CH-4 Plane problems in linear isotropic elasticity

1. CH-4 Plane problems in linear isotropic elasticity HUMBERT Laurent laurent.humbert@epfl.ch laurent.humbert@ecp.fr Thursday, march 18th 2010 Thursday, march 25th 2010

2. 4.1 Introduction Framework :linear isotropic elasticity, small strains assumptions, 2D problems (plane strain , plane stress) The basic equations of elasticity (appendix I) : - Equilibrium equations (3 scalar equations) : f : body forces (given) s : Cauchy (second order) stress tensor z, x3 Explicitly, x, x1 Cartesian basis symmetric y, x2

3. displacement imposed on Gu tractions applied on Gt - Linearized Strain-displacement relations (6 scalar equations) : Equations of compatibility: f because the deformations are defined as partial derivative of the displacements

4. 15 unknowns - Hooke’s law (6 scalar equations) : Isotropic homogeneous stress-strain relation Lamé’s constants Inversely, with → well-posed problem → find u 4

5. Young’s modulus for various materials :

6. but 1D interpretation : 1 Before elongation 2 traction 3

7. - Navier’s equations: 3 displacement components taken as unknowns - Stress compatibility equations of Beltrami - Michell : 6 stress components considered as unknowns

8. Boundary conditions : Displacements imposed on Su Surface tractions applied on St f noutwards unit normal to St → displacement and/or traction boundary conditions to solve the previous field equations

9. Thus, 4.2 Conditions of plane strain Assume that Strain components : “thick plate” functions of x1andx2only

10. Associated stress components and also (!), Inverse relations, rewritten as

11. n t - 2D static equilibrium equations : : body forces Surface forces t are also functions of x1andx2only : components of the unit outwards vector n

12. - Non-zero equation of compatibility (under plane strain assumption) implies for the relationship for the stresses: That reduces to by neglecting the body forces, Proof ?

13. Proof: - Introduce the previous strain expressions in the compatibility equation one obtains (1) - Differentiate the equilibrium equation and add (2) - Introduce (2) in (1) and simplify

14. 4.3 Conditions of plane stress Condition : thin plate From Hooke’s law, Similar equations obtained in plane strain : and also, functions of x1andx2only

15. Inverse relations, and the normal out of plane strain ,

16. Airy’s stress function : introduce the function as equations of equilibrium automatically satisfied ! then substitute in leads to Biharmonic equation Same differential equation for plane stress and plane strain problems Find Airy’s function that satisfies the boundary conditions of the elastic problem

17. : polar coordinate system Notch / crack tip Crack when , notch otherwise 4.4 Local stress field in a cracked plate : - Solution 2D derived by Williams (1957) - Based on the Airy’s stress function

18. Local boundary conditions : Find Remote boundary conditions stress field displacement field Concept of self-similarity of the stress field (appendix II) : Stress field remains similar to itself when a change in the intensity (and scale) is imposed Stress function in the form

19. with Consider the form of solution Biharmonic equation in cylindrical coordinates: Solutions of the quadratic equation : complex conjugate roots

20. Using Euler’s formula Consequently, : ci(complex) constants A, B, C and D constants to be determined … according to the symmetry properties of the problem !

21. F F natural crack Mode I loading F F Modes of fracture : A crack may be subjected to three modes More dangerous ! Notch, crack Example : Compact Tension (CT) specimen :

22. symmetric part of Mode I – loading with Stress components in cylindrical coordinates Use of boundary conditions,

23. Non trivial solution exits for A, C if For a crack, it only remains or … that determines the unknowns l infinite number of solutions n integer

24. Relationship between A and C For each value of n → relationship between the coefficients A and C → infinite number of coefficients that are written: From, with (crack)

25. Airy’s function for the (mode I) problem expressed by: Reporting

26. Expressions of the stress components in series form (eqs 4.32): Starting with, and recalling that,

27. From, and using,

28. Range of n for the physical problem ? The elastic energy at the crack tip has to be bounded but, is integrable if or

29. or Singular term when Mode - I stress intensity factor (SIF) :

30. In Cartesian components, → does not contain the elastic constants of the material → applicable for both plane stress and plane strain problems : n isPoisson’s ratio

31. Asymptotic Stress field: y Similarly, r y θ O x r θ O x singularity at the crack tip + higher–order terms (depending on geometry) fij: dimensionless function of q in the leading term Anamplitude , gijdimensionless function of qfor the nth term

32. Evolution of the stress normal to the crack plane in mode I : Stresses near the crack tip increase in proportion to KI If KI is known all components of stress, strain and displacement are determined as functions of r and q (one-parameter field)

33. Singularity dominated zone : → Admit the existence of a plastic zone small compared to the length of the crack

34. Units of Expressions for the SIF : Closed form solutions for the SIF obtained by expressing the biharmonic function in terms of analytical functions of the complex variable z=x+iy Westergaard (1939) Muskhelishvili (1953), ... Ex : Through-thickness crack in an infinite plate loaded in mode -I:

35. For more complex situations the SIF can be estimated by experiments or numerical analysis Y: dimensionless function taking into account of geometry (effect of finite size) , crack shape → Stress intensity solutions gathered in handbooks : Tada H., Paris P.C. and Irwin G.R., « The Stress Analysis of Cracks Handbook », 2nd Ed., Paris Productions, St. Louis, 1985 → Obtained usually from finite-element analysis or other numerical methods P

36. Examples for common Test Specimens B : specimen thickness

37. When a = c Circular: (closed-form solution) Semi-circular: Mode-I SIFs for elliptical / semi-elliptical cracks Solutions valid if Crack small compared to the plate dimension a≤c

38. Associated asymptotic mode I displacement field : shear modulus with Displacement near the crack tip varies with Polar components : Cartesian components : E: Young modulusn: Poisson’s ratio Material parameters are present in the solution

39. y=r sinq crack x= r cosq y=r sinq crack x= r cosq Ex: Isovalues of the mode-I asymptotic displacement: plane strain, n=0.38

40. Mode II – loading Same procedure as mode I with the antisymmetric part of Asymptotic stress field :

41. Cartesian components: Associated displacement field :

42. Mode III – loading Stress components : Displacement component :

43. Closed form solutions for the SIF Mode II-loading : Mode III-loading :

44. Withnapplied loadsin Mode I, Principe of superposition for the SIFs: Similar relations for the other modes of fracture But SIFs of different modes cannot be added ! Principe of great importance in obtaining SIF of complicated specimen loading configuration Example: (a) (b) (c)

45. Plane stress Plane strain 4.5 Relationship between KIand GI: Mode I only : When all three modes apply : Self-similar crack growth Values of G are not additive for the same mode but can be added for the different modes

46. Proof in load control (ch 3) Work done by the closing stresses : with slide 38, with but, and also slide 32 for and injecting in GI Calculating

47. → expressed in local frame ( ) Thus, Stress tensor components : 4.6 Mixed mode fracture in global frame ( ) biaxial loading Q = Rotation tensor

48. Mode I loading : → Principe of superposition : Mode II loading :

49. Propagation criteria Mode I Crack initiation when the SIF equals to the fracture toughness or Mixed mode loading Self-similar crack growth is not followed for several material Useful if the specimen is subjected to all three Modes, but 'dominated' by Mode I General criteria: explicit form obtained experimentally

50. Examples in Modes I and II m , n and C0parameters determined experimentally Erdogan / Shih criterion (1963): Crack growth occurs on directions normal to the maximum principal stress Condition to obtain the crack direction