Dent Mater 8:131-136, March, 1992

A simple model of crack propagation in dental restorations R.L. Sakaguchi 1, M. Cross 2, W.H. Douglas ~ IMinnesotaDental Research Centerfor Biomaterials and Biomechanics, Universityof Minnesota, Schoolof Dentistry, Minneapolis, MN, USA 2SchoolofMathematics, Statistics and Computing, ThamesPolytechnic,London, England

Abstract. Although natural teeth often exhibit microcracks, they rarely demonstrate bulk fracture. However, conventional fullcrown restorations periodically exhibit failure due to fracture. Presented here is evaluation of a simple model of crack propagation that estimates crack growth during cyclic loading. A finite element model of a premolar tooth provides the tensile stresses adjacent to cusp loading. If the crack propagation rates for natural teeth, porcelain-fused-to-metal crowns and composite crowns are compared with the wear rates of their respective materials as determined in an artificial mouth, it is evident that the low wear rate of composites may predispose them to fracture. Natural teeth disperse occlusal stresses throughout the dentin so that the effect of high occlusal stress is minimized. Porcelain tends to wear the opposing dentition, which reduces areas of high occlusal stress. Composite, however, demonstrates crack propagation rates higher than those of either natural teeth or porcelain. This, in addition to its low wear rate, might predispose the material to fracture. This model should be used only as a qualitative indicator of fracture tendency. The high calculated crack propagation rates in composites may explain the observed clinical failures and microchipping at the area of occlusal contact, as noted in SEM analysis. Natural teeth often demonstrate microcracks, or craze lines, in the dental enamel layer (Fig. 1). These cracks generally run along the long axis of the tooth; however, they rarely result in a failure of the tooth crown unless much of the tooth structure is undermined due to decay or the presence of a large restoration. Artificial materials that are designed for replacement of lost tooth structure, such as composites, also exhibit microcrack generation and propagation, but these materials often demonstrate significant bulk fracture, which requires replacement ofthe restoration (DiBenedetto, 1973). Inlaboratory studies that used an artificial mouth (DeLong and Douglas, 1983), it was found that the wear rates of new-generation composites were comparable with those of natural teeth (Sakaguchi et al., 1986a). In clinical practice, however, embrittlement and subsurface damage contribute to the overall deterioration of composite restorations (Wu et al., 1984). Although composite restorative materials may not possess optimal physical properties, they are used extensively in dentistry as direct filling materials because of their aesthetics and adhesion to natural tooth structure. Composites also demonstrate potential as a full-coveragecrown material. However, the problem ofcrack generation and propagation remains an issue. If composite materials are to be used to replace large amounts of lost tooth structure, three important issues need

Fig. 1. Craze lines in mature natural enamel.

consideration: (1) How do composite restorative materials behave under occlusal stress relative to natural enamel and dentin? (2) Why do composites demonstrate bulk fracture but exhibit low wear rates? (3) Is the crack propagation rate related to the wear rate of composites? This study examines these issues by use of a simple model for crack propagation, as proposed by Paris and Erdogen (1963) and modified by Forman et al. (1967), incorporating stresses derived from a validated two-dimensional plane strain finite element (FE) model of a maxillary premolar tooth (Sakaguchi et al., 1991). The followingmaterials are compared: (1) natural enamel and dentin, (2) porcelain-fused-to-metal (PFM), and (3) composite.

MATERIALSAND METHODS A Simple Crack Propagation Model. Small surface defects or flaws can propagate when subjected to sufficient stress, resulting in a surface crack. By use of a Linear Elastic Fracture Mechanics (LEFM) approach for a surface crack in a semiinfinite body (Caddell, 1980): K, = o ~x/~-

(1)

where K~ is the stress-intensity factor for opening mode (Mode I) crack propagation, a is the maximum allowable design stress and a is the allowable crack size. Crack propagation and brittle fracture can be prevented by alteration in: (1) the structure shape; (2) the stresses incurred by the material; (3) the initial flaw size; or (4) the critical

Dental Materials~March 1992 131

Fig. 3. Finite element model triangulation mesh. Load application indicated on

2. Nodes and material boundaries used in the finite element model.

buccal cusp,

ess intensity factor (plane strain) for static loading, K~c, o known as the fracture toughness. Generally, the struc'e shape, stresses, and flaw size are not easily controlled; ;refore, a material with adequate K~cneeds to be chosen •efully so that fracture will be minimized. It is only when reaches the critical value, K~o that the crack will propa-

;e. Rearrangement of Eq.(1) for K~c,the maximum size sure crack, a, that can be tolerated before crack propagation :urs at a given stress, g, is given by: a=

(2)

cis and Erdogen (1963) suggested that the crack propagation .e, da/dN, caused by cyclic loading in a structure, is marily a function of the stress-intensity factor range, AK, ere A is a material constant and N is a single cycle:

13) dN improved equation was offeredby Forman et al. (1967) who )posed that the crack growth rate could be determined from ~relationship: da A(AKI ) p

dN

(1- R,)KIc - AKI

(4)

ere A and p are numerical constants related to the tempera'e of the test environment and strain rate, respectively ldon, 1973); AK~,the range of the stress-intensity factor is en by AK~= K~m~- K~min;K~cis the critical stress-intensity tor for fracture; and R I -~--Kimin/gimax for a given cycle. Development of the Finite Element (FE) Model. The luired magnitudes and locations of critical tensile stresses the premolar crown crack propagation model were deed from a two-dimensional, plane strain finite element ,del. A maxillary premolar mathematical model was ~igned that used the mean dimensions of a large populan of natural teeth (Langsjoen and Noble, 1982). The )del consisted of 343 nodes and 586 elements. The external mdary apical to the cemento-enamel junction was fixed Sakaguchi et aL/Modeling crack propagation in dental restorations

in the horizontal, x, and axial, y, directions. The remainder of the nodes were given two degrees of freedom. The natural tooth model included two material regions: enamel and dentin. A typical crown preparation was adapted for incorporation of the restorative materials into the FE model. The crown preparation was designed with butt shoulders oriented perpendicular to the vertical axis and with uniform removal of the enamel (Fig. 2). For the composite crown model, the three regions included dentin, core composite, and veneer composite. The core composite was bonded to the dentin and veneer composite. Microfilled composite was layered as a veneer over a posterior composite core. The veneered design was chosen to optimize strength from the posterior composite core and polishability and esthetics from the microfilled veneer. The PFM crown model incorporated an outer veneer of porcelain over a gold alloy coping, which was bonded to dentin. Materials' properties are listed in Table 1 (Morin et al., 1988). A microcomputer-basedtwo-dimensional FE software package, Interactive Finite Element Computing System (IFECS) (Lewis and Cross, 1978), was used. It provided an interactive computing environment for FE modeling application, including pre-processing, automatic mesh generation, input data verification, and post-processing. An optimaltriangnlar mesh was generated with the prescribed nodes used as vertices, sorting out the relevant material properties for each element. Any errors incurred in defining the boundary were determined by the mesh- generation procedure. Several assumptions were made in the FE model: (1) absolute bonding between composite layers and dentin, between enamel and dentin, and among porcelain, metal alloy, and dentin; (2) negligible effect of the pulp tissue on the overall performance of the model; (3) material properties for the experimental composite materials represented by conventional materials in the same category; and (4) enamel, dentin, and composite represented as isotropic, homogeneous materials. A plane strain FE model was implemented because of the assumption of uniform loading and uniform material properties in the mid-buccal-lingual cross-section of interest. The

TABLE 1: MATERIALPROPERTIESFOR NATURALTOOTH

Material Dentin

AND CROWN MODEL Poisson's Modulusof Ratio Elasticity (GPa) 0.25 15,4

TABLE 2: MAXIMUMPRINCIPALTENSILESTRESSES FOR A 13.2-N

OCCLUSAL LOAD AT THE BUCCALCUSP TIP StressIntensity Factor, K~c (MNm-3/2) 3.08

Enamel

0.30

46.9

0.7-1.27

Core Composite

0.25

22.1

1.5

Veneer Composite

0.25

8.0

1.0

Porcelain

0.30

83.0

1.0

Ceramometal Alloy

0.30

86.0

strain normal to the x-y plane (Ez) and the shear strains (Yxz and Yyz)are assumed to be zero. The assumption of plane strain is realistic for bodies with constant cross-sectional area along the z axis subjected to loads that act only in the x- and/or y-directions and do not vary in the z-direction. Only a unit thickness of the structure must be considered, because each unit thickness behaves identically (except near the ends). Since the behavior of the tooth crown at the mesial and distal marginal ridges (the ends) was not of interest, this representation was appropriate. The mathematical model was validated by a physical model of a natural premolar tooth with strain gauges at the buccal and lingual heights of contour. Results from the experimental study were in good agreement with the FE model (correlation coefficient r > 0.95) (Sakaguchi et al., 1991). Loads were placed on the incline of the buccal cusp to simulate contact by the opposing tooth (Fig. 3). Vertical point loads (13.9 N) were applied at discrete locations along the lingual incline ofthe buccal cusp between the area ofintercuspal position and cusp tip. Point loads of the same magnitude were also applied at the same positions, but were oriented normal to the surface. The load magnitude is consistent with physiologic values, as reviewed by DeLong and Douglas (1983). Identical loading and boundary conditions were applied to the natural tooth, PFM crown, and composite crown models.

Material

Maximum PrincipalTensileStress (MPa)

Natural Tooth

1.06

Porcelain-fused-to-Metal

1.20

Composite Crown

1.56

RESULTS Maximum principal tensile stresses developed in the local area of load application are listed in Table 2 for the three materials. Stress contours displaying the maximum principal tensile and compressive stresses are shown in Figs. 4-6. Isochromatic areas on the contour plots represent isostress areas within the limits shown in the legends. The standard convention of representing tension by positive values and compression by negative values is used. DISCUSSION

The three crown designs exhibit different patterns of stress distribution. The natural tooth is distinguished from the composite and porcelain crowns in the dissipation of stress into the underlying layers. This is a result of an elastic material (dentin) supporting a higher modulus material (enamel) interfaced through an intimate stress distributor (dentinoenamel junction). The composite crown demonstrates the highest tensile strains and stresses immediately adjacent to the area of contact (Fig. 4). This was particularly evident when the load was placed near the tip of the buccal cusp. Since composites are weaker in tension than in compression, this area of high tensile strains and stresses is a potential site for crack growth. The stresses provided by the FE model can be used as a preliminary estimate for use in a simple stable crack growth model. Stable crack growth can be obtained in a cyclic fatigue test (DiBenedetto, 1973) and offers the ability to be modeled. Given the restoration design and materials presented above, the crack propagation rate initiated by a surface flaw o f l ~m

....~::: - 3.62 - 2.76 - 1.9 ~i~i:- 1.04 ~ - 0.184 S 0.674 i)i)~))ii; 1.53 m 2.39 m 3.25 m 4.11 R 4.96 ............. 5.82 6.68 : 7.54 i:iiiii~ 8.4

Fig. 4. Stress distribution within composite crown, indicating maximum principal stresses.

Dental Materials~March 1992

133

:i:i~i~i~i~i:-5.14 .....-.. 4.32 :-3.5 ;~:~- 2.68 m - 1.86 t - 1.04 iiiiiiiiiiiii~-0.218 am 0.603 I 1.42 i 2.24 3.07 iiiiiiiiiiii~ 3.89 ~!~i~ii; 4.71 iii; 5.53 ~iiiilii! 6.35 ::::::::::::r

Fig. 5. Stress distribution within porcelain-fused-to-metal crown, indicating maximum principal stresses.

d. The initial surface flaw of i ~m was chosen as Live estimate of a random surface defect due to inclusion. The following parameters were used: [)6N/m3/2for filled resin composites (Waters, 1980) ce gmin = 0 n cycling from ami, = 0 to (~max 108 N/m 2, which is the maximum tensile stress for ite restoration design as a result of a 13.2 N load al cusp tip. s that: AK = ~ / ~ a

= 2189.8 N / m 3j2

(5)

ck propagation rate when Eq.(4) is used is: da

- 5.26 x 10-4 ~m/cycle

(6)

dN formed in an artificial mouth indicate that 250,000 cycles represent one year of chewing (Coffeyet al., ,n this number of cycles is used, the crack propagaLhecomposite crown is 131.5 pm/year. Although it oted that this is a worst-case estimate, because it at each contact is at the same location and has the [tude, the locations ofthe loads and magnitudes are physiologic limits. The propagation rate suggests defect on the order ofa micron can initiate a fatigue ;hin the restoration under physiologicloads. ~arison, the stresses for a conventional PFM crown [tooth are listed in Table 3 and shown in Figs. 5 and of the stresses developed from a 13.2 N load the same location as in the composite and the ~:~ovalues (de Groot, 1986; Marshall and Lawn, m flaw in the PFM crown propagated at a rate of OK cycles and in the natural tooth at a rate of OK cycles. These values are less than half the •in the composite crown design. The values for A lssumed to be the same for all three materials, hi et al./Modefing crack propagation in dental restorations

since the loading conditions are identical (Radon, 1973). If the wear rates of composite, porcelain, and natural enamel in terms of maximum depth of wear are considered, the experimental simulations on an artificial mouth indicate that the wear rate of composite (49 _+ 10 ~m/250K cycles) (Sakaguchi et al., 1986a) is approximately one-half that of porcelain (115 + 20 ]~m/250Kcycles) (DeLonget al., 1986) and slightly lower than that of enamel (92 +_51 ~m/250K cycles) (Sakaguchi et al., 1986b). The wear rates for enamel and porcelain are greater than their respective crack propagation rates. In composites, however, the reverse is the case and the wear rate is less than the crack propagation rate. This predisposes composites to catastrophic failure, because the material does not wear, which might minimize areas of high occlusal stress. The early wear-in period is critical for dental restorations, because placement of slightly elevated restorations will concentrate high occlusal stresses at those areas. Materials such as porcelain will wear the opposing tooth or demonstrate wear of the porcelain surface, and silver amalgam restorations will creep and flow to alter the occlusal contact area. However, composite restorations do not exhibit these properties; therefore, they are subject to high stresses at occlusal contact areas if they are not adjusted properly. Some of the inherent properties of composite restorations also predispose them to fracture. Composites are particularly sensitive to microdefects that can be produced by bubbles, dust, and other contaminants, which act as sites for stress concentration (Spanoudakis andYoung, 1984). These small TABLE 3: CONTACTWEAR RATES AND CRACK PROPAGATION RATES Contact Crack Dominant Wear Rate* Propagation Deterioration Material

(pm/y)

Natural Tooth

92 + 51

Porcelain-fused-to-Metal

115 + 20

60.3

wear

Resin Composite

49 + 10

131.5

crack propagation

Rate (IJm/y) 42.4

* Wear rate determinedfrom artificialmouth studies.

Mechanism wear

- 5.85 '- 5.04 : - 4.22 i:~i:::- 3.4 - 2.58 t1.76 ........ 0 94 m - 0.121 m 0.698 i 1.52 I 2.34 ::! 3.16 3.97 4.79 ~!~!~" 5.61 :::::::::::::,

_

Fig. 6. Stress distribution within natural tooth indicating maximum principal stresses.

defects initiate cracks that propagate when stressed through occlusal loading. Cyclic loading due to mastication further contributes to wear and generation of subsurface microdefects (McKinney andWu, 1985). In addition, debonded fillers may also act as stress concentrators, which significantly multiplies the number of potential crack growth sites. Although mature natural enamel (which is also a composite material) often demonstrates craze lines, as shown in Fig. 1, these microcracks rarely result in bulk fracture of the tooth unless the dentinal support has been lost to caries or large restorations. Although composite restorations and porcelain crowns sometimes fracture under occlusal loads, the natural tooth responds to the same load without failing. The rod structure of the enamel is interfaced through a tight, intimate dentino-enamel junction to the elastic, supportive dentin. This allows occlusal stresses to be dispersed into the dentin rather than remain localized at the area of load application.

sufficiently supported by natural tooth structure.

ACKNOWLEDGMENT The authors would like to thank Mr. Peter Chow for his work on the PC version of IFECS, to Mr. Michael Bramwell for his suggestions on the project, and to Ms. Marilynn Erickson for her work on the processing of the manuscript. Received February 27, 1991/Accepted July 9, 1991 Address correspondence and reprint requests to: R.L. Sakaguchi Minnesota Dental Research Center for Biomaterials and Biomechanics University of Minnesota, School of Dentistry, 16-212 Moos Tower 515 Delaware Street, SE Minneapolis, MN 55455 USA

CONCLUSIONS The simple model described here demonstrates the significance of the ability of natural teeth to disperse occlusal stresses. This is reflected in the ability of natural teeth to limit crack propagation relative to restorative dental materials, such as porcelain and composite. Although the model is quite simple (in that it assumes homogeneous materials, simplifies the actual composition of the materials and does not examine the behavior at the crack tip), the model appears to rank the materials in correct order in terms of their tendency to fracture. This model should be used only as a qualitative indicator of fracture tendency. The high calculated crack propagation rates in composites may explain the observed clinical failures and microchipping at the area of occlusal contact, as noted in SEM analysis. Two clinical implications follow from this study. Since composite materials may not wear sufficiently for areas of heavy occlusal contact to be eliminated, extreme care should be taken to establish proper occlusion on composite restorations. In areas of chronically high occlusal stress, such as areas adjacent to cusp tips, composite materials may not be indicated. This is particularly true where the composite is not

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