Сraig. Dental Materials

can be bent and adjusted with less chance of fracturing.

Consider a bar of material subjected to an applied force, F. We can measure the magnitude of the force and the resulting deformation

If we next take another bar of the same material, but different dimensions, the force-deformation characteristics change (Fig. 4-3, A). However, if we normalize the applied force by the crosssectional area A (stress) of the bar, and normalize the deformation by the original length (strain) of

the bar, the resultant stress-strain curve now becomes independent of the geometry of the bar (Fig. 4-3, B). It is therefore preferential to report the stress-strain relations of a material rather than the force-deformation characteristics. The stressstrain relationship of a dental material is studied by measuring the load and deformation and then calculating the corresponding stress and strain.

The testing of many materials necessitates loads of 2220 N or more and the measurement of deformations of 0.02 mm or less. A further requirement may be that the load should be applied at a uniform rate or that the deformation should occur at a uniform rate. A typical machine that permits testing of tension, compression, or

shear is shown in Fig. 4-4.In the figure, a rod is clamped between two jaws and the tensile properties are measured by pulling the specimen. The load is measured electronically with a force transducer and the deformation is measured with an extensometer clamped over a given length of the specimen. One obtains a plot of load versus deformation, which can be converted to a plot of stress versus strain (Fig. 4-5) by the simple calculations described previously.

In the calculation of stress, it is assumed that the cross-sectional area of the specimen remains constant during the test. The resulting stressstrain curve is called an enginee&zgstress-strain cume, and stresses are calculated based on the

original cross-sectional area. For many materials, significant changes in the area of the specimen may occur as it is being deformed. A stress-strain curve based on stresses calculated from a nonconstant cross-sectional area is called a true stress-strain curve. A true stress-strain curve may be quite different from an engineering stressstrain curve at high loads because significant changes in the area of the specimen may occur. For example, if a specimen is being tested in tension and the area decreases, the engineering stress will be lower than the true stress. The engineering stress-strain curve is used throughout the remaining chapters.

A stress-strain curve for a hypothetical mate-

Fig, 4-4 Servo-hydraulic mechanical testing machine capable of applying axial, shear, bending, andlor torsional loads to a material.

rial that was subjected to increasing tensile stress until fracture is shown in Fig. 4-5.The stress is plotted vertically, and the strain is plotted horizontally. As the stress is increased, the strain is increased. In fact, in the initial portion of the curve, from 0 to A, the strain is linearly proportional to the stress, and as the stress is doubled, the amount of strain is also doubled. When a stress that is higher than the value registered at A is achieved, the strain changes are no longer linearly proportional to the stress changes. Hence the value of the stress at A is known as the proportional limit.

PROPORTIONAL AND ELASTIC LIMITS

The proportional limit is defined as the greatest stress that a material will sustain without a deviation from the linear proportionality of stress to strain. Below the proportional limit, no permanent deformation occurs in a structure. When the stress is removed, the structure will return to its original dimensions. Within this range of stress application, the material is elastic in nature, and if the material is stressed to some value below the proportional limit, an elastic or reversible strain will occur. The region of the stress-strain curve before the proportional limit is called the elastic region. The application of a stress greater than the proportional limit results in a permanent or irreversible strain in the specimen; the region of the stress-strain curve beyond the proportional limit is called the plastic region.

The elastic limit is defined as the maximum stress that a material will withstand without permanent deformation. Therefore for all practical purposes the proportional limit and elastic limit represent the same stress within the structure, and the terms are often used interchangeably in referring to the stress involved. Keep in mind, however, that they differ in fundamental concept, in that one deals with the proportionality of strain to stress in the structure, whereas the other describes the elastic behavior of the material. The proportional limit of the material in Fig. 4-5is approximately 330 MPa. The proportional and elastic limits are quite different for different materials. Values for proportional or elastic limits in either tension or compression can be determined, but the values obtained in tension and compression will differ for the same material.

The concepts of elastic and plastic behavior can be illustrated with a simple schematic model of the deformation of atoms in a solid under stress (Fig. 4-6).The atoms are shown in Fig. 4-6,A, with no stress applied, and in Fig. 4-6,B, with an applied stress that is below the value of the proportional limit. When the stress shown in B is removed, the atoms return to their positions shown in A. When a stress is applied that is greater than the proportional limit, the atoms can move to a position as shown in

Fig. 4-6, C, and, on removal of the stress, the atoms remain in this new position. The application of a stress less than the proportional or elastic limit results in a reversible strain, whereas a stress greater than the proportional or elastic limit results in an irreversible or permanent strain in the specimen.

The model described in Fig. 4-6 is considerably oversimplified; a more realistic but more complicated model of plastic deformation is shown in Fig. 4-7. In this schematic the atoms can move to new stable positions, resulting in plastic deformation by the movement of dislocations or

imperfections (as indicated by the black circles in Fig. 4-7) in the structure of the solid. These imperfections allow the consecutive movement of atoms without the need for an entire row or plane of atoms to move.

YIELD STRENGTH

Stress-strain curves determined in the laboratory are rarely as ideal as the curve shown in Fig. 4-5.Therefore it is not always feasible to explicitly measure the proportional and elastic limits. The yield strength or yield stress (YS) of a

Fig. 4-7 Sketch of an atomic model showing plastic deformation taking place by the movement of dislocations.

(Adapted from Cottrell AH: Sci Am 217[3]:90,1967.)

material is a property that can be determined readily and is often used to describe the stress at which the material begins to function in a plastic manner. At this stress, a limited permanent strain has occurred in the material. The yield strength is defined as the stress at which a material exhibits a specified limiting deviation from proportionality of stress to strain. The amount of permanent strain is arbitrarily selected for the material being examined and may be indicated as 0.1%, 0.2%, or 0.5% (0.001, 0.002, 0.005) permanent strain. The amount of permanent strain may be referred to as the percent offset. Many specifications use 0.2% as a convention.

The yield stress is determined by selecting the desired offset and drawing a line parallel to the linear region of the stress-strain curve. The point at which the parallel line intersects the stressstrain curve is the yield stress. On the stress-strain curve shown in Fig. 4-5, for example, the yield strength is represented by the value B. This represents a stress of about 360 MPa at a 0.25% offset.

This yield stress is slightly higher than that for the proportional limit and also indicates a specified amount of deformation. Again, note that when a structure is permanently deformed, even to a small degree (such as the amount of deformation at the yield strength), it does not return completely to its original dimensions when the stress is removed. For this reason, the proportional limit, elastic limit, and yield strength of a material are among its most important properties.

Any dental structure that is permanently deformed through the forces of mastication is usually a functional failure to some degree. For example, a bridge that is permanently deformed through the application of excessive biting forces would be shifted out of the proper occlusal relation for which it was originally designed. The prosthesis becomes permanently deformed because a stress equal to or greater than the yield strength was developed. Recall also that malocclusion changes the stresses placed on a restoration; a deformed prosthesis may therefore be subjected to greater stresses than originally intended. Usually a fracture does not occur under such conditions, but rather only a permanent deformation results, which represents a destructive example of deformation. A constructive example of permanent deformation and stresses in excess of the elastic limit is observed when an appliance or dental structure is adapted or adjusted for purposes of design. For example, in the process of shaping an orthodontic appliance or adjusting a clasp on a removable partial denture, it may be necessary to introduce a stress into the structure in excess of the yield strength if the material is to be permanently bent or adapted. Values of yield strength for some partial denture alloys are listed in Table 4-1.

ULTIMATE STRENGTH

In Fig. 4-5, the test specimen is subjected to its greatest stress at point C. The ultimate tensile strength or stress is defined as the maximum stress that a material can withstand before failure in tension, whereas the ultimate compressive strength orstressis the maximum stress a material can withstand in compression. If the direction

*O.1%offset

of loading has previously been specified, then the term ultimate strength (stress) is often used. The ultimate stress is determined by dividing the maximum load in tension (or compression) by the original cross-sectional area of the test specimen. The ultimate tensile strength of the material in Fig. 4-5 is about 380 MPa.

The ultimate strength of an alloy is used in dentistry to give an indication of the size or cross section required for a given restoration. Note that an alloy that has been stressed to near the ultimate strength will be permanently deformed, so a restoration receiving that amount of stress during function would be useless. Therefore, although data on materials used in dentistry usually specify values for ultimate strength, the use of ultimate strength as a criterion for evaluating the relative merits of various materials should not be overemphasized. The yield strength is of greater importance than ultimate strength because it is a gauge of when a material will start to deform.

FRACTURE STRENGTH

In Fig. 4-5 the test specimen fractured at point D. The stress at which a material fractures is called the fracture strength or fracture stress. Note that a material does not necessarily fracture at the point at which the maximum stress occurs. Af- ter a maximum stress is applied to some materials, they begin to elongate excessively, and the stress calculated from the force and the original cross-sectional area may drop before final fracture occurs. Accordingly, the stress at the end of

the curve is less than at some intermediate point on the curve. Therefore in the most general case the ultimate and fracture strengths are different. However, for the specific cases of many dental alloys subjected to tension, the ultimate and fracture strengths are the same, as is seen later.

ELONGATION

The deformation that results from the application of a tensile force is elongation. Elongation is extremely important because it gives an indication of the workability of an alloy. As may be observed from Fig. 4-5, the elongation of a material during a tensile test can be divided conveniently into two parts: (1) the increase in length of the specimen below the proportional limit (from 0 to A), which is not permanent and is proportional to the stress applied; and (2) the elongation beyond the proportional limit and up to the fracture strength (from A to D), which is permanent. The permanent deformation may be measured with an extensometer while the material is being tested and calculated from the stressstrain curve. A common method to express total elongation is in percentage, such as 20% elongation for a 5-cm test specimen. The percent elongation would be calculated as follows:

Increase in length

% Elongation = Original length x 100%

We see that elongation and axial strain are similar.

The total percent elongation includes both the elastic elongation and the plastic elongation. The plastic elongation is usually the greater of the two, except in materials that are quite brittle or those with very low elastic moduli. A material that exhibits a 20% total elongation at the time of fracture has increased in length by one fifth of its original length. Such a material, as in many dental gold alloys, has a high value for plastic or permanent elongation and, in general, is a ductile type of alloy, whereas a material with only 1% elongation would possess little permanent elongation and be considered brittle.

Values of percent elongation of some crown and bridge a i d partial denture alloys are compared in Table 4-2. An alloy that has a high value for total elongation can be bent permanently without danger of fracture. Clasps can be adjusted, orthodontic appliances can be prepared, and crowns or inlays can be burnished if they are prepared from alloys with high values for elongation. When selecting alloys for specific clinical purposes, therefore, it is necessary to recognize that because they may be subjected to permanent deformation and adaptation during the construction or assembly of the restoration, it is necessary to have an acceptable amount of elongation. In other restorations in which permanent deformation is not anticipated, it is possible to employ materials with a lower value for elongation. A relationship exists between elongation and yield strength for many materials, including dental gold alloys, where, generally, the higher the yield strength, the less the elongation.

ELASTIC MODULUS

The measure of elasticity of a material is described by the term elastic modulus, also referred to as modulus of elasticity or Young's modulus, and denoted by the variable E. The elastic modulus represents the stiffness of a material within

the elastic range. The elastic modulus can be determined from a stress-strain curve (see Fig. 4-51 by calculating the ratio of stress to strain or the slope of the linear region of the curve. The modulus is calculated from the equation

Stress o

Elastic modulus = -o r E = -

Strain E

Because strain is dimensionless, the modulus has the same units as stress, and is usually reported in MPa or GPa (1 GPa = 1000 MPa).

The elastic qualities of a material represent a fundamental property of the material. The interatomic or intermolecular forces of the material are responsible for the property of elasticity (see Fig. 4-6). The stronger the basic attraction forces, the greater the values of the elastic modulus and the more rigid or stiff the material. Because this property is related to the attraction forces within the material, it is usually the same when the material is subjected to either tension or compression. The property is generally independent of any heat treatment or mechanical treatment that a metal or alloy has received, but is quite dependent on the composition of the material.

The elastic modulus is determined by the slope of the elastic portion of the stress-strain curve, which is calculated by choosing any two stress and strain coordinates in the elastic or linear range. As an example, for the curve in Fig. 4-5 the slope can be calculated by choosing the following two coordinates:

o, = 150 MPa, e, = 0.005; and

o, = 300 MPa, E, = 0.010

The slope is therefore

(0, - o1)/(e2- el) = (300 - 150)'

(0.010 - 0.005) = 30,000 MPa = 30 GPa

Stress-strain curves for two hypothetical materials, A and B, of different composition are shown in Fig. 4-8. Inspection of the curves shows that for a given stress, A is elastically deformed less than B, with the result that the elastic modulus for A is

Strain ( x I O - ~ )

Fig. 4-8 Stress-strain curves of two hypothetical materials subjected to tensile stress.

greater than for B. This difference can be demonstrated, numerically, by calculating the elastic moduli for the two materials subjected to the same stress of 300 MPa. At a stress of 300 MPa, material A is strained to 0.010 (1%) and the elastic modulus is

E=-- 300 MPa - 30,000 MPa = 30 GPa

0.010

On the other hand, material B is strained to 0.02 (2%)) or twice as much as material A for the same stress application. The equation for the elastic modulus for B is

300 MPa

The fact that material A has a steeper slope in the elastic range than material B means that a greater stress application is required to deform material A to a given amount than for material B. From the curves shown in Fig. 4-8, it can be seen that a stress of 300 MPa is required to deform A to the same amount elastically that B is deformed by a stress of 150 MPa. Therefore A is said to be stiffer or more rigid than B. Conversely, B is more

*1 gigapascal (GPa) = 103 MPa

flexible than A . Materials such as rubber and plastics have low values for the elastic modulus, whereas many metals and alloys have much higher values, as shown in Table 4-3.

POISSON'S RATIO

During axial loading in tension or compression there is a simultaneous axial and lateral strain. Under tensile loading, as a material elongates in the direction of load, there is a reduction in cross section. Under compressive loading, there is an increase in the cross section. Within the elastic range, the ratio of the lateral to the axial strain is called Poisson's ratio (v). In tensile loading, the Poisson's ratio indicates that the reduction in cross section is proportional to the elongation during the elastic deformation. The reduction in cross section continues until the material is fractured.

Poisson's ratios of some dental materials are listed in Table 4-4.Brittle substances such as hard gold alloys and dental amalgam show

little permanent reduction in cross section during a tensile test. More ductile materials such as soft gold alloys, which are high in gold content, show a high degree of reduction in crosssectional area.

DUCTILITY AND MALLEABILITY

Two significant properties of metals and alloys are ductility and malleability. These properties cannot always be determined with certainty from a stress-strain curve. Ductility is the ability of a material to be plastically deformed; it is indicated by the plastic strain.

A high degree of compression or elongation indicates good malleability and ductility, although certain metals show some exception to this rule. The reduction in area in a specimen, together with the elongation at the breaking point, is, however, a good indication of the relative ductility of a metal or alloy.

The ductility of a material represents its ability to be drawn into wire under a force of tension. The material is subjected to a permanent deformation while being subjected to these tensile forces. The malleability of a substance represents its ability to be hammered or rolled into thin sheets without fracturing.

Ductility is a property that has been related to the workability of a material in the mouth. It has also been related to burnishability of the margins of a casting. Although ductility is important, the amount of force necessary to cause permanent deformation during the burnishing operation also must be considered. A burnishing index has

Note: Some authorities consider tungsten to be the most ductile metal.

been used to rank the ease of burnishing alloys and is equal to the ductility (elongation) divided by the yield strength.

The relative malleability and ductility of 10 n~etalsused in dentistry and industry are given in Table 4-5. It is interesting that gold and silver, used extensively in dentistry, are the most malleable and ductile of the metals, but other metals do not follow the same order for both malleability and ductility. In general, metals tend to be ductile, whereas ceramics tend to be brittle.

RESILIENCE

Resilience is the resistance of a material to permanent deformation. It indicates the amount of energy necessary to deform the material to the proportional limit. Resilience is therefore measured by the area under the elastic portion of the stress-strain curve, as illustrated in Fig. 4-9, A. Resilience can be measured by idealizing the area of interest as a triangle and calculating the area of the triangle, (1/2)bh. The resilience of the material in Fig. 4-5, for example, would be 1/2 x 0.011x 330 = 1.82 m M N / ~ The~ . units are m M N / (meter~ ~ x megaNewtons per cubic meter), which represents energy per unit volume of material.

Fig. 4-9 Stress-strain curves showing A, the area indicating the resilience, and B, the area representing the toughness of a material.

Resilience has particular importance in the evaluation of orthodontic wires because the amount of work expected from a particular spring in moving a tooth is of interest. There is also interest in the amount of stress and strain at the proportional limit because these factors determine the magnitude of the force that can be applied to the tooth and how far the tooth will need to move before the spring is no longer effective. For example, Fig. 4-10 illustrates the loaddeflection curve for a nickel-titanium (Ni-Ti) orthodontic wire. Note that the loading (activation) portion of the curve is different from the unloading (deactivation) portion. This difference is called hysteresis.

TOUGHNESS

Toughness, which is the resistance of a material to fracture, is an indication of the amount of energy necessary to cause fracture. The area under the elastic and plastic portions of a stressstrain curve, as shown in Fig. 4-9, B, represents the toughness of a material. Toughness is not as easy to calculate as resilience, and the integration is usually done numerically. The units of toughness are the same as the units of resilience- m M N / or~ mMPa/m~ . Toughness therefore represents the energy required to stress the material

Strain-,

to the point of fracture. Note that a material can be tough by having a combination of high yield and ultimate strength and moderately high strain at rupture, or by having moderately high yield and ultimate strengths and a large strain at rupture.

FRACTURE TOUGHNESS

Recently the concepts of fracture mechanics have been applied to a number of problems in dental materials. Fracture mechanics characterizes the behavior of materials with cracks or flaws. Flaws or cracks may arise naturally in a material or nucleate after a time in service. In either case, any defect generally weakens a material, and, as a result, sudden fractures can arise at stresses below the yield stress. Sudden, catastrophic fractures typically occur in brittle materials that don't have the ability to plastically deform and redistribute stresses. The field of fracture mechanics provides an analysis of and design basis against these types of failures.

Two simple examples illustrate the significance of defects on the fracture of materials. If one takes a piece of paper and tries to tear it, greater effort is needed than if a tiny cut is made in the paper. Analogously, it takes a considerable force to break a glass bar; however,