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Prevention and Control of Fracture in Metal Structures
Dr. Ayman A. Shama PhD, PE
Summary of the Cracking Problem in Structures
Types of Cracks
Corrosion Fatigue
Environmental Effects
Corrosion-Fatigue Cracking
Stress-Corrosion Cracking
Crack growth
Member Fracture Structural Failure
Material Properties related to Crack Propagation
Ductile Versus Brittle Behavior of Material
Possible Failure modes for metals
Ductile Brittle
General Comparison
Ductile Fracture Process
Brittle Fracture Process
Toughness of Material
Defined as the energy of mechanical deformation per unit volume prior to fracture
Units: Inch. Pound force per cubic inch
It is usually characterized by the area under a stress-strain curve in a slow tension test.
f
0
dvolume
energy = strain
f = strain at failure
= stress
Notch Toughness of Material
Charpy V-notch (CVN) impact test is widely used to evaluate notch toughness
Ductile-to-Brittle Transition
One of the primary functions of the Charpy test is to determine whether a material experiences a ductile-to-brittle transition with decreasing temperature.
Evaluation of Crack Growth using Principles of Fracture Mechanics
What is Fracture Mechanics? Fracture mechanics is an approach to evaluate the fracture
behavior of structural members characterized by pre-existing flaws or notches.
It is based on a stress analysis in the vicinity of the notch.
Practical design information required are:
The fracture toughness of the material as obtained from fracture-mechanics tests
The nominal stress on the structural member being analyzed
Flaw size and geometry of the structural member being analyzed
The three basic modes of crack surface displacement are:
Mode I: opening or tensile mode
Mode II: sliding mode
Mode III: tearing mode
In isotropic materials, brittle fracture usually occurs in Mode I
Brittle Fracture of Cracked Members
An applied stress may be amplified or concentrated at the tip of a crack, the magnitude of this amplification depends on crack orientation.
Flaws are sometimes referred to as stress raisers
Stress Concentration
Stress Concentration
t
m
a2
= the magnitude of the nominal applied tensile stress t= the radius of curvature of the crack tip a = the length of a surface crack, or half the length of an internal crack
Stress Concentration
The ratio m/ is denoted the stress concentration factor kt:
t
mt
a2k
1. The main principle of fracture mechanics is to establish parameters to represent the driving force or applied stress.
2. For linear fracture mechanics a parameter namely stress intensity factor (K) relates the stress field magnitude in the vicinity of a crack tip; size; shape; and orientation of the crack
3. Since brittle fracture usually occurs in Mode I for isotropic materials then K will be further denoted as KI
4. The calculation of KI is analogous to the stress demand in traditional and represent the driving force.
5. Another parameter needs to be evaluated , which is a measure of fracture toughness, analogous to the member capacity and represents the resistance force to crack extension Kc.
6. To prevent brittle fracture, the engineer keeps the calculated applied stress intensity factor , KI, below the measured fracture toughness, Kc
7. Other parameters such as the J-integral (J) and the crack-tip-opening displacement (CTOD) are also established for the elastic-plastic regime.
Fracture Mechanics Methodology
Irwin (1957) defined the stress intensity factor by means of the following limit for mode crack I:
m0
I2
limK
If we consider the line crack to be the limiting case of the elliptical hole then
a2m
therefore
aa
22
limK0
I
Stress Intensity factor
KI has units of : mMPa.inksi
All Structural members that have flaws can be loaded to various levels of KI
KI should always be kept below a limiting (critical) value Kc to prevent fracture
The solutions of the stress intensity factors have been obtained for a wide variety of problems.
aKI
Infinite-width plate containing a through-thickness crack
)b/a(faKI
Finite-width plate containing a through-thickness crack
32ba525.1ba288.0ba128.01)b/a(f
a12.1KI
Infinite-width plate containing an edge crack
)b/a(faKI
Finite-width plate containing an edge crack
432ba39.30ba72.21ba55.10ba231.012.1)b/a(f
)b/a(faKI
Finite-width plate containing a double edge crack
32ba93.1ba197.1ba203.012.1)b/a(f
4/1
2
2
22
I cosc
asin
Q
aK
Embedded Elliptical Crack in Infinite Plate
Surface Crack in Infinite Plate
kI MQ
a12.1K
5.0
t
a2.10.1Mk
)b/a(faKI
Edge crack in beam in bending
432ba14ba08.13ba33.7ba40.112.1)b/a(f
tb
M62
The stress intensity factor for the centrally applied tension load is
)b/a(fabt
PK aa
I
Superposition of stress Intensity factor Edge crack subjected to tension plus bending loads
The stress intensity factor for the bending moment is:
)b/a(fatb
M6K b
2
bI
The total stress intensity factor of this case is:
a)b/a(fb
e6)b/a(f
bt
PKKK bab
IaII
2
ysy
K
2
1r
For plane-stress conditions For plane-strain conditions
2
ysy
K
6
1r
Crack-Tip Deformation and Plastic Zone size
Adopting Irwin approach
2/12
ys
app
appIeff
5.01
aK
for app = 0.75 ys
a18.1K appIeff
for app = ys
a40.1K appIeff
A large plate containing a 0.2 in. center crack is subjected to a tensile stress of σ0 = 30 ksi .
a.) Determine the plastic zone size for (σys = 80 ksi ) steel and (σys = 40 ksi ) steel
inksi81.161.030aK5.0
0I
Example
.in007.080
81.16
2
1K
2
1r
22
ys
Iy
Step-1 evaluate the stress intensity factor
for (σys = 80 ksi ) steel app = 0.375 ys no need to use K Ieff Step-2
.in039.040
835.19
2
1K
2
1r
22
ys
Ieffy
for (σys = 40 ksi ) steel app = 0.75 ys use K Ieff
inksi835.1981.16X18.1K18.1K IIeff
Plastic zone size is about 20% of the crack size !!
The J-integral represents a way to calculate the strain energy release rate , or work per unit fracture surface area, in a material.
The J-Integral
the strain energy stored in the member
P2
1U
the strain energy release rate
a
U
t
1
A
UG
For linear-elastic behavior, the J-integral is identical to G:
E
K1GJ
2I
2
II
For nonlinear behavior, the J-Integral is determined using computational methods such as the finite element method
The Crack-Tip Opening Displacement (CTOD)
sys
2I
IE
K
The CTOD relationship for a center crack in a wide plate is
Estimation of Force Resistance Parameters-Kc- Jc- c )
For LEFM we are concerned with SIF therefore emphasis will be first on Kc
Fracture toughness, Kc, is defined as the resistance to the propagation of a crack in structural member.
Factors affecting Fracture Toughness Temperature
As the temperature increases, fracture toughness increases
Loading rate
)ondsecper.in/.in(ratestrain
Fracture toughness of structural materials increases with decreasing loading rate
Loading rate and temperature combined
The temperatures at which the fracture toughness levels begins to increase significantly depend upon the loading rate
Constraint
Constraint
As the thickness is increased, the constraint increases, and the flow stress curve is raised.
Fracture toughness of structural materials increases with decreasing constraint.
Under conditions of low temperature, rapid loading, or high constraint, ductile materials may not exhibit any deformations before fracture.
Conclusion
Fracture Toughness Testing
The maximum KIC and KJC capacities that can be measured for a specimen of thickness B are:
sysIC5.2
BK
30
EBK
sysJC
ASTM Standard Fracture Tests
ASTM Test Method E-399: Standard Test Method for Plane Strain Fracture Toughness of Metallic Materials
For the determination of KIC Plane-strain critical fracture toughness value is
obtained at slow loading rates. Fracture is sudden, resulting in unstable brittle
fracture with little or no deformation.
ASTM Standard Fracture Tests
ASTM Test Method E-399--Annex A7. Special Requirements for Rapid Load Plane-Strain Fracture Toughness KIC(t) Testing
For the determination of KIC(t) Plane-strain critical fracture toughness value is
obtained at intermediate loading rates, where t = time to maximum load in seconds.
Constraint is maximum and failure is sudden, resulting in unstable brittle fracture with little or no deformation.
ASTM Standard Fracture Tests
ASTM Test Method E-1221: Standard Test Method for Determining Plane-Strain Crack-Arrest Fracture Toughness, K la, of Ferritic Steels
For the determination of KIa (KID) (crack-arrest toughness).
Linear-elastic behavior during dynamic or impact loading results in rapid unstable brittle fracture.
ASTM Standard Fracture Tests
ASTM Test Method E-813: Standard Test Method for JIc, A Measure of Fracture Toughness
For the determination of JIc JIc is a measure of the fracture toughness at the
onset of slow stable crack extension. Behavior is non-linear elastic plastic.
ASTM Standard Fracture Tests
ASTM Test Method E-1737: Standard Test Method for J-Integral Characterization of Fracture Toughness
A new test method has been developed to cover all J-integral test results such as Jc, JIc, in one standard.
Behavior would be elastic-plastic with or without stable crack extension.
ASTM Standard Fracture Tests ASTM Test Method E-1820-96: Standard Test Method for Measurement of Fracture Toughness
For the determination K, J, CTOD (): This test is developed for materials where the
type of behavior and thus the type of test needed also is not known before testing.
A bend or compact specimen is tested and the P-DCMOD and records are analyzed to determine either K, J, or values.
KIC Critical Stress Intensity Factor (Fracture Toughness)
A Fracture criterion determines how much fracture toughness is necessary for a particular structural application.
Fracture criteria are related to the three levels of fracture performance, namely plane strain, elastic plastic, or fully
plastic
Fracture-Criteria
Originally established for large welded ship hull structures and then adopted for usage in other classes of structures.
Assumes brittle fracture is not likely to occur in an element if the material absorbs more than 15 ft-lb Charpy V-notch impact energy at the anticipated operating temperature
The 15-ft-lb CVN Impact Criterion
Transition-Temperature Criterion Fracture characteristics is described in terms of the transition
from brittle to ductile behavior as measured by Charpy V-notch impact test
In addition to notch toughness this criterion uses also a transition-temperature to specify the level of performance.
Through-Thickness Yielding Criterion
This criterion is based on two observations :
First, increasing the design stress in a particular application results in more stored energy in a structure
Second, increasing plate thickness promotes a more severe state of stress, namely, plane strain.
This criterion requires that in the presence of a large sharp crack in a large plate, through-thickness yielding should occur before fracture
For through-thickness yielding to occur in the presence of a large sharp crack in a large plate, plane stress conditions are required:
.in2tfortK ysc
The equation defines the plane stress condition at which considerable through-thickness yielding begins to occur.
To ensure linear elastic (plane strain) state of stress at the crack tip and KI = KIc at initiation of crack propagation:
2
ys
IcK5.2t,a
KIc = Critical value of SIF (fracture toughness)
ys = Yield strength of the material
the leak-before-break criterion assumes that a crack of twice the wall thickness in length should be stable at a stress equal to the nominal design stress
Leak-Before-Break Criterion
2ys
22I
)/(5.01
tK
At fracture , KI = Kc and because standard material properties were usually obtained in terms of KIc, the following relation between Kc and KIc has to be used:
)4.11(KK 22Ic
2C and
2
ys
IcK
t
1
Therefore
2
2ys
2Ic2
Ic2ys
2
t
K4.11K
)/(5.01
t
Examples
1-fracture load for a cracked beam under bending
A rectangular section beam has a depth 2c = 150 mm, width t = 25 mm, and length L = 2.0 m. The beam is loaded as simply supported with a concentrated load P at the center. A notch is machined into the beam on the tension side opposite the point of application of P. The depth of the notch was increased by fatigue loading until a = 15 mm. The beam is made of 17-7PH precipitation hardening steel with fracture toughness 77 MPa (m)0.5 and yield stress 1145 MPA. a. Determine whether or not plane strain conditions are satisfied for the beam. b. Determine the fracture load P.
04.1ba14ba08.13ba33.7ba40.112.1)b/a(f432
10.0150
15
b
a
mm31.111145
1000775.2
K5.2t,a
22
ys
Ic
Plane strain conditions are satisfied
)b/a(fa
Kor)b/a(faK Ic
Ic
tb2
PL3
tb4
PL6
tb
M6222
Therefore
)b/a(faL3
tKb2P
)b/a(fa
K
tb2
PL3 Ic2
Ic
2
kN9.65N6585604.1(15)1000X2(3
)100077)(25()150(2
)b/a(faL3
tKb2P
2Ic
2
2: Evaluation of the Fracture Load for a Mechanical Tool A mechanical tool as shown in Fig. is made of AISI 4340. The dimensions of the tool are d = 250 mm, b = 60 mm, and the width t = 25 mm. Determine the magnitude of the fracture load P for the crack length of a = 5 mm. Describe the stress field condition in the vicinity of the crack.
By using the superposition method for :
)b/a(faKI
Case of finite-width plate containing an edge crack
432ba39.30ba72.21ba55.10ba231.012.1)b/a(f
and
)b/a(faKI
Case of Edge crack in beam in bending
432ba14ba08.13ba33.7ba40.112.1)b/a(f
tb
M62
Therefore,
For the crack length of a = 5 mm, a / b = 5 / 60 = 0.0833
163.1ba39.30ba72.21ba55.10ba231.012.1)b/a(f432a
047.1ba14ba08.13ba33.7ba40.112.1)b/a(f432b
a)b/a(fb
e6)b/a(f
bt
PKKK bab
IaII
a)b/a(fb
e6)b/a(f
bt
PKKK bab
IaII
From table the critical fracture toughness factor of this material is 59 MPa (m)0.5
Therefore,
1000/5047.160
250x6163.1
25x60
10Px100059
6
kN8.25P
The total maximum stress is
tb
Pe6
bt
P2m
25x60
250x1000x8.25x6
25x60
1000x8.252
MPa1503MPa2.447 ys
3- Prioritize structural repairs at a site An engineer inspected two structures in a site. The first structure is a truss made up of Aluminum tubular sections each has an outside diameter of 4 in. and wall thickness of 0.25 in. The inspection revealed circumferential through-thickness cracks of 0.60 in. length. A recent load rating of this structure showed that these members are currently carrying 47 kips tension loads. The second structure is a high strength steel pressure vessel that is carrying 5000 psi of internal pressure and has 30 in. nominal diameter, and 0.60 in. wall thickness. Inspection of this structure revealed a surface flaw of length 2 in. and an a/2c ratio of 0.25. Which structure should be repaired first?
Material ys (ksi ) Kic (ksi.in0.5)
Aluminum alloy 64 21
High strength steel 180 220
)b/a(faKI
a= 0.30 in.
b= 5.8875 in.
a/b= 0.30/5.8875=0.051
Truss Structure
01.1ba525.1ba288.0ba128.01)b/a(f32
Therefore
ksi4.21)01.1(30.0
21
)b/a(fa
KIc1max
since
Therefore
)b/a(faKI
222 .in94.25.344
area
kips36.51)4.2(4.21AP
Therefore the maximum tension load the member can sustain is:
Factor of safety against fracture: 093.147
36.51FOS 1structure
kI MQ
a12.1K
The general relation among KI , , and a for a surface flaw
k
Ic2max
M.a12.1
QK
and
Pressure Vessel Structure
from the figure try a value of 1.4 for a//2c=0.25
396.1)5.083.0(2.115.0t
a2.10.1Mk
.in5.0a.in2c2for25.0c2/a
psi5320)30(
)1000)(60.0)(133(2
d
t2p 2max
For the first trial assume Mk =1.396 and Q=1.4, therefore
ksi133396.1.)5.0)(14.3(12.1
4.1220
M.a12.1
QK
k
Ic2max
The maximum internal pressure p that can be sustained
t2
pd2max
For p = 5320 psi 74.0180
133
ys
2max
Therefore from figure Q=1.35
Using Mk=1.396 and Q=1.35, try a second iteration
ksi5.130396.1.)5.0)(14.3(12.1
35.1220
M.a12.1
QK
k
Ic2max
psi5220)30(
)1000)(60.0)(5.130(2
d
t2p 2max
For p = 5220 psi 725.0180
5.130
ys
2max
Therefore from figure Q=1.36
Using Mk=1.396 and Q=1.36, try a third iteration
ksi8.130396.1.)5.0)(14.3(12.1
36.1220
M.a12.1
QK
k
Ic2max
psi5232)30(
)1000)(60.0)(8.130(2
d
t2p 2max
For p = 5232 psi 726.0180
8.130
ys
2max
Therefore from figure Q = 1.365 no further iterations are required
Factor of safety against fracture:
0464.15000
5232FOS 2structure
093.1FOS0464.1FOS 1structure2structure
Therefore structure-2 should be repaired first
Fracture-Control Plans
A fracture-control plan is a specific set of guidelines and recommendations developed for a particular structure. They include but not limited to:
1. Knowledge of the service conditions to which the structure will be subjected.
2. The use of structural materials with adequate fracture toughness.
3. Elimination or minimization of stress raisers.
4. Control of welding procedures,.
5. Proper inspection plans of the structure.
References
1. Anderson, T.L. “ Fracture Mechanics: Fundamentals and Applications”, Taylor and Francis, 2005
2. Tada, H., Paris, P., and Irwin, G. “The Stress Analysis of Cracks Handbook”, ASME Press, 2000
3. Paris, P. and Sih, G. “Stress Analysis of Cracks”, Research Report, Department of Mechanical Engineering, Lehigh University, 1967
4. Irwin, G. “ Analysis of stresses and Strains Near the End of a Crack Transversing a Plate”, Journal of Applied Mechanics, Vol. 24, 1957
5. Wells, A., “Unstable Crack Propagation in Metals:”, Cranfield Crack Propagation Symposium, Vol 1, 1961
7. Rolfe, S. and Barsom, J, “ Fatigue and Fracture and Fatigue Control in Structures” ASTM, 1999
8. E 399-90 (Reapproved 1997). Standard Test Method for Plane-Strain Fracture Toughness of Metallic Materials, ASTM, Vol. 03.01.
9. E 399-A-7. Special Requirements for Rapid-Load Plane-Strain Fracture Toughness KIc(t ) Testing, ASTM, Vol. 03.01.
10. E 1221-96. Standard Test Method for Determining Plane-Strain Crack-Arrest Fracture Toughness of Ferritic Steels, ASTM, Vol. 03.01.
11. E 813-89. Standard Test Method for JIc, A Measure of Fracture Toughness, ASTM, Vol. 03.01.
12. E 1737-96. Standard Test Method for J-Integral Characterization of Fracture Toughness, ASTM, Vol. 03.01.
References-continued