## Highlights

- •
Residual stress in the veneers was quantified by means of Vickers indentation.

- •
Effect of

*σ*_{R }on Hertzian cone crack propagation was determined. - •
The CTE mismatch had a dominant role in generation of

*σ*_{R }gradients. - •
Fast cooling did not significantly affect the

*σ*_{R }gradients. - •
High compressive

*σ*_{R }hindered crack propagation toward the interface.

## Abstract

## Objectives

The present study evaluated the effect of the coefficient of thermal expansion (CTE) mismatch and the cooling protocol on the distribution of residual stresses and crack propagation in veneered zirconia bilayers.

## Methods

Ceramic discs with two different CTEs (Vita VM9 and Lava Ceram) were fired onto zirconia plates and cooled following a slow (0.5 °C/s) or a fast (45 °C/s) cooling protocol. The residual stress distribution throughout the veneer thickness was assessed by means of depth-wise Vickers indentation after sequentially sectioning the bilayers parallel compared to normal to the interface. A mathematical solution for the residual stress distribution was used as reference. Additionally, Hertzian cone crack propagation in the veneers was induced by cyclic contact loading and measured at different number of cycles to estimate the crack growth rate.

## Results

The higher CTE mismatch of the VM9 group generated an important stress gradient with high compressive residual stresses near the interface, hindering the crack propagation. The low CTE mismatch group (Lava Ceram) developed only a slight stress gradient and higher cone crack growth rates. No differences were observed between the two cooling protocols applied regarding stress magnitude and crack propagation behavior.

## Significance

The CTE mismatch has a predominant role in the generation of residual stress gradients within the veneer, which directly influences contact-induced crack propagation. Based on the results, the cooling protocol had no significant effect on the residual stress distribution in zirconia-veneer bilayers.

## 1

## Introduction

Zirconium dioxide (ZrO _{2 }) has been promoted during the last decade as the most reliable framework material for all-ceramic dental restorations. Nevertheless, the Achilles’ heel of this system seems to be the high incidence of fractures (or *chipping *) of the glassy veneer that is fused onto the ZrO _{2 }. Prospective clinical evaluations report values up to 30% of veneer fractures over 5–10 years of service . Both clinical observations and in vitro experiments have shown that these fractures originate from occlusal contact areas in the veneer and propagate within this layer, thereby not reaching the interface . Even though different factors may trigger chipping events, the build-up of internal residual stresses has been speculated to play a key role in contact induced unstable cracking .

Among the reasons leading to the development of internal residual stresses, mismatch of the coefficients of thermal expansion (CTE or *α *) between the framework and veneer materials, as well as the effect of cooling rate, have been extensively studied in the last years . Positive CTE mismatches ( *α *_{core }− *α *_{veneer }= +Δ *α *) generate compressive residual stresses in the veneer close to the framework, potentially hindering local crack extension. As a rule of thumb, DeHoff et al. suggested a safe range of Δ *α *between −0.61 ppm K ^{−1 }and +1.02 ppm K ^{−1 }for veneered-ZrO _{2 }systems. However, depending on the thickness ratio between both materials, stress gradients arise throughout the veneer layer , even reaching tensile stresses at the surface of thick veneers. To counterbalance these latter stresses and to prevent the onset of surface cracks , tempering protocols have been used in the past to induce surface compression. Above its glass transition temperature ( *T *_{g }) the veneering ceramic melt is in a viscoelastic state, which allows relaxation of all generated stresses . As the temperature drops through the *T *_{g }interval, solidification gradually takes place and any unrelaxed stress becomes locked inside the solid material . If sufficiently fast cooling is applied, external areas (i.e., veneer surface) rapidly contract and solidify first, while the interior remains viscous over a relatively longer period. When the temperature in this region decreases, contraction is hindered by the already solid surface and tensile stresses arise in the inner (subsurface) zone. As a consequence, the surface is put into compression. The magnitude of these residual stress gradients has been described to be directly related to the applied cooling rate from above *T *_{g }. Nevertheless, the low thermal diffusivity of zirconia accounts for higher temperature gradients inside the veneer layer , potentially increasing subsurface tensile stresses, which have been pointed out to be responsible for accelerating the veneer chipping . Slow cooling protocols have therefore been proposed .

Characterization of residual stresses is a current challenge in dental materials science. Different methods have been proposed and applied , with varying degrees of success. Among them, the use of Vickers indentations to evaluate local variations in residual stress states has been shown to be a suitable technique for dental ceramics . This approach is anchored in the science of contact mechanics and based on the effect of residual stresses on the propagation of cracks induced from sharp indents; while tensile stresses facilitate crack extension, leading to longer cracks, compressive stresses will hinder their growth . The method is definitely very convenient, since measurement of stress states in the material only implies minimal surface damage , allowing multiple testing in the same specimen, with potential for application on more complex geometries . On the other hand, the use of Hertzian contact testing with spheres on flat-layered structures has proven to be an ideal simplified model to study the damage modes in ceramic materials . Owing to their brittleness, glass-ceramics and glasses are particularly susceptible to cone cracks that characteristically develop in the vicinity of the contact circle. At reduced loads (200 N) the effect of the Hertzian contact stress field is diminished and the local residual stress becomes dominant , governing the propagation of cone cracks. Under cyclic blunt contact loading, through-thickness variations in fracture toughness, due to the presence of residual stresses, are this way sure to affect crack growth rates in the veneer layer .

The aim of the present study was to characterize the residual stress distribution within the veneers of two ZrO _{2 }-veneer bilayer systems having different CTE mismatches and cooled from firing temperatures at different rates. The residual stresses generated by these two variables were quantified using Vickers indentation and their effect on crack propagation rates was measured by means of Hertzian contact loading.

## 2

## Material and methods

## 2.1

## Mathematical solution for thermal stress distribution in bilayers

A large number of solutions for the stress distributions in multilayers due to thermal mismatch have been proposed. Stoney’s analysis was the first to relate film residual stresses with the bilayer bending. The presence of asymmetric residual stresses during cooling, due to CTE mismatch of the components, conduces to a complete system bending . When the resulting mismatch stresses in the film are tensile, the substrate becomes concave, whereas compressive stresses give rise to a convex curvature . His equation has been extensively used and adapted to describe the stress distribution in multilayers under different geometries and conditions. To date, one of the most accepted solutions for the residual stresses in film-substrate multilayer systems, is the one formulated by Hsueh . In his model, the stress/strain distributions can be correctly approximated by solving three parameters: the uniform strain component ( *c *), the location of the bending axis ( *t *_{b }) and the radius of curvature of the system ( *r *). Finding the solutions for these parameters, under given boundary conditions, leads to an exact close-form expression for the elastic deformation due to residual stresses . This model was applied by Hsueh et al. to a dental ceramic bilayer system, in order to study the stress distribution due to bending and thermal mismatch between layers. The thickness of each of the two layers, core ( *t *_{2 }) and veneer ( *t *_{1 }) materials, as well as the relative position in the coordinate system ( *z *), are presented in Fig. 1 . The solution for the biaxial in-plane thermal stresses in the veneer layer ( *σ *_{1 }) is expressed as:

σ 1 = E ′ 1 E ′ 2 t 2 ( α 2 − α 1 ) Δ T E ′ 1 t 1 + E ′ 2 t 2 + κ z − E ′ 1 t 1 2 + E ′ 2 t 2 2 + 2 E ′ 2 t 1 t 2 2 ( E ′ 1 t 1 + E ′ 2 t 2 ) for 0 ≤ z ≤ t 1

where the elastic modulus ( *E *) is expressed as the biaxial modulus, *E *′ = *E */(1 − *ν *), with *ν *being the Poisson’s ratio. The difference in temperature (Δ *T *) is considered from *T *_{g }to room temperature (25 °C). *α *_{1 }and *α *_{2 }correspond to the CTE of veneer and core materials, respectively, and *κ *stands for the curvature of the bilayer (1/ *r *), which is given by:

κ = 6 E ′ 1 E ′ 2 t 1 t 2 ( t 1 + t 2 ) ( α 2 − α 1 ) Δ T E ′ 1 t 1 4 2 + E ′ 2 t 2 4 2 + 2 E ′ 1 E ′ 2 t 1 t 2 ( 2 t 1 2 + 2 t 2 2 + 3 t 1 t 2 )