Computational Materials Science 64 (2012) 289–294

Contents lists available at SciVerse ScienceDirect

Computational Materials Science

journal homepage: www.elsevier .com/locate /commatsci

Finite element analysis of thermo-mechanical loaded teeth

Krzysztof Pałka a,⇑, Jarosław Bienias a, Hubert Debski b, Agata Niewczas c

a Department of Materials Engineering, Lublin University of Technology, Nadbystrzycka 36, 20-618 Lublin, Polandb Department of Machine Construction, Lublin University of Technology, Nadbystrzycka 36, 20-618 Lublin, Polandc Department of Conservative Dentistry, Medical University of Lublin, Karmelicka 7, 20-081 Lublin, Poland

a r t i c l e i n f o a b s t r a c t

Article history:Received 28 October 2011Received in revised form 8 May 2012Accepted 9 May 2012Available online 4 June 2012

Keywords:Finite element methodThermal stressTemperature distributionStress analysisDental filling

0927-0256/$ - see front matter � 2012 Elsevier B.V. Ahttp://dx.doi.org/10.1016/j.commatsci.2012.05.037

⇑ Corresponding author. Tel./fax: +48 81 5384214.E-mail addresses: k.palka@pollub.pl (K. Pałka), j.b

h.debski@pollub.pl (H. Debski), agatan117@wp.pl (A.

This paper presents the possibility of applying the finite element method for the analysis of stress level inhard dental tissues, restored with class I dental filling and exposed to thermal and mechanical load. Thestudies were made on a geometrical model imitating the real geometry of a premolar tooth obtainedusing the X-ray microtomography technique and CAD software. The distributions of reduced stressdefined in accordance with the Huber–Mises–Hencky (H–M–H) hypothesis in hard dental tissues wereanalyzed, and assessment of the degree of strength of the adhesive layer at the border of the compositerestoration and biological tissue was attempted. The application of numerical simulations (Abaqus)enables real assessment of the tooth tissue strength, which allows assessing the risk of unsuccessful den-tal treatment, and helps prepare rational methods of preventing tooth damage resulting from load. Max-imum reduced stresses were located in areas of the external load attachment and exceeded 668.8 MPawith the force loading and 34 MPa and 58.4 MPa with temperature loading of 55 �C and 5 �C respectively.Superposition of loadings has produced maximum stresses of 669.4 MPa in the case of 5 �C.

� 2012 Elsevier B.V. All rights reserved.

1. Introduction

There is an increasing demand for dental restorations that visu-ally match the tooth and which bond to dental tissue effectively.The use of composite dental restorations allows dentists to restoreteeth, joining functional and anatomic aspects to aesthetic consid-erations. The restoration systems using these materials are basedon adhesive dentistry, requiring an effective and durable connec-tion to the dental tissues. Composite restorations connected tothe dental structure are subjected to a number of differentmechanical loads [1]. The polymerization contraction that occursduring curing of the resin matrix can lead to stresses at the inter-face between the restoration and dental tissue. Additional interfa-cial stresses are subsequently superimposed by mechanicalloading on the tooth during mastication, and by the thermal load-ing at drinking and eating of hot and cold foods [1,2]. The investi-gation of the effects of these thermal changes on restored teeth,and the associated bond failure and microleakage, either in vitroor in vivo, present serious experimental difficulties. Mathematicalmodeling of the process using finite element analysis offers analternative approach to the problem [3–5]. The analysis of the ef-fect of thermal fluctuations on restored teeth should be considered,as it has direct impact on the stress in the interface due to the

ll rights reserved.

ienias@pollub.pl (J. Bienias),Niewczas).

difference between the thermal properties in the tooth and inthe restoring materials, affecting durability of the interface. Theclinical consequences of cracks in the adhesive layer are marginalleak, postoperative sensitivity, and the occurrence of recurrentcaries [6].

Due to the universal character of the finite element method,which enables modeling of complex physical phenomena, itsincreasingly broad use in an interdisciplinary context, connectedwith the analysis of mechanical parameters of modern materials,can be observed [7]. Among the present day areas of applicationof numerical analyses involving FEM, bioengineering and dentistrycan be included, with respect to the assessment of stress levels andqualification of strength hypotheses in bone tissues. It allowsdesigning and optimizing modern materials used for dental recon-structions, such as crowns or crown-root fillings, as well as assess-ing the risk of unsuccessful dental treatment resulting in damageto the hard dental tissue structure or the filling material [8].Numerical simulations using the finite elements method in den-tistry can constitute a stage of preclinical tests connected with bio-mechanical aspects of the design and optimization of dentalfillings. The knowledge of stress distribution in the tooth-fillingarea, which is a configuration of three different materials, allowsassessing the degree of strength for individual tissues of the inves-tigated configuration, and illustrates two-way interactions be-tween them. Correct evaluation of the degree of strength of harddental tissues can be a basis for the right choice of treatment meth-od, thus minimizing the risk of mechanical damage to the dental

290 K. Pałka et al. / Computational Materials Science 64 (2012) 289–294

tissue of the filling material. It is shown by increasingly frequentreports in present-day literature concerning the use of numericalanalyses for assessing the degree of strength of the hard dental tis-sues and evaluating mechanical properties of dentistry materials[1–3].

In this article, numerical analyses were performed and de-scribed, comparing the response of a human molar with compositerestoration of class I. Initially, a transient thermal model was ana-lyzed to obtain the thermal loading due to temperature distribu-tion at different time steps. Afterwards, the stresses arising fromthermal changes were computed. Attention was given to the resto-ration–tissue interface, where problems are observed in clinicalpractice. This work is a complement of previous research on ther-mal fatigue of teeth with restoration.

2. Materials and methods

The subject of the studies was a human premolar tooth whichwas an exact reconstruction of the geometry of a real object usingX-ray microtomography (12 lm resolution). The images of subse-quent tooth sections formed the basis for preparing dimensionalcoordinates of the tooth geometry (spline type parametric curves,dimensional surfaces) using CAD software (Fig. 1a).

The purpose of the numerical calculations was to determine thedistribution of stresses generated in hard dental tissues, for themaximum load of 400 N [3,4]. The second purpose was to deter-

Fig. 1. (a) Virtual cross section of the tooth model, (b) coupling interactions andmechanical load in the FEM-model; arrows indicates the points of mechanicalloading.

mine the distribution of temperature and stresses generated bychanges of temperature in range of 5–55 �C [9,10] with dwell timeof 30 s [11]. The distributions of reduced stresses, determined onthe basis of the H–M–H (Huber–Mises–Hencky) strength hypothe-sis, were analyzed.

The numerical calculation process in mechanical load case thatwas applied was geometrically non-linear in character using theincremental–iterative Newton–Raphson method. For all the ele-ments (enamel, dentine and filling), material models with the lin-ear-resilient characteristics in the whole range of the external loadwere defined. The calculations allowed assessing the degree ofstrength of hard tissues with respect to the examined tooth, as wellas comparative analysis of a healthy tooth and a tooth with a den-tal filling. The process of discretization of hard dental tissues andthe filling material was based on the solid body elements of thetetragonal type – C3D4, that is four-node cubic elements with a lin-ear shape function, having three translation degrees of freedom, inevery node each.

The mechanical load of the tooth in the form of concentratedforce of 400 N was introduced through interactions of the couplingtype, allowing its distribution across the group of nodes on the en-amel surface, as shown in Fig. 1b. The border conditions of thenumerical model were defined by clamping the dentine nodes be-low the border of the enamel, blocking all translation degrees offreedom in those nodes. For the model of a tooth with a dental res-toration, in which the properties of the adhesive layer between thebiological tissue of the tooth and the dental filling material wereallowed for, the adhesive layer model based on the contact interac-tions of the surface-based cohesive behavior type was used. Thedescription of the adhesive layer requires a definition of the initia-tion point, as well as the character of the evolution of the layerdamage consisting in gradual degradation of the junction rigiditytogether with the increased external load. The basic constitutivelaw that is applied to describe the cohesive contact layer is thecriterion of the traction–separation damage (disruptive strength– separation limit value), by which both the normal effect (separa-tion) and the damage caused by the contact effect (shear) can beallowed for in the damage analysis. For the numerical calculationsthere were used the following values of adhesive layer’s proper-ties: Young’s modulus E = 1000 MPa, Poisson’s ratio m = 0.3, peelstrength Knn = 25 MPa, shear strength Kss and Ktt = 12.5 MPa (basedon [4]).

For a thermal-displacement analysis coupled with the linearfunction of shape for displacements and temperature, a C3D4Ttype finite element was utilized – a four-node tetragonal solidbody element. The coupled temperature-displacement analysiswas made to expose the outer surface of enamel to an increasedtemperature of 55 �C, as well as a lowered temperature of 5 �Cfor 30 s [10,11]. The temperature of 37 �C was adopted as the initialtemperature for the whole model of the tooth. The following valuesfor the boundary conditions on the outer surface of the enamelwere assumed:

– for the temperature of 55 �C: film coefficient 7.37 � 10�4 [J/(s mm2 �C)], sink temperature 55 �C,

– for the temperature of 5 �C: film coefficient 5.95 � 10�4 [J/(s mm2 �C)], sink temperature 5 �C.

The adhesive layer between the filling material and the hard dentaltissue has been taken into consideration in the model; for that pur-pose, the contact interactions of the surface-based cohesive behav-ior type in the adhesive layer were modified by adding parametersconcerning conditions of the heat flow between individual dentaltissues [7]. It necessitated complementing the contact propertieswith the thermal conductivity coefficient value k = 0.4 � 10–3 [J/(s mm �C)], as well as the width of the adhesive layer was 10 lm.

Table 1Properties of materials used in FE analysis [9].

Properties of material Enamel Dentine Composite restoration Adhesive system

Young’s modulus E (MPa) 84,100 18,600 15,000 4500Poisson’s ratio m 0.33 0.31 0.3 0.3Density q (g/mm3) 2.90 � 10�3 4.0 � 10�3 1.95 � 10�3 –Thermal expans. coeff. a (1/�C) 17 � 10�6 10.6 � 10�6 25 � 10�6 –Thermal conductivity k [J/(s mm �C)] 0.92 � 10�3 0.63 � 10�3 1.5 � 10�3 0.4 � 10�3

Specific heat cp (J/g �C) 0.7536 1.1724 1.4 –

Fig. 2. H–M–H stress distribution in the treated tooth: (a) in whole tooth, (b) inrestoration.

K. Pałka et al. / Computational Materials Science 64 (2012) 289–294 291

Elastic models of materials were used for dental tissues and therestoration material. Because of the coupled analysis, the mechan-ical properties of the materials, i.e. Young’s modulus, Poisson’s ra-tio, density, as well as the thermal properties, such as: specificheat, thermal expansion coefficient and thermal conductivity, weredefined for every element. The properties of dental tissues and fill-ing materials adopted for the numerical analysis have been pre-sented in Table 1. As a numerical tool, ABAQUS/Standardsoftware was used.

3. Results

3.1. Mechanical load

The results of the numerical calculations have been presentedas maps of reduced stresses rred determined on the basis of thestrength hypothesis of Huber–Mises–Hencky. An assessment ofthe degree of strength in the adhesive layer between the biologicaltissue of the tooth and the dental filling material was attempted aswell. In Fig. 2, the distribution of the reduced H–M–H stress intooth and in the restoration has been presented.

The maximum reduced H–M–H stress values in the enamel ele-ments are located in areas of the external load attachment and arerz = 668.8 MPa. The areas are characterized by rapid reduced stressgradients to the maximum value, with the general level of stress inthe enamel material of the mean value of rz = 20–33 MPa. Themaximum reduced stress values in the dentine model arerz = 23.7 MPa. Also in this case, near the dental filling edge, an in-creased reduced stress level was observed.

Analyzing the stress level in the composite filling material andon the surface of the adhesive layer, an area of increased value ofreduced stress of rz = 15 MPa was observed. This is also confirmedby the adhesive layer strength maps (Fig. 2b). The visible troubleareas are found in a short segment of the upper edge of the fillingand locally on its side surface.

3.2. Thermal load

The numerical analysis of temperature and displacement al-lowed determining the distribution of temperature and the result-ing distribution of stress in hard dental tissues (Fig. 3). The stressanalysis was made based on the reduced stress determined withthe Huber–Mises–Hencky strength hypothesis. As a result of expo-sure to the environmental temperature of 55 �C, the enamel sur-face temperature changed from the initial value T0 = 37 �C to thevalue of 34.2 �C. The value of temperature in the dentine was54.3 �C in the ‘‘top’’ elements of the dentine geometry, directly incontact with the inner layer of the enamel, while in the area of di-rect contact with the filling material the temperature did not ex-ceed the value of 50 �C. In the restoration material the maximumtemperature value reached 52.9 �C on the upper edges of the filling.

The maximum values of the reduced H–M–H stress located inthe enamel elements on their outer edge are 20–30 MPa. Severestress gradients up to 34 MPa were observed in this area as wellas near the sharp edges of contact between the enamel and the fill-ing on the tooth surface. This area can involve a potential point of

damage to the filling material. The remaining outer surface of thecrown is characterized by an even distribution of the reducedstress of about 7.5 MPa, jointly with the surface of restoration onwhich only insignificantly lower stress values were observed. Themaximum values of the reduced stress in the dentine model are10 MPa and are located near the dentine–restoration contact wall.Also in this case, in the vast area of the dentine model the distribu-tion of stress is even, while in the area of contact with the lowersurface of the restoration the stress reaches 7.5 MPa.

Fig. 4 presents the distributions of reduced stress and tempera-ture in tissues of the tooth structure with a restoration exposed tothe environmental temperature of 5 �C. As a result of the exposureof the crown part to the temperature of 5 �C, the enamel tempera-ture decreased from the initial value of 37–6.62 �C. On the outer

Fig. 3. (a) Distribution of reduced stress by H–M–H hypothesis and (b) distributionof temperature on cross section of the tooth model (55 �C).

Fig. 4. (a) Distribution of stress and (b) distribution of temperature on cross sectionof the tooth model (5 �C).

292 K. Pałka et al. / Computational Materials Science 64 (2012) 289–294

surface of the enamel, an even distribution of temperature can beobserved. The distribution of temperature of a cooled tooth exhib-ited great similarity to that of heating. A gradual increase in tem-perature up to the value of 36.9 �C in the root part was observed.The filling material, in spite of the differences in thermal proper-ties, with a cooling time of 30 s, does not constitute a significantthermal barrier.

The reduced H–M–H stresses on the outer surface of the enamelwere characterized by even distribution, and the highest stress va-lue was observed on the outer edges and was 58.4 MPa, while thelowest – in the nodules and it was about 5 MPa. On the masticatingsurface, a marked difference in stresses between the enamel(30 MPa) and the filling (5 MPa) was observed. For the restorationmaterial, an even distribution of reduced stress, not exceeding10 MPa, was found. The value of stress in the elements of dentinecontacting the enamel was about 5 MPa. The maximum values ofreduced stress in the dentine model are 10 MPa and are locatedin the outer part of the tooth near the border of the enamel–den-tine contact as well as in the bottom of restoration.

3.3. Superposition of mechanical and thermal load

On the basis of two-step mechanical analysis and coupledthermal-displacement analysis there was obtained distribution of

H–M–H residual stress generated by loading force of 400 N and ex-posed to the temperature of 55 �C for 30 s. The second part ofsuperposition stresses analysis was conducted with loading forceof 400 N and exposition to the temperature of 5 �C for 30 s. InFigs. 5 and 6 respectively, the distribution of the reduced H–M–Hstress in tooth tissues has been presented.

In case of temperature 55 �C the maximum reduced H–M–Hstress values in the enamel elements are located in areas of theexternal load attachment and are rz = 668.5 MPa and are slightlyhigher than obtained without temperature interaction. The areasare characterized by rapid reduced stress gradients to the maxi-mum value, with the general level of stress in the enamel materialof the mean value of rz = 75 MPa. The maximum reduced stressvalues in the dentine model are rz = 38.5 MPa. Also in this case,near the dental filling edge, an increased reduced stress level wasobserved. Analyzing the stress level in the composite filling mate-rial and on the surface of the adhesive layer, an area of increasedvalue of reduced stress of rz = 25.1 MPa was observed. The visibletrouble areas are found in a short segment of the upper edge of thefilling and locally on its side surface.

Generally, it may be said that maximum value of stresses insuperposition of mechanical and thermal load (55 �C) are very sim-ilar. The distribution of stresses obtained in superposition showedsimilar behavior as in mechanical loading.

Fig. 5. (a) Distribution of H–M–H stress on the tooth surface and (b) on the crosssection of the tooth model (400 N, 55 �C).

Fig. 6. (a) Distribution of H–M–H stress on the tooth surface and (b) on the crosssection of the tooth model (400 N, 5 �C).

K. Pałka et al. / Computational Materials Science 64 (2012) 289–294 293

In the case of temperature 5 �C the average H–M–H stresses hada value of 67.2 MPa with an uniform distribution. Maximum stres-ses were observed in areas of the external load attachment and are669.4 MPa. It is a similar value to that obtained in heating. How-ever, the lower temperatures appears more dangerous becausethe superposition of compression stresses occurred. Hard tooth tis-sues exhibit a high compression strength however the value andstate of stresses involved in that conditions may impair the adhe-sive layer of restoration effecting in microleakage. The thermalstress values are relatively low however observed decohesionand increasing of marginal gap may be caused by cyclic deforma-tion [9].

4. Discussion and conclusions

The level of stress resulting from the occlusion load is one of themost essential factors affecting the degree of load for both the den-tal tissues, as well as the strength of the adhesive layer of the fillingmaterial to the biological tissue. The results allow revealing thetrouble areas occurring in the short segment of the upper edge ofthe filling and locally on its side surface (Fig. 2). These areas can

constitute a potential source of damage to the adhesive layerexposed to a complex state of load, thereby leading to the propaga-tion of the border crevice [9]. In most cases, this phenomenon wasa direct cause of damage to the dental filling [4].

In clinical practice the problem of spots and leaks observed onthe border of the restoration is enhanced by the use of compositematerials, which exhibit significant difference in thermal expan-sion compared with dental tissues. Adding up thermal andmechanical stresses in the bonding layer can lead to cracks andresult in microleakage. The results obtained for both the loweredand the increased temperatures are convergent with the resultsobtained in works [1,5,8]. Cornacchia et al. [1] defined also thetype of stresses for cooling and heating of the tooth; coolingdown causes tensile stresses in the enamel and compressivestresses in the dentine; in heating, the stress directions are re-versed. For that reason, the use of filling materials from the samegroup is recommended for treatment due to lack of differences inthe thermal expansion coefficients and generation of low thermalstress.

The stress values determined in the FEM analysis should not betreated as exact, as the properties of dental tissues and fillingmaterials are broad in range. Also, tooth geometry is an individual,

294 K. Pałka et al. / Computational Materials Science 64 (2012) 289–294

ontogenetic characteristic. Based on the results of this research, itcan be stated that variable loads caused by temperature changesranging from 5 to 55 �C can generate reduced stresses of themaximum value of 50–90 MPa, which are considerably lower incomparison to the strength of tooth tissues (for enamel 350 MPa,for dentine 247 MPa) but near the strength of filling material[1,3–5]. However the variable stress gradient can contribute todamage both the weakest element – the bonding system andcracking of the filling material and dental tissues as well.

Acknowledgment

Presented work was financed from the scientific funds in theyears 2008-2011 as a research project.

References

[1] T.P.M. Cornacchia, E.B. Las Casas, C.A. Cimini Jr., R.G. Peixoto, Med. Biol. Eng.Comput. 48 (2010) 1107–1113.

[2] Y. Arman, M. Zor, M.A. Güngör, E. Akan, S. Aksoy, J. Biomech. 42 (2009) 2104–2110.

[3] P. Ausiello, S. Rengo, C.L. Davidson, D.C. Watts, Dent. Mater. 20 (2004) 862–872.

[4] P. Ausiello, A. Apicella, C.L. Davidson, Dent. Mater. 18 (2002) 295–303.[5] D.N. Fenner, P.B. Robinson, P.M-Y. Cheung, Med. Eng. Phys. 20 (1998) 269–275.[6] D. Ehrenberg, G.I. Weiner, S. Weiner, J. Prosthet. Dent. 95 (2006) 230–236.[7] O.C. Zienkiewicz, The Finite Element Method, McGraw-Hill, Blacklick, Ohio,

1971.[8] M. Toparli, H. Aykul, S. Sasaki, J. Oral. Rehabil. 30 (2003) 99–105.[9] F.K. Wahab, F.J. Shaini, S.M. Morgano, J. Prosthet. Dent. 90 (2) (2003) 168–174.

[10] C.P. Ernst, K. Canbek, T. Euler, B. Willershausen, Clin. Oral Invest. 8 (2004) 130–138.

[11] M.S. Gale, B.W. Darvell, J. Dent. 27 (1999) 89–99.