KL Murty
MSE 450
Fracture
(7 : 1-8, 14 : 1-4)
Brittle vs Ductile ⇔ Relative terms - ductile fracture implies appreciable plastic
deformation prior to fracture
- in general, bcc and hcp metals exhibit brittle fracture while fcc are ductile some fcc metals can be so ductile ⇒ point fracture akin to superplastic materials
Fracture in tensile testing : (Fig.7-1) and fracture characterization
• strain to fracture --------------------- ductile vs
brittle
• mode (crystallographic) ------------ shear
vs cleavage
• appearance of fracture surface ----- fibrous vs granular
(Figs. 7-7,8,9)
------dimpled vs
faceted
Fracture stress (σf) ⇒ Cohesive strength (σm) -- similar to theoretical yield strength
(except here in tension)
cohesive E
˜
(Eq. 7-5)
see text for derivation : σmax
π
In a perfect brittle material, all the elastic strain energy is expended in creating the
Eγs
cohesive
˜
two new surfaces: σmax
ao , ao is interatomic spacing (Eq. 7-8)
• Stresses in a Cracked Body : presence of cracks reduces σf due to σ-concentration at
the crack tip (recall MAT 201 / see section 2-15)
define : c =
{half crack length (interior crack)
crack length of surface crack
(Fig. 7-4)
ρ = radius of curvature
so that,
σmax = σ [1 + 2
c
]~2σ
ρ
c
,
ρ
A
P
here σ is the applied nominal stress (A )
cohesive
Fracture occurs when σmax = σmax
c
2c
Eγs
c
=
ao
ρ
With ρ ~ ao(sharpest crack possible),
Or, σ at fracture, σf : 2 σf
σf =
Eγs
4c
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KL Murty
MSE 450
Griffith Theory (of Brittle Fracture) : (7-4)
A crack will propagate when the decrease in
elastic strain energy is at least equal to the
energy required to create new crack surface
Increase in surface energy = decrease in elastic strain energy (expended in fracture)
d∆U
Or, UE + Us = ∆U ⇒ dc = 0. Or σf =
stress) : as c increases, σf decreases.
for thicker plate or plane strain case, σf =
2Eγs
Eq. 7-15 for a thin plate (plane
πc
2Eγs
(1-ν2)πc
• note these are valid for completely brittle material with no plastic deformation
& γs may be altered by environment (corrosion, etc)
If there is some plastic deformation, Griffith Model needs to be modified ⇒
Orowan suggested _ γt = γs + γp ,p for plastic work required to extend the crack
so that σf =
2E(γs +γp)
˜
πc
2Eγp
since
πc
γ
p
>> γs
• Read 7-6 and 7-7 : Metallographic Aspects of Fracture / SEM Fractography •
In general, cracks are formed during plastic flow ⇒ dislocation pile-ups (Fig. 7-10)
lead to cracks - fracture involves 3 stages :
(a) plastic deformation to produce cracks
(b) crack initiation (nucleation)
(c) crack propagation
3 Types of Fracture :
Quasi-cleavage
Dimpled-Rupture
Cleavage
brittle
ductile
flat facets
dimples / tear sides
dimples around inclusions
around facets
(microvoid coalescence)
Fig. 7-7
Fig. 7-8
Fig. 7-9
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KL Murty
MSE 450
Effects of Notch & Crystallography on Fracture :
Notch effects (7-10) : p. 268 increased tendency for brittle fracture
• produce high local stresses
• introduce triaxial tensile stress state
• produce high local strain hardening and cracking
Temperature Effects : depend on crystal structure - all exhibit ductile fracture at HT
while bcc metals exhibit - DBTT
effect of T on σf vs σy : bcc metals show temperature sensitive of σy at low Ts
How to determine fracture energy & DBTT ?
• Charpy and • Izod impact tests ( recall MAT 201) ..... Cv vs T (Fig. 14-3)
Cv vs T for bcc vs fcc & hcp metals _
Instrumented Charpy Testing - fracture initiation / propagation
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KL Murty
MSE 450
Fracture Mechanics
(Ch. 11)
Recall Griffith Criterion for brittle fracture ⇒
2Eγs
σf =
, a is half crack-length
πa
from competition between the decrease in elastic strain energy and increase in surface
πa2σ2
σ2
energy : Uel = - E
per unit plate thickness {= 2E x volume (= 2(πa2))}
and Us = 2 (2a) γs
dU
⇒ U = Uel + Us , & {
da }af= 0 − condition for stable crack growth
This implies that at a given σ , if a = af , cracks start propagating as long as the load or
stress is applied (i.e. what stable means).
Orowan modification _ σf =
2E(γs+γp)
˜ σf =
πa
2Eγp
˜
πa
E γp
a
plane stress vs plane strain (a factor of 1-ν2 in the denominator)
•Unfortunately, γp is not a measurable quantity ⇒ Irwin’s Energy Balance Approach
Fracture occurs at a σ corresponding to a Critical Crack Extension Force (G)
also known as Strain Energy Release Rate:
πaσ2
∂U
where G = E = 2E ; 2 appears since crack extends both sides for a central crack.
∂a
When G = G c , crack extension occurs.
Thus σf =
EGc
πa
⇒ compare with
δ
c
P
a2
Griffith criterion: G c = 2(γs+γp) ˜ 2γp
Can determine G as in Fig. 11-1.
a2 > a1
a1
δ
single-edge-notch specimen (Fig. 11-1)
P = M δ or δ = C P where M is elastic stiffness and C is elastic compliance
Measure Pmax at which unstable crack starts ⇒ G c =
2
Pmax
∂C
(Eq. 11.8)
2 ∂a
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KL Murty
MSE 450
Crack Tip Stresses (11-3)
Crack Deformation Modes (Fig. 11-3):
Mode I - Opening mode •
Mode II - Sliding mode
Mode III - Tearing mode
Stresses around the crack tip (Mode I): Eq. 11-9 / Fig. 11-2 using elasticity
Plane stress σx = σ
a
2r f1(θ)
σy = σ
a
2r f2(θ)
a
2r f3(θ)
τxy = σ
these stresses are valid near the crack tip for a > r > ρ
note : straight ahead of the crack tip (θ=0) : σx = σy = σ
a
2r and τxy = 0.
note - as r → 0 , σx and σy → ∞ implies there exists a zone where these elastic
stress field equations are not valid
indeed, at r = rc where σx , σy = σo (yield stress), yielding occurs and
elasticity does not hold (plastic or process zone close to the crack tip)
Irwin defines stress intensity factor
K=σ
πa
P
, units of K
w
P
σ (net section stress) = (w-a)B
here, mode I → KI = σ
B
a
πa
Now, can rewrite the stress fields in Eq. 11-9 in terms of K
K is a convenient way of describing the stress distribution around a flaw:
if two flaws of different geometry have the same values of K then the stress fields
around them are identical
In general, K = Y σ
πa , where Y is a geometry factor (α in the text)
Y = 1 for a crack in an infinite size body & Y = (
πa
w
tan w ) - Eq. 11-13
πa
• can find equations for Y in ASTM standards for various geometries •
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KL Murty
MSE 450
Now, we can relate K to G (energy release rate / crack extension force):
πaσ2
recall G = E and since K = σ πa ⇒ K2 = GE for plane stress (Eq. 11-14)
GE
plane strain (Eq. 11-15)
& K2 =
(1-ν2)
Fracture Criterion : (basic concept)
K-zone vs Process (or plastic) zone - near the crack-tip plastic flow - microvoids
since cannot get σtip → ∞ due to plastic
deformation at σ = σo → processzone (rp) occurs when
K
= σo
σx or σy =
2πrp
or rp =
Process-zone
K-zone
K2
2πσ o2
so, K-zone is the zone around the crak-tip where σ’s are scaled by K.
If K-zone >> process-zone (i.e, rp is small), failure (or bond rupture) in process zone
will be determined by stresses in K-zone, i.e., by K or LEFM
⇒ fracture criterion is : K = KIc ⇒ critical fracture toughness
KIc is a material parameter - depends on T and strain-rate but not on geometry (for
large thickness - Plane Strain Critical Fracture Toughness) - Fig. 11-7
That means in terms of fracture stress (σf) :
KIc = Y σf πa
• Fracture Design Basis : for a given load or stress (σ) in a material with a crack or
flaw of size ‘a’ , K = Y σ πa and if K < KIc _ no failure
Or, can find ac for given KIc and σ so that for a < ac no failure would occur.
if ac > B (thickness) , then have the condition of Leak Before Crack - implying that
the crack will pass through the wall without catastrophic brittle fracture
- if ac < B, get brittle fracture ⇐ avoid / bad design
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KL Murty
MSE 450
Plastic Zone Size : The above approach using K-fields is not valid for ductile
materials where process-zone is large
since rp corresponds to where σy = σo
(see fig.)
K
at rp , σy = σo =
2πrp
1 K 2
Or, rp =
{ }.
2π σo
Note : as σo ⇑ rp ⇓ - approach LEFM
Otherwise, we need to use EPFM
or may use plasticity corrections
Fig. 11-10
Before we discuss plasticity corrections, consider how to determine KIC: (11-5)
• using CT, 3-point bend, or notched round specimen
Plane Strain
(Thick Plate)
rp =
vs
Plane Stress
(Thin Sheet)
1 K2
6π σ2
rp =
o
1 K2
2π σ2
o
(see View Graph)
Relation between σο and KIc
Plasticity Corrections :
1. Irwin : replace ‘a’ by
so that
aeff = a + rp
Keff = σ
π aeff = σ
π (a+rp) , rp = f(K,σo)
• K appears both sides ⇒ solve by iterative process
πK2
2. Dugdale : plastic zone in the form of narrow strips (Fig. 11-11) of size R= 8σ ,
o
so that
aeff = a + R ⇒ calculate Keff
• There is a limit to the extent to which K can be adjusted •
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KL Murty
MSE 450
Need EPFM (Elastic Plastic Fracture Mechanics) :
1.
CTOD : Crack-Tip-Opening-Displacement
considers material ahead of the crak as a no. of miniature tensile specimens and
crack propagates as each of these tensile specimens fractures (Fig. 11-12)
δc δc
since εf = l ~
2ρ
{Gauge length ~2ρ} • fracture criterion is : δc = 2ρεf
for plane-stress (thin-plate) : δc = εf t (t = thickness)
• can show that G = σo δ (Eq. 11-32)
or Gc = λ σo δc (Eq. 11-33); here, λ depends on where CTOD is measured
unfortunately, CTOD (or COD) method is very sensitively dependent on the
specimen geometry - cannot have a single number !
• better approach is to use J-Integral Method (11-8)
2. J-Integral Mehod :
• Rice has shown that J-integral defined
as in Eq. 11-37 is path independent
around the crack
[J] = MN/m {or MPa-m}
• J is the potential energy difference
between two identically loaded
specimens with different crack
lengths (Fig. 11-14) :
J=
K2
∂U
= G = E (Eq. 11-38)
∂a
J= ⌠
⌡ (W dy - T
Γ
_u
ds) (Eq. 11-37)
_x
y
Fig. 11-13
Γ
x
ds
ASTM E813-81 : J-Test Procedure 2A
for a 3-point bend specimen, J = Bb
A = area under Load vs displacement curve
b = W - a (remaining ligament)
B = specimen thickness
page 8
KL Murty
MAT 450
Spring ‘96
• Multiple specimens with different crack lengths :
measure crack growth (∆a) due to loading to a given displacement level and
noting the load (P) : calculate J vs ∆a (see fig. 11-15b)
∆a is meaured after the test by heat tinting and fracturing
different initial crack sizes, ‘a’, would give different ‘b’ values and give the
J-R curve (R - resistance) (Fig. 11-15a) ⇒
determine the critical crack initiation elastic-plastic fracture toughness (JIC)
• Single Specimen and determine ‘a’ as a function of P
‘a’ is measured from compliance or using Potential Drop technique or
unloading compliance method
In either case, obtain data as in Fig. 11-15 & find JIC : still dimensional criterion is
checked for the test to be valid (but size limitation is not as stringent as in KIC) Eq. (pg. 367) for 3-point bend specimen : b ≥ 25
• for A533B steel : Valid JIC - b = 0.5 in
JIC
.
σo
vs
Valid KIC - B ~ 2 ft (!)
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