# Relationship between strain energy and fracture pattern morphology of thermally tempered glass for the prediction of the 2D macro-scale fragmentation of glass

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## Abstract

This work deals with the prediction of glass breakage. A theoretical method based on linear elastic fracture mechanics (LEFM) merged with approaches from stochastic geometry is used to predict the 2D-macro-scale fragmentation of glass. In order to predict the fragmentation of glass the 2D Voronoi tesselation of distributed points based on spatial point processes is used. However, for the distribution of the points influence parameters of the fracture structure are determined. The approach is based on two influencing parameters of fragment size \(\delta \) and fracture intensity \(\lambda \), which are described in this paper. The Fragment Size Parameter describes the minimum distance between the points and thus the size of a fragment. It is derived from the range of influence of the remaining elastic strain energy in a single fragment taking into account the LEFM based on the energy criterion of *Griffith*. It considers the extent of the initial elastic strain energy \(U_0\) before fragmentation obtained from the residual stress as well as a ratio of the released energy \(\eta \) due to fragmentation. The Fracture Intensity Parameter describes the intensity of the fragment distribution, and thus the empirical reality of a fracture pattern. It can be obtained by statistical evaluation of the fracture pattern. In this work, the fracture intensity is determined from the experimental data of fracture tests. The intensity of a fracture is the quotient of the number of fragments in an observation field and its area and is assumed to be constant in the observation filed. The fracture intensity and the correlation between a constant intensity and the Fragment Size Parameter was determined. The presented methodology can also generally be used for the prediction of fracture patterns in brittle materials using a Voronoi tesselation over random fields.

## Keywords

Fragmentation Tempered glass Fragment size Fracture intensity Elastic strain energy Spatial point process Voronoi tessellation Fracture pattern## List of symbols

- \(\sigma _{ij}\)
Stress tensor

- \(\epsilon _{ij}\)
Strain tensor

- \(r, \theta \)
Cylinder coordinates (radius and polar angle)

- \(\tau \)
Shear stress

- \(\sigma (z)\)
Stress function along the z-axis of the plate

- \(\sigma _s\)
Surface stress

- \(\sigma _m\)
Mid-plane stress

*t*Glass plate thickness

- \(\nu \)
Poisson’s ratio

*E*Young’s modulus

- \(\mathbf {P}\)
Set of points \(\mathbf {p}_i, i=1\ldots n\) in the domain

*K**K*The 2D domain of the glass fracture pattern

- \(\theta \)
Point process parameters

- \(\mathcal {P}\)
Probability of obtaining

*n*points \(\mathbf {p}_i, i=1\ldots n\) in the domain*K**n*,*N*Random variable and obtained realization for the number of points \(\mathbf {p}_i, i=1 \ldots n\) in the domain

*K**HC*Hardcore Process

*MHC*Matérn Hardcore Process

*SR*Strauss process

- \(f_n(P)\)
Probability density of a point pattern \(\mathbf {P}\)

- \(c_i\)
Normalizing constants of a probability density

- \(s(\mathbf {P})\)
Number of point pairs to be penalized in HC

- \(\delta _{HC}\)
Minimum Euclidean distance of points in HC

- \(\beta \)
Intensity of the original SR

- \(\gamma _{SR}\)
Inhibition parameter of SR

- \(\delta \)
Fragment Size Parameter

- \(\lambda \)
Fragment/point process intensity parameter

- \(\varGamma \)
Fracture surface energy

- \(\gamma \)
Specific fracture surface energy

- \(A_{fr}\)
Fracture surface of a fragment

- \(\rho ^*\)
Correction factor of the fracture surface

- \(G_c\)
Critical energy release rate

- \(K_{Ic}\)
Critical stress intensity factor

- \(\eta \)
Energy relaxation factor

- \(A_{\eta }\)
Energy relaxation zone of a fragment

- \(U_{\eta }\)
Elastic strain energy in the relaxation zone \(A_\eta \)

- \(U_0\)
Initial strain energy

- \(U_1\)
Remaining strain energy

- \(U_R\)
Relative remaining strain energy

- \(N_D\)
Fragment density in the observation field with the length of

*D*

## 1 Introduction

Glass is one of the most popular building materials today. However, the tensile strength is governed by small flaws in the surface which reduce the actual engineering strength of annealed float glass to 30–100 MPa (Schneider et al. 2016). Due to the residual stress state thermally tempered glass shows a greater resistance to external loads and in case of failure it is quite safe in terms of cutting and stitching due to the small, blurred fragments. Therefore, thermally tempered glass is also known as tempered safety glass. The residual stress state is obtained by the tempering process and is approximately parabolic distributed along the thickness with the compressive stress on both surfaces and an internal tensile stress in the mid-plane. By imposing a compressive residual stress at the surface, the surface flaws will be in a permanent state of compression which has to be exceeded by externally imposed stress before failure can occur (Schneider 2001; Nielsen et al. 2010; Pourmoghaddam et al. 2016; Pourmoghaddam and Schneider 2018a). The amount of the residual surface compressive stress largely depends on the cooling rate and therefore on the heat transfer coefficient between the glass and the cooling medium (Gardon 1965; Aronen and Karvinen 2017).

Thus, for the same residual stress distribution a higher stored energy is determined in the thicker plates. This has been investigated and proved experimentally by several authors (Akeyoshi and Kanai 1965; Barsom 1968; Gulati 1997; Lee et al. 2012; Mognato et al. 2017; Pourmoghaddam and Schneider 2018b).

Several models for relating the fragment size to the residual stress state have been suggested in the literature, e.g. Acloque (1956), Barsom (1968), Gulati (1997), Shutov et al. (1998), Warren (2001) and Tandon and Glass (2005). Some of these works have proposed models for the fragments size based on an energy approach. Most of the works try to establish an analytical model for the fragment size considering the release of the so-called tensile strain energy defined as the part of the strain energy resulting from the mid-plane tensile stress alone. A theoretical method based on fracture mechanics was developed by e.g. Warren (2001), Molnár et al. (2016). The Voronoi tesselation of randomly distributed points in the glass plane leading to a cell structure similar to glass breakage was mentioned in Molnár et al. (2016). However, the statistical distribution method of the points in the plane itself is significant when 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 is therefore important in order to find the best method of point distribution.

The motivation is that Voronoi tessellation of points distributed in the plane results in a fracture structure based on both fracture mechanical and stochastic parameters. Hence, the main objective is to determine the significant influence parameters of the glass fracture in order to develop a method for the prediction of 2D macro-scale fragmentation of glass. In this work, two influence parameters of the fracture structure, the *Fragment Size Parameter*\(\delta \) and the *Fracture Intensity Parameter*\(\lambda \) are described and determined. Using these two parameters the spatial point process can be influenced and calibrated by the energy density and empirical reality of a fracture pattern.

In order to determine the Fragment Size Parameter \(\delta \) a method was developed with which the minimum distance between two points in a point cloud can be determined by selecting the energetic criterion (Griffith 1920) from the linear fracture mechanics with respect to the strain energy state before and after the fragmentation. In other words; the present work aims at determining the minimum distance between randomly distributed points in a plane representing fragments based on the strain energy state which is stored in the glass plate due to the thermal tempering. Subsequently, the fragment density is estimated using Hexagonal Close Packed (HCP) and a uniformly distributed point group.

The Fracture Intensity Parameter \(\lambda \) describes the degree of intensity of the fracture in a limited observation field and is a value for the fragment number within the observation field. This parameter was determined from the experimental data of fracture tests.

## 2 Energy conditions

*U*stored in a deformed linear elastic, isotropic body is obtained by integrating the energy per unit volume over the volume of the body:

*t*(Fig. 3). This parabolic stress distribution \( \sigma (z) \) can be written in terms of the surface stress \( \sigma _s \) as:

*t*.

*r*for the radius and \( \theta \) for the polar angle the constitutive relationship for plane stress is defined by:

*E*and the Poisson’s ratio \( \nu \). The equilibrated stress state is also assumed to be hydrostatic for field stresses. A planar hydrostatic stress state means that no shear stresses occur \( (\tau _{r\theta }=0) \) and the normal stresses are always principal stresses, which are equal in the plate plane \( (\sigma _r=\sigma _\theta =\sigma (z)) \) and zero in the direction of the thickness \(( \sigma _z=0) \). Hence, the total initial elastic strain energy \( U_0 \) can be written as:

*R*, thickness

*t*and the residual surface stress of \( \sigma _s \).

## 3 Voronoi tesselation over spatial point processes

This section provides a brief introduction to the two mathematical tools that play a central role in the latter algorithm: Spatial point processes and Voronoi Diagrams. Further details and exhaustive presentations of these topics are covered in Baddeley et al. (2016), Wiegand and Moloney (2014), Schmidt (2015), Møller (1994), Okabe et al. (1992) and Ohser and Schladitz (2009).

### 3.1 Spatial point processes

*K*(which is in the case of this paper the 2D domain of the glass fracture pattern with area content |

*K*|). The number of points

*n*is itself a random variable

*N*that typically follows a discrete Poisson distribution. The least complex point patterns exhibit

*complete spatial randomness (CSR)*, i.e. the point locations \({\mathbf {p_i}}\) occur independently and uniformly (equal likelihood) over the domain

*K*. The probability density of a point pattern

*P*with point process parameters \(\theta \) and distribution of the point locations \(f_n^\theta ({\mathbf {p_1}},\ldots ,{\mathbf {p_n}})\) is given by

*n*points are equally likely, the different points are furthermore independent and each point is uniformly distributed over the domain

*K*. Further point processes can be defined by probability densities \(f_P (P|\theta )\) which differ in their mathematical form from Eq. (12) (i.e. a CSR process), this will serve as the basis for two more point processes presented later in this section.

*intensity*\(\lambda (\mathbf {p})\) (mean number of events per unit area at the point \(\mathbf {p}\)) and the second-order property reflects the spatial dependence in the process (clustering or repulsion). More complex spatial point processes are able to enforce either clustering or repulsion of the points, which is observable in the second-order statistics of these processes. Figure 4 shows examples of point patterns which exhibit CSR and two examples violating CSR due to regularity (repulsion) and clustering.

Based on inspection of typical fracture patterns, such as shown in Figs. 1 and 2 a repulsion behaviour of the underlying spatial point process can be concluded. Repulsion processes possess a repulsion domain, which is located around each seed point to avoid points being too close to each other. When \(\dim K=2\), this domain is a disk with radius \(\delta _{HC}\), which is a model parameter of the underlying spatial point process. This kind of pairwise interaction of the points can be modelled via repulsion/inhibition processes. In the context of this paper *Matérn Hardcore Processes (MHC)* (Matérn 1960) as well as *Strauss (repulsion) processes (SR)* (Strauss 1975) are investigated, which belong to the family of the *Gibbs processes* (Baddeley et al. 2016).

*Strauss (repulsion) processes SR*(Strauss 1975) are investigated, which belong to the family of the

*Gibbs processes*. Without going in detail here, c.f. (Stoyan et al. 1995), the density of a Strauss point process reads

According to Baddeley et al. (2016), \(\beta \) is not the ‘intensity’ of the HC or SR model, instead \(\beta \) should be seen as the spatially varying ‘fertility’ which is counterbalanced by the ‘competitive’ effect of the hard core to give the final intensity, thus the ‘intensity’ is the product of fertility and competition. \(\beta \) is referred to as the (first-order) trend which is of more interest than the resulting ‘intensity’ of a point process.

The observed number of seeds in the inspection window *n* can be interpreted as a further parameter of the processes, however conditionally on *n*, the processes possess the number of parameters as stated before. The sampling of a point process is in general not a simple task (as e.g. the normalizing constant *Z* may be unknown as in the Strauss process case), thus usually Monte Carlo methods have to be applied as they do not require normalized probabilities for sampling (Baddeley et al. 2016; Møller et al. 1999).

### 3.2 Voronoi tesselation

*K*, called the

*seeds*, the Voronoi cell associated to the seed \({\mathbf {p_i}} \in \mathbf {P}\), denoted as \(V({\mathbf {p_i}})\), corresponds to the region in which the points are closer to \({\mathbf {p_i}}\) than to any other seed in \(\mathbf {P}\):

*K*without region overlap. By using the Euclidean distance (\(q=2\)) in Eq. (16), Voronoi cells are guaranteed to be convex polygons. Refer to Okabe et al. (1992) for a thorough treatment of Voronoi tessellations and their properties.

## 4 Methodology of Fracture Structure Parameter determination

### 4.1 Basic idea

The Fragment Size Parameter \(\delta \) is equal to the hardcore distance of the hardcore and Strauss spatial point process and thus can be understood as a local minimum point distance. For the determination of the Fragment Size Parameter \(\delta \) not only in dependence of the residual stress based on the thermal tempering, but on any kind of stress (e.g. stress resulting from an external load), the linear elastic fracture mechanics (LEFM) based on the energy criterion introduced by Griffith (1920) is used. Part of the energy is released by new surfaces generating from cracking and branching of progressive cracks. The fracture pattern can be estimated with the help of the energy concept in fracture mechanics, taking into account stochastic fracture pattern analyses. This method is compared with the results of the finite element simulations in Nielsen (2017).

In high-speed images (Nielsen et al. 2009), it was observed that so-called “whirl-fragments” were generated by a whirl-like crack propagation. It was also observed that progressing cracks branched at an angle of 60 and formed a hexagonal fracture. A hexagonal fracture structure is assumed to be the “perfect” fracture pattern or “perfectly” broken glass plate. Predicting such a “perfect” fracture structure is easy, provided \(\delta \) is known. All points in the plane have the same distance to each other and can be distributed by Hexagonal close packing (HCP) of points (see Fig. 6a). The Voronoi tessellation takes place via the Delaunay triangulation of the HCP-distributed points (Fig. 6b) and results in the cell structure of a honeycomb (Fig. 6c). Thus the number of fragments in an observation field can be predicted well.

The Fracture Intensity Parameter \(\lambda \), which is a characteristic value for the number of fragments in an observation field, was determined using the experimental data of the fracture tests in carried out in Pourmoghaddam and Schneider (2018b). In Fig. 7, three samples of fracture patterns are shown for the same glass thickness but different residual stresses respectively elastic strain energy density \(U_D\). Fracture intensity is determined by placing \(50\,\hbox {mm} \times 50\,\hbox {mm}\) observation fields on the fracture patterns and determining the average number of fragments in correlation to the energy density for each sample (Pourmoghaddam and Schneider 2018b).

### 4.2 Fragment Size Parameter \(\delta \)

The determination of the Fragment Size Parameter \(\delta \) is based on the energy conditions described in Sect. 2. As it is shown in Fig. 6a, the Fragment Size Parameter \(\delta =2r_0\) is the distance between two neighboured points. For the development of the approach a HCP distributed point cloud is assumed. This is necessary because we do not want to go into the intensity of the fracture pattern at first and want to describe the point distance purely physically. There are three assumptions which were necessary for the determination of the Fragment Size Parameter respectively for the minimum distance of two distributed points.

### Assumption 1

The first assumption is that the glass plate will break into cylindrical fragments. It was assumed that the sphere of influence for the stored elastic strain energy *U* of each point representing a fragment is a circle, which is here called “Energy circle” respectively a cylinder in 3D with a radius \(r_0\) before fragmentation. As can be seen in Fig. 6a, the energy circles touch but do not overlap. Consequently, all energy in the observed field is distributed in these energy circles. Thus, each radius \(r_0\) depends on the elastic strain energy in the influence area of the respective point. The free gaps between the energy circles are zero energy areas and have no influence on the further calculation. This is legitimate, as we are distributing the total strain energy in the observed field through the energy circles.

### Assumption 2

### Assumption 3

*N*, in dependence on the residual stress respectively the elastic strain energy

*U*. The method significantly depends on the relaxation factor \(\eta \) (see Fig. 10), which can have values from 0 to 1. The relaxation factor of \(\eta =0\) would mean that there is no fracture and \(\eta =1\) that the energy is relaxed completely. Thus the fragment size or the minimum distance between two points for a hardcore spatial point process is mainly dependent on the energy relaxed in the fracture state.

As described before and shown in Fig. 10, the fragment density is affected by the relaxation factor \(\eta \). The greater \(\eta \), the higher the percentage of the stored elastic strain energy that is released during the fragmentation. A larger energy produces more crack surfaces and thus also a finer fracture pattern or in other words a larger fragment number within an observation field. For example for a residual surface compressive stress of 100 MPa we calculate in an observation field of size \(50\,\hbox {mm}\times 50\,\hbox {mm}\) a fragment density of \(N_{50}=14\) for a relaxation factor of \(\eta =0.05\) and \(N_{50}=132543\) for a relaxation factor of \(\eta =0.9\).

*t*the area of the base shape of the fragment (fragment volume divided by thickness) can be recalculated from the weight. Using the energy density \(U_D\) for the determination of the correlation, the curves of base area and thus the Fragment Size Parameter \(\delta \) for the different thicknesses coincide to one line. In Fig. 11, the elastic strain energy density calculated from the different residual stresses of the specimens is applied over the Fragment Size Parameter \(\delta \). Each of the black triangles represents the average of more than 130 fragments per specimen. For the calculation of \(\delta \) from the experimental data a fragment with the number of edges \(n \rightarrow \infty \) (cylindrical fragment) is assumed which contains other edge numbers and the Fragment Size Parameter \(\delta \) is determined over the radius of the fragment. The red line represents the fragment size calculated analytically using Eq. (25). For the analytical calculations the measured residual stress values of the specimens used in Pourmoghaddam and Schneider (2018b), a Young’s modulus of \(E=70000\) MPa, a correction factor of \(\rho ^*=1\) and the plane stress state have been taken into account.

### 4.3 Fracture Intensity Parameter \(\lambda \)

#### 4.3.1 Deterministic fragmentation process

*D*the fragment density \(N_D\) can be described as:

*K*which further implies, that the expected number of points falling in an observation region with the side length

*D*is proportional to its area:

*Hexagonal Close Packing (HCP)*. Combining Eqs. (27) with (28), an approximation for the expected fracture intensity \({\lambda _{HCP}}\) of a honey comb, motivated from deterministic fracture mechanics, can be expressed in terms of the Fragment Size Parameter \(\delta =2r_0\) as:

In Fig. 14a–c the Voronoi tesselation of HCP distributed points with the point distance \(\delta \) is shown for a plate with the residual mid-plane tensile stress of 50 MPa. The respective fragmentation density is shown in an observation field of size \(50\,\hbox {mm} \times 50\,\hbox {mm}\). The relaxation factor \(\eta \) increases from \(\eta =0.04\) in Fig. 14a to \(\eta =0.2\) in Fig. 14c. It is shown that for the same residual stress of 50 MPa the point distance decreases with higher values for \(\eta \).

*Hardcore Process (HC)*. The fragment density \(N_D\) is now connected to the spatial point process intensity. Analogously to Eq. (29), using Eq. (14) and combining it with Eq. (28), an improved estimation of the expected Fracture Intensity Parameter \(\check{\lambda }_{HC}\) can be derived in terms of the Fragment Size Parameter \(\delta =2r_0\) as:

#### 4.3.2 Stochastic fragmentation process

If fracture tests are conducted, it can be observed, that the center locations of fragments do vary stochastically within an object under investigation due to different reasons such as inhomogeneous thermal pre-stressing of 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 locations in an observation field of size \(50\,\hbox {mm} \times 50\,\hbox {mm}\) with a uniform distribution for the point locations instead of HCP. This kind of randomly distribution led to a varying distance between points. However, the distance determined on the basis of the relaxation factor \(\eta \) was given as the minimum distance for the point process. In Fig. 15a–c the Voronoi tesselation of uniformly distributed points with the minimum point distance \(\delta _{min}\) is shown for a plate with the residual mid-plane tensile stress of 50 MPa. In comparison to the fragment density for the case of HCP distributed points, the number of fragments is underestimated for uniformly distributed points. Until \(\eta =0.04\) the difference is not significant yet. However, the difference is more pronounced for the higher relaxation factor of \(\eta =0.2\). This is because on the one hand, Eqs. (29)–(31) are derived as approximations for the underlying stochastic point process as prior expectations motivated from the deterministic fracture process and on the other hand, in the deduction of Eqs. (29)–(31) a minimum energy requirement was not introduced for the randomly distributed points.

#### 4.3.3 Experimental determination of the Fracture Intensity Parameter \(\lambda \)

Now assuming a constant intensity in each observation field the average Fracture Intensity Parameter \(\lambda \) can be calculated for each specimen using Eq. (28).

Using the energy density \(U_D\) for the determination of the correlation, the curves of the fragment density \(N_{50}\) and thus the fracture intensity \(\lambda \) for the different thicknesses coincide to one line. In Fig. 17, the correlation between the elastic strain energy density \(U_D\), calculated from the measured residual stresses of each specimen, and the Fracture Intensity Parameter \(\lambda \) is presented. It can be observed that the accuracy of the experimental results of the fracture intensity decreases with lower energy density respectively for larger fragments.

## 5 Summary, conclusion, outlook and future research

### 5.1 Conclusion

*Hardcore Process*, e.g., possesses two parameters, the intensity of the points in an observation region and the (minimal) hardcore distance between point pairs, thus it is a two-parametric model. In this work, due to comprehensive investigations of fracture patterns and analytical considerations based on the LEFM the parameters characteristic for the fracture intensity called

*Fracture Intensity Parameter*\(\lambda \) and the hardcore distance between point pairs called Fragment Size Parameter \(\delta \) have been determined. These two parameters are applied over the elastic strain energy \(U_D\) in Fig. 18.

The fracture mechanical parameters were laid and the connections to the parameters of the spatial point processes were highlighted. Under assumption of a cylindrical fragment with the radius \(r_0\) of the base area, a Fragment Size Parameter \(\delta \) was generated as a function of the elastic strain energy remaining in the fragment after the fragmentation considering the relaxed energy. In this work, the energy relaxation factor \(\eta \) was fitted to the results of the fracture tests. The minimum distance between two points (Hardcore distance \(\delta =2r_0\)) can be determined on the basis of the LEFM.

*K*as a function of the temper stresses. The Fracture Intensity Parameter \(\lambda \) to consider the intensity of the fracture structure was also discussed. A correlation between the elastic strain energy density and the intensity of the fracture pattern was established based on the experimental data of fracture tests. An estimator for the constant Fracture Intensity Parameter was derived for two cases: the deterministic Hexagonal Close Packed (HCP) setting as well as a Hardcore stochastic point process, both acting on an observation field

*K*. The generation of fracture patterns under the assumption of HCP leads on the one hand to a visually observable poorer depiction of the glass fracture pattern and on the other side a poorer estimate of the fracture intensity compared to the Voronoi tesselation of the observation domain over spatial randomly distributed point locations \(\mathbf {p}\). Thus, further research will head in the direction of assuming point locations for the Voronoi tiling as random variables. The underlying probability distribution has to be estimated from fracture experiments.

### 5.2 Outlook and future research

In order to predict the 2D macro-scale fragmentation of glass considerations from Linear Elastic Fracture Mechanics (LEFM) with approaches from stochastic geometry in terms of spatial point patterns have to be merged. This method is called “Bayesian Reconstruction and Prediction of Glass Fracture Patterns (BREAK)”. In Pourmoghaddam et al. (2018a, b), numerical studies on the probability distributions of for different geometrical properties of tesselation over the CSR, MHC and SP with different parameter settings were conducted and evaluated. As no analytical expressions for the posterior distribution of e.g. the number of edges, the area content or the circumference of a typical Voronoi cell of a MHC or SP exist and these quantities are of interest in a fracture mechanical setting, numerically intensive Monte Carlo simulations of these processes were used to numerically infer the respective distributions to allow further fracture mechanical processing of these information. Besides the fracture mechanical relevance a general mathematical value can be addressed to the obtained results, as the posterior distributions for the geometrical properties can be used in other applications without loss of generality.

An example for the estimation of the spatial point process intensity and thus fragment density is depicted in Fig. 19 (note, that in this computation no edge correction was applied as the choice of a correction method itself is non-trivial, c.f. Baddeley et al. (2016); Møller (1994). In Fig. 19 the estimation of the intensity \(\lambda _{HC}\) of the underlying point process as well as the induced Voronoi Tesselation for the fracture pattern of one of the tested glass panes is shown as 3D view and a top view. The process calibration as well as simulation of further fracture patterns will be highlighted in part two of this paper in depth. In order to determine the statistical values of the fracture structure such as fragment edge number, the fragment perimeter, fragment base area, etc. first the fracture image has to be recorded and morphologically processed. Then a spatial point process model fed with the informations from the fracture mechanics (fragment size respectively the hardcore distance between point pairs) and the experimental data of fracture tests (fracture intensity) has to be matched to the fracture image and calibrated to evaluate a candidate model. Subsequently, the fracture pattern can be simulated. The overall methodology with the incorporated theories is depicted in Fig. 20. The results presented in this paper, the incorporated theories as well as further experimental investigations of the fracture structure will flow into future research in the field of 2D macro-scale fracture structure prediction. An upcoming publication deals with the further deduction, implementation and calibration of the method “BREAK” in order to simulate fracture patterns which conserve different statistical properties of the underlying glass fracture patterns such as the distribution of interpoint-distances or the distribution of remaining fracture particle area. The assessment of different spatial point processes as well as the properties of the induced Voronoi tesselations have to be studied and compared against the empirical statistics obtained from the images of the fracture tests on glass plates. The formalisation of the findings in terms of parameter tables as basis for the simulation of the spatial point processes with their Voronoi tesselation will be provided in the second part of this paper.

## Notes

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

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