Finite Elements and Fracture Mechanics Leslie BanksSills The
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
Outline • Introduction to fracture mechanics (homogeneous material). • The finite element method. • Methods for calculating stress intensity factors. • Interface fracture mechanics. 2
Dreszer Fracture Mechanics Laboratory 3
4
Liberty Ships-World War II • The hulls of Liberty Ships fractured without warning, mainly in the North Atlantic. • 1945. Cracks propagated in 400 of these ships including 145 catastrophic failures; only 2 exist today which are seaworthy. 5
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. 6
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). 7
The Aloha Boeing 737 Accident 8
Modes of Fracture mode III m = I, III i, j=1, 2, 3 9
Asymptotic Stress Field in Mode I 10
Stress Intensity Factor m = I, III units 11
Fracture Toughness ASTM 399 Standard compact tension specimen material parameter, depends on environment 12
J -- integral strain energy density tractions J is a conservative integral 13
Griffith’s Energy G 14
J vs G 15
The Finite Element Method For a static problem: 16
The Element Lagrangian shape functions for a four noded element 17
The Element (continued) isoparametric element 18
Special Crack Tip Elements quarter-point elements Henshell and Shaw, 1975, quadrilateral elements Barsoum, 1974, 1976, triangular elements 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
Eight Noded Isoparametric Element shape functions 23
Eight Noded Isoparametric Element shape functions (continued) 24
Square-Root Singular Element Banks-Sills and Bortman (1984) 25
Methods of Calculating KI • Direct Methods • Stress extrapolation • Displacement extrapolation • Indirect Methods J – integral • Griffith’s energy • Stiffness derivative • 26
Displacement Extrapolation 27
Displacement Extrapolation (continued) for plane strain 28
Displacement Extrapolation (continued) for 29
J -- integral strain energy density tractions J is a conservative integral 30
J -- integral (continued) 31
Area J -- integral 32
Griffith’s Energy 33
Stiffness Derivative Technique 34
Results (central crack) method no. of elements % diff. (regular) % diff. (1/4 point) disp. ext. 100 5. 4 – 1. 8 J – integral (line) 100 2. 2 0. 37 J – integral (area) 100 Griffith’s energy 121 1. 6 0. 37 stiffness derivative 100 2. 0 0. 21 35
Results (edge crack) method no. of elements % diff. (1/4 point) disp. ext. 100 – 0. 6 J – integral (line) 100 0. 43 J – integral (area) 100 0. 32 stiffness derivative 100 0. 32 36
Mixed modes: M – integral 37
Auxiliary Solutions solution (2 a) solution (2 b) 38
Interface Fracture Mechanics 39
Interface Fracture Mechanics (continued) phase angle or mode mixity energy release rate 40
Interface Fracture Mechanics (continued) 41
M – integral (1) (2) 42
Auxiliary Solutions solution (2 a) solution (2 b) 43
Results 44
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 homogeneous notches and bimaterial wedges including thermal stresses. Student wanted for extending these methods to piezoelectric materials 45
- Slides: 45