5. fundamental fracture mechanics - tokushima u...small scale yieldingⅠ plastic zone is very...
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5. Fundamental fracture mechanics
Stress concentrationⅠ
Fracture New free surface formation
Force
Material Environment
Stress concentration
◎ Notch … Cross section suddenly changes
P
P
Stress line bypath
P
P
Stress line
Growth
Initiation ◎ Exact crack growth
driving force?
◎ Resistance to fracture
of material?
Fracture mechanics
Thick stress line Stress
concentration
Stress concentration factorⅠ
θ＋σ
＋＋σ
σθ 2cos3
12
12 4
4
2
2
r
a
r
a
At point（r，θ）, stress of θdirection
（ 5.1）
n
tKSσ
σmaxfactor ion concentrat tress
Stress concentration factor
（ 5.3）
Stress concentration of hole
σ∞= σ
Infinite plate
Tension of plate with hole
Stress concentration Ｋt
33max
σ
σ
σ
σtK
σy is maximum at hole edge
σ＋σ
＋＋σ
σ 30cos312
112
max
（θ＝0°、a＝ｒ）
Stress concentration factorⅡ
σ∞=1 Infinity plate with ellipse hole(2a and 2b)
applied sressσ
Infinite plate
Stress concentration factor of
ellipse hole
ρ＋
σ
σ a
b
aKt 2121max
Stress concentration factor （ellipse hole）
（ 5.2）
32121 ＋ρ
aKt
For circle hole
a = ρ
Stress concentration factorⅢ
Application
σ∞
2a = 20
2
10
ρ=1
y
x
Infinite plate
Consider ellipse hole （a = 10、ρ=1）
32.7102121 ρ
aKt
Stress concentration factor
2
2aA
ρ
AKt 21
Stress concentration factor
Stress concentration factorⅣ
2
4aA
ρ
AKt 21
Equation
Exercise Obtain the stress concentration factor
for followed notch
Infinite plate
2a
a a
σ
σ
aa
A 22
4 ρ＝a
83.32
2121 a
aAKt
ρ
Stress concentrationfactorⅤ
Stress concentration factor
Infinite plate
σ∞＝１
3
Kt=3
a （Radius a）
σy
x
(a) Kt of circle hole
2a
（Radius 2a）
For radius a and 2a,
Same Ｋt = 3
Stress is nominal stress, σn at
Minimum cross section
P＝2（b-a）σn
Finite plate
(b) Stress concentration factor
for finite plate
σn
ModeⅠ
Deformation mode of crack
x
y
z
Crack
Component with crack forms under applied load.
x
y
z
Crack
ModeⅡ ModeⅢ
Deformation mode of crack
x
y
z
Crack
Stress concentration factor of ellipse hole
2
22222
23
1221
1311121
n
n
nssnssn
snnnsny σσ …（ 5.4）
2
2
2
32
32
11
32
11
32
n
n
nn
n
n
n
nn
nnσσ
2
2
2
3
1
21
1
32
nn
nn
nn
nnσσ
n
21σ
2
32
2
max11
31121
n
n
nn
nnnnσσ
an ρ …（ 5.2） ρσ
σ aKt 21max
0 axsNotch tip of ellipse ⇒ x=0 ⇒
Stress distribution at crack tip
Stress in vicinity of crack,σ∞
Distance from crack tip x／a
Str
ess
σy／
σ∞
ρ／a＝0
Kt=∞
（Crack）
ρ／a＝0.01
Kt=21
ρ／a＝0.1
Kt=7.32 ρ／a＝0.5
Kt=3.83
0.1 0.2
10
20
0 0
Stress distribution at ellipse notch tip
2b
2a
ρ
Kt
σy y
σ∞=1
x
Kt
…（ 5.2） ρσ
σ aKt 21max
For case of crack (ρ= 0),
tK（Independent of crack length
and crack type）
Stress intensity factorⅠ
Stress at notch tip
Max. stress at notch root
Kt
Different notch and stress
Express stress at notch root
Stress concentration factor …
OK
For crack
Maximum stress ∞
（Independent of crack type and length）
NO
Stress intensity factor KⅠ
Intensity of stress field in vicinity of crack
aK πσⅠ
Stress intensity factor of infinite plate with crack
Next
Stress intensity factorⅡ
Stress distribution at crack tip
…（ 5.4）
2
22222
23
1221
1311121
n
n
nssnssn
snnnsny σσ
For crack
Notch radius ρ→ 0 ⇒ 0a
nρ
212
1
22
11
22
3
ss
s
ssss
ssy
σσσ
xx
a
sss
s 1
22212
1
σσσ …（ 5.5）
Proportion to square root a x Inverse proportion
～ Stress distribution in vicinity of crack ～
In vicinity of crack
x is near 0
1a
xs
Stress intensity factorⅢ
x x
2a x
Ky
πσ Ⅰ
2
yσ
（x → 0）
σ∞
y
Stress distribution in vicinity of crack tip
and stress intensity factor
Stress intensity factor of infinite plate with a crack
x
K
x
a
x
ay
ππ
πσσσ Ⅰ
222
KⅠ：stress intensity factor
Mode I
aK πσⅠ One crack
and infinite plate
※ Unit mMPa[ ]
ⅠⅠ ・ πσ F aK
◎ For finite plate and three dimension
Corrected factor depends on
crack geometry
Stress intensity factor Ⅳ
Stress distribution of different
crack length
x x
2a
yσ
σ∞
y
2a’
σ’∞
Long crack Short crack
＜
σy
A
Crack length and stress distribution
aK πσⅠ
aK πσⅠ ''
Long crack length 2a、stress σ∞
Short crack length 2a’、stress σ’∞
KⅠ equals to KⅠ’
⇒ Same intensity of
stress field
Intensity of stress in vicinity of crack
is decided by only stress intensity factor
Stress intensity factorⅤ
Summary
◎ Stress at crack tip is ∞
◎ Stress in vicinity of crack is inverse proportion 、
to square root x( distance from crack tip)
◎ Intensity of stress at crack tip is decided by
only stress intensity factor
（No relation between external force,
specimen dimension and crack length）
Stress intensity factor of
different type cracks
Small scale yieldingⅠ
Plastic zone is very smaller than crack length
In elastic zone around plastic zone, it can consider the same as
non plastic deformation.
It called as small scale yielding state.
Stress can be evaluated by stress intensity factor, K.
Small scale yielding
Plastic deformed zone
Elastic body ～ Under applied load, stress is ∞
Instant fracture
Non linear、 invalid of K Practice
Valid K
Small scale yieldingⅡ
Plastic zone at crack tip
x
Ky
πσ Ⅰ
2 …（ 5.6）
2
2
1
s
p
Kr
σπⅠ
2
2
1
y
Kx
σπⅠ
Elastic perfect plastic body Applied load is not changed by
plastic deformation.
Plastic zone extends until two areas
are the same.
22
1
2
122
ss
p
KKrR
σπσπⅠⅠ
2
ⅠK
After correction, plastic zone size R
…（ 5.13）
R=2rp
O O’ D
A
C B
E
F
x
Yield stress σs
a
φ
rp rp
Imaginary
elastic crack
Crack
Elastic stress ditribution
Plastic zone corrected
elastic stress distribution
Stress distribution
After yielding
σy
Small scale yielding in plane stress state
Small scale yielding Ⅲ
R=2rp
O O’ D
A
C B
E
F
x
Yield stressσs
a
φ
rp rp
Imaginary
elastic crack
Crack
Elastic stress distribution
Stress distribution corrected
by plastic zone
Stress distribution after
plastic deformation
σy
Crack opening displacement
22
4Ⅰ
Ⅰ
σπφ K
E
K
s
Crack opening displacement by
plastic deformation
Condition of small scale yielding
Plastic zone size R
Crack tip opening displacement
φ
Proportion to KⅠ over two
and
Small scale yielding Ⅳ
To disappear singularity,
decision of R／a
（Dagdale model ）
22
8
s
Ky
σ
π
Small scale yielding
応力比 x = σ∞／σs
Pla
stic
zo
ne
size
y
= R／
a
Plastic zone size at crack tip
Range of small
Scal yielding
12
sec
Sa
R
σ
σπ…（ 5.15）
222
8
SSa
R
σ
σ
σ
σπ
For small scale yielding
Applied stressσ ＜ yield stress σS
21sec 2xx When x << 1,
…（ 5.16）
Equal to （ 5.13）
Relative applied load σ∞／σS = 0.4
Relative plastic zone R／a = 0.2
Small scale yield
Small scale yielding Ⅵ
Thickness B
crack
Plastic
zone
δ
Crack opening
displacement
Plastic zone in thick plate
Stress state Plane stress （thin plate） … ２dimension
Plane strain （thick plate） … ３dimension
Plane stress
2
2
1
s
p
Kr
σπⅠ
Surface, plane stress
Plane strain
Inside material, plane strain 2
6
1
s
p
Kr
σπ≒ Ⅰ
Fracture toughnessⅠ
Fracture toughness means the resistance to crack propagation
of material under static load
Fracture toughness?
For cracked body to plastic deform,
When tress intensity factor is over the critical value,
Crack suddenly propagates
and fracture occurs
Fracture toughnessⅡ
KⅠC Plane strain
fracture toughness
Thickness B
Fract
ure
tou
gh
nes
s K
C
Plane strain region
Region（Ⅲ）
notck
Fatigue crack
Unstable growth
Transition region
Region（Ⅱ）
Shear lip
Vertical
Plane stress region
Region（Ⅰ）
Slant Stable growth
Fracture toughness Ⅲ
KⅠC Plane strain
Fracture toughness
Thickness B
Fractu
re t
ou
gh
nes
s K
C
・ Plane stress in plastic zone
・ Stable growth,
Slant type fracture surface
・ High fracture toughness
Thin plate
・ Plane strain in plastic zone
・ Unstable crack growth、
Vertical fracture surface
・ A constant fracture toughness
Thick plate
Plane strain state
Small scale yielding
＋
2
C5.2 ,
S
KaB
σⅠ
Fracture toughness Ⅳ
材料降伏応力
σs（MPa）
平面ひずみ破壊靭性
KⅠC
アルミニウム合金
2024－T4
7075－T651
324
540
49.5
36.3
チタン合金
Ti－6Al－4V 921 78.0
鋼
AISI 4340
A 5 3 3 B
1656
343
61.5
186
KⅠC at room temperature
Stress and displacement
2a
y
x
θ r
Plate
σ∞
E，ι
u
v
σy
σx
τxy
τxy
2
3sin
2sin1
2cos
2
θθθ
πσ Ⅰ
r
Ky
2
3sin
2sin1
2cos
2
θθθ
πσ Ⅰ
r
Kx
2
3sin
2sin
2cos
2
θθθ
πτ Ⅰ
r
Kxy
Stress in the vicinity of crack tip（r，θ）
2sin21
2cos
22
2θκθ
πⅠ r
G
Ku
2cos21
2sin
22
2θκθ
πⅠ r
G
Kv
x，y direction
displacement Plane stress
Plane strain
νν 13
ν43
Small yieldingⅤ （Plastic zone in vicinity of crack tip）
R
Small ellipse a =ρ/4
Circle a =ρ
Large ellipse a =4ρ
Plastic zone R／ρ
Str
ess
σs／
σm
ax
Due toσmax
Region of same
Plastic zone
σmax（Elastic maximum stress）
ρ （Notch radii） Same
R／a ≦0.4
Applied stress is the same
Concept of linear fracture mechanics
At crack tip, the same fracture phenomenon occurs
σ
ρ=0
Plastic zone
（b）Elastic-plastic stress filed
σ
a1 a2
KⅠ1 KⅠ2 =
（a）Same elastic stress field
Concept of fracture mechanics
Elastic stress is the same,
and then elastic and plastic stress is also the same
For different crack length, if stress intensity factor is the same,
Concept of linear notch mechanics
At each notch tip, the same fracture phenomenon occurs.
σ
t1
t2 Plastic zone
ρ一定
（b）Same elastic plastic stress field
σ
t1
t2
ρ1= ρ2
σmax1=σmax2
ρ1=ρ2
（a）Same elastic stress field
Concept of linear notch mechanics
Addition to elastic stress, elastic-plastic stress is also the same.
For two notches、notch radii ρand elastic max. stress are the same
Strain energy release rate I（Griffith’s theory）
Energy release rate calculated strain energy
σ
σ
2a ρ
Free surface
Crack grows 2⊿a
Strain energy when crack
grows unit length E
a 2σπ
EE
aπσ
E
aG
222
Ⅰσπ K
Strain energy release rate
Griffith’s equation
2
1
2
a
E
π
γσ γⅠ 2
2
E
KG
Strain energy release rate II （Irwin’s study①）
Change of strain energy with increasing crack growth
a x
0
y
x
x
Ky
πσ Ⅰ
2
（a）
0
a Δa
y π
ΔⅠ xa
E
Kv
4
x
（b）
Energy release rate calculated from stress at crack tip
Crack grows ⊿a Elastic strain energy releases ΔU
Released elastic
Strain energy ΔU
After cracking, load applies to Δa
And reversed displacement, v , occurs
Before cracking, working,ΔW ＝
Strain energy release rate III
x dx
x
y
0 Before growth
After growth
（c）
Dsplacement
Str
ess
Working at x
＝Strain energy
σydx
（d）
Strain energy change with increasing crack growth
Energy release rate calculated from stress at crack tip ②
Consider crack upper side
22
4
2
2 0
dxxa
E
K
x
KU a
π
Δ
π
⊿ ΔⅠⅠ
dxx
xa
E
K a
Δ
π
ΔⅠ
0
2
aGaE
KU ΔΔ⊿ Ⅰ
2
G ； Strain energy release rate
（Driving force of crack）
Stress concentration
σx （Compressive
stress）
σy （Tensile stress） Stress concentration
Stress distribution by concentrated force
σy
a
y
σ∞ 0
σ∞
σ∞
σ∞ Unlimited plate
x
（a）
=
（b）For non hole plate appledσ
σ∞
σｙ =σ∞
σ∞
σ∞
σ∞
y
a
x
（b）
＋
＋
（c）Circumference of a small hole
many concentrated force, Pi distribute
a P1
Pn
（c）
Tensile stress based on σ∞
Is the same as compressive stress
based on Pi
（Notch）
Distance from crack tip x／ρ
Circle a =ρ
Large ellipse a =4ρ
Small ellipse a =ρ/4
When Notch radii ρand σmax are the same, Stress distribution
Stress distribution in the vicinity of crack tip
Stress distribution along x axis of ellipse applied σ∞
2
22222
23
1221
1311121
n
n
nssnssn
snnnsny σσ
Agreement of
stress
3.0ρ
x
If ρandσmax are the same
For different notch,
Stress distribution is equal