Chapter-2
Linear Elastic Fracture
Mechanics
INTRODUCTION TO LEFM
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Concepts of fracture mechanics before 1960; applicable only to
materials that obey Hooke’s law
Since 1960; fracture mechanics theories developed to account for
various types of nonlinear material behavior (plasticity and
viscoplasticity) as well as dynamic effects
All of these more recent results; extensions of Linear Elastic
Fracture Mechanics (LEFM)
This chapter
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Energy, and stress intensity approaches to linear fracture
mechanics
Early work of Inglis and Griffith
Introduction to energy release rate and stress intensity parameters.
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Later chapters about LEFM
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Chapter 7 and Chapter 8; laboratory testing of linear elastic
materials
Chapter 9; application of LEFM to structures
Chapter 10 and chapter 11; apply LEFM to fatigue crack
propagation and environmental cracking,
Chapter 12; numerical methods for computing stress intensity factor
and energy release rate.
ATOMIC VIEW OF FRACTURE
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A material fractures when sufficient stress and work are applied at
atomic level to break bonds that hold atoms together
Bond strength is supplied by attractive forces between atoms
Figure 2.1; schematic plots of potential energy and force vs separation
distance between atoms
Equilibrium spacing occurs where potential energy is at a minimum
A tensile force is required to increase separation distance from the
equilibrium value
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This force must exceed the cohesive force to sever the bond
completely
Figure 2.1 Potential energy and force
as a function of atomic separation. At
equilibrium separation xo potential
energy is minimized, and attractive and
repelling forces are balanced
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It is of interest to estimate cohesive strength at the atomic level; it can
be shown that cohesive stress σ0 is given by
E : Young’s modulus
 or x0: atomic spacing (approximately)
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Surface energy can be estimated as
Surface energy per unit area γs is equal to one-half of fracture energy
because two surfaces are created when a material fractures. Therefore
STRESS CONCENTRATION EFFECT OF FLAWS
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Previous section; theoretical cohesive strength of a material is
approximately E / π
However; experimental fracture strengths for brittle materials are
typically three or four orders of magnitude below this value
Experiments by Leonardo da Vinci, Griffith, and others; discrepancy
between actual strengths of brittle materials and theoretical estimates
was due to flaws in these materials
Fracture cannot occur unless stress at atomic level exceeds cohesive
strength of material
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First quantitative evidence for stress concentration effect of flaws;
Inglis who analyzed elliptical holes in flat plates
Elliptical hole 2a long by 2b wide with an applied stress perpendicular
to major axis of ellipse; Fig 2.2
He assumed that hole was not influenced by plate boundary
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Plate width >> 2a, and plate height >> 2b
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Stress at tip of major axis (Point A) is given by
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Ratio σA/σ defined as stress concentration factor kt
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When a >> b, hole is circular and kt = 3.0; well-known result in most
strength-of-materials textbooks
Figure 2.2 Potential energy and force
as a function of atomic separation. At
equilibrium separation xo potential
energy is minimized, and attractive and
repelling forces are balanced
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As major axis a increases relative to minor axis b, elliptical hole begins
to take on the appearance of a sharp crack; better to use radius of
curvature ρ
When a >> b
Good approximation for stress concentration due to a notch that is not
elliptical except at the tip
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Previous equation; infinite stress at tip of sharp crack; ρ = 0
Major concern; no material can withstand infinite stress
Material with sharp crack should theoretically fail even at near-zero
load
This paradox motivated Griffith to develop fracture theory based on
stress rather than local stress
Infinitely sharp crack is not possible in real materials
Metals deform plastically; blunting of sharp crack
In the absence of plastic deformation, minimum crack tip radius is of
the order of atomic radius
Substituting ρ = 0 in previous equation
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Assuming that fracture occurs when σA = σc, and combining previous
equations
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Last equation; rough estimate of failure stress
Continuum assumption of Inglis is not valid at atomic level
Gehlen and Kanninen; similar result from numerical simulation of a
crack in a 2D lattice; discrete atoms connected by nonlinear springs
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α is a constant of the order of unity, based on atomic forcedisplacement law.
GRIFFITH ENERGY BALANCE
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First law of thermodynamics
System goes from non-equilibrium to equilibrium state; decrease in
energy
1920; Griffith applied this idea to crack formation
Plate with crack of length 2a, subjected to constant stress σ (Fig)
Assume that plate width >> 2a; with plane stress valid
Note that, plates are same if a >> b
For crack to grow, sufficient potential energy needed in plate to
overcome surface energy of material
Griffith energy balance for incremental increase in crack area dA
under equilibrium conditions:
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Serious distinction between crack area and surface area
Crack area; projected area of crack; 2aB in current example
Crack includes two matching surfaces; so crack area is 2A
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Griffith approach also applicable to other crack shapes
Fracture stress for penny-shaped flaw (circular) embedded in material:
a is crack radius and υ is Poisson’s ratio
Comparison with Critical Stress Criterion
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Griffith model based on global energy balance
For fracture to occur, energy stored in structure must be sufficient
to overcome surface energy of material.
Fracture involves breaking of bonds
Stress on atomic level must be equal to cohesive stress
This local stress intensification can be provided by flaws in material
Similarity between stress-concentration and energy-balance equations
for is obvious
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Predictions of global fracture stress from Griffith approach and local
stress criterion differ by less than 40%.
The two approaches are consistent with one another for the case of
sharp crack in an ideally brittle solid
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Apparent contradiction when the crack-tip radius is significantly
greater than atomic spacing.
Griffith model implies that fracture stress is insensitive to ρ
Inglis stress analysis; σf must vary with 1/√ρ for σc to be attained at
notch tip
Consider a crack with ρ = 5x10-6 m
Crack would appear sharp under a light microscope, but ρ would be
4 orders of magnitude larger than atomic spacing in a typical
crystalline solid
Local stress approach would predict global fracture strength 100
times larger than Griffith equation
Actual material behavior is somewhere between these extremes
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This apparent discrepancy can be resolved by viewing fracture as a
nucleation and growth process
When global stress and crack size satisfy Griffith energy criterion,
there is sufficient thermodynamic driving force for crack growth, but
fracture must first be nucleated
Nucleation of fracture can come from a number of sources, eg
Microscopic surface roughness at tip of flaw could produce sufficient
local stress concentration to nucleate failure
Another possibility; Fig 2.5; sharp microcrack near tip of a
macroscopic flaw with a finite notch radius
Macroscopic crack magnifies stress in the vicinity of microcrack,
which propagates when it satisfies Griffith equation
Microcrack links with large flaw, which then propagates if Griffith
criterion is satisfied globally
This type of mechanism controls cleavage fracture in ferritic steels;
Chapter 5
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Figure 2.5 A sharp microcrack at the tip of a macroscopic crack
Modified Griffith Equation
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Griffith equation is valid only for ideally brittle solids
Griffith obtained good agreement with experimental fracture
strength of glass
However, Griffith equation severely underestimates fracture
strength of metals
Irwin and Orowan independently modified Griffith expression to
account for materials that are capable of plastic flow
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γp is plastic work per unit area of surface created and is typically much
larger than γs.
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Although Irwin and Orowan originally derived their equation for metals,
it is possible to generalize Griffith model for any type of energy
dissipation
wf is fracture energy, which could include plastic, viscoelastic, or
viscoplastic effects, depending on the material
Fig 2.6; various types of material behavior and corresponding fracture
energy
Word of caution in the case of materials that exhibit nonlinear
deformation
Griffith model applies only to linear elastic material behavior; global
behavior of structure must be elastic
Any nonlinear effects, such as plasticity, must be confined to a small
region near crack tip.
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Figure 2.6 Crack propagation in
various types of materials, with
corresponding fracture energy:
(a) ideally brittle material
(b) quasi-brittle elastic-plastic
material
(c) brittle material with crack
meandering and branching
Example 2.1
A flat plate made from a brittle material contains a macroscopic throughthickness crack with half length a1 and notch tip radius ρ. A sharp
penny-shaped microcrack with radius a2 is located near tip of the larger
flaw; Fig 2.5. Estimate minimum size of microcrack required to cause
failure in the plate when Griffith equation is satisfied by the global stress
and a1.
Example 2.1
A flat plate made from a brittle material contains a macroscopic throughthickness crack with half length a1 and notch tip radius ρ. A sharp
penny-shaped microcrack with radius a2 is located near tip of the larger
flaw; Fig 2.5. Estimate minimum size of microcrack required to cause
failure in the plate when Griffith equation is satisfied by the global stress
and a1.
Solution
Nominal stress at failure is obtained by substituting a1 into Eq 2.19. Stress
in the vicinity of the microcrack can be estimated from Eq 2.11, which is
set equal to Griffith criterion for the penny-shaped microcrack (Eq 2.20)
Solving for a2 gives
ENERGY RELEASE RATE
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1956; Irwin energy approach for fracture; essentially equivalent to
Griffith model, but more convenient for solving engineering problems
Energy release rate G; measure of energy available for an increment of
crack extension
G is rate of change in potential energy with crack area;
Also called crack extension force or crack driving force
Energy release rate for a wide plate in plane stress with a crack of
length 2a (Fig 2.3) is given by
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Crack extension occurs when G reaches a critical value
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where G c is a measure of fracture toughness of the material
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It is convenient at this point to introduce compliance; inverse of plate
stiffness
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It can be shown that
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where P is applied load, and B is plate thickness
Example 2.2
Determine the energy release rate for a double cantilever beam (DCB)
specimen; Fig 2.9
Figure 2.9 Double cantilever beam (DCB) specimen
INSTABILITY AND R CURVE
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Crack extension occurs when G = 2wf
This crack growth may be stable or unstable, depending on how G and
wf vary with crack size
Convenient to replace 2wf with R, material resistance to crack
extension
Plot of R vs crack extension; resistance curve or R curve
Corresponding plot of G vs crack extension; driving force curve
Wide plate with through crack of initial length 2a0 (Fig 2.3)
At a fixed remote stress σ, energy release rate varies linearly with
crack size
Fig 2.10; schematic driving force vs R curves for two types of
material behavior
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INSTABILITY AND R CURVE
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Crack extension occurs when G = 2wf
This crack growth may be stable or unstable, depending on how G and
wf vary with crack size
Convenient to replace 2wf with R, material resistance to crack
extension
Plot of R vs crack extension; resistance curve or R curve
Corresponding plot of G vs crack extension; driving force curve
Wide plate with through crack of initial length 2a0 (Fig 2.3)
At a fixed remote stress σ, energy release rate varies linearly with
crack size
Fig 2.10; schematic driving force vs R curves for two types of
material behavior
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Figure 2.10 Schematic driving force vs R curve diagrams
(a) flat R curve and (b) rising R curve
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First case, Fig 2.10(a); flat R curve
Material resistance is constant with crack growth
Crack is stable when stress is σ1
Fracture occurs when stress reaches σ2; crack propagation is
unstable because driving force increases with crack growth, but
material resistance remains constant
Fig 2.10(b); material with a rising R curve
Crack grows a small amount when stress reaches σ2, but cannot
grow further unless stress increases
When stress is fixed at σ2, driving force increases at a slower rate
than R
Stable crack growth continues as stress increases to σ3
Finally, when stress reaches σ4, driving force curve is tangent to the
R curve; plate is unstable with further crack growth because rate of
change in driving force exceeds slope of the R curve
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Conditions for stable crack growth:
and
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Unstable crack growth occurs when
Reasons for R Curve Shape
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Shape of R curve depends on material behavior and, to a lesser
extent, on configuration of cracked structure
R curve for an ideally brittle material is flat
When nonlinear material behavior accompanies fracture, R curve can
take on a variety of shapes
Ductile fracture in metals usually results in a rising R curve
If cracked body is infinite (plastic zone is small compared to
relevant dimensions of the body), R curve becomes flat with further
growth
Some materials can display a falling R curve
When a metal fails by cleavage, R curve would be relatively flat if
crack growth were stable
However, cleavage propagation is normally unstable
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Load Control vs Displacement Control
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Stability of crack growth depends on the rate of change in G; second
derivative of potential energy
Driving force G is the same for both load control and displacement
control
However, rate of change of driving force curve depends on how the
structure is loaded
Displacement control tends to be more stable than load control
With some configurations, driving force actually decreases with
crack growth in displacement control
When R curve is determined experimentally, specimen is usually
tested in displacement control,
Since most of the common test specimen geometries exhibit falling
driving force curves in displacement control, it is possible to obtain a
significant amount of stable crack growth
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Example 2.3
Evaluate the relative stability of a DCB specimen (Fig 2.9) in load control
and displacement control
STRESS ANALYSIS OF CRACKS
Figure 2.13 Definition of coordinate axis
ahead of a crack tip. The z direction is normal to the page
Stress Intensity Factor
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Each mode of loading produces singularity at the crack tip
It is convenient to represent it by the stress intensity factor K
There are three types of loading that a crack can experience; Fig 2.14
Mode I loading; principal load is applied normal to crack plane;
tends to open the crack
Mode II; in-plane shear loading; tends to slide one crack face with
respect to the other
Mode III; out-of-plane shear
A cracked body can be loaded in any one of these modes, or a
combination of two or three modes
It is usually given a subscript to denote the mode of loading; Fig 2.14;
KI, KII, KIII
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Figure 2.14 The three modes of loading that can be
applied to a crack
Relationship between K and Global Behavior
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Consider a through crack in an infinite plate subjected to a remote
tensile stress; Fig 2.3
Remote stress σ is perpendicular to the crack plane; loading is pure
Mode I
Linear elastic bodies must undergo proportional stressing
Thus crack-tip stresses must be proportional to remote stress
It can be shown that relationship between KI and global conditions has
the form
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Amplitude of crack-tip singularity for this
configuration is proportional to the remote
stress and square root of crack size
Figure 2.3
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A related solution is that for a semi-infinite plate with an edge crack;
Fig 2.16; infinite height, finite width
This configuration can be obtained by slicing the plate in Fig 2.3
through the middle of the crack
Stress intensity factor for edge crack can be shown to be given by
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12% increase for edge crack is caused by different boundary
conditions at the free edge
Figure 2.16 Edge crack in a semi-infinite plate
subjected to a remote tensile stress
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Penny-shaped crack in an infinite medium (Fig 2.4) is another
configuration for which a closed-form solution for KI exists:
where a is the crack radius
More general case of an elliptical or semielliptical flaw is illustrated in
Fig 2.19
Two length dimensions are needed to characterize the crack size:
2c and 2a, major and minor axes of the ellipse
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Figure 2.4 A penny-shaped (circular) crack embedded
in a solid object subjected to a remote tensile stress
Figure 2.19 Mode I stress
intensity factors for elliptical
and semielliptical cracks. These
solutions are valid only as long
as crack is small compared to
plate dimensions and a << c
Q: flaw shape parameter
s: surface correction factor
Effect of Finite Size
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Closed-form K solution generally for crack of simple shape (rectangle
or ellipse) in infinite plate
Crack dimensions are small compared to plate size
As crack size increases, or as plate dimensions decrease, outer
boundaries begin to exert an influence on crack tip
Closed-form K solution is usually not possible
Cracked plate subject to remote tensile stress; Fig 2.20
Effect of finite width on crack tip stress distribution, represented by
lines of force
Local stress is proportional to spacing between lines of force
Infinite plate; line of force at a distance W from crack centerline has
force components in x and y directions
If plate width is restricted to 2W, force must be zero on free edge;
this boundary condition causes the lines of force to be compressed;
higher stress intensification at crack tip
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Figure 2.20 Stress concentration effects due to a through crack in finite
and infinite width plates: (a) infinite plate and (b) finite plate.
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Mode I stress intensity factor for finite width is given by
K I   a f (a / W )
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Stress intensity approaches infinite-plate value as a/W approaches
zero
More accurate solutions for through crack in finite plate have been
obtained from finite-element analysis; one such solution is
Table 2.4 lists stress intensity solutions for several common
configurations
These KI solutions are plotted in Fig 2.23
Several handbooks are devoted solely to stress intensity solutions
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Figure 2.23 Plot of stress intensity solutions from Table 2.4
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Although stress intensity solutions are given in a variety of forms, K
can always be related to the through crack using an appropriate
correction factor
σ: characteristic stress
a: characteristic crack dimension
Y: dimensionless constant that depends on crack and plate geometry
and mode of loading
Example 2.4
Show that KI solution for the single edge notched tensile (SENT) panel
reduces to Equation (2.42) when a << W.
Example 2.4
Show that KI solution for the single edge notched tensile (SENT) panel
reduces to Equation (2.42) when a << W.
Solution
Equation 2.42
All of the KI expressions in Table 2.4 are of the form
P: applied force
B: plate thickness
f(a/W): dimensionless function
Principle of Superposition
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Linear elastic materials; similar individual components of stress, strain,
and displacement are additive
Similarly, stress intensity factors are additive as long as mode of
loading is consistent
Principle of superposition allows stress intensity solutions for complex
configurations to be built from simple cases for which solutions are
well established
Example
An edge-cracked panel (Table 2.4) subject to combined membrane (axial)
loading Pm and three-point bending Pb
Solution
Both types of loading impose pure Mode I conditions, so KI values can be
added
where fm and fb are geometry correction
factors for membrane and bending loading,
respectively
Listed in Table 2.4 and plotted in Fig 2.23
RELATIONSHIP BETWEEN K AND G
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Two parameters that describe the behavior of cracks
Energy release rate G: quantifies net change in potential energy
that accompanies an increment of crack extension
Stress intensity factor K: characterizes stresses, strains, and
displacements near the crack tip
Energy release rate describes global behavior, while K is a local
parameter
For linear elastic materials, K and G are uniquely related
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Through crack in an infinite plate subject to a uniform tensile stress
(Fig 2.3), G and KI are given by Equation (2.24) and (2.41)
Combining these two equations leads to the following relationship
between G and KI for plane stress
For plane strain conditions, E must be replaced by E/(1-ν2).
Notation in this book
CRACK-TIP PLASTICITY
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Linear elastic stress analysis of sharp cracks predicts infinite stresses
at the crack tip
In real materials; stresses at crack tip are finite because crack-tip
radius must be finite
Elastic stress analysis becomes increasingly inaccurate as inelastic
region at crack tip grows
Simple corrections to linear elastic fracture mechanics (LEFM) are
available for moderate crack-tip yielding
Size of crack-tip-yielding zone can be estimated by two methods
Irwin approach; elastic stress analysis is used to estimate elasticplastic boundary
Strip-yield model
Both approaches lead to simple corrections for crack-tip yielding
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Figure Examination of crack tip plasticity using Thermoelastic Stress
Analysis (TSA); phase difference may be used to size the plastic zone
Figure Maps of (a)TSA signal and (b) phase difference, and (c) line plots through
the crack tip of an Aluminium 2024 CT specimen of fatigue crack length = 31.2 mm
with mean load = 1000N; load amplitude = 500N; frequency 30Hz. (d)
Shape of the plastic zone found using the phase difference map
is superimposed on the line plot
Irwin Approach
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Fig 2.13; on the crack plane (θ = 0), normal stress σyy in a linear elastic
material is given by
As a first approximation, we can assume that boundary between
elastic and plastic behavior occurs when above stresses satisfy a yield
criterion
Plane stress conditions; yielding occurs when σyy = σYS, uniaxial yield
strength of material
Substituting into above equation and solving for r gives a first-order
estimate of plastic zone size
 YS
KI

2r
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This simple analysis is not strictly correct
It is based on an elastic crack-tip solution
When yielding occurs, plastic zone must increase in size
A simple force balance leads to a second-order estimate of plastic
zone size which is twice as large as the first-order estimate; Fig 2.29
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Irwin accounted for softer material in the plastic zone by defining an
effective crack length that is slightly longer than the actual crack size
where ry for plane stress is given on previous slide
Figure 2.29 First-order and second-order estimates of plastic zone size (ry
and rp). Crosshatched area represents load that must be redistributed,
resulting in a larger plastic zone.
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In plane strain
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Effective stress intensity is obtained by
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An iterative solution is usually required to solve for Keff
K is first determined in the absence of a plasticity correction
First-order estimate of aeff is then obtained from equation (2.64) or
(2.68)
This is used to estimate Keff
A new aeff is computed from Keff estimate
Process is repeated until successive Keff estimates converge
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In certain cases, this iterative procedure is unnecessary because a
closed-form solution is possible
Examples
Effective Mode I stress intensity factor for a through crack in an infinite
plate in plane stress is given by
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For embedded elliptical flaw
Qeff is the effective flaw shape parameter
Not recommend to use Irwin plastic zone adjustment for practical
applications; chapter 9 gives recommended approaches for handling
plasticity effects
Strip Yield Model
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Strip-yield model was first proposed by Dugdale and Barenblatt
They assumed a long, slender plastic zone at the crack tip in a nonhardening material in plane stress for a through crack in an infinite
plate
Assuming a crack of length 2a + 2ρ
ρ is length of plastic zone
Closure stress equal to σYS applied at each crack tip
This strip-yield model is a classical application of the principle of
superposition
For σ << σYS, it yields
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Note similarity between this equation (2.79) and earlier eq (2.66)
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1/π = 0.318, and π/8 = 0.392; Irwin and strip-yield approaches
predict similar plastic zone sizes
Burdekin and Stone obtained a more realistic estimate of Keff for the
strip-yield model
Comparison of Plastic Zone Corrections
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Figure 2.33; comparison between a pure LEFM analysis (Eq 2.41),
Irwin correction for plane stress (Eq 2.70), and strip-yield correction on
stress intensity (Eq 2.81)
Effective stress intensity, nondimensionalized by σYS√πa, is plotted
against normalized stress
LEFM analysis predicts a linear relationship between K and stress
Both Irwin and strip-yield corrections deviate from LEFM theory at
stresses greater than 0.5 σYS
The two plasticity corrections agree with each other up to
approximately 0.85 σYS
In 1970s, strip-yield model was used to derive a practical methodology
for assessing fracture in structural components
Approach called failure assessment diagram (FAD)
Higher course; described in Chapter 9
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Figure 2.33 Comparison of plastic zone corrections for a
through crack in plane strain
Figure 2.34 Crack-tip plastic zone shapes estimated from elastic
solutions (Table 2.1 and Table 2.3) and von Mises yield criterion
for Mode I loading
End Chapter-2