numero_59_art_20 S. Cao et alii, Frattura ed Integrità Strutturale, 59 (2022) 265-310; DOI: 10.3221/IGF-ESIS.59.20 265 Cracking patterns of brittle hemispherical domes: an experimental study Siwen Cao Dept. Mechanics, Materials and Structures, Budapest University of Technology and Economics, Hungary caosiwen.china@gmail.com András A. Sipos Dept. Morphology and Geometric Modeling & MTA-BME Morphodynamics Research Group, Budapest University of Technology and Economics, Hungary siposa@eik.bme.hu, http://orcid.org/0000-0003-0440-2165 ABSTRACT. Crack formation in hemispherical domes is a distinguished problem in structural mechanics. The safety of cracked domes has a long track record; the evolution of the cracking pattern received less attention. Here, we report displacement-controlled loading tests of brittle hemispherical dome specimens, including the evolution of the meridional cracking pattern. The 27 investigated specimens, 20 cm in diameter, were prepared in 3D printed molds, and their material is one of the three mixtures of gypsum and cement. We find that neither the (limited) tensile strength nor the exact value of the thickness significantly affects the statistical description of the cracking pattern, i.e., the cracking phenomenon is robust. The maximal number of the meridional cracks never exceeds seven before the fragments’ disintegration (collapse). We find that the size distribution of the fragments exhibits a lognormal distribution. The evolution is reflected in the load-displacement diagrams recorded in the test, too, as significant drops in the force are accompanied by an emergence of one or more new cracks, reflecting the brittle nature of the phenomenon. A simple, stochastic fragmentation model, in which a segment is fragmented at either in the middle or at the fourth point, fairly recovers the observed size distribution. KEYWORDS. Cracking pattern; Brittle material; Hemispherical dome; Evolution; Experimental study. Citation: Cao, S., Sipos, A.A., Cracking patterns of brittle hemispherical domes: an experimental study, Frattura ed Integrità Strutturale, 59 (2022) 265-310. Received: 10.09.2021 Accepted: 03.11.2021 Published: 01.01.2022 Copyright: © 2022 This is an open access article under the terms of the CC-BY 4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. INTRODUCTION ransferring loads with thin shells is highly efficient because normal forces dominantly balance the external loads, and internal bending is marginal. With a favorable geometry (and boundary supports), internal bending mostly vanishes as long as the spatial distribution of the external loads does not vary in time [1]. The resulting material efficiency manifests in a wide range of applications, e.g., sheet metal forming, crash-worthiness test, structural design, pressure vessel T https://youtu.be/rHrnffLVzRY S. Cao et alii, Frattura ed Integrità Strutturale, 59 (2022) 265-310; DOI: 10.3221/IGF-ESIS.59.20 266 liability, and shipbuilding, to name a few [2,3]. The application of thin shells in the form of domes and other double-curved surfaces boosted the construction of large public spaces in the history of architecture. As they were built from masonry or concrete, the limited tensile strength of these materials favored compression in the shell. So, masonry or concrete shells are particularly vulnerable to internal tension and often develop cracks perpendicular to the maximal (positive) principal stress. At first sight, the stress trajectories of principal stresses seem to be an efficient tool to predict the emerging cracking pattern of the shell. Note that upon the appearance of a crack, the stress distribution, hence the direction of the stress trajectories, significantly alters. This influence of the former cracks on the formation of new ones undoubtedly influences the final cracking pattern. Since the appearance of meridional cracks on the dome of the St. Peter’s Basilica in Rome [1,4], it has become the most documented evaluation of a cracked thin shell. The debate around the origin and danger of the cracked state founded structural mechanics in its modern form [5], and it still provides a fundamental motivation to basic research [6]. Beyond this famous example, the significant number of existing structures explain that the mechanics of a cracked hemispherical dome has been widely analyzed (see [7] and the references therein). In general, a masonry dome with a sufficiently large opening angle under self-weight is subject to hoop tensile forces in their lower portion, which lead to vertical cracks appearing along the dome’s meridian planes [5,8,9]. A close inspection reveals such hoop tension cracks in most of the cases. Nonetheless, a varying thickness of the shell might limit the length of these cracks. In theory, there exists a solution for a crack-free geometry [10], but it is far from practical. The cracks of the dome open wide along with a large band and break up into fragments (called slices) that behave as independent pairs of semi-arches leaning on each other [7]. The dome stands as a series of arches with a standard keystone at the final extent of cracking. The collapse happens by lowering the top’s weight, accompanied by significant horizontal thrusts on the bearing elements. In a monitored in-scale experiment [11,12], the classical solution to the primeval dome’s thrust finds its confirmation. Some local effect often causes the initiation of cracking. E.g., Ginovart et al. [13] present the evolution of the rupture of the oval dome in Tortosa, where a failure of a roof beam led to the bending of the lantern, and the resulting asymmetric load distribution is associated with crack initiation. Beyond static loads, chemical effects and temperature changes also play a crucial role in the cracking process. Masi et al. [8] investigate the reason and period of formation of the meridional cracks on the dome of the Pantheon in Rome. They found that concrete shrinkage, together with gravity, may have been the leading mechanical causes of the cracks in the early phases of the building’s life. Qasim et al. [3] focus on fractures initiated at the remote support margins. Margin cracks can become dominant when loading forces are distributed over broad contact areas. Bartoli et al. [14] point out that the emerging cracks have a significant effect on further damage propagation and result in a complex crack pattern that fundamentally modifies the structural behavior of the dome. Instead of a hemispherical shell, the dome now behaves like four drifting half-arches. To reduce the complexity of the problem, researchers tend to focus on shallow domes, i.e., fragments of hemispherical domes [15,14], or they provide pure numerical investigations [6]. Understanding the cracking phenomenon invokes the membrane theory of shells [17]. It is a textbook exercise to show that the hoop stress in domes with a sufficiently high opening angle must be tensile [5]. Although the membrane theory of shells can be used to explain the principal reason behind the emerging cracking pattern, it can hardly serve as a framework for understanding the evolution of the cracks. The significant change in the stress field upon cracking mentioned above is just a partial reason. Note that the assumptions of membrane theory, especially the negligible internal bending criterion, are lost as soon as the first crack starts to propagate in the shell. Nonetheless, advanced fracture mechanics techniques provide such a modeling framework; let us mention only variational brittle fracture, which theory was successfully applied to predict the observed cracking pattern of the Panthéon in Paris [18]. Despite the long track record of the problem, the evolution of crack formation is still partially unrevealed. This paper focuses on the experimental investigation of crack development of hemispherical domes made of a homogeneous, brittle material, a gypsum-cement mixture, with limited tensile strength. After a summary of the performed experiments, the final, fragmented state and morphology development are discussed. Relation to the recorded load-displacement relationship at the top of the dome is provided. In the end, we introduce a simple stochastic model that recovers the observed size distribution of the fragments. MATERIALS AND METHODS ur experiments aim to reveal the evolution of the cracking pattern of complete hemispherical domes loaded with a distributed load around the top of the dome. We focus on the effect of two control parameters, the slenderness of the dome and the material’s tensile strength. The =t/R slenderness of the dome with radius R and thickness t O S. Cao et alii, Frattura ed Integrità Strutturale, 59 (2022) 265-310; DOI: 10.3221/IGF-ESIS.59.20 267 varies between  =0.0511 and  =0.105. The ft tensile strength is varied via the gypsum/cement ratio of the material. Here we summarize the experimental setup and the execution of the testing program. Preparation of the specimens Hemispherical dome specimens were made of a concrete-like material without reinforcement or fiber content. During the preparation, the material was poured into a custom-made plastic mold manufactured by 3D printing. The mold consists of 3 parts, the inner mold, the outer mold, and the inner mold support (Fig.1). We aimed a constant thickness for each specimen; hence both the inner and the outer molds have a spherical surface. With Rout=100mm radius to the outer surface, the external mold is sufficiently braced with stiffening plates. As domes with different thickness values are tested, three internal molds with different diameters were prepared. The adopted 3D printing technology used polylactic acid (PLA) filaments as a printing material [19]. After pouring, the specimens are dried at room temperature for two times 12 hours. Then the specimen is removed from the mold, and the mold is cleaned and reused for a new production cycle. Outer mold Inner mold Inner mold support Dome specimen Figure 1: The 3D printed plastic mold for the preparation of the dome specimens. The dome specimens have the following geometric properties: the outer diameter is 200 mm, the t design thicknesses are 5 mm, 7.5 mm, and 10 mm. Some of the manufactured domes are depicted in Fig.2. Note that all specimens possess an identical outer diameter. Let Rout denote the half of the outer diameter and Rin the internal radius. As the domes are with constant thickness, at any surface point t = Rout − Rin. As the thickness is constant, the R radius of the mid surface is simply Figure 2: Some dome specimens after their removal from the mold. S. Cao et alii, Frattura ed Integrità Strutturale, 59 (2022) 265-310; DOI: 10.3221/IGF-ESIS.59.20 268 + = 2 out inR R R (1) The  slenderness of the dome is defined via  = t R (2) The relevant geometric data are summarized in Tab.1. t (mm) Rout (mm) Rin (mm) R (mm)  10.0 100 90.0 95.00 0.105 7.5 92.5 96.25 0.078 5.0 95.0 97.50 0.051 Table 1: The geometry of the tested specimens. Material As cracks in domes are mainly tensile cracks [20], it is worthy of investigating the effect of the tensile strength on the cracking evolution. Following the ideas and practices of other researchers [21], we exploit that different ratios of cement and gypsum produce materials with different tensile strengths. After many attempts, the ratio of cement and gypsum was finally settled, as it is shown in Tab.2. Label (Gypsum: Cement): Water ft (MPa) S2.5 (2.5: 1): 0.67 0.316 S3.0 (3.0: 1): 0.67 0.293 S3.5 (3.5: 1): 0.67 0.233 Table 2: The material parameters (by weight) and the mean of measured tensile strength. To obtain the ft tensile strength of the material, the widely used Brazilian Disc tests were carried out [22]. In specific, simultaneously with making the dome, cylinders with a diameter of 50 mm and a height of 50 mm were also produced using the same mixture. It means the Brazilian Disc and the dome specimens possessed identical material parameters. The strength of the material was tested on these cylinders. In order to ensure that the specimens with different material parameters have a monotonous strength relationship, a total of 3 sets of tests were done, each containing nine specimens. We found that the mean of the ft tensile strength of the materials S2.5, S3.0, and S3.5 equals 0.316MPa, 0.293MPa, and 0.233MPa, respectively. Labeling the specimens We use a unique identifier for each specimen. Let the letters S, T, and L refer to strength, thickness, and loading angle (explained below), respectively. For instance, with these in hand, T10S2.5L30_1 means that for this dome t=10 mm, the ratio of gypsum and cement reads 2.5:1 (see Tab.2), and the loading angle is θ=30° (see the Fig.4(b) and this letter is not varied in this study). Number 1 at the end of the label is a counter to identify the specimen uniquely. In summary, the recommended manufacturing process for the domes (Fig.3) is as follows: 1. Scribble the vaseline to the surface of dome mold; 2. Weigh the required water, cement, and gypsum, mix them and stir for a few minutes; 3. Dump the mixture of water, cement, and gypsum into the outer mold first; 4. Assemble the inner mold: assemble, during this phase, vibrate; 5. Fasten screws while vibrating; 6. Vibrate the mold for at least 5 minutes to excrete bobbles inside the mixture; S. Cao et alii, Frattura ed Integrità Strutturale, 59 (2022) 265-310; DOI: 10.3221/IGF-ESIS.59.20 269 7. Wait for 12 hours; 8. Disassemble the inside mold holder; 9. Scrape the top surface to remove extra part of the model (i.e., make the bottom surface of the model smooth); 10. Remove the inner mold; 11. Wait for 12 hours; 12. Remove the outer mold; 13. Dry it for two days; 14. Airtight the model with a plastic bag and wait for the experiment. 15. Make three Brazilian discs with the same material simultaneously (Fig.3). Figure 3: Preparation of the specimens. Dome molds and Brazilian Discs molds with the mixture inside, after vibration. Experimental setup Domes were loaded vertically along their axis of symmetry. Displaced-controlled vertical pressure was applied to the upper surface of each hemispherical dome. The half-angle θ characterizes the loaded area (see Fig.4 (a)). In this investigation, the value of that angle is fixed at θ=30°. The pressure from the testing machine was distributed to the loaded area by a custom- made plastic spacer obtained by 3D printing. One side of the spacer is a curved surface, which can be fully attached to the dome’s outer surface; the other side is a flat surface, which can be fully attached to the loading indenter (Fig.4 (b)). The spacer realizes even surface loading whole on the loaded area. At the bottom of the dome, a frictionless restraint was applied; in specific, vaseline was greased to the bottom of the dome to reduce friction. 2 distributed load 30 =  R100 mm 105 mm (a) (b) Figure 4: Loading of hemispherical specimens. (a) Half-angle θ determines the loaded area on the surface of the dome. (b) The half section schematic and size of the 3D printed plastic spacer used to distribute the load on the surface of the specimen (distances are in mm). The experiment was performed in a Zwick/Roell Z-150 testing machine. The loading speed was 0.25 mm/min in each test. An iron plate was placed between the testing machine and the bottom of the dome. There was a positioning dial on the iron plate. The dial is used to conveniently record the relative position of the cracks (Fig.5). S. Cao et alii, Frattura ed Integrità Strutturale, 59 (2022) 265-310; DOI: 10.3221/IGF-ESIS.59.20 270 Each specimen is placed on the iron plate so that it should be concentric with the dial, and the top of the dome should be aligned with the loading axis of the testing machine. Figure 5: A specimen placed into the testing machine. Observe the plastic spacer between the specimen and the head of the testing machine. The loading was stopped if either of two conditions was met: 1. A total collapse of the dome or complete separation of any fragment appeared due to the intersections of cracks (disintegration of the body). 2. The vertical deformation at the top of the dome exceeded 5 mm. To obtain meaningful results, each test case was repeated three times. As we have three values for the slenderness and three material mixtures, a total of 27 specimens were investigated. RESULTS The geometry of the cracking pattern at the state of collapse n the performed experiments(see Fig. 6), the maximal number Nmax of meridional cracks never exceeded 7; the Nmin minimal number was 4. The average number of cracks is N=6.30, with sample variance sN=0.87. Two-sided student t- tests show that the observable number of the cracks is robust, i.e., influenced neither by the material quality nor the shell thickness. This latter seems slightly affecting N tough so that smaller t/R results in marginally more cracks on average. (The measured data, including a photograph of the final stage, is provided as supplementary material.) Figure 6: A cracked dome in the final stage. (a) Top view. (b) The 0 degrees angle view. (c) The 180 degrees angle view. The numbers show the order in which the crack appeared. Annexes 1 and 2 contain the photographs for each specimen. Let Φmin denote the minimal angle between two cracks along the dome’s perimeter at the final stage. Here we find, that the average reads Φmin=28.3° with a sample variance of 12.9°. Two-sided student t-tests show that the material has no significant I S. Cao et alii, Frattura ed Integrità Strutturale, 59 (2022) 265-310; DOI: 10.3221/IGF-ESIS.59.20 271 influence on the minimal angle; however, the thickness matters: increasing t/R produces a higher minimal angle: while at t=5 mm the average of the minimal angle is Φmin=20.7°, then for t=10mm it reaches Φmin=37.9°. Observe that in the case of a slender dome, the closest cracks have about 1/8 of the entire perimeter. Similarly, let Φmax denote the maximal angle between two cracks along the perimeter of the dome at the final stage. Here we find, that the average reads Φmax=92.9° with a sample variance of 16.9°. Two-sided student t-tests show that neither the material nor the thickness significantly influences the maximal angle. Observe that the fragment with the maximal area without a vertical crack is close to 1/4 of the whole perimeter. Finally, let Φavg denote the average angle between two cracks along the dome’s perimeter at the final stage. Here, in accordance with the average number of cracks, we find that the average reads Φavg=58.4° with a sample variance of 9.7°. Two-sided student t-tests show that neither the material nor the thickness significantly influences the average angle. Observe that the fragment with the average area without a vertical crack is close to 1/6 of the whole perimeter. Given all the performed experiments, a sum of 170 fragments was produced. The distribution of the fragments concerning the angle measured on the base follows a lognormal distribution (see Fig.7) with parameters Φ=3.947 and Φ=0.488. Figure 7: Cumulative distribution of the length of the fragments at the final stage. Experimental data (blue), a best-fit normal (grey), and lognormal (brown) distributions. The evolution of the cracking pattern Now we turn to the evolution of the cracking pattern. Observing the cracking pattern suggests that a new crack forms at the vicinity of the half point of a fragment between two cracks in many cases. However, in some other cases, the crack is close to the 1/4 point and sometimes appears close to one of the 1/8, 3/8, 5/8, 7/8 points of the fragment. Let L be a fixed positive integer and 0≤x≤1 a rational number. Let Δ:=2-(L+1), and we define the following sets: A:     1 2 x B:          1 3 4 4 x C1:          3 5 8 8 x C2:          1 7 8 8 x D:       − 0.. 1 ..1x (3) 0,00 0,10 0,20 0,30 0,40 0,50 0,60 0,70 0,80 0,90 1,00 123,75101,2578,7556,2533,7511,25 angle (degrees) LOGNORMAL EXPERIMENT NORMAL S. Cao et alii, Frattura ed Integrità Strutturale, 59 (2022) 265-310; DOI: 10.3221/IGF-ESIS.59.20 272 0 1/8 2/8 3/8 4/8 5/8 6/8 7/8 1 AD B BC2 C2C1 C1 Figure 8: The sets defined by eq. (3) visualized on a line segment with unit length. Here set A and B are in the vicinity of the middle point and a fourth of a fragment with a length of 1, respectively. Similarly, sets C1 and C2 denote the neighborhood of the eighth-points of the fragment, C1 is close to the middle of the fragment, C2 to the origin, respectively. Finally, set D is in the vicinity of the starting and endpoints of the fragment. The first crack appearing in the dome cannot be classified with this system (due to the rotational symmetry of the uncracked dome). The classification from the second crack can be carried out so that the position of the emerging new crack is evaluated based on the positions of the neighboring cracks. Here the angle difference between the adjacent cracks is normed to 1. In an unscaled version, we compute the above-defined sets for each experiment and classified the 2nd, 3rd, 4th, 5th, and 6th crack at L=3. The results are summarized in Tab. 3. Observe that the second crack typically appears between the midpoint and the one- fourth point of the perimeter (region C1). The location of the consecutive cracks gradually moves in the direction of the middle point of the actual fragment. It is also worthy to note that in the vicinity of an existing crack, we never observe the formation of new ones (as the region D is empty), and the middle point itself is not that much favored, especially by cracks 2 and 3, as one would expect. crack2 crack3 crack4 crack5 crack6 A 18.5% 11.1% 26.9% 30.8% 18.2% B 11.1% 40.7% 38.5% 30.8% 22.7% C1 44.4% 33.3% 34.6% 26.9% 59.1% C2 25.9% 14.8% 0.0% 11.5% 0.0% D 0.0% 0.0% 0.0% 0.0% 0.0% Table 3: Location-distribution of the cracks inside the fragment they appear (unscaled version). The rows of the table refer to the sets in Fig. 8. The proportions indicated in the table are calculated based on the 27 specimens. The most probable occurrence is typeset in boldface. Observe that cracks either appear close to the half of the fragment (sets A and C1) or around the fourth point (set B). crack2 crack3 crcak4 crack5 crack6 A 18.5% 22.2% 61.5% 65.4% 86.4% B 11.1% 70.4% 26.9% 11.5% 4.5% C1 44.4% 0.0% 0.0% 0.0% 0.0% C2 25.9% 0.0% 0.0% 0.0% 0.0% D 0.0% 7.4% 11.5% 23.1% 9.1% Table 4: Location-distribution of the cracks inside the fragment they appear (scaled version). The rows of the table refer to the sets in Fig. 8. The proportions indicated in the table are calculated based on the 27 specimens. The most probable occurrence is typeset in boldface. Observe the tendency that cracks appearing later tend to be close to the half of the fragment (set A). Note that in the previous investigation, we did not consider that the length of the fragments decreases during the fragmentation. One way to remove the effect of the length from the results above is reducing the value of L in the definition of the sets above after a fragmentation event. This evaluation is called a scaled version. Let Lact denote the value of L associated with a fragment, where the new crack forms. Nonetheless, the value of Lact cannot be smaller than 1 (in this case, only set A is meaningful). Again, starting with a single crack and L=3, in the experimental data, we find the following distribution S. Cao et alii, Frattura ed Integrità Strutturale, 59 (2022) 265-310; DOI: 10.3221/IGF-ESIS.59.20 273 of the location of the new cracks, summarized in Tab. 4. Observe the strong tendency towards the emerging location in the middle of the fragments from crack 4. Note that this result does not contradict the findings of the unscaled version for the sixth crack: the actual length of the neighborhood of the middle point gradually decreases as the length of the fragments reduces in the process. To hit this neighborhood is more and more difficult as it shrinks towards a point. In the scaled version, we associate the neighborhood of the middle point with the 2^(-Lact); hence, the probability of hitting this set is higher. During the experiments, the testing machine measured the deformation, the vertical displacement of the dome crone in specific. The recorded force-displacement curves exhibit a characteristic behavior: Whenever a new crack appears, the dome deforms suddenly in the vertical direction, and the loading force drops. As the dome continuously produces cracks during the loading process, the force-deformation curve resembles a saw, as peaks appear one after another (see Fig.9). The numbers 1 to 6 in Fig. 9 represent the numbers for the consecutive cracks for the specimen T10S3.5L30_2. Between the peaks, the structure exhibits a dominantly elastic behavior reflected in a close-to-linear load-displacement curve. In about one-third of the specimens, some of the cracks appear simultaneously, i.e., two or more cracks occur at the same time. Nonetheless, this phenomenon is reflected in a more significant stress drop on the force-deformation curve. After the appearance of several cracks (but before total collapse), elasticity seems to be lost; as the increment in the load- bearing vanishes, the structure exhibit a plastic-like behavior; however, this ductility stems from the significant movements of the fragments produced by the preceding cracking process, not the material itself. Hence, we identify two phases in the loading process (see Fig. 9): in the beginning, we observe a dominantly shell-like behavior called the dome stage. We come to the arch stage after reaching the maximal load (and the emergence of several cracks). Nonetheless, an exact definition to separate these two stages is impossible to give, as the transformation between the two is gradual. Figure 9: A load-displacement curve of three specimens. T10S3.5L30_2 (gray), T7.5S3.5L30_2 (brown), T5S3.5L30_2 (blue). The numbers show the consecutive meridional cracks of the specimen T10S3.5L30_2. Observe the linearly elastic behavior between the occurrence of two subsequent cracks and the significant drop in the force during the rapid formation of the crack. The load-bearing capacity of the model in the arch stage is much lower than that of the dome stage. Note that the dome is a complete body; its bearing capacity depends on the t/R slenderness and the material quality; while the arch is composed of several fragments, its bearing capacity depends on the two parts, self-carrying capacity and the bottom friction. For the arch stage itself, its bearing capacity is determined by the weakest fragment. When cracks occurred in the experiments, the minimum and maximum vertical displacement were between 0.09mm and 4.68mm. The critical vertical displacement that separates the dome and arch stages is approximately 2.0mm. The average vertical displacement at the occurrence of the first crack is about 0.55mm. A 0.37mm displacement is needed to open the second crack at a gradual increase of the load. The next cracks require respective 0.35mm, 0.29mm, 0.26mm, and 0.26mm increments in the top displacement. We see that although in some cases the appearance of two or more cracks is -100 0 100 200 300 400 500 600 700 0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5 S ta n d ar d f o rc e (N ) Deformation (mm) T10S3.5L30_2 T7.5S3.5L30_2 T5S3.5L30_2 1 2 3 4 5 6 dome stage arch stage S. Cao et alii, Frattura ed Integrità Strutturale, 59 (2022) 265-310; DOI: 10.3221/IGF-ESIS.59.20 274 simultaneous, on average, the cracks occur in well-separated intervals, and it is also meaningful that the first crack requires a bigger increment than any other. DISCUSSION Effect of the slenderness and the material quality s we have seen, the fracture pattern is marginally affected by the material properties and the slenderness of the dome. This means that the geometric pattern produced by the fracture process is robust. Nonetheless, the maximal force is significantly differing as either the tensile strength or the slenderness of the dome is varied (Tab.5). S2.5 S3.0 S3.5 T10 855.9 1440.0 630.7 T7.5 725.4 720.9 778.9 T5 237.9 350.7 255.3 Table 5: Maximum loading force (N), average values. Nonetheless, beyond the load-bearing capacity, the energy needed for the emergence of a new crack is indeed influenced by the thickness and the material quality of the specimen. In Tab. 6, the average area under the load-displacement diagram between consecutive cracks is given. Observe that the first crack requires the highest energy regardless of the thickness or the material quality in most cases. The smallest jump appears either for the fifth or the sixth crack. As the length of the cracks is, in general, equal, it shows that the phenomenon at the structural level is quasi-brittle, as the formation of a new crack is easier for domes with several pre-existing cracks. It is also worthy to note that the second, third and fourth crack requires more or less identical energy to form. crack1 crack2 crack3 crack4 crack5 crack6 T10 average 246.5 144.8 217.3 175.5 81.6 50.7 variance 215.3 117.3 104.3 109.2 27.9 24.1 T7.5 average 197.9 62.6 87.1 88.2 75.9 93.6 variance 138.3 38.5 42.0 60.2 34.4 87.6 T5.0 average 36.1 63.7 48.8 28.3 14.9 20.1 variance 27.2 27.6 25.5 17.0 12.6 14.6 S2.5 average 158.7 122.2 92.6 120.0 54.0 55.8 variance 177.5 125.2 65.6 123.9 41.7 44.2 S3.0 average 231.5 87.8 158.1 72.3 61.5 69.6 variance 201.5 45.3 131.5 57.2 45.8 92.9 S3.5 average 90.3 61.2 102.6 76.8 50.1 27.5 variance 101.0 34.3 80.5 65.2 38.5 19.6 Table 6: External work (N⸱mm) needed for the crack formation. Entries in boldface denote the maximal, in italic the minimal jumps in the strain energy. A S. Cao et alii, Frattura ed Integrità Strutturale, 59 (2022) 265-310; DOI: 10.3221/IGF-ESIS.59.20 275 Location of the emerging cracks From the result of the experiments, the locations of the cracks are counted. Cracks 3 to 7 represent the order of cracks. Sort the fragment size such that 1st to 7th represents the length of the fragments from large to small. For example, the third crack (i.e., crack 3) either appears on the longer fragment (denoted to 1st) or on the smaller part (the 2nd). Similarly, crack 4 appears either on the most extended fragment (1st), the middle fragment (2nd), or the smallest fragment (3rd). The experimental distribution of the crack occurrence is given in Tab. 7. It shows that cracks tend to appear on the longer fragments, as expected. crack3 crack4 crack5 crack6 crack7 1st 74.07% 61.54% 70.37% 72.73% 40% 2nd 25.93% 38.46% 22.22% 18.18% 40% 3rd - 0.00% 7.41% 9.09% 13.33% 4th - - 0.00% 0.00% 6.67% 5th - - - 0.00% 0.00% 6th - - - - 0.00% Table 7: The empirical probability of the next crack appearing on the fragments ordered based on their size (i.e., the 1st is the longest). To decrease the effect of fragments number and show the tendency of a new crack occurring on bigger fragments exactly, we define Mt: ( ) ( )= − −: /t crack Min Max MinM L L L L (4) where, Lcrack is the length of the fragment at which a new crack occurs, LMax is the maximum length of the current fragments, and LMin denotes the minimum length of current fragments. Tab. 8. shows the value of Mt for each crack, averaged for all tested specimens. Mt is not affected by the tensile strength or the thickness of the dome, and the overall mean of Mt is around 0.82. crack3 crack4 crack5 crack6 crack7 Mt average 0.74 0.83 0.90 0.90 0.79 Table 8: The mean of Mt for each crack. A simple model of the cracking evolution A simple stochastic process might be associated with the fragmentation process observed in our experiments. As in the experiments, we measured the cracks’ distance in degrees, took a fragment with a length of L=360, and fix a threshold 0≤ pt≤1. Let Nact denote the actual number of the fragments (we start with Nact=1). Let L1, L2…, LNact represent the length of the fragments. With a probability proportional to the size of the fragments, take a number from the set {1,…, Nact} to select fragment i to be fragmented. Choose a random number, denoted to r, between 0 and 1. If r