An asymptotic homogenization procedure is employed to obtain effective elastic properties of the composite tooth filling, a homogenized macro– stress field and a first approximation to the micro-stress field, from properties of the components and applied macro–loads. Using the approximate micro–stress field, a non–local initial strength and fatigue durability macro–conditions for the composite filling material is expressed in terms of the homogenized macro–stresses. An illustrative example with the stress singularity on the tooth–filling interface is presented showing the need in the non-local analysis. Effective elastic properties of the tooth filling is numerically simulated for some size distributions of the reinforcing particles

Mikhailov, SE

Orlik, J

English

Brunel University Research Archive

Computer Methods in Biomechanics and Biomedical Engineering - 4 (Ed.: J. Middleton, N. G. Shrive &M. L. Jones), University of Wales College of Medicine, ISBN 1 903847095, 2002HOMOGENIZATION IN STRENGTH AND DURABILITYANALYSIS OF REINFORCED TOOTH FILLINGSergey E. Mikhailov1, Julia Orlik21. ABSTRACTAn asymptotic homogenization procedure is employed to obtain effective elasticproperties of the composite tooth filling, a homogenized macro– stress field and afirst approximation to the micro-stress field, from properties of the components andapplied macro–loads. Using the approximate micro–stress field, a non–local initialstrength and fatigue durability macro–conditions for the composite filling material isexpressed in terms of the homogenized macro–stresses. An illustrative example withthe stress singularity on the tooth–filling interface is presented showing the need inthe non-local analysis. Effective elastic properties of the tooth filling is numericallysimulated for some size distributions of the reinforcing particles.2. INTRODUCTION AND MOTIVATIONTooth filling materials are considered consisting of a polymer matrix filled with glassparticles. The volume fraction, size, shape and distribution of the particles influ-ence mechanical properties of the filling material. Since 70-es, the homogenizationtechniques were widely used for obtaining effective elastic properties of the com-posite and the homogenized macro-stress field. The simplest averaging theories aremixture theory and Representative Volume Element (RVE) method (Hashin, 1983).In [1], the approximation to the micro-stress-field is derived from properties of thecomponents, micro–geometry of the composite and applied macro–loads, using aKeywords: Asymptotic homogenization, Strength, Fatigue, Singularity, Non-local conditions1Professor, Dept. of Mathematics, Glasgow Caledonian University, Cowcaddens Road, Glasgow,G4 0BA, U.K.2Doctor, Fraunhofer Institut Techno- und Wirtschaftsmathematik, Gottlieb-Daimler-Str. 49,67663 Kaiserslautern, Germany1formal asymptotic expansion. Convergence of the micro–solution, as the structureperiod tends to zero, is proved in [2] by the two–scale homogenization technique. Thepresent work is based on the fact that this limit for the micro–stress field is exactlythe first term of its asymptotic expansion, which is a product of the homogenizedmacro–stress tensor and the so–called stress concentration tensor, related only tothe micro–geometry and stiffness tensors of the composite components.Fatigue caused by normal chewing and temperature changes seems to be the mainreason of tooth filling fracture. We use here the Wo¨hler durability functions andthe linear damage accumulation rule to analyze durability under fatigue. It is alsomerged with the non-local approach of [3]. This generalizes to fatigue some applica-tion of the homogenization to micro-strength analysis presented in [4]. As a result,the initial micro-strength and durability conditions (for each component of the com-posite) are presented in terms of the elastic, strength and durability micro-propertiesof the composite components and of the macro-stresses.3. ELEMENTS OF STRENGTH AND FATIGUE DURABILITY ANALYSISFor a bounded stress field σij(y), any local strength condition at a point y ∈ Ω canbe written in the form Λ (σ(y), y) < 1, where Λ is a normalised equivalent stressfunction, a material characteristic. For some materials Λ is associated with the vonMises equivalent stressΛM(σ(y), y) =√[(σ1(y)− σ2(y))2 + (σ2(y)− σ3(y))2 + (σ3(y)− σ1(y))2]/(2σ2c (y)),or with the Tresca equivalent stress ΛT (σ(y), y) = maxk,m |σk(y) − σm(y)|/σc(y),where σ1, σ2, σ3 are the principal stresses and σc is a known uniaxial strengthof material. The local strength conditions, however, are generally not applicableto unbounded (singular) stresses since the conditions will predict fracture undervirtually any singular stress field.To illustrate that such singular stress fileds are relevant to the tooth filling simu-lation, a finite element calculation was done for a simple modell of a half-sphericaltooth (radius a = 5mm) with a half-spherical filling (radius a/2) presented on figures1 and 2. It is supposed that the tooth with filling is loaded by a uniform pressureFigure 1: Equivalent von MisesstressesFigure 2: Equivalent von Misesstresses1MP from above and interaction of the tooth with the jaw was described by the2Winkler contact condition. The Young modulus and the Poisson ratio are takenE = 90GPa, ν = 0.3 for the tooth tissue and E = 8GPa, ν = 0.3 for the fillingmaterial. One can observe that the equivalent von Mises stresses in the filling andtooth grow sharply near the singular line, which is the intersection of the tooth-filling interface with the external boundary. Applying the techniques described in[5], it was shown that the stresses in the both materials have indeed a singularasymptoics σij(ρ, θ) = KFij(θ)ρ−γ + σ¯ij(ρ, θ), where ρ, θ are local polar coordinateswith the origin in a singular point, constant K is a stress intensity factor, Fij(θ) isa bounded function of the local polar angle θ, σ¯ij(ρ, θ) is a smaller term. Solvingthe corresponding transcendental equation given in [5], we obtain the singularityexponent γ ≈ 0.207 for the particular materials and geometry.For general, especially singular stress fields, e.g. belonging to L2(Ω), a (point) non-local strength condition Λ(σ, y) < 1 can be applied. Here Λ(σ, y) is a normalisedequivalent stress functional, see [3]. Particularly, Λ can be connected with somekind of weighted averaging of σij(x), x ∈ Ω in some surrounding of the point y,Λ(σ, y) = Λ (σ˜(y), y) , σ˜ij(y) =∫Ωϕijkl(x, y)σkl(x)dx, (1)where σ˜ij are components of an auxiliary non-local stress tensor, and ϕ(x, y) is amaterial characteristic, such as∫Ω ϕijkl(x, y)dx = δikδjl, and Λ is a function, asabove. Then the strength condition for the whole body isΛΩ (σ) := supy∈ΩΛ (σ˜, y) < 1, (2)where ΛΩ(σ) is the body normalized equivalent stress functional.Example (i) If ϕijkl(x, y) = δikδjlϕ(x, y), ϕ(x, y) ={34pid3|x− y| < d0 |x− y| ≥ d for 3D,where d is a material constant, then σ˜(y) = 34pid3∫ ∫ ∫|x−y|<dσ(x)dx.Example (ii) If ϕ(x, y) is the Dirac-function, then σ˜(y) = σ(y), and the non-localstrength-condition coincides with the local one.The Wo¨hler diagram (Wo¨hler function) for a material under a regular uniaxialcycling with constant stress range ∆σ = σmax − σmin and mean stress σm =(σmax+σmin)/2, describes the critical number of cycles N∗(∆σ, σm) needed to reachrapture. For an in-fase multiaxial cycling, we consider ∆σ = ∆σij and σm = σmij(i, j = 1, 2, 3) as tensors. For simplicity, suppose further that σmij = 0. If the ma-terial fatigue properties and/or stress field vary with the coordinate, one can writefor a body Ω an (initial) durability condition in the form N < infy∈ΩN∗(∆σ(y), y)where N∗ := N∗(∆σ, y) is the Wo¨hler curve for a homogeneous material with thefatigue properties as at the point y, under the cycling ∆σij homogeneous in spacecoordinates. However the local fatigue durability condition is generally not applica-ble to singular stress fields. For the more general classes of stress fields, a non-localin space fatigue durability condition can be applied, e.g.,N < infy∈ΩN∗(∆σ; y), N∗(∆σ; y) = N∗(∆σ˜(y), y), (3)∆σ˜ij(y) =∫Ωϕijkl(x, y)∆σkl(x)dx (4)where ϕ(x, y) is as above.34. ELEMENTS OF HOMOGENISATION TECHNIQUEConsider the elasticity problem for the composite material with a large number ofperiodically distributed inclusions,∂σεij(x)∂xj= fi(x) x ∈ Ω, σεij(x) = aijkl(xε)∂uεk(x)∂xl(i, j = 1, ..., 3) (5)+ boundary and transmission conditionsHere, (uεi )3, (σεij)3×3 and (aijkl)3×3×3×3 are displacements, stresses and elastic coef-ficients respectively, ε is a small parameter related to the periodicity cell εY of thecomposite. Let Nq = (Npjq)3×3×3 ∈ Hper[0](Y ) be a solution to the auxiliary periodicweak problem of elasticity:∫Yaihjk(ξ)∂∂ξk(Npjq(ξ) + ξpδjq)∂vi(ξ)∂ξhdξ = 0 ∀vi ∈ Hper[0](Y ) (6)which depends only on the micro-structure of the composite, stiffness of its com-ponents and is independent of the boundary conditions. The homogenized dis-placement and stress fields, u0i , σˆij are given by a solution to the uniquelysolvable homogenized problem coinciding with (5) after replacement there thevariable elastic coefficients aihjk by their homogenized counterparts: aˆihjk =1|Y |∫Yaihqp(ξ)[δjqδkp +∂∂ξpNkqj(ξ)]dξ. Using results of [2], one can show that asthe structure period ε tends to zero , the sequence of stress fields σε(x) contains asubsequence, which two-scale converges to a function σ0(x, x/ε),where σ0ih(x, ξ) = aihjk(ξ)∂∂ξk(Njpq(ξ) + ξqδjp)∂u0p(x)∂xq=: Bihjk(ξ)σˆjk(x). (7)Similar to [1], Chap.9, Sec.4, we call Bihγδ(ξ) = aihjk(ξ)∂∂ξk(Njpq(ξ) + ξqδjp)aˆ−1pqγδ thestress concentration tensors. Here (aˆ−1pqγδ)3×3 is the homogenized compliance tensor,which is the inverse to the homogenized stiffness tensor (aˆγδαβ)3×3.5. HOMOGENIZATION OF NORMALISED EQUIVALENT STRESSAND WO¨HLER FUNCTIONIn a periodic medium, all becomes dependent on the period ε. FunctionsΛ (σ(y), y) and N∗(∆σ(y), y) in the local strength and durability conditions be-come Λε (σε(y); y) and N∗ε (∆σε(y); y) respectively; particular non-local represen-tations (1), (4) become Λε (σε; y) = Λε (σ˜ε(y); y) , N∗ε (∆σε; y) = N∗ε (∆σ˜ε(y); y) ,σ˜εij(y) =∫Ωϕεijkl(x, y)σεkl(x)dx, for all y ∈ Ω. Suppose Λε(σ˜ε(y); y) = Λ(σ˜ε(y);yε),N∗ε(∆σ˜ε(y); y) = N∗(∆σ˜ε(y);yε),σ˜εij(y) :=∫Ωϕijkl(x, y,xε,yε)σεkl(x)dx, ∆σ˜εij(y) :=∫Ωϕijkl(x, y,xε,yε)∆σεkl(x)dx, (8)Our aim is to derive initial macro-strength and macro-durability conditions similarto (2) and (3), i.e., the homogenized normalized equivalent stress function Λˆ and theWo¨hler function Nˆ∗ for the composite from the micro-strength and micro-durabilityconditions. This will allow to estimate the macro-strength/durability in terms ofaveraged stresses and material micro-characteristics. One can prove the following4Proposition 1 (Homogenization of non-local strength and fatigue durability). Letσε ∈ L2(Ω) be a sequence of solutions to (5). Suppose the body is macro-homogeneousin strength and fatigue durability and its point micro-strength and micro-durabilityconditions are Λε (σε; y) = Λ(σ˜ε; yε)< 1, N < N∗ε (∆σε; y) := N∗(∆σ˜ε; yε), whereΛ (σ˜ε; ζ) and N∗−1(∆σ˜ε; ζ) are continuous in the first arguments and are periodicand bounded in the second argument; σ˜ε and ∆σ˜ε have form (8). Then the limit (asε→ 0) initial sufficient non-local macro-strength and macro-durability conditionsare supy∈ΩΛˆ (σˆ; y) < 1, N < infy∈ΩNˆ∗(∆σˆ; y), (9)where Λˆ(σˆ; y) := supζ∈YΛ(∫Ωϕˆijkl(y, ζ, x)σˆkl(x)dx; ζ), (10)Nˆ∗(∆σˆ; y) := infζ∈YN∗(∫Ωϕˆijkl(y, ζ, x)∆σˆkl(x)dx; ζ), (11)ϕˆijkl(y, ζ, x) =1|Y |∫Yϕ(x, y, ξ, ζ)Bijkl(ξ)dξ.For a finite dimension of the periodicity cell (ε > 0), the strength and durabilityconditions (9) can be considered as approximate.Example: In the particular case when the non-local weight function is independentof the cell characteristics, i.e. ϕε(x, y) = ϕ(x, y), we have ϕˆihγδ(x, y, ζ) = ϕ(x, y) andΛˆ(σˆ; y) = supζ∈Y Λ (∫Ω ϕ(x, y)σˆ(x)dx, ζ), Nˆ∗(∆σˆ; y) = infζ∈Y N∗ (∫Ω ϕ(x, y)∆σˆ(x)dx, ζ).That is, the cell stress concentration does not influence the composite strength anddurability for sufficiently small cells obeying the non-local strength/durability con-dition.Let us consider now an in-face multi-axial loading, homogeneous in space coordi-nates, with varying cycle parameters such that closed loops can be always identified.Let n = 1, 2, ... be the number of a closed loop with the stress range ∆σij(n) in theloading history. Let N∗(∆σ) be the Wo¨hler function for the considered material.The discrete linear damage accumulation rule gives the durability condition in theformN∑n=11N∗[∆σ(n)]< 1. Taking into account that the function ∆σij(n) can changeonly at integer values of the cycle number n, the linear damage accumulation rulecan be rewritten also in the integral form∫ N0dnN∗[∆σ(n)]< 1.Note that a more general phenomenological concept based on normalized equivalentstress functionals and allowing non-linear damage accumulation is introduced in [6]to estimate strength and durability under creep or fatigue.For a periodically inhomogeneous composite and varying macro-stress, one can provethe followingProposition 2 Let ∆σεij(y, n) be a sequence of solutions to (5), which belong toL2(Ω) for any loop number n < N . Suppose the body micro-durability condi-tion is supy∈Ω∫ N0dnN∗ε(∆σ(., n); y)< 1, where N∗ε(∆σ; y) is as in Proposition 1.Then the limit (as ε → 0) initial sufficient non-local macro-durability conditionis supy∈Ω∫ N0dnNˆ∗(∆σˆ(., N); y)< 1, where Nˆ∗is given by (11).For ε > 0, the last durability condition can be considered as approximate.5Computer Methods in Biomechanics and Biomedical Engineering - 4 (Ed.: J. Middleton, N. G. Shrive &M. L. Jones), University of Wales College of Medicine, ISBN 1 903847095, 20026. NUMERICAL EXAMPLETo illustrate the influence of the micro–geometry on the macro–stiffness and macro–strength, a numerical example with the following characteristics was considered:bulk modulus, k, for cement is 7.2 · 109Pa, for glass is 78.7 · 109Pa; shear modulus,µ, for cement is 3.3 ·109Pa, for glass is 32.2 ·109Pa; uniaxial strength, σc, for cementis 50 · 106Pa, for glass is 200 · 106Pa.The averaged hydrostatic strength is determined from the solution to an auxiliaryproblem for a unit-cell hydrostatically extended from all sides by the normal dis-placements ±1/2 with zero shear tractions. For the averaged uniaxial strength,an auxiliary problem with uniaxial displacements ±1/2 on two opposite sides ofthe unit cell and zero normal displacement on other sides with zero shear tractionsshould be solved. The Finite Element code ANSYS was used to solve the auxiliaryperiodic problems (6). We suppose that the periodicity cell is symmetric and willconsider 1/8–th of the cell, structured as in the picture below: two glass spheres atthe opposite corners and another glass sphere on the cube diagonal between them.Inclusion radii are 0.124b, a = b/4, then the volume fraction of inclusions is 1.597%.The calculated composite properties are presented in the table.Bulk Mod.,[Pa] 7.23 · 109Shear Mod.,[Pa] 3.53 · 109Uniax. str.,[Pa] 43.91 · 106Hydr. str.,[Pa] 95.89 · 106Maximal von Mises stress arises for both hydrostatic and uniaxial tensions on thesurface of the middle glass ball at the point nearest to the corner ball. Nevertheless,the most dangerous appears to be the neighboring point of the matrix, due to thelarge difference in the strengths of the matrix and inclusion. Note that, unlike tothe components strength, the homogenized hydrostatic strength for the compositeis not infinite. The composite tensile initial strength is lower than strengths of theboth composite components.7. REFERENCES1. Pobedrya, B. E., Mechanics of composite materials, Moscow State UniversityPublishing, 1984.2. Allaire, G., Homogenization and Two-Scale Convergence, SIAM J. Math. Anal.,1992, Vol. 23, 1482–1518.3. Mikhailov, S. E., A functional approach to non-local strength conditions andfracture criteria: I&II, Eng. Fract. Mech., 1995, Vol. 52, No.4, 731–754.4. Mikhailov, S. E. and Orlik, J., Homogenization Methods and Macro-Strength ofComposites, Proceedings of the annual meeting GAMM 2001.5. Bogy, D.B., Two edge-bonded elastic wedges of different materials and wedge an-gles under surface tractions. Trans. ASME, Ser. E: J. Appl. Mech., 1971, Vol. 38,377–386.6. Mikhailov, S.E., Theoretical Backgrounds of Durability Analysis by NormalisedEquivalent Stress Functionals, Preprint PP/MAT/SEM/00-122, Glasgow Cale-donian University, 2000.6

A functional approach to non-local strength conditions and fracture criteria:

Homogenization and Two-Scale Convergence,

Mechanics of composite materials,

Theoretical Backgrounds of Durability Analysis by Normalised Equivalent Stress Functionals,

Two edge-bonded elastic wedges of diﬀerent materials and wedge angles under surface tractions.

Homogenization in strength and durability analysis of reinforced tooth filling

Homogenization in strength and durability analysis of reinforced tooth filling

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