Abstract
Objective
To design a protocol for the determination of the flexural modulus of elasticity of wire with high accuracy and precision.
Methods
Cantilever bending was used at long span and low loads, using a laser displacement sensor. Various wires were tested for proof-of-concept: stainless steel, Elgiloy, Wiptam and a gold alloy, with primary attention being paid to the source and magnitude of errors.
Results
The moduli of elasticity were determined with a median relative error of ∼0.15%, with a worst case of 0.56%, taking into account all measurement errors. The precision was such as to permit distinctions between the values for the wires of different thermomechanical history.
Significance
The determination of the elastic modulus of wires may be made with good precision using cantilever bending providing attention is given to all sources of error. These results represent the most reliable measures of modulus of elasticity of orthodontic archwires to date.
1
Introduction
Wires are routinely used in orthodontic appliances, primarily for generating the forces required to move teeth. These forces depend on the shape and dimensions of the spring that is the essence of the component, but in terms of characterizing the wire itself the flexural rigidity, EI z is the key descriptor. I z , the second moment of inertia ( i.e. , of the wire cross-section), represents the shape factor, while the modulus of elasticity, E , represents the controlling material property, its stiffness .
Knowledge of these values, and the associated device mechanics, is important in the selection of archwires, especially now that the emphasis is on establishing the force system to be used before designing the appliance . Indeed, as numerical simulation of tooth movement becomes commonplace, it is of greater importance than ever that values be accurate.
However, the measurement of even such an elementary property as the value of E has been associated with a number of difficulties. Thus, ADA Specification No. 32 requires a cantilever bending test, but in using this it has been reported that the determined flexural modulus depends on the specimen geometry. For example, Brantley et al. found values to increase with span length, which is contrary to the concept of a material property. Goldberg et al. and Yoshikawa et al. resorted to analytical techniques to correct this apparent dependence, but notably failed to correct for load-cell deflection. Unfortunately, all of these studies have used small test span lengths, in the range 12–50 mm, and this reduces the relative accuracy of the span length measurement, while the choice of large deflections conceivably introduced some risk of plastic deformation, but in any case is geometrically problematic as the Hooke’s Law interpretation of results depends on deflections being small such that the geometry is essentially unchanged. Three- , four- and even five-point bending (adding one support on each side to a three-point bending test to represent the dimension of a slot) tests have had several systematic errors in addition to the specimen geometry dependence: in particular, failure to account for machine deflection, excessive wire deflection , and friction that caused more scatter. In addition, the modest experimental accuracy of the above test designs prevents them from being able to approach the true values of the test materials’ moduli of elasticity. For comparison, it may be noted that the ISO standard has as a requirement that the measured modulus of elasticity (for linear materials) is within the range claimed by the manufacturer, using the same test, which involves three-point bending on a 10 mm span, at 23 °C. This makes no pretence of accuracy in any absolute sense and in effect merely examines labeling. It cannot therefore be taken as a reference method for the material property as such. If such values are required, other techniques must be used.
Given these problems, and the fundamental importance of the value of the modulus of elasticity, an improved test apparatus and protocol were designed, based on cantilever bending, with a view to increased accuracy and precision of the determination. The approach is very simple, but attention was paid to reducing and quantifying the errors arising from every source.
2
Materials and methods
2.1
Apparatus
The principal features of the apparatus are indicated in Fig. 1 . A massive (17 kg) T-slotted steel surface-plate as a base was carried on a critically-damped concrete bench top (400 kg) to minimize vibration. The work area, at 23 ± 2 °C, 50 ± 10%RH, was shielded from draughts. Local station gravity, g s , was 9.787641 ms −2 .
A square-nosed, hollow-handled, four-jaw pin vice (Cat. No. 120, Eclipse, Sheffield, UK), size selected to match the test wire it carried, was held securely in a V-groove on the top of the aluminum cylinder, which was itself bolted to the steel base. A traveling microscope (model 2158, 11495-HB, The Precision Tool & Instrument Co., Thornton Heath, Surrey, England), horizontal range 168 mm, the vernier scale reading to 10 μm, was mounted on the base parallel to the test wire so that both the nose of the pin vice and the load point of the wire could be observed in order to determine span length ( L ). A laser displacement sensor (CCD, LK-031 series, Keyence, Osaka, Japan), mounted on a three-axis, micrometer-adjustable stand, also bolted to the base plate, and arranged to be accurately vertical (maximum angular error <0.25°), was used to determine load-point deflection.
A weighing pan taken from a microbalance, and weighing approximately 1 g, was suspended at the load point. A 2-mm square flat stainless steel tab was laser-welded to the top of the weighing pan loop, to lie normal to the load axis. Loads were provided by standard analytical balance weights, 10 mg to 2 g, whose actual masses were determined to 0.01 mg on a single pan, 160-g capacity, analytical balance (H542, Mettler Instruments, Zurich, Switzerland), the mean of eight separate determinations being used (the standard deviations of these measurements were of the order of 2–10 × 10 −2 mg). These weights were supplemented by stainless steel wire ‘riders’, similarly calibrated, for use when the weighing pan itself was too great a tare load. The corrections for air buoyancy (pressure, temperature, and humidity) were calculated and found to be well below detection and thus no further account of this was taken.
All wires were straight and tested as-received ( Table 1 ). Wire diameters ( D ) were determined with a digital micrometer gauge (Mitutoyo, Japan) reading to 1 μm. A very small notch was cut in the wire near one end with a fine triangular file, and carefully deburred, to prevent the weighing pan or riders from slipping. A thin film of light machine lubricating oil was applied to the notch.
| Wire alloy | Composition (mass%) | Manufacturer |
|---|---|---|
| Co–Cr (Wiptam) | Co 40 | Krupp, Widia, Essen, Germany |
| Cr 20 | ||
| Ni 16 | ||
| Fe 14 | ||
| Mo 7.25 | ||
| Mn 2 | ||
| + Other elements | ||
| Co–Cr (Elgiloy) | Co 40 | Rocky Mountain Orthodontics, Denver, CO, USA |
| Cr 20 | ||
| Ni 15 | ||
| Fe 15.8 | ||
| Mo 7 | ||
| Mn 2 | ||
| C 0.15 | ||
| Be 0.04 max. | ||
| Austenitic stainless steel | Cr 17–20 a | K.C. Smith, Monmouth, Gwent, Wales |
| Ni 8–12 | ||
| C 0.15 | ||
| Balance mainly Fe | ||
| Ag–Au–Pt | Au 61 | Cendres + Métaux SA, Biel/Bienne, CH |
| Pt 13.5 | ||
| Ag 16.5 | ||
| Cu 9 | ||
2.2
Procedure
Four separate measurements of the wire specimen diameter were taken at four orientations, approximately 0°, 45°, 90° and 135°, at four roughly equally spaced points along the span length, the mean of which 16 values was taken as the effective diameter ( D ); the standard deviation of those measurements was recorded.
The wire was secured in the pin vice such that the notch was uppermost in approximately the intended position. The traveling microscope was used to observe that the wire was correctly located and gripped, and that there was no detectable movement when the free end of the wire was deflected by the maximum test amount (5 mm). The fiducial reading for span was then taken using the traveling microscope at the nose of the pin vice. The distance to the center of the V-notch (nominal load point) was then determined, ignoring self-weight deflection. This distance was taken as the span, L . The span was arbitrary, but controlled to limit the maximum deflection (which ranged from 0.1 to 5 mm, but typically 1–3 mm).
The weighing pan was then suspended at the notch and the laser sensor adjusted so that the illuminated point lay at the center of the tab, directly over the notch. The vertical distance of the sensor was adjusted to give a suitably small fiducial reading (0 < y < 0.030 mm); stability was confirmed. A weight was then placed at the center of the pan, to minimize tilt, and when stability was again obtained, the deflection recorded. Five nominally equally-spaced values of load were applied, so that six displacements were recorded in each run including the fiducial reading. Runs were of two kinds: ‘single’, for an instance of a wire being set up, and ‘replicate’, in which redetermination of the load-deflection behavior was made without demounting the wire, in order to assess the reproducibility of that part of the process.
2.3
Analysis
The slope ( b = y / F ), and its standard error, of the assumed straight-line load–displacement plot, including the fiducial point (but without constraint), was determined for each set of six readings by linear regression in software (SigmaPlot for Windows, version 10, Systat, Chicago, IL, USA). E was then found from
E = 64 L 3 3 π D 4 F y
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