FATIGUE ALGORITHM FOR DENTAL IMPLANT

F O U N D A T I O N S O F C I V I L A N D E N V I R O N M E N T A L E N G I N E E R I N G

No. 7 2006

Marcin WIERSZYCKI*, Witold KĄKOL, Tomasz ŁODYGOWSKI

Institute of Structural Engineering, Poznań University of Technology, ul. Piotrowo 5, Poznań, Poland

FATIGUE ALGORITHM FOR DENTAL IMPLANT

Received: 11 September 2006 Accepted: 26 September 2006

Restorations with the application of implants are effective and commonly used in

dental treatment. The computer simulation of implant fatigue life employing FEA is considered in the paper. For the simulation of implant structure behavior a 3D model including a spiral thread is applied, which allows for the full simulation of the kinematics of the implant, describing the multiaxial state of stress and, in consequence, the possibility of screw loosening. The cyclic scheme of the physiological occlusal loading and fatigue changes of dental material, bone loss phenomenon and changeability of boundary conditions are investigated. The valuable results for fatigue life which may be applied in modern prosthodontics are presented.

Key words: fatigue, biomechanics, dental implant 1. INTRODUCTION Implants are commonly applied methods of treatment in construction of dental restorations. Unfortunately, numerous clinical observations point the occurrence of both early and late complications (Goodacre et al., 2003). In many cases, these problems are caused by mechanical fractures of the implants themselves (Hędzelek et al., 2003, Kąkol et al., 2002). One of the most serious complications are fracture and cracking of the dental implant parts. Damage of dental implant components makes the further treatment very difficult. The ground of understanding the observed mechanical complications and their

* Corresponding author. Tel.: (48 61) 665 21 03; fax: -.

© Publishing House of Poznan University of Technology, Poznań 2006 ISSN 1642-9303

E-mail address: [email protected] (M. Wierszycki)

364 Marcin Wierszycki, Witold Kąkąl, Tomasz Łodygowski

reasons is the knowledge of stress and strain fields and the nature of mechanical behavior of implant structure. The stress and strain fields as well as the mechanical behavior of implant may be obtained with the help of the Finite Element Analysis (Zienkiewicz and Taylor, 2000).

The finite element method (FEM) plays an important role today in solving engineering problems in many fields of science and industry and can also be successfully applied in the simulations of biomechanical systems and dental implants (Będziński, 1997, Merz et al., 2000, Sakaguchi et al., 1993). The simulation with the help of the Finite Element Analysis (FEA) allows for taking the key features into the consideration: like material inhomogeneity, anisotropy and changeability of mechanical properties of material as well as a very complicated geometry, boundary and load conditions. Therefore, the FEM is an efficient tool for testing biomechanical sets, like dental implants (Alkan et 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 have unambiguous cyclic character. This enables us to indicate material fatigue as the source of the observed damage of implants (Genna, 2004, Khraisat et al., 2002, Teoh, 2000,). Also the examination of the fracture surface confirms the supposition of fatigue character of dental implant damage (Zagalak, 2003). The aim of this study was to confirm this proposition by means of 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 three stage process. The first stage is a crack initiation. The fatigue cracks are initiated from surface of the cyclic loaded components by the local stress and strain. The second stage is the crack propagation. The cracks are propagated in a direction perpendicular to the direct stress. The final stage is the damage of the loaded component. These are described by fracture mechanics (Draper, 1999). The earlier theories described fatigue failure as single process and fatigue calculations were based on engineering stresses developed in structure. These calculations of fatigue life used a relationship between amplitude of engineering stresses and the number of cycles to failure. This relationship is well known as a so-called S-N curve. The modern fatigue theories express the endurance curve in terms 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 365

strain locally achieve very high level in comparison with their values calculated with the help of engineering stress approach (Kocańda, 1985, Schutz, 1996).

Stress state from FEA

Characteristic of load changing

Fatigue material data

Analysis of stress and strain state

Fatigue cycles analysis

Fatigue strength

Material data

0,00000 0,00200 0,00400 0,00600 0,008000255075100125150175200225250275300325350375400

s, e

Pk E, K', n'∆σ, ∆ε

b, cNf

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 – Young's modulus, K' – cyclic strain hardening coefficient, n' – cyclic

strain 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 is that cracks initiated in the first stage of fatigue phenomenon are caused by local plasticity at notches. This first stage determines also the limit of fatigue life. The fatigue behavior of material in these small volumes of real structure can be compared with fatigue life of smooth specimens of the same material under corresponding stress-strain sequence (Draper, 1999). For this approach it is necessary to have the information about strain field at each point of structure. This information can be obtained during experimental investigations (by using strain gauges) or estimated with the help of numerical simulations. The attractive and useful technique of simulation is the finite element method (Bishop and Sherratt, 2000). The algorithm of the modern fatigue analysis, which is based on results of the FEA, and used in advanced strain-life fatigue, is shown in Fig.1. This algorithm, incorporating a multiaxial plasticity model, that uses stress results from the FEA, variations in loading, hysteresis loop cycle closure, and cyclic material properties, enables us to estimate the fatigue life and fatigue reliability factors (fe-safe User's 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 model of implant was created on the basis of the Polish, commonly used, implantological system OSTEOPLANT. This is a two-component system which consists of root and abutment. They are connected with a non-rotational hexagonal 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 fact the geometry of numerical model was simplified to axisymmetric description (Fig.2). The external and internal threads of implant body and screw were simplified to axisymmetric, parallel rings. Since the loads and deformations of implant are asymmetric, it was necessary to use the semi-analytical discretization technique. The applied axisymmetric solid elements use standard isoparametric interpolation in the radial – symmetry axis plane, combined with the Fourier interpolation with respect to the angle of revolution. This formulation enables them to describe a nonlinear asymmetric deformation. In order to simplify this formulation and reduce the number of variables, the asymmetric deformation was assumed to be symmetric with respect to the radial – symmetry axis plane at an angle equal 0 or π (ABAQUS Manuals, 2005). This approach significantly reduces the size of the problem (ca. 75 000 dof) in comparison with a full three dimensional model (ca. 500 000 dof). The costs of the calculation were also significantly reduced by this fact itself (Wierszycki et al., 2006). All the components of an implant system are made of medical alloys of titanium. For general stress-strain analyses, the isotropic, non-linear elastic-plastic characteristics of material models were taken into account. The material properties were based on the certificate of conformity and the literature (Wang, 1996). The mechanical properties of titanium alloys are shown in Table 1. For fatigue calculations, the model of material had to be simplified to a linear elastic description.

Table 1. Mechanical properties of implant materials

For a two-component implant, one of the most crucial aspects of numerical modeling is the simulation of mechanical assembly – tightening of the implant screw process (Lang et al., 2003, Wierszycki et al. 2006). For axisymmetric model, tightening simulations cannot be performed as a real physical process. A work-around approach is necessary. One of the possible solutions is the use of an artificial, non-physical temperature field and thermal expansion of screw material. In this approach, the middle part of the screw was subjected to thermal load. This temperature causes reduction of the screw length in its middle part. The implant body and abutment were tightened. The thermal expansion property of the screw material was defined as orthotropic in such a

368 Marcin Wierszycki, Witold Kąkąl, Tomasz Łodygowski

way that shrinking occurred only along the screw axis. The value of axial force in a tightened screw was calculated from the empirical equation (Bozkaya and Müfüt 2005, Merz et al., 2000, Lang et al., 2003). It is dependent on friction coefficient and torque moment. This force value varies from 80 to 850 N. They can be verified during full simulation of screw tightening with the help of a fully three dimensional FE model of an implant (Wierszycki et al., 2006b). The tightening simulation involves solving a complex contact problem. For this purpose, it is necessary to define three pairs of the contact surfaces between: root and abutment, root and screw, abutment and screw. The friction coefficient varying from 0.1 (as in a specially finished surface) to 0.5 (as in dry titanium to titanium friction) may be found in the literature. In the present analysis, three different friction coefficients (0.1, 0.2 and 0.5) were considered. The friction characteristics on these surface pairs is one of the key parameter influencing preload axial force. For the fatigue life analysis, this preloading generates significant initial conditions for stress field and increases the value of mean stresses 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 part of the jaw bone. The geometry of a small part of jaw surrounding the implant is very simplified, but it enables us to take into the consideration the changes in implant fixing conditions (Koca et al., 2005, Sevimay et al., 2005). The material model of bone was simplified to a linear elastic description as well. The jaw bone is constantly undergoing remodeling processes. Within a few months, the jaw bone can be completely renewed. The changing stiffness of the bone and bone loss phenomenon are very important (Cehreli et al., 2004,

Fatigue algorithm for dental implant 369

Goodacre et al., 2003, Taylor, 1998). Based on the classic theory of the fatigue calculations it is not possible to take into account the bone remodeling in a direct way. The selection of an appropriate approach for this phenomenon is not a trivial task. In this study, the following simplified approach was used. A series of simulations for varying stiffness of cancellous bone and levels of osseointegration were performed (Table 2). The bone loss has a significant influence on the implant behavior, stress distribution and the consequential fatigue damage. The degree of encasement and osseointegration of the implant dependents on the bone quality, the stresses developed during healing and function, and the location of the implant in the jaw. This percentage may decrease to as low as 50% which is caused by remodeling and resorption of the bone phenomena. In these analyses, three levels of osseointegration were considered. In the case of the first level, the implant body is fully fixed in the jaw bone. In the next two, the degree of implant body embedding decreases to 75% 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 geometry description and the semi-analytical discretization enables us to carry out numerous simulations in realistic time period. The three different values of friction coefficient, three boundary condition schemes and four values of cancellous bone stiffness were taken into the consideration (Wierszycki et al., 2006). The fatigue crack initiation process is a surface phenomenon. The fatigue calculations require the extrapolation of the stresses obtained from FEA in integration points to the nodes of finite elements. The shape functions are used for this purpose (ABAQUS Manuals, 2005). The key aspect of the accuracy in extrapolation procedure is the necessity of averaging stresses from the adjacent elements. This extrapolation can have significant influence on the results for fatigue calculations. The extrapolated nodal values of stresses are generally not as accurate as the stress values calculated at the integration points. This problem is especially crucial for the areas of high stress gradients. The fine and detailed meshes were necessary in the vicinity of notches in the implant model, where

370 Marcin Wierszycki, Witold Kąkąl, Tomasz Łodygowski

accurate nodal values of stresses are required for correct fatigue life calculation (fe-safe User's Manual, 2005). 2.2. Strain-life fatigue analysis The strain-life fatigue calculations are based on the material response to the cyclic elastic-plastic strain and the relationship between these strains and fatigue durability. If yielding occurs in one of the areas of model for cyclic loading, the material response at this node is a hysteresis loop of the true stress and the elastic-plastic strain. The strain-life fatigue calculations base on an analysis of the sequence of true stress and identification of the closed hysteresis loop – fatigue cycle. A rainflow cycle counting algorithm is used to extract fatigue cycles. The stable material cyclic response is approximated with the help of the cyclic stress-strain curve. The cyclic stress-strain curve is constructed through the peaks of hysteresis loops for a different constant amplitude of total strain. According to Masing’s hypothesis, for many homogenous materials, the hysteresis loop equation can be obtained with the help of re-scaling the cyclic stress-strain curve by factor 2. The hysteresis loop equation in terms of ranges is:

'1

'22

n

KE

∆

+∆

=∆σσε (2.1)

where ∆σ and ∆ε are the amplitudes of true stress and strain. The cyclic material properties are described by Young modulus E, cyclic strain hardening coefficient K' and cyclic strain hardening exponent n' (Bishop and Sherratt, 2000).

The equation of the strain-life relationship is obtained through considering the amplitude of the total strain which is expressed as a sum of the elastic 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:

cff

bf

f NNE

)2()2(2

,,

εσε

+=∆

, (2.2)

The elastic component of this equation is expressed by the fatigue strength coefficient σ’f and the Basquin’s exponent. The plastic component depends on the fatigue ductility coefficient ε’f and the Coffin-Manson exponent. For multiaxial fatigue analysis, some modifications are needed. The approach to fatigue bases on two assumptions: first is that the damage occurs mostly on the plane of the maximum shear strain amplitude, while the second says that the damage 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 371

relationship, the strain-life equation can be written as function of the maximum shear strain γmax and strain normal to the maximum shear strain εn:

cff

bf

fn NNE

)2(75.1)2(65.122

,,

max εσεγ

+=∆

+∆

. (2.3)

The change of constants on the right-hand side of the Eqn. 2.3, corresponding to the basic uniaxial strain-life equation (2.2), is made by taking into the consideration the uniaxial plane stress condition. For elastic component of strain-life relationship in Eqn. 2.3, the approximate value of Poisson’s ratio amount to 0.3 is used. For the plastic component, assuming purely plastic strains, 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 calculate the in-plane principal stresses and their orientation. If the direction of principal stresses is not constant, a critical plane procedure is used to calculate the orientation of the most damaged plane at the node. These elastic stresses are multiplied by the loading sequence to form a stress- and strain-time history at each node. The time history of the principal elastic stress and strain needs to be translated into the elastic-plastic stress-strain. This elastic-plastic conversion, performed with the help of biaxial Neuber's rule, uses assigned cyclic material properties. The stress concentration factor and scale factor can be applied at this stage. Neuber's rule equates the total strain energy for the fully elastic and elastic-plastic stress-strains. This relationship is defined as Neuber hyperbola and is expressed by the following equation:

22 ,kt

sKE

σ ε∆ ∆ = (2.4)

where ∆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, nominal elastic stress sk and Young modulus E. For this approach, stresses at each node are treated as a separate entity and not as a discrete description of field. This technique of true elastic to elastic-plastic stresses conversion cannot take into the consideration the stress redistribution phenomena as a result of yielding. For most cases of fatigue damage, this approximation can be accepted, because yielding occurs only in small areas of the structure, such as notches. The cyclic strain-time history was used for the strain-life fatigue analysis. The value of mean stresses has significant influence on the fatigue life. It is necessary to take into the consideration the correction for mean stress effect in fatigue calculations. In our study, the Morrow’s correction was used. The elastic term of strain-life equation is corrected by subtracting the value of mean stresses of the cycle from the fatigue strength coefficient. The cyclic stress-strain curves are

372 Marcin Wierszycki, Witold Kąkąl, Tomasz Łodygowski

modified. For each node of FE model, the full set of simultaneous equations was solved and the fatigue life was calculated (Draper, 1999).

The basic results of fatigue calculations are the fatigue lives. In the FEA post-processor, the contour plots of the fatigue life can be displayed. In order to estimate the failure-free term of implant, a design life is defined and the fatigue strength reserve factor (FOS – Factor Of Strength) is calculated. This is defined as the scale factor, by means of which the stresses at each node can be increased or reduced, in order to give the required fatigue life. This very interesting and vivid result, which is displayed as the contour plot, can show how much the structure is over- or under-strength for expected designed life. The iterative approach is necessary to calculate the FOS at each node. In the first step, the current 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 is lower or higher than the design life, the stresses at this node are re-scaled by factor adequately lower or higher than 1.0. For FOS value near to 1.0, the scale factor is 0.01. In another case, the scale factor is 0.1. At the next stage, the stress- and strain-time history are recalculated for new value of re-scaled stresses. The in-plane principal stresses and their orientation are recalculated. Next, the time history of the principal elastic stress and strain are translated into the elastic-plastic stress-strain time history. Finally, the fatigue life is recalculated and compared with the assumed design life. If it is necessary, the next iteration is performed (fe-safe User's Manual, 2005). 2.3. Fatigue material data The fatigue calculations with the use of the strain-life fatigue analysis need accurate the fatigue material data to provide the correct fatigue life estimation. The best source of the fatigue material characteristics are the experimental tests of smooth specimens for constant strain amplitudes between fixed strain limits (Draper, 1999). Naturally, this source needs many technically advanced resources, and it is expensive and complex. The easiest attainable sources of fatigue material data are the literature and manufacturers’ certificates. Commonly published by titanium manufacturers and researchers, the fatigue characteristics often provide information only about basic fatigue strength for a specific endurance. These data are useful for the stress-life analysis but are insufficient for the strain-life approach. For the strain-life fatigue analysis, the six additional material properties are required: the cyclic strain hardening coefficient K', the cyclic strain hardening exponent n', the fatigue strength exponent b, the fatigue ductility exponent c, the fatigue strength coefficient σ’f, and the fatigue ductility coefficient ε’f. There are many approximated relationships available, in order to obtain these data. All of them base on some physical interpretations of the fatigue properties or the relationships between the fatigue properties and other well known physical parameters of materials such as

Fatigue algorithm for dental implant 373

the 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, the Seeger’s method was used. The fatigue strength coefficient σ’f and cyclic strain hardening coefficient K' are approximated with the help of the re-scaling conventional monotonic ultimate tensile stress σu. For titanium alloys:

uf σσ 67.1, = ,

uK σ61.1'= . (2.5)(2.6)

The Seeger’s method is a modification of the method of universal slopes and assumes that the slopes of elastic and plastic asymptotes of strain-life curve are the same for some specific kinds of alloys. For titanium alloys b = -0.095 and c = -0.69. Similarly, the cyclic strain hardening exponent and fatigue ductility coefficient 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 0.35

2.4. Characteristics of load The external loads of implant model were applied in the second step of the simulation. The loading of an implant is never axial. The maximum values of the vertical 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 was taken into consideration. For the fatigue life calculations, it is necessary to define the nature of the load changeability as a curve load-time history. The loading sequence definition is the primary condition for the correct stress- and strain-time history forming. It is not easy to determine these time history characteristics and typical values of occlusal forces. The experimental investigations such as measurements of loads or strain in oral cavity environment are extremely difficult and impossible to perform in practice. Therefore, there is no information about occlusal forces sequence useful for fatigue calculations in recognized literature. For this study, a physiologically proven scheme of occlusal forces a sequence was assumed, see Fig. 4.

374 Marcin Wierszycki, Witold Kąkąl, Tomasz Łodygowski

The maximum and average values and directions of the loading were defined on the grounds of the information from the literature. The sequence of the loads was approximated 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 contacts for 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 time period between meals are 2 · 105 per year. The assumed number of days corresponding 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 axial forces in the screw and the changes in the scheme of boundary conditions have the greatest influence on fatigue changes.

For axial forces up to 400 N, the fatigue changes occur in 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 assumed designed life. For axial forces above 600 N, there is a noticeable increase in the areas endangered by the fatigue failure. The degree of required stress reduction 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 and a very small friction coefficient on a screw thread (Kąkol et al., 2003). In two-component implants, the high tightening force has a biological and medical motivation: 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 is important to pay attention to the fatigue results of this increase in tightening force which significantly reduces the fatigue life of implant components (Kąkol et al.,2003).

For different bone density and, at the same time, divergent stiffness of boundary conditions, significant differences of stress distributions present in the screw are noticeable. Yet, it does not lead to any serious fatigue changes (Wierszycki et al., 2006).

Fatigue algorithm for dental implant 375

Fig.

4. S

eque

nce

of th

e lo

ads t

ime-

hist

ory

376 Marcin Wierszycki, Witold Kąkąl, Tomasz Łodygowski

The fatigue life is strongly dependent on the values of mean stress. The high mean stress causes a shorter fatigue life for the same amplitude of strain. The change of stiffness of boundary conditions does not cause a significant change of mean stress, which is produced by the tightening force. This is especially visible in the screw.

The simulations performed for four different levels of osseointegration proved significant influence of the bone loss phenomenon on distribution and values of stress in individual implant components and fatigue life of whole structure as well. The boundary condition schemes, corresponding to different osseointegration levels, change the mechanical response of an implant. This causes large differences in values of mean stress and rage of amplitude of strain. Finally, the fatigue changes appear in various areas of implant components, 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 377

4. EXPERIMENTAL VERIFICATION The confirmation of the numerical results is a case report of implant fracture and examination of fractured surface. The mechanical failure occurred in a 55-year-old male patient after one-year period of using single crown replacing the first maxillar premolar (Zagalak, 2003). The fracture line passed through the upper part of implant body and abutment screw. The failure was located transversally to the long axis between smooth and threaded part of the fixture. The radiograph showed also the bone loss down to the third thread of the implant. The examination with the use of scanning electron microscope presented the characteristic for fatigue damage changes like fatigue bands, indicating the advancement of the crack front under cyclic loading. In Fig. 8 fracture surface of dental implant is presented. The fractography revealed obliteration and incrustation of some of the fracture features most probably because of postfracture wear of contacting surfaces, which were the result of the component remain joined together by abutment screw. It is also possible to 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 dental implant system. The results of the numerical computations performed in the

378 Marcin Wierszycki, Witold Kąkąl, Tomasz Łodygowski

environment of finite element code depend significantly on initial the stress states, boundary conditions that reflect the level of osseointegration of the implant with the bones and daily sequence of loading. The computations verified by the experimental observations prove the fatigue character of final failure of implant screw. The numerical results can serve as a good predictor of the time to change this element. BIBLIOGRAPHY 1. ABAQUS Analysis User's Manual, ABAQUS, Inc. Pawtucket, 2005. 2. ABAQUS Theory Manual, ABAQUS, Inc. Pawtucket, 2005. 3. feSafe User's 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

Fatigue algorithm for dental implant 379

cycled abutments are replaced with as-manufactured abutments, The Journal of Prosthetic Dentistry, 95, 1 (2006) 26-32.

15. Hędzelek W., Zagalak R., Łodygowski T., Wierszycki M., The effect of marginal bone loss and bone density on the risk of late implant components failures, 27th Annual Conf. of the European Prosthodontic Association, Geneva, 2003.

16. Khraisat A., Stegaroiu R., Nomura S., Miyakawa O., Fatigue resistance of two implant/abutment joint designs, The Journal of Prosthetic Dentistry, 88, 6 (2002) 604-610.

17. Kąkol W., Łodygowski T., Wierszycki M., Hędzelek W., Zagalak R., Numerical Analysis of the Behavior of Dental Implant, ABAQUS Users' Conference 2002, Newport RI, 2002.

18. Kąkol W., Łodygowski T., Wierszycki M., Estimate of tooth implant fatigue under cyclic loading, Conf. Computer Methods in Mechanics, Gliwice, 2003.

19. Khraisat A., Hashimoto A., Nomura S., Miyakawa O., Effect of lateral cyclic loading on abutment screw loosening of an external hexagon implant system, The Journal of Prosthetic Dentistry, 91, 4 (2004) 326-334.

20. Koa C-C., Swiftb J. Q., DeLonga R., Douglasa W. H., Kima Y., Anc K-N., Changd C-H., Huangd H-L., An intra-oral hydraulic system for controlled loadingof dental implants, Journal of Biomechanics, 35 (2002) 863-869.

21. Koca O. L., Eskitascioglu G., Usumez A., Three-dimensional finite-element analysis of functional stresses in different bone locations produced by implants placed in the maxillary posterior region of the sinus floor, The Journal of Prosthetic Dentistry, 93, 1 (2005) 38-44.

22. Kocańda S., Zmęczeniowe pękanie metali (in Polish), Wydawnictwo Naukowo-Techniczne, Warszawa 1985.

23. Lang L. A., Kang B., Wang R. F., Lang B. R., Finite element analysis to determine implant preload, The Journal of Prosthetic Dentistry, 90, 6 (2003) 539-546.

24. Lee K.-S., Song J.-H., Estimation methods for strain-life fatigue properties from hardness, International Journal of Fatigue, 28 (2006) 386–400.

25. Merz B. R., Hunenbart S., Belser U. C., Mechanics of the implant-abutment connection: an 8-degree taper compared to a butt joint connection, The International Journal of Oral & Maxillofacial Implants, 15, 4 (2000) 519-526.

26. Park, J.-H., Song J.-H., Detailed evaluation of methods for estimation of fatigue properties, International Journal of Fatigue, 17, 5 (1995) 365–373.

27. Roessle M. L., Fatemi A., Strain-controlled fatigue properties of steels and some simple approximations, International Journal of Fatigue, 22 (2000) 495–511.

380 Marcin Wierszycki, Witold Kąkąl, Tomasz Łodygowski

28. Schutz W. A history of fatigue, Engineering Fracture Mechanics, 54, 2 (1996) 263-300.

29. Sakaguchi R. L., Borgersen S. E., Nonlinear finite element contact analysis of dental implant components, The International Journal of Oral & Maxillofacial Implants, 7, 1 (1993) 655-661.

30. Sevimay M., Turhan F., Kilicxarslan M. A., Eskitascioglu G., Three-dimensional finite element analysis of the effect of different bone quality on stress distribution in an implant-supported crown, The Journal of Prosthetic Dentistry, 93, 3 (2005) 227-234.

31. Taylor, T, Prosthodontic problems and limitations associated with osseointegration, The Journal of Prosthetic Dentistry, 79, 1 (1998) 74-78.

32. Teoh S. H., Fatigue of biomaterials: a review, International Journal of Fatigue, 22, (2000) 825–837.

33. Wang .K, The use of titanium for medical applications in the USA, Materials Science and Engineering, A213 (1996) 134-137.

34. Wierszycki M., Kąkol W., Łodygowski T., Numerical complexity of selected biomechanical problems, Journal of Theoretical and Applied Mechanics (in press).

35. Wierszycki M., Kąkol W., Łodygowski T., The screw loosening and fatigue analyses of three dimensional dental implant model, ABAQUS Users' Conference 2006, Boston MA, 2006.

36. Zagalak R. Ocena własności mechanicznych dwuczęściowych wszczepów stomatologicznych Osteoplant (in Polish), PhD Thesis, Fundacja Akademii Medycznej w Poznaniu, Poznań 2003.

37. Zienkiewicz O.C., Taylor R. L., The Finite Element Method, Elsevier, 2000.

Dedicated to our teacher and friend Prof. Andrzej Garstecki on His 70 anniversary