Orthodontic Materials (eBook)

2 Mechanics and Mechanical Testing of Orthodontic Materials

William A. Brantley

Theodore Eliades

Alan S. Litsky

Appearance of the fracture surface for a notched tensile test specimen as the fracture mode changes from a mixture of plane stress and plane strain to purely plane strain conditions

Introduction

Bending Deformation

Torsional Deformation

Mechanical Testing Methods for Orthodontic Materials

Mechanical Testing Machines and Experimental Procedures

Specific Tests for Evaluation of Mechanical Properties of Orthodontic Materials

References

Introduction

Basic knowledge of solid mechanics is important for an appreciation of biomechanical principles used in tooth movement and for understanding the important bending and torsion tests used to measure the mechanical properties of clinical relevance for orthodontic wires. Mechanical testing is employed in the biomaterials laboratory to evaluate the important properties of metallic, ceramic, and polymeric orthodontic materials that were discussed in Chapter 1.

In this chapter the major concepts for bending and torsional deformation of solids, and the origin of mathematical relationships that have been used to evaluate the mechanical properties of archwires, will be presented. The two types of universal mechanical testing machines will be described, followed by consideration of experimental procedures for laboratory measurement of some mechanical properties of orthodontic materials. Particular emphasis will be placed upon bending tests, the evaluation of adhesive bond strength, and the measurement of fracture toughness. The diametral compression test and the measurement of fatigue behavior will also be discussed. Extensive information about other mechanical property measurements for dental materials is available in the Craig textbook.

Bending Deformation

Figure 2.1 shows a segment of a uniform rectangular beam subjected to pure elastic bending, i.e., acted upon only by a bending moment that is of insufficient magnitude to cause permanent deformation. The beam may be considered to represent an orthodontic archwire. In order to have static mechanical equilibrium, the ends of this segment are acted upon by equal and opposite bending moments. From examination of Figure 2.1 it is apparent that, at the top and bottom outermost surfaces, the length of the beam has experienced an increase (tensile strain) and a decrease (compressive strain) parallel to the axis, respectively. For a symmetric beam (round, rectangular, or square cross section), the material at the midplane does not experience any deformation. The un-deformed midplane of the elastically bent beam is termed the neutral surface, and the trace of the neutral surface on the cross section perpendicular to the beam axis is termed the neutral axis. Both the neutral surface and neutral axis are indicated in Figure 2.1.

Fig. 2.1 A portion of a symmetric rectangular beam subjected to pure elastic bending. The location of the neutral surface is indicated, along with the position of the neutral axis on an axial cross section of the beam. An element of area (ΔA) located a distance (y) from the neutral surface is also shown

Figure 2.2 shows that the stress in the beam varies linearly with distance from the mid-plane, reaching maximum values at the outermost surface. The corresponding strain is obtained by dividing the stress by the modulus of elasticity (Young's modulus) of the beam material, which has the same value for tensile and compressive stress. In textbooks on solid mechanics, force and moment balances are performed on a section of the beam (free-body diagram) to derive the elastic flexure formula. It is also assumed that plane surfaces perpendicular to the undeformed beam axis remain planar after the elastic deformation. The relationship between the stress (σ) developed in the beam as a function of the bending moment (M) and distance from the neutral axis (y) is

where I represents the moment of inertia of the cross section. The moment of inertia is a geometric quantity that corresponds to the resistance of a particular cross section to bending and is given by the relationship:

where the yi are the distances of the elemental areas (ΔAi) from the neutral axis and the summation is over all the elemental areas comprising the cross section of the beam. The contributions to the moment of inertia are greatest for the elemental areas farthest from the neutral axis, since each area is multiplied by the square of its distance from this centerline of the cross section (for a symmetric beam). This principle is exploited with the rectangular I-beam used in the construction of buildings, where the maximum amount of material is located farthest from the center (neutral axis) of the beam.

Since the largest value of y corresponds to the greatest distance (c) from the neutral axis to the surface of the beam, the maximum stress developed by the bending moment is given by

The section modulus (Z) is often defined as

so that the maximum stress in the beam can be written alternatively as

This expression is convenient to use when comparing the maximum stress developed in a series of orthodontic wires of varying cross-sectional dimensions. The relationship is analogous to the equation that defines normal stress (tensile or compressive) as the quotient of force and cross-sectional area. For bending, the role of force is assumed by the moment, and the section modulus provides information about the cross-sectional geometry of the beam.

Writing an integral expression for the moment of inertia as the elemental areas become infinitesimally small, it can be shown that for a round beam of diameter (d):

Fig. 2.2 Variation of the compressive and tensile stress with distance from the midplane of the bent symmetric beam in Figure 2.1

a Diameter of round wire.

b Cross-sectional dimensions of rectangular wire.

c Bent flatwise.

d Bent edgewise.

For a rectangular beam of width w and thickness t in the direction of bending, the moment of inertia is given by

Because of the dependence of I on the fourth power of diameter and the cube of thickness, there can be considerable differences in the resistance to bending of orthodontic archwires having different cross-sectional dimensions.

For comparison, the values of moment of inertia for several archwire sizes are listed in Table 2.1. The enormous differences in I are evident for the four round wires shown. The greater value of I for a square wire, compared to a round wire of the same cross-sectional dimensions, can be seen from the examples of the 0.406 mm diameter round wire and the 0.406 mm × 0.406 mm square wire. The bending direction has a substantial effect on the value of I for a rectangular wire, as shown for the 0.457 mm × 0.635 mm wire bent in the edgewise (t = 0.635 mm) and flatwise (t = 0.457 mm) directions.

For a given value of moment or load, it can be shown that the elastic bending deflection of a beam (or archwire segment) is inversely proportional to its length. Consequently, the stiffness in bending for an archwire segment of length l is given by the following proportionality

where the elastic modulus E represents the alloy contribution and I/l represents the segment geometry (cross section and length) contribution to stiffness. This relationship is useful for comparing the relative values of stiffness in bending for round and rectangular archwires of different alloys, using the preceding expressions for I and values of E given in Chapter 4.

Torsional Deformation

Figure 2.3 shows the stress distribution over the cross section of a circular beam (round orthodontic wire) of radius c subjected to torsional loading in the elastic range. It can be seen that the beam experiences shear stress, which varies linearly from zero at the center to a maximum value (τmax) at the surface. With the assumption that plane sections...