Typical values for the fracture toughness are. Ice. 0,2 MNm-3/2 or MPa m0,5. Glass. 2 MNm-3/2 or MPa m0,5. Cast iron 70 MNm-3/2. Steel. 50 â 200 MNm-3/2.
Fracture Mechanics Applications
Lennart Elfgren Div. of Structural Engineering Luleå University of Technology
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Contents • History and Notation • • • • •
Stress Intensity Factor KI [Nm-3/2] Fracture toughness KIc [Nm-3/2] Fracture Energy Gf [Nm/m2] Characteristic length lch [mm] Size Effect and Brittleness
• Concrete and Fatigue • Ice • Rock 2
Forces • Ancient builders as Archimedes (287-212 BC), • Vitruvius (ca 20 BC) • Leonardo da Vinci (14521519)
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Stresses • Galilei Galileo (1564-1642)
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Theory of Elasticity • Robert Hooke (1635-1703) • Claude Louis Navier (1785-1836)
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Fracture Mechanics Alan Arnold Griffith (1893-1963) Fracture mechanics is the field of mechanics concerned with the study of the formation of cracks in materials. Fracture Mechanics was introduced during World War I to explain the failure of brittle materials. The stress needed to fracture bulk glass is around 100 MPa but the theoretic stress for breaking atomic bonds is approximately 10 000 MPa Experiments on glass fibers that Griffith conducted suggested that the fracture stress increases as the fiber diameter decreases. Griffith suggested that this was due to the presence of microscopic flaws in the bulk material and introduced an energy approach 6
Linear Elastic Fracture Mechanics
KI [Nm-3/2] is denoted Stress Intensity Factor 7
Stress Intensity Factor KI
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Fracture Toughness KIC The crack propagates when KI > KIC or when so > scr = KIC / sqr(pa) Typical values for the fracture toughness are Ice 0,2 MNm-3/2 or MPa m0,5 Glass 2 MNm-3/2 or MPa m0,5 Cast iron 70 MNm-3/2 Steel 50 – 200 MNm-3/2 Concrete 1 – 2 MNm-3/2
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Fracture Energy Gf The Fracture energy or surface energy quantifies the disruption of intermolecular bonds that occurs when a surface is created. For a softening material as concrete, rock or clay we get
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Relation KIC and Gf When a crack is formed in a plate (with thickness = 1) the following energy is needed W = 2a G released U = -(so2/E)pa2 The critical crack length acr can be found from
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Fracture Energy Gf Ice 1 - 15 Nm/m2 Glass ca 50 Nm/m Cement Paste ca 20 Nm/m2 Concrete 60 – 200 Nm/m2 Granite ca 70 Nm/m2 Pine ca 500 Nm/m2 Steel Gf = KIc2/E = 1002/ 200 000 MN/m2 = = 0,05 MN/m2 = 50 000 Nm/m2 12
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Material Properties Material
ft [MPa]
E [GPa]
GF [Nm/m2]
KIc [Nm-1,5]
lch [mm]
Glass
3500
75
50
2
0,0003
Ice
2
5
10
0,2
12,5
Cement Paste
4
7
20
0,4
9
Concrete
3
30
100
1,7
330
Cast Iron
130
190
25000
70
280
HS Steel
1500
195
50000
100
4,3
Granite
200
60
70
2
0,1
Pine
45
12
500
2,5
3
CFRP
1000
150
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Arne Hillerborg • 4 Jan 1923 • M Sc 1945 • Ph D 1951 • Prof LTH 1968, 1973 Fictitious crack model Characteristic length lch = EGF/fct2
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Kent Gylltoft 1983
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Test
FEM
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FEM Model
Wöhler Curve
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Abisko June 28-30 1989
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Lars Stehn 1993
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Ice Mechnics
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Test Specimen
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S = 0,59 MPa B = 0,37 m
G = 13 Nm/m2 E = 4,7 GPa
K = (GE)0,5 = = 250 kPa m0,5
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Björn Täljsten 1994
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Keivan Noghabai 1998
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Ulf Ohlsson 1995
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Håkan Thun 2006
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Conclusions • There is a size effect in structures made of materials with tensile softening. This implies that small structures may be robust and tough while large structures of the same geometry may be brittle. • The fracture energy and the characteristic lengths are important parameters. • The importance of modelling of the softening behavior of materials (= Fracture Mechanics) will probably grow in importance with time
