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MSBS ch8

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Chapter 8. Fracture of
Cracked Members
Mechanical Strengths and Behavior of Solids
2015/8/9
Seoul National University
-1-
Contents
1
Introduction
2
Preliminary Discussion
3
4
Relationship between 𝐺𝐺 and 𝐾𝐾
5
𝐾𝐾 for various cases
Safety Factors
6
Additional topics on K
7
Trends of Fracture Toughness 𝑲𝑲𝑰𝑰𝒄𝒄
8
2015/8/9
Fracture mechanics under Plasticity
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-2-
8.1 Introduction
• Unexpected failures below the material’s yield strength
initial cracks
Figure 8.1
A propane tank truck explosion due to fracture from initial cracks in welds
• Fracture mechanics
<Notch-impact test>
Rough guide for choosing materials
2015/8/9
<Fracture mechanics>
Specific analysis of strength and life for various cracks
Seoul National University
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8.2 Preliminary Discussion
• Cracks as Stress Raisers
(stress concentration factor)
πœŽπœŽπ‘¦π‘¦ = 𝑆𝑆 1 + 2
2015/8/9
𝑐𝑐
𝑐𝑐
= 𝑆𝑆 1 + 2
𝑑𝑑
𝜌𝜌
Seoul National University
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8.2 Preliminary Discussion
• Behavior at Crack Tips in Real Materials
– Large plastic deformations near the crack tip (plastic zone)
– High stress is spread over a region (stress redistribution)
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8.2 Preliminary Discussion
• Behavior at Crack Tips in Real Materials
<Polymer>
<Ceramic>
craze zone
Plastic Zone
*Measurement of Cohesive Parameters of Crazes in Polystyrene Films (Experimental and Applied Mechanics) http://what-when-how.com/
**Enhancement mechanisms of graphene in nano-58S bioactive glass scaffold: mechanical and biological performance
http://www.nature.com/srep/2014/140416/srep04712/full/srep04712.html
2015/8/9
Seoul National University
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8.2 Preliminary Discussion
• Effects of Cracks on Strength
Stress intensity factor K
- a measure of crack severity
- affected by size, stress, and geom.
- linear-elastic assumption (LEFM)
Fracture toughness Kc
- criteria for brittle fracture
- affected by material, temperature,
loading rate, thickness
Plane strain fracture toughness KIc
- thicker plate: a lower value of Kc
- a worst-case value of Kc
- material-dependent property
2015/8/9
𝐾𝐾 = 𝑆𝑆 πœ‹πœ‹πœ‹πœ‹ (π‘Žπ‘Ž β‰ͺ 𝑏𝑏)
𝐾𝐾 < 𝐾𝐾𝑐𝑐 ∢ elastic deformation
𝐾𝐾 > 𝐾𝐾𝑐𝑐 ∢ brittle fracure
Material
Toughness Kic
MPa m
Seoul National University
Steel
Polymer
Ceramic
AISI 4130
ABS
Concrete
110
3.0
1.19
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8.2 Preliminary Discussion
• Effects of Cracks on Strength
Area loss (
)
: area loss due to crack
𝑆𝑆 =
𝑃𝑃
2𝑑𝑑(𝑏𝑏−π‘Žπ‘Ž)
= 𝜎𝜎0 1 − π‘Žπ‘Ž/𝑏𝑏
Critical stress (
)
: fracture due to stress intensity
𝑆𝑆𝑐𝑐 = 𝐾𝐾𝑐𝑐 / πœ‹πœ‹πœ‹πœ‹
Stress deviation (
)
: due to plastic deformation
violating LEFM assumption
Figure 8.5
2015/8/9
Failure data for cracked plates of 2014-T6 Al at
−195°C.
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8.2 Preliminary Discussion
• Effects of Cracks on Brittle vs. Ductile behavior
Transition crack length 𝒂𝒂𝒕𝒕
: critical size b/w yielding and brittle fracture
<Ductile material>
2
<Brittle material>
Ductile
Yield strength
𝜎𝜎0
↓
Fracture toughness
𝐾𝐾𝑐𝑐
↑
Transition crack
length π‘Žπ‘Žπ‘‘π‘‘
Brittle
↑
↓
↓
Material
2015/8/9
1 𝐾𝐾𝑐𝑐
π‘Žπ‘Žπ‘‘π‘‘ =
πœ‹πœ‹ 𝜎𝜎0
↑
*above π‘Žπ‘Žπ‘‘π‘‘ is valid for wide and center-cracked plate
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8.3 Relationship between 𝐺𝐺 and 𝐾𝐾 (1)
• Strain Energy Release Rate 𝐺𝐺
: Energy per unit crack are to extend the crack
𝐺𝐺 = −
1 𝑑𝑑𝑑𝑑
𝑑𝑑 𝑑𝑑𝑑𝑑
• Stress Intensity Factor 𝐾𝐾𝑰𝑰
: stresses near the ideal sharp crack (linear-elastic & isotropic)
𝐾𝐾𝐼𝐼
πœƒπœƒ
πœƒπœƒ
3πœƒπœƒ
πœŽπœŽπ‘¦π‘¦ ≅
𝑐𝑐𝑐𝑐𝑐𝑐 1 + sin sin
2
2
2
2πœ‹πœ‹πœ‹πœ‹
𝐾𝐾𝐼𝐼 = lim πœŽπœŽπ‘¦π‘¦ 2πœ‹πœ‹πœ‹πœ‹
π‘Ÿπ‘Ÿ,πœƒπœƒ→0
𝐾𝐾𝐼𝐼 = 𝐹𝐹𝐹𝐹 πœ‹πœ‹πœ‹πœ‹
2015/8/9
𝐹𝐹 ∢ a dimensionless function that depends on the geometry and loading configuration
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8.3 Relationship between 𝐺𝐺 and 𝐾𝐾 (1)
2015/8/9
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8.3 Relationship between 𝐺𝐺 and 𝐾𝐾 (2)
• Energy-balance approach
Fracture energy = released strain energy π‘ˆπ‘ˆ + bond-breaking energy 𝑆𝑆
Released strain energy 𝑼𝑼
1
πΈπΈπœ–πœ– 2 𝜎𝜎 2
=
π‘ˆπ‘ˆ = οΏ½ 𝑓𝑓 𝑑𝑑𝑑𝑑 =
𝑉𝑉
2
2𝐸𝐸
∗
𝛽𝛽𝛽𝛽 ∗ π‘Žπ‘Ž ∗
𝜎𝜎 2 2
π‘ˆπ‘ˆ = −
πœ‹πœ‹π‘Žπ‘Ž
π‘ˆπ‘ˆ = −2
2
2𝐸𝐸
for plane stress loading 𝛽𝛽 = πœ‹πœ‹
Bond-breaking energy 𝑺𝑺
𝑆𝑆 = 𝐺𝐺𝑐𝑐 ∗ π‘Žπ‘Ž
Energy-balance
πœŽπœŽπ‘“π‘“2
πœ•πœ•(π‘ˆπ‘ˆ + 𝑆𝑆)
= − πœ‹πœ‹π‘Žπ‘Žπ‘π‘ + 𝐺𝐺𝑐𝑐 = 0
For π‘Žπ‘Ž = π‘Žπ‘Žπ‘π‘ ,
πœ•πœ•πœ•πœ•
𝐸𝐸
𝐸𝐸𝐺𝐺𝑐𝑐
∴ πœŽπœŽπ‘“π‘“ =
πœ‹πœ‹π‘Žπ‘Žπ‘π‘ β‘ 
𝑆𝑆 = 𝐺𝐺𝑐𝑐 π‘Žπ‘Ž
*Introduction to Fracture Mechanics, David Roylance, 2001
http://ocw.mit.edu/courses/materials-science-and-engineering/3-11-mechanics-of-materials-fall-1999/modules/frac.pdf
2015/8/9
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8.3 Relationship between 𝐺𝐺 and 𝐾𝐾(3)
• Stress criteria for Rupture
𝐾𝐾𝑐𝑐 = πœŽπœŽπ‘“π‘“ πœ‹πœ‹π‘Žπ‘Žπ‘π‘
β‘‘
• Relationship b/w 𝐺𝐺 and 𝐾𝐾
From β‘  & β‘‘,
𝐾𝐾𝑐𝑐 =
𝐸𝐸𝐺𝐺𝑐𝑐
∗ πœ‹πœ‹π‘Žπ‘Žπ‘π‘ =
πœ‹πœ‹π‘Žπ‘Žπ‘π‘
𝐾𝐾𝑐𝑐2 = 𝐺𝐺𝑐𝑐 𝐸𝐸
𝐸𝐸𝐺𝐺𝑐𝑐
for plane stress (πœŽπœŽπ‘§π‘§ = 0)
𝐾𝐾𝑐𝑐2 = 𝐺𝐺𝑐𝑐 𝐸𝐸/(1 − 𝜈𝜈 2 ) for plane strain (πœ–πœ–π‘§π‘§ = 0)
*Introduction to Fracture Mechanics, David Roylance, 2001
http://ocw.mit.edu/courses/materials-science-and-engineering/3-11-mechanics-of-materials-fall-1999/modules/frac.pdf
2015/8/9
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8.4 𝐾𝐾 for various cases (1)
• Cracked plates under tension
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8.4 𝐾𝐾 for various cases (2)
• Cracked plates under bending
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8.4 𝐾𝐾 for various cases (3)
• Round shaft with circumferential crack
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8.4 𝐾𝐾 for various cases (4)
• Plate w/ forces to the crack faces
• ASTM standard compact specimen
𝐹𝐹𝑃𝑃 =
2015/8/9
2 + 𝛼𝛼
(0.886 + 4.64𝛼𝛼
1 − 𝛼𝛼 3/2
−13.32𝛼𝛼 2 + 14.72𝛼𝛼 3 − 5.6𝛼𝛼 4 )
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8.4 𝐾𝐾 for various cases (5)
• Safety factors against brittle fracture
𝐾𝐾𝐼𝐼𝐼𝐼
𝐾𝐾𝐼𝐼𝐼𝐼
=
𝑋𝑋𝐾𝐾 =
𝐾𝐾
𝐹𝐹𝑆𝑆𝑔𝑔 πœ‹πœ‹πœ‹πœ‹
𝐾𝐾𝐼𝐼𝐼𝐼 = 𝐹𝐹𝑐𝑐 𝑆𝑆𝑔𝑔 πœ‹πœ‹π‘Žπ‘Žπ‘π‘
π‘‹π‘‹π‘Žπ‘Ž =
2015/8/9
π‘Žπ‘Žπ‘π‘
𝐹𝐹
=
𝑋𝑋
π‘Žπ‘Ž
𝐹𝐹𝑐𝑐 𝐾𝐾
2
𝑋𝑋𝐾𝐾 ∢ safety factor for fracture toughness
π‘‹π‘‹π‘Žπ‘Ž ∢ safety factor for crack size
𝑆𝑆𝑔𝑔 ∢ applied stress
π‘Žπ‘Ž ∢ crack size
π‘Žπ‘Žπ‘π‘ ∢ critical crack length
𝐾𝐾 ∢ fracture toughness
𝐾𝐾𝐼𝐼𝐼𝐼 ∢ plane strain fracture toughness
𝐹𝐹𝑐𝑐 ∢ fracture toughness at π‘Žπ‘Žπ‘π‘
– Elements for assigning safety factors
1) statistical information of crack shape, stress, material prop.
2) safety factor set by design code, company policy, government regulation
– Crack size π‘Žπ‘Ž should be quite smaller than π‘Žπ‘Žπ‘π‘ to satisfy reasonable 𝑋𝑋𝐾𝐾
– In general, 𝑋𝑋𝐾𝐾 is set to be large due to great variance of 𝐾𝐾𝐼𝐼𝐼𝐼
– Safety factors on crack length must be rather large to achieve reasonable
safety factors on 𝐾𝐾 and stress
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8.5 Additional topics on K (1)
• Practical applications: Complex 3-D crack cases
– Useful cases include cracked plates, shafts, cracked tubes, discs, stiffened
panels, etc., including three-dimensional cases
– 𝐹𝐹 values are elevated for points where the crack front intersects the surface
and max. 𝐾𝐾 (b)-(d).
(a)
(b)
𝐾𝐾 = 𝐹𝐹𝐹𝐹 πœ‹πœ‹πœ‹πœ‹
(c)
(d)
𝑆𝑆𝑑𝑑 ∢ tension, P
𝑆𝑆𝑏𝑏 ∢ bending, M
Figure 8.17 Stress intensity factors for (a) an embedded circular crack, (b) half-circular surface crack, (c)
quarter-circular corner crack, and (d) half-circular surface crack in a shaft
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8.5 Additional topics on K (2)
• Half-circular surface crack
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8.5 Additional topics on K (3)
• Elliptical crack
(a)
πœ‹πœ‹πœ‹πœ‹
𝑓𝑓 ,
𝐾𝐾 = 𝑆𝑆
𝑄𝑄 πœ™πœ™
πœ‹πœ‹/2
𝑄𝑄 = οΏ½
0
1−
π‘“π‘“πœ™πœ™ =
π‘Žπ‘Ž
𝑐𝑐
2
2
2
cos πœ™πœ™ + sin πœ™πœ™
π‘˜π‘˜ 2 sin2 𝛽𝛽 𝑑𝑑𝑑𝑑,
1/4
π‘Žπ‘Ž
π‘˜π‘˜ = 1 −
𝑐𝑐
2
2
π‘Žπ‘Ž/𝑐𝑐 ≤ 1
𝑄𝑄: flow shape factor
(b)
πœ‹πœ‹πœ‹πœ‹
,
𝐾𝐾𝐷𝐷 = 𝐹𝐹𝐷𝐷 𝑆𝑆
𝑄𝑄
2015/8/9
π‘Žπ‘Ž
𝑄𝑄 ≈ 1 + 1.464
𝑐𝑐
Seoul National University
1.65
(π‘Žπ‘Ž/𝑐𝑐 ≤ 1)
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8.5 Additional topics on K (4)
• Crack growing from notches (holes, fillets, rivets, etc.)
(by numerical analysis)
<Short crack>
<Long crack>
For small 𝑙𝑙/𝑐𝑐,
For large 𝑙𝑙/𝑐𝑐, 𝑀𝑀hole ≈ π‘Žπ‘Žcrack .
𝐾𝐾𝐴𝐴 = 1.12𝑆𝑆𝑆 πœ‹πœ‹πœ‹πœ‹
𝑆𝑆 ′ = π‘˜π‘˜π‘‘π‘‘ 𝑆𝑆
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𝐾𝐾𝐡𝐡 = 𝐹𝐹𝐹𝐹 πœ‹πœ‹πœ‹πœ‹
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8.5 Additional topics on K (5)
• Superposition for combined loading
𝑃𝑃
𝑏𝑏𝑏𝑏
6𝑀𝑀 6𝑃𝑃𝑃𝑃
𝐾𝐾2 = 𝐹𝐹2 𝑆𝑆2 πœ‹πœ‹πœ‹πœ‹,
𝑆𝑆2 = 2 = 2
𝑏𝑏 𝑑𝑑 𝑏𝑏 𝑑𝑑
𝑃𝑃
6𝐹𝐹2 𝑒𝑒
𝐾𝐾 = 𝐾𝐾1 + 𝐾𝐾2 =
𝐹𝐹 +
πœ‹πœ‹πœ‹πœ‹
𝑏𝑏𝑏𝑏 1
𝑏𝑏
𝐾𝐾1 = 𝐹𝐹1 𝑆𝑆1 πœ‹πœ‹πœ‹πœ‹,
<Eccentric loading of a plate>
𝐾𝐾1 = 𝐹𝐹𝑃𝑃𝑃
𝑆𝑆1 =
𝑃𝑃
𝑑𝑑 𝑏𝑏
𝐾𝐾3 = 𝐹𝐹3 𝑆𝑆 πœ‹πœ‹πœ‹πœ‹
𝐾𝐾2 =
𝐾𝐾1 =
1
𝑃𝑃
𝐹𝐹3 πœ‹πœ‹πœ‹πœ‹
𝐾𝐾1 + 𝐾𝐾3 =
𝐹𝐹𝑃𝑃𝑃 +
2
2
2𝑑𝑑 𝑏𝑏
𝑃𝑃
1
𝑑𝑑 𝑏𝑏 sin πœ‹πœ‹πœ‹πœ‹
𝐾𝐾3 = 𝑆𝑆 2𝑏𝑏 tan
<Single/row of a cracks (bolt, rivet)>
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𝐾𝐾2 =
πœ‹πœ‹πœ‹πœ‹
2
1
𝑃𝑃
𝐾𝐾1 + 𝐾𝐾3 =
2
2𝑑𝑑 𝑏𝑏
1
sin πœ‹πœ‹πœ‹πœ‹
+
1
πœ‹πœ‹πœ‹πœ‹
tan
2
2
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8.5 Additional topics on K (6)
• Inclined or parallel cracks to an stress
– Alternating crack direction
: It does not grow in its original plane.
Toughness for mixed-mode
– Interactive stresses
are generally unknown.
: Fracture modes are not independent.
– Possible approach
: Projection of crack normal to the stress direction
Opening
𝐾𝐾𝐼𝐼 = 𝑆𝑆 cos 2 πœƒπœƒ πœ‹πœ‹πœ‹πœ‹
𝐾𝐾𝐼𝐼𝐼𝐼 = 𝑆𝑆 cos πœƒπœƒ sin πœƒπœƒ
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Sliding
πœ‹πœ‹πœ‹πœ‹
Tearing
≈
𝐾𝐾 = 𝑆𝑆 πœ‹πœ‹ a cos πœƒπœƒ
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8.5 Additional topics on K (7)
• Leak-Before-Break (LBB) design of pressure vessels
– Pressure vessels should be designed to leak before fracture.
– A through-wall crack length 2𝑐𝑐 ≈ 2𝑑𝑑
– Critical crack size 𝑐𝑐𝑐𝑐
𝐾𝐾𝐼𝐼𝑐𝑐 = 𝐹𝐹𝐹𝐹 πœ‹πœ‹π‘π‘π‘π‘
𝐹𝐹 = 1 (∡ wide plate)
2015/8/9
1 𝐾𝐾𝐼𝐼𝑐𝑐
𝑐𝑐𝑐𝑐 =
πœ‹πœ‹ πœŽπœŽπ‘‘π‘‘
2
Seoul National University
𝑑𝑑 < 𝑐𝑐𝑐𝑐 : leak before break
𝑑𝑑 > 𝑐𝑐𝑐𝑐 : brittle fracture
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8.6 Trends of Fracture Toughness 𝑲𝑲𝑰𝑰𝒄𝒄 (1)
• Fixtures for a fracture toughness with crack growth bend specimen
Figure 8.27 Fixtures for a fracture toughness test on a bend specimen. The dimension W corresponds to our b.
(Adapted from [ASTM 97] Std. E399; copyright © ASTM; reprinted with permission.)
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8.6 Trends of Fracture Toughness 𝑲𝑲𝑰𝑰𝒄𝒄 (2)
• Fracture Toughness
– A deviation from linearity on the P-v plot, or a sudden drop in force due to rapid
cracking, identifies a point 𝑃𝑃𝑄𝑄 corresponding to an early stage of cracking
– The value of 𝐾𝐾, denoted 𝐾𝐾𝑄𝑄 , is the stress intensity factor corresponding to 𝑃𝑃𝑄𝑄
– 𝐾𝐾𝑄𝑄 may be somewhat lower than the value 𝐾𝐾𝑐𝑐 corresponding to the final
fracture of the specimen.
– Fracture toughness testing of metals based on LEFM principles governed by
several ASTM standards, notably Standard Nos. E399 and E1820.
– Standards No. D5045 (polymers) and No. C1421 (ceramics)
95% of slope
𝐾𝐾𝑄𝑄 decreases with thickness
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8.6 Trends of Fracture Toughness 𝑲𝑲𝑰𝑰𝒄𝒄 (3)
• Material
– Material-dependent 𝐾𝐾𝐼𝐼𝑐𝑐
Material
Toughness
[MPa m]
Metal
Polymer
Ceramic
20~200
1~5
1~5
– Large CoV of 𝐾𝐾𝐼𝐼𝑐𝑐 : 10~20%
• Microstructural influences
– Chemical composition
: sulfide inclusions facilitate fracture.
– Processing (forging, rolling, extruding)
: anisotropy and planes of the flattened grains
– Neutron radiation (radiation embrittlement)
: large numbers of point defects
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*CoV: coefficient of variation (CoV = 𝜎𝜎/πœ‡πœ‡)
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8.6 Trends of Fracture Toughness 𝑲𝑲𝑰𝑰𝒄𝒄 (4)
• Temperature
– Cleavage @ low temp.
: fracture with little plastic deform. along the crystal planes of low resistance
– Dimples rupture @ high temp.
: fracture with plasticity-induced formation, growth, and joining of tiny voids
Upper shelf
Cleavage
Dimpled rupture
(microvoid coalescence)
Lower shelf (𝐾𝐾𝐼𝐼𝑐𝑐 )
<Fracture mechanics shift>
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8.6 Trends of Fracture Toughness 𝑲𝑲𝑰𝑰𝒄𝒄 (5)
• Temperature and Loading rate
– A higher rate of lading lowers the fracture toughness 𝐾𝐾𝐼𝐼𝑐𝑐 . (temperature shift)
– Statistical variation of 𝐾𝐾𝐼𝐼𝑐𝑐 is especially large within the temperature transition.
2015/8/9
Seoul National University
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8.7 Fracture mechanics under Plasticity (1)
• The size of plastic zone π‘Ÿπ‘Ÿπ‘œπ‘œ
–
–
Yielding at crack tips (plastic zone) will be studied.
Plastic zone may not be large if the LEFM theory is to be applied.
For plane strain (πœ€πœ€π‘§π‘§ = 0, πœŽπœŽπ‘§π‘§ = 2πœˆπœˆπœŽπœŽπ‘¦π‘¦ ),
For plane stress (πœŽπœŽπ‘§π‘§ = 0),
𝜎𝜎π‘₯π‘₯ = πœŽπœŽπ‘¦π‘¦ =
π‘Ÿπ‘Ÿπ‘œπ‘œπ‘œπ‘œ =
πœŽπœŽπ‘¦π‘¦ ≅
𝐾𝐾
2πœ‹πœ‹πœ‹πœ‹
lim πœŽπœŽπ‘¦π‘¦ =
πœƒπœƒ→0
2015/8/9
1 𝐾𝐾
2πœ‹πœ‹ πœŽπœŽπ‘œπ‘œ
𝑐𝑐𝑐𝑐𝑐𝑐
𝐾𝐾
𝐾𝐾
2
2πœ‹πœ‹πœ‹πœ‹
οƒ  2π‘Ÿπ‘Ÿπ‘œπ‘œπ‘œπ‘œ =
πœƒπœƒ
πœƒπœƒ
3πœƒπœƒ
1 + sin sin
2
2
2
1 𝐾𝐾
πœ‹πœ‹ πœŽπœŽπ‘œπ‘œ
2
For octa. or max. shear stress yield criterion,
𝜎𝜎0
≈ 2.5𝜎𝜎0 → 3𝜎𝜎0
𝜎𝜎π‘₯π‘₯ = πœŽπœŽπ‘¦π‘¦ =
1 − 2𝜈𝜈
1 𝐾𝐾
2π‘Ÿπ‘Ÿπ‘œπ‘œπ‘œπ‘œ =
3πœ‹πœ‹ πœŽπœŽπ‘œπ‘œ
2
2πœ‹πœ‹πœ‹πœ‹
Seoul National University
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8.7 Fracture mechanics under Plasticity (2)
• The size of plastic zone π‘Ÿπ‘Ÿπ‘œπ‘œ
– For thin specimen, Poisson contraction in the thickness occurs.
– It results in yielding on shear planes inclined through the thickness.
Plane stress
2015/8/9
Seoul National University
Plane strain
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8.7 Fracture mechanics under Plasticity (3)
• Plasticity limitations on LEFM
– LEFM is valid for small plastic zone compared with crack tip-to-boundary dist.
– 8π‘Ÿπ‘Ÿπ‘œπ‘œ is generally considered to be sufficient οƒ  4 times of crack zone size
– Since π‘Ÿπ‘Ÿπ‘œπ‘œπ‘œπ‘œ is larger than π‘Ÿπ‘Ÿπ‘œπ‘œπœ€πœ€ , an overall limit of the use of LEFM is
π‘Žπ‘Ž, 𝑏𝑏 − π‘Žπ‘Ž , β„Ž ≥ 8π‘Ÿπ‘Ÿπ‘œπ‘œπ‘œπ‘œ
4 𝐾𝐾
=
πœ‹πœ‹ πœŽπœŽπ‘œπ‘œ
2
(LEFM applicable)
or
Region of K-dominance
<Crack specimen geometry>
2015/8/9
<LEFM applicable region, K-field>
Seoul National University
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8.7 Fracture mechanics under Plasticity (3)
• Plasticity limitations on LEFM
– LEFM is valid for small plastic zone compared with crack tip-to-boundary dist.
– 8π‘Ÿπ‘Ÿπ‘œπ‘œ is generally considered to be sufficient οƒ  4 times of crack zone size
– Since π‘Ÿπ‘Ÿπ‘œπ‘œπ‘œπ‘œ is larger than π‘Ÿπ‘Ÿπ‘œπ‘œπœ€πœ€ , an overall limit of the use of LEFM is
π‘Žπ‘Ž, 𝑏𝑏 − π‘Žπ‘Ž , β„Ž ≥ 8π‘Ÿπ‘Ÿπ‘œπ‘œπ‘œπ‘œ
2015/8/9
4 𝐾𝐾
=
πœ‹πœ‹ πœŽπœŽπ‘œπ‘œ
2
(LEFM applicable)
Seoul National University
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8.7 Fracture mechanics under Plasticity (4)
• Fracture mechanics beyond linear elasticity
– Condition of "π‘Žπ‘Ž, 𝑏𝑏 − π‘Žπ‘Ž , β„Ž ≥ 8π‘Ÿπ‘Ÿπ‘œπ‘œπ‘œπ‘œ " is not satisfied, (under excessive yielding)
– LEFM and 𝐾𝐾 are not applicable due to excessive yielding.
– Following three approaches are available.
(1) Plastic zone adjustment
– The stress outside of the plastic zone is similar to elastic stress.
– Hypothetical crack (π‘Žπ‘Žπ‘’π‘’ =π‘Žπ‘Ž + π‘Ÿπ‘Ÿπ‘œπ‘œπ‘œπ‘œ ) with its tip near the center of the plastic
zone.
– Not applicable for large stress to cause yielding on whole section; 80% of the
fully plastic force or moment.
𝐾𝐾 = 𝐹𝐹𝐹𝐹 πœ‹πœ‹πœ‹πœ‹
𝐾𝐾𝑒𝑒 = 𝐹𝐹𝑒𝑒 𝑆𝑆 πœ‹πœ‹π‘Žπ‘Žπ‘’π‘’ = 𝐹𝐹𝑒𝑒 𝑆𝑆 πœ‹πœ‹ π‘Žπ‘Ž + π‘Ÿπ‘Ÿπ‘œπ‘œπ‘œπ‘œ
2015/8/9
𝐹𝐹𝑒𝑒 = 𝐹𝐹 π‘Žπ‘Žπ‘’π‘’ /𝑏𝑏
where
1 𝐾𝐾𝑒𝑒
π‘Ÿπ‘Ÿπ‘œπ‘œπ‘œπ‘œ =
2πœ‹πœ‹ πœŽπœŽπ‘œπ‘œ
Seoul National University
2
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8.7 Fracture mechanics under Plasticity (5)
(2) J-Integral
– 𝐽𝐽 is the generalization of the strain energy release rate, 𝐺𝐺, to nonlinear-elastic.
– It reains significance as a measure of the intensity of the elasto-plastic stress
and strain fields around the crack tip.
– Two different and independent P-v curves are required.
– Basis of fracture toughness tests, ASTM Standard No. E1820.
𝐾𝐾𝐼𝐼2𝑐𝑐 = 𝐺𝐺𝐼𝐼𝑐𝑐 𝐸𝐸′
Eq. (8.10)
𝐾𝐾𝐼𝐼𝑐𝑐 𝐽𝐽 =
𝐽𝐽𝐼𝐼𝑐𝑐
𝐸𝐸 ′
for plane stress (πœŽπœŽπ‘§π‘§ = 0)
𝐸𝐸 ′ = 𝐸𝐸
𝐸𝐸 ′ = 𝐸𝐸/(1 − 𝜈𝜈 2 ) for plane strain (πœ–πœ–π‘§π‘§ = 0)
𝐾𝐾𝐽𝐽 =
𝐽𝐽𝐽𝐽
πœ€πœ€
𝐾𝐾𝐽𝐽 ≈ 𝐾𝐾 1 + πœ€πœ€π‘’π‘’π‘π‘π‘›π‘›
2015/8/9
Seoul National University
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8.7 Fracture mechanics under Plasticity (6)
(2) J-Integral: Fracture toughness tests for 𝐽𝐽𝐼𝐼𝑐𝑐
– Complexity encountered in 𝐽𝐽𝐼𝐼𝑐𝑐 testing is that nonlinearity in P-v behavior is
due to a combination of crack growth and plastic deformation
– 𝐽𝐽 calculation for the standard bend and compact specimens
𝐽𝐽 =
𝐽𝐽𝑒𝑒𝑒𝑒
+
𝐽𝐽𝑝𝑝𝑝𝑝
πœ‚πœ‚π΄π΄π‘π‘π‘π‘
𝐾𝐾 2 1 − 𝜈𝜈 2
+
=
𝐸𝐸
𝑑𝑑 𝑏𝑏 − π‘Žπ‘Ž
πœ‚πœ‚ = 1.9
for bend specimen
πœ‚πœ‚ = 2 + 0.522(1 − π‘Žπ‘Ž/𝑏𝑏) for compactspecimen
<Bend specimen>
<Unloading compliance method>
2015/8/9
Seoul National University
<Compact specimen>
<Potential drop test>
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8.7 Fracture mechanics under Plasticity (7)
(3) Crack-Tip Opening Displacement (CTOD; 𝜹𝜹)
– 𝐾𝐾 can be used to estimate the displacement separating the crack faces.
– CTOD is also used as the basis of fracture toughness tests; ASTM Standards
E1290 and E1820
𝐾𝐾 2
𝐽𝐽
𝛿𝛿 ≈
≈
πΈπΈπœŽπœŽπ‘œπ‘œ πœŽπœŽπ‘œπ‘œ
• Summary
2015/8/9
Seoul National University
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