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Pre-overloading to Extend Fatigue Life of Cast Clasps

Department of Conservative Dentistry and Prosthodontics, Dental School, University of Jordan, Amman 11942, Jordan; anabtawime{at}yahoo.com

	ABSTRACT

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Permanent deformation in bending is associated with the development of residual stresses. The objective of this study was to characterize those residual stresses and test whether they can be manipulated to extend the fatigue lives of cast clasps. Simulations with validated non-linear finite element models were used to characterize the residual stresses in clasps cast from Ti-6Al-7Nb, Co-Cr, and Type IV gold alloys. In addition, two groups of as-cast and pre-overloaded (subjected to a load that produced 20 µm of permanent deformation) Ti-6Al-7Nb clasps (10 specimens each) were subjected to cyclic 0.5-mm deflections at 5 Hz until fatigue. Pre-overloaded specimens demonstrated significantly longer fatigue lives (32,200 ± 17,300 cycles) than did those tested in the as-cast condition (17,900 ± 7600 cycles), consistent with the maximum tensile stress values revealed by finite element analysis.

KEY WORDS: fatigue • finite element analysis • non-linear analysis • residual stresses • clasp

	INTRODUCTION

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Among many other metals, Ti-6Al-7Nb and Type IV gold alloys offer flexible alternatives to the conventional Co-Cr alloy for fabrication of removable partial denture frameworks. Because of the reduced possibility of traumatic overloading, clasps made from flexible materials can be designed to engage deeper undercuts on abutment teeth than is conventionally accepted for Co-Cr alloys (Mahmoud et al., 2005). This is preferable when esthetics or periodontal health is of primary concern. However, the high strains associated with engaging deeper undercuts can increase the possibility of permanent deformation and fatigue fracture (Mahmoud et al., 2005).

Approaches—including the development of new materials (Lin et al., 2005), heat or chemical treatment (Akahori et al., 2002), and the optimization of clasp designs (Sato et al., 1995)—have been undertaken to reduce the possibilities of such failures.

It is also widely accepted that the formation of favorable compressive residual stresses at the surface can improve fatigue resistance. In the industrial field, shot peening and surface rolling are practical examples of such mechanical treatment (Dieter, 1986).(AQ) Permanent deformation in bending is also associated with residual stresses (Beer et al., 2002). Using the same logic, we hypothesized that deliberate induction of minute permanent deformations can be used to increase fatigue resistance.

To test the hypothesis, we conducted a bending test with finite element simulation for cast clasps made from Co-Cr, Type IV gold, and Ti-6Al-7Nb alloys. The residual stress patterns associated with different levels of permanent deformation were characterized from experimentally verified mathematical models.

The stress analysis results were used for the design of a fatigue test for Ti-6Al-7Nb clasps, as a test of the hypothesis that residual stresses accompanying minute permanent deformations can extend the fatigue lives of cast clasps.

	MATERIALS & METHODS

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Bending and Fatigue Test Specimens
The basic test specimen was a cast clasp adapted in one plane to a cylinder having a 5-mm radius. The angle subtended by the clasp was 120 degrees (Fig. 1). Similar to a previous report, casting patterns for test specimens were made in a duplicating mold (Mahmoud et al., 2005). The average width and thickness of the casting patterns at 30 and 120 degrees were 0.75/1.01 mm and 0.54/0.65 mm, respectively.

View larger version (37K): [in this window] [in a new window]   Figure 1. Schematic representation of the basic test specimen (right). Each specimen was deflected by being loaded through a sphere located at its tip. The loading was radial to the outside in a direction that made a 30-degree angle with the cylinder’s cross-sectional plane (rectangular insert). To the left is a picture of a representative finite element model. The models were reproduced based on the dimensions of the real specimens.

Finite Element Analysis
By the use of the pre-processor of finite element computer software (ANSYS 8.0 FEM; ANSYS Inc, Canonsburg, PA, USA), a three-dimensional finite element model for each bending test clasp was created, based on width, thickness, and cross-sectional shape of each specimen, and measured at 7 representative locations by means of a measuring microscope (MM-60; Nikon, Tokyo, Japan). Each model was meshed by 8 nodes-hexahedral elements (Fig. 1).

Similar to a previous report, a multi-linear material property model was used for each alloy (Mahmoud et al., 2007). Prediction of permanent deformation was based on the von Mises equivalent stress criterion for yielding (Dieter, 1986; Beer et al., 2002), which assumes isotropic behavior independent of loading rate. To account for geometric non-linearity, we used the "large displacement" option (Zeinkiewicz and Taylor, 2000; Zhang, 2004). The multi-linear stress-strain curves for the 3 alloys were based on the results of a previous study (Mahmoud et al., 2007) (Fig. 2A). The friction coefficient for each test was estimated from the energy dissipation ratio (Mahmoud et al., 2007), and a Poisson’s ratio of 0.33 was used for the 3 alloys (Dieter, 1986).

View larger version (28K): [in this window] [in a new window]   Figure 2. Flow curves, mathematical models validation, and results of the bending tests and finite element simulations. (A) Plot of the "multi-linear material property" flow curves (in which each stress-strain curve is represented by connected successive lines of decreased slope) used in the finite element analysis for clasps made from the 3 alloys. The 458-MPa, 685-MPa, and 883-MPa underlined values represent the proportional limit values used for Gold Type IV, Co-Cr, and Ti-6Al-7Nb alloys, respectively. The elastic modulus values were 229 GPa, 92.6 GPa, and 115.8 GPa for Co-Cr, Gold Type IV, and Ti-6Al-7Nb alloys, respectively. (B) Plot of real deflection-predicted deflection corresponding to different levels of permanent deformation for the 9 bending test specimens and their corresponding simulation models (3 specimens for each alloy). The straight line represents an ideal relation when the predicted and real values match perfectly. (C) Plot of [ Permanent deformation/ von Mises (VM) plastic strain] ratios as a function of von Mises (VM) plastic strain on the inner upper corner of 3 representative models for the 3 alloys. (D) Plot of [ von Mises (VM) residual elastic strain/ von Mises (VM) plastic strain] ratios as a function of von Mises (VM) plastic strain on the inner upper corner of 3 representative models for the 3 alloys. (E) Plot of permanent deformation as a function of deflection for the 9 bending test specimens (3 specimens for each alloy). The average amount of deflection at which permanent deformation started to appear was 275 ± 12 µm, 490 ± 18 µm, and 656 ± 13 µm for Co-Cr, Gold Type IV, and Ti-6Al-7Nb alloy clasps, respectively. (F) This Fig. shows the changes in the first principle stress value (S1 -maximum tensile stress) for a representative Ti-6Al-7Nb model. The maximum S1 in the clasp and the S1 at the inner upper corner were recorded when the model was deflected from its rest position for 0.5 mm after each permanent deformation increment. The amount of reduction in the S1 value at the inner upper corner of the clasp was almost equal to the absolute value of residual 3rd principle stress (S3 - maximum compressive stress) there.

Fatigue Test
Two groups of Ti-6Al-7Nb clasps (10 specimens each) were used. Clasps of one group were tested in the as-cast condition, while those of the other were pre-overloaded before the fatigue test to produce a permanent deformation of 20 µm in the same direction as the fatigue test’s.

As in the bending test, each specimen was fixed to the testing machine and subjected to a sinusoidal unidirectional cyclic deflection of 0.5 mm by force at the tip of each clasp arm. The frequency was 5 Hz, and the direction of loading was similar to that of the bending test in a direction that makes a 30-degree angle with the cylinder’s cross-sectional plane. The test was continued until fracture, and a t test was performed for comparison of the mean number of cycles for the two groups.

	RESULTS

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Mathematical Model Verification
The deflection values corresponding to different levels of permanent deformation from 5 µm to 800 µm were recorded for the 9 bending tests and their corresponding simulation models. Linear regression analysis was performed for the real-predicted deflection plot (Fig. 2B). The r² value was 0.983, and the linear regression equation was:

Deflection_Predicted (µm) = 0.911 (Deflection_Real) + 85 which approached an ideal (Y = X) relation for the deflection range reported in this study (250 to 1800 µm).

Finite Element Analysis Results
We used the finite element models to examine the following variables: the maximum values of the von Mises equivalent stress, first principal stress (S1 or maximum tensile stress), third principal stress (S3 or maximum compressive stress), von Mises plastic and elastic strains, and the accompanyingpermanent deformation. The stresses and strains recorded after load release were all described as residual.

The maximum von Mises stress first exceeded the proportional limit at the inner upper corner of each clasp model; this was associated with the development of plastic strains. As deflections increased further, the plastic strains started to involve the lower outer clasp surface opposing the inner upper corner as well (Fig. 3).

View larger version (47K): [in this window] [in a new window]   Figure 3. von Mises equivalent stress distribution patterns for a representative Ti-6Al-7Nb model. The scale at the bottom describes the ranking of colors. The dark blue indicates the lowest stress values close to zero, and the red indicates the highest; areas in gray have stress values exceeding the proportional limit and therefore represent areas undergoing plastic deformation. A and C represent the stress distributions on loading for deflections that resulted in 25 µm and 350 µm permanent deformations, respectively, while B and D represent those on unloading. The inner upper corner and the opposing outer side, both showing the highest stress values, were especially projected. Each model was split at an 80-degree location to show the stress distribution inside the clasp. When unloaded, the residual stress patterns had a four-layer configuration: The 2 surface layers had stresses opposite those when loaded, while the deep 2 layers had the same stresses (blue arrows, compressive; red arrows, tensile). The inner upper corner showed the highest von Mises residual stress values (compressive). For clarity, different scales were used for the 4 Figs.

The relationship between changes in permanent deformation, von Mises plastic, and residual elastic strains at the inner upper corner of the models is demonstrated in Figs. 2C and 2D. Initially, developing plastic strains showed hardly any accompanying permanent deformations, but were associated by a high ’residual elastic strain increase/plastic strain increase’ ratio. As the value of plastic strain increased, the ’permanent deformation increase/plastic strain increase’ ratio rose, with a simultaneous drop in the ’residual elastic strain increase/plastic strain increase’ ratio.

The ’residual elastic strain increase/plastic strain increase’ ratio was generally higher for the Ti-6Al-7Nb alloy, followed by Gold type IV, and being the least in the Co-Cr alloy (Fig. 2C).

Fatigue Test, Rationale behind Design, and Results
In the bending test, it was revealed that the Ti-6Al-7Nb alloy had significantly higher resistance to permanent deformation, followed by type IV gold, with the least being for the Co-Cr alloy (Fig. 2E). With its high flexibility and resistance to permanent deformation, this alloy has the potential to be used in situations where high strains are expected.

In the finite element analyses, similar to the bending test, Ti-6Al-7Nb clasp models were loaded incrementally. After each incremental loading, the models were deflected for 0.5 mm, and the maximum S1 values were plotted as a function of permanent deformation (Fig. 2F). With the initial development of permanent deformations, the maximum tensile stress exhibited a steep decrease; however, as permanent deformation exceeded 50 µm, it started to increase.

Initially, the maximum tensile stress (S1) was at the surface of the inner upper corner of each clasp and kept decreasing throughout the bending test (Fig. 2F). The decrease was almost equal to the maximum compressive residual stress (S3), which was located there as well throughout the test. Maximum residual S3 was directly correlated with the maximum residual von Mises elastic strain.

As permanent deformation increased beyond 50 µm, the rise in maximum S1 could be tracked to the tensile residual stresses developing in the deep layer below the inner upper corner (Fig. 3D). This rise was also associated with the movement of maximum S1 location from the surface to that deep layer.

The fatigue test was designed based on the bending test and stress analysis results explained above. The mean number of cycles for the pre-overloaded clasps was (32,200 ± 17,300) cycles, which was significantly greater than that for clasps tested in the as-cast condition (17,900 ± 7600) (P < 0.05). There was no significant difference in recorded permanent deformations between the two groups (13 and 11 µm, respectively).

	DISCUSSION

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Non-linear finite element analysis provided a powerful tool for study of the stress and strain patterns accompanying permanent deformation. We found that initially developed plastic strains caused almost equal values of residual elastic strains of opposite sign, without appreciable permanent deformation. On load release, the elastic core worked to bring the clasp back to its original position. Resisted by the plastically deformed areas, the structure took a new position, satisfying a new force equilibrium state (Beer et al., 2002). This new position defined the permanent deformation amount. Due to their relative size, areas undergoing initial plastic deformations were too small to change the final position of the structure appreciably, and, when unloaded, were compressed or extended back to their original position before yielding, producing almost equal amounts of elastic strains (Beer et al., 2002). However, as more areas became involved in plastic strain, the ratio of ’permanent deformation increase/plastic strain increase’ started to rise. Simultaneously, the ’residual elastic strain increase/plastic strain increase’ ratio fell.

In the fatigue test, clasps subjected to the pre-overloading treatment exhibited significantly longer fatigue lives, satisfying our proposed hypothesis. The influence of pre-overloading on stress distributions demonstrated a directional nature. Being applied in the same direction as the anticipated cyclic loading, areas first showing plastic deformation were the same as those having the highest stress values in the fatigue test. The residual stresses developed in the plastically deformed areas were of opposite sign, and led to a proportional reduction in the maximum and mean stress parameters of the loading cycles, even though the stress amplitude, which is governed by the cyclic deflection amount, was basically the same.

The models were assumed to be homogeneous and defect-free. However, it is widely known that fatigue cracks tend to nucleate at stress-raisers like defects and inclusions (Dieter, 1986). In reality, the overloading effect will be specifically greater at those high stress areas, leading to the creation of compressive residual stresses exactly where they are needed. Compressive residual stresses will reduce the crack-driving force (Suresh, 1998), and as the crack advances, the wake of material that has previously been deformed plastically will contribute to crack growth retardation, or even arrest, through plasticity-induced crack closure (Elber, 1970, 1971).

Besides their beneficial influence when applied before cyclic loads, transient tensile overloads interrupting cyclic loading also have the potential to retard crack advance or even arrest it completely (Suresh, 1998). Transient tensile overloads are believed to act through their plasticity-induced crack closure (Elber, 1970, 1971), crack-tip blunting (Lankford and Davidson, 1982), residual compressive stresses (Allison, 1979; Taira and Tanaka, 1979), and deflection or bifurcation of the crack (Suresh, 1983), among other fatigue-retardation mechanisms. If the overloading treatment tested in this study is expanded to be applied on a periodic basis during removable partial denture maintenance visits, this might have the potential to extend the fatigue life even more. The same mechanism may work in accidental overloading episodes expected to affect clasps when the prosthesis is dislodged or inserted in a path different from that designed.

Fatigue in the oral cavity is expected to be influenced by the corrosive nature of oral fluids (Anusavice and Brantley, 2003). Corrosion works synergistically with loading to accelerate crack growth in corrosion fatigue (Dieter, 1986). Tensile overloading-induced crack closure will reduce the ingress of corrosive materials and reduce their influence at this critical area (Dieter, 1986).

Compressive overloading in a direction opposite that of cyclic loading will produce effects opposite to those discussed above (Suresh, 1998). The consequence will be exacerbation of crack growth and apparent shortening of fatigue life. Unfortunately, such compressive overloading happens whenever a clasp is activated by being bent toward the tooth surface, a potentially destructive practice.

In conclusion, overloading can lead to the production of residual stresses of high magnitudes before the appearance of significant permanent deformations. In addition to their effect on the mean and maximum nominal stress values, those residual stresses will specifically target stress-raising areas, like defects and cracks. When applied in the same direction as that of cyclic loading, they will exhibit a favorable effect, leading to fatigue life extension.

Fractographic analysis and testing of the relative effects of overloading treatments on fatigue resistance under different cyclic-loading levels and scenarios are topics for future research.

	ACKNOWLEDGMENTS

 
Special thanks to Dr. Wakabayashi, Dr. Takahashi, and technician Iwasaki for their help. This study was supported by Grants-in-Aid for Scientific Research, 16591936 (N.W.) and 14571840 (N.W.), from The Ministry of Education, Science, and Culture of Japan, and by Scientific Research Grant (125, 2005–2006), from the University of Jordan, Amman, Jordan.

Received January 9, 2006; Last revision April 12, 2007; Accepted May 7, 2007

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