Abstract Distance fields are scalar functions defining theminimum distance of a given point in the space from theboundary of an object. Crack surfaces are geometric entitieswhose shapes can be arbitrary, often described with ruledsurfaces or polygonal subdivisions. The derivatives of dis-tance functions for such surfaces are discontinuous acrossthe surface, and continuous all around the edges. These prop-erties of the distance function were employed to build intrin-sic enrichments of the underlying mesh-free discretisationfor efficient simulation of three-dimensional crack propaga-tion, removing the limitations of existing criteria (reviewedin this paper). Examples show that the proposed approachis able to introduce quite convoluted crack paths. The incre-mental nature of the developed approach does not requirere-computation of the enrichment for the entire crack sur-face as advancing crack front extends incrementally as a setof connected surface facets. The concept is based on purelygeometric representation of discontinuities thus addressingonly the kinematic aspects of the problem, such to allow forany constitutive and cohesive interface models to be used.
E. Barbieri (B)School of Engineering and Materials Science, Queen MaryUniversity of London, Mile End Road, London E1 4NS, UKe-mail: [email protected]
N. PetrinicDepartment of Engineering Science, University of Oxford, Parks Road,Oxford OX1 3PJ, UKe-mail: [email protected]
1 Introduction
Meshfree methods for the solution of partial differentialequations in elasticity have come a long way since the veryfirst papers of Libersky and Petschek [27] on smoothed par-ticle hydrodynamics (SPH), where a meshless method wasapplied for the first time in solid mechanics.
The original versions of SPH, however, lacked of theproperty of consistency (or reproducibility), especially atthe boundaries. The landmark papers by Belytschko and co-workers [7–9] on element-free Galerkin (EFG), contempo-rary with the papers by Liu and coworkers [11,28–30] onReproducing Kernel Particle Method (RKPM), opened theway to the widespread diffusion of meshfree methods forlinear and non-linear solid mechanics [34].
Both methods are substantially equivalent, though orig-inated from different points of view: EFG from computergraphics, RKPM from wavelet theory. In EFG, the shapefunctions are computed through moving least squares (MLS),a popular technique for surface reconstruction from a cloud ofscattered points [26]; in RKPM, shape functions are numeri-cal discretisations of a convolution integral between a kerneland polynomial basis, and the term reproducing derives fromtheir property of being able to reproduce all the monomialsin the basis.
Since then, many other meshfree methods were devel-oped. An incomplete list includes: the hp-clouds [14], finitepoint method [32], the free-mesh method [38], the meshlesslocal Petrov–Galerkin (MLPG) [2], local boundary integralequation (LBIE) [40], natural element method [36], meshlessfinite element method (MFEM) [23], the cracking particlesmethod [33]. For a complete review of these methods and thehistory, the interested reader can refer to [18,31].
This paper focuses on the EFG/RKPM as the basis forintroduction of a new efficient algorithm for introduction
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Fig. 1 Criteria fordiscontinuities in weightfunctions
(a) (b) (c)
Fig. 2 Differences indiscretisation and shapefunctions
(a) (b) (c)
of discontinuities into continuous interpolation functionsfor discretisation of three-dimensional domains. Early algo-rithms were only two-dimensional. Chronologically, the firstwas the visibility criterion [7], based on reciprocal visibilityof nodes hampered by a crack, where the boundaries of thebody and any interior lines of discontinuity are consideredopaque. In the visibility criterion the domain of influence forthe weight function from a node to a point is imagined to belike a ray of light, i.e. if the ray reaches an opaque line, forexample an interior discontinuity, the point is not includedin the support (Fig. 1a).
A major shortcoming of this method is that non-convexboundaries could not be treated properly (Fig. 1b), since itexcludes nodes or points that instead should be included. Amodification of this criterion is the diffraction method [6].When the ray encounters a crack line, the distance betweenthe two points is lengthened. The resulting support is showedin Fig. 1c.
Another technique for obtaining discontinuous approxi-mations is the transparency method [6], which is a combi-nation of the visibility and the diffraction. The transparencyis not uniform for the whole length but it changes so that itis completely transparent at the tip and becomes completelyopaque at a short distance from the tip. In this manner, therange of vision for each node near the crack tip is not sud-denly truncated when it reaches the crack tip, but diminishessmoothly to zero at a short distance from the tip. One incon-venience of the transparency method is that it does not workwell when nodes are placed near the crack surface.
A different way of introducing discontinuities, withoutusing the above mentioned criteria, falls into the category ofenriched weight functions [15–17]. These methods rely onmodification of the kernel for selected nodes, which requires
very little or no modification of the structure of the underlyingcomputer codes.
Following the philosophy of modified weight function,we proposed an intrinsic enrichment for meshfree methodsin [5], which is able to incorporate the finite nature of thecrack in the weight functions associated with the discreti-sation nodes, showing good accuracy for multiple arbitraryand branching cracks in two dimensions in an elastic isotropicbody. The novel aspect of the proposed method is due to thenature of the enriching function, which derives from the com-putation of distance fields and their derivatives for polygonallines. This approach is capable of handling heavily kinkedand branching cracks, with accurate stress intensity factors(SIFs) and optimal convergence rates.
In this paper, we will extend it to three dimensional cracksurfaces, following the same philosophy.
We will show its effectiveness and ease of treatment forthree-dimensional crack propagation, which to date remainsone of the most challenging aspects in numerical simulationfor industry. A similar philosophy is presented in [19], wheresigned distance functions are used to build extrinsic enrich-ments for extended finite element method (XFEM), with thedistinguishable differences clearly pointed out in this paper.
2 Discretisation of the governing equations
We will assume, for simplicity, an isotropic linear elas-tic body: however, the enrichment presented in this paperis applicable to shape functions in general, therefore com-pletely independent from the particular constitutive model.The choice of linear elasticity is due to the vast availabilityof reference solutions, experimental evidences and numeri-
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cal simulations, therefore it facilitates comparison of results.For the elastic problem, the equilibrium equations and theboundary conditions are, in absence of body forces:
∇ · σ = 0 x ∈ Ω (1)
σ · n = t = λt0 x ∈ Γt (2)
u = u x ∈ Γu (3)
where σ is the Cauchy stress tensor, n is the normal unity vec-tor of the boundary Γt where the traction λt0 is prescribedand Γu is the boundary where the displacement u is pre-scribed. The traction t0 is a unitary reference traction fieldwith magnitude λ.
Using the displacement u as a test function for Eqs. (1),(2) and (3), the variational form can be written as∫
Ω
δεT σdΩ−∫
Γt
δuT tdΓt +α
∫
Γu
δ (u−u)T (u−u) dΓu =0
(4)
where α is a penalty parameter (usually a large number) usedto enforce the essential boundary conditions. The field vari-able u(x) is approximated by the function uh which is dis-cretised by N particles
uh(x) =N∑I
φI (x)UI (5)
where φI : Ω → R is the I th shape function and UI is thenodal value located at the position xI .
Equation (5) is formally identical to a finite elements (FE)expansion: differently from FE, meshfree shape functionsare not Lagrangian interpolations defined element-wise, butreproducing kernels centered on each discretisation node,with compact support of radius ρI and without using anymesh (Fig. 2).
The discretisation used in this paper is known as repro-ducing kernel particle method (RKPM) [28], although it isequivalent with the MLS approximation used in the elementfree Galerkin (EFG) method [8].
The Ith shape function is given in RKPM by the formula
φI (x) = CI (x)w
(xI − x
ρI
)ΔVI (6)
where the corrective term CI (x) is [30]
CI (x) = pT (0)M(x)−1p(
xI − xρ
)(7)
Note that CI (x) is normally evaluated in a scaled andtranslated version to prevent ill-conditioning of the moment
Fig. 3 Example of a kernel: the 2kth order spline
matrix M(x)
M(x) =N∑
I=1
p(
xI − xρ
)pT(
xI − xρ
)w
(xI − x
ρ
)ΔVI
(8)
where ρ in Eqs. (7) and (8) is the average value of the compactsupport radii.
Equation (6) derives from a numerical discretisation of aconvolution integral [28], therefore the term ΔVI is a mea-sure (i.e. length, area or volume) of a sphere centered in xI .
The vector p is called basis function and it contains all thepolynomial functions reproduced by the approximation (6),i.e.
N∑I=1
φI (x)p j (xI ) = p j (x) j = 1, . . . , k (9)
with k the number of polynomial functions in the basis. Forinstance, a common basis function is usually the linear
p(x) = [1 x y]
(10)
Equation (9) expresses the reproducibility conditions: theability of reproducing p1(x) = 1 is called partition of unity.
The function w in (6) and (8) is called kernel or win-dow function: Examples of kernel functions are the 3rd orderspline
w(ξ) =
⎧⎪⎪⎨⎪⎪⎩
23 − 4ξ2 + 4ξ3 0 ≤ ξ ≤ 1
2
43 − 4ξ + 4ξ2 − 4
3ξ3 12 < ξ ≤ 1
0 ξ > 1
(11)
which is C2 or more generally the 2kth order spline
w(ξ) ={(
1 − ξ2)k 0 ≤ ξ ≤ 1
0 ξ > 1(12)
which is Ck−1 (Fig. 3 ). The order of continuity of a ker-nel function is important because it influences the order ofcontinuity of the shape functions.
Substituting Eq. (5) in (4) and assuming linear elasticstress–strain relationships, leads to
(K + αV) U = λFext + αFu (13)
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Fig. 4 Isosurfaces for d(3)b (x)
(a) (b) (c) (d)
Fig. 5 Isosurfaces for d(3)(x)
(a) (b) (c) (d)
where K is the stiffness matrix,
KI,J =∫
Ω
BTI DBJ (14)
where B is the linear strain–displacement matrix
BI =
⎡⎢⎢⎢⎢⎢⎣
∂φI
∂x0
0∂φI
∂y∂φI
∂y
∂φI
∂x
⎤⎥⎥⎥⎥⎥⎦
(15)
D is the tensor of elastic moduli (in Voigt notation),
σ = Dε (16)
V is the penalty matrix
VI,J =∫
Γu
φTI (x)φJ (x)dΓu (17)
Fext is the generalized force vector
FextI =
∫
Γu
φTI t0dΓu (18)
FuI =
∫
Γu
φTI udΓu (19)
The interested reader can refer to [4,13,31] for insights onpractical computer programming of the EFG or RKPM.
The key idea of the method is to substitute the kernelfunction w(x) in Eqs. (6) and (7) with a modified weightfunction w(x)h(x) where h(x) is the enrichment.
φI (x) = CI (x)w
(xI − x
ρ
)h(x, xI ) (20)
CI (x) = pT (0)M(x)−1p(
xI − xρ
)(21)
M(x)=N∑
I=1
p(
xI −xρ
)pT(
xI −xρ
)w
(xI −x
ρ
)h(x, xI )
(22)
3 A first example: the penny-crack enrichment
In this section, we introduce the philosophy of a distance-fieldbased enrichment through a very simple example: the pennycrack (or penny-shaped crack), i.e. a crack with a circular(or elliptical) front. Let us consider for simplicity a circularfront, as in Fig. 18, although it will become clear later thatthe generalization to elliptical crack fronts is straightforward.The crack is contained in a plane, with normal vector n,centre in C = [xC , yC , zC ] and radius a. Let us then choosetwo orthonormal vectors t and b, both orthogonal to n: for acircular penny-crack, t and b are two arbitrary vectors in theplane of the crack, for elliptical penny-crack, t and b are thedirections of the major and minor axis.
For a generic query point x = [x, y, z], the distance fromthe penny is calculated as follows: firstly, coordinates aretransformed⎡⎣ ξ
η
ζ
⎤⎦ =
⎡⎣ tT
bT
nT
⎤⎦⎡⎣ x − xC
y − yC
z − zC
⎤⎦ (23)
then, the signed distance field can be immediately computed
d (2)s (x) =
√ξ2 + η2 − a (24)
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where the superscript (2) refers to the two-dimensional dis-tance field. Two different distance functions can be derivedfrom (24): the absolute (or unsigned distance) d(2) and thepositive part (of the signed distance function) d (2)
+
d (2)(x) =∣∣∣d (2)
s (x)
∣∣∣ (25)
d (2)+ (x) =
d (2)s (x) +
∣∣∣d (2)s (x)
∣∣∣2
(26)
Equations (25) and (26) generate two different three-dimens-ional distance functions. These functions are the minimumdistance from the crack front d(3)
b (Eq. 27; Fig. 4) and theminimum distance from the penny crack d(3) (Eq. 28; Fig. 5).The first one is the focus of this paper, since it is useful toderive an enrichment; the second one is instead useful forcalculating the J-integral.
d(3)b (x) =
√d(2)(x)2 + ζ 2 (27)
d(3)(x) =√
d (2)+ (x)2 + ζ 2 (28)
Let us now define a phase function, as
ϕ(3) = ζ
d(3)(x)(29)
As it can be seen from the isosurfaces in Fig. 6, ϕ(3) isa function that is discontinuous over the penny crack, sinceon the upper side its value is one and on the lower side isinstead −1: in between the two sides, ϕ smoothly varies from−1 to 1. The phase function ϕ(3) in Eq. (29) is almost thesought function. Nonetheless, it is a suitable enrichment foran extrinsic approach, as in XFEM, but, in order to createa discontinuity with an intrinsic approach, the enrichmentshould value zero on the negative side of the crack. Never-theless, this is simply achieved by scaling and translating thefunction
h+(x) = ϕ(3)(x) + 1
2(30)
The function h+ provides the smooth transition from 0 to 1and it reflects the angular transition from the upper face tothe lower face of the crack.
It should be remarked that the expression (30) can be onlyused to enrich the nodes in the upper side of the crack. Indeed,applying (30) for the nodes on the opposite side of the crack,would simply result in zeroing their kernels. Therefore, inorder to apply effectively the enrichment, for these nodes it issufficient to apply the complement to one of (30) (Figs. 7, 8).
This operation is a sort of switch, as it selects for eachnode the appropriate enrichment: the final expression of htakes the following form
h(x, xI )=h+(x)H(ϕ(3)(xI )
)+(1 − h+(x)
)H (−ϕ(3)(xI ))
(31)
4 The enrichment for a planar polygonal crack face
Let us now extend the idea in Sect. 3 to a crack face of anarbitrary polygonal shape. The important aspect to remark isthat this extension does not necessarily require a subdivisionof the polygon in more elementary shapes, such as trianglesor quadrangles. The only difference is in the computation ofthe absolute distance function, as in Eq. (28). There is a vastamount of literature for the computation of the distance func-tion of three-dimensional objects [24]: for the scope of thiswork, a naive approach is sufficient, which is nonetheless fastenough for polygons with low number of edges. An impor-tant part of the algorithm is the computation of the signeddistance in the plane of the polygon, which, it is important tonote, does not need the computation of the level-sets throughthe numerical solution of the Eikonal equation [35]. Instead,it requires only the absolute distance function and an addi-tional information, which is if a generic query point is insideor outside the polygon. There are many algorithms availablein the literature to solve this problem, known as particle-in-polygon (the interested reader can refer to any textbook ofcomputational geometry, for example [12,21] ): here, againfor the scopes of this paper, we use a criterion based on thephase function, as introduced in Sect. 3.
Assuming that the polygon is contained in a plane withnormal n, assuming t parallel to one of the edges and usingb = t × n, the reference transformation in Eq. (23) is used.Then, assuming that the polygon has n edges, d (2)
+ (x) is com-puted using coordinates ξ and η, according to the followingalgorithm:
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
for i = 1, . . . , n
d(2)(ξ, η) = d(2)i (ti , si ) if i = 1
d(2)(ξ, η) = min(
d(2), d(2)i (ti , si )
)if i = 2, . . . , n
ϕ(2)(ξ, η) = si
d(2)i (ti ,si )
if i = 1
ϕ(2)(ξ, η) = m
(ϕ(2),
si
d(2)i (ti ,si )
, d(2), d(2)i (ti , si )
)if i = 2, . . . , n
end f or
(32)
where m(a, b, f, g) is a generalized min function as inEq. (53), d(2)
i (ti , si ) is the two-dimensional distance functionof the ith edge, where coordinates have been transformed inthe local system (ti , si ).
Indeed, let us consider the ith edge having endpoints ξ1
and ξ2. The tangent vector is
t = ξ2 − ξ1
|ξ2 − ξ1| = [ t1 t2]T
(33)
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Comput Mech
Fig. 6 Isosurfaces for ϕ(3)
(a) (b) (d)(c)
(e) (f) (g) (h)
Fig. 7 Isosurfaces for d(3)(x)
(a) (b) (c) (d)
Fig. 8 Isosurfaces for ϕ(3)(x)
(a)
(e) (f) (g) (h)
(b) (c) (d)
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Fig. 9 One-dimensional signeddistance function ds(t) and itspositive part d+
1 (t)
(a) (b)
Fig. 10 Surfaces for d(2)+ (ξ, η)
and ϕ(2)(ξ, η) at each step
0
0.5
1
1.52
2.5
33.5
4
(a)−1−0.8−0.6−0.4−0.200.20.40.60.81
(b)0
0.5
1
1.5
2
2.5
3
3.5
4
(c)−1−0.8−0.6−0.4−0.200.20.40.60.81
(d)
0.511.522.533.54
(e)−1−0.8−0.6−0.4−0.200.20.40.60.81
(f)
0.511.522.533.54
(g)−1−0.8−0.6−0.4−0.200.20.40.60.81
(h)
0.511.522.533.54
(i)−1−0.8−0.6−0.4−0.200.20.40.60.81
(j)
0.511.522.533.54
(k)00.10.20.30.40.50.60.70.80.91
(l)
with normal vector
n = [ t2 −t1]T
(34)
With the following reference transformation[
ts
]=[
tT
nT
] [ξ − ξm
η − ηm
](35)
where (ξm, ηm) is the midpoint of the edge.it is possible to define the one-dimensional signed distance
(Fig. 9a) as
ds(t) =∣∣∣∣t − t1 + t2
2
∣∣∣∣−∣∣∣∣ t1 − t2
2
∣∣∣∣ (36)
The positive part of the signed distance function d+1 (t)
(Fig. 9b)
d+1 (t) = ds(t) + |ds(t)|
2(37)
Similarly to Eq. (28), the two-dimensional distance func-tion of the ith edge is
d(2)i =
√d+
1 (ti ) + s 2i (38)
However, Algorithm (32) provides only the two-dimensi-onal distance function of the polygon, in absolute value, i.e.unsigned (Fig. 10i). Instead, Eq. (28) requires the positivepart of the signed distance function, shown in Fig. 10k. Nev-ertheless, Algorithm (32) describes a procedure for a gener-alized two-dimensional phase field ϕ(2), which has the impor-tant property of being positive outside the polygon, and neg-ative inside the polygon, if the polygon’s edges have normalvectors pointing outwards (otherwise the opposite is true).
This generalized two-dimensional phase field ϕ(2) for thefirst edge, is the directional derivative (in the direction nor-mal to the segment) of the distance function d(2)
i (Fig. 10b).It is then updated through the generalized min function
m(a, b, f, g) as in Eq. (59), which assigns respectively eitherthe current ϕ(2) or the new phase function wherever the(absolute) distance function is minimum: for example, inFig. 10d, the function s2
d(2)i (t2,s2)
is assigned to the points clos-
est to the second edge.Instead, for all the other points, it is assigned the function
ϕ(2) computed at the previous step ( the one for the first edges1
d(2)1 (t1,s1)
). This procedure is repeated for all the edges, with
the final result shown in Fig. 10j.Therefore, the sought positive part of the signed distance
function can be calculated as
d(2)+ (ξ, η) = d(2)(ξ, η) sign
(ϕ(2)(ξ, η)
)(39)
where d(2)+ is shown in Fig. 10k, d(2) is shown in Fig. 10i and
sign(ϕ(2))
in Fig. 10l.
Finally, derivatives of the enrichment (necessary for theenrichment of the derivatives of the shape functions) can becomputed by formally differentiating all the equations in thesection, through the application of the derivatives of the minoperator as reported in the Appendix.
5 The enrichment for multiple non-planar polygonalcrack faces
The algorithms described in Sect. 4 can be extended to sets ofnon-planar crack-faces of arbitrary shape. A complex multi-faceted non-planar crack can be decomposed in one of thesesituations:
– two non-planar faces sharing an edge– many non-planar faces sharing a vertex
Motivated by Eq. (28), one could define a generalizedabsolute Z-coordinate, from the following
Zabs =√
d(3)(x)2 − d(2)+ (x)2 (40)
where d(3)(x) is the three-dimensional of the non-planarcrack faces (figure), and d(2)
+ (x) is the two dimensional posi-tive distance function computed with an algorithm similar to(32). Assuming n as the number of the faces, three fields areupdated:
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
f or i = 1, . . . , n
d(3)(x) = d(3)i (x) if i = 1
d(3)(x) = min(
d(3), d(3)i (x)
)if i = 2, . . . , n
ϕ(3)(x) = ζi
d(3)i
if i = 1
ϕ(3)(x) = m
(ϕ(3),
ζi
d(3)i (x)
, d(3), d(3)i (x)
)if i = 2, . . . , n
d(2)+ (x) = d(2)
i,+(x) if i = 1
d(2)+ (x) = m
(d(2)+ , d(2)
i,+(x), d(3), d(3)i (x)
)if i = 2, . . . , n
end f or
(41)
where d(3)i (x) is the three-dimensional distance function as
in Eq. (28) for the i th face, d(2)i,+(x) is the distance function
as in Eq. (39) for the i th face, ϕ(3) is the three-dimensionalgeneralization of the phase function defined in Sect. 4 andFig. 10.
A generalized signed Z-coordinate is computed after com-pleting the Algorithm 41
Z = Zabs sign(ϕ(3))
(42)
and finally the phase function for the multi-planar crack as:
φ = Z
d(3)(43)
It must be remarked that a function similar to generalizedsigned Z-coordinate could be computed using the approachdescribed in [3], as for example in [19] where it was applied inan XFEM context. In this work we use a different approach,since (42) is calculated in a inverse manner, rather thandirectly using pseudo-normals (Figs. 11, 12, 13, 14).
Fig. 15 Definition of active edge; black thick line crack front; red crossactive edges; blue dots inactive edges. (Color figure online)
6 Crack growth strategy and algorithm
Before proceeding further, it is worth to introduce some ter-minology that will be used in the rest of the section. We callcrack front the set of edges referenced only by one face. Avertex is considered active if inside the domain of analysis,whilst an edge on the crack front is considered active if itsmidpoint is inside (Fig. 15).
Once an active edge is detected, a local reference systemis established (Fig. 16a), the length da is kept fixed, whilethe orientation θ is decided according to the maximum hoopstress criterion.
θ = 2 arctan
⎛⎝ K I
4K I I− sign(K I I )
4
√(K I
K I I
)2
+ 8
⎞⎠ (44)
where K I and K I I are respectively mode I and mode II SIFs.The SIFs are computed from the stress fields using a domainformulation of the interaction integral, as described in [22]and [37].
The algorithm works in the following way: a new mid-point is generated, using the angle θ extracted from the SIFs,and two new vertices are added (green crosses and dots inFig. 16b). The colour green indicates that they still require apoint-in-polygon test to decide whether they are active (red)or inactive (blue). Nevertheless, the two existing vertices
Fig. 17 Crack growth strategy:red dots and crosses arerespectively active vertices andedges, black dots and lines arerespectively active internalvertices and edges. (Color figureonline)
(a)
(c) (d) (e)
(b)
of the active edge and the added vertices form an addedface (Fig. 16b). Then, the new vertices (Fig. 16d) are added,through linear interpolation of the coordinates of the addedmidpoints. If the active edge has an adjacent active edge, thenonce the new vertices are generated, a triangular tessellation
is created in order to consider the non-planarity of the addedfaces. Otherwise, if the active edge does not have adjacentactive edges, (as for example in Fig. 15), the new face is allcontained in a plane, therefore the points can be connectedby a quadrangle.
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Comput Mech
Fig. 18 Inclined penny-shaped crack in a cube under tension
−180 −150 −120 −90 −60 −30 0 30 60 90 120 150 180
0
0.1
0.2
0.3
0.4
0.5
SIFs
Fig. 19 Stress intensity factors (SIFs) for β = 0: continuous line forthe three fracture modes: analytical; blue dots numerical Mode I; reddiamonds numerical Mode II; green squares numerical Mode III. (Colorfigure online)
−180 −150 −120 −90 −60 −30 0 30 60 90 120 150 180
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
SIFs
Fig. 20 Stress intensity factors (SIFs) for β = π/10: continuous linefor the three fracture modes: analytical; blue dots numerical Mode I;red diamonds numerical Mode II; green squares numerical Mode III.(Color figure online)
Finally, for both cases, the list of the active edges andvertices is updated. In addition, two lists of the vertices andedges are updated: these lists refer to edges and vertices fromwhere the growth occurred.
The reason is that in this manner, the appropriate enrich-ment can be selected (Fig. 17), since, as mentioned in Sect. 5,a multifaceted non-planar crack can be decomposed in either
−180 −150 −120 −90 −60 −30 0 30 60 90 120 150 180
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
SIF
s
Fig. 21 Stress intensity factors (SIFs) for β = π/6: continuous linefor the three fracture modes: analytical; blue dots numerical Mode I;red diamonds numerical Mode II; green squares numerical Mode III.(Color figure online)
−180 −150 −120 −90 −60 −30 0 30 60 90 120 150 180
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
SIF
s
Fig. 22 Stress intensity factors (SIFs) for β = π/4: continuous linefor the three fracture modes: analytical; blue dots numerical Mode I;red diamonds numerical Mode II; green squares numerical Mode III.(Color figure online)
pairs of two non-planar faces sharing an edge or many non-planar faces sharing a vertex. The list of pairs of non-planarfaces sharing an edge is referred as list of active internaledges, whilst the list of non-planar faces sharing a vertex iscalled list of active internal vertices. These lists are main-tained and updated simply with their addresses in a globallist of vertices and edges. It must be stressed that the enrich-ments are only calculated for the new entries in this list, anddo not need to be re-calculated for the entire crack face. Thisincremental properties is one of the greatest advantages ofthis approach. For example, let us refer to Fig. 17a, wherethree steps are illustrated: at the first step, the enrichment isintroduced in the kernel
w1I = w0
I (x)h1I (x) (45)
where w0I (x) is the uncracked weight function for node xI .
At subsequent steps (k > 1), the enriched weight function atcrack step k + 1 is
wk+1I = wk
I hk+1I (x) (46)
where hk+1I (x) is the enrichment of the v-crack which over-
laps for a face with the previous crack step. For the case in
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Fig. 23 Arrea–Ingraffea beam: test definition and boundary condi-tions, with initial crack (grey)
Fig. 17a, when k = 3, only the light blue faces (sharing theactive internal edge) are considered in the enrichment, sincethe grey face has been already considered in the step withk = 2.
7 Numerical examples
In this section, several examples are shown to demonstratethe effectiveness of the philosophy of distance functionsfor intrinsic enrichments, particularly in three-dimensionalcases, where a crack can become multi-faceted as a conse-quence of their propagation under mixed mode loading. Weshall start with the computation of the SIFs for a circularpenny crack 7.1, for which the enrichment was introducedas introductory example in Sect. 3, by using immediatelycomputable distance functions. We will then proceed withexamples of three-dimensional crack propagation that showthe true advantage of using this approach: a curved crackpath under mixed shear-bending loading (hereafter referredas the Ingraffea beam [1]), two disaligned edge cracks in adog-bone specimen under tensile loading (hereafter referredas the Yates test [39]) and, finally, a helical crack path of an
Fig. 25 Yates test: tensile loading and cantilevered at the other end(black face), with two initial offset cracks (grey)
initially inclined crack in a beam loaded in torsion (hereafterreferred as the Brokenshire test [10]). For all these tests, weanticipate, there is concordance with existing experimentaland numerical results [19,20,25].
7.1 Inclined planar penny-shaped crack under remotetension
The first example is for a single inclined circular penny-shaped crack under remote tension (Fig. 18), as showed inSect. 3. The radius of the crack is a and the inclination towardsthe vertical axis is β. The angle θ is the azimuthal anglearound n in the plane of the crack, assuming as θ = 0 theplane including n and the vertical axis. Analytical solutionsexist for all three failure modes for a crack in an infinitemedium under remote tensile loading σ0:
K I = σ0
√2
π
√πa cos2(β) (47)
K I I = σ0
√2
π
2
2 − νsin β cos β cos(θ)
√πa (48)
K I I I = σ0
√2
π
2
(2 − ν)(1 − ν)sin β cos β sin(θ)
√πa
(49)
Fig. 24 Crack path for theArrea–Ingraffea beam
(a) (b)
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Fig. 26 Crack paths for thedouble edge-crack problem fordifferent initial offsets
(a)
(b) (c) (d)
Fig. 27 σx stress plot for theYates test right before completefracture (displacements aremagnified with the same scalingfactor)
(a)
(b)
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Fig. 28 Specimen and crackpaths for the Brokenshire test
(a)
(b) (c)
(d) (e)
The infinite medium is mimicked by assuming a = L/10,with L = 2 m. The material is assumed isotropic linear elas-tic with ν = 0.3 and Young’s modulus E = 1 GPa. The dis-cretisation nodes are regularly placed in a structured manner20 ×20 ×20 i.e. not conforming to the penny crack, withoutdividing the penny crack into triangles. Moreover, no meshrefinement was used near the crack. It is worth to remind thata structured mesh is used here for simplicity and it is not arequirement: the same enrichment is applicable to unstruc-tured meshes as well, with more or less equally accurateresults.
The background mesh is composed of tetrahedral integra-tion cells, with a second order Gaussian quadrature (eightintegration points per cell). In order to provide a compari-son with analytical solutions, SIFs are calculated with thedomain formulation of the interaction integral, as reportedin [22] and [37]. A good accuracy can be achieved with arelatively coarse mesh (only four nodes per diameter of the
penny crack): Figs. 19, 20, 21 and 22 show close matchingwith the analytical solutions for a crack in an infinite domain(Eqs. 47, 48, 49), with slight differences due to the finitenessof the specimen.
7.2 The Arrea–Ingraffea beam
The example in this section is the crack growth for a simplysupported beam under bending, loaded with a force non-collinear to the edge crack, positioned in the middle of thebeam (Fig. 23).
The input parameters are L = 916 mm, H = 306 mm,W = 519 mm, b = 397 mm, c = 61 mm (with b+c = B/2),a = 82 mm, F1 = 1 N. A structured background mesh isused, with mesh size 10 mm.
As a result, the crack grows under mixed mode loading,propagating in a curved path, and eventually aligning in adirection parallel to the force. This is a classic benchmark
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Fig. 29 Stress plots of the finalconfiguration for theBrokenshire test
(a)
(c) (d)
(b)
case for which numerical and experimental results can befound in [1]. The input parameters are B = 916 mm, H =306 mm, W = 519 mm, b = 397 mm, c = 61 mm (withb + c = B/2), a = 82 mm, F1 = 1 N, t = 100 mm , withmesh size h = 16.6 mm. Conversely to the test in [1], cracksare supposed traction-less, since we are only interested herein the crack path.
As it can be seen from Fig. 24, the crack path is similar tothe one observed in [1]: there is a major kink at the beginningand then it gradually aligns with the direction of the loadingforce.
7.3 The Yates test
This example shows the interaction of the non-collinear edgecracks under fatigue loading, as reported experimentally andnumerically (using finite element software with re-meshing)in [39]. The input parameters are (Fig. 25) L = 40 mm,H = 140 mm, a = 8 mm, R = B = 15 mm, h = 80 mm,W = 100 mm and t = 6 mm. The background mesh is inthis case unstructured, with a mesh size of 3 mm.
Four different values of the offset distance 2b are showedin Fig. 26: 16 mm, reported in [39] with experimental andnumerical (two-dimensional finite element simulations withre-meshing) values, and 4, 2 and 1 mm.
Yates et al. [39] reports onset of influence of the twocracks, for 2b = 8 mm. Therefore, we varied 2b for smallerlengths, not reported in [39], with the aim of observing amore pronounced interaction and ultimately coalescence ofthe cracks, and showing the predictive ability of the enrich-ment. Figures 26b, 26c and 26d show the generation of a
lenticular fragment, precisely at 2b = 4 mm the joining ofthe cracks happens near the boundaries (Fig. 26b), whilst forsmaller lengths the two crack faces intersect each other wellbefore the boundaries: for 2b = 1 mm (Fig. 26c), the lentic-ular shape is less pronounced than 2b = 2 mm, as expected,since for smaller offset the cracks tend to join in a collineardirection.
Figure 27 show the deformed configuration of the crackedspecimen, one step before the cracks either intersect or reachthe boundary. It is clearly demonstrated the ability of theenrichment of introducing a curved discontinuity in the dis-placements and the high regions of stress concentrations nearthe crack fronts, for the σx stress.
7.4 The Brokenshire test
The final example is a skew crack propagated under torsionalloading [10]. For this test (28a) the parameters are W = H =100 mm, L = 400 mm, t = 25 mm and b = 75 mm.
The background mesh is structured with a mesh size of8.33 mm. The four pyramids at the end are supposed rigid:two at one end have their respective apices pinned, the othertwo one is pinned, the other apex is loaded with a verti-cal applied displacement. To replicate these conditions, thespecimen is replaced with a rectangular cuboid, cantileveredat one end, and a torsional displacement field applied at theopposite face:
⎧⎨⎩
u(x, y, z) = 0v(x, y, z) = −γ (z − H/2)
w(x, y, z) = γ (y − W − b)
(50)
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with γ = π/12. The inclination of the crack face (grey inFig. 28a) is π/4 with respect of the length of the beam, and atthe first step the corresponding enrichment is introduced as arectangle. In the subsequent steps, before activating the crackgrowth algorithm, it is discretised in a structured manner with8 × 8 segments.
The specimen is thus loaded in torsion, and the crackgrows in an helical pattern [10]. Figure 28 show the computedcrack face from different views, where the helical shape isdistinctly visible. Figure 29 show the deformed configurationin the last step, where the crack opening generates a mixedflexural–torsional stress state.
8 Conclusion
This paper presented a new approach for modelling three-dimensional crack propagation for solids, in a meshlessdisplacement-based formulation. The method is based ongenerating derivatives of distance functions suitable forintrinsic enrichments of the kernels. The derivatives of suchdistance fields have the important properties of being simul-taneously discontinuous across the surface, and continuousall around the edges. Therefore, a polygonal crack surfacecan be introduced exactly, without spurious effects.
The derivatives of this function are computed through asuccessive decomposition of geometrical entities (a poly-gon), in lower-dimensional geometrical entities (segments),for which an explicit expression is provided in this paper.
The algorithms presented provide an alternative look toproblems known in computational geometry as point-in-polygon and signed distance function, without requiring theintroduction of the Eikonal equation as in level-sets-basedmethods. Instead, multifaceted non planar crack paths canbe treated in an incremental manner, not requiring the re-computation of the enrichment for the entire propagatedcrack, but only for the newly introduced facets.
Since it is distance-based, the approach is purely geometri-cal, therefore it is material-independent. In principle, it mightprove useful for introducing discontinuities in the approxi-mation of any partial differential equations (thermal, fluid,magnetic etc...) solved with a weak form where test and trialfunctions are meshfree-based.
Acknowledgments This work was supported by the UK Engi-neering and Physical Sciences Research Council (EPSRC, GrantEP/G042586/1) and Defense Science and Technology Laboratory(DSTL), both of which are gratefully acknowledged.
Explicit expression for min and max functions
It is useful to define explicit expressions for the min and maxfunctions, where the min (max) function is the minimum
(a)
(b)
Fig. 30 max and min functions with f (x) = cos(x) (dash-dotted thinline) and g(x) = sin(x) (dashed thick line))
(maximum) between two functions f : Rk → R and g :
Rk → R with k = 1, 2, 3
m ( f (x), g(x)) = min ( f (x), g(x)) ={
f (x) if f (x) − g(x) ≤ 0g(x) if f (x) − g(x) > 0
(51)
M ( f (x), g(x)) = max ( f (x), g(x)) ={
g(x) if f (x) − g(x) ≤ 0f (x) if f (x) − g(x) > 0
(52)
Equations (51) and (52) can be rewritten using the Heavisidefunction H(x)
m( f, g) = g H( f − g) + f H(g − f ) (53)
M( f, g) = f H( f − g) + g H(g − f ) (54)
with the obvious commutative property
m( f, g) = m(g, f ) M( f, g) = M(g, f ) (55)
Particular cases of Eqs. (53) and (54) are the positive andnegative part of a function (Figs. 30, 31)
f+ = M( f, 0) (56)
f− = m( f, 0) (57)
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(a)
(b)
Fig. 31 Derivatives of min and max functions with f (x) = cos(x)
and g(x) = sin(x); continuous line min and max functions, dashed linederivatives
and
f (x) = f+(x) + f−(x) = M( f, 0) + m( f, 0) (58)
The functions in Eqs. (53) and (54) can be generalized to
m(a, b, f, g) = b H( f − g) + a H(g − f ) (59)
M(a, b, f, g) = a H( f − g) + b H(g − f ) (60)
where m assigns function a whenever f ≥ g and b other-wise; M instead assigns function b whenever f ≥ g and aotherwise. The following property holds true:
Equation (61) allows the calculation of the derivatives of themin and max functions:
∂ min( f, g)
∂x= m
(∂ f
∂x,∂g
∂x, f, g
)(62)
∂ max( f, g)
∂x= M
(∂ f
∂x,∂g
∂x, f, g
)(63)
References
1. Arrea M, Ingraffea A (1982) Mixed-mode crack propagation inmortar and concrete. Technical report. Cornell University, Ithaca,pp 81–13
2. Atluri S, Zhu T (1998) A new meshless local Petrov–Galerkin(MLPG) approach in computational mechanics. Comput Mech22(2):117–127
3. Baerentzen JA, Aanaes H (2005) Signed distance computationusing the angle weighted pseudonormal. IEEE Trans Vis ComputGraph 11(3):243–253
4. Barbieri E, Meo M (2012) A fast object-oriented matlab implemen-tation of the reproducing kernel particle method. Comput Mech49(5):581–602
5. Barbieri E, Petrinic N, Meo M, Tagarielli V (2012) A new weight-function enrichment in meshless methods for multiple cracks inlinear elasticity. Int J Numer Methods Eng 90(2):177–195
6. Belytschko T, Fleming M (1999) Smoothing, enrichment andcontact in the element-free Galerkin method. Comput Struct71(2):173–195
7. Belytschko T, Gu L, Lu Y (1994) Fracture and crack growthby element-free Galerkin methods. Model Simul Mater Sci Eng2:519–534
8. Belytschko T, Lu Y, Gu L (1994) Element-free Galerkin methods.Int J Numer Methods Eng 37(2):229–256
9. Belytschko T, Lu Y, Gu L (1995) Crack propagation by element-free Galerkin methods. Eng Fract Mech 51(2):295–315
10. Brokenshire DR (1996) A study of torsion fracture tests. Ph.D.thesis, Cardiff University, Cardiff
11. Chen JS, Pan C, Wu CT, Liu WK (1996) Reproducing kernel parti-cle methods for large deformation analysis of non-linear structures.Comput Methods Appl Mech Eng 139(1):195–227
12. De Berg M, Cheong O, Van Kreveld M (2008) Computationalgeometry: algorithms and applications. Springer, Berlin
13. Dolbow J, Belytschko T (1998) An introduction to programmingthe meshless element free Galerkin method. Arch Comput MethodsEng 5(3):207–241
14. Duarte CA, Oden JT (1996) Hp clouds: an Hp meshless method.Numer Methods Partial Differ Equ 12(6):673–706
15. Duflot M (2006) A meshless method with enriched weight func-tions for three-dimensional crack propagation. Int J Numer Meth-ods Eng 65(12):1970–2006
16. Duflot M, Nguyen-Dang H (2004) A meshless method withenriched weight functions for fatigue crack growth. Int J NumerMethods Eng 59:1945–1961
17. Duflot M, Nguyen-Dang H (2004) Fatigue crack growth analysisby an enriched meshless method. J Comput Appl Math 168(1–2):155–164
18. Fries T, Matthies H (2003) Classification and overview of meshfreemethods. Informatikbericht Nr 3. Institute of Scientific Computing,Technical University Braunschweig, Brunswick, Germany
19. Fries TP, Baydoun M (2012) Crack propagation with the extendedfinite element method and a hybrid explicitimplicit crack descrip-tion. Int J Numer Methods Eng 89(12):1527–1558
20. Gasser TC, Holzapfel GA (2006) 3D crack propagation in unrein-forced concrete: a two-step algorithm for tracking 3D crack paths.Comput Methods Appl Mech Eng 195(37):5198–5219
21. Goodman JE, O’Rourke J (2010) Handbook of discrete and com-putational geometry. Chapman and Hall/CRC, Boca Raton
22. Gosz M, Moran B (2002) An interaction energy integral methodfor computation of mixed-mode stress intensity factors along non-planar crack fronts in three dimensions. Eng Fract Mech 69(3):299–319
23. Idelsohn SR, Onate E, Calvo N, Del Pin F (2003) The meshlessfinite element method. Int J Numer Methods Eng 58(6):893–912
123
Comput Mech
24. Jones MW, Baerentzen JA, Sramek M (2006) 3D distance fields:a survey of techniques and applications. IEEE Trans Vis ComputGraph 12(4):581–599
25. Kaczmarczyk L, Nezhad MM, Pearce C (2013) Three-dimensionalbrittle fracture: configurational-force-driven crack propagation.arXiv, arXiv:1304.6136 (preprint)
26. Lancaster P, Salkauskas K (1981) Surfaces generated by movingleast squares methods. Math Comput 37(155):141–158
27. Libersky LD, Petschek A (1991) Smooth particle hydrodynam-ics with strength of materials. In: Advances in the free-Lagrangemethod including contributions on adaptive gridding and thesmooth particle hydrodynamics method. Springer, Berlin, pp 248–257
28. Liu W, Jun S, Zhang Y (1995) Reproducing kernel particle meth-ods. Int J Numer Methods Fluids 20(8–9):1081–1106
29. Liu WK, Jun S, Li S, Adee J, Belytschko T (1995) Reproducingkernel particle methods for structural dynamics. Int J Numer Meth-ods Eng 38(10):1655–1679
30. Liu WK, Li S, Belytschko T (1997) Moving least-square repro-ducing kernel methods (i) methodology and convergence. ComputMethods Appl Mech Eng 143(1):113–154
31. Nguyen V, Rabczuk T, Bordas S, Duflot M (2008) Meshless meth-ods: a review and computer implementation aspects. Math ComputSimul 79:763–813
32. Onate E, Idelsohn S, Zienkiewicz O, Taylor R (1996) A finite pointmethod in computational mechanics: applications to convectivetransport and fluid flow. Int J Numer Methods Eng 39(22):3839–3866
33. Rabczuk T, Belytschko T (2004) Cracking particles: a simplifiedmeshfree method for arbitrary evolving cracks. Int J Numer Meth-ods Eng 61(13):2316–2343
34. Ren B, Qian J, Zeng X, Jha A, Xiao S, Li S (2011) Recent devel-opments on thermo-mechanical simulations of ductile failure bymeshfree method. Comput Model Eng Sci 71(3):253
35. Sethian JA (1999) Level set methods and fast marching methods:evolving interfaces in computational geometry, fluid mechanics,computer vision, and materials science, vol 3. Cambridge Univer-sity Press, Cambridge
36. Sukumar N (1998) The natural element method in solid mechanics.Ph.D. thesis, Northwestern University, Chicago
37. Walters MC, Paulino GH, Dodds RH (2005) Interaction integralprocedures for 3-D curved cracks including surface tractions. EngFract Mech 72(11):1635–1663
38. Yagawa G, Yamada T (1996) Free mesh method: a new meshlessfinite element method. Comput Mech 18(5):383–386
39. Yates J, Zanganeh M, Tomlinson R, Brown M, Garrido F(2008) Crack paths under mixed mode loading. Eng Fract Mech75(3):319–330
40. Zhu T, Zhang JD, Atluri S (1998) A local boundary integral equa-tion (IBIE) method in computational mechanics, and a meshlessdiscretization approach. Comput Mech 21(3):223–235
