## Abstract

## Objectives

Piston-on-three-ball tests have been selected by the International Organization for Standardization to establish ISO 6872 for the evaluation of the biaxial strength of dentistry–ceramic materials. However, the formula adopted in ISO 6872 for the fracture load-biaxial strength relationship was an approximate equation originally derived for piston-on-ring tests of monolayered discs. This formula was modified and extended to the case of multilayered discs subjected to piston-on-ring loadings recently. The purpose of the present study is to evaluate the adequacy of applying the formula for piston-on-ring to piston-on-three-ball tests for both monolayered and multilayered discs.

## Methods

Finite element analyses were performed to simulate both piston-on-three-ball and piston-on-ring tests. Different degrees of friction between the specimen supporting surface and the loading fixture were considered in the simulations. The simulated biaxial stress distributions through the disc thickness for both tests were then compared to the formula to examine how the predictions agree.

## Results

Examples of monolayered, bilayered and trilayered discs subjected to piston-on-three-ball and piston-on-ring loadings were simulated for comparison with the formulae. The results depended on friction when the disc was supported by a ring, however the results became insensitive to friction when the disc was supported by three balls. For the frictionless contact, both loading tests yield almost the identical biaxial stress distribution through the disc thickness and agree well with the formula. On the other hand, the maximum tensile stress on the surface of the disc decreased when the friction increased. As a result the biaxial tensile strength was overestimated for piston-on-ring tests if friction was neglected.

## Significance

The results of our finite element analyses demonstrate how the friction between the specimen supporting surface and the loading fixture affects the biaxial strength evaluation in piston-on-ring and piston-on-three-ball tests. It is critical to have frictionless contact between the disc and the supporting ring when evaluating the biaxial strength by piston-on-ring tests. Otherwise, the application of the approximate formula, which was derived for piston-on-ring, to piston-on-three-ball is even better than piston-on-ring. This is true for not only monolayered discs but also multilayered discs.

## 1

## Introduction

All-ceramic dental prostheses have been used extensively for clinical restorations because of their superior appearance, inertness and biocompatibility. Ceramic materials generally have high compressive strength, however they are brittle and unable to withstand high tensile stresses. Therefore tensile strength is an important factor that influences the clinical success of dental restorations . Standard test methods for determining the flexural strength of ceramic materials are either uniaxial (e.g. three- or four-point bending of beams) or biaxial flexural tests (e.g. piston-on-ring, piston-on-three-ball, ball-on-ring and ring-on-ring tests). Biaxial flexural strength tests have several advantages over uniaxial tests because multiaxial stress states are produced and edge failures are eliminated . As a result, biaxial flexure tests become increasing popular as a means of measuring the strength of dental ceramics . In such tests, a thin disc is supported by a ring (or three balls) near its periphery and loaded through a smaller coaxial ring, a piston, or a ball in its central region. The disc is subjected to a biaxial moment in its central region and the stresses are biaxial in this region.

The International Organization for Standardization has selected piston-on-three-ball tests to establish ISO 6782 for dentistry–ceramic materials . This selection is justified because the piston-on-three-ball fixture does not requires the two disc surfaces to be absolutely planar and slightly warped specimens can be accommodated by the three-ball supports . However, the biaxial strength formulae of ISO 6872 were originally derived for piston-on-ring tests. A question is hence raised as to whether it is satisfactory to apply the stress formulation for piston-on-ring to piston-on-three-ball tests. In addition, the application of ISO 6872 has been limited to the case of monolayered disc. While dental prostheses are usually fabricated into bilayered structures with good appearance but relatively weak veneer porcelains on stiff and strong supporting cores, an extended test standard for biaxial strength of bilayered and multilayered discs is essential.

Recently, Hsueh et al. derived closed-form solutions for stresses in multilayered discs subjected to ring-on-ring and piston-on-ring biaxial flexure tests . However, it should be noted that all the formulae from ISO 6872 and that of Hsueh et al. were based on a ring-support loading fixtures. Hence, the purpose of the present study is to examine the adequacy of applying the formulae for piston-on-ring to piston-on-three-ball tests for both monolayered and multilayered discs. First, ISO 6872 and Hsueh et al.’s formulae are reviewed to provide general background. Then, finite element analyses (FEA) are carried out to simulate piston-on-ring and piston-on-three-ball loading on monolayered, bilayered and trilayered discs. Friction between the specimen supporting surface and the loading fixture is considered with different coefficients of friction in simulations. Specific results are calculated for Cerinate porcelain and Empress glass ceramic monolayered, zirconia/porcelain bilayered and zirconia/porcelain/Empress-II trilayered discs subjected to piston-on-three-ball and piston-on-ring loadings. Finally, a comparison between the analytical results and finite element results is made to examine the applicability of the formulae for piston-on-ring to piston-on-three-ball biaxial flexure tests.

## 2

## Materials and methods

## 2.1

## Multilayered discs subjected to biaxial moment loading

A diametrical section through the axis of symmetry of a thin multilayered disc is shown in Fig. 1 . The cylindrical coordinate system, *r *, , and *z *, are used to describe the spatial coordinates. For general cases, the disc consists of *n *layers of individual thickness *t *_{i }where the subscript *i *denotes the layer number. Furthermore, layer one is at the bottom of the disc and its lower surface is located at *z *= 0. The interface between layers *i *and *i *+ 1 is located at *z *= *h *_{i }, and the upper surface of layer *n *is located at *z *= *h *_{n }which also represents the thickness of the disc. The relationship between *h *_{i }and *t *_{i }is:

The disc is subjected to biaxial moment loading through contact with the loading fixture at the support and the loading surfaces, *z *= 0 and *z *= *h *_{n }. The interfaces between layers are assumed to remain bonded during loading. Schematic drawings of piston-on-ring and piston-on-three-ball loadings are shown in Fig. 2 . For piston-on-ring loading, the disc with radius *R *is supported by a support ring with radius *a *, and loaded by a piston at the center with a radius *c *. For piston-on-three-ball loading, the disc is supported on three balls that are evenly spread on a circle with a radius *a *.

## 2.2

## ISO 6872 standard

According to the ISO 6872 standard, a monolayered disc specimen is positioned in the sample holder on top of the three supporting balls. The load at fracture is recorded and the biaxial flexure strength for the specimen is calculated using the following equation :

Y = ( 1 + ν ) 1 + ln a 2 R 2 + ( 1 − ν ) a 2 R 2

where *σ *_{BI }is the maximum tensile stress ( *σ *_{BI }) in megapascals (MPa), *P *is the failure load in Newtons (N), *ν *is the Poisson’s ratio, and other parameters about dimensions are in millimeters (mm). It is instructive to note that the stress given by Eq. (2) is independent of the Young’s modulus for monolayered discs.

## 2.3

## Hsueh et al. formula

For multilayered discs, the continuity conditions at the interfaces between layers are required for analyses. Hsueh et al.’s formulas were obtained by (1) finding the correlation between monolayered and multilayered discs subjected to biaxial moment and (2) converting the existing solutions for monolayered discs to those for multilayered discs. The stress-moment relation is described by:

σ i = E i ( z − z ) M ( 1 − ν i ) ( 1 + ν ave ) D ( for i = 1 to n )