Failure stress criteria for composite resin

Failure stress criteria for composite resin

Citation for published version (APA):

Groot, de, R., Haan, de, Y. M., Dop, G. J., Peters, M. C. R. B., & Plasschaert, A. J. M. (1987). Failure stress criteria for composite resin. Journal of Dental Research, 66(12), 1748-1752.

Document status and date: Published: 01/01/1987

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Failure Stress Criteria for

Composite

Resin

R. DE GROOT, M.C.R.B. PETERS,Y.M. DE HAAN', G.J. DOP2, and A.J.M. PLASSCHAERT

DepartmentofCariology andEndodontology, Subfaculty ofDentistry, University ofNijmegen,PO Box9101,6500 HBNijmegen, The Netherlands; 'MaterialsScienceGroup, Departmentof Civil Engineering, DelftUniversityofTechnology, Stevinweg 1, 2628 CNDelft, TheNetherlands; and 2DivisionofFundamentals of Mechanical Engineering, DepartmentofMechanicalEngineering, Eindhoven University ofTechnology, PO Box

513, 5600MBEindhoven, TheNetherlands

In previous work (Peters andPoort, 1983), thestressdistribution in

axisymmetricmodels of restoredteethwasanalyzed by finite element analysis (FEA). To comparethetri-axialstress state atdifferentsites,

they calculated the Von Mises equivalent stress and used itas an

indication for weaksites. However, theuseofVon Mises'theory for materialfailurerequiresthatthecompressiveandtensilestrengthsbe equal, whereas forcomposite resin thecompressivestrength values

are, onthe average,eighttimeslarger thanthetensilestrength values.

Theobjective ofthis study was to investigate theapplicability ofa

modified Von Mises and the Drucker-Prager criterion to describe mechanicalfailure ofcomposite resin. Inthesecriteria, thedifference betweencompressiveandtensile strengthisaccountedfor. The stress criteriaappliedto anuni-axial tensilestress state arecompared with

thoseappliedto atri-axial tensilestress state. Theuni-axialstate is

obtainedinaRectangularBar(RB)specimenand thetri-axialstate inaSingle-edge NotchedBend(SENB)specimenwithachevron notch atmidspan. Both typesofspecimens, madeoflight-cured composite,

werefractured in a three-pointbendtest. The size ofthe specimens was limited to 16 mm x 2 mm x 2 mm (span, 12 mm).

Load-deflectioncurves wererecordedand usedforlinearelasticFEA. The

resultsshowed thatthe Drucker-Prager criterion is a moresuitable criterionfordescribing failure ofcomposite resinsdue tomulti-axial

stress states than arethe Von Mises criterion and the modifiedVon Mises criterion.

JDent Res 66(12):1748-1752, December, 1987

Introduction.

Composite resin as restorative material in posterior teeth re-quires special properties, such as high mechanical strength, high abrasion resistance, andgoodadhesion totooth structure to withstand chewing forces, as well as low polymerization shrinkage, stability inwater, and color stability. Inthisstudy, weshallconcentrate onthemechanical strength. Recently, the stressdistribution in loaded teeth, restored with amalgam,was studied (Peters and Poort, 1983) by FEA. Principal stresses

(('l,

u2,

03)

and VonMises'

equivalent

stress

(ureqi)

were

com-pared at different sites. The equivalent stress was obtained from theequation ofcriterion #1 (Appendix):

9eql = (-3J2 )112 (1)

withJ2' the second invariant ofthedeviatoric stresstensor:

J2' = -1/6 [(r -02)2+(U2

-3)2

+(03-o.l)2] (2)

where ul,

02,

andU3 are the principal stresses.

Thecriticalvalue of

creql

(identicalwith yield stress

o,)

can be determined fortheuni-axial state with a tensile test.In the case of brittle failure, the tensile strength is assumed to be identicalwiththeyield stress. According to criterion #1, fail-urewill occur in athree-dimensional structure in a region where thecalculatedCreql exceeds the yield stress

cr..

Because com-posite resinsfracture in a more or less brittle way, the validity of a yield criterion is uncertain. Moreover, the compressive

Received for publication January 10, 1986 Accepted for publication August 4, 1987 1748

strength ofcomposite resins is about eight times larger than the tensile strength, whereas they are treated equally by cri-terion #1. The aim of this studywas toinvestigate the appli-cability of three stress failure criteria (Von Mises, criterion #1; modified Von Mises, criterion #2; and Drficker-Prager,

criterion #3) tobrittle fracture ofacomposite resin. Criterion #1 was modified to obtain criterion #2 by addition of the hydrostatic stress. This addition results in a criterion which takes intoaccountthe difference betweencompressive and ten-sile strength ofa composite. The equivalent stress(0eq2) was obtained from theequation of criterion #2 (Appendix):

_ (k

1)

J

[(k-1)2J12-12J2'k]1/2

0eq2 2k 2k

whereJ1 is the firstinvariant of the stresstensor:

J-1 (91+ 2+ 3)

(3)

(4)

and k the ratio betweencompressiveand tensile strength. Criterion #2

(Williams,

1973)is suitable fordescribingthe failure ofpolymers, which are like the resinpart of the com-posite without fillerparticles. Another criterion (Driicker-Pra-ger,criterion #3; Drficker and Prager, 1952)may accountfor the ratio of compressive to tensile strength. It is commonly used in the field of soil mechanics to describe failure or de-formation ofabody consistingof soilparticles. The filler par-ticles of the compositewithout the resincanbeconsideredas such. Theequivalentstress

(O.

3)

wasobtained from the equa-tion of criterion #3 (Appendixt:

k-1 k+l1

C0eq3 ,

~(

k

3J2')

1/2

2k 2k (5)

Afailurecriterion, describing onlymaterialproperties,should for one single set of material parameters be able to predict failure in an uni-axial as well as any tri-axialstress state. A failure criterion can berepresented by a surface in the three-dimensional stress space with the three principal stresses on the co-ordinate axes. Forsucha surfaceto bedetermined

ex-perimentally, a large number of different stress states should beinvestigated.Inthis study, however,acomparison has been made betweenfailure described according tocriteria #1, #2, and #3 in an uni-axial and a tri-axial stress state (all three principalstresses arepositive).ARectangularBar(RB) speci-men(Fig. la), fractured inathree-point bendtest,has an uni-axial stress state atthe site where failure starts, and a Single-edge Notched Bend (SENB) specimen with achevron notch (Fig. lb) hasatri-axialstress state. Ingeneral,the stress state inastructure (atooth in the clinical case) is more likely to be tri-axial thanuni-axial; therefore, the SENB specimen seems to bemore consistent with clinical stress states. This type of specimen for controlled fracture experiments is often used to determine parameters of fracture mechanics such as work-of-fracture (TattersallandTappin, 1966; Rasmussen et al., 1973; Rasmussenetal., 1976; Rasmussen, 1978).This study is aimed at determination of a failure criterion forcomposite resin. A

FAILURE STRESSCRITERIAFORCOMPOSITERESIN

ulus (E)wascalculated according to:

E=

4B=

_3 W2

W3

U(

+

3(1+

v/2)

SO)

(Williams, 1973). The correction factor has a value of 0.096 (using, for Poisson's ratio, v = 0.3, W = 2, and S = 12) and isnecessarybecause the width is not small compared with the span. The valuesE of each specimenwere averaged. For RBtesting, 12 specimenswereused.

A reduced number of SENB specimens (5) was caused by the complicated manufacturing of these specimens.

The fracturesurfaces were examined, and the distance from the notch tip to the outer surface (c) was measured with a measuring microscopesothat the accuracy of the cutting could be evaluated.

FEAcalculations.-For reasonsof symmetry, themodeling and the calculationof the three-dimensional (3-D) stress dis-tribution with FEA canberestricted to one-quarter(6 mm x 1 mm x 2 mm) of the RB and SENB bars (element mesh shown in Figs. 2a and 2b) by introduction of the appropriate boundary conditions. For analysis of the SENB specimen, 3-Dmodeling isnecessary. The numberof elementswas 12and 111 for the RB and SENB models, respectively. Care was takento increase the number of elements for theSENBmodel in the notch tip region forreasons ofaccuracy. Fig. 3 depicts the distribution of elements as a function of distance from

-%

(6)

Fig.1-(a)Rectangular Bar in three-point bend experiment. (b)

Single-edge Notch Bend specimen with chevron notch. P. load; S, span; B,

thickness;W,width.

failure criterion isnecessaryto supportprediction of mechan-ical failure when FEAofacomposite-restored tooth is used.

Materials

and

methods.

Experiments.-Composite (Silux®, a bis-GMA resin with

colloidal silicaparticles; averagesize, 0.04 pxm; fillercontent, 51% by weight, accordingtothemanufacturer; batch 060884, univ4XY1, 3M Co., St. Paul, MN)wasinserted intoa

stain-lesssteel mold and covered with aglass plate, thereafterthe specimens (16 mm x 2mm x 2mm) werepolymerized by

visiblelight(Translux®, Kulzer & Co. GmbH, Bereich Dental, D-6382Friedrichsdorf1, Federal Republic of Germany).After five min, thespecimenswerestored intapwater atroom

tem-perature forfrom 24 to 28hr, duringwhich period theywere

takenfrom thewaterfor about fiveminsothatachevron notch couldbe cut bymeans ofa diamond disk (537/220 H

super-diaflexHorico®, HopfRingleb& Co. GmbH,Berlin45, Fed-eralRepublic of Germany) (diameter, 22mm;thickness, 0.15 mm) and watercoolant. The bars were fractured in a three-pointbend test bymeansof anInstron testing machine, at a

cross-head speedof0.5mm/min.The spanof the support (S) of approximately 12 mm was determined with a measuring

microscope. Load-deflection (P,u)curves wererecorded. The

loadatfracture(Pc)wasdefinedasthehighestload.The

thick-ness (B) and the width (W) of the specimensweremeasured

with amicrometer. For eachRBspecimen, theYoung's

mod-q

Fig.2-(a)RBmodel:quarterof thebar,asused for FEA calculations.

(b)SENB model: quarterof thebar, asusedfor FEA calculations.

b

I

1750 DEGROOT et al. 1000I hsle per mm3 100 10 +

R5

.,

I

SENB model 0 1 2 3 4 5 6

distance frommidspmn (mm)

Fig. 3-Distributionofnumberof elements usedtomodel the RB and

SENBspecimens, as afunction of distancefrommidspan.

t

P

midspan. The type of element usedwas a3-D isoparametric 20-node brick. Theassigned modulus ofelasticity(E = 3.70

GN/m2)resulted from the experiments,whereas the Poisson's ratiowastaken tobe 0.3, which value issupported byvalues for composites reported by Finger (1974) and Whiting and Jacobson(1980)ranging from 0.23to0.33.Byassumptionof linearelasticmaterial properties andgeometric linearity,a lin-earrelationship exists between both calculateddeflections and stresses vs. applied load. The deflection wasprescribed, and the reaction force in theloading point wasderived fromFEA calculations. Theequivalent stresses accordingtocriteria #1, #2, and #3 were calculated by FEA for the region where failure initiationwasobserved in theexperiments. For theRB and SENB specimens, this regionwas situatedatmidspan, at the side opposite the applied load. By replacement of the re-action force with the measured fracture load (Pc), the critical values

oreqlc,

Oeq2cq

and e3c of the equivalent stresses were obtained. The ratio (k = 8) between compressiveandtensile strength was computed from the compressive and tensile strengthsasprovided by the manufacturer. The influence ofk onthecalculatedequivalent stress wasinvestigated by varying k from 5 to 10. Because the thickness (B) and width (W)

measured on the experimental specimens deviated from the values used in the FEA calculations (B = 2 mmand W = 2

mm), acorrection termbased onthis deviationwasappliedto the stressescalculated by FEA.

Results.

Experiments.-The

load-deflection curves (example inFig. 4) showedalinear elastic behaviorofthespecimens until frac-ture. FortheRBspecimens,theloadsuddenly dropped to zero. FortheSENB specimens, controlled fracture curves were ob-tained. In this way, it was possible to quantify the work-of-fracturebycalculatingthe area under theload-deflectioncurve (TattersallandTappin, 1966).

The measured value of the span (S) was 11.98 mm. The loadat fracture (Pc) is given in the Table. The average values of the thickness (B) and the width (W)ofthe specimens are given in the Table to show the deviation from the values as-sumed in theFEA calculation. The highest value of the mea-sured distance (c) from the notch tip to the surface of the specimen was 0.074 mm. The calculation of the modulus of

elasticity is based on equation (6) and yields E = 3.70 + 0.35 GPa.

RB

U

SENB

U

Fig. 4-Qualitative example of load-deflection curves of aRectangular

Bar andSingle-edge Notch Bend specimenshowing linear elastic behavior until fracture.

TABLE

THE RESULTS OF MEASUREMENTS ANDCALCULATIONS

Specimens RB SENB N=12 N=5 Measurements P, (N) 23.6 + 2.3 4.63 + 0.11 B (10-3m) 2.043 + 0.033 2.031 + 0.013 W (10-3m) 2.102 ± 0.058 2.121 + 0.063 FEA calculations (Teqlc (MPa) 44.4 + 4.4 * 34.9 + 4.5 Orq2c (MPa) 48.3 + 4.8 * 71.5 + 10.7 ;eq3c (MPa) 46.6 ± 4.7 -o- 54.3 + 8.1

*significantly differentat alevel of p = 0.005.

onotsignificantlydifferentat alevelofp = 0.005.

Fractureload (P.),thickness(B),width(W),and criticalvalues(Jeqic, Oreq2c,and(eq3c)ofequivalentstressesaccordingtocriteria#1(Von Mises), #2(modifiedVonMises),and #3(Driicker-Prager)for RB(rectangular

bar)andSENB(single-edge notched bend) specimens. Averagevalues + standarddeviation;N number ofspecimens.

FEA calculations. -The critical values of these equivalent

stresses

(Oeqicti

req2ci and

(eq3c)

in the fracture

region,

accord-ingtocriteria#1, #2,and#3(obtained fromFEA ontheRB and SENB models) aregiven in the Table. A Student t test showed that criticalvalues of theequivalentstressesaccording toVonMisesand modifiedVonMises criteria measured with RBspecimensweresignificantly different from criticalvalues of equivalent stresses measured with SENB specimens at a level of p=0.005. This was not the case for the Drucker-Pragercriterion. The influence ofvariation in k on the calcu-lated equivalent stresses was small. The equivalent stress

ac-cording tocriterion #2 for k = 10 was8% largerthan fork

= 5, while for criterion #3 this differencewasonly4%.

FAILURE STRESS CRITERIAFORCOMPOSITE RESIN

Discussion.

The objective of this study was to investigate the applica-bility ofseveral failurecriteria tocomposite resin. Two stress stateswererealized inRB and SENBspecimens. So that a fair comparison would be ensured, the two types of specimens were stored under identical conditions. For clinical use of an established criterion, more realistic conditions are necessary for determination of the parameters. For example, attention should be paidto watersorption and hydrolytic degradation of

agingcomposite resin.

In general,the strength of dental materialsistested by

ap-plication of uni-axial stress states (tension or compression)to the test specimens. In 3-D dental

structures,

atri-axial stress state is encountered. Therefore, a failure criterion for

com-positeshould be essentially tri-axial.

Examinationof the fracture surface of the SENB specimens revealed that thetip of the chevron notch was always less than 0.074mmfromtheedge, whichis4%of thewidth. Therefore, itwasconcluded that the experimentalgeometry of the speci-menscould be represented by the SENB model as used in the FEA. Cutting the chevron notches in bars will never produce exact SENB specimens. Theload-deflectioncurvessupported thecorrectness of the assumptionof the linear material prop-erties for theFEAcalculations. The experimentallydetermined modulus of elasticity falls into the range (3.3-5.3 GPa)

re-portedby Reinhardt and Vahl (1983).

Anyvalid failure criterion for composite resins dependson anumberof material parameters which should be thesamefor allpossiblestress states.Thecriteria investigated in thispaper are all two-parameter criteria for which the ratio k and the critical values of the various equivalent stresses have been chosen. Materials suchascomposite resins havealarger com-pressive than tensile strength, which means that k> 1. In the Von Misescriterion, theparameter k mustbe equal to 1. For thisreason,thiscriterion is basicallynotsuitable forcomposite resins. Composite resins typically have k-values within the range5<k<10.Aproper stress failure criterion shouldyield thesamevalue for the critical values of the equivalent stress for all possible tri-axial stress states, including our uni-axial andtri-axialstress states(Table).Theremainingcriteria

(mod-ified Von Mises and Drucker-Prager)can use realisticvalues for k. The resultsobtained for the considered criteria show that the difference in critical equivalent stresses for thetwostress states(uni-axialvs. tri-axial)is minimal for theDrucker-Prager

criterion. Forthis reason, the Driicker-Prager criterion seems to be a more suitable criterion for use as a general criterion for mechanical failure of compositeresins subjected to com-plex stresspatterns.

Acknowledgments.

The authors would like to thank Dr. Ir. P.G.Th. van der Varst for hiscontribution to the textandDr. H.A.J. Reukers for hisexperimental assistance. This researchwaspartof the researchprogram: Restorations and Restorative Materials.

Appendix.

Von Mises' yield criterion states that yielding will occur when the second invariant of the deviatoric stress tensor

(J2')

reaches a criticalvalue, or:

P(-J2')

= 1

(Al)

with:

J2=

-1/6

[(aR

-r2)2+

(U2

3)

+

(Gr3

U)2] (A2)

and

o,,

02,

and r3 principalstresses and p aparameter. In the case of simple tension, yielding (uy is yield strength) will occurwhen:

(A3)

01 = (Tyand 92 = 0r3 = 0-Substitution in eq. Al gives:

p = 3I ay. (A4)

Now, knowing p, the interpretation of Von Mises becomes clear. Substitution of A4 intoAl leadsto:

Sy2

= 3 (-J2'). (A5)

Therefore, the aforementioned criticalvalue isequalto0ry2.

By definingan equivalentstress

(orq,)

as:

9eql = (-3

J21)1/2,

(A6)

Von Mises' criterion states that yieldingwilloccur when the equivalent stressreaches acriticalvalue

(ueqlc

= cy). Notice

that J2' in eq. A5 denotes the value of J2' at the momentof failure, where J2' in eq. A6 can have any value below the critical value.

According to Williams (1973), formaterials with different compressive and tensile strength values, Von Mises' yield cri-terion canbe modifiedby adding the hydrostatic stress:

qJ,+P(-J2')

= 1, (A7)

where p and q are parametersand

J1

the first invariant of the stresstensor:

Ji = (9 +F2

+Or3).

(A8)

pandq canbeexpressed intermsof theyieldstressesinsimple

tension andsimplecompression, cy, andcryc, respectively:

q crYt + p 1/3

cyt2

= 1 -qcYc + p 1/3 cy,2 = 1 Hence: 0rYc Cryt -__3 q = and p = CrYc0Yt CrYccrYt For knownratio ofcompressive totensile strength:

k =

cry/cy-

,

(A9a) (A9b)

(AlOa/b)

(All)

and substitution of eq. Al1a, AlOb, and All into eq. A7 gives:

k y2

-(k-l)ry1J1

+3J2' =

0-Solvingeq. A12for

cy,:

0Yt -=

(k-

1)Jl

±

[(k-

1)2J12

- 12

J2'k]1/2

2k

(A12)

(A13)

(because strength values need to be

positive, only

the urys

value can be accepted).

Again, anequivalentstress

(Ueq2)

canbe defined:

(0eq2 =

(k

-)J1

±

[(k-1)2J12-122J2'k]"12

2k (A14)

Notice that J1 and

J2'

in eq. A13 denote the valuesofJ1 and

J2'

at the momentoffailure, whereJ1 and

J2'

in eq. A14can

have any valuebelowthecritical value.

The Drucker-Prager criterion

(Driicker

and

Prager,

1952)

reads asfollows:

qJj

+

r(-J2')112

= 1.

(A15)

1752 DEGROOTetal.

Ananalogue derivation, asgivenfor the modifiedVonMises criterion, leads to:

k-1 k+l Ue3= 2k + 2k

List of

Symbols.

symbol unit B (ii c (m) E (Pa) FEA

J,

(Pa)

J2'

(Pa2)

k p (Pa-2) p (N)

PC

(N)

q (Pa-1) r (Pa-1) RB S (m) SENB u (m) W (m) 0reqI (Pa) Ueq2 (Pa) O'eq3 (Pa) Oeqlc (Pa) Ueq2c (Pa) 0Feq3c Uy UYc (A16) UYt (Jo (02 (T3 description

thicknessofthe bar

distance from surface to notch tip

modulus ofelasticity

finite element analysis

first invariant ofstress tensor

second invariant ofdeviatoricstress tensor ratio ofcompressive and tensile strength

parameterin stresscriterion loadatmidspan

fracture load

parameterin stresscriterion

parameter instresscriterion Rectangular Barspecimen

span of thesupport

Single-edge Notched Bend specimen

deflection of bar atmidspan

width of the bar

equivalent stress criterion #1 (VonMises)

equivalentstresscriterion#2(modifiedVon Mises)

equivalentstress criterion #3 (Driicker-Pra-ger)

critical value ofueql

critical value of(Teq2

(Pa) (Pa) (Pa) (Pa) (Pa) (Pa) (Pa)

critical value of

O;eq3

yield stress

yield stressin simplecompression

yieldstressin simpletension principalstress

principal stress

principal stress

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