Fracture Mechanics
Prof. Bal Annigeri
Script
Mon 8/18 3:00 PM – 4:30 PM
The beginnings of fracture mechanics – Griffith 1920’s
Types of Fracture
Brittle Fracture
Breakage without prior plastic deformation and very fast crack propagation no
shear lips present
Ductile Fracture
General comments about Failure of Materials
Fatigue : Breakage following cyclic loading
Yielding : Breakage following plastic deformation
Buckling: Failure due to Elastic instability
Excessive Deflection
Corrosion
Creep/Creep Rupture
Wear
Ref. p. 4 text for examples of material failure
Concept of Notch Toughness Testing
Engineering Stress-Strain Curve
Notch Toughness = Area under Stress-Strain Curve
Toughness = Ability of a smooth member to absorb energy usually when loaded slowly
Notch Toughness = Ability of a flaw containing member to absorb energy usually when
loaded dynamically
Notch Toughness Testing
Charpy V-Notch
Dynamic Tear
KIc (fracture toughness)
Kid (dynamic fracture toughness)
CTOD
Jc Integral
Materials Issues: Cracks, Corrosion Pits, Flaws …
Environment Issues: Loading, Temperature, Load rate, Constraints (strain/load), Environment
Constitutive Laws of Material behavior
Linear Elastic solid (Hooke’s Law): Stress proportional to strain
Elastic-Plastic solid : Increment in stress proportional to strain
Plot: Energy Absorbed vs Temperature at different rates of loading
Concept of Nil-Ductility Temperature (NDT): Below NDT -> Material is Brittle
Primary Factors that Control Brittle Fracture)
Material Toughness (Fracture Toughness) K Stress Intensity Factor (Prof. Kies)
Kc - Critical Stress Intensity Factor (Plane Stress)
KIc - Critical Stress Intensity Factor (Plane Strain)
KID - Critical Stress Intensity Factor (Plane Strain, Dynamic Loading)
KIscc - Critical Stress Intensity Factor (Plane Strain, Corrosion)
CTOD - Critical Crack Tip Opening Displacement
J Integral
J-R Curves (J-Resistance)
K-R Curves (K Resistance)
Fatigue Loading – Importance in Practice
Converting a Complex Loading History into a Constant Amplitude Loading
Rain-Flow Cycle Pairing
Plot Flaw Size vs Number of Load Cycles -> Three regions
Region I – Micro-mechanics (Fracture Initiation)
Region 2 – Meso-mechanics
Region 3 – Macro-mechanics (Fracture Propagation)
Inglis-Kolosov – Stresses in an Elliptical Hole in a large plate
Fracture Modes
Mode I - Opening
Mode II – In plane Shear
Mode III – Out of Plane Shear
Stresses at the Crack Tip
Mon 8/18, 4:30-4:45 PM Break
Mon 8/18, 4:45-6:10 PM
Stress Intensity Factors Examples
Center through thickness crack, size 2a in a large plate uniaxial tension
KI = sigma sqrt(Pi a)
Plate with Semi-Elliptical Surface Crack (Not through Thickness; Depth a , Size 2c)
KI = 1.12 sigma sqrt(Pi a/q) ; q=shape factor (Sneddon) , p. 39 text
Important: Crack Growth Occurs WHEN K -> Kc
Note: Kc is a material Property and K is a measure the strength of the stress singularity
Compare with applied stress -> Yield Stress as criterion for yield failure
Plot of sigma vs flaw size for constant Kc
Comparison with Euler Buckling Curve
Fracture Instability
Stress Tensor Components in a small Square 2D region under a General State of Stress
Conservation of Linear Momentum – Equilibrium Equations
Review of Solution of Stress Analysis Problems
Equations of Equilibrium
Compatibility
Stress-Strain Law
Boundary Conditions
Components of Stress Tensor and Tractions acting on an Inclined Plane (Coordinate Axes
Rotation)
Mohr Circle for Stress (2D)
Example: p. 25 Dieter
Comments on paper by Annigeri 2007 on Kinematics
Mon 8/18, 6:10-6:25 PM Break
Mon 8/18, 6:25- PM
Exercise: Identify five examples of catastrophic fracture failures. Make a brief description of the
situation in each case and list the factors involved.
Silver Bridge (St Mary’s) Collapse – Baek Nakgon
Helikopter Service Flight 451 – TaeHwan
DeHavilland Comet Failures – Yong
UA811 Cargo Door Failure – Jae Chul
Sikorsky-S76C Failure – Jung Geun
Other Examples:
Boston Molasses Disaster
UA Flight 232 – Sioux City Crash
Hyatt Regency Walkway Collapse
I-35W Mississippi River Bridge Collapse
Markham Colliery Mine Disaster
Sinking of the Titanic
WTC Buildings Collapse
Liberty Ship Fatigue Failures
Sea Gen Oil Off-shore Drilling Rig Collapse
Tues 8/19 2:00 PM – 3:30 PM
Strain: From 1D to 2D to 3D (See Paper 1, Eqn 2)
Solution of Boundary Value Problems
Equilibrium
Compatibility
Constitutive Law
Boundary Conditions
Principle of Linear Superposition – Widely Used in LEFM
Equivalence of the cracked body in Tension with the sum of a uncracked body in Tension
plus the stress in a body containing a crack stressed enough to give the zero stress at the crack
surfaces in the cracked body
Through this the focus shifts entirely on the crack tip
The Williams eigensolution for Fracture Cracks in Modes I, II, II
Singular solutions : stress ~ 1/sqrt(r)
(i.e. stress -> infinity at crack tip)
Survey of some Simple Solutions
Crack of length 2a in an infinite domain
Plate of Finite Width under Uniaxial Tension with Crack of length 2a
Plate of Finite Width under uniaxial tension with Double Edge Notch
Example: Finite Plate, b = 1, sigma = 50 ksi, KIc = 100 ksi sqrt(in)  Find critical a
Ans: ac = 0.7
Elliptical Crack (Sneddon)
Compact Tension Specimen
Crack Mouth Opening (Proportional or not with the Load)
Tues 8/19 3:30 PM – 3:45 PM Break
Tues 8/18 3:45 PM – 5:20 PM
Stresses at a Crack Tip (Plane Stress)
Recall sigma_yy  infinity as x  0.
However there is the yield stress sigma_ys
For an elastic-perfectly plastic material a Plastic Zone of radius r_Y develops
For Plane Strain, the yield stress is bigger because of the constraint (Irwin)
Improved estimate of the plastic zone radius by extension of zone  r_p = 2 r_Y
Actual Shape of the Plastic Zone
Plane Strain – Bean – Smaller
Plane Stress – Circle – Larger Zone
However: The Plastic Zone Size is Much smaller than the Crack Size
Note: ~ 70% of Fracture problems can be handled by LEFM
Griffith’s Observations on Fracture of GLASS Fibers (Size and Surface Finish Effects)
Recall: At the Crack Tip the Stress goes to Infinity, REGARDLESS of the size of the load
Griffith Theory – When a crack forms, surfaces and surface energy appear whilst elastic
energy decreases
Uo - Elastic energy of the Uncracked Body
Ua – Decrease in Elastic Energy of the Body when introducing the Crack = Pi s^2 a^2/E
U_gamma - Increase in Surface Energy when introducing the Crack = 4 a gamma
Energy Balance: U = Uo – Ua + U_gamma
Energy is Minimum when dU/da = 0  sigma sqrt(Pia) = sqrt(2 E gamma)
Energy Release Rate : G = Pi s^2 a/E = KI^2/E = 2 gamma
Conclusion: Crack is Unstable when sigma sqrt(Pia) > sqrt(2 E gamma)
Orowan: Addition of Energy of Plastic Deformation
Energy Release Rate : G = Pi s^2 a/E = 2 (gamma + gamma_P)
Conversion Factors for SIF from SI to English 1 ksi sqrt(in) ~ 1 MN m^-3/2
Experimental Determination of Fracture Toughness (ASTM Standard)
Plot Kc vs Specimen Thickness  Asymptotic to KIc
P. 81 Text
Experimental precautions: Specimen size (Importance of B/r_Y ~ 47)
Load Strategy: First Cycle the load then Increase it Monotonically (Fig. 3.20, 3.21)
Tues 8/19 5:20 PM – 5:35 PM Break
Tues 8/19 5:30 PM – 7:00 PM
Exercise – Paper 2 , p. 878 from Reading Assignment – Angle Crack
How to Design Structures with Cracks: What is the maximum flaw size the structure can
tolerate under the given loading conditions? a_crit = (KIc/C s)^2
Discussion of Non-Destructive-Evaluation (NDE) of materials
One wants to know the biggest crack size that the inspection will miss
Probability of Flaw Detection + Confidence Level
Exercise: Design a Pressure Vessel Diameter = 30 in , Wall Thickness => 0.5 in for p = 5000 psi
Exercise: Graph on p. 39 text
Wed 8/20 2:00 PM – 3:30 PM
Fractured samples distributed with a magnifying glass. Students asked to describe the
component, loading and fracture type.
Example: p. 150 text - Pressure Vessel Problem from yesterday. Do all the calculations and
reproduce the results in table on p. 157.
Constitutive Relations
To connect kinetic variable to kinematic variable (force to motion – stress to strain)
Hooke’s Law for 3D for Isotropic Solids (Component Form and Index Notation)
Normal Strains - Normal Stresses
Shear Strains – Shear Stresses
Concept of Strain Energy Density W
Concept of Separation of the Stress Tensor into Hydrostatic and Deviatoric components
The Yield Condition
Von Mises
Tresca
Wed 8/20 3:30 PM – 3:45 PM Break
Wed 8/20 3:45 PM – 5:15 PM
Energetics of Cracked Bodies
Energy release rate for prescribed loading
Energy release rate for prescribed displacement
Energy release rate for Prescribed Load P of an Edge Cracked Specimen:
P=Load ; Delta = Load Point Displacement ( = P C where C = compliance)
Compliance C = Delta/P
; Stiffness K = P/Delta
Plot P-Delta
Elastic Strain Energy U = (1/2) P Delta
Work Done by Load W = P Delta
Potential Energy Pi = U-W = -(1/2) P Delta
Energy Release Rate G = - dPi/da = d((1/2) P Delta)/da
= (1/2) (dP/da) Delta + (1/2) P d Delta/da = (1/2) P d Delta/da
= (1/2) P^2 (d C/da)
Plot P vs Delta for a then for a+da, for a constant load, the area under P between the
lines =Gda
Energy release rate for Prescribed Displacement Delta of an Edge Cracked Specimen:
Imagine first sample without crack, then introduce crack (displacement does not change
since it is prescribed)
Elastic Strain Energy U = (1/2) P Delta
Work Done by Load W = 0
Potential Energy Pi = U-W = (1/2) P Delta
Energy Release Rate G = - dPi/da = -d((1/2) P Delta)/da
= -(1/2) (dP/da) Delta - (1/2) P d Delta/da = -(1/2) (dP/da) Delta =
= (1/2) (P/C) Delta (d C/da) = (1/2) P^2 (dC/da)
Plot P vs Delta for a then for a+da, for a constant load, the area under P between the
lines =Gda
Energy release rate is the same for both cases
J-Integral
Crack containing body – Part of Boundary subjected to traction, part to displacement
J = int(W dy) is a measureof the strain on the notch tip
Example: infinite band of height 2h containing a semi-infinite crack
JIc = KIc^2 (1 – nu^2)/E = [Pi (1-nu^2)/E] s^2 a (small crack in a large body)
Dugdale-Barenblatt Yield Strip Model
Crack of size 2c flanked at both sides by yield strips of size d (where the stress is sY)
J-integral calculation
Concept of Crack Tip Opening Displacement (CTOD)
Conditions at the Crack Tip when Yielding takes place
Wed 8/20 5:15 PM – 5:30 PM Break
Wed 8/20 5:30 PM – 7:00 PM
Exercises: Problems 2f-2j Knott-Withey
Thurs 8/20 2:00 PM – 3:30 PM
Review of J Integral
Equal to Energy release rate G
Comparison Linear vs Nonlinear deformation
At the critical J, Jc , the crack becomes unstable
Lines of maximum shear stress
In Plane stress CT specimen
In Plane Strain CT specimen
The K-R curve (Resistance Curve)
Recall: Linear superposition to show the equivalence between
A crack (size 2a) containing plate under uniaxial load
A plate without crack under uniaxial load (traction sigma at the location of the crack) +
An unloaded crack containing plate with traction –sigma applied at the crack
Dugdale Model: Addition of length 2d to 2a (to make the cracked zone 2c) to account for
plastic zone at crack tip
Same as above except in the plastic zones ahead of the crack tip the traction is
-(sigma – sigmaY) (elastic-perfectly plastic)
so that on the actual plastic zone, the stress is sigmaY
Plastic Deformation leads to Crack Blunting
The singularity at the crack tip is eliminated
The size of the plastic zone d is found
J/Jssy = (8/Pi^2)(sY/s)^2 ln(sec(Pi s/2 sY))
Exercise: Expand ln(sec(x) in Taylor series and evaluate
Thurs 8/20 3:30 PM – 4:00 PM Break
Thurs 8/20 4:00 PM – 5:30 PM
Mode III (Antiplane Loading)
Must satisfy: Equilibrium, Compatibility, Stress-Strain, Boundary Conditions
Laplace’s equation for u3
Cauchy-Riemann Eqns
Elastic-Plastic Solution (McClintock-Irwin)
Equilibrium
Yield Condition
Constitutive Equations (Hencky)
Shear Strains, Displacement, Slip vs r, theta
Thurs 8/20 5:20 PM – 5:35 PM Break
Thurs 8/20 5:35 PM – 7:00 PM
Fatigue Crack Growth
Cyclic Loading – Pmax – Pmin , R ratio
Crack Size a may increase with the number of load Cycles N
Minimum detectable (initial) cracksize a_i
At some size a_f the part fails
For safety choose a_designLife < a_f
Stress Intensity factor K and Stress Intensity factor Range Delta K
Focus on the rate of growth of crack da/dN size vs Delta K
Measurements show three regions: Initial, Steady State, Final
Steady State Crack growth Rate - Delta K relationship (Paris Law)
Example: p. 212 text – Fatigue Crack Growth
Fri 8/21 12:00 PM – 1:00 PM Lunch
Fri 8/21 1:00 PM – 2:00 PM
Example: p. 212 text – Fatigue Crack Growth – contd
Exact and numerical integration
Linear Cummulative Damage D (Miner)
For two distinct load cycles
D = (N1/Nf1) + (N2/Nf2)
Use of Rainflow Cycle Counting to define Load Cycles
Fri 8/21 2:15 PM – 2:30 PM Break
Fri 8/21 2:30 PM – 3:45 PM
NASGRO code demonstration.
Explanation of the Portal Interface
NASFLA program
Select Crack Configuration
Select Material
Define Spectrum
Running
Examination of Results
Fri 8/21 3:45 PM – 3:50 PM Break
Fri 8/21 3:50 PM – 4:15 PM
NASGRO Exercises: Through Crack vs Surface Crack