{'fraction_non_alphanumeric': 0.02511669259835519, 'fraction_numerical': 0.00836296954878862, 'mean_word_length': 4.865873533246415, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Mechanical Design Principles of Avian Eggshells for Survivability

Biological materials exhibit complex structure-property relationships which are only beginning to be elucidated. Understanding the underlying physical mechanisms of the structure-property relationships is the key to designing bioinspired materials. The eggshell is an excellent example because many design trade-offs are well balanced by its seemingly simple but highly evolved structures. The first design trade-off: to break an eggshell, the internal breaking load needed is much lower than the external breaking load. The reason is that the dome-shaped structure leads to a special stress distribution which inhibits the propagation of the first crack. The second design trade-off: the eggshell is tough during the incubation but weakened at the time of hatching. The membrane of the eggshell significantly improves the toughness by bridging the primary crack and creating more secondary cracks. Moreover, the moisture content-controlled property of the membrane makes the toughness of the eggshell tunable. The third design trade-off: the eggshell must be thick enough to withstand the external forces, but the supply of the building material calcium is limited. The proposed three-index model indicates that the driving force in the evolution of eggshell thickness could be the balance between the protection and nutrient consumption to maximize survivability. The general methodologies presented here hold great potential for the development of eggshell-inspired structural materials that can be used in sports safety applications and the packaging industry.


INTRODUCTION
Egg-laying reproduction was one of the most successful evolutionary innovations that enabled vertebrate animals to leave the water and conquer dry land ~360 million years ago [1].
Calcified eggshell is produced by all modern birds and it is thought to have played an important part in the survival of birds through the Cretaceous-Palaeogene extinction [2]. The calcified eggshell is an important structure to the developing embryo for three reasons. It protects the embryo from the microbial and physical environment. Additionally, it regulates the water and gas exchange during the incubation. The calcified eggshell also serves as a calcium store and provides calcium for the embryo's skeleton growth [3]. The diverse functions of the eggshell result from its hybrid organic-inorganic composite materials and the sophisticated structures that developed in million years of evolution.
Avian eggshell is a complex bioceramic comprising a calcium-carbonate mineral constituent (~95% by weight) and an organic matrix (~3.5% by weight) [4]. For different bird species, the eggshell exhibits a variety of sizes and thicknesses. Also, the shape of the eggshell varies dramatically across different species: conical in shorebirds, spherical in owls, and elliptical in hummingbirds [5]. At the nanoscale, the avian eggshell is composed of two layers of collagenized fibrous eggshell membranes and multiple calcified layers with different crystal orientations and nanostructures [6]. Extensive studies have been carried out to explain the structure-function relationship of eggshells. For instance, egg-shape variation could be attributed to the flight strength and efficiency of birds [5]. The well-organized multi-layered nanostructures of the eggshell change during the incubation, providing tunable mechanical properties to the eggshell and therefore weakening the eggshell for hatching [7].
Nacre and exoskeleton of lobster are also bioceramics that have the function of protecting the living organisms inside. Their extraordinary mechanical properties, such as high strength and toughness, originate from well-designed microstructures [8,9]. For these bioceramics, the design objective is simple and straightforward: making the structure stronger and tougher. However, the eggshell has several design trade-offs that must be balanced. The eggshell must be sufficiently strong to resist the fracture caused by the load applied from the outside. Meanwhile, the eggshell must be easy to break from inside. In addition, the eggshell must be tough to delay the possible crack growth during the incubation, but the toughness must be weakened at the time of hatching.
Moreover, the eggshell must be thin to save the calcium that is limited in the natural diet of birds.
But the eggshell cannot be too thin to resist physical challenges from the environment. These trade-offs have already been balanced in the million years of evolution and the balance strategies are hidden in the sophisticated structures of the eggshell. However, little is known about the underlying mechanical design principles of the eggshell structure to balance the trade-offs. Here, we use experimental and computational approaches to elucidate such design principles in particular with regard to three key features of the eggshell structure: the dome-shaped shell, the membranes attached to the eggshell, and the exquisitely evolved thickness as shown in Figure   1A. More specifically, we will answer three questions: (i) Why the dome-shaped eggshell is hard to break from the outside but easy to break from the inside? (ii) How do membranes affect the toughness of the eggshell and make the toughness tunable? (iii) What is the possible driving force in the evolution of eggshell thickness? The general methodologies presented here hold great potential for the development of biologically inspired structural materials that can be used in sports safety applications and the packaging industry.

RESULTS AND DISCUSSION
Firstly, to access the breaking forces of the eggshell under the external load applied from the outside surface as well as the internal load applied from the inside surface, we performed two types of compression tests (Figure 2A and Figure 3A) on the half eggshell and compared the load-displacement curves. The typical load-displacement curves from two types of tests are shown in Figure 1B. The breaking force of the external compression test is much larger than the breaking force of the internal compression test. This large difference can be attributed to the two different fracture patterns induced by the dome-shaped structure. What are the two different fracture patterns and how do they affect the breaking force will be discussed later. Also, to evaluate the effect of membranes on the toughness of the eggshell, compression tests are performed on two types of eggshell samples: one with the membrane attached to the eggshell and the other one with the membrane removed. Two typical load-displacement curves from the two types of samples are shown in Figure 1C. The eggshell sample that has no membrane almost completely lost the load-carrying capacity after the breakage. However, the eggshell sample with the membrane maintains a certain amount of load-carrying capacity during the whole loading process. How the membranes dramatically increase the toughness of the eggshell will be discussed later. In the end, the data on the mass of eggs and thickness of eggshells from different species of birds are collected from previous research [10][11][12] and plotted in Figure 1D. The relationship between the shell thickness and mass tends to follow a special pattern (dashed line).
In the following discussion, the underlying mechanical design principle of this special pattern will be uncovered and the possible driving force in the evolution of eggshell thickness will be presented. (A) Three key features of the eggshell structure that balance the design trade-offs: the domeshaped shell, the membranes attached to the eggshell, and the exquisitely evolved thickness. (B) The typical load-displacement curves of eggshells with internal and external load acting on them. The breaking force of the external compression test is much larger than the breaking force of the internal compression test. (C) Two typical load-displacement curves from eggshell samples with and without membranes. The eggshell sample with the membrane is much tougher than the other one. (D) Data points on the mass of eggs and thickness of eggshells from different species of birds. The relationship between the shell thickness and mass tends to follow a special pattern.

Dome-shaped eggshell and two different fracture patterns
Eggshells are made up of 95% calcium carbonate and can be considered a brittle material.
For brittle material, in general, once the maximum principal stress exceeds the tensile strength, the small cracks initiate. Then these small cracks propagate rapidly and lead to a sudden catastrophic failure. Therefore, based on the theory of brittle fracture, a hypothesis can be made as follows. For eggshells in the compression tests, the cracks initiate at the position where the maximum principal stress (σ1) reaches tensile strength, then the cracks propagate rapidly and lead to catastrophic failure. To verify this hypothesis, FE simulations are performed to obtain the stress distribution on the eggshell at the breaking point. Note that, the geometric parameters, material properties, and the breaking load applied to the FE model are measured experimentally.
The detailed information can be found in the supplemental information.
For eggshells with internal compression, the cracked sample from the test is shown in Figure 2B. On the sample, 4 vertical cracks initiate from φ = 90° and propagate to φ ≈ 80°. The stress distribution results from the FE simulations are shown in Figures 2C and 2D. Note that, the stress varies through the thickness direction, therefore, the stress distributions on both the inside surface and outside surface are shown respectively. With the breaking internal load (2.6 N) applied to the model, the highest maximum principal stress can be found on the outside surface of the eggshell at φ = 90°. The value of the highest maximum principal stress at that point is 25.2 MPa which is slightly higher than the tensile strength (19.9 MPa) of the eggshell. According to the experimental and simulated results, the fracture follows the following process: with the internal load increasing gradually, maximum principal stress on the outside surface of the eggshell at φ = 90° exceeds the tensile strength firstly, then, cracks at that position initiate and propagate rapidly and lead to a sudden catastrophic failure. Conclusively, the hypothesis we made before is verified by the observed fracture pattern in the test with internal load applied to the eggshell. Secondly, the value of the highest maximum principal stress at that point is 480.0 MPa which is almost 24 times larger than the tensile strength (19.9 MPa) of the eggshell. If the crack initiated here leads to the failure, the breaking force would be much lower than the measured breaking force of 49.0 N. Three interesting findings from the results may provide a reasonable explanation for this paradox. Firstly, with the breaking internal load (49.0 N) applied to the model, the position that has the highest maximum principal stress on the outside surface is located at φ ≈ 80°, while the crack initiation point observed from the test is at φ ≈ 75°. Secondly, the value of the highest maximum principal stress at that point is 17.3 MPa which is just slightly lower than the tensile strength (19.9 MPa) of the eggshell. Lastly, at that point, the maximum principal stress 1  equals the stress component   which leads to a vertical crack, while the crack observed from the test is vertical. All these findings lead to an assumption that two cracks initiate at different locations and different load magnitudes. One crack initiates on the inside surface at φ = 90°, and the load is way lower than the breaking load when its initiation. More importantly, this crack does not lead to catastrophic failure. The other crack initiates on the outside surface at φ ≈ 80°, the load is the breaking load when its initiation. This crack propagates rapidly after initiation and leads to catastrophic failure. This assumption seems plausible, however, an important question needs to be answered: why the first crack does not lead to catastrophic failure? To answer the question, the detailed stress distribution in the through-thickness direction near φ = 90° is obtained from the FE simulated results as shown in Figure 4A. The gray region which has the highest maximum principal stress is very narrow and surrounded by the lower maximum principal stress region. Especially in the thickness direction, when the crack propagates upward, it will almost immediately reach the compression-dominated region and could stop there. In fracture mechanics, the energetic stability determines whether a crack propagation is 'catastrophic' or 'controlled'. More specifically, if 0 dK dC  the crack is stable, and if 0 dK dC  , the crack is unstable and leads to catastrophic failure. Where K is the stress intensity factor and C is crack length. Therefore, a 3D FE model with a semi-elliptical surface crack on the inside surface is created to calculate the relationship between stress intensity factor and crack length as shown in Figures 4B and 4D. The stress intensity factors at two crack tips A and C are calculated to represent the whole crack front. Two crack aspect ratios are considered which are a/c = 3 and 6. The results are shown in Figures 4E and 4F, where all four curves follow the same 2-stage pattern. In the first stage, 0 dK dC  , the crack is small and unstable. The crack propagates rapidly until it reaches the peak of the curve and into the second stage. In the second stage, 0 dK dC  , stress intensity decreases as the crack length increases. The crack propagates slowly and stably and finally stops. In a previous study [13], a micro-crack (not detectable by routine visual inspection) at a load less than breaking force was found on the inside surface of the eggshell at φ = 90° as shown in Figure 4C. This experimental result strongly supports the simulation result. To conclude, the eggshell with external load has a special two-crack fracture pattern. The first crack initiates on the inside surface at φ = 90° and rests very soon while the crack is very small. The second crack initiates on the outside surface at φ ≈ 80°. It propagates rapidly after initiation and leads to catastrophic failure. The rest of the first crack can be attributed to the dome-shaped structure-induced special stress distribution at the loading point.
Moreover, to further verify this conclusion, a validation test is performed as shown in Figure   S4. The basic idea of the validation test is to apply geometric imperfections at the crack initiation positions to create stress concentration and accelerate the crack initiation. Two types of samples with geometric imperfections are prepared by drilling small holes at φ = 90° (first crack) and φ ≈ 80° (second crack) respectively on the eggshell. For samples with a drilled hole at φ = 90°, the drilled hole does not affect the breaking force. However, the breaking force of samples with a drilled hole at φ ≈ 80° is much lower than the average breaking force of 49.0 N. This validation test further proves that the second crack leads to the failure of the eggshell. (A) Maximum principal stress distribution in thickness direction near loading point. The gray region which has the highest maximum principal stress is very narrow and surrounded by a lower maximum principal stress region. (B) Schematic diagram of a semi-elliptic surface crack on the inside surface. Stress intensity factors will be calculated at crack tips A and C to determine if the crack is stable. (C) Micro-cracks near loading point at loads less than the breaking load from a previous study [13]. Experimental observations indicate this first crack does not lead to a catastrophic failure. (D) FE model and detailed mesh of a semi-elliptic surface crack on the inside surface. (E-F) FE simulated results of stress intensity factors at crack tips A and C respectively for a propagating crack. The second stage, 0 dK dC  , indicates that the crack propagates slowly and stably and finally stops.
Using the validated two-crack fracture pattern proposition, the breaking force of eggshell with load acting on different positions is predicted as shown in Figure S5. The predicted results match well with the experimental results. The results can also explain the reason why the eggshell is easier to break on the equator than on the two ends. Moreover, the effect of the two key geometric parameters, thickness, and curvature of the dome, on the breaking load of the thinshell structure is systematically studied as shown in Figure S6.

Membrane and the toughness of the eggshell
The eggshell membrane is a collagenized fibrous tissue attached to the inside surface of eggshells. It has two layers of membranes: a thin inner membrane and a thicker outer membrane.
The total thickness of the two layers of membranes is approximately 0.1mm [14,15]. The eggshell membrane has been considered to have 3 essential functions for avian reproduction. It provides a structural foundation for the calcification of the eggshell [16]. It controls the gas exchange and helps the embryo breathe inside of the eggshell [17]. Also, it prevents bacterial penetration during the hatching process [18]. All these functions are based on the eggshell membrane's special structural, chemical, and physical properties, while the other hand, these properties have been utilized by many researchers in various fields. Eggshell membrane has been used as a template to fabricate hierarchically ordered microporous networks composed of oxide materials [19]. It also has been used as a polysulfide reservoir for highly reversible Li-S batteries [20]. However, the mechanical property of the eggshell membrane seems to be overlooked and few studies uncover the significant effect of eggshell membrane on the toughness of the eggshell.
To study the effect of membrane on the toughness of the eggshell, two groups of eggshell samples are prepared. For the first group of eggshells, the membranes are removed from the eggshell. The second group is the control group, and the membranes are still attached to the inside surface of the eggshell. The results of the two groups of samples from the compression tests are shown in Figure 5A. The breaking forces for the two groups of eggshells are nearly the same. The membrane has no significant effect on the strength of the eggshell considering it is so soft and thin compared to the eggshell.
Two distinctly different types of after-breakage force-displacement curves of the two groups of eggshell samples are shown in Figure 5B. Eggshells that have no membrane almost completely lose the load-carrying capacity after the breakage. However, eggshells with membranes maintain a certain amount of load-carrying capacity during the whole loading process. The initiation and propagation of cracks on the two groups of eggshells at different displacements are shown in Figures 5C and 5D. The effect of membrane on the toughness of eggshells can be summarized into two aspects.  Figure S7). Just like organic matrix bridging in nacre [21] and collagen-fibril bridging in bone [22]. Eggshell membrane can bridge a developing crack and carry the load that would drive the propagation of the crack. It lowers the stress intensity factor at the vertical crack tip and stops the propagation of the primary vertical crack. Eggshells that have no membrane or dry membrane almost completely lose the load-carrying capacity after the breakage. However, eggshells with wet membranes maintain a certain amount of load-carrying capacity during the whole loading process. (C-D) Crack initiation and propagation of eggshell with and without membrane attached. For eggshell with membrane, the primary vertical crack stops halfway to the bottom. While for eggshell that has no membrane, the primary vertical crack propagates all the way to the bottom. Because of the membrane, the high stress that cannot be effectively relieved by the primary crack, the membrane changes the stress distribution and creates many secondary cracks that improve the toughness and energy absorption ability of the eggshell.
Secondly, because the membrane stops the propagation of primary vertical cracks, the overall stress distribution and secondary cracks are changed accordingly. After the primary vertical crack, multiple vertical and circumferential secondary cracks alternately initiate and propagate on the eggshell. These two types of cracks are driving by two different types of stress components:   that controls the vertical crack and   that controls the circumferential crack.
These two stress components are competing during the entire loading process. The membrane dramatically affects the competing results as shown in Figures 6A and 6B. For eggshell with membrane, the three drops in the force-displacement curves represent three different crack initiations. From points 1 to 3, the winners of the competition are   ,   , and   . Therefore, the three cracks initiated sequentially are vertical crack, circumferential crack, and vertical crack.
However, for the eggshell that has no membrane, the stress distributions are completely different after the initiation of the first vertical crack. Therefore, the three cracks initiated sequentially are vertical crack, vertical crack, and circumferential crack. Moreover, the simulated results also show that eggshell without membrane has much lower load-carrying capacity.
Except for the static compression test, dynamic impact tests are also performed on these two types of eggshells as shown in Figures S8A, S8B. An impactor with a speed of 2.7m/s hits the eggshell, and this impact process is captured by a high-speed camera. The whole impact process lasts for about 6ms. The eggshell with the membrane remains in one piece after the impact while the eggshell has no membrane breaks into many pieces. Using the recorded pictures, the velocity change of the impactor, as well as the reaction force-displacement curve, are calculated and shown in Figures S8C and S8D. Clearly, the eggshell with the membrane absorbs more impact energy. In conclusion, because of the membrane, the high stress that cannot be effectively relieved by the primary crack, the membrane changes the stress distribution and creates many secondary cracks that improve the toughness and energy absorption ability. The membrane makes the eggshell tougher and protects the embryo inside, but it also leads to a problem: does the membrane impede the hatching process and make it much hard for the chick to break out from the eggshell? Previous studies revealed many physical and biological transitions of the eggshell during the incubation process that will help the chick to break the eggshell. For example, the incubated eggshell is 7.3% thinner than the unincubated eggshell [23], and the pore density of the eggshell increases by about 12% [24]. During the incubation, avian eggs lose approximately 20% water, and the membranes dry out accordingly [25]. However, no one realized that the change of moisture content of the membrane is also an evolutionary strategy that makes the eggshell weaker during the hatching process. To uncover the effect of the Why is the toughness of the eggshells with dry and wet membranes distinctly different?
That is because the moisture content can dramatically affect the mechanical property of the membrane. Dry and wet membrane specimens ( Figure S9A) are prepared and tested; the stressstrain curves are shown in Figure 6C. The failure strain of the dry membrane is about 0.03 but the failure strain of the wet membrane is about 0.3, 10 times higher than the dry membrane.
Therefore, when the eggshell breaks, the dry membrane breaks too, but the wet membrane does not break with the eggshell, instead, it provides a closure force along the crack path and stops the crack propagation.
Three typical mechanical properties of the three groups of eggshells: stiffness (E, slope of the force-displacement curve), strength (S, the breaking force), toughness (T, the area under the static force-displacement curve) are calculated and compared in Figure 6D. The results indicate that the wet eggshell membrane with a large breaking strain greatly increases the toughness of the eggshell. During the incubation period, the moisture content of the membrane decreases, and the breaking strain decreases correspondingly which leads to a dramatic decrease of the toughness of the eggshell and helps the chick to break out of the eggshell easily.

A possible driving force in the evolution of eggshell thickness
For different species of birds, the mass of the eggs and the thickness of the eggshells are different. The mass of the ostrich (Struthio camelus) egg is about 1500g and the thickness is around 2 mm [26]. Where the mass and thickness of chicken (gallus domesticus) egg are about 50g and 0.38mm respectively [27]. Why the eggshell of chicken is not thicker, i.e., 0.6mm, or thinner, i.e., 0.2mm? From an evolutionary standpoint, the process of evolution is the survival of the fittest, so 0.38mm could be the best thickness in natural selection. Many efforts have been made trying to explain such natural selection. A general explanation is that the eggshell is created to satisfy opposing goals. It must be thick enough to withstand the external forces imposed on it during the incubation. On the other hand, it cannot be too thick to waste the building material or too thick to inhibit the hatchling to break its way out [10,28]. To uncover what parameters that may affect the eggshell thickness, further studies have been carried out.
These parameters including the egg mass [29], incubator mass [30], breeder age [31]. These studies provided some regression equations to describe the relationship between those factors and eggshell thickness. However, these studies summarized the phenomenon observed in many experiments, but still, they didn't touch the essence of what is the driving force in the evolution of eggshell thickness.
The thickness of the eggshell seems to be controlled by a balance between protection and nutrient consumption, to maximize survivability. Reproduction is energetically costly and requires a sufficient supply of macronutrients and micronutrients. Among these, calcium is probably the most limiting micronutrient [32]. To quantitatively study the contradictory relations between the protection and calcium consumption, a three-index model is created as shown in Figure 7A. The first one is the production index, PI, which is defined as: Where t is the thickness of the eggshell; egg V the volume of the egg which is considered as the limited calcium that can be used to produce eggshell; and eggshell V is the volume of the eggshell itself and it is a function of thickness. The thicker the eggshell is, the fewer eggs can be produced by the limited building material. For example, as shown in Figure   7A, with a small thickness, ts, 20 eggs are produced, but with a larger thickness, tl, only 5 eggs are produced. More eggs produced doesn't necessarily mean that more offspring can survive because eggs are fragile, and they could be broken under environmental loads. So here, the second parameter is introduced, BI, breakage index which defines how many eggs break under the environmental loads. In our example, for eggs with small thickness, ( ) 14 s BI t = , which means 14 eggs break under the environmental load. The last parameter is the survival index, SI PI BI =−, which indicates how many eggs survive. Among all three thicknesses, medium thickness, tm, has the highest survival index and wins the natural selection. In conclusion, our hypothetical model assumes that the driving force in the evolution of eggshell thickness is to maximize the survival index. Therefore, the optimal thickness of eggshells is the thickness that has the maximum survival index. To validate this hypothetical model, the relationship between SI and eggshell thickness needs t to be uncovered.
Considering SI is the difference between PI and BI, and PI can be easily obtained using Equation 1, determining BI is the key to finding the optimal thickness. BI can be calculated with the following 3 steps. The first step is to determine the relationship between the breaking force,  Figure 7B.
After having the function F(t) that describes the relationship between breaking force and thickness. The second step is to calculate the breaking possibility using F(t). Here we assume the maximum environmental loads of different eggs follow a normal distribution which can be described using the following equation: Where y is the probability density, x is the maximum environmental load,  and  can be calculated using the following equation: Where m is the mass of the egg, g is the gravitational acceleration, and k is a load factor which is set as 100 here. x0 can be solved using the following equation: For instance, assume an eggshell has a breaking force of 60 N in a mild environment, E = 0.9, the value of breaking probability p is the area of the green region. In a harsh environment, E = 0.5, the value of breaking probability p is the area of the green, blue and red regions together.
To this point, the breaking index BI can finally be calculated: With E = 0.8, the three indexes are calculated for eggshells with different thicknesses from 0.2 mm to 1.0 mm as shown in Figure 7D. Production index and breaking index both decrease as the thickness increases, and the breaking index decreases much faster. The survival index increases first and then decreases, which creates a peak point. The thickness corresponding to the peak value of the survival index is t ≈ 0.4 mm which is very close to the average thickness of eggshell, t ≈ 0.38mm, observed from the tests.
To further validate the hypothetical model, survival indexes for different environmental factors are calculated and the results are shown in Figure 7E. The red line connects all highest survival index for different environmental factors is the corresponding optimal thickness. The optimal thickness is located in a very narrow range, from 0.34mm to 0.44mm. Also, the optimal thickness is thinner for mild environmental conditions and thicker for harsh environmental conditions which is reasonable.
All the discussion above is focused on chicken eggshells, but the hypothetical model itself should also be able to apply to other species of birds. Therefore, the relationship between optimal thickness and the mass of the egg is calculated using the discussed calculation procedure. The gray region in Figure 7F represents the optimal thickness for different environmental factors from 0.5 to 0.99. Evidently, the majority of the observed data points are in or very near to the gray region which strongly validates the 3-index hypothetical model. To conclude, the possible driving force in the evolution of eggshell thickness is the balance between protection and calcium consumption, to maximize survivability

CONCLUSION
Due to its special function, the eggshell has three main trade-offs in the reproduction which are: 1) the eggshell should be easy to break from the inside but hard to break from the outside; 2) the eggshell should be tough during the incubation but be weakened at the time of hatching; 3) the eggshell should be thin to save the building material but not too thin to be easily broken. In this paper, we have investigated the structure of the eggshell and the underlying mechanism that balance the trade-offs. More specifically, we studied the crack initiation and propagation of the eggshell under two different types of loads, the internal and external loads.

Specimen Preparation
The eggs were obtained from a retail outlet, described as "Large Grade AA Cage Free Organic eggs". The external dimensions (a, b, e) of the eggshell were measured before testing and the thickness t was measured after testing.
To obtain the compression specimens, the eggs were cut in half using a hand drill with a diamond saw rotating disc. The cutting edge was then sanded with 1000 Grit Sandpaper. The half eggshell specimens were washed with distilled water and checked under the light.
Specimens with any visible cracks were discarded. Three types of specimens were prepared.
Specimens with wet membranes attached to them were stored in water. Specimens with dry membranes were dried in air for 24 hours. For specimens without membranes, the membranes were removed using clamps.
To obtain the membrane specimens for tensile tests, the eggshells were cut in half first.
Then the eggshells were rinsed in 1 M HCl to dissolve CaCO3, leaving the organic shell membranes. After the shell membranes were thoroughly washed with water, the dogbone shape specimens were cut out of the membrane in the latitudinal direction. Half of the specimens were stored in water and the other half were dried in air for 24 hours.

Mechanical Testing
To capture the mechanical response of the eggshell and membrane, compression test and uniaxial tensile tests were performed using an MTS mechanical tester (C43) with 1kN and 100N load cells. Both the compression test and tensile test were conducted in a quasi-static regime.
The displacement rate for the compression test was 0.5 mm/min and the strain rate for the tensile test was 0.001/sec. The toughness was measured from the load-displacement which is defined as the area under the force-displacement curve which represents the energy absorbed and dissipated. This definition is wildly used in rigid biological composites. Images of the specimens during the loading procedures were captured at a rate of 1 FPS.
The impact tests were performed on self-assembled impact test equipment as shown in Figure S10. The impactor moves downwards in the guide rail. Before contacting the eggshell, the trigger activates the high-speed camera's shutter. The impact duration is then recorded by a high-

ACKNOWLEDGMENTS
The author would like to thank Professor Toshio Nakamura for his helpful advice on various technical issues in this Paper.

DECLARATION OF INTERESTS
The authors declare no competing financial interests.  Figure S2. The geometric parameters of the tensile specimen of the eggshell that used for obtaining the tensile strength. Note that, unlike the regular tensile specimen which is flat, the specimen used here is curved. The curvature shape makes the stress distribution more complex and therefore the tensile strength cannot be calculated directly from the experimental test results. Therefore, a FE model of the curved tensile specimen is created and the result of the tensile strength is obtained from both the breaking load from the experimental test and the stress distribution from the FE simulation. Figure S3. Extra external compression test with a spherical indenter. The two types of compression tests described in the main manuscript used indenters with different shapes: spherical indenter for internal compression test and flat indenter for external compression test. It is reasonable to doubt that the shape of the indenter may affect the breaking force. However, the breaking force obtained from the external compression tests with a spherical indenter is almost the same as the breaking force obtained from the external compression tests with a flat indenter in the main manuscript. So, the shape of the indenter does not have an apparent effect on the test results of breaking force.   Note that the shape of these models can be described by the equations shown in Figure S1 but e is set to 0 for all models. Four different breaking forces are calculated in the FE simulation. F1 is the external breaking force at the top. F2 is the internal breaking force at the top.
F3 is the external breaking force at the equator. F4 is the internal breaking force at the equator.    Figure S10. Self-assembled impact test equipment.