Size-dependent strength of dental adhesive systems

DENTAL-2355; No. of Pages 13

ARTICLE IN PRESS d e n t a l m a t e r i a l s x x x ( 2 0 1 4 ) xxx.e1–xxx.e13

Available online at www.sciencedirect.com

ScienceDirect journal homepage: www.intl.elsevierhealth.com/journals/dema

Size-dependent strength of dental adhesive systems Marc Campillo-Funollet a,∗ , Gary F. Dargush b , Richard A. VanSlooten b , Joseph C. Mollendorf b , Hyeongil Kim a , Steven R. Makowka b a b

Department of Restorative Dentistry, SUNY at Buffalo, Buffalo, NY, USA Department of Mechanical and Aerospace Engineering, SUNY at Buffalo, Buffalo, NY, USA

a r t i c l e

i n f o

a b s t r a c t

Article history:

Objective. The aim of this study is to explain the influence of peripheral interface stress singu-

Received 17 June 2013

larities on the testing of tensile bond strength. The relationships between these theoretically

Received in revised form

predicted singularities and the effect of specimen size on the measured bond strength are

18 January 2014

evaluated.

Accepted 26 March 2014

Methods. Finite element method (FEM) and boundary element method (BEM) analyses of

Available online xxx

microtensile bond strength test specimens were performed and the presence of localized

Keywords:

dicted by the models was compared to previously published experimental data.

high stress concentrations and singularities was analyzed. The specimen size effect preTensile bond strength

Results. FEM analysis of single-material trimmed hour-glass versus cast cylindrical spec-

Size-dependent behavior

imens showed different theoretical stress distributions, with the dumbbell or cylindrical

Adhesion

specimens showing a more homogeneous distribution of the stress on the critical symme-

Finite element method

try plane. For multi-material specimens, mathematical singularities at the free edge of the

Boundary element method

bonded interface posed a computational challenge that resulted in mesh-dependence in the

Stress singularities

standard FEM analysis. A specialized weighted-traction BEM analysis, designed to eliminate

Generalized fracture mechanics

mesh-dependence by capturing the effect of the singularity, predicted a specimen size effect

Mesh dependence

that corresponds to that published previously in the literature. Significance. The results presented here further support the attention to specimen dimensions that has already broadened the empirical use of the microtensile test methods. FEM and BEM analyses that identify stress concentrations and especially marginal stress singularities must be accounted for in reliable bonding strength assessments. Size-dependent strength variations generally attributed to the effects of flaw distributions throughout the interfacial region are not as relevant as the presence of singularities at bonded joint boundaries – as revealed by both FEM and BEM analyses, when interpreted from a generalized fracture mechanics perspective. Furthermore, this size-dependence must be considered when evaluating or designing dental adhesive systems. © 2014 Academy of Dental Materials. Published by Elsevier Ltd. All rights reserved.

∗

Corresponding author at: 215 Squire Hall, 3435 Main Street, Buffalo, NY 14214, USA. Tel.: +1 716 816 5561; fax: +1 716 829 2114. E-mail address: [email protected] (M. Campillo-Funollet).

http://dx.doi.org/10.1016/j.dental.2014.03.010 0109-5641/© 2014 Academy of Dental Materials. Published by Elsevier Ltd. All rights reserved.

Please cite this article in press as: Campillo-Funollet M, et al. Size-dependent strength of dental adhesive systems. Dent Mater (2014), http://dx.doi.org/10.1016/j.dental.2014.03.010

DENTAL-2355; No. of Pages 13

ARTICLE IN PRESS

xxx.e2

d e n t a l m a t e r i a l s x x x ( 2 0 1 4 ) xxx.e1–xxx.e13

1.

Introduction

Bonding of dental restorations to tooth structure was one of the major advances in dentistry during the past century. Since the first dental adhesive systems were developed, numerous modifications have been performed to enhance their bonding ability and increase longevity of the restorations. Along with these bonding agents, test methods have been developed to determine the efficacy of the bonded interface under in vitro conditions. The most widely used methods to test bond strength to dental substrata are the tensile bond strength test (TBS) and the shear bond strength test (SBS) [1]. Different versions of these two methods exist regarding the size and shape of the specimens. For example, TBS is also known as microtensile bond strength test (␮TBS) when the bonded area is below 1 mm2 . A similar differentiation exists for the SBS test versus the microshear bond strength test (␮SBS). On the other hand, the TBS test can be performed using different processes to prepare the samples, which result in different shapes of the final tested specimens. The ‘stick’, ‘dumbbell’, and ‘hourglass’ shapes are the most popular in the literature [2]. Due to the different procedures used in the in vitro bond strength tests, results obtained in different studies can differ largely and the same materials are ranked differently by ␮TBS and ␮SBS [3]. Being able to reproduce the results obtained in a laboratory setting is key in order to determine objectively the properties of the materials and their behavior after the bonding process. Without an agreement among the test results, trying to establish a relationship between the laboratory data and the clinical outcomes becomes meaningless. In 1994, the first ISO document describing a standardized process to test adhesion to dental tissues was released as ISO/TR 11405 – Dental Materials – Guidance on testing of adhesion to tooth structure. This document was updated in 2003 as the ISO/TS 11405 – Dental Materials – Testing of adhesion to tooth structure. Despite the details described in the ISO document regarding the sample preparation, some parameters such as the bonding area are not clearly defined. The large variability of the bond strength test results has been pointed out in several reviews and some authors even question that any clinically relevant data can be obtained from such tests [4–9]. The lack of reproducibility of some methods and the high cost and complexity level of others seem to prevent any practical solution from arising [2]. In an attempt to discern the mechanical factors contributing to the scattering of the results in the in vitro setting, finite element methods (FEMs) have been used in the dental field since the 1970s [10–17]. These models allow estimation of the stress distribution on the bonded surface under an applied load. The present paper focuses on the evaluation of stress concentration and generalized stress intensity factors that have been overlooked in the past and should be considered to understand better the results of the TBS and ␮TBS tests. Models of bonded joints present several challenges. In the theoretical stress evaluation, singularities occur at the borders of the specimen due to the shape of the bonded surface and the different stiffness of the materials. A singularity is a

mathematical effect caused by the geometry of the specimen and the mismatch in elastic material properties. Discontinuities such as re-entrant corners, cracks, and/or the interface between two dissimilar materials can lead to singularities. When modeled using standard FEM formulations, the stress solutions become mesh dependent, with localized stresses diverging to infinity as the mesh is refined by using smaller finite elements. In fact, when the most frequently used linear elastic models are selected, the mentioned geometric discontinuities almost invariably produce singularities. Specifically, when modeling a bonded interface between dentin and a composite resin, a singularity will appear due to the inherent difficulty to maintain continuity of displacements and tractions across the interface. One can consider the effect of the singularity as a finite force applied on a very small area (the edge of the bonded surface in the case of a bonded joint), leading to increasingly high stress values because, mathematically, the calculated maximum stress at the point where the force is concentrated tends to infinity. In a series of papers, Reedy and Guess conducted experiments and developed novel approaches to characterize the mechanical tensile strength of metal–epoxy butt joints [18–22]. Although materials and sample sizes in their work are very different from those in the TBS and ␮TBS specimens, the mechanics is quite similar and many of the ideas on generalized fracture mechanics, size-dependent strength and the corresponding extensions of FEM originating from Reedy and Guess can be applied to the dental adhesive problem. Other relevant literature includes the theoretical work on stress singularities by Williams [23], Bogy [24], Carpinteri [25,26] and Dunn et al. [27–29]. In fact, here we will adopt several of these conceptual developments, but will use a different analysis approach. As an alternative to energy based approaches, such as finite element methods, one can develop integral equations to solve linear elastic boundary value problems, including those involving bonded interfaces between two materials. In simple terms, starting from a basic reciprocal theorem, the displacement at any point in an elastic body can be written in terms of the displacements and tractions everywhere on the boundary. This traditional boundary integral formulation uses knowledge of the elastostatic fundamental representation of the displacement at any point in an elastic body (of infinite extent) due to a point force applied at the origin, as summarized in the beginning of Appendix 1. Standard BEM formulations also suffer from mesh dependence when applied to problems with stress singularities. However, several different approaches exist to address problems with cracks in homogeneous specimens. The most popular uses quarter point and traction singular elements [30]. For mechanical problems involving bonded joints of dental materials, more general singularities must be considered, including those associated with bi-material free edges and cracks [31,32]. Here we show that weighted-traction boundary element models (BEM) provide accurate, mesh-independent results for bonded joints, and recommend these approaches be used in studies of bonded dental materials. The presence of singularities does not impact weighted traction BEM as much as conventional BEM or FEM, so a better representation of the

Please cite this article in press as: Campillo-Funollet M, et al. Size-dependent strength of dental adhesive systems. Dent Mater (2014), http://dx.doi.org/10.1016/j.dental.2014.03.010

ARTICLE IN PRESS

DENTAL-2355; No. of Pages 13

d e n t a l m a t e r i a l s x x x ( 2 0 1 4 ) xxx.e1–xxx.e13

Fig. 1 – Basic bonded joint model (R = 2 mm, H = 4.0 mm, r = 0.60 mm, h = 0.025 mm; dumbbell specimen adapted from Phrukkanon et al. [39]).

characteristics of the bonded interface can be obtained by treating the mathematical complexities of infinite stresses analytically, rather than numerically. The weighted traction BEM is described in further detail in Appendix 1. A more complete treatment of singularities, material failure characteristics and size-dependent strength is provided in Appendix 2, where the limitations of all linear elastic theories also are discussed, along with further details on the bi-material interface work by Reedy and Guess [18–22]. In the present work, the weighted traction boundary element formulation developed by Dargush and Hadjesfandiari is adopted in order to compute the generalized stress intensity factors Kg [31,32]. The main goal of the present work is to evaluate the influence of singularities in TBS specimens and their effects on the outcomes from standard FEM and weighted traction BEM for predicting dental adhesive system failures. We should emphasize that finite element methods also can be extended to address such singular problems [33–37,27]. However, in the present work, only standard FEM approaches are employed. This is done to illustrate the mesh dependence that one should expect when modeling problems with stress singularities using standard FEM, an approach that has dominated the dental adhesive literature. As we will show, this mesh sensitivity must be considered carefully, if any meaningful conclusions are to be reached from standard FEM analyses of the dental adhesive problem.

2.

Materials and methods

Fig. 1 shows a representation of the basic bonded dental joint considered here, along with the elastic modulus and Poisson’s ratio values used for each of the materials involved. Additional models were also created to study the behavior of the stresses and the singularities in different situations as described below. As a first example, the effects of the geometry on the stresses imposed by a tensile load on hourglass and dumbbell specimens were analyzed using FEM. Two FEM models were created using typical TBS specimen shapes and dimensions, but eliminating the adhesive layer and replacing the

xxx.e3

dentin–adhesive interface by a symmetry plane. The elastic modulus and Poisson’s ratio of dentin, as defined in Fig. 1, were used for these models. The geometries and meshes for these two models are shown in Fig. 2. Although the test specimens are axisymmetric, we chose to use fully three dimensional models, primarily for display purposes. Both models were created in ANSYS Meshing, v14 using hexahedral elements. The elements chosen were twenty node isoparametric elements with three degrees of freedom at each node and the finite element equations were developed assuming linear elastic material behavior. A uniaxial tensile load was applied to the left end of each specimen in the normal direction to produce a nominal stress of 20 MPa on the minimum cross-section. On the symmetry plane, at the right end in Fig. 2, a frictionless support boundary condition was imposed in order to constrain the axial displacement while allowing free transverse displacements. The finite element equations for the two models were solved in double precision using ANSYS Mechanical, v14. We should note that in Van Noort et al. [38], a similar investigation comparing hourglass and dumbbell specimens has been conducted. Next, the problem of a bonded interface was studied using an FEM cylindrical model, such as the one shown in Fig. 3.In constructing the corresponding finite element mesh, the smallest elements were located inside the adhesive layer at the specimen surface, as depicted in Fig. 3, where the geometry and mesh are shown. The mesh spacing for the model in the figure was set at 0.01 mm in both the axial and radial directions. A tensile load equivalent to 20 MPa for the nominal interface normal stress was applied to the specimen and the finite element equations were solved in double precision using ANSYS Mechanical, v14. To characterize the impact of the singularities on the stress distribution, the mesh was refined in four steps. In the first case, the axial and radial mesh spacing of the elements surrounding the singularity was set at 0.02 mm. In the three subsequent cases, the mesh size in this critical region wasreduced to 0.01, 0.005, and 0.002 mm, respectively. For each case, the same loading conditions were applied and the FEM equations were solved. The problem geometry, material properties and loading conditions for the BEM analyses were as defined in Fig. 1. In order to characterize the system more completely using BEM, two different approaches were adopted for the weighted traction boundary element analysis; the first with fully bonded interfaces and the second with a small crack on one of the interfaces at the free edge. In both cases, an axisymmetric formulation was used to model the problem allowed by the purely cylindrical geometries of the dentin and resin components, the bonding adhesive layer and uniaxial loading. The surfaces of the dentin and resin domains were modeled with 88 quadratic boundary elements, whereas 72 quadratic elements were used to represent the response of the adhesive. This boundary element mesh was graded such that very small elements of approximate length 0.0025 mm were used near the intersection of the interfaces and the free edge, where the stress singularities occur. Weighted traction boundary element solutions were obtained for 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, and 1.1 mm specimen radii and adhesive thicknesses of 0.025, 0.050, and 0.100 mm

Please cite this article in press as: Campillo-Funollet M, et al. Size-dependent strength of dental adhesive systems. Dent Mater (2014), http://dx.doi.org/10.1016/j.dental.2014.03.010

DENTAL-2355; No. of Pages 13

ARTICLE IN PRESS

xxx.e4

d e n t a l m a t e r i a l s x x x ( 2 0 1 4 ) xxx.e1–xxx.e13

Fig. 2 – Geometry and mesh representations for the FEM models of dentin.

Fig. 3 – Geometry and FE mesh representation of the cylindrical bonded specimen. A longitudinal view (left) and an axial view (right) are shown (arrow indicates location of adhesive layer).

in order to investigate the size dependence of the response. The results were compared with previously published experimental data [39]. With current understanding, one cannot predict the critical value of weighted traction from first principles, although promising damage mechanics based approaches do exist for certain material systems [40]. Instead, here the critical values of the weighted traction are estimated from physical testing. From a theoretical standpoint, this is equivalent to selecting a nominal tensile strength at any one particular specimen size. Based upon the failure data from Phrukkanon et al. [39], the nominal tensile strength for a specimen with 0.6 mm radius is estimated to be 19 MPa, which incidentally is quite close to the nominal value of 20 MPa assumed for all standard FEM analyses performed here. By using the estimated strength at this one particular radius, the size dependency of the tensile strength predicted by the weighted traction BEM models was computed and then compared with all of the experimental strength values presented in Phrukkanon et al. [39]. The size dependence of the tensile strength was also investigated by introducing a small crack near the outer surface of one of the cylinders on the interface. The same cases described above were analyzed for a crack location on the dentin–adhesive interface and, separately, on the resin–adhesive interface. A baseline crack length of 5 ␮m was selected to study the scaling behavior of this overall dental adhesive system and the sensitivity of the results to that particular crack length also was investigated.

To summarize our approach, we first apply standard FEM to monolithic specimens and study the resulting stress concentrations under uniaxial tensile loading. Next, we use standard FEM for adhesively bonded specimens and investigate mesh dependence due to the presence of stress singularities on the free edge of the interfaces. Then, we apply a weighted traction BEM to these interface problems, assuming perfect bonding, in an effort to obtain mesh independent solutions that could be useful in determining size-dependent strength of the specimens. Finally, we introduce small interface edge cracks and again apply the weighted traction BEM to study the influence of specimen size on strength. For all of the bonded dental specimens, a specific geometric configuration is considered, which permits comparison with previously published experimental data.

3.

Results

The axial stress contours on a longitudinal cutting plane and the right symmetry plane of the models of the monolithic dentin specimens are shown in Fig. 4. These plots show that the axial stress on the right symmetry plane is not uniform for the hourglass model while it is uniform for the dumbbell model, achieving the nominal stress of 20 MPa. Fig. 5 shows the axial stress for both specimens on the right symmetry plane along a radial line extending from the model center to its circumference. Clearly,

Please cite this article in press as: Campillo-Funollet M, et al. Size-dependent strength of dental adhesive systems. Dent Mater (2014), http://dx.doi.org/10.1016/j.dental.2014.03.010

DENTAL-2355; No. of Pages 13

ARTICLE IN PRESS d e n t a l m a t e r i a l s x x x ( 2 0 1 4 ) xxx.e1–xxx.e13

xxx.e5

Fig. 4 – Axial stress contours on a longitudinal plane (above) and from a view of the imposed symmetry plane (below) for the models using a single material with elastic properties of dentin. Results for both hourglass (left) and dumbbell (right) shapes are presented in the figure.

the dumbbell-shaped specimen provides a uniform axial stress distribution on the entire symmetry plane, while the hourglass specimen does not. Consequently, dumbbellshaped specimens are preferred for meaningful dental

testing protocols. For these two problems, standard FEM approaches provide accurate stress distributions, which converge with mesh refinement because no singularities are present.

Fig. 5 – Axial stress on a radial line at the symmetry plane extending from the center of the specimen to the edge. Please cite this article in press as: Campillo-Funollet M, et al. Size-dependent strength of dental adhesive systems. Dent Mater (2014), http://dx.doi.org/10.1016/j.dental.2014.03.010

DENTAL-2355; No. of Pages 13

ARTICLE IN PRESS

xxx.e6

d e n t a l m a t e r i a l s x x x ( 2 0 1 4 ) xxx.e1–xxx.e13

Fig. 6 – Axial stress distribution on an axial cutting plane through the bonded specimen (arrows indicate material interfaces).

Next, we consider the bonded dentin–adhesive–resin specimen of Fig. 1. The axial stress distribution on an axial cutting plane through a cylindrical specimen diameter is illustrated in the contour plot from standard ANSYS FEM analysis in Fig. 6. This figure shows that the presence of the singularities produces a completely different stress distribution at the adhesive interfaces than the homogeneous dumbbell-shaped model. We also observe that the surface profile of the specimen is distorted by a Poisson effect due to the large axial strains in the more flexible adhesive material. A highly zoomed view of the FEM axial stress contours on the outer surface of the cylindrical specimen near the adhesive layer is provided in Fig. 7. The surface mesh is superimposed on the contour plot to show the location of the singularities. The right singularity, shown in red, occurs at the resin composite–adhesive interface and the left singularity, shown in yellow, occurs at the dentin–adhesive interface. Also, the maximum axial stress from the FEM analysis for this baseline mesh is 30.63 MPa, which occurs at the resin composite interface. Fig. 8 shows a highly magnified view of the axial stress distribution on a cross-sectional cutting plane through the resin

Fig. 8 – Axial stress distribution on a cross-sectional cutting plane through the resin composite singularity.

composite singularity. This plot provides a view of the high radial gradient of the axial stress near the free edge interface singularity. The computed values of the maximum axial stress at the resin composite singularity using different surface mesh sizes are listed in Table 1. The second column in the table lists the estimated maximum axial stress at the singularity for the given finite element mesh dimension. These elements have a square cross-section in the axial-radial plane. Consequently, their characteristic size is simply the square root of their cross-sectional area. Notice that the maximum stresses are diverging to infinity with additional mesh refinement. The maximum axial stress is shown as a function of mesh size in the log–log plot in Fig. 9.The linear nature of the plot indicates that the stress data may be fitted by a power-law function of the form: =

K ˛ (L)

(1)

where L is the mesh size, K is a constant coefficient, ˛ is a constant exponent and  is the axial stress. Applying linear regression analysis in the log–log domain yields the following values: ˛ = 0.125

K = 17.2 MPa mm˛

(2a,b)

Table 1 – Calculated values of maximum axial stress for different sizes of the elements surrounding the singularity. Fig. 7 – Highly zoomed view of the axial stresses on the outer surface near the dentin–adhesive–composite interface of the bonded cylindrical specimen (arrows indicate material interfaces).

Surface mesh sizes (mm)

Maximum axial stress (MPa)

0.002 0.005 0.010 0.020

37.271 33.693 30.63 27.986

Please cite this article in press as: Campillo-Funollet M, et al. Size-dependent strength of dental adhesive systems. Dent Mater (2014), http://dx.doi.org/10.1016/j.dental.2014.03.010

DENTAL-2355; No. of Pages 13

ARTICLE IN PRESS d e n t a l m a t e r i a l s x x x ( 2 0 1 4 ) xxx.e1–xxx.e13

xxx.e7

Fig. 9 – Maximum axial normal stress versus mesh characteristic size of the elements around the singularity for the cylindrical model of a bonded joint.

and the equation: =

17.2 (L)

0.125

MPa

(3)

with L measured in mm. As the FEM mesh is refined, Eq. (3) suggests correctly that the stresses for this linear elastic analysis should increase continually toward an infinite value. Appendices 1 and 2 provide detail on the meaning and analysis of stress singularities. The powers of these elastic stress singularities for the fully bonded interfaces were calculated to be ˇ = 1 −  = 0.125 for the dentin–adhesive interface and ˇ = 1 −  = 0.161 for the interface between the resin and the adhesive, based on the analytical formulation by Bogy [24]. Notice that the maximum stresses obtained from a standard FEM analysis scale with characteristic mesh size to a similar power. Now we turn to the results obtained for the adhesively bonded dental specimen using the weighted traction BEM analysis, which is able to provide mesh independent solutions even when stress singularities occur. Fig. 10 presents  the scaling behavior of the weighted axial tractions tz associated with both the dentin–adhesive (D–A) and resin–adhesive (R–A) interfaces under the assumption of perfect bonding with a nominal interface tensile stress nom = 1 MPa. The lowest val ues of tz are associated with the strongest systems, which clearly occur for the smallest radius and thinnest adhesive layer. As the radius becomes larger or the layer thickness is increased, the weighted traction is magnified and, consequently, the allowable applied load will be reduced. Notice from Fig. 10 that both specimen radius and layer thickness have a strong influence on weighted tractions and thus on the predicted values of overall strength.

Fig. 10 – Top: weighted axial traction at the dentin–adhesive interface (D–A). Bottom: weighted axial traction at the resin–adhesive interface (R–A).

Introducing a crack at the edge of the dentin–adhesive and resin–adhesive interfaces yields complex powers of the stress singularities 1 −  = 0.5 + 0.039i and 1 −  = 0.5 + 0.042i, respectively, at the tip of the crack. This greatly complicates the analysis, but a more general form of the weighted traction BEM approach can be used [32]. From the weighted traction BEM analyses with adhesive thickness h = 0.025 mm and crack length ˛ = 0.005 mm, the scaling behavior of the magnitude of the complex stress intensity factor |K| was found to be as shown in Fig. 11, again with a nominal interface tensile stress nom = 1 MPa. Notice the significant dependence of this measure on the size of the overall specimen, as defined by the radius r. Therefore, we again would predict size-dependent strength. The actual location of the critical failure surface would depend upon |K| and the relative fracture toughness of the dentin–adhesive and resin–adhesive interface systems, which at present we do not know. The results in Fig. 11 are based on the assumption of the existence of an edge crack of length a = 0.005 mm. Different values of a will produce different levels of |K|. However,

Please cite this article in press as: Campillo-Funollet M, et al. Size-dependent strength of dental adhesive systems. Dent Mater (2014), http://dx.doi.org/10.1016/j.dental.2014.03.010

DENTAL-2355; No. of Pages 13

ARTICLE IN PRESS

xxx.e8

d e n t a l m a t e r i a l s x x x ( 2 0 1 4 ) xxx.e1–xxx.e13

Fig. 11 – Scaling behavior of the stress intensity factor after introducing a crack on one of the interfaces.

the scaling behavior will remain consistent, as long as the crack length is small compared to the other relevant geometric problem dimensions r and h. In particular, weighted traction BEM solutions with a crack length a = 0.0005 mm produced nearly identical scaling to that seen in Fig. 11, while small deviations in the scaling behavior occurred for a crack length a = 0.010 mm for cases with h = 0.025 mm. Thus, we find that the scaling relations become more complicated, as the crack length approaches the adhesive thickness. Based upon the results from Figs. 10 and 11, the effect of specimen size on the tensile strength is calculated and shown in Fig. 12, where the scaled BEM results obtained with and without the edge interface crack are presented together. Here all curves are scaled to provide a nominal strength ¯ f = 19 MPa for the case with r = 0.6 mm and h = 0.025 mm. For example, on the dentin–adhesive interface with a nominal stress nom =  1 MPa, the weighted axial traction t¯ z = 0.394 MPa mm0.125 . As a result, the critical weighted traction of the dentin–adhesive interface is estimated as 

tz,cr =

¯ f nom

 t¯ z = 7.47 MPa mm0.125

(4)

Then, for all other specimens, the nominal strength, based upon failure of the dentin–adhesive interface, is determined as f = ¯ f

 t¯ z

 tz



= nom

tz,cr 

tz

(5)

Thus, for the case with r = 0.9 mm and h = 0.025 mm, one  obtains tz = 0.446 MPa mm0.125 from the weighted traction BEM and the nominal strength is estimated to be  f = 16.8 MPa, as plotted in Fig. 12.  A similar calculation yields tz,cr = 7.15 MPa mm0.161 for the resin–adhesive interface, which allows estimation of the corresponding nominal strength for all other specimen

Fig. 12 – Strength of the bonded joint for the different specimen radii studied with and without introducing cracks on the interface. Also represented in the chart are the averaged experimental values obtained for the same radii by Phrukkanon et al. [39].

geometries. Again considering the case with r = 0.9 mm and h = 0.025 mm, the weighted traction BEM provides  tz = 0.427 MPa mm0.161 and, consequently, the nominal strength for resin–adhesive failure is estimated to be  f = 16.8 MPa, which is nearly identical to the strength of the dentin–adhesive interface, causing the two failure curves to be almost indistinguishable in Fig. 12. However, we must recognize that this is due partially to the assumption that the critical nominal strength is ¯ f = 19 MPa for both interfaces at the reference specimen configuration of r = 0.6 mm and h = 0.025 mm. Consequently, the proximity of the two curves is then the result of similar scaling behavior. In reality, the strength of one of the interfaces may be significantly larger, in which case the other interface will control failure. Fig. 12 suggests that whether failure occurs initially on the dentin–adhesive or resin–adhesive interface, the resulting scaling behavior will be very similar. Notice, in addition, from Fig. 12 that BEM analyses with and without small interface cracks provide nearly this same scaling behavior. The procedure to determine failure strength for the cracked specimens is similar to that discussed above for fully bonded specimens, except that stress intensity magni tude |K| replaces weighted traction tz . In addition, Fig. 12 presents the experimental tensile strength values reported by Phrukkanon et al. [39], which as expected display some variability. However, the generalized fracture BEM predictions capture well the overall scaling behavior of these physical tests. Moreover, all of the failure modes, identified by Phrukkanon et al. [39] for cylindrical specimens, initiate near the outer edge on the bonded interface, exactly as predicted by the proposed generalized fracture model.

Please cite this article in press as: Campillo-Funollet M, et al. Size-dependent strength of dental adhesive systems. Dent Mater (2014), http://dx.doi.org/10.1016/j.dental.2014.03.010

DENTAL-2355; No. of Pages 13

ARTICLE IN PRESS xxx.e9

d e n t a l m a t e r i a l s x x x ( 2 0 1 4 ) xxx.e1–xxx.e13

4.

Discussion

The validity of tensile bond strength testing has been challenged due to inconsistent test outcomes originating from heterogeneity in tooth specimens and the stress distribution at the bonded interface, hindering an accurate assessment of the interfacial bond strength. The microtensile bond strength test was introduced by Sano et al. to overcome these limitations with use of small bonding areas and improved homogeneity in specimen preparations [41]. The higher bond strength measured with the microtensile test compared to its macroscopic counterpart was attributed initially to the lesser probability to find a flaw of critical size in the smaller specimens. While this tendency is certainly true, the size dependence of the bond strength can be understood more completely through the concepts of generalized fracture mechanics, as will be discussed below. Consequently, size dependence is linked closely to the differing mechanical properties of the materials involved in the bonded interface and the associated stress singularities at the outer edge. In an attempt to discern the mechanical factors contributing to the scattering of the results in the in vitro setting, finite element models (FEMs) have been used by several authors [42–45]. The early models utilized 2D elements and fairly coarse meshes. Over the intervening years the models have become more sophisticated using new 3D element models and much finer meshes. These models allow for the possibility of non-uniform stress distribution on the bonded surface under the load applied during the tests. However, the one major shortcoming of all current finite element analyses is that under tensile loading these models fail to capture in a meaningful way the stress singularities, which occur at the substrata–adhesive interfaces in the mathematical formulation of the problem. Consequently, the maximum normal stress at the interface computed using standard finite element methods is not unique, but rather depends on the size of the mesh elements at the interface. Although standard finite element models can predict an approximate stress distribution in the test specimen, these models are unable to predict from first principles the stress condition for bond failure. Several different failure criteria

have been established from experimental data. However, it has not been demonstrated that these are all equivalent, nor physically meaningful, as discussed in Appendix 2. Thus, some variability in the reported tensile bond strength test results may also be due to the interpretations associated with the use of different failure criteria. The core of the typical solid mechanics equation solvers (finite difference, finite element, boundary element, etc.) used here and by others for assessing bond strength have a number of restrictions because of implicit assumptions embodied in the governing differential equations. The main assumption is linear elastic deformation in a homogeneous material, characterized by Young’s modulus, E, and Poisson’s ratio,  in the isotropic case. Use of different values for these properties also has contributed to inconsistent bond strengths reported in the literature. For more meaningful comparison, the values of E and ␯ used here were those of Phrukkanon et al. for: dentin, adhesive and resin; namely: 15 GPa and 0.31, 4 GPa and 0.35 and 20 GPa and 0.25, respectively [39]. These values are shown in Table 2, along with those from other investigators for comparative reference. The fracture toughness is also included, where findable. In the present study, typical cases of specimens used in bond strength testing were modeled using standard FEM and weighted traction BEM approaches in order to shed light on the size-dependent effect that has been reported previously by several authors [39,41,59]. In the literature, this behavior has been attributed to an effect of the flaw distribution over the entire cross-section. However, an accurate modeling of the bonded joint can explain this effect in terms of the presence of singularities caused by the different properties of the materials and the geometry of the interface. Furthermore, such an approach, can predict scaling behavior based on the dimensions of the specimens. Considering the effect of the singularities on the stress distribution, it becomes apparent that the way to solve the problem is to shift the focus from stress to stress intensity factor for problems modeled with cracks or to generalized stress intensity factor for models without cracks. In order to place these arguments on a quantitative basis, there is a need to use specialized computational methods, such as the weighted

Table 2 – Materials properties values reported in the literature. Kind of material

Specific material

Tooth Dentin used here Tooth Dentin other

Tooth Dentin used here Tooth Dentin other

Adhesive used here Adhesive other Adhesive other

Adhesive used here Opti Bond XTR (Kerr Corp) Clearfil SE (Kuraray)

Adhesive other Resin used here Resin other

AdheSE (Ivoclar-Vivadent) Resin used here Z100 (3M ESPE)

Young’s modulus, E (GPa) 15 [39] 15 [46], 7.6–14 [47], 20 ± 2 [48], 16.5–18.6 [49] 4 [39] 3.1 [46] 5.87 [46], 5.62–7.61 [52], 15.81–21.71 [53], 12.387 and 27.384 [54] 5.3 [56] 20 [39] 23 [46], 21 [57]

Poisson’s ratio,  0.31 [39] 0.3 [46]

Fracture toughness, Kc (MN m−3/2 ) 1.0–4.0 [50]

0.35 [39] 0.25 [46] 0.3 [46], 0.22–0.32 [53]

0.47 [50], 0.63–0.82 [51] 0.43 ± 0.14 [50], 1.85 [55]

0.25 [39] 0.3 [46], 0.25 [57], 0.302 [58]

1.50 [55]

Please cite this article in press as: Campillo-Funollet M, et al. Size-dependent strength of dental adhesive systems. Dent Mater (2014), http://dx.doi.org/10.1016/j.dental.2014.03.010

DENTAL-2355; No. of Pages 13

ARTICLE IN PRESS

xxx.e10

d e n t a l m a t e r i a l s x x x ( 2 0 1 4 ) xxx.e1–xxx.e13

traction BEM, to solve the underlying non-smooth mechanics problems for the relevant stress intensity or generalized stress intensity factors. Remarkably, for the dental materials model presented in this paper, both approaches (with and without a small crack on the interface), as well as both interfaces, produce nearly identical strength scaling and predict reasonably well the experimental results obtained by Phrukkanon et al. [39]. We should note that alternatively one could develop a Weibull scaling law to represent the failure data in Fig. 12, with either the cross-sectional area or the circumference as the basis. However, unlike the generalized fracture mechanics approach, the Weibull law would not provide a physical explanation for the scaling, nor would it provide any information concerning the potential failure modes. Clearly, from the results presented in Fig. 12, the theory of stress singularities provides a very good prediction of the scaling behavior for these multi-material dental systems. However, it should be emphasized that the present theory cannot predict the precise level of the failure line, which today must be based on interface testing. As a final point, it should be noted that the proposed size dependent stress singularity approach predicts that failure is not randomly distributed on the interface due to inherent flaws, but rather initiates at the outer free edge of the cylinders on the interface, where the stress singularities appear. This is precisely the dominant failure mode and location indicated in Phrukkanon et al. [39], thus lending further support to the hypothesis that failure is controlled by the free edge interface singularities. Whether these interfaces are treated as perfectly bonded or with small hypothetical cracks makes little difference. Both models appear to be effective idealizations of the physical problem, which can be useful in evaluating and designing dental adhesive systems.

in their work on metal–adhesive butt joints and many others [27–29,33–35,37]. The key is to recognize that these mathematical singularities exist in the dental adhesive interface problem and then to use an appropriate formulation, whether it is BEM or FEM based, to provide bounded mesh-independent solutions. In any case, we advocate that failure be defined in terms of stress intensity factors that can be related directly to fracture toughness values for brittle materials utilized in dental restorations, allowing resolution of noted anomalies while adding new perspective. Most importantly, this permits a quantitative assessment and prediction of the observed sizedependency in failure strength. Furthermore, armed with this new understanding, one can begin to design new dental adhesive systems with improved strength by optimizing materials and geometries. Such an effort is now underway by the present interdisciplinary dental research group at the University at Buffalo, where a series of physical tests also are being conducted to gain additional insight into the behavior of these adhesive systems.

Acknowledgements The authors would like to thank Dr. Robert Baier (SUNY at Buffalo) for many useful discussions.

Appendix 1. Traditional and weighted traction boundary element method Symbolically, the boundary integral equation is written as c( )u( ) + T( , x) u( x) dS = S

5.

Conclusions

Prediction of the strength of dental adhesive systems represents a challenging problem. This is due primarily to the complications associated with the use of brittle materials and the presence of mathematical stress singularities at the freeedge of the bonded interfaces within a linear elastic model of these systems. Application of weighted traction boundary element method (BEM) analysis to bonded dumbbellor cylinder-shaped specimens is found here to be superior to application of the standard finite element method (FEM), better matching published experimental data because of its ability to assess accurately the role of stress singularities. The key new feature in this BEM approach is the utilization of stress-intensity-related weighting factors previously absent in all FEM analyses of dental adhesive joints. As a consequence, the weighted traction BEM approach provides mesh independent results and shifts the focus from usual stressbased failure criteria to stress intensity factor criteria that affords a better understanding of bonding problems, whether one models small hypothetical outer edge interface cracks or not. Although not attempted here, one could introduce extensions of the standard FEM approach to address stress singularities, as done previously by Reedy and Guess [18–22]

U( , x) t( x) dS S

(A.1)

where u and t denote the displacement and traction, respectively, for the problem of interest, whereas U and T are the tensorial kernels that represent the displacement and traction fields, respectively, from the fundamental solution. Meanwhile, the tensor c depends only upon the local geometry at the point  and reduces to the identity tensor for a point inside the body. The integrals in Eq. (A.1) are over the entire bounding surface S. If the point  is on the boundary, then the kernels U and T contain singularities when  and x coincide and the integral on the left-hand side requires treatment as a Cauchy principal value integral, as indicated by the bar through the integral sign in Eq. (A.1). By discretizing the surface of the elastic body into elements, similar to those used in the finite element method, and then writing Eq. (A.1) at each boundary node, one can develop a boundary element formulation for the problem. All of this has become standard within the boundary element community and a wide array of problems can be solved using this approach [60,61]. Advantages of a boundary element approach include a reduction in dimensionality of the problem due to the surface-only mesh requirement and the inclusion of tractions as a primary variable, which allows very accurate determination of stresses. However, for non-smooth problems, where there are cracks, notches, or bi-material interfaces, the tractions t in Eq.

Please cite this article in press as: Campillo-Funollet M, et al. Size-dependent strength of dental adhesive systems. Dent Mater (2014), http://dx.doi.org/10.1016/j.dental.2014.03.010

ARTICLE IN PRESS

DENTAL-2355; No. of Pages 13

d e n t a l m a t e r i a l s x x x ( 2 0 1 4 ) xxx.e1–xxx.e13

(A.1) become singular and require special treatment in order to obtain meaningful mesh-independent solutions. From a local analysis of a non-smooth linear elasticity problem, one finds that the stress components tend toward infinity at the critical location and that the variation takes the following form lim ␴(, )(2 )

→0

1−

= Kg ␶( )

(A.2)

where  and are cylindrical coordinates with origin at the singular point, ␴ is the stress tensor, Kg represents the generalized stress intensity factor, ␶ is a non-dimensional tensor that captures the angular variation of the stress field and 1 −  is the power of the stress singularity. For a crack with unrestrained edges, Re(1 − ) = 1/2. With this approach, Eq. (A.1) is rewritten as

c( )u( ) + T( , x) S

u

( x) u ( x) dS = U( , x)

t

( x) t ( x) dS

S

(A.3) where u and t are weighting functions, while u and t represent the weighted displacements and tractions, respectively. For bi-material cracks, t at the tip is directly proportional to the complex stress intensity factor Kg that controls failure. On the other hand, for bi-material free edge singularities, the value of t , and its proportional counterpart Kg , also may be used as a failure measure, although the full theoretical explanation for this remains an open issue. Finally, note that the power of the singularity is almost identical when calculated using mesh convergence characteristics in FE (˛ = 0.125, as minus the slope of the axial stress convergence data in Fig. 9) or analytically based on the formulation of Bogy [24] in the weighted traction BE models (ˇ = 1 −  = 0.125 for the dentin–adhesive interface and ˇ = 1 −  = 0.161 for the interface between resin and adhesive). It should be noted that both FE and BE are mathematically capable of solving for the singularities. However, using the weighted traction BE approach from [31,32] is more convenient and accurate, because of the boundary-only nature of BEM.

Appendix 2. Strength theories, stress singularities and size-dependence In order to understand the tensile strength variations and the parameters that have been used in this paper to quantify the response of the specimens it is important to describe certain general characteristics of the materials. In particular, we note that the mechanical behavior of a brittle material is quite different from that of a ductile material. Thus, it is important to give careful consideration before carrying ideas established for ductile materials over to brittle counterparts. In compression, the stress–strain response of a brittle material resembles that for a ductile material, although the underlying mechanism of energy dissipation in the nonlinear regime may be different. However, the more important difference is on the tension side, where linear elastic response of a brittle material

xxx.e11

terminates abruptly at a failure load designated as Pf . One may attempt to construct a stress–strain response curve by dividing force by cross-sectional area and elongation by length. As a result, a well-defined yield strength  y can be identified in compression, as in the case of the ductile material. However, the tensile failure strength  f remains ambiguous and a large standard deviation is typically observed when attempting to determine it experimentally. Tensile failure in brittle materials is very sensitive to the presence of cracks, notches and other flaws and, in particular, the calculated stress at failure is size dependent. Larger specimens tend to fail at lower levels of  f . More generally, one can argue that the concept of a tensile failure stress in a brittle material is meaningless. Instead, one needs to consider ideas from generalized fracture mechanics, which deals with non-smooth problems, such as cracks, notches, and bi-material interfaces (or bonded joints). The linear elasticity solutions of these problems all involve singularities (i.e., infinities) in the stress field at the tip of the crack or notch, or at the free edge on a bi-material interface. Consequently, failure needs to be defined in terms of (generalized) stress intensity factors, which in turn must be compared to (generalized) fracture toughness of the brittle material. Here, some discussion is in order concerning the presence of stress singularities. Does a physical crack with co-planar surfaces and a zero radius of curvature at the tip exist? In reality, does an infinite stress occur at such locations? The answer in both cases is no. These are mathematical idealizations. However, the important point is that these are useful mathematical idealizations. Furthermore, by bringing these fundamental concepts for the mechanical response of brittle materials into the field of dentistry, it would appear that many of the anomalies that have been reported in the literature can be resolved or, at the very least, viewed from a new perspective. As stated earlier, stress singularities occur at the adhesivesubstrata interfaces of a test specimen when it is subjected to a tensile load. In the case of an axisymmetric specimen, such as the hourglass or dumbbell-shaped specimens, the singularities occur at the intersection of specimen surface and the adhesive interfaces. Consequently, for the tensile bond strength test, the two singularities form as circular rings on the specimen surface, one at the dentin–adhesive interface and the other at the resin composite–adhesive interface. The power of these singularities depends on the material properties, the elastic moduli and Poisson’s ratios of dentin, adhesive, and resin composite. From Fig. 12, notice that the scaling behavior for the two interfaces is nearly identical and, furthermore, is reasonably well aligned with the experimental results, although the latter show slightly greater size dependence. Recall that, based upon the results presented in Fig. 10, the size dependent character of the strength also is influenced by the adhesive thickness and this was not controlled, nor even measured, in the Phrukkanon et al. specimens [39]. The presence of the factor 1− in Eq. (A.2) indicates that the stresses exhibit power law behavior with distance  from the singular point. The appearance of power laws is a feature in many critical phenomena. Here the stress-distance power law triggers size dependency in the tensile strength of brittle materials. In addition, the scalar Kg , which may be complex, is

Please cite this article in press as: Campillo-Funollet M, et al. Size-dependent strength of dental adhesive systems. Dent Mater (2014), http://dx.doi.org/10.1016/j.dental.2014.03.010

DENTAL-2355; No. of Pages 13

ARTICLE IN PRESS

xxx.e12

d e n t a l m a t e r i a l s x x x ( 2 0 1 4 ) xxx.e1–xxx.e13

anticipated to be a useful measure for estimating the tensile strength of components. Beyond these theoretical arguments, what foundation is there for such claims? In one comprehensive study, Reedy and Guess conducted experiments and developed novel approaches to characterize the mechanical tensile strength of metal–epoxy butt joints [18–22]. Their problem, which involved either steel or aluminum cylinders, is in essence a bonded joint in a cylindrical specimen similar to the one illustrated in Fig. 1. Based upon the elastic properties of the metals and epoxy, along with the local eigen solutions for fully bonded elastic bi-material 90◦ wedges derived by Bogy [24], one finds that the power of the stress singularity for aluminum-epoxy is ˇ = 1 −  = 0.27, while that for the steel-epoxy interfaces is ˇ = 1 −  = 0.30. Thus, stress singularities will be present at the free-edge of the metal–epoxy interface in an elastic analysis of the problem and, according to the theory of generalized fracture mechanics, size dependent response can be expected. This size dependence was confirmed in the Reedy and Guess experimental program, which showed that strength increased with thinner epoxy layers [18–22]. Of course, this resultis consistent with the everyday experience that thin layers of glue are strongest. Reedy and Guess also proposed the use of ideas from fracture mechanics and bi-material interface singularities to quantify the size dependency of tensile strength of the butt joints with bond thickness 2h [18–22]. Their finite element studies with or without free-edge cracks predicted that 1− strength should scale approximately as (2h) , which is quite close to the experimental trend. Dargush and Hadjesfandiari examined this problem using the weighted traction boundary element method with a fully bonded interface [31], whereas in a more recent paper a small crack with a = 0.01 mm was introduced on the interface at the free edge [32]. In both −1 cases, the (2h) scaling behavior of tensile strength versus bond thickness again was confirmed. Complete details for the models used in the boundary element method (BEM) analyses for the metal–epoxy butt joint can be found in those two references. Here we use this same conceptual approach to investigate the size-dependencies of the radius r and thickness h for the model dental adhesive system displayed in Fig. 1.

references

[1] Braga RR, Meira JBC, Boaro LCC, Xavier TA. Adhesion to tooth structure: a critical review of “macro” test methods. Dent Mater 2010;26:E38–49. [2] Armstrong S, Geraldeli S, Maia R, Raposo LHA, Soares CJ, Yamagawa J. Adhesion to tooth structure: a critical review of “micro” bond strength test methods. Dent Mater 2010;26:E50–62. [3] Bonifacio CC, Shimaoka AM, de Andrade AP, Raggio DP, van Amerongen WE, de Carvalho RCR. Micro-mechanical bond strength tests for the assessment of the adhesion of gic to dentine. Acta Odontol Scand 2012;70:555–63. [4] Oilo G. Bond strength testing – what does it mean. Int Dent J 1993;43:492–8. [5] Salz U, Bock T. Testing adhesion of direct restoratives to dental hard tissue – a review. J Adhes Dent 2010;12:343–71.

[6] Tagami J, Nikaido T, Nakajima M, Shimada Y. Relationship between bond strength tests and other in vitro phenomena. Dent Mater 2010;26:E94–9. [7] Roeder L, Pereira PNR, Yamamoto T, Ilie N, Armstrong S, Ferracane J. Spotlight on bond strength testing – unraveling the complexities. Dent Mater 2011;27:1197–203. [8] Heintze SD, Thunpithayakul C, Armstrong SR, Rousson V. Correlation between microtensile bond strength data and clinical outcome of Class V restorations. Dent Mater 2011;27:114–25. [9] Anusavice KJ. Standardizing failure, success, and survival decisions in clinical studies of ceramic and metal–ceramic fixed dental prostheses. Dent Mater 2012;28:102–11. [10] Farah JW, Craig RG, Sikarski DL. Photoelastic and finite-element stress analysis of a restored axisymmetric first molar. J Biomech 1973;6:511–4. [11] Anusavice KJ, Dehoff PH, Fairhurst CW. Comparative-evaluation of ceramic-metal bond tests using finite-element stress-analysis. J Dent Res 1980;59:608–13. [12] Borchers L, Reichart P. 3-Dimensional stress-distribution around a dental implant at different stages of interface development. J Dent Res 1983;62:155–9. [13] Rubin C, Krishnamurthy N, Capilouto E, Yi H. Stress-analysis of the human tooth using a 3-dimensional finite-element model. J Dent Res 1983;62:82–6. [14] Farah JW, Craig RG, Meroueh KA. Finite-element analysis of 3-unit and 4-unit bridges. J Oral Rehab 1989;16: 603–11. [15] Van Noort R, Cardew GE, Howard IC, Noroozi S. The effect of local interfacial geometry on the measurement of the tensile bond strength to dentin. J Dent Res 1991;70: 889–93. [16] Kelly JR, Tesk JA, Sorensen JA. Failure of all-ceramic fixed partial dentures in-vitro and in-vivo – analysis and modeling. J Dent Res 1995;74:1253–8. [17] Dellabona A, Van Noort R. Shear vs tensile bond strength of resin composite bonded to ceramic. J Dent Res 1995;74:1591–6. [18] Reedy Jr ED, Guess TR. Comparison of butt tensile-strength data with interface corner stress intensity factor prediction. Int J Solids Struct 1993;30:2929–36. [19] Reedy Jr ED, Guess TR. Butt-joint tensile-strength – interface corner stress intensity factor prediction. J Adhes Sci Technol 1995;9:237–51. [20] Reedy Jr ED, Guess TR. Interface corner stress states: plasticity effects. Int J Fract 1996;81:269–82. [21] Reedy Jr ED, Guess TR. Interface corner failure analysis of joint strength: effect of adherend stiffness. Int J Fract 1998;88:305–14. [22] Reedy Jr ED. Connection between interface corner and interfacial fracture analyses of an adhesively-bonded butt joint. Int J Solids Struct 2000;37:2429–42. [23] Williams ML. The stresses around a fault or crack in dissimilar media. Bull Seismol Soc Am 1959;49:199–204. [24] Bogy DB. Two edge-bonded elastic wedges of different materials and wedge angles under surface tractions. J Appl Mech 1971;38(2):377–86. [25] Carpinteri A. Stress-singularity and generalized fracture-toughness at the vertex of reentrant corners. Eng Fract Mech 1987;26:143–55. [26] Carpinteri A. An elastic–plastic crack bridging model for brittle-matrix fibrous composite beams under cyclic loading. Int J Solids Struct 2006;43:4917–36. [27] Dunn ML, Suwito W, Cunningham SJ. Fracture initiation at sharp notches: correlation using critical stress intensities. Int J Solids Struct 1997;34:3873–83. [28] Dunn ML, Cunningham SJ, Labossiere PEW. Initiation toughness of silicon/glass anodic bonds. Acta Mater 2000;48:735–44.

Please cite this article in press as: Campillo-Funollet M, et al. Size-dependent strength of dental adhesive systems. Dent Mater (2014), http://dx.doi.org/10.1016/j.dental.2014.03.010

DENTAL-2355; No. of Pages 13

ARTICLE IN PRESS d e n t a l m a t e r i a l s x x x ( 2 0 1 4 ) xxx.e1–xxx.e13

[29] Dunn ML, Hui CY, Labossiere PEW, Lin YY. Small scale geometric and material features at geometric discontinuities and their role in fracture analysis. Int J Fract 2001;110:101–21. [30] Blandford GE, Ingraffea AR, Liggett JA. Two-dimensional stress intensity factor computations using the boundary element method. Int J Numer Meth Eng 1981;17: 387–404. [31] Dargush GF, Hadjesfandiari AR. Generalized stress intensity factors for strength analysis of bi-material interfaces. Mech Adv Mater Struct 2004;11:1–15. [32] Hadjesfandiari AR, Dargush GF. Analysis of bi-material interface cracks with complex weighting functions and non-standard quadrature. Int J Solids Struct 2011;48:1499–512. [33] Barsoum RS. Application of quadratic isoparametric finite-elements in linear fracture mechanics. Int J Fract 1974;10:603–5. [34] Henshell RD, Shaw KG. Crack tip finite elements are unnecessary. Int J Numer Meth Eng 1975;9: 495–507. [35] Lin KY, Mar JW. Finite element analysis of stress intensity factors for cracks at a bi-material interface. Int J Fract 1976;12:521–31. [36] Sinclair GB, Okajima M, Griffin JH. Path independent integrals for computing stress intensity factors at sharp notches in elastic plates. Int J Numer Meth Eng 1984;20:999–1008. [37] Treifi M, Oyadiji SO, Tsang DKL. Computation of the stress intensity factors of sharp notched plates by the fractal-like finite element method. Int J Numer Meth Eng 2009;77:558–80. [38] Betamar N, Cardew G, Van Noort R. Influence of specimen design on the microtensile bond strength to dentin. J Adhes Dent 2007;9:159–68. [39] Phrukkanon S, Burrow MF, Tyas MJ. The influence of cross-sectional shape and surface area on the microtensile bond test. Dent Mater 1998;14:212–21. [40] Zou Z, Reid SR, Li S. A continuum damage model for delamination in laminated composites. J Mech Phys Solids 2003;51:333–56. [41] Sano H, Shono T, Sonoda H, Takatsu T, Ciucchi B, Carvalho R, et al. Relationship between surface-area for adhesion and tensile bond strength – evaluation of a micro-tensile bond test. Dent Mater 1994;10:236–40. [42] Neves AD, Coutinho E, Poitevin A, Van Der Sloten J, Van Meerbeeka B, Van Oosterwyck H. Influence of joint component mechanical properties and adhesive layer thickness on stress distribution in micro-tensile bond strength specimens. Dent Mater 2009;25: 4–12. [43] Neves Ade A, Coutinho E, Cardoso MV, Jaecques S, Lambrechts P, Sloten JV, et al. Influence of notch geometry and interface on stress concentration and distribution in micro-tensile bond strength specimens. J Dent 2008;36:808–15.

xxx.e13

[44] Van Noort R, Noroozi S, Howard IC, Cardew G. A critique of bond strength measurements. J Dent 1989;17: 61–7. [45] Dehoff PH, Anusavice KJ, Wang ZX. 3-Dimensional finite-element analysis of the shear bond test. Dent Mater 1995;11:126–31. [46] Le SY, Chiang HC, Huang HM, Shih YH, Chen HC, Dong DR, et al. Thermo-debonding mechanisms in dentin bonding systems using finite element analysis. Biomaterials 2001;22:113–23. [47] Black J, Hastings GW, editors. Handbook of biomaterial properties. Chapman & Hall: London; 1998. [48] Xu HHK, Smith DT, Jahanmir S, Romberg E, Kelly JR, Thompson VP, et al. Indentation damage and mechanical properties of human enamel and dentin. J Dent Res 1998;77:472–80. [49] Craig RG, Peyton FA. Elastic and mechanical properties of human dentin. J Dent Res 1958;37:710–8. [50] Soderholm KJ. Review of the fracture toughness approach. Dent Mater 2010;26:E63–77. [51] De Munck J, Van Landuyt K, Peumans M, Poitevin A, Lambrechts P, Braem M, et al. A critical review of the durability of adhesion to tooth tissue: methods and results. J Dent Res 2005;84:118–32. [52] Yamauti M, Nikaido T, Ikeda M, Otsuki M, Tagami J. Microhardness and Young’s modulus of a bonding resin cured with different curing units. Dent Mater J 2004;23:457–66. [53] Papadogiannis D, Tolidis K, Lakes R, Papadogiannis Y. Viscoelastic properties of low-shrinking composite resins compared to packable composite resins. Dent Mater J 2011;30:350–7. [54] Vanmeerbeek B, Peumans M, Verschueren M, Gladys S, Braem M, Lambrechts P, et al. Clinical status of 10 dentin adhesive systems. J Dent Res 1994;73:1690–702. [55] Ferracane JL, Condon JR. In vitro evaluation of the marginal degradation of dental composites under simulated occlusal loading. Dent Mater 1999;15:262–7. [56] Salerno M, Derchi G, Thorat S, Ceseracciu L, Ruffilli R, Barone AC. Surface morphology and mechanical properties of new-generation flowable resin composites for dental restoration. Dent Mater 2011;27:1221–8. [57] Della Bona A, Anusavice KJ, Mecholsky Jr JJ. Apparent interfacial fracture toughness of resin/ceramic systems. J Dent Res 2006;85:1037–41. [58] Chung SM, Yap AUJ, Koh WK, Tsai KT, Lim CT. Measurement of Poisson’s ratio of dental composite restorative materials. Biomaterials 2004;25:2455–60. [59] Shono Y, Terashita M, Pashley EL, Brewer PD, Pashley DH. Effects of cross-sectional area on resin–enamel tensile bond strength. Dent Mater 1997;13:290–6. [60] Banerjee PK, Butterfield R. Boundary element methods in engineering science. London: McGraw-Hill; 1981. [61] Brebbia CA, Telles JCF, Wrobel LC. Boundary element techniques. Berlin: Springer-Verlag; 1984.

Please cite this article in press as: Campillo-Funollet M, et al. Size-dependent strength of dental adhesive systems. Dent Mater (2014), http://dx.doi.org/10.1016/j.dental.2014.03.010