proceedings of - darwin - southwest research institute

Proceedings of ASME Turbo Expo 2004 Power for Land, Sea, and Air

June 14-17, 2004, Vienna, Austria

GT2004-53323

EFFICIENT FRACTURE DESIGN FOR COMPLEX TURBINE ENGINE COMPONENTS

R. Craig McClung Michael P. Enright Yi-Der Lee Luc J. Huyse

Southwest Research Institute San Antonio, Texas 78228-0510

Simeon H. K. Fitch

Mustard Seed Software 19751 Encino Way

San Antonio, Texas 78259

ABSTRACT

Many high-energy turbine engine components are fracture critical. However, the complex three-dimensional (3D) geometries and stress fields associated with these components can make accurate fracture analysis impractical. This paper describes a new computational approach to efficient fracture design for complex turbine engine components. The approach employs a powerful 3D graphical user interface (GUI) for manipulation of geometry models and calculated component stresses to formulate simpler 2D fracture models. New weight function stress intensity factor solutions are derived to address stress gradients that vary in all directions on the fracture plane.

INTRODUCTION

Many high-energy turbine engine components are fracture critical. However, both the geometries and stress fields associated with these components are often complex and fully three-dimensional, so accurate fracture analysis can be difficult.

Determining the stress intensity factor (SIF, commonly denoted as K) associated with a postulated crack in the 3D geometry is certainly possible with current technology. Several advanced numerical fracture tools—both finite element (FE) and boundary element (BE) codes are available commercially. These codes can perform a near-exact calculation for the given 3D cracked geometry. However, most of these codes require the user to generate a new geometric model or to substantially modify the existing model originally developed to perform stress analysis. Some codes provide adequate re-meshing to facilitate the crack size and shape changes associated with the modeling of subcritical fatigue crack growth (FCG), although these codes typically do not supply state-of-the-art models to calculate crack growth rates.

Furthermore, the resulting calculation of K is typically very numerically intensive and therefore requires substantial

computational time. Modeling fatigue crack growth substantially increases the computational requirements even further. The resulting computational burden is simply not practical for routine design calculations, when many different potential fracture critical locations must be analyzed. If probabilistic design is employed, further multiplying the computational demands, these numerical approaches become unmanageable.

The practical alternative is to employ simplified two-dimensional (2D) fracture models. Accurate stress intensity factor solutions have been derived for cracks with idealized shapes in simple geometries such as rectangular plates, cylinders, and plates with round holes. The K solutions are either available in closed form or require only limited numerical integration, and so they are computationally very fast. These solutions are readily available in a number of commercial and public domain FCG analysis codes, and many gas turbine engine companies have even developed their own FCG codes. These codes often have sophisticated models to calculate FCG rates.

However, these simple 2D approaches have two very substantial limitations. First of all, the derived K solutions typically do not address a general nonlinear stress gradient in all directions on the crack growth plane. Only simplified stress fields, such as uniform tension or bend, or perhaps a nonlinear stress gradient along one dimension of the fracture plane, are admitted. Therefore, the complex 3D stress fields often encountered in real components cannot be modeled, and the resulting stress intensity factor solution suffers a significant loss of accuracy.

Second, even if the available stress intensity factor solution is adequate to model the actual stress distributions in the component, the user is typically left with the daunting challenge of extracting the necessary stress information from

1 Copyright © 2004 by ASME

the original 3D finite element model and importing it to the simplified 2D fracture model. This challenge includes the identification of the initial crack location and the plane along which the crack will grow (a plane that is typically not coincident with the global coordinate system of the FE model), the calculation of the stresses on this plane (for multiple load cycles), and then the transfer of the extensive stress information into a completely different format and different computer program. This process is both time-consuming and error-prone.

This paper describes a new computational approach to efficient fracture design for complex turbine engine components. The approach links a powerful 3D graphical user interface (GUI) for manipulation of geometry models and calculated component stresses with new weight function stress intensity factor solutions that address bivariant stress gradients.

CRACK GROWTH ANALYSIS BASED ON THREE-DIMENSIONAL FINITE ELEMENT MODELS

For many years, finite element analysis of stresses in rotating gas turbine engine components has been based on 2D axisymmetric models of the rotor geometry. Three-dimensional features such as bolt holes have been modeled with hybrid techniques such as superposition of approximate stress gradients due to the stress concentration onto the 2D stress field for the rotor without the hole. However, recent advances in computer technology have made fully 3D finite element analysis a practical alternative. Users can now routinely create and analyze models consisting of 100,000 or more elements using a desktop personal computer. These 3D models can provide substantially better accuracy than the old 2D and pseudo-3D models.

As noted earlier, the extraction of the 3D stress information needed to support a simplified fracture analysis is a daunting task. In response to this challenge, a new methodology has been developed to model 2D crack growth using results from 3D finite element analysis.

The first step is to locate the crack origin on the 3D model. Fatigue cracks can originate at many locations on a component, but often originate on the surface at locations of highest stress. Current finite element visualization technology easily permits the display of contours associated with calculated stresses for a given FE model. Also commonly available is the basic functionality to rotate and zoom a 3D model to view the regions of greatest interest. The computer mouse can then be used to select a point of interest on the surface of the model.

The second step is to identify the plane on which the fatigue crack will grow. When fatigue cracks are very small, they may grow along planes that are influenced by the orientation of the local material microstructure and by the local stress multiaxiality. However, as fatigue cracks continue to grow in most common rotor materials, they usually grow along planes of maximum principal stress. Since the growth of the smallest cracks is not typically analyzed (these cracks are generally too small to be found via conventional inspection, and their analysis can be substantially complicated by microstructural considerations), the maximum principal stress plane approach is generally an acceptable simplification.

Calculation of this maximum principal stress plane is relatively straightforward, based on the six stress components (three normal and three shear) at the initial crack location. This plane is assumed to be invariant as the crack grows, which is a

reasonable and customary assumption for most configurations. Once the orientation of the maximum principal stress plane is calculated, the plane is extended throughout the 3D model to “slice” the model. The creation of this “slice plane” is illustrated schematically in Figure 1.

The third step is to construct a detailed two-dimensional model of the slice plane with its associated stresses. The points of intersection between the 2D slice plane and the 3D finite elements are identified, and the stresses are computed at the intersection points by interpolation from the 3D stress results. A new 2D finite element mesh of triangular elements is then constructed from the intersection points based on a Delauney triangularization algorithm [1-3]. This new 2D mesh is not used for any additional finite element stress computation, but instead is only used as a guide for the extraction of stress information from the original 3D model.

The fourth step is to superimpose a simplified 2D fracture model on this 2D fracture plane. The 2D fracture model is commonly based on a rectangular geometry. The crack origin is on one boundary of the rectangle (for a surface crack). If the fracture model is based on a one-dimensional stress gradient, then the gradient line is located passing through the crack origin and aligned perpendicular to the feature surface. The stresses on the crack plane along this gradient line are then extracted from the original 3D finite element model.

This methodology has recently been implemented in the DARWIN® probabilistic fracture mechanics computer program. DARWIN has been developed in the Turbine Rotor Material Design (TRMD) research program with the sponsorship of the Federal Aviation Administration and the partnership of four major aircraft engine manufacturers. The initial focus of TRMD and the DARWIN software was the threat of hard alpha anomalies in titanium engine rotors [4]. More recently, TRMD and DARWIN have been addressing the threat of surface damage due to manufacturing or maintenance anomalies [5].

Crack Location

Principal StressDirection

Slice Plane (Normal to Crack Principal Stress Direction)

Figure 1. Identification of slice plane based on initial crack

location and associated principal stress.


The DARWIN implementation provides a convenient means of illustrating the new methodology. See Figure 2. In the upper left panel (a), the user imports and manipulates (zooms/pans/rotates) a 3D finite element model directly in the GUI. In the lower left panel (b), the user has selected a surface crack location with the mouse—note the small white sphere in the (red) high stress region. The software has computed the principal stress direction at this surface location and then constructed a plane perpendicular to this principal stress direction. A portion of this plane is represented in the GUI, but the actual plane extends all the way across the 2D model.

In the upper right panel (c), the software has sliced the 3D model to create a 2D model with a new triangular mesh and

new stress contours based on the interpolation from the original 3D model. The lower right panel (d) zooms in on a portion of this 2D mesh. Here the user has employed the mouse to create the specific 2D fracture model. The crack location originally specified in panel (b) is indicated by the small white circle. A rectangular plate with a local coordinate system has been defined to coincide approximately with the physical boundaries of the 2D slice. In this particular example, a univariant weight function stress intensity factor solution [5] is being employed, and the appropriate stress gradient line through the crack origin is also shown in the GUI. The stresses needed as input to the K solution will be automatically extracted along this line.

(a)

(c)

(b)

(d)

Figure 2. Illustration of methodology for surface crack analysis using 3D finite element results: (a) import 3D finite element

model and reorient to locate surface crack, (b) select surface crack with mouse (GUI automatically identifies principal stress plane), (c) slice model along principal stress plane to reveal 2D crack propagation plane, and (d) define fracture mechanics plate.


Figure 3 illustrates the same methodology for a second rotor geometry with a more pronounced 3D character. In this case, the original 3D model is a unit segment of the solid of rotation and has a single bolt hole. The crack origin is located on the internal surface of the bolt hole. The fracture model employs a univariant stress intensity factor solution for an offset surface crack at an offset round hole in a rectangular plate. Additional GUI tools are provided to define the parameters associated with the hole and other geometric details.

One advantage to this 3D→2D approach is that many of the tools originally developed for 2D fracture mechanics assessment can be directly employed. For example, Figure 4 shows one of the existing DARWIN GUI tools used to visualize the size and shape of an advancing crack through animation. The propagating crack shown in Figure 4 was initially defined using the model shown in Figure 3(d).

(a)

(c)

(b)

(d) Figure 3. Illustration of methodology for analysis of offset surface cracks at offset holes using 3D FE results: (a) import 3D FE

model and reorient to locate surface crack, (b) select surface crack with mouse (GUI automatically identifies principal stress plane), (c) slice model along principal stress plane to reveal 2D crack propagation plane, and (d) define fracture mechanics plate.


Figure 4. Animation of growing fatigue crack

DEVELOPMENT OF BIVARIANT WEIGHT FUNCTION STRESS INTENSITY FACTOR SOLUTIONS

Most available bivariant solutions for embedded cracks in infinite bodies can be analytically derived [6,7]. However, the analytical procedure becomes much more complex when free surfaces are introduced, and few solutions are available. As a consequence, numerical solutions derived from finite element and boundary element methods are utilized.

The CC01 corner crack solution in NASGRO [8], based on the Raju-Newman finite element solutions [9] through interpolation, is a commonly used bivariant solution. However, the application is limited by the assumed linear stress variation across the plate thickness and width.

The NASCRAC software [10] permits the calculation of K for general bivariant stress fields using an RMS-averaged effective stress intensity factor for the entire crack boundary along the direction associated with the corresponding degree-of-freedom. However, this method does not provide information on the specific local variation in K, and it exhibits accuracy limitations, especially at deeper crack lengths [11].

Zhao et al [12] has developed an approach based on the slice synthesis method (SSM) in which SIFs obtained from two perpendicular cross-sectional slices in a three-dimensional (3D) body that intersect at the crack front are “synthesized” to determine the SIF corresponding to that specific elliptical angle. However, accuracy is again an issue for this approximate method, and the computational speed of the algorithm can become impractical for some fatigue crack growth applications.

Highly accurate SIF solutions for cracked structural components with general stress distributions are possible with a point weight function (PWF) formulation. Several PWFs have been proposed that can be used as the integrand in the surface integral for SIFs. The PWF for a circular crack in an infinite body has long been available [7,13]. Oore and Burns [14] and Orynyak [15,16] have developed PWFs for elliptical cracks. The formulation proposed by Oore and Burns represents the PWF in terms of a contour integral. In contrast, Orynyak’s approximate PWF is a much simpler functional form constructed similar to the PWF for circular cracks.

The current version of NASGRO [8] includes a limited number of bivariant SIF solutions based on a hybrid approach developed by Fujimoto [17], utilizing the PWF of Orynyak while extending the Petroski-Achenbach method [18] to three-dimensional cracks. The hybrid method utilizes a weight function formulation capable of correctly spatially weighting loads applied at any point on the crack surface while permitting existing solutions for elliptical or part-elliptical flaws to be used as reference solutions. However, some inconsistencies have been discovered for deep cracks when using different sets of reference solutions.

Glinka [19] has further extended the work done by Oore and Burns, and his new formulation is, in principle, applicable to cracks of arbitrary shape. However, some accuracy issues remain unresolved, and the integration scheme employed introduces substantial computation speed issues. Weight Function Formulation

A new bivariant weight function formulation is illustrated here for a corner crack in a rectangular plate of finite width and thickness (Figure 5). The corner crack is assumed to have a quarter-elliptical shape that can be characterized by two degrees of freedom, the crack dimensions c and a in the width and thickness directions, respectively.

x

y

• QR(ξ0, η)

• Q(ξ, η)

• Q*(0, η)

• Q’(ξ0, η0)yQ (-ξ, η) •

• xQ (ξ, -η)

c

at

W

Figure 5. Geometry configuration and nomenclature

for corner crack in plate.


Due to its simplicity and accuracy, the PWF proposed by

Onynyak for an elliptical crack in an infinite body was used as the basic weight function. This form can be written as

RrRW

QQQQ

ππ 2

22

′

′−

=l

(1) 0.

2φ/π

0 1 2 3 4

SIF

-1.2

-0.8

-0.4

0.0

4

0.8

1.2

1.6

Symbols: Orynyak PWFLines: Shah & Kobayashi

a/c=0.2a/c=0.4a/c=0.6a/c=0.8a/c=0.9

σ(x,y)=(x/c)3+(x/c)2(y/a)+(x/c)(y/a)2+(y/a)3

The equation represents the weighting effects on any point Q’ along the crack front contributed by a point unit load applied at Q on the elliptical crack surface. Here, R is the distance between and Q , r the distance between Q and Q , and

the distance between Q and . The definitions for Q, Q’,

, and are more concise using elliptical coordinate

notation. Respectively, they are given by

*Q

Q

R*

QQ ′l

RQ

Q′*

( )ηξ ,=Q ,

,Q and Q where ξ and η are the

parameters in the elliptical coordinate system to identify a point along the crack plane with its origin at the center of the crack. It can be seen that Q, , and Q are defined along the same

elliptical angle η. ξ

( )0η0,ξ=′Q (ξ ,0 ) )η=R

Q

( η,* 0=

*R

0 is the “elliptical radius” defining the elliptical crack front. This PWF equation is similar to the one for circular cracks derived by Sih [7] as well as Shah and Kobayashi [6] except for different definitions for the length parameters. The validity of this PWF is demonstrated by a comparison with Shah and Kobayashi’s analytical derivation [6] for embedded cracks subjected to crack opening stresses applied on the crack plane (Figure 6). Here 2φ/π is the normalized elliptical angular position around the perimeter of the embedded crack.

Figure 6. Comparison of Orynyak PWF method with Shah-Kobayashi analytical results.

′−Π+

′

−Π+−Π+

++

−=

′

′

′

′

′

′

xx

yy

Rr

RrRW

QQ

QQ

QQ

QQ

QQQQ

yx

1111

1

321

2

2

2

2

2

22

l

l

l

l

l ππ (3)

where x and y define the Cartesian coordinates of Q, and

221 axcx −=′ and 221 cxay −=′ . The SIF can therefore

be evaluated by performing surface integration across the crack area. For a given stress distribution σ(x,y) applied on the crack plane obtained from an uncracked body subjected to remote loadings, the stress intensity factor is written as

To account for the free boundary correction for a quarter-elliptical corner crack in a quarter-infinite body, Eq. (1) was modified to include two additional length parameters. Now the basic point weight function applicable at Q for a point unit

load applied at Q is given by ′

++

−=

′

′

′

′

′

′ 2

2

2

2

2

22

1QQ

QQ

QQ

QQ

QQQQ

yxR

rRWl

l

l

l

l ππ (2)

( ) dxdyWyxKa a

yc

QQ∫ ∫−

′⋅=0

1

0

2

2

, σ (4)

The parameters 1Π , 2Π , and are calibrated by reference

solutions at both a- and c-tips to characterize the finite boundary effects. Accordingly, in this approach, three reference stress solutions are required at each tip, and the stress intensity factor at a- and c-tips can be determined by

3Π

The additional length parameters QQx

l ′ and

QQy ′l correct

the free surface effects for a corner crack by assuming a symmetrical stress distribution for an imaginary prolonged crack extending into the other three quadrants. In reference to Figure 5,

QQx ′l is the distance between xQ and Q , and ′QQy ′l

the distance between yQ and Q . The locations, ′ xQ and

yQ ,

are points symmetrical to the location of point load Q with respect to the x- and y-axes.

( )

dxdyxx

yy

Rr

RrRyx

K

cacaca

a ay

c

QQ

QQ

QQ

QQ

QQca ca

y

ca

cax

ca

ca

′−Π+

′

−Π+−Π+

++

−

=∫ ∫

−

1111

1,

,3

,2

,1

0

1

02

2

2

2

2

22

,

2

2

,

,

,

,

, l

l

l

l

l ππσ

(5)

Additional correction terms are required to account for finite width and thickness effects. Following several exploratory investigations, a formulation was identified with reasonable numerical accuracy and stable convergent approach. The point weight function at Q for a point unit load applied at

Q for a quarter elliptical crack in a finite plate is thus provided by

′

where the superscripts a and c denote parameters associated with a- and c-tips, respectively.

To facilitate the computation, Eq. (5) is preferably written in terms of elliptical coordinate parameters instead of Cartesian.


For 1≤= caα , the relationships between Cartesian and

elliptical coordinates (ξ, η) are as follows:

ηξ coscoshbx = , ηξ sinsinhby = , 22 ac −=b (6) For any point along the crack front, the “elliptical radius” in elliptical coordinate system is the same and is given by

−+

=ααξ

11ln

21

0 (7)

gt

g

( )

33

33

32

23

23

32

3

13

22

22

3

31

3

03

2

12

2

21

3

30

2

0211

2

20

011000,

+

+

+

+

+

+

+

+

+

+

++

+++=

ty

Wxg

ty

Wxg

ty

Wxg

ty

Wxg

ty

Wxy

Wx

tyg

ty

Wxg

ty

Wxg

Wxg

tyg

ty

Wxg

Wxg

tyg

Wxggyxσ

The above equation is derived by eliminating the dependency on the elliptical angle η among coordinate transformation equations. The Cartesian coordinates for Q, , , RQ *Q Q′ , xQ ,

and yQ in terms of elliptical coordinate parameters, are as

follows:

(17)

( )ηξηξ sinsinh,coscosh bbQ = (8)

The coefficients for the polynomials pi, qi, and gi are determined by regressions to the actual stress variations in the component of interest.

( )ηξηξ sinsinh,coscosh 00 bbQR = (9) Reference Solutions

( 0,cos* ηbQ = ))

(10)

( ηξηξ sinsinh,coscosh 00 bbQ =′ (11)

( ηξηξ sinsinh,coscosh bbQx −= ) (12)

( ηξηξ sinsinh,coscosh bbQy −= ) (13)

and the infinitesimal area becomes

( ) ξηηξ ddbdxdy 222 sinsinh += (14)

The weight function method requires an accurate set of reference solutions for a matrix of crack geometries. These reference solutions were numerically generated using the FADD3D fracture mechanics software, a general boundary element code for three-dimensional linear elastic fracture analysis [20]. Development of BE models is much less labor-intensive than FE models, since only surfaces must be meshed. This was especially important for the large number of crack models required in the current effort. FADD3D has been shown to maintain exceptional accuracy even with relatively coarse meshes [21,22].

Furthermore, the integration limits are converted from 2210: cxay −→ and to cx →0: 00: ξξ → and

20: πη → . The integration sequence is now interchangeable.

To accelerate the evaluation of surface integral described in Equation (5), DARWIN provides two separate modules for crack opening stresses that can be simplified by polynomials in addition to modules for general stressing. These modules contain pre-integrated terms derived based on prescribed polynomial forms in one-dimensional and two-dimensional variations. The applicable polynomial for stress variation in one dimension is

( ) ( )i

ii W

xpxyx ∑=

==

6

0

, σσ (15)

Reference solutions were generated at 150 different combinations of geometrical aspect ratios: a/c=0.1, 0.2, 0.4, 0.6, 0.8, 1.0; c/W=0.1, 0.2, 0.5, 0.8, 0.9; and a/t=0.1, 0.2, 0.5, 0.8, 0.9. For a/c and a/t less than 0.1 and approaching zero, reference stress solutions for edge cracks were generated and used [23]. For a/c larger than 1, the reference solutions were obtained from the corresponding c/a ratios by reversing the stress field and the associated geometry dimensions. As a result, the expanded reference solution matrix constitutes a discrete database covering a complete range of aspect ratio combinations; i.e., a/c from 0 to ∞, a/t from 0 to 0.9 and c/W from 0 to 0.9. For any arbitrary combination of aspect ratios, the reference solutions are determined from the matrix of specific solutions through Hermite interpolation.

for stress variations along the x-direction only, or

( ) ( )i

ii t

yqyyx ∑=

==

6

0

, σσ (16) =σ −= ayσ

Three sets of reference solutions were determined for each crack geometry. These are denoted as solutions for unit tension, unit bending along the x-axis, and unit bending along the y-axis. The solutions were generated by applying the reference stresses on the crack plane in the corresponding uncracked body. The three reference stresses are given by

10 , 11 + , and 12 +−= cxσ .

Figure 7 depicts a sample boundary element mesh pattern used to simulate the crack surface for c/a=1, a/t=0.5, and c/W=0.5. To describe this crack surface, 64 9-node boundary surface elements were specified. Each element had four corner, one embedded, and four intermediate nodes, and four doubly curved sides.

for stress variations along the y-axis only, where t and W represent the plate thickness and width (ref. Figure 5). The polynomial for stress variation in two dimensions contains cross product terms and is given by


Figure 8 shows selected reference solutions numerically determined by FADD3D for various a/t ratios at c/a=1 and c/W=0.5. The variations were plotted against the parametric elliptical angle φ measured along the crack front from the c-tip, the surface tip along the plate width direction. Here 2φ/π =0 corresponds to the c-tip, and 2φ/π=1 to the a-tip. F(φ) is the normalized stress intensity factor, given by K / Qaπ , where Q

is the shape factor associated with the a/c ratio. Q is defined by

( )( )

>+≤+

=1for ,464.111for ,464.11

65.1

65.1

caaccacaQ (18)

2φ/π

0.0 0.2 0.4 0.6 0.8 1.0

F(φ)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

a/t=0.1a/t=0.2a/t=0.5a/t=0.8a/t=0.9

c/a=1c/W=0.5

σref=σ0=1

The FADD3D results, like many numerical solutions, often

exhibit sharp changes at the intersection of the crack perimeter with a free surface (2φ/π = 0 and 1) because of the complex nature of the singularity at that point. In order to minimize the effect of this numerical artifact, the reference solution values for the a-tip and the c-tip are generally taken at the first node inside the free surface, between 2° and 3° from the free surface.

(a)

Validation

Independent numerical results using FADD3D have been used to validate the bivariant weight function solutions. Selected comparisons are presented here to demonstrate the numerical accuracy achieved by the formulation.

For validation, two types of stress variation—one univariant, one bivariant—were applied on the crack plane. The univariant stress distribution was specified as

3

1

+−=

ay

uniσ (19)

2φ/π

0.0 0.2 0.4 0.6 0.8 1.0

F(φ)

0.0

0.4

0.8

1.2

1.6

2.0

a/t=0.1a/t=0.2a/t=0.5a/t=0.8a/t=0.9

c/a=1c/W=0.5

σref=σ1=-y/a+1

which varies from 1.0 at the plate surface to zero at the a-tip, the surface tip in the plate thickness direction. The bivariant stress distribution contains cross-product terms and is given by

(b)

33

11

+−

+−=

ay

cx

biσ (20)

2φ/π

0.0 0.2 0.4 0.6 0.8 1.0

F(φ)

0.0

0.4

0.8

1.2

1.6

2.0

2.4

a/t=0.1a/t=0.2a/t=0.5a/t=0.8a/t=0.9

c/a=1c/W=0.5

σref=σ2=-x/c+1

(c)

Figure 8. Selected normalized SIF results from FADD3D. Graphs correspond to three reference stresses: (a) unit tension, (b) bending along x-axis, and (c) bending along y-axis.

Figure 7. Boundary element mesh pattern used to simulate crack surface for FADD3D computation; c/a=1, a/t=0.5 and c/W=0.5.


This stress distribution varies from 1.0 at the center of the crack and bivariantly approaches zero at both surface tips. The distribution functions contain a third order dependency to represent a steep stress gradient. These two stress fields are illustrated in Figure 9. The specified stress functions can be transformed into expanded polynomials as described in Eqs. (16) and (17).

Bivariant K solutions for other geometries, including embedded and surface cracks in plates, and surface and corner cracks at offset round holes in plates, are under development.

2φ/π

0.0 0.2 0.4 0.6 0.8 1.0

Nor

mal

ized

SIF

0.0

0.2

0.4

0.6

0.8

1.0

Bivariant FieldUnivariant Field

Two aspect ratios were used for demonstration; a/c=0.4 and 1.0. The normalized SIF variations as a function of parametric elliptical angle along the crack front for two stress distributions are shown in Figures 10 and 11 for a/c=1.0 and a/c=0.4, respectively. The normalization factor is the same as previously defined, Qaπ .

The predicted results at both a- and c-tips using the new bivariant K solutions for corner crack in plate were plotted against these two sets of numerical results obtained from FADD3D. The predicted results were taken at 3 degrees below free surface, while the computed results were based on the next available data points below the free surface. The comparison, depicted in Figure 12, indicates very good agreement.

Figure 10. Normalized SIF from FADD3D for a/c=1.0. Univariant Solutions vs. Bivariant Solutions Are bivariant solutions really needed? Are conventional

univariant solutions adequate? The answers to these questions certainly depend on the specific stress fields involved, but the errors introduced by using a univariant solution for a highly bivariant stress field can be substantial. This is illustrated by the selected results for univariant and bivariant stress distributions shown in Figure 9. In this example, a univariant solution to the bivariant stress field would be identical to the solution for the univariant stress field, since the univariant stress field is identical to the bivariant stress field along one primary axis of the crack. However, the bivariant stress distribution decreases to zero at both surface tips, while the univariant stress distribution is uniform in the y-direction and applies a much larger resultant force on the crack plane than the bivariant stress field. The resulting univariant stress intensity factor solution dramatically overestimates the actual SIF by a factor of three at the a-tip (2φ/π =1) and nearly a factor of ten at the c-tip (2φ/π =0), as shown in Figure 10.

2φ/π

0.0 0.2 0.4 0.6 0.8 1.0

Nor

mal

ized

SIF

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Bivariant FieldUnivariant Field

Figure 11. Normalized SIF from FADD3D for a/c=0.4.

Normalized SIF by FADD3D

0.0 0.2 0.4 0.6 0.8 1.0

Nor

mal

ized

SIF

by

Wei

ght F

unct

ion

0.0

0.2

0.4

0.6

0.8

1.0

Figure 9. Surface plots showing stress distributions specified by univariant and bivariant stress functions.

Figure 12. Comparison of FADD3D and weight function SIF solutions for corner crack in plate.



CLOSING REMARKS The GUI-based analysis scheme described in the first half

of this paper is invaluable to the practical application of the bivariant stress intensity factor solutions described in the second half of the paper. The bivariant SIF solutions require as input a complete description of the stresses on the crack plane. Extraction and manipulation of this large volume of information would be extremely cumbersome manually, but can be achieved easily with the new type of user interface. ACKNOWLEDGEMENTS

The support of the Federal Aviation Administration (Grant 99-G-016) is gratefully acknowledged. Bruce Fenton and Joe Wilson of the FAA Technical Center and Tim Mouzakis of the FAA Engine and Propeller Directorate are thanked for their oversight and encouragement. The TRMD Steering Committee, currently comprising Darryl Lehmann (Pratt & Whitney), Jon Tschopp (General Electric), Ahsan Jameel (Honeywell), and Jon Dubke (Rolls-Royce Corp.), has made many valuable contributions. REFERENCES 1. Boivin, C. and Ollivier-Gooch, C., 2002, "Guaranteed-

quality triangular mesh generation for domains with curved boundaries", International Journal for Numerical Methods in Engineering, 55, pp. 1185 - 1213.

2. Borouchaki, H. and George, P.L., 1997, "Aspects Of 2-D Delaunay Mesh Generation", International Journal for Numerical Methods In Engineering, 40, pp. 1957-1975.

3. Waldhart, C., Enright, M.P., and Fitch, S.K., 2004, “Methodology for Application of 3-D Stresses to Surface Damage-Based Risk Assessment,” Proc., 45th Structures, Structural Dynamics, and Materials Conference, Palm Springs, CA, April 19-22, 2004.

4. Millwater, H., Wu, J., Riha, D., Enright, M., Leverant, G., McClung, C., Kuhlman, C., Chell, G., Lee, Y., Fitch, S., and Meyer, J., 2000, AA Probabilistically-based Damage Tolerance Analysis Computer Program for Hard Alpha Anomalies in Titanium Rotors,” Paper 2000-GT-0421, Proc. ASME Turbo Expo 2000, Munich, Germany.

5. Enright, M.P., Lee, Y.-D., McClung, R.C., Huyse, L., Leverant, G.R., Millwater, H., and Fitch, S.H., 2003, “Probabilistic Surface Damage Tolerance Assessment of Aircraft Turbine Rotors,” Paper GT-2003-38731, Proc. ASME Turbo Expo 2003, Atlanta, Georgia.

6. Shah, R.C. and Kobayashi, A.S., 1971, “Stress Intensity Factor for an Elliptical Crack under Arbitrary Normal Loading,” Engineering Fracture Mechanics, 3, pp. 71-96.

7. Kassir, M.K. and Sih, G.C., 1975, “Three-Dimensional Crack Problems,” Mechanics of Fracture, Vol. II, G.C. Sih Ed., Noordhoff, Holland.

8. NASGRO Reference Manual, 2003, Version 4.02, NASA Johnson Space Center and Southwest Research Institute.

9. Raju, I.S., and Newman, J.C., Jr., 1988, “Stress Intensity Factors for Corner Cracks in Rectangular Bars,” in Fracture Mechanics: 19th Symposium, ASTM STP 969, T.A. Cruse, Ed., American Society for Testing and Materials, Philadelphia, PA, pp. 43-55.

10. Harris, D. O., Bianca, C. J., Eason, E. D., Salter, L. D., and Thomas, J. M., 1987, “NASCRAC: A Computer Code for Fracture Mechanics Analysis of Fatigue Crack Growth,” Proc. 28th Structures, Structural Dynamics, and Materials Conference, Monterey, California, Part 1, pp. 662-667.

11. Favanesi, J. A., Clemmons, T. G., and Lambert, T. J., 1994, “Fracture Mechanics Life Analytical Methods Verification Testing,” NASA CR 4602.

12. Zhao, W., Newman J.C., Jr., Sutton, M.A., Wu X.R., and Shivakumar, K.N., 1995, “Analysis of Corner Cracks at Hole by a 3-D Weight Function Method with Stresses from Finite Element Method,” NASA TM 110144.

13. Smith, F.W., Kobayashi, A.S., and Emery, A.F., 1967, “Stress Intensity Factors for Penny-Shaped Cracks, Part I – Infinite Solid,” ASME Journal of Applied Mechanics, 34, pp. 947-952.

14. Oore, M. and Burns, D.J., 1980, “Estimation of Stress Intensity Factors for Embedded Irregular Crack Subjected to Arbitrary Normal Stress Fields,” Journal of Pressure Vessel Technology, 102, pp. 202-211.

15. Orynyak, I.V., Borodii, M.V., and Torop, V.M., 1994, “Approximate Construction of a Weight Function for Quarter-Elliptical, Semi-Elliptical and Elliptical Cracks Subjected to Normal Stresses,” Engineering Fracture Mechanics, 40, pp. 143-151.

16. Orynyak, I.V. and Borodii, M.V., 1995, “Point Weight Function Method Application for Semi-Elliptical Mode I Cracks,” International Journal of Fracture, 70, pp. 117-124.

17. Fujimoto, W.T., 2000, “A Weight Function Approach for Tracking the Growth of Surface and Radial Cracks through Residual Stress Fields or Bi-Variant Stress Fields,” Proc. 2000 USAF Aircraft Structural Integrity Program Conference, San Antonio, Texas.

18. Petroski, H.J. and Achenbach, J.D., 1978, “Computation of the Weight Function from a Stress Intensity Factor,” Engineering Fracture Mechanics, 10, pp. 257-266.

19. Glinka, G., and Reinhardt, W., 2000, “Calculation of Stress Intensity Factors for Cracks of Complex Geometry and Subjected to Arbitrary Nonlinear Stress Fields,” Fatigue and Fracture Mechanics, ASTM STP 1389, G.R. Halford and J.P. Gallagher, Eds., American Society for Testing and Materials, West Conshohocken, PA, pp. 348-370.

20. Xiao, L., and Mear, M.E., 1998, FADD3D User’s Manual, Dept. of Engineering Mechanics, University of Texas at Austin.

21. Li, S., Mear, M.E., and Xiao, L., 1998, “Symmetric Weak-Form Integral Equation Method for Three-Dimensional Fracture Analysis,” Computer Methods in Applied Mechanics and Engineering, 151, pp. 435-539.

22. Li, S. and Mear, M.E., 1998, “Singularity-Reduced Integral Equations for Discontinuities in Linear Elastic media,” International Journal of Fracture, 93, pp. 87-114.

23. Wu, X.-R., and Carlsson, A.J., 1991, Weight Functions and Stress Intensity Factor Solutions, Pergamon Press, New York.

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