talat - fracture

8/3/2019 Talat - Fracture

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TALAT Lecture 2403

Applied Fracture Mechanics

49 pages and 40 figures

Advanced Level

prepared by Dimitris Kosteas, Technische Universitt Mnchen

Objectives:

Teach the principles and concepts of fracture mechanics as well as providerecommendations for practical applications

Provide necessary information for fatigue life estimations on the basis of fracturemechanics as a complementary method to the S-N concept

Prerequisites:

Background in engineering, materials and fatigue required

Date of Issue: 1994 EAA - Europea n Aluminiu m Associati on


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TALAT 2403 2

2403 Applied Fracture Mechanics

Contents

2403.01 Historical Context ..................................................................................... 3Fracture and Fatigue in Structures ...........................................................................3

2403.02 Notch Toughness and Brittle Fracture ................................................... 5Notch-Toughness Performance Level as a Function of Temperature and LoadingRates.........................................................................................................................5Brittle Fracture .........................................................................................................8

2403.03 Principles of Fracture Mechanics............................................................ 9Basic Parameters......................................................................................................9

Material Toughness ............................................................................................ 9Crack Size ........................................................................................................... 9Stress Level ......................................................................................................... 9

Fracture Criteria .....................................................................................................13Members with Cracks ............................................................................................14Stress Intensity Factors ..........................................................................................16Deformation at the Crack Tip ................................................................................21Superposition of Stress Intensity Factors ...............................................................22

2403.04 Experimental Determination of Limit Values according to VariousRecommendations ................................................................................................... 23

Linear-Elastic Fracture Mechanics ........................................................................24 Experimental Determination of K Ic - ASTM-E399........................................... 24Test procedure: ................................................................................................. 25

Elastic-Plastic Fracture Mechanics ........................................................................27Crack opening displacement (COD) - BS 5762................................................ 27

Determination of R-Curves - ASTM-E561........................................................ 28 Determination of J Ic - ASTM-E813.................................................................. 30 Determination of J-R Curves - ASTM-E1152 .................................................. 33Crack Opening Displacement (COD) Measurements - BS 5762...................... 34

2403.05 Fracture Mechanics Instruments for Structural Detail Evaluation... 36Free Surface Correction F s ....................................................................................37Crack Shape Correction F e ....................................................................................37Finite Plate Dimension Correction F w ..................................................................38Correction Factors for Stress Gradient F g .............................................................38Remarks on Crack Geometry.................................................................................39

2403.06 Calculation of a Practical Example: Evaluation of Cracks Forming at aWelded Coverplate and a Web Stiffener .............................................................. 41

Coverplate..............................................................................................................42Web Stiffener .........................................................................................................43

2403.07 Literature/References ............................................................................... 472403.08 List of Figures.......................................................................................... 48


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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 appearedinadequate. 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 extensionin 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 spectacularfailures, has led to the development of various fracture criteria. Fracture criteria andfracture 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 plasticdeformation and at extremely high speeds. Brittle fractures are not so common as fatigue(the latter characterised by progressive crack development), yielding, or bucklingfailures, 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 andGallileo talked about fracture, too. The big break in fracture mechanics came in 1920with the Griffith theory, applicable mostly to brittle materials, as well as Orowan andIrwing and Williams in the 1940's.

Catastrophic brittle failures were recorded in the 19 th and early 20 th century. Therewere 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 byengineers. 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 throughthe 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 PleasentBridge 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:


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(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 withvery low fracture toughness at the failure temperature, and

(d) fracture resulted from a combination of factors and it would not have occurred inthe 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 theslowly 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 andloading as well as maintenance. Fractures cannot be eliminated in structures by merelyusing materials with improved notch toughness. The designer still has the fundamentalresponsibility for the overall safety and reliability of the structure.


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TALAT 2403 5

2403.02 Notch Toughness and Brittle Fracture

Notch-toughness performance level as a function of temperatureand 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 sothat the designer can determine the relative importance of each of them during designrather than during failure analysis.

Notch-Toughness Performance Level as a Function of Temperatureand 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 . Notchtoughness is defined as the ability of a material to absorb energy (usually when loadeddynamically) 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-notchimpact specimen, dynamic tear test specimen K Ic, pre-cracked Charpy, etc. Toughnessis 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 (thelatter arising often in welded components among other reasons due to residual stressesand the complexity of welds). Ductile fractures are generally preceded by large amountof plastic deformation occurring usually at 45 to the direction of the applied stress.Brittle or cleavage fractures generally occur with little plastic deformation and areusually 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 environmentalconditions.

Plane-strain behaviour refers to fracture under elastic stresses and is essentially brittle.Plastic behaviour refers to ductile failure under general yielding conditions accompaniedusually, but not necessarily, with large shear lips. The transition between these two isthe elastic-plastic region or the mixed-mode region. Higher loading rates move thecharacteristic transition curve to higher temperatures. A particular notch toughness valuecalled the nil-ductility transition (NDT) temperature generally defines the upper limitsof plane-strain behaviour under conditions of impact loading. In practice the questionhas 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 threedifferent 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.


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TALAT 2403 6

Although fully plastic behaviour would be a very desirable level of performance, it maynot be necessary or even economically feasible for many structures.

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Training in Aluminium Application Technologies

Notch-Thoughness Performance Levelsvs. Temperature

2403.02.01

L e v e

l s o f p e r f o r m a n c e a s m e a s u r e

d

b y a b s o r b e d e n e r g y

i n n o

t c h e d s p e c

i m e n s

Plastic

Elastic -Plastic

PlaneStrain

(Macro linear)Elastic

Impact loading

Static loading

Intermediateloading rate

NDT (Nil-Ductility Transition)

TemperatureD. Kosteas, TUM

Notch-Thoughness Performance Levelsvs. Temperature

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Relation between Performance andTransition Temperature for three different Steel Qualities

Plastic

Elastic-Plastic

PlaneStrain

L e v e

l s o f

P e r f o r m a n c e

Service Temperature

Steel 1Steel 2Steel 3

NDT(Steel 3)

NDT(Steel 2)

NDT(Steel 1)

TemperatureD. Kosteas, TUM

2403.02.02

Not all structural materials exhibit a ductile-brittle transition. For example, aluminiumas well as very high strength structural steels or titanium do not undergo a ductile-brittletransition. For these materials temperature has a rather small effect on toughness, seeFigure 2403.02.03 .


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TALAT 2403 7

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Notch-Toughness vs. Temperature

2403.02.03

+

++

++

+

xx

xx

xx x

ISO - ProbelongitudinaltransverseHAZfiller

parent metal

x+

Alloy 5083-H113

N o t c h

t o u g

h n e s s

k p m

/ c m

Temperature in C

6

4

2

0200-100-200

D. Kosteas, TUM

}

Notch-Toughness of Welded Aluminium Alloy 5083vs. Temperature

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 tomeasure 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 curveslike in Figure 2403.02.01 for various materials and specific applications. However,even if correlations are developed for existing structures, they do not necessarily holdfor certain designs or new operating conditions or new materials. Test results areexpressed in terms of energy, fracture appearance, or deformation, and cannot always betranslated 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 readilyrecognised and utilised by the engineer, namely stress and flaw size. Fracture mechanicsis 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 havesharp cracks.

Large and complex structures always have discontinuities of some kind. Dolan has madethe flat statement that " every structure contains small flaws " whose size and distributionare 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 structuralmembers, ranging from below 0.02 mm to several cm long. The significant point is that


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TALAT 2403 8

discontinuities are present in fabricated structures even though the structure may havebeen inspected. The problem of establishing acceptable discontinuities, for instance inwelded structures, is becoming an economic problem since techniques that minimize thesize 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 defectincreases 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 andflaw size that would be required to cause fracture. The engineer can then quantitativelyestablish allowable stress levels and inspection requirements so that fracture cannotoccur. Fracture mechanics can also be used to analyse the growth of small cracks, as forexample 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 betweenfracture-mechanics values and existing toughness test results. In many cases, because of the current limitations on test requirements for measuring fracture toughness K Ic exist-ing notch-toughness tests must be used to help the designer estimate K Ic values.

Brittle Fracture

The number of catastrophic brittle fractures have been very small generally. Especiallyfor aluminium, exhibiting rather high fracture toughness values over the wholetemperature 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.


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TALAT 2403 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, residualstresses, fatigue, constraint, etc., can contribute to brittle fractures in large structures.However, there are three primary factors in fracture mechanics that control thesusceptibility of a structure to brittle fracture:

1) Material toughness K c, KIc, KId 2) Crack size a3) Stress level

Material Toughnessis the ability to carry load or deform plastically in the presence of a notch and can bedescribed in terms of the critical stress-intensity factor under conditions of plane stressKc or plane strain K Ic for slow loading and linear elastic behaviour or K Id 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 thanlinear elastic behaviour, the material toughness is measured in terms of parameters suchas R-curve resistance, J Ic, and COD as described later on.

Crack SizeFracture initiates from discontinuities or flaws. These can vary from extremely smallcracks 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 growby fatigue or stress corrosion to a critical size.

Stress LevelTensile 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 formany years and controlled the above factors qualitatively through good design (adequate

sections, minimum stress concentrations) and fabrication practices (decreasedimperfection or discontinuity size through proper welding and inspection), as well as theuse of materials with sufficient notch toughness levels.


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TALAT 2403 10

A linear elastic fracture mechanics technology is based upon an analytical procedurethat relates the stress field magnitude and distribution in the vicinity of a crack tip to thenominal 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 theelastic stress field in the vicinity of a crack tip for tensile stresses normal to the plane of

the crack (Mode I deformation).

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Elastic Stress Field Distribution near a Crack

2403.03.01

Nominal

stress

y

x

Magnitude of stressalong x axis, y

y ( = 0)

y

x

Cracktip r

xy

y = cos (1 + sin sin )KI

(2 r)1/23 2

2

2

x = cos (1 - sin sin )KI

(2 r)1/23 2

2

2

Elastic Stress Field Distribution near a Crack

Source: Irwin and Williams, 1957

The equations describing the crack tip stress field distribution were formulated by Irwinand Williams (1957) as follows:

x

I K

r = 2 2

12

32

cos sin sin

y

I K

r = + 2 2

12

32

cos sin sin

xy

IKr

=2 2 2

32

sin cos cos

xz yz= =0 z = 0 Plane Stress (thin sheet)

( ) z x y= + Plane Strain (thick sheet)

The distribution of the elastic stress field in the vicinity of the crack tip is invariant in allstructural components subjected to this type of deformation. The magnitude of theelastic stress field can be described by a single parameter, K I, 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 affectthe 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 .


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TALAT 2403 11

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KI values for various crack geometries

2403.03.02

2 a

Through thickness crack

a

2 c

a

KI Values for Various Crack Geometries

K a I = K a I = 112 K aQ I

= 112

Surface crack

where Q = f(a/2c, )

Edge crack

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 K c. For mode I deformation and forsmall crack tip plastic deformation, i.e. plane strain conditions, the critical stress in-tensity factor for fracture for fracture instability is K Ic. The value K Ic represents thefracture toughness of the material (the resistance to progressive tensile crack extension

under plane strain conditions) and has units of MN/m 3/2 (or MPa/mm 1/2 or ksi in).

1ksi in = 34.7597 N/mm 3/2 or MPa/mm 1/2

{1ksi = 6.89714 N/mm 2 or MPa}

This material toughness property depends on the particular material, loading rate, andconstraint:

Kc critical stress intensity factor for static loading and plane stress conditions of variable constraint. This value depends on specimen thickness andgeometry, 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 strainconditions of maximum constraint.


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TALAT 2403 12

whereK K K C ac Ic Id , , =

C = constant, function of specimen and crack geometry = nominal stress

a = flaw size

Through knowledge of the critical value at failure for a given material of a particularthickness and at a specific temperature and loading rate, tolerable flaw sizes for a givendesign stress level can be determined. Or design stress levels may be determined thatcan be safely used for an existing crack that may be present in a structure, see Figure2403.03.03 .

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Training in Aluminium Application Technologies2403.03.03

Relation between material toughness, flaw

size and stress 2 a

(Fracture zone)

K C o f t o u g h e r s t e e l

KC

KC = critical value of K I

KI = f(! , a)

! f

! 0

a 0 a f

I n c r e a s

i n g s t r e s s ,

Increasing flaw size, 2a

Relation between Material Toughness, FlawSize and Stress

Increasing materialtoughness

(COD, J Ic, R)

In an unflawed structural member, as the load is increased the nominal stress increasesuntil an instability (yielding at ys) occurs. Similarly in a structural member with a flawas the load is increased ( or as the size of the flaw grows by fatigue or stress corrosion)the stress intensity K I increases until an instability, fracture at K Ic, 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 theactual 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 K c level.


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TALAT 2403 13

COLUMN RESEARCH COUNCILCOLUMN STRENGTH CURVE

EULER CURVE

YIELDING

(a) COLUMN INSTABILITY

P

P

L

L/r

! YS

!

! = ! YS

( )

c

E

L r =2

2 /

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YIELDING

(b) CRACK INSTABILITY

!

!

a

! YS

!

! = ! YS

cC K

C a=

2a

Source: Madison and Irwin

Analogy: Column Instability and Crack Instability 2403.03.04

The critical stress intensity factor K Ic represents the terminal conditions in the life of astructural component. The total useful life N T of the component is determined by thetime necessary to initiate a crack N I and by the time to propagate the crack N P fromsubcritical dimensions a 0 to the critical size a c. Crack initiation and subcritical crack propagation are localised phenomena that depend on the boundary conditions at thecrack tip. Subsequently it is logical to expect that the rate of subcritical crack propaga-tion depends on the stress intensity factor K I which serves as a single term parameter re-presentative of the stress conditions in the vicinity of the crack tip. Fracture mechanicstheory 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 K Iscc -value is used as the failure criterion rather than K Ic. This threshold value K Iscc is thevalue below which subcritical crack propagation does not occur under static loads inspecific environment. In the relationship between material toughness, design stress and

flaw size, Figure 2403.03.03, KIscc replaces K c as the critical value of K I.

Fracture Criteria

A careful study of the particular requirements of a structure is the basis in developing afracture control plan, i.e. the determination of how much toughness is necessary andadequate. 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


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TALAT 2403 14

Developing a fracture control plan for a complex structure is very difficult. All factorsthat may contribute to the fracture of a structural detail or failure of the entire structurehave 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 haveto 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 extendedindiscriminately 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 K Ic (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 K Ic or K c of the material at the temperature and

loading rate of the application. The fracture toughness can be modified bychanging 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 criticalcrack. These can be controlled by design changes, fabrication, andinspection.

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 inthe 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 evaluationof the safety and reliability of a structure is possible.

Fracture mechanics can be subdivided into two general categories namely linear-elasticand elastic-plastic. The following relationships and equations for stress intensity factorsare 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 aninfinitesimal element containing a crack form. Figure 2403.03.06 shows the coordinatesystem and stress components ahead of a crack tip


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TALAT 2403 15

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Basic cracking modes

Mode I

Mode II

Mode III

xy

z

xy

z

xy

z

Basic Modes of Cracking

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Coordinate System and Stress Components aheadof a Crack Tip 2403.03.06

Leading edge of the crack

r

y

x

z

x

y

yz

z xz

xy

Mode I is characterized by displacement of the two fracture surfaces perpendicular toeach 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 respectivestress fields in the vicinity of the crack tips, Figure 2403.03.07 . In any problem thedeformations 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 thethree basic types of stress fields. For practical applications Mode I is the most importantsince according to Jaccard even if a crack starts as a combination of different modes itis soon transformed and continues its propagation to a critical crack under Mode I.


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TALAT 2403 16

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Training in Aluminium Application Technologies2403.03.07Stress Field Equations in the Vicinity of Crack Tips

x

I K

r =

2 2

12

32

cos sin sin

y

I K

r =

+ 2 2

12

32

cos sin sin

xy

I K

r =

2 2 232

sin cos cos

xz yz= =0

uK G

r I = + 2 21 2

2

1 2

2

cos sin

vK G

r I = 2 22 2

2

1 2

2

sin cos

w = 0

x

II K

r =

+ 2 2

22

32

sin cos cos

xy

II K

r =

2 2

12

32

cos sin sin

y

II K

r =

2 2 232

sin cos cos

( ) z x y= + xz yz= =0

uK G

r II = + 2 22 2

2

1 2

2

sin cos

vK G

r II = + 2 21 2

2

1 2

2

cos sin

w = 0

xz

III K

r =

2 2sin

yz

III K

r =

2 2cos

x y z xy= = = =0

wK G

r III =2 2

1 2

sin

u v= =0

Stress Field Equations in the Vicinity of Crack TipsMode I Mode II Mode III

( ) z x y= +

Dimensional analysis of the equations shows that the stress intensity factor must belinearly 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 intensityfactor is given by

K f g 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 outof respective literature

[1] C.P. Paris and G.C. Sih, "Stress analysis of cracks" in "Fracture toughness testingand 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


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TALAT 2403 17

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Through-Thickness Crack

Finite Width Plate

2a

2b

SIF for Through-Thickness Crack 2403.03.08

Infinite Width Plate

Nominal Stress

2a

K a= ( ) ( )( )[ ]K a a b I = sec / 21

2

( ) ( )( )[ ]sec / a b21

2

Tangent correction for finite width

a b0.074 1.000.207 1.030.275 1.050.337 1.080.410 1.120.466 1.160.535 1.220.592 1.29

Stress intensity factor for a double-edge crack: Figure 2403.03.09

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Double-Edge Crack

a

2b

SIF for Double-Edge Crack 2403.03.09

a

( ) ( )( )[ ]sec / a b2 12Tangent correction for finite width

ab

0.074 1.000.207 1.030.275 1.050.337 1.080.410 1.120.466 1.16

0.535 1.220.592 1.29

( ) ( )( )[ ]K a a b I = 112 2 12. sec /


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TALAT 2403 18

Stress intensity factor for a single-edge crack: Figure 2403.03.10

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Single-Edge Crack

2b

SIF for Single-Edge Crack 2403.03.10

KI =1.12 (a) 1/2

a

For a semi-finite edge-cracked specimen:

For a finite width edge-cracked specimen:

KI = (a) f(a/b)

Correction factor for a single-edge crackin a finite width plate

a/b f(a/b)0.10 1.150.20 1.200.30 1.290.40 1.370.50 1.510.60 1.680.70 1.890.80 2.140.90 2.461.00 2.86

Stress intensity factor for cracks emanating from circular or elliptical holes: Figure2403.03.11

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!

!

bN

a N

a Fa

( )K FI = , a

where = aa

and = baN

N

N

( )F ,

1.1

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

01.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40

Cracks Emanating fromCircular or Elliptical Holes

SIF for Cracks Emanating from Circular or Elliptical Holes

The stress intensity factorin the case of a finite plate is

=1.0For a circular hole

2403.03.11

=aa N


19/49

TALAT 2403 19

Stress intensity factor for a single-edge crack in beam in bending: Figure 2403.03.12

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Single-Edge Crack in Beam in Bending

SIF for Single-Edge Crack in Beam in Bending 2403.03.12

a

wM M

( )K

M

W a f

aW I

=

632

W: Depth of the Beam

Correction factor for notched beamsa/W f(a/W)

0.05 0.360.10 0.490.20 0.600.30 0.660.40 0.690.50 0.72

>0.60 0.73

Stress intensity factor for an elliptical or circular crack in an infinite plate: Figure2403.03.13

alu


!

!

aa

c

c

"

Elliptical or Circular Crack in an Infinite Plate

SIF: Elliptical or Circular Crack in an Infinite Plate 2403.03.13

( )K

a ab I

= +

32

0

2

2

2

14

sin cos

02 2

22

3

2

0

12

= c ac d sin

The stress intensity factor at any point along the perimeter of elliptical orcircular cracks in an infinite body subjected to uniform tensile stress is

The point on the perimeter of the crackis defined by the angle and the elliptic integral #o

( )For circular cracks, c = a = K a I 21 2

When c and = 2 this case is reducedto a through- thickness crack.

,


20/49

TALAT 2403 20

Stress intensity factor for a surface crack: Figure 2403.03.14

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Surface Crack"Thumbnail Crack"

SIF: Surface Crack

!

2ca

!

2403.03.14

0

2 2

22

0

2

32

1= c ac

d sin

The stress intensity factor can be calculatedfrom the equations of elliptical crack using afree surface correction factor of 1.12and for the position = /2

K I =

1121

2

. a

Q

with and the elliptic integralQ = 0 2

Q is regarded as a shape factor because its valuesdepend on a and c.For values of Q see Figure 2402.03.15.

Flaw shape parameter: Figure 2403.03.15

alu


0.5

0.4

0.3

0.2

0.1

00.5 1.0 1.5 2.0 2.5

a/2cRatio

Flaw Shape Parameter, Q

Flaw Shape Parameter, Q 2403.03.15

ys =0

ys = 0 80. ys = 0 60.

ys =10.

2c a

Flaw Shape Parameter, Q


21/49

TALAT 2403 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


22/49

TALAT 2403 22

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Plane stressmode I

Planestrain

mode I

Cracktip

Midsection

Surface

Crack tip

y

zx

KI

2 $ ! yrIp (plane strain)

z

rp (Plane stress)

Plane strain region

Machine notch

Fatigue crackPlastic zone

Plastic Zone Dimensions

A - Overall view B - Edge view

Specimen cross-section

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 Figure2403.03.07 . Consequently the total stress intensity factor can be obtained byalgebraically adding the individual stress intensity factors corresponding to each load.


23/49

TALAT 2403 23

2403.04 Experimental Determination of Limit Values according toVarious Recommendations

Linear-Elastic Fracture Mechanics Experimental determination of K Ic - ASTM-E399

Elastic-Plastic Fracture Mechanics Crack opening displacement (COD) - BS 5762 Determination of R-Curves - ASTM-E561 Determination of J Ic - 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 ys is the limiting value for loadingstresses, the critical stress intensity factors K Ic, K Id or K c are the limiting values for thestress intensity factor K I. The critical stress intensity factor K c at which unstable crack growth occurs for conditions of static loading at a particular temperature depends onspecimen thickness or constraint, Figure 2403.04.01 .

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Effect of Thickness on K cPlane stress Plane strain

240

200

160

120

80

40

05.03.02.01.00.750.500.300.25

Thickness, in.

K c ,

k s i

i n .

K Ic

Thickness at the beginningof plane strain is:

Effect of Thickness on K c

t=2.5KIc

y

2

The limiting value of K c for plane strain (maximum constraint) conditions is K Ic (slowloading rate) or K Id (dynamic or impact load). The K Ic-value is the minimum value forplane strain conditions. The K Ic-value would also be a minimum value for conditions of maximum structural constraint (for example, stiffeners, intersecting plates, etc.) that


24/49

TALAT 2403 24

might lead to plain-strain conditions even though the individual structural membersmight be relatively thin.

Design procedures based on the K Ic-value provide a "conservative" approach to thefracture problem. The following chapter gives information on experimental require-ments and procedures for the measurement of K Ic-values.

For thin section problems further procedures based on a K c or R-curve analysis forelastic-plastic behaviour problems an analysis on the basis of J Ic- or COD-procedureswill be described in further chapters.

Linear-Elastic Fracture Mechanics

Experimental Determination of K Ic - ASTM-E399

The accuracy with which K Ic describes the fracture behaviour of real materials dependson 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 thetip of a crack where crack extension would originate. In this sense K I is exact only in thecase of zero plastic strain as in brittle materials. For most structural materials, asufficient degree of accuracy may be obtained if the plastic zone ahead of a crack tip issmall in comparison with the region around the crack in which the stress intensity factoryields a satisfactory approximation of the exact elastic stress field. The decision of whatis sufficient accuracy depends on the particular application. A standardised test method

must be reproducible, specimen size requirements are chosen so that there is essentiallyno question regarding this point. The standardised test method for determining K Ic ma-terial values is the ASTM-E399 test method.

alu

Training in Aluminium Application TechnologiesSE Bend Specimen and CT Tension Specimen

2.1W 2.1WB

=W/2

W

a

1.25 WW

0 . 6 W

0 . 6 W

0.275 W

0.275 W

0.25 W Dia.

aB

=W/2

Compact TensionSpecimen

SE(B) BendSpecimen

after ASTM-E399

2403.04.02


25/49

TALAT 2403 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 arereproduced in Figure 2403.04.02 according to ASTM-E399.

Test procedure:

Determine location and orientation of test specimen in respect to component to beanalysed: Specimen orientation L-S and L-T cover the most common crack cases instructural engineering components, such as welded structures, see Figure 2403.04.03 .Surface cracks grow initially in the direction of the plate thickness and propagate furtheron as through-thickness cracks.

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Specimen Orientation 2403.04.03

according to ASTM

Determine critical specimen size dimension: To fulfil requirements of maximum

constraint and small plastic zone in relation to specimen dimensions the followingrelations must be observed.

a = crack depth > 2.5 (KIc / ys)2 B = specimen thickness > 2.5 (KIc / ys)2 W = specimen depth > 5.0 (KIc / ys)2

This leads for instance to a specimen thickness of approximately 50 times the radius of the plane strain plastic zone.

Even before a K Ic test specimen can be machined, the K Ic value to be obtained mustalready be known or at least estimated. Three general rules may be used


26/49

TALAT 2403 26

overestimate the K Ic value on the basis of experience with similar materialsand judgement based on other types of notch-toughness tests

use specimens that have as large as thickness as possible, namely a thicknessequal to that of the plates to be used in service

use the following ratio of yield strength to modulus of elasticity to select aspecimen size. These estimates are valid for very high strength structuralmaterials, steels having yield strength of at least 1000 MPa and aluminiumalloys having yield strength of at least 350 MPa.

ys /E Minimum recommendedthickness and crack length

[mm]0.0050-0.0057 75,00.0057-0.0062 63,00.0062-0.0065 50,00.0065-0.0068 44,0

0.0068-0.0071 38,00.0071-0.0075 32,00.0075-0.0080 25,00.0080-0.0085 20,00.0085-0.0100 12,5

0.0100 or greater 6,5

Many low- to medium-strength structural materials in section sizes of interest for mostlarge 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 K Ic values is invalidated by general yieldingand the formation of large plastic zones. Alternative methods must be used for fractureanalysis as described in further chapters.

The following values are given as an example for common aluminium alloys

KI MPa

0,2 MPa

KI / 0,2 2.5(K Ic / 0,2 )2

AlMgSi1 50 245 0,0416 104AlZn4,5Mg1 73 370 0,0389 97

Select and prepare a test specimen: Most probably one of the two standard specimenshapes will be selected, slow-bend test specimen or compact-tension specimen. Theinitial machined crack length 'a' should be 0.45 W so that the crack can be extended byfatigue to approximately 0.5 W. Usually the selection of the specimen thickness B ismade first.

Perform test following requirements of ASTM-E399 procedure: This includes initialfatigue cracking of the test specimen. Measure and plot crack opening displacement v

against load P.


27/49

TALAT 2403 27

Analyse P- v record, calculate conditional K Ic (=K Q) values, perform validation check for K Ic: If the K Q values meet the above stated requirements, like a or B 2.5(K Q / 0.2 )and W 5.0(K Q / 0.2)2, the K Q=KIc. If not the test is invalid, the results may be usedto 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 sectionsizes of interest for large complex structures are of insufficient thickness to maintainplane 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 K Ic 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 couldbe 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 tothe concept of critical crack extension force and thus the COD values could be related tothe plane strain fracture toughness K Ic. COD measurements can be made even whenthere is considerable plastic flow ahead of a crack. Using a crack tip plasticity modelproposed by Dugdale it is possible to relate the COD to the applied stress and crack length.

As with the K I analysis the application of the COD approach to engineering structures

requires the measurement of a fracture toughness parameter c, the critical value of thecrack 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 c-test is regarded as an extension of the K Ic testing the british standardizedtest method after BS 5762 is very similar to the ASTM-E399 test method for K Ic.Similar specimen preparation, fatigue-cracking procedures, instrumentation, and testprocedures are followed. The displacement gage is similar to the one used in K Ic 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 c to the actual crack tip opening displacement of the component analyzed and characterized by geometrical dimensions of the component


28/49

TALAT 2403 28

and the existing flaw and its location, as well as the material used - for a specific servicetemperature and loading rate.

Determination of R-Curves - ASTM-E561

KIc is governed by conditions of plane strain ( z=0) with small scale crack tip plasticity.Kc is governed by conditions of plane stress ( z=0) with large scale crack tip plasticity.Kc values are generally 2-10 times larger than K Ic. KIc values depend on only twovariables, temperature and strain rate. K c 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. Planestress fracture toughness evaluations using an R-curve or resistance curve analysis asone of several extensions of linear elastic fracture mechanics into elastic-plastic fracturemechanics is envisaged. An R-curve characterizes the resistance to fracture of a materialduring incremental slow stable crack extension. An R-curve is a plot of crack growthresistance as a function of actual or effective crack extension. K R, also in MPa m unitsis the crack growth resistance at a particular instability condition during the R-curvetest, i.e. the limit prior to unstable crack growth. In Figure 2403.04.04 the solid linesrepresent the R-curves for different initial crack lengths. The dashed lines represent thevariation in K I with crack length for different constant loads P1


29/49

TALAT 2403 29

R-curves can be determined either by load control or displacement control tests. Theload control technique can be used to obtain only that portion of the R-curve up to theKc value where complete unstable fracture occurs. The displacement control techniquecan 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 y at fracture,relative to the test specimen in plane dimensions W and a, requires an elastic-plasticapproach. Here the crack-opening displacement at the physical crack tip is measuredand used in calculating the equivalent elastic K value. This elastic-plastic crack model isdesignated the crack-opening-stretch (COS) method, where and COS are equivalentterms. 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 thicknessof a test specimen is equal to the plate thickness considered for actual service. The otherdimensions are made considerably larger. The advantage of the displacement controltechnique is partially offset by the necessity for new or unique loading facilities andsophisticated instrumentation, whereas the load-control method in conjunction withrelatively 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 lowtoughness, 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 constructedon the load displacement record from a test sample. The compliance values /P fromthese second lines are used to determine the associated a eff /W-values that reflect the

effective crack length a eff = a 0 + a + r y using also an appropriate relationship for thegiven specimen between crack opening and effective crack length. Each a eff /W-value isthen used to determine a respective K eff -value, the latter is plotted against a eff 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 a cr required to cause fractureinstability. A normalised plot showing the general relationship of a cr to such designparameters, nominal design stress and yield stress, for a large center-cracked tensionspecimen 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 (K Ic,KId, Kc, KIscc ) are available under a given loading rate, temperature and state of stress.


30/49

TALAT 2403 30

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Normalized Plot for Critical Flaw Size

2403.04.05

10.08.0

5.0

3.0

2.0

1.00.80

0.50

0.30

0.20

= 0.10( )KCYS

100

40

20

10

4

2

1.0

0.4

0.2

0.10

0.04

0.02

0.010

0.004

0.0020.001

1.000.900.800.700.600.500.400.300.200.100

a c r ,

c r i t i c a

l f l a w s i z e , i

n c h e s

( )DYS

acr

2a cr

D

D

KI = D a

a cr =1

2

( )KCD

Normalized Plot for Critical Flaw Size

Determination of J Ic - ASTM-E813

Another means of directly extending fracture mechanics concepts from the linear-elasticbehaviour to the elastic-plastic behaviour is the path independent J-integral proposed byRice as a method of characterizing the stress strain field at the tip of a crack by anintegration path taken sufficiently far from the crack tip to be substituted for a path closeto the crack tip region.

For linear elastic behaviour the J-interal is identical to the energy release rate G per unitcrack extension. Therefore a J-failure criterion for the linear elastic case is identical tothe K Ic-failure criterion, under linear-elastic plane strain conditions.

( ) J G

K

E Ic Ic

Ic= = 12 2

The energy line integral J is defined for either elastic or elastic-plastic behaviour

J W dy T U x

dx R

=

where R=any contour around the crack tip. Figure 2403.04.06 shows the crack-tipcoordinate system and arbitrary line integral contour. Note the counterclockwiseevaluation starting from the lower flat notch surface.


31/49

TALAT 2403 31

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Note the counterclockwise evaluationstarting from the lower flat notch surface.

R

r

y

x

n

J - Integral 2403.04.06

J W dy T U x

dx R

=

The energy line integral J is defined for anarbitrary contour R around the crack tip foreither elastic or elastic-plastic behaviour.

W = the strain energy densityT = the traction vector according to

the outward normal n along R,U = displacement vectors = arc length along R

T ni ij ij=

J - Integral

W = the strain energy densityT = the traction vector according to the outward normal n along R, T i = ij n j U = displacement vectors = arc length along R

The actual testing procedure is standardized in ASTM-E813 and either a family of loaddisplacement records for different initial crack sizes or a single specimen (i.e. sameinitial crack size) may be used.

Using a compliance method several specimens of varying crack length are used toobtain P vs. curves. Values of energy per unit thickness (area under the P- curve) areobtained for different initial crack lengths at various values of deflection . The slopesof these curves are the changes in potential energy per unit thickness per unit change incrack length and thus are equal to values of

JB

Ua

= 1

See Figure 2403.04.07 for an illustrative interpretation of the J integral.

Specimen forms and dimensions (bend, bar, C(T) or WOL specimen), testing equipmentand procedure are given in ASTM-E813.


32/49

TALAT 2403 32

alu

Training in Aluminium Application TechnologiesInterpretation of the J - Integral

PB

a da

2403.04.07

P

a

a+da

JBda

J U = 1

B a

Using a compliance method several specimens of varying crack length are used toobtain P vs. curves.Values of energy per unit thickness (area under the P- curve) are obtained for differentinitial crack lengths at various values of deflection .The slopes of these curves are the changes in potential energy per unit thicknessper unit change in crack length and thus are equal to values of J.

Interpretation of the J - Integral

For the data analysis, Figure 2403.04.08 , calculate J values from the P vs. usingJ=A/(B (W-a 0)) f(a0 /W), where f(a 0 /W) is a correction factor for the given

specimen form, for a three-point bending specimen f(a 0 /W) = 2. Plot J vs. a. Constructthe 'blunting' line J=2r' a, with '=( ys+ult ). Draw the best fit line to the J vs.cack-extension points. Include only the points where actual crack extension has occured,see Figure 2403.04.09 . Where crack extension appears only as a stretch zone the point

should fall along the blunting line.

alu

Training in Aluminium Application TechnologiesProcedure of J Measurement

Load

Displacement, (Step 1)

PrecrackEnd a

(Step 2)

J

J IcFit of Data Points

(Step 4)(Step 3)

J = 2 flow a

a

J

a

2403.04.08


33/49

TALAT 2403 33

alu

Training in Aluminium Application TechnologiesEvaluation Procedure and Limits of J Measurement

Blunting Line

0.15 mm Offset

J -

I n t e g r a l

k J o u

l e s / m

R-Curve Regression Line

1.5mm Offset

( eliminated data )

0 0.5 1.0 1.5 2.0 2.5a p [ ]mm

( )a minp ( )a maxp

2403.04.09

Evaluation Procedure and Limits of J Measurement

Eliminate all data points which lie above J max =b0 '/15. Final verification of validresults is performed by comparing specimen dimensions a, B, b 0 = W-a 0 as follows

( )

= a B or b

J Q

,

'

0 25

Determination of J-R Curves - ASTM-E1152

The J-R curve characterizes the resistance of metallic materials to slow stable crack growth after initiation from a preexisting fatigue crack or other sharp flaw. The J-R canbe used as an index of material toughness for alloy design, material selection, and qual-ity assurance. The J-R curve from bend type specimens defines the lower bound esti-mates of J-capacity as a function of crack extension, and has been observed to beconservative in comparison with those obtained with tensile loading specimenconfigurations. The J-R curve can be used to assess the stability of cracks in structuraldetails in the presence of ductile tearing.

A testing procedure has been standardized in ASTM-E1152. The resistance curve ismeasured and the J integral is estimated in a way analogous to ASTM-E813.

The maximum J integral for a specimen is given by the smaller of:

Jmax = b0 '/20 or

Jmax = B '/20


34/49

TALAT 2403 34

Data points with J values exceeding Jmax should be eliminated. The maximum crack extension capacity is given by

amax = 0.1 b0

Crack Opening Displacement (COD) Measurements - BS 5762

A standardized testing procedure is given in BS 5762. The crack opening displacement is measured on three-point bend specimens, see Figure 2403.04.10 , and transformedto a crack tip opening displacement . The crack extension values are measured on thespecimen fracture surface. A plot of COD values vs. these crack extension values aallows the estimation of a critical value.

alu


S = 4W4.5W

W

P

a

vp

z

r(W-a) W

a

B =W2

after BS 5762

Hinge Model for a Standard Bending Specimen 2403.04.10

Measurements of COD

Values of are estimated by

c = u = m = i =K

EW a

W a z vyp

2 212

0 40 4 0 6

( ) , ( ), ,

+

+ +

where

K P S

B W f

aW

=

3 2 /


35/49

TALAT 2403 35

Depending on the form of the respective load-displaced curve 4 different critical values are interpreted:

c COD without stable crack extension, instable crack leading to fracture, partialbrittle fracture or pop-in in the P-v curve

u

COD with stable crack extension, instable crack leading to fracture fracture,partial brittle fracture or pop-in in the P-v curve

i COD at initiation of stable crack extensionm COD at maximum load for P-v curves with an extended region of stable crack ex-

tension

Use specimen dimensions, especially thickness, similar to dimensions in service.


36/49

TALAT 2403 36

2403.05 Fracture Mechanics Instruments for Structural DetailEvaluation

Free surface correction F s Crack shape correction F e

Finite plate dimension correction F w Correction factors for stress gradient F g Remarks on crack geometry

In the structural component to be evaluated the existence of a crack-like flaw of given orassumed dimensions is postulated. The stress field at the location of the flaw has to beconsidered. Information on flaw size and orientation and loading have to be consideredthrough a fracture mechanics parameter, a, K, J, or value. For the material and itsservice conditions, like temperature, environment, static or dynamic loading, the respec-

tive characteristic material value has to be estimated.

Failures in metal structures are most of the times due to the formation of one or morecracks as a result of the repeated application of loads, i.e. fatigue. Under certainenvironmental conditions, like corrosion, or due to fabrication conditions a flaw may bepresent which will act as an initiating crack for a fatigue failure.

If a large crack exists in a structure because of fabrication or some other event, the de-sign fatigue resistance curves are no longer applicable. The residual fatigue resistance inthese cases must be assessed by fracture mechanics models.

Complex details such as those in common use in most structural engineering structures,especially but not only in welded structures, the stress intensity factor for a surface crack of depth 'a', can be conveniently related to the well known expression for a centralthrough crack in an infinite plate by use of correction factors.

The correction factors modify a to account for effects of free surface F s, finitewidth F w, non uniform stresses acting on the crack F g, and the crack shape F e. The re-sulting stress intensity factor is expressed as

K F F F F ae s w g=

To evaluate fracture instability, the total sum of stresses due to residual welding or roll-ing stresses, dead load, and live loads must be considered. For cyclic fatigue loading, is the live load variation in stress which results in a K stress intensity value range.

For the correction factors, solutions both empirical and exact can be found in theliterature. For common cases in engineering practice the following expressions arestated.


37/49

TALAT 2403 37

Free Surface Correction F s

In a semi-elliptical surface crack in a plate subjected to uniform stress

F a cs = 1211 0186. .

The accuracy is 1.5% for 0.2 < (a/c) < 0.4. The ratio (a/c) 0.3 has been observed withthe welded aluminium beams of the TUM fatigue program. This value has been reportedas a lower limit for steel weldments as well.

Crack Shape Correction F e

Integral transformation of a 3-dimensional elliptical crack shape has resulted in theelliptical crack shape correction factor,

Fe = 1/E(k)

for the point of maximum stress intensity on the ellipse, where

( ) E k k d o

( ) sin= 1 2 22

with

k c a

c2

2 2

2=

i.e. dependent only upon the ratio of the minor to the major axis semi-diameter ratio a/c.

Values of F e for respective values of k 2 can be taken from the curve in Figure2403.05.01 .

alu

Training in AluminiumApplication Technologies

1

0.95

0.9

0.85

0.8

0.75

0.7

0.65

0 0.2 0.4 0.6 0.8 1k2

Fe

Crack Shape Correction Factor F e 2403.05.01

Crack Shape Correction Factor F e

( ) E k k d o

( ) sin= 1 2 22

k c a

c2

2 2

2=F E k e =1 ( ) where with


38/49

TALAT 2403 38

Relationships for a/c have been empirically determined for different structural geome-tries and are given in Figure 2403.05.02 . The lower boundary for stiffeners is givenwith c a= 1403 0 951. . mm. The lower boundary for coverplates is given withc a= 5451 1133. . mm. For the above mentioned mean value a/c = 0.3 the respective F e value is F e = 0.912.

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C = 1.296 a 0.946 (mm)

C = 1.403 a 0.951 (mm)

C = 3.355 + 1.29a (mm)

C = 1.489 a1.241

(mm)

C = 1.088 a 0.946 (in.)

C = 1.197 a 0.951 (in.)

C = 0.132 + 1.29a (in.)

C = 3.549 a 1.133 (in.)

C = 5.451 a 1.133 (mm)

C = 3.247 a 1.241 (in.)

StiffenersCoverplates

Crack Depth, a (mm)

a/c

1.0

0.8

0.6

0.4

0.2

0 2 4 6 8

0.1 0.2 0.3 a (in.)

Source: Fisher, steel structures

Crack Shape Measurements 2403.05.02

Finite Plate Dimension Correction F w

For a central crack in a plate of finite width the correction factor is (see also underLecture 2403.03 ):

( ) ( )( )[ ]F a bw = sec / 21 2

or for a double edge crack in a plate of finite width

( ) ( )( ) ( ) ( )( )[ ]F b a a bw = 2 21 2

/ tan /

with an accuracy of 0.3% for (a/b)


39/49

TALAT 2403 39

gration. An outline for this procedure is given in "Albrecht/Yamada: Rapid Calculationof SIF, Journal Struct. Division ASCE, Vol 103, No ST2, Feb 1977". Approximateequations for the factor F g have been derived for several details.

One approximate method which appears applicable to a number of details such as stiffe-ners, attachments, coverplates and gussets states that

F K

Ggtm=

+ 1

with = a/t and K tm = maximum SIF at weld toe. Characteristic values for G and andexpressions for K tm can be taken from Figure 2403.05.03 .

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Representative Values for G, " and K tm

Source: Albrecht/Yamada

Web Gusset

Type of Detail

Z

tf

tcp

tg

WWg

R > 25 mm

tf L

L

Wftf

tw

G "

Gusset Plateswith RadiusTransition

CoverplatedBeam

Stiffener andShort Attachment

2.776 0.2487

6.789 0.4348

0.862 0.60

0.88 0.576

2403.05.03

for web stiffeners:Ktm = 1.621*log(z/t f) + 3.963

for coverplates:Ktm = -3.539*log(z/tf)

+ 1.981*log(t cp /tf) + 5.798

for gussets with R>25mm:Ktm = -1.115*log(R/W)

+ 0.537*log(L/W)+ 0.138*log(W g /W)+ 0.285*log(t

g /t

f)+0.68

Ktm

For groove welds (butt welds) with the reinforcement in place (Gurney)

( )F ag b= 2 5.

where ( ) ( )b h= + 1 2 3 1 0 06. log . and a = 2a/t, h is the acute angle between the platesurface and the tangent to the weld profile.

Remarks on Crack Geometry

Cracks emanating from internal discontinuities in welds quickly assume a circularshape. Even very irregularly shaped discontinuities (pores, inclusions) may be modelled

as either an ellipse growing into a circular shaped crack or the initial discontinuity maybe considered as a circumscribed circle.


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TALAT 2403 40

Cracks forming and growing at the weld toe will be generally modelled as semi-ellipti-cal surface cracks. Multiple cracks will occur along the weld toe of a transverse weld,such as weld toes of coverplates and stiffeners welded to the flange of a girder. Thesesmall single cracks are usually located closely to each other and initially tend to growinto a semi-circular shape, but eventually the cracks begin to coalesce. Coalescence of single cracks into a merged crack has been observed to occur at crack depths as small as

1.27 mm. Typical initial discontinuities will be approximately 0.38 mm to 1.52 mm longand up to 0.76 mm deep, see Figure 2403.05.02 . The following expression for the crack length 'c' may be used an approximate lower boundary of measured coverplated beams

c a= 5462 1133. .

where 2c is the width of the coalesced multiple crack.

Finally Figure 2403.05.04 shows a plot of the correction factors as a function of crack size for a coverplated beam. It can be seen that small cracks are significantly affected by

the stress gradient correction factor F g. This also contributes to the spread in the crack width and the early coalescence of the individual flaws. F g tends to decay rapidly and isnot a significant factor for larger cracks.

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Training in Aluminium Application TechnologiesCorrection Factors vs. Crack Size for Coverplates

Correction Factors vs. Crack Size for Coverplates

C o r r e c t

i o n

F a c t o r s

Fg

Fs

Fw

Fe

7.0

6.0

5.0

4.0

3.0

2.0

1.0

0 0.5 1.0 = a/t f

2403.05.04


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TALAT 2403 41

2403.06 Calculation of a Practical Example: Evaluation of CracksForming at a Welded Coverplate and a Web Stiffener

Coverplate Web stiffener

For an existing crack in a structural component the residual fatigue resistance may beassessed by fracture mechanics. The crack is modelled as above and the stress intensityrange estimated as

K F F F F ae s w g e=

where e is the equivalent constant stress range calculated from random or spectrumstress amplitudes through a damage accumulation assumption.

The cycles and time required to propagate a crack ai to some larger crack af can be esti-mated as

N N dN da

C K p N

N

ma

a

i

f

i

f

= = =

It is also necessary to check the fracture resistance of the large crack. This is given by

K F F F F a K e s w g cmax max=

when the crack tip is in a residual tension region, max = y .

Two different details of a welded aluminium beam are investigated, a coverplate and aweb stiffener, shown in Figure 2403.06.01 .

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Trainingi n AluminiumApplication Technologies

2403.06.01

Cover Plate Web Stiffener

Influence of Cover Plates and Web Stiffeners

Source: D. Kosteas, TUM

In both cases the crack extends into the heat affected zone. The crack propagation beha-viour in this zone is assumed to be similar to the upper limit of the typical experimental


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TALAT 2403 42

values for base material given in Figure 2404.06.03 . Crack propagation da/dN in theregion of K max = K c above 10 -5 m/cycle is not taken into account, it corresponds to alife portion in the low cycle fatigue region and may be neglected. The limiting value isKc = K Ic = 60 MPa m in the case of base metal of 10 mm thickness. Actually a reduc-tion of 10% must be taken into account for values in the HAZ in 30 mm plate thickness.

Respective values for 15 mm flange thickness as in the above example are not available.The ultimate strength limit for the material is 360 MPa. The size of initial flaws may betaken as 0.1 mm, such as observed oxide inclusions in the fusion zone.

Coverplate

With the approximation c a= 5462 1133. . and a = 0.1 mm at the beginning of crack propagation we get c = 0.402 and a/c = 0.250 and k 2 = 0.938 and consequently with F e =0.933. With a = 15 mm and c = 117.5 mm at the approximate end of crack propagation

a/c = 0.128 and F e = 0.976. The surface correction F a cs = 1211 0186. . = 1.118 or1.144 for a/c = 0.25 and a/c = 0.128.

The finite plate width results in the following correction. The flange width is 300 mmand the flange thickness is 15 mm. We assume an initial crack at the coverplate weld toewith a depth a = 0.1 mm into the flange thickness and a width of 2c. In the beginning of crack extension with the average a/c = 0.3, c results in c = 0.33 mm. On the other handthe approximation for coverplates of c a= 5462 1133. . gives c = 0.40 mm. Such cracksalways develop in the middle of a flange above the web. So we have the configurationof a crack in the middle of a plate of finite width. Since this is not a through crack, the

geometric influence is covered already by F s as above and F w = 1.

At the approximate end of crack propagation we have a through thickness crack of a =15 mm and a respective c a= 5462 1133. . = 117.45 mm. Therefore the correction factorfor finite width is given as

Fw = [sec(( c)/(2b))] 1/2 = [sec(( 117.45)/(300))] 1/2 = 1.7295

The stress gradient correction factor F g depends on the actual crack length a. Weassume at the beginning of crack propagation a = 0.1 mm. Hereafter, we have thefollowing values for the constants a = a/t f = 0.1/15 = 0.00667 and G = 6.789 and b =0.4348.

Finally the stress concentration factor

Ktm = ( ) ( ) + +3539 1981 5798. log . log . Z t t t f cp f

and with t cp = 15 mm, t f = 15 mm and z = t cp = 15 mm we have K tm = 5.798.In the geometrical assumptions for the above constants the width of coverplate and

flange was identical. In the case of our example we have b f = 300 mm > 250 mm = b cp as well as a further stress concentration because of the beam shape and the web-to


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TALAT 2403 43

flange weldments. Strain gage measurements have indicated a stress concentration of approximately 30 %, so a further global stress concentration factor of 1.3 is assumed.

Web Stiffener

In a similar way as above the free surface correction factor is calculated. For thebeginning of the crack propagation a = 0.1 mm and at the approximate end of crack propagation, with a through thickness crack established a = 15 mm. From therelationship of Figure 2403.05.02 c = 1.403a 0.951 for stiffeners we get for the half crack-width c = 0.157 mm or c = 18.43 mm and accordingly a/c = 0.64 or a/c = 0.81; weassume an average value of a/c = 0.8.

F a cs = = =1211 0186 1211 0186 0 8 1045. . . . . .

the accuracy could be lower though, since a/c > 0.4.

For the beginning of crack propagation we assume again F w = 1 since concentrationeffects are covered by the free surface correction. At the end of crack propagation with athrough-flange-thickness crack of a = 15 mm and with a/c = 0.8 as assumed c =18.75mm. So the correction factor for finite width plate with an edge crack (as the

worst case for the component) is ( ) ( )( ) ( ) ( )( )[ ]F b c c bw = 2 21 2

/ tan / and with2b = 300 mm, F w = 1.013.

The shape correction factor F e is calculated for the beginning of crack propagation witha = 0.1 mm and a/c = 0.64, F e = 0.767. For the end of the elliptical crack shape, when

the crack is through the flange thickness with a = 15 mm and c = 18.75 mm we get F e =0.706.

The stress gradient correction factor F g depends on the actual crack length a. Weassume at the beginning of crack propagation a = 0.1. Hereafter we have the followingvalues for the constants a = a/t f = 0.1/15 = 0.00667 and G = 2.776 and b = 0.2487.

Finally the stress concentration factor K tm = ( )1621 3 963. log . + Z t f with t f = 15 mmand z = t f = 15 mm we have K tm = 3.963. The stress gradient correction factor for thebeginning of crack propagation is calculated as F g = 1.545 and for the end of crack propagation F g = 1.050.

Detail Crack Size F s Fg Fw Fe Coverplate Beginning 1,118 3,278 1,000 0,933

End 1,144 0,744 1,730 0,976Web Stiffener Beginning 1,045 1,545 1,000 0,767

End 1,045 1,050 1,013 0,706

Correction factors can be also estimated through the IIW Recommendation "The Fitnessfor Purpose of Welded Structures", SST-1141/89. A solution is provided for surfacecracks at weld toes in the general form of K M aK = .


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TALAT 2403 44

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Correction Factor Development with Progressing Crack

Coverplate F totWeb Stiffener M K

F tot, MK

a/B

5

4

3

2

1

00 0,2 0,4 0,6 0,8 1

2403.06.02

Correction Factor Development with Progressing Crack

The full correlation between correction factors F tot for the coverplate or M K for the webstiffener vs. the crack development expressed with the parameter a/B (where B = 15 mm= flange thickness) is given in the diagram of Figure 2403.06.02 . These values will beused together with the appropriate regions of crack propagation diagrams for therespective material zones, Figure 2403.06.03 .

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Crack Propagation vs. and eff


Crack Propagation vs. and eff

2403.06.03

For the estimation of number of cycles to failure the upper portion of the da/dN diagramabove 10 -5 m/cycle is not taken into account. The upper boundary da/dN curve actuallyused in the calculation is shown in Figure 2403.06.04 , where also the limit curve given

by IIW recommendation SST-1141-89 is plotted. These lead to the final estimated S-N


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TALAT 2403 45

curve for detail D2 = coverplate, Figure 2403.06.05 , and detail B2 = web stiffener,Figure 2403.06.06 , plotted against respective experimental results.

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Calculation Curves for da/dN Lines

2403.06.04Calculation Curves for da/dN Lines

AlZn4.5Mg1 (7020)Base materialR = 0.0, Thickness = 8.0 mm

10 -3

10 -4

10 -5

10 -6

10 -7

10 -8

10 -9

10 -10

10 -111 2 4 6 10 20 40 60 100

Boundary curvefor calculation

d a / d N [ m / c y c

l e ]

Keff [MPa m]

KC = 60 MPa m 1/2

IIWLimit curve

+ R= 0

x R = -1

o R = +0.5


mI

m II

m III

m IV

mV

m Im IIm IIIm IVmV

Keff da/dN

=15=2.5=0.94=8.0

=2.7

0.7812.84.656.1660

1x10 -114x10 -105.25x10 -98.46x10 -9

8.08x10 -83.37.10 -5

Boundary curve data for calculation:

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S-N Curve for Cover Plate

x+ + +

+

+

+ +

+ +

x x

x x

++

+

x

29.5

21.319.9

P 50P 50P 50

500

400

300

200

100

8070605040

30

20

10

8765

10 4 10 5 10 6 10 7 10 82 4 6 2 4 6 2 4 6 2 4 6 2 4 6

S t r e s s r a n g e

i n M P a

Cycles to failure N

S-N Curve for Cover Plate

Detail D2 - TUMAlZn4,5Mg1

R = -1R = 0,1R = +0,6

Calculation result

IIW boundarycurve

Test results



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TALAT 2403 46

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S-N Curve for Web Stiffener

54.9

P 50

500

400

300

200

100

8070605040

30

20

10

8765

10 4 10 5 10 6 10 7 10 82 4 6 2 4 6 2 4 6 2 4 6 2 4 6

S t r e s s r a n g e

i n M P a

Cycles to failure N

S-N Curve for Web Stiffener

Detail D2 - TUMAlZn4,5Mg1

R = -1

R = 0,1

Calculation result

IIW boundarycurve

Test results



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TALAT 2403 47

2403.07 Literature/References

Introductory chapters to fracture mechanics have been taken mainly out of the text book by

Rolfe/Barsom : "Fracture and Fatigue Control in Structures"

further information has been utilised from

Hertzberg : "Deformation and fracture mechanics of engineering materials", JohnWiley & Sons, New York, 1976

Schwalbe : "Bruchmechanik metallischer Werkstoffe", Hanser Verlag, Mnchen,1980

F. Ostermann (Hrsg.): "Betriebssicherheit von Aluminium Konstruktionen", SeminarMeckenheim, 1990

Graf, U .: "Bruchmechanische Kennwerte und Verfahren fr die Berechnung derErmdungsfestigkeit geschweiter Aluminium Bauteile", Berichte ausdem konstruktiven Ingenieurbau, 3/92, TU Mnchen, 1992

Kosteas : "Vorlesungsmanuskript zur angewanden Bruchmechanik", TU Mnchen,1991

Rossmanith: "Grundlagen der Bruchmechanik", Springer Verlag, Wien/New York,1982


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TALAT 2403 48

2403.08 List of Figures

Figure Nr. Figure Title (Overhead)2403. 02 .01 Notch-Toughness Performance Levels vs. Temperature2403.02.02 Relation between Performance and Transition Temperature for Three

Different Steel Qualities2403.02.03 Notch-Toughness of Welded Aluminium Alloy 5083 vs. Temperature

2403. 03 .01 Elastic Stress Field Distribution near a Crack 2403.03.02 K I Values for Various Crack Geometries2403.03.03 Relation between Material Toughness, Flaw Size and Stress2403.03.04 Analogy: Column Instability and Crack Instability2403.03.05 Basic Modes of Cracking2403.03.06 Distribution of Stress in the Crack Tip Region2403.03.07 Stress Field Equations in the Vicinity of Crack Tips

2403.03.08 SIF for Through-Thickness Crack 2403.03.09 SIF for Double-Edge Crack 2403.03.10 SIF for Single-Edge Crack 2403.03.11 SIF for Cracks Emanating from Circular or Elliptical Holes2403.03.12 SIF for Single-Edge Crack in Beam in Bending2403.03.13 SIF: Elliptical or Circular Crack in an Infinite Plate2403.03.14 SIF: Surface Crack 2403.03.15 Flaw Shape Parameter, Q2403.03.16 Distribution of Stress in the Crack Tip Region2403.03.17 Plastic Zone Dimensions

2403. 04 .01 Effect of Thickness on K c 2403.04.02 SE Bend Specimen and CT Tension Specimen2403.04.03 Specimen Orientation2403.04.04 R-Curves and Critical Fracture Toughness Values K R for Different Initial

Cracks a 0 2403.04.05 Normalized Plot for Critical Flaw Size2403.04.06 J-Integral2403.04.07 Interpretation of the J-Integral2403.04.08 Procedure of J Measurement

2403.04.09 Evaluation Procedure and Limits of J Measurement2403.04.10 Hinge Model for a Standard Bending Specimen


49/49

Figure Nr. Figure Title (Overhead)2403. 05 .01 Crack Shape Correction Factor F e 2403.05.02 Crack Shape Measurements2403.05.03 Representative Values for G, and K tm 2403.05.04 Correction Factors vs. Crack Size for Coverplates

2403. 06 .01 Influence of Cover Plates and Web Stiffeners2403.06.02 Correction Factor Development with Progressing Crack 2403.06.03 Crack Propagation vs. K and Keff 2403.06.04 Calculation Curves for da/dN Lines2403.06.05 S-N Curve for Cover Plate2403.06.06 S-N Curve for Web Stiffener

talat - fracture

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