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Al-Khwarizmi 

Engineering   
Journal 

 

               Al-Khwarizmi Engineering Journal, Vol. 15, No. 2, June , (2019) 

P.P. 44- 49 

 

  
Study of Transverse and Longitudinal Crack Propagation in Human 

Bone Using the Finite Element Method with MATLAB 
 

 Abdullah Dhayea Assi   
 Institute of Technology / Baghdad  

Email: drabduallh_dhayeaa@yahoo.com 

 
(Received 7 August 2018; accepted 7 January 2019) 

https://doi.org/10.22153/kej.2019.01.001 

 
 

Abstract 

 
A finite element is a study that is capable of predicting crack initiation and simulating crack propagation of human 

bone. The material model is implemented in MATLAB finite element package, which allows extension to any 

geometry and any load configuration. The fracture mechanics parameters for transverse and longitudinal crack 

propagation in human bone are analyzed. A fracture toughness as well as stress and strain contour are generated and 

thoroughly evaluated. Discussion is given on how this knowledge needs to be extended to allow prediction of whole 

bone fracture from external loading to aid the design of protective systems. 

 

Keywords: Bone fracture, Fracture toughness, Stress intensity factor. 

 
 

1. Introduction 
 

Bone studies have been a subject of interest in 

biomechanics for more than 100 years. Many 

authors have studied in this field starting by 

(Melvin, 1993) [1] Who studied the surface 

cracks and found that the surface crack 

significantly reduces absorbed energy during the 

fractured bone fracture, where the fractured 

transverse fractures in the bone marrow and 

fractured bone are measured. The work in this 

filed still continued and updated, also evaluated 

the fracture toughness value experimentally in 

transverse and longitudinal directions. 

Studies of mechanical properties of bones are 

important for many reasons. 

First, predict how you expect the bone to behave 

in the body. 

Second, to explain the behavior of bone as a 

substance, resulting in an understanding of why 

a given bone structure gives you its properties. 

Finally, to ensure a viable system in case of 

substituted material for bone (Moyle & 

Bowden,1984)[2]. 

This studied will be taken in the consideration 

previous experiment studied and finite element 

analysis approach will be used for comparison 

reason and to approve that this technique can be 

used in future work instead of experiment 

methods which finally will take us to faster 

improvement and development. 

 

 

2. Fracture Mechanics of Bone 
 

The aim of fracture mechanics is to describe 

the fracture process in the structural material by 

determining the relationship between the stress 

field at the tip of the crack and the spread of the 

crater. In the other meaning the resistance of the 

material against rapid crack propagation which 

known as critical stress intensity factor (Kc) or 

fracture toughness [1]. Stress intensity factor is 

depending on the load configurations and 

geometry of the body as shown ie Eq.1: 

� � �√� ∗ ��	�ˎ��                                   ... (1) 
 



Abdullah Dhayea Assi                  Al-Khwarizmi Engineering Journal, Vol. 15, No. 2, P.P. 44- 49 (2019)  

 

45 

 

At the fracture, stress intensity factor is 

considered as material property and Eq.1 will be 

rewrite to be as shown in Eq.2: 

�
 � ��� ∗ �� �	�ˎ��                               … (2) 
Also there is very important parameter for this 

characterizing is called critical strain energy 

release rate (Gc), which means the energy release 

due to crack propagation per unit of thickness. 

Also, it is considered as material property at the 

fracture and can be calculated based on Eq.3 and 

4 for plane stress and plane strain respectively: 

G� � K
�/E    Plane Stress                                 … (3)  

G� � K
�/E	1 − ν��    Plane Strain             … (4) 

In order to study the fracture mechanics of 

any material there are several types of tests have 

been established for this purpose. The composite 

material can be considered as two-phase, one-

phase mineral, collagen and terrestrial content as 

another stage. From the other side bone is 

considered as viscoelastic material that means it 

is sensitive to the speed at which the load is 

applied. In general, the faster the strain rate the 

higher the stiffness. For these reasons bone has 

some properties as creep, stress relaxation and 

hysteresis. 

Also the bone is classified as anisotropic 

material that means it has different mechanical 

properties when loaded along different axes due 

to structure of the bone is dissimilar in the 

transverse and longitudinal directions. Most 

fracture in bone structure occurred as a results of 

several loading tension, compression, bending, 

torsion and shear or combination between two 

types of them. For example, the type of fracture 

that occurs in tensile is the transverse fracture 

while in the oblique fractional compression [3]. 

 

 

3. Bone Fracture Experiments 
 

There is several number of standard test for 

this purpose, for example Three points bend test, 

Four points bend test, and Crack bridging 

experiment. 

A number of experimental studies have 

investigated the fracture toughness (Kc) of the 

human bone. As the results of these experiments 

the fracture toughness (Kc) of the human bone is 

to be in the range of 2-8 MPa m1/2. In 

experimental study, the fracture toughness value 

(Kc) for transverse crack is about 5 MPa m1/2 

while it in longitudinal crack is about 4 MPa m1/2 

(Nalla, Kinney & Ritchie, 2003) [4]. 

 

 

4. Finite Element Analysis 
 

In this study, the finite element analysis 

approach is used to model and simulate the 

behavior of the bone by using MATLAB package 

[5]. Two cases are considered in this study, three 

points bend and compact tension specimen. 

 
a) Three Points Bend 
 

The objective of this case is to study the 

longitudinal crack where the load is parallel to 

crack direction as shown in Figure-1, the model 
used by Akram et al. [6] was employed. 
 

 
 

Fig. 1. Longitudinal crack case. 

 

 

In this case three forces act on the structure, 

produce two equal moments, each being the 

product of one of the two peripheral force and 

distance to the axis of rotation. 

After the material is implanted into MATLAB 

program and run this model for several loads as 

shown in Table-1 the result of fracture toughness 

value (Kc) is 5.98 MPa m1/2. Figure-2 shows of 

one case run by MATLAB program. 

 

 

 



Abdullah Dhayea Assi                  Al-Khwarizmi Engineering Journal, Vol. 15, No. 2, P.P. 44- 49 (2019)  

 

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Fig. 2. Longitudinal crack case from MATLAB program. 

 

 

Table 1, 

List of fracture toughness value longitudinal crack. 

S* Load (P) 

KN 

Kc MPa.m0.5 Stress (σ) 
MPa 

1 6 1.947  3.7 

2 10 3.841 6.16 

3 14 5.72   8.63 

4 20 7.689 12.3 

5 28 10.703 17.2 

 15.6 5.98 9.598 

 

 

 
 

Fig. 5. Stress at crack tip versus with Kc for case 

(a)  

 
b) Compact Tension 

 

The objective of this case is to study the 

transverse crack where the load is perpendicular 

to crack direction as shown in Figure-3. 

 

 
 

Fig. 3. Transverse crack case. 
 

 

In this case two opposite forces act on the 

structural to open the initial crack on the 

structural. 

Also after the material is implanted in 

MATLAB program and run this model for 

several loads as shown in Table-2, the result of 

fracture toughness value (Kc) is 7.15 MPa m1/2. 

Figure -4 shows of one case run by MATLAB 

program. 

 

 



Abdullah Dhayea Assi                  Al-Khwarizmi Engineering Journal, Vol. 15, No. 2, P.P. 44- 49 (2019)  

 

47 

 

 
 

Fig. 4. Transverse crack case from MATLAB program. 

 
Table 2, 

List of fracture toughness value for transverse 

crack. 
S* Load (P) 

KN 

Kc 

MPa.m0.5 
Stress (σ) 
MPa 

1 40 4.77 206 

2 50 5.96 258 

3 60 7.16 310 

4 70 8.34 362 

5 80 9.54 413 

 60 7.15 309.8 

  
 

Fig. 6. Stress at crack tip versus with Kc for case 

(b). 

 

Isotropic material properties were used 

follows: σyield = 85, σultimate =120, E=18000 
(MPa). The weight was regarded as a boundary 

condition. Segment weights expressed in 

percentage of total body weight. Were used [3]. 

In comparison to peer studies such as Jaafar et al. 

[7], We note that the results are very close 

 

  

 

 
 

Fig. 7. Stress-Strain Curve of Human Bone.  
 



Abdullah Dhayea Assi                  Al-Khwarizmi Engineering Journal, Vol. 15, No. 2, P.P. 44- 49 (2019)  

 

48 

 

5. Conclusion 
 

As mentioned previously, the main objective 

of this study is to evaluate the by the fracture 

toughness by using Finite Element Analysis 

approach (MATLAB program). The results of 

this study are compared with the previous 

experimental results to check the possibility of 

using this approach in future work. 

The results of Finite Element Analysis using 

MATLAB program found are consistent with the 

experimental results that give the fracture 

toughness of the bone to be between 2-8 MPam1/2 

and to be higher in transverse direction. The 

highest fracture toughness direction is when the 

crack path deflects at 90 degree to the plane of 

maximum tensile stress. Also as shown in Figure 

(5,6) the stress at the crack tip in Transverse 

crack is higher than in longitudinal crack. 

 

 

Nomenclature 
 

a Crack Length 

E Modulus of Elasticity 

KC Stress Intensity Factor (Fracture 

Toughness) 

P Load 

 

Greek Symbol 
 

σ Stress at any time 
σf Fracture Stress 
υ Poisson's Ratio 
af Crack Length at fracture 

6. References 
 

[1] J.W.Melvin, "Fracture Mechanics of Bone", 
Journal of Biomechanical Engineering, Nov 

1993. 

[2] P.D.Moyle & R.W.Bowden "Fracture of 
Human Femoral Bone", Journal of 

Biomechanical Engineering, 1984. 

[3] Thalamann NM Kang MJ, Goto T: 
Problems and Solutions for the Accurate 3D 

Functional Modeling of the Hip and 

Shoulder. Proc. of IEEE Computer Graphics 

International. 2002; 
[www.miralab.unige.ch/papers/135.pdf] 

[4] R.K Nalla, J.H.Kinney & R.O. Ritchie, 
"Mechanistic Fracture Criteria for the 

Failure of Human Cortical Bone", Neutral 

Publishing Group, 2003. 

[5] Alfio Quarteroni and Fausto Saleri, 
"Scientific Computing with MATLAB and 

Octave" Second Edition, Springer-Verlag 

Italia, Milano, 2006. 

[6] Jaffar AA, Abbas SJ & Abdullah HM: 
Stress analysis of the hip bone. Al-

Khwarizmi Engineering Journal. 2005; 
1:61-72. 

[7] Dr. Akram Abood Jaffar , Dr. Sadiq Jaffar 
Abass , Mustafa Qusay Ismae: 

Biomechanical Aspects of Shoulder and Hip 

Articulations: A Comparison of Two Ball 

and Socket Joints, Khwarizmi Engineering 

Journal. 2006; 1: 1-14. 
 

 

 



 )2019( 44- 49، صفحة 2د، العد15دجلة الخوارزمي الهندسية المجلم                                         عبد هللا ضايع عاصي                 

  

49 

 

  

  

  العناصر المحدودةدراسة نمو الشق الطولي والعرضي لعظم االنسان باستخدام طريقة 
  طة الماثالبابوس 

  

  عبدهللا ضايع عاصي 
  تكنولوجيا/ بغدادال معهد

drabduallh_dhayeaa@yahoo.com :البريد االلكتروني  

 
 

  

  الخالصة 
  

ان النموذج المادي مطبق  ته في العظم االنساني.أتوليد الشق ونش ةالشق ومحاكاان اي دراسة باستخدام طريقة العناصر المحدودة قادرة على توقع نمو 
حمل مما يمكن عوامل ميكانيك الكسر  هطة طريقة العناصر المحدودة التي تسمح لالمتداد الى اي شكل هندسي واي ترتيبافي رزمة من برنامج الماثالب بوس

مخطط االجهاد واالنفعال يولدان ويقيمان كليا. ان المناقشة اعطت  فضالً عنمن تحليل توليد الشق الطولي والمستعرض في العظم االنساني .ان صالبة الكسر 
ً كمَ    الخارجي لمساعدة تصميم االنظمة الوقائية.من المعرفة الضرورية التي تمدد للسماح لتنبؤ بالكسر العظمي الكامل من التحميل  ا