## Abstract

## Objectives

To provide a brief summary of the background theory of interfacial fracture mechanics and develop an analytical framework that identifies the critical factors for the analysis of the initiation and propagation of adhesion failure in composite restorations.

## Methods

A conceptual framework utilizing interfacial fracture mechanics and Toya’s solution for a partially delaminated circular inclusion in an elastic matrix, which can be applied (with caution) to approximate polymer curing induced cracking about composite resins for class 1 cavity restorations.

## Results

The findings indicate that: (1) most traditional shear tests are not appropriate for the analysis of the interfacial failure initiation; (2) material properties of the restorative and tooth material have a strong influence on the energy realize rate; (3) there is a strong size effect; and (4) interfacial failure once initiated is characterized by unstable propagation along the interface almost completely encircling the composite.

## Significance

The work is important for the analysis of the reliability of composite class I restorations and provides an adequate interpretation of recent adhesion debonding experimental results utilizing tubular geometry of specimens. The approach clearly identifies the critical parameters including; curing strain, material modulii, size and interfacial strain energy release rate for reliable development of advanced restorative materials.

## 1

## Introduction

Polymerization contraction of dental composite produces a shrinkage stress that may be capable of propagating fracture in the vicinity of the restoration–dentin/enamel interface from pre-existing flaws. The contraction stresses are capable of inducing several clinical phenomena, namely deformation of the resin composite and tooth , debonding of the resin composite from the tooth , and cracking in the tooth . The contraction stress may be one of the factors causing postoperative problems and contributing to recurrent caries with resin composite restorations .

Dental adhesives are usually tested in shear or tension even though neither of these approaches recreates the local conditions such as superimposed composite curing strains in tooth restorations triggering failure. Furthermore, it is now well recognized that traditional stress-based calculations are not the appropriate tool for studying dental adhesion . The proper way and basis in other technologies ranging from aerospace structures to electronic thin films is the use of interfacial fracture mechanics. The latter provides a framework that can be used to assess the structural integrity of interfaces that are relevant to dental composite restorations. The purpose of this paper is to briefly present the background theory of interfacial fracture mechanics and develop an analytical equation for the initiation of adhesion failure and crack propagation applicable for qualitative analysis of composite class I restorations and appropriate to recent experimental test results on the adhesive strength utilizing tubular geometry of specimens .

## 2

## Materials and methods

Materials and structures, including interfaces, are generally accepted to contain pre-existing flaws that determine their strength. Shapes and dimensions of these pre-existing flaws in dental restorations can significantly be affected by the restorative material, cavity preparation and placement techniques. The most critical flaws with respect to the possible fracture mechanisms are the interfacial cracks, which are located in the high stress concentration area . Subsequently, in dental restorations the weak link is the composite–dentin/enamel interface, which in addition may contain defects such as air bubbles and contaminants . In this section we briefly summarize the general methods for the analysis of interfacial cracks.

## 2.1

## Mechanics of interface cracks

An analytical framework of interface cracks in isotropic bi-materials is well documented in the literature ; therefore, in this review only some of the basic relations required are presented.

Consider the local Cartesian coordinate system ( *x *, *y *) and polar coordinate system ( *r *, ) at the interface crack tip as shown in Fig. 1 . The bi-material elastic properties that govern the stress state are the Dundurs’ parameters defined as:

α = μ 1 ( κ 2 − 1 ) − μ 2 ( κ 1 − 1 ) μ 1 ( κ 2 + 1 ) + μ 2 ( κ 1 + 1 ) , | α | ≤ 1 2 β = μ 1 ( κ 2 − 1 ) − μ 2 ( κ 1 − 1 ) μ 1 ( κ 2 + 1 ) + μ 2 ( κ 1 + 1 ) , | β | ≤ 1 2

and a related parameter

where *μ *_{j }is the shear modulus and *κ *_{j }the Kolosov’s constants of material, the subscripts *j *= 1, 2 refer to the two materials, *κ *_{j }= 3 − 4 *ν *_{j }and *κ *_{j }= (3 − *ν *_{j })/(1 − *ν *_{j }) respectively for plane strain and plane stress state, and *ν *_{j }is the Poisson’s ratio. Parameter *β *vanishes for identical materials; and non-zero values of *β *cause the linear elastic crack tip stress and displacement field to oscillate.

The near-tip stress along the bonded interface part ( = 0) are expressed as:

σ x ( r , 0 ) + i τ x y ( r , 0 ) = K ( r / l ) i ε 2 π r + O ( 1 )

where *r *is the distance from the tip, i=√−1

i = − 1

, and *K *= *K *_{I }+ *iK *_{II }is the complex stress intensity factor (SIF), associated to a reference length *l *for opening *K *_{I }and shear *K *_{II }crack tip stress intensity factors. The stress intensity factor, *K *, depends on the problem geometry and applied loading.

The complex stress intensity factor can be related to the traditional strain energy release rate, *G *, by the following equation:

where *c *_{jj }= (1 + *κ *_{j })/ *μ *_{j }.

In many cases, the strain energy release rate, *G *, can be determined analytically from straightforward energy balance considerations bypassing complex analytical or numerical techniques.

Another important parameter significantly affecting the interfacial fracture is the phase angle of loading, *ψ *, which is a measure of the degree of mixity of crack tip shear to opening mode experienced by the interface crack. It is defined as

The bounds at both *π */2 and − *π */2 represent interface crack having zero opening: whereas *ψ *= 0 refers to a crack subjected to zero shear loading. It has to be mentioned that *ψ *is often non-zero even when the external loading is normal to the interface crack. This situation arises because of the elastic mismatch across the interface. Furthermore, the magnitude of the phase angle of mixity may be a function of crack length. Consequently, when analyzing crack interface problems, it is essential that *ψ *be calculated. This second parameter controlling the interface fracture represents the major difference from the classical fracture mechanics dealing with an elastically homogeneous material.

Both parameters, the energy release rate and phase angle of loading, can be determined either from corresponding finite element calculations or using analytical or semi-analytical integral equation methods. Later in this paper we will provide an analytical equation for the evaluation of the complex stress intensity factor, which can be reasonably applied for the analysis of the dental adhesive in class I restorations as well as composite shrinkage tests, where the test specimen has a simple tubular geometry

The near-tip relative displacement across the crack, Δ *u *_{k }= *u *_{k }( *r *, *π *) − *u *_{k }( *r *, − *π *) are expressed by:

Δ u y ( r ) + i Δ u x ( r , 0 ) = 8 1 + 2 i ε K ( r / l ) i ε ( c 11 + c 22 ) cosh ( π ε ) + O ( r ) , for r → 0