## Abstract

## Objective

Strength is one of the preferred parameters used in dentistry for determining clinical indication of dental restoratives. However, small dimensions of CAD/CAM blocks limit reliable measurements with standardized uniaxial bending tests. The objective of this study was to introduce the ball-on-three-ball (B3B) biaxial strength test for dental for small CAD/CAM block in the context of the size effect on strength predicted by the Weibull theory.

## Methods

Eight representative chairside CAD/CAM materials ranging from polycrystalline zirconia (e.max ZirCAD, Ivoclar-Vivadent), reinforced glasses (Vitablocs Mark II, VITA; Empress CAD, Ivoclar-Vivadent) and glass-ceramics (e.max CAD, Ivoclar-Vivadent; Suprinity, VITA; Celtra Duo, Dentsply) to hybrid materials (Enamic, VITA; Lava Ultimate, 3M ESPE) have been selected. Specimens were prepared with highly polished surfaces in rectangular plate (12 × 12 × 1.2 mm ^{3 }) or round disc (Ø = 12 mm, thickness = 1.2 mm) geometries. Specimens were tested using the B3B assembly and the biaxial strength was determined using calculations derived from finite element analyses of the respective stress fields. Size effects on strength were determined based on results from 4-point-bending specimens.

## Results

A good agreement was found between the biaxial strength results for the different geometries (plates vs. discs) using the B3B test. Strength values ranged from 110.9 MPa (Vitablocs Mark II) to 1303.21 MPa (e.max ZirCAD). The strength dependency on specimen size was demonstrated through the calculated effective volume/surface.

## Significance

The B3B test has shown to be a reliable and simple method for determining the biaxial strength restorative materials supplied as small CAD/CAM blocks. A flexible solution was made available for the B3B test in the rectangular plate geometry.

## 1

## Introduction

Strength is the preferred material property used by manufacturers to classify, compare, rank and advertise dental restoratives. Possibly because their target audience are dentists, usually laymen in the affairs of mechanics, fracture strength (σ * _{f }*) may get through more easily because of its simple relationship to load (

*F*) capacity (i.e.,

*σ*∝

_{f }*F*), a concept of ease clinical applicability: “stronger” materials are indicated for regions of higher masticatory loads. More relevant for describing fracture in brittle dental ceramics and composites, properties like fracture toughness – though more and more present in advertising brochures – include additional parameters such as crack size/geometry and inverse exponential relationships,

^{1 }

1 Strength, *σ *, is inversely related to the length of pre-existing cracks or defects in the material through the Griffith/Irwin relation: *σ *= *K *_{Ic }/ *Y *√ *pa *_{c }, with *K *_{Ic }being the fracture toughness in mode I loading, *Y *a geometrical factor related to the crack geometry, specimen geometry and type of loading, and *a *_{c }the length of the critical crack at failure. Defects can be artificially produced by damage of components/specimens, leading to a severe drop in strength if large enough, or belong to the natural flaw population(s) of the material . In ceramics natural flaws are mainly constituted of pores, inclusions (impurities), agglomerates or poorly sintered regions, usually formed during the powder processing and sintering steps. In well-controlled processing environments defects can be minimized, increasing the strength and reliability of the material, where critical crack sizes tend to scale to the size of microstructural features.

more complex ideas to make sense of in practice. For many engineers and materials scientists, strength testing provides more than stress-at-failure values: strength data is a window into the nature of the microstructure, residual stresses and defect size, type and distribution in the material. Associated with fractography, strength data can guide processing routes toward eliminating strength-limiting defects and increasing material reliability.

Today, dental indirect restorative materials (i.e., reinforced glasses, glass-ceramics, polycrystalline ceramics and composites) are increasingly supplied in pre-processed block (cuboid) or blank (disc) geometries for machining in CAD/CAM systems. The processing of materials in a controlled industrial environment undoubtedly reduces possible sources of defects created during intermediary laboratorial steps (such as sintering of ceramic powders, polymerization of resin composites, etc.). This also benefits the testing of strength, once samples are sectioned from the actual components that are later placed in service. Strength testing gains thereby clinical relevance and may serve for accurate quality control.

Because many indirect materials are indicated for single-unit restorations of diminute size only (i.e., crowns, onlays, inlays), they are supplied exclusively as blocks of small dimensions, such as 18 mm × 16 mm × 18 mm ^{3 }(e.g., C16) down to I8 blocks (8 mm × 8 mm × 15 mm ^{3 }). These limitations in size have great implications in the testing of such materials, as they fall short regarding minimal size requirements of most standards for unixial flexural strength testing using beams. Most national and international engineering standards dealing with strength testing in three-point and four-point bending set fixture spans lengths *L *of 40+ mm preferably, allowing a minimum of *L *= 20 mm, as in ASTM C 1161, ISO 17565 and ENV 843-1 for ceramics and ISO 4049 for dental resin composites. For that, at least 22 mm long beams are required, a condition not fulfilled for materials available only in the aforementioned block dimensions. To accommodate limited sized specimens, e.g., prepared out of small CAD/CAM blocks, the standard ISO 6872 for dental ceramics admits span lengths as small as 12 mm for three-point bending and 16 mm for four-point bending. The use of shorter spans in three- and four-point bending configurations is feasible not without reservations, as intrinsic problems of flexural test set-ups get amplified through miniaturization . Inaccuracies in specimen-fixture articulation, positioning and parallelism, along with friction and wedging effects, as thoroughly pointed out in Refs. , may lead to several percent error in the determination of strength using regular 20 mm or 40 mm support span lengths, let alone with shorter fixtures. For miniaturized set-ups extra precision has been advised, particularly concerning alignment of loading points and refinement of testing jigs . Added to these fundamental sources of error, smaller specimens are more prone to damage during production; corner chippings may be too large to be removed through chamfering of the edges, and skewed strength distributions may result thereof. This has been shown to be the case for miniaturized alumina specimens tested in Ref. , a problem of severe repercussions for fragile glassy dental ceramics. For resin composites, short-beam uniaxial flexural tests have also been criticized in a recent finite element analysis study due to similar shortcomings . Any remark in these regards are missing in ISO 6872, a standard widely used by manufacturers to test and advertise products, and the most used standard for testing dental ceramics by the dental scientific community .

A resolution for small specimens is found in biaxial flexure tests, such as the piston-on-ring test, first adopted in ASTM F 394-78 for 32 mm-diameter discs under a three-ball support and later adapted for specimens having diameters between 12–16 mm in ISO 6872. Apart from allowing small-sized specimens to be tested, biaxial flexural tests benefit from edges outside any important stress field contributing to failure initiation, a common problem in uniaxial tests. The biaxial stress state developed on the tensile side of the specimen simulates to a better extent multiaxial stress conditions in real applications, and does not discriminate cracks in particular orientations. Yet, loading and support fixtures consisting of rings or multiple balls require perfectly flat parallel surfaces for the stress state to be described by the available solutions. Errors are further introduced as friction is generated between the specimen and the fixed supports.

In 2002 a new test based on a ball-on-three-balls (B3B) configuration has been devised to address some drawbacks of conventional biaxial bending tests. The set-up is based on three balls that load the specimen over one supporting ball; friction is avoided by allowing the three loading balls to roll as the specimen bends and fractures. The four contact points tolerate slight warping of the surface without invalidating the assumptions of the numerical solution. Validations against other uniaxial bending tests with various technical ceramics have attested the flexibility of the B3B test regarding specimen size and geometry. Through the use of finite element analysis the stress state in the piece can be determined for complex 2-dimensional specimen shapes, opening the possibility for testing material samples of non-circular cross-sections.

In this contribution, we introduce the B3B method for biaxial strength testing of dental restorative CAD/CAM materials. The materials to be tested here have been thoroughly characterized in a previous contribution in terms of microstructure and the elastic constants needed for the B3B test solution. Two alternatives of dealing with small CAD/CAM blocks are demonstrated herein, using either discs or rectangular specimen geometries. The solution for the function needed to calculate the biaxial strength for the plate geometry is then made available with the Poisson’s ratio being the only variable to be inserted. The size effect on strength in dental CAD/CAM restoratives is further illustrated by comparing flexural strength values obtained in biaxial (B3B test) and uniaxial 4-point bending (4-PB) configurations.

## 2

## Background

## 2.1

## The ball-on-three-balls (B3B) test

The B3B test was introduced by Börger et al. to overcome some disadvantages of traditional biaxial strength assemblies (i.e., ring-on-ring, piston-on-ring, piston-on-balls, etc.), highly sensitive to deviations in plane-parallelism of the specimen’s faces and to the effect of unknown friction between the sample and the loading/supporting fixtures . The well-defined load transfer in the B3B (four point contact) allows testing of specimens with flatness deviations up to 16% , and small warpings present in as-sintered specimens. The influence of friction is considerably reduced when compared to fixed supports, as the loading balls roll outward during bending of the specimen . Unlike rotationally symmetric set-ups (e.g., ring-on-ring), the stress field at the tensile side of the specimen has a threefold symmetry in the B3B assembly . The load is transferred through a small contact area to the specimen by the central ball, with generated tensile stresses being highly dependent upon the radius of this contact area, which is, in turn, a function of the elastic moduli of the tested material and balls . Consequently, materials with low elastic modulus may show large deformations during loading that change the contact and support radii, influencing the stress state and increasing the test error. This has more severe implications for low-modulus dental composites than for stiff dental ceramics, and will be addressed in detail later in the text.

The maximum principal stress takes place at the center top of the specimen, and scales with the force applied *F *, and the inverse of the square root of the thickness *t *, viz:

σ max = f ( α , β , υ ) F t 2 .

The function *f *, derived using finite element (FE) analysis , has been shown to be determined by three main independent geometrical parameters influencing the stress state in the disc (see diagram in Fig. 1 a): the dimensional ratio formed by the support radius *R *_{a }(with *R *_{a }= (2√3 *R *_{b })/3), and the radius of the specimen *R *, ( *R *_{a }/ *R *, or *β *in Eq. (1) ); the thickness to specimen radius ratio ( *t */ *R *, or *α *in Eq. (1) ) and; the Poisson’s ratio of the tested material, *ν *. Accordingly, for a disc geometry with thickness *t *, radius *R *and support radius *R *_{a }, the function *f *can be determined using :

f ( α , β , υ ) = c 0 + ( c 1 + c 2 α + c 3 α 2 + c 4 α 3 ) 1 + c 5 α × ( 1 + c 6 β )

where *c _{i }*(

*i*= 0,…6) are constants obtained from fitting procedures of FE analysis results. Values for these constants can be found for the range of

*ν*ratios studied in Ref. , or directly calculated with a web-Mathematica tool available at www.isfk.at . This renders the B3B test scalable for small specimen sizes and different specimen geometries, only requiring some adjustment of the function

*f*. By using rectangular plates instead of discs as specimens, the test loses the three-fold symmetry and acquires a mirror-symmetry that intersects one of the loading (supporting) points. Because the extra material on the edges of plates stiffens the specimen, an important effect on the bending strain is observed, requiring a new numerical analysis for the function

*f*. Strength obtained using the B3B test with rectangular plates have been validated extensively for alumina, silicon nitride and a capacitor ceramic in Refs. . Rectangular plate testing has been shown to yield values comparable to those obtained with discs described by the prediction of Weibull scaling theory.

## 2.2

## Size effect on strength

Unlike fracture toughness, a material property that – in theory – should always yield the same value regardless of specimen size, geometry and testing method (at least for non-R-curve materials), strength is rather described by the distribution of flaw sizes in a set of specimens. This invariably leads to a dependence of the strength distribution on the size of the specimens and type of applied stress (i.e., uniaxial or biaxial). Implied in the Weibull theory, the most accepted statistical concept dealing with fracture of brittle materials, larger specimens have a higher probability of containing large strength-limiting defects, yielding lower strength values than small specimens tested in the same loading conditions. A direct comparison of strength values between specimens of different sizes violates this fundamental assumption and should be avoided. A solution to this problem is to scale the strengths obtained from two different specimens sizes ( *σ *_{1 }and *σ *_{2 }, for example) by considering their effective volumes ( *V *_{eff,1 }and *V *_{eff,2 }) using the well-known relation:

σ 1 σ 2 = ( V eff 2 V eff 1 ) 1 / m

being *m *the Weibull modulus. Only through this procedure strength values obtained from different test configurations (e.g., 3- or 4-point bending) and specimens sizes using the same test set-up can be compared. The right-hand side of Eq. (3) is determined based on the location of the defect-initiating fracture in the specimens, being whether volume defects (such as pores, agglomerates) or surface defects (usually artificially-induced cracks from surface finishing procedures). If the latter is defining the fracture mode, *V *_{eff }gives room to the effective surface *S *_{eff }, which is also dependent on size and test geometry. Solutions for *V *_{eff }and *S *_{eff }of standardized uniaxial bending (e.g., 3- and 4-PB) assemblies have been thoroughly summarized in Ref. . For the 4-PB test having the distance between the loading and supporting rollers being ¼ *L *(the configuration used in the present study, see Fig. 1 b), *V *_{eff }and *S *_{eff }are determined by:

V eff, 4PB = V 4 [ ( m + 2 ) ( m + 1 ) 2 ]

and

S eff, 4PB = L [ h + b ( m + 1 ) ] { ( m + 2 ) [ 2 ( m + 1 ) 2 ] }

where *V *is the specimen’s volume ( *b × h *× *L *) and *m *the Weibull modulus. The effective volume, *V *_{eff }, in discs and plates tested using the B3B test takes place at the center of the top surface and is determined by :

V eff, B3B = ∫ σ > 0 ( σ eq ( r → ) σ ref ) m d 3 r