364 Marcin Wierszycki, Witold Kąkąl, Tomasz Łodygowskireasons is the knowledge of stress and strain fields and the nature of mechanicalbehavior of implant structure. The stress and strain fields as well as themechanical behavior of implant may be obtained with the help of the FiniteElement Analysis (Zienkiewicz and Taylor, 2000). The finite element method (FEM) plays an important role today in solvingengineering problems in many fields of science and industry and can also besuccessfully applied in the simulations of biomechanical systems and dentalimplants (Będziński, 1997, Merz et al., 2000, Sakaguchi et al., 1993). Thesimulation with the help of the Finite Element Analysis (FEA) allows for takingthe key features into the consideration: like material inhomogeneity, anisotropyand changeability of mechanical properties of material as well as a verycomplicated geometry, boundary and load conditions. Therefore, the FEM is anefficient tool for testing biomechanical sets, like dental implants (Alkanet al., 2004, Eskitascioglu et al., 2004, Koca et al., 2005, Lang et al., 2003,Wierszycki et al. 2006). The physiological lateral and occlusal loads of implants haveunambiguous cyclic character. This enables us to indicate material fatigueas the source of the observed damage of implants (Genna, 2004,Khraisat et al., 2002, Teoh, 2000,). Also the examination of the fracture surfaceconfirms the supposition of fatigue character of dental implant damage(Zagalak, 2003). The aim of this study was to confirm this proposition by meansof numerical simulations of the dental implant system (Kąkol et al., 2002)carrying out fatigue calculations by the fe-safe program (Safe Technology, Ltd)and mechanical simulations by the ABAQUS/Standard (ABAQUS, Inc).2. ALGORITHM OF FATIGUE CALCULATIONS Many current theories describe the metal fatigue phenomenon as a threestage process. The first stage is a crack initiation. The fatigue cracks are initiatedfrom surface of the cyclic loaded components by the local stress and strain.The second stage is the crack propagation. The cracks are propagated in adirection perpendicular to the direct stress. The final stage is the damage of theloaded component. These are described by fracture mechanics (Draper, 1999).The earlier theories described fatigue failure as single process and fatiguecalculations were based on engineering stresses developed in structure. Thesecalculations of fatigue life used a relationship between amplitude of engineeringstresses and the number of cycles to failure. This relationship is well known as aso-called S-N curve. The modern fatigue theories express the endurance curve interms of local true strains at the small locations in the loaded components –notches, such as: holes, grooves or fillet radii. In these areas values of stress and
Fatigue algorithm for dental implant 365strain locally achieve very high level in comparison with their values calculatedwith the help of engineering stress approach (Kocańda, 1985, Schutz, 1996). Stress state from FEA s, e Material data Characteristic of load changing 400 375 350 Analysis of stress and strain state 325 300 275 250 225 200 175 150 125 100 75 50 25 0 0,00000 0,00200 0,00400 0,00600 0,00800 Pk E, K, n ∆σ, ∆ε Fatigue cycles analysis Fatigue material data Nf b, c Fatigue strength Fig. 1. Algorithm of fatigue calculations based on FEA results. s - nominal stress, e - nominal strain, Pk = P(t) – load signal, ∆σ – true stress amplitude, ∆ε – true strain amplitude, E – Youngs modulus, K – cyclic strain hardening coefficient, n – cyclicstrain hardening exponent, b – fatigue strength exponent, c – fatigue ductility exponent,σ’f – fatigue strength coefficient, ε’f – fatigue ductility coefficient, Nf – number of cycles
366 Marcin Wierszycki, Witold Kąkąl, Tomasz Łodygowski One of the fundamental assumptions of the local stress-strain theory isthat cracks initiated in the first stage of fatigue phenomenon are caused by localplasticity at notches. This first stage determines also the limit of fatigue life. Thefatigue behavior of material in these small volumes of real structure can becompared with fatigue life of smooth specimens of the same material undercorresponding stress-strain sequence (Draper, 1999). For this approach it isnecessary to have the information about strain field at each point of structure.This information can be obtained during experimental investigations (by usingstrain gauges) or estimated with the help of numerical simulations. The attractiveand useful technique of simulation is the finite element method (Bishopand Sherratt, 2000). The algorithm of the modern fatigue analysis, which isbased on results of the FEA, and used in advanced strain-life fatigue, is shown inFig.1. This algorithm, incorporating a multiaxial plasticity model, that usesstress results from the FEA, variations in loading, hysteresis loop cycle closure,and cyclic material properties, enables us to estimate the fatigue life and fatiguereliability factors (fe-safe Users Manual, 2005).2.1 FEA analysis The numerical analysis of the dental implant structure is not trivial.Nevertheless, it is a purely mechanical problem. In contrast to structure,the environmental conditions, such as external loads or boundary conditions,have already more complex, biomechanical reasons. The numerical modelof implant was created on the basis of the Polish, commonly used,implantological system OSTEOPLANT. This is a two-component systemwhich consists of root and abutment. They are connected with a non-rotationalhexagonal slot, assembled by a screw. Fig. 2. Axisymmetric model of dental implant with a part of jaw bone
Fatigue algorithm for dental implant 367 The majority of implant parts are axisymmetric. Due to this factthe geometry of numerical model was simplified to axisymmetric description(Fig.2). The external and internal threads of implant body and screw weresimplified to axisymmetric, parallel rings. Since the loads and deformationsof implant are asymmetric, it was necessary to use the semi-analyticaldiscretization technique. The applied axisymmetric solid elements use standardisoparametric interpolation in the radial – symmetry axis plane, combined withthe Fourier interpolation with respect to the angle of revolution. Thisformulation enables them to describe a nonlinear asymmetric deformation. Inorder to simplify this formulation and reduce the number of variables,the asymmetric deformation was assumed to be symmetric with respectto the radial – symmetry axis plane at an angle equal 0 or π (ABAQUSManuals, 2005). This approach significantly reduces the size of the problem (ca.75 000 dof) in comparison with a full three dimensional model (ca. 500 000dof). The costs of the calculation were also significantly reduced by this factitself (Wierszycki et al., 2006). All the components of an implant system are made of medical alloys oftitanium. For general stress-strain analyses, the isotropic, non-linear elastic-plastic characteristics of material models were taken into account. The materialproperties were based on the certificate of conformity and the literature(Wang, 1996). The mechanical properties of titanium alloys are shown inTable 1. For fatigue calculations, the model of material had to be simplified toa linear elastic description. Table 1. Mechanical properties of implant materials For a two-component implant, one of the most crucial aspects ofnumerical modeling is the simulation of mechanical assembly – tightening of theimplant screw process (Lang et al., 2003, Wierszycki et al. 2006). Foraxisymmetric model, tightening simulations cannot be performed as a realphysical process. A work-around approach is necessary. One of the possiblesolutions is the use of an artificial, non-physical temperature field and thermalexpansion of screw material. In this approach, the middle part of the screw wassubjected to thermal load. This temperature causes reduction of the screw lengthin its middle part. The implant body and abutment were tightened. The thermalexpansion property of the screw material was defined as orthotropic in such a
368 Marcin Wierszycki, Witold Kąkąl, Tomasz Łodygowskiway that shrinking occurred only along the screw axis. The value of axial forcein a tightened screw was calculated from the empirical equation (Bozkaya andMüfüt 2005, Merz et al., 2000, Lang et al., 2003). It is dependent on frictioncoefficient and torque moment. This force value varies from 80 to 850 N. Theycan be verified during full simulation of screw tightening with the help of a fullythree dimensional FE model of an implant (Wierszycki et al., 2006b). Thetightening simulation involves solving a complex contact problem. For thispurpose, it is necessary to define three pairs of the contact surfaces between: rootand abutment, root and screw, abutment and screw. The friction coefficientvarying from 0.1 (as in a specially finished surface) to 0.5 (as in dry titaniumto titanium friction) may be found in the literature. In the present analysis, threedifferent friction coefficients (0.1, 0.2 and 0.5) were considered. The frictioncharacteristics on these surface pairs is one of the key parameter influencingpreload axial force. For the fatigue life analysis, this preloading generatessignificant initial conditions for stress field and increases the value of meanstresses in the whole implant (Kąkol et al., 2002). a) b) c) Fig. 3. Boundary conditions of implant model – levels of osseointegration: a) 100%, b) 75%, c) 50% The boundary conditions of implants are modeled as a small partof the jaw bone. The geometry of a small part of jaw surrounding the implant isvery simplified, but it enables us to take into the consideration the changes inimplant fixing conditions (Koca et al., 2005, Sevimay et al., 2005). The materialmodel of bone was simplified to a linear elastic description as well. The jawbone is constantly undergoing remodeling processes. Within a few months, thejaw bone can be completely renewed. The changing stiffness of the bone andbone loss phenomenon are very important (Cehreli et al., 2004,
Fatigue algorithm for dental implant 369Goodacre et al., 2003, Taylor, 1998). Based on the classic theory of the fatiguecalculations it is not possible to take into account the bone remodeling in a directway. The selection of an appropriate approach for this phenomenon is not atrivial task. In this study, the following simplified approach was used. A seriesof simulations for varying stiffness of cancellous bone and levelsof osseointegration were performed (Table 2). The bone loss has a significantinfluence on the implant behavior, stress distribution and the consequentialfatigue damage. The degree of encasement and osseointegration of the implantdependents on the bone quality, the stresses developed during healing andfunction, and the location of the implant in the jaw. This percentage maydecrease to as low as 50% which is caused by remodeling and resorption of thebone phenomena. In these analyses, three levels of osseointegration wereconsidered. In the case of the first level, the implant body is fully fixed in thejaw bone. In the next two, the degree of implant body embedding decreases to75% and 50% respectively (Fig.3) (Hędzelek et al., 2003). Table 2. Stiffness of cortical and cancellous bone Bone Cortical Cancellous 1st scheme 13 000 9 500 2nd scheme 13 000 5 500 3rd scheme 13 000 1 600 4th scheme 13 000 690 The modeling approach with the use of the axisymmetric geometrydescription and the semi-analytical discretization enables us to carry outnumerous simulations in realistic time period. The three different valuesof friction coefficient, three boundary condition schemes and four valuesof cancellous bone stiffness were taken into the consideration (Wierszyckiet al., 2006). The fatigue crack initiation process is a surface phenomenon. The fatiguecalculations require the extrapolation of the stresses obtained from FEA inintegration points to the nodes of finite elements. The shape functions are usedfor this purpose (ABAQUS Manuals, 2005). The key aspect of the accuracy inextrapolation procedure is the necessity of averaging stresses from the adjacentelements. This extrapolation can have significant influence on the results forfatigue calculations. The extrapolated nodal values of stresses are generally notas accurate as the stress values calculated at the integration points. This problemis especially crucial for the areas of high stress gradients. The fine and detailedmeshes were necessary in the vicinity of notches in the implant model, where
370 Marcin Wierszycki, Witold Kąkąl, Tomasz Łodygowskiaccurate nodal values of stresses are required for correct fatigue life calculation(fe-safe Users Manual, 2005).2.2. Strain-life fatigue analysis The strain-life fatigue calculations are based on the material responseto the cyclic elastic-plastic strain and the relationship between these strains andfatigue durability. If yielding occurs in one of the areas of model for cyclicloading, the material response at this node is a hysteresis loop of the true stressand the elastic-plastic strain. The strain-life fatigue calculations base on ananalysis of the sequence of true stress and identification of the closed hysteresisloop – fatigue cycle. A rainflow cycle counting algorithm is used to extractfatigue cycles. The stable material cyclic response is approximated with the helpof the cyclic stress-strain curve. The cyclic stress-strain curve is constructedthrough the peaks of hysteresis loops for a different constant amplitude of totalstrain. According to Masing’s hypothesis, for many homogenous materials,the hysteresis loop equation can be obtained with the help of re-scalingthe cyclic stress-strain curve by factor 2. The hysteresis loop equation in termsof ranges is: 1 ∆σ ∆σ n ∆ε = + 2 (2.1) E 2K where ∆σ and ∆ε are the amplitudes of true stress and strain. The cyclic materialproperties are described by Young modulus E, cyclic strain hardening coefficientK and cyclic strain hardening exponent n (Bishop and Sherratt, 2000). The equation of the strain-life relationship is obtained throughconsidering the amplitude of the total strain which is expressed as a sum of theelastic and plastic strain amplitude. These relationships between strain amplitude∆ε and fatigue endurance 2Nf can be assumed as linear on log10-log10 axes .The basic uniaxial strain-life equation is: ∆ε σ f , = (2 N f ) b + ε ,f (2 N f ) c , (2.2) 2 EThe elastic component of this equation is expressed by the fatigue strengthcoefficient σ’f and the Basquin’s exponent. The plastic component dependson the fatigue ductility coefficient ε’f and the Coffin-Manson exponent.For multiaxial fatigue analysis, some modifications are needed. The approachto fatigue bases on two assumptions: first is that the damage occurs mostly onthe plane of the maximum shear strain amplitude, while the second says that thedamage is a function of this shear strain and also strain normal to this plane.Based on the well known strain equations for plane stress and stress-strain
Fatigue algorithm for dental implant 371relationship, the strain-life equation can be written as function of the maximumshear strain γmax and strain normal to the maximum shear strain εn: ∆γ max ∆ε n σf , + = 1.65 (2 N f ) b + 1.75ε ,f (2 N f ) c . (2.3) 2 2 EThe change of constants on the right-hand side of the Eqn. 2.3, corresponding tothe basic uniaxial strain-life equation (2.2), is made by taking into theconsideration the uniaxial plane stress condition. For elastic componentof strain-life relationship in Eqn. 2.3, the approximate value of Poisson’s ratioamount to 0.3 is used. For the plastic component, assuming purely plasticstrains, Poisson’s ratio is taken as amounting to 0.5 (Draper, 1999). For each node of the FE model, the 6-stress tensor is used to calculatethe in-plane principal stresses and their orientation. If the direction of principalstresses is not constant, a critical plane procedure is used to calculatethe orientation of the most damaged plane at the node. These elastic stresses aremultiplied by the loading sequence to form a stress- and strain-time historyat each node. The time history of the principal elastic stress and strain needsto be translated into the elastic-plastic stress-strain. This elastic-plasticconversion, performed with the help of biaxial Neubers rule, uses assignedcyclic material properties. The stress concentration factor and scale factor can beapplied at this stage. Neubers rule equates the total strain energy for the fullyelastic and elastic-plastic stress-strains. This relationship is defined as Neuberhyperbola and is expressed by the following equation: 2 sk ∆σ ∆ε = K t2 , (2.4) Ewhere ∆σ and ∆ε are the amplitudes of true stress and strain. The nominal stress-strain product is defined by the elastic stress concentration factor Kt, nominalelastic stress sk and Young modulus E. For this approach, stresses at each nodeare treated as a separate entity and not as a discrete description of field. Thistechnique of true elastic to elastic-plastic stresses conversion cannot take into theconsideration the stress redistribution phenomena as a result of yielding. Formost cases of fatigue damage, this approximation can be accepted, becauseyielding occurs only in small areas of the structure, such as notches. The cyclicstrain-time history was used for the strain-life fatigue analysis. The value ofmean stresses has significant influence on the fatigue life. It is necessary to takeinto the consideration the correction for mean stress effect in fatiguecalculations. In our study, the Morrow’s correction was used. The elastic term ofstrain-life equation is corrected by subtracting the value of mean stresses of thecycle from the fatigue strength coefficient. The cyclic stress-strain curves are
372 Marcin Wierszycki, Witold Kąkąl, Tomasz Łodygowskimodified. For each node of FE model, the full set of simultaneous equations wassolved and the fatigue life was calculated (Draper, 1999). The basic results of fatigue calculations are the fatigue lives. In the FEApost-processor, the contour plots of the fatigue life can be displayed. In order toestimate the failure-free term of implant, a design life is defined and the fatiguestrength reserve factor (FOS – Factor Of Strength) is calculated. This is definedas the scale factor, by means of which the stresses at each node can be increasedor reduced, in order to give the required fatigue life. This very interesting andvivid result, which is displayed as the contour plot, can show how much thestructure is over- or under-strength for expected designed life. The iterativeapproach is necessary to calculate the FOS at each node. In the first step, thecurrent calculated fatigue life is compared to the assumed designed life. The 5%tolerance is used. Next, if the current value of fatigue life for a given node islower or higher than the design life, the stresses at this node are re-scaled byfactor adequately lower or higher than 1.0. For FOS value near to 1.0, the scalefactor is 0.01. In another case, the scale factor is 0.1. At the next stage, thestress- and strain-time history are recalculated for new value of re-scaledstresses. The in-plane principal stresses and their orientation are recalculated.Next, the time history of the principal elastic stress and strain are translated intothe elastic-plastic stress-strain time history. Finally, the fatigue life isrecalculated and compared with the assumed design life. If it is necessary, thenext iteration is performed (fe-safe Users Manual, 2005).2.3. Fatigue material data The fatigue calculations with the use of the strain-life fatigue analysisneed accurate the fatigue material data to provide the correct fatigue lifeestimation. The best source of the fatigue material characteristics are theexperimental tests of smooth specimens for constant strain amplitudes betweenfixed strain limits (Draper, 1999). Naturally, this source needs many technicallyadvanced resources, and it is expensive and complex. The easiest attainablesources of fatigue material data are the literature and manufacturers’ certificates.Commonly published by titanium manufacturers and researchers, the fatiguecharacteristics often provide information only about basic fatigue strength for aspecific endurance. These data are useful for the stress-life analysis but areinsufficient for the strain-life approach. For the strain-life fatigue analysis,the six additional material properties are required: the cyclic strain hardeningcoefficient K, the cyclic strain hardening exponent n, the fatigue strengthexponent b, the fatigue ductility exponent c, the fatigue strength coefficient σ’f,and the fatigue ductility coefficient ε’f. There are many approximatedrelationships available, in order to obtain these data. All of them base on somephysical interpretations of the fatigue properties or the relationships between thefatigue properties and other well known physical parameters of materials such as
Fatigue algorithm for dental implant 373the modulus of elasticity, hardness or tensile properties (Lee K.-S. and Song J.-H., 2006, Park and Song, 1995, Roessle and Fatemi, 2000). In this study, theSeeger’s method was used. The fatigue strength coefficient σ’f and cyclic strainhardening coefficient K are approximated with the help of the re-scalingconventional monotonic ultimate tensile stress σu. For titanium alloys: σ ,f = 1.67σ u , (2.5) K = 1.61σ u . (2.6)The Seeger’s method is a modification of the method of universal slopes andassumes that the slopes of elastic and plastic asymptotes of strain-life curve arethe same for some specific kinds of alloys. For titanium alloys b = -0.095 andc = -0.69. Similarly, the cyclic strain hardening exponent and fatigue ductilitycoefficient were assumed as constants, n = 0.11 and ε’f = 0.35 (Draper, 1999).The applied fatigue data of implant titanium alloys are shown in Table 3. Table 3. Fatigue material data Implant body Abutment Screw K [MPa] 1195.26 1561.7 1616.44 n 0.11 0.11 0.11 b -0.095 -0.095 -0.095 c -0.69 -0.69 -0.69 σ’f [MPa] 1239.81 1619.9 1676.68 ε’f 0.35 0.35 0.352.4. Characteristics of load The external loads of implant model were applied in the second step of thesimulation. The loading of an implant is never axial. The maximum values of thevertical components of it are estimated at 600 N and the horizontal ones at 100 N(Hędzelek et al., 2003). In fatigue analyses, only the horizontal component wastaken into consideration. For the fatigue life calculations, it is necessary todefine the nature of the load changeability as a curve load-time history. Theloading sequence definition is the primary condition for the correct stress- andstrain-time history forming. It is not easy to determine these time historycharacteristics and typical values of occlusal forces. The experimentalinvestigations such as measurements of loads or strain in oral cavityenvironment are extremely difficult and impossible to perform in practice.Therefore, there is no information about occlusal forces sequence useful forfatigue calculations in recognized literature. For this study, a physiologicallyproven scheme of occlusal forces a sequence was assumed, see Fig. 4.
374 Marcin Wierszycki, Witold Kąkąl, Tomasz ŁodygowskiThe maximum and average values and directions of the loading were defined onthe grounds of the information from the literature. The sequence of the loads wasapproximated with the help of random number generator. In an applied high-cycled scheme of 24-hour loads, the following assumptions were done:the average values of occlusal forces are 20 N, the number of occlusal contactsfor one meal are 20 per minute, the average time of meal is 15 minutes,the number of meals are three per day, the number of occlusal contacts for timeperiod between meals are 2 · 105 per year. The assumed number of dayscorresponding to four years was taken as the number of cycles (Zagalak, 2003)3. SELECTED RESULTS OF FATIGUE CALCULATIONS The FOS distribution analysis for particular cases indicates the axialforces in the screw and the changes in the scheme of boundary conditions havethe greatest influence on fatigue changes. For axial forces up to 400 N, the fatigue changes occurin the neighborhood of the first thread twist and the notch under the screw head.For these areas, the stresses should be 5-10 % reduced to achieve the assumeddesigned life. For axial forces above 600 N, there is a noticeable increasein the areas endangered by the fatigue failure. The degree of required stressreduction reaches ca. 30% as well. In the most unfavorable load case,the maximal axial force value (ca. 900 N) is the result of a high torque anda very small friction coefficient on a screw thread (Kąkol et al., 2003). In two-component implants, the high tightening force has a biological and medicalmotivation: assurance of tightness of abutment to implant body connection,reduction of mobility of implant components and screw loosening resistance(Gratton et al., 2001, Hecker et al., 2006, Khraisat et al., 2004). However, it isimportant to pay attention to the fatigue results of this increase in tighteningforce which significantly reduces the fatigue life of implant components(Kąkol et al.,2003). For different bone density and, at the same time, divergent stiffnessof boundary conditions, significant differences of stress distributions presentin the screw are noticeable. Yet, it does not lead to any serious fatigue changes(Wierszycki et al., 2006).
Fatigue algorithm for dental implantFig. 4. Sequence of the loads time-history 375
376 Marcin Wierszycki, Witold Kąkąl, Tomasz ŁodygowskiThe fatigue life is strongly dependent on the values of mean stress. The highmean stress causes a shorter fatigue life for the same amplitude of strain. Thechange of stiffness of boundary conditions does not cause a significant change ofmean stress, which is produced by the tightening force. This is especially visiblein the screw. The simulations performed for four different levels of osseointegrationproved significant influence of the bone loss phenomenon on distribution andvalues of stress in individual implant components and fatigue life of wholestructure as well. The boundary condition schemes, corresponding to differentosseointegration levels, change the mechanical response of an implant.This causes large differences in values of mean stress and rage of amplitudeof strain. Finally, the fatigue changes appear in various areas of implantcomponents, see Fig. 5 (Wierszycki et al., 2006). a) b) Fig. 5. FOS distribution in screw for 4-year design life and two different levels of osseointegration: 75% (a), 100% (b)
Fatigue algorithm for dental implant 3774. EXPERIMENTAL VERIFICATION The confirmation of the numerical results is a case report of implantfracture and examination of fractured surface. The mechanical failure occurredin a 55-year-old male patient after one-year period of using single crownreplacing the first maxillar premolar (Zagalak, 2003). The fracture line passedthrough the upper part of implant body and abutment screw. The failure waslocated transversally to the long axis between smooth and threaded part of thefixture. The radiograph showed also the bone loss down to the third thread of theimplant. The examination with the use of scanning electron microscopepresented the characteristic for fatigue damage changes like fatigue bands,indicating the advancement of the crack front under cyclic loading. In Fig. 8fracture surface of dental implant is presented. The fractography revealedobliteration and incrustation of some of the fracture features most probablybecause of postfracture wear of contacting surfaces, which were the resultof the component remain joined together by abutment screw. It is also possibleto identify area with plastic deformation corresponding to final fracture. a) b) Fig. 6. Fatigue fracture surface of dental implant (SEM pictures): a) 150x, b) 900x.5. CONCLUSIONS The work presents the algorithm used for the fatigue analysis of the dentalimplant system. The results of the numerical computations performed in the
378 Marcin Wierszycki, Witold Kąkąl, Tomasz Łodygowskienvironment of finite element code depend significantly on initial the stressstates, boundary conditions that reflect the level of osseointegration of theimplant with the bones and daily sequence of loading. The computations verifiedby the experimental observations prove the fatigue character of final failure ofimplant screw. The numerical results can serve as a good predictor of the time tochange this element.BIBLIOGRAPHY1. ABAQUS Analysis Users Manual, ABAQUS, Inc. Pawtucket, 2005.2. ABAQUS Theory Manual, ABAQUS, Inc. Pawtucket, 2005.3. feSafe Users Manual, Safe Technology Ltd, Sheffield, 2005.4. Alkan I., Sertgöz A., Ekici B., Influence of occlusal forces on stress distribution in preloaded dental implant screws, The Journal of Prosthetic Dentistry, 91, 4 (2004) 319-325.5. Będziński R., Biomechanika inżynierska (in Polish), Oficyna Wydawnicza Politechniki Wrocławskiej, Wrocław, 1997.6. Bishop N. W. M., Sherratt F., Finite Element Based Fatigue Calculations, NAFEMS, Glasgow, 2000.7. Bozkaya D., Müfüt S., Mechanics of the taper integrated screwed-in (TIS) abutments used in dental implants, Journal of Biomechanics, 38 (2005) 87– 97.8. Cehreli .M, Sahin .S, Akca K., Role of mechanical environment and implant design on bone tissue differentiation: current knowledge and future contexts, Journal of Dentistry, 32 (2004) 123–132.9. Draper J., Modern metal fatigue analysis, HKS, Inc. Pawtucket, 1999.10. Eskitascioglu G., Usumez A., Sevimay M., Soykan M., Unsal E. The influence of occlusal loading location on stresses transferred to implant- supported prostheses and supporting bone: A three-dimensional finite element study, The Journal of Prosthetic Dentistry, 91, 2 (2004) 144-150.11. Genna F., Shakedown, self-stresses, and unilateral contact in a dental implant problem, European Journal of Mechanics A/Solids, 23 (2004) 485– 498.12. Goodacre C. J., Bernal G., Rungcharassaeng K., Kan J. Y. K., Clinical complications with implants and implant prostheses, The Journal of Prosthetic Dentistry, 90, 2 (2003) 121-132.13. Gratton D. G., Aquilino S. A., Stanford C. M., Micromotion and dynamic fatigue properties of the dental implant–abutment interface, The Journal of Prosthetic Dentistry, 85, 1 (2001) 47-52.14. Hecker D. M., Eckert S. E., Choi Y.-G., Cyclic loading of implant-supported prostheses: Comparison of gaps at the prosthetic-abutment interface when
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