## Abstract

## Objectives

The reliability and longevity of ceramic prostheses have become a major concern. The existing studies have focused on some critical issues from clinical perspectives, but more researches are needed to address fundamental sciences and fabrication issues to ensure the longevity and durability of ceramic prostheses. The aim of this paper was to explore how “sensitive” the thermal and mechanical responses, in terms of changes in temperature and thermal residual stress of the bi-layered ceramic systems and crown models will be with respect to the perturbation of the design variables chosen (e.g. layer thickness and heat transfer coefficient) in a quantitative way.

## Methods

In this study, three bi-layered ceramic models with different geometries are considered: (i) a simple bi-layered plate, (ii) a simple bi-layer triangle, and (iii) an axisymmetric bi-layered crown.

## Results

The layer thickness and convective heat transfer coefficient (or cooling rate) seem to be more sensitive for the porcelain fused on zirconia substrate models.

## Significance

The resultant sensitivities indicate a critical importance of the heat transfer coefficient and thickness ratio of core to veneer on the temperature distributions and residual stresses in each model. The findings provide a quantitative basis for assessing the effects of fabrication uncertainties and optimizing the design of ceramic prostheses.

## 1

## Introduction

Fabrication of multilayered ceramics signifies an important topic of research in many technological fields, e.g. aerospace, defense, manufacturing and biomedical engineering. In prosthetic dentistry, more and more patients are tending to receive all-ceramic restoration for its outstanding esthetics and excellent biocompatibility . A more reliable core veneered all-ceramic restoration is expected to integrate the strength of ceramic cores with the superior esthetics but weaker outer layer of veneer ceramics to achieve the desirable strength and appearance of the layered structure throughout the entire life of fabrication, implementation and application.

Although all-ceramic dental restorations have significant advantages in biocompatibility and esthetics, the structural integrity and clinical longevity are still a foremost concern of dentists and patients. Various clinical studies have been conducted to understand the possible failure mechanisms for layered alumina or zirconia based ceramic devices . The presence or development of tensile stresses in multiple material dental prostheses may cause porcelain cracking and/or chipping , because in general, ceramic materials are relatively brittle and can be hard to withstand high tensile stresses that are often yielded from functional loading. Consistent findings have been reported that bi-layered dental ceramics may fail during fabrication process at certain elevated temperatures . The mismatches of thermal expansion and thermal gradients inevitably make the layered structures subject to a high residual stresses when cooled from a furnace temperature to the room temperature. Fischer et al. explored the influence of thermal properties of veneering ceramics on the fracture load of bi-layered single crowns. They found that the highest fracture load occurred in veneering ceramics when the product of mismatched thermal expansion coefficients (Δ *α *) and temperature difference (Δ *T *) between veneering ceramic and substrate is higher than 580 × 10 ^{−6 }. Fischer et al. further investigated the effects of thermal misfit on the shear strength of zirconia/veneering ceramic composites. They suggested that a veneering ceramic with Y-TZP should have a value of Δ *α *× Δ *T *≈ 1000 × 10 ^{−6 }for providing adequate shear strength of veneering ceramic fused on zirconia restorations. Aboushelib et al. improved the bond strength of resin composite to zirconia by combining surface treatment and selective infiltration etching. To a certain extent, these research efforts attempted to address the sophisticated mechanics issues involved in the layered ceramic structures.

The thermal process involved in the layered ceramic design can be modeled mathematically in terms of partial differential equations. In such a system, a number of physical parameters or design variables can be chosen to modify (either manually or automatically) for reducing the thermal gradient and residual stress. The primary concern would be how to select and/or modify the variables so that the design of layered structures can follow a right direction toward an optimal solution, i.e. minimizing the thermal gradient and residual stress in bi-layered dental restorations. This becomes a problem to determine how sensitive the thermal and mechanical responses would correspond to a certain variation or perturbation of these design variables. For example, when analyzing core-veneered all-ceramic dental crown, one may be interested to know the sensitivity of the temperature response built inside the materials with regard to small changes in the thickness of each layer or heat transfer coefficient during the cooling process. In this sense, the sensitivity analysis actually represents a mathematical operation that provides a quantitative means to determining the gradient of response with respect to the change in selected design variables.

In engineering fields, the sensitivity analysis for multiple layered materials has drawn some attention in the literature. Ferraiuolo et al. presented to re-size the coupling pin and corresponding contact area consisting of ZrB _{2 }–SiC and C/SiC–graphite dome to ensure the structural integrity of the components connected in space engineering. In their study, a sensitivity analysis was implemented for design optimization with an aim of reducing stress concentration next to the coupling hole of the tip component and decreasing the large volume loss due to spark erosion. The key parameters considered for the sensitivity analysis were the bend radius, length, and the diameter of the hole. Corbin performed a finite element analysis (FEA) to evaluate the thermal strain sensitivity with respect to structural variables that develop in the solder ball interconnection during thermal cycling. Kitada et al. also adopted finite element method (FEM) to evaluate the stress sensitivity for multi-stacked thin Si wafers composed of copper trough silicon via (TSV) and copper/low-k BEOL structure in electronic engineering, aiming to reduce the thermal–mechanical stress. It is believed that the resultant highly stressed material is the critical region for the initiation of the cracking or de-lamination that affects the mechanical integrity and reliability. The thickness of the adhesive layer was chosen as the key parameter for sensitivity analysis. Although these engineering applications signify the importance of sensitivity analysis involving thermomechanical coupling in multilayer systems, there has been lack of systematic study on sensitivity of layered dental ceramic structures. It is unclear which design variables lead to most or least significant changes in thermal gradient and residual stress and how much they would be, quantitatively, though such information is of decisive implication to assessment of fabrication uncertainty and design optimization.

The objective of this study was to explore how “sensitive” the thermal and mechanical responses (e.g. temperature and thermal residual stress) will be with respect to the possible design and fabrication variables (e.g. layer thickness and heat transfer coefficient) during the cooling process of fabricating ceramic dental prostheses.

## 2

## Materials and methods

In this study, the gradient information is derived to evaluate the dependence of thermal–mechanical responses on structural and fabrication parameters, respectively. If a relatively small perturbation in a key design variable or parameter leads to a relatively great change in mechanical and/or thermal responses, the performance can be regarded to be more sensitive to this variable or parameter . As a consequence, such design variable and/or parameter should be modified very carefully.

## 2.1

## Computational procedure

A general schematic description of the steps to perform a sensitivity analysis within the finite element analysis framework can be outlined as follows :

- 1.
Establish what the objective of design analysis is and then relate the thermal and mechanical responses to some design variables in terms of explicit or implicit functions or some form of approximation surrogate model.

- 2.
Decide which design variables should be investigated in the analysis. At this step, one of the key design variables is perturbed by a certain amount each time, allowing that the structural or fabrication parameters are modified in a certain way. Moreover, perturbation of a specific variable should be chosen properly, not too small and not too big for capturing the gradient information precisely and stably.

- 3.
Perform FEA for the control model and perturbed model, respectively, in order to determine the different responses of thermal and/or mechanical fields.

- 4.
The finite difference method is used to calculate the sensitivity in terms of objective gradient with respect to the perturbation of design variables.

## 2.2

## Material properties

In this study, the felspathic porcelain VITA VM7 or VM9 was selected as a veneering material. The core ceramics were the conventional alumina and zirconia (Y-TZP). Materials of coating and substrate were both assumed to be homogeneous and isotropic. Temperature-dependent nonlinear material properties are adopted herein from the literature and the manufacturer, as summarized in Tables 1–3 . Three bi-layered ceramic scenarios are taken into account for the sensitivity analyses below.

Elastic modulus [GPa] | 70 |

Poisson’s ratio | 0.26 |

Density [kg/m ^{3 }] |
2431 |

Tensile strength [MPa] | 45.8 |

Temperature | Conductivity | Specific heat | CTE ^{Alumina } |
CTE ^{Zirconia } |
Shear modulus |
---|---|---|---|---|---|

[°C] | [W/m °C] | [J/kg °C] | [10 ^{−6 }/°C] |
[10 ^{−6 }/°C] |
[GPa] |

25 | 1.37 | 734 | 6.48 | 9.05 | 25.8 |

100 | 1.35 | 841 | 6.65 | 9.15 | 25.5 |

200 | 1.32 | 947 | 6.78 | 9.29 | 25.1 |

300 | 1.28 | 1021 | 6.85 | 9.40 | 24.6 |

400 | 1.23 | 1071 | 6.94 | 9.52 | 23.6 |

500 | 1.19 | 1106 | 7.13 | 9.65 | 22.8 |

550 | 1.14 | 1120 | 7.23 | 9.77 | 17.2 |

Temp. | Conduct. | Specific heat | Density | Elastic modulus | CTE | Shear modulus | Tensile strength |
---|---|---|---|---|---|---|---|

[°C] | [W/m °C] | [J/kg °C] | [kg/m ^{3 }] |
[GPa] | [10 ^{−6 }/°C] |
[GPa] | [MPa] |

25 | 2.92 | 466 | 6095 | 210 | 10.17 | 80.0 | 237.9 |

100 | 2.88 | 513 | 6080 | 208.7 | 10.42 | 79.5 | 228.4 |

200 | 2.83 | 546 | 6060 | 205.4 | 10.62 | 78.3 | 215.8 |

300 | 2.78 | 568 | 6040 | 200.7 | 10.87 | 76.6 | 218.3 |

400 | 2.73 | 584 | 6019 | 194.6 | 11.10 | 74.3 | 220.7 |

500 | 2.68 | 597 | 5999 | 187.4 | 11.31 | 71.7 | 237.9 |

550 | 2.66 | 603 | 5989 | 183.3 | 11.52 | 70.3 | 224.1 |

Temp. | Conduct. | Specific heat | Density | Elastic modulus | CTE | Shear modulus | Tensile strength |
---|---|---|---|---|---|---|---|

[°C] | [W/m °C] | [J/kg °C] | [kg/m ^{3 }] |
[GPa] | [10 ^{−6 }/°C] |
[GPa] | [MPa] |

25 | 32.82 | 781 | 3998 | 399.9 | 6.96 | 163.0 | 260.6 |

100 | 25.37 | 937 | 3991 | 395.8 | 7.23 | 161.2 | 252.7 |

200 | 19.64 | 1026 | 3982 | 390.3 | 7.71 | 158.8 | 242.2 |

300 | 16.12 | 1082 | 3972 | 384.8 | 8.08 | 156.4 | 231.7 |

400 | 13.72 | 1124 | 3962 | 379.3 | 8.38 | 154.0 | 253.8 |

500 | 11.98 | 1157 | 3953 | 373.8 | 8.75 | 151.6 | 275.8 |

550 | 11.27 | 1172 | 3947 | 371.0 | 8.91 | 150.4 | 269.6 |

## 2.3

## Basic models

- 1
**Bi-layered ceramic plate:**The dimension of this model is 15 mm in length, 4 mm in the total thickness according to our experimental study . The thickness of substrate is 0.6 mm and that for porcelain layer is 3.4 mm, as shown in Fig. 1 (a) . - 2
**Simplified triangular bi-layered model:**The typical cusp has an angular shape around crown edge and varies from tooth to tooth. To make such a more clinical oriented scenario to a simple physical model, a simplified triangular model is created as illustrated in Fig. 1 (b). It has a base length of 15 mm, a total height of 4 mm. The core layer is in 0.6 mm thick and the characteristic thickness of veneering layer is 3.4 mm in the thickest point. - 3
**Axisymmetric crown model:**The crown model was sectioned to a 2D axisymmetric model as shown in Fig. 1 (c). The core layer with a thickness of 0.7 mm was taken to ensure the proper veneer appearance according to the manufacturing specifications. Correspondingly, the thickness of veneering layer*t*_{v }is around 1.1 mm. From clinical perspective, the thickness of veneering layer should range from 1.0 to 2.0 mm, depending on the location of tooth selected.

## 2.4

## General equations

These abovementioned three scenarios are exemplified to explore how sensitive the thermal (temperature) and mechanical (thermal stress) responses are relative to the changes in design variables (i.e. layer thickness *t *and heat transfer coefficient *h *, respectively), according to the gradients defined in Eqs. (1)–(4) .