## Abstract

## Purpose

This study evaluated failure behavior when resin-composite cylinders bonded to dentin fractured under traditional “shear” testing. Failure was assessed by scaling of failure loads to changes in cylinder radii and fracture surface analysis. Three stress models were examined including failure by: bonded area; flat-on-cylinder contact; and, uniformly-loaded, cantilevered-beam.

## Methods

Nine 2-mm dentin occlusal dentin discs for each radii tested were embedded in resin and bonded to resin-composite cylinders; radii (mm) = 0.79375; 1.5875; 2.38125; 3.175. Samples were “shear” tested at 1.0 mm/min. Following testing, disks were finished with silicone carbide paper (240–600 grit) to remove residual composite debris and tested again using different radii. Failure stresses were calculated for: “shear”; flat-on-cylinder contact; and, bending of a uniformly-loaded cantilevered beam. Stress equations and constants were evaluated for each model. Fracture-surface analysis was performed.

## Results

Failure stresses calculated as flat-on-cylinder contact scaled best with its radii relationship. Stress equation constants were constant for failure from the outside surface of the loaded cylinders and not with the bonded surface area or cantilevered beam. Contact failure stresses were constant over all specimen sizes. Fractography reinforced that failures originated from loaded cylinder surface and were unrelated to the bonded surface area.

## Conclusions

“Shear bond” testing does not appear to test the bonded interface. Load/area “stress” calculations have no physical meaning. While failure is related to contact stresses, the mechanism(s) likely involve non-linear damage accumulation, which may only indirectly be influenced by the interface.

## 1

## Introduction

Advancements in dental adhesive technology have greatly changed restorative dentistry. Bond strength testing is essential for the analysis of new products and understanding testing and clinical variables . In spite of many advances in adhesive technology, the one area needing further research has been assessing the bonded interface . Van Meerbeek et al. describe the various methods and techniques for measuring dentin bonding interfaces, with the macro-shear bond-strength test reported as being the most popular (26% of scientific papers reporting on bond strength) . In spite of the fact that the shear bond-strength tests are the most easy and fastest material testing technique to perform, current literature questions that the shear stress tests: (1) use a meaningless calculation, (2) the origination of most failures likely does not involve the interface, and (3) failures are more consistent from external surfaces in tension than interfacial surfaces in shear based from Weibull scaling analysis .

In his opinion piece on adhesive strength testing, Darvell rightly challenged that mode II (shear) failure cannot occur in practice . He argues that stress gradients will invariably be induced due to such effects as the discontinuity in elastic moduli across the interface and levering. He predicts that stress concentrations and gradients should be expected no matter how the loading is accomplished and therefore a uniform shear stress field is very unlikely.

Such opinion was validated when Braga et al. demonstrated through 3D finite element analysis (FEA) that the stress distributions upon loading were not uniformly distributed across the measured surface area and that tensile stresses exceeded shear ( Fig. 1 ) . Many others have questioned the use of the simplistic stress equation, load/area. Thus the calculation by this simple equation has long been understood to give false results as interfacial stresses are invariably non-uniform and not described by load/area .

As a result, shear test calculations should not be determined through the measured surface area, but rather focus should be on understanding the non-uniform stress distributions found upon failure and the real mechanism(s) of failure. Stress distribution diagrams, illustrated by Braga et al., at the dentin side of the dentin/composite interface displayed stresses close to the loading area than the whole surface area . In addition, the tensile stresses where much higher than the shear stresses, implying tensile stresses are the more probable causes of failure in “shear” testing. This confirmed earlier work by DeHoff and Anusavice that stresses were quite concentrated and had no relationship to the simplistic “stress” calculated by dividing the load by the bonded area .

In addition, Braga et al. demonstrated through 3D finite element analysis (FEA) that the stress distributions upon failure were not necessarily focused at the true interface . Through failure mode data collected from 37 studies published between 2007 and 2009, Braga et al. reported cohesive or mixed failure modes in 40–70% of specimens . This form of failures implies that the fracture paths never involved the interface, from beginning through final propagation. Fracture paths not involving the interface (i.e., “cohesive failure”) have been addressed by many, including van Noort et al. and Versluis et al. .

Besides the apparent fact that uniform shear stresses likely do not exist and cannot be achieved, is the reality that brittle and quasi-brittle materials do not fail by shear stresses but only in tension. Metals can fail due to yielding along lines of shear stresses and in fact the most widely used model for the yielding of metals are von Mises stresses, being a combination of shear stresses. Mode I opening, not Mode II (shear), is the only stress intensity of interest in analysis of brittle fracture.

The purpose of this study was to evaluate a variety of stress states during “shear” bond testing. Failure from the bonded surface has been traditionally assumed. Failure from the loaded cylinder could be due to (1) contact stress in flat-on-cylinder loading or (2) bending stresses in a cantilevered beam, uniformly loaded. Failure modes were assessed by examining scaling effects of failure loads relative to known changes in bonding surface area (by *r *^{2 }), resin cylinder surface area ( *r *^{0.5 }) and for a uniformly-loaded cantilevered beam ( *r *^{3 }). Contact stresses (flat-on-cylinder) would appear to validly represent loading with the device used which loads the entire 5 mm length of the composite cylinder with a flat piston. Should the cylinder beam bend slightly in compliance, this would then turn the stress state into a uniformly-supported beam in bending. This analysis was reinforced by fractography of the fracture surfaces to determine failure origin.

## 2

## Mathematical background

- (A)
For typical “shear stress” testing, bond strength (

*σ*) is assumed to follow this simplistic formula: - (B)
For flat-on-cylinder loading, contact stress (

*σ*) is given by<SPAN role=presentation tabIndex=0 id=MathJax-Element-2-Frame class=MathJax style="POSITION: relative" data-mathml='σ=2Lπblwithb=KL’>σ=2Lπblwithb=KL−−√σ=2Lπblwithb=KL

σ = 2 L π b l with b = K Lwith

*b*as the half contact width and*K*[which has units of length/load—so*b*has units of length (i.e.,*r*)]=where<SPAN role=presentation tabIndex=0 id=MathJax-Element-3-Frame class=MathJax style="POSITION: relative" data-mathml="L=loadl=cylinderlengthK=2πl1−v12/E1+1−v22/E21/d1+1/d21/2v1andv2=Poisson’sratiosE1andE2=elasticmoduli”>L=loadl=cylinderlengthv1andv2=Poisson’sratiosE1andE2=elasticmoduliK=(2πl(1−v21/E1)+(1−v22/E2)(1/d1)+(1/d2))1/2L=loadl=cylinderlengthK=2πl1−v12/E1+1−v22/E21/d1+1/d21/2v1andv2=Poisson’sratiosE1andE2=elasticmoduli

L = load l = cylinder length K = 2 π l 1 − v 1 2 / E 1 + 1 − v 2 2 / E 2 1 / d 1 + 1 / d 2 1 / 2 v 1 and v 2 = Poisson’s ratios E 1 and E 2 = elastic moduli*d*_{1 }and*d*_{2 }= diameter of cylinder and flat (=∞) [so*K*≈ 1/[*π*l(1/2*r*)] - (C)
Treating the system as a uniformly loaded, cantilevered cylinder the highest bending stress is given by

<SPAN role=presentation tabIndex=0 id=MathJax-Element-4-Frame class=MathJax style="POSITION: relative" data-mathml='σ=L×l22×Z’>σ=L×l22×Zσ=L×l22×Z

σ = L × l 2 2 × Zwhere

*l*is the cylinder length,*Z*is the section modulus = 0.78·*r*^{3 }.By evaluating the stress equations in

*r*^{2 },*r*^{1/2 },*r*^{3 }and load (*L*and √*L*/*L*), and assuming that failure stress (*σ*) is a constant the following relationships are derived which should be constant over the four radii used:<SPAN role=presentation tabIndex=0 id=MathJax-Element-5-Frame class=MathJax style="POSITION: relative" data-mathml='r2L=1π×σ=constant’>r2L=1π×σ=constantr2L=1π×σ=constant

r 2 L = 1 π × σ = constant<SPAN role=presentation tabIndex=0 id=MathJax-Element-6-Frame class=MathJax style="POSITION: relative" data-mathml='LLr1/2=2π×l4/πl1−v12/E1+1−v22/E21/2σ’>L√Lr1/2=2π×l(4/πl((1−v21/E1)+1−v22/E2))1/2σLLr1/2=2π×l4/πl1−v12/E1+1−v22/E21/2σ

L L r 1 / 2 = 2 π × l 4 / π l 1 − v 1 2 / E 1 + 1 − v 2 2 / E 2 1 / 2 σ### You may also need