Finite Elements and Fracture Mechanics Leslie Banks-Sills The Dreszer Fracture Mechanics Laboratory Department of Solid Mechanics, Materials and Systems Tel Aviv University ISCM-15, October, 2003

2 Outline Introduction to fracture mechanics (homogeneous material). The finite element method. Methods for calculating stress intensity factors. Interface fracture mechanics.

3 Dreszer Fracture Mechanics Laboratory

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5 Liberty Ships-World War II The hulls of Liberty Ships fractured without warning, mainly in the North Atlantic. There were 2,751 Liberty Ships manufactured between Cracks propagated in 400 of these ships including 145 catastrophic failures; only 2 exist today which are sea-worthy.

6 Liberty Ships-(continued) The low temperatures of the North Atlantic caused the steel to be brittle. These are the first ships mass produced with welds. Fractures occurred mainly in the vicinity of stress raisers. The problem may be prevented by employing higher quality steels and improvement of the design of the ship.

7 The Aloha Boeing 737 Accident On April 28, 1988, part of the fuselage of a Boeing 737 failed after 19 years of service. The failure was caused by fatigue (multi-site damage).

8 The Aloha Boeing 737 Accident

9 Modes of Fracture mode Imode IImode III m = I, II, III i, j=1, 2, 3

10 Asymptotic Stress Field in Mode I

11 Stress Intensity Factor m = I, II, III units

12 Fracture Toughness ASTM 399 Standard material parameter, depends on environment compact tension specimen

13 J -- integral strain energy density tractions J is a conservative integral

14 Griffith ’ s Energy G

15 J vs G

16 The Finite Element Method For a static problem:

17 The Element Lagrangian shape functions for a four noded element

18 The Element (continued) isoparametric element

19 Special Crack Tip Elements quarter-point elements Henshell and Shaw, 1975, quadrilateral elements Barsoum, 1974,1976, triangular elements

20 Special Crack Tip Elements quarter-point elements Henshell and Shaw, 1975, quadrilateral elements Barsoum, 1974,1976, triangular elements

21 Special Crack Tip Elements quarter-point elements Henshell and Shaw, 1975, quadrilateral elements Barsoum, 1974,1976, triangular elements

22 Special Crack Tip Elements quarter-point elements Henshell and Shaw, 1975, quadrilateral elements Barsoum, 1974,1976, triangular elements

23 Eight Noded Isoparametric Element shape functions

24 Eight Noded Isoparametric Element shape functions (continued)

25 Square-Root Singular Element Banks-Sills and Bortman (1984)

26 Methods of Calculating K I Direct Methods Stress extrapolation Displacement extrapolation Indirect Methods J – integral Griffith’s energy Stiffness derivative

27 Displacement Extrapolation

28 Displacement Extrapolation (continued) for plane strain

29 Displacement Extrapolation (continued) for

30 J -- integral strain energy density tractions J is a conservative integral

31 J -- integral (continued)

32 Area J -- integral

33 Griffith ’ s Energy

34 Stiffness Derivative Technique

35 Results (central crack) % diff. (1/4- point) % diff. (regular) no. of elements method – disp. ext J – integral (line) J – integral (area) Griffith’s energy stiffness derivative

36 Results (edge crack) % diff. (1/4- point) no. of elements method –0.6100disp. ext J – integral (line) J – integral (area) stiffness derivative

37 Mixed modes: M – integral

38 Auxiliary Solutions solution (2a)solution (2b)

39 Interface Fracture Mechanics

40 Interface Fracture Mechanics (continued) phase angle or mode mixity energy release rate

41 Interface Fracture Mechanics (continued)

42 M – integral (1) (2)

43 Auxiliary Solutions solution (2a)solution (2b)

44 Results

45 Summary Accurate methods have been presented for calculating stress intensity factors based on energy methods. The best methods are the area J – integral, stiffness derivative and area M – integral for mixed modes and interface cracks. The J and M – integrals can be extended for thermal stresses, body forces and tractions along the crack faces. Conservative integrals have been derived for homo- geneous notches and bimaterial wedges including thermal stresses. Student wanted for extending these methods to piezo- electric materials