Ceramic bracket design: An analysis using the finite element method

ORIGINAL ARTICLES

Ceramic bracket design: An analysis using the finite element method Joydeep Ghosh, BDS, MS," Ram S. Nanda, DDS, MS, PhD, b Manville G. Duncanson, Jr., DDS, MS, PhD, ° and G. Fr~ins Currier, DDS, MSD, MEd d Oklahoma Ci~ Oklal This investigation was designed to generate finite element models for selected ceramic brackets and graphically display the stress distribution in the brackets when subjected to arch wire torsion and tipping forces. Six commercially available ceramic brackets, one monocrystalline and five polycrystalline alumina, of twin bracket design for the permanent maxillary left central incisor were studied. Three-dimensional computer models of the brackets were constructed and loading forces, similar to those applied by a full-size (0.0215 × 0.028 inch) stainless steel arch wire in torsion and tipping necessary to fracture ceramic brackets, were applied to the models. Stress levels were recorded at relevant points common among the various brackets. High stress levels were observed at areas of abrupt change in geometry and shape. The design of the wire slot and wings for the Contour bracket (Class One Orthodontic Products, Lubbock, Texas) and of the outer edges of the wire slot for the Allure bracket (GAC, Central Islip, N.Y.) were found to be good in terms of even stress distribution. The brackets with an isthmus connecting the wings seemed to resist stresses better than the one bracket that did not have this feature. The design of the isthmus for the Transcend (Unitek/3M, Monrovia, Calif.) and Lumina (Ormco, Glendora, Calif.) brackets were found to be acceptable as well. The Starfire bracket ("A" Company, San Diego, Calif.) showed high stresses and irregular stress distribution, because it had sharp angles, no rounded corners, and no isthmus. The finite element method proved to be a useful tool in the stress analysis of ceramic orthodontic brackets subjected to various forces. This analysis provides key information to the development of an optimum bracket design. (AM J ORTHOD DENTOFACORTHOP 1995;108:575-82.)

T h e fracture of ceramic brackets from arch wire tipping and torquing forces has been reported to be a problem by the orthodontic profession. 1-3 Bracket fracture contributes to increased chair time, patient discomfort, and the potential health hazard of aspirating a bracket fragment. Studies on the brittle fracture behavior of ceramic brackets have shown that surface cracks and flaws greatly reduce their fracture resistance. Because there is virtually no plastic deformation in ceramic materials that could relieve these stresses at the tips of the cracks, the cracks propagate until total structural failure occurs. 4 Loading tests of actual ceramic brackets with arch wire torquing and tipping forces have indicated that there are significant differences in fracture strengths among the currently available ceramic From the University of Oklahoma. "Visiting Assistant Professor and Research Associate, Department of Orthodontics. bprofessor and Chair, Department of Orthodontics. CProfessor and Chair, Department of Dental Materials. dprofessor, Department of Orthodontics. Copyright © 1995 by the American Association of Orthodontists. 0889-5406/95/$5.00 + 0 8/1/53941

brackets. 5-9Differences have been seen between the polycrystalline and the single crystal alumina brackets and among the various polycrystalline ones. The fracture strengths of ceramic brackets were found to be significantly less than their metal counterparts, but adequate for clinical use? 9 The currently available polycrystalline and single crystal ceramic brackets not only have different manufacturing processes and physical properties but also different design configurations. Several investigators have hypothesized that bracket design could be a factor in the different fracture strengths among the commercially available ceramic brackets. The finite element analysis, which has been widely used in the aircraft industry, is an effective method for determining stress distribution patterns for structures of complex design and known material properties. According to this method, the actual structure is subdivided into a finite number of independent discrete elements. These elements are interconnected at points called nodes and are superimposed onto a coordinate grid system, to which every node is referenced. The finite elements retain the properties of the original material. Forces are 575

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a p p l i e d o n a r e a s o f the m o d e l to s i m u l a t e t h e r e a l life situation. S u p p o r t s o r k i n e m a t i c constraints, called boundary conditions, a r e i m p o s e d t h a t will e n s u r e e q u i l i b r i u m o f t h e loads. W i t h matrices, t h e d i s p l a c e m e n t s of t h e n o d e s a r e r e l a t e d to t h e e x t e r n a l forces. T h e strains a r e t h e n d e t e r m i n e d with t h e s t r a i n - d i s p l a c e m e n t relations, a n d finally t h e stresses a r e e v a l u a t e d with t h e stress-strain r e l a t i o n s h i p s . This m e t h o d of finite e l e m e n t analysis has n o t b e e n a p p l i e d to the study of t h e intera c t i o n b e t w e e n t h e design of c e r a m i c b r a c k e t s a n d t h e i r s t r e n g t h s w h e n s u b j e c t e d to t o r s i o n a l a n d t i p p i n g forces o f an a r c h wire. T h e p u r p o s e s o f this study w e r e (1) to use t h e finite e l e m e n t analysis in d e v e l o p i n g m o d e l s o f s e l e c t e d c e r a m i c b r a c k e t s a n d to display g r a p h i c a l l y t h e stress d i s t r i b u t i o n with a r c h wire t o r s i o n a n d t i p p i n g forces, a n d (2) to m e a s u r e a n d c o m p a r e t h e stress levels at c o m m o n l y s e l e c t e d p o i n t s a m o n g t h e six d i f f e r e n t designs o f b r a c k e t s . MATERIALS AND METHODS

Six commercially available ceramic brackets of twin bracket design for the permanent maxillary left central incisor were studied. All brackets had a 0.022-inch slot and were similarly preadjusted in torque and angulation with positive 12° palatal root torque and positive 5 ° mesial crown angulation. Of the brackets studied, one was single crystal and five were polycrystalline alumina. The polycrystalline brackets tested, with their respective suppliers, were Allure (GAC, Central Islip, N.Y.), Ceramaflex (TP Orthodontics, LaPorte, Ind.), Contour (Class One Orthodontic Products, Lubbock, Texas), Lumina (Ormco, Glendora, Calif.), and Transcend (Unitek/3M, Monrovia, Calif.). The only monocrystalline bracket was Starfire ("A" Company, San Diego, Calif.). The description of each bracket is given by crystallinity-manufacturer's abbreviation, i.e., P-A means Polycrystalline-Allure bracket. For each bracket type by manufacturer, an attempt was made to evaluate the variations in the thicknesses of their dimensions. Measurements of eight brackets from each manufacturer were made with a vernier caliper accurate to 0.0001 inch (Mitutoyo Model no. 500-351; MTI Corp., Paramus, N.J.). It was discovered that the variations in size within each bracket type were minimal. Since the focus of this study was to evaluate the effect of differences in shape of the bracket types, the use of bracket dimensions taken from a representative bracket from each group would suffice in developing a model. These values were then accurately transferred onto a computer (Zenith Data General 386, Under-Ware Electronics, Wichita, Kan.) to construct three-dimensional models of the brackets with the SuperDraw software (Algor Interactive Systems, Pittsburgh, Pa.). Once a

American Journal of Orthodontics and Dentofacial Orthopedics December 1995

closed geometric diagram of the bracket was defined, a mesh was generated with a network of intersecting lines to divide the model into quadrilateral elements. All elements used were three-dimensional, six-sided brick elements with eight corner nodes. The number of elements on the bracket models ranged between 1480 and 1904, whereas the number of nodes ranged between 2188 and 2657. A typical bracket model is shown in Fig. 1, which depicts a three-dimensional view of the Ceramaflex (P-Cx) bracket. The values for properties of single crystal ceramic material were obtained from its manufacturer, and the same values were applied to all six bracket models. This eliminated material properties as a variable and allowed bracket design alone to be studied for its effect on stress distribution patterns. Single crystal ceramic properties were chosen because they have a higher tensile strength and modulus of elasticity than polycrystalline ceramics. The weight density value used was 0.14 pounds per cubic inch, Young's modulus of elasticity was 5.5 x 107 pounds per square inch, Poisson's ratio was 0.29, and coefficient of thermal expansion was 5 x 106 per degrees Celsius. The ceramic material was treated as isotropic. Loading forces in torsion and tipping were then applied to the bracket models as described in the following: The force values were obtained from previous research wherein forces applied by a full-size (0.0215 inch x 0.028 inch) stainless steel arch wire in torsion and tipping necessary to fracture ceramic brackets were determined. With torsion, Holt et al. 7 reported the range of mean torquing forces required for fracturing six different types of ceramic brackets to be from 0.168 to 0.281 pounds (76.2 to 127.5 gm) with the overall mean at 0.235 pounds (106.6 gm). This force, applied at a distance of 1.91 inches (48.5 mm) from the center of the wire, produced a couple in the bracket slot. The value of each force of the couple was calculated to be 6.358 pounds (2884 gm). This force value was applied simultaneously at the incisal edge of the base of the wire slot and gingival outer edge of the wire slot, to simulate a palatal root torquing movement. With tipping, Rhodes et al. 8 reported the range of mean forces required for fracturing five different types of ceramic brackets (0.022-inch wire slot) to be from 0.664 to 1.43 pounds (301.3 to 648.6 gm). The overall mean of these forces Was 0.961 pounds (435.8 gm). This force value was applied simultaneously at the line angle where the gingival wall of the wire slot meets the mesial bracket' surface and at the line angle where the incisal wall of the wire slot meets the distal bracket surface to simulate a distal root tipping movement. Because the bracket is bonded onto the tooth surface through the base and is not free to move, the nodes on the base of all the bracket models were held fixed. No other nodes in the models were restrained. A linear static analysis was performed using the linear static stress analysis processor (Algor Interactive Systems, Inc., Pittsburgh, Pa.). Convergence testing was done through a process of trial and error wherein the mesh design was

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Fig, 1. Typical bracket model showing three-dimensional view of P-Cx bracket.

refined in the models to increase mesh density in areas of high stress gradients. The yon Mises stresses (psi) were then recorded at areas of interest between the final bracket models.

RESULTS Tipping Fig. 2 shows the 13 key points on the brackets where stresses generated with tipping and torquing forces were compared. A typical stress distribution pattern with tipping forces is shown in Fig. 3, which depicts the three-dimensional, mesial, distal and facial views of the Lumina (P-L) bracket. To the right of the figure is a spectrum of colors reflecting stress levels ranging from red (very high) to purple (lowest). Table I shows the stress values at the points common among the various brackets. Each section of the table reflects the stress levels at a particular point on each of the brackets. The minimum stresses on the brackets ranged from 0.13 psi (0.0009 MPa) for the Starfire (S-S) bracket to 1.13 psi (0.008 MPa) for the Transcend (P-T) bracket and were located near the bracket base farthest away from the sites of force application. The maximum stresses ranged from 8,460 psi (58.3 MPa) for the P-A bracket to 18,200 psi (125.5 MPa) for the S-S bracket and were located near the points of force application. The major stresses on all the brackets were concentrated along two lines: one,

at the gingival wall of the wire-slot where it meets the mesial bracket surface (Fig. 2, line 1-3), and the other, at the incisal wall of the wire-slot where it meets the distal surface (Fig. 2, line 2-4). This is consistent with the points Of force application, which cause distal root tipping. The stresses gradually decreased, moving away from these edges. The stresses at the "points" as shown in Fig. 2 are described as follows. The sites of highest stress for S-S, P-Cx, and P-L brackets was at the mesiogingival outer point of the wire slot (point 1). The overall maximum stress was recorded at this point for the S-S bracket at 18,200 psi (125.5 MPa) (Table I, point 1). Overall, P-A showed relatively the lowest maximum stress value at 8460 psi (58.3 MPa) (Table I, point 3). High stresses were also concentrated at the distoincisal outer point of the wire slot (point 2), the corners of the base of the wire slot, at the mesiogingival end (point 3), and the distoincisal base point of the wire slot (point 4). The stresses radiated from the wire slot, gradually decreasing as they progressed toward the base of the tying slot. High stress values were also recorded at the base of the tying slots (Table I, points 7 and 8). The stresses also radiated from the edges of the wire slot along the edges of the base of the wire slot and toward the junction of the bracket wing and the isthmus.

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1995

Gingival Mesia~l ~~

Distal

1 Gingival ~ S l o t

TyingS l e ~ Inclsal Mesio-Gin~ivOalbliouV eiew

Mesial lncisal Disto-i~dsO albliquV eiew

Fig. 2. Thirteen points of comparison for tipping and torquing stresses on edgewise ceramic orthodontic bracket. Surfaces of bracket, gingival, incisal, mesial and distal, are indicated. 1, Mesiogingival outer wire slot point: Point formed at junction of gingival wall of wire slot, and mesial and facial bracket surfaces. 2, Distoincisal outer wire slot point: Point formed at junction of incisal wall of wire slot, and distal and facial bracket surfaces. 3, Mesiogingival base wire slot point: Point formed at junction of gingival wall and base of wire slot, and mesial bracket surface. 4, Distoincisal base wire slot point: Point formed at junction of incisal wall and base of wire slot, and distal bracket surface. 5, Mesioincisal base wire slot point: Point formed at junction of incisal wall and base of wire slot, and mesial bracket surface. 6, Distogingival base wire slot point: Point formed at junction of gingival wall and base of wire slot, and distal bracket surface. 7, Mesiogingival tying slot point: Point formed at deepest concavity at mesial end of gingival tying slot. 8, Distoincisal tying slot point: Point formed at deepest concavity at distal end of incisal tying slot. 9, Mesiogingival isthmus point: Point at facial end of junction of mesiogingival wing and isthmus. 10, Distoincisal isthmus point: Point at facial end of junction of distoincisal wing and isthmus. 11, Distogingival outer wire slot point; Point formed at junction of gingival wall of wire slot, and distal and facial bracket surfaces. 12, Midgingival isthmus point: Point at middle of gingival isthmus at junction of gingival wall of wire slot and facial bracket surface. 13, Distogingival tying slot point: Point formed at deepest concavity at distal end of gingival tying slot.

Torquing A typical stress distribution pattern for torquing is shown in Fig. 4, which shows the three-dimensional, mesial, distal and facial views of the P-T bracket model. Table II indicates the Stress values at the commonly selected points between the brackets. Similar to Table I, each section of Table II reflects the stress level at a particular point on each of the brackets. The minimum stress values on the brackets ranged from 0.15 psi (0.001 MPa) for the P-T bracket to 1.03 psi (0.007 MPa) for the P-Cx bracket and were located near the bracket base farthest away from the points of force application, whereas the maximum stress values ranged from 16,810 psi (115.9 MPa) for the P-A bracket to 67,890 psi (468.1 MPa) for the S-S bracket and were located near the points of force application. The major stresses on all the brackets were concentrated along two lines: one, at the junction of

the gingival wall of the wire slot and frontal bracket surface (Fig. 2, Line 1-11), and the other, at the junction of the incisal wall of the slot and the base of the slot (Fig. 2, Line 5-4). This is consistent with the points of force application which cause palatal root torquing. The stresses gradually decreased moving away from these edges. The stresses at the points as shown in Fig. 2 are described as follows. The sites of highest stress for the S-S, P-A, P-Cx, P-L, and P-T brackets was at the mesiogingival outer point on the wire-slot (point 1). The overall maximum stress was recorded at this point for the S-S bracket at 67,890 psi (468.1 MPa) (Table II, point 1). The site of highest stress for the Contour bracket (P-C) was at the midgingival isthmus point (point 12), with the value recorded at this site being 23,930 psi (165 MPa) (Table II, point 12). High stresses were also concentrated at the distogingival outer point on the wire slot (point 11),

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108,

13

I

M

F

579

r 18185.6 ~[~i,~i 16670.2 15154.7 13639.2 12123.7 10608.3 9092.84 i!~i!i~!!6061.89 7577.37

--

4546.42 3030.94 1515.47

_20 I

........ ~ "

(3

D

Fig. 3. Typical stress distribution pattern with tipping forces showing (clockwise, from top left) three-dimensional, mesial, distal and facial views of P-L bracket. Letters in each of views indicate mesial (M), distal (D), facial (F), incisal (I), or gingival (G) surfaces.

Fig. 4. Typical stress distribution pattern with torquing forces showing (clockwise, from top left) three-dimensional, mesial, distal and facial views of P-T bracket. Letters in each of views indicate mesial (M), distal (D), facial (F), incisal (I) or gingival (G) surfaces.

the mesioincisal base point on the wire slot (point 5), and the distoincisal base point on the wire slot (point 4). The stresses from the wire slot gradually decreased, moving toward the base of the tying slots. As with tipping, high stresses were also re-

corded at the base of the tying slots (Table II, points 7 and 13). Stresses radiated along the edges of the wire slot throughout the width of the bracket and concentrated near the corners and the middle of the isthmus.

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Table h Comparison of stress measurements between the brackets when subjected to tipping forces (0.961 lbs or 435.8 gin; from Rhodes et al. s) at selected points (stress levels at points 1 through 10 are shown) Point 1

Point 2

Point 3

Point 4

Point 5

Mesiogingival outer

Distoincisal outer

Mesiogingival base

wire slot point

wire slot point

Distoincisal base wire slot point

Mesioincisal base

wire slot point

wire slot point

Bracket Stress-psi(MPa)

Bracket Stress-psi(MPa) Bracket Stress-psi (MPa) Bracket Stress-psi (MPa) Bracket Stress-psi (MPa)

S-S P-L P-Cx P-C P-T P-A

P-L P-C S-S P-Cx P-T P-A

18,200" (125.5) 16,700" (115.1) 11,000' (75.8) 11,000 (75.8) 8,710 (60.1) 7,900 (54.5)

12,700 (87.6) 11,900" (82.1) 11,000 (75.8) 10,400 (71.7) 7,500 (51.7) 6,320 (43.6)

8,790 (60.6) 8,460* (58.3) 8,390 (57.8) 7,780 (53.6) 7.600 (52.4) 7,550 (52.1)

S-S P-A P-Cx P-T P-C P-L

S-S P-T P-Cx P-C P-L P-A

9,770 (67.4) 9,030* (62.3) 8,670 (59.8) 8,480 (58.5) 7,620 (52.5) 7,300 (50.3)

P-T S-S P-Cx P-A P-L P-C

1,210 1,210 1,210 1,040 936 403

(8.3) (8.3) (8.3) (7.2) (6.5) (2.8)

*Indicates maximum stress value in bracket.

Table II. Comparison of stress measurements between the brackets when subjected to torquing forces (6.358 Ibs or 2884 gm; from Holt et al. 7) at selected points (stress levels at points 1 through 13 are shown) Point 1

Point 11

Point 5

Point 4

Point 3

Mesiogingival outer

Distogingival outer

Mesioincisal base

Distoincisal base

wire slot point

wire slot point

wire slot point

wire slot point

Mesiogingival base wire slot point

Bracket ] Stress-psi (MPa)

Bracket Stress-psi(MPa) Bracket Stress-psi(MPa) Bracket Stress-psi(MPa) Bracket Stress-psi (MPa)

S-S P-L P-Cx P-C P-T P-A

S-S P-L P-Cx P-C P-T P-A

67,890* (468.1) 56,150" (387.2) 25,710" (177.3) 21,780 (150.2) 19,840' (136.8) 16,810" (115.9)

66,050 55,420 25,360 21,360 19,320 16,410

(455.4) (382.1) (174.9) (147.3) (133.2) (113.1)

S-S P-T P-L P-Cx P-A P-C

20,900 (144.1) S-S 16,160 (58.3) P-T 15,030 (103.6) P-L 12,820 (88.4) P-Cx 12,530 (86.4) P-A 11,450 (78.9) P-C

21,530 (148.4) 16,600 (114.48) 15,500(106.9) 13,040(89.9) 12,740 (87.8) 11,060 (76.3)

P-Cx S-S P-C P-A P-L P-T

9,223 8,881 8,023 7,394 6,674 6,325

(63.6) (61.2) (55.3) (51.0) (46.0) (43.6)

*Indicates maximum stress value in bracket.

DISCUSSION

The design of a structure must allow prescribed functions as well as maintain structural integrity, i.e., each part of the system must have sufficient strength to carry out its designated function. Stress analysis attempts to ensure that each element of a system will meet the structural requirements during the course of its performance. Stresses with tipping forces

The highest stress levels were seen for three of the six brackets (S-S, P-Cx, and P-L) at the mesiogingival outer point of the wire slot (Fig. 2, point 1). The design of the preadjusted brackets makes this corner less bulky and hence more prone to high stresses. Because the outer edges of the wire-slot in the P-A bracket were bevelled, the lowest stress levels were evident at that point. At the base of the wire slot, two special design considerations were seen. First, whenever there was a squared base of the wire slot, the base corners of the slot were points of high stress.

Whenever there was a rounded base design, the flow of stress was smoother, and it was distributed over a larger area, resulting in less stress as was seen in the P-C bracket. Second, the thickness of the material between the base of the wire slot and tying slot affected the resistance to orthodontic stresses-the greater this thickness, the less the stress values. The results of the finite element analysis support the results of laboratory experiments by Rhodes et al. 8 The mesiogingival and distoincisal wings appear to be high stress areas and therefore were more predisposed to fracture. The greater disposition of the mesiogingival wing fracture reported by Rhodes may be due to higher stress levels at the junction of the mesiogingival wing and the isthmus. Stresses with torquing forces

The stresses at the mesiogingival outer point of the wire slot were the highest in five of six brackets. The brackets S-S and P-L, which lack the bevel or

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Point 6

Point 7

Point 8

Point 9

Point 10

Distogingival base wire slot point

Mesiogingival tying slot point

Distoincisal tying slot point

Mesiogingival isthmus point

Distoincisal isthmus point

581

Bracket Stress-psi (MPa) Bracket Stress-psi (MPa) Bracket Stress-psi (MPa) Bracket Stress-psi (MPa) Bracket Stress-psi (MPa)

P-Cx S-S P-L P-T P-A P-C

889 616 535 443 443 163

P-Cx P-A P-C S-S P-T P-L

3,540 2,670 2,370 2,150 2,010 1,720

(24.4) (18.4) (16.3) (14.8) (13.9) (11.9)

P-Cx P-C P-A S-S P-T P-L

3,780 3,080 2,790 2,710 2,560 2,250

(26.1) (21.2) (19.2) (18.7) (17.7) (15.5)

P-C P-Cx P-A P-T P-L S-S

1,150 (7.9) 1,000 (6.9) 997 (6.9) 642 (4.4) 544 (3.8) No Isthmus

P-C P-Cx P-A P-L P-T S-S

1,180 (8.1) 1,000 (6.9) 749 (5.2) 476 (3.3) 440 (3.0) No Isthmus

Point 6

Point 7

Point 13

Point 12

Distogingival base wire slot point

Mesiogingival tying slot point

Distogingival tying slot point

Midgingival isthmus point

Bracket

P-Cx P-C S-S P-A P-L P-T

(6.1) (4.3) (3.7) (3.4) (3.1) (1.1)

Stress-psi(MPa)

8,915 7,687 7,555 7,090 6,273 5,925

(61.5) (53.0) (52.1) (48.9) (43.3) (40.9)

Bracket

P-Cx P-A S-S P-L P-C P-T

I

-psi (MPa)

10,160 9,217 7,001 6,927 6,653 6,468

(70.1) (63.6) (48.3) (47.8) (45.9) (44.6)

the roundness of the wire slot, had more than double the amount of stress at this point than the remaining four brackets. The finding by Holt et al. 7 that the whole incisal half of the bracket broke off frequently with torsional forces could be explained by the finding that stresses concentrated at the base of the wire slot on the incisal, from where they radiated toward the base of the incisal tying slot and base of the bracket, causing vulnerability of the incisal half. On the other hand, stresses seemed to dissipate over a larger area on the gingival half. The results of this study also support the findings of the laboratory experiment conducted by Aknin 9 who studied the resistance to fracture of ceramic brackets manufactured after 1989, when subjected to torsional forces and also found the incisal half of the bracket to be more prone to fracture.

Comparison of bracket designs The design of the S-S bracket seemed to allow for high stresses and an uneven stress distribution.

Bracket

P-Cx P-A S-S P-C P-L P-T

Stress-psi(MPa)

10,320 9,163 7,019 6,999 6,983 6,474

(71.2) (63.2) (48.4) (48.3) (48.1) (44.6)

Bracket

Stress-psi (MPa)

P-L P-C P-Cx P-A P-T S-S

27,270 (188.0) 23,930* (165.0) 11,310 (78.0) 10,590 (73.0) 10,740 (74.1) No Isthmus

The high stress levels were due to sharp lines and angles and the lack of an isthmus between the twin brackets. However, in a recent experiment 9 to test fracture strength with torsional forces, the wire completely rotated outside the brackets without a single bracket failure. In the other brackets, which were polycrystalline, the presence of isthmus between the twin brackets was an important design consideration. More fully contoured brackets, the presence of a well-designed isthmus (P-T and P-L brackets), and roundness in the base of the wire slot (P-C bracket) appear to strengthen these brackets by better stress distribution. The concept of flow analogy is helpful in reducing stress concentrations, a° There is a similarity between the velocity of the flow of an ideal fluid in a channel and the flow of stresses in an element with the same geometry as the channel. An abrupt change in geometry causes stress concentrations and a disruption in stress flow similar to the turbulence it would cause in a fluid flow.

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Limitations of the study

The results of the finite element analysis must be interpreted with care. The accuracy of the analysis is dependent on modeling the structure as closely as possible to the actual. However, a certain amount of approximation, manifested chiefly in terms of type, number, and arrangement of elements, is inevitable with complex designs. One must also be aware of the assumptions used in the formulation, materials characterizations used, nature of boundary conditions, and the representation of loads. All these factors affect the validity of the results. In this study, the monocrystalline material properties were kept the same for all the brackets for the purpose of comparison. Therefore the stress values cannot be used directly for the polycrystalline brackets. To evaluate performance on an individual basis, it would be best to obtain the mechanical properties of the ceramic material used from the manufacturer and subject the model to various loading conditions. Combinations of torquing and tipping loads applied at the same time, as commonly performed in final stages of treatment, may result in stresses at certain areas far greater than seen in this study. It should be beneficial to analyze such a situation with the finite element method. A limitation of the finite element method in the analysis of solid mechanics is that a few complex phenomena, including cracking and fracture behavior, are not accommodated adequately by the method. I1 These p h e n o m e n a are currently under research, and refinements can be expected. It would then be interesting to include the presence of microcracks into the bracket models and repeat this study, so as to more closely simulate the real life situation. CONCLUSIONS

T h e stress distribution patterns of the ceramic brackets showed that stresses tended to concentrate at or near points of application of force, and stress fields were generally not uniform in these areas. Stresses were concentrated at corners, edges, and other areas of abrupt change in the shape of the bracket. The stress distribution was not uniform where the cross-section of the structure changed suddenly, and had large gradients at localized points. In the bracket models, this nonuniform stress pattern was mainly seen at the junction of the wings and the isthmus, the junction of the wall of the wire slot and the facial bracket surface, the

junction of the walls of the wire slot with its base, and at the junction of the wings as well as the bracket base to the tying slot. The various brackets studied showed several design features that demonstrated good as well as poor stress distribution. Combining the better features of design could c o m p e n s a t e for a lesser material and aid in developing an optimum bracket that would provide a quality product to profession. The finite element analysis reduces the need for prototypes and laboratory experimentation and allows more design options to be tested in a given amount of time. Physical testing can then be reserved for only the most promising designs. We thank the University of Oklahoma College of Dentistry for the seed grant that made this research study possible. REFERENCES

1. Scott GE. Fracture toughness and surface cracks-the key to understanding ceramic brackets. Angle Orthod 1988;58: 5-8. 2. Swartz ML. Ceramic brackets. J Clin Orthod 1988;22:82-8. 3. Lindquist JT. Letter: ceramic bracket survey. St. Louis: AAO, 1989. 4. Freiman SW. Brittle fracture behavior of ceramics. Ceramic Bull 1988;67:392-401. 5. Flores DA, Caruso JM, Scott GE, Toufic Jeiroudi M. The fracture strength of ceramic brackets: a comparative study. Angle Orthod 1990;60:269-76. 6. Gunn S, Powers JM. Strength of ceramic brackets in shear and torsion tests. J Clin Orthod 1991;25:355-8. 7. Holt MH, Nanda RS, Duncanson MG. Fracture resistance of ceramic brackets during archwire torsion. AM J ORTHOD DENTOFACORTHOP1991;99:287-93. 8. Rhodes RK, Duncanson MG Jr, Nanda RS, Currier GF. Fracture strengths of ceramic brackets subjected to mesiaP distal archwire tipping forces. Angle Orthod 1992;62:67-76. 9. Aknin PC. Fracture strengths of ceramic brackets during archwire torsion. [Master's thesis.] Oklahoma City: University of Oklahoma, 1993. 10. Budynas RG. Advanced strength and applied stress analysis. New York: McGraw-Hill, 1977. 11. Desai CS, Abel JF. Introduction to the finite element method- a numerical method for engineering analysis. New York: Van Nostrand Reinhold, 1972. Reprint requests to:

Dr. Ram S. Nanda Professor and Chair Department of Orthodontics College of Dentistry University of Oklahoma 1001 S.L. Young Blvd. Oklahoma City, OK 73190