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Shear in Flexure of Fiber Composites with Different End Supports
1 Center for Biomaterials, MC-1615, Correspondence: *corresponding author, Goldberg{at}nso1.uchc.edu
The integrity of fiber-reinforced composite (FRC) prostheses is dependent, in part, on flexural rigidity. The object of this study was to determine if the flexure behavior of uniform FRC beams with restrained or simply supported ends and various length/depth (L/d) aspect ratios could be more accurately modeled by correcting for shear. Experimental results were compared with three analytical models. All models were accurate at high L/d ratios, but the shear-corrected model was accurate to the lowest, more clinically relevant, L/d values. In this range, more than 40% of the beam deflection was due to shear.
Key Words: fiber-reinforced • composites • flexure • shear • restraints
As fiber-reinforced composite procedures and materials evolve, one of the fundamental issues will continue to be rigidity or stiffness of the prosthesis. Not only is stiffness important for clinical function, and the inevitable comparison with metal-ceramics, but also the success of the adhesive interfaces between the FRC lamina, between the substructure and the veneer (Rosentritt et al., 2001), and between the prosthesis and tooth structure (Loose et al., 1998) are dependent in large part on the deflection and deformation of the restoration. Most studies of FRC rigidity have used three-point flexural loading (Freilich et al., 1999; Vallittu, 1999; Behr et al., 2000; Karmaker and Prasad, 2000) of standard beams (International Standard, 1992; ASTM Standard, 1997), and it was implicit that prosthesis rigidity would be proportional to the measured flexure modulus of the material. However, in vitro tests of FRC fixed partial denture (FPD) substructures (Haser, 1999) show that prosthesis rigidity cannot be accurately predicted from these earlier reports. While several factors are probably involved, it seems reasonable that the restraint of the prosthesis retainer and abutment, in contrast to the freely supported three-point specimen, contributes to this discrepancy. Accordingly, one purpose of this study was to develop and evaluate a laboratory flexure test of FRC beams based on restrained-end loading.
In addition, the classic flexure formula (Popov, 1976)
Restrained-end Test Apparatus A restrained-end test apparatus (Fig. 1  ) was designed and fabricated. Specimens of fiber-reinforced composites with uniform rectangular cross-sections were clamped at both ends with a load applied in the center. Screws used to secure the samples in the end-restraints were tightened to 2200 gm*cm with a torque watch (Waters Manufacturing, Inc., Wayland, MA, USA). To accommodate various sample dimensions and span lengths, the clamps were adjustable on the base of the testing jig. A central load was applied until failure with an Instron testing machine (model 1113, Instron Corp., Canton, MA, USA) by means of a 10-mm-wide blade indenter with a 1.6-mm end radius, at a crosshead speed of 0.05 cm/min. For comparison purposes, flexure behavior of the FRC was evaluated with the same test parameters, but with three-point loading. Tests for both loading conditions were conducted over a range of length/depth (L/d) values.
Fabrication and Testing of FRC at Various L/d Strips of continuous, unidirectional, pre-impregnated fiber-reinforced composite (FibreKorTM, Pentron, Inc., Wallingford, CT, USA) were used to fabricate the test specimens. In this composite, 6- to 10-µm-diameter glass fibers are uniformly distributed in an unpolymerized dimethacrylate matrix (Karmaker et al., 1997; Freilich et al., 1998). Fiber volume fraction, Vf, was 42.5%, as confirmed by ashing. Each 3 x 0.3 mm strip was placed with the fibers parallel to the long axis of a Teflon mold and condensed to eliminate voids between layers. The samples were visible-light-cured for 9 min (Cure LiteTM Plus, Pentron, Inc., Wallingford, CT, USA), carefully removed from the mold, light-cured for an additional 9 min, and finally post-cured in a vacuum/heat-curing unit for 30 min at 110°C (Conquest Curing Unit, Pentron, Inc., Wallingford, CT, USA). Specimens were stored at ambient conditions for fewer than 4 days prior to being tested. Specimen dimensions were 2 x 2 x 80 mm3 (width x thickness x length) or 2 x 1 x 80 mm3. The former were tested at span lengths of 40, 30, 20, and 10 mm, and the latter were tested at span lengths of 60, 50, 40, 30, and 20 mm (n = 3 for each L/d group). This resulted in L/d ratios of 5 to 60. Experimental rigidity values (load/deflection) were calculated within the elastic region. The load at the elastic limit, determined graphically as the limit of the initial linear region of the load-deflection curve, and the maximum load were recorded.
Analytical Models of Flexural Rigidity
The deflection,
The longitudinal modulus of the composite, E, was calculated to be 34.2 GPa from the rule of mixtures (Agarwal and Broutman, 1980):
 were used to calculate classic small deflection theoretical beam rigidity, P/ , for a restrained-end beam with various ratios of L/d. An analogous arrangement of the more widely recognized relationship,
The second analytical model of beam rigidity included the contribution of shear. In this model, total deflection is the sum of the deflection produced from flexure plus the deflection due to shear,
The third analytical model was based on flexure, shear, and the assumption of softening at the clamps due to fiber damage (Jancar et al., 1994). These earlier investigators calculated values for an apparent modulus, E*, that included all three effects at various ratios of L/d. In the present study, we substituted these values of E* into Eqs. 2 and 4 to calculate the theoretical rigidity of the FRC beams based on the assumption of flexure, shear, and damage at the clamp. All three models of flexure behavior were statistically compared with the experimental results by means of the chi-square goodness-of-fit test.
The experimentally determined rigidity of the FRC beams at L/d ratios of less than 30 and tested with restrained ends was compared with the three analytical models (Fig. 2A ). Similar comparisons were made for three-point loading (Fig. 2B). Theoretical and measured values of stiffness up to L/d of 60 were tabulated (Appendix 2, www.dentalresearch.org). As anticipated, all three models, in both restrained-end and three-point loading, were accurate at higher values of L/d. At values below approximately 15, the standard flexure formula increasingly overestimated stiffness. The soft zone model overestimated rigidity in restrained-end loading, but is comparable with the shear-corrected formula in three-point loading. The model correcting for only shear was predictive down to the lowest values of L/d. The difference between the model and the experimental data became statistically significant (p < 0.05) below L/d values of 20 and 10, for restrained-end and three-point loading, respectively.
Comparing measured rigidity (Figs. 2A and 2B or Appendices 2A and 2C, www.dentalresearch.org), at high values of L/d, the beams in the restrained-end device are 2-4 times more rigid than those tested with simply supported ends, consistent with the coefficients of 12/192 and 48/192 for the different loading conditions in the flexure formula. At lower values of L/d, the difference decreases, consistent with the decreasing influence of the (L/d)3 term and the common shear term in Eq. 6.
While the focus of this study was rigidity, the loads at the elastic limit are also quite important clinically. These values are shown for each loading condition as a function of L/d (Fig. 3
All analytical models of rigidity became more accurate with increasing values of L/d. This is consistent with generally accepted theory and previous reports of flexure behavior of fiber-reinforced composites in the materials science and dental literature. This is because even under central loads, long slender beams deform primarily due to flexure or bending, there is little contribution to deflection from shear, and end effects are small.
Even though the difference between the shear-corrected restrained-end model and experimental values was statistically significant (p < 0.05) below L/d values of 10, the model was still reasonably predictive in this clinically important region. This is consistent with the relative contributions of shear and flexure to total deflection, which were calculated with Eq. 6 as a function of L/d (Fig. 4
Analysis of the data in this study suggests that the rigidity of three-unit FRC prostheses is more dependent on the shear modulus, G, than previously acknowledged. Investigators frequently associate restoration stiffness with elastic modulus, E. For either restrained-end or three-point loading at L/d above about 30, this is true, but the L/d for most three-unit FRC prostheses is well below 10. Accordingly, if an increase in appliance stiffness is desired, attention needs to be directed to the shear modulus as well as to E. The in-plane shear modulus for unidirectional composites with fiber volume fractions below 50% are largely dependent on the matrix properties (Kaw, 1997). Since most dental FRC systems use similar dimethacrylate matrices, it is likely that the G values for most products are comparable. Differences in products are primarily due to fiber content, type, and orientation. Analysis of the current data indicates that independent measurement of G will be necessary for accurate modeling of flexural behavior. Literature suggests that G values for these fiber composites are in the range of 0.5 to 5.0 (O’Brien and Ryge, 1978; ASM International Handbook Committee, 1987), but with clinically relevant dimensions, this range translates to an approximately six- and three-fold difference in rigidity for restrained-end and three-point loaded beams, respectively.
The shear deformation probably also explains the dependence of FRC strength on span length. At higher span lengths, the load is supported primarily by the higher-strength fibers. As L/d is decreased, the load is supported by the fibers in tension and compression as well as the matrix in shear. Therefore, while the clinically important maximum loads (Appendix 3, www.dentalresearch.org) and loads at the elastic limit (Fig. 3
It is interesting to compare the rigidity of the restrained-end and three-point loading models in the L/d region below 10 (Fig. 2
One reason for the current study was to evaluate the use of a restrained-end test arrangement instead of the more traditional three-point flexure test. Several researchers have studied the in vitro flexure behavior of FRC fixed partial dentures (Loose et al., 1998; Vallittu, 1998; Behr et al., 1999), but these reports focused on the effects of fiber reinforcement and strength or maximum load of the complete prosthesis. Haser (1999) studied non-veneered FRC substructures and reported rigidity as well as load, which allows for some comparison with the present data. The rigidity of FRC substructures bonded to steel dies was reported to be 939 N/mm. The L/d of the FRC section between the retainers was 5.0 and approximately uniform. The rigidity of beams with comparable L/d predicted with the present shear-corrected models for restrained-end and three-point loading are 767 and 607 N/mm, respectively (Fig. 2 In the clinical situation, physiological tooth movement (Parfitt, 1960), deformation of the luting agent, and possibly other factors would reduce the apparent rigidity of the prosthesis. It is possible that the shear-corrected three-point test is more representative when clinical factors allow for greater mobility of the prosthesis, such as failure of the luting cement (Loose et al., 1998). The shear-corrected restrained-end model may be more accurate for cases of less prosthesis movement, possibly such as restorations over implants. Of course, neither test method is intended as a replacement for more complex approaches that attempt to reproduce effects of anatomy, periodontal ligament, and other clinical factors. The three-point and restrained-end tests are most useful for developing analytical models and for evaluating material properties, formulation changes, environmental challenges, etc. The present data can be compared with earlier work by others who studied similar formulations and storage conditions by calculating flexure modulus with Eq. 4 for samples with nominal cross-sections of 2 mm x 2 mm tested at a span length of 20 mm. Under these comparable conditions, Lassila et al. (2002) and Karmaker and Prasad (2000) reported flexure modulus values of 24.0 and 27.5 GPa, compared with the present value of 27.1 GPa.
This study is supported by a grant from the Whitaker Foundation awarded through the Biomedical Engineering Alliance for Connecticut; by Connecticut Innovations, Inc.; and by Pentron Laboratory Technologies, LLC. During part of the time that this work was conducted, Drs. Goldberg and Burstone were University-approved consultants for Pentron, and patents assigned to the University of Connecticut were licensed to Pentron. Potential conflicts were managed according to University policy. This paper is based on a thesis submitted in partial fulfillment of the requirements for the degree of Master of Science at the University of Connecticut. Portions of this study were presented as abstracts at the 1999 & 2002 IADR General Sessions (Vancouver, Canada, and San Diego, CA, USA, respectively).
A supplemental appendix to this article is published electronically only at http://www.dentalresearch.org. Received for publication May 13, 2002. Revision received October 7, 2002. Accepted for publication January 10, 2003.
Journal of Dental Research, Vol. 82, No. 4,
262-266 (2003)
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ABSTRACT
INTRODUCTION
) and three-point loading (
). Error bars indicate standard deviations of the calculated means (n = 3).