

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

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

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

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

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

17. Airy Function: Polynomials
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. A Hole Under Remote Shear
Boundary Value Problems: 2-dimensional
A Hole Under Remote Shear
BCs :
Chapter 5.5 48

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

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

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

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

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

25. Boundary Value Problems: 2-dimensionalor
Chapter 6.3 22