Experimental evaluation of fracture properties of bovine hip cortical bone using elastic

Abstract

BACKGROUND:

Understanding the fracture mechanics of bone is very important in both the medical and bioengineering field. Bone is a hierarchical natural composite material of nanoscale collagen fibers and inorganic material.

OBJECTIVE:

This study investigates and presents the fracture toughness of bovine cortical bone by using elastic plastic fracture mechanics.

METHODS:

The J-integral was used as a parameter to calculate the energies utilized in both elastic deformation (J _el) and plastic deformation (J _pl) of the hipbone fracture. Twenty four different types of specimens, i.e. longitudinal compact tension (CT) specimens, transverse CT specimens, and also rectangular unnotched specimens for tension in longitudinal and transverse orientation, were cut from the bovine hip bone of the middle diaphysis. All CT specimens were prepared according to the American Society for Testing and Materials (ASTM) E1820 standard and were tested at room temperature.

RESULTS:

The results showed that the average total J-integral in transverse CT fracture specimens is 26% greater than that of longitudinal CT fracture specimens. For longitudinal-fractured and transverse-fractured cortical specimens, the energy used in the elastic deformation was found to be 2.8–3 times less than the energy used in the plastic deformation.

CONCLUSION:

The findings indicate that the overall fracture toughness measured using the J-integral is significantly higher than the toughness calculated by the stress intensity factor. Therefore, J-integral should be employ to compute the fracture toughness of cortical bone.

Keywords

Fracture toughness bovine hip bone compact tension specimen

1. Introduction

Bone is a natural complex hierarchical structure composite material that contains three constituents: water, inorganic (mineral crystal), and organic material 43% by volume (cells, lipids) [1]. Quantitatively, bone primarily contains about 70% minerals, nanoscale hydroxyapatite, 22% proteins (90% collagen fiber) and 8% water by weight [2]. The collagen plays an important role in the toughness of bone, while the minerals content is involved in bone stiffness [3,4]. Hipbone in the human and bovine body is a very important bone as it supports the whole body and is very fragile. In medical science and bioengineering, the understanding of fracture mechanics and fracture behavior of bone is very important. For the evaluations of bone fracture risk, the mechanical behavior of cortical bone is crucial [5–7]. Norman et al. investigated the fracture toughness (K _c) of tibia human bone and compared it with the bovine cortical bone. They cut the longitudinal CT specimens from both bovine and human medial of tibia bone according to the American Society for Testing and Materials (ASTM) E-399. The results indicate that the human cortical bone fracture toughness is 60–68% is considerably weaker than the bovine cortical bone [8]. Libonati et al. investigated experimentally bone fracture toughness and crack propagation. They conducted three pointing bending tests and single edge three point bending tests. They showed that the fracture mode of bone is different from conventional material steel [9]. Tanabe et al. performed quasi-static fracture toughness tests on the bovine hip cortical bone. The fracture toughness (K _c) and strain energy (G _c) are the functions of specimen thickness. The K _c and G _c were estimated with respect to the crack length. Both transverse and longitudinal specimens showed that the K _c and G _c are decreased with the increase of the thickness of the specimens [10].

The elastic modulus (E) and shear modulus of bovine cortical bone were experimentally investigated using different span-depth ratios ranging from 6 to 32 by Lefevre et al. [11]. They prepared 29 plates specimens of different dimensions and performed three point bending test using MTS device using crosshead speed of 0.5 mm/min and compute elastic modulus of wide range difference from 6–21 GPa. An et al. studied fracture toughness (K _c) and J-integral using linear elastic fracture mechanics (LEFM) and elastic plastic fracture mechanics (EPFM) of the five different ages range from 25 to 51 years of male human tibia cortical bone and 18 months old of bovine cortical. The K _c and J _c were computed both experimentally and numerically using ABAQUS software. It has been observed that K _c and J _c of the bovine cortical are 111% and 108% greater than the human cortical bone [12]. The elastic plastic fracture toughness of human dentin was investigated by Yan et al. using ASTM standard E1820 single edge notched specimens. The J-integral of longitudinal direction is less than transverse direction. The stress intensity factor is 32% less than the fracture toughness K _jc calculated by EPFM [13].

In the last three decades, researchers have studied the mechanical properties, such as fracture toughness, stiffness, the density of different bone using theories of linear elastic fracture mechanics (LEFM). The mechanical properties of cortical bone are immensely significant to prevent the fracture of a different bone. The mechanical properties vary with location of bone in a body, the effect of disease, age and also change with region to region [4,14]. Through the parameters of LEFM, such as K _c and G _c are used to compute the toughness of bone [15,16]. The fundamental assumption of LEFM is based on there is no or slight yielding occurs before fracture. But bones have exhibit some amount of post yield deformation, so the assumption of LEFM could not be applied to bone [17]. For materials, which have crack growth propagation at tip plastically, the theories of LEFM cannot be used to compute their fracture toughness. To accurately investigate the bone fracture, elastic-plastic fracture mechanics is required. The parameter J-integral is used in EPFM to compute the energies used in both elastic and plastic deformations of the hipbone [18,19].

In this paper, the mechanical properties of the male bovine cortical bone in both longitudinal and transverse directions were investigated based on the EPFM. The main goal of this paper is to estimate fracture toughness (K _jc) of the male bovine hip cortical bone using EPFM. The rest of the paper is organized as follows. The experimental setup for fracture toughness tests and tensile tests are illustrated in Section 2. The results of our experimental work are presented in Section 3. Section 4 summarizes the discussions and the conclusion is given in Section 5.

2. Material and methods

Six fresh male bovine hip bones of different ages (1 year to 3.5 years) and the same area, were collected from the market’s slaughterhouse as shown Fig. 1(a). Twenty four different types of specimens, i.e. longitudinal CT specimens, transverse CT specimens, rectangular (unnotched) longitudinal specimens and transverse specimens for tensile test, were cuts using hacksaw from the mid diaphysis of the bovine hip bone (6 longitudinal CT, 6 transverse CT and 12 for unnotched specimens for both Longitudinal and Transverse tensile test) according to the ASTME-1820 Standard [20]. We supplied a constant spray of clean water during machining; the bone is kept hydrated during processes to retain its moisture contents. For CT tests, 12 CT specimens were prepared in the longitudinal transverse direction to the fiber orientation. The dimensions of CT specimens are 12 × 3 ×14 mm (width × thickness × height) with a crack length (a _o) of 5 mm as shown in Fig. 1(b). In the longitudinal CT specimens, the collagens fibers are parallel to the specimen axis and collagens fiber are perpendicular to the specimen axis in the transverse CT specimens [21,22]. A crack length (a _o) of 5 mm was made in the each specimen using a sharp razor blade. All the machined and unmachined specimens were stored in the freezer until the test. All the CT specimens were tested in the Universal Testing Machine (UTM) as shown in Fig. 2. Unnotched rectangular specimens were prepared to perform the tension test to obtain the elastic modulus. The test was performed through UTM at the same loading rate. The ends of the specimens were covered with scotch tape. The elastic modulus was calculated from the slope of the stress-strain curve [23,24]. All tests were performed at normal room temperature (28 °C). The specimen was fixed in the steel fixture and held in UTM jaws as shown in Fig. 1(d). The speed of the cross head of the UTM is 0.5 mm/min. The tensile load is applied by the machine on the specimens. The applied load on the specimen and its corresponding deformation recorded by a PC connected to the machine during the test. The specimen was loaded until the crack started to grow and the test was stopped at this stage.

Fig. 1.

(a) Hip bone of male bovine. (b) The geometry of CT specimens according to ASTM-1820 standard. (c) CT specimen, and (d) The specimen is held with the help of the fixture in UTM.

Fig. 2.

Experiment setup of fracture toughness.

The J-integral indicates the resistance to crack propagation of the material under tensile deformation. J-integral contained two components, i.e. elastic J-integral and plastic J-integral. Figure 3 illustrates these components of J-integral on a load and loading-line displacement diagram for CT specimen. After fracture the J-integral was computed by Eq. (1) for each specimen [20]: $\begin{eqnarray}\displaystyle J_{\text{total}}=J_{el}+J_{pl}={\displaystyle \frac{K_{c}^{2}}{E^{\prime }}}+{\displaystyle \frac{2A_{pl}}{B_{N}b_{o}}}. & & \displaystyle\end{eqnarray}$ (1) Where K _c is the critical stress intensity factor, A _pl is the area of plastic region on load-loading-line displacement graph, B _N is net thickness of the specimen, uncracked specimen is b _o = w − a _o. For plane stress effective Young’s modulus E ^′ is equal to E, while for plane strain $E^{\prime }=\frac{E}{(1-{\nu}^{2})}$ , where 𝜈 is the Poisson’s ratio and is assumed to be 0.30 [25].

Fig. 3.

Load (N) vs loading-line displacement (mm) graph of CT specimens for J-integral measurement.

The critical stress intensity factor K _c is calculated as follows [20]: $\begin{eqnarray}\displaystyle K_{c}={\displaystyle \frac{F}{(BB_{N}W)^{1/2}}}f({\gamma}). & & \displaystyle\end{eqnarray}$ (2) Where F is maximum load and B _N net thickness of the specimens, ${\gamma}=\frac{a_{o}}{w}$ and $\begin{eqnarray}\displaystyle f({\gamma})=\frac{[(2+{\gamma})(0.886+4.64{\gamma})-13.32{\gamma}^{2}+14.72{\gamma}^{3}-5.6{\gamma}^{4}]}{(1-{\gamma})^{3/2}}. & & \displaystyle \nonumber\end{eqnarray}$ Elastic modulus (E) is calculated from the stress-strain curve of unnotched bovine cortical specimens: $\begin{eqnarray}\displaystyle B\leq 2.5\left({\displaystyle \frac{K_{C}}{{\sigma}_{ys}}}\right)^{2}. & & \displaystyle\end{eqnarray}$ (3) In Eq. (3), B shows the thickness of specimens, and 𝜎_ys is the yield stress of the cortical bone, which is calculated by the stress-strain curve of the unnotched specimens in both directions. If B is less than the 2.5(K _c∕𝜎_ys)² then the elastic modulus is considered plane stress, otherwise, it is considered as plane strain [20].

From the J-integral, linear elastic equivalent fracture toughness (K _jc) of the hip bone is also calculated as: $\begin{eqnarray}\displaystyle K_{jc}=\sqrt{J_{\text{total}}E^{\prime }}. & & \displaystyle\end{eqnarray}$ (4) Then the value K _c obtained from Eq. ((2)) can be compared to the value K _jc calculated from Eq. ((4)).

3. Results

As mentioned earlier, the ex vivo load deform data obtained from the UTM computer control, were used to analyze Young’s modulus, stress intensity factor, and J-integral which show good consensus with the previously published data [26]. The K _c was calculated by using Eq. (2) based on the LEFM, as shown in Table 1. It was found that the average K _c of the CT specimens in longitudinal and transverse directions is 1.92 MPam^1∕2, and 2.5 MPam^1∕2, respectively. Furthermore, the average Young’s modulus (E) of unnotched specimens in the longitudinal and transverse directions was found to be 2.11 GPa and 3.03 GPa, respectively.

Table 1
Fracture toughness in longitudinal and transverse CT fracture specimens

Sr. no Longitudinal CT fracture toughness (MPa⋅m^1∕2) Transverse CT fracture toughness (MPa⋅m^1∕2)

1 1.40 2.28

2 2.28 4.03

3 1.57 1.75

4 2.63 1.92

5 1.35 2.28

6 2.20 2.45

Sr. no	Longitudinal CT fracture toughness (MPa⋅m^1∕2)	Transverse CT fracture toughness (MPa⋅m^1∕2)
1	1.40	2.28
2	2.28	4.03
3	1.57	1.75
4	2.63	1.92
5	1.35	2.28
6	2.20	2.45

After performing fracture toughness tests; the authors used plane stress assumption because the thickness of the CT specimens is less than the standards used in Eq. (3). Despite this, plastic behavior during the CT test was also observed. Therefore, ASTM E-1820 was used to determine J-integral [20]. The average J-integral of an elastic region of longitudinal and transverse specimens were computed 1.51 kPa⋅m and 1.968 kPa⋅m, respectively by Eq. (1). The average plastic deformation component of J-integral is 4.33 kPa⋅m and 5.96 kPa⋅m in longitudinal and transverse directions, respectively. The combining actions of elastic and plastic results of J-integral in both longitudinal and transverse direction are the total J-integral of 5.840 and 7.91 kPa⋅m, respectively. The linear elastic equivalent fracture toughness calculated using Eq. (4) is 3.33 and 4.45 MPa⋅m^1∕2 in longitudinal and transverse directions, respectively. Table 2 presents all the average mechanical properties of the hip bone.

Table 2

Average mechanical properties of the hip bone

	E (N = 6)	K _c (N = 6)	J _el (N = 6)	J _pl (N = 6)	J _total (N = 6)	K_jc (N = 6)
	GPa	MPa⋅m^1∕2	kPa⋅m	kPa⋅m	kPa⋅m	MPa⋅m^1∕2
Longitudinal fracture	2.11	1.9	1.510	4.33	5.840	3.330
Transverse fracture	3.79	2.5	1.968	5.96	7.928	4.45

N = Number of specimens, (E) = Elastic modulus, (K _c) = fracture toughness, (J _el) = elastic J-integral deformation, (J _pl) = plastic J-integral deformation, (J _total) = total J-integral deformation, (K _jc) = linear elastic equivalent fracture toughness (K _jc).

4. Discussion

This paper investigates both longitudinal and transverse fractures of the male bovine cortical bone. The stress intensity factor (K _c) and Young’s modulus (E) are computed on the base of the LEFM [27] by the plane stress assumption. In the transverse direction, the average value of K _c and E is 1.3, 1.7 times the value of K _c and E in the longitudinal direction respectively. In the longitudinal direction, the values of K _c and E are smaller than that in the transverse direction [25]. Bone is a complex hierarchical structure, in transverse specimens, the collagen fiber is perpendicular to the crack growth propagation, so more energy is required to overcome the collagen fiber strength in the transverse direction than longitudinal as shown in Fig. 4 [1,28,29]. Hydration of collagen fiber can affect the mechanical properties of cortical bone. Bone consists of about 43% of organics by volume. The organics constitute of bone like bone matrix, have a strong bonding force, and fill the gap between the layers of bone [30].

Fig. 4.

(a, c) The transverse and longitudinal CT specimens before load application and (b, d) Fracture crack propagation in transverse and longitudinal CT specimens.

The J-integral (J _total) represents the energy absorption in originally unnotched ligaments. The J-integral determines the bone toughness in both directions (longitudinal and transverse) of collagens fibers of CT specimens of bovine cortical bone. The energy used in the plastic zone is ignored in the LEFM, but the ex vivo experimental results showed that 70–75% of energy was spent in the plastic zone deformation.

To date, the toughness of bone was less investigated by J-integral. Zioupos and Currey computed the J-integral of the 35 years old human femur bone by using a graphical approach and found the J _total value is 1.2 kPa⋅m [31]. The G _c values reported in plenty of literature are very high than our values [4]. The hoof wall of equine was investigated at different cross head speeds and showed the J-Integral value from 11 to 13 kJ/m² [32]. In this paper, the elastic deformation J _el is calculated by Eq. ((1)) and average values are 1.51 and 1.9 kPa⋅m in a longitudinal and transverse direction, respectively. The average energy in the longitudinal direction is 26% less than the transverse direction. In 2007 Yan et al. studied the single edge notches beam specimens of the bovine femur cortical bone and found that the plastic part of J-integral deformation J _pl (5.3 kPa⋅m) in the transverse CT specimens is 4.1 times greater than the average value of elastic J-integral deformation J _el (1.3 kPa⋅m) [18]. An et al. prepared CT specimens of bovine tibia cortical bone and their results showed that the plastic J integral deformation J _pl (6.635 kPa⋅m) is 2 times larger than the elastic J-integral deformation J _el (3.203 kPa⋅m). They found that the J _el is smaller than the J _pl of the CT specimens of human and cortical bone [12]. In this paper, the average value of J _pl is 26% greater than the J _el value for CT specimens of bovine cortical bone. The total J-integral of the hip bone is 11–20% less than the bovine femur bone. The average value of linear elastic equivalent fracture toughness (K _jc) in both directions is about 42% greater than the stress intensity factor K _c, respectively.

5. Conclusions

In this paper, elastic-plastic fracture mechanics were applied in both longitudinal and transverse directions to examine the mechanical properties of the bovine cortical bone, since the bone exhibits a slight plastic deformation before fracture. The experimental study of fracture toughness of the bovine cortical bone using ASTM E1820 CT specimens showed that the J-integral is 31% greater in the transverse direction than in the longitudinal direction. The overall fracture toughness (K _jc) computed using EPFM is considerably greater than the stress intensity factor (K _c). Therefore, EPFM theories should be employed to estimate the fracture toughness of human and bovine cortical bone.

Footnotes

Acknowledgement

This work was supported by the Research Program of the Department of Mechanical Engineering, International Islamic University Islamabad.

Conflict of interest

The authors declare that they have no conflict of interest.

References

Rho

J.-Y.

Kuhn-Spearing

and Zioupos

, Mechanical properties and the hierarchical structure of bone, Medical Engineering & Physics 20 (1998), 92–102.

Fratzl

Gupta

Paschalis

and Roschger

, Structure and mechanical quality of the collagen–mineral nano-composite in bone, Journal of Materials Chemistry 14 (2004), 2115–2123.

Viguetcarrin

Garnero

and Delmas

P.D.

, The role of collagen in bone strength, Osteoporosis International 17 (2006), 319–336.

Wang

and Puram

, The toughness of cortical bone and its relationship with age, Annals of Biomedical Engineering 32 (2004), 123–135.

Wang

Shen

and Agrawal

C.M.

, Age-related changes in the collagen network and toughness of bone, Bone 31 (2002), 1–7.

Gauthier

Follet

Langer

Gineyts

Rongiéras

Peyrin

, Relationships between human cortical bone toughness and collagen cross-links on paired anatomical locations, Bone 112 (2018), 202–211.

Leng

Reyes

M.J.

Dong

X.N.

and Wang

, Effect of age on mechanical properties of the collagen phase in different orientations of human cortical bone, Bone 55 (2013), 288–291.

Norman

T.L.

Vashishth

and Burr

D.B.

, Fracture toughness of human bone under tension, Journal of Biomechanics 28 (1995), 309–320.

Libonati

and Vergani

, Bone toughness and crack propagation: An experimental study, Procedia Engineering 74 (2014), 464–467.

10.

Tanabe

and Bonfield

, Effects of initial crack length and specimen thickness on fracture toughness of compact bone, JSME International Journal Series C-mechanical Systems Machine Elements and Manufacturing 42 (1999), 532–538.

11.

Lefevre

Baron

and Pithioux

, Evaluation of the elastic modulus of cortical bone: Adaptation of experimental protocols to small samples, Computer Methods in Biomechanics and Biomedical Engineering 16 (2013), 328–329.

12.

Liu

Arola

and Zhang

, Fracture toughening mechanism of cortical bone: An experimental and numerical approach, Journal of the Mechanical Behavior of Biomedical Materials 4 (2011), 983–992.

13.

Yan

Taskonak

Platt

J.A.

and Mecholsky

J.J.

, Evaluation of fracture toughness of human dentin using elastic–plastic fracture mechanics, Journal of Biomechanics 41 (2008), 1253–1259.

14.

Osterhoff

Morgan

E.F.

Shefelbine

S.J.

Karim

McNamara

L.M.

and Augat

, Bone mechanical properties and changes with osteoporosis, Injury 47 (2016), S11–S20.

15.

Behiri

J.C.

and Bonfield

, Fracture mechanics of bone: The effects of density, specimen thickness and crack velocity on longitudinal fracture, Journal of Biomechanics 17 (1984), 25–34.

16.

Hansen

Zioupos

Simpson

Currey

J.D.

and Hynd

, The effect of strain rate on the mechanical properties of human cortical bone, Journal of Biomechanical Engineering-Transactions of The Asme 130 (2008), 011011.

17.

Currey

J.D.

, Sacrificial bonds heal bone, Nature 414 (2001), 699.

18.

Yan

Mecholsky Jr

J.J.

and Clifton

K.B.

, How tough is bone? Application of elastic–plastic fracture mechanics to bone, Bone 40 (2007), 479–484.

19.

Ritchie

R.O.

Buehler

M.J.

and Hansma

P.K.

, Plasticity and toughness in bone, Physics Today 62 (2009), 41–47.

20.

Standard

, Standard test method for measurement of fracture toughness, ASTM (2001), 1–46, E1820-01.

21.

Georgiadis

Mueller

and Schneider

, Techniques to assess bone ultrastructure organization: Orientation and arrangement of mineralized collagen fibrils, Journal of the Royal Society Interface 13 (2016), 20160088.

22.

Boyde

Bianco

Barbos

M.P.

and Ascenzi

, Collagen orientation in compact bone: I. A new method for the determination of the proportion of collagen parallel to the plane of compact bone sections, Metabolic Bone Disease and Related Research 5 (1984), 299–307.

23.

Spatz

H.-C.

and Vincent

, Young’s moduli and shear moduli in cortical bone, Proceedings of the Royal Society of London. Series B: Biological Sciences 263 (1996), 287–294.

24.

Rahman

Ur Rahman

and Khan

, Effect of orientation and age on the crack propagation in cortical bone, Bio-Medical Materials and Engineering 29 (2018), 601–610.

25.

Reilly

D.T.

and Burstein

A.H.

, The elastic and ultimate properties of compact bone tissue, Journal of Biomechanics 8 (1975), 393–405.

26.

Kahler

Swain

M.V.

and Moule

, Fracture-toughening mechanisms responsible for differences in work to fracture of hydrated and dehydrated dentine, Journal of Biomechanics 36 (2003), 229–237.

27.

Nalla

R.K.

Kruzic

J.J.

Kinney

J.H.

and Ritchie

R.O.

, Mechanistic aspects of fracture and R-curve behavior in human cortical bone, Biomaterials 26 (2005), 217–231.

28.

Rho

Kuhnspearing

and Zioupos

, Mechanical properties and the hierarchical structure of bone, Medical Engineering & Physics 20 (1998), 92–102.

29.

Weiner

and Wagner

H.D.

, The material bone: Structure-mechanical function relations, Annual Review of Materials Science 28 (1998), 271–298.

30.

Yan

, Elastic-Plastic Fracture Mechanics of Compact Bone, University of Florida, 2005.

31.

Zioupos

and Currey

, Changes in the stiffness, strength, and toughness of human cortical bone with age, Bone 22 (1998), 57–66.

32.

Vashishth

Behiri

and Bonfield

, Crack growth resistance in cortical bone: Concept of microcrack toughening, Journal of Biomechanics 30 (1997), 763–769.

Experimental evaluation of fracture properties of bovine hip cortical bone using elastic–plastic fracture mechanics

Abstract

BACKGROUND:

OBJECTIVE:

METHODS:

RESULTS:

CONCLUSION:

Keywords

1. Introduction

2. Material and methods

Table 1 Fracture toughness in longitudinal and transverse CT fracture specimens Sr. no Longitudinal CT fracture toughness (MPa⋅m1∕2) Transverse CT fracture toughness (MPa⋅m1∕2) 1 1.40 2.28 2 2.28 4.03 3 1.57 1.75 4 2.63 1.92 5 1.35 2.28 6 2.20 2.45

Footnotes

Acknowledgement

Conflict of interest

References

Table 1
Fracture toughness in longitudinal and transverse CT fracture specimens

Sr. no Longitudinal CT fracture toughness (MPa⋅m^1∕2) Transverse CT fracture toughness (MPa⋅m^1∕2)

1 1.40 2.28

2 2.28 4.03

3 1.57 1.75

4 2.63 1.92

5 1.35 2.28

6 2.20 2.45