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Strength Determination of Brittle Materials as Curved Monolithic Structures P. Hooi, O. Addison and G.J.P. Fleming J DENT RES published online 10 February 2014 DOI: 10.1177/0022034514523621 The online version of this article can be found at: http://jdr.sagepub.com/content/early/2014/02/10/0022034514523621 A more recent version of this article was published on - Mar 17, 2014
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research-article2014
JDR
XXX10.1177/0022034514523621
RESEARCH REPORTS Biomaterials & Bioengineering
P. Hooi1, O. Addison2, and G.J.P. Fleming1* 1
Materials Science Unit, Dublin Dental University Hospital, Trinity College Dublin, Dublin, Ireland; and 2Biomaterials Unit, School of Dentistry, University of Birmingham, Birmingham, UK; *corresponding author, garry.fleming@ dental.tcd.ie
Strength Determination of Brittle Materials as Curved Monolithic Structures
J Dent Res XX(X):1-5, 2014
Abstract
The dental literature is replete with “crunch the crown” monotonic load-to-failure studies of all-ceramic materials despite fracture behavior being dominated by the indenter contact surface. Load-to-failure data provide no information on stress patterns, and comparisons among studies are impossible owing to variable testing protocols. We investigated the influence of nonplanar geometries on the maximum principal stress of curved discs tested in biaxial flexure in the absence of analytical solutions. Radii of curvature analogous to elements of complex dental geometries and a finite element analysis method were integrated with experimental testing as a surrogate solution to calculate the maximum principal stress at failure. We employed soda-lime glass discs, a planar control (group P, n = 20), with curvature applied to the remaining discs by slump forming to different radii of curvature (30, 20, 15, and 10 mm; groups R30-R10). The mean deflection (group P) and radii of curvature obtained on slumping (groups R30-R10) were determined by profilometry before and after annealing and surface treatment protocols. Finite element analysis used the biaxial flexure load-to-failure data to determine the maximum principal stress at failure. Mean maximum principal stresses and load to failure were analyzed with one-way analyses of variance and post hoc Tukey tests (α = 0.05). The measured radii of curvature differed significantly among groups, and the radii of curvature were not influenced by annealing. Significant increases in the mean load to failure were observed as the radius of curvature was reduced. The maximum principal stress did not demonstrate sensitivity to radius of curvature. The findings highlight the sensitivity of failure load to specimen shape. The data also support the synergistic use of bespoke computational analysis with conventional mechanical testing and highlight a solution to complications with complex specimen geometries.
KEY WORDS: maximum principal stress, radius of curvature, finite element analysis, soda-lime glass, biaxial flexure, load to failure.
DOI: 10.1177/0022034514523621 Received September 30, 2013; Last revision November 25, 2013; Accepted January 15, 2014 © International & American Associations for Dental Research
Introduction
T
he dental literature is replete with studies investigating the fracture resistance of all-ceramic materials, and “crunch the crown” monotonic load-tofailure tests are routinely used (Scherrer and de Rijk, 1993; Burke and Watts, 1998; Tinschert et al., 2001; Attia and Kern, 2004; Borges et al., 2009). The fracture behavior in “crunch the crown” tests is dominated by indenter contact surface damage and fails to replicate subsurface radial crack extension observed clinically in monolithic crowns (Kelly et al., 2012). Additionally, load-to-failure data provide no information on stress patterns within complex crown geometries and comparisons among studies are impossible (Kikuchi et al., 1997; Qasim et al., 2007; Kelly et al., 2012) owing to variable testing protocols (loading rates, test bar dimensions, loading vectors, crown support substrates). It is important to differentiate “crunch the crown” tests from fatigue studies on equivalent specimen geometries that provide insight by replicating clinical fracture behavior of all-ceramic materials when performed carefully (Rekow and Thompson, 2007; Zhang et al., 2013). Testing geometric planar specimens allows calculation of failure stresses using theories for the deflection of plates (Timoshenko and Woinowsky-Krieger, 1959). By employing standardized planar geometries, sample preparation, and loading protocols, comparisons between materials and investigators are possible. However, the data generated can provide a conservative measure of fracture behavior (Qasim et al., 2005) but cannot account for geometric factors that may act to modify the stressing patterns or defect populations within a complex body. Finite element analyses (FEAs) have been widely used in dentistry as virtual case studies (Rekow et al., 2006; Dejak et al., 2012; Cuddihy et al., 2013). FEA employs the adaptive capabilities of parametric simulations to assess sensitivities to discrete changes in the variables that define a model, including material, dimensional, and/or geometric parameters. While virtual case studies are insightful, the usefulness of FEA simulations are governed by the limitations of specific features of the model being investigated. By implication, features that are not imputed into the simulation cannot be determinants of any factors investigated. The idealizations employed in an FEA simulation may therefore be detrimental by inaccurately representing material behavior. Importantly, the actual strength of a dental ceramic depends on the statistical distribution and configuration of defects; therefore, a probabilistic approach to modeling fracture behavior is required. Whilst FEA simulations can incorporate statistical elements, empirical measurements are essential to firstly define how a statistical defect population can vary in the fabricated complex geometric shapes. May et al. (2012) compared an FEA model that included predicted failure criteria derived from experimental data to investigate representative “crunch the crown” monotonic load-to-failure testing. The assumed failure criteria
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2
Hooi et al.
J Dent Res XX(X) 2014 following stress-relief-inducing processes. One surface of the planar discs (n = 10) and the convex surface of the R10 curved discs (n = 10) were profilometrically measured on a multiaxis leveling device. Profilometric scans were obtained across a rectangular area (1-mm width) coincident with the center and extended radially to the disc periphery. Scans (n = 101) were performed with an inductive-gauge conisphere-diamond stylus tip (2-µm radius) at 1 mm/s and 0.75 mN, with a 10-µm step size (y-direction) and data capture every 10 µm (x-direction), providing a 42-nm resolution in the z-direction via a Talysurf CLI 2000 (Taylor-Hobson, Leicester, UK). Radii of curvature (R) were calculated in the x- and y-directions with Heron’s formula:
Figure 1. Disc-shaped specimens are illustrated from flat (P) to curved with decreasing radii of curvature (R30-R10). The section view taken through the axes of the disc-shaped specimens in the front view shows the radii of curvature through the center of each specimen.
were established experimentally by Yi and Kelly (2008) but associated with an “uncertainty” in the FEA results, which limited the authors’ confidence in the approach. However, simplified nonplanar geometries (curved discs), with a single radius of curvature, have been proposed to examine the impact of geometric factors on radial fracture (Qasim et al., 2006). In the absence of the analytic solutions required to describe the stress distribution in curved discs tested in biaxial flexure (Pecanac et al., 2011), we aimed to investigate the influence of nonplanar geometries on the maximum principal stress defined by FEA simulation. Radii of curvature analogous to elements of complex dental geometries (cusp inclines or occlusal surfaces) were investigated, and the FEA simulation was integrated with load-to-failure testing, which acted as a surrogate solution to calculate the maximum principal stress at failure. Our null hypotheses were that mean load to failure and maximum principal stress were insensitive to radii of curvature.
Materials & Methods Generic 3-dimensional geometries (Figure 1) of 4 curved discs (12.0-mm diameter, 1.0-mm thickness) with different radii of curvature (30, 20, 15, and 10 mm) were created with a parametric computer-aided-design system (Pro/Engineer, Wildfire 2.0, PTC, Needham, MA, USA). One hundred soda-lime planar discs were scribe cut from planar glass stock (percentage weight: SiO2 72%, Na2O 14%, CaO 7%, MgO 4%, Al2O3 2%; AGC Glass Europe, Brussels, Belgium), and 20 planar discs (radius of curvature = ∞) served as the control (group P). Curvature was applied to the remaining discs by slump forming to 575°C, rotated 4 times through 90° every 60 s to complete the slumping process. The curved discs were cooled to ambient temperature at 1°C/s to produce clinically relevant radii of curvature of 30, 20, 15, and 10 mm assigned to groups R30-R10, respectively (n = 20 for each group). To identify whether residual stresses were introduced during manufacture, modifications in the geometric flatness of planar discs (P) and the curvature of curved discs (R10) were measured
R=
abc 4. s ( s − a ) ( s − b ) ( s − c )
,
(1)
where a, b, and c were the distances between 3 points on a trace with s the semiperimeter defined by the points and calculated through a+b+c (2) . 2 Repeat measurements of the identical measurement area were performed following alumina particle air abrasion of the nonprofiled surface of the planar discs (PABRADED, n = 5) and the concave surface of the curved discs (R10ABRADED, n = 5) with 50-µm alumina at 2 cm with 4 bar for 5 s (GOBI-2, Wassermann Dental-Maschinen GmbH, Hamburg, Germany) (Hooi et al., 2013). Further repeat measurements were performed following annealing at 510°C: intermediary between the glass-softening (490°C) and glass-transition (535°C) temperatures for both planar discs (PANNEALED, n = 5) and curved discs (R10ANNEALED, n = 5). Specimens were heated at 20°C/min to 510°C, held for 40 min, and cooled at 5.5°C/min to room temperature (Shelby, 2005). The radii of curvature of discs before and after stress relief were compared with paired t tests employed at α = 0.05 (SPSS 14.0, SPSS Inc, Chicago, IL, USA). To approximate clinically relevant surface condition and a representative flaw population, one surface of planar discs and the concave surface of group R30-R10 curved discs were alumina particle air abraded and acid etched for 120 s with 9.6% hydrofluoric acid (Porcelain Etch Gel, Pulpdent Corporation, Watertown, MA, USA) before rinsing with distilled water (10 s) and drying with oil-free compressed air (Hooi et al., 2013). Random discs (n = 3) from each group were then profilometrically evaluated to determine the actual radii of curvature via the scanning protocol outlined. In addition, measurements of the contralateral surfaces were measured to give an insight into any differences from the introduced surface roughness (Ra value). The central thickness of each disc was then measured with a screw-gauge micrometer (Mitutoyo Corporation, Tokyo, Japan) and a ball-on-ring testing configuration used to determine the load to failure (N) of the discs with the modified surface tested in tension. The discs were positioned on a 10-mm-diameter knife-edge ring support and loaded with a 4-mm-diameter stainless-steel ball indenter at 1 mm/min. The biaxial flexure stress at failure was calculated in the planar discs (group P) by Equation 3 (Shetty et al., 1980): s=
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J Dent Res XX(X) 2014 3 Strength Determination of Brittle Materials Table 1. Mean (SD) Deflection and Radius-of-Curvature Data Determined by Profilometry for the Planar and Curved Discs Condition
Planar Discs: Deflection, µm
As received Annealed Abraded
0.878 (0.603)a
Curved Discs (R10): Radius of Curvature, mm
0.783 (0.527)a 0.921 (0.305)a 1.253 (0.916)b
1.280 (0.407)b
10.190 (0.366)c 10.180 (0.353)c
10.305 (0.373)c,d 10.433 (0.393)d 10.309 (0.375)c,d
Different superscript denominators within columns denote significant differences (p ≤ .05). R10, 10 mm.
σ=
−3W (1 + ν ) a 1 −ν 1 + 2 ln + 4π t b 1 +ν
b2 − 1 2 2a
a2 2 , (3) R
where W was the load to failure (N); ν, the Poisson ratio (0.22) (Kolli et al., 2009); t, the thickness; a, R, and b, the radii of the knife-edge support, disc, and contact zone (t/3), respectively (Hsueh et al., 2005). Without a closed-form analytic solution to calculate the biaxial flexure stress for each curved disc (groups R30-R10), the maximum principal stress was determined by inputting the measured central thickness, measured radius of curvature, and load to failure into the FEA model (n = 80). A validated 2-dimensional axisymmetric FEA simulation (Pro/MECHANICA, Wildfire 2.0, PTC) (Fleming et al., 2012) was adapted to model the curved discs. The FEA simulation was assigned a maximum polynomial value of 6 and convergence criteria of 3% for a specified measure parameter of maximum principal stress at an axial position (r = 0) in the curved discs on the surface in tension. Analysis of mean deflection and the radii-of-curvature group means with and without annealing were performed utilizing a general linear model univariate analysis at α = 0.05 (SPSS 14.0). The mean measured Ra values and radii of curvature (groups P, R30-R10) were assessed before load to failure via one-way analyses of variance (ANOVAs) and post hoc Tukey tests at α = 0.05. One-way ANOVAs were performed (α = 0.05) to determine whether load-to-failure or maximum principal stress data were significantly modified by radii of curvature.
Results Profilometric measurements to identify stress-induced changes in specimen curvature for planar discs identified a significant effect of alumina particle air abrasion (p ≤ .05) but not annealing (p > .05). For curved discs, no significant effect was evident following alumina particle air abrasion or annealing (Table 1). Following surface modification (alumina particle air abrasion and hydrofluoric acid etching) the mean radii of curvature of the curved discs (groups R30-R10) differed significantly (Table 2). One-way ANOVA and post hoc Tukey tests demonstrated that discs with the minimum radius of curvature (R10) had significantly reduced surface roughness (Table 2) compared with discs from groups P, R20, and R15 (p ≤ .008). One-way ANOVAs further demonstrated that the magnitude of the load to failure was significantly modified by the radius of curvature of the discs (Figure 2). The actual radii of curvature for the curved discs were defined in the parametric geometry of the FEA model for each
Figure 2. Mean maximum principal stress and mean load to failure plotted against the radius-of-curvature data, highlighting the increasing load to failure with reduced radius of curvature and the insensitivity of the maximum principal stress to the radius of curvature.
disc. The FEA model included the measured central thickness and load at failure for individual curved discs to determine the maximum principal stress resulting from the applied load to failure. Comparison of the mean maximum principal stress and the mean load-to-failure data (Table 2, Figure 2) demonstrated that the load to failure increased as the radius of curvature reduced, which was in contrast to the pattern evident for the maximum principal stress. This indicated that the more curved geometry increased the load to failure without modifying the maximum principal stress determined with the FEA simulation.
Discussion Testing planar discs in biaxial flexure is widely employed to investigate the fracture behavior of dental ceramics (Cattell et al., 1997; Addison et al., 2008). FEA simulations (Anusavice and Hojjatie, 1992) and quantitative fractography (Kelly, 1999; Quinn et al., 2005) of failed all-ceramic restorations identified that clinical failure is often manifest by the tensile extension of surface flaws. The observed clinical failure mode of radial fracture in monolithic restorations is reproduced with the biaxial flexure stress test (Fleming et al., 2006); however, the fracture behavior of planar specimens is somewhat unrepresentative of the geometric complexities evident in full-contour restorations (Rekow et al., 2006). Previously, the validity of an FEA simulation for planar discs tested in biaxial flexure was confirmed with the bending solutions for equivalent geometry (Fleming et al., 2012). The current parametric FEA model was conducive to shape modification
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4
Hooi et al.
J Dent Res XX(X) 2014
Table 2. Finite Element Analysis Simulation for the Planar (Group P) and Curved Discs (Groups R30-R10): Mean (SD)
Group P R30 R20 R15 R10
Thickness, mm 0.93 0.92 0.93 0.93 1.05
(0.01)a (0.02)a (0.02)a (0.02)a (0.08)b
Inner Surface Ra Value, µm 1.511 1.383 1.437 1.449 1.289
(0.053)c (0.052)d,e (0.048)c,d (0.048)c,d (0.041)e
Actual Radius of Curvature, mm Flat (∞) 28.21 (1.62)f 19.86 (3.20)g 15.43 (0.44)h 8.25 (0.54)i
Load to Failure, N 40.6 46.0 50.8 64.1 93.7
(3.5)j (6.0)j,k (5.9)k (5.4)l (15.5)m
Maximum Principal Stress, MPa 97.7 98.7 98.9 112.8 105.0
(8.0)n (11.1)n (9.8)n (9.2)o (15.4)n,o
The mean thickness, inner surface Ra-values, actual radius-of-curvature, load-to-failure in BFS testing and maximum-principal-stress determined using the FEA simulation for the planar (Group P) and curved-discs (Groups R30-R10) examined. Different superscript denominators within columns denote significant differences (p ≤ .05). R30, 30 mm; R20, 20 mm; R15, 15 mm; R10, 10 mm.
by varying the radius of curvature and the thickness of the individual specimens tested in biaxial flexure. The experimentally determined load-to-failure data avoided the necessity to incorporate an arbitrary probabilistic failure criterion (Yi and Kelly, 2008) in the FEA simulation. In the current study, the radius of curvature was not observed to impact on the maximum principal stress in the curved discs; therefore, this hypothesis was rejected. Interestingly, significantly increased loads to failure were observed in the curved discs when compared with planar discs; therefore, this hypothesis was accepted. The pronounced sensitivity of the load-to-failure changes with radii of curvature provides definitive evidence of the problems associated with comparisons to the widely reported “crunch the crown” monotonic load-to-failure test data in monolithic crowns (Scherrer and de Rijk, 1993; Burke and Watts, 1998; Tinschert et al., 2001; Attia and Kern, 2004; Borges et al., 2009). While expert opinion has frequently challenged the use of such techniques (Kikuchi et al., 1997; Qasim et al., 2007; Kelly et al., 2012), “crunch the crown” data are consistently published within the literature. Specimens employed in “crunch the crown” testing contain multiple geometric features to represent the full-contour restoration under investigation. It is reasonable to expect that variability among test specimens is exacerbated with increasing geometric complexity. In the absence of stringent quality control comparisons among test specimens, batches and studies are limited (Kelly et al., 2012). While computer-aided design and manufacturing can reduce the geometrical variation in such specimens, there is evidence that the process can introduce significant differences in the critical defect populations when test specimen batches are prepared from a single bur set (Addison et al., 2012). The current investigation highlights that load-to-failure testing can be performed when complemented by numeric simulations that are specimen specific and where specimen failure, as in the current study, is determined to be representative of the clinically operative failure mechanism. However, even for a simplistic model, the sensitivities are high to shape variation among individual specimens from an “identical” series. The fabrication of the nominally identical disc-shaped specimens with a single radius of curvature with a standardized protocol was monitored for consistency in the current study by profilometric inspection to verify that the specified dimensional and geometric parameters were controlled and any identified deviations from the specifications were factored into the simulation models. The findings demonstrated that the adapted flat-planar solution adequately described the failure stress of all curved structures in
the radii of curvature examined, which would be representative of most curvatures observed in a full-contoured restoration other than at a cusp tip. The surface state responsible for the specimen at failure was assumed to be equivalent for all specimen groups given that a common surface treatment was undertaken. However, for specimens with the smallest radius of curvature (R10), significant differences in the microscopic surface topography were observed and were assumed to result from the complexities of abrasive particle interactions with the curved surface. Although this raises a cautionary note about the accuracy of the measured values, it further highlights the need for modeling more complex geometries where identical and clinically relevant experimental treatments may yield different results. The current approach, although laborious, measured the actual load to failure while accounting for the geometric influence on maximum principal stress and the unique defect populations in each of the tested curved discs, without the requirement to predict the probabilistic pattern of fracture. The current study demonstrated the complications associated with interpreting load-to-failure monotonic data and provides support for the relevance of testing simple geometric specimens. While fatigue studies based on full-contour restorations add insight (Zhang et al., 2013), monotonic testing is clearly redundant. The data reported support the synergistic use of bespoke computational analysis with conventional mechanical testing.
Acknowledgments This work was funded by a Trinity College Dublin Ussher Fellowship for Paul Hooi and carried out at Dublin Dental School and Hospital. The authors declare no potential conflicts of interest with respect to the authorship and/or publication of this article.
References Addison O, Marquis PM, Fleming GJ (2008). Quantifying the strength of a resin-coated dental ceramic. J Dent Res 87:542-547. Addison O, Cao X, Sunnar P, Fleming GJ (2012). Machining variability impacts on the strength of a chair-side CAD/CAM ceramic. Dent Mater 28:880-887. Anusavice KJ, Hojjatie B (1992). Tensile stresses in glass-ceramic crowns: effect of flaws and cement voids. Int J Prosthodont 5:351-358. Attia A, Kern M (2004). Fracture strength of all-ceramic crowns luted using two bonding methods. J Prosthet Dent 91:247-252. Borges GA, Caldas D, Taskonak B, Yan J, Sobrinho LC, de Oliveira WJ (2009). Fracture loads of all-ceramic crowns under wet and dry fatigue conditions. J Prosthodont 18:649-655.
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J Dent Res XX(X) 2014 5 Strength Determination of Brittle Materials Burke FJ, Watts DC (1998). Effect of differing resin luting systems on fracture resistance of teeth restored with dentine-bonded crowns. Quintessence Int 29:21-27. Cattell MJ, Clarke RL, Lynch EJ (1997). The biaxial flexural strength and reliability of four dental ceramics-Part II. J Dent 25:409-414. Cuddihy M, Gorman CM, Burke FM, Ray NJ, Kelliher D (2013). Endodontic access cavity simulation in ceramic dental crowns. Dent Mater 29:626634. Dejak B, Młotkowski A, Langot C (2012). Three-dimensional finite element analysis of molars with thin-walled prosthetic crowns made of various materials. Dent Mater 28:433-441. Fleming GJ, Maguire FR, Bhamra G, Burke FM, Marquis PM (2006). The strengthening mechanism of resin cements on porcelain surface. J Dent Res 85:272-276. Fleming GJ, Hooi P, Addison O (2012). The influence of resin flexural modulus on the magnitude of ceramic strengthening. Dent Mater 28:769-76. Hooi P, Addison O, Fleming GJ (2013). Can a soda-lime glass be used to demonstrate how patterns of strength dependence are influenced by pre-cementation and resin-cementation variables? J Dent 41:24-30. Hsueh CH, Lance MJ, Ferber MK (2005). Stress distributions in thin bilayer discs subjected to ball-on-ring tests. J Am Ceram Soc 88:1687-1690. Kelly JR (1999). Clinically-relevant approach to failure testing of allceramic restorations. J Prosthet Dent 81:652-661. Kelly JR, Benetti P, Rungruanganunt P, Della Bonna A (2012). The slippery slope - critical perspectives on in vitro research methodologies. Dent Mater 28:41-51. Kikuchi M, Korioth TW, Hannam AG (1997). The association among occlusal contacts, clenching effort, and bite force distribution in man. J Dent Res 76:1316-1325. Kolli M, Hamidouche M, Bouaouadja N, Fantozzi G (2009). HF etching effect on sandblasted soda-lime glass properties. J Eur Ceram Soc 29:2697-2704. May LG, Kelly JR, Bottino MA, Hill T (2012). Effects of cement thickness and bonding on the failure loads of CAD/CAM ceramic crowns: multiphysics FEA modeling and monotonic testing. Dent Mater 28:e99e109.
Pecanac G, Bause T, Malzbender J (2011). Ring-on-ring testing of thin, curved bi-layered materials. J Eur Ceram Soc 31:2037-2042. Qasim T, Bush MB, Hu XZ, Lawn BR (2005). Contact Damage in Brittle Coating Layers: Influence of Surface Curvature. J Biomed Mater Res Part B: Appl Biomater 73:179-185. Qasim T, Bush MB, Hu XZ (2006). The influence of complex surface geometry on contact damage in curved brittle coatings. Int J Mech Sci 48:244-248. Qasim T, Ford C, Bush MB, Hu XZ, Malament KA, Lawn BR (2007). Margin Failures in Brittle Dome Structures: Relevance to Failure of Dental Crowns. J Biomed Mater Res Part B: Appl Biomater 80:78-85. Quinn JB, Quinn GD, Kelly JR, Scherrer SS (2005). Fractographic analyses of three ceramic whole crown restoration failures. Dent Mater 21:920-929. Rekow D, Thompson VP (2007). Engineering long term clinical success of advanced ceramic prostheses. J Mater Sci Mater Med 18:47-56. Rekow ED, Harsono M, Janal M, Thompson VP, Zhang G (2006). Factorial analysis of variables influencing stress in all-ceramic crowns. Dent Mater 22:125-132. Scherrer SS, de Rijk WG (1993). The fracture resistance of all-ceramic crowns on supporting structures with different elastic moduli. Int J Prosthodont 6:462-467. Shelby JE (2005). Mechanical properties. In Introduction to glass science and technology; 2nd edition. Cambridge: The Royal Society of Chemistry, pp 199-200. Shetty DK, Rosenfield AR, McGuire PA, Bansal GK, Duckworth WH (1980). Biaxial fracture tests for ceramics. Ceram Bull 59:1193-1197. Timoshenko S, Woinowsky-Krieger S (1959). Symmetrical bending of circular plates. In: Theory of plates and shells; 2nd edition. New York, NY: McGraw Hill, pp 87-121. Tinschert J, Natt G, Mautsch W, Augthun M, Spiekermann H (2001). Fracture resistance of lithium disilicate-, alumina-, and zirconia-based three-unit fixed partial dentures: a laboratory study. Int J Prosthodont 14:231-238. Yi YJ, Kelly JR (2008). Effect of occlusal contact size on interfacial stresses and failure of a bonded ceramic: FEA and monotonic loading analyses. Dent Mater 24:403-409. Zhang Y, Sailer I, Lawn BR (2013). Fatigue of dental ceramics. J Dent 41:1135-1147.
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Strength Determination of Brittle Materials as Curved Monolithic Structures P. Hooi, O. Addison and G.J.P. Fleming J DENT RES published online 10 February 2014 DOI: 10.1177/0022034514523621 The online version of this article can be found at: http://jdr.sagepub.com/content/early/2014/02/10/0022034514523621 A more recent version of this article was published on - Mar 17, 2014
Published by: http://www.sagepublications.com
On behalf of: International and American Associations for Dental Research
Additional services and information for Journal of Dental Research can be found at: Email Alerts: http://jdr.sagepub.com/cgi/alerts Subscriptions: http://jdr.sagepub.com/subscriptions Reprints: http://www.sagepub.com/journalsReprints.nav Permissions: http://www.sagepub.com/journalsPermissions.nav
Version of Record - Mar 17, 2014 >> OnlineFirst Version of Record - Feb 10, 2014 What is This?
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research-article2014
JDR
XXX10.1177/0022034514523621
RESEARCH REPORTS Biomaterials & Bioengineering
P. Hooi1, O. Addison2, and G.J.P. Fleming1* 1
Materials Science Unit, Dublin Dental University Hospital, Trinity College Dublin, Dublin, Ireland; and 2Biomaterials Unit, School of Dentistry, University of Birmingham, Birmingham, UK; *corresponding author, garry.fleming@ dental.tcd.ie
Strength Determination of Brittle Materials as Curved Monolithic Structures
J Dent Res XX(X):1-5, 2014
Abstract
The dental literature is replete with “crunch the crown” monotonic load-to-failure studies of all-ceramic materials despite fracture behavior being dominated by the indenter contact surface. Load-to-failure data provide no information on stress patterns, and comparisons among studies are impossible owing to variable testing protocols. We investigated the influence of nonplanar geometries on the maximum principal stress of curved discs tested in biaxial flexure in the absence of analytical solutions. Radii of curvature analogous to elements of complex dental geometries and a finite element analysis method were integrated with experimental testing as a surrogate solution to calculate the maximum principal stress at failure. We employed soda-lime glass discs, a planar control (group P, n = 20), with curvature applied to the remaining discs by slump forming to different radii of curvature (30, 20, 15, and 10 mm; groups R30-R10). The mean deflection (group P) and radii of curvature obtained on slumping (groups R30-R10) were determined by profilometry before and after annealing and surface treatment protocols. Finite element analysis used the biaxial flexure load-to-failure data to determine the maximum principal stress at failure. Mean maximum principal stresses and load to failure were analyzed with one-way analyses of variance and post hoc Tukey tests (α = 0.05). The measured radii of curvature differed significantly among groups, and the radii of curvature were not influenced by annealing. Significant increases in the mean load to failure were observed as the radius of curvature was reduced. The maximum principal stress did not demonstrate sensitivity to radius of curvature. The findings highlight the sensitivity of failure load to specimen shape. The data also support the synergistic use of bespoke computational analysis with conventional mechanical testing and highlight a solution to complications with complex specimen geometries.
KEY WORDS: maximum principal stress, radius of curvature, finite element analysis, soda-lime glass, biaxial flexure, load to failure.
DOI: 10.1177/0022034514523621 Received September 30, 2013; Last revision November 25, 2013; Accepted January 15, 2014 © International & American Associations for Dental Research
Introduction
T
he dental literature is replete with studies investigating the fracture resistance of all-ceramic materials, and “crunch the crown” monotonic load-tofailure tests are routinely used (Scherrer and de Rijk, 1993; Burke and Watts, 1998; Tinschert et al., 2001; Attia and Kern, 2004; Borges et al., 2009). The fracture behavior in “crunch the crown” tests is dominated by indenter contact surface damage and fails to replicate subsurface radial crack extension observed clinically in monolithic crowns (Kelly et al., 2012). Additionally, load-to-failure data provide no information on stress patterns within complex crown geometries and comparisons among studies are impossible (Kikuchi et al., 1997; Qasim et al., 2007; Kelly et al., 2012) owing to variable testing protocols (loading rates, test bar dimensions, loading vectors, crown support substrates). It is important to differentiate “crunch the crown” tests from fatigue studies on equivalent specimen geometries that provide insight by replicating clinical fracture behavior of all-ceramic materials when performed carefully (Rekow and Thompson, 2007; Zhang et al., 2013). Testing geometric planar specimens allows calculation of failure stresses using theories for the deflection of plates (Timoshenko and Woinowsky-Krieger, 1959). By employing standardized planar geometries, sample preparation, and loading protocols, comparisons between materials and investigators are possible. However, the data generated can provide a conservative measure of fracture behavior (Qasim et al., 2005) but cannot account for geometric factors that may act to modify the stressing patterns or defect populations within a complex body. Finite element analyses (FEAs) have been widely used in dentistry as virtual case studies (Rekow et al., 2006; Dejak et al., 2012; Cuddihy et al., 2013). FEA employs the adaptive capabilities of parametric simulations to assess sensitivities to discrete changes in the variables that define a model, including material, dimensional, and/or geometric parameters. While virtual case studies are insightful, the usefulness of FEA simulations are governed by the limitations of specific features of the model being investigated. By implication, features that are not imputed into the simulation cannot be determinants of any factors investigated. The idealizations employed in an FEA simulation may therefore be detrimental by inaccurately representing material behavior. Importantly, the actual strength of a dental ceramic depends on the statistical distribution and configuration of defects; therefore, a probabilistic approach to modeling fracture behavior is required. Whilst FEA simulations can incorporate statistical elements, empirical measurements are essential to firstly define how a statistical defect population can vary in the fabricated complex geometric shapes. May et al. (2012) compared an FEA model that included predicted failure criteria derived from experimental data to investigate representative “crunch the crown” monotonic load-to-failure testing. The assumed failure criteria
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J Dent Res XX(X) 2014 following stress-relief-inducing processes. One surface of the planar discs (n = 10) and the convex surface of the R10 curved discs (n = 10) were profilometrically measured on a multiaxis leveling device. Profilometric scans were obtained across a rectangular area (1-mm width) coincident with the center and extended radially to the disc periphery. Scans (n = 101) were performed with an inductive-gauge conisphere-diamond stylus tip (2-µm radius) at 1 mm/s and 0.75 mN, with a 10-µm step size (y-direction) and data capture every 10 µm (x-direction), providing a 42-nm resolution in the z-direction via a Talysurf CLI 2000 (Taylor-Hobson, Leicester, UK). Radii of curvature (R) were calculated in the x- and y-directions with Heron’s formula:
Figure 1. Disc-shaped specimens are illustrated from flat (P) to curved with decreasing radii of curvature (R30-R10). The section view taken through the axes of the disc-shaped specimens in the front view shows the radii of curvature through the center of each specimen.
were established experimentally by Yi and Kelly (2008) but associated with an “uncertainty” in the FEA results, which limited the authors’ confidence in the approach. However, simplified nonplanar geometries (curved discs), with a single radius of curvature, have been proposed to examine the impact of geometric factors on radial fracture (Qasim et al., 2006). In the absence of the analytic solutions required to describe the stress distribution in curved discs tested in biaxial flexure (Pecanac et al., 2011), we aimed to investigate the influence of nonplanar geometries on the maximum principal stress defined by FEA simulation. Radii of curvature analogous to elements of complex dental geometries (cusp inclines or occlusal surfaces) were investigated, and the FEA simulation was integrated with load-to-failure testing, which acted as a surrogate solution to calculate the maximum principal stress at failure. Our null hypotheses were that mean load to failure and maximum principal stress were insensitive to radii of curvature.
Materials & Methods Generic 3-dimensional geometries (Figure 1) of 4 curved discs (12.0-mm diameter, 1.0-mm thickness) with different radii of curvature (30, 20, 15, and 10 mm) were created with a parametric computer-aided-design system (Pro/Engineer, Wildfire 2.0, PTC, Needham, MA, USA). One hundred soda-lime planar discs were scribe cut from planar glass stock (percentage weight: SiO2 72%, Na2O 14%, CaO 7%, MgO 4%, Al2O3 2%; AGC Glass Europe, Brussels, Belgium), and 20 planar discs (radius of curvature = ∞) served as the control (group P). Curvature was applied to the remaining discs by slump forming to 575°C, rotated 4 times through 90° every 60 s to complete the slumping process. The curved discs were cooled to ambient temperature at 1°C/s to produce clinically relevant radii of curvature of 30, 20, 15, and 10 mm assigned to groups R30-R10, respectively (n = 20 for each group). To identify whether residual stresses were introduced during manufacture, modifications in the geometric flatness of planar discs (P) and the curvature of curved discs (R10) were measured
R=
abc 4. s ( s − a ) ( s − b ) ( s − c )
,
(1)
where a, b, and c were the distances between 3 points on a trace with s the semiperimeter defined by the points and calculated through a+b+c (2) . 2 Repeat measurements of the identical measurement area were performed following alumina particle air abrasion of the nonprofiled surface of the planar discs (PABRADED, n = 5) and the concave surface of the curved discs (R10ABRADED, n = 5) with 50-µm alumina at 2 cm with 4 bar for 5 s (GOBI-2, Wassermann Dental-Maschinen GmbH, Hamburg, Germany) (Hooi et al., 2013). Further repeat measurements were performed following annealing at 510°C: intermediary between the glass-softening (490°C) and glass-transition (535°C) temperatures for both planar discs (PANNEALED, n = 5) and curved discs (R10ANNEALED, n = 5). Specimens were heated at 20°C/min to 510°C, held for 40 min, and cooled at 5.5°C/min to room temperature (Shelby, 2005). The radii of curvature of discs before and after stress relief were compared with paired t tests employed at α = 0.05 (SPSS 14.0, SPSS Inc, Chicago, IL, USA). To approximate clinically relevant surface condition and a representative flaw population, one surface of planar discs and the concave surface of group R30-R10 curved discs were alumina particle air abraded and acid etched for 120 s with 9.6% hydrofluoric acid (Porcelain Etch Gel, Pulpdent Corporation, Watertown, MA, USA) before rinsing with distilled water (10 s) and drying with oil-free compressed air (Hooi et al., 2013). Random discs (n = 3) from each group were then profilometrically evaluated to determine the actual radii of curvature via the scanning protocol outlined. In addition, measurements of the contralateral surfaces were measured to give an insight into any differences from the introduced surface roughness (Ra value). The central thickness of each disc was then measured with a screw-gauge micrometer (Mitutoyo Corporation, Tokyo, Japan) and a ball-on-ring testing configuration used to determine the load to failure (N) of the discs with the modified surface tested in tension. The discs were positioned on a 10-mm-diameter knife-edge ring support and loaded with a 4-mm-diameter stainless-steel ball indenter at 1 mm/min. The biaxial flexure stress at failure was calculated in the planar discs (group P) by Equation 3 (Shetty et al., 1980): s=
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J Dent Res XX(X) 2014 3 Strength Determination of Brittle Materials Table 1. Mean (SD) Deflection and Radius-of-Curvature Data Determined by Profilometry for the Planar and Curved Discs Condition
Planar Discs: Deflection, µm
As received Annealed Abraded
0.878 (0.603)a
Curved Discs (R10): Radius of Curvature, mm
0.783 (0.527)a 0.921 (0.305)a 1.253 (0.916)b
1.280 (0.407)b
10.190 (0.366)c 10.180 (0.353)c
10.305 (0.373)c,d 10.433 (0.393)d 10.309 (0.375)c,d
Different superscript denominators within columns denote significant differences (p ≤ .05). R10, 10 mm.
σ=
−3W (1 + ν ) a 1 −ν 1 + 2 ln + 4π t b 1 +ν
b2 − 1 2 2a
a2 2 , (3) R
where W was the load to failure (N); ν, the Poisson ratio (0.22) (Kolli et al., 2009); t, the thickness; a, R, and b, the radii of the knife-edge support, disc, and contact zone (t/3), respectively (Hsueh et al., 2005). Without a closed-form analytic solution to calculate the biaxial flexure stress for each curved disc (groups R30-R10), the maximum principal stress was determined by inputting the measured central thickness, measured radius of curvature, and load to failure into the FEA model (n = 80). A validated 2-dimensional axisymmetric FEA simulation (Pro/MECHANICA, Wildfire 2.0, PTC) (Fleming et al., 2012) was adapted to model the curved discs. The FEA simulation was assigned a maximum polynomial value of 6 and convergence criteria of 3% for a specified measure parameter of maximum principal stress at an axial position (r = 0) in the curved discs on the surface in tension. Analysis of mean deflection and the radii-of-curvature group means with and without annealing were performed utilizing a general linear model univariate analysis at α = 0.05 (SPSS 14.0). The mean measured Ra values and radii of curvature (groups P, R30-R10) were assessed before load to failure via one-way analyses of variance (ANOVAs) and post hoc Tukey tests at α = 0.05. One-way ANOVAs were performed (α = 0.05) to determine whether load-to-failure or maximum principal stress data were significantly modified by radii of curvature.
Results Profilometric measurements to identify stress-induced changes in specimen curvature for planar discs identified a significant effect of alumina particle air abrasion (p ≤ .05) but not annealing (p > .05). For curved discs, no significant effect was evident following alumina particle air abrasion or annealing (Table 1). Following surface modification (alumina particle air abrasion and hydrofluoric acid etching) the mean radii of curvature of the curved discs (groups R30-R10) differed significantly (Table 2). One-way ANOVA and post hoc Tukey tests demonstrated that discs with the minimum radius of curvature (R10) had significantly reduced surface roughness (Table 2) compared with discs from groups P, R20, and R15 (p ≤ .008). One-way ANOVAs further demonstrated that the magnitude of the load to failure was significantly modified by the radius of curvature of the discs (Figure 2). The actual radii of curvature for the curved discs were defined in the parametric geometry of the FEA model for each
Figure 2. Mean maximum principal stress and mean load to failure plotted against the radius-of-curvature data, highlighting the increasing load to failure with reduced radius of curvature and the insensitivity of the maximum principal stress to the radius of curvature.
disc. The FEA model included the measured central thickness and load at failure for individual curved discs to determine the maximum principal stress resulting from the applied load to failure. Comparison of the mean maximum principal stress and the mean load-to-failure data (Table 2, Figure 2) demonstrated that the load to failure increased as the radius of curvature reduced, which was in contrast to the pattern evident for the maximum principal stress. This indicated that the more curved geometry increased the load to failure without modifying the maximum principal stress determined with the FEA simulation.
Discussion Testing planar discs in biaxial flexure is widely employed to investigate the fracture behavior of dental ceramics (Cattell et al., 1997; Addison et al., 2008). FEA simulations (Anusavice and Hojjatie, 1992) and quantitative fractography (Kelly, 1999; Quinn et al., 2005) of failed all-ceramic restorations identified that clinical failure is often manifest by the tensile extension of surface flaws. The observed clinical failure mode of radial fracture in monolithic restorations is reproduced with the biaxial flexure stress test (Fleming et al., 2006); however, the fracture behavior of planar specimens is somewhat unrepresentative of the geometric complexities evident in full-contour restorations (Rekow et al., 2006). Previously, the validity of an FEA simulation for planar discs tested in biaxial flexure was confirmed with the bending solutions for equivalent geometry (Fleming et al., 2012). The current parametric FEA model was conducive to shape modification
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Table 2. Finite Element Analysis Simulation for the Planar (Group P) and Curved Discs (Groups R30-R10): Mean (SD)
Group P R30 R20 R15 R10
Thickness, mm 0.93 0.92 0.93 0.93 1.05
(0.01)a (0.02)a (0.02)a (0.02)a (0.08)b
Inner Surface Ra Value, µm 1.511 1.383 1.437 1.449 1.289
(0.053)c (0.052)d,e (0.048)c,d (0.048)c,d (0.041)e
Actual Radius of Curvature, mm Flat (∞) 28.21 (1.62)f 19.86 (3.20)g 15.43 (0.44)h 8.25 (0.54)i
Load to Failure, N 40.6 46.0 50.8 64.1 93.7
(3.5)j (6.0)j,k (5.9)k (5.4)l (15.5)m
Maximum Principal Stress, MPa 97.7 98.7 98.9 112.8 105.0
(8.0)n (11.1)n (9.8)n (9.2)o (15.4)n,o
The mean thickness, inner surface Ra-values, actual radius-of-curvature, load-to-failure in BFS testing and maximum-principal-stress determined using the FEA simulation for the planar (Group P) and curved-discs (Groups R30-R10) examined. Different superscript denominators within columns denote significant differences (p ≤ .05). R30, 30 mm; R20, 20 mm; R15, 15 mm; R10, 10 mm.
by varying the radius of curvature and the thickness of the individual specimens tested in biaxial flexure. The experimentally determined load-to-failure data avoided the necessity to incorporate an arbitrary probabilistic failure criterion (Yi and Kelly, 2008) in the FEA simulation. In the current study, the radius of curvature was not observed to impact on the maximum principal stress in the curved discs; therefore, this hypothesis was rejected. Interestingly, significantly increased loads to failure were observed in the curved discs when compared with planar discs; therefore, this hypothesis was accepted. The pronounced sensitivity of the load-to-failure changes with radii of curvature provides definitive evidence of the problems associated with comparisons to the widely reported “crunch the crown” monotonic load-to-failure test data in monolithic crowns (Scherrer and de Rijk, 1993; Burke and Watts, 1998; Tinschert et al., 2001; Attia and Kern, 2004; Borges et al., 2009). While expert opinion has frequently challenged the use of such techniques (Kikuchi et al., 1997; Qasim et al., 2007; Kelly et al., 2012), “crunch the crown” data are consistently published within the literature. Specimens employed in “crunch the crown” testing contain multiple geometric features to represent the full-contour restoration under investigation. It is reasonable to expect that variability among test specimens is exacerbated with increasing geometric complexity. In the absence of stringent quality control comparisons among test specimens, batches and studies are limited (Kelly et al., 2012). While computer-aided design and manufacturing can reduce the geometrical variation in such specimens, there is evidence that the process can introduce significant differences in the critical defect populations when test specimen batches are prepared from a single bur set (Addison et al., 2012). The current investigation highlights that load-to-failure testing can be performed when complemented by numeric simulations that are specimen specific and where specimen failure, as in the current study, is determined to be representative of the clinically operative failure mechanism. However, even for a simplistic model, the sensitivities are high to shape variation among individual specimens from an “identical” series. The fabrication of the nominally identical disc-shaped specimens with a single radius of curvature with a standardized protocol was monitored for consistency in the current study by profilometric inspection to verify that the specified dimensional and geometric parameters were controlled and any identified deviations from the specifications were factored into the simulation models. The findings demonstrated that the adapted flat-planar solution adequately described the failure stress of all curved structures in
the radii of curvature examined, which would be representative of most curvatures observed in a full-contoured restoration other than at a cusp tip. The surface state responsible for the specimen at failure was assumed to be equivalent for all specimen groups given that a common surface treatment was undertaken. However, for specimens with the smallest radius of curvature (R10), significant differences in the microscopic surface topography were observed and were assumed to result from the complexities of abrasive particle interactions with the curved surface. Although this raises a cautionary note about the accuracy of the measured values, it further highlights the need for modeling more complex geometries where identical and clinically relevant experimental treatments may yield different results. The current approach, although laborious, measured the actual load to failure while accounting for the geometric influence on maximum principal stress and the unique defect populations in each of the tested curved discs, without the requirement to predict the probabilistic pattern of fracture. The current study demonstrated the complications associated with interpreting load-to-failure monotonic data and provides support for the relevance of testing simple geometric specimens. While fatigue studies based on full-contour restorations add insight (Zhang et al., 2013), monotonic testing is clearly redundant. The data reported support the synergistic use of bespoke computational analysis with conventional mechanical testing.
Acknowledgments This work was funded by a Trinity College Dublin Ussher Fellowship for Paul Hooi and carried out at Dublin Dental School and Hospital. The authors declare no potential conflicts of interest with respect to the authorship and/or publication of this article.
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