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 jqzhang2@shu.edu.cn

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

Boundary Value Problems: Basic equations 15 unknown mechanical variables

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Boundary Value Problems: 2-dimensional Stresses: BCs:

Boundary Value Problems: 2-dimensional or Chapter 6.3 22