Introduction: The avian egg is a unique mechanical structure that must satisfy several conflicting demands; the shell must be strong enough to prevent it from being crushed during natural incubation by the weight of the bird, but yet must not be too strong to prevent the hatchling from breaking out of egg at the end of the incubation[1],[2]. Most eggshell studies described allometric equations based on experiments, but did not try to explain the origin of the relationship. Here, we study 12 bird species across three orders of magnitude in body mass by static compression test and finite element analysis to understand the principle behind the design of avian eggshell.
Materials and Methods: We carried out static compression test to measure the deformation of the egg under an applied force. The initial slope of the load-displacement curve is defined as the rigidity K of the eggshell. The profile, mass, width a, length b, and thickness are measured and used to create finite element models (Fig.1(b)). We carried out finite element analysis to obtain elastic modulus of each eggshell.
Results: We define a dimensionless number as a universal parameter to characterize the avian eggshells. is the egg weight; the shape index[3] indicates the influence of ellipsoidal shell geometry on K . Fig. 2 shows the dimensionless number for the 12 bird species with a body mass ranging from 64 g to 68920 g. Despite this wide range of mass, the dimensionless numbers, ranging from 75 ~ 214, remain nearly constant. We fit the data to a clear scaling law shown by the blue solid line .
Discussion: Is there a certain rule that the dimensionless number of every species would fall in a certain range in terms of the design of eggshells? In this study, we predict the lower bound of the dimensionless number base on the following scenario: When an eggshell is subjected to a force equivalent to the weight of its bird body, the maximum thickness that induces the eggshell to buckle is the critical thickness[4] which determines the minimum dimensionless number. Compression simulations are conducted on the new model with critical thickness to obtain the minimum dimensionless number. By plotting allometric relation from dimensionless numbers derived from the above scenario, we generate the lower bound (Fig. 2), which predicts that the dimensionless number of every eggshell in the world is not only roughly the same, but also bigger than this lower bound.
Conclusion: We conducted experimental and numerical studies to characterize the rigidity of avian eggshell. We discover that the dimensionless numbers for the bird species all fall within a narrow range, suggesting that there exists an “optimal design” of the eggshell as a result of a long journey of evolution. The design of eggshells plays a crucial role in the reproduction of bird species. To rationalize the experimental results, we perform numerical simulations to quantify elastic modulus of eggshells. We also propose a lower bound of the dimensionless number.
Fig. 1 Eggshells (a) Specimens (b) 3D eggshell models generated by simulation
Fig. 2 The dimensionless number versus bird body mass. (The blue line represents the experimental result; the red line represents the prediction of the lower bound)
Ministry of Science and Technology 104-2815-C-002 -141 -E
References:
[1] A. Ar, H. Rahn, and C. V Paganelli, “The avian egg: mass and strength,” Condor, vol. 81, no. May 2015, pp. 331–337, 1979.
[2] Kunt Schmidt Nielsen, Scaling: Why is animal size so important? Cambridge University Press, 1984.
[3] A. Lazarus, H. Florijn, and P. Reis, “Geometry-Induced Rigidity in Nonspherical Pressurized Elastic Shells,” Phys. Rev. Lett., vol. 109, no. 14, pp. 1–5, Oct. 2012.
[4] L. D. Landau and E. M. Lifshitz, Theory of Elasticity, 3rd ed. Elsevier, 1986.