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Int J Fract (2018) 210:113–136https://doi.org/10.1007/s10704-018-0265-z

ORIGINAL PAPER

Achieving pervasive fracture and fragmentation inthree-dimensions: an unstructuring-based approach

Daniel W. Spring · Glaucio H. Paulino

Received: 19 August 2017 / Accepted: 23 January 2018 / Published online: 3 February 2018© Springer Science+Business Media B.V., part of Springer Nature 2018

Abstract There are few methods capable of captur-ing the full spectrum of pervasive fracture behavior inthree-dimensions. Throughout pervasive fracture sim-ulations, many cracks initiate, propagate, branch andcoalesce simultaneously. Because of the cohesive ele-ment method framework, this behavior can be cap-tured in a regularizedmanner. However, since the cohe-sive element method is only able to propagate cracksalong element facets, a poorly designed discretizationof the problem domain may introduce artifacts into thesimulated results. To reduce the influence of the dis-cretization, geometrically and constitutively unstruc-tured means can be used. In this paper, we present andinvestigate the use of three-dimensional nodal pertur-bation to introduce geometric randomness into a finiteelement mesh. We also discuss the use of statisticalmethods for introducing randomness in heterogeneousconstitutive relations. The geometrically unstructuredmethod of nodal perturbation is then combined witha random heterogeneous constitutive relation in threenumerical examples. The examples are chosen in orderto represent some of the significant influencing factors

D. W. SpringDepartment of Civil and Environmental Engineering,University of Illinois at Urbana-Champaign, Urbana, IL,USAe-mail: [email protected]

G. H. Paulino (B)School of Civil and Environmental Engineering, GeorgiaInstitute of Technology, Atlanta, GA, USAe-mail: [email protected]

on pervasive fracture and fragmentation; including sur-face features, loading conditions, and material grada-tion. Finally, some concluding remarks and potentialextensions are discussed.

Keywords Pervasive fracture · Fragmentation · Nodalperturbation · Cohesive elements · Extrinsic PPRmodel · Weibull distribution

1 Introduction

The pervasive fracture and fragmentation of structuresoccurs at various scales and in various contexts. Whena structure is struck with either a high-velocity directimpact or a blast load, the damage incurred typicallypervades a region of the structure. Pervasive damageinvolves the entire spectrum of fracture behavior; fromcrack initiation, through crack propagation, branchingand coalescence, all the way to complete fragmenta-tion. At any one time, there may be hundreds or thou-sands of micro- to macro-cracks in the structure. Beingable to model and understand the factors which influ-ence the fragmentation of structures can lead to bet-ter design practices in many fields of engineering. Forexample, understanding how a ceramic plate fragmentswhen struckwith a high velocity projectile can improvethe design of armor for personnel carriers. Alterna-tively, understanding the process by which a kidneystone fragmentsmay help in the design of surgical toolsand procedures. However, due to the inherent complex-

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114 D. W. Spring, G. H. Paulino

ity ofmodeling pervasive damage, there are fewnumer-ical methods capable of capturing the full spectrum ofbehavior in three-dimensions.

Currently, there are many methods which have beenproposed for simulating fracture problems. Some of thepopular methods consist of the extended (or general-ized) finite element method (FEM); meshless or parti-cle based methods; peridynamics (Silling and Bobaru2005; Bažant et al. 2016), and the cohesive elementmethod. The extended FEM uses discontinuous shapefunctions to represent the profile of a crackwithin a con-tinuum. The discontinuity in the element’s shape func-tion effectively splits the element; which requires a lotof heuristic manipulations, and can present numericalintegration and time-stepping issues in dynamic frac-ture simulations (Park et al. 2012). In three dimensions,the extended FEM has proven useful in modeling frac-ture problems dominated by a single crack (Sukumaret al. 2000; Areias and Belytschko 2005). However,when faced with the full spectrum of fracture behavior,the extended FEM can become prohibitively compli-cated (Bishop 2009).

Rather than using elements to discretize the prob-lem domain, meshless methods represent the unknownfields with nodal information (Nguyen et al. 2008). Byincreasing the number of nodes dispersed in a region,a higher resolution of information in that region isobtained. Thus, nodes are typically clustered aroundthe critical portions of the domain; which, in frac-ture problems, is the crack-tip. Each node is containedwithin a nodal-support region and a weight function isdefined over each nodal-support region. The spacingof the nodes is such that each point in the domain iscovered by at least three distinct weight functions. Theresulting global fields are then interpolated from nodalinformation using a Moving Least Squares fitting pro-cedure (Lancaster and Salkauskas 1981). Examples ofmeshlessmethods include theDiffuse ElementMethod(Nayroles et al. 1992), the Element Free Galerkinmethod (Belytschko et al. 1994a, b; Belytschko andFleming 1999), the Reproducing Kernel Method (Liuet al. 1995), the Meshless Local Petrov–Galerkinmethod (Atluri and Zhu 1998), the cracked parti-cles method (Rabczuk and Belytschko 2004), and theMaterial Point Method (Sulsky and Schreyer 2004).Althoughmeshless methods have shown some promisein modeling problems where fracture is dominated bya single crack, many of them still suffer from vol-ume deletion during pervasive fracture and fragmen-

tation problems. As the continuummaterial fragments,it evolves into a collection of spheres; which have a the-oreticalmaximumpacking limit of 74% (Bishop 2009),leading to a nonphysical loss of volume in the model.

The cohesive element method, motivated by thework of Dugdale (1960) and Barenblatt (1959), usescohesive zonemodels to explicitly represent the inelas-tic zone of damage in front of the crack tip. In dynamicfracture simulations, extrinsic cohesive elements areadaptively inserted ahead of a propagating crack tipand resist the separation of the adjacent bulk elementsthrough cohesive forces (Ortiz and Pandolfi 1999;Zhang et al. 2007). The insertion of cohesive elementsis restricted to element facets. This restriction allowsthe cohesive element method some unique capabilities.The mesh topology regularizes the domain and natu-rally handles the branching and coalescence of fracturesurfaces. The elements are not split, thus the minimumfacet size is fixed, and time-stepping issues related tosmall facet sizes do not occur. Additionally, the restric-tion of fracture surfaces to element facets results in acontinuous volume of bulk material as it fragments.However, restricting fracture to occur along elementfacets can result in mesh dependent behavior, and thusthe mesh discretization in dynamic cohesive fracturesimulations is important. If one were to choose a struc-tured mesh, or to use mesh smoothing techniques on arandomly generated mesh, the resulting discretizationcould bias the fracture patterns. Thus, it isworth explor-ingmeans of introducing randomness into the problem.

Randomness can be introduced to a finite elementproblem through either topological, geometric, or con-stitutive means. The primary topologically unstruc-tured methods consist of: remeshing, element split-ting and adaptive refinement. With remeshing or adap-tive refinement, the internal state variables of the finemesh need to be interpolated from those of the coarsemesh. For refinement around a propagating crack tip,the repeated application ofmesh-to-mesh transfer oper-ators may result in significant numerical diffusion(Mosler and Ortiz 2009). Additionally, these meth-ods are not particularly suitable to pervasive frac-ture problems, as the entire domain often needs to berefined, quickly losing the cost-saving advantages ofadaptive refinement. Alternatively, some researchersapply element splitting techniques to increase the num-ber of crack paths in three-dimensional tetrahedralmeshes. The advantage of element splitting is thatit may be combined with an edge-collapse operator

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Achieving pervasive fracture and fragmentation in three-dimensions 115

(Kallinderis and Vijayant 1993; Molinari and Ortiz2002), to coarsen regions far from the crack tip. Thedownside of this technique is that edge-splitting onlyprovides a small amount of refinement, and the qualityof the elements can deteriorate once split.

In this paper, we present a combined geometric andconstitutive approach for introducing randomness intootherwise structured numerical models, and apply it tothe investigation of pervasive fracture and fragmenta-tion behavior. First, a nodal perturbation technique ispresented to alleviate the structure in meshes gener-ated using automatic mesh generators. A study on theeffect of the nodal perturbation is presented, where themetrics of Lo’s parameter andmaximum andminimuminterior angles are used to quantify the effect of vari-ous nodal perturbation factors on mesh quality. As aresult of the study on mesh quality, a maximum valuefor the nodal perturbation factor is recommended and isused throughout the remainder of the paper. Addition-ally, a Weibull distribution is motivated as a means ofgenerating a random heterogeneous constitutive rela-tion. The combined effect of the geometrically random(via nodal perturbation) and constitutively random (viaa random assignment of material strength) methods isinvestigated in a series of examples. The remainder ofthe paper is organized as follows. Section 2 outlines thenumerical framework for our study. The cohesive zonemodel selected to represent the dynamic failure behav-ior is the extrinsic Park–Paulino–Roesler (PPR)model.An overview of the model is presented in Sect. 3, alongwith a discussion on using random heterogeneous con-stitutive relations for introducing randomness into amodel. Section 4 outlines the technique of nodal per-turbation, to introduce geometric unstructuredness intoa finite element mesh. Additionally, in this section, weoutline the series of geometric studies used to inves-tigate the influence of nodal perturbation on the qual-ity of the mesh. In Sect. 5 we present three numericalexamples which highlight many of the significant fac-tors influencing the pervasive fracture and fragmenta-tion of structures. Finally, we provide some concludingremarks in Sect. 6.

2 Numerical framework

In this work we use cohesive elements, within the finiteelement framework, to model fracture. There are twoclasses of cohesive elements. Intrinsic cohesive ele-

ments are inserted to the model prior to the simulation(Zhang and Paulino 2005) and are convenient whenfracture is restricted to occur along a specified line orin a specified region. However, this approach is knownto alter the effective properties of the bulk materialin the zone of fracture (Falk et al. 2001; Klein et al.2000). Alternatively, extrinsic cohesive elements canbe inserted to the mesh on the fly, where and whenneeded. Extrinsic elements are convenient for simu-lating problems where the location of fracture is notknown a priori. The dynamic mesh connectivity, nec-essary for the adaptive insertion of extrinsic cohesiveelements between bulk elements, is handled through atopological data structure (Celes et al. 2005; Paulinoet al. 2008). The constitutive relation of the cohesiveelements corresponds to the chosen cohesive model, aswill be discussed in the next Section.

In fracture simulations it is important to take intoconsideration the effect of finite deformations. Whencohesive elements are first activated, they display zerothickness, but as fracture progresses the adjacent bulkelements separate and rotate. By not taking into con-sideration these finite rotations the mechanics of theproblem may not be accurately captured. In this workfinite deformations are taken into account by means ofthe total Lagrangian formulation. In this formulation,the deformation is described with respect to the unde-formed configuration. The expression of the principalof virtual work with respect to the undeformed config-uration relates the sum of the virtual strain energy andvirtual kinetic energy to the sum of the virtual workdone by external and cohesive tractions:∫

�

(S : δE + ρu · δu) d� =∫

�coh

Tcoh · δ�ud�coh

+∫

�ext

Text · δud�ext , (1)

where � is the domain in the reference configuration,S = JF−1σF−T is the second Piola-Kirchoff stresstensor, E is the Green-Lagrange strain tensor, ρ is thematerial density, u is the displacement vector, u is theacceleration vector, Text is the traction applied alongthe external surface, �ext , in the reference configura-tion, F is the deformation gradient, J = detF is theJacobian, and the internal displacement jump,�u, pro-duces a cohesive traction, Tcoh , over the internal cohe-sive surface �coh in the reference configuration. Forimplementation, Eq. (1) is expressed in matrix nota-tion as:

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Ku + Mu − Rcoh − Rext = 0. (2)

where K is the stiffness matrix, M is the mass matrix,Rcoh is the cohesive force vector, andRext is the exter-nal force vector.

Numerically, to progress the dynamic simulation,time is discretized using the explicit central differencemethod (Newmark 1959). The computation of nodaldisplacements, velocities and accelerations at timen+1are computed from those at time n through the follow-ing scheme:

un+1 = un + �t un + 1

2(�t)2 un (3)

un+1 = M−1 (Rext − Rintn+1 + Rcohn+1

)(4)

un+1 = un + �t

2(un + un+1) (5)

where�t denotes the time step.Note thatRint andRcoh

are the internal and cohesive force vectors obtainedfrom the separate contribution of the bulk and cohesiveelements, respectively. To compute themassmatrix,M,we apply a standard mass lumping technique in whichthe diagonal terms of the consistent mass matrix arescaled, preserving the totalmass and resulting in a diag-onal mass matrix (Hinton et al. 1976; Hughes 2000).

3 Random heterogeneous constitutiverelation by means of statistical distribution

This section outlines the constitutive relation selectedfor the cohesive elements and discusses the use of ran-dom methods for capturing microscale heterogeneity.

3.1 Park–Paulino–Roesler cohesive model

In the dynamic simulations performed in this work,cracks are represented through the use of cohesive zoneelements. As mentioned previously, we use the extrin-sic cohesive element approach. In this approach, thecriterion to insert the cohesive element is external tothe formulation. A cohesive element is inserted oncethe averaged normal or tangential stress along a facetexceeds the normal or tangential cohesive strength ofthe material, respectively. Once the element is inserted,its behavior is governed by the traction-separation rela-tion of the extrinsic Park–Paulino–Roesler (PPR) cohe-sivemodel (Park et al. 2009). The extended formulation

of the model can be found in the principal publication;below, we just include a brief summary for complete-ness.

The PPR cohesive model is potential-based, mean-ing that the traction-separation relation is derived froma potential and the unloading/reloading relation andcontact formulation are independent of the model. Thepotential is given as:

� (�n,�t ) = min (φn, φt )

+[�n

(1 − �n

δn

)α

+ 〈φn − φt 〉]

[�t

(1 − |�t |

δt

)β

+ 〈φt − φn〉]

, (6)

where the Macaulay bracket 〈·〉 is defined such that〈x〉 = (|x | + x) /2.

The normal, Tn , and tangential, Tt , tractions aredetermined by taking the derivative of the potentialwith respect to the normal opening, �n , and tangen-tial opening, �t , respectively. Hence,

Tn (�n,�t ) = −α�n

δn

(1 − �n

δn

)α−1

[�t

(1 − |�t |

δt

)β

+ 〈φt − φn〉]

(7)

Tt (�n,�t ) = −β�t

δt

(1 − |�t |

δt

)β−1

[�n

(1 − �n

δn

)α

+ 〈φn − φt 〉]

�t

|�t | (8)

where φn and φt are the mode I and mode II fractureenergies, and α and β are the mode I and mode II shapeparameters. The energy constants�n and�t are definedas:

�n = (−φn)〈φn−φt 〉/(φn−φt ) ,

�t = (−φt )〈φt−φn〉/(φt−φn) (9)

for different fracture energies (φn �= φt ), and as:

�n = −φn, �t = 1 (10)

if the fracture energies are the same for both mode Iand mode II separation, (φn = φt ). In total, there aresix user inputs to the extrinsic PPR cohesivemodel: φn ,φt , α, β, and the mode I and mode II cohesive strengthsσmax and τmax , respectively. A set of sample traction-separations relations are illustrated in Fig. 1.

For this study, the unloading/reloading relation iscoupled, meaning that the unloading in the normal

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Fig. 1 Traction separationrelations for a normalopening (φn = 100N/m,σmax = 40 MPa, α = 3.0),and b tangential opening(φt = 200 N/m,τmax = 30 MPa, β = 5.0)

( )μm

01

23

02

46

40

30

20

10

06

42

0 0 132

Tangential Opening Normal Opening ( )μm

(a)

( )μm( )μm

00.005

0.01

02

46

Normal Opening (mm)

30

20

10

0

0 0Normal Opening Tangential Opening

64

2 0.0050.01

(b)

T Δ tΔn,nν ( )

T Δ tΔn,nν ν( )δn

δtΔn

Δ t

T Δ tΔn,n ( )ν

Δnmax Δ tmax2 2+ηmax =

ηmax

(a)

T Δ tΔn,t ( )ν

Tt ( )Δ tΔn,ν ν

ΔnΔ t

δnδt

ηmax

Tt ( )ν Δ tΔn,

Δnmax Δ tmax2 2+ηmax =

(b)

Fig. 2 Depiction of the coupled unloading schemes for a normal and b tangential interactions (Park 2009), with linear unload-ing/reloading to the origin

direction is coupled to that in the tangential direction.In addition, we assume that the unloading occurs lin-early back to the origin, as seen in Fig. 2. Contact isassumed to occur when the normal separation within acohesive element becomes negative.We use the penaltystiffness approach, inwhich a high stiffness counteractsthe interpenetration of elements. This approach is pop-ularly implemented in conjunction with cohesive ele-ments, however, others are available (Simo et al. 1986;Espinosa et al. 2000; Falk et al. 2001; Spring et al.2016).

3.2 Statistical methods for capturing materialheterogeneity

While the focus of this work is on the investigation ofhomogeneous or homogenizedmaterials, we recognizethat all materials contain heterogeneity (or defects) atthe microscale. Defects naturally arise in materials dueto grain boundaries, voids, or inclusions (Becher et al.

1998; Sun et al. 1998). As well, defects can be intro-duced through the act of processing or machining thematerial (Levy 2010). These microscale defects con-stitute regions where stresses can concentrate and leadto damage or failure. In this work, the representationof this heterogeneity is achieved by means of a statis-tical distribution of material properties, specifically adistribution of the strength of the material.

Over the years, there have beenmanyproposedmod-els for capturing the distribution of defects in amaterial.One of the simplest models for incorporating defectsin the material is to distribute the material parame-ters based on a constant probability density function(PDF); which is equivalent to a random perturbation tothe material’s properties (Ostoja-Starzewski and Wang2006; Wang et al. 2008; Song and Belytschko 2009).While this model is simple, it has no physically moti-vated basis. Alternatively, one could take a statisticalapproach, distributing the material strength based ona probabilistic model (Bažant and Chen 1996). Peirce(1926) developed a probabilistic failuremodel based on

123


Fig. 3 Effect of inputparameters on the Weibulldistribution: a λ, and b m.Here we assumeσmin = 264 MPa andV0 = 1

250 300 350 4000

0.02

0.04

0.06

0.08

Strength (MPa)PD

F

50λ=

10m=

5m=2m=

(a)

200 300 400 500 600 7000

0.005

0.01

0.015

0.02

Strength (MPa)

PDF

2m=

50λ=

100λ=

200λ=

(b)

theweakest link theory and extremevalue statistics.Hismodel was later refined by Fréchet (1927), among oth-ers (Fischer and Tippett (1928) and vonMises (1936)).However, the most popular probabilistic model for thestudy of material failure is that of Weibull (1939).

Weibull’s probabilistic failure model was motivatedby randomness he observed in the ultimate failurestrength of material specimens tested in an identicalmanner. He explained his motivation through a simplethought experiment. Consider a series of rods of lengthL , with cross-sectional area A, loaded to failure by anexternal load P . If one were to repeat this experimentthe ultimate failure load would not be a constant, butwould differ each time. The failure loads could then begrouped around a mean and a statistical analysis couldbe completed.Based on a set of experiments, conductedon a variety of materials with multiple loading condi-tions, he proposed the well knownWeibull distributionwith probability of failure Pf given by:

Pf = 1 − e−N (σ,V ) (11)

where V is the volume of the material, σ is the measureof stress, and N (σ, V ) is a material function, indepen-dent of the position. The specific form of N (σ, V ) isoften the topic of debate; however,Weibull noted that itmust be a monotonically increasing function of σ anddetermined that an effective relation for most homoge-neous materials is:

N (σ, V ) = 1

V0

(σ − σmin

λ

)m

(12)

where m is the Weibull modulus, V0 is a normalizingvolume (often taken as V0 = 1, Danzer 1992), λ is ascale parameter, and σmin is the lower bound of mate-rial strength. To illustrate the influence of the Weibullmodulus and scale parameter on the probability den-

sity function, some sample functions are illustrated inFig. 3. The mean, median and variance of the Weibulldistribution are calculated as

σmean = σmin + λ�

(1 + 1

m

), (13)

σmedian = σmin + λ ln (2)1m , (14)

Var(σ ) = λ2

[�

(1 + 2

m

)−

(�

(1 + 1

m

))2]

.

(15)

In the years since Weibull presented his distribu-tion, many researchers have proposed alternate formsof N (σ, V ). Freudenthal (1968) proposed a general dis-tribution for homogeneous and brittle materials. Heassumed that the flaws do not interact, and that theprobability of failure only depends on the number ofcritical flaws, Nc,S , present in a specimen of size S:

N (σ, V ) = Nc,S(σ ). (16)

Later, Danzer (1992) extended this distribution forinhomogeneousmaterials.Alternatively, Jayatilaka andTrustrum (1977) proposed a model based on flaw sizedistribution and material strength:

N (σ, V ) = Ncn−1

n!

(πσ 2

K 2IC

)n−1

(17)

where N is the number of cracks, KIC is the criticalstress intensity factor of the material, and n and c arecharacteristic constants. The above mentioned modelsassume that there is no interaction between defects.However, Afferrante et al. (2006) demonstrate that theWeibull distribution of material strength applies in ageneral sense, even if there is interaction between thedefects. They also note that the Weibull modulus doesnot necessary correspond to a material constant, andthat it may be influenced by the interaction among

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cracks, or the interaction of cracks and the stress field.Thus, in the examples we investigate in Sect. 5, wewill use the Weibull distribution (12) and will vary theWeibull modulus to illustrate its influence on the globalfracture behavior. Although it is recognized that a spa-tial correlation may exist in the strength of a material,no presumption of such a correlation is made in theexamples presented in Sect. 5.

4 Unstructured geometry by means of nodalperturbation

The most popular technique for discretizing a volumeinto tetrahedral elements may be the Delaunay trian-gulation (Lo 1991; Delaunay 1934; Cavendish et al.1985; Schroeder and Shephard 1988). Delaunay tri-angulation is the technique of choice for most auto-matic mesh generators (Schröberl 1997; Geuzaine andRemacle 2009). However, automatic mesh generatorsoften conduct additional post-processing of the mesh;to remove elements with degenerate edges and sliverelements. In some cases this additional post-processingleads these (initially random) meshes to contain anunderlying structure. To remove this structure, we pro-pose using the technique of nodal perturbation (NP).

To implement the NP algorithm, we apply the fol-lowing steps. First, all the nodes in the mesh are tra-versed and restrictions are placed on boundary nodes.Corner nodes are fixed, edge nodes are restricted to theoriginal line of the edge and face nodes are restricted tothe boundary face. At each node, theminimumdistancebetween the node and the opposite faces of the adjacenttetrahedrons is computed. The node is then perturbed ina random direction; where the magnitude of the pertur-bation corresponds to the computed distancemultipliedby a constant perturbation factor, NP ≤ 1. After per-turbing the nodes, a Laplacian smoothing technique isused to improve the mesh quality (Field 1988; Paulinoet al. 2010). In this step, each element in the mesh isvisited and the quality of the element is assessed. Theelement quality is quantified by Lo’s parameter (Lo1991), γ :

γ = 72√3 Vtetrahedron(∑

square of edges)3/2 (18)

where Vtetrahedron is the volume of the tetrahedron.Equivalently:

γ (A, B,C, D)

= 12√3 (AB × AC) · AD(‖AB‖2 + ‖BC‖2 + ‖CA‖2 +‖AD‖2 + ‖CD‖2 + ‖BD‖2)3/2 .

(19)

where A, B, C , and D correspond to the nodal posi-tions of the tetrahedron, and × and · correspond tothe cross and dot products, respectively. The higherthe Lo’s parameter, the higher the quality of the ele-ment. For example, equilateral tetrahedra have a Lo’sparameter of 1.0. If an element fails tomeet aminimumspecified Lo’s parameter, the position of each node inthe element is displaced by the average of the distancevectors to the neighboring nodes on the edges incidentto the node (Paulino et al. 2010). This procedure is iter-ated until all elements in the mesh meet the minimumquality requirements.

Here, we conduct a series of geometric studies on amesh before and after NP. The studies are conductedon a cubic domain randomly discretized with 47,924tetrahedral elements, as illustrated in Fig. 4. An illus-tration of the effect of NP on a typical Delaunay meshis shown in Fig. 4b–d. The NP factors used in this studyrange from0.1 to 0.6 in increments of 0.1. Three uniquemeshes are generated for each random NP factor, andthe results of the studies are averaged. To quantify theeffect of NP, we track three metrics: element quality(Lo’s parameter), minimum interior angle and maxi-mum interior angle.

The desired average mesh quality parameter isselected as 0.7; which is generally acceptable for finiteelement simulations (Paulino et al. 2010). The mini-mum Lo’s parameter for an unperturbed mesh is 0.434,and decreases with increasing NP factor. A histogramof the mesh quality for increasing NP factors is illus-trated in Fig. 5. As the NP factor increases, the rangeof Lo’s parameters broadens and skews to lower val-ues. Regardless of the NP factor, the maximum Lo’sparameter is approximately 1.0.

To gain additional insight into the influence of NPon the mesh, we investigate the minimum and maxi-mum interior angles in the elements. The interior anglescan give insight into the initial distortion of the ele-ments. For example, the commercial software Abaqusqualifies elements with interior angles less than 10◦or greater than 160◦ as distorted (ABAQUS 2011).A histogram of the minimum interior angle is pre-sented in Fig. 6a, while one for the maximum interiorangle is presented in Fig. 6b. For the unperturbed case,the smallest minimum interior angle is approximately

123


Fig. 4 Influence of nodalperturbation on meshesgenerated using a Delaunaytriangulation (Schröberl1997). a Unperturbed, bnodal perturbation factor of0.2, c nodal perturbationfactor of 0.4, and d nodalperturbation factor of 0.6

(a) (b)

(c) (d)

25.3◦, and decreases with increasing NP factor. Simi-larly, the largest maximum interior angle for the unper-turbed case is approximately 120.7◦ and increases withincreasing NP factor. These results are in line withexpectations, as perturbing the nodes distorts the ele-ments and the higher the perturbation factor the higherthe level of distortion.

A summary of the results of the geometric studyis presented in Table 1. Based on the results of thestudy, the NP factor of 0.4 is determined to be themaximum factor which still maintains a high qualitymesh and conforms to the recommended interior angles(ABAQUS 2011). Thus, this is the NP factor we use inthe remainder of the paper.

The above-mentioned geometric studies discuss theinfluence of the nodal perturbation factor on the qual-ity of the mesh, but some comments can be madeon the influence of the nodal perturbation factor onthe stable time step. When conducting a dynamic

fracture simulation, the time step is controlled bythe Courant-Friedrichs-Lewy (CFL) stability condition(Bathe 1996):

�t ≤ leCd

(20)

where le is the shortest distance between any two nodesin themesh, andCd is the dilatational wave speed of thematerial. Typically, the time step is recommended to befurther reduced to 10%of that required by theCFL con-dition, when conducting a dynamic fracture simulation(Zhang 2003). The nodal perturbation technique doesincrease the coefficient of variation for element facetsand thus increases the likelihood for a small, time-step-controlling facet to be introduced to themesh.However,since the meshing algorithms considered here are ran-dom Delaunay triangulations, there is always a chancethat there could be a small edge introduced, regard-less of whether or not nodal perturbation is used. Thus,

123


Fig. 5 Histograph of meshquality for various nodalperturbation factors

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2000

4000

6000

8000

10000

12000

Mesh Quality (Lo’s Parameter)

Num

ber o

f Occ

urre

nces

NP = 0.0 (Unperturbed)NP = 0.2NP = 0.4NP = 0.6

Fig. 6 Results of the studyon the interior angles in themesh: a histograph ofminimum angles, and bhistograph of maximumangles

0 5 10 15 20 25 30 35 40 45 50 55 600

5000

10000

15000

Minimum Angle ( ° )

Num

ber o

f Occ

urre

nces


(a)

60 70 80 90 100 110 120 130 140 150 160 1700

5000

10000

15000

Maximum Angle ( ° )

Num

ber o

f Occ

urre

nces


(b)

the assessment of the time step must be conducted on acase-by-case basis. For themesh investigated here, con-taining 47,924 linear tetrahedral elements, the small-est facet was calculated to be approximately twice assmall when a nodal perturbation factor of 0.4 wasused, compared to when no nodal perturbation wasused.

5 Examples

In this section, three example problems are investi-gated. The first example considers the centrifugal load-ing of a spinning disk. The second example investigatesthe impulse loading of a hollow sphere, with a focus onthe influence of surface features on the fragmentation

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Table 1 Summary of the results for the mesh quality study

Lo’s parameter (γ ) Minimum angle (◦) Maximum angle (◦)NP factor Min Mean Max Min Mean Max Min Mean Max

0.0 0.434 0.858 0.999 25.3 43.7 58.1 61.5 81.8 120.7

0.1 0.419 0.848 0.998 24.3 43.0 58.4 61.9 82.6 123.1

0.2 0.292 0.820 0.998 19.9 41.1 58.1 62.0 84.8 132.5

0.3 0.225 0.782 0.997 15.2 38.8 57.9 62.5 87.8 142.0

0.4 0.132 0.738 0.996 11.5 36.5 57.3 62.5 90.9 152.4

0.5 0.064 0.695 0.996 7.2 34.4 57.6 63.3 94.2 162.7

0.6 0.027 0.656 0.998 3.7 32.6 58.0 61.6 97.1 171.0

The quantities for each metric are averaged over three random instances

behavior of the sphere. The third example considers thefragmentation of a kidney stone under direct impact.Since kidney stones often display a radial gradationof material properties, we consider both homogeneousand functionally graded materials. Unless noted oth-erwise, we consider a fragment to be a mass of bulkelements (and cohesive elements which have not fullyseparated) completely surrounded by boundary facetsand/or fully separated cohesive elements. A pseudocode of the procedure we use to determine the frag-ments is provided in Appendix A. For each example,a mesh refinement study was conducted such that thenumber of fragments whose volume exceeded 1% ofthe volume of the model converged, for the case ofa homogeneous material. For a more rigorous and in-depth discussion on the challenges and issues relatedto the mesh convergence of models for fragmentationsimulation, the interested reader is referred to Bishopand Strack (2011) and Bishop et al. (2016). Compar-isons to experiments are made where possible. In all ofthe examples, we consider the influence of a randomlyassigned cohesive strength on the fragmentation behav-ior. The Weibull function (12) is selected to describethe distribution of cohesive strength, with the volumeparameter assumed to be equal to 1.0 (Danzer 1992).The impact of the volume parameter on fracture pat-terns has been discussed elsewhere by Brannon et al.(2007).

5.1 Centrifugal loading of a spinning disk

This example considers the fragmentation of a spin-ning disk; which is motivated by the use of structural

ceramics in the high stress environment of a spinningturbine. We consider two different geometries for thedisk, as illustrated in Fig. 7. The two disks, from thispoint forward, will be referred to as the small disk andthe large disk, respectively. Each disk is ceramic, con-stituted of silicon nitride (Si3N4); the elastic modulusof which is 300 GPa, the Poisson’s ratio is 0.3, andthe density is 3250 kg/m3. The mode I fracture energy(φn) and shape parameter (α) are set as 180 N/m and2, respectively. The minimum cohesive strength, σmin ,is set as 425 MPa. Two Weibull moduli, m = 2, 5 andtwo scale parameters λ = 40 MPa, 80 MPa are consid-ered. The mode II fracture properties are assumed to bethe same as the mode I fracture properties. The mean,median and variance of the cohesive strength for eachcase is listed in Table 2.

Experimentally, the small disk specimen was inves-tigated by Swank and Williams (1981); however, tothe best of the author’s knowledge, there have been nonumerical investigations using this geometry and load-ing condition. To simulate this problem, we use a ran-domly generated mesh containing 124,882 linear tetra-hedral elements (30,421 nodes), and a NP factor of 0.4.The spinning of the disk is represented as a centrifugalforce (applied as a body force), with an angular velocityof 4π × 103 rad/s. The angular velocity is ramped uplinearly over 100 µs and held constant thereafter. Foreach Weibull modulus (m) and scale parameter (λ) werun three simulations (a total of 12 simulations). Forthis example, we illustrate each of the results in Fig. 8.Each result displays the final fragmented shape of thespecimen.

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Fig. 7 Geometriesinvestigated in the spinningdisk example: a small disk,and b large disk

41.3 mm6.4 mm

3.8 mm

(a)

80.0 mm

50.0 mm

3.0 mm

(b)

Table 2 Summary of thecohesive strengths used inthe analysis of the spinningdisks

Scale parame-ter (λ) (MPa)

Weibull modu-lus (m)

Mean strength(MPa)

Medianstrength (MPa)

Variance(MPa)

40 2 460.5 458.3 343.4

40 5 461.7 462.2 70.8

80 2 495.9 491.6 1373.5

80 5 498.5 499.3 283.1

Based on the results, it is clear that the random dis-tribution of material properties, and the random pertur-bation of nodes, produces a random result in each sim-ulation. However, qualitatively there are many similar-ities among the fracture patterns. For the most part, thesmall disk specimen fractures into approximately fourlarge fragments and 1–3 medium fragments. The largefragments are defined as the bulk of material betweentwo cracks which span the entire thickness of the disk.A medium fragment is defined as the bulk of materialthat results after the branchof a through-thickness crackreaches the outer boundary of the specimen. For exam-ple, the result illustrated in Fig. 8f displays four largefragments and two medium fragments. In addition tothe qualitative fracture patterns, we also calculate thecrack speed through the specimen. The crack speed isdetermined by simply documenting the time it takesfor a crack to propagate through the entire width of thedisk. A summary of the results is presented in Table 3.

The crackvelocity varies between2769and3181m/s,depending on the distribution of material strength. Forthe wider distribution of strength, m = 5, the crackvelocity is higher than that observed for a more homo-geneous, or narrow, distribution of strength, m = 2.The Rayleigh wave speed, CR , for Si3N4 is 5600m/s,thus the computed values fall in the range of 0.49CR

to 0.57CR ; which is consistent with the experimentallyexpected crack velocity.

The large disk specimen was investigated experi-mentally by Hashimoto et al. (1996). Numerically, thisproblem has been simulated by Zhou and Molinari(2004). They simulate fracture using a linear cohe-sive traction-separation relation, and also consider theinfluence of material heterogeneity. Their investigationdetermined that the greater the heterogeneity in thematerial (i.e. the wider the range of cohesive strength),the fewer fragments produced. In addition, they com-pute an average crack velocity of 5500m/s, or 98%

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Fig. 8 Results for the smalldisk geometry with Weibullparameters: a-c λ = 40MPa, and m = 2; d-f λ = 40MPa, and m = 5; g-i λ = 80MPa, and m = 2; and j-lλ = 80 MPa, and m = 5. Allthe results are illustrated intheir final fragmented form,which occurred in the rangeof 118–127µs. For visualclarity, the small fragments(those comprised of fewerthan 10 bulk elements) havebeen removed from thedisplayed results

of the Rayleigh wave speed; a result which they noteis inconsistent with experiments (Zhou and Molinari2004).

To simulate this problem, we use a model contain-ing 140, 339 elements (40,768 nodes). Similarly tothe small disk, we represent the spinning of the largedisk as a centrifugal force, with an angular velocity of2.8π × 103 rad/sec. The angular velocity is ramped

up linearly over 100µs and held constant thereafter.For each Weibull modulus (m) and scale parameter(λ) we run three simulations (a total of 12 simula-tions). For this example, we only illustrate a typicalresult for each combination of Weibull modulus andscale parameter in Fig. 9. Each result displays the finalfragmented shape of the specimen. The resulting num-ber of large and medium fragments is summarized in

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Table 3 Summary of the small disk results

Scale parameter(λ) (MPa)


Large fragments Medium fragments Crack velocity (m/s)

40 2 4.33 2.67 2888

40 5 4.00 2.67 3181

80 2 4.00 2.33 2769

80 5 3.67 3.33 3179

Each result is the average of three simulationsA large fragment is defined as the bulk of material between two cracks which span the entire thickness of thecylinder. A medium fragment is defined as the bulk of material that results after the branch of a through-thicknesscrack reaches the outer boundary of the specimen

Table 4. For the smaller scale parameter,λ = 40MPa, agreater number of large and medium fragments is pro-duced than in the case with a larger scale parameter,λ = 80 MPa. Overall, the fracture patterns correspondwell to those observed experimentally (see Fig. 4 inHashimoto et al. 1996), and the distribution of frag-ments corresponds well to those in alternate numericalinvestigations (Zhou and Molinari 2004).

The velocity of the through-thickness cracks isalso tracked, and the results are included in Table 4.The crack velocity varies between 3165 and 3390m/s,depending on the distribution ofmaterial strength. Sim-ilar to the small disk geometry, for a wider distributionof strength, m = 5, the crack velocity is higher thanthat observed for a more homogeneous distribution ofstrength, m = 2. For comparison, the computed valuesfall in the range of 0.56CR to 0.61CR .

5.2 Impulse Loading of a Hollow Sphere

This example investigates the fragmentation of a hol-low sphere loaded with a radial impulse. We considerboth a smooth sphere (Fig. 10) and a sphere contain-ing surface features (Fig. 11). Numerically, the smoothsphere geometry was investigated by Levy (2010) andVocialta et al. (2016). Levy investigated the effect of thethickness of the sphere on the shape and distribution offragments while Vocialta et al. extended Levy’s workfor parallel implementation. In this study, we set theinner radius of the sphere to be 9.25mm, and the outerradius to be 10mm, as illustrated in Fig. 10a. The geom-etry is discretized with approximately 100,000 lineartetrahedral elements (approximately 25,000 nodes), as

illustrated in Fig. 10b. The sphere is constituted of alu-minum oxide (Al2O3); the elastic modulus of which is370 GPa, the Poisson’s ratio is 0.22, and the densityis 3900 kg/m3. The mode I fracture energy and shapeparameter are set as 50 N/m and 2, respectively. Theminimum cohesive strength, σmin , is set as 264 MPa.Two Weibull moduli, m = 2, 5, and two scale parame-ters λ = 50 MPa, 100 MPa are considered. The mean,median and variance of the cohesive strength for eachcase is listed in Table 5.

The sphere is loaded with a radial impulse. Sincewe assume the sphere to be centered on the origin, theinitial nodal velocities are prescribed as:

vx (x, y, z) = εx, vy (x, y, z) = εy,

and vz (x, y, z) = εz, (21)

where x , y, and z, are the nodal coordinates, and ε is theapplied rate of strain. In this study wewill consider twodifferent strain rates, ε = 2500 s−1 and ε = 5000 s−1.Once again, for each geometry, loading rate, Weibullmodulus and scale parameter, we run three simulations(a total of 72 simulations). For visual clarity, the frag-ments will be individually colored in each of the resultswe illustrate.

For a strain rate of ε = 2500 s−1, we illustratesome typical results in Fig. 12 for a Weibull mod-ulus of m = 2 and scale parameter λ = 50 MPa.As shown in Fig. 12a, the smooth sphere fragmentsinto large pieces with no discernible pattern. Alter-natively, each case with surface features displays adistinct pattern by which it fragments. For the casewith surface dimples, fracture was observed to initi-ate at the centroid of the dimples and to radiate out-wards, as illustrated in Fig. 13. This is not surprising,

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Fig. 9 Typical results forthe large disk geometry withWeibull parameters: aλ = 40 MPa, and m = 2; bλ = 40 MPa, and m = 5; cλ = 80 MPa, and m = 2;and d λ = 80 MPa, andm = 5. All the results areillustrated in their finalfragmented form, whichoccurred at approximately87µs. For visual clarity, thesmall fragments (thosecomprised of fewer than 10bulk elements) have beenremoved from the displayedresults

Table 4 Summary of the large disk results

Scale parameter (λ) (MPa) Weibull modulus (m) Large fragments Medium fragments Crack velocity (m/s)

40 2 12.33 6.67 3165

40 5 13.67 7.00 3390

80 2 11.33 4.33 3203

80 5 10.67 5.67 3346

Each result is the average of three simulationsA large fragment is defined as the bulk of material between two cracks which span the entire thickness of thecylinder. A medium fragment is defined as the bulk of material that results after the branch of a through-thicknesscrack reaches the outer boundary of the specimen

as the geometry is thinnest at this location, causingstresses to concentrate here. Additionally, the place-ment of the dimples appears to control the geometryof the fragments. In Fig. 12b, one can see that theedges of the fragments align with neighboring dim-

ples. When we consider the case with bumps (Fig. 12c)there is clear evidence that the edges of the fragmentsare deflected away from the bumps. Thus, the bumpsremain intact during fragmentation.While we only dis-play selected results here, this behavior was shown

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Fig. 10 Model of a smooth,hollow sphere; a geometry,and b mesh

9.25mm

0.75mm

XY

Z

(a) (b)

Fig. 11 Geometry of ahollow sphere with differentsurface features, a dimples,and b bumps. The dimplesare generated by removingmaterial that intersects withspheres of radius,r = 1.1 mm, placed at aradial distance of 10.5 mmfrom the origin. The bumpsare generated through theunion of the smooth sphereand spheres of radius,r = 1.1 mm, placed at aradial distance of 9.5 mmfrom the origin

XY

Z

(a) (b)

Table 5 Summary of thecohesive strengths used inthe analysis of the hollowsphere

Scale parame-ter (λ) (MPa)


Mean strength(MPa)

Medianstrength (MPa)

Variance (MPa)

50 2 308.3 305.6 536.5

50 5 309.9 310.5 110.6

100 2 352.6 347.3 2146.0

100 5 355.8 356.9 442.3

across all the Weibull moduli and scale parameters weconsidered.

If we increase the applied strain to ε = 5000 s−1,fragmentation becomes more pervasive, as illustratedin Fig. 14. Similar to the previous case, the fragmen-tation of the smooth sphere does not display any dis-cernible pattern. However, fracture continues to initiate

at, and to radiate outwards from, the dimples; and eachbump is wholly contained in a single fragment. Onceagain, this behavior was shown across all the Weibullmoduli and scale parameters we considered.

To further examine the fragmentation behavior, weplot the evolution of fragmentation through time inFig. 15. Results are shown for a variation in scale

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Fig. 12 Final fragmented geometries, after impacted with animpulse load at a rate of ε = 2500 s−1, for: a a smooth sphere, b aspherewith dimples, and c a spherewith bumps. The results showtypical fracture patterns, but are for the case with λ = 50 MPa

and m = 2. The fragments are colored for visual clarity, andthe small fragments (those comprised of fewer than 10 bulk ele-ments) have been removed from the displayed results

Fig. 13 Initial stages offracture in a sphere withdimples. a Contour ofprincipal stress, illustratingthe concentration of stress atthe centroids of the dimples,immediately prior tofracture; and b fractureinitiating at the dimples

2.7E 08+2.5E 08+2.3E 08+2.1E 08+1.9E 08+1.7E 08+1.5E 08+1.3E 08+

XY

Z

(a) (b)

parameter, λ, with a constant Weibull modulus of m =2. In each scenario, fragmentation initiates quickly(after approximately 0.5 to 2µs), and takes approxi-mately 20µs to complete. In all the cases with a higherapplied rate of strain, ε = 5000 s−1, the majority of thefragmentation occurs in the first 10µs, and fragmen-tation evolves slower in cases with λ = 100 MPa thanthey do in cases with λ = 50 MPa. When we examinethe cases with a lower rate of strain, ε = 2500 s−1, littledifference is observed in the cases with λ = 50 MPaand λ = 100 MPa. However, for the case with dimples(Fig. 14b) fragmentation evolves at a faster rate thanwhen we consider a smooth sphere, or a sphere withbumps. From this investigation, we see that the distri-

bution of material parameters does have an impact onthe manner in which fragmentation occurs; however,the fragmentation patterns are also significantly influ-enced by surface features. This investigation providessupport to the idea of being able to control fragmenta-tion behavior through simple design features.

An investigation of energy evolution is conductedfor the case of the smooth sphere, as illustratedin Fig. 16. The energy is comprised of the strain(Eint ), kinetic (Ekin) and fracture (E f ra) energies. Theimpulse load on the structure results in a large initialkinetic energy, which is converted into strain energy inthe system. The fracture energy remains equal to zeroup until the point of crack initiation, and monotoni-

123


Fig. 14 Final fragmented geometries, after impacted with animpulse load at a rate of ε = 5000 s−1, for: a a smooth sphere, b aspherewith dimples, and c a spherewith bumps. The results showtypical fracture patterns, but are for the case with λ = 50 MPa

and m = 2. The fragments are colored for visual clarity, andthe small fragments (those comprised of fewer than 10 bulk ele-ments) have been removed from the displayed results

0 10 200

0.2

0.4

0.6

0.8

1

Time (μs)

Nor

mal

ized

Vol

ume

of F

ragm

ents

0 10 200

0.2

0.4

0.6

0.8

1

Time (μs)0 10 20

0

0.2

0.4

0.6

0.8

1

Time (μs)

(a) (b) (c)

ε = 2500s−1, λ = 50MPa ε = 2500s−1, λ = 100MPa ε = 5000s−1, λ = 50MPa ε = 5000s−1, λ = 100MPa

Fig. 15 Evolution of fragmentation for: a a smooth sphere, b a sphere with dimples, and c a sphere with bumps. The results shownconsider a Weibull modulus of m = 2

cally increases over time. The summation of energyin the system remains constant over time, indicating aconservation of energy. For a higher rate of strain, theinitial kinetic energy in the system is larger, as illus-trated in Fig. 16b. This higher rate of strain also leads toan earlier onset of fracture, however, the relative mag-nitudes of potential and fracture energy in each systemremains unchanged.

While not pursued here, this example motivates adeeper investigation into the effects of surface featureson fragmentation patterns and distributions. The frag-mentation patterns illustrated here indicate a strongdependence on the kind of surface feature, but it is alsolikely that the fragmentation patterns would be a func-tion of the spacing and sizing of each surface feature.Statistically quantifying such a functional dependence

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0 0.5 1 1.50

0.02

0.04

0.06

0.08

0.1

0.12

Time (μs)

Ener

gy E

volu

tion

(Nm

)Strain Energy, EintKinetic Energy, EkinFracture Energy, Efra

0 0.5 1 1.50

0.1

0.2

0.3

0.4

0.5

Time (μs)

Ener

gy E

volu

tion

(Nm

)

Strain Energy, EintKinetic Energy, EkinFracture Energy, Efra

(a) (b)

Fig. 16 Evolution of energy for a hollow sphere under an impulse load with an applied rate of strain of: a 2500 s−1, and b 5000 s−1

of the distribution of fragments on surface featuresmaybe pursued as a future work.

5.3 Direct impact of a kidney stone

This example considers the direct impact of a kidneystone through the use of a Lithoclast. The Lithoclast isa surgical tool which propels a metallic probe againstthe stone. The kinetic energy of the probe is dissi-pated throughout the stone, forming cracks and frag-ments, allowing for easier removal or passage. Thestructure of the kidney stone has long been known tobe nonhomogeneous (Zhong et al. 1992, 1993; Pit-tomvils et al. 1994), however, most attempts at sim-ulating such problems have assumed continuous mate-rial properties (Mota et al. 2006; Caballero and Moli-nari 2010, 2011). Mota et al. (2006) studied the frag-mentation of manufactured, cylindrical kidney stonesmade from gypsum. They load the stone using the tech-nique of Lithotripsy, wherein a pulse of pressure isapplied to the exterior of the stone. They also con-ducted experiments and displayed the cohesive ele-ment method’s ability to capture fragmentation pat-terns which agreed with those observed experimen-tally. Caballero andMolinari (2010) and Caballero andMolinari (2011) simulate the direct impact of the kid-ney stone through the Lithoclast, but only consideredthe simplified case of a homogeneous stone structure.In this example, we investigate the influence of a radi-ally heterogeneousmicrostructurewithin the stone, andinclude the influence of randomly distributed materialstrength.

The model we use to simulate this problem has aradius of 10mmand contains, contains 51,746 elements(10,424 nodes), as illustrated in Fig. 17. The elementsat the site of impact have a maximum element size of0.1 mm, and those in the bulk of the stone have a max-imum size of 0.65 mm. In this investigation we willconsider the three distinct sets of material propertieslisted in Table 6. We will first investigate the fragmen-tation behavior for a homogeneous stone; then considerthe behavior when the core of the stone is comprisedof a different material than the outer layer of the stone.The numerical framework described in Sect. 2 appliesto both homogeneous and functionally graded materi-als. To numerically capture the linear gradation of thematerial between distinct zones, we use graded finiteelements; which incorporate the material gradation atthe size-scale of the element. There have been multiplemethods designed to allow for graded material proper-ties at the element scale (Santare and Lambros 2000;Kim and Paulino 2002), however, in this work, we usethe generalized isoparametricmethod proposed byKimand Paulino (2002). In the generalized isoparametricmethod the element shape functions are used to inter-polate the material parameters from the Gauss pointsto the nodes. Between distinct zones we grade all bulkand cohesive material parameters, as well, we continueto assume a random distribution of cohesive strength.

The impact of the probe is described by displacingthe nodes in the impact site by a prescribed velocity.The velocity of the probe is not known precisely, butwe assume that it decreases monotonically. CaballeroandMolinari (2010) andCaballero andMolinari (2011)suggest that the impact velocity follows the relation:

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Fig. 17 Geometry andmesh of the kidney stonemodel

Table 6 Summary of material properties in common kidney stones

Stone composition (%) φ (N/m) σ (MPa) ρ (kg/m3) E (MPa)

COM 0.735 1.500 2038 25.162

CA 0.382 0.500 1732 8.504

CA(50)/COM(50) 0.553 1.000 1885 16.833

COM calcium oxalate monohydrate; CA carbonate apatite (Zhong et al. 1993)

uC (t) = uC0

[1 − e−η t

tmax

1 + (e−η − 1

) ttmax

], (22)

where η ∈ (−∞,+∞), uC0 is the initial velocity of theprobe, and tmax is the total time of contact. The totaltime of contact is determined through the followingrelation:

tmax = ut=tmax

uC0

(e−η − 1

)21 − e−η (1 + α)

, (23)

where ut=tmax is the maximum displacement achievedby the probe after impact. The velocity relation used inthis investigation is illustrated in Fig. 18.

First, we examine the fragmentation behavior of thehomogeneous kidney stone.Weconsider all threemate-rials listed in Table 6 and plot the evolution of frag-

Fig. 18 Impact velocity of metal probe on kidney stone

mentation through time in Fig. 19. Results are shownfor a variation in both scale parameter, λ, and Weibull

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0 50 1000

500

1000

1500

2000

2500

3000

3500

4000

Time (μs)

Vol

ume

of F

ragm

ents

( mm

3 )

0 50 1000

500

1000

1500

2000

2500

3000

3500

4000

Time (μs)0 50 100

0

500

1000

1500

2000

2500

3000

3500

4000

Time (μs)

λ = 0.2MPa, m = 2 λ = 0.2MPa, m = 5 λ = 0.4MPa, m = 2 λ = 0.4MPa, m = 5

(a) (b) (c)

Fig. 19 Evolution of fragmentation for a homogeneous kidney stone constituted of: a CA, b CA(50)/COM(50), and c COM

modulus, m, for each material. In general, the frag-mentation evolves slower for the most compliant mate-rial (CA), than it does for the least compliant mate-rial (COM). When considering the random distribu-tion of material strength, the scale parameter is shownto have a much greater influence on the evolution offragmentation that the Weibull modulus. For a smallerscale parameter, λ = 0.2 MPa, fragmentation occurssooner and is more pervasive than in the case of alarge scale parameter, λ = 0.4 MPa. Additionally,the final volume of fragments indicates that the stonedoes not fully fragment when the range of materialstrength increases. The results shown are for a sin-gle instance of the simulated results, however, eachscenario was simulated three times, and the presentresults and conclusions are typical across each sim-ulation.

When we assume that the stone contains a differ-ent constitutive make-up in the inner core than it doesin the outer layer, the fragmentation behavior changessignificantly. In this investigation, we consider sce-narios with both a more compliant and less compli-ant material in the inner core. Based on experimen-tal observations of the microstructure of kidney stones(Zhong et al. 1992, 1993; Pittomvils et al. 1994), theouter core was assumed to be 15% the thickness of theradius, and the zone transitioning between the innercore and outer layer was assumed to be 20% the thick-

ness of the radius. Multiple values were investigatedfor the thickness of both the outer layer and zone oftransition, but these values did not significantly influ-ence the global response. Typical results are illustratedin Fig. 20. When we have a more compliant innercore, as in Fig. 20a, b, complete fragmentation evolvesrapidly (in approximately 40–45µs) regardless of thematerial in the core. Additionally, the distribution ofmaterial strength has a much smaller effect than inthe homogeneous stones. Alternatively, when a lesscompliant material is present in the inner core, as inFig. 20c, fragmentation is significantly restricted. Themaximum volume of fragments produced was approx-imately 125mm3, significantly less than the volume ofthe stone (4189mm3). This small volume of fragmentsdemonstrates the limited ability of the softer outer layerto transmit enough energy to the inner core, from theimpacting probe, to cause fragmentation to occur. Sim-ilar behavior was observed across all scenarios we con-sidered, including scenarios with different materialsand thicker and thinner outer layers.

Similarly to the the previous example, we investi-gate the evolution of energy throughout the simulation.In this case, we examine the evolution of energy for twohomogeneousmaterials, as illustrated in Fig. 21. In thisexample, the external energy from the impacting probeis converted into strain, kinetic and fracture energy. Inthe initial stages of impact, the external energy is pri-

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0 50 1000

1000

2000

3000

4000

Time (μs)

Vol

ume

of F

ragm

ents

(mm

3 )

0 50 1000

1000

2000

3000

4000

Time (μs)0 50 100

0

20

40

60

80

100

120

Time (μs)(a) (b) (c)

λ = 0.2MPa, m = 2 λ = 0.2MPa, m = 5 λ = 0.4MPa, m = 2 λ = 0.4MPa, m = 5

COM

CA(50)/COM(50)

COM

CA(50)/COM(50)

COM

CA

Fig. 20 Evolution of fragmentation for a graded kidney stone constituted of: a an inner core of CA, and an outer layer of COM; b aninner core of CA(50)/COM(50), and an outer layer of COM; and c an inner core of COM, and an outer layer of CA(50)/COM(50)

Fig. 21 Evolution ofenergy for a homogeneouskidney stone constituted of:a CA, and b COM

0 10 20 30 40 50 60 70 800

0.01

0.02

0.03

0.04

0.05

0.06

Time (μs)

Strain Energy, EintKinetic Energy, EkinFracture Energy, EfraExternal Energy, Eext

Ener

gy E

volu

tion

(Nm

)

(a)

0 10 20 30 40 50 60 70 800

0.02

0.04

0.06

0.08

0.1

0.12

Time (μs)

Ener

gy E

volu

tion

( Nm

) Strain Energy, EintKinetic Energy, EkinFracture Energy, EfraExternal Energy, Eext

(b)

marily converted into strain energy. The strain energyincreases rapidly, but decreases as the damaged regionin the stone expands. As the stone begins to fracture,the kinetic energy and fracture energy increase. Overtime, the problem reaches equilibrium, and the fractureof the stone stops. At equilibrium, the external energyis converted into kinetic and fracture energy, each ofwhich remain constant with time.

6 Concluding remarks

The cohesive element method’s ability to capture thefull range of fracture behavior: from crack initiation,through crack propagation, branching and coalescence,all the way to complete fragmentation, has allowed usto study the pervasive fracture problemsherein. In orderto reduce mesh induced artifacts on fracture behav-

ior, often caused by the use of smoothing operatorsin automatic mesh generators, we propose the nodalperturbation operator to introduce geometric random-ness. We perform a geometric analysis on Lo’s meshquality parameter and minimum and maximum inte-rior angles in the mesh before and after nodal pertur-bation. Through the investigation, we demonstrate thatthe use of a nodal perturbation factor around 0.4 isable to produce a highly randomized mesh while stillmaintaining high quality elements satisfying require-ments on Lo’s parameter and interior angles. Thus, thisis the value of the nodal perturbation factor we rec-ommend, and is the value used throughout the exam-ples.

To further alleviate the influence of the mesh on thefracture patterns, we discuss the use of random hetero-geneous constitutive relations. Random heterogeneousconstitutive relations are motivated by the idea that no

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material is truly homogeneous, but contains hetero-geneities at the microscale. We provide a summary ofmany of the prominent methods, for representing thesemicroscale heterogeneities, including the well-knownWeibull distribution. We use these statistical meansto distribute the strength of the material in the prob-lems we investigate. To capture the failure responseof the material, we implement the extrinsic PPR cohe-sive model, and adaptively insert cohesive elements infront of each crack tip to capture the nonlinear fracturebehavior. This paper serves as an instance of the use ofthe extrinsic PPR model in three-dimensional fracturesimulations.

Three numerical examples are presented, which arespecifically selected to highlight many of the promi-nent features effecting the fragmentation of struc-tures. The first example considers high-speed spin-ning of a ceramic disk. The combination of geo-metric and constitutive randomness results in randomfracture behavior; which corresponds well to experi-mentally observed fragmentation patterns and crackspeeds. The second example demonstrates the abil-ity to use surface features on structures undergoingblast or impact loads to control fragmentation pat-terns. The third example demonstrates the significanceof accounting for material gradation in the investiga-tion of structures under direct impact. By using thecase of a graded kidney stone, we demonstrate thatthe gradation of the material can significantly influ-ence the fragmentation behavior, and in some casescan even inhibit fragmentation from occurring alto-gether.

Acknowledgements We acknowledge support from the Natu-ral Sciences and Engineering Research Council of Canada andfrom the U.S. National Science Foundation (NSF) through Grant#1624232 (formerly #1437535). The information presented inthis publication is the sole opinion of the authors and does notnecessarily reflect the views of the sponsors or sponsoring agen-cies.

Appendix A. Pseudo code for fragment definition

The following pseudo code outlines the method used todelineate the fragments in the pervasive fragmentationresults presented in the paper. The objective of the codeis to create a list of fragments and all bulk elementsbelonging to that fragment.

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