Rheinisch-Westfälische Technische Hochschule Aachen Institut für Allgemeine Mechanik der RWTH Aachen. Univ.-Prof. Dr.-Ing.

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1 Rheinisch-Westfälische Technische Hochschule Aachen Institut für Allgemeine Mechanik der RWTH Aachen Univ.-Prof. Dr.-Ing. Bernd Markert Bachelor project Variation in the mechanical properties of the mandibular bone Submitted to: The German University in Cairo Submitted by: Marina Boshra Ayoub ID: Supervised by: M.Sc. Mohammad Shehadeh 1

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3 Rheinisch-Westfälische Technische Hochschule Aachen Institut für Allgemeine Mechanik der RWTH Aachen Univ.-Prof. Dr.-Ing. Bernd Markert Bachelor project Variation in the mechanical properties of the mandibular bone Submitted to: The German University in Cairo Submitted by: Marina Boshra Ayoub ID: Supervised by: M.Sc. Mohammad Shehadeh 3

4 Certificate This is to certify that I-The thesis comprises only my original work towards the Bachelor Degree. II- Due acknowledgement has been made in the text to all other material used. Marina Boshra 18 August,

5 Acknowledgments First, I am so grateful to God for giving me strength and enlightenment to finish this book. I would like to thank my supervisor M.Sc. Mohammad Shehadeh for his patience and encouragement in every step, and I really appreciate all the experiences shared with him in this project. I want to give thanks to all of my friends from the other faculties who lend me a hand in programming, simulation and understanding biological issues. Finally, I want to thank my parents for believing in me and for giving me the opportunity to make my bachelor project in RWTH Aachen University. 5

6 Abstract The aim of this study is to determine the effect of the variation in the mechanical properties on the human mandible. The mechanical properties in the lower jaw bone were investigated from different literatures. It was necessary to divide the mandible into three parts: the symphysis, the corpus and the ramus, to study the variation in the mechanical properties based on the location. The bone was treated as non-homogenous material, which is different in the three parts of the mandible, and orthotropic material, which is different in the three orthogonal dimensions. Finite element method software (ABAQUS) is used to simulate the deformation and different behavior of the mandibular jaw on applying external load (premolar bite). The hinge movement of the temporomandibular joint, during opening and closing of the mouth, was taken in consideration. After simulation, maximum stresses took place in the ramus region and maximum strains took place in the premolar region. 6

7 Contents 1 Introduction Background Composition Mechanical properties Elastic modulus Coritcal bone Trabecular bone Density Stresses acting on the mandible Muscle forces Joint reaction forces Occlusion forces Resistance of deformation Sagittal bending Lateral transverse bending and Torsion of the mandibular corpus Materials and methods Working with programmed ABAQUS Modeling and analysis using ABAQUS/CAE The part module Thickness Depth Horizontal distance The property module Material Section assignment Orientations The assembly module The step module The interaction module Tie constraints Coupling constraints The load module Load Boundary conditions

8 3.2.7 The mesh module Mesh controls Element type Units Results Trial Trial Trial Trial Discussion Sandwich theorem Limitation Strength Conclusion References

9 List of Figures Figure 1 The human mandible...13 Figure 2 Bones with different function differ strongly in shape. Long bones (such as the femur, left) provide stability against bending and buckling. Short bones (such as the vertebra, center) provide stability against compression along the vertical axis, in the case of the vertebra).plate-like bones (such as the skull, right) protect vital organs Figure 3 Hierarchical structural organization of bone: (a) cortical and cancellous bone; (b) osteons with Haversian systems; (c) lamellae; (d) collagen fiber assemblies of collagen fibrils; (e) bone mineral crystals, collagen molecules, and non-collagenous proteins Figure 4 The human mandible Figure 5 The trabecular bone Figure 6 Mandible sections in region 1(incisors and canines), region 2(premolars) and region 3(molars) Figure 7 The Elastic modulus values for the three regions...21 Figure 8 CT image for the mandible Figure 9 Mathematical model of the mandible. The origin is taken in the center of the condylar, the x-axis is parallel to the occlusion bite plane. t.p= posterior temporalis, t.a= anterior temporalis, pt.l.=lateral pterygoid, j= joint force, m.+pt.m.= masseter +medial pterygoid,o.=opener, b= bite force, P1, M1,M2 = first premolar,first molar, second molar Figure 10 The muscles of mastication Figure 11 The temporomandibular joint...27 Figure 12 The bending and torsional moments acting on the mandible

10 Figure 13 Stress in a curved beam as a result of a lateral transverse bending load Figure 14 The three spatial reference planes of the mandible...31 Figure 15 (a) Running script in Abaqus Figure 15 (b) Running script in Abaqus Figure 16 Running macro files in Abaqus Figure 17 (a) Top view of the model Figure 17 (b) Side view of the model Figure 18 (a) Creating partition in Abaqus...38 Figure 18 (b) The three partitioned sections of the human mandible.38 Figure 19 The x-axis (radial direction), y-axis (circumferential direction) [2] (a) lingual side of the mandible, (b) buccal side of the mandible and (c) the inferior border of the mandible Figure 20 (a) Assigning material orientation in Abaqus Figure 20 (b) The assigned orientation.. 39 Figure 21 Monitoring the analysis 40 Figure 22 The applied constraints on the model Figure 23 The applied loads and boundary conditions on the model...43 Figure 24 (a) Ramus mesh before modification Figure 24 (b) Ramus mesh after modification Figure 25 The meshed model Figure 26 (a) The model before modification...46 Figure 26 (b) The model after modification Figure 27 (a) The model with both cortical and trabecular bones...47 Figure 27 (b) The model with trabecular bone.. 47 Figure 28 (a) Stresses of the mandibular bone..48 Figure 28 (b) Strains of the mandibular bone Figure 29 (a) Stresses of the mandibular bone Figure 29 (b) Strains of the mandibular bone

11 Figure 29 (c) The hinge movement of the temporomandibular joint 50 Figure 30 Stress-strain curve of cortical and trabecular bone Figure 31 Stress concentration region in ramus

12 List of Tables Table 1.The origin, insertion and the action of the muscles contributing in The mastication process 26 Table 2.The elastic coefficients for cortical and trabecular bone of human mandible...36 Table 3.Stiffness coefficients for human mandible...37 Table 4.Density of cortical and trabecular bone in human mandible 38 12

13 Chapter 1 Introduction The mandible, a U-shaped bone, is the only mobile bone of the facial skeleton; and since it houses the lower teeth, its motion is essential for mastication. The mandible is composed of 2 hemimandibles joined at the midline by a vertical symphysis (see Fig.1). The hemimandibles fuse to form one single bone by the age of 2 years. Each hemimandible is composed of a horizontal body, which is called the corpus and a posterior vertical extension, which is called the ramus [10]. Fig.1 The human mandible [16] In this study, the mandible is divided into three parts: the symphysis, the corpus, and the ramus; to make the study of the variation in the mechanical properties easier. The mandible is assumed to be a rigid body, in which deformation is neglected, but due to external forces, some stresses and strains are introduced to it. Bone composition, density and mechanical properties are the main factors that determine the distribution of these stresses and strains [20]. Bones are created with an extremely interesting ceramic composite whose components are primarily collagen (organic protein) and hydroxyapatite (inorganic mineral).they are combined to provide a mechanical and supportive role [26,28,30,33]. Depending on the orientation of collagen fibers, two types of bones can be distinguished: lamellar bone (cortical bone) and non-lamellar (trabecular or cancellous bone) [27, 28].The composition of the cortical bone is different than that of the trabecular bone. The cortical bone is a stiff and brittle material, which is composed of unidirectional uniform oriented fibrils. Its density is high, with porosity ranging between five to ten percent. 13

14 In contrast, the trabecular bone is less dense, with porosity ranging from 50% to 90%, and it can be considered as a foam-like network of bones. It is formed of less wellorganized packets of fibrils to form a network of rods and plates, interspersed with large marrow spaces. These different structures are the main reasons for having different densities within human jaw [5, 26, 33]. The density also varies along the mandible, its value was measured to be 1.18 gm/cm3 [19]. The apparent density in the anterior human mandible is higher than that of the posterior [12, 19]. Some researchers explained the importance of this feature in withstanding forces and moments acting on the mandible [20, 22]. Previous reviews showed a change in density as well as other mechanical properties of the human mandible with age and gender, which are neglected in this paper [12]. Because of the variations of the bone composition and densities along the mandible, it is difficult to treat bone as a homogenous, isotropic material. The mechanical properties of each part is studied, and showed significant difference in the three parts of the mandible in the three orthogonal directions. In general, the cortical bone showed increase in stiffness values as it approaches the ramus, and the trabecular showed increase in stiffness values as it approaches the symphysis [15, 18]. Both cortical and trabecular bones have higher stiffnesses that reach 24.4 and GPA respectively, in the longitudinal direction E3 (mesio-destal direction) compared to the other two directions E2 and E1 (circumferential and radial directions) [59]. As mentioned, the mandible is subjected to external forces that introduce strains in the human mandible. The forces that act on the bones are muscle forces, joint forces and biting forces [20]. According to the law of statics, the three vectors of the total muscle force resultant, the joint force and the bite force must meet in one point to satisfy the equilibrium condition of the mandible. Since the directions of the joint forces and the bite forces are fixed by means of splint construction, the direction of the vector of total muscle force resultant will depend only upon the position of the biting force [37]. The magnitude of the biting force shows increase by a factor of nine from the anterior teeth (100.08N) to the posterior teeth (900.76N). The mean maximum moment shows linear relation with the occlusion forces from incisors to first molar except for the second molar [40]. The highest bite forces are exerted when biting takes place in the position of first molar [37, 39, 44]. Maximum joint forces are found, while biting, in 14

15 the first premolar and first molar positions. The joint forces vary from 399 to 1118N per side [37]. Regarding the muscle forces, muscles of mastication include four pairs of muscles: masseter, temporalis, medial pterygoid, and lateral pterygoid muscles, in the right and the left side of the skull [36, 43]. The masseter, temporalis, and medial pterygoid muscles are the main closers of the human jaw [40]. The temporalis muscles, positioning muscles, do not participate in biting [47]. They elevate the mandible to the position of having real forces, such as biting, by the other two closure muscles (masseter and medial pterygoid).the total maximum force exerted by the temporalis muscle equals 1120N [37].The other two closure muscles exert total force, equivalent to them, equals 1279N [37].The opening movement of the jaw depends mainly on the digastric and the lateral pterygoid muscles. The maximum forces were measured for the lateral pterygoid and the other opener muscle and were found to be 757 and 230 N, respectively [37]. During biting and power stroke of mastication, a combination of sagittal bending, corpus rotation, and transverse bending occurs [20]. This paper will mainly concentrate on the effect of the sagittal moment on introducing strains in the mandible. In the sagittal plane, the mandible is bent by the effect of vertical components of muscle forces, the reaction forces of the condylar and the bite forces; the highest moments occur in the ramus and adjoining areas of the corpus. Strain data on human mandible is scarce, but some studies managed to come up with results from monkeys' samples and FEM models. The model of Koroith predicted that the symphyseal region experiences higher magnitudes of the tensile strain on the lingual cortical surface ( µ ) than on the labial surface, and that the strains on the molar region are smaller than those at the symphysis [57]. Also, Throckmorton loaded the human mandible with artificial muscle forces of 600N and measured the strains along the corpus and it was up to 800 µ [58]. In vivo strain gauge studies on monkeys, strain values were demonstrated and it were equal to 1122, 1442, and2564 µ for maximum tensile, compression and shear strain values, respectively, during molar biting on transducer [6]. 15

16 A simulation is carried out using the finite element method software (ABAQUS). A simple model is used, with given dimensions based on previous literatures. The mandible is treated as a composite which is composed of three layers: two cortical layers and a trabecular layer. Also, the bone is treated as a non-homogenous material, which is different in the three parts of the mandible, and an orthotropic material, which is different in the three orthogonal dimensions. The orthotropic behavior is variable in each part of the mandible for the cortical bone and constant for the trabecular bone. Biting force (premolar biting) is the external load acting on the mandible to study the reaction forces in the ramus and the deformation on the mandible. Boundary conditions are adjusted to express the hinge movement of the temporomandibular joint during the opening and closing of the mouth. 16

17 Chapter 2 Background 2.1 Composition The bone is the principle structural tissue of the skeletal system, which is used to carry loads among vertebrates. Bones can have different shapes depending on their specific functions such as the ones shown in Fig. 2 [29]. For its special role, it was created with an extremely interesting ceramic composite, whose components are primarily collagen (organic protein) and hydroxyapatite (inorganic mineral). They combine to provide a mechanical and supportive role in the body [26, 28, 30, 33]. Fig.2 Bones with different function differ strongly in shape. Long bones (such as the femur, left) provide stability against bending and buckling. Short bones (such as the vertebra, center) provide stability against compression along the vertical axis, in the case of the vertebra). Plate-like bones (such as the skull, right) protect vital organs [29] Regarding the weight, around 30% of bone is composed of organic compounds. Collagen is a fibrous protein, which provides the bone with strength and flexibility, and is an important component of many other tissues, including skin and tendon. The other 70% of the bone is made up of 60% inorganic mineral hydroxyapatite and 10% water, which is very important determinant of mechanical behavior to bone according to Currey study [35].The inorganic mineral hydroxyapatite includes calcium phosphate, calcium carbonate, calcium fluoride, calcium hydroxide and citrate. 17

18 Calcium is considered bone cement, which makes bone stronger [27, 31, 33]. Depending on the orientation of collagen fibers, two types of bone can be distinguished: lamellar bone (cortical bone) and non-lamellar (trabecular or cancellous bone) [27, 28]. Bone has a complex structure that is described as a hierarchical structure in number of reviews [29, 33, 34]. On the other hand, Rho used Scaling method for discussing the bone architecture at the various levels of structural organization.these levels and structures are: (1) the macrostructure: cancellous and cortical bone; (2) the microstructure (from 10 to 500 mm): Haversian systems, osteons, single trabeculae; (3) the sub-microstructure (1 10 mm): lamellae; (4) the nanostructure (from a few hundred nanometers to 1 mm): fibrillar collagen and embedded mineral; and (5) the subnanostructure (below a few hundred nanometers): molecular structure of constituent elements, such as mineral, collagen, and non-collagenous organic proteins. This hierarchically organized structure makes the material of bone heterogeneous and anisotropic [30] (see Fig.3). Fig.3 Hierarchical structural organization of bone: (a) cortical and cancellous bone; (b) osteons with Haversian systems; (c) lamellae; (d) collagen fiber assemblies of collagen fibrils; (e) bone mineral crystals, collagen molecules, and non-collagenous proteins[30] On the nanometer scale, collagen molecules are arranged in parallel with each other (head to tail) with a gap or hole zone of approximately 40 nm between each molecule. These holes act as a template for the crystallization of hydroxyapatite nanocrystals, known as Mineralization, which extends also into other intermolecular spaces, forming mineralized fibril [33,34] (see Fig.3). 18

19 The collagen fibrils can arrange themselves to be either unidirectional uniform oriented fibrils to form lamellae or a block of randomly oriented woven fibrils. Woven bone is temporary; it is always replaced by lamellar bone by age. The most dominant type of cortical bone in adult humans is osteonal or Haversian bone, where about 10 to 15 lamellae are arranged in concentric cylinders about a central Haversian canal, a vascular channel about 50µm in diameter that contains blood vessel capillaries, nerves and a variety of bone cells (see Fig.3). The basic functional unit of mature cortical bone is the osteon or Haversian system. On the sub-micro scale, there are two types of bones: cortical bone which comes as tightly packed lamellar, Haversian, or woven bone; and trabecular bone, which comes as a highly porous cellular solid. In the latter, the lamellae are arranged in a less wellorganized packets to form a network of rods and plates interspersed with large marrow spaces. The cortical bone shell (found at the outer surface of each bone) can reach a thickness between several tenths of a millimeter (in vertebra) to several millimeters or even centimeters (in the mid-shaft of long bones). The thickness of the struts in the spongy trabecular bone is fairly constant between one and three hundred micrometers [26, 33] (see Fig 4 and 5). Fig.4 The human mandible [24] Fig.5 The trabecular bone [29] The distinction between cortical and trabecular bone is most easily made based on porosity. A cortical bone is dense with porosity ranges from 5% and 10%.The trabecular bone is less dense with porosity ranging from 50% to 90% and can be considered as a foam-like network of bone trabeculae. Lighter areas indicate more densely mineralized regions [5, 33].The bone is strongest in compression than in shear and tension [5, 20]. 19

20 2.2 Mechanical properties Understanding the mechanical properties in the mandible will be useful in the identification of the different behaviors of the bone under stresses and loads. The mandible has been treated as a rigid structure, in which deformation is neglected, but due to the external stress factor, some stresses and strains are introduced to it. The ranges and distributions of these stresses and strains depend on certain factors: the nature of external loading, material properties and geometry of the mandible, including the distribution of the bone tissue [20] Elastic modulus 1-Cortical bone Schwartz-Dabney measured the cortical bone elastic modulus by using ultrasonic velocities technique and its range was GPA with a mean value of 17.9 GPA [18]. Ashman, Dechow, Erkmen, and Verplancke showed also very similar results to that range [2, 6, 8, 21]. The stiffest cortical bone is found in the ramus part with range GPA with average value 20.5 GPA [8, 18]. Some different results for elastic modulus of cortical bone were found [14, 17]. The corpus has an intermediate elastic modulus of the cortical ranging between GPA with an average value of 18.9 GPA [18, 21], and symphysis with least stiffness range GPA with an average value of 16.9 GPA. From the previous deduced results, it is obvious that the elastic modulus of cortical bone increases as it approaches the ramus. The trabecular bone shows opposite results; it increases as it approaches the symphysis as shown in Fig.6 and 7. 2-Trabecular bone The cancellous (compact) bone appears to be highly anisotropic according to Giesen and Eijden studies [7, 20]. The cancellous bone is less dense than cortical bone [5]. Its Mechanical properties strongly depend on its apparent density [4, 20] and structure [9, 20].This will make it important to study the density of trabecular bone in order to understand its mechanical properties and also counting on the age parameter of the cadaver [12]. 20

21 The stiffness and the strength of the bone are proportional to apparent density. The bone is stiffest and strongest in the direction in which the trabeculae are aligned [7, 20]. Although many researchers tried to study trabecular microstructure, only a few have been conducted on human Jaw bone because high resolution scanner, particularly micro-ct, might be difficult to be applied on the trabecular bone [11]. The elastic modulus of compact bone can vary with huge ranges depending on the bone architecture, type of loading, trabecular density and orientation [20]. Also, Mazess relates the large accuracy errors in measuring trabecular bone to the marrow composition [23]. AM et al. used compression testing to determine elastic modulus of human trabecular mandible bone in three orthogonal directions [1]. It has greatest value in the mesiodistal (mean 907, SD±849, range MPA), due to the horizontal trabeculae orientation, followed by the bucco -lingual (mean 511, SD ±565, range MPA) and then infero-superior (mean 114, SD ±78, range MPA) directions. Another study divided the human mandible to three regions: region 1(incisors and canines), region 2(premolars) and region 3(molars) [15]. It was found that the elastic modulus increases towards symphysis in the range from MPA; the values were MPA, 47.30MPA, and 67.48MPA for the three regions, respectively [13, 15]. The elastic modulus of the condylar is equal to 438 MPA vertically and 157 MPA horizontally with small significant differences between researches [7, 13]. Van Eriden and Giesen relate the higher elastic modulus, in the axial direction, to the vertical orientation of the plate-like trabeculae in the axial direction. Fig.6 Mandible sections in region 1(incisors and canines), region 2(premolars) and region (molars) Density [15] Fig.7 The elastic modulus values for the three regions [15] 21

22 2.2.2 Density Archimedes principle was used to measure bone apparent density by using bone samples, which contain both cortical and trabecular bones. The apparent density of the mandible was measured to be 1.18 gm/cm 3 [19]. The apparent density ranges from 1.85 to 2.0 gm/cm 3 for the cortical bone and 0.33 to 0.55 gm/cm 3 for the trabecular bone [1, 7, 19]. For trabecular bones with bone marrow, higher range was found from 0.85 to 1.53 g/cm3 [15, 19, 23]. The apparent density in the anterior human mandible is higher than that of the posterior as the lateral bending moment increases from posterior to anterior, and reaches its highest magnitude near the symphysis [12, 19, 20]. Also, Blok suggested that forces in the anterior region are mainly oriented in one direction while in the posterior region forces from two or more directions have an influence on the trabecular morphology. The different structures in the human jaw are the main reasons for having different density within jaw. Another important parameter, on which the density depends, is the age. With age, the cross sectional area decreases and there is increase in the density as well as the porosity in the bones. Although, it is expected to have lower density with increasing the porosity in the bone, Kingsmill stated that this did not merely reflect an increasing proportion of cortex as the bone became smaller but the bone becomes consolidated with increasing age [12]. Fig.8 CT image for the mandible [25] 22

23 2.3 Stresses acting on the mandible Since the mandible is the only movable bone in the human skull, it is interesting to study the different forces acting on it. The mandible is subjected to muscles forces, joint reaction forces and biting forces [20]. According to the law of statics, the three vectors of the total muscle force resultant, the joint force and the bite force must meet in one point to satisfy the equilibrium condition of the mandible. Since the directions of the joint forces and the bite forces are fixed by means of splint construction, the direction of the vector of total muscle force resultant will depend only upon the position of the biting force [37]. Fig.9 Mathematical model of the mandible. The origin is taken in the center of the condylar, the x-axis is parallel to the occlusion bite plane. t.p= posterior temporalis, t.a= anterior temporalis, pt.l.=lateral pterygoid, j= joint force, m.+pt.m.= masseter +medial pterygoid,o.=opener, b= bite force, P1,M1,M2= first premolar, first molar, second molar [37] Muscle forces A Muscle is made up by a various number of parallel fibril bundles. These fibril bundles can contract about 57% of its fully stretched length. Muscles pull the mandible in the direction of its fibers. The activation of the fibers depends on the type of the effort required from the muscle; only few fibers contract (to its fullest extent) for weak effort. Many fibers contract for greater power needed. Every single movement requires an 23

24 activation of a number of muscles [43]. For example, high masseter activity is accompanied by low temporal muscle activity or vice versa at a particular bite force level. From this information, it is expected that some muscles can reach their maximum force activation before others. The forces exerted by contracting muscles can be represented by vectors, its direction can be defined by the connecting lines between the centroids of the origins and the insertions of the muscles (See Table1) [37]. Muscles of mastication include four pairs of muscles: masseter, temporalis, medial pterygoid, and lateral pterygoid muscles, in the right and the left side of the skull [36, 43]. The masseter, temporalis, and medial pterygoid muscles are the main closers [40]. The temporalis muscles, positioning muscles, do not participate in biting [47]. They elevate the mandible to the position of having real forces, such as biting, by the other two closure muscles (masseter and medial pterygoid).the temporalis muscles have two parts anterior and posterior part. The anterior part has vertical fibers, it is responsible for pulling the mandible up in its direction and close the mouth. The posterior part has horizontal fibers that pull the mandible back in its direction [43]. The maximum force exerted by temporalis anterior and posterior parts equals 724N and 395N for the left and right side together [37].The masseter and the medial pterygoid act as a sling, the former from the lateral side and the latter from the medial side of the ramus [43]. They provide powerful force together on closures. Pruim found that the maximum muscle force for both of them equals 1279N [37]. The opening movement of jaw depends mainly on the digastric and the lateral pterygoid muscles [41]. Unlike the other three pairs of muscles, the lateral pterygoid muscle has vertical fibers, which is responsible for pulling the mandible forward and down. The lateral pterygoid main action is to draw forward the condyle and articular disk so that the mandible is protruded and the inferior incisors are projected in front of the upper. It is also responsible for depressing the mandible [43, 46]. From the changes in the ICR trajectory, it was clear that the swing movement in the onset of the jaw-opening is controlled by the digastric muscle and the slide movement by lateral pterygoid muscle [41]. The maximum forces of the lateral pterygoid and other opener muscles equal 757 and 230 N, respectively [37]. 24

25 From previous values, it is obvious that the jaw-closing muscles are more powerful than the jaw-opening muscles; consequently, a relatively small amount of muscle activity is needed to close the jaw [41]. The average total muscle forces, from both sides, was examined by Mainland in three heads, it equals 2814 N [39]. Fig.10 The muscles of mastication [38] 25

26 Muscle Origin Insertion Action Temporalis Temporal fossa of Coronoid -Elevates the (vertical and temporal bone process of the mandible(vertical horizontal mandible fibers) fibers) -Retracts the mandible(horizontal fibers) Masseter Zygomatic arch Lateral(ramus- -Elevates the angle) of mandible mandible Medial Sphenoid bone Medial ramus -Elevates the pterygoid of mandible mandible Lateral Sphenoid bone Anterior side of -Depresses the pterygoid mandibular mandible condyle -Protrude the mandible One lateral Sphenoid bone Anterior side of -Excursive pterygpid mandibular movements condyle Digastric anterior belly - Intermediate -Depresses the digastric fossa tendon (hyoid mandible (mandible); bone) posterior belly - mastoid process of temporal bone Table 1. The origin, insertion and the action of the muscles contributing in the mastication process [43, 46] 26

27 2.3.2 Joint reaction forces The TMJ forces are highest when biting takes place in the first premolar. Maximum joint forces is found in the first premolar and first molar positions, it varies from 399 to 1118N per side. These forces seem rather high, but they are exerted on the articular disk not on the mandibular fossa [37]. The articular disk consists of collagenous fibers that helps in withstanding excessive pressure in the TMJ. The maximum compressive stress is MPA, it is located in the most dense area of the disk indicated. The maximum tensile stress is 3.97 MPA at the upper boundary of the middle portion of the disk. The mean principle stresses are ,-0.543, 0.664, and 0.521MPA in the anterior, middle, posterior, lateral and medial areas of the condylar, respectively [44]. The condylar movement is a combination of two types of motion: rotation and translation. The movement depends mainly on the instantaneous center of rotation (ICR). If the ICR is located close to the joint, the rotation motion takes place. Otherwise, the translation motion dominates [41]. The side of the mandible, in which biting takes place, is called the working side, while the other side is called the balancing side [43]. Fig.11 The temporomandibular joint [45] 27

28 2.3.3 Occlusion forces According to Mansour study, the magnitude of biting force increases by a factor of nine from the anterior teeth (100.08N) to the posterior teeth (900.76N). The mean maximum moment shows linear relation with the occlusion forces from incisors to first molar [40]. The second molar does not follow that relation; it has relatively low moment and biting force. In second molar position, the muscle force resultant is near to the bite position, and this case violates the mechanical equilibrium. Also, this may lead to decrease in muscle activity and bite force [37]. The highest bite forces are exerted when biting takes place in the position of first molar [37, 39, 44]. Pruim found that biting forces are 633N, 965N, and 756N for first premolar, first molar and second molar, respectively [37]. In other study, the ratios of occlusal forces to that of the first molars were 0.80, 0.66, 0.51, and for the second molar, premolars, canine and incisors, respectively [44].On the other hand, there are some studies who contradict this results; Mansour stated that the force at the second molar is approximately 10% more than at the first molar [40]. The maximum biting effort is found to be 304 N [51]. The forces exerted during swallowing are reported to be higher (293.2N) than chewing forces (132.4 N), due to the intercuspal occlusion position in swallowing, in which the cusps of the teeth of both arches fully interpose themselves with the cusps of the teeth of the opposing arch [50, 51]. Strain data on human mandible is scarce, but some studies managed to come up with results from monkeys' samples and FEM models. The model of Koroith predicted that the symphyseal region experiences higher magnitudes of the tensile strain on the lingual cortical surface ( µ ) than on the labial surface, and that the strains on the molar region are smaller than those at the symphyseal region [57]. Also, Throckmorton loaded the human mandible with artificial muscle forces of 600N and measured the strains along the corpus and it was up to 800 µ [58]. In vivo strain gauge studies on monkeys, strain values were demonstrated and they were equal to 1122, 1442, and 2564 µ for maximum tensile, compression and shear strain values, respectively, during molar biting on transducer [6]. 28

29 During biting and powerstroke of mastication, a combination of sagittal bending, corpus rotation, and transverse bending occurs [20]. Fig. 12 The bending and torsional moments acting on the mandible [20] Sagittal bending In the sagittal plane, the mandible is bent by the effect of vertical components of muscle forces, the reaction forces of the condylar and the bite forces. The magnitude of the sagittal bending moment depends on the point of the application and moment arm length. At the working side, where biting take place, the maximum shear stresses are in the region between bite force and muscle force. At the balancing side, the maximum stresses are in the region between the joint force and the muscle force. During symmetrical loading, as in incisal biting, the amount of sagittal bending is equal in both mandibular corpus. The result of the sagittal bending is that, at the balancing side, the lower margin of the mandible is in compression and the upper margin is in tension; at the working side the reverse bending moment occurs. On both sides, the highest moments occur in the ramus and adjoining areas of the corpus. 29

30 Torsion of the mandibular corpus The mandible rotates around its horizontal axis by the effect of the resultant elevator muscle force acting laterally to this axis and the bite force acting medially to this axis. The torsion results in an eversion of the lower border of the corpus and in an inversion in the upper border, leading to a narrowing in the dental arch [20, 54]. The torsion also results in a shear stress ττ = TT rr JJ, where T, r, j are the applied torque, the radius of the cross section, and the polar moment of inertia of the cross section. Lateral transverse bending The corpus is subjected to lateral bending by the effect of the laterally directed force component of jaw elevator muscles (masseter and temporal) and lateral component biting force. The lateral bending moment increases from posterior to anterior, and reaches its maximum value close to the symphysis [20, 52]. It results in a compressive stress at the buccal cortical surface of the mandible and to a tensile stress at the lingual surface. The amount of the stress can be approximated by σσ = KK MM DD/II, where M, D, I are the bending moment, the distance between the center of mass and the inner or outer surface, and the second moment of inertia, respectively. K is a factor that is applicable to curved beams. It has different values for the inner and outer surfaces. The value of K depends further on the radius of curvature (R) divided by the distance (d) from the center of mass to the inner surface [20]. Fig.13 Stress in a curved beam as a result of a lateral transverse bending load [20] 30

31 2.4 Resistance to deformation The material properties are not the only factors that matter in withstanding deformation, but the mandible has special geometrical features that also serve its mechanical purposes. The second moment of inertia (I) is the measure of the resistance of a crosssection to bending in both the sagittal plane (Iz) and horizontal plane (Ix). The polar moment of inertia (J) is the measure of the resistance of a cross-section to torsion. The bone cross sectional area is the measure of the ability of a cross-section to resist shear stresses parallel to its section, and axial stresses normal to cross-section. The relative amount of cortical bone within a cross-section can be expressed as cortical index. The cortical index measures the ability of the mandible to resist stresses and strains [20]. Fig.14 The three spatial reference planes of the mandible [55] Sagittal bending Since the second moment of inertia is the main responsible for resisting a bending moment in a certain plane, so increasing moment of inertia in sagittal plane Iz will be the best way to counteract the sagittal bending moment acting on the mandible. Increasing Iz can be done by increasing the cortical bone distribution in the superior and inferior regions of the mandible. Deagling 's study shows that the Iz value (57.7) is approximately three times larger than Ix value (17.8),which makes this cross sectional shape design a favor for resisting the sagittal bending moment during mastication and biting. Also, the large vertical dimension of the ramus can be considered as an efficient design to withstand that large moment [3, 20]. 31

32 2.4.2 Lateral transverse bending and Torsion of the mandibular corpus The lower margin of the symphysis and corpus are thicker than the upper margin. Also, the cortical bone is stiffer at the lower border of the corpus than that of the alveolar. This makes the tensile stresses acting on the lower margin less than the compression stresses acting on the upper margin, and this is important because bone is weaker in tensile than in compression [20]. The medial and the lateral sides of the corpus experience the maximum shear stresses. The ability of the corpus to resist torsion is directly proportional to the ratio between Iy and Ix. The most efficient design to resist torsion is circular cross section, but the elliptical cross sectional geometry is a more favorable design for the mandible, since the sagittal bending loads are larger than the transverse bending loads and the torsional loads [20, 52]. The symphysis is will prepared, geometrically and mechanically, to resist the large value of the lateral bending moment. It has the largest thickness in the human mandible equals mm (SD=4.0) [56]. Also, it has the largest density ranges from 0.85 to 1.53 g/cm3 [15]. Longitudinal elastic modulus increases from molar to symphysis to reach a value equals 19.4 GPA [6].The lingual cortex is stiffer than the buccal cortex in the symphysis and the premolar region, so it can resist the high tensile stresses acting on it [20]. Also, the teeth contribute with special geometrical shape to facilitate the mastication process. The posterior teeth have larger bearing areas of their roots. Also, the surface areas of the posterior teeth are greater than those of anterior teeth. The posterior teeth accept large axial forces and protect the anterior teeth from the large forces of closure. The anterior teeth guide the mandible as it carries out various excursive movements with tooth to tooth contact [40]. 32

33 Chapter 3 MATERIALS AND METHODS 3.1 Working with programmed ABAQUS Each time while entering data manually to Abaqus or any simulation program, it takes a lot of time and concentration. Sometimes it is not a real problem, when the model is simple with few variables, but it is not the case in a complicated model with many variables. On the other hand, using programmed Abaqus is much easier; it takes fewer seconds to run a script in Abaqus and less time to concentrate in the script, while writing it (see Fig.15). One other advantage of using a script is the possibility to change the parameters any time so easily directly from the script, for example, material properties: elastic modulus, poison ratio and density. Also, all the simulation setup will be saved in the form of small readable text files only a few kilobytes size, which makes it easier to transfer these files with coworkers. Fig.15 (a) Running script in Abaqus Fig.15 (b) Running script in Abaqus 33

34 When python script is not available, there is another alternative way, which is Macro files. Macro files record every action done in Abaqus, it can be run after that from Macro manager (see Fig.16). Also, every Abaqus simulation is accompanied by files with.rpy extension that can be run directly in Abaqus by modifying its extension to.py extension. Both python script files and macros files are used to run the simulation in this study [63]. Fig.16 Running macro files in Abaqus 3.2 Modeling and analysis The Part module using ABAQUS/CAE The mandible is assembled from three Solid 3D deformable parts: symphysis, corpus and ramus (see Fig.17). Dimensions of the parts are taken from previews reviews [48, 60, 61]. 1-Thickness: Materials and methods: Standard radiologic software is used to measure thicknesses in clinical landmark areas of the dentate mandibles of young men and women. A total of 150 dentate men and 75 dentate women aged from 18 to 30 years had undergone computed tomography of the head and neck region [48]. 34

35 Assumptions in this study: In three regions: 1) Midline of symphysis: 16 mm. 2) Beginning-end of the corpus: mm. 3) Ramus: 8.44mm. 2-Depth: The depth of the mandible is constant in the model and equals 29 mm [60]. 3-Horizontal distance: Materials and methods: Direct measurement is used on dry mandibles or by using radiographs of dry mandibles or patients. Ninety dry dentulous mandibles of both sexes were examined for position, size, and shape of mental foramen [60, 61]. Assumptions in this study: The horizontal distance between: 1) Symphysis menti and mental foramen is 27 mm. 2) Mental foramen and posterior border of ramus is mm. Fig.17 (a) Top view of the model Fig.17 (b) Side view of the model 35

36 3.2.2 The property module 1-Material Orthotropic material is used in this simulation; it has three mutually perpendicular axes, in which it has different material properties. Stiffness and compliance matrices for orthotropic materials are presented in terms of Young's moduli, shear moduli, and Poisson's ratios. The equations of compliance matrix requires twelve different variables (E1, E2, E3, G12, G31, G23, υ12, υ13, υ21, υ23, υ31, υ32). Assumptions in this study: Cortical bone is changing along the human mandible, but trabecular bone is constant due to the lack of information. Material x10 3 (MPA) E 1 E 2 E 3 G 12 G 31 G 23 Cortical Cortical Cortical Trabecular Material υ 12 υ 13 υ 21 υ 23 υ 31 υ 32 Cortical Trabecular Table 2. The elastic coefficients for cortical and trabecular bone of human mandible [18] 36

37 Stiffness matrix: Ϭ= C (3.1) Equations used to get stiffness matrix: υ12 υ21- υ31 υ13- υ32 υ23- υ12 υ23 υ31- υ21 υ32 υ13 V=1/1- C11= (1- υ12 υ 21) VE3 C22= (1- υ32 υ23) VE1 C33= (1- υ13 υ31) VE2 C12= (υ13+ υ12 υ23) VE3 C13= (υ23+ υ13 υ21) VE3 C23= (υ21+ υ31 υ23) VE1 C44=G31 C55=G32 C66=G12 (3.2) (3.3) (3.4) (3.5) (3.6) (3.7) (3.8) (3.9) (3.10) (3.11) Materialx10 3 (MPA) C11 C12 C22 C13 C23 C33 C66 C55 C44 Cortical Cortical Cortical Trabecular Table3. Stiffness coefficients for human mandible [18] 37

38 Density: Cortical Trabecular 1.8E-9 tonne /mm3 0.55E-9 tonne/mm3 Table4. Density of cortical and trabecular bone in human mandible [1, 7, 19] 2-Section assignment First, the parts are partitioned into three partitions: trabecular bone located between two cortical bones. The ratios for the three layers are 0.25: 0.5: 0.25, respectively (see Fig.18). Second, four sections are added to the section item: Cortical1, Cortical2, Cortical3, Trabecular. Section's type is set to homogenous and its category is set to solid. Third, the four sections are assigned to the parts: Cortical1 and Trabecular section to symphysis part, Cortical2 and Trabecular sections to corpus part, and Cortical3 and Trabecular to ramus part. Fig.18 (a) Creating partition in Abaqus Fig.18 (b) The three partitioned sections of the human mandible 38

39 3-Orientations The global coordinate system determines the default material orientations, however specific orientation can be assigned to the assembly parts by selecting datum coordinate systems from the viewport [62]. A rectangular datum coordinate system is used to define the local coordinates in the model according to previous literatures [2, 17, 59] (see Fig.19). Stacking direction is always chosen in the normal direction to the layup orientation (thickness direction); it is chosen in this simulation along the first axis (see Fig.20).It is possible also to orient the stacking direction from mesh module [62]. Fig.19 The x-axis (radial direction), y-axis (circumferential direction) [2]. (a) Lingual side of the mandible, (b) buccal side of the mandible and (c) the inferior border of the mandible Fig.20 (a) Assigning material orientation in Abaqus Fig.20 (b) The assigned orientation 39

40 3.2.3 The assembly module The following steps cannot be performed on parts but instances. Six instances (two symphysis, two corpus and two ramus are created to form the assembly. Assumptions in this study: Instance type is set to independent, and if any change is needed, it can be performed on the instances in the assembly. The auto-offset option is chosen to import parts apart from each other The step module Step sequence is defined within the model; it provides a convenient way to capture changes in the loading and boundary conditions of the model, changes in the way parts of the model interact with each other, the removal or addition of parts, and any other changes that may occur in the model during the course of the analysis. Also, it is allowable to change the analysis procedure, the data output, and various controls within the steps. In general, static step is selected. Also, automatic time increments is chosen; it starts the incrementation using initial value entered in the initial increment size.the size of subsequent time increments are adjusted based on how quickly the solution converges [62] (see Fig.21). Assumptions in this study: In step-1, the time period is set to 1, the maximum increments are set to increments, the initial time increment is initially set to 1E-5 second, the maximum time increment is initially set to second, and the minimum time increment is set to 1E-10 second. Fig 21. Monitoring the analysis 40

41 3.2.5 The interaction module 1-Tie constraints Applying constraints to the assembly is one step that must be done before applying loads, otherwise, the parts will get detached from each other. By clicking on constraints item, it is possible to choose between different types of constraints. In this study, the assembled parts are tied surface-surface; one part act as a master part and the other one as a slave part. There are two options either to tie the parts surfaces or nodes; tying parts surfaces is the option chosen in this simulation (see Fig.22). Tying parts surface to surface has some benefits: it gives more accurate contact stresses and it gives better convergence. It is preferable to adjust the master surface to the stiffer part (coarser mesh and higher elastic modulus). On the other hand, there is another way to merge parts together in mesh module, but it has a problem that it treats the whole model as one part with same mechanical properties, that is why it is not chosen in this study. Assumptions in this study: In ramus-corpus tie, ramus is chosen to be the master part and corpus is consequently chosen to be the slave part (as it has finer mesh and lower stiffness). In corpussymphysis tie, corpus is chosen to be the master part and symphysis is chosen to be the salve part. Fig.22 The applied constraints on the model 41

42 2-Coupling constraints A kinematic coupling constraint is used to transmit rotation to the structure (ramus) while permitting radial motion. A kinematic coupling constraint requires the specification of a reference node, coupling nodes, and the constrained degrees of freedom at these nodes. The reference node has both translational and rotational degrees of freedom. The coupling nodes, the nodes that are constrained to the rigid body motion of a single node and the degrees of freedom that participate in the constraint, are selected individually in a local coordinate system [62]. Assumptions in this study: The two reference points are chosen at the two upper points of the temporomandibular joint. The constraints regions are the surfaces of the condylar (see Fig.23). Rotation of the structure around the x-axis is introduced in the initial step and suppressed in step-1 to fulfill the hinge movement of the temporomandibular joint and then the closure of the mouth, while biting. The other degrees of freedom are restricted The Load module 1-Load Regarding the biting forces, Pruim found that they are 633N, 965N for first premolar, first molar [37].Since these forces are applied in this model on the mandibular bone, not on the teeth, so it must be lower than these values because teeth are stiffer than mandibular bone with approximate ratio 4:1 [42]. Assumptions in this study: Biting forces is set to 150 N in step-1 in first premolar region. 42

43 2-Boundary conditions Displacement and rotation boundary condition is used in this study. Two boundary conditions are introduced to the model. The first boundary condition is set to the two reference nodes of the coupling constraints. The model is allowed to rotate around the global x-axis. The boundary condition is introduced in the initial step to express the rotational hinge movement of the temporomandibular joint around the global x-axis, and it is suppressed in step-1 to express the closure of the mouth after that. The second boundary condition is set to the back of the ramus as shown in Fig.23. The model is allowed to rotate around the x-axis and translate up and down. The boundary condition is introduced in step-1 to prevent the rotation of the model when the external load (biting force) is applied. Fig.23The applied loads and boundary conditions on the model 43

44 3.2.7 The Mesh module 1-Mesh controls Element shape is set to hexahedral elements. Two techniques of meshing are used in this simulation: swept meshing and structured meshing. Structured meshing generates structure meshes using simple predefined mesh shapes; it transforms the mesh of a regular shape, such as square or a cube, onto the geometry of the region required for meshing. Structured meshing gives better results in the regions, where there is an interest in the deformation rate. Swept meshing is used to mesh complex shapes; it forms mesh on one side (source side) and copies the nodes of that mesh till it reaches the final side (target side) [62]. Assumptions in this study: Corpus and symphysis parts are meshed by structure mesh, as they have simple geometries. Ramus part is meshed by both, structured and swept mesh, due to complexity of the shape. Improving mesh quality: Ramus part is partitioned into several four dimensional shapes on one face, to have more structured parts as shown in Fig.22. These partitions are extruded along the whole thickness by using "Extrude/sweep edges". Also, medial axis algorithm is chosen for swept parts (parts of curvature) with minimization of mesh transitions. Fig.24 (a) Ramus mesh before modification Fig.24 (b) Ramus mesh after modification 44

45 2-Element type Assumptions in this study: Family is set to 3D stress. Element library is set to explicit option; it updates the stiffness matrix based on geometry changes and material changes after each increment. Geometric order is set to first order element. Fig.25 The meshed model Units There is not inherent set of units in ABAQUS, but the user has to choose certain set of units in the simulation. The set used in this simulation is mm for length, Newton for force, Second for time, Tonne/mm3 for density, N/mm2 (MPA) for stress and N/mm2 (MPA) for Young's Modulus. 45

46 Chapter 4 Results 4.1 Trial 1 In the first trials, there were some errors involving the material orientation, which were fixed after adjusting the orientation according to the previous literatures as shown in Fig.19. Encastre boundary condition was chosen and set to the lower surface of the working side. The applied forces were 1200, 900, and 700 N for joint force of working side, balancing side and premolar biting force, respectively. The seeding density was set to 1.9 in the whole model. The figures below show the difference in the results after the material orientation is modified in the model. Fig.26 (a) The model before modification Fig.26 (b) The model after modification Comments: The first observation was the low value of the deformation scale factor in the modified model. The reason for that behavior is that the model became stiffer after modifying the material orientation. The second observation was that the maximum stresses took place in the lower borders of the model. The reason for that behavior can be the 46

47 boundary condition that was set to the bottom surface of the model. The third observation is that the trabecular bone has lower deformation compared to the cortical bone in the model. 4.2 Trial 2 In this trial, a simulation is performed on three models with different material composition. The three models are composed of cortical bone, trabecular bone, both cortical and trabecular bones, respectively. This trial was performed to see how the core material will behave in a model of different material composition and in a model of the same material composition, when external load is applied. Fig.27 (a) The model with both cortical and trabecular bones Fig.27 (b) The model with trabecular bone Comments: It was observed that the core material behaves normally, when the model is composed of one material. Therefore, the reason for having low deformation, in the trabecular bone in the previous trial, is related to the different material composition in the model (two cortical bones and trabecular bone). This behavior will be discussed later in chapter 5. Also, it was found that the maximum stresses take place in the tied locations in this trial and the previous one. 47

48 4.3 Trial 3 In this trial, the tying is modified and performed on the surfaces that joins symphysis with symphysis, symphysis with corpus, and corpus with ramus as shown in Fig.22. The master surface is set to the part of more stiffness (higher elastic modulus and coarser mesh), and the slave surface is set to the other corresponding surface. The mesh of each part is changed to make ramus stiffer and the other two parts finer. The seeding density is set to 2.25, 1.9, and 1.5 for ramus, corpus and symphysis, respectively. The external load (premolar biting) is set to 150 N and the other loads are removed. Displacement and rotation boundary condition is chosen. The first boundary condition is set to the back of ramus as shown in Fig.22. It has two degrees of freedom: UR1 to rotate around the x-axis, and U3 to move up and down. Also, a coupling constraints is applied to two reference nodes shown in Fig.23, and the second boundary condition is applied to these nodes that allow the structure to rotate around the x-axis (hinge movement). The initial time step is set to 1E-05 second, the minimum time step is set to 1E-10 second, the maximum time step is set to second, the maximum number of increment is set to increments, and the boundary conditions are identified in the initial step. Fig.28 (a) Stresses of the mandibular bone Fig. 28 (b) Strains of the mandibular bone 48

49 Comments: In this trial, convergence is judged unlikely in an early time step equals 6.17E-04 second. The reaction forces are observed in the ramus part. Maximum stresses are generated between condylar and coronoid process, its value equals 3.937E-01MPA. Maximum strain is formed in the premolar region, its value equals 1.135E-03mm, which is a very small value compared to literature. 4.4 Trial 4 In this trial, the same boundary condition in initial step is kept unchanged for the two coupling points, and it is restricted in step-1 to demonstrate the real movement of the mouth during closure. Finer mesh is applied for better convergence, the seeding density is set to 2.1, 1.5, and 1.3 for ramus, corpus and symphysis, respectively. After many trials, the initial time step is set to 1E-05 second, the minimum time step is set to 1E-10 second and the maximum time step is set to 0.01 second for saving time. Fig.29 (a) Stresses of the mandibular bone Fig.29 (b) Strains of the mandibular bone 49

50 Comments: In this trial, convergence is judged unlikely too but in later time step equals 0.8 second. Maximum stress value is higher in this simulation, it equals 2.039E+03MPA. Maximum strain value equals 3.03E-01mm (approaching literature values). The hinge movement of the temporomandibular joint is shown in Fig. 29 (c). Fig. 29 (c) The hinge movement of the temporomandibular joint 50

51 Chapter 5 Discussion 5.1 Sandwich theorem While trial, it was found that the trabecular bone is not affected by the stresses, and so it does not deform or it has lower deformation compared to cortical bone. This unexpected behavior can be explained by applying sandwich composite theory to the model. Unlike typical composite structures, which have stiff fibers embedded in a less stiff matrix, sandwich composite (honeycomb sandwich composite) is composed of two thin faces of high stiffness combined with low-density cores. Its importance is that it can give high stiffness with lower weight compared to a model of three cortical layers. Also, it has high durability and high bending strength. In addition to the efficiency between stiffness and strength, honeycomb sandwich panels are fairly fatigue resistant and great insulators or radiators depending on the core material selection [64, 66]. Fig.30 Stress-strain curve of cortical and trabecular bone [67] 51

52 Sandwich structures mainly carry applied bending moments as tensile and compressive stresses in the two face-sheets, whereas applied transverse forces predominantly are carried by the core material as shear stresses, and this feature can possibly be the explanation for the unexpected behavior of the trabecular bone in this study [65]. Figure 30 shows that the cortical bone has brittle behavior and low ultimate strain, while trabecular bone has more ultimate strain and lower stiffness. The combination of both mechanical behaviors in the human mandible increases the durability and the bending strength, when stresses are applied. 5.2 Limitation The geometry has some limitations: first, the sagittal dimension is constant along the whole model; it was difficult to draw it because as the complexity of the geometry increases, the errors increase in meshing. Some parts, like ramus and corpus, were not of high precision due to the lack of information. Dimensions of the human mandible is changeable with age, gender, edentulous and dentate samples, but average values are taken in this model [61]. The thicknesses of the lower and upper border of the human mandible is not the same, as stated before, but it is assumed to be constant in this model [20]. Regarding the material properties, the information was limited for the trabecular bone and it was assumed to be homogenous along the whole model. Also, the viscoelastic behavior of the bone is neglected in this study, which results in giving inaccurate results and unexpected behavior during simulation. When a load is applied, the material enters the plastic region and deform, however, the stress in a viscoelastic material depends not only on the strain but also on the time history of the strain. The mandible has different mechanical properties, for example, it is stiffer in the lingual side than the lateral side and also it is stiffer in the lower border than the upper border. In this study, average values of elastic modulus are taken. Loads and Boundary conditions have some limitations: there are different forces and moments that act on the mandible that were discussed previously. This study focused on the stresses and strains generated in the mandible, when biting takes place in the premolar region. The muscle forces, which take place while biting, are neglected in this 52

53 study. The temporomandibular joint has two motions: rotation and translation; translation motion is neglected in this study. Approximate value of the biting force is assumed equal to 150N. The maximum stresses took place in the region between the condylar and the coronoid process due to stress concentration zone generated of the sudden change in geometry (see Fig.31). Fig 31 Stress concentration region in ramus 5.3 Strength With respect to technical capabilities, the hinge movement of the temporomandibular joint, during the opening and closing of the mouth, was well expressed by the boundary conditions. Working with programmed Abaqus saved a lot of time for more simulations. Also, the surface-surface tie constraint was one of the strengths in this study. It tied two different surfaces together in the duration of the simulation; it was useful for mesh refinement purposes; it allowed rapid transitions in mesh density in the model. With respect to mechanical capabilities, the non-homogenous and orthotropic material properties of the mandible were well expressed by using simple model. Also, this study provides a way of investigating abnormal biting occlusions, such as unilateral and bilateral occlusions, in which the teeth are not aligned properly. 53