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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
3
To my mother and father who have always supported me in all that I have done.
4
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.
5
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
6
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
7
B AUXILIARY DISPLACEMENT AND STRESS STATES........................................ 111
Homogeneous Crack ............................................................................................ 111
Bi-material Crack .................................................................................................. 112
LIST OF REFERENCES ............................................................................................. 116
BIOGRAPHICAL SKETCH .......................................................................................... 129
8
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
9
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
10
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
A-7. Comparison of the xx
σ contours for a hard inclusion in a finite plate. A)
ANSYS, B) MXFEM. ......................................................................................... 101
A-8. Comparison of the xx
σ contours for a hard inclusion in a finite plate. A)
ANSYS, B) MXFEM. ......................................................................................... 102
A-9. Representative geometry for a void in an infinite plate. ...................................... 102
A-10. Comparison of the xx
σ contours for a void in an infinite plate. A) Theoretical,
B) MXFEM. ....................................................................................................... 104
A-11. Comparison of the xy
σ contours for a void in an infinite plate. A) Theoretical,
B) MXFEM. ....................................................................................................... 104
A-12. Comparison of the yy
σ contours for a void in an infinite plate. A) Theoretical,
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
11
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
12
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.
13
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.
14
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
15
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
17
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.
18
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
19
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
20
[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
21
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
22
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.
23
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)
24
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
25
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
26
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].
The material properties used in the analysis were chosen to be Young’s modulus
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
σ contours for a hard inclusion in a finite plate. A)
ANSYS, B) MXFEM.
A
0 0.02 0.04 0.06
B
Figure A-7. Comparison of the xx
σ contours for a hard inclusion in a finite plate. A)
ANSYS, B) MXFEM.
102
0.85 0.9 0.95 1 1.05 1.1
Figure A-8. Comparison of the xx
σ contours for a hard inclusion in a finite plate. A)
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
Figure A-10. Comparison of the xx
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
σ contours for a void in an infinite plate. A)
Theoretical, B) MXFEM.
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
σ contours for a void in an infinite plate. A)
Theoretical, B) MXFEM.
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.
The material properties used in the analysis were chosen to be Young’s modulus
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.