Hoop Stress and the Conical Connection

The conical, shank and socket connection between the implant fixture and the abutment has become a popular dental implant design feature.^1,2  The connection between these two elements is, in effect, a conical wedge that may exert lateral forces under load that may result in fracture of the coronal implant fixture walls known as hoop stress. The conical connection is held together by a connecting abutment screw that extends from the internal of the abutment into the coronal socket of the implant fixture.

Rarely will the implant coronal socket fracture (Figures 1 through 3). Occlusal loading appears to be a factor. However, it is not known whether the force load direction, axial or off axial, or alloy of the implant is more important than the magnitude of the load. The alloy of the implant components may be a factor in that pure titanium may not adequately resist occlusal loading. When the implant coronal fractures, it is referred to as “flowering” because the fracture configuration resembles the pedals of a flower (Figure 4).

The conical connection is sometimes referred to as a Morse taper. The Morse taper was invented by Stephen Morse in the 1860s. It is a conical shank and socket design and is produced in eight sizes, 1–8°. The Morse taper is a friction fit between the shank and socket portions to hold the two pieces together. Other sized designs are the Brown and Sharp, Bridgeport Machines, National Machine Tool Builders Association, and Jacobs and Jarno tapers. A cold weld may form between the shank and socket.

The forces in the anterior maxilla are off axial by virtue of the anatomical configuration of the human jaws. The forces in the anterior jaws are approximately one-third of those in the posterior jaws and are generally not beyond the yield strength of the labial bone.³  These off-axial forces, however, may be detrimental to a conical implant connection because they are directed at the thin portion of the connection. The coronal rim of the fixture socket may fracture under the off-axial load.

Implant crest bone loss has been reduced due to design developments with close spaced cutting threads designed to create implant-bone hoop-stress stability when the implant is seated in the supporting bone.⁴ 

When an implant fails, the metal failure appears to be catastrophic in that there are no waves of metal on the fracture faces. That is, there was one sudden complete break and not a gradual separation. A fracture of this type requires that the implant be removed.

The objective of this study was to evaluate the axial and off-axial load fatigue failure tolerance of a laboratory-designed conical connection with respect to the failure of dental implants. Constraints have been identified for the project, such as the biting forces of human beings, and the unavailability of dental implants for testing. The project was approached from an analytical standpoint, and the theoretical data were validated through experimental results.

A theoretical analysis of the fatigue of dental implants was performed using various proprietary engineering simulation software program products including ANSYS, Working Model, and SolidWorks. These programs were combined with theoretical calculations; the background for the fatigue of dental implants was developed. These programs helped simulate what would be expected during a fatigue experiment. Estimations of how long each test would take were developed, which proved important because high-cycle fatigue testing can range in the millions of cycles. The analytical results formed were used to create an appropriate experimental design. The experimental results should then reflect those of the computational analysis.

Fatigue is the progressive and localized structural damage that occurs when a material is subjected to cyclic loading. Fatigue test results in data presented as a plot of stress (S) against the number of cycles to failure (N), known as an S-N curve. The data are typically obtained by cycling smooth or notched specimens until failure. This curve is then used as the standard characterization means for fatigue properties of materials. All equations when not referenced are from Norton.⁵ 

To design the fatigue test of the dental implant, fatigue theories must be used to decide upon testing limits. To begin, a Goodman diagram for a standard test specimen for titanium 6Al-4V (Ti 6AL-4V) was created. Goodman diagrams are used to predict failure of an object undergoing numerous fluctuating stresses. A Goodman diagram to predict failure at infinite life is found using the following equation:

where is the alternating stress, the stress that placed on the test specimen cyclically during the fatigue test; is the mean stress, or average stress on the test specimen throughout test; is the endurance strength for infinite life specific to the material of the specimen; and is the ultimate tensile strength for the material of the specimen. To examine the endurance strength of a material, standard fatigue tests such as a rotating-beam test are done on a specimen with simple geometry. Empirically, S-N curves such as the one shown in Figure 5 are created.

It can be seen from the S-N graph (Figure 5) that high cycle fatigue typically occurs between 10³ and 10⁶ cycles, and after 10⁶ cycles the specimen will never fail. This stress is the endurance stress. The endurance and ultimate tensile strengths were found for Ti 6Al-4V, and Equation 1 was converted into the graph in Figure 6.

In the Goodman diagram, if the alternating and mean stresses on the test specimen are below the line in Figure 7, the test specimen should never fail; but above the line, the specimen should eventually fail. Equation 1 and Figure 7 use only data that are found empirically, using a rotating beam fatigue test, for a standard test specimen.⁶  No S-N curves have been made for dental implants because it requires hundreds of fatigue tests. The following section describes how an estimate for fatigue failure of a dental implant can be found.

To create an estimate for the fatigue life of the dental implant, the endurance limit for Ti-6Al-4V must be modified to include the differences between the dental implant and the standard test specimen. To create an estimate, the Marin equation was used:

where k_a, k_b, k_c, k_d, and k_e are the surface condition, size, load condition, temperature, and reliability factors, respectively. is the standard endurance limit for Ti 6Al-4V, and is the estimated endurance limit for the dental implant. Because there are no endurance data available for the dental implants in use, the k factors help to estimate the endurance limit for the tested implants. More miscellaneous k's can be added to Equation 2 but were limited to the 5 named here for simplicity. To find k_a, the equation below was used:

where s and c are constants for different surface finishes. The values for all other k-factors are described in the “Results” section.

To create a Goodman diagram for the dental implant, the S_e found in Equation 2 can replace the in Equation 1. The Goodman diagram found can only predict failure for 10⁶ cycles. The Goodman diagram can be altered as to predict failure at any number of cycles between 10³ and 10⁶ cycles using the following equations. The fatigue strength, S_f, is the strength at a specific amount of cycles in which failure will occur. To find the fatigue strength for a specific amount of cycles, the following equation was used:

where is the number of cycles to failure and and are constants defined below.

and

In Equations 5 and 6, f is a constant that can be found using Figure 7 in the Appendix. Goodman diagrams for a specific amount of cycles to failure can be found by replacing with S_f in Equation 1. Creating Goodman diagrams for fatigue strength is useful for specifications and limits for the fatigue test machine design. Once the alternating and mean stresses on the implant are found analytically, they can be plotted on a Goodman curve and using Equation 4, the number of cycles to failure can be estimated. The following sections provide the theory on how the alternating and mean stresses on the implant are found.

Before the mean stress on the implant can be determined, the residual stress caused by the interference fit between the abutment and implant must be found. Both the abutment and implant are tapered at the same degree, and the abutment is secured in the implant using a press fit. As the abutment is tapped down into the implant, deformation causes a tangential stress on the implant. Figure 8 and Equations 7 through 11 describe how the tangential stresses are found.

In a simple case, the pressure generated at the interference fit between a cylindrical shaft and hub is found as follows:

where p is the pressure, d is the outer diameter of the shaft, d_o is the outside diameter of the hub, d_i is the inner diameter of the shaft, E is the modulus of elasticity, and δ is the difference between inner diameter of the hub and outer diameter of the shaft. For the implant, it is assumed that because the taper is only 1–8°, the angle will not significantly affect results or change Equation 7, which uses nontapered elements. The following changes to Equation 7 were made for evaluation of an implant:

and

The displacement, δ, is caused as the abutment is tapped down into the implant. Also, the abutment is not hollow, so Equation 9 must be accounted for first. Therefore, the pressure becomes

where the implant is considered the hub and the abutment is equivalent to the shaft as described in Equation 7. Finally, the tangential stress on the implant is found in Equation 11:

The alternating stress is caused by the cyclic vertical loading on the top of the abutment. This stress, because of the geometry, is a combined load. For combined loading, the von Mises stress was found using finite element analysis (FEA), where stress concentrations can be determined. A 200 N force was applied to the abutment, and by using ANSYS (ANSYS, Catonsburg, Pa), an FEA design, the maximum von Mises stress on the implant was found. This applied stress was then used to find the alternating stress.

The mean stress for fatigue testing is the average stress between the maximum and minimum stresses. Normally, in reversed cyclic loading, the mean stress would be equal to 0, but in the fatigue testing of the implant, this is not the case. To find the mean and alternating stresses, the maximum and minimum stresses must be found. The residual stress caused by the press fit, σ_resid, of the abutment is the minimum stress, σ_min; thus,

The maximum stress, σ_max, is a combination of the residual stress and the applied stress caused by the 200 N force on the abutment:

For combination loading the alternating, σ_a, and mean, σ_m, stresses are determined as follows:

and

The above-mentioned equations can be summarized in Figure 9, where the vertical axis is stress and the horizontal axis is time. During testing, each period of the sinusoidal line above is considered a cycle.

Before FEA can be performed on the implant and abutment geometries, a simple case must be performed, where a known theoretical calculation can be compared to the FEA results. This test case helps validate the constraints and meshing when modeling the implant and abutment. Deflection analysis was performed on a cylindrical cantilever beam, seen in Figure 10, with similar dimensions as the implant.

One end of the cantilever beam was constrained, whereas a 200-N force was placed on the opposite end, compressing the bar (Figure 11).

The equation for uniaxial deflection of a beam with constant cross-sectional area and stiffness is as follows:

where δ is deflection, F is force applied, l is the overall length of the beam, A is the cross-sectional area of the beam, and E is the modulus of elasticity. The deflection found using Equation 16 is then compared to the “total deflection” analysis performed in ANSYS. In both the FEA evaluation for deflection and the calculation, the following values from Table 1 were used.

A deformation of 1.4085 × 10⁻⁶ m was found using Equation 16. For the FEA, a tetrahedron mesh was used, forming 21 600 elements and 31 400 nodes. The total deformation found through ANSYS was found to be 1.3905 × 10⁻⁶ m. The percent different between the two deformations was 1.28%; thus, the ANSYS model is acceptable and analysis of the implant can be performed.

For the analytical test setup, three pieces were modeled in the CAD program SolidWorks: the implant, an abutment, and a medium to hold the body of the implant. The assembly of parts can be seen in Figure 12, where the orange part is the abutment, the gray part is the implant, and the blue part is the structural steel.

Analysis of the general geometry of contemporary implants was used. Using the average dimensions for abutment taper, outer diameter, and length of four brands (Lifecore, Blue Sky Bio, Allfit, and Straumann as described by Junghoon⁷), the geometry of the implant was created. A second geometry with a smaller inner diameter for the implant was also created. Table 2 shows the dimensions of geometries 1 and 2. For the analysis, material properties for Ti 6Al-4V were used for the abutment and implant, and structural steel was used for the material in which the implant was placed. As a parameter described in ISO 14801, the implant was left 2 mm above the structural steel to mimic bone loss.^7,8 

A 200-N force was applied to the top surface of the abutment. The bottom surface was constrained to the bottom of the structural steel. A cold weld may be formed between the shank and socket surfaces; thus, a “bonded” contact was used. There should be no movement of the abutment. As in the FEA of the cylindrical beam, a tetrahedral mesh was used for the implant with 59 000 elements and 138 000 nodes (Figure 13).

A “total equivalent stress” or von Mises stress analysis was performed for on the system. The results yielded a maximum stress on the geometry 1 implant of 117 and 76.4 MPa for geometry 2. A stress plot of a cutaway of the implant was made (Figure 14).

The structural steel and abutment were removed as the focus of the analysis is on the implant. The maximum stress found on the implant was then used in Equation 13 of the “Theory” section to find the alternating and mean stresses on the implant.

A residual stress on the implant is caused by the press fit between the implant socket and the abutment shank. To find the tangential or hoop stress on the implant, Equations 7 through 11 were used. Table 2 shows the dimensions for each of the geometries that are used in the calculation of the tangential stress. In the calculations, the elongation of the implant could not be found without knowledge of how far the abutment is tapped down into the implant. Bozkaya and Müftü⁸  found that the abutment typically moves from 1 to 5 μm down into the implant during installation. With the vertical displacement known, the difference between the diameters of the implant and abutment can be found using the following equation:

where is the new diameter of the abutment after the abutment is tapped down, d_a is the original diameter of the abutment, D is the distance the abutment moves down, and θ is the angle of the taper. In Equation 7, d_abutement can be replaced by⁠. The tangential stress on the implant was found to be 39.3 MPa for geometry 1 and 40.5 MPa for geometry 2.

With the results from the last two sections, the alternating and mean stresses were found. Using Equations 14 and 15, the alternating and means stresses for geometry 1 were 97.8 and 58.5 MPa and for geometry 2 were 78.8 and 38.8 MPa.

To predict failure, a Goodman diagram for the specific geometry and material properties for the dental implants was created using Equations 1 through 6. For all the following factors, conditions and constants from Shigley's Mechanical Engineering Design were used.⁶  For the surface condition factor, constants for machined metal were used in Equation 3, as the abutment and implant will be machined. The surface condition factor was found to be 0.729. The outer diameter of the implant was used for the size condition factor and equaled 1.171. Because the implant is under combination loading, the loading condition factor for bending loading was used, which is a factor of 1. As for the temperature condition factor, a factor of 1 was used for two reasons. The implant will be tested at room temperature, where the use a factor of 1 at room temperature is advised. Furthermore, in Lee's research, where similar fatigue testing of dental implants was performed, it was concluded that fatigue testing on the implants did not increase the temperature of the titanium implant.⁹  Finally, the reliability factor was found. According to Haugen and Wirching,¹⁰  endurance strengths vary by less than 8%, and this factor is used when creating a conservative design. There should be a 99.9% confidence that the design will not fail. For the design of the test of the dental implants, there should 99.9% confidence that the implants will fail because the object is to create failures. Thus, a confidence level used for the reliability factor was 0.1%. The reliability factor used was 1.247.

The alternating and mean stresses found on the implant were plotted (red dot on diagram) on the implant-specific Goodman diagram for geometry 1 (Figure 15).

With an applied force of 200 N on the abutment, the implant should never fail (Figure 15). To create a failure, the applied force must be increased.

The mean and alternating stresses computed through theoretical calculations and the use of FEA suggested a 200-N vertical load was well below the infinite life Goodman line developed. Therefore, under this loading, failure would be expected to never occur. When doing experimental testing, the vertical force was kept, but the force was increased in magnitude from 200 to 2000 N. The force was increased to bring the alternating stress higher than the value needed for infinite life. It was then raised even higher to create failure in a reasonable amount of cycles for testing. As long as the failure is predicted to occur after more than 1000 cycles, the failure is still considered high cycle fatigue; therefore, the Goodman diagrams developed are still accurate.

A vertical force of 2000 N creates an alternating stress of 364 MPa. Through the use of the Goodman diagram, it is predicted that failure occurs after approximately 40 000 cycles. If failures are found to be consistent at this force, the applied force can then be lowered in an attempt to use more realistic chewing forces instead of 2000 N.

Testing was done on laboratory-machined test implants. These test implant coronas were titanium ally, fabricated at a commercial machine shop (Windham Industries, Inc., Windham Conn). Through the testing, two slightly different implant geometries were used. The wall thicknesses of the implant sockets were 0.25 and 0.50 mm (Figure 16).

The test rig delivered 200–3000 N force at a frequency of 10–40 Hz (Figure 17). Testing was done in a dry environment 20 ± 5°C.⁹ 

The rig design used the basics of Scott Russell straight line mechanism.¹¹  This creates a straight line through a set of linkages and uses three equal length segments to create straight line motion (Figure 17).¹²  This rig delivered the forces to the test implants at the desired newtons of force. The design and specifications of the test rig are available through one of us (D.F.).

The first test implants were machined with the specifications for geometry 2, 0.5 mm (Table 2). The test rig was set up using the (69.85 mm) radial position. A force of 2220 N was applied. The spring constant for the elastomer spring was rated at ±20%, so the output force was well within the limits (−4.5%). Using the ANSYS model and Goodman diagram, failure at this force was expected to occur at approximately 50 000 cycles. After 332 000 cycles and 32 h of axial testing, no failure occurred so the fatigue test was stopped (Figure 18).

The test implants machined to specifications for geometry 1, 0.5 mm, were tested. Four implants were tested axially at 2220 N. Each ran for 300 000 cycles, and again no failures occurred.

No failures occurred at 10 times human biting force; the load testing was then placed at 30°. In the human mouth, both vertical and horizontal force components create bending moments on the implant based on the crown's placement on the abutment (Figure 19). Five vertical loading tests were performed.

Two test implants with geometry 1, 0.25-mm walls, were tested at 30° off-axial 2200-N load. Both implants plastically deformed at 645 and 678 N.

A 1925-N force was applied to one implant at 5° off axially, and a failure occurred at 16 850 cycles.

This work is comprised of a theoretical analysis, model conceptionalization, and finally testing. Theoretical results showed that the axial force would not cause a hoop stress failure in the socket, so forces were increased past the human biting force that again resulted in no failure. The proposed 200-N vertical force was increased to 2220 N, but no failure occurred with cycles in excess of 300 000. Two implants were then tested at forces applied 30° off axially. These implants plastically deformed at 645 and 678 N with one cycle. Another test of one implant was performed at 5° off axially, causing a failure in 16 850 cycles.

The alloy of dental implants may affect the strength and resistance to fracture of the implant.⁷  The alloy tested here is a standard commercially available alloy used in industrial applications. Dental implant manufacturers use a variety of proprietary alloys whose proportions may not be provided. The axial loading here did not produce a material failure and was predicted under human bite force parameters. The off-axial load did produce a material failure, but was in the high range of human bite force. The small number of cycles and samples limit credibility. Further study is needed to define the range of human bite force that may induce a material failure. Many patients receive implant supported fixed restorations in the anterior maxilla. Prosthetics here are put in an off-axial load situation form the lingual. Most patients cannot generate enough bite force to cause a material failure. It may be necessary to measure the bite force of patients who will be treated with anterior implant supported fixed prostheses to predict a failure possibility. Any patient who the clinician suspects may generate high bite forces should be admonished as to a possible fatigue fracture complication. One of us (D.F.) is now testing off-axial loads on this connection in the human bite force range.

Commercially available implant fixtures were not used in this testing due to cost and our desire to avoid any semblance of commercial bias.

The photo in Figures 1 through 4 is of an internal hex (Implant Direct Screw Plant) that fractured supporting an anterior maxillary fixed partial denture. The popularity of the internal hex conical connection may have a weakness when used to support anterior fixed partial dentures. Here, the mandibular opposing teeth are directed to the lingual of the maxillary prosthesis and impart a cyclic off-axial load. This is a detrimental load that, in one of our experiences (D.F.), is the only situation where implant connection failure occurs. The connection thickness and alloy of the fixture may influence the appropriateness of this design in the anterior maxillary where off-axial loading is a given parameter. These parameters may induce implant design changes or admonish clinicians as to their use in the anterior maxilla.

Another factor involved in off-axial loading is the potential vulnerability of the facial bone in the anterior maxilla. Although the titanium metal connection may be at risk, the ability of the facial cortical bone to resist these loads is an unknown. Investigation of the load resistance of facial cortical bone is of interest and may reveal why some implants fail and others do not.

Axial-loaded forces do not appear to be of paramount concern for hoop stress fatigue fracture in dental implant-abutment conical design. However, off-axial loading may induce a fracture of the socket portion of the conical connection in the load range of human bite force. The resistance to fracture may depend on the alloy of the implant, magnitude of the load, and the number of loading cycles. Further testing is required to assess the most appropriate alloy, critical load, incident angle, and critical load cycles for the conical implant-abutment connection.

We acknowledge the technical aid in fabrication of the test rig of Tom Mealy, Peter Glaude, and Serge Doyon.