1. Introduction

2. Materials and Methods

Figure 1. Conical-Cylindrical dental bur: 1—the active part of the dental bur, 2—the neck of the dental bur, 3—dental bur foot.

Figure 1. Conical-Cylindrical dental bur: 1—the active part of the dental bur, 2—the neck of the dental bur, 3—dental bur foot.

Figure 2. Sample dental material (Co-Cr alloy).

Figure 2. Sample dental material (Co-Cr alloy).

3. Results

3.1. Estimation of wear and durability of dental burs

The experimental research had in mind, first of all, the determination of the mass, m_f of the specimen dental bur subjected to the research, respectively of the mass of the processed dental material, m_m before testing.

Table 2 and Table 3 show the mass and dimensions of the specimen dental bur (Table 2), respectively the mass and dimensions of the sample dental material subjected to testing (Table 3), before testing.

The results of the experiments show the dependence on time and the rotation speed of the dental bur, the geometric characteristics of its active part, and the mass of material lost through wear, m_w [10]. The list of these parameters was given in Table 1.

The method used to obtain the results is based on polynomial interpolation using the least squares technique. Thus, the general mathematical relationship of interpolation polynomials, written in canonical form, is as follows:

$$q\left(t,\omega \right)=\sum _{i=0}^3\sum _{j=0}^3{c}_{ij}{t}^i{\omega }^j,$$

(1)

which developed up to the third degree, becomes:

q(t, ω) = c₀₀t⁰ω⁰ + c₁₀t¹ω⁰ + c₂₀t²ω⁰+ c₃₀t³ω⁰+ c₀₁t⁰ω¹+ c₀₂t⁰ω² + c₀₃t⁰ω³,

(2)

as being sufficient to describe the wear phenomenon of the active part of the dental burs, and q the interpolated parameter (here, the mass lost by the wear of the active part of the dental bur, m_w).

Note: As interpolated parameters can be used any other characteristic/functional parameter of the active part of the researched dental bur, for example, seating angle, clearance angle, sharpening angle, the area, or radius of the tip circle, etc.

The synthesis of the results of the least squares interpolation for the measured parameter (see Table 1) during the experimental tests is given in Table 4 and they contain the coefficients of the polynomials, c₀₀…c₀₃ with two variables (time, t, and the rotation speed of the dental bur, ω, implicitly and of its active part), as well as the average of the squared errors, ε, which ranks the approximation performances.

Note: Polynomial coefficients were calculated with Mathcad using the least squares method.

It can be seen that the polynomial coefficients are double indexed, the first index (example: 1 from c₁₀) represents the power exponent of the temporal variable (time, t). the second index (example: 0 of c₁₀) represents the rotation speed, ω.

To calculate the error, ε, the relation was used:

$$\epsilon =100·\frac{\sqrt{{\left(y_w-y\left(x_w\right)\right)}^2}}{\overline{y}·N},$$

(3)

where: y_w,w = 1,…, N are the experimental data for the dependent variable, for example, m_w; x_w = t_w, ω_w is the data string of the independent variable, for example, t and ω; N is the number of experimental data; $\overline{y}$ is the mean value of the experimental data for the string of the dependent variable.

For cutting, chipping, etc. instruments/tools, in general, and dental burs in particular, the assessment of wear and lifetime (durability) is done in relation to one or more geometric or physical parameters (such as the lamellas angles of the active part of the dental burs, the area or radius of the peak circle), as described in the literature of specialty [11,12].

Thus, the main indicator of dental bur wear is the mass removed (lost) through wear, m_w. For the experiments carried out, the variation of this parameter, m_w, was recorded for the five operating times, established in advance (1, 2, 3, 3.5, and 4 hours of operation) and for the four values of the speed rotation of the dental bur (7000, 12000, 20000, and 35000 rpm). The results were stored in the Excel numerical database and a synthetic representation of them can be seen in Figure 4.

Similar results can be obtained and for other characteristic/functional parameters of the active part of the researched dental bur (the lamellas angles: of seating, clearance, sharpening; the area or the radius of the peak circle, etc.).

Figure 4. Evolution of the mass, m_w of the active part of dental burs removed by wear in the work process, as a function of time, t and of rotation speed, n.

Figure 4. Evolution of the mass, m_w of the active part of dental burs removed by wear in the work process, as a function of time, t and of rotation speed, n.

Estimation of the lifetime (durability) of the active part of the tested dental bur, cylindrical-conical type (see Figure 1), manufactured from a metal mixture in which tungsten predominates (58.14%), in relation to the mass lost through wear of the active part of the dental bur, m_w is shown in Table 5. To estimate the lifetime of the dental bur, in relation to m_w at the four revolutions used (see Table 5) in the experimental tests, were take account considered the polynomial interpolation functions of the first, second, and third degree, as specified above.

Thus, to estimate the durability or lifetime of the active part of a dental bur, by calculation, the working time calculation relationship was used [13]:

$$t_{\mu }=\frac{\ln \frac{1}{1-\mu }}{\omega ·d},$$

(4)

established based on the conventional determination of a fraction, μ of the total mass of the active part of the dental bur, which is lost during the working process, ω the angular velocity of the researched burs, and d model parameter, which is determined by the method of least squares applied to a non-linear function dependent on m_w and the angular space (friction length) traveled by the dental bur.

To establish the critical fraction, μ of material of the active part of the dental bur, which can be lost by the milling process, one can start from about 65 % of the mass of the approximate circular crown of the active part of the dental bur, determined experimentally through the total wear of the lamellas of the active part of the dental bur, followed of weighing and by the difference to the weight of the specimen dental bur (Figure 5) the value of 0.522 g was obtained.

Thus, for the dental bur tested in the laboratory, the results are those given in Table 6, from the column of mass lost through wear, m_w.

Using the experimental data whose evolution can be seen in Figure 6, the values obtained are for the time variation of the mass of the dental bur, m_w, and of the milled dental material, m_m. Thus, the initial mass of the specimen dental bur was m_f = 4.701 g, and the mass of the processed dental material, m_m = 6.180 g, the final values of the mass of the dental bur, m_w varied between 4.700 g and 4.360 g and of the processed dental material varied between 6.102 g and 4.160 g. Figure 6 shows the average variations in time of the masses of the two bodies in interaction/contact with relative movement (in the work process): the dental bur and the dental material, subject to milling. The processes that take place by friction on the contact surfaces and in the surface layers, respectively in the micro volumes of asperities in contact are of a different nature and depend on a series of mechanical, physical, chemical, and geometric factors.

It is observed that both bodies (dental bur and dental material), which interact (are in contact with relative movement), lose mass. The loss of mass over time has a variation according to an exponential curve (after one hour of operation), even if, in the prescribed working interval, due to the values of the parameters, their curvature is not sufficiently accentuated.

For the mass lost by the dental bur and the mass lost by the milled dental material, the efficiency ratio of the dental bur was additionally considered, e(t), i.e., the ratio between the dental material lost (removed, in grams) to the material lost (removed, in grams) by the dental bur.

The graph of the variation of the mass of material lost by the dental bur and of the milled dental material as a function of time, measured during the tests, can be seen in Figure 7, and the time variation of the efficiency ratio e(t) between the mass loss of the dental material milled and the mass loss of the dental bur material (on the active area) is represented in Figure 8.

Based on the graphical representations in Figure 7 and Figure 8 it can be seen that the material and configuration of the dental bur allow it to be efficient, meaning that a small amount of dental bur material lost removes more material from the milled dental material. However, this effect is visible for a time much longer than four hours of operation, after which the inefficiency of the dental burs was observed (see Figure 5).

4. Discussions

Conclusions

• The study presented in this paper was limited to the analysis and the consideration of only one parameter, m_w, to establish and evaluate the durability and reliability of a type of dental bur, but it can be extended/applied, to other functional parameters of the active part of dental burs (seating angle, clearance angle, sharpening angle, tip circle area, etc.).
• The wear phenomenon of the active part of a dental bur limits the durability and reliability of dental burs and is studied based on the experimental results obtained in the work process and validated by interpolation and led to polynomial functions which approximate very well the dependent parameter, m_w, considered in the experimental program.
• Additionally, these functions allowed the determination of the limit (extreme) wear values, where dental bur becomes ineffective, although the material and configuration allow it to be efficient (i.e., a small amount of dental bur material lost, removes an amount more from the milled dental material), in the period of time determined analytical and experimental.
• At the same time the interpolation of the experimental data/results led to practical applications, useful for the extension of the life of the dental bur, even for the optimization of the parameters of operation, execution, and maintenance; establishing some possible criteria for replacing the worn ones; the provision of ideas for constructive solutions, or the extension of the lifetime of the materials from which the dental burs are made.
• The complete solution of the problem without experimental information becomes an approximate one because the experiments were performed on only one type of dental bur. The results closest to reality are those given by the interpolation function of the third degree, and they are in good agreement with the experimental results.
• The results obtained by interpolation showed that the dental bur during the work process has a high efficiency (mw is very low) during the first 11 hours of operation when the life increased by about 10 %. After 11 hours of operation, mw increases relatively exponentially; thus, it is recommended that the dental bur is normally replaced with a new one to ensure dental high material processing standards.
• The researched parameter, m_w measured by weighing, was observed to take different values and randoms between two limits, each value has its own frequency (a number of repetitions), and the distribution of these values (with their frequency) can be represented by a normal distribution law, based on Gauss's function.
• A similar study can be justified by other functional parameters of the active part of the researched dental bur, with the goal to check if the method used is validated, regarding the extension of the lifetime (durability) or even optimization of their operation (reliability).
• It is proposed, the continuation of the research method by obtaining several experimental data and supplementary information for a more complete solution to the problem of wear of the active part of dental burs, method extension on other functional parameters that were not monitored, and other types of dental burs in order to record the parameters, using new deterministic or even statistical models of the phenomenon.
• In this way, will be validated the influence (importance) of the parameter, mw, of the work regime together with the other parameters mentioned in the paper. The mathematical model obtained based on the structured and discussed experimental data in this paper will be used to investigate the possibilities of improving or optimizing the working regimes of dental burs and other characteristic/functional parameters.

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