VOL. 9, PP. 237-252 (1975)
J. BIOMED. MATER. RES.
-
Finite Element Analysis of Dental Structures Axisymmetric and Plane Stress Idealizations
LAWRENCE G. SELNA, Mechanics and St?ytures Department, University of California, Los Angeles, HERBERT T. SHILLINGBURG, JR.,* School of Dentistry, University of California, Los Angeles, and PETER A, KERR, Mechanics and Structures Department, University of California, Los Angeles, California 90024
Summary The finite element method used to study stress generated in a maxillary second premolar MIa result of occlusal forces. This mathematical technique has been applied extensively in structural engineering and structural mechanics. It is well suited to the analysis of stress in teeth and dental restorations because it can closely simulate the geometries, loads, and material inhomogeneitiesin the system being studied.
INTRODUCTION A knowledge of the intensity and distribution of stress can be helpful in avoiding or minimizing the failure of dental restorations. The effects of stress on dental restorations and upon teeth themselves has been a topic of some interest to dental researchers over the past twenty years. Early published reports indicate that this research concentrated on cavity configurations of Class I1 amalgam restorations. One of the earliest photoelastic studies applied to dentistry dealt with the stress distribution in tooth structure surrounding restorations. The general usefulness of the photoelastic method and the ways of applying this technique to dentistry were described by Mahler and Peyton.2 Studies have been done on the effects of preparation geometry on the restoration while other studies of the Class I1 configura*Present address : College of Dentistry, University of Oklahoma, Oklahoma City, Oklahoma 73190
237 @ 1975 by John Wiley & Sons, Inc.
238
SELNA, SHILLINGBURG, AND KERR
tion examined effects on both tooth structure and the re~toration.~-s Photoelastic techniques have been employed to study inlays,g crowns,lo proximal margins,l 1 convergence of axial walls, l 2 occlusal reduction, l3 proximal reduction, l 4 fixed partial dentures,15pin placement for amalgam retention, l6 and impression distortion during removal from the mouth.” The proSome three dimensional studies have been cedures used for studying three dimensional models require great care and considerable time because of the complex shapes being studied. Consequently, two dimensional “slice” models are usually used. In the field of engineeripg, other methods of stress analysis are also utilized. A technique which has been extensive application since 195621in the fields of civil and aerospace engineering is the finite element m e t h ~ d . Z ~ -In ~ ~this approach, mathematical idealizations are used, rather than physical models. A study of preparation geometries requires the examination of many configurations. Consequently, mathematical representation of the physics of tooth deformation and stress would be useful since it could be used for different tooth geometries by changing only a few input parameters to model. Finite element analysis permits the study of as many tooth and restorative materials as might be present in any given problem: enamel, dentin, cement, gold, etc. With this method, many types of preparation geometries and restorative materials can be studied. This can provide some refinement of the photoelastic technique, where only two materials have been studied: undifferentiated tooth structure and the restorative material.
MATERIALS AND METHODS Two types of idealizations were used in this study, and different grid sizes were tried to determine what combinations would give the best approximations of tooth shape and behavior. The basic concept of the finite element method is the subdivision of the body to be studied into many smaller elements of a simpler geometric shape. For two dimensional studies, triangles or quadrilaterals are often used, while tetrahedra or hexahedra are frequently used in three dimensional studies. These elements may be assembled like buiIding blocks to approximate a body of any given shape. The
STRESS IDEALIZATION IX IIEXTAL STRUCTURES
239
elements are designated so that apices of adjoining elements meet to form (‘node”’. The apex of one element cannot fall on the side of a n adjacent element. When analysis is performed, a system of simultaneous equations results, relating the forces and displacements, which are evaluated at the nodes. Aftcr solving the equations for displacements, the stresses can be computed within each element. Development of the finite element method has been closely tied to the development of high speed electronic digital computers. The large systems of simultaneous equations which result from the formulation can be solved feasibly only by using a computer. The computer can also be programmed to draw contour plots, showing the stress contours on sections of the tooth being studied. The essential information required for stress analysis encompasses the geometry, the boundary loads and supports, and the material properties. The geometry describes the spacial configuration of the mathematical idealization of the tooth. The occlusal forces are represented as force boundary conditions and the tooth mobility is represented by spring-type support conditions. The material properties include the constitutive or stress-strain laws, the range of validity of those laws, and the failure criteria. The relationship between stress and strain is complicated by these factors: a) the tooth is composed of different material subregions and is therefore inhomogenous; b) in each of the subregions the stressstrain law may be dependent upon direction since the tissue is anisotropic; and c) in each of the subregions the stress-strain behavior may be a function of time and rate of loading, i.e. the tissue is viscoelastic. Failure criteria prescribe the stress combinations and levels which will cause fracture in the material subregions. The formulation and solution procedure is established using virtual work methods. The virtual work in the tooth is expressed in terms of displacements at a finite number of nodal points within the tooth. The virtual work performed on the tooth by external loads is expressed a t these same points. If the internal and external virtual work are equated, the displacements a t the nodal points can be evaluated . The virtual strain energy for the tooth structure is given by
V
SELNA, SHILLINGBURG, AND KERR
240
in which 6 U = virtual strain energy; r = stress tensor; 6a = virtual strain tensor; V = volume of the tooth. The external virtual work is given by 6W
=
I
T6uds
S
in which 6 W = external virtual work; T = surface traction vector; 6u = virtual displacement vector; S = tooth surface. Referring to a finite element embedded within the tooth, interpolation functions may be used to express the displacement field within each element in terms of nodal displacement values. This is usually written as u =
(3)
CPU
in which CP = interpolation function; u = nodal coordinate displacements. The virtual strain tensor for each element is given by 8E
in which CP,, are given by
=
CP,, 6u
=
(4)
gradient of the interpolation function. The stresses 7
=
CCP,, u
(5)
in which C = generalized Hooke’s law for anisotropic elastic materials. This stress-strain law can be extended to plastic materials by using the incremental methods.25 If eqs. (4) and ( 5 ) are substituted into eq. (l),and the integration is performed over element i, the result is 6Ui =
s
vi
6uCP,,
cq, udV
(6)
The properties of the interpolation functions permit a simple integration over the element volume, Vi. This yields the scalar form 6Ui
=
6u
ki u
(7)
in which ki = stiffness matrix for the element i. The virtual strain energy for all the elements which model the tooth may then be obtained by summing the virtual energies for all the elements to yield
STRESS IDEALIZATION IN DENTAL STRUCTURES
241
where there are n elements comprising the tooth. An alternative form of eq. (8) is 6 U = 6r Kr
(9)
in which r, 6r = real and virtual global coordinate displacements; K = global stiffness matrix. The corresponding external virtual work is
6W=6rF
(10)
in which F = global force components which are evaluated from eq. (2). Using the Divergence Theorem, eqs. (9) and (10) can be equated, and since 6r is arbitrary, the result is
Kr
=
F
(11)
Equation (11) is a set of linear simultaneous equations which may be solved for the nodal displacements, r. Using these displacements, eq. (5) may be employed to find the element stresses. Equations (5), (7), and (11) which are the essential ingredients. in this finite element procedure can be programmed on the digital computer. Output from the computer program will yield the displacement and stress fields throughout the tooth structure. This theory, written in general terms, is valid for all finite element representations in one, two, or three dimensions. In this study, attention will be focused on two types of two dimensional representations: plane stress and axisymmetric. The plane stress model most closely approximates the model used in two dimensional photoelasticity. A midsagittal section of a maxillary premolar (Fig. 1A) was divided into a finite number of triangular subregions (Fig. 1B). The mathematical solid used for analysis was produced by expanding the section in the third (depth) dimension (Fig. 1C). Occlusal forces were applied as three uniformly distributed line loads totalling 100 lb. They were placed in the depth dimension at the desired nodes of the crown in a vertical direction (Fig. 2 ) . The support provided by the periodontal membrane and the underlying bone was a little more difficult to model mathematically. A
SELNA, SHILLINGBURG, AND KERR
242
A
0
C Fig. 1. Generation of a plane stress solid.
rigid support on the roots was deemed unsatisfactory because of the mobility experienced by teeth under load. A more suitable representation was considered to be a continuous elastic foundation. To approximate this condition, springs, both vertical and horizontal, were placed over the interfacial region on exterior nodes (Fig. 2). The spring coefficients were evaluated from studies relating displacement and applied force.26 The high cusp axisymmetric model was generated by taking the lingual half of a midsagittal section (Fig. 3A) and dividing it into subregions (Fig. 3B). This half section was rotated about an axis corresponding to the center axis of the tooth to produce a mathematical solid (Fig. 3C). The low cusp model was similarly formed by taking one-half of a mesiodistal section through the central groove and marginal ridge, and rotating it about the central axis. The component elements of these solids are three-sided concentric rings (Fig. 4). Occlusal forces and spring reactions were distributed uniformly as line loads around the concentric rings formed by the exterior nodes on the crown and root (Fig. 5 ) . All three test cases were studied with both coarse and fine element subdivisions (Fig. 6 ) . The greater the number of elements in the
STRESS IDEALIZATION IN DENTAL STRUCTURES
243
* 33.3 LBS
+ 0
Ib/in
1 0 Ib/in
Fig. 2. Loading and support conditions for the plane stress idealization.
244
SELNA, SHILLINGBURG, AND KERR
-\ 1
A
0
Fig. 3. Generation of an axisymmetric solid.
idealization, the more closely the actual continuum of the physical body can be approximated. Consequently, the results of increasingly finer grid sizes converge on the true solution. It is wise to concentrate elements in the regions of greatest interest and stress concentration. The decision of how many elements to use rests with the designer who must optimize between computer time, computer capacity, and an acceptable degree of accuracy.
For the material properties, dentin and enamel were considered to be homogenous and isotropic. The modulus of elasticity for enamel was 5.0 X lo6 psi27-2e and that for dentin was 2.0 X lo6 psi.3o Poisson’s ratio was taken as 0.25 for both enamel and dentin. To simplify this initial work, the pulp was not considered and was treated as a void. Failure was also not considered. RESULTS The results of the analysis showing minimum and maximum principal stress, are presented in the form of contour plots prepared by the computer (Figs. 7-12). The different intensity of contours are identified by their characteristic symbols. The magnitude and direction
STRESS IDEALIZATION I N DENTAL STRUCTURES
i7
7
245
i
M
.I
cr
SELNA, SHILLINGBURG, AND KERR
246
Fig. 6. Coarse and fine grid plane stress idealizations.
SYMBOLS 0 t5uXF.u
h -4.900
+ -.ooo I - U W
8
E I - 0 W 2 0
a’
?ii
Y
800
n
1600
x P
2400
3
3200
E
I
4000
a
4800
g
Fig. 7. Stress contours-plane
stress, coarse grid.
STRESS IDEALIZATION IN DENTAL STRUCTURES
247
248
SELNA, SHILLINGBURG, AND KERR
STRESS IDEALIZATION I N DENTAL STRUCTURES
249
SYMBOLS' 0 tymc
+ -um I -Mo 0
-2.00
a
-MX)
* -1600 2
0
7
000
n
1600
W
2400
P
3200
1 4 w o R
a00
J
1
MINIMUM PRINCIML STRESS
MAXIMUM PRINCIPAL STRESS
Fig. 12. Stress contours-axisymmetric, low cusp, fine grid.
of the principal stresses at different locations in the tooth can be obtained directly from the principal stress contours (Fig. 13).
DISCUSSION The contour plots for principal stress were chosen because they give a good visual impression of the stress distribution, the regions of high intensity, and the stress gradients. As would be expected, the highest intensity occurs directly beneath the applied loads. Since the contours are defined at equal intervals of intensity, the closeness of the contour spacing indicates the stress gradient. In the roots of the teeth, the contours are widely spaced, indicating that the stress distribution is nearly uniform. The plot of minimum principal stress records the high negative or compressive stresses. The maximum
250
SELNA, SHILLINGBURG, AND KERR PLANE SECTION
SCALE I
*=6000 PSI
Fig. 13. Magnitude and direction of principal stresses at selected locations.
STRESS IDEALIZATION I N DENTAL STRUCTURES
251
principal stress plot which records the accompanying stress in the orthogonal direction, remained close to the zero value in this case. The finer grid size (Figs. 8, 10, and 12) yielded results which agree closely with the “coarse” grid results (Figs. 7, 9, and 11). The magnitude and direction of principal stresses shown for the plane stress idealization (Fig. 13) demonstrates the effect of eccentric load application on internal stress distribution in the crown and roots. This form of presentation lends itself to interpretation: a) locations of cleavage planes in enamel or dentin can be deduced, and b) zones of Luders line formation in restorative materials can be readily found. I n the present case a failure plane is likely to occur on the lingual face running through the enamel-dentin junction and well down into the root.
CONCLUSIONS The finite element method, an often used engineering technique, is well suited for application to dental stress analysis. Axisymmetric and plane stress idealizations are two approaches that have been investigated. Further work with these mathematical models, as well as with three dimensional idealizations is called for, as well as correlation studies with photoelastic and clinical findings. The authors wish t o express their appreciation to Mr. Charles L. Frederickson, Aeromechanics Division, Weapons Development Department, U. S. Naval Weapons Center, China Lake, California, for the preliminary work in this study; t o Dr. Manville G. Duncanson, Jr., Dental Materials Department, University of Oklahoma College of Dentistry, for his criticism and sdvice; and to Mr. Richard Beal, UCLA School of Dentistry Art Department, for the illustrations. The Campus Computing Network, UCLA, made a significant contribution by providing the computing facilities necessary for this study. This study was supported in part by U.S.P.H.S. General Research Grant No. RR-053-04.
References 1. M. A. Noonan, J . Dent. Child., 16, 24 (1949). 2. D. B. Mahler and F. A. Peyton, J . Dent. Res., 34, 831 (1955). 3. R. C. Haskins, D. C. Haack, and R. L. Ireland, J . Dent. Res., 33,757 (1954). 4. W. F. Guard, D. C. Haack, and R. L. Ireland, J . Arner. Dent. ASSOC., 57, 631 (1958). 5. L.-E. Granath, Odontal. Revy, 14, 278 (1963).
252 6. 7. 8. 9. 10. 11.
12. 13.
14. 15. 16. 17.
18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30.
SELNA, SHILLINGBURG, AND KERR L.-E. Granath, Odontal. Revy, 15, 169 (1964). L.-E. Granath, Odontal. Revy, 15, 290 (1964). L.-E. Granath, Oduntal. Revy, 15, 349 (1964). R. G. Craig, M. K. El-Ebrashi, P. J. LePeak, and F. A. Peyton, J . Prosth. Dent., 17,277 (1967). R. G. Craig, M. K. El-Ebrashi, and F. A. Peyton, J. Prosth. Dent., 17, 292 (1967). M. K. El-Ebraahi, R. G. Craig, and F. A. Peyton, J . Prosth. Dent., 22, 333 (1969). M. K. El-Ebrashi, R. G. Craig, and F. A. Peyton, J . Prosth. Dent., 22, 346 (1969). M. K. El-Ebrashi, R. G. Craig, and F. A. Peyton, J. Prosth. Dent., 22, 565 (1969). M. K. El-Ebrashi, R. G. Craig, and F. A. Peyton, J . Prosth. Dent., 22, 663 (1969). M. K. El-Ebrashi, R. G. Craig, and F. A. Peyton, J . Prosth. Dent., 23, 177 (1970). J. P. Standlee, A. A. Caputo, and E. W. Collard, Dent. Pract., 21,417 (1971). E. W. Collard, A. A. Caputo, and J. P. Standlee, J. Prosth. Dent., 29, 498 (1973). M. L. Lehman and E. L. Hampson, Brit. Dent. J., 113,337 (1962). C. B. Walton and M. M. Leven, J. Amer. Dent. Assoc., 50,44 (1955). E. W. Johnson, C. R. Castaldi, D. J. Gau, and G. P. Wysocki, J . Dent. Res., 47, 548 (1968). C.IMartin, . and L. J. Topp, J . Aeronaut. Sci., M. J. Turner, R. W. Clough, € 23, 805 (1956). Proceedings of the Conference on Application of Finite Element Method to Civil Engineering, Vanderbilt University, Nashville, Tenn., 1970. Proceedings of the Conferences on Matrix Methods in Structural Mechanics, Wright Patterson Air Force Base, Ohio, First Conference-1965, Second Conference-1968, Third Conference-1971. 0. C. Zienkiewicz, The Finite Element Method in Engineering Science, 2nd ed., McGraw-Hill, London, 1971. J. A. Stricklin and W. E. Haisler, Formulation, Computation, and Solution Procedures for Material and/or Geometric Non-linear Structural Analysis by the Finite Element Method, Sandia Contract Report No. PO B2-5617, 1972. H. R. Muhlemann, J . Periodontol., 38, 114 (1967). R. G. Craig, F. A. Peyton, and D. W. Johnson, J . Dent. Res., 40,937 (1961). J. W. Stanford, G. C. Paffenbarger, J. W. Kumpula, and W. T. Sweeney, J . Amer. Dent. ASSOC.,57, 487 (1958). J. W. Stanford, K. V. Weigel, G. C. Paffenbarger, and W. T. Sweeney, J . Amer. Dent. ASSOC., 60, 746 (1960). F. A. Peyton, D. B. Mahler, and B. Hershenov, J. Dent. Res., 31,366 (1952).
Received June 4, 1974
J. BIOMED. MATER. RES.
-
Finite Element Analysis of Dental Structures Axisymmetric and Plane Stress Idealizations
LAWRENCE G. SELNA, Mechanics and St?ytures Department, University of California, Los Angeles, HERBERT T. SHILLINGBURG, JR.,* School of Dentistry, University of California, Los Angeles, and PETER A, KERR, Mechanics and Structures Department, University of California, Los Angeles, California 90024
Summary The finite element method used to study stress generated in a maxillary second premolar MIa result of occlusal forces. This mathematical technique has been applied extensively in structural engineering and structural mechanics. It is well suited to the analysis of stress in teeth and dental restorations because it can closely simulate the geometries, loads, and material inhomogeneitiesin the system being studied.
INTRODUCTION A knowledge of the intensity and distribution of stress can be helpful in avoiding or minimizing the failure of dental restorations. The effects of stress on dental restorations and upon teeth themselves has been a topic of some interest to dental researchers over the past twenty years. Early published reports indicate that this research concentrated on cavity configurations of Class I1 amalgam restorations. One of the earliest photoelastic studies applied to dentistry dealt with the stress distribution in tooth structure surrounding restorations. The general usefulness of the photoelastic method and the ways of applying this technique to dentistry were described by Mahler and Peyton.2 Studies have been done on the effects of preparation geometry on the restoration while other studies of the Class I1 configura*Present address : College of Dentistry, University of Oklahoma, Oklahoma City, Oklahoma 73190
237 @ 1975 by John Wiley & Sons, Inc.
238
SELNA, SHILLINGBURG, AND KERR
tion examined effects on both tooth structure and the re~toration.~-s Photoelastic techniques have been employed to study inlays,g crowns,lo proximal margins,l 1 convergence of axial walls, l 2 occlusal reduction, l3 proximal reduction, l 4 fixed partial dentures,15pin placement for amalgam retention, l6 and impression distortion during removal from the mouth.” The proSome three dimensional studies have been cedures used for studying three dimensional models require great care and considerable time because of the complex shapes being studied. Consequently, two dimensional “slice” models are usually used. In the field of engineeripg, other methods of stress analysis are also utilized. A technique which has been extensive application since 195621in the fields of civil and aerospace engineering is the finite element m e t h ~ d . Z ~ -In ~ ~this approach, mathematical idealizations are used, rather than physical models. A study of preparation geometries requires the examination of many configurations. Consequently, mathematical representation of the physics of tooth deformation and stress would be useful since it could be used for different tooth geometries by changing only a few input parameters to model. Finite element analysis permits the study of as many tooth and restorative materials as might be present in any given problem: enamel, dentin, cement, gold, etc. With this method, many types of preparation geometries and restorative materials can be studied. This can provide some refinement of the photoelastic technique, where only two materials have been studied: undifferentiated tooth structure and the restorative material.
MATERIALS AND METHODS Two types of idealizations were used in this study, and different grid sizes were tried to determine what combinations would give the best approximations of tooth shape and behavior. The basic concept of the finite element method is the subdivision of the body to be studied into many smaller elements of a simpler geometric shape. For two dimensional studies, triangles or quadrilaterals are often used, while tetrahedra or hexahedra are frequently used in three dimensional studies. These elements may be assembled like buiIding blocks to approximate a body of any given shape. The
STRESS IDEALIZATION IX IIEXTAL STRUCTURES
239
elements are designated so that apices of adjoining elements meet to form (‘node”’. The apex of one element cannot fall on the side of a n adjacent element. When analysis is performed, a system of simultaneous equations results, relating the forces and displacements, which are evaluated at the nodes. Aftcr solving the equations for displacements, the stresses can be computed within each element. Development of the finite element method has been closely tied to the development of high speed electronic digital computers. The large systems of simultaneous equations which result from the formulation can be solved feasibly only by using a computer. The computer can also be programmed to draw contour plots, showing the stress contours on sections of the tooth being studied. The essential information required for stress analysis encompasses the geometry, the boundary loads and supports, and the material properties. The geometry describes the spacial configuration of the mathematical idealization of the tooth. The occlusal forces are represented as force boundary conditions and the tooth mobility is represented by spring-type support conditions. The material properties include the constitutive or stress-strain laws, the range of validity of those laws, and the failure criteria. The relationship between stress and strain is complicated by these factors: a) the tooth is composed of different material subregions and is therefore inhomogenous; b) in each of the subregions the stressstrain law may be dependent upon direction since the tissue is anisotropic; and c) in each of the subregions the stress-strain behavior may be a function of time and rate of loading, i.e. the tissue is viscoelastic. Failure criteria prescribe the stress combinations and levels which will cause fracture in the material subregions. The formulation and solution procedure is established using virtual work methods. The virtual work in the tooth is expressed in terms of displacements at a finite number of nodal points within the tooth. The virtual work performed on the tooth by external loads is expressed a t these same points. If the internal and external virtual work are equated, the displacements a t the nodal points can be evaluated . The virtual strain energy for the tooth structure is given by
V
SELNA, SHILLINGBURG, AND KERR
240
in which 6 U = virtual strain energy; r = stress tensor; 6a = virtual strain tensor; V = volume of the tooth. The external virtual work is given by 6W
=
I
T6uds
S
in which 6 W = external virtual work; T = surface traction vector; 6u = virtual displacement vector; S = tooth surface. Referring to a finite element embedded within the tooth, interpolation functions may be used to express the displacement field within each element in terms of nodal displacement values. This is usually written as u =
(3)
CPU
in which CP = interpolation function; u = nodal coordinate displacements. The virtual strain tensor for each element is given by 8E
in which CP,, are given by
=
CP,, 6u
=
(4)
gradient of the interpolation function. The stresses 7
=
CCP,, u
(5)
in which C = generalized Hooke’s law for anisotropic elastic materials. This stress-strain law can be extended to plastic materials by using the incremental methods.25 If eqs. (4) and ( 5 ) are substituted into eq. (l),and the integration is performed over element i, the result is 6Ui =
s
vi
6uCP,,
cq, udV
(6)
The properties of the interpolation functions permit a simple integration over the element volume, Vi. This yields the scalar form 6Ui
=
6u
ki u
(7)
in which ki = stiffness matrix for the element i. The virtual strain energy for all the elements which model the tooth may then be obtained by summing the virtual energies for all the elements to yield
STRESS IDEALIZATION IN DENTAL STRUCTURES
241
where there are n elements comprising the tooth. An alternative form of eq. (8) is 6 U = 6r Kr
(9)
in which r, 6r = real and virtual global coordinate displacements; K = global stiffness matrix. The corresponding external virtual work is
6W=6rF
(10)
in which F = global force components which are evaluated from eq. (2). Using the Divergence Theorem, eqs. (9) and (10) can be equated, and since 6r is arbitrary, the result is
Kr
=
F
(11)
Equation (11) is a set of linear simultaneous equations which may be solved for the nodal displacements, r. Using these displacements, eq. (5) may be employed to find the element stresses. Equations (5), (7), and (11) which are the essential ingredients. in this finite element procedure can be programmed on the digital computer. Output from the computer program will yield the displacement and stress fields throughout the tooth structure. This theory, written in general terms, is valid for all finite element representations in one, two, or three dimensions. In this study, attention will be focused on two types of two dimensional representations: plane stress and axisymmetric. The plane stress model most closely approximates the model used in two dimensional photoelasticity. A midsagittal section of a maxillary premolar (Fig. 1A) was divided into a finite number of triangular subregions (Fig. 1B). The mathematical solid used for analysis was produced by expanding the section in the third (depth) dimension (Fig. 1C). Occlusal forces were applied as three uniformly distributed line loads totalling 100 lb. They were placed in the depth dimension at the desired nodes of the crown in a vertical direction (Fig. 2 ) . The support provided by the periodontal membrane and the underlying bone was a little more difficult to model mathematically. A
SELNA, SHILLINGBURG, AND KERR
242
A
0
C Fig. 1. Generation of a plane stress solid.
rigid support on the roots was deemed unsatisfactory because of the mobility experienced by teeth under load. A more suitable representation was considered to be a continuous elastic foundation. To approximate this condition, springs, both vertical and horizontal, were placed over the interfacial region on exterior nodes (Fig. 2). The spring coefficients were evaluated from studies relating displacement and applied force.26 The high cusp axisymmetric model was generated by taking the lingual half of a midsagittal section (Fig. 3A) and dividing it into subregions (Fig. 3B). This half section was rotated about an axis corresponding to the center axis of the tooth to produce a mathematical solid (Fig. 3C). The low cusp model was similarly formed by taking one-half of a mesiodistal section through the central groove and marginal ridge, and rotating it about the central axis. The component elements of these solids are three-sided concentric rings (Fig. 4). Occlusal forces and spring reactions were distributed uniformly as line loads around the concentric rings formed by the exterior nodes on the crown and root (Fig. 5 ) . All three test cases were studied with both coarse and fine element subdivisions (Fig. 6 ) . The greater the number of elements in the
STRESS IDEALIZATION IN DENTAL STRUCTURES
243
* 33.3 LBS
+ 0
Ib/in
1 0 Ib/in
Fig. 2. Loading and support conditions for the plane stress idealization.
244
SELNA, SHILLINGBURG, AND KERR
-\ 1
A
0
Fig. 3. Generation of an axisymmetric solid.
idealization, the more closely the actual continuum of the physical body can be approximated. Consequently, the results of increasingly finer grid sizes converge on the true solution. It is wise to concentrate elements in the regions of greatest interest and stress concentration. The decision of how many elements to use rests with the designer who must optimize between computer time, computer capacity, and an acceptable degree of accuracy.
For the material properties, dentin and enamel were considered to be homogenous and isotropic. The modulus of elasticity for enamel was 5.0 X lo6 psi27-2e and that for dentin was 2.0 X lo6 psi.3o Poisson’s ratio was taken as 0.25 for both enamel and dentin. To simplify this initial work, the pulp was not considered and was treated as a void. Failure was also not considered. RESULTS The results of the analysis showing minimum and maximum principal stress, are presented in the form of contour plots prepared by the computer (Figs. 7-12). The different intensity of contours are identified by their characteristic symbols. The magnitude and direction
STRESS IDEALIZATION I N DENTAL STRUCTURES
i7
7
245
i
M
.I
cr
SELNA, SHILLINGBURG, AND KERR
246
Fig. 6. Coarse and fine grid plane stress idealizations.
SYMBOLS 0 t5uXF.u
h -4.900
+ -.ooo I - U W
8
E I - 0 W 2 0
a’
?ii
Y
800
n
1600
x P
2400
3
3200
E
I
4000
a
4800
g
Fig. 7. Stress contours-plane
stress, coarse grid.
STRESS IDEALIZATION IN DENTAL STRUCTURES
247
248
SELNA, SHILLINGBURG, AND KERR
STRESS IDEALIZATION I N DENTAL STRUCTURES
249
SYMBOLS' 0 tymc
+ -um I -Mo 0
-2.00
a
-MX)
* -1600 2
0
7
000
n
1600
W
2400
P
3200
1 4 w o R
a00
J
1
MINIMUM PRINCIML STRESS
MAXIMUM PRINCIPAL STRESS
Fig. 12. Stress contours-axisymmetric, low cusp, fine grid.
of the principal stresses at different locations in the tooth can be obtained directly from the principal stress contours (Fig. 13).
DISCUSSION The contour plots for principal stress were chosen because they give a good visual impression of the stress distribution, the regions of high intensity, and the stress gradients. As would be expected, the highest intensity occurs directly beneath the applied loads. Since the contours are defined at equal intervals of intensity, the closeness of the contour spacing indicates the stress gradient. In the roots of the teeth, the contours are widely spaced, indicating that the stress distribution is nearly uniform. The plot of minimum principal stress records the high negative or compressive stresses. The maximum
250
SELNA, SHILLINGBURG, AND KERR PLANE SECTION
SCALE I
*=6000 PSI
Fig. 13. Magnitude and direction of principal stresses at selected locations.
STRESS IDEALIZATION I N DENTAL STRUCTURES
251
principal stress plot which records the accompanying stress in the orthogonal direction, remained close to the zero value in this case. The finer grid size (Figs. 8, 10, and 12) yielded results which agree closely with the “coarse” grid results (Figs. 7, 9, and 11). The magnitude and direction of principal stresses shown for the plane stress idealization (Fig. 13) demonstrates the effect of eccentric load application on internal stress distribution in the crown and roots. This form of presentation lends itself to interpretation: a) locations of cleavage planes in enamel or dentin can be deduced, and b) zones of Luders line formation in restorative materials can be readily found. I n the present case a failure plane is likely to occur on the lingual face running through the enamel-dentin junction and well down into the root.
CONCLUSIONS The finite element method, an often used engineering technique, is well suited for application to dental stress analysis. Axisymmetric and plane stress idealizations are two approaches that have been investigated. Further work with these mathematical models, as well as with three dimensional idealizations is called for, as well as correlation studies with photoelastic and clinical findings. The authors wish t o express their appreciation to Mr. Charles L. Frederickson, Aeromechanics Division, Weapons Development Department, U. S. Naval Weapons Center, China Lake, California, for the preliminary work in this study; t o Dr. Manville G. Duncanson, Jr., Dental Materials Department, University of Oklahoma College of Dentistry, for his criticism and sdvice; and to Mr. Richard Beal, UCLA School of Dentistry Art Department, for the illustrations. The Campus Computing Network, UCLA, made a significant contribution by providing the computing facilities necessary for this study. This study was supported in part by U.S.P.H.S. General Research Grant No. RR-053-04.
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Received June 4, 1974
