American Institute of Aeronautics and Astronautics

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XFEM Toolkit for Crack Path and Life Prediction of

Aluminum Structures

Jim Lua1, Jay Shi

2, Philip Liu

3, and Navin Thammadi

4

Global Engineering and Materials, Inc., Princeton, NJ, 08530

This study is focused on the development and demonstration of the capability of the

XFEM toolkit for ABAQUS (XFA) to predict fatigue crack growth and remaining life of 2D

and 3D aluminum structures. The new technique couples the level set based crack

description with the fast marching method (FMM) for crack growth. Both the jump and tip

enrichment functions are used to accurately capture the crack tip driving force during the

fatigue crack growth prediction. To facilitate the crack insertion and definition within an

existing FEM model without a crack, a customized ABAQUS CAE is developed to allow the

user to specify the location, orientation, and size of the crack that is independent of the

existing finite element mesh. Verification and validation studies are performed for the

curvilinear crack growth and life prediction of 2D modified compact specimens and a 3D T-

weldment. The developed XFA package will allow the user to produce quantifiable metrics

relating fatigue and fracture calculations to structural performance. In addition, the tool can

be used to interpret the structural healthy monitoring data for risk informed decision

making on repair, maintenance, and life extension options.

Nomenclature

Nf = fatigue life

σ0.2 = 0.2% offset proof stress

Pmax, Pmin = maximum and minimum apply load in a cycle

R = applied load ratio (Pmin/Pmax)

∆P = applied load range (Pmax – Pmin)

A(a) = crack length

σy = yield stress

H(x) = Heaviside function

F(x) = crack growth speed along its front

KI, KII = stress intensity factor

φ, ψ = level set functions for a 3D crack

θC = crack growth angle

C, m = Paris fatigue parameters

I. Introduction

he use of aluminum for the hulls of high-speed maritime vessels has increased significantly in the recent years.

The design of large aluminum high-speed vessel operating under hostile environments requires the welded

structure to withstand sub-critical growth of manufacturing flaws and service-induced defects against failure.

Traditional analysis approach for these structures is performed using simplified models with empirical parameters or

handbook lookup. These approaches are not adequate for welded structures featuring complex geometric details and

loading conditions. Given the escalating costs associated with the test-driven certification and qualification

1 Senior Principal Scientist, GEM-NJ Office, Princeton, NJ, 08540, AIAA Member.

2 Senior Developer, GEM-NJ Office, Princeton, NJ, 08540, AIAA Member.

3 Senior Scientist, GEM-MD Office, Baltimore, MD 21224.

4 Research Scientist, GEM-NJ Office, Princeton, NJ, 08540.

T

50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference <br>17th4 - 7 May 2009, Palm Springs, California

AIAA 2009-2616

Copyright © 2009 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

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procedures, there is an immediate need for verified computational software to perform crack growth simulation

under monotonic and cyclic loading.

The research needs in aluminum structure haven been summarized by Sielski1. Research needs are in the

following key areas: 1) material property and behavior; 2) fatigue design and analysis; 3) performance metrics,

reliability, and risk; and 4) structural healthy monitoring. The US Navy’s Office of Naval Research has established a

multi-year research program on Aluminum Structural Reliability2. The primary objective of the program is to

improve the technology for design construction, operation, and maintenance of high-speed aluminum naval vessels.

The compounding effects from material heterogeneity and residual stress will make the weldment prone to

fatigue crack initiation. The effects of welding on the ultimate strength and fatigue life have been studied by

Collette3 and Ye and Moan

4, respectively. The complexity in component geometry, material heterogeneity, and the

initial stress distribution, will likely initiate a general 3D crack with an arbitrary crack growth pattern. Failure to

account for the 3D crack geometry may lead to poor prediction of the crack tip driving force which will have a

pronounced impact on the fatigue life prediction due to a power law used in the crack growth model. A summary of

technical challenges for the life prediction of an aluminum weldments are listed below.

• Difficulty in measuring an initial residual stress field and monitoring its evolution during the fatigue crack

growth

• Curvilinear crack growth path driven by mixed mode loading and discontinuous structural geometry feature

• Inability to characterize mixed Mode I & II and Mode I & III fatigue crack growth behavior using da/dN

data from Mode I testing

• Lack of experimental data for fatigue crack growth validation in 3D Al-weldments

• Variation of fatigue crack growth behavior with grain structure and environmental aging

• Effects of tip plasticity under variable amplitude loading

• Simulation of crack growth under random spectrum loading

• Effect of local fracture on the global stiffness variation

Typically, a crack advances in homogenous materials mostly in Mode I. While mixed mode state can happen in

homogeneous materials, the crack will immediately try to kink into a path where pure model I exists. Moreover,

even if a crack has been initiated as a planar crack with a simple geometry (circular/ellipse), the crack will quickly

become non-planar and its front will evolve into an arbitrary shape during its growth. In spite the successes of using

conventional finite element methods in computational fracture mechanics, mesh generation in three-dimensions for

crack growth simulations is still a formidable task. This is due to the growth of the crack which requires special re-

meshing along the crack front to obtain accurate solutions for crack tip driving force. The presence of multiple

cracks will make the current state-of-the-art remeshing module intractable. The use of the extended finite element

methodology developed by Belytschko and Black5 with its mesh topology independent of any arbitrary crack surface

has proven to have great potential in automating the process as the crack grows with time yet the mesh is kept fixed.

To ensure the structural integrity and durability of an aluminum vessel, a fracture mechanics based damage

tolerance assessment tool has to be developed. Given the unavoidable initial defect in a welded joint, an accurate

prediction of the time evolution of the crack length and its growth path is essential to produce quantifiable metrics

relating fatigue and fracture calculations to structural performance. In addition, the use of the crack length/path as a

key input to the structural performance metrics can provide a direct link to the structural health monitoring system

(SHM). By integrating a structural health monitoring system with an advanced fatigue and fracture analysis tool, we

can effectively address the following three questions:

• If a crack is found via SHM, what is the implication for residual strength and time to repair?

• Where are the critical crack locations? and

• Locations where SHM efforts should be focused?

To greatly reduce the computational burden associated with insertion of an arbitrary crack into an existing

model and simulate the crack growth without re-meshing, an XFEM toolkit for ABAQUS has been developed by

Global Engineering and Materials, Inc. for both 2D and 3D delamination onset and growth prediction6-8

under the

sponsorship of Air Force. Its extension to metallic fatigue crack growth has been accomplished under the ONR

funding for an aluminum weldment with a material heterogeneity and residual stress field9. In this paper, a

curvilinear crack path and life prediction is performed using the XFEM toolkit for modified 2D compact test

specimens and a representative 3D T- weldment. The key components of the XFEM based fatigue crack growth

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prediction tool along with its interaction with other ONR sponsored research components are summarized in Fig. 1.

The initial location and size of the crack can be determined from a NDT technique associated with its data

interpretation module. Using the initial crack configuration, operational profile, and an existing FEM model without

the crack, the XFEM-based damage assessment tool is applied to compute the fatigue crack growth path and its

associated life. Design iteration is performed for a given selection of the repair option to seek a cost effective repair

and maintenance action. Tool validation and verification can be performed at the component and structural level

using the test data before and after the repair.

Figure 1. Illustration of Key Components of Flaw Assessment and Life Prediction Tool

II. Summary of Key Solution Modules in XFEM Toolkit

The XFEM methodology developed by Belytschko and Black5, Moës et al.

10, and Sukumar et al.

11, provides an

efficient way to simulate the arbitrary crack growth in an existing mesh without remeshing. To greatly enhance its

modeling capability and commercial viability, we have implemented the XFEM in ABAQUS via its user-defined

elements. Key solution modules along with the numerical procedures in crack growth prediction are displayed in

Fig. 2. A customized ABAQUS CAE is developed to insert an arbitrary crack into an existing FEM model without

the crack. Given the crack location and size, both the level set values and enrichment types are defined for all the

nodes associated with the user-defined elements. An incremental crack growth simulation is performed based on a

given fatigue crack growth law and definition of its tip drive force. The stress intensity factor is computed for the

linear elastic fracture mechanics problems while either a crack opening displacement (COD) or a crack tip opening

angle (CTOA) is employed for an elastoplastic material. Given the crack growth increment and its growth direction,

a fast marching method (FMM) is applied to determine the new crack front position and its new level set values.

Accurate numerical integration used in stiffness matrix formation is achieved for cracked elements via the slicing

technique as shown in Fig. 2. A brief description of each solution model is given below.

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Figure 2. Display of Key Components of XFEM’s Fatigue Analysis Module via ABAQUS’ UEL

A. Customized ABAQUS CAE for Crack Definition

In order to insert an arbitrary crack into an existing FEM model and automatically generate the XFEM input files

for fatigue and fracture analysis, a customized ABAQUS CAE is developed using Python script as shown in Fig. 3.

Figure 3. Illustration of Crack Insertion via a Customized ABAQUS CAE

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After importing an existing or creation of a FEM model without a crack shown in Fig. 4, user can define a crack

plane by selecting its location and normal direction. Using sketching tools in ABAQUS/CAE, an arbitrary crack

front can be created on the selected plane along with the mesh seeds assignment on the cracked domain as shown in

Fig. 5. A non-structural mesh is automatically generated on the cracked plane where the crack geometry is defined

via the level set values at these nodal points. Meshed crack plane is inserted into the base model via an assemble

process and the XFEM input files are created as illustrated in Fig. 6.

Figure 4. Importation of an Existing FEM Model without a Crack

Figure 5. Definition of Crack Front Along with Mesh Seeds for the Crack Domain

Figure 6. Assemble and Generation of XFEM Input Files

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B. Description of Crack Geometry via Its Two Level Set Values

A 3D crack is defined by two almost orthogonal level sets (signed distance functions). One of them describes the

crack as a two-dimensional surface in a three-dimensional space (φ), and the second is used to describe the one-

dimensional crack front (ψ) as shown in Figure 7. Both the crack surface and its font are determined from its zero

level set values. By discretizing the level set function via its FEM shape functions, the crack geometry can be fully

defined via the level set values at nodal points associated with an original mesh that is not in conformation with the

crack geometry. An example application of the level set description of a planar crack in a 3D joint is shown in Fig.

8. The layered colored contours in Φ indicate the distance between the cracked plane and a sampling plane.

Similarly, the annular colored contour in ψ represents the distance between the crack front and a nearby circle

surrounding the crack front.

Figure 7. Representation of a 3D Crack via Its Two Level Set Values

Figure 8. Representation of a Planar Crack in a 3D Joint with Its Two Level Set Values (ΦΦΦΦ, ψψψψ)

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C. XFEM Formulation for a 3D Crack via Jump and Tip Enrichment

In XFEM, the approximation of displacement field ( )hu x can be expressed as

( ) ( ) ( ) ( ) ( ) ( )1~4

h

i i i i i i j ij

i i i j

N N H N=

= + + Ψ

∑ ∑ ∑ ∑u x x a x x b x x c (1)

where i

a , i

b , ij

c are the relevant coefficients to the continuous, discontinuous and partially discontinuous

displacement components. ( )iN x is the shape function at node i; ( )iH x are the shifted jump functions, which

are different from node to node and defined as

( ) ( ) ( )i iH H H= −x x x (2)

The branching enrichment functions are listed as follows

( ) ( )1 2 3 4sin , cos , sin sin , cos sin2 2 2 2

r r r rθ θ θ θ

θ θ

Ψ = Ψ = Ψ = Ψ =

(3)

For a 3D solid element, there are 15 DOFs in total at each tip enriched node. It includes 3 DOFs for the continuous

displacement components (ai), and 12 DOFs for the tip enrichment components (cij). For a jump enriched node, only

6 DOFs (ai and bi, i=1, 3) are required to characterize the displacement discontinuity on the wake of the crack. The

crack opening displacement can be written as

[ ]( ) [ ]( ) + −

−

+ -

a b c a b c

a

[u] = u - u = N N N x N N N x b

c (4)

Because the jump functions and the first term of branching functions are discontinuous across the crack, Eq. (4) can

be reduced to

1 12i i i

i

N b rc

= +

∑+ -

b c

b[u] = u - u = N N

c1

(5)

Using the principal of virtual work, we have

[ ] S

dv u ds b u dv u dsσε τδ δ τδΩ Ω ∂Ω

+ = +∫ ∫ ∫ ∫ (6)

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where [u] is displacement jump from Eq. (5) and [ ]S

u dsτδ∫ is the surface interaction term defined as the virtual

work associated with the crack opening displacement jump. This surface interaction denoted by INT

δπ can be

written in the local crack coordinate system as

[ ] dsINT

S S

u dsδπ τδ τδ= = ∆∫ ∫ (7)

where δ∆ denotes the displacement jump along the normal and tangential direction and τ is the corresponding

traction components associated with the local crack coordinate system. Substituting Eq. (1) as the displacement test

function and Eq. (7) in Eq. (6), a tangent stiffness matrix can be derived and implemented in ABAQUS via its user-

defined subroutines. Both the contact and a cohesive model has been included in computing the surface interaction

term given in Eq. (7). With the added capability, the effects from the local fracture process zone and crack closure

can be addressed explicitly. The construction of sub-elements for visualization is also built within the UEL. The X-

FEM update module is implemented within UEXTERNALDB. In order to visualize X-FEM subdomain data, a post-

processing module is developed within UEXTERNALDB to generate ABAQUS output database using its ODB

API.

D. Crack Growth Model

The linear elastic fracture mechanics has been widely used in the current damage tolerance analysis and fatigue

certification of both metallic and composite structures. To comply with the current design guideline and well

documented certification practice, the stress intensity factor is selected as the crack tip driving force for our

problems with the assumption of small scale yielding. To simulate a curvilinear crack growth, a crack growth

direction law based on the maximum hoop stress criterion (Erdogan and Sih12

) is applied. With respect to the local

crack tip coordinate system (x’-y’), the crack will propagate from its tip in the direction θc where the hoop stress

(σθθ) is maximum.

In order to account for the stress ratio dependent crack growth, the Walker’s equation (Walker13

) is employed as

shown below.

[ ] ;)1( 1 m

effeff RKCdN

da −−∆= γ );/()( maxmin resreseff KKKKR ++= openingeff KKK −=∆ max (8)

In Eq. (8), Kmax and Kmin is the maximum and minimum stress intensity factors associated with the applied cyclic

load, Kres is the stress intensity factor associated with the inherent residual stress field, and Kopening is the smallest

applied stress intensity factor value that makes the crack open. To be consistent with the common assumption that

the crack can propagate only during that fraction of the fatigue loading cycle in which the crack faces are separated,

the effective stress intensity factor (∆Keff) is regarded as being responsible for the fatigue crack growth. It is assumed

that no fatigue crack damage occurs when compression is transmitted across crack faces. The Paris type fatigue

crack growth model can be recovered from the Walker’s model if we set γ=0 and ∆Keff = Kmax – Kmin.

E. Fast Marching Method for Crack Propagation

Given the level set description of crack surface and fatigue crack growth law, the use of fast marching method

(FMM) has shown to be computationally attractive for tracking and advancing the crack front11

. We integrate FMM

with XFEM in ABAQUS to propagate the crack without remeshing. A structured finite difference grid is constructed

to cover the local crack domain. A hyperbolic equation in terms of the level set function (ψ(x)) is solved using a

second order upwind finite difference scheme. The front speed F(x) which is equivalent to the fatigue crack growth

rate is computed from a fatigue law. By assigning an arbitrary set of velocity components along the crack front, a

non-planar crack growth or arrest can be modeled. An illustration of FMM implementation for a planar crack growth

is shown in Fig. 9. The algorithm for crack growth simulation via the coupled XFEM and FMM is described below.

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Figure 9. Illustration of Fast Marching Method (FMM) via Its Application to a Planar Circular Crack

• Use XFEM to compute the front velocity, F, based on a given fatigue crack growth law and level set

functions (φ, ψ);

• Compute the distance, r, to the crack front;

• Solve 0=∇⋅∇ ρF to extend the crack velocity;

• Update crack surface, ψ, and advance crack front, φ;

• Reinitialize, φ, so that ψ=0 and φ=0 are orthogonal.

F. Module Integration

Given the current crack tip driving force (∆Keff), stress ratio Reff, and number of cycle increments (dN), the crack

growth module (Eq. (8)) is used to determine the crack extension (da) along the current crack front. Using the crack

growth direction determined from KIImax(θ)=0 and the crack growth increment (da), the F vector shown in Fig. 9 is

determined. Using FMM implemented in ABAQUS, both the new crack surface and the corresponding level set

updates (φ, ψ) is computed for next crack configuration. Two analysis termination criteria are used: 1) Kmax(Nf) >

KIC, where KIC is the fracture toughness; and 2) a(Nf) > ac, where ac is the maximum crack size. If either of these

criteria is satisfied, the final accumulative fatigue life (Nf) is recorded as the ultimate structural life. In addition,

crack arrest is evaluated if Kmax < Kth indicating the crack may come to a full stop under the compressive residual

stress field.

III. Verification and Validation of XFEM Toolkit

Given the power law nature in the Paris type fatigue law, an accurate evaluation of the near tip stress intensity

factor is critical for the reliable life and crack path prediction. At present, multiple fatigue tests on modified compact

specimens are performed at Navy Lab (NSWCCD) for 5083-H116 aluminum alloys. Upon completion of all the

tests, both the curvilinear crack growth path and associated fatigue life predicted from the XFEM toolkit will be

compared with the experimental data. Our simulation results for the modified compact specimen are presented

below in addition to the XFEM verification study using double node fracture models. The 3D capability of the

XFEM toolkit is also demonstrated via its application to the crack growth simulation of a T-weldment.

A. XFEM Verification Study Using a Standard Compact Specimen

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To evaluate the accuracy of the XFEM prediction on the stress intensity factor (SIF) of a stationary crack, a

standard compact specimen shown in Fig. 10 is used where the component geometry and material properties are the

same as those defined in the Reference14

. The initial crack length is 21.01 mm and the vertical applied load is 11KN.

To verify the accuracy of the XFEM model prediction, a double node fracture model is also crated and the SIF is

computed from the J-integral and the near tip crack opening displacement.

Figure 10. Problem Statement for a Standard Compact Specimen

Figure 11. Comparison of ABAQUS’ Double Node and XFEM Model Prediction

A comparison of von Mises stress contour using double node and XFEM model is shown in Fig. 11. Under the

same scale factor, the near tip stress distribution from the double node and XFEM model is almost identical. Using

the plane stress assumption, the Mode I stress intensity factor (KI) is computed from its near tip crack opening

displacement and J-integral in the double node model. Our XFEM prediction in KI is almost identical to the double

node KI value converted from the J-integral. In addition, the XFEM based KI value also agrees well with the curve-

fit formula given by Lei12

. The slight discrepancy is mainly due to the plane strain condition used in deriving the

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curve-fit formula given by Lei14

. Because of the use of linear element in the double node model, the near tip

singularity cannot be captured. The resulting KI computed from the near tip crack opening displacement has a

slightly higher error of 2%.

B. Mesh Sensitivity and Stationary Crack Analysis Using Modified Compact Specimens

Modified compact test specimens have been fabricated by the Navy Lab (see Fig. 12) to examine the curvilinear

crack growth path. Different from the standard compact specimen, an additional hole is introduced and a “miss hole”

or a “sink hole” crack growth pattern can be simulated experimentally with a slight change in the hole position. The

material properties of 5083-H116 aluminum alloy published in SSC-44815

are used for 5383-H116.

Figure 12. Definition of Modified Compact Specimen Fabricated by NSWCCD

Prior to the fatigue crack growth analysis, a mesh sensitivity study is performed using the “miss hole” specimen

with an initial crack length of 11.176 mm (0.44 in). Double node fracture models are created for this stationary crack

subjected to a pull load of 5604.76 N (1260 lb). As shown in Fig. 13, three mesh densities are used to examine the

mesh sensitivity and solution convergence in SIF prediction. The near tip element size for the fine, medium, and

coarse mesh is 0.0501 mm, 0.1003 mm, and 0.341 mm, respectively. A comparison of deformed shape and von

Mises stress contour is depicted in Fig. 14 for these three mesh densities. In general, the values of J-integral

predicted from these three mesh densities are consistent and the SIFs computed from the corresponding J-integral

are very close to each other. Given this dominant Mode I loading, the Mode II contribution to J is ignored in

computing KI.

Different from the relatively mesh insensitive J-integral, the SIF computed from the local crack tip opening

displacement (CTOD) does show its mesh dependence. Despite the use of the fine mesh, the SIF computed from

CTOD still have about 5% difference in comparison with the convergent SIF value determined from the J-integral.

As shown in Fig. 14, a convergent solution in SIF can be achieved from using the medium mesh.

An XFEM analysis is performed for the medium mesh model and a comparison of its model prediction with the

corresponding double node model is shown in Fig. 15. Both Mode I and II SIF is determined from the XFEM model

prediction despite this dominant Mode I loading. The presence of an additional hole in the modified compact

specimen has introduced the mode mixity and will result in a curvilinear crack growth path. Since the smaller KII

contribution has been ignored in converting J to KI, the KI value from the double node model is slightly higher than

the KI prediction from the XFEM toolkit.

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Figure 13. Mesh Sensitivity and Convergence Study Using Miss Hole Specimen with a Stationary Crack

Figure 14. Comparison of SIF Prediction Using Double Node Model with Different Mesh Densities

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Figure 15. Comparison of Double Node and XFEM Prediction for a Modified Compact Specimen Using the

Medium Mesh

C. Fatigue Crack Path and Life Prediction Using Modified Compact Specimens

After verifying the XFEM model for a stationary crack using the medium mesh, crack path and life prediction is

performed for both the “miss hole” and “sink hole” specimen shown in Fig. 12. The cyclic fatigue load is defined in

Fig. 16 along with a curve-fit Paris law using the fatigue test data in SSC-44815

at the load ratio (R) of 0.1.

Snapshots of crack growth path and the associated number of cycles are displayed in Fig. 17 for both the

Figure 16. Summary of Fatigue Load and Fatigue Crack Growth Model for Life Prediction of Modified

Compact Specimen

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“sink hole” and “miss hole” specimen. As can be seen from Fig. 17 and 18, the crack tip propagates towards and

Figure 17. Snapshots of Crack Path and Fatigue Cycles for the Sink and Miss Hole Model

Figure 18. Snapshots of Progressive Crack Growth in Miss and Sink Hole Specimen

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enters the hole in the sink hole model while the crack path propagates towards the hole but doesn’t enter the hole in

the miss hole model. In addition, for the same crack length shown in Fig. 17, the cumulative fatigue cycles for the

miss hole model is always higher than the corresponding sink hole model. Under the load control fatigue loading,

the intensified stress zone near the crack tip increases during the fatigue crack growth. Once the crack hits the edge

of the hole in the sink hole model, the stress concentration zone propagates along the edge of the circular hole.

To further explore the fatigue crack driving force and its associated number of cycles, a comparison of fatigue

response curves is shown in Fig. 19 for the sink and miss hole specimen. The short life associated with the sink

specimen is mainly due to its higher crack tip driving force measured by its SIF. Once the crack is very close to the

hole, a sudden increase in the SIF can be observed from Fig. 19.

Figure 19. Comparison of Cycle and Stress Intensity Factor Curves

D. 3D XFEM Analysis for a T-Weldment Subjected to a Mixed Mode Loading

To demonstrate the capability of the XFEM toolkit for the crack insertion and growth prediction for a 3D

structural component, a representative T-joint subjected to mixed loading is considered here. As shown in Fig. 20,

Figure 20. Problem Statement of a 3D T-Weldment Subjected to a Mixed Mode Loading

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three material models are used for the base, heat affect zone (HAZ), and welding zone. Two concentrated forces in

the horizontal and vertical direction are applied on the top web surface via the multiple point constraints (MPCs).

The joint is simply supported at the top surface of the base plate as shown in Fig. 21. An initial fabrication induced

defect is embedded directly underneath the web in the welding zone. A circular shape is assumed for the initial crack

with its radius of 6.35 mm (0.25 in).

A 3D FEM model without the circular initial flaw is developed first with three distinct material zones shown in

Fig. 20. Using our customized ABAQUS/CAE, a crack plane is inserted into the existing FEM model where a

circular crack front is defined using CAE’s sketching toolkit (see Fig. 21). Level set initiation is performed using the

non-structural mesh for the cracked plane to identify the crack location and shape within the XFEM framework.

Given the level set description of the initial crack, the XFEM input files are generated where the jump and tip

enrichment zones are defined as shown in Fig. 21.

Figure 21. Crack Definition and Insertion in a T-Weldment via the Customized CAE

Figure 22. Deformed Shape and Slicing View of Crack Growth Path

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The deformed shape and crack growth path from the 3D XFEM prediction is shown in Fig. 22. Since the crack is

internal and embedded within the welding zone, the crack growth configuration cannot be directly observed from the

base model. Using the slicing view, four snapshots of the crack growth configuration are shown in Fig. 22. The

crack growth changes from the planer to the non-planar pattern. The maximum crack size associated with these four

snapshots along the crack front is 7.35 mm, 8.35 mm, 10.35 mm, and 12.35 mm. The projection view of the

corresponding crack front on the xz plane is shown in Fig. 23. Because of the non-uniform distribution of SIFs along

the crack front and the non-planar crack growth pattern under the mixed mode loading, the crack front evolves from

its circular shape to an ellipse.

Figure 23. A Projection View of Crack Front at Its Growth Step Size of 1 mm, 2 mm, 4 mm, and 6 mm

The variation of the SIF along the evolving crack front is shown in Fig. 24. The maximum value of KI is about

Figure 24. Variation of SIFs Along the Crack Front During the Fatigue Crack Growth

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four times higher than the KII and KIII. The presence of both KII and KIII component perturbs the crack growth

direction resulting in a non-planar crack growth pattern as shown in Fig. 22. Since the initial defect is embedded in

the lower stress zone (see Fig. 22), the magnitude of SIFs is low as can be seen from Figure 24. Given its extremely

slow crack growth rate, it is unlikely the fatigue life will be driven form the propagation of this initial defect. It is

expected that a new crack can be initiated at the triangular welding tip and will propagate through the base plate.

New simulation results will be published in a forthcoming paper.

IV. Summary of Conclusions

An advanced fatigue analysis toolkit for ABAQUS was developed based on the mesh independent XFEM

technology. The presence of the crack is characterized by level set initiation while its growth is determined from the

level set update at the nodal points associated with the original mesh. An efficient crack front tracking algorithm was

implemented based on the fast marching method (FMM). To facilitate the insertion of an arbitrary crack into an

existing mesh without a crack, a customized ABAQUS CAE was developed for crack definition, level set

initialization, automatic selection of enrichment zone and types, and generation of XFEM input files. A stress ratio

dependent crack growth model has been used for the crack path and life prediction.

Verification and validation studies were performed using standard and modified compact tension specimens. For

a stationary crack, the fracture parameters predicted from the XFEM toolkit agree very well with the J-integral

solution obtained from the double node fracture model. Using the miss and sink hole specimen designed by the

Navy Lab (NSWCCD), the distinct fatigue crack growth behavior was simulated and will be used in the future

validation study after completion of all fatigue tests at the Navy Lab. For the sink hole specimen, the analysis

predicted that the fatigue crack will enter the hole. While for the miss hole the crack avoids the hole and continues to

propagate after passing it. Under the load control fatigue load, both the crack tip driving force and the intensified

near tip stress zone increases during the crack growth. Shorter fatigue life was found for the sink hole specimen

from our numerical prediction.

The applicability of the XFEM toolkit for a 3D T-weldment is demonstrated via the fatigue assessment of an

embedded circular flaw under a mixed mode loading. The presence of the mode mixity and non-uniform SIF

distribution along the crack front drives the crack growth in a non-planar and non self-similar manner. A

prohibitively large computational burden can result in the double node model from arbitrary crack insertion, FEM

model generation, and remeshing during the crack growth.

The simulation results have revealed the low criticality of this initial defect. Because of the lower stress level at

the initial crack site, the resulting slow crack growth rate would not cause the fatigue failure of the joint. A new

crack could be initiated at the higher stress concentration zone and its fast propagation could result in final fatigue

failure. A fatigue analysis using an edge crack near the tip of the welding zone will be performed in the near future.

Given the unavoidable initial defect in a welded joint, the developed XFEM toolkit can produce quantifiable

metrics relating fatigue and fracture calculations to structural performance. In addition, the use of the crack

length/path as a key input to the structural performance metrics can provide a direct link to the structural health

monitoring system (SHM).

Acknowledgments

The authors gratefully acknowledge the support from ONR 331 under contract N00014-08-C-0443 with Dr. Paul

Hess as the program monitor.

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