mirzaei-fracturemechanicslecture.pdf

M. Mirzaei, Fracture Mechanics

1

Fracture Mechanics Theory and Applications by

Majid Mirzaei

These lecture notes have been prepared as supplementary material for a graduate level course on Fracture Mechanics. You are welcome to read or print them for your own personal use. All other rights are reserved.

Little originality is claimed for these notes other than selection, organization, and presentation of the material.

The major references that have been used for preparation of the lecture notes are as follows:

References:

1. Anderson, “Fracture Mechanics Fundamentals and Applications.” 2. Meguid, “Engineering Fracture mechanics” 3. Kanninen, "Advanced Fracture Mechanics" 4. Broek, “Elementary Engineering Fracture Mechanics” 5. Dowling, "Mechanical Behavior of Materials" 6. “Structural Integrity Lecture Notes”, by Prof. Gray


2

Fracture Mechanics Lecture Notes Majid Mirzaei, PhD Associate Professor Dept. of Mechanical Eng., TMU [email protected] http://www.modares.ac.ir/eng/mmirzaei

Chapter 1 Some Fundamental Aspects of Structural Design and Failure Analysis The ultimate goal in the field of applied solid mechanics is to be able to design structures or components that are capable of safely withstanding static or dynamic service loads for a certain period of time. In general, we may summarize the activities usually carried out during design, manufacture, and testing of a product as shown in Fig.1.1.

User Requirements AcceptanceUser Tests

SystemRequirements

VerificationSystem Tests

Architecture AssemblyIntegrationTests

Detail Design Sub-SystemsSub-SystemsTests

Manufacture

Partitioning Integration

Fig. 1.1


3

In the above process, most of the engineering decisions are based on semi-empirical design rules, which rely on phenomenological failure criteria calibrated by means of standard tests. The failure criteria are derived based on extensive observations of failure mechanisms, together with theoretical models that have been developed to describe these mechanisms. In the view of the above arguments, one of the most important questions in the design process is: What Constitutes Mechanical Failure? In general, various failure mechanisms may be classified into the two broad fields of Deformation and Fracture. A more detailed list is: Excessive Elastic Deformation Unstable Elastic Deformation (Buckling) Plastic Deformation Fracture Fatigue Creep Stress Corrosion Cracking The occurrence of each failure mode depends on various factors as depicted in Fig.1.2.

DeformationFracture

ElasticPermanent

Load:Static, Dynamic

Time ScaleShort, Long Environment

Corrosive?

Temperature

StaticTime Dependent

StaticTime Dependent

Fig. 1.2


4

Evolution of Structural Design The art of structural design has drastically evolved through the history of mankind. A rather broad, but somehow specific, classification is:

I. Design Based on Previous successful designs II. Introduction of the Stress and Strain Concepts

III. Strength of Materials Approach IV. Theory of Elasticity Approach V. Fracture Mechanics Approach

Design Philosophies For many applications, it is sufficient to determine the maximum static or dynamic stress that the material can withstand and then design the structure to ensure that the stresses remain below acceptable limits. This involves fairly routine constitutive modeling and numerical or analytical solution of appropriate boundary value problems. More critical applications require some kind of defect tolerance analysis. In these cases, the material or structure is considered to contain flaws and we must decide whether to replace the part or leave it in service under a more tolerable loading for a certain period of time. This kind of decision is usually made using the disciplines of Fracture Mechanics. In general, we may distinguish three different design philosophies as follows: Safe Life: The component is considered to be free of defects after fabrication and is designed to remain defect-free during service and withstand the maximum static or dynamic working stresses for a certain period of time. If flaws, cracks, or similar damages are visited during service, the component should be discarded immediately. Fail Safe: The component is designed to withstand the maximum static or dynamic working stresses for a certain period of time in such a way that its probable failure would not be catastrophic. For example a pressure vessel designed to work under the leak-before-burst (LBB) condition should show leakage as a result of crack propagation. The aim is to prevent catastrophic failure by detecting the crack at its early stages of growth and also reducing the internal pressure. Damage Tolerance: The component is designed to withstand the maximum static or dynamic working stresses for a certain period of time even in presence of flaws, cracks, or similar damages of certain geometry and size.


5

Fracture Mechanics Fracture mechanics is a field of solid mechanics that deals with the mechanical behavior of cracked bodies. The wide range of applicability of the Engineering Fracture Mechanics disciplines is shown in Fig. 1.3.

a

Engineering Fracture Mechanics

10-8 10-4 10-2 100 102

Fig. 1.3

Historically, the first major step in the direction of quantification of the effects of crack-like defects was taken by C. E. Inglis (a professor of Naval Architecture). In 1913 he published a stress analysis for an elliptical hole in an infinite linear elastic plate loaded at its outer boundaries. By making the minor axis very much less than the major, a crack-like discontinuity can be modeled.

σ0

σ0

σ0

σ0

X2

X1

ξ0

η

2C

ab

Fig. 1.4


6

The complete solution of the above problem can be found in the Elasticity textbooks (also in my lecture notes on the theory of elasticity). The extremums of the stresses can be shown to be:

( )

( )

max

00,

min3 0,

2 2

2

2

ab

ba

η η π

π πη η

σ σ

σ σ

=

=

=

= (1-1)

The above equations show that as 0b → (the ellipse becomes a crack) a stress singularity develops at the crack tip. A. A. Griffith, who was studying the effects of scratches and similar flaws on aircraft engine components, transformed the Inglis analysis by calculating the effect of the crack on the strain energy stored in an infinite cracked plate. He proposed that this energy, which is a finite quantity, should be taken as a measure of the tendency of the crack to propagate. Griffith also carried out tests on cracked glass spheres and showed that the simple elastic analysis could be applied to describe the propagation of different size cracks at different stress levels. The mechanics of fracture progressed from being a scientific curiosity to an engineering discipline primarily because of the major failures that occurred in the Liberty ships during World War II. The Liberty ships had an all-welded hull, as opposed to the riveted construction of traditional ship designs. Of the roughly 2700 liberty ships build during World War II, approximately 400 sustained fractures, of which 20 were essentially total.

Fig. 1.5 After World War II, the fracture mechanics research group at the Naval Research Laboratory was led by Dr. G.R. Irwin. Having studied the early work of Inglis, Griffith, and others, Irwin found out that the basic tools needed to analyze fracture were already available. Irwin’s first major contribution was to extend the Griffith approach to metals by including the energy dissipated by local plastic flow.


7

In 1956, Irwin developed the energy release rate concept, which is related to the Griffith theory but is in a form that is more useful for solving engineering problems.

dG RdAΠ

= − ≥ (1-2)

Next, he used the Westergaard approach to show that the stresses and displacements near the crack tip could be described by a single parameter that was related to the energy release rate. This crack tip characterizing parameter later became known as the stress intensity factor.

a1

σ

σMP

P

a2

σ

σ

Fig. 1.6

In practice, all this work was largely ignored by engineers as it seemed too mathematical and it was only in the 1970's that fracture mechanics came to be accepted as a useful and even essential tool. There were many reasons for this, for example, the development of non-destructive examination methods which revealed hidden cracks in structures, the demand of space industry for high-strength high integrity pressure vessels, the increasing use of welding and the severe conditions of offshore structures, etc. Hence, most of the practical development of fracture mechanics has occurred in the last thirty years. The fundamental aspects of Fracture Mechanics are reviewed in Chapter 2.


8

Elasto-Plastic Fracture Mechanics In the regime where the global stress-strain response of the body is linear and elastic (LEFM), the elastic energy release rate, G, and the stress intensity factor K can be used for characterizing cracks in structures. These aspects will be discussed in Chapter 2.

In the elastic-plastic region (EPFM) also called yielding fracture mechanics (YFM), the fracture characterizing parameters are the J-integral and the crack-tip-opening displacement, CTOD. These aspects will be discussed in Chapter 3.

The J contour integral is extensively used in fracture mechanics, as both the energy and the stress based criteria, for determining the onset of crack growth. Referring to Fig. 1.7, the original form of the J-Integral for a line contour surrounding the crack tip can be written as:

∫Γ

⎟⎠⎞

⎜⎝⎛

∂∂

−= dsxuTwdyJ i

i (1-3)

x

y

a

Ti

M

PT

Fig. 1.7

In cases where fracture is accompanied by substantial plastic deformation, an alternative description of the crack tip state has been established, designated the “crack-tip-opening displacement (CTOD) approach”. This idea is based on the experimental finding that cracks tend to open up under load, as shown below in the magnified view. The basis of the CTOD approach is that forward propagation of the crack, as shown in the right figure, should only occur when the CTOD reaches a specific value which is characteristic of the material (see Chapter 3).


9

CTODc

Fig. 1.8

Fatigue It has long been known that a component subjected to fluctuating stresses may fail at stress levels much lower than its monotonic fracture strength, due to a process called Fatigue. Fatigue is an insidious time-dependent type of failure which can occur without any obvious warning. It is believed that more than 95 percent of all mechanical failures can be attributed to fatigue. There are normally three distinct stages in the fatigue failure of a component, namely: Crack Initiation, Incremental Crack Growth, and the Final Fracture.

Fig. 1.9


10

Classical Fatigue The classical approach to fatigue, also referred to as Stress Controlled Fatigue or High Cycle Fatigue (HCF), through S/N or Wöhler diagrams, constitutes the basis of the SAFE LIFE philosophy in design against fatigue. In order to determine the strength of materials under the action of fatigue loads, specimens with polished surfaces are subjected to repeated or varying loads of specified magnitude while the stress reversals are counted up to the destruction point. The number of the stress cycles to failure can be approximated by the WOHLER or S-N DIAGRAM, a typical example of which is given below.

Fig. 1.10

Low Cycle Fatigue Based on the LCF local strain philosophy, fatigue cracks initiate as a result of repeated plastic strain cycling at the locations of maximum strain concentration.

Fig. 1.11


11

Fatigue Crack Propagation If a crack exists in the component before it goes into service, for example due to weld fabrication or from some other cause, the “initiation” stage is by-passed and the fatigue failure process is taken up entirely with incremental growth and final fracture. Most fatigue failures in practice are in the low stress region, much less than the yield stress, where the LEFM is likely to be valid. Hence, the LEFM principles can be applied to predict incremental fatigue crack propagation. These aspects will be discussed in Chapter 4.

Fig. 1.12

Creep Creep can be defined as a time-dependent deformation of materials under constant load (stress). The resulting progressive deformation and the final rupture, can be considered as two distinct, yet related, modes of failure. For metals, creep becomes important at relatively high temperatures, i.e., above 0.3 of their melting point in Kelvin scale. However, for polymers substantial creep can occur at room temperature.


12

Fig. 1.13

Creep Crack Growth The two major parameters used for correlating creep crack growth data are the stress intensity factor K and the integral C*. The time-dependent energy Integral, C* , is similar to the J-Integral, but is written in terms of strain rates instead of strain:

* iij j

uC wdy n dsx

σΓ

∂⎛ ⎞= −⎜ ⎟∂⎝ ⎠∫&

& (1-4) These aspects will be discussed in Chapter 4.

Fig. 1.14


13

Failure Analysis One of the most significant applications of fracture mechanics is in the process of Failure Analysis of components. Fig. 1.15 shows the cracked Girth-Gear of an industrial Ball-Mill. These gears are up to 12 meters in diameter and over 90 tones in weight, with a manufacturing cost exceeding $500,000. These types of gears are expected to have fatigue lives of 20 years and more. In this case history, within the first two years of operation, a few cracks initiated from certain locations between the gussets and the gear flange, and propagated towards the lightening holes, as shown in Fig. 1.15b. Since the premature occurrence of several similar cracks in certain locations could be interpreted as the possibility of a faulty design, it was decided to perform a complete stress analysis of the mill using the finite element method. The analysis results clearly revealed the cause of failure, i.e., high stress built-up in specific locations adjacent to the gear flange, and conformed to various characteristics of the existing cracks, including their propagation paths. This work also concerned the assessment of the remaining life of the gear through modeling of crack growth in the high-stress region. In these analyses, semi-elliptical cracks were modeled and grown through a variable stress field, and the crack driving forces were calculated. The calculated crack growth rates were used to estimate the remaining life of the gear. Such information is vitally important as it gives the mill operators a timeframe to order a new gear and plan the replacement procedure. Meanwhile they can safely operate the mill and save thousands of dollars per working day. More details are reported in: Mirzaei, M., Razmjoo,A., and Pourkamali, A., "Failure Analysis of the Girth Gear of an Industrial Ball Mill," Proceedings of the 10th International Congress of Fracture, 2001, (ICF10) USA.

Fig. 1.15


14

Failure Analysis of an Exploded Gas Cylinder This investigation involved the determination of the cause of the explosion of a gas cylinder containing hydrogen. The general cracking pattern of the cylinder, the fractographic features, and the stress analysis results were all indicative of an internal gaseous detonation. Consequently, a number of specific features of the detonation-driven fracture of cylindrical tubes with closed ends were identified. These features included: flap bulging, crack curving and branching adjacent to the bulged area, formation of special markings on shear lips, and multiple cracking at the cap center. The investigations indicated that the initial cracks were created by local excessive shear and the crack propagations were almost entirely governed by structural waves. Based on the above observations and using the results of the dynamic stress analysis of the cylinder, the main characteristics of the gaseous detonation such as the initial and peak pressures and the traveling speed were estimated and the composition of the initial gas mixture was specified. The results showed that the content of the cylinder was a detonable mixture of hydrogen and oxygen (35-45 vol% hydrogen) with an approximate pressure of 2 MPa (9 percent of the working pressure of the cylinder). The details are reported in: Mirzaei, M., "Failure analysis of an exploded gas cylinder", Engineering Failure Analysis , 15/7 (2008), pp. 820-834. doi:10.1016/j.engfailanal.2007.11.005

Fig. 1.16


15

Chapter 2 Linear Elastic Fracture Mechanics Fundamental Concepts Fracture is the separation of a component into, at least, two parts. This separation can also occur locally due to formation and growth of cracks. Let us investigate the force required for such a separation in a very basic way. A material fractures when sufficient stress and work are applied on the atomic level to break the bonds that hold atoms together. Fig. 2.1 shows a schematic plot of the potential energy and force versus separation distance between atoms.

Repulsion

Attraction

PotentialEnergy

Distance

Bond Energy

EquilibriumSpacing

Tension

Compression

BondEnergy

Cohesive Force

AppliedForce

λ

K

--

-

--

-

--

---

-- -

--

--

--X0

Fig. 2.1

The bond energy is given by:

0

bx

E Pdx∞

= ∫ (2-1)


16

where x0 is the equilibrium spacing and P is the applied force. A reasonable estimate of the cohesive strength at the atomic level can be obtained by idealizing the interatomic force-displacement relationship as one half the period of a sine wave, so we may write:

sincxP P πλ

⎛ ⎞= ⎜ ⎟⎝ ⎠

(2-2)

where λ is defined in Fig.2.1. For small displacements, we may consider further simplification by assuming:

cxP P πλ

⎛ ⎞= ⎜ ⎟⎝ ⎠

(2-3)

Hence, the bond stiffness (i.e., the spring constant) can be defined by:

cPk πλ

= (2-4) If both sides of this equation are multiplied by the number of bonds per unit area and the equilibrium spacing x0 (gage length), then k can be converted to Young’s modulus E and P to the cohesive stress cσ . Solving for cσ gives:

0c

Exλσ

π= (2-5)

Assuming 0xλ ≈ , we may write Eqn (2-5) as:

cEσπ

≈ (2-6) For steels with the Young’s modulus of 210 GPa, the above equation estimates a fracture stress of 70000MPa, which is almost 25 times the strength of the most strong steels!! The reason behind the above huge discrepancy is the existence of numerous defects in ordinary materials. These defects can be quite diverse by nature. Starting from the atomic scale, they may include point defects (for example vacancies: atoms missing), line defects, extra atomic planes (dislocations). On the microstructural level we may consider defects due to grain boundaries, porosity, etc. Some of these defects may also evolve during the processes of deformation and fracture. For instance, plastic deformation involves various movements of dislocations which may interact and eventually result in local damages. Plastic deformation may also lead to the formation of microvoids which may coalesce and evolve to microcracks.


17

On the other hand, the term fracture mechanics has a special meaning: description of fractures which occur by propagation of an existing sharp crack. Hence, the assumption of a preexisting crack is essential in fracture mechanics. It is evident that numerous microscopic defects and/or microcracks naturally exist in ordinary materials. However, the scope of Engineering Fracture Mechanics is almost entirely concerned with macrocracks which are either present in the components (as a result of manufacturing processes like welding) or develop during the service by various failure mechanisms such as fatigue or creep. The crack propagation can occur in many ways. For instance we may have fast-unstable and slow-stable crack growth under monotonic loading, or a cycle by cycle growth under alternating loads. In general, the resistance to crack growth can be defined by a special term called the toughness of the material. Ductile Versus Brittle By definition, ductile fracture is always accompanied by a significant amount of plastic deformation, while brittle fracture is characterized by very little plastic deformation (see Fig.2.2). Both types of fracture have distinctive features on macro and micro levels.

BrittleDuctile

Fig. 2.2 To a large extent, ductility and brittleness depend on the intrinsic characteristics of materials such as chemical composition and microstructure. Nevertheless, extrinsic parameters like temperature, state of stress, and loading rate may also have substantial influences on the fracture properties of materials. In general, materials show brittleness at low temperatures, high strain rates, and triaxial state of tensile stress. Let us consider the deformation and fracture mechanisms of a ductile material subject to a simple tension test during which several strength levels can be defined (see Fig.2.3). The first level is the proportional limit, below which there is a linear relationship between the stress and strain (point A). The second one is the elastic limit which defines the stress level below which the deformation is totally reversible (point B).


18

Strain

StressTest

Specimen Gauge length

AB

C

D

E

F

Fig. 2.3 The third level is the yield strength which marks the beginning of irreversible plastic deformation (point C). Some materials show a clear yield point and also a lower yield point like point D. For others the yield strength is a point that is difficult to define and, in practice, it is defined as the intersection of the σ-ε curve and a line parallel to the elastic portion of the curve but offset from the origin by 0.2% strain. Beyond this point and up to the next level, which is the ultimate tensile strength (point E), the plastic deformation is uniform along the gage length of the specimen. The inflection point in the σ-ε curve is due to the onset of localized plastic flow or necking as depicted in the Fig. 2.4(A). Finally, the point F represents the final fracture.

A B C D

Fig. 2.4

The occurrence of necking results in a triaxial state of stress. Accordingly the plastic deformation becomes more difficult and small particles within the microstructure start to fracture or separate from the matrix causing microvoids, as depicted in Fig. 2.4(B). The microscopic appearance of microvoid formation is depicted in Fig.2.5.


19

Fig. 2.5

The resulting microvoids will eventually coalesce and form an internal disk-shaped crack. The final fracture occurs by a shearing-off process of the internal crack towards the specimen surface. The result is a typical cup-cone fracture depicted in Figures 2.4(D) and 2.6(left).

Fig. 2.6

As mentioned before, brittle fracture is characterized by very little plastic deformation which usually results in flat fracture surfaces as depicted in Fig. 2.6 (right). The most important type of brittle fracture is called Cleavage. In this type of fracture, which is also called transgranular, cracking occurs through separation of certain crystalloghraphic planes within grains. The result is a very shiny and flat fracture surface. Another type of fracture, called intergranular, occurs through separation of grains from each other and can be attributed to those mechanisms that weaken the grain boundaries. Creep fracture is a typical example.


20

Linear Elastic Fracture Mechanics (Energy Approach) The aim of this section is to present the fundamental aspects of Linear Elastic Fracture Mechanics (LEFM) using an Energy approach. Let us start with a brief review of some of the relevant material from the theory of elasticity. Consider a deformable body in an equilibrium state under the influence of surface tractions and body forces as depicted in Fig. 2.7. The virtual work can be defined as “the work done on a deformable body, by all the forces acting on it, as the body is given a small hypothetical displacement which is consistent with the constraints present”. The virtual displacements are represented by the symbol “δ”, known as the variation operator. In general, the loadings consist of body forces and surface tractions. The later are prescribed over a part of the boundary designated by Sσ . Over the remaining boundary, designated by uS , the displacement field u is prescribed. However, it must be ensured that

0δ =u on uS to avoid violating the constraints.

Su

Su

S

T

fFig. 2.7

Accordingly, the virtual work may be defined by:

virtS V

i i i iS V

W

T u dS f u dV

δ δ δ

δ δ

= +

= +

∫ ∫

∫ ∫

T. u f . u

(2-7)

in which T and f are the surface-traction and body-force vectors, respectively. Now we may invoke the stress boundary relations and implement the divergence theorem to obtain:


21

( )

( ) ( )

,

0

, ,

i i i i ij j i i iS V S V

ij i i ijV

ij j i i ij i jV

T u dS f u dV n u dS f u dV

u f u dV

f u u dV

δ δ σ δ δ

σ δ δ

σ δ σ δ=

+ = +

⎡ ⎤= +⎣ ⎦

⎡ ⎤= + +⎢ ⎥

⎣ ⎦

∫ ∫ ∫ ∫

∫

∫

(2-8)

In the above, ijσ are the components of the Cauchy stress tensor and jn are the components of the outward unit normal to the surface. The first grouping of terms within the last integral equals zero because of the equilibrium. The product of the symmetric stress tensor with the skew-symmetric part of ,i juδ is also zero. Since the symmetric part of ,i juδ is nothing but a variation in the strain tensor, we may write the following expression known as the principle of virtual work, PVW:

i i i i ij ijS V V

T u dS f u dV dVδ δ σ δε+ =∫ ∫ ∫ (2-9)

The general form of the constitutive expressions for an elastic continuum can be written as:

ijij

ψσ ρε∂

=∂

(2-10)

in which ρ is the density, andψ is the strain energy density. If we substitute the above expression for stresses into Eq. (2-9), we will have:

i i i i ijijS V V

i i i iS V V V

T u dS f u dV dV

T u dS f u dV dV dV U

ψδ δ ρ δεε

δ δ ρδψ δ ρψ δ

∂+ =

∂

+ = = =

∫ ∫ ∫

∫ ∫ ∫ ∫

(2-11)

where U is the total strain energy stored in the body. The left side of Eq. (2-11) may be defined as a variation in the potential energy, V− , so we may write:

, ( ) 0, 0V U U Vδ δ δ δ− = + = Π = (2-12) in which Π is the total potential energy of the body. The above expression states that during the elastic deformation the external work is converted to stored elastic strain energy, and vice versa, so that the variation of the total potential energy is zero.


22

In Fracture Mechanics, however, the total potential energy is the only source for crack growth. Accordingly, an energy criterion for the onset of crack growth can be defined in the following general form:

dG RdAΠ

= − ≥ (2-13)

in which G is called the energy release rate (also known as the crack driving force), A is the cracked area, and R is the resistance of the material to crack growth. The energy release rate, G, can be considered as the energy source for the crack growth and may be obtained from the stress analysis of the cracked geometry. On the other hand, the resistance to crack growth (R) can be considered as the energy sink and depends on the operating fracture mechanism. It should be mentioned that the latter depends on many factors including: the chemical composition and microstructure of the material, temperature, environment, loading rate, and the state of stress. Fixed Displacement Condition In continuation of our discussions concerning the energy approach, we will investigate the behavior of cracked components under two distinct loading conditions. First, suppose that we have stretched a cracked component by the amount ∆ and have constrained it as depicted in Fig. 2.8.

aa

P

a+da

-dP

Δ

A

B

C

D

Δ

Fig. 2.8


23

The amount of elastic strain energy stored in the component is equal to the triangle ABD, and the slope of the load-displacement curve represents the stiffness of the component. Let us initially assume that the stored energy is sufficient to maintain an incremental crack growth (da) under the fixed displacement condition. Since the component with a longer crack has a lower stiffness, the stored elastic energy decreases to a new level equal to the triangle ACD. Since there is no externally applied load in the system, the total potential energy is equal to the strain energy (the only source to provide the required energy for the crack growth). Hence, we may write:

0

( 0), ( ),2

12

PV U U Pd

dU dPGB da B da

Δ

Δ Δ

Δ= Π = = Δ =

Δ⎛ ⎞ ⎛ ⎞= − = −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

∫ (2-14)

where B is the thickness of the component.

Constant Load Condition: Here we consider a cracked component under a constant external load P as depicted in Fig. 2.9.

aa

P

P

a+da

dΔ

A

B C

DE

Δ

F

Fig. 2.9 The amount of elastic strain energy stored in the component is equal to the triangle ABE. Now we assume that the available energy is sufficient to maintain an incremental crack growth (da) under the constant load condition. The component with a longer crack has a lower stiffness but, in this case, the stored elastic energy increases to a new level equal to


24

the triangle ACD. The reason is that an excess amount of energy, provided by moving the constant load P through the distance d∆ (equal to rectangle BCDE), has now been added to the system. Hence, we may write:

0 2

2

21

2P P

V P

PU Pd

PU V P

P U

dU P dGB da B da

Δ

= Δ

Δ= Δ =

ΔΠ = − = − Δ

Δ⇒Π = − = −

Δ⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

∫

(2-15)

Note that in both cases the energy release rate is provided by the stored energy U and is equal to:

1 dUGB da⎛ ⎞= ⎜ ⎟⎝ ⎠

(2-16)

Moreover, for both cases we may write:

2

2

P

CPP CGB a

dU dUda da Δ

Δ=

∂⎛ ⎞= ⎜ ⎟∂⎝ ⎠⎛ ⎞ ⎛ ⎞= −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

(2-17)

In the above C is the compliance of the component. The above equation can be used to obtain G provided that the variation of compliance with the crack length is available. In practice, various analytical, numerical, and experimental techniques are available for this purpose. Assignment 1: Find the critical load for the Component Shown in the figure in terms of the resistance R. Assume a >> b , a >> h.


25

Example Consider a large plate under remote uniaxial tensile stress. The plate has a central through-thickness crack of the length 2a, as depicted in Fig. 2.10.

X

av

σ

σ

Fig. 2.10 The strain energy of the above system consists of two parts: the elastic energy of the plate without crack, plus the strain energy required to introduce the crack. The latter is equal in magnitude to the work required to close the crack by the stresses acting in its position.

0

00

14 v( )2

C

a

U U U

B x dx Uσ

= +

= +∫ (2-18)

The expression for v can be obtained from a complete stress analysis of this cracked geometry. As will be shown later, this expression is:

2 22v a xEσ

= − (2-19) which shows that the crack-opening is maximum at the center of the crack and zero at its tip. Substituting for v in Eq.(2-18) we have:

2 20

BU a UE

π σ= + (2-20) which in combination with Eq.(2-15) results in:


26

2

1

2

dUGB da

aE

σ

π σ

⎛ ⎞= ⎜ ⎟⎝ ⎠

=

(2-21)

The above equation was derived for two crack tips. Accordingly, the G expression for each crack tip is:

2aGE

πσ= (2-22)

The above equation is remarkable as it shows how the energy release rate increases with increasing the far-field stress and the crack length. We may generalize the above equation for different components by writing it as:

2aGE

πσβ= (2-23) in which, β is a parameter that depends on the geometry and loading condition. Now suppose that we have an experimental specimen (similar to the one shown in Fig. 2.10 but with finite dimensions) made of a specific alloy. Also consider a real component, made of the same material, but quite different with respect to geometry, size, loading, and crack length (see Fig. 2.11). Also assume that we have managed to obtain the β parameter for both components and called them β1 and β2 respectively. We may perform a fracture toughness test on the experimental specimen by gradually increasing the stress and noting the critical stress level σc at which the crack starts to grow. Accordingly, we may obtain the critical energy release rate as:

21

1c

caG R

Eπσβ= = (2-24)

This quantity represents a material property called fracture toughness. On the other hand, if the applied loading on the real component creates a far-field tensile stress around the crack tip, there exists a similitude condition between the two components. Since the two components are made of the same material, we may calculate the fracture stress for the real component as follows:

22

2

2 2

fc

cf

aG R

EEG

a

πσβ

σπβ

= =

⇒ = (2-25)


27

Finally, the obtained fσ can be used to calculate the amount of the external load and/or moment associated with the onset of crack growth.

a1

σ

σMP

P

a2

σ

σ

Fig. 2.11

In practice, however, crack growth may occur in very complicated stress fields. In general, we consider three basic modes for crack growth, although mixed-mode growth is also possible. Mode I is the opening or tensile mode where the crack faces separate symmetrically with respect to the x1-x2 and x1-x3 planes. In Mode II, the sliding or in-plane shearing mode, the crack faces slide relative to each other symmetrically about the x1-x2 plane but anti-symmetrically with respect to the x1-x3 plane. In the tearing or anti-plane mode, Mode III, the crack faces also slide relative to each other but anti-symmetrically with respect to the x1-x2 and x1-x3 planes. The energy release rates related to these modes are termed GI, GII, and GIII respectively. In mixed mode problems we simply add the energy release rates of different contributing modes to obtain the total energy release rate.

III

mode III

II

mode II

I

mode I

Fig. 2.12: Three basic loading modes for a cracked body: (a) Mode I, opening mode; (b) Mode II, sliding mode; (c) Mode III, tearing mode.


28

Assignment 2: A cylindrical pressure vessel, with a diameter of 6.1 m and a wall thickness of 25.4 mm, catastrophically failed (fractured) when the internal pressure reached 17.5 MPa. The steel of the pressure vessel had E = 210 GPa, a yield strength of 2450 MPa, a value of GC = 131 kJ/m2.

a) Show that failure would not have been expected if the Von Mises yield criterion had served for design purposes. b) Using the energy approach determine the size of crack that might have caused this failure.

Crack Growth Instability Analysis In the previous section we used the energy approach to define a criterion for the onset of crack growth. The energy approach can also be used for the analysis of different aspects of further crack growth such as instability, dynamic crack growth, and crack arrest. For this purpose we use the energy diagrams as depicted in Fig. 2.13.

σ1 σ1 σ1

σ2

a1 Δa1 Δa2a2a3

R

G1

G3

G2

Fig. 2.13 In these diagrams, the left horizontal axis is used to define the original crack lengths from which we may draw different lines whose slopes are related to the far-field stress. The intersections of these lines with the vertical axis represent the energy release rate, G. The vertical axis also represents the crack growth resistance, GC or R. In general, this


29

resistance may vary as the crack grows because of different reasons such as the formation of shear lips in the plane stress condition or non-homogeneities in material. However, for a component made of homogenous isotropic material (with no thermal gradient) under plane strain condition the R-Curve may be represented by a horizontal line. Fig. 2.13 shows that the crack with the length a1 under the far field stress σ1 and the corresponding energy release rate G1 cannot grow because G1 < R. In order to make it grow, we have to increase the stress level to σ2. Nevertheless, longer cracks like a2 or a3 can grow even under σ1. In the above figure, a2 represents a crack under constant load condition whose energy release rate increases with further growth. In this case, the excess energy may accelerate the crack and the resulting unstable growth can cause a catastrophic failure. On the other hand, a3 represents a crack under fixed displacement condition whose energy release rate decreases with further growth. In this case the crack may stop after a stable growth equal to ∆a1. In practice the crack arrest may occur at some further distance like ∆a2 since the crack has some stored energy which may provide the required driving force even if the apparent G is less than R. Fig. 2.14 shows the energy diagram with a rising R curve which usually occurs in plane stress conditions.

σ1 σ1 σ1

a1 Δa1 Δa2a2a3

R

G1

G3

G2

Fig. 2.14 The increase in resistance can be attributed to the formation of shear lips, which in turn results from plastic deformations at the crack tip. We will elaborate on this issue later when we discuss the crack tip plasticity. As depicted in Fig. 2.14, the criteria for unstable crack growth under constant load for plane stress can be defined as:

, G RG Ra a

∂ ∂≥ ≥

∂ ∂ (2-26)


30

Crack Speed In this section we will obtain an estimation of the crack speed in an unstable growth condition. The modeling is considered for an idealized situation of an infinite sheet with a central crack of length 2a under remote tensile stress σ. The idea is that the surplus of energy, represented by triangle ABC in Fig. 2.15, can be converted to the kinetic energy of the material elements in the crack path as they move apart from each other. As will be shown later, the horizontal and vertical displacements of these elements can be written as:

2 ( )

2 ( )

u

v

u ar fE

v ar fE

σ θ

σ θ

⎧ =⎪⎪⎨⎪ =⎪⎩

(2-27)

σ

ac

a

r

R

A

B C

θr

Y

X

v

u

Fig. 2.15 in which ( )uf θ and ( )vf θ are geometric parameters. As the crack grows, its tip would be further from the considered elements. Hence, we may assume r a∝ and combine the constants in the form of C1 and C2 and write:

1 1

2 1

a au C u CE Ea av C v C

E E

σ σ

σ σ

⎧ = ⇒ =⎪⎪⎨⎪ = ⇒ =⎪⎩

&&

&&

(2-28)


31

in which the “dot” means differentiation with respect to time. Accordingly, the kinetic energy for the plate can be defined and calculated as follows:

( )

( )

2 2

22 2 2

1 22

22 2

2

121212

KE u v dxdy

a C C dxdyE

k a aE

ρ

σρ

σρ

= +

= +

=

∫∫

∫∫

& &

&

&

(2-29)

In the above equations ρ is the density and we have combined all the constants into a single term k. Moreover, as “a” is the only characteristic length in the infinite plate, the value of the integral was considered proportional to 2a . On the other hand the surplus of energy, which can be converted to kinetic energy for two crack tips, can be defined and calculated as follows:

( )

2

22

2

2 ( ) 2

( )

c

c

a

Sa

a

ca

c

E G R da

aR a a daE

a aE

πσ

πσ

= −

= − − +

= −

∫

∫ (2-30)

in which ac is the initial crack length and a is the crack length at every instant. Equating the two energies we may find the crack growth rate as:

2 1 caEak aπ

ρ⎛ ⎞= −⎜ ⎟⎝ ⎠

& (2-31)

A more detailed analysis of the crack tip stress field has given an estimation of 0.38 for the first term on the right hand side of the above equation. The second term on the right is the speed of propagation of longitudinal waves in the material, so we have:

0.38 1 cS

aa Va

⎛ ⎞= −⎜ ⎟⎝ ⎠

& (2-32)

Based on the above expression, it is clear that there is a limit to the crack speed in every material. Nevertheless, the speed of unstable crack growth is comparable with the speed of propagation of sound waves in the material. This means that an unstable growth of an initial crack with a few millimeters length may destroy several kilometers of a pipeline in a few minutes!


32

Crack Branching Another interesting aspect of a growing crack is branching. As depicted in Fig. 2.16, under constant load where the energy release rate increases with further crack growth, there might be a point where the available energy becomes twice the energy required to grow a single crack. As mentioned before, this surplus of energy usually accelerates the crack, but if the material permits, the situation may change in favor of crack branching. In general, when we observe that a component has been shattered into numerous pieces, we may think of too much energy available and/or too little energy required for crack to grow. For example we may think of the fracture caused by an explosion, or the shattering of a glass of water slipping from your hand!

σ1

a1 Δa1

R

R

Fig. 2.16

Crack Arrest As mentioned before, unstable crack growth may lead to catastrophic failure and must be prevented at all cost. One practical remedy is to use riveted patches or other types of stiffeners to simulate a fixed-displacement condition and arrest the crack. The location of the arrester must be chosen properly by considering the kinetic energy of the crack. As depicted in Fig. 2.17, the patch may decrease the energy down to the point C where the crack is expected to stop after a growth equal to ∆a2. In practice, however, the crack may grow further until its kinetic energy is consumed. This point has been specified in Fig. 2.17 by considering the area CDE roughly equal to ABC.


33

σ1

a1 Δa1

Δa2

Δa3

RG1

A

BC D

E

Fig. 2.17

LEFM, Stress Approach Investigation of the crack-tip stress and displacement fields is important because these fields govern the fracture process occurring at the crack tip. In subsequent sections the stress analyses of the three major modes of crack growth will be presented.

The Mode III Problem The analysis of Mode III is relatively simple because we may assume that u1 = u2 = 0 and u3 = u3(x1 , x2). Accordingly, we may only consider the following nonzero strain components:

3 3,12

uα αε = (2-33)

where the Greek subscripts have the range 1, 2. Therefore, the only nontrivial stress components are: 3 32α ασ με= (2-34)


34

where μ is the shear modulus. Accordingly, the only relevant equation of equilibrium in the absence of body forces is: 3 , 0α ασ = (2-35)

The above three equations can be combined to give the Laplace's equation in terms of

displacements:

23, 3 0u uαα = ∇ = (2-36)

In order to solve the above equation we use the complex variable method which provides a powerful technique for establishing the solutions of plane elasticity problems. The complex variable z is defined by z = x1 + ix2 or, equivalently, in polar coordinates z = reiθ. The overbar denotes the complex conjugate, z = x1 − ix2. It can be shown that:

1

2

( ) ( ) / 2( ) ( ) / 2

x e z z zx m z z z i=ℜ = += ℑ = −

(2-37)

where eℜ and mℑ denote the real and imaginary parts respectively. Let f(z) be a holomorphic (analytic) function of the complex variable z. A complex function is holomorphic in a region if it is single valued and its complex derivative exists in the region. For such a function the Cauchy-Riemann equations can be written as:

1 2 1 2

,u v v ux x x x∂ ∂ ∂ ∂

= = −∂ ∂ ∂ ∂

(2-38)

Differentiating the Cauchy-Riemann equations twice and combining them, we have: 2 2 0u v∇ =∇ = (2-39)

Thus, the real and imaginary parts of any holomorphic function are solutions to Laplace's equation. Therefore, the solution of Eq. (2-36) can be written as:

31 ( ) ( )u f z f zμ⎡ ⎤= +⎣ ⎦ (2-40)

Introducing Eq. (2-40) into Eq. (2-33) we may write:

31

32

1 ( ) ( )2

( ) ( )2

f z f z

i f z f z

εμ

εμ

⎡ ⎤′ ′= +⎣ ⎦

⎡ ⎤′ ′= −⎣ ⎦

(2-41)

Combining Eqns (2-34) and (2-41) we have:


35

31 32 2 ( )i f zσ σ ′− = (2-42)

Now, let the origin of the coordinate system be located at the tip of a crack lying along the negative x1 axis as shown in Fig. 2.18.

σx

σy

θr

Y

X

τxy

Fig. 2.18: Crack-tip region and coordinate system

Next, we will focus our attention upon a small region D containing the crack tip and consider the following holomorphic function: 1( ) ,f z Cz C A iBλ+= = + (2-43)

where A, B, and λ are real undetermined constants. For finite displacements at the crack tip we must have: ( 0), 1z r λ= = −f ). The substitution of Eq. (2-43) into Eq. (2-42) yields:

31 32 2( 1) 2( 1) ( )(cos sin )i Cz r A iB iλ λσ σ λ λ λθ λθ− = + = + + + (2-44) whence,

31

32

2( 1) ( cos sin )2( 1) ( sin cos )

r A Br A B

λ

λ

σ λ λθ λθσ λ λθ λθ

= + −= − + +

(2-45)

The boundary condition that the crack surfaces remain traction free requires that 32 0σ = on θ π= ± . Consequently, we have:

sin cos 0sin cos 0

A BA B

λπ λπλπ λπ

+ =− =

(2-46)

To avoid the trivial solution, the determinant of the coefficients of the above equations must vanish. This requires that sin 2 0λπ = , which for λ> −1 has the following roots:


36

1 , / 2, 0,1,2,...2

n nλ = − = (2-47) Of the infinite set of functions of the form of Eq. (2-43) that yield traction-free crack surfaces within D, the function with λ = –1/2 for which A = 0, provides the most significant contribution to the crack-tip fields. For this case Eq. (2-45) becomes,

311/ 2

32

sin( / 2)cos( / 2)(2 )

IIIKr

σ θσ θ

−⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭ (2-48)

which is usually written as:

311/ 2

32

sin( / 2)cos( / 2)(2 )

IIIKr

σ θσ θπ

−⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭ (2-49)

It should be noted that the origin of the above, rather unfortunate, modification is that K was originally considered as a way to calculate G, so the π term was artificially added to the denominator to facilitate the calculations. Also note that B has been chosen such that: { }1/ 2

32 00lim (2 )III r

K r θπ σ =→= (2-50)

We will also have:

12

32 sin( / 2)

2IIIK ru θ

μ π⎛ ⎞= ⎜ ⎟⎝ ⎠

(2-51)

The quantity KIII is referred to as the Mode III stress intensity factor, which is established by the far field boundary conditions and is a function of the applied loading and the geometry of the cracked body. Whereas the stresses associated with the other values of λ are finite at the crack tip, the stress components of Eq. (2-49) have an inverse square root singularity at the crack tip. It is clear that the latter components will dominate as the crack tip is approached. In this sense, Eqs (2-49) and (2-51) represent the asymptotic forms of the elastic stress and displacement fields.

The Mode I and Mode II Problems For the Mode I problem, we assume the displacement field as u1 = u1(x1, x2), u2 = u2(x1, x2), and u3 = 0. Accordingly we may write:

3

3 33

10,

0,

i Eαβ αβ αβ γγ

α αα

νε ε σ νδ σ

σ σ νσ

+ ⎡ ⎤= = −⎣ ⎦

= = (2-52)


37

In the absence of body forces the equilibrium equations reduce to , 0αβ ασ = , and the nontrivial compatibility equation becomes:

, , 0αβ αβ αα ββε ε− = (2-53)

The equilibrium equations will be identically satisfied if the stress components are expressed in terms of the Airy stress function, 1 2( , )x xΨ = Ψ , such that:

, ,αβ αβ γγ αβσ δ= −Ψ +Ψ (2-54)

After the insertion of Eq. (2-54) into Eq. (2-52), the compatibility equation requires that the Airy function satisfy the biharmonic equation: 2 2, ( ) 0ααββΨ = ∇ ∇ Ψ = (2-55)

Note that 2∇ Ψ satisfies the Laplace's equation so we can write the following expression (analogous to the antiplane problem): 2 ( ) ( )f z f z∇ Ψ = + (2-56)

where f(z) is a holomorphic function. Eq. (2-56) can be integrated to yield the following real function. [More details can be found in my lecture notes on the theory of elasticity, (Chapter 3. 2D Static Boundary Value Problems: Plane Elasticity)].

1 ( ) ( ) ( ) ( )2

z z z z z zω ω⎡ ⎤Ψ = Ω + Ω + +⎣ ⎦ (2-57) where ( )zΩ and ( )zω are holomorphic functions. Now, we may substitute the above expression into Eq. (2-54) to write:

11 22

22 11 12

2 ( ) ( )

2 2 ( ) ( )

z z

i z z z

σ σ

σ σ σ ω

⎡ ⎤′ ′+ = Ω +Ω⎣ ⎦⎡ ⎤′′ ′′− − = Ω +⎣ ⎦

(2-58)

Due to symmetry with respect to the crack plane we choose a solution of the form: 1 1,Az Bzλ λω+ +Ω = = (2-59)

where A, B, and λ are real constants. To avoid singular displacements at the crack tip, we should set λ > –1. The introduction of Eq. (2-59) into Eq. (2-58) yields:


38

[ ]{[ ]

22 12 ( 1) 2cos cos( 2)

cos sin( 2) sin }

i r A

B i A B

λσ σ λ λθ λ λ θ

λθ λ λ θ λθ

− = + + −

+ − − + (2-60)which must vanish for θ = ± π. Consequently, we may write:

(2 )cos cos 0

sin sin 0A B

A Bλ λπ λπ

λ λπ λπ+ + =

+ = (2-61)

The existence of a nontrivial solution for the above set of equations requires that sin 2 0λπ = , which for λ> −1 gives the following roots:

1 , / 2, 0,1,2,...2

n nλ = − =

Again, the dominant contribution to the crack-tip stress and displacement fields occurs with λ = –1/2, for which A = 2B. Similar to the antiplane problem, an inverse square root singularity in the stress field exists at the crack tip. Substituting Eq. (2-59), with A = 2B and λ = –1/2, into Eqs (2-58) and (2-60), we find that:

11

12 1/ 2

22

1 sin( / 2)sin(3 / 2)cos( / 2) sin( / 2)cos(3 / 2)

(2 )1 sin( / 2)sin(3 / 2)

IKr

σ θ θσ θ θ θ

θ θσ

−⎧ ⎫ ⎧ ⎫⎪ ⎪ ⎪ ⎪=⎨ ⎬ ⎨ ⎬⎪ ⎪ ⎪ ⎪+⎩ ⎭⎩ ⎭

(2-62)

which is usually written as:

11

12 1/ 2

22

1 sin( / 2)sin(3 / 2)cos( / 2) sin( / 2)cos(3 / 2)

(2 )1 sin( / 2)sin(3 / 2)

IKr

σ θ θσ θ θ θ

πθ θσ

−⎧ ⎫ ⎧ ⎫⎪ ⎪ ⎪ ⎪=⎨ ⎬ ⎨ ⎬⎪ ⎪ ⎪ ⎪+⎩ ⎭⎩ ⎭

(2-63)

Accordingly, the Mode I stress intensity factor K, is defined as: { }1/ 2

22 00lim (2 )I r

K r θπ σ =→= (2-64)

The displacements can be written as:

1 221

22

cos( / 2) 1 2sin ( / 2)

2 2 sin( / 2) 1 2cos ( / 2)I

u K ru

θ κ θ

μ π θ κ θ

⎧ ⎫⎡ ⎤− +⎧ ⎫ ⎪ ⎣ ⎦⎪⎛ ⎞=⎨ ⎬ ⎨ ⎬⎜ ⎟⎝ ⎠ ⎡ ⎤+ −⎩ ⎭ ⎪ ⎪⎣ ⎦⎩ ⎭

(2-65)

When the foregoing is repeated with A and B being pure imaginary, the following equations can be obtained for the Mode II problem:


39

[ ][ ]

11

12 1/ 2

22

sin( / 2) 2 cos( / 2)cos(3 / 2

cos( / 2) 1 sin( / 2)sin(3 / 2)(2 )

sin( / 2)cos( / 2)cos(3 / 2)

IIKr

θ θ θσσ θ θ θ

πσ θ θ θ

⎧ ⎫− +⎧ ⎫⎪ ⎪⎪ ⎪ = −⎨ ⎬ ⎨ ⎬

⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎩ ⎭

(2-66)

1 221

22

sin( / 2) 1 2cos ( / 2)

2 2 cos( / 2) 1 2sin ( / 2)II

u K ru

θ κ θ

μ π θ κ θ

⎧ ⎫⎡ ⎤+ +⎧ ⎫ ⎪ ⎣ ⎦ ⎪⎛ ⎞=⎨ ⎬ ⎨ ⎬⎜ ⎟⎝ ⎠ ⎡ ⎤− − −⎩ ⎭ ⎪ ⎪⎣ ⎦⎩ ⎭

(2-67)

{ }1/ 2

12 00lim (2 )II r

K r θπ σ =→= (2-68)

Finally, we may summarize the expressions derived for different modes by considering the following general expression for the stresses in a cracked body:

( )2ij ijK f

rσ θ

π= +L (2-69)

It is clear that the first term is dominant very near to the crack tip. As we move further from the crack tip the singular term weakens and the additional terms become significant. As depicted in Fig. 2.19, equal stress intensity factors for two different cracks with different lengths in different geometries under different loadings ensure similar crack tip stress fields. Hence, the critical stress intensity factor Kc, obtained at the onset of crack growth for a specific material and geometry, can be interpreted as a mechanical property named fracture toughness.

Fig. 2.19


40

Williams General Solution The most general solution for cracks under generalized in-plane loading was provided by Williams. The solution starts by considering stresses at the corner of a plate under different boundary conditions and corner angles ψ. Here, a crack is treated as a special case where the angle of the plate corner is 2π and the surfaces are traction free, as depicted in Fig. 2.20. The general stress function proposed by Williams is:

( ) ( ) ( ) ( )( )

11 2 3 4

1

sin 1 * cos 1 * sin 1 * cos 1 *

*,

r c c c c

r F

λ

λ

λ θ λ θ λ θ λ θ

θ λ

+

+

Φ = + + + + − + −⎡ ⎤⎣ ⎦=

(2-70)

where c1 , c2 , c3 , and c4 are constants. Using the relevant expressions in polar coordinates, the following expressions can be obtained for the stresses:

( )

( )

( )

1, ,2

1,

1, ,2

1 1 1

1

1 1

rr r

rr

r r

r F Fr r

r F

r Fr r

λθθ

λθθ

λθ θ θ

σ λ

σ λ λ

σ λ

−

−

−

′′= Φ + Φ = + +⎡ ⎤⎣ ⎦

= Φ = +⎡ ⎤⎣ ⎦

′= − Φ + Φ = −

(2-71)

In the above expressions, primes denote derivatives with respect to θ.

θ*

θψ

r

Fig. 2.20 It can be shown that the displacements vary with rλ . Thus, in order to have finite displacements everywhere, we must have λ > 0. Also, the crack faces are traction free, which implies the following equalities: (0) (2 ) (0) (2 ) 0F F F Fπ π′ ′= = = = (2-72)

To ensure a nontrivial solution we must set sin 2 0λπ = , which for λ> 0 has the roots:


41

/ 2, 1, 2,...n nλ = = (2-73)

Accordingly, the most general form for the stress function and the resulting stresses can be written as:

( )

12

1

2

0

*,2

1*,2 *,

nN

n

mij M

ij ijm

nr F

fr f m

r

θ

θσ θ

+

=

=

⎡ ⎤⎛ ⎞Φ = ⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦

⎛ ⎞−⎜ ⎟ ⎡ ⎤⎝ ⎠= + ⎢ ⎥⎣ ⎦

∑

∑

(2-74)

where f is a function of F and its derivatives. Eventually, the above expressions result in general expressions in the form of Eqs. (2-69) for the Mode I and the Mode II crack problems. Again, it is clear that the first term is dominant very near to the crack tip. As we move further from the crack tip the singular term weakens and the additional terms become significant.

Design Philosophy Based on LEFM So far we have learned that cracks may start growing when the stress intensity factor (SIF) reaches a critical value Kc, called “fracture toughness”. Later we will show that the SIF can be related to the far-field stress and crack length by the following general expression: cK Y a Kσ π= ≥ (2-75)

in which Y is a geometric factor (see Fig.2.26). Thus for the design of a cracked or potentially cracked structure we have to decide what design variables can be selected, as only two of these variables can be fixed and the third must be determined. For example we may select a special steel to resist a corrosive liquid, so Kc is fixed, and the design stress level may also be fixed due to weight considerations. In this case we may calculate the maximum size of tolerable cracks using Eq. (2-75). Based on the above arguments it is clear that the application of LEFM in design procedures usually involves the following activities: 1. Measurement of the critical stress intensity factors that cause fracture for the material. 2. Determination of the size and location of cracks in the structure or component. 3. Calculation of the stress intensity factors for the cracks in the structure or component for the anticipated loading conditions.


42

Fig. 2.21

The first item will be discussed after the concept of crack tip plasticity is introduced. The second task can be performed using some kind of non-destructive test techniques. Examples of such techniques are ultrasound and x-ray techniques, and inspection with optical microscopy. If the crack is detected, most of these techniques will provide an estimation of the crack length. If not, one should assume for design purposes that the structure contains cracks that are just too short to be detected. The third activity, i.e. calculation of the stress intensity factors, can be performed using various techniques, including: 1. Finding the analytic solution to the full linear elastic boundary value problem, and deducing stress intensities from the asymptotic behavior of the stress field near the crack tips. 2. Deducing the stress intensity factors from energy methods (In the next section we will discuss the related relationships). 3. Using experimental techniques 4. Using numerical methods such as boundary integral and finite element methods.

K-G Relationship So far we have discussed two different criteria, based on energy considerations and crack tip stress field, for the onset of crack growth. Naturally, there should be a relationship between the two. In this section we will discuss this relationship. Fig. 2.22 depicts a crack of initial length a subject to Mode I loading with the origin located at a distance ∆ behind the crack tip.


43

Δ

Y

XV

Fig. 2.22

Now assume that we may partially close the crack through application of a compressive stress field to the crack faces between x = 0 and x = ∆. The required work is:

0

v2

2yW dr

σΔ= ∫

(2-76)

The factor of 2 on work is required because both crack faces are displaced. In the above

yσ is the compressive stress distribution and v is the crack opening displacement. As this work will be released as energy, the energy release rate G can be written as:

0

00

lim

v2lim2y

WG

drσ

Δ→

Δ

Δ→

=Δ

=Δ ∫

(2-77)

We may define the stresses and displacements in terms of the stress intensity factor as:

2 2

2 2

2

22v

Iy

I

Kr

K a xa xE E a

σπ

σπ

=

−= − =

(2-78)

Noting that x r a= + −Δ and neglecting the second order terms in our calculations, we may rewrite the above as:


44

( )

22 2v 2 2

2 2

I

I

K r rra aE

K rE

π

π

Δ= Δ − + −

= Δ −

(2-79)

Substituting the above into Eq. (2-77), we have:

2

00

2lim IK rG drE rπ

Δ

Δ→

Δ −=

Δ ∫ (2-80)

The result of integration is:

2IKG

E=

(2-81)

which can be modified for plane strain as follows:

( )2

21 IKGE

ν= − (2-82)

The above expressions are general relationships between K and G for Mode I. However, the analysis procedure can be repeated for other modes of loading. When all three modes of loading are present, the energy release rate is given by:

2 2 22 2(1 ) (1 ) (1 )

I II III

I II III

G G G G

K K Kv v vE E E

= + +

= − + − + + (2-83)

As mentioned before, the three modes are additive with respect to energy release rate because it is a scalar quantity. However, it should be noted that the above equation assumes self similar crack growth, i.e., a planar crack is assumed to remain planar and maintain a constant shape as it grows. This is usually not the case for mixed-mode fractures. Assignment 3: A thin-walled cylinder contains a crack forming an angle 30 degree to the longitudinal axis. Numerical data: radius R = 0.2 m, crack half-length a and wall thickness t are a = t = 0.005m, fracture toughness Kc = 50 MPa m . Determine at which torque T crack may start to grow.


45

Mixed Mode Fracture When we are dealing with an angled crack (similar to that depicted in Fig. 2.23), the first question is naturally about the magnitude of the far-field stress σ at which the crack starts to grow. The second question is how to determine the direction of further crack growth. σ

σ

Y

X

θ

r

β

Fig. 2.23 The first question can be answered by the methods described in the previous section, while there are two general approaches for predicting the direction of crack growth. In the first approach, it is assumed that the crack growth occurs in the direction perpendicular to the maximum tangential stress at (or near) the crack tip. The second approach considers crack growth in the direction for which the strain energy density is minimal, on the basis that this corresponds to a maximum in energy release rate. The strain energy density in the vicinity of the crack tip may be written as:

( )

[ ]

[ ]

2 211 12 22

11

12

22

1 ( )2

1 (1 cos )( cos )16

1 sin (2cos 1)16

1 ( 1)(1 cos ) (1 cos )(3cos 1)16

I I II IISa K a K K a K

r r

a k

a k

a k

θψ

θ θμ

θ θμ

θ θ θμ

= + + =

= + −

= − +

= + − + + −

(2-84)

where (3 4 )k ν= − for plane strain, and (3 ) /(1 )k ν ν= − + for plane stress.


46

According to this criterion the crack starts to grow when S reaches a critical value Sc, and the direction of crack growth is given when S is a minimum:

2

20, 0dS d Sd dθ θ

= > (2-85)

For example, for pure mode I we have:

( ) ( ) 2

11min

2

0

2( 1)16

I I c

c c

S S a K

kS K

θ θ

μ

= = =⎡ ⎤⎣ ⎦−

⇒ = (2-86)

For the angled crack shown in Fig. 2.23, the strain energy density can be obtained based on the following expressions for the stresses:

2

2

5 1 3 5 3 3cos cos sin sin cos4 2 4 2 4 2 4 22 23 1 3 3 3 3cos cos sin sin sin4 2 4 2 4 2 4 22 21 1 3sin sin4 2 4 22

I IIrr t

I IIt

Ir

K Kr r

K Kr r

Kr

θθ

θ

θ θ θ θσ σ θπ π

θ θ θ θσ σ θπ π

θ θσπ

⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞= − + − + +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞= + + − − +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠⎛ ⎞⎛ ⎞= + +⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

1 3 3cos cos sin cos4 2 4 22

IIt

Kr

θ θ σ θ θπ

⎛ ⎞⎛ ⎞+ −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

(2-87)

where we have;

2

2

sin , sin cos

sinI II

t

K a K aσ π β σ π β β

σ σ β

= =

=(2-88)


47

Stress Intensity Factor A major activity in the design process based on fracture mechanics is the determination of the stress intensity factor for the particular problem. In the following sections we will discuss some of the pertinent analytical, experimental, and numerical methods. Analytical Determination of SIF In this section we drive the stress intensity factor expression for an infinite plate with a central crack of length 2a, under remote stress σ0 (see Fig. 2.24).

aσx

σy

σ0

σ0

σ0

σ0θ

r

Y

X

τxy

Fig. 2.24 We use the general form of the Westergard stress function as follows: e y mΦ =ℜ Φ+ ℑ Φ (2-89)

for which we can define the integration and differentiation operations with respect to z as:

ddzdzddzdz

ddzdz

ΦΦ = Φ = Φ

ΦΦ = Φ = Φ

Φ′ ′Φ = Φ = Φ

∫

∫

∫

(2-90)


48

Writing the Cauchy-Riemann equations we have:

m e dey x dze m dmy x dz

∂ℑ Φ ∂ℜ Φ Φ= ≡ℜ

∂ ∂∂ℜ Φ ∂ℑ Φ Φ

− = ≡ ℑ∂ ∂

(2-91)

Using Eq. (2-91), we can differentiate Eq. (2-89) with respect to “y” and write: ,y m m y e y eΦ = −ℑ Φ+ℑ Φ + ℜ Φ ≡ ℜ Φ (2-92)

Accordingly, we can calculate the stresses using the following Equations:

,

,xx yy

yy xx

e y m

e y m

σ

σ

′= Φ =ℜ Φ − ℑ Φ

′= Φ =ℜ Φ + ℑ Φ (2-93)

Now, we should choose a stress function that satisfies the local and global boundary conditions of this problem. Let us check the following stress function:

02 2

zz aσ

Φ =−

(2-94)

The global and local boundary conditions are:

0B.C.1 ,

B.C.2 0, ,

B.C.3 0, , 0 0

xx yy

xx yy

xx yy

x y z

y x a

y a x a e

σ σ σ

σ σ

σ σ

→∞⇒ →∞⇒ = =

= = ± ⇒Φ = ∞ ⇒ = = ∞

= − < < ⇒ℜ Φ = ⇒ = =

(2-95)

Note that the proposed stress function satisfies these conditions. Now we translate the origin to the crack tip.

02 2

( )( )

z az a aσ +

Φ =+ −

(2-96)

Since we are interested in very near field stresses where z a<< , we neglect the z2 term and write,

12

02a zσ

−Φ =

(2-97)

Next we switch to polar coordinates with iz re θ= and write Eq. (2-97) as,


49

0[cos( / 2) sin( / 2)]2a irσ θ θΦ = −

(2-98)

Hence, for the normal stress component on the crack line near the tip we have:

22 0 0 02aerθ θσ σ= == ℜ Φ =

(2-99)

Finally, using the original definition of KI for the Mode I problem, we may write:

{ }1/ 222 00

1/ 200

0

lim (2 )

lim (2 )2

I r

I r

I

K r

aK rr

K a

θπ σ

π σ

σ π

=→

→

=

⎧ ⎫⎪ ⎪= ⎨ ⎬⎪ ⎪⎩ ⎭

=

(2-100)

In practice, the far field stress in the x direction does not have any effect on the crack tip stress field. Hence, the above expression for K is also applicable when the sheet is under uniaxial far-field stress in the Y direction. It is possible to obtain the K-expression using the principle of superposition. Thus, for the geometries shown in Fig. 2.25 we may write:

0Ia Ib

Ia

Ia

K K a

K a

K a

σ π

σ π

σ π

+ =

+ =

=

(2-101)

Fig. 2.25


50

Fig. 2.26


51

Elliptical Cracks For many real components cracking starts at free surfaces. These cracks often have semi-elliptical or quarter-elliptical shapes. There may also be elliptical cracks embedded in components. Because of their importance, significant research has been done on modeling and quantification of the effect of elliptical cracks in different structures under various types of loadings. A solution for an embedded elliptical crack for an infinite domain derived by Irwin is:

12 4

2 22sin cosI

a aKb

σ π θ θ⎛ ⎞

= +⎜ ⎟Φ ⎝ ⎠ (2-102)

in which a and b are defined in Fig. 2.27, θ is measured counterclockwise from the point B, and Φ is an elliptical integral of the second kind defined by:

12 2 2/ 2 2

201 sinb a d

bπ

θ θ⎛ ⎞−

Φ = −⎜ ⎟⎝ ⎠

∫ (2-103)

Values for Φ are reported in the form of tables and graphs. It is also possible to find an approximation to the above integral using a series expansion and write the K expression as:

12 4

2 22 2

2

sin cos38 8

Ia aKa bb

σ π θ θπ π

⎛ ⎞= +⎜ ⎟

⎝ ⎠+ (2-104)

From the above expression it is obvious that IK varies along the crack front. Its magnitude is largest at the end of the minor axis and lowest at the end of the major axis as follows:

2

(max)

(min)

I

I

aK

abK

σ π

πσ

=Φ

=Φ

(2-105)

A correction factor of 1.12 is usually considered for the part-through semi-elliptical cracks. For the quarter-elliptical corner crack the factor is 1.2. We may also use the general expression IK Y bσ π= and find the Y factor from the following diagrams.


52

Fig. 2.27


53

Experimental Determination of SIF The different methods used for experimental determination of stress intensity factors belong to a broader domain called experimental stress analysis. Here, we briefly review the usage of electrical resistance strain gages and the method of photoelasticity. An excellent source for detailed study of these and other methods, such as laser interferometry and the method of Caustics, is: Dally, J. W., Riley, W. F., Experimental Stress Analysis, 3rd edition, 1991. Also are the lecture notes on experimental stress analysis, provided by James W. Phillips, Department of Theoretical and Applied Mechanics, University of Illinois at Urbana-Champaign. Strain Gage Method The stress intensity factor can be determined experimentally by placing one or more strain gages near the crack tip. However, to avoid severe strain gradients, the gages should not be placed at very near field. On the other hand, as we move further from the crack tip, we need more terms to be able to express the field parameters correctly. The three-term representation of the strain field is:

( ) ( )

( ) ( )

( ) ( )

( ) ( )

12

0 0

122

1

12

0 0

122

1

1 12 2

0 1

3cos 1 1 sin sin 22 2 2

cos 1 1 sin2 2

3cos 1 1 sin sin 22 2 2

cos 1 1 sin2 2

32 sin cos sin cos2 2

xx

yy

xy

E A r B

A r

E A r B

A r

A r A r

θ θ θε ν ν

θ θν ν

θ θ θε ν ν ν

θ θν ν

θ θμγ θ θ

−

−

−

⎡ ⎤= − − + +⎢ ⎥⎣ ⎦

⎡ ⎤+ − − +⎢ ⎥⎣ ⎦

⎡ ⎤= − − + −⎢ ⎥⎣ ⎦

⎡ ⎤+ − − +⎢ ⎥⎣ ⎦

⎡ ⎤ ⎡ ⎤= −⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

(2-106)

where A0 , B0 , and A1 are unknown coefficients which depend on loading and the geometry of the specimen. For instance we have:

0 2IKAπ

= (2-107)

In general, we need three strain gages to be able to determine the above three unknowns. However, it can be shown that it is possible to use only one gage oriented at angle α and positioned along the Px′ axis, as shown in Fig. 2.28.


54

Fig. 2.28 Accordingly, the stress intensity factor can be determined from:

1 3 1 32 cos sin sin cos 2 sin cos sin 22 2 2 2 22

Ix x

KE kr

θ θ θμ ε θ α θ απ′ ′

⎡ ⎤= − +⎢ ⎥⎣ ⎦ (2-108)

where,

11

k νν

−=

+ (2-109)

The choice of the angles α and θ depend on the Poisson’s ratio and can be determined from the table below.

Table 2.1 For the choice 60α θ= = , the required expression is very simple:

83I x xK E rπ ε ′ ′=

(2-110)


55

Photoelasticity Method Photoelasticity is a whole-field stress analysis technique based on an optical-mechanical property called birefringence, possessed by many transparent polymers. A loaded photoelastic specimen, combined with other optical elements, and illuminated with an ordinary light source exhibits fringe patterns that are related to the difference between the principal stresses in a plane normal to the light propagation direction. A polariscope is needed for viewing the fringes induced by the stresses (see Fig. 2.29). Two types of pattern can be obtained: isochromatics and isoclinics. The former is related to the principal-stress differences and the latter to the principal stress directions.

Fig. 2.29 The sensitivity of a photoelastic material is characterized by its fringe constant fσ . This constant relates the value “N” associated with a given fringe to the thickness h of the specimen in the light-propagation direction and the difference between the principal stresses in the plane normal to the light-propagation direction as follows:

1 2Nfhσσ σ− =

(2-111)

In practice, the principal stresses are obtained from the stresses defined in Eq. (2-63) and combined with Eqn. (2-111) to give:


56

2211 22

1 2 1222

sin2

2

I

I

Kr

r f NKh

σ

σ σσ σ σ

θπ

π

−⎛ ⎞− = +⎜ ⎟⎝ ⎠

=

⇒ =

(2-112)

For example, let us calculate the value of the stress intensity factor for the specimen shown in Fig. 2.30.

Fig. 2.30

Suppose that the fringe designated with number 5 is located at the distance 0.23 in. from the crack tip, the material fringe value is 43, and the specimen thickness is 0.213 in. The magnitude of the stress intensity factor can be calculated as:

2

2 (0.23)(43)(5)1.2 ksi in

0.213

Ir f NKh

σπ

π

=

= =

(2-113)


57

Numerical Determination of SIF For most practical problems, either there is no analytical solution, or the available solutions are only crude approximations. Hence, numerical techniques play a very important role in determination of SIFs for real components. Numerical methods for fracture mechanics can be categorized in many different ways. We will explore some aspects of the so called “computational fracture mechanics” as we proceed in this course. A brief explanation of different approaches to numerical determination of SIF follows: Ordinary and Extended Finite Element Methods In the finite element method (FEM), the structure of interest is subdivided into discrete shapes called elements. Different Element types can be used to cover the problem domain. The elements are connected at node points where continuity of the displacement fields is enforced. The displacements at the nodes depend on the element stiffness and the nodal forces. For structural problems, the solution of the problem consists of nodal displacements. The stress and strain distribution throughout the body, as well as the crack parameters such as SIF, can be inferred from the nodal displacements. A number of commercial FEM packages have the ability for crack modeling and fracture mechanics calculations. There is also a very useful noncommercial code, called FRANC2D, which is developed in Cornell University. The code is surprisingly easy to learn and work with and has many capabilities, This is an ideal code for education and research. It can be downloaded at http://www.cfg.cornell.edu/software/software.htm. In the Extended Finite Element Methods (X-FEM), a discontinuous function and the two-dimensional asymptotic crack-tip displacement fields are added to the finite element approximation to account for the crack, using the notion of partition of unity. This enables the domain to be modeled by finite elements with no explicit meshing of the crack surfaces. The initial crack geometry is represented by level set functions, and subsequently signed distance functions are used to compute the enrichment functions that appear in the displacement-based finite element approximation. The method has basically been developed in Northwestern University. Boundary Element Methods The problems in science and engineering are usually solved mathematically through specifying appropriate boundary conditions. Given these boundary conditions, it is theoretically possible to solve for the tractions and the displacements on the boundary, as well as the stresses, strains, and displacements within the body. The boundary integral equation method (BIE) is a very powerful technique for solving for unknown tractions and displacements on the surface. This approach can also provide solutions for internal field quantities, but finite element analysis is more efficient for this purpose. The BIE formulation leads to a set of integral equations that relate surface displacements to surface tractions. In order to solve for the unknown boundary data, the surface must be subdivided into segments (i.e., elements), and the boundary integrals are approximated by a system of algebraic equations. Once all of the boundary quantities are known,


58

displacements at internal points can be computed. The boundary elements have one less dimension than the body being analyzed, i.e., the boundary of a two-dimensional problem is surrounded by one-dimensional elements, while the surface of a three-dimensional solid is paved with two-dimensional elements. Consequently, boundary element analysis can be very efficient, particularly when the boundary quantities are of primary interest. This method is inefficient, however, when solving for internal field quantities. The noncommercial FRANC3D code, along with OSM (solid modeler) and BES (3D BIM solver) codes provide a powerful set for analyzing fracture mechanics problems. These codes are developed in Cornell University can be downloaded at http://www.cfg.cornell.edu/software/software.htm. Mesh Free Methods These methods are relatively new and still in the stage of development, but they possess some interesting features that make them strong candidates for computational fracture mechanics in the future. In these methods the problem domain is represented by a set of arbitrarily distributed nodes. There is no need to use meshes or elements for field variable interpolation. The density of nodes can be chosen high near discontinuities like cracks. In contrast to FEM, there is no need for conformation of mesh with cracks and this makes the modeling of evolving cracks easier. Determination of SIF using FEM In general, we may consider two different approaches for determination of SIF using FEM, i.e., the point matching and the energy methods. The former technique is based on inferring the stress intensity factor from the stress or displacement fields in the body. In the latter the energy release rate is first computed and the stress intensity is obtained using Eqs. (2-81) and (2-82). Displacement Extrapolation Following a liner elastic analysis, the stress intensity factor can be determined by equating the numerically obtained displacements with their analytical expression in terms of the SIF. For example, the mode-I displacement equations are:

( , )2

Ii i

K ru fG

θ νπ

= (2-114)

Using the above expression we may obtain a quantity IK ∗ using the nodal point displacement iu∗ at some point ( , )r θ close to the crack tip as follows:


59

[ ]2 ( , )I i iK G f urπ θ ν∗ ∗=

(2-115)

The stress intensity factor can be inferred by plotting the quantity IK ∗ against the distance from the crack tip and extrapolating to r = 0. This technique was the predominant one prior to the discovery of the quarter point singular element. In early finite element studies of LEFM, it was recognized that unless singular elements were used, it would be necessary to have a very fine mesh at the crack tip to approximate the stress singularity with non-singular elements. Singular Elements It is interesting to note that certain element/node configurations can produce strain singularities. Forcing the elements at the crack tip to exhibit a strain singularity greatly improves the accuracy and reduces the need for a high degree of mesh refinement at the crack tip. It has been shown that ordinary quadratic isoparametric elements can be degenerated to give desired singularity by moving the mid-side nodes to the l/4 points.

Fig. 2.31 Energy Approach, Elemental Crack Advance We have already shown that the energy release rate can be obtained from the rate of change in the total potential energy with crack growth. Assume a two-dimensional body with unit thickness. If two separate numerical analyses are performed (one with the crack length a and the other with the crack length a + ∆a) the energy release rate is given by:

Ga

ΔΠ⎛ ⎞= −⎜ ⎟Δ⎝ ⎠

(2-116)


60

This technique is easy to implement because the total strain energy is a natural output by many commercial analysis codes. This technique is also more efficient than the point matching methods because the global energy estimates do not require refined meshes. One disadvantage of this method is that it requires multiple solutions, while other methods infer the desired crack tip parameter from a single analysis. Assignment 4: Consider a center-cracked plate of the AISI 4340 steel (E = 210 GPa, ν = 0.3), which has dimensions, as follows: width 2W = 76 mm; length-to-width ratio 2H/2W = 5; and thickness B = 6 mm. The plate contains an initial crack of length (half) a = 1 mm. It is subjected to a tension loading of P = 240 kN. Compute the SIF using the available closed form solutions and FEM. Compare the results.

Energy Method, Virtual Crack Extension, Stiffness Derivative Formulation In this section we will briefly explain the method developed by Parks and Hellen. Consider a two-dimensional cracked body with unit thickness, subject to Mode I loading. The potential energy of the body, in terms of the finite element solution, is given by:

[ ] [ ][ ] [ ] [ ]12

T TKΠ = −u u u F (2-117)

Accordingly, it can be shown that the energy release rate is proportional to the derivative of the stiffness matrix with respect to crack length.

[ ] [ ] [ ]12

T KdGda a

∂Π= − = −

∂u u

(2-118)

The implementation of the above expression does not require changing all of the elements in the mesh. Instead, we may accommodate the crack growth by moving only the elements near the crack tip as illustrated in Fig. 2.32.


61

Fig. 2.32 Each of the elements between Γ0 and Γ1 is distorted such that its stiffness changes. The energy release rate is related to this change in element stiffness as follows:

[ ] [ ] [ ]1

12

CNT i

i

KG

a=

⎧ ⎫∂= − ⎨ ⎬∂⎩ ⎭

∑u u (2-119)

where [ ]iK are the elemental stiffness matrices and NC is the number of elements between Γ0 and Γ1. It should be noted that in this analysis it is sufficient merely to calculate the change in elemental stiffness matrices corresponding to shifts in the nodal coordinates and there is no need to generate a second mesh with a longer crack. One problem with the stiffness derivative approach is that it involves cumbersome numerical differentiation. A more recent formulation of the virtual crack extension method overcomes these difficulties as discussed below. Energy Method, Virtual Crack Extension, Continuum Approach In this section we will briefly explain the method developed by deLorenzi. Fig. 2.33 illustrates a virtual crack advance in a two-dimensional continuum. Material points inside Γ0 experience rigid body translation a distance ∆a in the x1 direction, while points outside of Γ1 remain fixed. In the region between contours virtual crack extension causes material points to translate by ∆x1. For an elastic material, deLorenzi showed that energy release rate is given by:

11

1

1 jij i

iA

u xG w dAa x x

σ δ∂⎛ ⎞ ∂Δ

= −⎜ ⎟Δ ∂ ∂⎝ ⎠∫ (2-120)

where w is the strain energy density and the crack growth is assumed in the x1 direction. The integration need only be performed over the annular region between Γ0 and Γ1.

mirzaei-fracturemechanicslecture.pdf

Documents

fracture mechanics theory

advanced fracture mechanics

design process

evolution of structural

art of structural design

mechanical failure

failure analysis

failure mode