To report analytic solutions capable of identifying failure stresses from the biaxial flexure testing of geometries representative of endodontic access cavities prepared through dental restorative materials.
The ring-on-ring biaxial flexure strength of annular discs with a central circular hole supported peripherally by a knife-edge support and loaded evenly at the upper edge of the central hole were solved using general expressions of deformations, moments and shears for flat plates of a constant thickness. To validate the solutions, finite element analyses were performed. A three-dimensional one-quarter model of the test was generated using a linear P-code FEA software and the boundary conditions represented the experimental test configuration whereby symmetry planes defined the full model. To enable comparison of the maximum principal stresses with experimental derived data, three groups of nominally identical feldspathic ceramic disks ( n = 30) were fabricated. Specimens from Group A received a 4 mm diameter representative endodontic access cavity and were tested in ring-on-ring. Group B and C specimens remained intact and were tested in ring-on-ring and ball-on-ring, respectively, to give insight into strength scaling effects. Fractography was used to confirm failure origins, and statistical analysis of fracture strength data was performed using one-way ANOVAs ( P < 0.05) and a Weibull approach.
The developed analytical solutions were demonstrated to deviate <1% from the finite element prediction in the configuration studied. Fractography confirmed the failure origin of tested samples to coincide with the predicted stress maxima and the area where fracture is observed to originate clinically. Specimens from the three experimental groups A–C exhibited different strengths which correlated with the volume scaling effects on measured strength.
The solutions provided will enable geometric and materials variables to be systematically studied and remove the need for load-to-failure ‘crunch the crown’ testing.
Clinical studies have demonstrated that the loss of pulpal vitality leading to a need for endodontic treatment is amongst the most common complications for teeth that have received indirect extra-coronal restorations . As a consequence, clinicians are faced with the challenge of accessing the root canal anatomy by either removing or machining through the indirect restoration to complete endodontic therapy. Currently there is little robust evidence to inform clinicians regarding the need to replace dental crowns with access cavities machined through the restorative material. A significant contributory factor to the lack of clinical evidence is the deficiency of in vitro testing methodologies to robustly model the material behavior in such clinical situations as they arise.
Ceramic restorations that have been perforated to allow endodontic access have been observed to fracture and the fracture origin may occur at the periphery of the access cavity . Unfortunately, no comprehensive fractographic study has been performed but anecdotal reporting from 751 clinicians has strongly demonstrated a perceived association between endodontic access and subsequent restoration fracture for many all-ceramic restorative materials . The anatomical considerations of endodontic access require the machined access cavity to be located in the regions of subsequent concentrated masticatory loading – namely at the cingulum or on the occlusal surface of the anterior and posterior teeth, respectively . Previous investigations into the mechanical behavior of endodontically accessed restorations have almost exclusively used load-to-failure testing of ‘clinically representative’ geometries . These are most commonly reported as series of crowns prepared with access cavities – the so called ‘crunch-the-crown’ tests . However, such load-to-failure methods have problems with both reproducibility and validity and do not represent the common stress patterns in ordinary masticatory cycles . To provide useful guidance for clinicians, in vitro mechanical testing of dental materials should mimic the clinical failure mode . All-ceramic restorations routinely fail from the extension of pre-existing defects resident on the inner ‘fit’ surface(s) of the restoration in contrast to failure originating at the loaded surface. Load-to-failure testing methodologies involve loading morphologically realistic restorations with flat or spherical indenters . However, the forces required to initiate failure greatly exceed both the masticatory and parafunctional forces and result in exaggerated contact stresses . Furthermore, failures observed clinically are associated with the production of a low number of fracture fragments (most commonly two) whereas ‘crunch-the-crown’ testing methodologies produce a large number of fragments which is indicative of the considerable energy stored prior to failure. The application of stress solutions to such complex structures is also not possible and therefore only the load bearing capacity of the system can be reported . As a result, there has been an increased use of finite element analyses (FEA) within the dental literature . However, FEA models used in isolation are unable to predict the stability and statistical distribution of the surface defect population which accounts for the variability in the fracture strength of brittle systems . Therefore, in vitro load to failure mechanical testing methodologies that enable the stress calculation, at the clinically relevant point of failure are required.
The tensile strength of dental ceramics has previously been determined from indirect tensile, pure tension and flexure testing . Of these techniques, flexure testing is favored as it creates a pure state of tension on one aspect of the specimen . Flexure testing is most commonly performed utilizing rectangular beam-shaped specimens in uniaxial flexure or discs-shaped specimens in biaxial flexure . Biaxial flexure testing is considered advantageous over uniaxial testing as the specimens are considered to be easier to prepare and closely match the size (volume and thickness) of clinical restorations . Under loading a greater volume of the specimen is tested when compared with the uniaxial bending of beams resulting in a searching examination of the strength limitations of the specimen .
Failure stresses for disk-shaped specimens loaded in biaxial flexure in a ball-on-ring or ring-on-ring configuration can be derived from analytical solutions for stress distributions in isotropic bodies described initially by Timoshenko’s analyses of thin plates . Timoshenko’s analyses have been extensively applied within the dental literature to calculate the failure stresses for monolithic ceramic specimens loaded under biaxial flexure in different geometries . However, to study more complex geometries such as plates with central holes the development of closed-form solutions are necessary. The aim of this study is to report analytic solutions capable of identifying failure stresses from the biaxial flexure testing of geometries representative of endodontic access cavities prepared through dental restorative materials.
Materials and methods
Test specimen preparation
Disk-shaped ceramic specimens (14 mm diameter, 0.85 mm thickness) were prepared from a feldspathic ceramic advocated for use in dentin-bonded crowns (DBCs). To ensure reproducible specimen geometry, composition and surface condition, accurately proportioned dentin porcelain powder and modeling liquid were condensed into a ring-mold assembly prior to controlled firing. Vita Modeling Liquid (0.2 mL; batch no. 29990; Vita Bad Sackingen, Germany) was added to 0.6 g of VitaVM7 Base Dentin powder (batch no. 25660; Vita Bad Sackingen, Germany) to produce a thick slurry consistency. The entire slurry was transferred to a ring-mold which was firmly secured to an aluminum base plate which possessed a mirror-like finish. The assembly was centered on a vibrating table and the excess liquid was removed with an absorbent tissue over 60 s. The overfilled mold was leveled carefully with a razor blade and following disassembly of the ring-mold, the condensed specimens were placed with a consistent orientation on a clean silicon nitride firing slab and fired in a VITA Vacumat 40 T vacuum furnace (Vita Zahnfabrik H. Rauter GmbH & Co. KG, Bad Säckingen, Germany) according to the manufacturers’ recommended cycle. Following firing, the resultant specimens possessed a “glaze-like” surface from firing against the silicon nitride firing slab and a “fit” surface resulting from leveling of the condensed slurry with the razor blade and contact with air during firing. Specimens with obvious imperfections were discarded and remade and then randomized to negate for the timing or sequence of sample manufacture.
Following manufacture of 90 disk-shaped test specimens, 30 were randomly allocated to each of three groups (A–C). Specimens from Group A were selected to receive a representative endodontic access cavity. A central circular area of 4 mm diameter was marked onto the center of the “glazed” surface of each specimen with indelible ink. An access cavity perforating the thickness of the disk-shaped specimen was cut using a dental air turbine with copious water irrigation using a coarse cylinder diamond burr; pattern 542 (MDT technologies, Israel). A new burr was used after cutting five cavities and the burr orientation was maintained as closely as possible to 90° to the specimen surface to create a parallel walled cavity. The inner diameter of the resultant annular shaped specimens was measured using digital calipers accurate to 0.01 mm.
Biaxial flexure strength determination
Closed-form analytical solutions for stress determination during the testing of disk-shaped specimens in both ring-on-ring and ball-on-ring geometries are widely reported within scientific literature . This study reports and applies analytical solutions for the determination of stress maxima in disk-shaped specimens with a symmetrical central hole derived from solutions for deflection of annular specimens in a ring-on-ring geometry described by Roark . Specimens from Group B were loaded in a ring-on-ring configuration and specimens from Group C in a ball-on-ring configuration.
The biaxial flexure strength (BFS) of each material group was determined at the cross-head speed of 1.0 mm/min in either a ring-on-ring (Groups A and B) or ball-on-ring (Group C) configuration using a universal testing machine (Model 5544, Instron LTD, High Wycombe, Bucks, England). For ring-on-ring testing a loading ring of 4 mm diameter was used to centrally load the annular (Group A) or intact disk-shaped specimens (Group B). In the ball-on-ring configuration, a 3 mm radius ball-indenter was used to centrally load the disk-shaped specimens which were concentrically positioned on a 10 mm diameter knife-edge support. The specimens were orientated so that the “fit” surfaces were subjected to tensile stressing. The load at failure (N) and mean specimen thickness (mm) measured at the point of fracture of each fragment with a screw-gauge micrometer (Moore and Wright, Sheffield, UK) accurate to 10 μm was recorded.
The ring-on-ring BFS for intact discs (Group B) was calculated according to Eq. (1) .
σ max = 3 W 4 π t 2 2 ( 1 + v ) ln a b + ( 1 − v ) ( a 2 − b 2 ) R 2
where σ max was the maximum tensile stress (MPa), W the measured load at fracture (N); a the radius of knife-edge support (mm); t the specimen thickness at the point of fracture (mm); b the loading ring radius (mm); R the radius of disk specimen (mm) and ν was the Poisson’s ratio for the material – assumed as 0.25 in the current investigation .
The ball-on-ring BFS for specimens from Group C were calculated according to Eq. (2) , where σ max , t , a , ν and W were similarly defined.
The ring-on-ring BFS for annular discs with a central circular hole (Group A), supported on a knife-edge support with an equal ring load applied to the inner edge of the central hole ( Fig. 1 ) can be solved using general expressions deformations, moments and shears for flat plates of constant thickness described in Roark .
For an annular plate, with the outer edge simply supported and the inner edge free and exposed to a uniform annular line load w (N/mm), at a radius r 0 (mm), the maximum moment M max is the tangential moment ( M tb ) at the inner edge ( Fig. 1 ).
The unit tangential bending moment M t is given by the expression
where θ is the radial slope of the plate, D is the plate constant, r is the radial location of the quantity being evaluated (mm) and M r is the unit radial bending moment. For the geometric conditions applied M rb = 0 and M rb = 0. Hence, Eq. (3.2) can be simplified to