Journal of Oral Rehabilitation. 1992, Volume 19, pages 371-383
Analysis of a central maxillary incisor by using a three-dimensional finite element method S. DARENDELiLER*, H. DARENDELILERt ant/ T. KINOGLU* ^Department of Operative Dentistry, Faculty of Dental Medicine, Gazi University and f Department of Mechanical Engineering, Faculty of Engineering, Middle East Technical University, Ankara, Turkey
Summary
An attempt is made for determining the stress distribution in a maxillary central incisor by using a three-dimensional finite element model. The tooth is assumed isotropic, homogeneous, elastic and unsymmetrical. A load of 450N, 26° to the longitudinal axis is applied on the incisal margin of the tooth. The distribution of compressive, tensile and shear stresses were plotted for the whole tooth structure. It is found that craeks or fractures occur under the given load. Introduction
The tooth structure is composed of enamel, dentine, pulp, cementum and periodontal ligament, which are non-homogenous, anisotropic and unsymmetrical with respect to any axis. For these reasons, conventional methods such as photoelasticity and strain-gauge method are inadequate to give the correct and reliable results about the true stress distribution in the tooth (Craig et al., 1971; Derand, 1977; Henry 1977). Recently, use of numerical methods in stress analysis have been extensively applied in dentistry. One and probably the most effective of the numerical methods is the finite element method, which is used in the solution of complex structures by the aid of computers (Zienkiewicz, 1977). The structures are divided into a 'finite' number of elements, and each element may have different physical properties. The solution is based on these 'finite' elements. Finally, the results are obtained with respect to the variables defined in these elements. The two-dimentional axisymmertic finite element modelling has been used in most of the previous researches (Dayangaf, 1978; Farah & Craig, 1974; Kavsaoglu, 1986). Although numerical results can be easily obtained in two-dimensional modelling, it has some significant shortcomings; the human tooth is highly irregular, such that, it cannot be represented in a two-dimensional space and the actual loading cannot be simulated without taking the third dimension into consideration. The distribution of various materials of the tooth structure does not show any symmetry. Therefore a three-dimensional modelling with the actual dimensions should be preferred for a reliable analysis (Borchers «& Reichart, 1983; Darendeliler, 1988; Rubin et al., 1983). The main aim of this study is to find the stress distribution in a three-dimensional model of maxillary central incisor by using the Finite Element Method (F.E.M.).
Correspondence: Dr Haluk Darendeliler, Department of Mechanical Engineering, Faculty of Engineering, Middle East Technical University, 06531 Ankara, Turkey.
371
372
5. Darendeliler et al.
Materials and methods Method In this study, the finite element method is used to determine the stress distribution in a central maxillary incisor. Since tooth dimensions differ widely from individual to individual, the geometry of the tooth is taken from Wheeler's study (1%9) where average values are used. The mechanical properties of the enamel and dentine are determined from previous research and are presented in Table 1 (Craig et al., 1975; Craig, 1979; Phillips, 1982). Assumptions In this study, the following assumptions have been made regarding the properties of tooth materials and its geometry. (i)
The tooth is considered to be composed of enamel and dentine since the thickness of periodontal ligament and cementum are very small. (ii) Young's modulus value of pulp is negligibly small when compared with enamel and dentine. So, the effect of pulp on stress distribution is neglected and the pulp is considered to be empty (Rubin et al., 1983). (iii) The tooth materials are assumed to be isotropic, homogeneous and elastic. (iv) The bone structure which supports the root is assumed to be rigid and a distributed external load of 450 N (Helkimo et al., 1977; Kampe et al., 1987; Van Steenberge & Vriesde, 1978) is applied at the incisal margin of the tooth (Fig. 1). The angle between the line of the force and the vertical axis is 26°, which represents the angle at the first contact of tooth during biting (Tylman & Malone, 1978). Finite element method The concept of the finite element method is based on discretisation of continuum into analytically modelled 'finite elements'. Each element has a predetermined number of nodes. Elements are connected with eachother at their nodes. For each element, the force-displacement relations are written in terms of the nodal variables (displacements) to obtain the element stiffness matrices. Later, the entire element stiffness matrices are assembled to obtain the global stiffness matrix. Based on the initial and boundary conditions, the displacement at the nodes are found for the given load. The stress distribution and directions of the principle stresses are calculated from these nodal displacements using the stress-strain and strain-displacement relations. Information required for the software used in the computer is as follows: (i)
Coordinates of the nodal points.
Table 1. Material properties
Compressive strength (Nmm ")
Shear strength
Material
Tensile strength (Nmm ")
Enamel Dentine
10 48
288 232
8 20
Young's modulus Poisson's ratio 13000 48000
0-31 0-33
Analysis of a central maxillary incisor
1
373
J
Fig. 1. The buccal and mesial views of the model of the maxillary central incisor which is divided to 12 parts.
(ii) Number of the nodes for each element. (iii) Young's modulus and Poisson's ratio of the material modelled by different elements. (iv) The initial and boundary conditions. (v) External forces applied on the structure. A three-dimensional model of a healthy central tooth was prepared to investigate the stress distribution. The prepared three-dimensional model was first divided into 12 parts along the longitudinal axis (Fig. 1). Each of the parts was then divided into smaller prismatic elements having the same height, which are equal to the thickness of each part. This way, 204 prismatic elements were obtained. Each element consisted of eight nodes with a total of 340 nodes (Fig. 2). Since the fracture generally occurs in the crown of the tooth, the crown was separated into eight parts, while the root was divided into four parts. Hence, larger number of elements were used in the crown to obtain reliable results where stress distribution is more critical.
374
S. Darendeliler et al.
Fig. 2. The shape of the tootli consisted of 204 elements (interior of the tooth is not shown for simplicity).
Results and discussion
In this study an attempt is made to investigate the stress distribution under the effect of the distributed force on the incisal margin for compressive, tensile and shear stresses separately, both for enamel and dentine by the finite element method. To the best of knowledge of the authors this is one of the first attempts in this field. During the evaluation of stress, the 12 parts of tooth structure are considered separately. The maximum values of stress for the elements in each of these parts are assumed to be the value represented in the figures showing the stress distribution along the longitudinal axis (Figs 3—8). For the figures which show the stress distribution in each of these parts between mesial and distal edges, the maximum stress values of the corresponding elements^ between the bucal and palatinal surfaces are selected and plotted (Figs 9 and 10). Figure 3 shows the distribution of compressive stress along the longitudinal axis of the tooth in enamel. It is apparent that the stress values are maximum around the cervical line and at the incisal margin. In the tooth structure, between the cervical line and the incisal margin, the stress values are nearly uniform and approximately half of the maximum value. The higher values are attributed to the termination of enamel at the cervical line, and the thin structure of enamel at the incisal margin where the distributed loacJ is applied. The tensile stress in the enamel of the tooth increases from the incisal margin towards the cervical line as illustrated in Fig. 4. Since the load is applied at an angle of 26° from the longitudinal axis, it results in bending increasing the tensile stress towards the cervical line where the tooth is fixed in the alveoler bone. The same increase can also be seen for the compressive stress as a result of bending, but its magnitude is smaller when compared with the stresses caused by the compressive load. Figure 5 shows the distribution of shear stress in the enamel of the tooth and it is clear that the maximum value is reached at the incisal margin. The shear values are nearly uniform in the other parts of enamel, though, they show a slight increase towards the cervical line. In dentine, the compressive stress increases towards the cervical line from the incisal margin where the highest values are attained (Fig. 6). This is due to the enamel structure which terminates at the cervical line and causes the compressive load to be
Analysis of a central maxillary incisor
375
Stress (N mm-2) 100 90 80 70 60
50 40
30 20 10 1
2
3
4
5
6
7
8
Longitudinal Axis (mm)
Fig. 3. Distribution of eompressive stress along the longitudinal axis in enamel from the ineisal margin towards the eervieal line by taking the first six parts into eonsideration, as shown in Eig. 1.
Stress (N mm" 70 60 50 40 30 20 10
1
2
3
4
5
6
7
Longitudinal Axis (mm)
Fig. 4. Distribution of tensile stress along the longitudinal axis in enamel from the ineisal margin towards the eervieal line by taking the first six parts into eonsideration, as shown in Eig. 1.
shared by the dentine. Towards the apex, a decrease is noted in the stress since the loads are transferred gradually to the alveolar bone. Figure 7 shows the same tendency for the distribution of the tensile stress in dentine while the maximum value reached is comparatively smaller than the values found for the eompressive stress. The termination of the enamel which results in the load being shared by the dentine, and the bending effect of the applied load with a component perpendieular to the longitudinal axis of the tooth, lead to an increase in the tensile stresses around the eervieal line.
376
S. Darendelller et al.
Stress (N mm'^)
Analysis of a central maxillary incisor by using a three-dimensional finite element method S. DARENDELiLER*, H. DARENDELILERt ant/ T. KINOGLU* ^Department of Operative Dentistry, Faculty of Dental Medicine, Gazi University and f Department of Mechanical Engineering, Faculty of Engineering, Middle East Technical University, Ankara, Turkey
Summary
An attempt is made for determining the stress distribution in a maxillary central incisor by using a three-dimensional finite element model. The tooth is assumed isotropic, homogeneous, elastic and unsymmetrical. A load of 450N, 26° to the longitudinal axis is applied on the incisal margin of the tooth. The distribution of compressive, tensile and shear stresses were plotted for the whole tooth structure. It is found that craeks or fractures occur under the given load. Introduction
The tooth structure is composed of enamel, dentine, pulp, cementum and periodontal ligament, which are non-homogenous, anisotropic and unsymmetrical with respect to any axis. For these reasons, conventional methods such as photoelasticity and strain-gauge method are inadequate to give the correct and reliable results about the true stress distribution in the tooth (Craig et al., 1971; Derand, 1977; Henry 1977). Recently, use of numerical methods in stress analysis have been extensively applied in dentistry. One and probably the most effective of the numerical methods is the finite element method, which is used in the solution of complex structures by the aid of computers (Zienkiewicz, 1977). The structures are divided into a 'finite' number of elements, and each element may have different physical properties. The solution is based on these 'finite' elements. Finally, the results are obtained with respect to the variables defined in these elements. The two-dimentional axisymmertic finite element modelling has been used in most of the previous researches (Dayangaf, 1978; Farah & Craig, 1974; Kavsaoglu, 1986). Although numerical results can be easily obtained in two-dimensional modelling, it has some significant shortcomings; the human tooth is highly irregular, such that, it cannot be represented in a two-dimensional space and the actual loading cannot be simulated without taking the third dimension into consideration. The distribution of various materials of the tooth structure does not show any symmetry. Therefore a three-dimensional modelling with the actual dimensions should be preferred for a reliable analysis (Borchers «& Reichart, 1983; Darendeliler, 1988; Rubin et al., 1983). The main aim of this study is to find the stress distribution in a three-dimensional model of maxillary central incisor by using the Finite Element Method (F.E.M.).
Correspondence: Dr Haluk Darendeliler, Department of Mechanical Engineering, Faculty of Engineering, Middle East Technical University, 06531 Ankara, Turkey.
371
372
5. Darendeliler et al.
Materials and methods Method In this study, the finite element method is used to determine the stress distribution in a central maxillary incisor. Since tooth dimensions differ widely from individual to individual, the geometry of the tooth is taken from Wheeler's study (1%9) where average values are used. The mechanical properties of the enamel and dentine are determined from previous research and are presented in Table 1 (Craig et al., 1975; Craig, 1979; Phillips, 1982). Assumptions In this study, the following assumptions have been made regarding the properties of tooth materials and its geometry. (i)
The tooth is considered to be composed of enamel and dentine since the thickness of periodontal ligament and cementum are very small. (ii) Young's modulus value of pulp is negligibly small when compared with enamel and dentine. So, the effect of pulp on stress distribution is neglected and the pulp is considered to be empty (Rubin et al., 1983). (iii) The tooth materials are assumed to be isotropic, homogeneous and elastic. (iv) The bone structure which supports the root is assumed to be rigid and a distributed external load of 450 N (Helkimo et al., 1977; Kampe et al., 1987; Van Steenberge & Vriesde, 1978) is applied at the incisal margin of the tooth (Fig. 1). The angle between the line of the force and the vertical axis is 26°, which represents the angle at the first contact of tooth during biting (Tylman & Malone, 1978). Finite element method The concept of the finite element method is based on discretisation of continuum into analytically modelled 'finite elements'. Each element has a predetermined number of nodes. Elements are connected with eachother at their nodes. For each element, the force-displacement relations are written in terms of the nodal variables (displacements) to obtain the element stiffness matrices. Later, the entire element stiffness matrices are assembled to obtain the global stiffness matrix. Based on the initial and boundary conditions, the displacement at the nodes are found for the given load. The stress distribution and directions of the principle stresses are calculated from these nodal displacements using the stress-strain and strain-displacement relations. Information required for the software used in the computer is as follows: (i)
Coordinates of the nodal points.
Table 1. Material properties
Compressive strength (Nmm ")
Shear strength
Material
Tensile strength (Nmm ")
Enamel Dentine
10 48
288 232
8 20
Young's modulus Poisson's ratio 13000 48000
0-31 0-33
Analysis of a central maxillary incisor
1
373
J
Fig. 1. The buccal and mesial views of the model of the maxillary central incisor which is divided to 12 parts.
(ii) Number of the nodes for each element. (iii) Young's modulus and Poisson's ratio of the material modelled by different elements. (iv) The initial and boundary conditions. (v) External forces applied on the structure. A three-dimensional model of a healthy central tooth was prepared to investigate the stress distribution. The prepared three-dimensional model was first divided into 12 parts along the longitudinal axis (Fig. 1). Each of the parts was then divided into smaller prismatic elements having the same height, which are equal to the thickness of each part. This way, 204 prismatic elements were obtained. Each element consisted of eight nodes with a total of 340 nodes (Fig. 2). Since the fracture generally occurs in the crown of the tooth, the crown was separated into eight parts, while the root was divided into four parts. Hence, larger number of elements were used in the crown to obtain reliable results where stress distribution is more critical.
374
S. Darendeliler et al.
Fig. 2. The shape of the tootli consisted of 204 elements (interior of the tooth is not shown for simplicity).
Results and discussion
In this study an attempt is made to investigate the stress distribution under the effect of the distributed force on the incisal margin for compressive, tensile and shear stresses separately, both for enamel and dentine by the finite element method. To the best of knowledge of the authors this is one of the first attempts in this field. During the evaluation of stress, the 12 parts of tooth structure are considered separately. The maximum values of stress for the elements in each of these parts are assumed to be the value represented in the figures showing the stress distribution along the longitudinal axis (Figs 3—8). For the figures which show the stress distribution in each of these parts between mesial and distal edges, the maximum stress values of the corresponding elements^ between the bucal and palatinal surfaces are selected and plotted (Figs 9 and 10). Figure 3 shows the distribution of compressive stress along the longitudinal axis of the tooth in enamel. It is apparent that the stress values are maximum around the cervical line and at the incisal margin. In the tooth structure, between the cervical line and the incisal margin, the stress values are nearly uniform and approximately half of the maximum value. The higher values are attributed to the termination of enamel at the cervical line, and the thin structure of enamel at the incisal margin where the distributed loacJ is applied. The tensile stress in the enamel of the tooth increases from the incisal margin towards the cervical line as illustrated in Fig. 4. Since the load is applied at an angle of 26° from the longitudinal axis, it results in bending increasing the tensile stress towards the cervical line where the tooth is fixed in the alveoler bone. The same increase can also be seen for the compressive stress as a result of bending, but its magnitude is smaller when compared with the stresses caused by the compressive load. Figure 5 shows the distribution of shear stress in the enamel of the tooth and it is clear that the maximum value is reached at the incisal margin. The shear values are nearly uniform in the other parts of enamel, though, they show a slight increase towards the cervical line. In dentine, the compressive stress increases towards the cervical line from the incisal margin where the highest values are attained (Fig. 6). This is due to the enamel structure which terminates at the cervical line and causes the compressive load to be
Analysis of a central maxillary incisor
375
Stress (N mm-2) 100 90 80 70 60
50 40
30 20 10 1
2
3
4
5
6
7
8
Longitudinal Axis (mm)
Fig. 3. Distribution of eompressive stress along the longitudinal axis in enamel from the ineisal margin towards the eervieal line by taking the first six parts into eonsideration, as shown in Eig. 1.
Stress (N mm" 70 60 50 40 30 20 10
1
2
3
4
5
6
7
Longitudinal Axis (mm)
Fig. 4. Distribution of tensile stress along the longitudinal axis in enamel from the ineisal margin towards the eervieal line by taking the first six parts into eonsideration, as shown in Eig. 1.
shared by the dentine. Towards the apex, a decrease is noted in the stress since the loads are transferred gradually to the alveolar bone. Figure 7 shows the same tendency for the distribution of the tensile stress in dentine while the maximum value reached is comparatively smaller than the values found for the eompressive stress. The termination of the enamel which results in the load being shared by the dentine, and the bending effect of the applied load with a component perpendieular to the longitudinal axis of the tooth, lead to an increase in the tensile stresses around the eervieal line.
376
S. Darendelller et al.
Stress (N mm'^)
