Download - Concept of Fracture Mechanics
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Fracture Avoidance with Proper Use of Material
Pyramid of Egypt Schematic Roman Bridge Design
h• The primary construction material prior to 19th were timber, brick and mortar• Arch shape producing compressive stress → stone have high compressive
strength
Riley; page 5Anderson; fig. 1-4, page 9Gordon; fig. 14, page 188
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Fracture Avoidance with Proper Use of Material (cont’)(cont )
• Roof spans and windows were arched to maintain compressive loading
Gordon; plate 1 (after page 224)Anderson; fig. 1-5
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Fracture Avoidance with Proper Use of Material (cont’)(cont )
• Mass production of iron and steel (relatively ductile construction materials) →feasible to build structures carrying tensile stressesy g
h lf d’ b d ( ) h b dThe Telford’s Menai suspension bridge (1819) The seven suspension bridge
(wrought iron suspension chains) (steel cable)Gordon; plate 11 & plate 12
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Stress Concentration, Fracture and Griffith TheoryGriffith Theory
• Stress distribution around a hole in aninfine plate was derived by G Kirsch ininfine plate was derived by G. Kirsch in1898 using the theory of elasticity
• The maximum stress is three times theuniform stress
• Kt = 3
Damage Tolerance Assessment Handbook; fig. 2-1, page 2-2
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Stress Concentration, Fracture and Griffith Theory (cont’)and Griffith Theory (cont )
• C. E. Inglis (1913) investigated in a plate with anelliptical holeelliptical hole
• He derived orbaKt /21+=ρ/aK 21+=
• Modeling a crack with a ellipse means ρ → 0 →Kt → ∞ → infinite stress
ρ/aKt 21+=
• Kt could not be used for crack problems
Damage Tolerance Assessment Handbook; fig. 2-2
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Stress Concentration, Fracture and Griffith Theory (cont’)and Griffith Theory (cont )
• A. A. Griffith (1920) used an energy balance analysis to explain the largereduction on the strength of glassreduction on the strength of glass
• Griffith proposed that the large reduction is due to the presence ofmicrocracks
• Griffith derived a relation between crack size and breaking strength byconsidering the energy balance associated with a small extension of a crack
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Stress Concentration, Fracture and Griffith Theory
Damage Tolerance Assessment Handbook; fig. 2-3 a & b
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Stress Concentration, Fracture and Griffith Theory (cont’)and Griffith Theory (cont )
))(( εσ LAPxWork 11== ))(( εσ LAPxWork22
==
1 ))(( ALWork σε21
=
))(( VWork σε21
=2
densityenergystrain=)(σε21 ygy)(2
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Stress Concentration, Fracture and Griffith Theory (cont’)
Damage Tolerance Assessment handbook; fig. 2-4 a & b
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Stress Concentration, Fracture and Griffith Theory (cont’)
• Crack length increase → plate becomes less stiff (more flexible) → slope of P vs xdecreases → applied load drop
• Change in energy stored is the difference in the shaded area
• Release of elastic energy is used to overcome the resistance to crack growth
• Rate of strain energy release = rate of energy absorption to overcome resistance tocrack growth
Damage Tolerance Assessment Handbook; fig. 2-4b
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Stress Concentration, Fracture and Griffith Theory (cont’)
• Energy balance :
Energy stored in the body before crack extension = Σ (energy remaining in thebody after crack extension + work done on the body during crack extension +
• Energy balance :
energy dissipated in irreversible processes)
Damage Tolerance Assessment Handbook; fig. 2-4b
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Stress Concentration, Fracture and Griffith Theory (cont’)
• Analyze a simplified geometry with a hole D = 2a
• σy = σ everywhere outside the hole
Damage Tolerance Assessment Handbook; fig. 2-5
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Stress Concentration, Fracture and Griffith Theory (cont’)
• Strain energy density =
E2σ2
• Total energy = volxE2
2σ
After crack extension of ∆a (assume σ is constant)
[ ]taWLtE
U 22
1 2πσ
−=
2
Elastic energy released
( )[ ]taaWLtE2
U 22
2 ∆+π−σ
=
2
Per unit of new crack areaE
ataUU2
21∆σπ
≅−
Damage Tolerance Assessment Handbook; fig. 2-5E2a
at2UUG
221 σπ≅
∆−
=
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Stress Concentration, Fracture and Griffith Theory (cont’)and Griffith Theory (cont )
E l d i d t b k t i b d f• Energy released is used to break atomic bonds → surface energy
• Surface energy (γe) is a material property
E b l > k th if• Energy balance ˙> crack growth if
γ≥ e2G
πγ
=σ eE4a
• Griffith analysis based on Inglis solution yield
E2 a2σπandπ
γ=σ eE2a E
aG σπ=
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Stress Concentration, Fracture and Griffith Theory (cont’)and Griffith Theory (cont )
• In 1957 Irwin reexamined the problem of stress distribution around a crackLinear Elastic Fracture Mechanics (LEFM)
p• He analyzed an infinite plate with a crack• Using the theory of elasticity the stresses are dominated by
]sinsin[cos23
21
22θθθ
πσ −=
rK
x
3θθθK]sinsin[cos
23
21
22θθθ
πσ +=
rK
y
23
22θθθτ coscossin
Kxy=
assumption r << a
2222πrxy
LEFM valid if plasticity remains small compared to the over all dimensions of crack and cracked bodies
)(θππσσ ijij fra
2=
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Stress Concentration, Fracture and Griffith Theory (cont’)and Griffith Theory (cont )
• The term is given the symbol K (stress intensity factor)• The term is given the symbol K (stress intensity factor)
for an infinite plateaπσ
aK I πσ=
• The relation of K to G is
for plane stress conditionK 2 for plane stress condition
• The use of G and KI leads to fracture criterion i e G and Ki i e fracture occur
EKG I=
• The use of G and KI leads to fracture criterion i.e. Gc and Kic i.e. fracture occurif
G = Gc or KI = KIc
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Stress Concentration, Fracture and Griffith Theory (cont’)and Griffith Theory (cont )
Stress Intensity FactorStress Intensity Factor
f faK πσ= for infinite plate
for other geometry
aK πσ=
aK πβσ= g y
β can be obtained from : 1. handbook solution
2. approximate method2. approximate method
3. numerical method
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Stress Concentration, Fracture and Griffith Theory (cont’)and Griffith Theory (cont )
Bannantine, fig. 3-4, page 92
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Stress Concentration, Fracture and Griffith Theory (cont’)
Bannantine; fig. 3-4, page 93 & 94
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Stress Concentration, Fracture and Griffith Theory (cont’)
Loading ModesLoading Modes
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Stress Concentration, Fracture and Griffith Theory (cont’)
Loading Modes (cont’)g ( )Loading stresses terms for mode II
]coscos[sin 32 θθθσ +−=K II
x ][2222πrx
3θθθσ coscossinK IIy = 2222πry
]sinsin[cos 31 θθθτ −=K II
• Stresses terms for mode III
]sinsin[cos22
122π
τrxy
θK22θ
πτ sin
rK III
xz −=
θK22θ
πτ cos
rK III
yz −=
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Extension of LEFM to Metals
• Griffith energy theory and Irwin’s stress intensity factor could explain thefracture phenomena for brittle solid
• For metals, beside surface energy absorption, the plastic energy absorption(γp) has to be added
γγ )(E +2
• For typical metal γ ≅ 1000 γ thus γ can be neglected
πγγ
σ)( peE
a+
=2
• For typical metal, γp ≅ 1000 γe, thus γe can be neglected
• It was not easy to translate energy concept into engineering practice
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Extension of LEFM to Metals (cont’)
• K concept was seen as the basis of a practical approach
• However K is an elastic solution while at the crack tip plastic zone developed• However, K is an elastic solution while at the crack tip plastic zone developed
• If it is assumed that the plastic zone at the crack tip is much smaller than thecrack dimension → K is still valid
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Extension of LEFM to Metals (cont’)Plastic zone size Monotonic LoadingPlastic zone size Monotonic Loading
for θ = 0
]sinsin[cos22
122
θθθπ
σ −=r
Ky
for θ 0
If σ is equal to yield strengthr
Ky π
σ2
=
If σy is equal to yield strength
orp
ysr
K*π
σ2
=p
Kr 2
2
2σ
π =*
plane stress
ysσ2
21
⎟⎟⎠
⎞⎜⎜⎝
⎛=
ys
pK
rσπ
*
plane stress
Corrected due to stress redistribution2
21 ⎟
⎠
⎞⎜⎜⎝
⎛=p
Kr
σπ p a e st ess
plane strain
2 ⎠⎜⎝ ysσπ 2
31
⎟⎟⎠
⎞⎜⎜⎝
⎛=
ysp
Kr
σπ
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Plane Strain Fracture Toughness Testing
Plane Strain Fracture Toughness Testing
f• Standard test method include ASTM E399: “Standard Test Methods forPlane Strain Fracture Toughness of Metallic Materials”.
• Stringent requirement for plane strain condition and linear behaviour of thespecimen.
• Specimen type permitted: CT, SENB, arc-shaped and disk shape.
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Plane Strain Fracture Toughness Testing (cont’)
Fracture Mechanics Testing
Specimen Configurations
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Plane Strain Fracture Toughness Testing (cont’)
Clevis for Compact Tension Specimen
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Plane Strain Fracture Toughness Testing (cont’)
• Use an extensometer (e.g. clip gage) to detect the beginning of crackextension from the fatigue crack.
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Plane Strain Fracture Toughness Testing (cont’)
• Calculation of KQ for compact tension specimen
)(/ Waf
BWP
K QQ 21=
whereWBW
( ) ( ) ( ) ( ) ( )( 657214321364488602 432( ) ( ) ( ) ( ) ( )(( )231
657214321364488602 432
Wa
Wa
Wa
Wa
Wa
Wa
Waf
−
−+−++=
.....)(
• This KQ has to be checked with previous requirements
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Plane Strain Fracture Toughness Testing (cont’)
Damage Tolerance Assessment Handbook; fig. 2-13
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Plane Strain Fracture Toughness Testing (cont’)
ASTM Standards; fig. 1, page 410
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Plane Strain Fracture Toughness Testing (cont’)
Fatigue Pre-cracking• Perform to obtain natural crack• Fatigue load must be chosen :
0 such that the time is not very longplastic zone at the crack tip is small0 plastic zone at the crack tip is small
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Plane Strain Fracture Toughness Testing (cont’)
Instrumentation for Displacement and Crack Length Measurements
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Plane Strain Fracture Toughness Testing (cont’)
• Crack front curvature
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Plane Strain Fracture Toughness Testing (cont’)
• Measure a1, a2 and a3 → 3321 aaaa ++
=Measure a1, a2 and a3 →
• Any two of a1, a2 and a3 must not differ more than 10% from
• For straight notch → asurface differ not more than 15% from and (asurface)left
3
aa
g surface ( surface)leftdoes not differ more than 10% from (asurface)right
a
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Plane Strain Fracture Toughness Testing (cont’)
• Load displacement curves to determine PQ
Additional Criteria
» Pmax/PQ < 1.1Pmax/PQ < 1.1
KQ <⎞
⎜⎛
2
52» ays
Q
⎞⎛
<⎟⎠
⎜⎜⎝
2
52σ
.
»B
K
ys
Q <⎟⎟⎠
⎞⎜⎜⎝
⎛2
52σ
.ys ⎠⎝
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Plane Strain Fracture Toughness Testing (cont’)
Damage Tolerance Assessment handbook; table 2-1, page 2-31
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Plane Strain Fracture Toughness Testing (cont’)
Damage Tolerance Assessment Handbook; table 2-1, page 2-32
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Plane Strain Fracture Toughness Testing (cont’)Thi k Eff tThickness Effect
• Plane strain condition occur for thick components
F t ti t i l ti l t i diti d t h i fl• For static material properties plane strain condition does not have influence
• For fracture toughness thickness have a strong influence
Damage Tolerance Assessment Handbook; fig. 2-16
Thickness effect on fracture strength
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Plane Strain Fracture Toughness Testing (cont’)
Thickness Effect (cont’)
• Specimen thicker than 1/2 inch →plane strain
• For thinner stock KQ increasesreaching a peak at thickness aboutreaching a peak at thickness about1/8 inch
• The peak KQ can exceed five timesKKic
Thickness effect on fracture strength
• After reaching the peak KQ declines at thickness lower than 1/8 inch
Thickness effect can be e plained ith ene g balance• Thickness effect can be explained with energy balance
Damage Tolerance Assessment Handbook; fig. 2-16
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Plane Strain Fracture Toughness Testing (cont’)Thickness Effect (cont’)
• σZ = 0 at free surface → plane stresson the surface → large plastic zoneon the surface → large plastic zone
• In the inside elastic materialrestrains deformation in Z direction
• For thick specimen interiordeformation is almost totallyrestraint (σZ ≈ 0) → plane strainrestraint (σZ ≈ 0) → plane straincondition
Three-dimensional plastic zones shape
• Going inward from the surface, plastic zone undergoes transition from largersize to smaller size
Damage Tolerance Assessment Handbook; fig. 2-17a
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Plane Strain Fracture Toughness Testing (cont’)
Thickness Effect (cont’)
• For decreasing thickness, ratio ofplastic volume to total thicknessplastic volume to total thicknessincrease
• Consequently energy absorption rateq y gy palso increases for thinner plates
• While elastic strain energy isindependent of thicknessindependent of thickness
• Thus for thinner plates more appliedstress is needed to extend the crack
Plastic volume versus thickness
Damage Tolerance Assessment Handbook; fig. 2-17b
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Plane Strain Fracture Toughness Testing (cont’)Thickness Effect (cont’)
• Plane stress condition results in fracturesurface having 45o angle to z axis →g gshear lips
• For valid Kic test (plane strain condition)→ little or no evidence of shear lips→ little or no evidence of shear lips
Damage Tolerance Assessment Handbook; fig 2-18
Typical Fracture Surface
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Plane Strain Fracture Toughness TestingTemperature Effect
• Fracture toughness depends on temperature
• However Al alloys are relatively insensitive over the range of aircraft servicetemperature condition
• Many alloy steels exhibit a sharp transition in the service temperature range
Damage Tolerance Assessment Handbook; fig. 2-21
Fracture toughness versus temperature
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KIc of Aircraft MaterialsTypical Yield Strength and Plane Strain Fracture ToughnessTypical Yield Strength and Plane Strain Fracture ToughnessValues for Several Al Alloys
ASM Vol. 19; table 5, page 776
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KIc of Some Materials (cont’)Al Alloys 2124 and 7475 vs 2024 and 7075Al Alloys 2124 and 7475 vs. 2024 and 7075
Application of Fracture Mechanics; fig. 6-9, page 180
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KIc of Some Materials (cont’)
Effect of Purity on KIc
ASM Vol. 19; table 6, page 777
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KIc of Aircraft Materials (cont’)
Typical Yield Strength and Fracture Toughness of High-Strength Titanium Alloy
ASM Vol. 19; table 3, page 831
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Failure in Large Scale Yielding
• Strength assessment for structures do not meet small scale yielding condition:
1. R-curve method
2. Net section failure
3. Crack tip opening displacement
4. J-integral
5. Energy density → mixed mode loading
6. Plastic collapse → for 3D cracks
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The Net Section on Failure Criterion
St t ti i d til t i l i ldi hi h th d t• Stress concentration in ductile materials causes yielding which smoothed outthe stress as applied load increased
• Failure is assumed to occur when stress at the net section was distributeduniformly reaching σu
Net section failure criterion
• For a plate width w containing a center crack of length 2a, the critical stress is
Damage Tolerance Assessment Handbook; fig. 2-34fc wa2wσ
−=σ
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Kc of Aircraft Materials
Plane Stress Fracture Toughness (Kc) for Several Al Alloys
ASM Vol. 19; fig. 10, page 779
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Crack Opening Displacement (COD)
• Applied load will cause a crack to open, the crack opening displacement canbe used as a parameter
• At a critical value of COD fracture occur
• Developed for steels
J-IntegralJ Integral
• J-integral is an expression of plastic work (J) done when a body is loaded
• J-integral can be calculated from elastic plastic calculation
• At a critical value of J fracture occur
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