Elements of Solid Mechanics Zhang Junqian jqzhang2@shu

1. Elements of Solid MechanicsZhang Junqianjqzhang2@shu.edu.cn 1

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

3. Stress and Kinetics Stress vector at a point Non-uniformly distributed Stress Uniformly distributed Stress Outward normal Dimension of stress: [force] / [length]2 = N / m2 (Pa) = 10-6 MPa 3

4. Stress and Kinetics Stress tensor at a point Stress vectors on the plane perpendicular to x-axis to y-axis to y-axis, respectively : Stress tensor: The first subscript indicates the direction of the plane normal upon the stress acts, the second subscript the direction of stress component. Positive stress rule: The directions of stress component and of the plane are both positive, or both negative. 4

5. Stress and Kinetics Cauchy’s formula Stress vectors on the plane with normal vector n Relationship: stress vector - stress tensor Stress tensor σ is symmetric Remark: Cauchy’s formula assures us that the nine components of stress are necessary and sufficient to define the traction on any surface element across a body. Hence the stress state in a body is characterized completely by the set of stress tensor σ 5

6. Stress and Kinetics Index notation and transformation of coordinates Coordinates: Subscript: (x, y, z) index (1, 2, 3) Stress vector: (tx, ty, tz) (t1, t2, t3) Stress tensor: Summation convention: The summation is implied by the repeated index, called dummy index. Use of any other index instead of i does not change the meaning. 6

7. Stress and Kinetics Index notation and transformation of coordinates Transformation of coordinates: Transformation of stress vector: Transformation of stress tensor: 7

8. x1 O f1 x2 f2 Stress and Kinetics Equations of motion (equilibrium) 2-dimensional 3-dimensional --- derived from the linear momentum balance orNewton’s second law of motion Symmetry: --- derived from the angular momentum balance 8

9. z S P Displacement vector: y x The change of volume (volumetric strain): Strain and Kinematics Strain tensor (deformation measure) 3 components: Strain-displacement relationship: , Normal strain or Shear strain 9

10. x O u dx P A v dy B y Strain and Kinematics Properties of Strain tensor Elongationof PA (Normal strain): Elongationof PB (Normal strain): Distortion of the right angle between two lines (Shear strain)：

11. Strain and Kinematics Transformation of coordinates Transformation of coordinates: Transformation of strain tensor: 11

12. Constitutive Model : Isotropic, linear elastic materials Uniaxial tension:  Thermoelastic constitutive equations in multiaxial-stress state: Poisson’s ratio Young’s modulus Coefficient of thermal expansion Typical materials: polycrystalline metal, polymers and concrete etc.

13. Constitutive Model : Anisotropic linear elastic materials Coefficient of thermal expansion Strain energy density: Stiffness matrix compliance matrix

14. Constitutive Model : Linear elastic orthotropic materials 9 independent elastic constants; 3 CTE constants

15. Constitutive Model :Transversely isotropic materials 5 independent elastic constants; 2 CTE constants

16. Constitutive Model : Rate independent plasticity Features of the inelastic response of metals Yield: If the stress exceeds a critical magnitude, the stress-strain curve ceases to be linear. Bauschinger effect: If the specimen is first deformed in compression, then loaded in tension, it will generally start to deform plastically at a lower tensile stress than an annealed specimen.  Decomposition of strain into elastic and plastic parts:

17. Constitutive Model : Rate independent plasticity Yield Criteria are the components of the `von Mises effective stress’ and  `deviatoric stress tensor’ respectively. • A hydrostatic stress (all principal stresses equal) will never cause yield, no matter how large the stress; • Most polycrystalline metals are isotropic. 

18. Constitutive Model : Rate independent plasticity Isotropic hardening model Perfectly plastic solid: Linear strain hardening solid Power-law hardening material

19. Constitutive Model : Rate independent plasticity Plastic flow law is the slope of the plastic stress-strain curve.

20. Constitutive Model : Rate independent plasticity Complete incremental stress-strain relations

21. Constitutive Model : Rate independent plasticity Typical values for yield stress of some materials

22. Constitutive Model : Viscoplasticity Features of creep behavior (constant stress) • If a tensile specimen of a solid is subjected to a time independent stress, it will progressively increase in length. • The length-time plot has three stages • The rate of extension increases with stress • The rate of extension increases with temperature  Features of high-strain rate behavior • The flow stress increases with strain rate • The flow stress rises slowly with strain rate up to a strain rate of about 106 , and then begins to rise rapidly.

23. Constitutive Model : Viscoplasticity Strain rate decomposition: Flow potential for creep: Flow potential for High strain rate: Plastic flow rule:

24. Material Failure : Introduction The mechanisms involved in fracture or fatigue failure are complex, and are influenced by material and structural features that span 12 orders of magnitude in length scale, as illustrated in the picture below

25. Material Failure : Mechanisms Failure under monotonic loading • Brittle • Very little plastic flow occurs in the specimen prior to failure • The two sides of the fracture surface fit together very well after failure •  In many materials, fracture occurs along certain crystallographic planes.  In other materials, fracture occurs along grain boundaries • Ductile • Extensive plastic flow occurs in the material prior to fracture • There is usually evidence of considerable necking in the specimen • Fracture surfaces don’t fit together • The fracture surface has a dimpled appearance, you can see little holes, often with second phase particles inside them.

26. Material Failure : Mechanisms Failure under cyclic loading • S-N curve normally shows two different regimes of behavior, depending on stress amplitude • At high stress levels, the material deforms plastically and fails rapidly.  In this regime the life of the specimen depends primarily on the plastic strain amplitude, rather than the stress amplitude.  This is referred to as `low cycle fatigue’ behavior • At lower stress levels life has a power law dependence on stress,  this is referred to as `high cycle’ fatigue behavior • In some materials, there is a clear fatigue limit, if the stress amplitude lies below a certain limit, the specimen remains intact forever.  In other materials there is no clear fatigue threshold.  In this case, the stress amplitude at which the material survives 108 cycles is taken as the endurance limit of the material. (The term `endurance’ appears to refer to the engineer doing the testing, rather than the material)

27. Material Failure : Stress and strain based failure criteria Failure criteria for isotropic materials: Tsai-Hill criterion for brittle fiber-reinforced composites and wood: . Ductile Fracture Criteria: Criteria for failure by high cycle fatigue: Criteria for failure by low cycle fatigue :

28. Boundary Value Problems:Basic equations 15 unknown mechanical variables 28

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

30. 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 can not 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).

31. 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。

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

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

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

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

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

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

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

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

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

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

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

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

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

45. 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

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

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

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

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

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