Materials Performance andCharacterization

N. K. Sharma,1 Ruchita Pal,2 D. K. Sehgal,3 and R. K. Pandey3

DOI: 10.1520/MPC20130069

Application of Elastic-Plastic FractureMechanics to Determinethe Locational Variationin Fracture Properties ofCortical Bone

VOL. 3 / NO. 3 / 2014

N. K. Sharma,1 Ruchita Pal,2 D. K. Sehgal,3 and R. K. Pandey3

Application of Elastic-Plastic FractureMechanics to Determine the LocationalVariation in Fracture Properties ofCortical Bone

Reference

Sharma, N. K., Pal, Ruchita, Sehgal, D. K., and Pandey, R. K., “Application of Elastic-Plastic

Fracture Mechanics to Determine the Locational Variation in Fracture Properties of Cortical

Bone,” Materials Performance and Characterization, Vol. 3, No. 3, 2014, pp. 429–447,

doi:10.1520/MPC20130069. ISSN 2165-39924

ABSTRACT

The complex nature of bone material results in a locational variation of fracture

and mechanical properties. The heterogeneity associated with bone material

and complex hierarchical assembly results in several toughening mechanisms,

such as plasticity, micro-cracking, viscoplasticity, etc. These toughening

mechanisms and presence of water in bone material makes the linear elastic

fracture mechanics (LEFM) inapplicable in such materials. The present work is

focused on the elastic-plastic fracture mechanics (EPFM) approach to estimate

the locational variation in fracture properties of buffalo cortical bone for

longitudinal, as well as transverse orientation of cracking. Samples from upper,

middle, and lower locations of bone diaphysis were tested using compact

tension and single-edge notch-bending testing methods for longitudinal and

transverse orientation of cracking, respectively. The crack-tip opening

displacement (CTOD) approach was applied to determine fracture properties,

such as CTOD toughness (dc), J integral (Jcd), and equivalent fracture

toughness (Kdc) at different locations of bone diaphysis. The effect of

orientation and location on mechanical properties of cortical bone, such as

elastic modulus (E) and yield strength (rys), was also analyzed with the help of

tensile testing. The equivalent fracture toughness values (Kdc) obtained in the

present work were found to be three times higher than the corresponding

Manuscript received October 7,

2013; accepted for publication

February 10, 2014; published

online June 7, 2014.

1

School of Technology, The Glocal

University, Mirzapur Pole,

Saharanpur, India (Corresponding

author),

e-mail: enksharma@yahoo.com;

nitin@theglocaluniversity.in

2

AIRF, Jawaharlal Nehru Univ., New

Delhi, India.

3

Dept. of Applied Mechanics, Indian

Institute of Technology Delhi, New

Delhi, India.

4

This paper is a contribution to a

Special Issue of Materials

Performance and Characterization

on “Fracture Toughness,” Guest

Editors, Bojan Podgornik and

Votjeh Leskovsek, Institute of

Metals and Technology, Ljubljana,

Slovenia.

Copyright VC 2014 by ASTM International, 100 Barr Harbor Drive, P.O. Box C700, West Conshohocken, PA 19428-2959 429

Materials Performance and Characterization

doi:10.1520/MPC20130069 / Vol. 3 / No. 3 / 2014 / available online at www.astm.org

values reported in the previous reports where the LEFM approach was applied

favoring the application of EPFM for bone materials. The mechanical

properties, as well as the fracture properties, were found to be maximum at

middle location and minimum at lower location of bone diaphysis. The

locational variation in fracture and mechanical properties observed in the

present work are considered to be because of locational distribution of

collagen fibrils, minerals, porosity, and density at different locations of bone

diaphysis.

Keywords

bone, fracture toughness, crack-tip-opening displacement, elastic-plastic fracture

mechanics, linear-elastic fracture mechanics

Introduction

Bone is a composite-like material consisting of minerals, carbonated hydroxyapatite

crystals, type I collagen, non-collagenous proteins, and water. Bone material is

treated as an anisotropic, heterogeneous material with hierarchical structure that

changes at different length scales. The complex nature of bone material results in a

wide range of fracture properties according to the orientation of crack, loading con-

dition, type of testing, and the location from where the specific sample is taken out.

Fracture mechanics of bone has been a very important field of study for researchers

to understand the risk of bone fracture because of age-related changes and several

diseases like osteoporosis, bone cancer, or osteogenesis imperfecta.

In different studies, longitudinal and transverse-direction fracture properties of

cortical bone have been obtained. To evaluate longitudinal fracture toughness, pre-

crack was oriented parallel to the long axis of the bone, whereas for transverse frac-

ture toughness the same was developed perpendicular to the long axis of the bone.

Both the single-edge notch-bend testing [1–3] and compact tension (CT) specimen

testing were used [4–14] to find the longitudinal and transverse fracture toughness

of cortical bone. A few researchers have also applied chevron–notched-beam testing

method to determine fracture toughness [15,16]. Nalla et al. [17] used the double-

notched test to determine toughness mechanism in bone. Feng et al. [18] determined

the fracture toughness of bone under different modes. It is noted that in all the above

tests of cortical bone the fracture toughness was evaluated in terms of critical-stress-

intensity factor (Kc) and the critical-strain energy-release rate (Gc) based on linear

elastic fracture mechanics (LEFM). Apart from these, in some studies [6,10–14] it

was suggested that the fracture toughness of bone increases with crack length and,

therefore, fracture resistance of bone materials cannot be described by a single pa-

rameter. These studies highlight the rising crack-growth-resistance behavior of corti-

cal bone and indicate that the increment of fracture toughness may be because of

micro-cracking, osteon pullout, fiber bridging, and crack deflection. Some research-

ers also tried to find out the effect of bone composition, mineral density, porosity,

crack orientation, loading condition, and storage media on fracture toughness of

cortical bone [3,7,9,18].

SHARMA ET AL. ON ELASTIC-PLASTIC FRACTURE MECHANICS 430

Materials Performance and Characterization

Over the last few years, Yang et al. [19,20] have shown that in case of cortical

bone fracture, the LEFM analysis is not sufficient to characterize fracture toughness.

They suggested a non-linear fracture model to accurately predict the fracture prop-

erties of cortical bone. This non-linear behavior of bone material during deforma-

tion may be considered because of involvement of several toughening mechanisms

like plasticity, micro-cracking, visco-plasticity, etc. The water content in bone mate-

rial may also be responsible for this non-linearity as cortical bone becomes brittle

and has reduced toughness after the removal of water in a vacuum oven at increased

temperature. This influence of water removal on toughness of cortical bone was

shown by Nyman et al. [21]. Yan et al. [22] in their studies applied elastic-plastic

fracture mechanics (EPFM) using the J-integral approach to study the fracture

toughness of cortical bone. In one of our recent studies [23], EPFM was applied

using a crack-tip opening displacement (CTOD) approach to find out the CTOD

toughness (dc), the equivalent fracture toughness (Kdc), and J toughness values for

buffalo tibia cortical bone.

From the above review, it is noted that the analysis of fracture in cortical bone

using EPFM approach has been conducted only to a very limited extent. Further lit-

tle work has been carried out to find out the locational variation in fracture proper-

ties of cortical bone. Taking these aspects into consideration, the EPFM approach is

applied in the present investigation to estimate the locational variation in fracture

properties such as crack-tip opening displacement (CTOD), equivalent critical-

stress-intensity factor and J integral for longitudinal as well as transverse orientation

of cracking in the case of a buffalo femoral cortical bone. The fracture micro-

mechanism of such bones is also investigated using scanning electron microscopy

(SEM).

Materials and Methods

The present work is performed on the femoral cortical bones obtained from two

young buffalos about 24 months of age. These bones were obtained under institu-

tional permission from the farm raised just after an animal’s natural death and with-

out causing any harm to the animal. After removal of bone tissue from the body, the

surrounding soft tissue was removed and bone tissue was wrapped in gauze, soaked

in normal saline, wrapped with plastic wrap, and placed in sealed, airtight plastic

bags. These plastic bags were placed in a freezer and stored at �20�C within 1 h after

the bone tissues had been harvested. The bones were kept hydrated in saline upon

removal from the freezer and during all stages of tissue preparation.

The material orientations for femoral cortical bone were defined using a cylin-

drical coordinate system with longitudinal (1), circumferential (2), and radial (3)

axis. The whole bone diaphysis was divided into three equal segments, namely,

upper, middle, and lower bone diaphysis. The material orientations and different

locations of the bone diaphysis are shown in Fig. 1. Further, different anatomic quad-

rants (A¼ anterior, M¼medial, P¼ posterior, and L¼ lateral) of bone diaphysis

were identified and marked accordingly on each one-third segment of the diaphysis.

Each segment of the diaphysis was then subsequently sectioned into a number of

specimens according to different anatomic quadrants. The preparation of specimens

from different anatomic locations of the bone diaphysis is shown in Fig. 2. The

SHARMA ET AL. ON ELASTIC-PLASTIC FRACTURE MECHANICS 431

Materials Performance and Characterization

specimen obtained from a particular length segment and anatomic quadrant was

suitably labeled (e.g., UA¼ specimen from upper length segment and anterior quad-

rant) and arranged in different groups according to their labels. The number of

specimens obtained from different anatomic locations of the bone diaphysis is listed

in Table 1. As per Table 1, number six from the quadrant represents four specimens

for fracture testing and two specimens for tensile testing, whereas number four rep-

resents two specimens for fracture and two for tensile testing. Comparatively, more

specimens were prepared from the medial quadrant because of limited availability of

bone material from other quadrants.

The fracture properties of cortical bone were evaluated with the help of a crack-

tip opening displacement (CTOD) method using compact tension (CT) and single-

edge notch-bend (SENB) specimens. This method is an alternative approach of

elastic-plastic fracture mechanics apart from the J-integral approach. CTOD is con-

sidered as a fracture property of the material and is a measure of the crack opening

at the vicinity of the crack tip. The CTOD value can be determined with the help of

a crack mouth opening displacement (CMOD). The values of CMOD can be

FIG. 2

Schematic diagram showing

sectioning of cortical bone

diaphysis for preparation of

mechanical and fracture tests

specimens from different

anatomic locations of the bone

diaphysis. (a) The flattened

cortical bone diaphysis was

sectioned into three equal

length (L/3) segments. (b)

Each segment of the diaphysis

was subsequently sectioned

into four parts according to

different anatomic quadrants

(A¼anterior, M¼medial,

P¼posterior, and L¼ lateral)

for further preparation of

different samples.

FIG. 1

Diagram showing different

locations of the femoral

diaphysis from where the

specimens were cut for SENB

and CT testing.

SHARMA ET AL. ON ELASTIC-PLASTIC FRACTURE MECHANICS 432

Materials Performance and Characterization

measured easily with the help of a clip gauge attached to the crack mouth of the

SENB and CT specimens. The CTOD toughness obtained from the CMOD values

can be further converted into equivalent J toughness and K toughness with the help

of different correlations [24].

In the present investigation, CT and SENB specimens were prepared to undergo

longitudinal (crack advances parallel to the long axis of bone) and transverse (crack

advances perpendicular to the long axis of bone) fracture, respectively. All of the

specimens for CTOD testing were prepared following BS 7448 [24]. In all, 30 speci-

mens were cut from upper, middle, and lower locations of the femoral bone diaphy-

sis out of which 15 specimens with dimensions 3mm (thickness)� 15mm

(width)� 60mm (length) were obtained for SENB testing and the other 15 speci-

mens with dimensions 3mm (thickness)� 20mm (width)� 19mm (length) for CT

testing. A very fine slit of appropriate length as per the British standard simulating

the fine crack is induced in the sample using a diamond wheel (Isomet 4000). Differ-

ent locations and orientations of the CTOD specimens prepared from the femoral

bone are shown in Fig. 1. To avoid buckling/twisting of the SENB specimen, metallic

(Al) strips were attached on both sides at the two ends of the specimen with the help

of screws as shown in Fig. 3.

The uniaxial tensile properties at three different locations of the bone diaphysis

were evaluated using dumbbell-shaped strip-type specimens. In all, 24 specimens

were obtained from upper, middle, and lower portions of the bone diaphysis out of

which 12 specimens with thickness 2.5mm, gauge length 25mm, gauge width 4mm,

and total length 80mm were prepared for conducting the tensile test in the longitu-

dinal direction (load being applied along the long axis of femur), whereas the other

FIG. 3

Schematic diagram showing

attachment of metallic strips to

the SENB specimen of cortical

bone.

TABLE 1

Breakdown of specimens according to different anatomic locations of bone diaphysis.

Lower Diaphysis Middle Diaphysis Upper Diaphysis

L M A P L M A P L M A P

4 6 4 4 4 6 4 4 4 6 4 4

Note: L, lateral; M, medial; A, anterior; P, posterior.

SHARMA ET AL. ON ELASTIC-PLASTIC FRACTURE MECHANICS 433

Materials Performance and Characterization

12 specimens with thickness 2.5mm, gauge length 8mm, gauge width 4mm, and

total length 22mm for the transverse tensile test (load being applied perpendicular

to the long axis of femur).

All different specimens were stored at room temperature in a solution of 50 %

saline and 50 % ethanol at all time until testing. To keep the specimens wet and to

avoid heating during cutting and polishing, a constant spray of water was supplied.

The SENB and compact tension tests were performed on an MTS 858 table-top

machine. The crack-mouth opening displacement (CMOD) was measured with the

help of a clip gauge during the test. The load-CMOD (P-CMOD) diagrams were

recorded and analyzed for the evaluation of CTOD. A uniaxial tensile test was per-

formed on a Zwick 7250 universal testing machine. The stress–strain curves in the

case of a uniaxial tensile test for longitudinal as well as transverse specimens

obtained from different locations of the bone diaphysis are shown in Figs. 4 and 5,

respectively. The yield strength values were obtained corresponding to 0.2 % perma-

nent set.

FIG. 5

Stress–strain curves in the case

of transverse tensile testing for

specimens obtained from

different locations of the

buffalo cortical bone diaphysis.

FIG. 4

Stress–strain curve in the case

of longitudinal tensile testing

for specimens obtained from

different locations of the

buffalo cortical bone diaphysis.

SHARMA ET AL. ON ELASTIC-PLASTIC FRACTURE MECHANICS 434

Materials Performance and Characterization

Typical load (P-CMOD) diagrams for longitudinal and transverse orientation of

cracking at three different locations of bone diaphysis are shown in Figs. 6 and 7,

respectively. The first load maxima/pop-in point has been taken as the critical point

in P-CMOD diagram [24]. In the present case, no pop-in was noticed on P-CMOD

diagram and the CMOD value is found to increase steadily with load in the elastic-

plastic situation. Hence, the CMOD corresponding to the maximum load point was

employed for the evaluation of CTOD.

For the computation of CTOD from the CMOD value, the total CMOD corre-

sponding to maximum load point was divided in to two parts: the CMOD corre-

sponding to the elastic part (ve) and the one corresponding to the plastic part (vp) of

FIG. 6

Load-CMOD curves for

longitudinal fractured

specimens obtained from

different locations of the

buffalo femoral bone diaphysis.

FIG. 7

Load-CMOD curves for

transverse fractured specimens

obtained from different

locations of the buffalo femoral

bone diaphysis.

SHARMA ET AL. ON ELASTIC-PLASTIC FRACTURE MECHANICS 435

Materials Performance and Characterization

the crack-mouth opening. Figure 8 shows the elastic and plastic parts of CMOD on

the P-CMOD curve.

The elastic (de) and plastic (dp) parts of CTOD were calculated using Eq 1,

whereas Eqs 2a and 2b were used to determine the plastic part of CTOD for the

SENB and CT specimens, respectively [24],

de ¼K2I ð1� t2Þ2Erys

(1)

where:

KI¼ the stress-intensity factor corresponding to the critical load,

E¼ the elastic modulus,

rys¼ the yield strength, and

t¼ the Poisson’s ratio.

dp ¼vp

arðW � aÞ þ 1

(2a)

dp ¼vp

aþ C �WrðW � aÞ þ 1

(2b)

where:

vp¼ the plastic component of CMOD corresponding to the critical load,

a¼ the original crack length,

W¼ the width of the specimen,

C¼ the total height of the CT specimen, and

r¼ the rotation factor, which may be taken as 0.4 and 0.46, respectively, for

SENB and CT specimens as per the standard [24].

Total CTOD (dc) was calculated using Eq 3 as given below:

dc ¼ de þ dp(3)

FIG. 8

Analysis of the load-CMOD

diagram for the evaluation of

CTOD toughness.

SHARMA ET AL. ON ELASTIC-PLASTIC FRACTURE MECHANICS 436

Materials Performance and Characterization

Stress-intensity factors in the case of SENB and CT tests were calculated using Eqs 4

and 5, respectively [24].

KI ¼PS

BW3=2f1(4)

KI ¼P

BW1=2f2(5)

where:

P is the maximum load,

S is the span length, and

B is the thickness of the specimen.

The f1 and f2 are the function of a¼ (a/W) and given by Eqs 6 and 7 for the

SENB and CT geometry, respectively,

f1 ¼3a0:5 1:99� a 1� að Þ½ � 2:15� 3:93aþ 2:7a2½ �

2 1þ 2að Þ 1� að Þ1:5(6)

f2 ¼ 29:6a0:5 � 185:5a1:5 þ 655:7a2:5 � 1017a3:5 þ 639a4:5(7)

Specimens from the middle location of bone diaphysis have been selected for

scanning electron microscopic (SEM) analysis as higher values of fracture properties

are obtained at this location. The samples for SEM analysis were prepared with

cross-section 3mm� 8mm from one of the bisected fractured specimen. These

samples were cleaned in acetone and dried slowly at 60�C in a vacuum oven. Fur-

ther, these samples were placed in a vacuum for about 4 h with silicate gel to remove

the remaining moisture and then coated with gold for the SEM examination. The

SEM examination was conducted on a ZEISS-EV040 instrument at appropriate

magnifications.

Results and Discussion

The fracture toughness values for all the three selected locations of the femoral bone

diaphysis were calculated as described above for both longitudinal and transverse

orientation of fracture. The values of elastic modulus (E) and yield strength (rys) as

obtained from the experiments for different locations of the femoral bone diaphysis

in case of longitudinal and transverse testing are reported in Tables 2 and 3, respec-

tively. The elastic part of CTOD (de), plastic part of CTOD (dp), and total CTOD

(dc) values computed for different locations of the bone diaphysis in case of longitu-

dinal and transverse orientations of fracture are presented in Tables 4 and 5,

respectively.

TABLE 2

Elastic modulus (E1) and yield strength (rysl ) for longitudinal specimens of buffalo femoral cortical bone obtained

from different locations of the bone diaphysis.

Lower Diaphysis (n¼ 4) Middle Diaphysis (n¼ 4) Upper Diaphysis (n¼ 4)

E1 (GPa) 17.26 2.39 27.36 1.75 24.76 1.43

rysl (MPa) 77.46 7.57 114.36 2.53 102.76 5.41

SHARMA ET AL. ON ELASTIC-PLASTIC FRACTURE MECHANICS 437

Materials Performance and Characterization

The equivalent fracture toughness in terms of critical-stress-intensity factor

(Kdc) and J-toughness (Jcd) values were calculated employing the corresponding dc

values using Eqs 8 and 9, respectively [24],

Kdc ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2dcErys

q(8)

Jdc ¼ 2rys � dc(9)

and the corresponding Kdc and Jcd values are also reported in Tables 4 and 5.

Based on the CTOD and J approaches, to determine whether the specimen

thickness (B) and length of the uncracked ligament (W-a) were sufficient to meet

the requirement of plane-strain condition, Eqs 10 and 11 were used, respectively

[24],

B; ðW � aÞ � 25ðdcÞ(10)

B; ðW � aÞ � 25 JC=rys� �

(11)

Note that the minimum thickness requirement for plane strain in the present

case is about 1.1mm (based on CTOD approach) and 2.3mm (based on the J

approach), whereas the actual specimen thickness is 3.0mm. The lengths of the

uncracked ligament for the present investigation are 7.5mm (for SENB specimen)

and 8mm (for CT specimen). Therefore, all of the specimens are meeting the plane-

strain condition and reported dc and Jc values are the plane-strain values. The values

of Poisson’s ratio for all calculations are taken in between 0.32 to 0.47 based on a

separate study.

TABLE 3

Elastic modulus (E2), yield strength (ryst ) for transverse specimens of buffalo femoral cortical bone obtained from

different locations of the bone diaphysis.

Lower Diaphysis (n¼ 4) Middle Diaphysis (n¼ 4) Upper Diaphysis (n¼ 4)

E2 (GPa) 12.46 1.35 17.16 1.46 14.66 1.16

ryst (MPa) 45.46 3.78 80.66 3.24 68.476 3.93

Note: The results reported in Tables 2 and 3 are the average of four values and n gives the number of samples tested. Standard deviationsare also given.

TABLE 4

The elastic CTOD (de), plastic CTOD (dp), and the total CTOD (dc) along with the equivalent fracture toughness

(Kdc) and J-toughness (Jcd) values for longitudinal fractured specimens obtained from different locations of the

femoral bone diaphysis.

Lower Diaphysis (n¼ 5) Middle Diaphysis (n¼ 5) Upper Diaphysis (n¼ 5)

de (mm) 0.00096 0.0002 0.00136 0.0001 0.00116 0.0002

dp (mm) 0.0136 0.0019 0.0236 0.0021 0.0136 0.0028

dc (mm) 0.0146 0.0021 0.0256 0.0022 0.0146 0.0029

Kdc (MPa � m1/2) 3.936 0.29 8.256 0.37 5.366 0.56

Jcd (kJ/m2) 1.256 0.19 3.996 0.35 1.996 0.40

SHARMA ET AL. ON ELASTIC-PLASTIC FRACTURE MECHANICS 438

Materials Performance and Characterization

JUSTIFICATION FOR USING ELASTIC-PLASTIC FRACTURE MECHANICS

(EPFM) FOR CHARACTERIZATION OF BONE TOUGHNESS

The results obtained in the present study for different locations of bone diaphysis

and orientation of cracking show that in all cases the plastic part of CTOD (dp) is

higher as compared to the elastic part of CTOD (de). As per Tables 4 and 5, the plas-

tic part of CTOD (dp) is 11.8 to 14.4 times higher for longitudinal orientation and

2 to 9 times higher for transverse orientation of cracking as compared to the elastic

part of CTOD (de). This shows a considerable amount of plasticity associated with

bone material. The significant amount of non-linearity preceding instability/cracking

is also visible in the P-CMOD diagrams obtained in this study. Based on the analysis

of the P-CMOD diagram, it is noticed that the diagram does not meet other require-

ments for validity, i.e., the non-linearity at 0.8 PQ� 0.25 of the non-linearity at PQand Pmax/PQ� 1.1, where PQ is the load at which the 5 % secant line intersects the

P-CMOD diagram. In other words, non-linearity is large enough to make the K

approach inapplicable in the bone samples. This non-linearity may be the result of

several toughening mechanisms like plasticity, micro-cracking, and viscoelasticity,

and presence of water in bone material as observed in our previous study [23].

The average equivalent fracture toughness (Kdc) values for longitudinal and trans-

verse orientation of fracture at mid-diaphysis location are 8.25MPa � m1/2 and

14.97MPa � m1/2, respectively, as reported in Tables 4 and 5, whereas the average Kc

values reported in the previous studies for the same location of bone diaphysis are

2.6MPa � m1/2 and 5.1MPa � m1/2, respectively [18,25]. The comparison of these val-

ues show that the Kdc values obtained in the present study are about three times

higher than the corresponding values reported in the literature where the linear-

elastic fracture mechanics (LEFM) approach was applied. This difference in K-

toughness values is because of non-linearity associated with bone material that

causes the K approach inapplicable for such materials and any attempt to determine

K toughness with the LEFM approach will result in an unrealistically low value.

These observations favor the application of EPFM for the determination of fracture

properties of bone material.

EFFECT OF ORIENTATION AND LOCATION ONMECHANICAL PROPERTIES

OF CORTICAL BONE

The mechanical properties of cortical bone at different locations and for different

directions of loading are mentioned in Tables 2 and 3, respectively. It may be noted

from reported values that the maximum values of elastic modulus and yield strength

TABLE 5

The elastic CTOD (de), plastic CTOD (dp), and the total CTOD (dc) along with the equivalent fracture toughness(Kdc) for transverse fractured specimens obtained from different locations of the femoral bone diaphysis.

Lower Diaphysis (n¼ 5) Middle Diaphysis (n¼ 5) Upper Diaphysis (n¼ 5)

de (mm) 0.0036 0.0008 0.0116 0.0025 0.0106 0.0014

dp (mm) 0.0276 0.0056 0.0266 0.0067 0.0206 0.0066

dc (mm) 0.0306 0.0063 0.0376 0.0088 0.0306 0.0076

Kdc (MPa � m1/2) 8.966 0.90 14.976 1.88 12.226 1.55

Jcd (kJ/m2) 4.726 0.97 8.346 2.01 6.146 1.56

Note: The results reported in Tables 4 and 5 are the average of five values. Standard deviations are also given.

SHARMA ET AL. ON ELASTIC-PLASTIC FRACTURE MECHANICS 439

Materials Performance and Characterization

are obtained at the middle location and minimum values at the lower location of

bone diaphysis for both orientations of loading. The upper diaphysis gives the inter-

mediate values of these properties. The maximum and minimum values of elastic

modulus for longitudinal orientation of loading are 27.3 GPa and 17.2GPa, respec-

tively, as per Table 2, and for transverse orientation of loading are 17.1GPa and

12.4GPa, respectively, as per Table 3. Similarly, for yield strength, the maximum and

minimum values for longitudinal orientation of loading are found to be 114.3MPa

and 77.4MPa, respectively, and for transverse orientation of loading are 80.6MPa

and 45.4MPa, respectively. It may also be noticed from the above tables that the av-

erage values of elastic modulus and yield strength in longitudinal orientation are,

respectively, 1.4 to 1.7 times higher as compared to the corresponding values in

transverse orientation. The maximum ratio of elastic modulus for longitudinal to

transverse orientation (E1/E2) is found to be 1.7 for the upper location and mini-

mum, i.e., 1.4 for the lower location of the bone diaphysis. For the case of yield

strength, the maximum ratio (rysl /rys

t ) was found to be 1.7 for the lower location and

minimum, i.e., 1.4 for the middle location of the bone diaphysis. It is interesting to

note that the ratio of elastic modulus (E1/E2) for human cortical bone obtained in

various studies ranges from 1.5 to 1.7 [26–30]. Yan et al. [22], in their study,

obtained the ratio of elastic modulus (E1/E2) to be 1.4 for bovine femoral bone.

These values show a close resemblance with the results obtained in the present

investigation for buffalo femoral bone.

The variation in mechanical properties of cortical bone for different loading ori-

entations and locations of bone specimens is because of anisotropic and heterogene-

ous nature of bone material. The main cause of anisotropic nature of bone material

is considered to be the non-longitudinal axial distribution of orientation of bone

minerals [31]. The heterogeneity associated with bone material is because of orienta-

tion of collagen fibrils, distribution of minerals, porosity, and density at different

locations of bone diaphysis. This is evident from various studies that factors such as

variation in porosity, density, and distribution of collagen fibrils, mineral contents,

etc. may lead to variation in mechanical properties of cortical bone [32–40].

EFFECT OF ORIENTATION AND LOCATION ON FRACTURE PROPERTIES

OF CORTICAL BONE

The fracture properties of buffalo femoral cortical bone as obtained by the CTOD

approach for different locations of bone diaphysis and orientations of cracking are

listed in Tables 4 and 5, respectively. The fracture properties are found to be better

for transverse orientation as compared to the longitudinal orientation of cracking.

The CTOD toughness (dc) values are observed to be 1.5 to 2.1 times higher in trans-

verse orientation as compared to the longitudinal orientation of cracking. Similarly,

equivalent fracture toughness (Kdc) and J-toughness (Jcd) values are reported to be,

respectively, 1.8 to 2.3 times and 2.1 to 3.8 times higher in transverse orientation as

compared to the longitudinal orientation of cracking. The better values of fracture

properties for transverse orientation of cracking may be because of the orientation

of bone lamellae and arrangement of vascular networks in between these lamellae

and the arrangement of collagen fibrils.

The maximum values of CTOD toughness (dc), equivalent fracture toughness

(Kdc), and J toughness (Jcd) for longitudinal orientation of cracking as reported in

SHARMA ET AL. ON ELASTIC-PLASTIC FRACTURE MECHANICS 440

Materials Performance and Characterization

Table 4 are 0.025 mm, 8.25MPa � m1/2, and 3.99 kJ/m2, respectively, and for trans-

verse orientation of cracking as per Table 5 are 0.037 mm, 14.97MPa � m1/2, and

8.34 kJ/m2, respectively. The maximum values of fracture properties are obtained at

middle location of the bone diaphysis for both the cases. The minimum values of

fracture properties (dc, Kdc, and Jcd) as listed in Tables 4 and 5 are 0.014mm,

3.93MPa � m1/2, and 1.25 kJ/m2, respectively, for longitudinal orientation of cracking

and 0.030mm, 8.96MPa � m1/2, and 4.72 kJ/m2, respectively, for transverse orienta-

tion of cracking. These values are obtained at the lower location of bone diaphysis

for both cases. The results shown in Tables 4 and 5 reflect that the values of CTOD

toughness (dc) are 1.8 and 1.2 times higher as compared to the corresponding values

measured, respectively, at the lower and upper locations of the bone diaphysis. The

equivalent fracture toughness (Kdc) values at upper and lower diaphysis are found to

be decreased by a factor of 1.5 and 2.1, respectively, for longitudinal orientation of

cracking and by 1.2 and 1.7 times, respectively, for the transverse orientation of

cracking whereas the J-toughness values at these locations are reported to be

decreased by a factor of 2.0 and 3.2, respectively, for longitudinal orientation of

cracking and by factors of 1.3 and 1.9, respectively, for transverse orientation of

cracking as compared to the values measured at the middle location of bone

diaphysis.

The arrangement of collagen fibrils and mineral particles at different locations

of bone diaphysis and the amount of mineralization and crystallinity may result in

locational variation of fracture properties along the bone diaphysis. Nalla et al. [41]

observed that fracture occurs more readily when the crack is oriented parallel to the

direction of fibers in dentin. Goldman et al. [38] observed that an increase in the ho-

mogeneity of collagen fiber orientation may decrease the toughness of bone as crack

will move a greater distance before hitting a perpendicular fiber. As per the study of

Currey [40], the range of mineralization results in an even greater range of mechani-

cal properties and very high values of mineralization may result in low values of

work to fracture.

The other factors that may be responsible for this variation in fracture proper-

ties are apparent density and porosity. Orias et al. [42] in their study have shown the

locational distribution of apparent density along the entire length of human cortical

bone and found that the distribution of apparent density was at a maximum near

the mid-location of femoral bone diaphysis. Yeni et al. [9], from their study, sug-

gested that fracture properties increase with increasing apparent density in human

cortical bone. These two observations reflect that because of higher density at the

mid-diaphysis location fracture toughness values may be higher. The porosity in

bone material because of plexiform vascular spaces, blood vessels, etc. at different

locations of bone diaphysis may be different and also result in local variation of frac-

ture properties. Some researchers have observed the effect of porosity on mechanical

and fracture properties of bone [43–46]. Yeni et al. [43] observed that the fracture

toughness of human cortical bone in mode I and mode II conditions significantly

decreases with increasing porosity. In a recent study, Tang and Vashishth [46]

observed the effect of porosity on bone fragility and reported that the propagation

toughness (R-curve slope) of bone reduces with increasing porosity. The above

observations show that the combined effect of all these factors may be responsible

for the locational variation in fracture properties of cortical bone.

SHARMA ET AL. ON ELASTIC-PLASTIC FRACTURE MECHANICS 441

Materials Performance and Characterization

SEM ANALYSIS OF FRACTURE SURFACE IN TWOORIENTATIONS

OF CRACKING

The SEM images of fracture surfaces from longitudinal and transverse fractured

specimens at different magnifications are shown in Figs. 9 and 10, respectively. The

microstructure of these specimens is mainly plexiform type, which is found predom-

inantly in large and rapidly growing animals. The bone lamellae and vascular plex-

uses are clearly visible on the fracture surfaces of these specimens. The SEM images

show that the fracture surface of a longitudinally fractured specimen is much

smoother than that of the transverse fractured specimen. This is because the bone

lamellae and plane of vascular networks are mainly oriented along the long axis of

bone; therefore, in the case of longitudinal fracture, a crack can easily propagate

along the interfaces between lamella and in the plane of vascular networks produc-

ing a smooth fracture surface. This leads to a less amount of energy consumption in

this case. In the case of transverse fracture, the fracture surface is found to be

FIG. 9 SEM images of longitudinal fractured specimen obtained from mid-diaphysis of the cortical femoral bone at different

magnifications. White arrow in image (a) indicates the direction of crack propagation. In image (b) and (c), white arrows

indicate the vascular networks and blood vessels. Black arrows in image (b) indicate the micro-cracks, and in image (c)

different bone lamellae.

SHARMA ET AL. ON ELASTIC-PLASTIC FRACTURE MECHANICS 442

Materials Performance and Characterization

rougher and uneven. This is because the blood vessels, those visible on the fracture

surface, may serve as barrier to crack growth or even arrest the propagation of crack.

The other factors, such as micro-cracking, crack deflection because of weak interfa-

ces, and bonding between organic matrix and apatite crystals may also force the

crack to follow a tortuous, zigzag path leading to significant amount of energy con-

sumption in the case of a transverse orientation of fracture.

Conclusions

The present investigation was conducted to investigate the variation in fracture

properties at three different diaphysis locations of buffalo femoral cortical bone

using CTOD testing method based on the elastic-plastic fracture mechanics. As per

this study, the plastic part of CTOD (dp) was observed to be significantly higher as

compared to the elastic part of CTOD (de) for all three different locations of bone

FIG. 10 SEM images of transverse fractured specimen obtained from mid-diaphysis of the cortical femoral bone at different

magnifications: (a) is the angle-view image of the fracture surface, and the white arrow represents the direction of

crack propagation. In images (b) and (c,) white arrows indicate the blood vessels and black arrows indicate the

micro-cracks.

SHARMA ET AL. ON ELASTIC-PLASTIC FRACTURE MECHANICS 443

Materials Performance and Characterization

diaphysis. The minimum and maximum values of fracture properties such as CTOD

toughness (dc), equivalent fracture toughness (Kdc), and J toughness (Jcd) were found

at the lower and middle locations of bone diaphysis, respectively, for both the orien-

tations of fracture. The bone lamellae and plane of vascular networks were found

mainly along the long axis of bone producing a smooth fracture surface in case of

longitudinal orientation of cracking. The fracture surface of transversely fractured

specimen was found to be rough and uneven as compared to the longitudinally frac-

tured specimen. This was because the orientation of blood vessels, combined effect

of other factors such as micro-cracking, crack deflection because of weak interfaces

and bonding between organic matrix and apatite crystals. These factors may serve as

barrier to the crack growth in case of transverse orientation of cracking. The overall

results show that elastic-plastic fracture mechanics approach is a better technique to

determine the fracture properties of bone material as compared to linear elastic frac-

ture mechanics.

ACKNOWLEDGMENTS

The writers extend their thanks to the anonymous reviewers and the editorial board

members for their valuable comments and suggestions for the improvement of the

paper.

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