It is important to study the properties and mechanics of egg drop impacts in order to reduce egg loss during processing and logistics and to provide a basis for the protective packaging of egg products. In this paper, we present the results of our study of the effects of the structural parameters on the mechanical properties of an egg using a finite element model of the egg. Based on Fluid-Solid coupling theory, a finite element model of an egg was constructed using ADINA, a finite element calculation and analysis software package. To simplify the model, the internal fluid of the egg was considered to be a homogeneous substance. The egg drop impact was simulated by the coupling solution, and the feasibility of the model was verified by comparison with the experimental results of a drop test. In summary, the modeling scheme was shown to be feasible and the simulation results provide a theoretical basis for the optimum design of egg packaging and egg processing equipment.
An egg has the shape of an approximate ellipse whose curvature radius is larger at one side than at the other. Consequently, they are liable to roll around and are extremely fragile. In order to avoid the loss of eggs due to collisions or large vibrations, it is necessary to place the eggs in cushioning materials while they are collected, graded, packaged, processed, stored, transported, and traded. Egg damage is primarily caused by dynamic loading, such as drops, collisions, and vibrations. Statistics show that the damage rate of eggs during processing is 3.7%, and the loss rate caused by eggshell cracks is 6–8%, although some microcracks remain invisible. When the losses from other causes are also taken into account, such as deterioration due to improper storage, the economic loss to China amounts to RMB 500 million yuan every year [
Several studies have been conducted at home and abroad on the static characteristics of eggs. For example, as early as 1986, Upadhyaya et al. [
In terms of the dynamics, Liu and Wu [
In order to analyze the characteristics of the natural frequency of a watermelon, Nourain et al. [
The majority of the existing theoretical models or numerical simulations established for the dynamic characteristics of eggs failed to take the fluid within the egg into consideration. Instead, they have paid an excessive amount of attention to the eggshell and static compression and dropping force analyses. In this paper, we describe the results of an in-depth finite element analysis into the mechanical properties of the egg and its associated damage mechanisms in order to provide more comprehensive theoretical guidance on solving egg damage problems in practice.
An egg consists of an eggshell, eggshell membrane, egg white, and egg yolk, of which the eggshell and membrane together represent 10–13%, the egg white represents 55–60%, and the yolk represents 32–35% of the total weight of the egg.
The typical parameters used for modeling an egg are the weight: 50 g, eggshell thickness: 0.35 mm, diameter of the major axis: 58.5 mm, diameter of minor axis: 45 mm, and egg-shape index
The shape of egg amending in Pro/E.
The eggshell and cushioning material models were established in the ADINA Structure module using the shell element. The eggshell and cushioning material were divided into 3,575 and 1,600 units, respectively. The internal fluid model of the egg was built using the ADINA-CFD 3D-fluid element, and the internal fluid was divided into 53,760 units. In order to simplify the model, internal fluid was assumed to be a homogeneous substance. The finite element model is shown in Figure
The mapped meshing of the structure and fluid models.
Parameters have been defined for the cushioning material and eggshell. A universal biomechanical test machine (INSTRON 3369) was used to test the mechanical properties with an elastic modulus of 0.578 MPa. According to the calculation standards for a curved surface published by the American Society of Agricultural Engineers [
Data regarding the eggshell, egg yolks, and protein.
Specific heat |
Density |
Viscosity | |
---|---|---|---|
Egg shell | 888 | 2300 | — |
Egg yolks | 3560 | 1035 | 1.6–4.8 × 10−3 |
Egg protein | 3560 |
|
3.12–8.9 × 10−3 |
The contact between the eggshell and the plane of the cushioning material was defined, the internal curve of the eggshell was defined as a fluid-solid coupling boundary, the four edges of the cushioning material were fully constrained, the internal fluid was simplified by assuming it was homogeneous, and the fluid–structure interaction (FSI) fluid-solid coupling boundary and free surface were taken as boundary conditions.
Fresh eggs with sizes and shape values similar to the model characteristics were selected for the experiment. After measuring the numerical values of major and minor axes of those eggs, there were five eggs whose egg-shape indices
The equipment used for the experiment included a drop test machine, PCB tri-direction transducer, and TP3 data acquisition system (LANSMONT Corporation).
As shown in Figure
Principles of the test method.
A finite element model describing an egg drop from a height of 15 cm was established to simulate the stress changes while the egg dropped and to compare it with the experimental results.
When solutions to the fluid and structure models were found, the fluid-solid coupling boundary conditions in both models were recovered. In terms of the boundary conditions of the structure, the internal surface of the eggshell was assumed to be a fluid-solid coupling interface, the surface that contacted the eggshell was selected as a fluid-solid variable section, and the free surface was selected as the boundary condition for the liquid level at which the fluid model contacted the air chamber. Those two models were coupled into the ADINA fluid-solid coupling solver (ADINA-FSI) in order to find solutions and to determine the nephogram of the stress on the eggshell, as shown in Figure
The nephogram of the effective stress of the egg on the EPE from a height of 15 cm.
As shown in Figure
The meridional displacement nephogram of the eggshell.
Figure
Contrast of the simulation and test results.
The value of the initial drop speed was adjusted by changing the boundary conditions in the structure and fluid models. The simulation results of the eggshell when the egg was dropped and the speed value when the eggshell failed were calculated by the FSI solver when the drop height limit for the egg and the EPE cushioning material could be obtained.
When a drop test was conducted at a height of 40 and 45 cm, the corresponding initial impact speeds in the finite element model were 2.8 m/s and 2.97 m/s, respectively. The simulation result indicated that the egg did not crack when it was dropped at an impact speed of 2.8 m/s, whereas the egg cracked when the speed reached 2.97 m/s. When the initial impact speed was lowered to 2.89 m/s, which was the initial speed of the model, there was nonconvergence after the 15th iteration during the solution procedure, which indicated that the egg was subject to a maximum impact when it contacted the EPE cushioning material from a corresponding height of 42.6 cm. It was then observed that the egg cracked when it was dropped onto the EPE cushioning material from a height of 42.6 cm. As shown in Figure
The failure of the eggshell when dropped onto the EPE cushioning material from a height of 42.6 cm.
(1) The egg was assumed to be a linearly rotating thin-walled structure with isotropic properties and a uniform thickness based on an analysis of previous analytical and research work locally and abroad that focused on a numerical model of the egg. A 3DCaMega five-axis automatic optical scanning system was used to scan the contour of the egg. The complete contoured surface of the egg was generated by means of reverse engineering, which simplified the modeling process.
(2) ADINA, a finite element analysis software package, was used to build a finite element model of the egg based on the principles of fluid-solid coupling. With a coupled solution, the software was used to simulate a drop impact of the egg. The feasibility of this model for use in a mechanical property simulation of drop impact was verified by the drop experiment. A comparison between the simulation and experimental results showed that the error in the simulation was less than 6%, which means that the model can be used in a realistic simulation.
(3) Based on the simulation, the height limit that an egg could survive without cracking when it was dropped onto EPE cushioning material was 42.6 cm, which was determined by modifying the initial conditions of the model based on a fluid-solid coupling model. The simulation results of this model will serve as a theoretical foundation and provide guidance for studies on egg protective packaging and the optimization of the design of egg processing equipment.
Wang Fang is the coauthor. Song Haiyan and Wang Fang are both the first authors of this manuscript.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
The authors acknowledge National Sci-Tech Support Plan Program of the “12th Five-year Plan” (2015BAD16B05); Tianjin Application Foundation and Cutting-Edge Technology Research Program (13JCZDJC30900); Tianjin Fundamental Research Funds for the Comprehensive Investment in “13th Five-Year”; Tianjin Food Safety and Low-Carbon Manufacture Collaborative Innovation Center; China Key Laboratory of Food Packaging Materials and Technology in Light Industry.