Corresponding author: Chengwei Wu, Zhen Chen E-mail: cwwu@dlut.edu.cn, chenzh@missouri.edu

Journal of Bionic Engineering 9 (2012) 330–335

Numerical Study of the Mechanical Response of Turtle Shell

Wei Zhang1,3, Chengwei Wu1, Chenzhao Zhang1, Zhen Chen1,2

1. State Key Laboratory of Structural Analysis for Industrial Equipment, Faculty of Vehicle Engineering and Mechanics, Dalian University of Technology, Dalian 116024, P. R. China

2. Department of Civil and Environmental Engineering, University of Missouri, Columbia, MO 65211-2200, USA 3. School of Materials Science & Engineering, Dalian University of Technology, Dalian 116024, P. R. China

Abstract The turtle shell is an amazing structure optimized through the long-term evolution by nature. This paper reports the me-

chanical response of the shell (Red-ear turtle) to static and dynamic loads, respectively. It is found that the turtle shell under a compressive load yields the maximum vertical displacement at the rear end, but the vertical displacement at the front end is only half of that at the rear end. The maximum horizontal displacement of the shell also occurs at the rear end. It is believed that such a deformation pattern is helpful for protecting the turtle’s internal organs and its head. The principal stress directions in the inside surface of the shell under a compressive load are almost the same as those of the biofiber distribution in the inside surface, which results in the strong bending resistance of the turtle shell.

Keywords: biomaterial, turtle shell, mechanical response, static and dynamic loads Copyright © 2012, Jilin University. Published by Elsevier Limited and Science Press. All rights reserved. doi: 10.1016/S1672-6529(11)60129-7

1 Introduction

Turtle is one of the oldest animals that still exist in the world, which is believed to have marvellously evolved since 220 millions years ago[1–5]. The turtle shell, as the protective armor and a source for it surviving prolonged anoxic acidosis, plays an important role for the animal to survive and evolve[5,6]. The integrated optimization design of the material structure at different levels of the turtle shell yields not only the strong de-fense ability, but also an anchoring site for the muscles and the major mineral reservoir of the body[7]. Although the bone-like shell of the Chrysemys turtle accounts for over 30% of the body mass of the animal, the turtle can still have certain mobility. Domokos and Varkonyi[8], Rivera et al.[9] observed and studied the inversion and upturn processes of various kinds of turtles, and found that the structure shape of the turtle shell is an optimized geometric design for its self-righting (reversal action) when turned upside down.

The turtle shell structure, as a multifunctional sys-tem designed by nature, not only provides the animal with the optimal load-bearing capacity, but also results

in the maximum space for the animal to store and protect their internal organs[10]. Rhee et al.[11] studied the mi-crostructure and mechanical property of the Terrapene carolina carapace and found that the turtle shell consists mainly of three layers of materials, namely, the interior and exterior layers consisting of cortical bone, and the middle layer (foam-like) consisting of cancellous bone. The turtle shell comprises a series of connected indi-vidual plates covered with a layer of horny keratinized scutes. The scutes are made up of a fibrous protein called keratin that also comprises the scales of other reptiles. They measured the mechanical properties of the bone fibers using a nano indenter with the results of elastic modulus ~20 GPa and hardness ~1 GPa. The observa-tion via Scanning Electron Microscopy (SEM)[12] indi-cates that the turtle shell is a complex biomaterial with a fracture toughness as high as 36.4 MPa·m1/2. The turtle shell structure, which is reinforced by the ribs arranged by the nature according to the load-bearing requirement, is a typical biocomposite structure. Zhang et al.[13] showed that the outer layer of the turtle shell is a stiff cover, but the inner surface is covered by a thin layer (~28 μm thick) of biofiber-reinforced biocompo-

Zhang et al.: Numerical Study of the Mechanical Response of Turtle Shell 331

site. Although the turtle shell is quite stiff at the surface, the shell can still yield a small amount of deformation during the turtle movement so as to make both respira-tion and locomotion functions feasible. It is the soft sutures connecting the small plates that give rise to a small elastic deformation of the shell under a small load, but become considerably stiff under a large load[14].

Based on the previous work as summarized above on the experimental measurement of the mechanical property of the turtle shell, we perform a numerical study in this paper to investigate the mechanical response of the turtle shell to static and dynamic compressive loads, respectively. It is found that the turtle shell is an integrated system optimized to provide the turtle with multifunctions required for sur-vival.

2 Finite element model of the turtle shell

Turtle shell gradually forms its own special struc-ture and functions during the long evolution process. The shell configuration consists of a series of complex curves and thus makes it difficult to establish an actual geometrical model of the turtle shell using a traditional numerical method. The turtle shell used in the present study is Trachemys scripta (Red-ear turtle, native of North America), which was described in detail in the previous study[13]. The material properties of the turtle shell vary with position, and are given in Tables 1–4. A turtle shell may be anisotropic, especially the fiber re-inforced biofilm on the inside surface. However, the mechanical property does not differ very much in dif-ferent directions for the exterior layer and middle layer of materials. For the interior layer of the material, the tensile test was taken in the fiber direction. The me-chanical strength of the biofiber reinforced film (~28 μm) depends on the tensile direction, varying from 70 MPa (transversal) to 135 MPa (longitudinal).

Table 1 Mass density of the turtle shell specimen[13] Location Interior layer Exterior layer Middle layerDensity (g·cm−3) 1.52 1.47 1.03

Table 2 Tensile modulus and strength of the turtle shell material[13]

Location Interior Exterior Middle Elastic modulus (MPa) 985 530 315 Tensile strength (MPa) 40 28 19

Table 3 Elastic modulus and ultimate strength of the rib[13]

Location Interior Middle Initial modulus (MPa) 1190 700 Tensile strength (MPa) 51.8 34.5

Table 4 Elastic modulus and ultimate strength of the biofiber reinforced film on the interior surface of the bottom plate[13]

Mechanical property Averaged Range Initial modulus (MPa) 1660 1550–1700 Tensile strength (MPa) 98 70–135

Fracture strain 0.10 0.07–0.15

We first obtained 199 pieces of the scanning images

(Somatom Sensation 16) parallel with the symmetrical plane of the shell, and then established a 3D Finite Element (FE) model of the turtle shell based on these CT images and geometry reconstructing technology. Fig. 1 shows several typical CT images of the turtle shell.

Fig. 1 Several typical CT images of the turtle shell.

The original construction of the turtle shell from

these CT images might have many small spiny parts, especially on the inside surface. We used ProE software to reconstruct the original model and to make the surface smooth. The reconstructed model of the turtle shell is shown in Fig. 2. The whole turtle shell is then discretized by 1566382 finite elements as shown in Fig. 3, in which there are 863026 tetrahedron elements and 703356 prism elements.

Fig. 2 Reconstructed model of the turtle shell.

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In the numerical analysis, we did not consider de-tails of the soft junctures of the shell due to the huge calculation job. If we consider the microgeometry scale of the soft junctures, we have to first develop a multis-cale computation method. However, the deformation of the shell soft junctures only plays an important role in small deformation[14]. In the case studied here, the shell deformation is far beyond the deformation produced by the soft junctures. Analysis error resulting from omis-sion of the soft connection of the suture joining of shell plates will be discussed in section 3.3.

Fig. 3 FE mesh of the turtle shell.

3 FM analysis of the mechanical response

3.1 Static analysis under compression load The turtle shell is subjected to a total compressive

load of 800 N during static analysis, which is about 25% of the fracture compression force of the shell[13]. Due to the geometrical symmetry of the turtle shell, the analysis model is the half of the turtle shell subjected to a com-pressive load of 400 N. The turtle shell model was com-pressed under two rigid plates as shown in Fig. 4. The Mises stress is shown in Fig. 5a. Generally speaking, the Mises stress in the inside surface of the ribs of the turtle shell reaches the highest value, while the outside surface gives the second highest level of stress and the middle layer of the shell gives the lowest stress. The maximum Mises stress occurs at the inside surfaces of the four ribs, reaching the highest value of 11.85 MPa at the given load mentioned above. The maximum of the absolute value of the principal stress is the compressive stress occurring at the four ribs, reaching in −10.55 MPa (maximum of absolute value of the stress, see Fig. 5b). There is no big difference between the maximum Mises stress and the maximum of the absolute value of principal stress. This indicates that what a fracture criterion should be used for a turtle shell rib seems not so important. The Mises stress in other areas ranges from 0.023 MPa to 7.57 MPa. It is

found that all the high stress areas are reinforced by the high strength biofibers, which is consistent with our previous results[13]. Due to the optimized shape, most parts of the turtle shell structure play an important role in supporting the compressive load, and no high stress concentration area is found.

Fig. 4 FE model for the static analysis.

Fig. 5 Stress distribution of the turtle shell: (a) Mises stress; (b) the maximum of the absolute value of compression stress. Note: the deformation is enlarged.

In terms of the deformation field, it is found that the turtle shell gives the maximum vertical displacement at the rear end, reaching up to 7.45 mm under the given load of 800 N. However, the vertical displacement at the front end ranges from 2.33 mm to 3.72 mm, only half of that at the rear end. The maximum horizontal displace-ment of the shell is 1.2 mm, also occurring at the rear end. It is believed that such a deformation arrangement is helpful for protecting the internal organs and its head. The principal stress direction at the inside surface is shown in Fig. 6. It is interesting to note that the principal stress directions at the inside of the shell are almost the same as those of the biofiber distribution directions as observed in the previous work[13].

Zhang et al.: Numerical Study of the Mechanical Response of Turtle Shell 333

Fig. 6 Principal stress vector plot inside surface of the turtle shell. 3.2 Vibration mode analysis

We have also performed the vibration modal analysis of the first 14 modes for the turtle shell, as shown in Table 5 without considering the rigid move-ment of the shell. Vibration modes from 1 to 10 are the local vibration mode. The first 4 vibration modes with frequencies ranging from 109.73 Hz to 214.21 Hz are shown in Fig. 7. All these four vibration modes occur at the edge or the end of the turtle shell, which are far from the internal organs. Those frequencies are at least one order higher than the first three vibration modal fre-quencies of the human body ranging from 4.9 Hz to 13 Hz[15], which is the sensitive frequency of human body[16–18]. This shows that the anti-vibration ability of the turtle shell is much better than that of human body. The last four vibration modes are global vibration modes with the frequencies from 498.9 Hz to 596.5 Hz. Those high natural frequencies are helpful for protecting the shell structure from resonance at low frequency.

3.3 Comparison with experiment

For the turtle shell studied here, the vertical elastic displacement at the loading point is 3.2 mm under 800 N of compression load, which is 11% less than the meas-ured value. It should be pointed here that the measured value in fact is the total displacement of the measuring system including the very small elastic deformation of the machine structure, which is slightly larger than the real displacement of the turtle shell at the measured point. This indicates that our numerical analysis agrees well with experiment. Also omission of the soft junctures of the shell does not bring a too big analysis error under large load. However, if we study the small deformation under a light load, the soft junctures deformation should be studied in details.

Table 5 The vibration modes and frequencies of the turtle shell Vibration

modeFrequency

(Hz) Description Vibration mode

Frequency(Hz) Description

1 109.73 Local 8 361.2 Local 2 130.97 Local 9 425.3 Local 3 177.25 Local 10 426.8 Local 4 214.21 Local 11 498.9 Global 5 216.71 Local 12 572.5 Global 6 284.37 Local 13 587.7 Global 7 314.56 Local 14 596.5 Global

(a)

(c)

Fig. 7 The first four vibration modes of the turtle shell: (a) the 1st modal response; (b) the 2nd modal response; (c) the 3rd modal response; (4) the 4th modal response.

Journal of Bionic Engineering (2012) Vol.9 No.3 334

Due to the linear relationship between the com-pression load and the shell stress, we can obtain the stresses when the shell damage occurs under the maxi-mum compression load of 3330 N (when a crack noise was heard from the shell) using the predicted stresses under the load of 800 N. Table 6 gives the comparison between the predicted stresses under load of 3330 N and the fracture stresses given from Table 1 to Table 4. To our surprise the turtle shell is almost an equal strength structure designed by nature although the first damage occurred at the middle layer in the shell rear area. The shell makes full use of the biomaterials. In addition, the predicted stresses at the inside surface are far from the tensile strength of the biofiber-reinforced film. This special design is also helpful for protecting the internal organs from injury if the shell is subjected to large compression load. Table 6 Comparison between the predicted stresses under load of 3330 N and the measured fracture strength[13]

Location Layer Max stress (MPa)

Strength (MPa) Failure?

Interior 49.3 51.8 no Rib

Middle 26.6 34.5 no Interior 30.9 39.9 no Exterior 24.7 28.4 no Shell Middle 19.7 18.6 yes

4 Conclusions

The turtle shell is an amazing structure optimized through the long-term evolution by nature. The shell material and structure are an integrated optimized sys-tem giving rise to an optimal protecting capacity from injury of the internal organs. The major findings from the numerical study are summarized as below:

(1) It is found that the turtle shell consists of three layers of materials. The inner layer is covered by a thin layer of reinforcing biofibres, which results in the highest strength. Under compressive load, the distribu-tion directions of the inner surface biofibres are almost the same as the principal stress directions obtained by the numerical analysis.

(2) The turtle shell yields the maximum vertical displacement at the rear end under a compressive load. However, the vertical displacement at the front end is only about half of that at the rear end. The maximum horizontal displacement of the shell also occurs at the rear end. It is believed that such a deformation ar-

rangement is helpful for protecting the internal organs and the animal head.

(3) Vibration modes of the turtle shell from the 1st to 10th are the local vibration mode. The first 4 vibration modes with frequencies ranging from 109.7 Hz to 214.2 Hz occur at the edge or the end of the turtle shell, which are far from the internal organs. Those frequencies are at least one order higher than the first three vibration modal frequencies of the human body ranging from 4.9 Hz to 13 Hz. This indicates that the anti-vibration ability of the turtle shell is much better than that of human body. The last four vibration modes are global vibration modes with the frequencies from 498.9 Hz to 596.5 Hz. Those high natural frequencies are helpful for protecting the shell structure from resonance at low frequency.

(4) The turtle shell is found to be an equal strength structure designed by nature, making full use of the biomaterials. This will inspire us to design more ad-vanced composite materials.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No.10972050 and No. 90816025).

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