Glass Struct. Eng.https://doi.org/10.1007/s40940-018-00091-1

RESEARCH PAPER

Relationship between strain energy and fracture patternmorphology of thermally tempered glass for the predictionof the 2D macro-scale fragmentation of glass

N. Pourmoghaddam · M. A. Kraus ·J. Schneider · G. Siebert

Received: 15 May 2018 / Accepted: 9 November 2018© The Author(s) 2018

Abstract This work deals with the prediction of glassbreakage. A theoretical method based on linear elasticfracture mechanics (LEFM) merged with approachesfrom stochastic geometry is used to predict the 2D-macro-scale fragmentation of glass. In order to pre-dict the fragmentation of glass the 2D Voronoi tes-selation of distributed points based on spatial pointprocesses is used. However, for the distribution of thepoints influence parameters of the fracture structure aredetermined. The approach is based on two influenc-ing parameters of fragment size δ and fracture inten-sity λ, which are described in this paper. The Frag-ment Size Parameter describes the minimum distancebetween the points and thus the size of a fragment. Itis derived from the range of influence of the remain-ing elastic strain energy in a single fragment takinginto account the LEFM based on the energy criterionof Griffith. It considers the extent of the initial elasticstrain energy U0 before fragmentation obtained fromthe residual stress as well as a ratio of the releasedenergy η due to fragmentation. The Fracture IntensityParameter describes the intensity of the fragment dis-

N. Pourmoghaddam (B) · J. SchneiderTechnical University of Darmstadt, Franziska-Braun-Str. 3,64287 Darmstadt, Germanye-mail: pourmoghaddam@ismd.tu-darmstadt.de

M. A. Kraus · G. SiebertUniversity of German Armed Forces Munich,Werner-Heisenberg-Weg 39, 85577 Munich, Germany

M. A. Krause-mail: m.kraus@unibw.de

tribution, and thus the empirical reality of a fracturepattern. It can be obtained by statistical evaluation ofthe fracture pattern. In this work, the fracture intensityis determined from the experimental data of fracturetests. The intensity of a fracture is the quotient of thenumber of fragments in an observation field and itsarea and is assumed to be constant in the observationfiled. The fracture intensity and the correlation betweena constant intensity and the Fragment Size Parameterwas determined. The presented methodology can alsogenerally be used for the prediction of fracture patternsin brittle materials using aVoronoi tesselation over ran-dom fields.

Keywords Fragmentation · Tempered glass ·Fragment size · Fracture intensity · Elastic strainenergy · Spatial point process · Voronoi tessellation ·Fracture pattern

List of symbols

σi j Stress tensorεi j Strain tensorr, θ Cylinder coordinates (radius and polar angle)τ Shear stressσ(z) Stress function along the z-axis of the plateσs Surface stressσm Mid-plane stresst Glass plate thicknessν Poisson’s ratio

123

N. Pourmoghaddam et al.

E Young’s modulusP Set of points pi , i = 1 . . . n in the domain KK The 2D domain of the glass fracture patternθ Point process parametersP Probability of obtaining n points pi , i = 1 . . . n

in the domain Kn, N Random variable and obtained realization for

the number of points pi , i = 1 . . . n in thedomain K

HC Hardcore ProcessMHC Matérn Hardcore ProcessSR Strauss processfn(P) Probability density of a point pattern Pci Normalizing constants of a probability densitys(P) Number of point pairs to be penalized in HCδHC Minimum Euclidean distance of points in HCβ Intensity of the original SRγSR Inhibition parameter of SRδ Fragment Size Parameterλ Fragment/point process intensity parameterΓ Fracture surface energyγ Specific fracture surface energyA f r Fracture surface of a fragmentρ∗ Correction factor of the fracture surfaceGc Critical energy release rateKIc Critical stress intensity factorη Energy relaxation factorAη Energy relaxation zone of a fragmentUη Elastic strain energy in the relaxation zone Aη

U0 Initial strain energyU1 Remaining strain energyUR Relative remaining strain energyND Fragment density in the observation field with

the length of D

1 Introduction

Glass is one of the most popular building materialstoday. However, the tensile strength is governed bysmall flaws in the surface which reduce the actual engi-neering strength of annealed float glass to 30–100MPa(Schneider et al. 2016). Due to the residual stress statethermally tempered glass shows a greater resistance toexternal loads and in case of failure it is quite safein terms of cutting and stitching due to the small,blurred fragments. Therefore, thermally tempered glassis also known as tempered safety glass. The residualstress state is obtained by the tempering process and

is approximately parabolic distributed along the thick-ness with the compressive stress on both surfaces andan internal tensile stress in the mid-plane. By imposinga compressive residual stress at the surface, the sur-face flaws will be in a permanent state of compressionwhich has to be exceeded by externally imposed stressbefore failure can occur (Schneider 2001; Nielsen et al.2010; Pourmoghaddam et al. 2016; Pourmoghaddamand Schneider 2018a). The amount of the residual sur-face compressive stress largely depends on the cool-ing rate and therefore on the heat transfer coefficientbetween the glass and the cooling medium (Gardon1965; Aronen and Karvinen 2017).

However, thermally tempered glass will fragmen-tize completely into many pieces, if the equilibratedresidual stress state within the glass plate is disturbedsufficiently and if the elastic strain energy in the glassis large enough, Fig. 1. The so-called Rupert’s dropwith bulbous head and thin tail can withstand highimpact or pressure applied to the head, but explodesimmediately into small particles when the tail is bro-ken (Silverman et al. 2012). The fragmentation is thedirect consequence of the elastic strain energy that isstored inside thematerial due to the residual stress state.The fragment size depends on the amount of the storedenergy. Small fragments are caused by a high storedstrain energy due to the high residual stress state orig-inating from the extremely rapid cooling. And lowerresidual stress states result in larger fragments due tolower stored strain energy (see Fig. 2). Thus, not onlythe stress but also the thickness of the glass plate playsa role in determining the strain energy.

Thus, for the same residual stress distribution ahigher stored energy is determined in the thicker plates.This has been investigated and proved experimentallyby several authors (Akeyoshi and Kanai 1965; Barsom1968;Gulati 1997; Lee et al. 2012;Mognato et al. 2017;Pourmoghaddam and Schneider 2018b).

The fragmentation pattern, i.e. the fracture structure,the fragment size and thus the fragment density are thedirect consequence of the amount of the initial strainenergy U0 which is available before the fragmentationsets in and can be written as:

U0 = UResidual stress +UExternal loading (1)

where it should be noted that in this study, it is assumedthat the glass is only subjected to residual stress fromthe thermal tempering U0 = UResidual stress .

123

Relationship between strain energy and fracture pattern morphology

Fig. 1 Fragmentedtempered glass(fragmentation initiated bydrill)

Fig. 2 Fragmentation ofglass plates with the samethickness of 12mm for alow stored strain energy(U = 78.1 J/m2), b highstored strain energy(U = 354.7 J/m2)

When the glass is fragmentized the internal energyconverts into a number of forms of energies (Reichet al. 2013; Nielsen 2017), e.g. for creating new frac-ture surfaces (cracking and crack branching), kineticsand sound energy. Hence, the remaining strain energyin a fragment U1 after the fragmentation sets in mustbe expressed as:

U1 = U0 − (Ucrack +Ukinetic +Usound +Uother ) (2)

whereUother represents other energy consumingeffectssuch as heat. In this study, only the relative remainingstrain energy UR as the ratio between the remainingstrain energy and the initial energy is considered. Therelative remaining strain energy can be written as:

UR,Rem = U1

U0(3)

The other forms of energy are not included here.The remaining stress state and the resulting remainingelastic strain energy in a single fragment has been cal-culated numerically by Nielsen (2017). Further exper-imental and numerical investigations of fragment’sdeformation and strain energy were carried out by

Nielsen and Bjarrum (2017). The change in strain wasdetermined by comparing the surface shape of a frag-ment before and after fracture.

Several models for relating the fragment size to theresidual stress state have been suggested in the lit-erature, e.g. Acloque (1956), Barsom (1968), Gulati(1997), Shutov et al. (1998), Warren (2001) and Tan-don and Glass (2005). Some of these works have pro-posedmodels for the fragments size based on an energyapproach. Most of the works try to establish an analyti-cal model for the fragment size considering the releaseof the so-called tensile strain energy defined as the partof the strain energy resulting from the mid-plane ten-sile stress alone.A theoreticalmethod based on fracturemechanics was developed by e.g. Warren (2001), Mol-nár et al. (2016). The Voronoi tesselation of randomlydistributed points in the glass plane leading to a cellstructure similar to glass breakage was mentioned inMolnár et al. (2016). However, the statistical distribu-tionmethod of the points in the plane itself is significantwhen it comes to the prediction of the fracture pattern.With regard to the parameters of the fracture structure,the comparison of different spatial point processes istherefore important in order to find the best method ofpoint distribution.

123

N. Pourmoghaddam et al.

Fig. 3 Stress distribution inthe far field area of atempered glass plate and asketch showing a contourline at zero stress (dottedline)

The motivation is that Voronoi tessellation of pointsdistributed in the plane results in a fracture struc-ture based on both fracture mechanical and stochasticparameters. Hence, the main objective is to determinethe significant influence parameters of the glass frac-ture in order to develop a method for the prediction of2D macro-scale fragmentation of glass. In this work,two influence parameters of the fracture structure, theFragment Size Parameter δ and the Fracture IntensityParameter λ are described and determined. Using thesetwo parameters the spatial point process can be influ-enced and calibrated by the energy density and empir-ical reality of a fracture pattern.

In order to determine the Fragment Size Parameterδ a method was developed with which the minimumdistance between two points in a point cloud can bedetermined by selecting the energetic criterion (Griffith1920) from the linear fracture mechanics with respectto the strain energy state before and after the fragmen-tation. In other words; the present work aims at deter-mining the minimum distance between randomly dis-tributed points in a plane representing fragments basedon the strain energy state which is stored in the glassplate due to the thermal tempering. Subsequently, thefragment density is estimated using Hexagonal ClosePacked (HCP) and a uniformly distributed point group.

The Fracture Intensity Parameter λ describes thedegree of intensity of the fracture in a limited observa-tion field and is a value for the fragment number withinthe observation field. This parameter was determinedfrom the experimental data of fracture tests.

2 Energy conditions

When external forces or residual stresses deform anelastic body, these stresses performwork as their pointsof application are displaced. This work is stored in the

body as elastic strain energy. The total strain energyU stored in a deformed linear elastic, isotropic bodyis obtained by integrating the energy per unit volumeover the volume of the body:

U = 1

2

∫V

σi jεi j dV (4)

where σi j is the stress tensor and εi j is the strain ten-sor. The residual stress in thermally tempered glassis distributed parabolically along the glass thick-ness t (Fig. 3). This parabolic stress distributionσ(z) can be written in terms of the surface stressσs as:

σ(z) = 1

2σs(1 − 3ζ 2), ζ = 2z

t(5)

using symbols defined in Fig. 3. The parabolic stressdistribution is in equilibrium and symmetric aboutthe mid-plane. The magnitude of the surface stress isapproximately twice the tensile stress (2σm = −σs).The zero stress level is at a depth of approximately 20%of the thickness t .

The stress state σ is assumed to be planar. Usingcylinder coordinates r for the radius and θ for the polarangle the constitutive relationship for plane stress isdefined by:

⎡⎣ σr

σθ

τrθ

⎤⎦ = E

1 − ν2

⎡⎣1 ν 0

ν 1 00 0 1−ν

2

⎤⎦

⎡⎣ εr

εθ

γrθ

⎤⎦ (6)

with the Young’s modulus E and the Poisson’s ratioν. The equilibrated stress state is also assumed to behydrostatic for field stresses.Aplanar hydrostatic stressstate means that no shear stresses occur (τrθ = 0) andthe normal stresses are always principal stresses, whichare equal in the plate plane (σr = σθ = σ(z)) and zero

123

Relationship between strain energy and fracture pattern morphology

in the direction of the thickness (σz = 0). Hence, thetotal initial elastic strain energy U0 can be written as:

U0 = 1

2

∫V(σrεr + σθεθ ) dV = 1 − ν

E

∫V

σ 2(z) dV

(7)

with

εr = εθ = 1 − ν

Eσ(z) (8)

Inserting the residual stress field from Eq. (5) andintegrating over the cylinder we find:

U0 = 1 − ν

E

z=0.5t∫

z=−0.5t

r=R∫

r=0

θ=2π∫

θ=0

σ 2(z) rdθ dr dz

= πR2 (1 − ν)

5Etσ 2

s

(9)

which is the initially stored strain energy in a cylindri-cal body of the radius R, thickness t and the residualsurface stress of σs .

The Eq. (9) can be written in terms of the strainenergy per unit surface area of any given base shapeof a body by dividing with the base area for the cylin-der (Barsom 1968; Gulati 1997; Warren 2001; Nielsen2017):

U0 = (1 − ν)

5Etσ 2

s (10)

The energy can also be written in terms of the energydensity UD:

UD = 1

5

(1 − ν)

Eσ 2s = 4

5

(1 − ν)

Eσ 2m (11)

which is the amount of elastic strain energy stored inthe system per unit volume and thus only depends onthe residual stress and the material properties.

3 Voronoi tesselation over spatial point processes

This section provides a brief introduction to the twomathematical tools that play a central role in the latteralgorithm: Spatial point processes and Voronoi Dia-grams. Further details and exhaustive presentations of

these topics are covered in Baddeley et al. (2016), Wie-gand and Moloney (2014), Schmidt (2015), Møller(1994), Okabe et al. (1992) and Ohser and Schladitz(2009).

3.1 Spatial point processes

A point process describes random configurations ofpoints P = {p1, . . . ,pn} in a continuous bounded setK (which is in the case of this paper the 2D domainof the glass fracture pattern with area content |K |).The number of points n is itself a random variable Nthat typically follows a discrete Poisson distribution.The least complex point patterns exhibit complete spa-tial randomness (CSR), i.e. the point locations pi occurindependently and uniformly (equal likelihood) overthe domain K . The probability density of a point pat-tern P with point process parameters θ and distributionof the point locations f θ

n (p1, . . . ,pn) is given by

fP (P|θ) = P[N (K ) = n|θ ]n! f θn (p1, . . . ,pn)

= λn(p) exp((1 − λ)|K |) (12)

Here,P is one of the possible point patterns and f (P)

can be interpreted as ‘the relative chance of obtainingthe point pattern P’ (Baddeley et al. 2016). Note, thatin Eq. (12), the probability density of a CSR does notdepend on the locations of the points pi , i.e. all pos-sible patterns of n points are equally likely, the differ-ent points are furthermore independent and each pointis uniformly distributed over the domain K . Furtherpoint processes can be defined by probability densi-ties fP (P|θ) which differ in their mathematical formfrom Eq. (12) (i.e. a CSR process), this will serve asthe basis for two more point processes presented laterin this section.

The behavior of spatial point processes can be char-acterized in terms of first-order and second-order prop-erties. Thefirst-order property is described by the inten-sity λ(p) (mean number of events per unit area at thepoint p) and the second-order property reflects the spa-tial dependence in the process (clustering or repul-sion). More complex spatial point processes are ableto enforce either clustering or repulsion of the points,which is observable in the second-order statistics ofthese processes. Figure 4 shows examples of point pat-terns which exhibit CSR and two examples violatingCSR due to regularity (repulsion) and clustering.

123

N. Pourmoghaddam et al.

Fig. 4 Spatial pointprocesses with sameintensity λ: a homogeneousPoisson point process(CSR), b regular pointprocess (MHC) and ccluster point process (CPP)

Basedon inspection of typical fracture patterns, suchas shown in Figs. 1 and 2 a repulsion behaviour ofthe underlying spatial point process can be concluded.Repulsion processes possess a repulsion domain,whichis located around each seed point to avoid points beingtoo close to each other. When dim K = 2, this domainis a disk with radius δHC , which is a model parame-ter of the underlying spatial point process. This kindof pairwise interaction of the points can be modelledvia repulsion/inhibition processes. In the context ofthis paperMatérn Hardcore Processes (MHC) (Matérn1960) as well as Strauss (repulsion) processes (SR)(Strauss 1975) are investigated, which belong to thefamily of the Gibbs processes (Baddeley et al. 2016).

In contrast to the CSR process with one parameter(intensityλ), theHCprocess possesses two parameters,the intensity λHC and the (minimal) hardcore distanceδHC , thus it is a two-parametric model with probabilitydensity (Baddeley et al. 2016):

f θn (p1, . . . ,pn) =

{cHCλ

n(p)

HC if n(p) ≥ δHC

0 if else(13)

where cHC is the normalization constant. Matérn(1960) proposed three schemes, called Matérn type-I, type-II, and type-III hardcore point processes, forconstructing repulsive point processes. A type-I pro-cess is gathered by sampling a primary process froma homogeneous Poisson process with intensity λ, andthen deleting all pointswith distance less than δHC . TheMatérn type-II process assigning each point an ‘age’,which defines, in case of point-distances less than δHC ,that the older point is kept while the younger pointis deleted in the sampling process. A Matérn type-IIIprocess lets a newer event be thinned only if it fallswithin radius δHC of an older event thatwas not thinnedbefore. According to Baddeley et al. (2016), the hard

core process is a CSR conditional on the event, that itsatisfies the hard core constraint. For the modelling ofthe fracture pattern, thismeans, that the intensityλHC isequivalent to the intensity of a CSR λ while only keep-ing points, that are not closer than δHC . In the latter ofthis paper it is highlighted, that the hardcore constraintintroduces significant differences in the second-orderstatistics of a point process and it is further shown, thatfracture patterns from experiments are in favour of ahardcore constraint.

Acc. to Illian et al. (2008), the intensity of a HC λHC

can be estimated from the intensity of a CSR λ via

λHC = (1 − exp(−λ · V ))/V (14)

where V = π · δ2HC . Analogously to the derivation ofEq. (29), combination of Eqs. (27) and (14).

In general, the hard core process is an appropri-ate model given, that it is physically impossible fortwo points to lie closer than a distance δHC . If closepairs of points are not impossible but unlikely to occur,the Strauss (repuslion) process (SR) ‘penalises’ ratherthan ‘forbids’ close pairs of points. In the context ofthis paper Strauss (repulsion) processes SR (Strauss1975) are investigated, which belong to the family ofthe Gibbs processes. Without going in detail here, c.f.(Stoyan et al. 1995), the density of a Strauss point pro-cess reads

f θn (p1, . . . ,pn) = cSRβn

SRγs(p)

SR (15)

where cSR is a constant, βSR > 0 is the ‘fertility’parameter, 0 ≤ γSR ≤ 1 is the interaction parame-ter, and s(p) is the number of point pairs, of whichthe Euclidean distance is less than δHC . The MatérnHardcore Process is a special case of the Strauss pro-cess with γSR ≡ 0, where γSR can be interpreted

123

Relationship between strain energy and fracture pattern morphology

as the inhibition parameter. The considerations beforelead to a total of two respectively three parameters,which are necessary to characterise either the Hard-core Process θHC = {λHC ; δHC } or the Strauss Pro-cess: θSR = {βSR; δHC ; γSR}. However, in Sect. 4 itis derived, that the parameters θHC as well as θSR aremotivated from and are connected to distinct parame-ters from Linear Elastic Fracture Mechanics (LEFM)considerations.

According to Baddeley et al. (2016), β is not the‘intensity’ of the HC or SR model, instead β should beseen as the spatially varying ‘fertility’which is counter-balanced by the ‘competitive’ effect of the hard core togive the final intensity, thus the ‘intensity’ is the prod-uct of fertility and competition. β is referred to as the(first-order) trend which is of more interest than theresulting ‘intensity’ of a point process.

The observed number of seeds in the inspectionwin-dow n can be interpreted as a further parameter of theprocesses, however conditionally on n, the processespossess the number of parameters as stated before. Thesampling of a point process is in general not a sim-ple task (as e.g. the normalizing constant Z may beunknown as in the Strauss process case), thus usuallyMonte Carlo methods have to be applied as they do notrequire normalized probabilities for sampling (Badde-ley et al. 2016; Møller et al. 1999).

3.2 Voronoi tesselation

For a given configuration of points P in K , called theseeds, the Voronoi cell associated to the seed pi ∈ P,denoted as V (pi), corresponds to the region in whichthe points are closer to pi than to any other seed in P:

V (pi) = {x ∈ K/||x − pi||q ≤ ||x − pj||q , ∀pj ∈ P, i �= j

}(16)

The Voronoi diagram generated by P is the set ofthe Voronoi cells {V (p1), . . . , V (pn)}. In Fig. 5, theVoronoi tessellation of randomly distributed points isshown. Voronoi diagrams entirely partition the domainK without region overlap. By using the Euclidean dis-tance (q = 2) in Eq. (16), Voronoi cells are guaranteedto be convex polygons. Refer to Okabe et al. (1992) fora thorough treatment of Voronoi tessellations and theirproperties.

Matérn Hardcore and Strauss repulsion processesin the context of this paper are used to generate the

Fig. 5 Example of a Voronoi diagram

seeds of Voronoi diagrams. From already conductedexperiments of breaking glass panes with distinct ther-mal residual stresses images of the associated fracturepatterns have been taken and used as samples for theidentification of the intensity and theHardcore distanceof the underlying spatial point processes. In this workis also proposed an analytical method for determiningthe Hardcore distances.

4 Methodology of Fracture Structure Parameterdetermination

4.1 Basic idea

The basic idea for the theoretical prediction and simula-tion of the fracture pattern is to distribute points with acertain distance δ to each other based on the locally act-ing stress respectively the stored elastic strain energy,and with a fracture intensity of λ obtained by evalua-tion of fracture patterns in the plane. The final fracturepattern is created by Voronoi tessellation of the result-ing point cloud. Points can be distributed using dif-ferent distribution methods (see Sect. 3). Taking intoaccount the size and intensity parameter, the follow-ing point distribution dependency results as the esti-mation parameters for the theoretical prediction of the2D macro-scale fracture structure:

θ = {δ, λ} (17)

This work focus on the elastic strain energy resultingfrom the residual stress state from the thermal temper-ing process. In order to predict the fragmentation andfor the recognition and the generation of the fracture

123

N. Pourmoghaddam et al.

Fig. 6 a Hexagonal close packing distribution of points with an “energy circle” of radius r0 as the area of influence of each point; thepoint distance is δ, b Delaunay triangulation of HCP, c honeycomb as the result of the Voronoi tesselation of the HCP

pattern, the Fragment Size Parameter δ and the FractureIntensity Parameter λ are determined.

The Fragment Size Parameter δ is equal to the hard-core distance of the hardcore and Strauss spatial pointprocess and thus can be understood as a local minimumpoint distance. For the determination of the FragmentSize Parameter δ not only in dependence of the residualstress based on the thermal tempering, but on any kindof stress (e.g. stress resulting from an external load),the linear elastic fracture mechanics (LEFM) based onthe energy criterion introduced by Griffith (1920) isused. Part of the energy is released by new surfacesgenerating from cracking and branching of progressivecracks. The fracture pattern can be estimated with thehelp of the energy concept in fracture mechanics, tak-ing into account stochastic fracture pattern analyses.This method is compared with the results of the finiteelement simulations in Nielsen (2017).

In high-speed images (Nielsen et al. 2009), it wasobserved that so-called “whirl-fragments” were gen-erated by a whirl-like crack propagation. It was alsoobserved that progressing cracks branched at an angleof 60 and formed a hexagonal fracture. A hexagonalfracture structure is assumed to be the “perfect” frac-ture pattern or “perfectly” broken glass plate. Predict-ing such a “perfect” fracture structure is easy, providedδ is known. All points in the plane have the same dis-tance to each other and can be distributed by Hexag-onal close packing (HCP) of points (see Fig. 6a). TheVoronoi tessellation takes place via the Delaunay tri-angulation of the HCP-distributed points (Fig. 6b) andresults in the cell structure of a honeycomb (Fig. 6c).Thus the number of fragments in an observation fieldcan be predicted well.

However, since the fracture pattern will not be per-fect in reality (see Fig. 7), due to stochastic materialproperties and stress distribution from the productionprocess, another parameter λ, which can be obtainedfrom experimental data and describes the intensity ofthe fragment distribution, and thus the empirical realityof a fracture pattern, must be added to the method.

The Fracture Intensity Parameter λ, which is a char-acteristic value for the number of fragments in an obser-vation field, was determined using the experimentaldata of the fracture tests in carried out in Pourmoghad-dam and Schneider (2018b). In Fig. 7, three samples offracture patterns are shown for the same glass thicknessbut different residual stresses respectively elastic strainenergy densityUD . Fracture intensity is determined byplacing 50mm×50mm observation fields on the frac-ture patterns and determining the average number offragments in correlation to the energy density for eachsample (Pourmoghaddam and Schneider 2018b).

4.2 Fragment Size Parameter δ

The determination of the Fragment Size Parameter δ isbased on the energy conditions described in Sect. 2. Asit is shown in Fig. 6a, the Fragment Size Parameter δ =2r0 is the distance between two neighboured points.For the development of the approach a HCP distributedpoint cloud is assumed.This is necessary becausewedonot want to go into the intensity of the fracture patternat first and want to describe the point distance purelyphysically. There are three assumptions which werenecessary for the determination of the Fragment Size

123

Relationship between strain energy and fracture pattern morphology

Fig. 7 Fracture patterns ofglass specimens of size1100mm × 360mm and thethickness of t = 12mm fordifferent residual stresses: aσm = 26.9MPa(UD = 6367.8 J/m3), bσm = 30.0MPa(UD = 7920.0 J/m3) and cσm = 38.1MPa(UD = 12774.2 J/m3)

Parameter respectively for the minimum distance oftwo distributed points.

Assumption 1 The first assumption is that the glassplate will break into cylindrical fragments. It wasassumed that the sphere of influence for the stored elas-tic strain energy U of each point representing a frag-ment is a circle, which is here called “Energy circle”respectively a cylinder in 3D with a radius r0 beforefragmentation. As can be seen in Fig. 6a, the energy cir-cles touch but do not overlap. Consequently, all energyin the observed field is distributed in these energy cir-cles. Thus, each radius r0 depends on the elastic strainenergy in the influence area of the respective point. Thefree gaps between the energy circles are zero energyareas and have no influence on the further calculation.

This is legitimate, as we are distributing the total strainenergy in the observed field through the energy circles.

Assumption 2 In the case of fracture the stored elasticstrain energy will release and there will be a relaxationof energy in the fragment. However, we assume thatin the case of fracture the elastic strain energy is notrelaxed completely but just by a relaxation factor η:

η = r0 − r1r0

= 1 − r1r0

(18)

This means, that there will be a remaining strainenergyU1 in the fragment after the fragmentation. Thiswas shown numerically in Nielsen (2017) by FE simu-lations on a fragment before and after the fracture setsin. For the energy sphere described under Assump-

123

N. Pourmoghaddam et al.

Fig. 8 Energy circle before fragmentation with radius r0 andafter fragmentationwith radius r1; Aη is the surface of the energyrelaxation zone

tion 1, this means that the energy circle shrinks to asmaller circle with a radius r1 after fragmentation (seeFig. 8). The area in which the energy relaxes is calledthe Energy relaxation zone. The area of the Energyrelaxation zone Aη can be easily calculated as follows:

Aη = πr02η(2 − η) (19)

In addition, the elastic strain energyUη in the relaxationzone can be calculated as follows:

Uη = σ̂ (1 − ν)

Et Aη (20)

wherein σ̂ describes the stress state as a factor of theintegrated stress function through the thickness. σ̂ canbe determined as σ̂m = 4

5σ2m by calculating with the

residual tensile mid-plane stress and σ̂s = 15σ

2s for

the residual compressive surface stress for a parabolicstress distribution from tempering. Uη is therefore theelastic strain energy released during fragmentation.

Assumption 3 It is assumed that in the event of a frac-ture, the elastic strain energy is released through thegeneration of new fracture surfaces. Other forms ofenergy such as acoustic or heat are neglected. There-fore, we assume that all released elastic strain energyUη is converted to the fracture surface energy Γ :

Uη = Γ = 2γ A f r (21)

where A f r = 2πr1tρ∗ is the fracture surface by assum-ing a cylindrical fragment. ρ∗ is a correction factor of

the fracture surface, since the fracture surface is not flatin reality but irregularly shaped (Quinn 2016).

ρ∗ = A f r,act

A f r,cyl(22)

Here A f r,act is the actual fracture surface and A f r,cyl

is the lateral surface of a cylinder. Therefore, for anassumption of a cylindrical fragment ρ∗ = 1. γ is thespecific fracture surface energy and can be describedin terms of the critical energy release rate:

Gc = 2γ (23)

Depending on the stress state, we obtain:

γ = Gc

2=

⎧⎪⎪⎨⎪⎪⎩

K 2I c

2E , Plane stress

K 2I c(1−ν2)

2E , Plane strain

(24)

where in KIc is the critical stress intensity factor. Insert-ing the Eq. (20) in Eq. (21) and resolving it to r0, weobtain:

r0 = 4Eγρ∗

σ̂ (1 − ν)

(1 − η)

η(2 − η)(25)

Furthermore, the relative remaining elastic strain energyUR,Rem can be described as:

UR,Rem = U1

U0= r21

r20= (1 − η)2 (26)

According to the first assumption, the glass plate willbreak into cylindrical fragments. In Fig. 9 the relativeremaining elastic strain energy UR,Rem in correlationwith the area of the base shape of the fragment A f r,base

normalized by t2 is shown in comparison with the FEresults in Nielsen (2017). As can be observed in Fig. 9,the analytical results of the fragment size from a valueof approximately A f r,base/t2 ≥ 1.5 corresponds to theFE results. The difference in the case of very small frac-ture surfaces is due to the fact that the FE simulationwas carried out on a fragment volume and thereforea multi-axial stress state was present. In the FE simu-lations stresses in the thickness direction occurred inthe fracture state, whereas the analytical solution wasbased on a continuous plane stress state.

123

Relationship between strain energy and fracture pattern morphology

Fig. 9 Relative remaining elastic strain energy UR,Rem (–) ver-sus fragment size given by A f r,base (mm2) for fractured soda-lime-silica glass, analytical result (black line) in comparison tothe FE results (dashed line) (Nielsen 2017)

Now we have described the distance between thedistributed points δ = 2r0, which directly affects thefragment size and so the fragmentation density N , independence on the residual stress respectively the elas-tic strain energy U . The method significantly dependson the relaxation factor η (see Fig. 10), which can havevalues from 0 to 1. The relaxation factor of η = 0would mean that there is no fracture and η = 1 that theenergy is relaxed completely. Thus the fragment size orthe minimum distance between two points for a hard-core spatial point process is mainly dependent on theenergy relaxed in the fracture state.

As described before and shown in Fig. 10, the frag-ment density is affected by the relaxation factor η. Thegreater η, the higher the percentage of the stored elas-tic strain energy that is released during the fragmen-tation. A larger energy produces more crack surfacesand thus also a finer fracture pattern or in other wordsa larger fragment number within an observation field.For example for a residual surface compressive stressof 100MPa we calculate in an observation field of size50mm× 50mm a fragment density of N50 = 14 for arelaxation factor of η = 0.05 and N50 = 132543 for arelaxation factor of η = 0.9.

The Fragment Size Parameter δ was determinedusing the experimental data of fracture tests. In Pour-moghaddam and Schneider (2018b) specimens withdifferent thicknesses were thermally tempered with

Fig. 10 Fragment density N50 in an observation field of size50mm × 50mm versus residual surface compressive stress fordifferent relaxation factors η = 0.05 to η = 0.9

different degree of tempering so that specimens withdifferent residual stresses were obtained for the frac-ture tests. After the fracture tests the average frag-ment weight was determined from more than 130 frag-ments per specimen. Knowing the density of glass(2500 kg/m3) and the glass thickness t the area of thebase shape of the fragment (fragment volume dividedby thickness) can be recalculated from the weight.Using the energy density UD for the determination ofthe correlation, the curves of base area and thus theFragment Size Parameter δ for the different thicknessescoincide to one line. In Fig. 11, the elastic strain energydensity calculated from the different residual stressesof the specimens is applied over the Fragment SizeParameter δ. Each of the black triangles represents theaverage of more than 130 fragments per specimen. Forthe calculation of δ from the experimental data a frag-ment with the number of edges n → ∞ (cylindricalfragment) is assumed which contains other edge num-bers and the Fragment Size Parameter δ is determinedover the radius of the fragment. The red line repre-sents the fragment size calculated analytically usingEq. (25). For the analytical calculations the measuredresidual stress values of the specimens used in Pour-moghaddam and Schneider (2018b), a Young’s modu-lus of E = 70000MPa, a correction factor of ρ∗ = 1and the plane stress state have been taken into account.

As it can be observed in Fig. 11 the analytical curvefits for a relaxation factor of η = 0.123. A relaxation

123

N. Pourmoghaddam et al.

Fig. 11 Elastic strain energy density UD (J/m3) versus Frag-ment Size Parameter δ (mm), black triangles with the black trendline show the average of more than 130 fragments per specimen(Pourmoghaddam and Schneider 2018b); the red line representsthe calculated fragment size for η = 0.123

factor of η = 0.123 means that the relative remain-ing elastic strain energy UR,Rem = 0.77 and thusapproximately 77% of the initial elastic strain energyU0 remains in the fragment and correspondingly 23%of the initial energy releases during the fracture pro-cess. In Fig. 12 the correlation between the elastic strainenergy density and the fragment density for the initialenergy density UD,0, remaining energy density UD,1

and the released energy density UD,η. Thus a directcorrelation between the energy in pre-fracture, fractureand post-fracture state is shown.

4.3 Fracture Intensity Parameter λ

4.3.1 Deterministic fragmentation process

The fracture intensity is another important parameterfor the theoretical prediction of the fragmentation ofglass. The fracture intensity is a characteristic valuefor the fragmentation behavior contains informationabout the fragment density (Number of fragments in agiven area) of the fracture structure. From a determinis-tic Fracture Mechanics point of view, the material canbe expected to be homogeneous and thus the spatiallocation of fragment centers does not depend on theactual location within an object under investigation.

Fig. 12 Elastic strain energydensityUD (J/m3)versusFragmentdensity N50 for the initial, remaining and released energy density,η = 0.123

As a starting point for an estimation of the locationsof the centers of the glass fragments this assumptionserves well and is called ‘deterministic fragmentationprocess’.

For the computation of the fragment density ND inan observation field a hexagonal fracture structure witha fragment side length a = 2r0/

√3 is now assumed

(Fig. 6c). In a square observation field with the sidelength D the fragment density ND can be described as:

ND = D2

2r20√3

(27)

Under the assumption homogeneity of the point pro-cess the intensity λ is constant over K which furtherimplies, that the expected number of points falling inan observation region with the side length D is propor-tional to its area:

ND = λ · D2 (28)

In order to obtain hexagonal Voronoi cell structure(honey comb), points can be distributed by HexagonalClose Packing (HCP). Combining Eqs. (27) with (28),an approximation for the expected fracture intensityλHCP of a honey comb, motivated from deterministicfracture mechanics, can be expressed in terms of theFragment Size Parameter δ = 2r0 as:

123

Relationship between strain energy and fracture pattern morphology

Fig. 13 Fracture intensity λ̃HCP (1/mm2) of a honey comb versus a Fragment Size Parameter δ (mm) b energy relaxation factor η (–)

Fig. 14 Fragmentation in an observation field of size 50mm ×50mm versus relaxation factor η with regard to the Voronoitesselation of HCP distributed points with the distance δ, theresidual mid-plane tensile stress σm = 50MPa, a η = 0.04;

δ = 17.89mm; N50 = 9; λ̃HCP = 0.0036 1/mm2 b η =0.1; δ = 6.92 hboxmm; N50 = 60; λ̃HCP = 0.0240 1/mm2

c η = 0.2; δ = 3.25mm; N50 = 274; λ̃HC = 0.1096 1/mm2

λ̃HCP = 2√3

· 1

δ2(29)

In Fig. 13a, a double logarithmic interrelation betweenfracture intensity of the honey comb and fragmentsize is shown. The fracture intensity decreases with acoarser fracture structure. λ̂HCP can further be rewrit-ten in terms of the energy relaxation factor η as:

λ̃HCP = 1

2√3r̂2

· η2(2 − η)2

(1 − η)2(30)

with

r̂ = 4Eρ∗

σ̂ (1 − ν)γ (31)

In Fig. 13b, a double logarithmic interrelation betweenfracture intensity of a honey comb and the energy relax-ation factor is shown. The fracture intensity increaseswith higher energy relaxation.

In Fig. 14a–c the Voronoi tesselation of HCP dis-tributed points with the point distance δ is shown fora plate with the residual mid-plane tensile stress of50MPa. The respective fragmentation density is shown

123

N. Pourmoghaddam et al.

Fig. 15 Fragmentation in an observation field of size 50mm ×50mm versus relaxation factor η with regard to the Voronoi tes-selation of uniform distributed points with theminimum distanceδmin , the residual mid-plane tensile stress σm = 50MPa, a η =

0.04; δmin = 17.89mm; N50 = 7; λ̃HC = 0.0028 1/mm2 (b)η = 0.1; δmin = 6.92mm; N50 = 44; λ̃HC = 0.0176 1/mm2 cη = 0.2; δmin = 3.25mm; N50 = 183; λ̃HC = 0.0732 1/mm2

in an observation field of size 50mm × 50mm. Therelaxation factor η increases from η = 0.04 in Fig. 14ato η = 0.2 in Fig. 14c. It is shown that for the sameresidual stress of 50MPa the point distance decreaseswith higher values for η.

The assumption of a hexagonal fracture structure iseased to the assumption of a polygonal fracture struc-ture induced by the Voronoi Tesselation over a Hard-core Process (HC). The fragment density ND is nowconnected to the spatial point process intensity. Analo-gously to Eq. (29), using Eq. (14) and combining it withEq. (28), an improved estimation of the expected Frac-ture Intensity Parameter λ̌HC can be derived in termsof the Fragment Size Parameter δ = 2r0 as:

λ̌HC = −4 ln

(1 −

√3π

6·δ2)

δ2π(32)

However, despiteEq. (32) takes into account themagni-fication of the observed intensity from the simulationsto estimate the underlying HC process intensity, theconnection to the HCP is still basis of the derivation ofthe estimator.

4.3.2 Stochastic fragmentation process

If fracture tests are conducted, it can be observed, thatthe center locations of fragments do vary stochasticallywithin an object under investigation due to differentreasons such as inhomogeneous thermal pre-stressingof a glass pane, inhomogeneous load application etc.Analogously to Sect. 4.3.1, the same investigation was

carried out but this time using random point locationsin an observation field of size 50mm × 50mm witha uniform distribution for the point locations insteadof HCP. This kind of randomly distribution led to avarying distance between points. However, the distancedetermined on the basis of the relaxation factor η wasgiven as the minimum distance for the point process.In Fig. 15a–c the Voronoi tesselation of uniformly dis-tributed points with the minimum point distance δmin

is shown for a plate with the residual mid-plane tensilestress of 50MPa. In comparison to the fragment densityfor the case of HCP distributed points, the number offragments is underestimated for uniformly distributedpoints. Until η = 0.04 the difference is not signifi-cant yet. However, the difference is more pronouncedfor the higher relaxation factor of η = 0.2. This isbecause on the one hand, Eqs. (29)–(31) are derivedas approximations for the underlying stochastic pointprocess as prior expectations motivated from the deter-ministic fracture process and on the other hand, in thededuction ofEqs. (29)–(31) aminimumenergy require-ment was not introduced for the randomly distributedpoints.

4.3.3 Experimental determination of the FractureIntensity Parameter λ

The Fracture Intensity Parameter λ has been deter-mined experimentally from fracture tests on thermallytempered glass specimens. In Pourmoghaddam andSchneider (2018b), 72 glass plates of size 1100mm ×360mm and three different thicknesses of 4mm, 8mm

123

Relationship between strain energy and fracture pattern morphology

Fig. 16 Square observation fields with the side length of D =50mm and the definition of the boundary for the fragmentationanalysis

Fig. 17 Elastic strain energy densityUD (J/m3) versus FractureIntensity Parameter λ (1/mm2) determined from fracture tests(Pourmoghaddam and Schneider 2018b)

and 12mmwere thermally temperedwith different heattreatment (Pourmoghaddam and Schneider 2018c) forachieving different residual stresses and then fracturedaccording to EN 12150-1. Subsequently, the averagefragment density ND of eight square observation fieldswith a side length D = 50mm taking a boundaryof fragmentation analysis into account, as shown inFig. 16, was determined by counting the fragments ofeach observation field. The fragmentation was influ-enced by the impact in an area under the impact position(impact influence zone). Therefore, only the fragmentsin the areas under an assumed angle of 45◦ left andright from the impact point were considered for theinvestigations.

Now assuming a constant intensity in each obser-vation field the average Fracture Intensity Parameter λ

can be calculated for each specimen using Eq. (28).

Using the energy density UD for the determinationof the correlation, the curves of the fragment densityN50 and thus the fracture intensity λ for the differentthicknesses coincide to one line. In Fig. 17, the cor-relation between the elastic strain energy density UD ,calculated from the measured residual stresses of eachspecimen, and the Fracture Intensity Parameter λ ispresented. It can be observed that the accuracy of theexperimental results of the fracture intensity decreaseswith lower energy density respectively for larger frag-ments.

5 Summary, conclusion, outlook and futureresearch

5.1 Conclusion

Based on fracture tests on glass plates with differentheat treatment for achieving different residual stresslevels (Pourmoghaddam and Schneider 2018c) thispaper deals with the determination of Fracture Struc-ture Parameters for the prediction of the 2D macro-scale fragmentation of glass. The Voronoi tesselationof distributed points using calibrated spatial point pro-cesses can be used in order to simulate the glass frac-ture structure. The probability density function of theHardcore Process, e.g., possesses two parameters, theintensity of the points in an observation region and the(minimal) hardcore distance between point pairs, thusit is a two-parametric model. In this work, due to com-prehensive investigations of fracture patterns and ana-lytical considerations based on the LEFM the parame-ters characteristic for the fracture intensity called Frac-ture Intensity Parameter λ and the hardcore distancebetween point pairs called Fragment Size Parameterδ have been determined. These two parameters areapplied over the elastic strain energy UD in Fig. 18.

The fracture mechanical parameters were laid andthe connections to the parameters of the spatial pointprocesses were highlighted. Under assumption of acylindrical fragment with the radius r0 of the base area,a Fragment Size Parameter δ was generated as a func-tion of the elastic strain energy remaining in the frag-ment after the fragmentation considering the relaxedenergy. In this work, the energy relaxation factor η

was fitted to the results of the fracture tests. The mini-mum distance between two points (Hardcore distanceδ = 2r0) can be determined on the basis of the LEFM.

123

N. Pourmoghaddam et al.

Fig. 18 Elastic strain energy densityUD (J/m3) versus FractureIntensity Parameter λ (1/mm2) (experimental: black triangles)and Fragment Size Parameter δ (mm) (experimental: black dia-monds); red line represents the Fragment Size Parameter deter-mined analytically

The number of fragments or the number of pointsrepresenting fragments is taken into account in anobservation field K as a function of the temper stresses.The Fracture Intensity Parameter λ to consider theintensity of the fracture structure was also discussed.A correlation between the elastic strain energy densityand the intensity of the fracture pattern was establishedbased on the experimental data of fracture tests. Anestimator for the constant Fracture Intensity Parameterwas derived for two cases: the deterministic Hexago-nal Close Packed (HCP) setting as well as a Hardcorestochastic point process, both acting on an observationfield K . The generation of fracture patterns under theassumption of HCP leads on the one hand to a visu-ally observable poorer depiction of the glass fracturepattern and on the other side a poorer estimate of thefracture intensity compared to the Voronoi tesselationof the observation domain over spatial randomly dis-tributed point locations p. Thus, further research willhead in the direction of assuming point locations forthe Voronoi tiling as random variables. The underly-ing probability distribution has to be estimated fromfracture experiments.

5.2 Outlook and future research

In order to predict the 2D macro-scale fragmen-tation of glass considerations from Linear Elastic

Fracture Mechanics (LEFM) with approaches fromstochastic geometry in terms of spatial point patternshave to be merged. This method is called “BayesianReconstruction and Prediction of Glass Fracture Pat-terns (BREAK)”. In Pourmoghaddam et al. (2018a, b),numerical studies on the probability distributions of fordifferent geometrical properties of tesselation over theCSR, MHC and SP with different parameter settingswere conducted and evaluated. As no analytical expres-sions for the posterior distribution of e.g. the number ofedges, the area content or the circumference of a typi-cal Voronoi cell of a MHC or SP exist and these quan-tities are of interest in a fracture mechanical setting,numerically intensiveMonteCarlo simulations of theseprocesses were used to numerically infer the respec-tive distributions to allow further fracture mechanicalprocessing of these information. Besides the fracturemechanical relevance a generalmathematical value canbe addressed to the obtained results, as the posterior dis-tributions for the geometrical properties can be used inother applications without loss of generality.

An example for the estimation of the spatial pointprocess intensity and thus fragment density is depictedin Fig. 19 (note, that in this computation no edge correc-tion was applied as the choice of a correction methoditself is non-trivial, c.f. Baddeley et al. (2016); Møller(1994). In Fig. 19 the estimation of the intensity λHC

of the underlying point process as well as the inducedVoronoi Tesselation for the fracture pattern of one ofthe tested glass panes is shown as 3D view and a topview. The process calibration as well as simulation offurther fracture patterns will be highlighted in part twoof this paper in depth. In order to determine the statis-tical values of the fracture structure such as fragmentedge number, the fragment perimeter, fragment basearea, etc. first the fracture image has to be recorded andmorphologically processed. Then a spatial point pro-cess model fed with the informations from the fracturemechanics (fragment size respectively the hardcore dis-tance between point pairs) and the experimental data offracture tests (fracture intensity) has to be matched tothe fracture image and calibrated to evaluate a candidatemodel. Subsequently, the fracture pattern can be simu-lated. The overall methodology with the incorporatedtheories is depicted in Fig. 20. The results presented inthis paper, the incorporated theories as well as furtherexperimental investigations of the fracture structurewill flow into future research in the field of 2D macro-scale fracture structure prediction. An upcoming pub-

123

Relationship between strain energy and fracture pattern morphology

Fig. 19 Estimated fracture intensity λ̃HC over the Voronoi tesselation of a Matérn Hardcore Process without edge-correction: a 3dview b top view

Fig. 20 Framework of the method “BREAK”

lication deals with the further deduction, implementa-tion and calibration of themethod “BREAK” in order tosimulate fracture patterns which conserve different sta-tistical properties of the underlying glass fracture pat-terns such as the distribution of interpoint-distances orthe distribution of remaining fracture particle area. Theassessment of different spatial point processes as wellas the properties of the induced Voronoi tesselationshave to be studied and compared against the empiri-cal statistics obtained from the images of the fracturetests on glass plates. The formalisation of the findings

in terms of parameter tables as basis for the simula-tion of the spatial point processes with their Voronoitesselation will be provided in the second part of thispaper.

Compliance with ethical standards

Conflict of interest The authors declare that they have no con-flict of interest.

Open Access This article is distributed under the terms of theCreative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/),whichpermits unrestricted

123

N. Pourmoghaddam et al.

use, distribution, and reproduction in any medium, provided yougive appropriate credit to the original author(s) and the source,provide a link to the Creative Commons license, and indicate ifchanges were made.

References

Acloque, P.: Deferred process in the fragmentation of tem-pered glass. In: Proceedings of the 4th International GlassCongress, pp. 279–291 (1956)

Akeyoshi, K., Kanai, E.: Mechanical properties of temperedglass. VII International Congress on Glass, no. paper 80(1965)

Aronen, A., Karvinen, R.: Effect of glass temperature beforecooling and cooling rate on residual stresses in temper-ing. Glass Struct. Eng. 3(1), 3–15 (2017). https://doi.org/10.1007/s40940-017-0053-6

Baddeley, A., Rubak, E., Turner, R.: Spatial Point Patterns:Methodology and Applications with R (2016)

Barsom, J.M.: Fracture of tempered glass. J. Am. Ceram. Soc.51(2), 75–78 (1968). https://doi.org/10.1111/j.1151-2916.1968.tb11840.x

Gardon, R.: The tempering of flat glass by forced convection.In: 7th Proceedings of the International Congress on Glass,p. 79. Institut.National duVerre, Charleroi, Belgique (1965)

Griffith, A.A.: The phenomena of rupture and flow in solids.Philos. Trans. R. Soc. Lond. 221(1921), 163–198 (1920)

Gulati, S.T.: Frangibility of Tempered Soda-Lime Glass Sheet,Glass ProcessingDays, pp. 72–76, Tampere, Finland (1997)

Illian, J., Penttinen, A., Stoyan, H., Stoyan, D.: Statistical Anal-ysis and Modelling of Spatial Point Patterns. Wiley, NewYork (2008)

Lee, H., Cho, S., Yoon, K., Lee, J.: Glass thickness and fragmen-tation behavior in stressed glasses. New J. Glass Ceram. 2,138–143 (2012). https://doi.org/10.4236/njgc.2012.24020

Matérn, B.: Spatial Variation, Meddelanden fran Statens Skogs-forskingsinstitut (Reports of the Forest Research Instituteof Sweden), vol. 49 (1960)

Mognato, E., Brocca, S., Barbieri,A.: Thermally processed glass:correlation between surface compression, mechanical andfragmentation test. In: Glass Performance Days, pp. 8–11(2017)

Møller, J.: Lectures on Random Voronoi Tesselations. Springer,Berlin (1994)

Møller, J., Barndorff-Nielsen, O., Kendall, W.S., Lieshout, M.V.:Markov Chain Monte Carlo and Spatial Point Processes.Chapman and Hall, London (1999)

Molnár, G., Ferentzi, M., Weltsch, Z., Szebényi, G., Borbás, L.,Bojtár, I.: Fragmentation of wedge loaded tempered struc-tural glass. Glass Struct. Eng. 1(2), 385–394 (2016). http://link.springer.com/10.1007/s40940-016-0010-9

Nielsen, J.H.: Remaining stress-state and strain-energyin tempered glass fragments. Glass Struct. Eng.2(1), 45–56 (2017). http://link.springer.com/10.1007/s40940-016-0036-z

Nielsen, J.H., Bjarrum, M.: Deformations and strain energy infragments of tempered glass: experimental and numeri-cal investigation. Glass Struct. Eng. 2(2), 133–146 (2017).http://link.springer.com/10.1007/s40940-017-0043-8

Nielsen, J.H., Olesen, J.F., Stang, H.: The fracture process oftempered soda-lime-silica glass. Exp. Mech. 49(6), 855–870 (2009). https://doi.org/10.1007/s11340-008-9200-y

Nielsen, J.H., Olesen, J.F., Poulsen, P.N., Stang, H.: Finite ele-ment implementation of a glass tempering model in threedimensions. Comput. Struct. 88(17—-18), 963–972 (2010).https://doi.org/10.1016/j.compstruc.2010.05.004

Ohser, J., Schladitz, K.: 3D Images of Materials Structures: Pro-cessing and Analysis. Wiley, New York (2009)

Okabe, A., Boots, B.N., Kokichi, S.: Spatial Tessella-tions: Concepts and Applications of Voronoi Diagrams.Wiley, New York (1992). http://doi.wiley.com/10.1002/9780470317013

Pourmoghaddam, N., Schneider, J.: Finite-element analysis ofthe residual stresses in tempered glass plates with holes orcut-outs. Glass Struct. Eng. 3(1), 17–37 (2018a). https://doi.org/10.1007/s40940-018-0055-z

Pourmoghaddam, N., Schneider, J.: Experimental investiga-tion into the fragment size of tempered glass. GlassStruct. Eng. 3(2), 167–181 (2018b). https://doi.org/10.1007/s40940-018-0062-0

Pourmoghaddam, N., Schneider, J.: Determination of the enginepower for quenching of glass by forced convection : sim-plified model and experimental validation of residual stresslevels. Glass Struct. Eng. 1969 (2018c). https://doi.org/10.1007/s40940-018-0078-5

Pourmoghaddam, N., Nielsen, J.H., Schneider, J.: Numericalsimulation of residual stresses at holes near edges andcorners in tempered glass: a parametric study. In: Engi-neered Transparency International Conference at Glasstec,pp. 513–525. Ernst & Sohn GmbH & Co. KG. (2016)

Pourmoghaddam,N.,Kraus,M.A., Schneider, J., Siebert,G.: Thegeometrical properties of random 2D Voronoi tesselationsfor the prediction of the tempered glass fracture pattern. In:Engineered Transparency 2018 Glass in Architecture andStructural Engineering, 1st ed, vol. 2, no. 5–6, pp. 325–339.Ernst & Sohn GmbH & Co. KG. (2018a). https://doi.org/10.1002/cepa.934

Pourmoghaddam, N., Kraus, M.A., Schneider, J., Siebert, G.:Prediction of the 2Dmacro-scale fragmentation of temperedglass using randomVoronoi tesselations. In: Forschungskol-loquium 2018 Grasellenbach - Baustatik-Baupraxis e.V.Springer Vieweg, pp. 56–59 (2018b). https://doi.org/10.1007/978-3-658-23627-4_21

Quinn, G.D.: Fractography of Ceramics and Glasses, vol. 191.National Institute of Standards and Technology (2016).http://nvlpubs.nist.gov/nistpubs/specialpublications/NIST.SP.960-16e2.pdf

Reich, S., Dietrich, N., Pfefferkorn, S., Weller, B.: Elastic strainenergy of thermally tempered glass increases its residualstrength. In: Glass Performance Days 2013, pp. 348–352(2013)

Schmidt, V.: Stochastic Geometry, Spatial Statistics and RandomFields: Models and Algorithms. Springer, Berlin (2015)

Schneider, J.: Festigkeit und Bemessung punktgelagerter Gläserund stoßbeanspruchter Gläser, Ph.D. dissertation, Technis-che Universität Darmstadt (2001)

Schneider, J., Kuntsche, J., Schula, S., Schneider, F., Wörner, J.-D.: Glasbau, Grundlagen, Berechnung, Konstruktion, 2ndedn. Springer Vieweg, Darmstadt (2016)

123

Relationship between strain energy and fracture pattern morphology

Shutov, A.I., Popov, P.B., Bubeev, A.B.: Prediction of the char-acter of tempered glass fracture. Glass Ceram. 55(1—-2),8–10 (1998). https://doi.org/10.1007/BF03180135

Silverman, M.P., Strange, W., Bower, J., Ikejimba, L.: Fragmen-tation of explosively metastable glass. Phys. Scripta 85(6)(2012). http://stacks.iop.org/1402-4896/85/i=6/a=065403

Stoyan, D., Kendall, W.S., Mecke, J.: Stochastic Geometry andIts Application. Wiley, New York (1995)

Strauss, D.J.: Amodel for clustering. Biometrika 62(2), 467–475(1975). http://biomet.oxfordjournals.org/content/62/2/467

Tandon, R., Glass, S.J.: Controlling the fragmentation behaviorof stressed glass. In: Fracture Mechanics of Ceramics—Active Materials, Nanoscale Materials, Composites, Glass,and Fundamentals, vol. 14, pp. 77–91. Springer, Berlin(2005)

Warren, P.D.: Fragmentation of thermally strengthened glass.Fractogr. Glasses Ceram. IV 12, 389–400 (2001)

Wiegand, T., Moloney, K.A.: Handbook of Spatial Point PatternAnalysis in Ecology. CRC Press, Boca Raton (2014)

Publisher’s Note Springer Nature remains neutral with regardto jurisdictional claims in published maps and institutional affil-iations.

123