This lecture demonstrates how to teach the principles and concepts of fracture mechanics as well as provide recommendations for practical applications; it provides necessary information for fatigue life estimations on the basis of fracture mechanics as a complementary method to the S-N concept. Background in engineering, materials and fatigue is required.

1. TALAT Lecture 2403
Applied Fracture Mechanics
49 pages and 40 figures
Advanced Level
prepared by Dimitris Kosteas, Technische Universität München
Objectives:
− Teach the principles and concepts of fracture mechanics as well as provide
recommendations for practical applications
− Provide necessary information for fatigue life estimations on the basis of fracture
mechanics as a complementary method to the S-N concept
Prerequisites:
− Background in engineering, materials and fatigue required
Date of Issue: 1994
 EAA - European Aluminium Association

2. 2403 Applied Fracture Mechanics
Contents
2403.01 Historical Context ..................................................................................... 3
Fracture and Fatigue in Structures ...........................................................................3
2403.02 Notch Toughness and Brittle Fracture ................................................... 5
Notch-Toughness Performance Level as a Function of Temperature and Loading
Rates.........................................................................................................................5
Brittle Fracture .........................................................................................................8
2403.03 Principles of Fracture Mechanics............................................................ 9
Basic Parameters ......................................................................................................9
Material Toughness ............................................................................................ 9
Crack Size ........................................................................................................... 9
Stress Level ......................................................................................................... 9
Fracture Criteria .....................................................................................................13
Members with Cracks ............................................................................................14
Stress Intensity Factors ..........................................................................................16
Deformation at the Crack Tip ................................................................................21
Superposition of Stress Intensity Factors...............................................................22
2403.04 Experimental Determination of Limit Values according to Various
Recommendations ................................................................................................... 23
Linear-Elastic Fracture Mechanics ........................................................................24
Experimental Determination of KIc - ASTM-E399........................................... 24
Test procedure: ................................................................................................. 25
Elastic-Plastic Fracture Mechanics ........................................................................27
Crack opening displacement (COD) - BS 5762 ................................................ 27
Determination of R-Curves - ASTM-E561........................................................ 28
Determination of JIc - ASTM-E813 .................................................................. 30
Determination of J-R Curves - ASTM-E1152 .................................................. 33
Crack Opening Displacement (COD) Measurements - BS 5762 ...................... 34
2403.05 Fracture Mechanics Instruments for Structural Detail Evaluation... 36
Free Surface Correction Fs ....................................................................................37
Crack Shape Correction Fe ....................................................................................37
Finite Plate Dimension Correction Fw ..................................................................38
Correction Factors for Stress Gradient Fg .............................................................38
Remarks on Crack Geometry.................................................................................39
2403.06 Calculation of a Practical Example: Evaluation of Cracks Forming at a
Welded Coverplate and a Web Stiffener .............................................................. 41
Coverplate ..............................................................................................................42
Web Stiffener .........................................................................................................43
2403.07 Literature/References ............................................................................... 47
2403.08 List of Figures.......................................................................................... 48
TALAT 2403 2

3. 2403.01 Historical Context
• Fracture and fatigue in structures
In his first treatise on "Mathematical Theory of Elasticity" Love, 100 years ago, discus-
sed several topics of engineering importance for which linear elastic treatment appeared
inadequate. One of these was rupture. Nowadays structural materials have been im-
proved with a corresponding decrease in the size of safety factors and the principles of
modern fracture mechanics have been developed, mainly in the 1946 to 1966 period.
Fracture mechanics is the science studying the behaviour of progressive crack extension
in structures. This goes along with the recognition that real structures contain disconti-
nuities.
Fracture mechanics is the primary tool (characteristic material values, test procedures,
failure analysis procedures) in controlling brittle fracture and fatigue failures in struc-
tures. The desire for increased safety and reliability of structures, after some spectacular
failures, has led to the development of various fracture criteria. Fracture criteria and
fracture control are a function of engineering contemplation taking into account econo-
mical and practical aspects as well.
Fracture and Fatigue in Structures
Brittle fracture is a type of catastrophic failure that usually occurs without prior plastic
deformation and at extremely high speeds. Brittle fractures are not so common as fatigue
(the latter characterised by progressive crack development), yielding, or buckling
failures, but when they occur they may be more costly in terms of human life and prop-
erty damage. Fatigue failures according to statistics is responsible for approx. 7% of
failures.
Aristotle talked about hooks on molecules, breaking them meant fracture. Da Vinci and
Gallileo talked about fracture, too. The big break in fracture mechanics came in 1920
with the Griffith theory, applicable mostly to brittle materials, as well as Orowan and
Irwing and Williams in the 1940's.
Catastrophic brittle failures were recorded in the 19th and early 20th century. There
were several failures in welded Vierendeel-truss bridges in Europe shortly after being
put into service before World War II. However, it was not until the large number of
World War II ship failures that the problem of brittle fracture was fully appreciated by
engineers. 1962 the Kings Bridge in Melbourne failed by brittle fracture at low tempera-
tures due to poor details and fabrication resulting in cracks which were nearly through
the flange prior to any service loading. Although this failure was studied extensively,
bridge-builders did not pay particular attention until the failure of the Point Pleasent
Bridge in West Virginia, USA on December 15, 1965. This was the turning point initia-
ting the possibilities of fracture mechanics in civil engineering.
This failure was unique in several ways, it was investigated extensively and its results
were characteristic for the procedures and possibilities of fracture mechanics analysis.
Therefore they are mentioned briefly here:
TALAT 2403 3

4. (a) fracture appeared in the eye of an eyebar caused by the growth of a flaw to a criti-
cal size under normal working stresses,
(b) the initial flaw was caused through stress-corrosion cracking from the surface of
the hole, hydrogen sulphide was probably the reagent responsible,
(c) the chemical composition and heat treatment of the eyebar produced a steel with
very low fracture toughness at the failure temperature, and
(d) fracture resulted from a combination of factors and it would not have occurred in
the absence of anyone of these
- the high hardness of the material made it susceptible to stress corrosion
cracking
- close spacing of joint components made it impossible to apply paint to high
stressed regions yet provided a crevice where water could collect
- the high design load of the eyebar resulted in high local stresses at the inside
of the eye greater than the yield strength of the steel.
- the low fracture toughness of the steel led to complete fracture from the
slowly propagating stress corrosion crack when it had reached a depth of
only 3.0 mm
It has been shown that an interrelation exists between material, design, fabrication and
loading as well as maintenance. Fractures cannot be eliminated in structures by merely
using materials with improved notch toughness. The designer still has the fundamental
responsibility for the overall safety and reliability of the structure.
TALAT 2403 4

5. 2403.02 Notch Toughness and Brittle Fracture
• Notch-toughness performance level as a function of temperature
and loading rates
• Brittle fracture
In the following chapters it will be shown how fracture mechanics can be used to de-
scribe quantitatively the trade-offs among stress, material toughness, and flaw size so
that the designer can determine the relative importance of each of them during design
rather than during failure analysis.
Notch-Toughness Performance Level as a Function of Temperature
and Loading Rates
The traditional mechanical property tests measure strength, ductility, modulus of elasti-
city etc. There are also tests available to measure some form of notch toughness. Notch
toughness is defined as the ability of a material to absorb energy (usually when loaded
dynamically) in the presence of a flaw. Toughness is defined as the ability of a smooth,
unnotched member to absorb energy (usually when loaded slowly).
Notch toughness is measured with a variety of specimens such as the Charpy-V-notch
impact specimen, dynamic tear test specimen KIc, pre-cracked Charpy, etc. Toughness
is usually characterised by the area under a stress-strain curve in a slow tension test.
Notches or other forms of stress raisers make structural materials susceptible to brittle
fracture under certain conditions.
The ductile or brittle behaviour of some structural materials like steels is well known,
depending on several conditions such as temperature, loading rate, and constraint (the
latter arising often in welded components among other reasons due to residual stresses
and the complexity of welds). Ductile fractures are generally preceded by large amount
of plastic deformation occurring usually at 45° to the direction of the applied stress.
Brittle or cleavage fractures generally occur with little plastic deformation and are
usually normal to the direction of principal stresses. Figure 2403.02.01 shows the var-
ious fracture states and the transition from one to another depending on environmental
conditions.
Plane-strain behaviour refers to fracture under elastic stresses and is essentially brittle.
Plastic behaviour refers to ductile failure under general yielding conditions accompanied
usually, but not necessarily, with large shear lips. The transition between these two is
the elastic-plastic region or the mixed-mode region. Higher loading rates move the
characteristic transition curve to higher temperatures. A particular notch toughness value
called the nil-ductility transition (NDT) temperature generally defines the upper limits
of plane-strain behaviour under conditions of impact loading. In practice the question
has to be answered regarding the level of material performance which should be re-
quired for satisfactory performance in a particular structure at a specific service tem-
perature, see Figure 2403.02.02. In this example and for impact loading the three
different steels exhibit either plane-strain behaviour (steel 1) or elastic-plastic behaviour
(steel 2), or fully plastic behaviour (steel 3) at the indicated service temperature.
TALAT 2403 5

6. Although fully plastic behaviour would be a very desirable level of performance, it may
not be necessary or even economically feasible for many structures.
Notch-Thoughness Performance Levels
vs. Temperature
Impact loading
by absorbed energy in notched specimens
Plastic
Static loading
Levels of performance as measured
Elastic - Intermediate
Plastic loading rate
NDT (Nil-Ductility Transition)
Plane
Strain
(Macro linear)
Elastic
Temperature
D. Kosteas, TUM
alu Notch-Thoughness Performance Levels
2403.02.01
Training in Aluminium Application Technologies vs. Temperature
Service Temperature
Plastic
Steel 3 Steel 2 Steel 1
Levels of Performance
Elastic-
Plastic
NDT NDT
Plane NDT
(Steel 2)
(Steel 3) (Steel 1)
Strain
Temperature
D. Kosteas, TUM
alu
Relation between Performance and
Training in Aluminium Application Technologies Transition Temperature for three different Steel Qualities 2403.02.02
Not all structural materials exhibit a ductile-brittle transition. For example, aluminium
as well as very high strength structural steels or titanium do not undergo a ductile-brittle
transition. For these materials temperature has a rather small effect on toughness, see
Figure 2403.02.03.
TALAT 2403 6

7. Notch-Toughness vs. Temperature
6
Alloy 5083-H113
+
+
+
+ +
Notch toughness kpm/ cm²
+
4
x
x x
x x
2 x ISO - Probe
x
longitudinal
}
transverse parent metal
+ HAZ
x filler
0
-200 -100 0 20
D. Kosteas, TUM Temperature in °C
alu Notch-Toughness of Welded Aluminium Alloy 5083
Training in Aluminium Application Technologies vs. Temperature 2403.02.03
Notch toughness measurements express the behaviour and respective laboratory test re-
sults can be used to predict service performance. Many different tests have been used to
measure the notch toughness of structural materials. These include
− Charpy-V notch (CVN) impact,
− drop weight NDT
− dynamic tear (DT)
− wide plate
− Battelle drop weight tear test (DWTT)
− pre-cracked Charpy, etc.
Notch toughness tests produce fracture under carefully controlled laboratory conditions.
Hopefully, test results can be correlated with service performance to establish curves
like in Figure 2403.02.01 for various materials and specific applications. However,
even if correlations are developed for existing structures, they do not necessarily hold
for certain designs or new operating conditions or new materials. Test results are
expressed in terms of energy, fracture appearance, or deformation, and cannot always be
translated to engineering parameters.
A much better way to measure notch toughness is with the principles of fracture mech-
anics, a method characterising the fracture behaviour in structural parameters readily
recognised and utilised by the engineer, namely stress and flaw size. Fracture mechanics
is based on a stress analysis as described in the next chapters and can account for the ef-
fect of temperature and loading rate on the behaviour of structural members that have
sharp cracks.
Large and complex structures always have discontinuities of some kind. Dolan has made
the flat statement that "every structure contains small flaws" whose size and distribution
are dependent upon the material and its processing. These may range from non-metallic
inclusions and microvoids to weld defects, grinding cracks, quench cracks, surface laps,
etc. Fisher and Yen have shown that discontinuities exist in practically all structural
members, ranging from below 0.02 mm to several cm long. The significant point is that
TALAT 2403 7

8. discontinuities are present in fabricated structures even though the structure may have
been inspected. The problem of establishing acceptable discontinuities, for instance in
welded structures, is becoming an economic problem since techniques that minimize the
size and distribution of discontinuities are available if the engineer chooses to use them.
Whether a given defect is permissible or not depends on the extent to which the defect
increases the risk of failure of the structure. It is quite clear that this will vary with the
type of structure, its service conditions, and the material from which it is constructed.
The results of a fracture-mechanics analysis for a particular application (specimen size,
service temperature, and loading rate) will establish the combinations of stress level and
flaw size that would be required to cause fracture. The engineer can then quantitatively
establish allowable stress levels and inspection requirements so that fracture cannot
occur. Fracture mechanics can also be used to analyse the growth of small cracks, as for
example by fatigue loading, to critical size.
Fracture mechanics have definite advantages compared with traditional notch-toughness
tests. The latter are still useful though, as there are many empirical correlations between
fracture-mechanics values and existing toughness test results. In many cases, because of
the current limitations on test requirements for measuring fracture toughness KIc exist-
ing notch-toughness tests must be used to help the designer estimate KIc values.
Brittle Fracture
The number of catastrophic brittle fractures have been very small generally. Especially
for aluminium, exhibiting rather high fracture toughness values over the whole
temperature regime, probability of failure is low. Nonetheless, when
− the design becomes complex,
− thick welded plates from high strength materials are used
− cost minimisation for the structure becomes more significant
− the magnitude of loading increases, and
− actual factors of safety decrease because of more precise computer designs,
the possibility of brittle fracture in large complex structures must be considered.
TALAT 2403 8

9. 2403.03 Principles of Fracture Mechanics
• Basic parameters
• Fracture criteria
• Members with cracks
• Stress intensity factors
• Deformation at the crack tip
• Superposition of stress intensity factors
Basic Parameters
Numerous factors like service temperature, material toughness, design, welding, residual
stresses, fatigue, constraint, etc., can contribute to brittle fractures in large structures.
However, there are three primary factors in fracture mechanics that control the
susceptibility of a structure to brittle fracture:
1) Material toughness Kc, KIc, KId
2) Crack size a
3) Stress level σ
Material Toughness
is the ability to carry load or deform plastically in the presence of a notch and can be
described in terms of the critical stress-intensity factor under conditions of plane stress
Kc or plane strain KIc for slow loading and linear elastic behaviour or KId under con-
ditions of plane strain and impact or dynamic loading, also for linear elastic behaviour.
For elastic-plastic behaviour, i.e. materials with higher levels of notch toughness than
linear elastic behaviour, the material toughness is measured in terms of parameters such
as R-curve resistance, JIc, and COD as described later on.
Crack Size
Fracture initiates from discontinuities or flaws. These can vary from extremely small
cracks within a weld arc strike to much larger or fatigue cracks, or imperfections of
welded structures like porosity, lack of fusion, lack of penetration, toe or root cracks,
mismatch, overfill angle, etc. Such discontinuities, though even small initially, can grow
by fatigue or stress corrosion to a critical size.
Stress Level
Tensile stresses (nominal, residual, or both) are necessary for brittle fractures to occur.
They are determined by conventional stress analysis techniques for particular structures.
Further factors such as temperature, loading rate, stress concentrations, residual stresses,
etc., merely affect the above three primary factors. Engineers have known these facts for
many years and controlled the above factors qualitatively through good design (adequate
sections, minimum stress concentrations) and fabrication practices (decreased
imperfection or discontinuity size through proper welding and inspection), as well as the
use of materials with sufficient notch toughness levels.
TALAT 2403 9

10. A linear elastic fracture mechanics technology is based upon an analytical procedure
that relates the stress field magnitude and distribution in the vicinity of a crack tip to the
nominal stress applied to the structure, to the size, shape, and orientation of the crack,
and to the material properties. In Figure 2403.03.01 are the equations that describe the
elastic stress field in the vicinity of a crack tip for tensile stresses normal to the plane of
the crack (Mode I deformation).
Elastic Stress Field Distribution near a Crack
σ
y
Magnitude of stress
along x axis, σy
σy
σx σx
Crack σy (θ = 0)
σ r
tip σy
Nominal θ
stress x
KI θ θ 3θ
σx = cos (1 - sin sin )
(2 π r)1/2 2 2 2
KI θ θ 3θ
σy = cos (1 + sin sin )
(2 π r)1/2 2 2 2
Source: Irwin and Williams, 1957
alu
Training in Aluminium Application Technologies
Elastic Stress Field Distribution near a Crack 2403.03.01
The equations describing the crack tip stress field distribution were formulated by Irwin
and Williams (1957) as follows:
KI θ θ 3θ 
σx = cos  1 − sin sin 
2πr 2 2 2
KI θ θ 3θ 
σy = cos  1 + sin sin 
2πr 2 2 2
KI θ θ 3θ
τ xy = sin cos cos
2 πr 2 2 2
τ xz = τ yz = 0
σz =0 Plane Stress (thin sheet)
σ z = µ ⋅ (σ x + σ y ) Plane Strain (thick sheet)
The distribution of the elastic stress field in the vicinity of the crack tip is invariant in all
structural components subjected to this type of deformation. The magnitude of the
elastic stress field can be described by a single parameter, KI, designated the stress in-
tensity factor. The applied stress, the crack shape, size, and orientation, and the struc-
tural configuration of structural components subjected to this type of deformation affect
the value of the stress intensity factor but do not alter the stress field distribution. This
allows to translate laboratory results directly into practical design information. Stress in-
tensity values for some typical cases are given in Figure 2403.03.02.
TALAT 2403 10

11. KI values for various crack geometries
a
σ σ σ
2a 2c a
σ σ σ
Through thickness crack Surface crack Edge crack
KI = σ ⋅ π ⋅ a a KI = 112 ⋅ σ ⋅ π ⋅ a
KI = 112 ⋅ σ ⋅ π ⋅
Q
where Q = f(a/2c, σ)
alu
Training in Aluminium Application Technologies
KI Values for Various Crack Geometries 2403.03.02
It is a principle of fracture mechanics that unstable fracture occurs when the stress in-
tensity factor at the crack tip reaches a critical value Kc. For mode I deformation and for
small crack tip plastic deformation, i.e. plane strain conditions, the critical stress in-
tensity factor for fracture for fracture instability is KIc. The value KIc represents the
fracture toughness of the material (the resistance to progressive tensile crack extension
under plane strain conditions) and has units of MN/m3/2 (or MPa/mm1/2 or ksi√in).
1ksi√in = 34.7597 N/mm3/2 or MPa/mm1/2
{1ksi = 6.89714 N/mm2 or MPa}
This material toughness property depends on the particular material, loading rate, and
constraint:
Kc critical stress intensity factor for static loading and plane stress conditions of
variable constraint. This value depends on specimen thickness and
geometry, as well as on crack size.
KIc critical stress intensity factor for static loading and plane strain conditions of
maximum constraint. This value is a minimum value for thick plates.
KId critical stress intensity factor for dynamic (impact) loading and plane strain
conditions of maximum constraint.
TALAT 2403 11

12. where
K c , K Ic , K Id = C ⋅ σ ⋅ a
C = constant, function of specimen and crack geometry
σ = nominal stress
a = flaw size
Through knowledge of the critical value at failure for a given material of a particular
thickness and at a specific temperature and loading rate, tolerable flaw sizes for a given
design stress level can be determined. Or design stress levels may be determined that
can be safely used for an existing crack that may be present in a structure, see Figure
2403.03.03.
Relation between material toughness, flaw
size and stress σ
2a
Increasing material
toughness σ
(COD, JIc, R)
Increasing stress, σ
(Fracture zone)
KC KC of
!f tough
er ste
el
!0 KI = f(! , a)
KC = critical value of KI
a0 af
Increasing flaw size, 2a
alu Relation between Material Toughness, Flaw
Training in Aluminium Application Technologies Size and Stress 2403.03.03
In an unflawed structural member, as the load is increased the nominal stress increases
until an instability (yielding at σys) occurs. Similarly in a structural member with a flaw
as the load is increased ( or as the size of the flaw grows by fatigue or stress corrosion)
the stress intensity KI increases until an instability, fracture at KIc, occurs. Another ana-
logy that helps to understand the fundamental aspects of fracture mechanics is the com-
parison with the Euler column instability, Figure 2403.03.04. To prevent buckling the
actual stress and L/r values must be below the Euler curve. To prevent fracture the act-
ual stress and flaw size must be below the Kc level.
TALAT 2403 12

13. COLUMN RESEARCH COUNCIL
COLUMN STRENGTH CURVE
P
L
! YS EULER CURVE
P
π 2⋅ E
σc =
( L / r) 2
!
YIELDING
! = ! YS
!
! YS
L/r
(a) COLUMN INSTABILITY KC
σc = 2a
C ⋅ π ⋅a
! !
YIELDING
! = ! YS
a
(b) CRACK INSTABILITY
Source: Madison and Irwin
alu
Analogy: Column Instability and Crack Instability 2403.03.04
Training in Aluminium Application Technologies
The critical stress intensity factor KIc represents the terminal conditions in the life of a
structural component. The total useful life NT of the component is determined by the
time necessary to initiate a crack NI and by the time to propagate the crack NP from
subcritical dimensions a0 to the critical size ac. Crack initiation and subcritical crack
propagation are localised phenomena that depend on the boundary conditions at the
crack tip. Subsequently it is logical to expect that the rate of subcritical crack propaga-
tion depends on the stress intensity factor KI which serves as a single term parameter re-
presentative of the stress conditions in the vicinity of the crack tip. Fracture mechanics
theory can be used to analyse the behaviour of a structure throughout its entire life.
For materials that are susceptible to crack growth in a particular environment the KIscc-
value is used as the failure criterion rather than KIc. This threshold value KIscc is the
value below which subcritical crack propagation does not occur under static loads in
specific environment. In the relationship between material toughness, design stress and
flaw size, Figure 2403.03.03, KIscc replaces Kc as the critical value of KI.
Fracture Criteria
A careful study of the particular requirements of a structure is the basis in developing a
fracture control plan, i.e. the determination of how much toughness is necessary and
adequate. Developing a criterion one should consider
− service conditions such as temperature, loading, loading rate, etc.
− the level of performance (plane strain, elastic-plastic, plastic)
− consequences of failure
TALAT 2403 13

14. Developing a fracture control plan for a complex structure is very difficult. All factors
that may contribute to the fracture of a structural detail or failure of the entire structure
have to be identified. The contribution of each factor and the combination effect of dif-
ferent factors have to be assessed. Methods minimising the probability of fracture have
to be determined. Responsibility has to be assigned for each task that must be underta-
ken to ensure the safety and reliability of a structure.
A fracture control plan can be defined for a given application and cannot be extended
indiscriminately to other applications. Certain general guidelines pertaining to classes of
structures (such as bridges, ships, vehicles, pressure vessels, etc.) can be formulated.
The fact that crack initiation, crack propagation, and fracture toughness are functions of
the stress intensity fluctuation ∆KI and of the critical stress intensity factor KIc (where-
by the stress intensity is related to the applied nominal stress or stress fluctuation) de-
monstrates that a fracture control plan depends on
− fracture toughness KIc or Kc of the material at the temperature and
loading rate of the application. The fracture toughness can be modified by
changing the material.
− the applied stress, stress rate, stress concentration, and stress fluctuation.
They can be altered by design changes and fabrication.
− the initial size of the discontinuity and the size and shape of the critical
crack. These can be controlled by design changes, fabrication, and
inspection.
Members with Cracks
Failure of structural members is caused by the propagation of cracks, especially in fati-
gue. Therefore an understanding of the magnitude and distribution of the stress field in
the vicinity of the crack front is essential to determine the safety and reliability of struc-
tures. Because fracture mechanics is based on a stress analysis, a quantitative evaluation
of the safety and reliability of a structure is possible.
Fracture mechanics can be subdivided into two general categories namely linear-elastic
and elastic-plastic. The following relationships and equations for stress intensity factors
are based on linear elastic fracture mechanics (LEFM).
It is convenient to define three types of relative movements of two crack surfaces. These
displacement modes, Figure 2403.03.05, represent the local deformation in an
infinitesimal element containing a crack form. Figure 2403.03.06 shows the coordinate
system and stress components ahead of a crack tip
TALAT 2403 14

15. Basic cracking modes
y
x
Mode I
z
y
x
Mode II
z
y
x
Mode III
z
alu
Training in Aluminium Application Technologies
Basic Modes of Cracking 2403.03.05
σy
τ xy
y τ yz
τ xz
σx σz x
r
θ
Leading edge of the crack
z
Coordinate System and Stress Components ahead
alu
of a Crack Tip 2403.03.06
Training in Aluminium Application Technologies
Mode I is characterized by displacement of the two fracture surfaces perpendicular to
each other in opposite directions. Local displacements in the sliding or shear Mode II
and Mode III, the tearing mode, are the other basic types corresponding to respective
stress fields in the vicinity of the crack tips, Figure 2403.03.07. In any problem the
deformations can be treated as one or combination of these local displacement modes.
Respectively the stress field at the crack tip can be treated as one or a combination of the
three basic types of stress fields. For practical applications Mode I is the most important
since according to Jaccard even if a crack starts as a combination of different modes it
is soon transformed and continues its propagation to a critical crack under Mode I.
TALAT 2403 15

16. Stress Field Equations in the Vicinity of Crack Tips
Mode I Mode II Mode III
KI θ θ 3θ  K II θ θ 3θ  KIII θ
σx = cos  1 − sin sin  σx =− sin  2 + cos cos  τ xz = − sin
2π ⋅ r 2 2 2 2π ⋅ r 2 2 2 2π ⋅ r 2
KI θ θ 3θ  KII θ θ 3θ KIII θ
σy = cos 1 + sin sin  σy = sin cos cos τ yz = cos
2π ⋅ r 2 2 2 2π ⋅ r 2 2 2 2π ⋅ r 2
KI θ θ 3θ KII θ θ 3θ 
τ xy = sin cos cos τ xy = cos  1 − sin sin 
2π ⋅ r 2 2 2 2π ⋅ r 2 2 2
σ z = ν (σ x + σ y ) σ z = ν (σ x + σ y ) σ x = σ y = σ z = τ xy = 0
τ xz = τ yz = 0 τ xz = τ yz = 0
θ θ θ θ θ
1 2 1 2 1 2
KI  r  K II  r  K III  r 
u= cos 1 − 2ν + sin 2  u= sin  2 − 2ν + cos2  w= sin
G  2π 
  2 2 G  2π 
  2 2 G  2π 
  2
θ θ θ θ
1 2 1 2
KI  r  K II  r 
v= sin  2 − 2ν − cos2  v= cos  −1 + 2ν − sin 2 
G  2π 
  2 2 G  2π 
  2 2
w=0 w=0 u=v=0
alu
Training in Aluminium Application Technologies
Stress Field Equations in the Vicinity of Crack Tips 2403.03.07
Dimensional analysis of the equations shows that the stress intensity factor must be
linearly related to stress and directly related to the square root of a characteristic length,
the crack length in a structural member. In all cases the general form the stress intensity
factor is given by
K = f ( g) ⋅ σ ⋅ a
where f(g) is a parameter that depends on the specimen and crack geometry.
Stress Intensity Factors
Various relationships between the stress intensity factor and structural component confi-
gurations, crack sizes, orientations, and shapes, and loading conditions can be taken out
of respective literature
[1] C.P. Paris and G.C. Sih, "Stress analysis of cracks" in "Fracture toughness testing
and its applications", ASTM STP No381, ASTM, Philadelphia 1965
[2] H. Tada, P.C. Paris and G.R. Irwing, ed., "Stress analysis of cracks handbook",
Del Research corporation, Hellertown, Pa. 1973
[3] G.C. Sih, "Handbook of stress intensity factors for researchers and engi-neers",
Institute of fracture and solid mechanics, Lehigh University, Bethlehem, Pa. 1973
Stress intensity factor for a through-thickness crack:Figure 2403.03.08
TALAT 2403 16

17. Through-Thickness Crack
Finite Width Plate Infinite Width Plate
σ
σ
Tangent correction for finite width
a
b [sec((π ⋅ a) / (2 ⋅ b))] 1
2
0.074 1.00
0.207 1.03
0.275 1.05
2a 0.337 1.08 2a
0.410 1.12
2b
0.466 1.16
0.535 1.22
0.592 1.29 σ
Nominal Stress
σ
[
KI = σ π ⋅ a ⋅ sec( (π ⋅ a) / ( 2 ⋅ b)) ] 1
2 K = σ π ⋅a
alu
SIF for Through-Thickness Crack 2403.03.08
Training in Aluminium Application Technologies
Stress intensity factor for a double-edge crack: Figure 2403.03.09
Double-Edge Crack
σ
Tangent correction for finite width
a
b [sec((π ⋅ a) / (2 ⋅ b))] 1
2
0.074 1.00
0.207 1.03
a a 0.275 1.05
0.337 1.08
0.410 1.12
2b 0.466 1.16
0.535 1.22
0.592 1.29
σ
. [
K I = 112σ π ⋅ a ⋅ sec( (π ⋅ a ) / ( 2 ⋅ b)) ] 1
2
alu
SIF for Double-Edge Crack 2403.03.09
Training in Aluminium Application Technologies
TALAT 2403 17

18. Stress intensity factor for a single-edge crack: Figure 2403.03.10
Single-Edge Crack
For a semi-finite edge-cracked specimen:
σ KI =1.12 σ(πa)1/2
For a finite width edge-cracked specimen:
KI = σ(πa)½ f(a/b)
Correction factor for a single-edge crack
in a finite width plate
a/b f(a/b)
a
0.10 1.15
0.20 1.20
2b 0.30 1.29
0.40 1.37
0.50 1.51
0.60 1.68
0.70 1.89
σ 0.80 2.14
0.90 2.46
1.00 2.86
alu
SIF for Single-Edge Crack 2403.03.10
Training in Aluminium Application Technologies
Stress intensity factor for cracks emanating from circular or elliptical holes: Figure
2403.03.11
1.1
Cracks Emanating from
1.0
Circular or Elliptical Holes
0.9
0.8
!
0.7
The stress intensity factor
0.6 in the case of a finite plate is
F( λ , δ ) 0.5
bN
ρ
K I = F( λ , δ ) ⋅ σ π ⋅ a
0.4
aN aF  a b 
0.3 a where λ =   and δ =  N 
 aN   aN 
0.2
0.1 ! For a circular hole δ = 1.0
0
1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40
 a
λ = 
 aN 
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Training in Aluminium Application Technologies
SIF for Cracks Emanating from Circular or Elliptical Holes 2403.03.11
TALAT 2403 18

19. Stress intensity factor for a single-edge crack in beam in bending: Figure 2403.03.12
Single-Edge Crack in Beam in Bending
a
M M
w
6M a Correction factor for notched beams
KI = f 
3
W
(W − a ) 2
a/W f(a/W)
0.05 0.36
0.10 0.49
0.20 0.60
0.30 0.66
0.40 0.69
0.50 0.72
W: Depth of the Beam
>0.60 0.73
alu
SIF for Single-Edge Crack in Beam in Bending 2403.03.12
Training in Aluminium Application Technologies
Stress intensity factor for an elliptical or circular crack in an infinite plate: Figure
2403.03.13
Elliptical or Circular Crack in an Infinite Plate
The stress intensity factor at any point along the perimeter of elliptical or
circular cracks in an infinite body subjected to uniform tensile stress is
!
3 1
σ (πa ) 2   a
2
4
KI = sin2 β +   cos2 β 
θ0 
  b 

The point on the perimeter of the crack
a
is defined by the angle ß
a and the elliptic integral #o
π 3
c
"
2
 c2 − a 2 2
c θ 0 = ∫ 1 − sin 2 θ  ⋅ dθ
0
 c2 
For circular cracks, c = a ⇒ K I = 2 ⋅ σ (a π )
12
When c → ∞ and β = π 2 , this case is reduced
! to a through - thickness crack.
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Training in Aluminium Application Technologies
SIF: Elliptical or Circular Crack in an Infinite Plate 2403.03.13
TALAT 2403 19

20. Stress intensity factor for a surface crack: Figure 2403.03.14
Surface Crack
! "Thumbnail Crack"
The stress intensity factor can be calculated
from the equations of elliptical crack using a
free surface correction factor of 1.12
and for the position ß=π/2
π ⋅ a 
1
2
K I = 112 ⋅ σ 
.
 Q  
with Q = θ 0 and the elliptic integral
a 2
2c 3
π 2
2
 c2 − a 2 
θ 0 = ∫ 1 − sin 2 θ  ⋅ dθ
0 c 2

Q is regarded as a shape factor because its values
depend on a and c.
For values of Q see Figure 2402.03.15.
!
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Training in Aluminium Application Technologies
SIF: Surface Crack 2403.03.14
Flaw shape parameter: Figure 2403.03.15
Flaw Shape Parameter, Q
0.5 2c a
0.4
σ σ ys = 0
Ratio
a/2c 0.3 σ σ ys = 0.60
σ σ ys = 0.80
σ σ ys = 10
.
0.2
0.1
0
0.5 1.0 1.5 2.0 2.5
Flaw Shape Parameter, Q
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Flaw Shape Parameter, Q 2403.03.15
Training in Aluminium Application Technologies
TALAT 2403 20

21. Deformation at the Crack Tip
The stress field equations show that the elastic stress at a distance r from the crack tip
(where r<<a) can be very large, see Figure 2403.03.16. In reality the material in this
region deforms plastically. A plastic zone surrounds the crack tip.
Elastic Stress Distribution
! YS Stress Distribution
after Local Yielding
!
x
Crack Tip 2rY
Plastic
Zone
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Training in Aluminium Application Technologies
Distribution of Stress in the Crack Tip Region 2403.03.16
The size of the plastic zone ry can be estimated from the stress field equations for plane
stress conditions and setting σy = σys
2
1  K 
ry = 
σ  
2π  ys 
Following a suggestion by Irwin that the increase in tensile stress for plastic yielding
caused by plane strain elastic constraint is of the order of √3, we can estimate the size of
the plastic zone under plane strain conditions as
2
1  K 
ry =  
6π  σ ys 
 
The plastic zone along the crack front in a thick specimen is subjected to plane strain
conditions in the center portion of the crack front where w = 0 and to plane strain condi-
tion near the surface of the specimen where σz = 0. That means that the plastic zone in
the center of a thick specimen is smaller than at the surface of the specimen, Figure
2403.03.17.
TALAT 2403 21

22. Midsection Plane stress
A - Overall view B - Edge view mode I
Plane
Surface strain
mode I
y
Cracktip
z
Crack tip
x
KI²
2 $! y²
rIp (plane strain)
Machine notch
Fatigue crack z
Plastic zone rp (Plane stress)
Plane strain region
Specimen cross-section
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Training in Aluminium Application Technologies
Plastic Zone Dimensions 2403.03.17
Superposition of Stress Intensity Factors
Stress components from such loads as uniform tensile loads, concentrated tensile loads,
or bending loads, all belonging to Mode I type loads, have the same stress field distribu-
tions in the vicinity of the crack tip according to the equations given in Figure
2403.03.07. Consequently the total stress intensity factor can be obtained by
algebraically adding the individual stress intensity factors corresponding to each load.
TALAT 2403 22

23. 2403.04 Experimental Determination of Limit Values according to
Various Recommendations
• Linear-Elastic Fracture Mechanics
− Experimental determination of KIc - ASTM-E399
• Elastic-Plastic Fracture Mechanics
− Crack opening displacement (COD) - BS 5762
− Determination of R-Curves - ASTM-E561
− Determination of JIc - ASTM-E813
− Determination of J-R Curves - ASTM-E1152
− Crack opening displacement (COD) measurements - BS 5762
Structural materials have certain limiting characteristics, yielding in ductile materials or
fracture in brittle materials. The yield strength σys is the limiting value for loading
stresses, the critical stress intensity factors KIc, KId or Kc are the limiting values for the
stress intensity factor KI. The critical stress intensity factor Kc at which unstable crack
growth occurs for conditions of static loading at a particular temperature depends on
specimen thickness or constraint, Figure 2403.04.01.
Effect of Thickness on Kc
240
Plane stress Plane strain
200
in.
160
Kc, ksi
120 K Ic
80 Thickness at the beginning
of plane strain is:
2
 K Ic 
40 t = 2.5  
σy
0
0.25 0.30 0.50 0.75 1.0 2.0 3.0 5.0
Thickness, in.
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Training in Aluminium Application Technologies
Effect of Thickness on Kc 2403.04.01
The limiting value of Kc for plane strain (maximum constraint) conditions is KIc (slow
loading rate) or KId (dynamic or impact load). The KIc-value is the minimum value for
plane strain conditions. The KIc-value would also be a minimum value for conditions of
maximum structural constraint (for example, stiffeners, intersecting plates, etc.) that
TALAT 2403 23

24. might lead to plain-strain conditions even though the individual structural members
might be relatively thin.
Design procedures based on the KIc-value provide a "conservative" approach to the
fracture problem. The following chapter gives information on experimental require-
ments and procedures for the measurement of KIc-values.
For thin section problems further procedures based on a Kc or R-curve analysis for
elastic-plastic behaviour problems an analysis on the basis of JIc- or COD-procedures
will be described in further chapters.
Linear-Elastic Fracture Mechanics
Experimental Determination of KIc - ASTM-E399
The accuracy with which KIc describes the fracture behaviour of real materials depends
on how well the stress intensity factor represents the conditions of stress and strain in-
side the actual fracture process zone. This is the extremely small region just ahead of the
tip of a crack where crack extension would originate. In this sense KI is exact only in the
case of zero plastic strain as in brittle materials. For most structural materials, a
sufficient degree of accuracy may be obtained if the plastic zone ahead of a crack tip is
small in comparison with the region around the crack in which the stress intensity factor
yields a satisfactory approximation of the exact elastic stress field. The decision of what
is sufficient accuracy depends on the particular application. A standardised test method
must be reproducible, specimen size requirements are chosen so that there is essentially
no question regarding this point. The standardised test method for determining KIc ma-
terial values is the ASTM-E399 test method.
W
a SE(B) Bend
Specimen
2.1W B
2.1W
=W/2
Compact Tension 0.25 W Dia.
0.6 W
Specimen 0.275 W
0.275 W
0.6 W
a
W B
1.25 W =W/2
after ASTM-E399
alu
SE Bend Specimen and CT Tension Specimen 2403.04.02
Training in Aluminium Application Technologies
TALAT 2403 24

25. Several different test specimen forms have been proposed in the course of the de-
velopment and standardization. Two of the most common for engineering applications,
the bend specimen or SE(B) specimen and the compact tension or C(T) specimen are
reproduced in Figure 2403.04.02 according to ASTM-E399.
Test procedure:
Determine location and orientation of test specimen in respect to component to be
analysed: Specimen orientation L-S and L-T cover the most common crack cases in
structural engineering components, such as welded structures, see Figure 2403.04.03.
Surface cracks grow initially in the direction of the plate thickness and propagate further
on as through-thickness cracks.
according to ASTM
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Specimen Orientation 2403.04.03
Training in Aluminium Application Technologies
Determine critical specimen size dimension: To fulfil requirements of maximum
constraint and small plastic zone in relation to specimen dimensions the following
relations must be observed.
a = crack depth > 2.5 ⋅ (KIc/σys)2
B = specimen thickness > 2.5 ⋅ (KIc/σys)2
W = specimen depth > 5.0 ⋅ (KIc/σys)2
This leads for instance to a specimen thickness of approximately 50 times the radius of
the plane strain plastic zone.
Even before a KIc test specimen can be machined, the KIc value to be obtained must
already be known or at least estimated. Three general rules may be used
TALAT 2403 25

26. − overestimate the KIc value on the basis of experience with similar materials
and judgement based on other types of notch-toughness tests
− use specimens that have as large as thickness as possible, namely a thickness
equal to that of the plates to be used in service
− use the following ratio of yield strength to modulus of elasticity to select a
specimen size. These estimates are valid for very high strength structural
materials, steels having yield strength of at least 1000 MPa and aluminium
alloys having yield strength of at least 350 MPa.
σys/E Minimum recommended
thickness and crack length
[mm]
0.0050-0.0057 75,0
0.0057-0.0062 63,0
0.0062-0.0065 50,0
0.0065-0.0068 44,0
0.0068-0.0071 38,0
0.0071-0.0075 32,0
0.0075-0.0080 25,0
0.0080-0.0085 20,0
0.0085-0.0100 12,5
0.0100 or greater 6,5
Many low- to medium-strength structural materials in section sizes of interest for most
large structures (ships, bridges, pressure vessels) are of insufficient thickness to main-
tain plane strain conditions under slow loading and at normal service temperatures.
Thus, the linear elastic analysis to calculate KIc values is invalidated by general yielding
and the formation of large plastic zones. Alternative methods must be used for fracture
analysis as described in further chapters.
The following values are given as an example for common aluminium alloys
KI σ0,2 KI/σ0,2 2.5(KIc/σ0,2)2
MPa MPa
AlMgSi1 50 245 0,0416 104
AlZn4,5Mg1 73 370 0,0389 97
Select and prepare a test specimen: Most probably one of the two standard specimen
shapes will be selected, slow-bend test specimen or compact-tension specimen. The
initial machined crack length 'a' should be 0.45 ⋅ W so that the crack can be extended by
fatigue to approximately 0.5 ⋅ W. Usually the selection of the specimen thickness B is
made first.
Perform test following requirements of ASTM-E399 procedure: This includes initial
fatigue cracking of the test specimen. Measure and plot crack opening displacement ∆v
against load P.
TALAT 2403 26

27. Analyse P-∆v record, calculate conditional KIc (=KQ) values, perform validation check
for KIc: If the KQ values meet the above stated requirements, like a or B ≥ 2.5(KQ/σ0.2)
and W ≥ 5.0(KQ/σ0.2)2, the KQ=KIc. If not the test is invalid, the results may be used
to estimate the material toughness only.
Elastic-Plastic Fracture Mechanics
Elastic-plastic fracture mechanics analysis is an extension of the linear elastic analysis.
As already mentioned low- to medium-strength structural materials used in the section
sizes of interest for large complex structures are of insufficient thickness to maintain
plane strain conditions under slow loading conditions at normal service temperatures.
Large plastic zones from ahead of the crack tip, the behaviour is elastic-plastic, invali-
dating the calculation of KIc values. There are three possible approaches into the elastic-
plastic region, through
− crack opening displacement (COD)
− R-curve analysis
− J-integral
Crack opening displacement (COD) - BS 5762
Proposed by Wells in 1961 the fracture behaviour in the vicinity of a sharp crack could
be characterized by the opening of the notch faces, namely the crack opening displace-
ment. He also showed that the concept of crack opening displacement was analogous to
the concept of critical crack extension force and thus the COD values could be related to
the plane strain fracture toughness KIc. COD measurements can be made even when
there is considerable plastic flow ahead of a crack. Using a crack tip plasticity model
proposed by Dugdale it is possible to relate the COD to the applied stress and crack
length.
As with the KI analysis the application of the COD approach to engineering structures
requires the measurement of a fracture toughness parameter δc, the critical value of the
crack tip displacement, which is a material property as a function of temperature,
loading rate, specimen thickness, and possibly specimen geometry, i.e. notch acuity,
crack length and overall specimen size.
Since the δc-test is regarded as an extension of the KIc testing the british standardized
test method after BS 5762 is very similar to the ASTM-E399 test method for KIc.
Similar specimen preparation, fatigue-cracking procedures, instrumentation, and test
procedures are followed. The displacement gage is similar to the one used in KIc testing
(Clip-Gage) and a continous load-displacement record is obtained during the test.
On the basis of the British Standard PD 6493 an analysis results is based on the
comparison of the critical COD value δc to the actual crack tip opening displacement of
the component analyzed and characterized by geometrical dimensions of the component
TALAT 2403 27

28. and the existing flaw and its location, as well as the material used - for a specific service
temperature and loading rate.
Determination of R-Curves - ASTM-E561
KIc is governed by conditions of plane strain (εz=0) with small scale crack tip plasticity.
Kc is governed by conditions of plane stress (σz=0) with large scale crack tip plasticity.
Kc values are generally 2-10 times larger than KIc. KIc values depend on only two
variables, temperature and strain rate. Kc values depend on 4 variables, temperature,
loading rate, plate thickness and initial crack length.
Plane stress conditions rather than plane strain conditions actually exist in service. Plane
stress fracture toughness evaluations using an R-curve or resistance curve analysis as
one of several extensions of linear elastic fracture mechanics into elastic-plastic fracture
mechanics is envisaged. An R-curve characterizes the resistance to fracture of a material
during incremental slow stable crack extension. An R-curve is a plot of crack growth
resistance as a function of actual or effective crack extension. KR, also in MPa√m units
is the crack growth resistance at a particular instability condition during the R-curve
test, i.e. the limit prior to unstable crack growth. In Figure 2403.04.04 the solid lines
represent the R-curves for different initial crack lengths. The dashed lines represent the
variation in KI with crack length for different constant loads P1<P2<P3. Each line is a
function of the crack length, KI = f(P,√a).
R-Curves
KR = Kc P1<P2<P3
P3
for a0 = a1
KR
Applied
K I = f (P, a ) KI Levels
P2
KR = Kc
for a0 = a2
KR < Kc
for a0 = a1
∆aactual ∆aactual
a1 a2
a
alu R-Curves and Critical Fracture Toughness Values KR 2403.04.04
Training in Aluminium Application Technologies for Different Initial Cracks a0
The two points of tangency represent points of instability, or the critical plane stress
intensity factor Kc = KR, at the particular crack length and, of course, for the given
conditions of temperature and loading rate. The KR value is always calculated by using
the effective crack length aeff and is plotted against the actual crack extension aact, that
takes place physically in the material during the test.
TALAT 2403 28

29. R-curves can be determined either by load control or displacement control tests. The
load control technique can be used to obtain only that portion of the R-curve up to the
Kc value where complete unstable fracture occurs. The displacement control technique
can be used to obtain the entire R-curve. The evaluation of R-curves for relatively low-
strength, high-toughness alloys exhibiting large scale crack tip plasticity σy at fracture,
relative to the test specimen in plane dimensions W and a, requires an elastic-plastic
approach. Here the crack-opening displacement δ at the physical crack tip is measured
and used in calculating the equivalent elastic K value. This elastic-plastic crack model is
designated the crack-opening-stretch (COS) method, where δ and COS are equivalent
terms. This method can be used with either a load-control or displacement control test.
A standardized testing procedure is available after ASTM-E561. Generally the thickness
of a test specimen is equal to the plate thickness considered for actual service. The other
dimensions are made considerably larger. The advantage of the displacement control
technique is partially offset by the necessity for new or unique loading facilities and
sophisticated instrumentation, whereas the load-control method in conjunction with
relatively simple measuring devices can be used with a conventional tension machine.
The limits of application of this technique are for materials with high strength with low
toughness, small plastic zone ahead of the crack tip. For materials with high levels of
toughness this analysis becomes increasingly less accurate.
The R-curve is determined by graphical means. A series of secant lines are constructed
on the load displacement record from a test sample. The compliance values δ/P from
these second lines are used to determine the associated aeff/W-values that reflect the
effective crack length aeff = a0 + ∆a + ry using also an appropriate relationship for the
given specimen between crack opening and effective crack length. Each aeff/W-value is
then used to determine a respective Keff-value, the latter is plotted against aeff
producing the R-curve for the given material.
The significance of the critical fracture toughness values obtained from an R-curve ana-
lysis is in the calculation of the critical flaw size acr required to cause fracture
instability. A normalised plot showing the general relationship of acr to such design
parameters, nominal design stress and yield stress, for a large center-cracked tension
specimen subjected to uniform tension is given in Figure 2403.04.05.
The plot can be used for any material for which valid fracture mechanics results (KIc,
KId, Kc, KIscc) are available under a given loading rate, temperature and state of stress.
TALAT 2403 29