Level Set Method,XFEM

1

ACCURATE INTEGRATION OF FATIGUE CRACK GROWTH MODELS THROUGH KRIGING AND REANALYSIS OF THE EXTENDED FINITE ELEMENT METHOD

By

MATTHEW JON PAIS

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2010

2

© 2010 Matthew Jon Pais

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To my mother and father who have always supported me in all that I have done.

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ACKNOWLEDGMENTS

I would first and foremost like to thank my advisor Dr. Nam-Ho Kim for his valuable

insights and patience with me as we faced our challenges.

Dr. Timothy Davis and Nuri Yeralan from the Computer Science Department for

their assistance in the creation and implementation of the reanalysis algorithm for quasi-

static growth and optimization.

I would also like to thank the members of the Multidisciplinary Design and

Optimization group at the University of Florida for their invaluable comments during my

rehearsals. Extra thanks go to Alexandra Coppé and Felipe Viana for our collaboration

during my time as part of the group.

Dr. Jorg Peters from the Computer Science Department for his comments which

lead to a conference paper discussing enrichment functions for weak discontinuities

independent of the finite element mesh.

All of the members of my committee for their thoughtful insights as I worked

toward this dissertation.

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TABLE OF CONTENTS page

ACKNOWLEDGMENTS .................................................................................................. 4

LIST OF TABLES ............................................................................................................ 8

LIST OF FIGURES .......................................................................................................... 9

LIST OF ABBREVIATIONS ........................................................................................... 11

ABSTRACT ................................................................................................................... 12

CHAPTER

1 INTRODUCTION .................................................................................................... 14

Motivation and Scope ............................................................................................. 14

Outline .................................................................................................................... 22

2 THE LEVEL SET METHOD .................................................................................... 23

Introduction ............................................................................................................. 23

Level Set Method for Closed Sections .................................................................... 23

Level Set Method for Open Sections ...................................................................... 25

Summary ................................................................................................................ 27

3 THE EXTENDED FINITE ELEMENT METHOD ..................................................... 28

Introduction ............................................................................................................. 28

Displacement Approximation .................................................................................. 29

Enrichment Functions ............................................................................................. 32

Crack Enrichment Functions ............................................................................ 33

Inclusion Enrichment Functions ........................................................................ 38

Void Enrichment Function ................................................................................ 42

Integration of Element with Discontinuity ................................................................ 42

XFEM Software ....................................................................................................... 44

Abaqus ............................................................................................................. 44

MXFEM ............................................................................................................ 45

Others............................................................................................................... 49

Summary ................................................................................................................ 49

4 CRACK GROWTH MODEL .................................................................................... 51

Introduction ............................................................................................................. 51

Stress Intensity Factor Evaluation .......................................................................... 52

Crack Growth Direction ........................................................................................... 56

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Crack Growth Magnitude ........................................................................................ 58

Finite Crack Growth Increment ......................................................................... 58

Fatigue Crack Growth ...................................................................................... 59

Summary ................................................................................................................ 63

5 KRIGING FOR INCREASED FATIGUE CRACK GROWTH STEP SIZE ................ 65

Introduction ............................................................................................................. 65

Kriging ..................................................................................................................... 66

Kriging for Integration of Fatigue Crack Growth Law .............................................. 68

Example Problems .................................................................................................. 71

Center Crack in an Infinite Plate Under Tension .............................................. 71

Edge Crack in a Finite Plate Under Tension..................................................... 74

Summary and Future Work ..................................................................................... 75

6 REANALYSIS OF THE EXTENDED FINITE ELEMENT METHOD ........................ 77

Introduction ............................................................................................................. 77

Cholesky Factorization............................................................................................ 78

Reanalysis of the Extended Finite Element Method ............................................... 79

Example Problems .................................................................................................. 82

Reanalysis of an Edge Crack in a Finite Plate .................................................. 83

Optimization for Finding Crack Initiation in a Plate with a Hole ........................ 86

Summary ................................................................................................................ 88

7 SUMMARY AND FUTURE WORK ......................................................................... 90

Summary ................................................................................................................ 90

Future Work ............................................................................................................ 92 APPENDIX

A MXFEM BENCHMARK PROBLEMS ...................................................................... 95

Crack Enrichments ................................................................................................. 95

Center Crack in a Finite Plate ........................................................................... 95

Edge Crack in a Finite Plate ............................................................................. 97

Inclined Edge Crack in a Finite Plate ................................................................ 98

Bi-material Center Crack in an Infinite Plate ..................................................... 99

Inclusion Enrichment............................................................................................. 100

Hard Inclusion in a Finite Plate ....................................................................... 100

Void Enrichment ................................................................................................... 102

Void in an Infinite Plate ................................................................................... 102

Other ..................................................................................................................... 105

Angle of Crack Initiation from Optimization..................................................... 105

Crack Growth in Presence of an Inclusion...................................................... 107

Fatigue Crack Growth .................................................................................... 108

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B AUXILIARY DISPLACEMENT AND STRESS STATES........................................ 111

Homogeneous Crack ............................................................................................ 111

Bi-material Crack .................................................................................................. 112

LIST OF REFERENCES ............................................................................................. 116

BIOGRAPHICAL SKETCH .......................................................................................... 129

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LIST OF TABLES

Table page 5-1. Accuracy of integration for chosen a∆ for a center crack in an infinite plate. ....... 73

5-2. Accuracy of integration for chosen N∆ for a center crack in an infinite plate. ...... 73

5-3. Accuracy of integration for chosen a∆ for an edge crack in a finite plate. ............ 74

5-4. Accuracy of integration for chosen N∆ for an edge crack in a finite plate. ........... 75

6-1. Crack initiation angle and time comparisons with and without reanalysis. ............ 88

A-1. Convergence of normalized stress intensity factors for a full and half model for a center crack in a finite plate. ............................................................................ 96

A-2. Convergence of normalized stress intensity factors for an edge crack in a finite plate. ................................................................................................................... 98

A-3. Convergence of normalized stress intensity factors for an inclined edge crack in a finite plate. ................................................................................................... 99

A-4. Comparison of the theoretical and MXFEM value for maximum energy release angle as a function of average element size. ................................................... 107

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LIST OF FIGURES

Figure page 1-1. Representative S-N curve for a material subjected to cyclic loading. .................... 14

1-2. Representation of the Mode I (left), Mode II (center) and Mode III (right) opening mechanisms. ......................................................................................... 17

2-2. Example of the signed distance functions for an open section. ............................. 26

3-1. The nodes enriched with the Heaviside and crack tip enrichment functions. ........ 34

3-2. One-dimensional bi-material bar problem. ............................................................ 40

3-3. Comparison of the various inclusion enrichment functions. .................................. 41

3-4. Examples of elements containing a discontinuity and the subdomains for integration. .......................................................................................................... 43

3-5. MXFEM GUI for automated input file creation. ...................................................... 46

3-6. Example problem to show plots generated by MXFEM. A) The geometry being considered, B) Example of the mesh output from MXFEM where blue circles and squares denote the Heaviside and crack tip enriched nodes and the black circles denote the inclusion enriched nodes. ............................................. 47

3-7. Example of the level set functions output from MXFEM. A) ( )φ x , B) ( )ψ x , C)

( )χ x , D) ( )ζ x . .................................................................................................. 47

3-8. Example of the stress contours output from MXFEM. A) XX

σ , B) XY

σ , C) YY

σ ,

D) VM

σ . ............................................................................................................... 48

4-1. Plate with a hole subjected to tension. A) Geometry, B) Mesh (h = 1/10) ............ 62

4-2. Convergence of crack path. A) Mesh density, B) a∆ , C) N∆ . ............................ 62

4-3. Close up view of the crack paths for a∆ around the hole. .................................... 63

5-1. Kriging model with an arbitrary set of five points and the uncertainty in gray. ....... 68

6-1. Edge crack in a finite plate for assessment of the reanalysis algorithm. ............... 84

6-2. Comparison of the assembly time for the stiffness matrix with and without the reanalysis algorithm. ........................................................................................... 85

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6-3. Comparison of the sensitivity of the assembly time of the reanalysis algorithm. A) Fixed DOF∆ , B) Fixed a∆ . .......................................................................... 86

6-4. Plate with a hole for crack initiation assessment of reanalysis algorithm. ............. 87

A-1. Representative geometry for a center crack in a finite plate. ................................ 95

A-2. Representative geometry for an edge crack in a finite plate ................................. 97

A-3. Representative geometry for an inclined edge crack in a finite plate. ................... 98

A-4. Representative geometry for a bi-material center crack in an infinite plate. .......... 99

A-5. Representative geometry for a hard inclusion in a finite plate. ........................... 100

A-6. Comparison of the xx

σ contours for a hard inclusion in a finite plate. A)

ANSYS, B) MXFEM. ......................................................................................... 101



ANSYS, B) MXFEM. ......................................................................................... 101



ANSYS, B) MXFEM. ......................................................................................... 102

A-9. Representative geometry for a void in an infinite plate. ...................................... 102


σ contours for a void in an infinite plate. A) Theoretical,

B) MXFEM. ....................................................................................................... 104

A-11. Comparison of the xy


B) MXFEM. ....................................................................................................... 104

A-12. Comparison of the yy


B) MXFEM. ....................................................................................................... 105

A-13. Representative geometry for a crack initiating at an angle θ for a plate with a hole. .................................................................................................................. 106

A-14. Comparison of crack paths to those predicted by Bordas for a soft (left) and a hard (right) inclusion. ........................................................................................ 108

A-15. Comparison between Paris model and XFEM simulation for fatigue crack growth for a center crack in an infinite plate. .................................................... 110

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LIST OF ABBREVIATIONS

AFRL Air Force Research Laboratory

CTOD Crack Tip Opening Displacement

FEM Finite Element Method

GUI Graphical User Interface

LSM Level Set Method

PUFEM Partition of Unity Finite Element Method

XFEM Extended Finite Element Method

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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

ACCURATE INTEGRATION OF FATIGUE CRACK GROWTH MODELS THROUGH

KRIGING AND REANALYSIS OF THE EXTENDED FINITE ELEMENT METHOD By

Matthew Jon Pais

May 2011

Chair: Nam-Ho Kim Major: Mechanical Engineering

The numerical modeling of fatigue crack growth is a challenging engineering

problem. Fatigue crack growth occurs as the result of repeated cyclic loading well below

the stress levels which typically would cause failure. The number of cycles to failure for

high-cycle fatigue is commonly of the order of 104-108 cycles to failure. Fatigue is

characterized by a differential equation which gives the crack growth rate as a function

of material properties and the stress intensity factor. Analytical relationships for the

stress intensity factor are limited to simple geometries. Thus, a numerical method is

commonly used to find the stress intensity factor for a given geometry under certain

loading. Here the extended finite element method is used to this end.

As there is no sense of the future stress intensity factor, the forward Euler method

is typically used to approximate the solution of the differential equation governing

fatigue crack growth as no information is available for a higher order approximation

based on future stress intensity factor values. This limits the allowable step size to very

small increments for the forward Euler method to retain accuracy. This leads to a very

large number of analyses to be performed as the number of cycles to failure is so large.

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The use of kriging to assist higher-order approximations is introduced. Here all

stress intensity factor data is fit using a kriging surrogate, which allows for stress

intensity factor values beyond the current data point to be approximated. These

extrapolated points enable the use of midpoint and Runge-Kutta methods for the

approximation of the differential equations governing fatigue crack growth. This enables

larger step sizes to be taken without a loss in accuracy for the solution of the governing

differential equation.

There is still a large amount of simulations required even with the increased time

step from the use of kriging to help approximate the solution of the differential equation

governing fatigue crack growth. It was observed that for the extended finite element

method a small portion of the global stiffness matrix is changed as a result of crack

growth. It is possible to use this small portion to save on both the assembly and solution

of the resulting system of linear equations. For the first simulation, the full stiffness

matrix is calculated and factored using the Cholesky algorithm. This results in a series

of triangular solves for the resulting system of linear equations. This factorization can

then be modified to account for the changes to the stiffness matrix. This results in

savings in both the assembly and factorization of the stiffness matrix for repeated

simulations reducing the computational cost associated with numerical fatigue crack

growth.

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CHAPTER 1 INTRODUCTION

Motivation and Scope

The nucleation and propagation of cracks in engineered structures is an important

consideration for the design of a structure. In particular fatigue fracture, caused by

repeated cyclic loading well below the yield stress of a material can cause sudden,

catastrophic failure. The relationship between the applied stress and the number of

cycles to failure is typically given by a material specific S-N curve as shown in Figure 1-

1. For fatigue failure once a small crack has formed, the cyclic loadings cause the

material ahead of the crack to slowly fail and the crack grows. Initially a crack grows

very slowly, maybe on the order of nanometers at a given cycle. Over time the crack

growth accelerates. Once the crack reaches a critical length ac a large amount of crack

growth occurs rapidly and without warning. There are many incidents caused by the

growth of fatigue cracks, many of which resulted in the loss of human lives.

Figure 1-1. Representative S-N curve for a material subjected to cyclic loading.

In 1952 the world’s first jetliner the de Havilland Comet [1] was entered into

service. In January and May of 1954 two of the Comets disintegrated during flights

between New York and London. The failure was caused by fatigue cracks which

Number of Cycles to Failure

Ap

plie

d S

tre

ss

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initiated near the front of the cabin on the roof. Over time the crack grew until it reached

a window, causing sudden catastrophic failure. In 1957 the 7th President of the

Philippines died [2] along with 24 others when fatigue caused a drive shaft to break,

subsequently causing power failure aboard the airplane. In 1968 a helicopter [3]

crashed in Compton, California due to fatigue failure of the blade spindle. Twenty one

lives were lost.

In 1980 the Alexander L. Keilland [4] oil platform capsized killing 123 people. The

main cause of the failure was determined to be a poor weld, which lead to a reduction in

the fatigue strength of the structure. In 1985 Japan Flight 123 [5] from Tokyo to Osaka

crashed in the deadliest plane crash in history. There were 4 survivors of the 524

people on the airplane. The fatigue failure was due to the incorrect repair of an impact

the tail had with a runway in 1978. Fatigue cracks slowly grew until causing sudden

rupture 7 years later. In 1988 an Aloha Airlines [6] flight between the Hawaiian islands

of Hilo and Honolulu suffered extensive damage after an explosive decompression

caused by the combined effects of a fatigue crack and corrosion. The plane safely

landed in Maui with 94 survivors, 65 injuries and 1 death. In this case, the fracture was

exacerbated by being in service well past it’s design life (89,000 service hours instead

of 75,000 design hours) as well as being subjected a corrosive environment caused by

exposure to high levels of humidity and salt. In 1989 a United Airlines flight [7] from

Denver, Colorado to Chicago, Illinois crashed due to a maintenance crew failing to find

a crack in a fan disk within the engine. There were 112 fatalities among the 285 people

aboard the airplane.

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In 1992 a Boeing 747 [8] crashed in to the Bijlmermeer neighborhood in

Amsterdam, Netherlands, killing 43. The plane crashed when fatigue failure did not

allow for the engine to cleanly separate from the wing as designed, leading to the

accident. In 1998 an InterCityExpress train [9] crashed in Eschede, Germany caused by

fatigue failure of the train wheels. Of the 287 passengers aboard, there were 101 deaths

and 88 injuries. It is the deadliest train disaster in German history.

In 2002 China Airlines Flight 611 [10] broke apart during a flight killing all 225

people aboard the airplane. Similar to Japan Airlines Flight 123, an incorrect repair

procedure allowed fatigue cracks to grow eventually causing failure. In 2005 metal

fatigue caused a Chalk’s Ocean Airways flight [11] from Fort Lauderdale, Florida to

Bimini, Bahamas to crash in Miami Beach, FL after metal fatigue broke off the right

wing. There were 20 casualties. In 2007 a Missouri Air National Guard F-15C Eagle [12]

crashed due to a structural part not meeting specifications, leading to a series of fatigue

cracks to develop and propagate. As recently as July 2009 a Southwest Airlines flight

[13] from Nashville, Tennessee to Baltimore, Maryland had to make an emergency

landing after a ‘football sized’ hole opened causing rapid decompression. Investigations

are still undergoing.

The series of accidents through history including those in the 1990s and 2000s

show that there is still much work to be done to prevent fatigue failures from taking

human lives. With the ever increasing speed of computers numerical methods such as

finite element methods are able to model problems with increasing fidelity and

increasing complexity. Fatigue crack growth models are empirical models which are

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generally created by performing one or a series of experiments and fitting the resulting

data to a function of the form [14]

( )da

f KdN

= ∆ (1.1)

where da/dN is the crack growth rate and K∆ is the stress intensity factor ratio, which is

a driving mechanism to crack growth. The stress intensity factor is used to describe the

state of stress at the tip of a crack and depends upon, crack location, crack size,

distribution and magnitude of loading, and specimen geometry. There are three modes

of fracture as shown in Figure 1-2. On the left is Mode I or opening mode. In the middle

is the Mode II or in-plane shear mode. Finally, on the right is the Mode III or out-of-plane

shear mode. In two-dimensions only Modes I and II may be considered, while Mode III

is also considered for three-dimensional problems. When multiple modes are occurring

simultaneously the stress intensity factors are referred to as being mixed-mode.

Figure 1-2. Representation of the Mode I (left), Mode II (center) and Mode III (right) opening mechanisms.

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For some simplified cases analytical expressions for the stress intensity factors

are available [15], but for a more general case a numerical method such as finite

element methods may be used to find the stress intensity factor. Specifically, the use of

a method such as the crack tip opening displacement (CTOD) [16], J-integral [17] or the

domain form of the contour integrals [18, 19] may be used to extract the mixed-mode

stress intensity factors from the finite element solution. Note that Eq. (1.1) does not

provide the crack size directly. Rather, it provides the rate of crack growth at a given

range of stress intensity factor, which also depends on the current crack size. Thus, a

numerical integration method should be employed to predict the crack growth. In

addition, the crack in a complex geometry may not grow in a single direction. The

direction of future growth is often considered to be governed by the maximum

circumferential stress criterion in the finite element framework as closed form solutions

for the direction of crack growth are available [20].

Even if the finite element method (FEM) has been developed drastically during the

last century, modeling crack growth in the classical FEM is not without its challenges.

As the finite element mesh must conform to the geometry, the mesh around the crack

tip must be recreated [21] whenever growth occurs. Even if a concept such as crack-

blocks [22] is used where a small region around a crack tip is remeshed at each

iteration, the computational demands of the remeshing can contribute significantly to the

simulations especially if a large number of iterations of fatigue crack growth are to be

modeled.

There are two main opinions on how to attempt to approximate the solution of the

governing differential equation in Eq. (1.1): controlling cycles or crack increment. The

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first model assumes that a fixed number of cycles N∆ is chosen prior to starting the

simulations [23]. Thus, a∆ is variable, being initially small and increasing with the

iteration number. At each simulation iteration, K∆ is calculated from the finite element

simulation, which is then used to approximate the solution of the ordinary differential

equation governing fatigue crack growth. Since the expression of K∆ is unknown as a

function of crack size, the differential equation cannot be directly integrated. Instead,

there are only values of K∆ at the current and all past simulation iterations, and an

explicit numerical integration method such as the forward Euler method [24] can be

used to approximate the current growth increment a∆ . As the crack increment is based

on K∆ , it is essential for the mesh to be sufficiently refined around the crack tip such

that K∆ has converged and an accurate representation of the localized state of stress

around the crack tip. The use of a higher-order integration method such as the midpoint

[24] or Runge-Kutta [24] methods requires information which is not available in the form

of simulations of crack increments between the current and next simulation iteration.

Furthermore, the required function evaluations may have no physical meaning,

especially considering that crack growth does not occur at all instances within a loading

cycle [25]. The forward Euler method requires small time steps to be accurate;

otherwise crack growth will be under predicted.

The other solution procedure assumes a fixed size of crack increment [21] at each

simulation iteration. Therefore, the number of elapsed cycles is initially large and

decreases with the iteration number. Here, the challenge is accurately back-calculating

the number of elapsed cycles from the fixed crack growth data. The forward Euler

method is commonly applied to the back-calculation of the number of elapsed cycles

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[26], but as with the approach of fixing N∆ there must be care taken in selecting a

sufficiently small value of N∆ for the forward Euler method to maintain accuracy. A

higher-order approximation could be applied here as well, but discrete values of K∆ are

not available for all data points required for the evaluation of the slope of the a-N curve

for these higher-order methods.

For a general mixed-mode crack growth simulation, it is also imperative that the

choice of fixed a∆ or N∆ be made with care. The path that the crack may take is also

influenced by the choice of a∆ or N∆ as having too large of a value of either can result

in deviation from the crack growth path that would be predicted with the use of smaller

growth increments. Once this deviation occurs the path of future crack growth is

definitely affected, but this deviation is not clear to a user unless a convergence study

with respect to crack path is performed. In the literature [21, 23], it is clear that the

chosen fixed crack increments influence the crack path under mixed-mode loading. In

addition to the localized mesh reconstruction around the crack tip care needs to be

taken to ensure accuracy in the crack path, displacement, and stress intensity factors

with FEM.

The extended finite element method (XFEM) [21] alleviates the challenges

associated with the mesh conforming to the geometry by allowing discontinuities or

other localized phenomena to be represented independent of the finite element mesh.

Additional functions, referred to as enrichment functions, are introduced into the

displacement approximation through the property of the partition of unity [27]. Additional

nodal degrees-of-freedom are also introduced that act to ‘calibrate’ the enrichment

functions as well as are used for interpolating within an element and in the calculation of

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stresses or stress intensity factors using a method such as the domain form of the

contour integrals. Without the need to worry about mesh construction, the XFEM still

requires for the convergence of the crack path, displacement, and stress intensity

factors for accurate crack modeling.

The goals and scope of this work are focused on accurate modeling of fatigue

crack growth under constant and variable amplitude loading for complex geometries

without sacrificing accuracy. In particular the following topics are addressed:

• Increasing the allowable step size for a given fixed increment of a∆ or N∆

without a loss in accuracy when approximating the solution to the governing

ordinary differential equation. The influence that this increased step size has

on the convergence of the crack path is also considered. The Kriging

surrogate model is exploited to enable the use of higher-order approximations

to the differential equations governing fatigue crack growth and to control the

accuracy of prediction.

• The fundamental formulation of the XFEM is also exploited to enable the

modeling of quasi-static crack growth with reduced computational time

through a reanalysis algorithm. When crack growth occurs, the changes to

the global stiffness matrix are limited to a very localized region about the

crack tip. Here, a supernodal Cholesky factorization is used to exploit these

properties. This factorization is modified to account for the changes to the

global stiffness matrix, allowing for large time savings to be realized for both

the assembly and factorization of the global stiffness matrix. This reanalysis

algorithm is also employed as a means to consider optimization problems in

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the XFEM framework where the location of a discontinuity is iteration

dependent.

Outline

Chapter 2 introduces the level set method for tracking closed and open sections.

This method is used to track the location of discontinuities in the XFEM as they do not

conform to the mesh. This includes cracks, inclusions and voids. Chapter 3 introduces

the extended finite element method. First the general form is considered. Then

enrichment functions are introduced for cracks, inclusions and voids. A discussion of the

commercial and open-source implementations of XFEM is presented. Chapter 4

considers the crack growth model that governs the crack growth at a given iteration.

Criterions which determine the direction of future crack growth are introduced. The

domain form of the contour integral is detailed for the extraction of the mixed-mode

stress intensity factors from a XFEM analysis of a cracked body. Finally, methods to

determine the amount of crack growth are given. Chapter 5 introduces the use of the

kriging surrogate model for increased accuracy in the integration of the ordinary

differential equation governing fatigue crack growth allowing larger step sizes to be

considered without loss of accuracy. Chapter 6 introduces and details the use of a

reanalysis algorithm to make the repeated simulations of crack growth in a quasi-static

environment affordable, allowing for additional simulations in a fixed amount of time.

This leads to the use of a smaller time step and thus, a more accurate approximation to

the fatigue crack growth model. Chapter 7 summaries the work and future work are

suggested.

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CHAPTER 2 THE LEVEL SET METHOD

Introduction

The level set method was introduced by Sethian and Osher [28] as a numerical

method which can be used to track the evolution of interfaces and shapes. The method

is based on evolving an interface subjected to a front velocity given by the physics of

the underlying problem which is being modeled. Level set methods have been used in a

wide range of engineering applications in topics such as compressible [29] and

incompressible [30] flow, computer vision [31], image processing [32], manufacturing

[33-35] and structural optimization [36, 37]. In this chapter, the general algorithm is

introduced for tracking either a closed or open section through the use of the level set

method. The level set method will be used for tracking the location of cracks, inclusions,

and voids in a structure for an extended finite element analysis. Inclusions and voids

represent closed sections, while cracks represent open sections in two-dimensions.

Level Set Method for Closed Sections

The level set method uses a discretization at grid points in the domain of interest.

Each of these points is assigned a signed distance value from that point to the nearest

intersection with the interface denoted Γ . A continuous level set function ( )φ x is

introduced where x is a point in the domain of interest Ω with interface Γ . The level set

function can be characterized as a function of the domain and time where

( )

( )

( )

, 0 for

, 0 for

, 0 for

t

t

t

φ

φ

φ

< ∈Ω

> ∉Ω

= ∈ Γ

x x

x x

x x

. (2.1)

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Thus, points inside the domain of interest are given negative signs, points outside of the

domain of interest are given positive signs, and points on the interface have no sign as

their signed distance is zero. An example of the signed distance function for a circular

domain is given in Figure 2-1. From Eq. (2.1) it can be noted that at any time t the

location of the interface can be found as the locations where

( ), 0tφ =x (2.2)

and is commonly referred to as the zero level set of φ .

Figure 2-1. Example of a signed distance function for a closed domain.

The evolution of the level set function is usually assumed to follow the Hamilton-

Jacobi equation [28] given as

vt

φφ

∂= ∇

∂ (2.3)

where v is the front speed and φ∇ is the spatial gradient of the level set function. The

solution of Eq. (2.3) is usually approximated using finite differencing techniques [24].

( ) 0φ >x

( ) 0φ <x

Γ

( ) 0φ =x

Ω

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When the forward finite difference technique is considered, the derivative of φ with

respect to time can be approximated as

1 0i ii i

t

φ φφ− −

+ ⋅∇ =∆

V (2.4)

where 1iφ + is the updated level set value,

iφ is the current level set value,

iV is the front

velocity vector, and t∆ is the elapsed time between i and 1i + . Equation (2.4) can be

rewritten in a more convenient form in two-dimensions as

1i i

i i i it u v

x y

φ φφ φ+

∂ ∂= − ∆ + ∂ ∂

(2.5)

where i

u is the front velocity in the x-direction and i

v is the front velocity in the y-

direction. In Eqs. (2.4) and (2.5) the time step t∆ is limited by the Courant-Friedrichs-

Lewy (CFL) condition [38] which ensures that the approximation to the solution of the

partial differential equation converges. The CFL condition is given as

( )

( )max ,

max ,

x yt

u v

∆ ∆∆ < (2.6)

where x∆ and y∆ are the grid spacing in the x and y-directions. In practice, the level

set function needs to only be defined in a narrow band [39-41] around the interfaces of

interest or can be represented using the fast marching method [42], a variant of the

level set method.

Level Set Method for Open Sections

The version of the level set method presented in the previous section is only valid

for a closed section. For an open section, the definition of the interior, exterior, and

interface as defined in Eq. (2.1) no longer have a physical meaning. Stolarska [39]

introduced a modified version of the level set method which allows for open sections to

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be tracked with the use of multiple level set functions. An open section as shown in

Figure 2-2 can be described by two level sets ( )φ x and ( )ψ x . The interface of interest

is given as the intersection of ( )φ x and ( )ψ x where

( ) ( )0 and 0φ ψ< =x x . (2.7)

Figure 2-2. Example of the signed distance functions for an open section.

An updating algorithm for these two coupled level set function is also given by

Stolarska [39]. For the case of the ( )φ x level set function, the update is identical to that

presented in Eqs. (2.3)-(2.5). Two regions are defined with respect to the ( )ψ x level set

function, ( )update 0φΩ = >x and no update 0Ω ≤ which correspond to the regions which will

and will not be updated. The level set function ( )ψ x is updated at the ith node

according to

( ) ( )

1 no update

1 update

in

in

n n

i i

yn xi i i

F Fx x y y

F F

ψ ψ

ψ

+

+

= Ω

= ± − − − Ω (2.8)

( )

( )

0

0

φ

ψ

>

<

x

x

( )

( )

0

0

φ

ψ

>

<

x

x( ) 0φ =x

( ) 0ψ =x

( )

( )

0

0

φ

ψ

>

>

x

x( )

( )

0

0

φ

ψ

<

>

x

x( )

( )

0

0

φ

ψ

>

>

x

x( )

( )

0

0

φ

ψ

<

<

x

x

( ) 0φ =x

27

where the crack tip displacement vector is given as ( ),x y

F F=F and the current crack tip

is given by the coordinates ( ),i ix y . The sign of the updated value 1n

iψ + is chosen to

correspond to the location of that node with respect to Figure 2-2.

Summary

The level set method allows for a closed or open section to be tracked by defining

signed distance values at fixed points in the domain of interest. The value of the level

set function at these points is then updated based on the front velocity at each point in

the domain using a finite difference technique to approximate the solution to the

governing partial differential equation. The level set method seems to be ideal for use in

a finite element environment where the nodes of the finite element mesh could be used

as the fixed points in the level set algorithm. The finite element shape functions could be

used to interpolate within an element to identify the values of the level set function if this

would be of interest.

28

CHAPTER 3 THE EXTENDED FINITE ELEMENT METHOD

Introduction

The extended finite element method (XFEM) allows for discontinuities to be

represented independent of the finite element mesh by exploiting the partition of unity

finite element method [27] (PUFEM). In this method additional functions, commonly

referred to as enrichment functions, can be added to the displacement approximation as

long as the partition of unity is satisfied, i.e. ( ) 1IN x =∑ where ( )IN x are the finite

element shape functions. The XFEM uses these enrichment functions as a tool to

represent a non-smooth behavior of field variables, such as stress across the interface

of different materials or displacement across cracks. In general, the enrichment

functions introduced into the displacement approximation are only defined over a small

number of elements relative to the total size of the domain. Additional degrees of

freedom are introduced in all elements where the discontinuity is present, and

depending upon the type of function chosen, possibly some neighboring elements

known as blending elements.

This chapter is divided into the following sections. First the incorporation of the

enrichment functions into the displacement approximation and effect these functions

have on the resulting system of equations are discussed. Then specific enrichment

functions are given with an emphasis on cracks, inclusions and voids for linear elastic

materials. The use of level set functions for the definition of the enrichment functions is

introduced. The integration of the enriched elements through the use of subdivision of

other methods is explored. Finally, a survey of the available commercial and open

29

source finite element codes which have incorporated the XFEM in various levels of

sophistication is provided.

Displacement Approximation

The additional functions used in the displacement approximation are commonly

called enrichment functions and the approximation takes the form:

( ) ( ) ( )h J J

I I I

I J

u x N x u x aυ

= +

∑ ∑ (3.1)

where I

u are the classical finite element degrees of freedom, ( )Jxυ is the thJ

enrichment function, and J

Ia are the enriched degrees of freedom corresponding to the

thJ enrichment function at the thI node. The enriched degrees of freedom introduced

by Eq. (3.1) generally do not have a physical meaning and instead can be considered

as a calibration of the enrichment functions which result in the correct displacement

approximation. Note that Eq. (3.1) does not satisfy the interpolation property,

( )h

I Iu u x= , due to the enriched degrees of freedom, instead additional calculations are

required in order to calculate the physical displacement using Eq. (3.1). The

interpolation property is important in practice in applying boundary or contact conditions.

Therefore, it is common practice to shift [43] the enrichment function such that

( ) ( ) ( )J J J

I Ix x xυ υϒ = − (3.2)

where ( )J

I xυ is the value of the Jth enrichment function at the thI node. As the shifted

enrichment function now takes a value of zero at all nodes, the solution of the resulting

system of equations satisfies ( )h

I Iu u x= and the enriched degrees of freedom can be

used for additional actions such as interpolation and post-processing. Here, the shifted

30

enrichment functions are referred to with upper case characters, and the unshifted

enrichment functions are referred to with lower case characters. The shifted

displacement approximation is given by

( ) ( ) ( )h J J

I I I I

I J

u x N x u x a

= + ϒ

∑ ∑ (3.3)

where ( )J

I xϒ is the Jth shifted enrichment function at the thI node. Hereafter, ( )IN x

and ( )J

I xϒ will be written as I

N and J

Iϒ .

The Bubnov-Galerkin method [44] may be used to convert the displacement

approximation given by Eq. (3.3) into a system of linear equations of form

=Kq f (3.4)

where K is the global stiffness matrix, q are the nodal degrees of freedom, and f are

the applied nodal forces. By appropriately ordering degrees of freedom, the global

stiffness matrix K can be considered as

=

uu ua

T

ua aa

K KK

K K (3.5)

where uu

K is the classical finite element stiffness matrix, aa

K is the enriched finite

element stiffness matrix, and ua

K is a coupling matrix between the classical and

enriched stiffness components. The elemental stiffness matrix, e

K for any member of

K may be calculated as

d , ,h

T

eu aα β α β

Ω= Ω =∫K B CB (3.6)

31

where C is the constitutive matrix for an isotropic linear elastic material, u

B is the

matrix of classical shape function derivatives, and a

B is the matrix of enriched shape

function derivatives. The general form of u

B and a

B is given by

( )

( )

( )

( ) ( )

( ) ( )

( ) ( )

,

,

,,

,,

, ,, ,

, ,

, ,, ,

, ,

0 0

0 00 0

0 0

0 00 0;

0 0

00

0

0

J

I I x

JI xI I y

I yJ

I I zI z

u a J JI z I y

I I I Iz y

I z I x J J

I I I Iz xI y I x

J J

I I I Iy x

N

NN

N

NN

N N N N

N NN N

N N

N N

ϒ

ϒ ϒ

= = ϒ ϒ

ϒ ϒ

ϒ ϒ

B B (3.7)

where ,I iN is the derivative of ( )IN x with respect to

ix and ( )

,

J

I I iN ϒ is the derivative of

( ) ( )J

I IN x xϒ with respect to i

x . In practice, ( ),

J

I I iN ϒ is calculated with the product rule

as

( ) ( )( ) ( )( )

( ) ( )( )( )J J

I I II J

I I

i i i

N x x xN xx N x

x x x

∂ ϒ ∂ ϒ∂= ϒ +

∂ ∂ ∂. (3.8)

Similarly, q and f in Eq. (3.4) are given by

TT =q u a (3.9)

where u and a are vectors of the classical and enriched degrees of freedom and

T T T=u a

f f f (3.10)

where u

f and a

f are vectors of the applied forces for the classical and enriched

components of the displacement approximation. The vectors u

f and a

f are given in

terms of applied tractions t and body forces b as

32

d dh ht

u I IN N

Γ Ω= Γ + Ω∫ ∫f t b (3.11)

and

d dh ht

J J

a I I I IN N

Γ Ω= ϒ Γ + ϒ Ω∫ ∫f t b . (3.12)

Stress and strain must be calculated with the use of the enrichment functions and

enriched degrees of freedom such that the effect of the discontinuity with a particular

element is considered. Therefore the strain and stress may be calculated as

[ ] T

u a=ε B B u a (3.13)

and

=σ Cε . (3.14)

Enrichment Functions

The XFEM has been used to solve a wide range of problems involving

discontinuities. In general, discontinuities can be described as either strong or weak. A

strong discontinuity can be considered one where both the displacement and strain are

discontinuous, while a weak discontinuity has a continuous displacement but a

discontinuous strain. There exist enrichment functions for a variety of problems in areas

including cracks, dislocation, grain boundaries, and phase interfaces [45-48]. Aquino

[49] has also studied the use of proper orthogonal decomposition to incorporate

experimental data into the displacement approximation for cases with no logical choice

of enrichment function. Fries [50] introduced the use of hanging nodes in the XFEM

framework with respect to inclusions, cracks, and fluid mechanics to allow for

automated mesh refinement around discontinuities.

33

Crack Enrichment Functions

The modeling of cracks in the XFEM has been thoroughly explored [45-48, 51, 52].

Belytschko [53] was the first to study cracks in the XFEM framework based on the

element-free Galerkin crack enrichment of Fleming [54]. Moёs [21] introduced the use of

the Heaviside enrichment function to simplify the representation of the crack away from

the tip. Works have been done in two [21, 39, 53, 55, 56] and three-dimensions [40-42,

57, 58] for linear elastic [21, 39-42, 53, 55-58], elastic-plastic [59, 60], and dynamic [61-

65] fracture.

The common practice is to incorporate two enrichment functions into the XFEM

displacement approximation to represent a crack. A Heaviside step function [21] is use

to represent the crack away from the tip and a more complex set of functions is used to

represent the crack tip asymptotic displacement field. The Heaviside step function is

given as

( )1, above crack

1, below crackh x

=

−. (3.15)

It can be noticed that the enrichment given by Eq. (3.15) introduces a discontinuity in

displacement across the crack. For a linear elastic crack tip, four enrichment functions

[54] are used to incorporate the crack tip displacement field into elements containing the

crack tip:

( ), 1 4

sin , cos , sin sin , sin cos2 2 2 2

x r r r rα α

θ θ θ θφ θ θ

= −

=

(3.16)

where r and θ are the polar coordinates in the local crack tip coordinate system the

origin it at the crack tip and 0θ = is parallel to the crack. Note that the first enrichment

function in Eq. (3.16) is discontinuous across the crack behind the tip in the element

34

containing the crack tip, acting as the Heaviside enrichment does. Should a node be

enriched by both Eqs. (3.15) and (3.16), only Eq. (3.16) is used as shown in Figure 3-1

where the Heaviside nodes are denoted by filled circles, while the crack tip nodes are

open.

Figure 3-1. The nodes enriched with the Heaviside and crack tip enrichment functions.

Another useful set of crack tip enrichment functions are those introduced by

Sukumar [66] for the modeling of cracks located at the interface between materials,

which is commonly referred to as a bi-material crack. The more complicated state of

stress caused by dissimilar materials on either side of the crack necessitates an

increased number of functions to span to asymptotic crack tip displacement field at the

crack tip. The bi-material crack tip enrichment functions are given as:

35

( ) ( ) ( )

( ) ( )

( ) ( )

( ) ( )

( ) ( )

, 1 12cos log e sin , cos log e cos ,

2 2

cos log e sin , cos log e cos ,2 2

cos log e sin sin , cos log e sin cos ,2 2

sin log e sin , sin log e cos ,2 2

sin log e sin , sin log e2

x r r r r

r r r r

r r r r

r r r r

r r r r

εθ εθα α

εθ εθ

εθ εθ

εθ εθ

εθ εθ

θ θφ ε ε

θ θε ε

θ θε θ ε θ

θ θε ε

θε ε

− −

= −

− −

=

( ) ( )

cos ,2

sin log e sin sin , sin log e sin cos2 2

r r r rεθ εθ

θ

θ θε θ ε θ

(3.17)

where ε is the bi-material constant

1 1

log2 1

βε

π β

−= +

(3.18)

given in terms of the second Dundurs parameter [67] β as

( ) ( )( ) ( )

1 2 2 1

1 2 2 1

1 1

1 1

µ κ µ κβ

µ κ µ κ

− − −=

+ + + (3.19)

where i

µ and i

κ are the shear modulus and Kosolov constant for the thI material. The

Kosolov constant given in terms of Poisson’s ratio as

3plane stress

1

3 4 plane strain

i

ii

i

ν

νκ

ν

−

+= −

. (3.20)

In elements which are cut by the crack but not the crack tip the Heaviside enrichment

given by Eq. (3.15) is still used.

Because the mesh does not conform to the domain, a method must be used to

track of the location of the cracks. To this end the use of the open segment level set

method introduced by Stolarska [39] and detailed in Chapter 3 is used. Two level set

36

functions are used to track the crack, the zero level set of ( )xψ represents the crack

body, while the zero level sets of ( )xφ , which is orthogonal to the zero level set of

( )xψ , represents the location of the crack tips. The two enrichment functions given in

Eqs. (3.15) and (3.16) can be calculated in terms of ( )xφ and ( )xψ such that

( ) ( )( )( )( )

1 for 0

1 for 0

xh x h x

x

ψψ

ψ

>= =

− <. (3.21)

Furthermore, the polar crack tip coordinates are given as

( ) ( )( )( )

2 2 and arctanx

r x xx

ψψ φ θ

φ= + = . (3.22)

The enriched nodes corresponding to the crack tip enrichment can also be determined

through the use of the level set functions defining the crack. Consider an element where

the maximum and minimum values of ( )xψ and ( )xφ are given as maxψ , minψ , maxφ , and

minφ . Then an element is enriched with the Heaviside enrichment when

max max min0 and 0φ ψ ψ< ≤ (3.23)

and the crack tip enrichment when

max max min0 and 0min

φ φ ψ ψ≤ ≤ . (3.24)

Therefore, the extended finite element and level set methods complement one another

well for the tracking of the location of the cracks. The representation of cracks in three-

dimensions [40, 42, 68] follows a similar methodology. In practice the level sets are

defined in only a narrow band about the crack as discussed in Chapter 3 or the fast

marching method [42, 68] is used.

37

The convergence rate of XFEM with crack enrichment functions has been an area

of interest [47, 69-74], particularly with respect to the challenges presented by the

partially enriched or blending elements caused by the crack tip enrichment. No blending

issues exist with the Heaviside function as it vanishes along all element boundaries. It

was noticed by Stazi [71] that the convergence rate for the XFEM was lower than the

equivalent traditional finite element problem. Chessa [75] identified that the partially

enriched crack tip elements lead to parasitic terms in the displacement approximation

and introduced an enrichment dependent assumed strain model to increase the

convergence rate. Fries [69] introduced a linearly decreasing enrichment weight

function in the blending elements to increase convergence. An area [70, 76, 77] instead

of single element crack tip enrichment has also been shown to increase convergence.

Through the use of these methods the convergence rate of cracked domains with the

XFEM has become equivalent to the equivalent traditional finite element problem [47,

69, 74].

Alternative crack tip conditions have also been explored such as cohesive cracks

[78-80], branching cracks [81], cracks under frictional contact [82], fretting fatigue cracks

[83], interfacial cracks [60, 66, 84, 85], cracks in orthotropic materials [86], and cracks in

piezoelectric materials [87]. Mousavi [88] introduced a unified framework for the

enrichment of homogeneous, intersecting, and branching cracks through the use of

harmonic enrichment functions. The XFEM has also been used to study a variety of

problems involving cracks including: the effect of cracks in plates [89, 90], crack

detection and identification [91-93], shape optimization [94, 95], and optimization with

changing crack location using a reanalysis technique [23].

38

Inclusion Enrichment Functions

The modeling of material interfaces independent of the finite element mesh

through the element-free Galerkin [96] as well as partition of unity finite element method

[43, 45, 47, 48, 97-101] has been studied. The enrichment function should incorporate

the behavior of the weak discontinuity, i.e., continuous displacement, but discontinuous

strain. The Hadamard condition [99] given by

+ − +− = ⊗F F a n (3.25)

where F is the deformation gradient, +n is the outward normal material interface, and a

is an arbitrary vector in the plane. The Hadamard condition must be satisfied by the

chosen enrichment function.

Sukumar [99] first introduced the use of the absolute value enrichment in terms of

the level set function ( )xζ , which gives the shortest signed distance from a given point

to the interface between the two materials. Therefore, the enrichment function takes the

form:

( ) ( )x xυ ζ= . (3.26)

The enrichment function is assumed to be nonzero only over the domain of support for

the enriched nodes, as with the crack enrichment function. For a bi-material boundary-

value benchmark problem the absolute value enrichment given by Eq. (3.26) led to a

convergence rate less than the equivalent traditional finite element method problem

where the mesh conforms with the material interface. It was hypothesized that the poor

convergence was related to the blending elements containing a partial enrichment. In an

attempt to improve the convergence rate a smoothing algorithm was introduced to

39

reduce the effects of the blending elements which increased the convergence rate, but

did not equal the traditional finite element method.

Moёs [97] studied modeling complex microstructure geometries with the use of

level set defined material interfaces and introduced a new enrichment function. The

modified absolute value enrichment takes the form:

( ) ( ) ( )I I I I

I I

x N x N xζ ζϒ = −∑ ∑ . (3.27)

Note that the enrichment function given by Eq. (3.27) is zero at all nodes and thus, does

not need to be shifted such that traditional degrees of freedom are recovered. If an

interface corresponds to the mesh, then no nodes are enriched as the enrichment

function will be zero and the problem will be equivalent to the traditional finite element

problem. The same benchmark problem considered by Sukumar [99] was considered

as well as a similar problem in three-dimensions. In two-dimensions the convergence

rate was shown to equal the traditional finite element method. In three-dimensions the

convergence rate was slightly less than the traditional finite element method. This

method is considered the current state-of-the-art for modeling inclusions with the XFEM.

Pais [98, 101] considered an element-based enrichment instead of nodal

enrichment where the displacement approximation took the form

( ) ( ) ( )h

I I e

I

u x N x u x aυ= +∑ (3.28)

where ( )xυ is a piecewise linear enrichment function where

( )( )

( )( )0

0 1

I xx

x

υυ

υ ζ

==

= = (3.29)

40

and e

a are elemental degrees of freedom. Thus, the enrichment function vanishes at all

nodes and takes a value of one at the interface locations. The proposed method allows

for the number of elemental degrees of freedom to be equal to the number of

dimensions of the problem. The resulting system of equations needs fewer degrees of

freedom than either the traditional or extended finite element method to represent the

same domain. It was found that the convergence rate is comparable to the absolute

value enrichment given by Eq. (3.26), due to errors in the prediction of the shear stress

distribution in two and three-dimensions. A comparison of the enrichment functions for

Eqs. (3.26)-(3.28) is given for the case of a bi-material bar shown in Figure 3-2.

Figure 3-2. One-dimensional bi-material bar problem.

For this example problem, Young’s modulus for bars 1 and 2 are 1 Pa and 10 Pa,

the cross-sectional area is assumed to be 1 m2 and the applied load is 1 N. A

comparison of the enrichment functions and locations of the enriched degrees of

freedom for the absolute value, modified absolute value and element-based enrichment

functions is given in Figure 3-3.

Bar 1 P

0.25L 0.75L

Bar 2

41

Figure 3-3. Comparison of the various inclusion enrichment functions.

The problem given in Figure 3-2 can be solved using the absolute value

enrichment, modified absolute value enrichment, or element-based enrichment all of

which yield equivalent final answers. When additional elements are considered, the

smoothing of the absolute value enrichment presents a challenge not found with the

modified absolute value or element-based enrichment for recovering the theoretical

displacement. Due to the improved convergence rate for the modified absolute value

enrichment this method is the most popular approach in the literature for modeling

inclusions with XFEM.

Other work on modeling inclusions in the XFEM include a unified model for the

representation of arbitrary discontinues and discontinuous derivatives [43]. The

imposition of constraints along moving or fixed interfaces was considered by Zilian

[100]. Dirichlet and Neumann boundary conditions for arbitrarily shaped interfaces were

presented for a general enrichment function. Instead of Lagrange multiplier [102, 103] or

penalty method [104] approaches, a mixed-hybrid method was introduced. Constant

42

boundary tractions, prescribed displacement differences and prescribed interfacial

displacement states were applied to a bi-material problem. Hettich [105, 106] studied

the modeling and failure of the interface between fiber and matrix in composite

materials.

Void Enrichment Function

Daux [81] was the first to represent voids with the XFEM. Sukumar [99] later

extended the void enrichment to take advantage of the use of the ( )xχ level set

function to track the void. Unlike the other enrichment functions presented here, the void

enrichment function does not require additional degrees of freedom; instead the

displacement approximation for a domain with a hole takes the form

( ) ( ) ( )h

I I

I

u x V x N x u= ∑ (3.30)

where ( )V x takes a value of 0 inside the void and 1 anywhere else. In practice,

integration is simply skipped where ( ) 0xχ < . Additionally, nodes whose support is

completely within the void are considered fixed degrees of freedom.

Integration of Element with Discontinuity

An area where the XFEM differs from the classical finite element method is on the

scheme used to perform the numerical integration described by Eq. (3.6). Challenges

arise in elements which contain a discontinuity. Standard Gauss quadrature [24]

requires that the integrands are smooth, which is not the case for an element containing

a strong or weak discontinuity. The approach introduced by Moёs [21] was to divide a

two-dimensional element into a set of triangular subdomains, where the discontinuity

was placed along the boundary of one of the subdomains. Integration would then be

43

performed over each subdomain, resulting in a series of integrations over continuous

domains. An example of an element completely cut by a crack as well as containing a

crack tip and the associated subdomains for integration are shown in Figure 3-4. The

creation of the subdomains is straightforward with the use of Delaunay tesselation for

the nodal and zero level set of ( )xψ in the parametric space.

Figure 3-4. Examples of elements containing a discontinuity and the subdomains for

integration.

In three-dimensions, it is possible to decompose elements cut by planar cracks

into a series of tetrahedrons [58] in a similar fashion to that of two-dimensional

elements. Mousavi also explored integration over arbitrary polygons [107] with an

application to XFEM [107] and through the use of the Duffy transformation [108],

showing excellent accuracy in the presence of singularities. Sukumar [109] presented a

method for the integration of an arbitrary polygon based on Schwarz-Christoffel

conformal mapping. Yamada [110] presented a hybrid numerical quadrature scheme

based on a modified Newton-Cotes quadrature scheme. Park [111] introduced a

44

mapping method for integration of discontinuous enrichments. Other methods are

detailed by Belytschko [45] and Fries [48].

XFEM Software

Due to the relatively short history of the XFEM, commercial codes which have

implemented the method are not prevalent. There are however, many attempts to

incorporate the modeling of discontinuities independent of the finite element mesh by

either a plug-in or native support [48, 112-115]. As most of these implementations are

works in progress there are various limitations on their practical use.

Abaqus

In 2009, the Abaqus 6.9 release [114] introduced basic XFEM functionality to the

Abaqus CAE environment. The Abaqus implementation of the XFEM is somewhat

different from that which was previously presented in this chapter. The implementation

is based on the phantom node method which was introduced by Hansbo [116] and

subsequently modified by Song [117] and Rabczuk [118]. The fundamental difference

between this implementation and the original XFEM is that the discontinuity is described

by superimposed elements and phantom nodes. In effect, an element is only defined in

an area where an element is continuous. Several elements are combined together such

that the total behavior of a discontinuous element is described. The method

implemented by Abaqus considers a cohesive crack model, which is only enriched with

the Heaviside function given by Eq. (3.21). Note that it has been shown repeatedly that

the use of only the Heaviside enrichment leads to poor accuracy of the resulting J-

integral calculation [115]. As a result of this enrichment scheme, all enriched elements

must be completely cut by the crack, as no crack tip field is considered.

45

Some of the limitations and challenges with modeling crack growth within Abaqus

6.9 using the XFEM follow:

• Only the STATIC analysis procedure is allowed

• Only linear continuum elements are allowed with or without reduced integration

• No parallel processing of elements is allowed

• No fatigue crack growth models are available

• No intersecting or branching cracks are allowed

• A crack may not turn more than 90° within a particular element

The Abaqus 6.9: Extended Functionality [113] update allows for energy release rate and

stress intensity factors to be evaluated for three-dimensional cracked domain. There is

currently no method available for the extraction of stress intensity factors in two-

dimensions. In practice, the modeling of crack growth within Abaqus with the current

implementation of the XFEM is challenging. As the formulation is based on the cohesive

model the same challenges exist of solving the system of equations as with cohesive

elements and methods such as viscous regularization must be applied to solve the

system of equations. Care must be taken to choose the regularization parameters in a

way that has a minimal impact on the resulting solution.

MXFEM

Basic functionality of the XFEM was implemented by Pais [119] in MATLAB for

two-dimensional plane stress and plane strain. A domain may be defined with a

structured grid of linear square quadrilateral elements with arbitrary loading and

boundary conditions. Enrichments provided include the homogeneous [21] and bi-

material [66] cracks, inclusions [97] and voids [81, 99]. All discontinuities are tracked

using the level set method detailed in Chapter 3. The ( )φ x and ( )ψ x level set functions

46

track the crack, the ( )χ x level set function tracks the voids, and the ( )ζ x level set

function tracks the inclusions. Integration of enriched elements is done through

subdivision of elements into triangular regions [21, 55].

A variety of plotting outputs may be requested including: level set functions, finite

element mesh, deformed finite element mesh, elemental and contours of stress, and

stress-intensity factor history for growing cracks. A graphical user interface (GUI) is

available which offers simplified functionality compared to the direct modification of the

input file. The GUI writes an input file based on the values of the GUI and then solves

the problem. An example of the GUI is given in Figure 3-5. Examples of some of these

plots are given for the geometry shown in Figure 3-6, Figure 3-7, and Figure 3-8, which

contains a circular inclusion below the crack and a void above the crack.

Figure 3-5. MXFEM GUI for automated input file creation.

47

A B

Figure 3-6. Example problem to show plots generated by MXFEM. A) The geometry being considered, B) Example of the mesh output from MXFEM where blue circles and squares denote the Heaviside and crack tip enriched nodes and the black circles denote the inclusion enriched nodes.

-1 0 1 2A

-4 -2 0 2 B

0 2 4C

0 2 4 D

Figure 3-7. Example of the level set functions output from MXFEM. A) ( )φ x , B) ( )ψ x ,

C) ( )χ x , D) ( )ζ x .

48

0 2 4A

-2 0 2 B

0 5 10C

σvm

0 5 10 D

Figure 3-8. Example of the stress contours output from MXFEM. A) XX

σ , B) XY

σ , C)

YYσ , D)

VMσ .

The domain form of the contour integrals [18, 19] is used to calculate the mixed-

mode stress intensity factors. For the bi-material crack case the algorithm presented by

[66] is used to identify the stress intensity factors. The mixed-mode stress intensity

factors are used in the maximum circumferential stress criterion [18] to give the direction

of crack extension. Crack growth may be modeled using either a constant increment of

growth or a fatigue crack growth law, such as Paris Law [120]. All crack growth

problems with constant a∆ or N∆ are solved using the reanalysis algorithm presented

in Chapter 5.

Additional functionality includes an optimization algorithm for finding some

optimum crack location and the ability to define a variable load history. Benchmark

problems for the various enrichment functions are provided in Appendix A including:

center crack in an infinite plate, center crack in a finite plate, center bi-material crack in

an infinite plate, edge crack in a finite plate, hard inclusion in a finite plate, void in a

infinite plate, crack growth in the presence of an inclusion, fatigue crack growth, and

49

optimization to identify the initial crack location with maximum energy release rate for a

plate with a hole.

Others

Bordas [51] implemented the XFEM as a object-oriented library in C++. Cenaero

[121] has implemented XFEM functionality into the Morfeo finite element environment.

Giner [115] implemented the XFEM in ABAQUS with the traditional crack tip enrichment

scheme for modeling two-dimensional growth through the use of user defined elements.

Global Engineering and Materials, Inc. [112, 122] is developing a XFEM-based failure

prediction tool as an ABAQUS plug-in which includes a small time scale fatigue model

[123] for variable amplitude loading. Sukumar [55, 56] implemented the XFEM in

Fortran, specifically the finite element program Dynaflow. Wyart [124] discussed the

implications of the implementation of the XFEM in general commercial codes. Some

amount of functionality is also available in getfem++ [125] and openxfem++ [126].

Summary

The XFEM allows for strong and weak discontinuities to be represented

independent of the finite element mesh by incorporating discontinuous functions into the

displacement approximation through the partition of unity finite element method.

Additional nodal degrees of freedom are introduced at the nodes of elements cut by a

discontinuity. An assortment of enrichment functions are available for a variety of crack

tip conditions and the Heaviside step function is used to model the crack away from the

tip. Inclusions are modeled using the modified absolute value enrichment, which shows

convergence equivalent to that of the classical finite element method. Voids can be

incorporated through the use of a step function and without additional nodal degrees of

freedom. Integration of elements containing enriched degrees of freedom, and therefore

50

a discontinuity require special integration treatment through the use of either a

subdivision or equivalent algorithm.

As the discontinuities do not correspond to the finite element mesh, some other

method must be used to track the discontinuities. The level set method for open

sections is used to track any cracks in the domain. The level set method for closed

sections is used to track inclusions and voids. The level set values are used as part of

the definition of the enrichment functions, leading to a symbiotic relationship between

the extended finite element and level set methods.

A version of the XFEM for the modeling of cohesive cracks has been introduced in

Abaqus 6.9, but it is not without its limitations. An implementation within MATLAB had

been completed here. The XFEM is also available through the use of plug-ins or some

open source finite element codes.

51

CHAPTER 4 CRACK GROWTH MODEL

Introduction

There are many models which attempt to predict the magnitude and direction of

crack growth caused by some arbitrary loading. The stress intensity factors, which give

the crack tip stress conditions, are accepted as the driving force behind crack growth.

The major focus here is on modeling fatigue driven crack growth. In general two cases

have been considered in the literature: where the increment of growth is fixed and the

number of elapsed cycles is back-calculated a posteriori, and where the number of

elapsed cycles is constant and the magnitude of growth is calculated at a given

iteration. The direction of crack growth is given in terms of the stress intensity factors. If

a constant number of elapsed cycles is used the magnitude of growth at a given

iteration is also given in terms of the stress intensity factors. For a fixed crack growth

increment, the stress intensity factors can be used to back-calculate the number of

elapsed cycles from one growth iteration to another. Thus, the critical component for the

modeling of fatigue crack growth is the evaluation of stress intensity factors such that

the direction and magnitude of growth are accurate.

In order for the crack growth prediction to have meaning, there are several

considerations about the use of the XFEM before the crack path is considered. First and

foremost, the finite element mesh must be sufficiently refined such that the stress

intensity factors may be calculated accurately, as they are the critical parameters which

predict the growth direction and magnitude. Also, care needs to be taken when

selecting the given increment of growth. As shown by Edke [127], there is a definite

relationship between the mesh size and crack growth path. While the presented

52

discussion was about removing oscillations by increasing mesh density, the underlying

cause was most likely incorrect stress intensity factor evaluation, leading to a non-

smooth crack growth curve.

This chapter is divided into the following sections. First, the extraction of mixed-

mode stress intensity factors is discussed. Then, methods for determining the direction

of crack growth in terms of the mixed-mode stress intensity factors are presented.

Finally, a discussion about the methods available for predicting the amount of crack

growth at a given iteration are compared and contrasted.

Stress Intensity Factor Evaluation

The most common way to extract the mixed mode stress intensity factors is

through the use of the domain form of the interaction integrals [18, 19, 128] in two [21,

55, 81] or three-dimensions [41, 57, 58]. The domain form of the interaction integrals is

an extension of the J-integral originally introduced by Cherepanov [129] and Rice [17].

In the domain form, the line integral specified by the J-integral is converted to an area

integral which is much more amenable to use with finite element simulations. As the J-

integral is used to calculate the energy release rate for a given crack, the interaction

integrals are used to extract the mixed-mode stress intensity factors. This method has

been shown to have excellent accuracy when applied to suitable meshes for many

crack conditions including homogenous [21], bi-material [66], and branching [81] cracks.

Other methods [45, 46, 48] have also been explored.

For a general mixed-mode homogeneous crack in two-dimensions the energy

release rate G can be expressed in terms of the relationship between the J-integral,

stress intensity factors and effective Young’s modulus eff

E as

53

2 2

I II

eff eff

K KJ

E E= + (4.1)

where eff

E is defined by the state of stress as

2

plane stress

plane strain1

eff

E

E E

ν

= −

(4.2)

where E is Young’s modulus and ν is Poisson’s ratio. The J-integral takes the form

dki jk j

i

uJ Wn n

xσ

Γ

∂= − Γ

∂ ∫ (4.3)

where W is the strain energy density. Equation (4.3) can be rewritten in the equivalent

form using the Dirac delta which is easier to implement in finite element code as

1

1

dij ij j

uJ W n

xδ σ

Γ

∂= − Γ

∂ ∫ . (4.4)

In order to calculate the mixed-mode stress intensity factors, two displacement

and stress states are superimposed onto one another. Auxiliary stress and

displacement states are superimposed onto the stress and displacement solution from

XFEM. The auxiliary stress and displacement states at the crack tip introduced by

Westergaard [130] and Williams [131] for a homogenous crack and by Sukumar [66] for

a bi-material crack are used in the calculation of the mixed-mode stress intensity factors

and are given in Appendix B. Hereafter, the XFEM states are given as ( )1

iju , ( )2

ijε , and

( )2

ijσ while the auxiliary states are given as ( )2

iju , ( )2

ijε , and ( )2

ijσ . The superposition of

stress states into Eq. (4.4) leads to

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

1 2 1 2 1 2 1 2

1

1

1d

2

i i

ij ij ij ij j ij ij j

u uJ n

xσ σ ε ε δ σ σ+

Γ

∂ + = + + − + Γ

∂

∫ . (4.5)

54

The expansion of the terms of Eq. (4.5) allows for the J-integral to be separated to the

auxiliary state ( )2J , XFEM state ( )1

J , and interaction state ( )1,2I given by

( ) ( ) ( )( )

( )( )2 1

1,2 1,2 1 2

1,

1 1

di ij ij ij j

u uI W n

x xδ σ σ

Γ

∂ ∂= − − Γ

∂ ∂ ∫ (4.6)

where ( )1,2W is the interaction strain energy density given as

( ) ( ) ( ) ( ) ( )1,2 1 2 2 1

ij ij ij ijW σ ε σ ε= = . (4.7)

The two superimposed stress states can be expressed using Eq. (4.1) after

rearrangement as

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

1 2 1 22

I I II II

eff

K K K KJ J J

E

++

= + + . (4.8)

Therefore, from Eqs. (4.6) and (4.8), the interaction state is given as

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

1,22

I I II II

eff

K K K KI

E

+= . (4.9)

Therefore the stress intensity factors for the XFEM state ( )1

IK and ( )1

IIK are given by

selecting ( )21

IK = and ( )2

0II

K = , followed by ( )20

IK = and ( )2

1II

K = such that ( )1

IK and ( )1

IIK

are given by:

( )( )1,Mode

1

2

I

eff

I

I EK = (4.10)

where ( )1,Mode II is the interaction integral for ( )2

1I

K = and ( )20

IIK = and

( )( )1,Mode

1

2

II

eff

II

I EK = (4.11)

where ( )1,Mode II is the interaction integral for ( )2

0I

K = and ( )21

IIK = .

55

The calculation of stress intensity factors for a bi-material crack follows a similar

algorithm as that for a homogenous crack. The more complicated state of stress caused

by the crack being located at the interface between two materials leads to a different

energy release rate relationship than the homogenous case given in Eq. (4.1). For the

bi-material case the relationship between the energy release rate and mixed-mode

stress intensity factors is given as

( ) ( )

2 2

* 2 * 2cosh cosh

I II

eff eff

K KG

E Eπε πε= + (4.12)

where *

effE is given by

1, 2,*

1, 2,

2eff eff

eff

eff eff

E EE

E E=

+ (4.13)

where ,i effE is given by Eq. (4.2) and the subscripts 1 and 2 denote the materials in the

upper and lower half-planes. Following the steps for the homogenous crack case, the

mixed-mode stress intensity factors may be extracted as

( )( ) ( )1,Mode * 2

1 cosh

2

I

eff

I

I EK

πε= (4.14)

and

( )( ) ( )1,Mode * 2

1 cosh

2

II

eff

II

I EK

πε= . (4.15)

For either the homogenous or bi-material case the interaction integral in Eq. (4.6)

is converted from a line integral into an area integral and a smoothing function s

q which

takes a value of 1 on the interior of the line integral defined by Eq. (4.6) and a value of 0

outside of the integral. Elements for the integration are selected by choosing a radius

from the crack tip. When this radius becomes sufficiently large the integral becomes

56

path-independent; i.e., any path larger than the path-independent radius ind

r will yield

equivalent solutions to Eq. (4.6). In practice a radius of three elements about the crack

tip is typically sufficient for path-independence. The divergence theorem is used to

create the equivalent area integral

( ) ( )( )

( )( )

( )2 1

1,2 1 2 1,2

1

1 1

di i sij ij j

Aj

u u qI W A

x x xσ σ δ ∂ ∂ ∂

= − − ∂ ∂ ∂

∫ (4.16)

which is simpler to implement in the finite element environment.

Other methods have also been used to extract the mixed-mode stress intensity

factors using XFEM. Duarte [132] used a least squares fit of the localized stress state

around the crack tip to extraction the stress intensity factors. Karihaloo [46] included

higher-order terms of the asymptotic crack tip expansion in two-dimension which

allowed for the stress intensity factors to be obtained directly without the use of

interaction integrals. No equivalent extension has been performed in three-dimensions

[45]. Lua [112, 123] introduced nodes along the crack fronts which were used with the

crack-tip opening displacement [16] to calculate the mixed-mode stress intensity factors.

This method is only valid for linear elastic cracks. Sukumar [42] attempted to evaluate

the stress intensity factors directly from the enriched degrees of freedom corresponding

to the crack tip enrichment function for pure mode I problem, but the accuracy of the

stress intensity factors was found to be insufficient. A similar method was introduced by

Liu [133] with much better results for homogeneous and biomaterial cracks.

Crack Growth Direction

The direction of crack propagation is accepted to a function of the mixed-mode

stress intensity factors present at a crack tip. While there are several criteria available

57

for both two [20, 134-137] and three-dimensions [137], in general they only differ in the

angle of the initial kink, but then converge to similar crack paths [55]. In two-dimensions,

these methods will give the angle of crack extension, which in general is the direction

which will minimize II

K .

In two-dimensions the main criteria for the crack growth direction are the maximum

circumferential stress [135], maximum energy release rate [136] and the maximum

strain energy density criterion [20]. Other criterion available in the literature are the

criterion of energy release rates and the generalized fracture criterion [137]. The

criterion which is most amenable to modeling crack growth with finite elements is the

maximum circumferential stress, as the growth direction c

θ is given in a closed form

solution in terms of the mixed-mode stress intensity factors. The maximum

circumferential stress criterion is given as the angle c

θ given by

2 2 2

2 2

2 8arccos

9

II I I II

c

I II

K K K K

K Kθ

+ + = − +

. (4.17)

Alternative, but equivalent formulas are given by Moёs [21] as

( )2

12arctan 8

4

I Ic II

II II

K Ksign K

K Kθ

= − +

(4.18)

and Sukumar [55]

( )

( )2

22arctan

1 1 8

II I

c

II I

K K

K Kθ

− = + +

. (4.19)

58

Crack Growth Magnitude

At each cycle of fatigue loadings, the amount of crack growth is in the order of

nano-meters, which is impractical to simulate. In practice, the governing differential

equation in Eq. (1.1) is solved at discrete points, which will be referred to as simulation

iteration in this work. There are two main approaches presented in the literature for the

amount of crack growth at a given simulation iteration. The first method assumes that a

known and finite amount of growth will occur at a given iteration. The second method

assumes that some governing law, such as a fatigue crack growth law can be used to

find the corresponding increment of growth at a particular iteration.

Finite Crack Growth Increment

In the literature, the selection of a constant crack growth increment is very popular

[23, 39, 51, 53, 56, 78, 82, 115, 138]. At a given cycle the amount of crack growth is

generally very small, approximately on the order of 10E-8 [127]. Therefore, it is more

computationally attractive to choose a small increment of growth to represent many

cycles instead of modeling them independently. The choice of a∆ is almost always

made a priori. It has been shown in the literature by Moёs [21] and Pais [23] that the

path of crack growth is related to the assumed increment

Challenges with this method include selecting a a∆ such that the crack path

converges to the appropriate path. With a single analysis it is unclear how one can

know whether or not the predicted crack path has converged. It is also unclear how to

interpolate between the data points since the a N− crack growth curve is nonlinear.

For applications where a variable load history may be considered, the use of a

finite increment of crack growth increment may no longer be accurate. The interaction

between the different loading magnitudes such as overloads, underloads, and their

59

interaction between one another would not be captured through the use of a finite

growth increment.

Fatigue Crack Growth

The slow progressive failure of a structure caused by crack growth under cyclic

loading is called fatigue. This source of failure is one of the most common affecting

engineering structures. There are many fatigue crack growth models of varying

complexity [14]. Most fatigue crack growth laws are of the form

( ),da

f K RdN

= ∆ (4.20)

where da/dN is the crack growth speed, K∆ is the stress intensity factor range, and R is

the stress ratio. Many different laws attempt to consider the effects of phenomenon [14]

such as crack growth instability when the stress intensity factor approaches it’s critical

value, threshold stress intensity factor, changing crack tip geometry, and elastic crack

tip conditions. For a variable load history, the interactions between overloads and

underloads are a crucial component of any fatigue crack growth model. An overload is a

load which is substantially greater than the mean, while an underload is a load which is

substantially less than the mean. Overloads increase crack tip plasticity which leads to

less crack growth in resulting loading cycles. An underload has the opposite effect and

leads to increased growth. The interaction of a overload and underload in subsequent

cycles is not well understood.

The first fatigue model was presented by Paris [120]. The relationship between the

crack growth rate and the stress intensity factor range is given as

mdaC K

dN= ∆ (4.21)

60

where C is the Paris model constant and m is the exponent; both of them are material

properties which can be derived from fatigue testing data. Limitations of the Paris model

include that it was originally designed for use only for Mode I loading. The stress ratio is

also neglected, as is any sort of crack closure model. Despite these limitations, the

Paris model is still commonly used in academia and industry.

If a fixed N∆ is to be used for numerical integration, then the corresponding

increment of growth a∆ can be approximated using the forward Euler method as

ma NC K∆ = ∆ ∆ . (4.22)

Care must be taken such that N∆ is not too large, otherwise the forward Euler method

will have insufficient accuracy.

In order to apply the Paris model to mixed-mode loading, several relationships

have been proposed for the calculation of a single effective stress intensity factor range

which can be used in Eq. (4.21). Tanaka proposed the relationship

4 44 8I IIK K K∆ = + (4.23)

based on curve fitting observed experimental data. Yan proposed a different correction

from the maximum circumferential stress criterion as

( )1

cos 1 cos 3 sin2 2

I IIK K Kθ

θ θ

∆ = + −

(4.24)

where θ is given by Eqs. (4.17)-(4.19). Finally, a relationship based on energy release

rate is given as

2 2

I IIK K K∆ = + . (4.25)

With regards to the XFEM, crack growth models have been limited primarily to the

Paris model. Belytschko [43] used it to drive the crack growth for a plate with holes, but

61

no specifics about how the Paris model was used is given. Gravouil [40] coupled the

level set method to the Paris model, but no clear definition of the number of elapsed

cycles was given. Instead the growth was governed by assuming N to be equivalent to

the traditional time for the level set method. Sukumar [42] calculated the increment of

crack growth a∆ as

max

max

m

I

I

a K

a K

∆=

∆ (4.26)

where maxa∆ was taken as 00.05a for initial crack length 0a for a value of max

IK

corresponding to the maximum value of IK along the crack front. Note that here a

value for maxa∆ acts very similarly to that of a fixed a∆ such that a small value is

required to ensure convergence.

A fixed crack growth increment a∆ can be used to model fatigue crack growth and

then the corresponding number of elapsed cycles can be back-calculated through the

use of the forward Euler approximation for the Paris model as

m

aN

C K

∆∆ =

∆. (4.27)

If the forward Euler method is going to be used as in Eq. (4.27) then care must be taken

such that a∆ is not too large, or accuracy will be lost.

An example is provided here where an XFEM analysis was performed on a plate

with a hole as shown in Figure 4-1. The chosen plate dimensions were a width of 4 m

and a height of 8 m with a hole with radius 1 m centered at (2,2) m. The edge crack has

an initial length of 0.5 m. The material properties for the plate were chosen as Young’s

modulus of 10 MPa, Poisson’s ratio of 0.33, Paris model constant and exponent of

62

1.5E-10 and 3.8 respectively. The applied stress was 14.5 MPa. The purpose of this

study was to attempt to assess the effect that the mesh density and crack growth

increments a∆ and N∆ on the predicted crack path. The results are shown in Figure

4-2.

A B

Figure 4-1. Plate with a hole subjected to tension. A) Geometry, B) Mesh (h = 1/10)

A B C

Figure 4-2. Convergence of crack path. A) Mesh density, B) a∆ , C) N∆ .

σ

σ

a

63

From Figure 4-2 it is clear that the choice of mesh size is very important as

expected. Without a sufficiently refined mesh, the accuracy of the stress intensity

factors will be poor leading to an inaccurate prediction of the crack growth direction. The

crack path is relatively insensitive to the choice of a∆ in this choice, but there is

deviation between the steps if a close up image of the crack paths near the hole is

examined as in Figure 4-3. The most sensitive case is that of N∆ where the cases of

N∆ = 10, 25, and 50 cycles actually miss the hole as predicted by N∆ = 1.

Figure 4-3. Close up view of the crack paths for a∆ around the hole.

Summary

Crack growth is modeled using mixed-mode stress intensity factors which are

extracted with the use of the domain form of the contour integrals. The mixed-mode

stress intensity factors are used in the maximum circumferential stress criterion in order

to find the direction of future crack growth. If a fixed increment of growth is considered,

then the stress intensity factors are used to back-calculate the number of elapsed

cycles. For a fixed number of elapsed cycles, the stress intensity factors are used to

determine the magnitude of crack growth at a given iteration. If the Paris model is to be

64

used, then an effective K∆ must be calculated to convert the mixed-mode stress

intensity factors into a single effective stress intensity factor range.

65

CHAPTER 5 KRIGING FOR INCREASED FATIGUE CRACK GROWTH STEP SIZE

Introduction

Fatigue crack growth models are accepted to be a function of the stress intensity

factor range K∆ [14]. In general there is no analytical equation for K∆ as it is a function

of the boundary conditions, geometry and loading. This creates a challenge for

modeling fatigue crack growth because there is a limited amount of information

available to approximate the solution to the nonlinear crack growth governed by the

chosen fatigue crack growth model. Therefore the common practice is to use the

forward Euler approximation to the fatigue crack growth model, which only requires K∆

at the current iteration. This limits the step size of crack growth increment a∆ [21] or

elapsed cycle increment N∆ [23] for the forward Euler approximation [24] to be

accurate. It is not possible to use a higher order approximation to the fatigue crack

growth law such as the midpoint or 4th order Runge-Kutta approximation [24] without

additional simulations at the locations needed for the additional slope evaluations of

these higher order approximations to the governing differential equation.

The objective of this chapter is to use a surrogate model to overcome the

abovementioned difficulty in using higher-order integration methods without additional

simulations to calculate K∆ . Surrogate models are commonly used to reduce the cost

associated with the evaluation of expensive functions. Types of surrogate modes

include polynomial response surface [139], kriging [140-142], radial basis neural

networks [143], and support vector regression [144]. Kriging is used here instead of

other surrogate models for several reasons. Kriging honors the observed data points

66

and interpolates between them. Further, kriging provides a measure of the uncertainty

at any point where it is defined in the terms of prediction variance.

In this chapter, two cases are considered for utilizing the kriging surrogate model.

First, when a∆ is fixed, the relationship between K∆ and a∆ is fitted using a kriging

surrogate. This surrogate is then used to approximate the corresponding N∆ for a given

a∆ such that the cyclic crack growth history can be obtained. As all function evaluations

will be interpolating the kriging surrogate, results are very accurate even for very large

increments of a∆ .Second, when N∆ is fixed, the relationship between K∆ and N∆ is

fitted using a kriging surrogate. This surrogate is then used to extrapolate forward in

time so that a higher order approximation to the chosen fatigue crack growth law can be

used to find the corresponding a∆ . In the future, the prediction variance will be used to

control the size of a∆ or N∆ to limit the allowable error in calculation.

Kriging

Kriging [140-142] can be used to approximate a function of interest ( )y x . As this

function is expensive to evaluate, it may be approximated by a cheaper model ( )y x

based on assumptions on the nature of ( )y x and on the observed values of ( )y x at a

set of p data points called experimental design. More explicitly,

( ) ( ) ( )ˆ ,y y ε= +x x x (5.1)

where 1[ , , ]T

dx x=x … is a real d -dimensional vector and ( )ε x represents both the error

of approximation and random errors.

Kriging estimates the value of the unknown function ( )y x as a combination of

basis functions ( )if x such as a polynomial basis and departures by

67

( ) ( ) ( )1

ˆ ,m

i i

i

y f zβ=

= +∑x x x (5.2)

where ( )z x satisfies ( ) ( ) ( )1

m

k k i i k

i

z y fβ=

= −∑x x x for all sample points ( )kx and is

assumed to be a realization of a stochastic process ( )Z x with mean zero,

( ) ( )( ) ( )2cov , , ,i j i j

Z Z Rσ=x x x x (5.3)

and process variance 2σ , and spatial covariance function given by

( ) ( )2 11 ,

T

pσ −= − −y Xb R y Xb (5.4)

where ( ),i j

R x x is the correlation between ( )iZ x and ( )jZ x , y is the value of the

actual responses at the sampled points, X is the Gramian design matrix constructed

using the basis functions at the sampled points, R is the matrix of correlations ( ),i j

R x x

among sample points, and b is an approximation of the vector of coefficients i

β of Eq.

(5.2). Figure 5-1 shows the prediction and the error estimates of kriging. It can be

noticed that since the kriging model is an interpolator, the error vanishes at sampled

data points.

68

Figure 5-1. Kriging model with an arbitrary set of five points and the uncertainty in gray.

Kriging for Integration of Fatigue Crack Growth Law

One of the first attempts to create a model to represent fatigue crack growth was

that of Paris [120]. The Paris model is well known and takes the form

( )mda

C KdN

= ∆ (5.5)

where da/dN is the crack growth rate, C is the Paris model constant, m is the exponent

and K∆ is the stress intensity factor range.

The simplest approach to integrating the Paris model for fatigue crack growth is

the use of the forward Euler method. Here the stress intensity factor range at the current

iteration is the only information needed to find the increment of growth between the

current and future iterations. For fixed number of cycles, the magnitude of crack growth

at a given increment may be calculated as

( )m

i ia N C K ∆ = ∆ ∆

(5.6)

where i is the current increment. The corresponding number of elapsed cycles can be

approximated as

( )

i m

i

aN

C K

∆∆ =

∆ (5.7)

for a fixed increment of crack growth. As the forward Euler method only uses the slope

at the current point and linearly interpolates to the next crack size it can lead to large

inaccuracies even if crack growth increments are relatively small.

The next approach to approximating the solution of the Paris model is the use of

the midpoint method where the growth increment may be calculated as

69

( )1/2

m

i ia N C K + ∆ = ∆ ∆

(5.8)

where i +1/2 is the midpoint between the current and next increment. Similarly, i

N∆ is

given as

( )1/2

i m

i

aN

C K +

∆∆ =

∆. (5.9)

Here the slope at the midpoint of the current cycle is use to extrapolate ahead to i +1.

This leads to a second-order accuracy in estimating the crack size at the next increment

as a more accurate approximation of the slope over the entire interval of the chosen

growth increment is used. This method, however, requires the evaluation of K∆ at

i +1/2 for each iteration being considered, effectively doubling the number of function

evaluations needed for the given simulation. For some situations a function evaluation

at i +1/2 may not have physical meaning such as non-integer values of N . The

implications of evaluating K∆ while a cycle is in progress are unclear as crack growth

does not occur at all instances during a cycle of loading.

Another approach is to use the 4th order Runge-Kutta method where the growth

increment is given as

( ) ( ) ( )1/2 146

m m m

i i i

Na C K K K+ +

∆ ∆ = ∆ + ∆ + ∆

(5.10)

where 1i + is the next increment. Similarly, N∆ can be back-calculated as

( ) ( ) ( )1/2 146

m m m

i i i

aN C K K K+ +

∆ ∆ = ∆ + ∆ + ∆

. (5.11)

Here the slope at four points over the interval are combined using weight functions so

that a more accurate representation of the average slope over a given interval may be

considered. The 4th order Runge-Kutta requires three function evaluations over a

70

particular interval and as with the midpoint method may require evaluations of K∆ at

non-integer values of N .

The solution of Eq. (5.5) with respect to modeling crack growth with finite element

simulations can take two main approaches. First, a fixed crack growth increment a∆

can be considered. Once the simulations have completed, the history between a∆ and

K∆ can be used to find the corresponding N∆ for each iteration. Second, a fixed

increment of elapsed cycles N∆ can be considered. Here, at each iteration a∆ is

approximated based on the selected N∆ and the iteration dependent K∆ . Either

method is capable of yielding an accurate representation of the fatigue crack growth

behavior for a given structure should an appropriate step size of a∆ or N∆ be chosen,

which becomes extremely small for the forward Euler method.

Here the kriging surrogate model is used to approximate the response of K∆ as a

function of either the crack length a or the number of elapsed cycles N . The kriging

approximation is then used for a higher-order integration to the Paris model, such as the

midpoint method given by Eqs. (5.8)-(5.9) or 4th-order Runge-Kutta method given Eq.

(5.10)-(5.11). For the case of a fixed a∆ , N∆ is calculated a posteriori to the

simulations using interpolation between data points. Equally spaced data points

between i

a and the final crack length are fitted using kriging. This surrogate is then

used to interpolate between data points and evaluate the corresponding N∆ for each

a∆ . For a fixed N∆ , kriging is used to fit the data up to the current cycle and

extrapolate approximate values of K∆ for the use of the midpoint and 4th order Runge-

Kutta methods. Three initial data points are provided before the use of kriging begins.

71

To get the second and third data points, the forward Euler method can be used as its

accuracy is good away from the critical crack length.

Example Problems

First two numerical results based purely upon accepted theoretical values are

used. This allow for an estimate of the amount of error that can be expected by the use

of kriging to provide data points instead of performing a simulation. The effect on the

choice of a∆ or N∆ is considered for each model. Results are presented for each case

as the final approximate value normalized by the exact final value of crack size or cycle

number. A value of one denotes perfect agreement with the exact solution. Finally two

results are provided for mixed-mode crack growth modeled using the XFEM where the

increment of growth also influences the convergence of the crack path.

The chosen geometry is a cylindrical pressure vessel with radius of 3.25 m, width

of 0.2 m, and thickness of 0.0025 m. The panel is subjected to a pressure of 0.06 MPa

and contains an initial half crack of length 0.01 m. The material is assumed to be AL

7071 which has Young’s modulus E of 71.7 GPa, Poisson’s ratio ν of 0.33, a critical

stress intensity factor of about 30 MPa m , Paris model constant C of 1.5E-10 and

exponent m of 3.8. Data is simulated to 1200 cycles, at which point eq

K∆ has exceeded

ICK . The results are presented for each case as a function of the ratio between the

exact final and predicted final crack size or cycle number such that a value of one

denotes the exact solution.

Center Crack in an Infinite Plate Under Tension

For the case of a center crack in an infinite plate under uniaxial tension the Mode I

stress intensity factor [16] is

72

I

K aσ π= (5.12)

where σ is the nominal tensile stress, and a is the half crack length. By substituting Eq.

(5.12)into Eq. (5.5) for Paris Law, rearranging terms, and performing the following

integration

( ) ( )

2 2

2 2

2

2

N

i

m m

aN i

m ma

da a aN

mC a Cσ π σ π

− − −

= = −

∫ (5.13)

where i

a is the initial crack size. Solving for N

a yields an expression for crack size in

terms of N such that

( )2

2

12

mm

N i

ma a NC aσ π

− = + −

. (5.14)

This allows for the midpoint and 4th order Runge-Kutta methods to be applied without

error as N

a and I

K can be directly evaluated and used in the Paris model.

For a fixed a∆ , the results are given in Table 5-1. Recall that the common a∆

used in the literature [21] is ai/10. For that crack growth increment, the Euler

approximation has over 5 percent error. Through the use of the midpoint or Runge-Kutta

method that error is reduced to less than 1 percent. Larger crack growth increments for

the Euler method lead to very large errors, which can be drastically reduced though the

use of a higher-order method. Note also that the use of kriging to fit data and provide

estimates for the necessary function evaluations for the higher-order method results is

no loss of accuracy.

73

Table 5-1. Accuracy of integration for chosen a∆ for a center crack in an infinite plate.

Napprox/Nexact a∆ Euler Midpoint Runge-Kutta KRG + MP KRG + RK

ai/160 1.0035 1.0000 1.0000 1.0000 1.0000 ai/80 1.0070 1.0000 1.0000 1.0000 1.0000 ai/40 1.0141 0.9999 1.0000 0.9999 1.0000 ai/20 1.0285 0.9998 1.0000 0.9998 1.0000 ai/10 1.0579 0.9991 1.0000 0.9991 1.0000 ai/5 1.1194 0.9964 1.0000 0.9964 1.0000 ai/2 1.3244 0.9786 1.0005 0.9787 1.0005 ai 1.7216 0.9271 1.0052 0.9367 1.0052

For a fixed N∆ , the results are given in Table 5-2. The first observation is that the

accuracy of Euler approximations is much more sensitive to changes in fixed N∆ when

compared to a fixed a∆ . However, the higher-order approximations are largely

insensitive to the chosen crack growth increment. As before, the kriging assisted

midpoint approximation is very accurate and comparable to using the exact formula.

The kriging assisted Runge-Kutta formulation tends to infinity with the default kriging

settings. It was found that extrapolating for 1iK +∆ leads to very poor predictions of the

crack growth path where the final crack path approaches infinity.

Table 5-2. Accuracy of integration for chosen N∆ for a center crack in an infinite plate.

aapprox/aexact N∆ Euler Midpoint Runge-Kutta KRG + MP KRG + RK

1 0.9980 1.0013 1.0013 1.0000 ∞ 5 0.9903 1.0013 1.0013 1.0000 ∞ 10 0.9809 1.0013 1.0013 1.0000 ∞ 25 0.9545 1.0012 1.0013 0.9990 ∞ 50 0.9155 1.0009 1.0013 0.9998 ∞

100 0.8519 0.9999 1.0013 0.9906 ∞

74

Edge Crack in a Finite Plate Under Tension

For the case of an edge crack in a finite plate under uniaxial tension the Mode I

stress intensity factor [15] is

2 3 4

1.12 0.281 10.55 21.72 30.39I

a a a aK a

W W W Wσ π

= − + − +

(5.15)

where a is the crack length and W is the plate width. As there is no analytical

expression for N

a as there is for the case of a center crack in an infinite plate the crack

length after 1200 cycles was approximated using forward Euler with 105 steps, which

leads to a difference of less than 0.01 percent from forward Euler with 104 steps.

For a fixed a∆ , the results are given in Table 5-3. Here the corresponding Euler

approximation is off by 7 percent, while the midpoint and Runge-Kutta approximations

have less than 1 percent error.

Table 5-3. Accuracy of integration for chosen a∆ for an edge crack in a finite plate.

Napprox/Nexact

∆a Euler Midpoint Runge-Kutta KRG + MP KRG + RK

ai/160 1.0045 1.0000 1.0000 1.0000 1.0000 ai/80 1.0091 1.0000 1.0000 1.0000 1.0000 ai/40 1.0183 0.9999 1.0000 0.9999 1.0000 ai/20 1.0370 0.9997 1.0000 0.9997 1.0000 ai/10 1.0754 0.9987 1.0000 0.9987 1.0000 ai/5 1.1558 0.9949 1.0000 0.9949 1.0000 ai/2 1.4255 0.9703 1.0006 0.9743 1.0033 ai 1.9515 0.8994 1.0073 1.0088 1.0801

For a fixed elapsed number of cycles in each simulation, the exact results for

midpoint and 4th order Runge-Kutta are not available as there is no explicit value for

Na . As with the case of a center crack in an infinite plate, the kriging 4th order Runge-

75

Kutta results are very poor and are not shown. From Table 5-4 it is apparent that once

again, the kriging assisted midpoint method allows for larger step sized compared to the

forward Euler method. In this case, for a step size of 100, the errors for the Euler and

kriging assisted midpoint methods are 17 and 3 percent. The similar trend of the kriging

assisted midpoint method being less accurate for larger N∆ continues, and that

approximation is still better than Euler alone. The increased errors in the approximation

compared to the case of a center crack in an infinite plate can most likely be explained

by the increased nonlinearity caused by the edge and finite effects present in this

geometry.

Table 5-4. Accuracy of integration for chosen N∆ for an edge crack in a finite plate.

aapprox/aexact N∆ Euler Midpoint Runge-Kutta KRG + MP KRG + RK 1 0.9971 N/A N/A 1.0001 ∞ 5 0.9858 N/A N/A 0.9995 ∞ 10 0.9756 N/A N/A 0.9990 ∞ 25 0.9381 N/A N/A 0.9976 ∞ 50 0.8922 N/A N/A 0.9924 ∞

100 0.8264 N/A N/A 0.9721 ∞

Summary and Future Work

Kriging has been used to enable the use of more accurate numerical integration

schemes in the integration of the partial differential equations governing fatigue crack

growth. Through two numerical examples, a center crack in an infinite plate and an

edge crack in a finite plate, it was shown that the use of kriging and the midpoint

method lead to large increases in the allowable step size without loss of accuracy.

Further work need to be completed to study the underlying cause of the errors for

the kriging assisted Runge-Kutta errors. With some calibration of the kriging parameters

76

a better solution to the fatigue crack growth model may be achieved. Based on the

numerical theory the approximation based on the Runge-Kutta algorithm should be

superior to the midpoint rule and it is possible that the kriging model can be adjusted to

increase this accuracy. It may also end up being the case that the interpolation needed

to the next simulation point may be too far for the kriging model to produce an accurate

approximation.

Kriging also provides an estimate of the uncertainty between data points. The use

of the uncertainty in kriging to allow for a variable step size based on a statistical model

will also be explored. At the beginning of fatigue crack growth, larger steps may be

taken as the curve is largely flat, but as the critical crack length is approached smaller

steps should be taken to account for the nonlinear crack growth behavior. This

approach should allow for a better allocation of resources for the numerical simulation of

fatigue crack growth.

Another area of study is to consider cases of mixed-mode crack growth. In this

case the path of crack growth will be curved. Thus, the allowable step size based on the

approximation of the solution of the fatigue crack growth law may be limited by the need

to have a converged crack path. This is an important interaction which will need to be

considered for practical use of the kriging assisted methods for use in a finite element

environment.

77

CHAPTER 6 REANALYSIS OF THE EXTENDED FINITE ELEMENT METHOD

Introduction

When crack growth is modeled in the quasi-static framework with the XFEM, there

are small changes to the global stiffness matrix associated in the crack growth between

subsequent iterations. Reanalysis methods [145] have been developed primarily for use

in the fields of design and optimization to efficiently solve problems where small

perturbations to an existing finite element domain are made as part of the solution

procedure. These small perturbations may be changes in the location of existing

elements, the addition of new elements, or a combination of both to the finite element

mesh. Instead of reconstructing the global finite element stiffness matrix from scratch,

only the changed portion is considered. These strategies can result in substantial

computational savings for the repeated analysis of a structure, in this case the modeling

of quasi-static crack growth in the XFEM framework.

These methods may be considered to be of two groups, methods which achieve

the exact solution and methods which achieve an approximate solution. For the

approximate solution there are methods based on the direct modification to the inverse

of a matrix [146] and methods which directly modify a factorized version of the matrix

[147]. Exact methods are usually applied to situations with relatively little change to the

original matrix. Examples of exact methods include the Sherman-Morrison-Woodbury

formula [148] or the more general binomial inverse theorem [149] for direct updating of

the matrix inverse and the direct updating of a Cholesky factorization [24] through the

use of row modifications [150]. The approximate methods are based on iterative solvers

[151-153] and are usually used when a large number of changes to the original matrix

78

are occurring. For optimization applications the approximate applications are typically

more efficient when a pre-conditioner such as the pre-conditioned conjugate gradient

(PCG) approach is used prior to starting the reanalysis [152]. Here the Cholesky

factorization for the initial XFEM stiffness matrix is modified in subsequent iterations to

solve the of the system of linear equations.

Cholesky Factorization

The factorization of a matrix corresponding to a system of linear equations before

solving the system of equations can result in a more efficient solution procedure than

directly solving the system of equations. The Cholesky factorization [24] is a special

case of LU decomposition for a matrix which is square, symmetric, and positive definite.

A more efficient version of the LU decomposition is the LDL decomposition which

avoids the need to take square roots during the factorization. As calculated the global

stiffness matrix is positive semi-definite until boundary conditions are applied, thus

boundary conditions are applied before the LDL decomposition algorithm is applied to

the solution of the linear system of equations generated by XFEM.

The LDL decomposition algorithm can be introduced through the use of some

matrix A which is square, symmetric, and positive definite which will be factorized as

T T

T T

11 21 21

21 22 32

31 32 33

A A A

A A A

A A A

= =

A LDL (6.1)

where L is the lower triangular matrix and D is the diagonal matrix given as

11

21 22

31 32 33

1 0 0 0 0

1 0 , 0 0

1 0 0

D

L D

L L D

= =

L D (6.2)

79

where the components of L and D are given by

for 1

1

1,

j

ij ij ik jk kkj

L A L L D i jD

−

=

= − > ∑ (6.3)

and

12

1

i

i ii ik kk

D A L D−

=

= −∑ . (6.4)

Reanalysis of the Extended Finite Element Method

The XFEM allows for discontinuities to be arbitrarily inserted into a finite element

mesh though the use of enrichment functions and additional nodal degrees of freedom.

For a domain which contains voids, inclusions and homogeneous cracks the

displacement approximation is of the form

( ) ( ) ( ) ( ) ( ) ( )4

1

h J J

I I I I I I I

I J

u x V x N x u x i H x a x bυ=

= + + + Φ

∑ ∑ (6.5)

where ( )IN x are the classical finite element shape functions, ( )V x is the void

enrichment function [81, 99], ( )xυ is the modified absolute value enrichment function

[97], ( )H x is the shifted Heaviside enrichment function [21], ( )J

IxΦ is the Jth crack tip

enrichment function [54], Iu are the classical degrees of freedom,

Ii are the additional

degrees of freedom associated with the modified absolute value enrichment, Ia are the

additional degrees of freedom associated with the Heaviside enrichment, and Ib are the

additional degrees of freedom associated with the crack tip enrichment functions.

Through the use of the Bubnov-Galerkin method [44] the global stiffness matrix is

80

T T T

T T

T

uu iu au bu

iu ii ai bi

au ai aa ba

bu bi ba bb

=

K K K K

K K K KK

K K K K

K K K K

. (6.6)

Note that in general the stiffness matrix for a particular element will only contain

components of the classical stiffness matrix uuK as the enrichment functions are

defined in a small portion of the finite element mesh. The structure of the global stiffness

matrix given in Eq. (6.6) is achieved through appropriate choice of numbering for nodes.

Notice from Eq. (6.6) that as a result of crack growth in a quasi-static approach,

which is commonly used to simulate fatigue crack growth, that only the region of the

global stiffness matrix associated with the Heaviside and crack tip enrichment functions,

denoted with subscripts a and b in Eq. (6.6), will change after the initial iteration. The

change to that region of the stiffness matrix is also small compared to the full stiffness

matrix. A method is introduced here to take advantage of the large constant portion of

the finite element stiffness matrix to allow for repeated simulations of crack growth in a

quasi-static modeling framework.

First the non-changing portion of the global stiffness matrix is separated from the

region where changes are occurring such that for the initial iteration

T

11 21

21 22

=

A AA

A A (6.7)

where the components of A are given in terms of the components of Eq. (6.6) as

T T

T T11 21 22

,

uu iu au bu

iu ii ai bi bb

au ai aa ba

= = =

K K K K

A K K K A K A K

K K K K

. (6.8)

81

For subsequent iterations the A matrix takes a modified form from Eq. (6.7) as

T T11 21

21 22

ba

ba bb cb

cb

=

T

A a A

A a a a

A a A

(6.9)

where the a matrices correspond to new Heaviside components of the stiffness matrix

which will become a part of 11A for future iterations. Note that for this approach only the

new Heaviside and crack tip nodes need be computed at a given iteration after the first.

Then the new Heaviside stiffness matrix components are retained for future use while

the dynamic crack tip stiffness matrix components are discarded and recalculated at the

next iteration. In this case the number of elements requiring calculation of stiffness

components at the i th growth iteration can be approximated as 3h a∆ where h is the

average element size and a∆ is the amount of crack growth from the prior iteration.

Aside from the assembly of the global stiffness matrix the corresponding

factorization of the global stiffness matrix can be modified based on direct calculation of

the new portions of the factorization. The LDL Cholesky factorization for Eq. (6.7) is

T TT

T

11 11 11 2111 21

21 22 2221 22 22

0 0

0 0

= =

L D L LA AA

L L DA A L. (6.10)

In this form only 21L and

22L need to be calculated at each iteration. For the case

presented in Eq. (6.9), the new Heaviside terms can simply be included in 21L and

22L

and then any factorized terms associated with the Heaviside enrichment function may

be included in 11A for future iterations.

82

Implementation of this algorithm for the modification of the Cholesky factorization

presents several challenges within the MATLAB environment. The solution of 21L and

22L required sparse matrix division to occur within MATLAB. Within the MATLAB

environment the sparse matrices are converted to dense matrices, then the division is

performed and the solution is converted back to a sparse matrix [154]. Furthermore, as

a fill-reducing ordering such as AMD [155-158] or METIS [159, 160] is not considered,

thus the 21L and

22L factorizations are basically dense matrices in order for the proper

factorization to be recovered. The lack of sparsity in the factorizations of the dynamic

portion of the finite element stiffness matrix along with the sparse to dense to spare

operations in MATLAB result in a fresh factorization being the least computationally

intensive option available. Thus, only the assembly savings have been realized at this

time.

While the main objective of this work is to use the reanalysis algorithm to allow for

more accurate approximation of the differential equations governing fatigue crack

growth through an increased number of simulations in a fixed amount of time there are

other instances where this approach could be beneficial. The other logical set of

problems which could take advantage of this algorithm are optimization algorithms

involving cracks or other discontinuities.

Example Problems

Three example problems for the use of the proposed reanalysis algorithm are

presented. The main objectives of these problems are to explore the effect that the

proposed reanalysis algorithm has on the computational cost of crack growth in a quasi-

static framework. First, the effect of the proposed algorithm with respect to the general

83

trend of total computational time is explored. Then the effects of the mesh density and

crack growth increment on the computational time are presented. Second, a simple

optimization problem is considered to identify the angle of maximum energy release rate

for a crack emanating from a plate with a hole under several combinations of tensile and

shear loading. Note that the results presented here are only for the savings in the

assembly of the stiffness matrix.

All simulations were performed on a Pentium 4 3.0 GHz CPU with 4 GB of

memory in 32-bit MATLAB R2009a on a 64-bit Windows XP operating system.

Reanalysis of an Edge Crack in a Finite Plate

In order to assess the proposed reanalysis algorithm a simple test case of pure

Mode I crack growth for an edge crack in a finite plate is considered. The effect of the

repeated assemblies of the stiffness matrix with respect to time is considered. Then the

effect of the mesh density for a constant change in the number of degrees of freedom

that are changing is investigated. Finally, the more realistic case of a fixed increment of

crack growth on various meshes is considered. The plate with edge crack geometry is

given in Figure 6-1. The material properties were assumed to be Young’s modulus of

10E6 and Poisson’s ratio of 0.3. The dimensions of the plate were 4 m x 4 m with an

assumed edge crack of length 0.5 m. The applied stress was 1 Pa. A structured mesh

of square quadrilateral elements was used for the analysis and plane strain conditions

were assumed.

84

Figure 6-1. Edge crack in a finite plate for assessment of the reanalysis algorithm.

For the edge crack specimen, 30 crack growth iterations were simulated, where in

each iteration the amount of crack growth was assumed to be constant and a∆ was

chosen as 0.1 m. The average element size for this simulation was 1/80. This

corresponds to initially about 200,000 classical degrees of freedom and about 150

enriched degrees of freedom. After the 30 iterations, there are about 1000 enriched

degrees of freedom. For the simulation without the reanalysis algorithm the total

stiffness matrix assembly time for the 30 iterations was 588 seconds while for the same

simulation with the reanalysis algorithm the total stiffness matrix assembly time for the

30 iterations was 39 seconds. A more detailed comparison of the iteration by iteration

differences are shown in Figure 6-2. The major conclusions from this analysis are that

the reanalysis algorithm has a very low cost after the initial iteration for the assembly of

the stiffness matrix relative to the method without reanalysis. Also, the trend for the

σ

a

σ

W

85

assembly without reanalysis is linearly increasing as a result of the additional degrees of

freedom being introduced into the linear system of equations. Note that the reanalysis

algorithm is horizontal with respect to the new enriched degrees of freedom.

0 5 10 15 20 25 300

5

10

15

20

25

30

Iteration

Ass

em

bly

Tim

e [

s]

No Reanalysis

Reanalysis

Figure 6-2. Comparison of the assembly time for the stiffness matrix with and without

the reanalysis algorithm.

Another important criterion to consider with respect to the algorithm is how

sensitive the reanalysis algorithm is with respect to the mesh density. Two cases were

considered, one with a crack growth increment equal to the average element size. This

represents a constant change in the number of degrees of freedom independent of the

mesh density. Then a second case is considered where a fixed increment of crack

growth is selected and the mesh density is changed. Two iterations were considered to

mimic the first two data points for the curves presented in Figure 6-2. The results are

presented in Figure 6-3. As expected the assembly without the reanalysis algorithm is a

quadratic function, whereas the reanalysis algorithm is largely insensitive to changes in

the mesh density other than for a very dense mesh.

86

10 20 30 40 50 60 70 80 90 1000

5

10

15

20

25

Number of Elements Per Unit Length

Ass

em

bly

Tim

e [

s]

Initial Iteration (No Reanalysis)

Future Iterations (Reanalysis)

A10 20 30 40 50 60 70 80 90 100

0

5

10

15

20

25

Number of Elements Per Unit Length

Ass

em

bly

Tim

e [

s]

Initial Iteration (No Reanalysis)

Future Iterations (Reanalysis)

B

Figure 6-3. Comparison of the sensitivity of the assembly time of the reanalysis algorithm. A) Fixed DOF∆ , B) Fixed a∆ .

Optimization for Finding Crack Initiation in a Plate with a Hole

An optimization algorithm is introduced using fminbnd in MATLAB to determine the

angle of crack initiation optθ which corresponds in the maximum energy release rate. As

the crack geometry is changing at each simulation used in the optimization algorithm, all

enriched stiffness matrix components corresponding to the Heaviside and crack tip

enrichments are discarded at the end of each simulation. Compared to the quasi-static

example, there is a larger cost associated with the reanalysis algorithm simply based on

the fact that more enriched elements must be accounted for at each simulation.

The material properties were assumed to be Young’s modulus of 10E6 and

Poisson’s ratio of 0.3. The example of an 8 m x 8 m plate with a hole with 1 m radius

subjected to a combination of tension and shear loading as shown in Figure 6-4 is used

for this analysis. A structured mesh of square quadrilateral elements was used for the

analysis and plane strain conditions with average element size 1/20 was assumed. The

optimization problem is given as

87

( )

2 2

min

3s.t.

2 2

I II

eff eff

K KG

E Eθ

π πθ

− = +

≤ ≤

(6.11)

where G is the energy release rate, IK and

IIK are the Mode I and II stress intensity

factors, Lθ and

Uθ are the lower and upper bounds on the angle and

effE is given as

2

plane stress

plane strain1

eff

E

E E

ν

= −

. (6.12)

A comparison of the total simulation time for the case with and without reanalysis is

presented in Table 6-1. For this problem the results of the comparison in total time see

an average savings of about 25% for the reanalysis method when only the assembly of

the stiffness matrix is considered.

Figure 6-4. Plate with a hole for crack initiation assessment of reanalysis algorithm.

σ

a

r

θ

τ

88

Table 6-1. Crack initiation angle and time comparisons with and without reanalysis.

σ [Pa] τ [Pa] optθ [rad] Fun Evals No Reanalysis [s] Reanalysis [s]

0 1 3.14 28 392 301

1 0 2.41 28 394 302

1 1 2.58 31 434 305

1 5 2.90 32 448 293

1 10 2.90 31 434 297

5 1 2.44 31 433 336

10 1 2.41 27 378 293

Summary

In this chapter the framework for a reanalysis algorithm for modeling quasi-static

crack growth with the XFEM is introduced with an extension to optimization problems

which involve discontinuities. The proposed method takes advantage of the small

amount of change to the global stiffness matrix as a result of crack growth in the XFEM

framework. This results in savings in terms of both the assembly of the modified

stiffness matrix and theoretically the solution of the resulting modified system of linear

equations by a direct modification of the original LDL Cholesky factorization of the

global XFEM stiffness matrix. Due to limitations in MATLAB and a large amount of fill-in

with respect to the factorization of the dynamic nodes the performance is not currently

satisfactory. Even without realizing the full saving which are possible a large savings

has been shown both in terms of the assembly time for the global stiffness matrix and

the total simulations time for both quasi-static and optimization problems.

In the future work an approach will be created to address the major bottlenecks

associated with the current reanalysis implementation. First, all possible enriched

degrees of freedom will be considered in the global stiffness matrix. Inactive nodes will

89

simply have corresponding rows and columns which are identity. A permutation to

recover the fill-reducing ordering will be introduced using either AMD or METIS. Instead

of directly recalculating large portions of the LDL Cholesky factorization, rows and

columns will be modified according to row add and row delete functions available in

CHOLMOD [147, 150, 161-163]. Furthermore, the direct update of the solution to the

prior iterations system of linear equations based on the modifications to the factorization

will be explored as a measure to further reduce the computational time of the reanalysis

algorithm. Finally, the use of the parallel computing toolbox within MATLAB will be

introduced to allow for further savings with respect to the assembly of the stiffness

matrix both for the case with and without reanalysis.

90

CHAPTER 7 SUMMARY AND FUTURE WORK

Summary

Fatigue is the process by which materials fail due to repeated loading well below

the levels that they would fail under static loading conditions. It is not uncommon for 104

– 108 cycles to be needed for failure to occur. This presents a challenge for

computational simulation as this number of cycles is not feasible for modeling in a finite

element environment. With classical finite element methods the mesh corresponds to

the domain of interest, which means that the mesh must be recreated locally around the

crack tip each time crack growth occurs. Fatigue crack growth has been one of the

leading causes of aircraft accidents throughout history and still causes incidents today.

Better computational tools for the modeling of fatigue crack growth will help to reduce

the risk of accidents from the initiation and propagation of a fatigue crack. The goal of

this work is to create strategies such that fatigue crack growth can be numerically

modeled with greatly reduced computational requirements.

The extended finite element method is introduced in Chapter 3. Through the use of

the partition of unity finite element method discontinuous functions are introduced into

the traditional continuous finite element displacement relationship. Enrichment functions

were presented for homogeneous and bi-material cracks, inclusions, and voids. The

elemental stiffness matrix is a combination of the traditional elemental stiffness matrix

and additional terms associated with the enrichment function. The incorporation of

discontinuous enrichment functions to represent cracks allows for cracks to grow

without remeshing, which can be a challenge in the traditional finite element framework.

As the discontinuities no longer correspond to the finite element mesh, the level set

91

method for open sections is used to track the location of cracks and the level set

method for closed sections is used to track the location of inclusions and voids.

Two cases are considered for the modeling of fatigue crack growth in Chapter 4.

The first choice is to fix the increment of crack growth a∆ at each simulation iteration.

The other option is to select a fixed number of elapsed cycles N∆ per simulation. The

mixed-mode stress intensity factors are calculated using the domain form of the

interaction integrals. The direction of crack growth at a given simulation is given by the

maximum circumferential stress criteria. When modeling crack growth the solution is

very sensitive to the convergence of not only the displacement as with conventional

finite element simulations, but also the stress intensity factors, rate of crack growth, and

path of crack growth.

A challenge with respect to modeling crack growth is the lack of information about

the stress intensity factors for a specific geometry subjected to loading and boundary

conditions. Other than for very simple geometries there are not analytical solutions for a

cracked body. Instead a numerical technique such as the finite element method and J-

integral must be used to calculate the stress intensity factors. As no information about

the future crack growth is known the forward Euler is the only option to predict the future

crack growth. This requires very small time steps. Kriging was used to fit the data and to

extrapolate into the future in order to allow for higher-order numerical methods to be

used in the approximation of the fatigue crack growth model. This allows for large step

sizes to be taken without a loss of accuracy in the prediction of the fatigue crack growth.

In Chapter 6 a reanalysis algorithm for the extended finite element method which

is applicable to modeling quasi-static crack growth and can be modified for optimization

92

problems involving cracks represented independent of the finite element mesh. The

algorithm takes advantage of the small change to the global finite element stiffness

matrix associated with crack growth from one growth iteration to another. For the first

iteration, the global stiffness matrix is calculated. For subsequent iterations all crack tip

enrichment components of the global stiffness matrix are cleared and then the modified

portion consisting of new Heaviside and the crack tip enrichment is calculated and

added to the global stiffness matrix. Due to restrictions in MATLAB involving algebra of

sparse matrices it is not computationally affordable to directly modify the initial

factorization. Instead the full matrix is factorized at each iteration with the use of the

backslash command within MATLAB. Initial saving from only the change in assembly

time shows a marked decrease in the computational resources necessary for repeated

simulations.

The goal of this work is to make fatigue crack growth simulation more affordable.

First, the XFEM is used to model crack growth without the need to remesh as the crack

grows. The use of kriging was introduced to enable the use of higher-order

approximations for the numerical integration of the differential equations governing

fatigue crack growth which reduces the number of simulations needed to model fatigue

crack growth without a loss in accuracy. Finally, a reanalysis algorithm was introduced

which allows for modification of the global stiffness matrix based on crack growth

between two iterations in a quasi-static simulation to save computational time for fatigue

crack growth.

Future Work

The use of kriging with higher-order numerical algorithms for solving differential

equations will be expanded. Only constant step sizes were considered so far as a part

93

of this method. The use of the uncertainty structure [140-142] for kriging will be

incorporated into the results in order to allow for a variable step size to be used. This

should allow for a more structured allocation of the available computational resources

and lead to similar results to the test cases presented with the need for less simulations.

Mixed-mode crack growth will also be used to study the competition between the

allowable step size for convergence of the crack path and the allowable step size for

accurate integration of the governing fatigue crack growth model.

The reanalysis algorithm will be expanded such that the computational savings

from the direct modification to the initial factorized matrix will be realized. A fully

populated stiffness matrix will be considered where all active and inactive enriched

degrees of freedom are preallocated. Inactive nodes will have only identity along the

diagonal of the corresponding rows and columns. A modified version of CHOLMOD

[147, 150, 161-164] will be used for the initial factorization of the global stiffness matrix,

modification to the factorization, and solution of resulting linear system of equations.

The fill-reducing ordering will be acquired from METIS [159, 160]. In addition the use of

the parallel computing toolbox in MATLAB will be explored for the parallelization of

MXFEM. This will allow for significantly reduced computational time for the simulation of

fatigue crack growth in the XFEM framework.

Finally, a new fatigue crack growth model [14, 25] will be introduced allowing for a

variable amplitude load history to be considered. The Paris model is limited in assuming

a constant cyclic load history. The chosen model will consider the effects of overloading

and underloading in the load history. An overload acts to retard future crack growth by

extra plasticity forming at the crack tip, while the opposite behavior is caused by an

94

underload. Modeling the direct cycle-by-cycle load history through the use of the

reanalysis algorithm will be compared to more common methods such as the rain-flow

counting method [165].

As the proposed reanalysis algorithm represents a substantial reduction in the

computational time for modeling quasi-static crack problems a real life application will

be explored which would have previously been unwieldy. The Air Force Research

Laboratory (AFRL) will supply acceleration data from flight histories. A finite element

model of an airplane will be used to apply these accelerations to the airplane model and

extract a localized variable load history. The variable load history will then be applied to

a XFEM analysis of the localized panel. The analysis will be quasi-static and the fatigue

crack growth model allowing for a variable loading history to be considered will be used

to propagate the crack. Through the use of flight data and the computational analysis a

more accurate approximation of the life of the cracked aircraft panel can be realized

through the use of a digital twin.

95

APPENDIX A MXFEM BENCHMARK PROBLEMS

Crack Enrichments

Center Crack in a Finite Plate

Figure A-1. Representative geometry for a center crack in a finite plate.

The theoretical stress intensity factor for a center crack in a finite plate [15] is

2 43

sec 12 40 50

t

IK a

πλ λ λσ π

= − +

(A.1)

where a Wλ = , σ is the applied stress, and a is the half crack length. The material

properties used in the analysis were chosen to be Young’s modulus of 10 MPa and

Poisson’s ratio of 0.3. The full domain was a plate with height 10 m and width 6 m with a

center crack of length 2 m. The applied stress was 1 Pa. Square plane strain

quadrilateral elements with a structured mesh were used. Both a full and half model

σ

2a

σ

2W

96

were considered based on the symmetry in the problem. A comparison of the

convergence as a function of the average element size h is given in Table A-1 where

the normalized stress intensity factors are given as

XFEM

n II t

I

KK

K= (A.2)

where t

IK is given by Eq. (A.1) and XFEM

IK is the value calculated by the XFEM analysis

using the domain form of the contour integral.

Table A-1. Convergence of normalized stress intensity factors for a full and half model for a center crack in a finite plate.

h n

IK Full, Right Tip n

IK Full, Left Tip n

IK Half, Right Tip

1/5 0.954 0.952 0.962

1/10 0.977 0.977 0.967

1/20 0.989 0.989 0.980

1/40 0.996 0.996 0.990

1/80 0.999 0.999 0.996

97

Edge Crack in a Finite Plate

Figure A-2. Representative geometry for an edge crack in a finite plate

The theoretical stress intensity factor for an edge crack in a finite plate [15] is

( )2 3 41.12 0.231 10.55 21.72 30.39t

IK aλ λ λ λ σ π= − + − + (A.3)

where a Wλ = , σ is the applied stress, and a is the crack length. The material

properties used in the analysis were chosen to be Young’s modulus of 10 MPa and

Poisson’s ratio of 0.3. The full domain was a plate with height 6 m and width 3 m with a

center crack of length 1 m. The applied stress was 1 Pa. Square plane strain

quadrilateral elements with a structured mesh were used. Both a full and half model

were considered based on the symmetry in the problem. A comparison of the

convergence as a function of the average element size h is given in Table A-2 where

the normalized stress intensity factors are given by Eq. (A.2).

σ

a

σ

W

98

Table A-2. Convergence of normalized stress intensity factors for an edge crack in a finite plate.

h n

IK

1/5 0.884

1/10 0.991

1/20 1.001

1/40 1.002

1/80 1.002

Inclined Edge Crack in a Finite Plate

Figure A-3. Representative geometry for an inclined edge crack in a finite plate.

The material properties used in the analysis were chosen to be Young’s modulus

of 10 MPa and Poisson’s ratio of 0.3. The full domain was a plate with height 2 m and

width 1 m with an edge crack from (0,1) to (0.4,1.4). The applied stress was 1 Pa.

σ

a

σ

W

W

99

Square plane strain quadrilateral elements with a structured mesh were used. Both a

full and half model were considered based on the symmetry in the problem. A

comparison of the convergence as a function of the average element size h is given in

Table A-3 where the normalized stress intensity factors are given by Eq. (A.2). The

reference solution [166] is t

IK = 1.927 and t

IIK = 0.819.

Table A-3. Convergence of normalized stress intensity factors for an inclined edge crack in a finite plate.

h n

IK n

IIK

1/10 0.958 1.028

1/20 0.980 1.022

1/40 0.987 1.018

1/80 0.990 1.016

Bi-material Center Crack in an Infinite Plate

Figure A-4. Representative geometry for a bi-material center crack in an infinite plate. σ

2a

σ

Material 2

Material 1

100

( ) ( ) ( )1 2 1 2 2 2it

K K iK i i aε

σ τ ε π−

= + = + + (A.4)

Inclusion Enrichment

Hard Inclusion in a Finite Plate

Figure A-5. Representative geometry for a hard inclusion in a finite plate.

For an inclusion in a plate, there are no known theoretical values for the

displacement or stress state that are known. So the inclusion enrichment is calibrated

based on results from the commercial finite element code ANSYS [167].


of 50 GPa and Poisson’s ratio of 0.3 for the plate and Young’s modulus of 70 GPa and

Poisson’s ratio of 0.3 for the inclusion. The full domain was a plate with height 10 m and

σ

σ

a

2W

2H

101

width 6 m with a center inclusion of radius 0.5 m. Through symmetry only the top right

hand quarter of the plate was modeled. The applied stress was 1 Pa. Square plane

strain quadrilateral elements with a structured mesh with average element size h of 1/30

m were used for comparison against ANSYS. The stress contours for the two cases are

presented in Figure A-6 ,Figure A-7 , and Figure A-8.

A

-0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1B

Figure A-6. Comparison of the xx


ANSYS, B) MXFEM.

A

0 0.02 0.04 0.06

B



ANSYS, B) MXFEM.

102

0.85 0.9 0.95 1 1.05 1.1



ANSYS, B) MXFEM.

Void Enrichment

Void in an Infinite Plate

Figure A-9. Representative geometry for a void in an infinite plate.

σ

σ

a

103

The stress fields in terms of polar coordinates from the center of the void in an

infinite plate[168] follow the relationships

( )

( )

( )

2 4

2 4

2 4

2 4

2 4

2 4

1 3cos2 cos4 cos4

2 2

3 31 cos2 cos4 cos4

2 2

1 3sin 2 sin 4 sin 4

2 2

xx

yy

xy

a a

r r

a a

r r

a a

r r

σ θ θ θ

σ θ θ θ

σ θ θ θ

= − − −

= − + +

= − + +

(A.5)

where r and θ are polar coordinates from the center of the circle and a is the hole

radius. The material properties used in the analysis were chosen to be Young’s

modulus of 10 MPa and Poisson’s ratio of 0.3. The full domain was a plate with height

10 m and width 10 m with a center hole of radius 0.4 m. The applied stress was 1 Pa.

Square plane strain quadrilateral elements with a structured mesh with average element

size h of 1/20 m was used. The stress contours based on the theoretical values from

Eq. (A.5) are compared to the XFEM stress contours in Figure A-10, Figure A-11, and

Figure A-12. The xx

σ and yy

σ agree very well with the theoretical values, while the

shear stress shows poorer agreement directly around the hole. Away from the hole the

shear stress values show good agreement with theoretical values.

104

-1.5 -1 -0.5 0 0.5 1

-1.5 -1 -0.5 0 0.5 1


σ contours for a void in an infinite plate. A)

Theoretical, B) MXFEM.

-1 -0.5 0 0.5 1

-1 -0.5 0 0.5 1

Figure A-11. Comparison of the xy



105

0 0.5 1 1.5 2 2.5

0 0.5 1 1.5 2 2.5

Figure A-12. Comparison of the yy



Other

Angle of Crack Initiation from Optimization

An optimization algorithm is introduced using fminbnd in MATLAB to determine the

angle of crack initiation which results in the maximum energy release rate. The example

of a 8 m x 8 m plate with a hole with 1 m radius subjected to uniaxial tension as shown

in Figure A-13 is used as the benchmark for the optimization implementation. Square

plane strain quadrilateral elements with a structured mesh with an average element size

of h equal to 1/20 m was used. For this problem the location of maximum stress

corresponds to the angle of crack initiation for a 0.15 m crack which will result in the

maximum energy release rate. For a plate under uniaxial tension this angle is 0 or π

radians. The optimization problem is given as

106

( )

2 2

min

s.t.

I II

eff eff

L U

K KG

E Eθ

θ θ θ

− = +

≤ ≤

(B.6)

where G is the energy release rate, IK and

IIK are the Mode I and II stress intensity

factors, Lθ and

Uθ are the lower and upper bounds on the angle and

effE is given as

2

plane stress

plane strain1

eff

E

E E

ν

= −

. (B.7)

Figure A-13. Representative geometry for a crack initiating at an angle θ for a plate with a hole.

The results are presented in Table A-4 for two sets of bounds - 2π ≤ θ ≤ 2π and

2π ≤ θ ≤ 3 2π . Results are presented as the optimum angle identified optθ by fminbnd

σ

σ

a

r

θ

107

with the function tolerance set to 1E-9 and the angle tolerance set to 1E-6. The

normalized energy release rate is

XFEM

n

t

GG

G= (B.8)

where tG is the energy release rate corresponding to an initiation angle of 0 or π

radians and XFEMG is the energy release rate from the optimization with the XFEM. For

this example the theoretical maximum energy release rate is given as tG = 3.5166E-7

J/m2.

Table A-4. Comparison of the theoretical and MXFEM value for maximum energy release angle as a function of average element size.

- 2π ≤ θ ≤ 2π 2π ≤ θ ≤ 3 2π

h optθ nG opt

θ nG

1/20 -1.715E-5 1.001 3.142 1.000

Crack Growth in Presence of an Inclusion

Bordas [51] presented the case of edge crack growth under mixed-mode loading

in a plate with either a hard or soft inclusion. The material properties used in the

analysis were chosen to be Young’s modulus of 1 GPa and Poisson’s ratio of 0.3 for

material 1 and Young’s modulus of 10 GPa and Poisson’s ratio of 0.3 for material 2. For

the case of a hard inclusion, material 2 is for the inclusion and material 1 is for the plate.

For the case of a soft inclusion, material 1 is for the inclusion and material 2 is for the

plate. The full domain was a plate with height 8 m and width 4 m with an edge crack of

length 0.5 m centered on the left edge. The circular inclusion of radius 1 is placed at the

horizontal midpoint and the vertical quarter point. There were 34 iterations of growth

108

where at each iteration the amount of growth was equal to a∆ = 0.1 m and in the

direction given by the maximum circumferential stress predicted by a unit tensile load.

Square plain strain quadrilateral elements with a structured mesh with average element

size h of 1/20 m were used. It is important to note that no boundary conditions were

given by Bordas so here it is assumed that edge crack boundary conditions are to be

used. This may explain the slight differences between the results of Bordas and those

presented in Figure A-14.

Bordas

MXFEM

Figure A-14. Comparison of crack paths to those predicted by Bordas for a soft (left) and a hard (right) inclusion.

Fatigue Crack Growth

The implementation of the Paris model [120] for fatigue crack growth was verified

within the XFEM framework. The Paris model predicts that crack growth occurs

according to the differential equation

109

( )mda

C KdN

= ∆ (A.9)

where da dN is the crack growth rate, C is the Paris model constant, m is the Paris

model exponent, and K∆ is the stress intensity factor range under cyclic loading. Here

the geometry is chosen to mimic a center crack in an infinite plate where K∆ is

K aσ π∆ = ∆ (A.10)

where σ∆ is the applied stress range and a is the half crack length.


of 71.7 GPa, Poisson’s ratio of 0.33, Paris model constant of 1.5E-10, and Paris model

exponent of 3.8. The full domain was a plate with height 0.4 m and width 0.6 m with a

center crack of length 0.02 m. The applied stress was 78.6 MPa. Square plane strain

quadrilateral elements with a structured mesh with average element size h of 1/500 m

was used. Through symmetry, only the right half of the geometry was modeled. Fatigue

crack growth was measure to 2450 cycles with step sizes of N∆ equal to 50 at each

simulation. A comparison of the theoretical and simulated crack growth is given in

Figure A-15.

110

0 500 1000 1500 2000 250010

15

20

25

30

35

40

45

50

Number of Cycles

Cra

ck L

en

gth

[m

m]

Paris Model

XFEM

Figure A-15. Comparison between Paris model and XFEM simulation for fatigue crack growth for a center crack in an infinite plate.

111

APPENDIX B AUXILIARY DISPLACEMENT AND STRESS STATES

Homogeneous Crack

The auxiliary displacement and stress states given by Westergaard [130] and

Williams [131] for a homogenous crack that are used in the extraction of the stress

intensity factors though interaction integrals [18, 19, 128] are given as:

( ) ( )1

1cos cos sin 2 cos

2 2 2 2I II

ru K K

θ θκ θ κ θ

µ π

= − + + +

(B.1)

( ) ( )2

1sin sin cos 2 cos

2 2 2 2I II

ru K K

θ θκ θ κ θ

µ π

= − + − +

(B.2)

3

2sin

2 2III

ru K

θ

µ π= (B.3)

11

1 3 3cos 1 sin sin sin 2 cos cos

2 2 2 2 2 22I II

K Kr

θ θ θ θ θ θσ

π

= − − +

(B.4)

22

1 3 3cos 1 sin sin sin cos cos

2 2 2 2 2 22I II

K Kr

θ θ θ θ θ θσ

π

= + +

(B.5)

( )33 11 22σ ν σ σ= + (B.6)

23

1cos

22III

Kr

θσ

π= (B.7)

31

1sin

22 r

θσ

π

−= (B.8)

12

1 3 3sin cos cos cos 1 sin sin

2 2 2 2 2 22I II

K Kr

θ θ θ θ θ θσ

π

= + −

(B.9)

where r and θ is the distance from the crack tip in polar coordinates, µ is the shear

modulus, ν is Poisson’s ratio, and κ is the Kosolov constant given by

112

3plane stress

1

3 4 plane strain

i

ii

i

ν

νκ

ν

−

+= −

(B.10)

where iν is Poisson’s ratio for the ith material.

Bi-material Crack

The auxiliary displacement and strain states in two-dimensions given by Sukumar

[66] for a bi-material crack that are used in the extraction of the stress intensity factors

through interaction integrals [18, 19, 128] are given as:

( )

( )

( )( )

1

1

2

2

1, , , upper-half plane

4 cosh 2

1, , , lower-half plane

4 cosh 2

i

i

i

rf r

ur

f r

θ ε κµ πε π

θ ε κµ πε π

=

(B.11)

where ε is the bi-material constant given by Eq. (3.18). For the extraction of I

K , the

functions 1f and 2f are

1 12 sin sinf D D Tδ θ ϕ= + = + (B.12)

2 22 sin cosf C C Tδ θ ϕ= − − = − − (B.13)

and for the extraction of II

K the functions 1f and 2f are

1 22 sin cosf C C Tδ θ ϕ= − + = − + (B.14)

2 12 sin sinf D D Tδ θ ϕ= − + = − + . (B.15)

In Eqs. (B.11) - (B.15), C , D , δ , and ϕ are defined as

' cos 'sin2 2

Cθ θ

β γ βγ= − (B.16)

cos ' 'sin2 2

Dθ θ

βγ β γ= + (B.17)

113

( )

( )

e upper-half plane

e lower-half plane

π θ ε

π θ εδ

− −

−

=

(B.18)

log2

rθ

ϕ ε= + (B.19)

and β , 'β , γ , and 'γ in Eqs. (B.16) and (B.17) are given by

( ) ( )

2

0.5cos log sin log

0.25

r rε ε εβ

ε

+=

+ (B.20)

( ) ( )

2

0.5sin log cos log'

0.25

r rε ε εβ

ε

−=

+ (B.21)

1

γ κδδ

= − (B.22)

1

'γ κδδ

= + (B.23)

where κ is the Kosolov constant in the given half plane. The strains are given in terms

of the derivatives of the displacements in Eq. (B.11) as

( ), ,

1

2ij i j j i

u uε = + . (B.24)

Thus, the displacement derivatives are given in the following form

,1 1

1,1 1,14

r fu A Bf

Bπ

= +

(B.25)

,2 1

1,2 1,24

r fu A Bf

Bπ

= +

(B.26)

,1 2

2,1 2,14

r fu A Bf

Bπ

= +

(B.27)

,2 2

2,2 2,24

r fu A Bf

Bπ

= +

. (B.28)

114

In Eqs. (B.25) - (B.28), A , B , and the derivatives of 1f , 2f , and r are given as:

( )

( )

1

2

1upper half-plane

4 cosh

1lower half-plane

4 cosh

Aµ πε

µ πε

=

(B.29)

2

rB

π= (B.30)

1,1 1, ,1 1, ,1rf f r f θθ= + (B.31)

1,2 1, ,2 1, ,2rf f r f θθ= + (B.32)

2,1 2, ,1 2, ,1rf f r f θθ= + (B.33)

2,2 2, , 2, ,2rf f r f θθ= + (B.34)

For the extraction of I

K the derivatives of 1f and 2f with respect to r and θ are

1, , 1,f D Tα α α= + (B.35)

2, , 2,f C Tα α α= − − (B.36)

and for the extraction of II

K the derivatives of 1f and 2f with respect to r and θ are

1, , 2,f C Tα α α= − + (B.37)

2, , 1,f D Tα α α= − + . (B.38)

The derivatives of C , D , 1T and 2T with respect to r and θ are

,r

DC

r

ε= (B.39)

,2

FC Eθ ε= − + (B.40)

21,r

TT

r

ε= (B.41)

115

21, 1 3

2

TT T Tθ ε= + + (B.42)

12,r

TT

r

ε= − (B.43)

12, 2 4

2

TT T Tθ ε= − + (B.44)

where E , F , 3T and 4T are given as

' ' cos sin2 2

Eθ θ

β γ βγ= − (B.45)

' cos ' sin2 2

Fθ θ

βγ β γ= + (B.46)

3 2 cos sinT δ θ ϕ= (B.47)

4 2 cos cosT δ θ ϕ= . (B.48)

The auxiliary stress states used in the interaction integrals are calculated using Hooke’s

law.

116

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BIOGRAPHICAL SKETCH

Matthew Jon Pais graduated from the University of Missouri in 2003 Magna Cum

Laude and Honors Scholar with a Bachelor of Science in Mechanical Engineering. As

an undergraduate student he participated in a National Science Foundation sponsored

Research Experience for Undergraduates (REU) at the University of Missouri at Rolla.

He also worked as an undergraduate research assistant which resulted in the

undergraduate honors thesis, Hydroxyapatite Reinforced Dental Composites. He joined

the University of Florida in 2007 after receiving the Alumni Fellowship. His research

interests include: finite element methods, fracture mechanics, optimization, and

surrogates.

Documents

Level Set Method,XFEM