Fracture Mechanics EML6570.001S16Fracture toughness of a Turtle Shell

Mohammed Yunus Khan Sami U45230529

Abstract

Turtle shells are one of the most natural fascinating objects created by nature which provides maximum protection against external forces. This paper talks about the microstructure and fracture toughness of the shell of a Trachemys scripta (Red ear turtle) found in most parts of the world. Maximum displacement was found at the shell’s rear end. This unique type of deformation patterns helps to protect the turtle’s internal organs [2] Bio fiber distribution is same at the inner surface under compressive loading with the principal stresses and this results in the resistance to strong internal loads [2]. Protection offered by the shell is studied by many researchers in order to implement that into a human body armor.

Introduction

It’s remarkable to think that a structure like a natural shell has the ability to protect the animal from danger. The presence of these vertebrates since 200 million years, the Mesozoic era confirms the protection offered by the shell [1]. The shell is like an intricate shield for the ventral and dorsal parts of the body and walls all the important organs. The shell of this specific species i.e. Trachemys scripta represents just about 32% of the body mass [1]. The shell's two defensive covers for the dorsal and ventral part are called as carapace and plastron separately and they are associated along the side by a district known as the scaffold or the bridge [4]. Researchers have found that there are three distinct layers of materials in the carapace, inside and the outside layer including cortical bone, and the middle layer including a cancellous bone. There is about 90% of collagen helices in the shell bone which makes it look like a boney structure and inorganic compounds of hydroxyapatite nanocrystals and minor measures of water [5].The shell of the turtle is a vital study, not just because of the protection it provides, but it also acts like a recognizable proof apparatus, specifically with fossils as the shell is one of the reasonable parts of a turtle to survive fossilization. Henceforth understanding the structure of the shell in living species gives us similar material with fossils.

Fig 1 [4]

How shells break

Shells can break when a compressive stress falls upon it and when it surpasses the yield stress. The most well-known activities which can break a turtle shell are when predators chase for sustenance they tend to nibble the shell. Any high force impact can bring about shell cracking. For instance humans can break them by watercraft propeller strikes or when it's sprinkling hailstones. A percentage of the diseases which happen because of nourishing deficiencies or because of absence of UV light and poor water quality can likewise influence the shell. However shells have enough resistance to withstand large forces. The link shown below is to a page which shows how an Eastern River Cooter's turtle survived a crocodile assault which failed to break its defensive shell. This South American croc bit the shell with a power of 2900 pounds, almost equivalent to 12899.84N. So I ended up calculating the fracture toughness of this turtle shown in the article. The toughness was approximately 89 Mpa√ (mm). Different turtle species have different fracture toughness. I will show the calculations done on Trachemys scripta as we go by.

Carapace appeared in dorsal perspective

Plastron appeared in ventral perspective

Carapace (top) and plastron (base) associated by the scaffold (bolt), appeared in side perspective. Front is to one side.

Fig 2 Image source: http://www.dailymail.co.uk/sciencetech/article-2314541/Turtle-shell-withstands-15-minute-attack-alligator-fails-crack-shell.html

Case Study

Fig 3 [6]

• Scientific name : Trachemys scripta• Common name : Red-eared turtle• Life span : 30 years in the wild• Mass : 8.5oz (adult)• Conservation status : Least Concern (Population stable)• Higher classification : slider turtles• Rank : Species

Mechanical properties of the shellThe carapace is an intense organ secured by keratinous epidermal scales called scutes [5]. It has both natural and inorganic constituents which act like building blocks of the bone. Fibrillar structures are then used, for occasion, parallel fiber and woven that show the richest, a plywood-like structure, which is found in concentric lamellae that shape the osteonal unit. At the unmistakable level cortical and trabecular bone is composed together to shape entire bone of different sorts changed as per diverse mechanical purposes.

Epidermal scutes cover the outer layer of the carapace and dead cornified cells included by ß-wrinkled sheet keratin is the main constituent of the hard and inflexible outer surface (stratum corneum) [5]. Beneath the stratum corneum, keratinocytes are displayed in an undefined keratin system, neighboring melanocytes, tones and lipids. The epidermal scutes are joined where it goes through the dermis. The covered bone and the epidermal layers are protected by the collagenous strands. The keratin scutes are waterproof and add mechanical confirmation, being the focal line of protection amidst loading [5].

Researchers found out the mechanical properties of the shell using a nano-indentor which measures the hardness of materials. The micro structure was observed using a scanning electron microscope which showed a Haversian system as the main structure of shell for load bearing in the cortical bone. The Volkmann's tube which is situated at the focal point of the Haversian framework supplies supplements for the shell to develop and concrescence [1]. We might think that the shell is hard with no ductility, the shell actually has some yielding properties which can cause it to deform a little before cracking. This is helpful for locomotion and respiration. This yielding property of the shell is because of the structure of the shell which comprises of little plates associated with delicate sutures which offer ascent to little versatile deformation of the shell under a load, yet it turns out to be impressively firm under an expansive burden. Figure 4 and 5 demonstrates the macrostructure of the shell. On every side of the shell, there are two fortifying ribs secured with bio fibers stretching out to the base plate surface see Fig 4b. They offer ascent to the most astounding quality in the turtle shell [1].

Fig 4 [1]

Fig 5 [1]photo images of the strengthen rib and the connection pattern with the bottom plate

Figure 6 and 7 show the mass density and the elastic modulus of the 3 layers of the carapace respectively. Mass density of the carapace was measured by using BT-25S digital balance (Sartorius, Germany) with resolution of 0.01 mg. Every example was cut, as a thin plate of thickness 1 mm, from the shell at various positions [1].

Density / (g/cm3

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Chart Title

Interior layer Exterior layer Middle layer Fig 6 [1] Mass density

Initial modulus/Mpa Final modulus/Mpa Tensile strength/Mpa

0

200

400

600

800

1000

1200

Chart Title

Interior Exterior Middle Fig 7 [1] Elastic modulus

As seen in the table the values for mass density and elastic modulus vary with respect to the layers. For my calculation of fracture toughness I have taken the first layer since I have considered surface crack on mode 1 condition. The papers I have referred spoke only about the microstructure and mechanical properties; I wanted to find out the stress intensity with THOSE values which were found in the paper.

Fracture mechanism

The sutures have critical part under compression conditions. Under flexural semi static physiological low loads, the interlocking morphology was seemed to engage reasonably broad deformations of the shell for breathing, development and metabolic limit of the reptile [6]. It has also been found that the impact energy increases 3 times on suture free regions [6]. Most of the research has been done on static response compared to dynamic and cyclic loading response.

Considering the fact of the turtle getting hurt by an animal attack, the stress acting on the carapace is subjected to cyclic loading conditions.

[1]Compressive loading test was performed on the shell specimen of

Length : 220mm Width : 145mm Thickness compression speed : 2mm/min Compression failure load : 3330N Maximum displacement : 10.8mm

3mm

4mm

2.8mm

Fig 9. [1]

The electronic extensometer of resolution 0.01 mm supplied by New SANS Inc. in Shanghai, China was used to measure the force– displacement curve [1]

Calculations (My work)

I have done the calculations to find the fracture toughness using the values obtained by the experiment done by the researchers. I calculated the bending moment using 3 point bending test.

Elastic beam theory can be used to calculate the bending stress which is nothing but the peak stress before failure [4].

σ = (M * c) / I

M is the bending moment, I is the second moment of area for the test specimen, and c is half the mean thickness of the sample. [4]

Bending moment M= F * d

Where F is the compressive force and d is the distance between loading posts

The second moment of inertia with respect to central axis is I = (w * hᶟ) / 12 [4]

Where w is the width if the specimen, h is the thickness of the specimen.

The strain energy is calculated by ε = (M * h) / 2EI [4]

Finally the fracture toughness or the stress intensity is calculated by Griffiths theoryK IC= σ√ (πa)

Fig 10 [4] radiograph of the shell showing location of failure (a) upper view (b) visible crack approximately 2cm in length.

Data

F = 3330N

t = 3mm; c = 1/t =1.5mm

W= 45mm

L = 220mm; d = L/2 = 110mm

h = 25mm (assumed)

a= 20mm

Calculations

E = 530Mpa (external layer)

M = F * d = 3330 x 110 = 366300 N-mm

I = (w *hᶟ) / 12 = (145 x 25ᶟ) / 12 = 188802.08mm⁴σ = Mc / I = (366300 x 1.5) / 188802.08 = 2.91 Mpa

ε = (M * h) / 2EI = (366300 x 25) / (2 x 530 x 188802.08) = 0.045

K IC = σ√ (πa) = 23.06 Mpa√ (mm)

Energy absorbed = (1/2) x 4.5 x 5.9= 13.27 J

The energy absorbed is actually the area under the stress strain curve [1]

Fig 11

Finite element model of the shell

Finite element analysis of the turtle shell helps us in identifying the strength and stiffness offered by the carapace. This will thus help us to make human body defensive armor. Examination information from compressive tests were created to give understanding into the scute through-thickness conduct of the turtle shell. Three administrations can be described similarly as constitutive showing: straight adaptable, perfectly inelastic, and densification regions, where solidifying happens [7]. By and large, the phenomenological stress-strain conduct is like that of metallic foams [7]. So a turtle carapace is like a sandwich of three layers as explained earlier, the inner layer, out layer and the middle layer. The middle layer is actually the core which is protected by the two outer layers i.e. inner and outer layer. The inside network of the carapace was nondestructively tried utilizing x-beam processed tomography [8]. Images can be seen in the below figures

Fig 12

The x-ray computer tomography was done by using a phoenix x-ray apparatus [8]. Images show how close the cells are located within the shell. Which provides the stiffness. Its basic molecular physics. If the cells are attached together compactly, stiffer is the specimen (solids). If the cells are attached loosely, the specimen is flowy like (liquids). If the cells are not attached it resembles gases. The image also shows the porosity offered by the shell. The large denser outside, inner foam like layer, and entire turtle shell carapace including each of the three layers showed porosity levels of 6.86%, 65.5%, and 48.9%, individually [8]. Fig 14 shows another CT image of the exterior part.

The carapace had small tiny spikes on the layers, experimentalists used ProE software to smoothen the layers and they re-made the model. Fig 15 shows the image of the reconstructed model [2]

Fig 13 (a) Side view (b) carapace showed in a top sectional view acquired from x ray beam Computer Tomography sole cut output demonstrating haphazardly dispersed closed cell pores inside the foam like inner lamina. [8]

Fig 14 several typical Computer Tomography images [2]

The model was then discretized by 1566381 elements as found in figure 16 in which there are 863025 tetrahedron components and 703355 crystal components [2]

In the numerical analysis, they didn't consider subtle elements of the delicate points of the shell because of the colossal calculation work. On the off chance that they take the geometry scale of the meticulous points, they had to first build up a multiscale calculation strategy. However, the distortion of the carapace at soft junctures is very less [2]. For the circumstance considered here, the shell distortion is more prominent than the disfigurement delivered by the delicate points [2]

Finite method analysis of the mechanical response

An aggregate force of 800 N was put on the carapace, which came upto 25% of the break pressure power of the carapace [2].Since the shell is symmetrical, the examination subject is half of the carapace under a compressive force of 400 N. Figure 17 shows how the shell was compressed under two plates in an analysis mode.

Fig 15 reconstructed model of the shell [2]

Fig 16 FE mesh [2]

Figure 16 [2] finite element model for fixed static analysis

Von misses loads on the shell is shown in figure 17

Figure 17 [2] stress distribution (Von mises)

As found in the figure the stress achieves the most elevated quality inside the surface of the ribs, while the external surface has the second largest amount of stress and the center layer has the least. The estimation of most extreme misses stress which happen at the four ribs has an outright estimation of - 10.55Mpa as found in the following figure.

Figure 18 [2] maximum of the absolute estimation of pressure stress

The most extreme of the absolute estimation of main stress is like greatest von misses stresses. This demonstrates what crack strategies should be utilized for a carapace rib. Scopes of misses' stresses at various regions fluctuate between 0.023 MPa to 7.57 MPa [2].

High stress areas are reinforced with bio fires which give extra strength [2]. The shell as a whole plays an important part under compressive loads. There is no one concentrated area where the stress is extremely high. To the extent disfigurement is concerned, the rear end has the most amazing vertical distortion , which is around 7.46 mm with given load of 800 N. Regardless, the front end vertical separation ranges from 2.34 mm to 3.71 mm, which is around half of the backside. 1.2 mm is the greatest horizontal displacement which happens at the backside. This type of game plan ensures the creature's interior organs [2]

The stress vector plot of principal stresses at the inner layer of the shell is seen in figure 19

Fig. 19 [2]

The principal stress bearings at the internal surface of the carapace and plastron are just about the same as those of bio fiber reinforcement, how intriguing! [2]

Conclusion

The turtle shell carapace resembles a sandwich composite structure having substantial outside lamellar bone layers and an inside hard settlement of closed cell fibrous foamy layer [8]. The compression experiments prove that the deformation happens in a linear elastic fracture mechanics. The inward closed cells assume a vital part in the deformation conduct of the cell. This can be agreed with the Finite element analysis test experiments.

Also there are four connecting ribs which provide strength to the shell which connect s the shell top board and bottom plate. Under compressive force the ribs will be subjected to pressure and hence the four ribs will look like four columns bolstering the top board. Nonetheless, the turtle shell all in all will be subjected to a bending load so that the inner surface is under tension whereas the outer surface is in compression. Therefore the inner surface is infiltrated with bio -fiber reinforced composite giving strength of approximately 100 Mpa. Particularly, the strengthened bio-fibers go down from the rib surface and afterward spread into the surface of the

base plate. It is trusted that such a circulation of the bio-fibers may look after the stress bearing within surface of the base plate to oppose the advancement of breaking [1].

The turtle shell as a whole can be made stronger if it’s given enough exercise, they need unfiltered sunlight to get vitamin D. Food should be given which is rich in calcium.

References

[1] Microstructure and mechanical property of turtle shell Wei Zhang, Chengwei Wu, ChenzhaoZhang and Zhen Chen[2] Numerical Study of the Mechanical Response of Turtle ShellWei Zhang,Chengwei Wu, Chenzhao Zhang, Zhen Chen[3] https://en.wikipedia.org/wiki/Turtle_shell[4] Biomechanics of Turtle Shells:How Whole Shells Fail inCompression Paul M. Magweneand JohnJ. Socha[5] Micro-structure and mechanical properties of the turtle carapaceas a biological composite shield Ben Achrai, H. Daniel Wagner [6]The red-eared slider turtle carapace under fatigue loading: The effect ofrib–suture arrangement Ben Achrai, H. Daniel Wagner[7] Compressive behavior of a turtle's shell: experiment, modeling, and simulation Damiens R, Rhee H, Hwang Y, Park SJ, Hammi Y, Lim H, Horstemeyer MF.[8] A study on the structure and mechanical behavior of the TerrapeneCarolinaCarapace:A pathway to design bio-inspired synthetic composites H. Rhee a, M.F. Horstemeyera,b, Y. Hwang a, H. Lim c, H. El Kadiri a, W. Trim