This is the submitted version of the article: Sellés, Albert G.; Marcé Nogué, Jordi; Vila Ginestí, Bernat; [et al.]. «Computa- tional approach to evaluating the strength of eggs : implications for laying in organic egg production». Biosystems engineering, Vol. 186 (October 2019), p. 146-155. DOI 10.1016/j.biosystemseng.2019.06.017 This version is avaible at https://ddd.uab.cat/record/215300 under the terms of the license 1 A computational approach to evaluate strength in eggs: lay and crash implications 2 for organic eggs production 3 4 Albert G. Sellésa,b,*, Jordi Marcè-Noguéa,c, Bernat Vilaa,b,d, Marco A. Péreze, Lluís Gilf, 5 Àngel Galobarta,b, Josep Fortunya 6 7 a. Institut Català de Paleontologia Miquel Crusafont – Escola Industrial 25, E-08201 8 Sabadell, Barcelona, Spain. 9 b. Museu de la Conca Dellà, c/Museu 4, E-25650 Isona, Lleida, Spain. 10 c. Center of Natural History (CeNak), Universität Hamburg, Martin-Luther-King-Platz 3, 11 Hamburg 20146, Germany. 12 d. Departament de Geologia, Facultat de Ciències, Universitat Autònoma de Barcelona, 13 Carrer de l’Eix central, 08193, Cerdanyola del Vallès, Barcelona, Catalonia, Spain. 14 e. IQS School of Engineering, Universitat Ramon Llull, Via Augusta 390, 08017 15 Barcelona, Spain. 16 f. Department of Strength of Materials and Structures in Engineering. Universitat 17 Politècnica de Catalunya UPC-ESEIAAT, c/ Colom 11, 08222, Terrassa, Spain. 18 19 *Corresponding author: albert.garcia@icp.cat 20 1 21 Abstract 22 23 With the aim of characterizing the different stress patterns in several egg types and 24 evaluate the resistivity of the eggshell while impacting in free fall, experimental tests 25 and computational simulations have been performed on eggs of the species Gallus 26 gallus (domestic hen), Struthio camelus (ostrich), and Testudo sp. (tortoise). The 27 different types of failure were determined for each taxa and stress distribution maps 28 were recorded and correlated between experimental results and computational 29 simulations using Finite Element Analysis (FEA). For domestic hens, the numerical 30 results obtained in both tests have highly correlation with the results obtained in 31 previous works; for ostrich (Struthio camelus) and tortoise (Testudo sp.) eggshells, the 32 new data represent a very good tool to understand in further works the failure 33 mechanisms of the eggshells. More interestingly, the present study provides the first 34 empirical and mathematic results that allow establishing confident safety interval related 35 to the laying process in the poultry industry. 36 37 38 Key-words: eggshell, strength, reproduction, failure mechanism 2 39 1. Introduction 40 41 The current growing demand in modern society for the called “organic foods” is 42 shaping the way alimentary industry face the request of the markets. According to the 43 most recent data provided by several international organizations (i.e. FAO, OECD, and 44 FAPRI) a significant increase of the global demand for animal protein can be 45 extrapolated in the years to come, being eggs the second alimentary source on the line 46 just behind the poultry meat (Mulder, 2017). In this sense, organic eggs stand among the 47 most requested organic products, leading the global proliferation of organic egg farms, 48 places where animals can lay freely on the ground. Given the economical impact of this 49 sector (Sumner et al., 2011), any advance in understanding the egg and eggshell 50 mechanics helping to minimize the risks of egg failure during laying may represents a 51 significant step-forward for the organic egg production. 52 Egg is a high complex biological structure composed of several concentric organic 53 and bio-mineralized layers. In a simplistic way, it can be defined as a container that has 54 an air chamber and viscous liquids (the yolk and the albumen) in its core that are 55 surrounded by two membranes and a external covering, the eggshell. Eggshell strength 56 is regulated by a certain number of variables such as genetics, the age of the laying 57 female, feed composition, diseases, climatic conditions or management by the farmer 58 (Solomon, 1990). 59 The characterization of the mechanical impact that suffers an egg on the substrate 60 and its strength have certainly received little attention in literature (Nedomová et al., 61 2009a; Trnka et al., 2012; Juang et al., 2017). In experimental way, eggshell strength 62 has been evaluated from a non-destructive and quasi-static compression test (Nedomová 63 et al., 2009b; Voisey and Hunt, 1974; Nedomová et al., 2013a, b), using a transducer 64 (Castilla et al., 2009) or introducing a dynamical test method using modal analysis 65 (Coucke et al., 1994) in hen and ostrich eggs. The mechanics and mechanisms of failure 66 of hen eggs have been examined experimentally under contact loading conditions 67 (MacLeod et al., 2005). In computational mechanics, the dynamic mechanical behaviour 68 of the egg has been evaluated with Finite Element Analysis (FEA) from simple 69 structural models (Coucke et al., 1998; Upadhyaya et al., 1986) to highly non-linear 70 transient dynamic analysis, including the study of the rupture by impact loading 71 (Nedomová et al., 2009b). FEA has also been used to evaluate the effects of variations 72 in certain geometrical and material parameters of the egg on the structural and acoustic 3 73 frequency response functions (Perianu et al., 2010) or even to study the microstructure- 74 controlled stability of selected eggshells of Indian dinosaurs (Srivastava et al., 2005). 75 According to previous works performed in hen eggs, the eggshell breaks at a certain 76 stress point (Nedomová et al., 2009a), namely fracture stress. Fracture stress is 77 independent of egg properties (i.e. geometry, size, and eggshell thickness), and on the 78 loading force orientation and it is affected only by the eggshell material properties. In a 79 scenario where impact is produced below the fracture stress point the physic integrity of 80 the egg is poorly compromised, with minor or without cracking, while impacts 81 occurring up to the fracture stress point produce different types of failure 82 Herein, experimental tests and computational simulations were performed in order 83 to evaluate the resistivity of the eggshell while impacting in free fall. Different impact 84 tests in tortoises, domestic hen, and ostrich eggs were performed to determine the 85 different types of failure among different groups of animals. These experimental results 86 were confronted and correlated with those obtained from computational simulations 87 using Finite Element Analysis (FEA), providing stress distribution maps. Thus, the 88 main goal of the present study is to provide new key information to characterize the 89 different stress pattern in eggs of different groups. 90 91 2. Material and Methods 92 93 2.1. Sample 94 95 The experimental program was conducted on twelve tortoise (Testudo sp.) eggs, 96 seventeen hen (Gallus gallus) eggs, and six ostrich (Struthio camelus) eggs. They have 97 been selected for representing different eggshell-types structures (Mikhailov, 1997). 98 Free-range hen eggs were obtained from a local product gross-store, while ostrich 99 eggs were acquired from an ostrich farm. Non-fertilized tortoise eggs were provided by 100 the Centre de Recuperació d’Amfibis i Reptils de Catalunya (CRARC, Catalonia, 101 Spain), after careful examination of a large sample of specimens collected during the 102 brooding season of Testudo sp., while hen and ostrich eggs were obtained from 103 commercial sources. 104 105 2.2. Experimental tests 106 4 107 The impact test procedure consisted in releasing the egg from a height which 108 determined the incident kinetic energy and hence the incident velocity. Tests were 109 performed by crashing the specimens on a rigid support (E=10000 MPa). In order to 110 evaluate the impact damage resistance, specimens were dropped from different height 111 levels ranging from 50 to 1500 mm (Fig. 1). Prior to each test, specimens were 112 geometrically characterized (length, width, mass). Falling egg height and impact 113 velocity were recorded (Tables 1-3). After the impact, the thickness of eggshells were 114 measured, and values were used to create the computational model (see below). With 115 the aim of visualize the impact event details, a high-speed camera MotionBLITZ Cube4 116 was used. The frequency of the camera was 1878 Hz representing a time resolution of 117 529 μs per frame. The initiation and propagation of the failure modes were 118 characterized thanks to the recorded results. The crash of the eggs were divided into 119 three different types of failure (Fig. 2): 120 Type A: Total crash and spill of the yolk. 121 Type B: Crack and spill of the yolk. The egg bounces. 122 Type C: Partial crack and no spill of the yolk. The egg bounces. 123 124 2.3. Computational Simulations 125 126 A transitory Finite Element Analysis (FEA) was developed to simulate the 127 dynamical impact of the eggshell on the substrate (Fig. 3) using the Finite Element 128 Package ANSYS 14.5 in a Dell Precision™ Workstation T7600 with 32 GB (4X8GB) 129 and 1600 MHz. The maximum Von Mises stress was recorded in the eggshell 130 depending on the different impact velocities for the different substrate materials. 131 132 2.3.1 Egg geometry 133 134 The geometry of each egg type was 3D digitally recorded and following is described 135 in detail the approach used. Several authors have provided different equations in order 136 to express the mathematical function describing the vast variability of the contour of the 137 hen egg shapes (i.e. Narushin, 1997, 2001; Baker, 2002; Denys et al., 2003; Stoddard et 138 al., 2017; Severa et al., 2013; Troscianko, 2014; Attard et al., 2018; Duursma et al., 139 2018). Nevertheless, most of them fall back in assuming that an ellipsoid profile or 140 modified ellipse equations is the best method for describing this particular avian egg 5 141 shape. Herein the geometry of the hen eggshell was obtained from digital photographies 142 and translated to a CAD model using Rhinoceros® (McNeel & associates, version 5.0, 143 Seattle, WA). 144 As a result, the average values from the measured hen (Gallus gallus) egg 145 specimens (Table 1) were: 55.96 mm in length, 43.86 mm in maximum width, and an 146 average eggshell thickness of 0.44 mm. Tortoise egg can be described as a revolution 147 ellipsoid. In this case, the tortoise (Testudo sp.) eggshell was created using the CAD 148 interface of ANSYS FEA package. The average obtained from the measured egg 149 specimens (Table 2) was used: an average length of 34.4 mm, an average width of 27.97 150 mm, and an average thickness of 0.4 mm. Finally, the shape of the ostrich (Struthio 151 camelus) egg can be described as a revolution ellipsoid. As in the previous case, the 152 ostrich eggshell was created using the CAD interface of ANSYS FEA package, using 153 the average values obtained from the measurements of the analysed specimens (Table 3) 154 it was established an average length of 158.4 mm, and an average width of 126.0 mm. 155 Assuming that the flexible testacea membrane that surrounds the inner part of the 156 eggshell could be omitted for this simulation purpose, the thickness of the eggshell 157 considered was an average value of 1.75 mm. 158 159 2.4. Solid grounds selection and modelling 160 161 Different impact substrates were considered in the computational model. Each 162 substrate was selected for having different elastic values associated to different soils- 163 types (from hard to soft), and meshed with hexahedral solid elements in the FEA model. 164 The values were chosen taking into account that saturated clay soils have a Poisson 165 coefficient around 0.45 (Simeonovova and Buchar, 2002; Nedomova et al., 2009b). 166 However, the Elastic Modulus is around E=5 to 25 MPa for soft clays, while that of 167 hard clays ranges from 50 to 100 MPa (Bowles, 2006), and reaching up to 10000 MPa 168 in rigid floors. Because of that, we established four cases of study where the Case1 169 represents an impact on a hard soil of 10000 MPa, Case 2 of 1000 MPa, Case 3 of 100 170 MPa, and Case 4 is characterised by a soft soil of 10 MPa. 171 All the eggs were meshed with shell elements of constant thickness considering as a 172 homogeneous isotropic linear elastic material with E = 0.0035 GPa and  = 0.45 173 (Simeonovova and Buchar, 2002; Nedomova et al., 2009b). The yolk, the albumen and 174 the air inside the egg chamber were considered as an hydrostatic pressure of a liquid of 6 175 1.025 g/cm3 (Rahn and Paganelli, 1989). Different tests were considered modifying the 176 impact velocity. The value of the velocity is associated to a height according to the 177 energetic equilibrium between potential and kinetic energy (Fig. 1). 178 In order to model the interaction between the eggshell and the soil in the very 179 moment of the impact, a frictionless contact was defined between the bodies. A 180 frictionless contact assumes that a gap was allowed in the normal direction and sliding 181 was freely allowed in the tangential direction. In the gap direction, an augmented 182 Lagrange contact formulation with large Normal Stiffness symmetric behaviour were 183 assumed in which only the contact surfaces were constrained from penetrating the target 184 surfaces (Wriggers, 2002). This formulation implies a non-linear solution. A 185 convergence iterative procedure based on a Newton-Rhapson iterative algorithm was 186 used. 187 188 3. Results 189 190 The experimental impact tests, with a rigid support (E=10000 MPa), showed that in 191 the tortoise (Testudo sp.) eggs (Table 2) the boundary between the total crash of the egg 192 (type A) and its crack and spill of the yolk (type B) appears when the fall height is 193 around the 1000-750 mm with a velocity of 4000 mm/s, while the boundary between 194 failure type B and partial cracking of type C occurs around 125-110 mm of fall height, 195 producing a velocity around 1500 mm/s (Table 2). 196 In the case of the domestic hen (Gallus gallus), the boundary between the failure 197 type A and B appeared between a fall height of 300-400 mm and a velocity of 2500 198 mm/s (Table 1), while between type B and C occurred in a fall height of 125-110 mm 199 and a velocity of 1500 mm/s (Table 1). Finally, in the ostrich (Struthio camelus) eggs 200 the fall heights reduced: the boundary between type A and B was around 250-300 mm 201 and a velocity of 2300 mm/s, while the type B-C boundary occurred in a fall height 202 around 50-75 mm and a velocity of 1,200 mm/s (Table 3). Figure 2 shows frames of the 203 recorded videos of the experimental tests with the different casuistry for defined types 204 A, B and C (see also Supplementary Information). 205 Regarding the results obtained with computational simulations, results reveal the 206 tendency to level off certain value of threshold for high impact velocities, and abrupt 207 changes for very low values of velocity (Supplementary Tables S1-S3). Figure 3 shows 208 the resulting maximum von Mises stress values during the impact of eggs against rigid 7 209 soil (E=10000 MPa), whereas Figure 4 shows maximum Von Mises stress versus height 210 and impact velocity defined in the FEA model where the egg is dropped off depending 211 on the hardness of soil for hen (Fig. 4 A and D), tortoise (Fig. 4 B and E), and ostrich 212 (Fig. 4 C and F). Von Mises stress corresponds to the moment of the impact between 213 the eggshell and the soil, and the maximum Von Mises stress reached in the 214 computational model. Height and velocity are related according to the law of 215 conservation of energy, which balances the kinematic energy (E=1/2mv2; being m the 216 mass of a body and v its velocity) and the potential energy (E=mgh; where m is the 217 mass, g is the acceleration due to gravity, and h is the height). Regarding the ostrich 218 egg, the non-linear solution of the impact for a soil E=10 MPa did not reached the 219 mathematical convergence (see Fig. 4 C and F). 220 There is also a close dependence between eggshell rupture force and impact velocity 221 that can be described as a logarithmic function, as occurs in many engineering materials 222 and according to previous observations (Trnka et al., 2012). 223 The type of failure, based in the casuistry defined via experimental tests, was 224 correlated between the results obtained in both computational and experimental models. 225 Given that a detailed analysis of returned data requires complex considerations, the 226 following results focus in describing the aforementioned correlations under extreme 227 experimental conditions, which are those when the test is running simulating a hard soil 228 with an elastic modulus of E=10000 MPa. For hen eggs, the failure type C was only 229 obtained below an impact velocity of 1000 mm/s (Table 1), which can be correlated 230 with a value below the 554 MPa in the computational simulation (Fig. 4D). These 231 results agree with previous impact analyses obtained also using hen eggs (Nedomová et 232 al., 2009). In tests with tortoise eggs, the type of failure C was observed below an 233 impact velocity of 1500 mm/s (Table 2), correlated with a value of Von Mises stress 234 below the 520 MPa (Fig. 4E). Finally, in the case of the ostrich eggs, type C was 235 recorded below an impact velocity of 1200 mm/s (Table 3), related with a value below 236 the 420 MPa in the computational simulation (Fig. 4F), and representing the lowest 237 value in comparison with the hen and the tortoise cases. 238 Regarding the computational analysis, Figure 3 reveals different distribution of Von 239 Mises stress depending on the type of failure. These observed distributions of stress can 240 be correlated with the different types of failure observed in the experimental test when 241 the fracture of the eggshell is with a partial crack, a major crack, or a total crash of it. 8 242 According to that, in tortoise eggshell, in the failure type A, the maximum values of 243 stress were located in a wide point and important values of stress were located in the 244 bottom half of the eggshell generating the total crash of the eggshell. For the failure 245 type B the peak stress was concentrated but appearing important stress values in the 246 surrounds of the impact point. This different distribution generated a crack in the 247 eggshell without the total crash of the shell. Finally, the failure type C revealed very 248 low concentrated peak stress, only placed just in the point of impact. This value was 249 lower than the threshold of 520 MPa and for instance, the eggshell was not breaking due 250 to the impact. In the hen eggshell, for type A, the maximum values of stress were 251 located in a wider point and important values of stress were located in the bottom half 252 of the eggshell generating the total crash of the eggshell. In the failure type B the peak 253 stress was concentrated but appearing important stress values in the surrounds of the 254 impact point (as in the same case in tortoises). This different distribution generated a 255 crack in the eggshell without the total crash of the shell. Finally, in the failure type C, 256 very low concentrated peak stress was placed just in the point of impact. This value was 257 lower than the threshold of 550 MPa, and therefore the eggshell was not broken due the 258 impact. 259 Finally, for the ostrich eggshell in the type A, the maximum values of stress were 260 placed in a wide point and important values of stress were located in the bottom half of 261 the egg generating the total crash, while in the failure type B the peak stress was 262 concentrated but appearing important stress values in the surrounds of the impact point 263 (as in the same type in tortoises and hens). This different distribution generated a crack 264 in the eggshell without the total crash of the shell. Finally, the type C showed a very 265 low concentrated peak stress sited just in the point of impact. This value was lower than 266 the threshold of 420 MPa, and for instance, the eggshell did not break due to the impact. 267 268 4. Discussion 269 270 Reproduction is one of the fundamental biological processes of any organism, from 271 which depends the survivorship of a species. Of its success depends the proper 272 conditions for the development of the offspring, but also it is the basis of the poultry 273 industry. Several studies have emphasised on the increasing role that organic food play 274 in global market. Among them organic eggs represent an important economical sector, 275 which is valued on up to US$ 2700 Mn by the end of 2025 only in Europe (data from 9 276 Koncept Analytics, 2018; QYResearch Group, 2018). Forecasts indicate that this trend 277 will increase in the future. 278 Because of that, any advance in minimizing the loss in the production of free-cage 279 eggs may represent a major economical impact. The egg quality and incubation/nesting 280 environment conditions are crucial factors for success laying (Grant, 1982). In this 281 regard, the results of the present study set the bases for establishing confident intervals 282 of oviposition height, from which eggs can suffer or avoid critical damages. 283 The final aim of the egg is the functionality of the structure in land environments 284 and to provide the proper inner conditions and its security (e.g. mechanical impacts or 285 microbial invasion) during the embryo development. Theoretically, the optimal 286 morphology of a maximum volume with minimum eggshell perimeter is a sphere. 287 However, the great diversity of egg morphologies shows that other variables are playing 288 a key role in the final egg morphology and function. Several functional geometries are 289 known from living and extinct reptiles including circular to elongated eggs morphology 290 with a huge size variability. It should be remarked that the final egg is the meeting point 291 between the producer of the egg (e.g. pelvis morphology, size of the producer, cost for 292 calcite/aragonite production, potential to lay the egg in the substrate, to produce a nest 293 etc) and the requirements of the embryo to properly develop (quantity of required 294 calcite, inner egg microstructure, eggshell thickness, air transport between the 295 membrane and the eggshell, etc). Of particular interest, eggshell thickness probably 296 plays an important role, as revealed by Ar and colleagues (1979), who demonstrated 297 that eggs have a positive allometry regarding the eggshell thickness revealing that more 298 resources are required to build up the eggshell thickness as they grow. In particular, the 299 eggshells have a power law of 1.5 related to support the gravitational loads of their 300 mothers, but also to support other loads or impacts, as show the results herein presented. 301 Nesting-site selection is a common behaviour among turtles (Hays et al., 1995), 302 lizards (Trauth, 1983; Warner and Shine, 2008), crocodiles (Combrink et al., 2017), and 303 birds (Clark and Shutler, 1999). In all cases, female choses the most suitable soil to lay 304 its eggs, and often exploiting the surrounding resources in order to increase the chance 305 of the reproduction success. Depending on the nest type, clutch requirements or 306 incubation behaviour, the female shows preferences for softer or harder substrates. 307 Given that each organic farm can be build up on different substrates, here we simulated 308 this variability by using different substrate with distinct elastic modulus in order to 309 cover a wide spectrum of possibilities. 10 310 Some Asian and South American countries are known for being great consumers of 311 turtle eggs. Although there are some attempts in farming turtles, the current demand of 312 turtle eggs is mainly satisfied by poaching. It is out of our scope to discuss the ethic 313 basis or the risk for human health for consuming turtles’ meat or eggs, but the debate is 314 currently intense (D’Cruze, 2012). From a biological perspective, all turtles excavates 315 on the substrate to create a hole where lays the eggs, known as egg chamber. After that, 316 eggs are buried and covered with sand for incubation. Taking as example the herein 317 studied terrestrial tortoise of the genus Testudo, the female excavates the nest using the 318 hindlimbs, which in turn determines the maximum depth of the hole. The hindlimbs in 319 this genus range around 50-60 mm (Djurakíc et al., 2011) that in the light of our results 320 reveals that the eggs could support an impact two times higher before to be crashed (egg 321 cracking produced up to 110-120 mm). Interestingly, the tortoise eggs present an 322 important variability of sizes (Deeming and Ferguson, 1991; Deeming, 2004), with an 323 eggshell thickness comparable to the hens, although its mean size is clearly smaller (one 324 third part). It should be noted also that the turtle eggs must support the weight of the 325 substrate during the incubation period, although in this case the mechanic fatigue is 326 playing a role out of the scope of the present article. In fact, the resistivity of turtle egg 327 is remarkable being the one that supports the highest impact height (750-1000 mm) on 328 hard substrate (E=10000 MPa) before total failure (boundary between Type A and B) of 329 the parameterised egg types (see Fig. 4). 330 On the contrary, free-living ostrich commonly selects sandy dry riverbeds to nest 331 (Sauer and Sauer, 1966), thus tending to choose relatively hard substrate to lay the eggs. 332 Female sits directly in the substrate when laying, with a null or very minor nest 333 production (Bertram, 1992). Interestingly, these big and thick-shelled eggs are the first 334 to be broken as revealed by the results (between 50-75 mm to be crashed), 335 demonstrating the importance for the ostrich female to lay very close or directly placed 336 on the floor. In an intermediate position, the hens lay the eggs in a nest in a squat 337 position causing an impact of few centimetres. In this case, the eggs could support also 338 an impact higher than the ostrich but lower than the tortoises. The total rupture of the 339 hen egg (boundary between Type A and B failures) takes place between 30 and 40 cm 340 (Table 1 and Supplementary Information), which values are similar to the average 341 height of domestic hens in erected position. It is also worthy noting that the frontier 342 between safety (Type C) and failure (Type A) scenarios is established between 500 and 343 550 MPa, whatever the type of egg (see Fig. 4). These results led to speculate that these 11 344 values might represent a universal boundary for the safety integrity of all amniotes eggs, 345 but more analyses are needed to confirm this observation. 346 In sum, the parental decision of the optimal place for the laying is the result of 347 several key variables that could be divided on three different perspectives: A) from the 348 female producer and its (biomechanical) capabilities to create the nest or clutch, B) from 349 the egg to support the loads of the parent weight (or support the weight of the substrate), 350 and C) its capability to resist the impact on the substrate during the laying, considering 351 that different substrates (from softer to harder) could provide different conditions for the 352 nest (or clutch) and the egg incubation. 353 354 5. Conclusion 355 356 In organic egg production minimize the egg losses is a key factor that imply to 357 maximize the safety factors and reduce the number of failure eggs. In order to provide a 358 model helping to achieve those goals we evaluated the eggshell strength in a free-fall 359 scenario. Both experimental and computational tests were proved to be equally 360 effectives and useful, but computational simulation raises as a more preferable model 361 instead experimental tests given that it represents a non-destructive method. 362 From a more applicable perspective, the present study provides the first empirical 363 and mathematic results that allow establishing confident safety interval related to the 364 laying process. According to our models, tortoises and hens cannot lay their eggs up to 365 11-12 cm from the surface without compromising the structural integrity of the 366 eggshells. These values decrease dramatically for the large eggs of ostrich, which can 367 suffer severe damaging if the egg is laid up to 5-7 cm from the ground surface. These 368 results can have, indeed, a considerable impact in the organic egg industry, helping the 369 farmers to minimize the risk of losing eggs and therefore gaining more profits. 370 Furthermore, our models also set the bases for the development of a framework that 371 allow evaluating other scenarios without (i.e. wildlife conservation) compromising any 372 sample or even for assessing the biomechanical response of elements from the fossil 373 record (i.e. dinosaur eggs). 374 375 Acknowledgements 376 12 377 This research paper is a contribution to the CERCA program (Generalitat de Catalunya) 378 and was supported by the research projects CGL 2011-30069-C02-01 and CGL2017- 379 82654-P of the Ministerio de Ciencia e Innovación and Ministerio de Economía, 380 Industria y Competitividad, respectively, and the European Regional Development Fund 381 of the European Union. We are grateful to Alejandro Molina, Daniel Bracamonte, 382 Miguel Briz, and Sara Montalat for their valuable help in the experimental tests. Our 383 special thanks to Albert Martínez Silvestre and Joaquim Soler Massana from the 384 “Centre de Recuperació d’Amfibis i Reptils de Catalunya (CRARC)” for the tortoise 385 eggs and useful comments. J.F. is member of the consolidated research group 2017 386 SGR 86 GRC of the Generalitat de Catalunya. 13 387 References 388 Ar, A., Rahn, H., Paganelli, C.V., 1979. The avian egg: Mass and strength. The condor 389 81 (4), 331-337. 390 Attard, M.R.G., Sherratt, E., McDonald, P., Young, I., Vidal-García, M., Wroe, S., 391 2018. A new, three-dimensional geometric morphometric approach to assess egg 392 shape. PeerJ 6, e5052 393 Baker, D.E., 2002. A geometric method for determining shape of bird eggs. The Auk 394 199 (4), 1179–1186. 395 Bertram, B.C.R., 1992. The ostrich communal nesting System. Monographs in 396 Behaviour and Ecology. Princeton University Press. 206 pp. 397 Bowles, J.E., 2006. Foundation Analysis and Design. John Wiley Sons. McGraw-Hill. 398 807 pg. 399 Castilla, A.M., Herrel. A., Van Dongen, S., Furio, N., Negro. J.J., 2009. Determinants 400 of eggshell strength in endangered raptors. J. Exp. Zool. A. Ecol. Genet. Physiol. 401 311, 303–11. 402 Clark, R.G., Shutler, D., 1999. Avian habitat selection: pattern from process innest-site 403 use by ducks?. Ecology 80 (1), 272–287. 404 Combrink, X., Warner, J.K., Downs, C.T., 2017. Nest-site selection, nesting behaviour 405 and spatial ecology of female Nile crocodiles (Crocodylus niloticus) in South 406 Africa. Behaviour Processes 135, 101–112. 407 Coucke, P., Langenakens, J., De Baerdemaeker, J., Sas, P., 1994. Experimental modal 408 analysis on chicken eggs. Proc. 12th Int. Modal Anal. Conf. 12, 1258–1263. 409 Coucke. P., Jacobs, G., De Baerdemaeker, J., 1998. Comparative analysis of the static 410 and dynamic mechanical eggshell behaviour of a chicken egg. Proc. 23rd Int. Conf. 411 Noise Vib. Eng. 1469–1474. 412 D’Cruze, N. 2012. The Cayman Turtle Farm. A case for change. WASP. 24 pg WSPA 413 INTERNATIONAL Fifth Floor 222 Gray’s Inn Road London WC1X 8HB United 414 Kingdom. 415 Deeming, D.C., 2004. Reptilian Incubation. Environment, evolution and behaviour. 416 Nottingham University Press, Nottingham. 349 pp. 417 Deeming, D.C., Ferguson, M.W.J., 1991. Egg Incubation: Its effecsts on embryonic 418 development in birds and reptiles. Cambirdge Univeristy Press, Cambridge. 448pp. 14 419 Denys, S., Pieters, J.G., Dewettinck, K., 2003. Combined CFD and experimental 420 approach for determination of the surface heat transfer coefficient during thermal 421 processing of eggs. J. Food Sci. 68, 943–951. 422 Djurakíc, M., Djordjevíc, S., Bonnet, X., Tomovíc, L., Ajtíc, R., Golubovíc, A., 2011. 423 Sexual body size and body shape dimorphism of Testudo hermanni in central and 424 eastern Serbia. Amphibia-Reptilia 32 (2011), 445–458. 425 Duurmsa, D.E., Gallagher, R.V.,Priece, J.J., Griffith, S.C., 2018. Variation in avian egg 426 shape and nest structure is explained by climatic conditions. Scientific Reports 8, 427 4141. 428 Grant, G.S., 1982. Avian incubation: egg temperature, nest humidity, and behavioral 429 thermoregulation in a hot environment. Ornithological Monographs 30, 87pp. 430 Hays, G.C., MacKay, A., Adams, C.R., Mortimer, J.A., Speakman, J.R., Boerema, M., 431 1995. Nesting site selective by sea tortoises. Journal of marine biology association. 432 Of UK. 75, 667–674. 433 Juang , J-Y., Chen, P-Y., Yang, D-C., Wu S-P., Yen, A., Hsieh H-I., 2017. The avian 434 egg exhibits general allometric invariances in mechanical designe. Scientific 435 Reports 7, 14205. 436 Koncept Analytics. 2018. Market Research Report. Global Organic Food Market: 437 Industry Analysis & Outlook 2018-2022. 86 pg. 438 MacLeod, N., Bain, M., Solomon, S., Hancock, J., 2005. Failure mechanisms of hens’ 439 eggs. In: Proceedings of the International Conference on Fracture. Torino. 440 Mikhailov, K.F., 1997. Avian eggshells: an atlas of scanning electron micrographs. Br. 441 Ornithol. Club Occas. Publ. 3, 1–88. 442 Mulder, N-D., 2017. Outlook for the Global and EU industry: Shapen industry for 443 competitiviness. International Poultry Congress 2017. 444 Narushin, V.G., 1997. The Avian egg: Geometrical description and calculation of 445 parameters. Journal of Agricultural Engineering Research 68(3), 201–205. 446 Narushin, V.G., 2001. Shape Geometry of the Avian Egg. J. agric. Engng Res. 79 (4), 447 441–448. 448 Nedomová, Š., Buchar, J., 2013a. Ostrich eggs geometry. Acta Univ. Agric. Silvic. 449 Mendelianae Brun. 61, 735–742. 15 450 Nedomová, Š., Severa, L., Buchar, J., 2009a. Influence of hen egg shape on eggshell 451 compressive strength. Int. Agrophysics 249–256. 452 Nedomová, Š., Trnka, J., Dvořáková, P., Buchar, J., Severa. L., 2009b. Hen’s eggshell 453 strength under impact loading. J. Food Eng. 94, 350–357. 454 Nedomová, Š., Trnka. J., Buchar, J., Stoklasová, P., 2013b. Response of the Ostrich 455 Eggshell to Non Destructive Impact. Adv. Appl. Acoust. 2. 71–76. 456 Perianu, C., De Ketelaere, B., Pluymers, B., Desmet, W., DeBaerdemaeker, J., 457 Decuypere, E., 2010. Finite element approach for simulating the dynamic 458 mechanical behaviour of a chicken egg. Biosyst. Eng. 106, 79–85. 459 QYResearch Group. 2018 Global Organic Egg Market Insights, Forecast to 2025. 113 460 pp. 461 Rahn, H., Paganelli, C.V., 1989. The initial density of avian eggs derived from the 462 tables of Schönwetter. J. Ornithol. 130, 207–215. 463 Sauer, E.G., Sauer, E.M., 1966. The behaviour and ecology of the South African 464 ostrich. Living Bird, 5, 45–75. 465 Severa, L., Nedomová, Š., Buchar, J., Cupera, J., 2013. Novel Approaches in 466 Mathematical Description of Hen Egg Geometry, International Journal of Food 467 Properties, 16:7, 1472–1482. 468 Simeonovova, J., Buchar, J., 2002. On the identification of the eggshell elastic 469 properties under quasistatic compression. Acta Univ. Agric. Silvic. Mendelianae 470 Brun. v. 52(5) p. 471 Solomon, S.E., 1990. Egg and Eggshell Quality. Mosby. 472 Srivastava, R., Sahni, A., Jafar, S. A., Mishra. S. 2005. Microstructure-dictated 473 resistance properties of some Indian dinosaur eggshells: finite element modeling. 474 Paleobiology 31, 315–323. 475 Stoddard, M.C., Yong, E.H., Akkaynak, D., Sheard, C., Tobias, J.A., Mahadevan, L. 476 2017. Avian egg shape: Form, function, and evolution. Science 356, (6344), 1249– 477 1254. 478 Sumner, D.A., Gow, H., Hayes, D., Matthews, W., Norwood, B., Rosen-Molina, J.T., 479 Thurman, W., 2011. Economic and market issues on the sustainability of egg 480 production in the United States: Analysis of alternative production systems. Poultry 481 Science 90 (1), 241–250. 16 482 Trauth, S.E., 1983. Nesting habitat and reproductive characteristics of the lizard 483 Cnemidophorus sexlineatus (Lacerti1ia: Teiidae). The American midland naturalist, 484 109 (2), 289–299. 485 Trnka, J., Buchar, J., Severa, L., Nedomová, Š., Stoklasová, P., 2012. Effect of loading 486 rate on hen’s eggshell mechanics. J. Food Res. 1, 96–105. 487 Troscianko, J., 2014. A simple tool for calculating egg shape, volume and surface area 488 from digital images. Ibis 156 (4): doi.org/10.1111/ibi.12177 489 Upadhyaya, S.K., Cooke, J.R., Gates, R.S., Rand, R.H., 1986. A finite element analysis 490 of the mechanical and thermal strength of avian eggs. J. Agric. Eng. Res. 33, 57– 491 78. 492 Voisey, P.W., Hunt, J.R., 1974. Measurement of eggshell strength. J. Texture Stud. 5, 493 135–182. 494 Warner, D.A., Shine, R., 2008. Maternal nest-site choice in a lizard with temperature- 495 dependent sex determination. Animal behaviour, 75, 861-870. 496 Wriggers, P., 2002. Computational Contact Mechanics. Continuum. Wiley. 17 497 Eggshell Fall Height Impact velocity Maximum Egg Maximum Egg Mass [g] Thickness Type of failure [mm] [mm/s] length [mm] Width [mm] [mm] 1500 5424.94 54.37 44 58.3 0.47 A 1400 5240.99 57.75 43.36 61.6 0.49 A 1300 5050.35 56.14 44.32 61.2 0.44 A 1200 4852.22 55.85 44.69 61.3 0.41 A 1100 4645.64 56.59 44.3 61 0.41 A 1000 4429.45 55.72 43.16 57.6 0.44 A 900 4202.14 54.88 44.27 59.7 0.43 A 800 3961.82 52.56 42.98 56.5 0.47 A 700 3705.94 55.29 44.49 59.1 0.41 A 600 3431.03 55.48 43.76 59.3 0.41 A 500 3132.09 56.79 43.78 61 0.45 A 400 2801.43 55.26 44.71 61.5 0.42 A 300 2426.11 57.86 43.5 61.1 0.41 B 200 1980.91 55.29 44.54 61.6 0.44 B 150 1715.52 55.74 44.08 59.8 0.46 B 100 1400.71 56.86 42.87 57.9 0.47 B 50 990.45 59 42.94 59.9 0.4 C 498 Table 1. Results obtained in the impact of the hen eggshell 499 Fall Impact Maximum Maximum Mass [g] Eggshell Type of Height [mm] velocity Egg length Egg Width [mm] Thickness failure [mm/s] [mm] [mm] 1000 4429.45 40.5 29 11.8 0.41 A 1000 4429.45 34.5 29.9 18.4 0.44 A 750 3836.01 39 31.3 20.3 0.4 B 500 3132.09 33 29.7 17.6 0.43 B 250 2214.72 35.9 25.6 13.2 0.4 B 150 1715.52 32.7 29.7 15.5 0.4 B 125 1566.05 31.6 24.3 10.5 0.4 B 125 1566.05 31.8 23.3 10.3 0.36 B 110 1469.08 34.2 23.9 10.6 0.34 C 100 1400.71 31.8 29.3 16.5 0.41 C 100 1400.71 36.4 30.3 18.5 0.4 C 50 990.45 32.2 29.4 17.3 0.41 B 500 Table 2. Results obtained in the impact of the tortoise eggshell 501 502 503 504 505 506 18 507 Fall Height Impact Maximum Maximum Mass [g] Eggshell + Eggshell Type of [mm] velocity Egg length Egg Width membrane Thickness failure [mm/s] [mm] [mm] Thickness [mm] [mm] 1000 4429.45 152.2 129.9 1465.2 2.09 1.84 A 900 4202.14 154.1 125.2 1404.6 1.85 - A 500 3132.09 159.0 118.9 1330.4 1.88 - A 350 2620.50 165.7 126.0 1391.5 1.63 1.56 A 300 2426.11 156.8 131.6 1552.5 1.98 1.82 A 250 2214.72 162.8 124.3 1502.7 1.9 1.78 B 150 1715.52 159.0 118.9 1330.4 1.88 - B 100 1400.71 152.2 129.9 1465.2 2.09 1.84 B 75 1213.05 165.7 126.0 1391.5 1.63 1.56 B 50 990.45 156.8 131.6 1552.5 1.98 1.82 C 50 990.45 154.1 125.2 1404.6 1.85 - C 40 885.89 162.8 124.3 1502.7 1.9 1.78 C 508 Table 3. Results obtained in the impact of the ostrich eggshell. 19 509 FIGURE CAPTIONS 510 511 Figure 1. Scheme of the Transient dynamic Analysis performed for the tortoise, hen, 512 and ostrich eggs. Abbreviations: d-short axis; D-long axis; Hn-high of falling; s-shell; 513 su-shell units; t-thickness of the eggshell; tm- testaceous membrane; v-initial velocity. 514 515 Figure 2. Example of types A, B, and C of the casuistry of the crash of the egg in the 516 substrate for the hen (Gallus gallus) egg. 517 518 Figure 3. Von Mises stress distribution for computational cases defined via impact 519 velocities according to the type of fracture A, B, and C when the maximum value of 520 stress was reached. 521 522 Figure 4. Maximum Equivalent Von Mises stress versus height and defined in the FEA 523 model where the egg is dropped (A, B, C) and impact velocity depending on the type of 524 soil (D, E, F). Tests with performed by hen (Gallus gallus; A, D), tortoise (Testudo sp.; 525 B, E), and ostrich (Struthio camelus; C, F) eggs. Failure types are represented as a 526 continuous degraded colour transition, being the type A associated to “red”, type B to 527 “yellow”, and type C to “green”. 20