Сraig. Dental Materials

if a small notch is placed on the surface of the glass bar, less force is needed to cause fracture. If the same experiment is performed on a ductile material, we find that a small surface notch has no effect on the force required to break the bar, and the ductile bar can be bent without fracturing (Fig. 4-11). For a brittle material such as glass, no local plastic deformation is associated with fracture, whereas for a ductile material, plastic deformation, such as the ability to bend, occurs without fracture. The ability to be plastically deformed without fracture, or the amount of energy required for fracture, is the fracture toughness.

In general, the larger a flaw, the lower the stress needed to cause fracture. This is because the stresses, which would normally be supported by material, are now concentrated at the edge of the flaw. The ability of a flaw to cause fracture depends on the fracture toughness of the material. Fracture toughness is a material property and is proportional to the energy consumed in plastic deformation.

A material is characterized by the energy release rate, G, and the stress intensity factor, K. The energy release rate is a function of the energy involved in crack propagation, whereas the stress intensity factor describes the stresses

at the tip of a crack. The stress intensity factor changes with crack length and stress according to

where Y is a function which is dependent on crack size and geometry. A material fractures when the stress intensity reaches a critical value, Kc. This value of the stress intensity at fracture is called the fracture toughness. Fracture toughness gives a relative value of a material's ability to resist crack propagation. The units of Kc are units of stress (force/length2) x units of length1", or

nano-indentation techniques have recently been introduced as a means of measuring mechanical properties of micron-sized phases in a material. For brittle materials, one of the properties that may be measured is fracture toughness. By selectively indenting specific regions of a microstructure, the relative effects of different microstructural constituents may be identified. The spatial variation in properties may also be

82 Chapter 4 MECHANICAL PROPERTIES

Fig. 4-11 Schematic of different types of deformation in brittle (glass, steel file) and ductile (copper) materials of the same diameter and having a notch of the same dimensions.

(From Flinn RA, Trojan PK: Engineering materials and their applications, Boston, 1981, Houghton Mifflin, p 535.)

determined. For example, amalgams show significant differences in fracture toughness as a function of distance from the margin.

Fracture toughness (K,,) has been measured for a number of important restorative materials, including amalgams, acrylic denture base materials, composites, ceramics, and orthodontic brackets, cements, and human enamel and dentin. Typical values for composites, ceramics, enamel, and dentin are indicated in Table 4-6.

The presence of fillers in polymers substantially increases fracture toughness. The mechanisms of toughening are presumed to be matrixfiller interactions but are not yet established. Similarly,the addition of up to 50 wt% zirconia to ceramic increases fracture toughness. As with other mechanical properties, aging or storage in a simulated oral environment or at elevated temperatures can decrease fracture toughness, but there is no uniform agreement in the literature. Attempts to correlate fracture toughness with wear resistance have been mixed, and therefore it is not an unequivocal predictor of the wear of restorative materials. Also, numerical analysis techniques have been applied to composites and the tooth-denture base joint to determine energy release rates in the presence of cracks.

Fracture

Y 6 6

Glass rod

Natch

Plastic deformation

Grooves act as small notches

--

Hardened steel file

K,,, Fracture toughness.

PROPERTIES AND STRESS-STRAIN CURVES

The shape of a stress-strain curve and the magnitudes of the stress and strain allow classification of materials with respect to their general properties. The idealized stress-strain curves in Fig. 4-12 represent materials with various combinations of physical properties. For example, materials 1to 4 have high stiffness, materials 1, 2, 5, and G have high strength, and materials 1, 3, 5, and 7 have high ductility. If the only requirement for an application is stiffness, materials 1 to 4 would all be satisfactory. However, if the requirements were both stiffness and

strength, only materials 1 and 2 would now be acceptable. If the requirements were to also include ductility, the choice would be limited to material 1. It is clear that the properties of stiffness, strength, and ductility are independent, and materials may exhibit various combinations of these three properties.

OTHER MECHANICAL PROP

TENSILE PROPERTIES O F BRITTLE MATERIALS

A variety of brittle restorative materials, including dental amalgam, cements, ceramics, plaster and stone, and some impression materials, is important to dental practice. In many instances the material is much weaker in tension than in compression, which may contribute to failure of the material in service. Such material should therefore be used only in areas subjected to compressive stresses.

Previously, test methods were described for the development of stress-strain curves resulting from tensile measurements on ductile materials such as metals, alloys, and some types of plastics. Similar test methods have been applied to brittle materials. However, brittle materials must be tested with caution, and any stress concentrations at the grips or anywhere else in the specimen can lead to premature fracture. As a result, there has been large variability in tensile data on

brittle materials. Although special grips have been used to permit axial tensile loading with a minimum of localized stress concentrations, obtaining uniform results is still difficult, and such testing is relatively slow and time consuming.

An alternative method of testing brittle materials, in which the ultimate tensile strength of a brittle material is determined through compressive testing, has become popular because of its relative simplicity and reproducibility of results. The method is described in the literature as the diametral compression test for tension, the Brazilian test, or the indirect tensile test. In this test method, a disk of the brittle material is compressed diametrically in a testing machine until fracture occurs, as shown in Fig. 4-13. The compressive stress applied to the specimen introduces a tensile stress in the material in the plane of the force application of the test machine. The tensile stress is directly proportional to the load applied in compression through the following formula:

Note that if the specimen deforms significantly before failure or fractures into more than two equal pieces, the data may not be valid. Some materials yield different diametral tensile

Load

P

I

Thickness

Compression support

Fig. 4-13 Drawing to illustrate how compression force develo~stensile stress in brittle materials.

strengths when tested at different rates of loading and are described as being strain-rate sensitive. Strain-rate dependence is discussed later in the chapter. The diametral tensile test is not valid for these materials, and thus the strain-rate sensitivity of a material should be determined before this test is used to evaluate the tensile strength. Values of diametral and ultimate tensile strengths for some dental materials are listed in Table 4-7.

COMPRESSIVE PROPERTIES

Compressive strength is important in many restorative dental materials and accessory items used in dental techniques and operations. This property is particularly important in the process of mastication because many of the forces of mastication are compressive. Compressive strength is most useful for comparing materials that are brittle and generally weak in tension and that are therefore not employed in regions of the oral cavity where tensile forces predominate. It is somewhat less useful to determine the compressive properties of ductile materials such as gold alloys. Compressive strength is therefore a useful property for the comparison of dental amalgam, resin composites, and cements and for determining the qualities of other materials such as

Diametral Ultimate

Tensile Tensile

Strength Strength

Material

Gold alloy Amalgam Dentin

Resin composite Feldspathic porcelain Enamel

Zinc phosphate cement

High-strength stone Calcium hydroxide

liner

plaster, investments, and some impression materials. Typical values of compressive strength of some restorative dental materials are given in Table 4-8.

Certain characteristics observed in materials subjected to tension are also observed when a material is in compression. For example, a stressstrain curve can be recorded for a material in compression similar to that obtained in tension. Such a curve represents a material that has both elastic and plastic characteristics when subjected to compressive stress, although the plastic region is generally small. The modulus of elasticity of a material in compression can be determined from the ratio of stress to strain in the elastic region. Such a modulus value is usually similar for a material whether tested in compression or tension. A proportional limit or yield strength in compression can also be observed. The ultimate compressive strength is calculated from the original cross-sectional area of the specimen and the maximum applied force, in a similar manner to the ultimate tensile strength.

When a structure is subjected to compression, note that the failure of the body may occur as a result of complex stress formations in the

body. This is illustrated by a cross-sectionalview of a right cylinder subjected to compression, as shown in Fig. 4-14. It is apparent from Fig. 4-14 that the forces of compression applied to each end of the specimen are resolved into forces of shear along a cone-shaped area at each end and, as a result of the action of the two cones on the cylinder, into tensile forces in the central portion of the mass. Because of this resolution of forces in the body, it has become necessary to adopt standard sizes and dimensions to obtain reproducible test results. Fig. 4-14 shows that if a test specimen is too short, the force distributions become more complicated as a result of the cone formations overlapping in the ends of the cylinder. If the specimen is too long, buckling may occur. Therefore the cylinder should have a length twice that of the diameter for the most satisfactory results.

SHEAR STRENGTH

The shear strength is the maximum stress that a material can withstand before failure in a shear mode of loading. It is particularly important in the study of interfaces between two materials, such as porcelain fused to metal or an implanttissue interface. One method of testing the shear strength of dental materials is the punch or pushout method, in which an axial load is applied to

Fig. 4-14 Drawing of complex stress pattern developed in cylinder subjected to compressive stress.

push one material through another. The shear strength (7)is calculated by

Shear skrength(z)= F h d h

where F is the compressive force applied to the specimen, d is the diameter of the punch, and h is the thickness of the specimen. Note that the stress distribution caused by this method is not "pure" shear and that results often differ because of differences in specimen dimensions, surface geometry, composition and preparation, and mechanical testing procedure. However, it is a simple test to perform and has been used extensively. Alternatively, shear properties may be determined by subjecting a specimen to torsional

86 Chapter 4 MECHANICALPROPERTIES

loading. Shear strengths of some dental materials are listed in Table 4-9.

BOND STRENGTH

A variety of tests have been developed to measure the bond strength between two materials, such as porcelains or laboratory composites to metal; cements to metal; or polymers, ceramics, resin composites, and adhesives to human enamel and dentin. Most of the tests are designed to place the bond in tension, although a few, especially for ceramics to metals, place the bond in shear. To simulate oral conditions, many of the test specimens are subjected to cycles of varying temperature, ranging from 5" to 50" C,before measuring bond strength. These bond-strength values may not simulate the clinical situation because of differences between the geometry of the test specimens and the clinical application. Bond strength values typically overestimate the bond strength obtained in clinical usage and should therefore be viewed with caution.

BENDING

The bending properties of many materials are as or more important than their tensile or compressive properties. The bending properties of stainless steel wires, endodontic files and reamers,

Angular deflection (degrees)

Fig. 4-15 Bending moment-angular deflection curves for endodontic reamers sizes 20 through 70.

and hypodermic needles are especially important. For example, ANSI/ADA Specification No. 28 for endodontic files and reamers requires bending tests.

Bending properties are usually measured by clamping a specimen at one end and applying a force at a fixed distance from the face of the clamp. Specimens are subjected to conditions that resemble pure bending, and cantilever beam theory has been used to analyze the data. As the force is increased and the specimen is bent, corresponding values for the angle of bending and the bending moment (force x distance) are recorded. Graphic plots of the bending moment versus the angle of bending are similar in appearance to stress-strain curves. As an example, a series of plots for various sizes of endodontic reamers is shown in Fig. 4-15. An instrument will be permanently bent if the bending angle exceeds the value at the end of the linear portion of the curve. The larger instruments are stiffer, as shown by the initial steeper slope. The initial linear portion of the curve is shorter for the larger instruments and thus the deviation from linearity occurred at lower angular bends.

The maximum bending stress o in a wire is

where Mis the bending moment, y is the distance from the neutral axis (plane of the specimen which is stress free) to the outer surface of specimen, and Iis the moment of inertia, which indicates the distribution of forces relative to the specimen geometry.

The maximum angle of bending, em,,, of a wire fixed at one end may be determined by the following formula:

where 1 is the distance from the point of force application to the fixed end, and E is the modulus. For round wires with l/d - 15 and 1= 25 mm, the elastic modulus in bending (also called modulus of stiffness) approximates the elastic modulus in tension. The equation is

where 1 is the span length of the wire, d is the diameter of the wire, and M/0 is the slope of a plot of bending moment versus angular deflection in radians. The use of cantilever beam theory to calculate E results in values about two thirds those determined in tension.

TRANSVERSE STRENGTH

The transverse strength of a material is obtained when one loads a simple beam, supported at

each end, with a load applied in the middle (Fig.

4-16). Such a test is called a three-point bending (3PB) test and transverse strength is often described in technical, dental, and engineering literature as the modulus of rupture (MOR) or flexural strength. The transverse strengths for several dental materials are shown in Table 4-10. The transverse strength test is especially useful in comparing denture base materials in which a stress of this type is applied to the denture during mastication. This test determines not only the strength of the material indicated, but also the amount of distortion expected. The transverse strength test is a part of ANSI/ ADA Specification No. 12 (IS0 1567) for denture base resins. The transverse strength and accompanying deformation are important also in long bridge spans in which the biting stress may be severe.

The stresses and deflections in three-point bending can be determined as specific cases of the more general formulae presented in the last section. A beam having a rectangular cross sec-

88 Chapter 4 MECHANICALPROPERTIES

tion of width, b, and height, d, has a moment of inertia of

For a load, applied in the center, the bending moment is

Substituting these relations for I and M into the general equation 8 = My/I, the equation for the maximum stress developed in a rectangular beam loaded in the center of the span becomes

3 x Load x Length

Stress = 2 x Width x ~ h i c k n e s s ~

The resulting deformation or displacement in such a beam or bridge can be calculated from

Deformation =

Load x ~ e n ~ t h ~

supports is 89 mm. Because this is a static situation (i.e.,the beam does not move), the reactant forces at the supports in this symmetrical loading are 333 N each. The solution for this beam may be calculated as follows:

The moment of inertia is determined by

Thus at the lower surface of the beam, y = 12.7 mm and

or 21 . 5 MPa. The lower surface of the beam is under a tensile stress of 21.5 MPa, and the upper surface is under a compressive stress of 21 . 5 MPa. The maximum deflection is

4 x Elastic modulus x Width x ~ h i c k n e s s ~

The significance of the length, thickness, and width of the restoration in relation to the strength and deformation is evident from these formulae. Both the length and the thickness of the span are critical, because the deformation varies as the cube of these two dimensions.

As a numerical example, consider a simple beam, such as the one shown in Fig. 4 - 1 6 , with a rectangular cross section of 6 . 4 mm in thickness and 2 5 . 4 mm in height and a concentrated load of 666 N applied in the center. The total length of the beam is 102 mm, and the distance between

Fig. 4-17 Analysis of transverse bending. A, Photoelastic model with isochromatic fringes. B, Drawing to illustrate isochromatic fringe order.

The transverse strength of a beam can also be determined by the photoelastic method of analysis. A model of the simple beam used in this example is shown in Fig. 4-17, A. The isochromatic fringes, or lines of constant principle stress, are shown in Fig. 4-17, A, with the neutral axis, NA. The fringe order of the isochromatics is shown in Fig. 4-17, B, and the isotropic point can be seen in the center. Below the loading point and above the support points, the beam is in compression, whereas in the center of the lower portion of the beam it is in tension. Along any isochromatic fringe the difference in the principal stresses is constant, and the difference in the state of stress between fringes is 0.41 MPa/fringe.

PERMANENT BENDING

During fabrication, many dental restorations are subjected to permanent bending. The adjustment of removable partial denture clasps and the shaping of orthodontic appliances are two examples of such bending operations. Bends are also often introduced into hypodermic needles or root canal files in service. Comparisons of wires and needles of different compositions and diameters subjected to repeated 90-degree bends are often made. The number of bends a specimen will withstand is influenced by its composition and dimensions, as well as its treatment in fabrication. Such tests are important because this information is not readily related to standard mechanical test data such as tensile properties or hardness. Severe tensile and compressive stresses can be introduced into a material subjected to permanent bending. It is partly for this reason that tensile and compressive test data on a material are so important.

TORSION

Another mode of loading important to dentistry is torsion or twisting. For example, when an endodontic file is clamped at the tip and the handle is rotated, the instrument is subjected to torsion. Because most endodontic files and reamers are rotated in the root canal during endodontic treatment, their properties in torsion are of particular

interest. ANSI/ADA Specification No. 28 for endodontic files and reamers describes a test to measure resistance to fracture by twisting with a torque meter. Torsion results in a shear stress and a rotation of the specimen. In these types of applications, we are interested in the relation between torsional moment (M, = shear force x distance) and angular rotation 7c. A series of graphs in which the torsional moment was measured as a function of angular rotation is shown in Fig. 4-18. In this example, the instruments were twisted clockwise, which results in an untwisting of the instrument. As was the case with bending, the curves appear similar to stress-strain curves, with an initial linear portion followed by a nonlinear portion. The instruments should be used clinically so that they are not subjected to permanent angular rotation; thus the degrees of rotation should be limited to values within the linear portion of the torsional moment-angular rotation curves. The larger instruments are stiffer in torsion than the smaller ones, but their linear portion is less. The irregular shape of the curves at high angular rotation results from the untwisting of the instruments.

The resultant shear stress in a wire of radius r may be calculated from

T = Mt x r/I,

Angular rotation (degrees)

Fig. 4-18 Torsional moment-angular rotation curves for endodontic files sizes 15 through 60.

where I, is the polar moment of inertia. The angular rotation may be calculated from

MtL

+= -

GI,

where L is the length of the shaft, and G is the shear modulus.

FATIGUE STRENGTH

Based on the previous discussions, a structure that has been subjected to a stress below the yield stress and subsequently relieved of this stress should return to its original form without any change in its internal structure or properties. It has been found that a few such applications of stress do not appreciably affect a material. However, when this stress is repeated a great number of times, the strength of the material may be drastically reduced and ultimately cause failure. Fatigue is defined as a progressive fracture under repeated loading. Fatigue tests are performed by subjecting a specimen to alternating stress applications below the yield strength until fracture occurs. Tensile, compressive, shear, bending, and torsional fatigue tests can all be performed.

The fatigue strength is the stress at which a material fails under repeated loading. Failure under repeated or cyclic loading is therefore

dependent on the magnitude of the load and the number of loading repetitions. Fatigue data is often represented by an S-N curve, a curve depicting the stress (or strain) at which a material will fail as a function of the number of loading cycles. An example of such a curve is shown in Fig. 4-19. From this curve we see that when the stress is sufficiently high, the specimen will fracture at a relatively low number of cycles. As the stress is reduced, the number of cycles required to cause failure increases. Therefore when specifying fatigue strength, the number of cycles must also be specified. For some materials, a stress at which the specimen can be loaded an infinite number of times without failing is eventually approached. This stress is called the endurance limit.

The determination of fatigue properties is of considerable importance for certain types of dental restorations subjected to alternating forces during mastication. Structures such as complete dentures, implants, and metal clasps of removable partial dentures, which are placed in the mouth by forcing the clasps over the teeth, are examples of restorations that undergo repeated loading. It has been estimated that alternate stress applications occurring during mastication may amount to approximately 300,000 flexures per year, whereas the greater stress necessitated by removing restorations from the mouth or placing them in position probably amounts to