Abstract In this work, a simple and efficient XFEMapproach has been presented to solve 3-D crack prob-lems in linear elastic materials. In XFEM, displace-ment approximation is enriched by additional functionsusing the concept of partition of unity. In the proposedapproach, a crack front is divided into a number ofpiecewise curve segments to avoid an iterative solution.A nearest point on the crack front from an arbitrary(Gauss) point is obtained for each crack segment. Incrack front elements, the level set functions are approx-imated by higher order shape functions which assurethe accurate modeling of the crack front. The values ofstress intensity factors are obtained from XFEM solu-tion by domain based interaction integral approach.Many benchmark crack problems are solved by the pro-posed XFEM approach. A convergence study has beenconducted for few test problems. The results obtainedby proposed XFEM approach are compared with theanalytical/reference solutions.
H. Pathak · A. SinghDepartment of Mechanical Engineering, IIT Patna,Patna, India
I. V. Singh (B) · S. K. YadavDepartment of Mechanical and Industrial Engineering,IIT Roorkee, Roorkee, Indiae-mail:[email protected]
1 Introduction
Cracks/voids are found in all engineering componentsat micro or macro level. The fracture failure of thesecomponents is always preceded by multi-site cracks.The crack tip stress fields of all such cracks interactwith each other. This interaction of cracks leads to theformation of one dominant crack which results in thefailure of the components under complex loading.
Fracture mechanics is widely used for the failure ofthe components in the presence of cracks. An accu-rate evaluation of the fracture parameters such as stressintensity factors (SIFs) becomes important to accu-rately predict the failure life of a component. In general,all real life crack problems are 3-D in nature, and do nothave regular shapes. The closed form analytical solu-tions are available only for some standard shape cracks.Therefore, the numerical methods are the only choiceleft to analyze/simulate 3-D crack problems. Although,numerical methods can simulate these problems butthe accurate and efficient simulation of these cracksremains a challenging tasks for the fracture mechanicscommunity.
In past, finite element method (FEM) has been exten-sively used for the study of cracks (Henshell and Shaw1975; Akin 1976; Barsoum 1977; Nikishkov and Alturi1987; Rhee and Salama 1987). In FEM, the geome-try is generally modeled by an adequate mesh, and acrack must coincide with the edges of the finite ele-ments i.e. a conformal mesh is required apart fromthe requirement of special elements to handle crack
123
190 H. Pathak et al.
tip asymptotic stress fields. However, the generation ofgood quality meshes for complex geometries and adap-tive simulation is an expensive task. Hence, the finiteelement simulations of 3-D cracks become even moredifficult and time consuming task. This leads to inaccu-racy in the simulation of 3-D cracks. To cope-up withthese issues, extended finite element method (XFEM)has been grown as an alternative tool to solve 3-D crackproblems. In XFEM, the geometric discontinuities suchas cracks, inclusions, and holes are not considered apart of the finite element mesh/domain. These geomet-ric discontinuities are modeled by adding enrichmentfunctions in the displacement approximation of FEAthrough partition of unity (PU) (Belytschko and Black1999; Dolbow 1999; Moës et al. 1999; Daux et al.2000). The enrichment functions are obtained from thetheoretical background of the problem under consider-ation. Primarily two enrichment functions (Sukumar etal. 2001; Liu et al. 2004) are required to model a crack inXFEM: first one is discontinuous on the crack surfacewhile the second one is asymptotic at the crack front.XFEM removes the burden associated with the meshgeneration for the problems involving voids, cracks andinterfaces, and thus provides the precise modeling ofcracks.
Although, a lot of work has been done in 2-D onthe crack modeling using XFEM but in 3-D, very lit-tle work has reported so far e.g. Sukumar et al. (2000)presented a 3-D XFEM formulations for the model-ing of mode-I crack problems, and showed that XFEMresults are in good agreement with the analytical solu-tions. In the present work, XFEM has been extended tosimulate 3-D cracks. A non-iterative scheme has beenused to define the geometry of the crack surface. Non-planer 3-D cracks are defined and tracked by vectorlevel set approach (Gravouil et al. 2002; Moës et al.2002). For computation of mixed mode stress intensityfactors, the domain based interaction integral approachis adopted. In this approach, two states are consideredand super imposed. The first state is considered as anauxiliary known state while other one (actual) remainsan unknown. The gradient or higher order derivativesof auxiliary field can be easily calculated for regularshape cracks but the calculation of these fields for arbi-trary shape cracks becomes quite difficult. Therefore,in the present work, the auxiliary fields are approxi-mated using partition of unity so that the gradient orhigher order derivative of auxiliary fields can be easilycalculated. Various 3-D crack problems are solved by
Fig. 1 Domain with a discontinuity (crack)
the proposed approach, and the results are comparedwith the analytical/reference solutions available in lit-erature. A convergence study of the results has alsobeen performed for few benchmark problems.
2 XFEM formulation
Three-dimensional XFEM formulation for a crackproblem is described in the following sub-sections:
2.1 Governing equations
Consider a three dimensional domain � having aboundary�, consist of�c, �t and�u as shown in Fig. 1.The displacement boundary condition is imposed on�u , traction is applied on �t and crack surfaces areassumed as traction free. The equilibrium equationsalong with boundary conditions are written as
∇ : σ + b = 0 in �
σ .n = t on �t
σ .n = 0 on �c+
σ .n = 0 on �c− (1)
where, n is the unit outward normal, u is the displace-ment field, σ is the Cauchy stress tensor, b is the bodyforces per unit volume, ε is the strain matrix. The con-stitutive relation for the elastic material is given as
σ = C : ε (2)
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3-D cracks simulations 191
where, C is the constitutive matrix for three dimen-sional linear isotropic elastic material, which can bewritten as
E is the Young’s modulus and ν is the Poisson’s ratio.
2.2 XFEM approximation for cracks
In XFEM, a given domain is sub-divided into twodistinct parts i.e. mesh generation for the domain(excluding internal boundaries) and enriching the finiteelement approximation by additional functions tomodel the internal boundaries.
By using the concept of PU, the standard approx-imation is enriched with additional functions. In 3-Dcrack modeling, the enriched displacement approxima-tion can be written in general form as Zi and Belytschko(2003):
uh(x) = Ln∑
j=1
N j (x)
⎡⎢⎢⎢⎢⎢⎣
u j + (H(x)− H(x j ))
a j︸ ︷︷ ︸j∈nr
+4∑α=1
(φα(x)− φα(x j )
)bαj
︸ ︷︷ ︸j∈n A
⎤⎥⎥⎥⎥⎥⎦
(4)
where, u j are the nodal displacement vector associatedwith the continuous part of the finite element solution;a j is the additional degree of freedom vector associatedwith the Heaviside function, H(x); bαj is the additionaldegree of freedom vector associated with asymptoticfunctions,φα(x); n is the set of all nodes in the mesh; nr
is the set of nodes whose shape function is completelycut by the crack surface, and n A is the set of nodeswhose shape function support is partially cut by thecrack front. Heaviside function, H(x) is defined for
those elements which are completely cut by the cracksurface whereas the asymptotic functions, φα(x) aredefined for those elements which are partially cut by thecrack front. Four enrichment functions used to modelthe radial as well as the angular behavior of asymptoticcrack-tip stress fields, are given as Moës et al. (1999):
φα(x)={φ1, φ2, φ3, φ4
}
= [√r cos θ2 ,√
r sin θ2 ,
√r cos θ2 sin θ,
√r sin θ
2 sin θ]
(5)
2.3 Vector level sets for a crack
In vector level set, a crack is defined by two vectorfunctions (Sukumar et al. 2000; Gravouil et al. 2002).These functions are calculated at nodes so that they canbe easily approximated using finite element interpola-tion functions. In order to improve the computationalefficiency, the vector level set is evaluated only for thoseelements which have their mean centre lying in a nar-row band of the crack surface. In XFEM, no geomet-rical representation of the crack is required. A crack isentirely described by the nodal data. The entire com-putational geometry (cube/cuboid in the present work)is discretized using eight node linear solid elements,while the crack is approximated using quadratic twentynode hexahedral elements. The quadratic approxima-tion is chosen for an accurate representation of thecurved crack geometries. The handling of the arbitrarycrack using vector level set is explained in next para-graph.
Level set function, φ(x) is a signed distance func-tion, which can be computed from f(x) by the followingformula φ(x) = ||f(x)|| H
(ξ2.f(x)
), where H(x) is
the Heaviside function which takes +1 above the cracksurface and −1 below the crack surface. Here, f(x) isthe vector between a point x and its projection on thecrack surface (�cr ), and is directed from x to the cracksurface, i.e. f(x) = x f − x where x f is the closestpoint projection of x on the crack surface (�cr ). Thesecond level set function is a signed distance functionwith respect to the crack front, and is computed usingthe formula ψ(x) = ξ1.f(x). A crack surface inside afinite element is represented by
φapprox (x) =n∑
i=1
Ni (x)φ(xi ) (6)
ψapprox (x) =n∑
i=1
Ni (x)ψ (xi ) (7)
123
192 H. Pathak et al.
X
Y
Z
1ξ2ξ
3ξ
fx
)(xp
Fig. 2 Arbitrary three-dimensional cracks
The criterion for the finding whether an elementbelongs to crack front or crack surface, is given as:
For crack front enriched elements: ψapprox (x) = 0and φapprox (x) = 0
For crack surface enriched elements: ψapprox (x) <0 and φapprox (x) = 0.
2.4 Enrichment procedure
The enrichment procedure in 3-D XFEM is relativelymore complicated than 2-D XFEM but the basic prin-ciple remains the same. In general, it is quite tedious tofind nearest point on a curve from an arbitrary point. Inliterature, an iterative procedure was adopted to definethe geometry of the crack surface while in this work;a new approach has been used for the modeling of acrack. In this approach, a crack front is divided intohundreds of curved splines to avoid an iterative solu-tion. Initially crack front is divided into many curvedsplines as shown in Fig. 2 (only thirteen points areshown). The points on the piecewise splined curveare given by f1, f2, f3, f4. . .. . .. . .. . .. . .. . .fi−2, fi−1, fi .Any point on the spline segment of the crack front isgiven by fi,t = t ×fi +(1− t)×fi+1 where t is the frac-tional length on the interpolating curve. To check accu-racy and computational cost associated with the pro-posed approach, three benchmark problems are solvedby three different approaches namely iterative, linearsegment and curve segment. In iterative approach, anearest point on the crack front from an evaluation point(Gauss point) inside a crack front element, is calculatedby applying iteration procedure (Gravouil et al. 2002;Moës et al. 2002) whereas in linear and curve segmentapproaches, the crack front is divided into two hun-dred (present work) linear or curve segments, then for
each segment, a nearest point from a Gauss point isselected using a local coordinate system. This nearestpoint lies on the crack segment. An orthogonal curvi-linear coordinate system is used to define the nearestpoint on the crack front. The components of derivativeof level set functions are required with respect to theglobal coordinate system. To explain this, consider anarbitrary three-dimensional crack as shown in Fig. 2.The geometry of the crack front can be represented bythe position vector r(x) pointing from the origin of the(X,Y, Z) global Cartesian coordinate system to somepoint t on the crack front. The mathematical descrip-tion of the crack front and crack surface can be writtenas
r(x) = 0 (8)
f (x) = 0 (9)
A curvilinear coordinate system denoted by ξ1, ξ2 andξ3 is used, where ξ1 is taken along the outward normalto the crack front, and lie in the plane of crack, ξ3 istaken perpendicular to crack surface, and is named asthe gradient of the crack surface, whereas ξ2 is tangentto the crack front, and is calculated by vector multipli-cation of ξ1 and ξ3 (the direction of ξ2 is taken parallelto the tangent on the curve),
ξ2 = ξ3 × ξ1 (10)
Let ξ1 = a1i + b1 j + c1k, ξ2 = a2i + b2 j + c2k, ξ3 =a3i + b3 j + c3k where i, j, k are unit vectors in X,Yand Z directions, respectively. The foot of the perpen-dicular from a point/node on the surface is denoted byx f . Now, a local vector from the foot to a point P canbe written as
f(x) = (x f − x) (11)
Local coordinate of a point with respect to the crackfront can be represented by
x = f(x).ξ1, y = f(x).ξ2, z = f(x).ξ3 (12)
Due to orthogonal coordinate system, z automaticallybecomes zero, so r and θ can be calculated in terms ofx and y as
r =√(
x2 + y2), θ = tan−1
(y
x
)(13)
where, x and y are the function of (X,Y, Z). The cal-culation of ∂r
∂X ,∂θ∂X ,
∂r∂Y ,
∂θ∂Y and ∂r
∂Z ,∂θ∂Z is more com-
plicated for 3-D cracks as compared to 2-D cracks.In order to simplify the differentiation with respect to
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3-D cracks simulations 193
(X,Y, Z), interpolation of x and y is performed afterfinding the values of x and y at each node. It can be eas-ily approximated using partition of unity. This processis performed in small bend of elements
x (ξ, η, ζ )approx =i=n∑i=1
xi Ni (ξ, η, ζ ) (14a)
y (ξ, η, ζ )approx =i=n∑i=1
yi Ni (ξ, η, ζ ) (14b)
where, (ξ, η, ζ ) are the coordinates of a point in thenatural coordinate system. After creating a local polarcoordinate system, r and θ can be easily differenti-ated by using the derivatives of shape functions. In thepresent work, the quadratic shape functions are usedto approximate the level set functions while the linearshape functions are used to approximate the solution.
r =√
x (ξ, η, ζ )2approx
+ y (ξ, η, ζ )2approx
,
θ = tan−1
(y (ξ, η, ζ )approx
x (ξ, η, ζ )approx
)(15)
2.5 Weak formulation
The potential energy function for a solid mechanicsproblem can be expressed as
(u)= 1
2
∫
�
ε(u):C : ε(u)d�−∫
�
b.ud�−∫
�t
t.ud�
(16)
By substituting the trial and test functions from Eq. (4)in the above equation and using the arbitrariness of thenodal variables, a following set of discrete equations isobtained
[K] {d} = {f} (17)
where, d is the vector of nodal unknowns, K and f arethe global stiffness matrix and external force vector.The elemental K and f vectors are given as
fe = {fui fa
i fb1i fb2
i fb3i fb4
i
}T(18)
Kei j =
⎡⎢⎢⎣
Kuui j Kua
i j Kubi j
Kaui j Kaa
i j Kabi j
Kbui j Kba
i j Kbbi j
⎤⎥⎥⎦ (19)
The stiffness matrix and force vector are computedon element level, and then assembled into their globalcounterparts through standard finite element assemblyprocedure. The additional degrees of freedom arise dueto the enrichment, are handled by considering the ficti-tious nodes. The elemental contribution of K and f aregiven as
Kαβi j =
∫
�e
(Bαi)T CBβj d�(α, β = u, a, b) (20)
fui =
∫
�t
Ni td� +∫
�e
Ni bd� (21)
fai =
∫
�t
Ni (H(x)− H(xi )) td�
+∫
�e
Ni (H(x)− H(xi ))bd� (22)
fbαi =
∫
�t
Ni (φα(x)− φα(xi )) td�
+∫
�e
Ni (φα(x)− φα(xi ))bd� (23)
where, α = 1, 2, 3, 4 and Ni is finite element shapefunction.
In the present work, shifted enrichment (Zi andBelytschko 2003) is used along with partition of unity
Standard Gauss quadrature requires a smooth poly-nomial in the integrand. Hence, an enriched elementrequires special treatment for the integration due to dis-continuity in the integrand. The numerical integrationin these elements is performed by dividing it into sev-eral tetrahedrons, both above and below the crack sur-face. To achieve this, first a point of intersection of crackfront with element surfaces and element edges is calcu-lated (Sukumar et al. 2003), then a higher order Gaussquadrature is used for the integration in the enrichedelements. In the present simulations, seven and threepoints Gauss quadrature scheme is used in those ele-ments which are intersected by crack front and cracksurface, respectively. The rest of the elements in theproblem domain are integrated by two points Gaussquadrature rule.
2.7 Domain form of J -integral
The individual stress intensity factors (K I , K I I andK I I I ) are obtained under mixed-mode loading condi-tions using domain based interaction integral approach.The domain for J -integral is shown in Fig. 3. The evalu-ation of J -integral in 3-D is based on the virtual domainextension approach (Moran and Shih 1987). In thisapproach, a crack front contour integral is expressed interms of volume integral over the domain surroundingthe crack front. Two states of the stress are superim-
posed with each other to extract the individual SIFs.For convenience, one state (auxiliary state) is assumedto be known while other one is the actual state. TheSIF of the known auxiliary state is taken one for themode which is being evaluated while the SIFs of othertwo modes in auxiliary states are assigned zero values.Thus, the actual SIF remains unknown correspondingto that particular mode for which one value is assignedin auxiliary state, and can be obtained by solving an
Evaluation
b
b
Ve
Fig. 3 Hollow cuboidal domain around penny crack
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3-D cracks simulations 195
interaction integral (Moës et al. 2002; Moran and Shih1987; Shih and Asaro 1988; Gosz et al. 1998; Goszand Moran 2002). This process is repeated for the othermodes.
The path independent J -integral for a homogeneouscracked body is given as
J =∫
�
(Wδ1 j − σi j
∂ui
∂x1
)n j dΓ (28)
As the crack surface is traction free hence J -integralwill not be evaluated on the crack surface. The surfacecontour integral is virtually extended along the crackfront as shown in Fig. 3. But for the numerical cal-culation, this contour is used in different way. GreenDivergence theorem is used to convert a closed surfaceintegral into volume integral by taking its gradient asgiven below
J ′ =∫
�1
[Wδ1 j − σi j
∂ui
∂x1
]qn j d� (29)
J ′ =∫
�1
Pi j qn j d� wherePi j = Wδ1 j − σi j∂ui
∂x1(30)
Using divergence theorem on J ′ = ∫�1
Pi j qd�, Eq. (30)
can be written as
H ′ =∫
V
[(P. �∇q
)+( �∇.PT
).q]dV
−∫
C++C−(q.P.m) dC −
∫
S++S−(q.P.m)d S)
(31)
where, m is the outward normal to the closed contourfor H . It is assumed that crack surfaces are tractionfree therefore, the integration on crack surface becomeszero. The value of weight q is chosen such that itbecomes zero on the surface of virtually extended con-tour thus third and fourth term in Eq. (31) becomeszero. Hence, Eq. (31) becomes
H ′ =∫ [(
P. �∇q)
+( �∇.PT
).q]dV (32)
where, �∇.PT = 0 for straight crack fronts and �∇.PT �=0 for curved crack fronts. The main difficulty in calcu-lating the interaction integral lies in the evaluation ofthe gradients and higher order derivatives of the aux-iliary fields for curved crack front. The calculation ofauxiliary field is same as the calculation of enrichment
function. The auxiliary field can be calculated by twomethods, an analytical approach (applicable for somesimple shape cracks), and an approximate approachbased on partition of unity. In this work, auxiliary fieldis approximated using partition of unity approach
σ aux ( j, k) ≈n∑
i=1
σ auxi ( j, k)Ni (X,Y, Z) (33)
where, i is the node number
uaux ≈n∑
i=1
uauxi Ni (X,Y, Z) (34)
After approximating the gradient of auxiliary field, itcan be easily evaluated. The weight, q used in Eq. (32)is defined as
q =⎧⎨⎩
0 on S0
1 on St
arbi trary otherwise(35)
where, S0 is the outer contour at crack front and St isthe inner contour at crack front.
In 3-D crack problems, independent mesh is requiredaround a point on the crack front where SIF needs tobe computed. In this work, q is defined in such a waythat it can be easily interpolated with the help of shapefunctions:
q =N∑1
qi Ni (36)
The variation of weight function q can be taken linearor higher order along the length of J -contour. In thiswork, a hollow cuboidal contour is used for the evalua-tion of J integral. A hollow cuboidal contour for pennyshape crack is shown in Fig. 3. For the evaluation ofJ -integral, a mesh of 4×2×2 is created by eight nodebrick elements. Top and front views of the contour areplotted in Fig. 3. In this work, q is assumed to varylinearly in all the directions, q = 1 for the nodes lyingon the inner contour St and q = 0 for the nodes lyingon outer contour S0. Finally, J is computed as
J (t) = H
V e/2(37)
where, t is an evaluation point
2.8 Interaction integral
J−integral defined in Eq. (37) is equal to the energyrelease rate for elastic materials. For a general mixed-mode case, it is given as
123
196 H. Pathak et al.
J = G = 1
E ′(
K 2I + K 2
I I
)+ 1
2μK 2
I I I (38)
where,
E ′ = E
1 − ν2 and μ = E
2 (1 + ν)(39)
The individual stress intensity factors K I , K I I andK I I I are required to compute the direction of fatiguecrack growth. Consider two independent equilibriumstates of a cracked body. State 1 is taken as actual statefor the given boundary conditions while state 2 is anauxiliary state. The J -integral for two superposed statesis given as
J tot =∫
�
⎡⎣W tot δ1 j −
(σ(1)i j +σ (2)i j
) ∂(
u(1)i +u(2)i
)
∂x1
⎤⎦n j d�
(40)
where, W tot = 12
(σ(1)i j + σ
(2)i j
) (ε(1)i j + ε
(2)i j
)
Equation (40) can be expanded, and rearranged toyield
J tot = J (1) + J (2) + M (1,2) (41)
In Eq. (41), J (1), J (2) are the J -integral for states 1 and2 and M (1,2) is the interaction integral for two equilib-rium states.
M (1,2) (t) = 2(1 − ν2
)E
[K I K aux
I + K I I K auxI I
]
+ 1
2μK I I I K aux
I I I (42)
3 Numerical results and discussion
Ten different cases of 3-D cracks in linear elastic mate-rials are taken, and solved by XFEM using curvecrack segment approach. To accurately model thecrack front, initially three benchmark problems namelypenny crack, inclined penny crack and lens shape crackare solved by three different approaches namely iter-ative, linear segment and curved segment. Next, anedge crack in finite size cube is solved by XFEM.Four separate cases involving cuboid with differentcrack geometries: square shape, inclined surface semi-elliptical, cosine-wave and arbitrary shape spline havebeen solved by curved segment approach. Finally, twocases of pipe bend with axial elliptical cracks lying atdifferent locations are taken for the simulation. Extrin-sic enrichment technique based on partition of unity
MPa100=σ
XY
Z
Fig. 4 Penny shape crack in finite domain
has been used to capture the discontinuity present inthe computational domain. To accurately model thecurved crack front, it is divided into two hundred curvesegments in all the problems under consideration. Theconvergence study has been performed for four bench-mark crack problems by taking five different sets ofnodes. The error in terms of energy norms has also beenpresented for these problems. The simulations are per-formed on HP Z210 Workstation with Intel Xeon CPUE31245 @ 3.30 GHz (4 GB RAM) machine.
3.1 Penny shape crack in finite domain
A cuboid of size 2 m × 2 m × 2 m containing a pennyshape crack is shown in Fig. 4. The top surface of cuboidis subjected to a uniform traction of σ = 100 MPa inZ -direction. The radius (r) of the penny is taken as0.1 m. Young’s modulus (E) and Poisson ratio (υ) aretaken as 200 GPa and 0.3, respectively. The computa-tional domain is discretized by eight node linear brickelement. This problem is solved by iterative, linear seg-ment and curve segment approaches. The analyticalresults i.e. SIF values for this problem are available inReferences Green and Sneddon (1950), Irwin (1962).
K I = 2σ
√r
π, K I I = 0, K I I I = 0, (43)
To check convergence of iterative, linear segment,curve segment approaches, the problem is solvedfor five different sets of nodes i.e. (16 × 16 × 16),(20 × 20 × 20), (24 × 24 × 24), (28 × 28 × 28) and(32 × 32 × 32). The error in energy norm has beenpresented in Fig. 5 for all three approaches. The results
123123
3-D cracks simulations 197
0.5 1 1.5 2 2.5 3 3.5
x 104
-5.5
-5
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
Number of nodes
log e (
Err
or in
ene
rgy
norm
)
Energy Norm (Linear Segment)
Energy Norm (Curve Segment)
Energy Norm (Iteration)
Fig. 5 Energy norm in K I for penny shape crack
0 10 20 30 40 50 60 70 80 9034
35
36
37
38
39
40
angle (degree)
SIF
s (
MP
a-m
1/2 )
K I Analytical
KI XFEM (Linear Segment)
KI XFEM (Curve Segment)
KI XFEM (Iteration)
Fig. 6 Variation of K I (MPa-m1/2) with J -domain position
Fig. 7 Stress contour of σzz for a penny shape crack
MPa100=σ
θ
Fig. 8 Inclined penny crack in finite domain
0.5 1 1.5 2 2.5 3 3.5
x 104
-5.5
-5
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
Number of nodes
log e (
Err
or in
ene
rgy
norm
)Energy Norm (Linear Segment)
Energy Norm (Curve Segment)
Energy Norm (Iteration)
Fig. 9 Energy norm in K I for inclined penny shape crack
are converged for a mesh size of 32 × 32 × 32. Thisfigure also shows that the rate of convergence of thecurve segment approach is higher than the iterative andlinear segment approaches.
After performing the convergence study, the furtherresults are obtained for a mesh size of 32 × 32 × 32 tocompare the computational time. The values of mode-ISIF obtained from iterative, linear and curve segmentapproaches are plotted in Fig. 6 along with the analyt-ical values. Due to symmetry of crack geometry, thevalues of SIF are obtained and plotted for one quad-rant (0◦–90◦) of the crack front. From the SIF plot,it is observed that the numerical solution obtained bycurve segment approach is much closer to the analyti-cal solution as compared to iterative and linear segmentapproaches. The computational time required in solv-ing this problem by iterative, linear and curve segment
123
198 H. Pathak et al.
0.5 1 1.5 2 2.5 3 3.5
x 104
-5
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
Number of nodes
log e (
Err
or in
ene
rgy
norm
)
Energy Norm (Linear Segment)
Energy Norm (Curve Segment)
Energy Norm (Iteration)
Fig. 10 Energy norm in K I I for inclined penny shape crack
0.5 1 1.5 2 2.5 3 3.5
x 104
-5
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
Number of nodes
log e (
Err
or in
ene
rgy
norm
)
Energy Norm (Linear Segment)
Energy Norm (Curve Segment)
Energy Norm (Iteration)
Fig. 11 Energy norm in K I I I for inclined penny shape crack
approaches is found as 2,008, 106 and 141 min, respec-tively. The maximum error in SIF for curve segmentapproach is found to be 1.78 %. A stress field contourof σzz at the crack surface obtained by curve segmentapproach is plotted in Fig. 7.
These results show that the accuracy of curve seg-ment approach is better than the iterative and linearsegment approaches. Moreover, the time required byiterative scheme is found much higher than the linearand curve segment approaches.
3.2 Inclined penny shape crack in finite domain
An inclined penny shape crack in finite domain sub-jected to a uniform tensile load has been considered fornumerical simulation. The boundary conditions alongwith other parameters used in the crack modeling areshown in Fig. 8. The penny of radius r = 0.1 m is
0 50 100 150 200 250 300 35025
25.5
26
26.5
27
27.5
28
28.5
29
29.5
30
angle (degree)
SIF
s (
MP
a-m
1/2 )
K I Analytical
KI XFEM (Linear Segment)
KI XFEM (Curve Segment)
KI XFEM (Iteration)
Fig. 12 K I variation with angle or J -domain position
0 50 100 150 200 250 300 350-20
-15
-10
-5
0
5
10
15
20
angle (degree)
SIF
s (
MP
a-m
1/2 )
K II Analytical
KII XFEM (Linear Segment)
KII XFEM (Curve Segment)
KII XFEM (Iteration)
Fig. 13 K I I variation with angle or J -domain position
inclined by α = π/6 with the horizontal surface. Thevalues of Young’s modulus (E) and Poisson ratio (υ)are taken as 200 GPa and 0.3 for the simulation. Thedimensions of virtually extended volume integral aretaken as V e = 0.02 m and b = 0.015 m. To check theconvergence of iterative, linear segment, curve segmentapproaches, the problem is solved for five different setsof nodes i.e. 16 × 16 × 16, 20 × 20 × 20, 24 × 24 ×24, 28×28×28 and 32×32×32. The error in energynorm for mode-I, mode-II and mode-III are presented inFigs. 9, 10 and 11, respectively. The results presentedin these figures show that the rate of convergence ofproposed curve segment approach is higher than thelinear curve segment approaches, and is almost equal tothe iterative approach. Moreover, the results are almostconverged for a mesh size of 32 × 32 × 32.
After performing the convergence study, the furtherresults are obtained for a mesh size of 32 × 32 × 32 to
123123
3-D cracks simulations 199
0 50 100 150 200 250 300 350-15
-10
-5
0
5
10
15
angle (degree)
SIF
s (
MP
a-m
1/2 )
K III Analytical
KIII XFEM (Linear Segment)
KIII XFEM (Curve Segment)
KIII XFEM (Iteration)
Fig. 14 K I I I variation with angle or J -domain position
Fig. 15 Contour plot of σzz for an inclined penny shape crack
compare the computational time. The mode-I, mode-II and mode-III SIFs obtained by linear, iterative andcurve segment approaches for an embedded inclinedcircular crack are compared with the available analyti-cal solutions, and are presented in Figs. 12, 13 and 14.For the inclined penny shape crack, the analytical solu-tion is given in Reference Duflot (2006).
K I = 2σ cos2 α
√r
π(44a)
K I I = − 4
2 − υσ sin α cosα
√r
πsin θ (44b)
K I I I = −4 (1 − υ)
2 − υσ sin α cosα
√r
πcos θ (44c)
where, θ represents the crack front angular positionwhere SIFs need to be evaluated. From the results pre-sented in Figs. 13 and 14, it is found that the mode-II andmode-III SIFs vary similar to the variation of sine and
MPa
100=
σMPa100=σ
MPa
100
=σ
MPa100=σ
Fig. 16 Lens shape crack in finite domain
0.5 1 1.5 2 2.5 3 3.5
x 104
-5.5
-5
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
Number of nodes
log e (
Err
or in
ene
rgy
norm
)Energy Norm (Linear Segment)
Energy Norm (Curve Segment)
Energy Norm (Iteration)
Fig. 17 Energy norm in K I for lens shape crack
cosine functions. The values of SIFs obtained by curvesegment approach are found quite close to the analyti-cal solution as compared to iterative and linear segmentapproaches. The computational cost associated with theiterative, linear segment and curve segment approachesare found as 2,185, 1,316 and 1,515 min, respectively.The s,tress contour of σzz is obtained by curve segmentapproach is plotted in Fig. 15, which clearly shows asingularity in the stress field.
3.3 Lens shape crack in finite domain
A lens shape crack in finite cuboidal domain has beensimulated under hydrostatic tensile load. The bound-ary conditions along with the other parameters used
123
200 H. Pathak et al.
0.5 1 1.5 2 2.5 3 3.5
x 104
-5
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
Number of nodes
log e (
Err
or in
ene
rgy
norm
)
Energy Norm (Linear Segment)
Energy Norm (Curve Segment)
Energy Norm (Iteration)
Fig. 18 Energy norm in K I I for lens shape crack
0 10 20 30 40 50 60 70 80 9036
36.5
37
37.5
38
38.5
39
39.5
40
40.5
41
angle (degree)
SIF
s (
MP
a-m
1/2 )
K I Analytical
KI XFEM (Linear Segment)
KI XFEM (Curve Segment)
KI XFEM (Iteration)
Fig. 19 K I variation with angle or J -domain position
in modeling of lens shape crack are shown in Fig. 16.The dimensions of cube are taken as L = 2 m,W =2 m, H = 2 m. The radius and azimuthal angle forcrack surface are r = 0.2m, and α = π
4 . Young’smodulus E = 68.9 GPa and Poisson ratio, υ = 0.22are taken from the literature (Moës et al. 2002). Thefinite element mesh is created using eight node brickelements. The dimensions of cuboid shape volume inte-gral are taken as Ve = 0.03 m and b = 0.075 m. Toperform a convergence study, the problem is simulatedby iterative, linear crack segment and curve crack seg-ment approaches for five different sets of nodes i.e.16×16×16, 20×20×20, 24×24×24, 28×28×28 and32×32×32. The error in energy norm for mode-I andmode-II SIFs is presented in Figs. 17 and 18, respec-tively. These figures show the rate of convergence forproposed curve segment approach is better than the lin-ear segment approach and is comparable to the iterativeapproach.
0 10 20 30 40 50 60 70 80 905
6
7
8
9
10
11
12
13
14
angle (degree)
SIF
s (
MP
a-m
1/2 )
K II Analytical
KII XFEM (Linear Segment)
KII XFEM (Curve Segment)
KII XFEM (Iteration)
Fig. 20 K I I variation with angle or J -domain position
Fig. 21 Stress contour plot of σzz for lens shape crack
After performing the convergence analysis, thisproblem is simulated by iterative, linear crack segmentand curve crack segment approaches for a mesh sizeof 32 × 32 × 32, and the results are compared withavailable analytical solution (Gosz and Moran 2002).The analytical solution for a lens-shaped crack in aninfinite solid body (Gosz and Moran 2002) subjectedto a uniform hydrostatic tensile load is given as.
K I = 0.877 × 2σ
√a
π(45a)
K I I = 0.235 × 2σ
√a
π(45b)
where, a = r cosα.The values of K I and K I I are presented in Figs. 19
and 20, respectively for 0 ≤ θ ≤ 90. From the SIFsplot, it has been found that mode-I SIF is more criticalas compared to mode-II and Mode-III SIFs. It is alsoclear that mode-I results obtained by curve segment
123123
3-D cracks simulations 201
MPa1=σ
W
Lt
Fig. 22 Edge crack in finite domain
400 600 800 1000 1200 1400 1600 1800 20000.4
0.42
0.44
0.46
0.48
0.5
0.52
0.54
0.56
Number of nodes
KI (
MP
a-m
1/2 )
K I Analytical
KI XFEM
Fig. 23 K I variation with number of nodes for an edge cracklength of 5 cm
approach are more close to the analytical solution incomparison to iterative and linear segment approaches.The computational time involved in numerical simu-lation using iterative, linear crack segment and curvecrack segment approaches is found as 2,050, 966 and1,028 min, respectively. The stress field contour of σzz
at crack surface is shown in Fig. 21, which clearlyshows a singularity in the stress field at the crack front.
On the basis of results obtained for these threesproblems, it has been found that the iterative approachis most expensive as compared to linear and curvecrack segment approaches. The accuracy of curve seg-ment approach is found nearly same as that obtainedby iterative approach. Although, linear crack segment
400 600 800 1000 1200 1400 1600 1800 20003.8
3.85
3.9
3.95
4
4.05
4.1
4.15
4.2
Number of nodes
KI (
MP
a-m
1/2 )
K I Analytical
KI XFEM
Fig. 24 K I variation with number of nodes for an edge cracklength of 17 cm
0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.180
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Crack Length (m)
KI (
MP
a-m
1/2 )
K I Analytical
KI XFEM
Fig. 25 Variation of K I (in MPa-m1/2) with crack length
approach takes minimum time but its results are notas accurate as obtained by curve segment approach.Therefore, proposed curve segment approach is mostsuitable in terms of accuracy and computational time,and has been used further for rest of the problems.
3.4 Edge crack in finite domain
A rectangular borosilicate glass plate of dimensionW = 0.25 m and L = 0.50 m with thickness t = 0.05 mhas been simulated using curve segment approach fordifferent length of edge cracks. The problem geome-try along with boundary condition is shown in Fig. 22.The top surface of the geometry is subjected to a uni-form traction of σ = 1 MPa while the bottom surfaceis constrained in all the directions. Young’s modulusand Poisson ratio are taken E = 627 GPa and υ = 0.2
123
202 H. Pathak et al.
Fig. 26 Stress contour plot of σzz for an edge crack length of15 cm
MPa100=σ
Fig. 27 Square shape crack in finite domain
for simulation. The analytical solution of an edge crackproblem is obtained from the following reference Khoeiet al. (2013) as:
K I = f (a/W )σ√
a (46)
f (a/W ) = 1.99 − 0.41( a
W
)+ 18.7
( a
W
)2
−38.48( a
W
)3 + 53.85( a
W
)4(47)
0.9 0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08 1.1
25
30
35
40
45
Distance along one edge
SIF
s (
MP
a-m
1/2 )
K I XFEM (Curve Segment)
Fig. 28 K I variation with distance along one of the crack edge
Fig. 29 Stress contour plot of σzz for square shape crack
The problem geometry is discretized using eight nodelinear brick elements. The results obtained using 6 ×12×4, 8×16×4, 10×20×4, 12×24×4, 16×32×4 setof nodal data are presented in Figs. 23 and 24 for a cracklength of a = 0.05 m and a = 0.17 m, respectively.From the plotted SIFs values, it has been found thatXFEM solution starts converging towards the analyti-cal solution with the increase in mesh size/nodes. TheSIF values obtained using a mesh size of 16×32×4 arepresented in Fig. 25 for various values of crack length.Figure 25 shows that XFEM solution is found quiteclose to the analytical solution. Since, these simulationsare performed for a finite geometry; hence small devia-tion between XFEM solution and analytical results hasbeen observed. The stress field contour of σzz at thecrack surface is shown in Fig. 26 for a crack length
123123
3-D cracks simulations 203
MPa100=σ
Fig. 30 Inclined surface elliptical crack in finite domain
-80 -60 -40 -20 0 20 40 60 80
-40
-20
0
20
40
60
80
100
angle (degree)
SIF
s (
MP
a-m
1/2 )
K I XFEM (Curve Segment)
KII XFEM (Curve Segment)
KIII XFEM (Curve Segment)
Fig. 31 Variation of K I , K I I and K I I I with angle
of a = 0.15 m. From the stress contour, stress fieldsingularity can be clearly visualized.
3.5 Square shape crack in finite domain
A square shape crack in finite size cuboidal domain hasbeen simulated using curve segment approach under thetensile loading. The boundary conditions along with theother parameters (W = 2 m, H = 2 m and t = 2 m,and a = H/10) are shown in Fig. 27. Young’s mod-ulus and Poisson ratio are taken as E = 200 GPa areυ = 0.22 for the simulation. The geometry is uni-formly discretized by 32 × 32 × 32 nodes. A uniformtensile traction of σ = 100 MPa is applied over the
Fig. 32 Stress contour plot of σzz for inclined surface ellipticalcrack
MPa100=σ
Fig. 33 Cosine wave crack in finite domain
0.9 0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08 1.1
-10
-5
0
5
10
15
20
25
30
35
40
distance (m)
SIF
s (
MP
a-m
1/2 )
K I (Curve Segment)
KII (Curve Segment)
KIII (Curve Segment)
Fig. 34 Variation of mode-I, mode-II and mode-III SIFs alongcrack edge
123
204 H. Pathak et al.
Fig. 35 Stress contour plot of σzz for cosine wave shape crack
top surface. The virtually extended volume interactionintegral has been implemented to obtain stress intensityfactors. The dimensions of virtually extended cuboidvolume integral contour are taken as V e = 0.03 m andb = 0.05 m. The mode-II and mode-III SIFs are foundquite small as compared to mode-I SIFs, hence onlymode-I SIFs is plotted in Fig. 28. From Fig. 28, themaximum value of SIF is found at the middle of thecrack edge, while the minimum value of SIF is foundat the corner of the crack surface. The stress field con-tour of σzz is plotted in Fig. 29 at crack surface. Thestress contour shows the singularity in stress field atthe crack front; which also describes the accuracy ofthe proposed XFEM for square shape planner crack.
3.6 Inclined surface elliptical crack in finite domain
In this sub-section, a non-planar semi-elliptical sur-face crack lying in finite size cuboid is numericallysolved using curve crack segment approach under ten-sile loading. The boundary conditions along with otherparameters are shown in Fig. 30. For this non-planarcrack surface, semi minor axis and semi major axisare taken as 0.45 and 0.75 m, respectively. The cracksurface is tilted by 30◦ from horizontal. Young’s mod-ulus and Poisson ratio are taken as E = 200 GPa andυ = 0.3, respectively. The complete geometry is uni-formly discretized using 32×32×32 nodes. A uniformtensile stress of σ = 100 MPa has been applied overthe top surface. The dimensions of virtually extendedcuboid volume integral are taken as V e = 0.03 m andb = 0.05 m. The mode-I, mode-II and mode-III SIFsvalues are presented in Fig. 31 along the crack frontfrom one end of the crack front to other end of the crack
front. The maximum value of mode-I SIF is found at0◦. A symmetrical pattern is observed from the plots ofmode-I and mode-II, whereas mode-III SIFs increaseslinearly from −40.2353 MPa
√m to 39.7161 MPa
√m.
The stress field contour of σzz is plotted in Fig. 32 atthe crack surface.
3.7 Cosine wave crack in finite domain
A cuboid of dimensions W = 2 m, H = 2 m andt = 2 m having a cosine wave surface crack lying atthe center is considered for the simulation. This crackproblem is solved using curve crack segment approach.The amplitude and length of cosine wave is taken as0.04 and 0.2 m, respectively. The boundary conditionsalong with other parameters are shown in Fig. 33. Thedomain is uniformly discretized by 32×32×32 nodesusing eight node linear brick elements. Young’s mod-ulus and Poisson’s ratio are taken as 200 GPa and 0.3,respectively. A uniform tensile traction of 100 MPa isapplied at the top surface of the cuboid. The stressintensity factors are obtained along the edge of thecrack surface. Figure 34 presents the mode-I, mode-II and mode-III SIFs. Initially, there is no significantchange in K I near crack edge, but at distance of 0.06 mfrom the centre of crack surface along the edge, K I
starts increasing and reached to a maximum value atthe centre then start decreasing, whereas after 0.06 m,K I I and K I I I starts decreasing, and reached to min-imum value at the centre. The stress field contour ofσzz is plotted in Fig. 35 at the crack surface. Fromthis plot, the singularity in stress field can be clearlyvisualized.
3.8 Axial elliptical surface crack in pipe bend
Next, pipe bend is considered as an important indus-trial component, which is commonly used in nuclear,power, automobile, aerospace and fuel industries. Linand Smith (1998) found that during fatigue loading, aninitial crack of semi-elliptical shape grows up to thecritical size. Therefore, the analysis of semi ellipticalcrack in pipe bend is important from the failure point ofview. In this section, semi-elliptical cracks placed at theouter and inner surfaces of the circular pipe have beenconsidered. Outer diameter, inner diameter and thick-ness of the pipe bend are taken as 0.2, 0.16 and 0.02 m,
123123
3-D cracks simulations 205
m0. 61=φm0.2=φ
m0. 61=φ
m0.2=φ
θ
R=0.3 m
Fig. 36 Pipe bend crack at extrados
respectively. Young’s modulus and Poisson’s ratio aretaken as 200 GPa and 0.3, respectively. An eight nodelinear brick element has been used for meshing thegeometry. The dimensions of J -contour are taken asV e = 0.03 m and b = 0.0018 m. Twelve elementsare taken along the thickness of the pipe bend. Theanalysis of surface cracks in pipe bend subjected to aninternal pressure has been performed using curve seg-ment approach. Under this type of loading, mode-I SIFdominates the failure while mode-II and mode-III arefound almost negligible.
3.8.1 Pipe bend with axial elliptical surface crackat outer surface of extrados
A pipe bend with an axial semi-elliptical crack lying atouter surface has been simulated by proposed XFEMapproach. The complete geometry along with bound-ary conditions are shown in Fig. 36. Both ends of thepipe bend are hinged. The minor and major axis ofsemi-elliptical crack are taken as 0.015 and 0.024 m.The pipe bend is subjected to an internal pressure of
0 10 20 30 40 50 60 70 80 902
3
4
5
6
7
8
9
10
11
12
angle (degree)
SIF
s (
MP
a-m
1/2 )
K I XFEM (Curve Segment)
Fig. 37 Variation of K I with angle θ for crack at extrados
10 MPa. The mode-I SIF dominate in comparison tomode-II and mode-III SIFs. The mode-I SIF resultsare presented in Fig. 37. Due to symmetry in SIFs,the results are presented only for θ = 0◦–90◦. Themaximum values of SIF is found as 9.169 MPa
√m at
θ = 0◦. The variation of SIF with θ is shown in Fig. 37.
123
206 H. Pathak et al.
θ
m0. 61=φ
m0.2=φ
m0. 61=φ
m0.2=φ
m0.3=R
Fig. 38 Pipe bend crack at intrados
0 10 20 30 40 50 60 70 80 902
4
6
8
10
12
14
16
18
20
22
angle (degree)
SIF
s (
MP
a-m
1/2 )
K I XFEM (Curve Segment)
Fig. 39 Variation of K I with angle θ for crack at intrados
3.8.2 Pipe bend with axial elliptical surface crackat inner surface of extrados
A pipe bend, subjected to an internal pressure withsemi-elliptical crack lying at inner surface is solved byproposed XFEM. The geometry along with the bound-ary conditions is shown in Fig. 38. Both ends of the
pipe are hinged. The dimensions of crack are same asin the previous problem. The mode-I SIF results arepresented in Fig. 39. Due to symmetry of SIFs aboutθ = 0◦, the results are presented only for θ = 0◦–90◦.From the SIF plot, it is seen that there is a steep rise inSIF beyond θ = 50◦.
3.9 Arbitary spline shape crack in finite domain
Finally, a cuboid with arbitrary spline shape crack lyingat centre has been simulated by XFEM. The outer cracksurface has been created manually. Total 26 arbitraryvirtual points are created and joined to each other toform arbitrary spline shape. After Delaunay triangu-lation of closed cracked boundary, an arbitrary cracksurface has been formed. The crack front geometry hasbeen simulated using curve crack segment approach.The vector level set has been implemented to locateenriched nodes. The cracked geometry along withboundary conditions is shown in Fig. 40. Young’s mod-ulus (E) and Poisson’s ratio (υ) are taken as 200 GPaand 0.3, respectively. A uniform tensile traction (σ )
123123
3-D cracks simulations 207
0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
Fig. 40 Arbitrary spline shape crack in finite domain
0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1
510
15
20
Fig. 41 J -domain position along the crack surface
of 100 MPa is applied at the top surface. The dimen-sions of virtually extended cuboid volume are taken asV e = 0.04 m and b = 0.08 m. The geometry is uni-formly discretized by 32 × 32 × 32 nodes. The valuesof SIFs are obtained along the outer curve of the cracksurface. The position of J -domain along the crack edgehas been presented in Fig. 41. Twenty five points areselected on the outer curve of crack surface for the eval-uation of SIFs. The values of mode-I SIF are presentedin Fig. 42.
0 5 10 15 20 250
10
20
30
40
50
60
70
80
90
100
J-domain Position on crack edge
SIF
s (
MP
a-m
1/2 )
K I XFEM (Curve Segment)
Fig. 42 Variation of K I with J -domain position along crackedge
4 Conclusions
In the present simulations, a simple and efficientextended finite element approach has been presentedto solve three-dimensional crack problems. Few 3-Dcrack problems were simulated by proposed curvedcrack segment, linear crack segment and iterativeapproaches. The results obtained by curve segmentapproach were compared with those obtained by linear
123
208 H. Pathak et al.
segment and iterative approaches. In the curve segmentapproach, the crack front was divided into many cracksegments to avoid an iterative solution. Some vectoroperations were performed to make the present com-putations fast and accurate. The auxiliary fields wereapproximated using higher order interpolation func-tions for an easy calculation of gradient and higherorder derivatives. Planar, non-planar and arbitrary 3-Dcrack problems were solved by XFEM. A convergencestudy was conducted for few benchmark crack prob-lems. The results obtained by proposed curve cracksegment approach were found in good agreement withthe analytical solutions. Also, the curve crack seg-ment approach is found more accurate as comparedto iterative and linear segment approaches. The pro-posed curve crack segment approach takes much lesstime as compared to iterative approach, while it takesslightly more time as compared to proposed linear seg-ment approach. Although, the linear segment approachtakes minimum time but its results are less accurate ascompared to the curve crack segment approach.
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