If a bar of material is subjected to an applied force, F, the magnitude of the force and the resulting defor- mation (δ) can be measured. In another bar of the same material, but different dimensions, the same applied force produces different force-deformation characteristics (Fig. 4.3A). However, if the applied force is normalized by the cross-sectional area A of the bar (stress), and the deformation is normalized by the original length of the bar (strain), the resulting stress-strain curve is independent of the geometry of the bar (Fig. 4.3B). It is therefore preferred that the stress-strain relations of an object be reported rather than the force-deformation characteristics. The stress-strain relationship of a dental material can be studied by measuring the load and deformation and then calculating the corresponding stress and strain.
When testing materials, loads should be applied at a uniform rate, and deformation should occur at a uniform rate. A typical universal testing machine can analyze materials in tension, compression, or shear. In the scheme illustrated in Fig. 4.4, a rod is clamped between two jaws and a tensile force is applied. The load is measured with a force trans- ducer and the deformation is measured with an extensometer clamped over a specified length of the specimen. A plot of load versus deformation is produced, which can be converted to a plot of stress versus strain (Fig. 4.5) by the simple calculations
A B
FIG. 4.2 Stress distribution in an implant-supported restoration. (A) Stresses on the shoulder of the implant body from an oblique occlusal load. (B) Stresses within the implant abutment and alveolar bone. (Courtesy Dr. Svenn Borgersen, Eagan, MN and Dr. Ronald Sakaguchi.)
described previously. By convention, strain is plot- ted on the x-axis as an independent variable because most tests are operated in strain control, where a constant strain is applied to the specimen and the resulting force is measured as the dependent, or y-axis, variable.
In the calculation of stress, it is assumed that the cross-sectional area of the specimen remains con- stant during the test. Using this assumption, the stress-strain curve is called an engineering stress-strain curve, and stresses are calculated using the original
cross-sectional area. When large loads are applied, or the object is tested in tension, the cross-sectional area might change dramatically during testing. In that case, the true stress, calculated with the actual cross-sectional area in the denominator, is very dif- ferent than the engineering stress, calculated with the original cross-sectional area. If the cross-sectional area decreases during the test, the true stress will be higher than the engineering stress because the denominator is smaller. In most mechanical tests, particularly those of small specimen dimensions, the original cross-sectional area is used for calculat- ing stress because it is often very difficult to measure the cross-sectional area as it changes throughout the experiment. Engineering stress is used in the presen- tation of stress-strain curves obtained in tension in this chapter.
Proportional and Elastic Limits
A stress-strain curve for a hypothetical material subjected to increasing tensile stress until failure is shown in Fig. 4.5. As the stress is increased, the strain is increased. In the initial portion of the curve, from 0 to A, the stress is linearly proportional to the strain.
As the strain is doubled, the stress is also doubled.
After point A, the stress is no longer linearly propor- tional to the strain. Hence the value of the stress at A is known as the proportional limit (SPL or σPL), defined as the highest stress at which the stress-strain curve is a straight line; that is, stress is linearly proportional to strain. Below the proportional limit, no permanent deformation occurs in a structure. When the force is
Slope E –––F/A /L
F/A ()
/L ()
B
F 3
F
2 1
A
L
FIG. 4.3 Force-deformation characteristics. (A) Force-deformation characteristics for the same material but having different dimensions. (B) Stress-strain characteristics of the same group of bars.
The stress-strain curve is independent of the geometry of the bar.
Load cell
Extensometer
Specimen
Moving crosshead
FIG. 4.4 Universal testing machine.
removed, the object will return to its original dimen- sions. Below the proportional limit, the material is elastic in nature.
The region of the stress-strain curve before the proportional limit is called the elastic region. When an object experiences a stress greater than the pro- portional limit, permanent or irreversible strain occurs. The region of the stress-strain curve beyond the proportional limit is called the plastic region. This characterization refers to linearly elastic materials such as many metals in which the relation between stress and strain is linear up to the proportional limit, and nonlinear thereafter. There are exceptions to this general rule, however. Materials described as super- elastic exhibit nonlinear elastic behavior; that is, their relationship between stress and strain in the elastic region does not follow a straight line, but removal of the load results in a return to zero strain.
The elastic limit (SEL or σEL) is defined as the maxi- mum stress that a material will withstand without permanent deformation. For linearly elastic materi- als, 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. An exception is when superelastic materials are considered. It is important to remem- ber, however, that the two terms differ in funda- mental concept; one deals with the proportionality of strain to stress in the structure, whereas the other describes the elastic behavior of the material. For the same material, values for proportional or elastic limit obtained in tension versus compression will differ.
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 (Figs. 4.6 and 4.7). The atoms are shown in Fig. 4.6A, without stress, and in Fig. 4.6B, with a resulting 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), indicating that the deformation was reversible. When the stress is greater than the proportional limit, the atoms move to a position as shown in Fig. 4.7B, and on removal of the stress, the atoms remain in this new position, indicating an irreversible, permanent deformation.
When the stress is less than the proportional or elastic limit, the strain is reversible, whereas when the stress is greater than the proportional or elastic limit, there is an irreversible or permanent strain in the object.
Yield Strength
It is often difficult to explicitly measure the propor- tional and elastic limits because the precise point of 0
Stress (MPa)
Strain (mm/mm)6 8 10 12 4
2 14
0 20 10 30 40 50 60 70 80 90 100
A
B C
D
A B C D
Stress (MPa)
Strain (mm/mm) Plastic deformation Elastic
deformation PL
SF
A
B
FIG. 4.5 Plotting stress-strain curves. (A) Stress-strain curve for a material subjected to tensile stress. Specimens illustrate amount of deformation at each point (A–D). (B) Elastic deformation is exhibited up to the proportional limit (PL) and plastic deformation is exhibited from PL to the fail- ure point, where we register the stress at failure (SF).
Shear force Shear
force
Elastic shear strain Shear
stress A
B
B
Shear force
Shear force
d A–B
interface A
B
A
Shear stress
FIG. 4.6 Sketch of an atomic model showing atoms in original position (A) and after elastic deformation (B). (Modified from Anusavice KJ. Phillips’ Science of Dental Materials. 11th ed. St. Louis: Saunders; 2003:79.)
deviation of the stress-strain curve from linearity is difficult to determine. The yield strength or yield stress or yield point (YS or σY) of a 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 point, a small, defined amount of permanent strain has occurred in the material. The yield strength is defined as the stress at which a material deforms plastically and there is a defined amount of permanent 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, or 0.005) permanent strain. The amount of permanent strain is referred to as the percent offset. Many specifications use 0.2% as a convention, but this depends on the plastic behavior of the material tested. For stiff materials with small elongation, the calculation of yield stress will include greater offsets than those materials with larger elon- gation or deformation.
The yield stress is determined by selecting the desired offset or strain on the x-axis and drawing a line parallel to the linear region of the stress-strain curve. The point at which the parallel line intersects the stress-strain 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% off- set. This yield stress is slightly higher than that for the proportional limit because it includes a speci- fied amount of permanent deformation. 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 elastic limit and yield strength
of a material are among its most important proper- ties because they define the transition from elastic to plastic behavior.
Any dental restoration that is permanently deformed through the forces of mastication usually loses its functionality to some extent. For example, a fixed partial dental prosthesis (such as a three-unit prosthesis) that is permanently deformed by exces- sive occlusal forces would exhibit altered occlusal contacts. The restoration is permanently deformed because a stress equal to or greater than the yield strength was generated. It is important to remember that dysfunctional occlusal loading also changes the stresses placed on a restoration. A deformed resto- ration may therefore be subjected to greater stresses than originally intended because the occlusion that was distributed over a larger number of occlusal con- tacts may now be concentrated on a smaller number of contacts. Under these conditions, fracture does not occur if the material is able to plastically deform.
However, this permanent change in shape represents a destructive example of deformation. Permanent deformation and stresses in excess of the elastic limit are desirable when shaping an orthodontic arch wire or adjusting a clasp on a removable partial denture.
In these examples, the stress must exceed the yield strength to permanently bend or adapt the wire or clasp. Elastic deformation occurs as the wire or clasp engages and disengages a retentive region in the cer- vical area of the tooth. Retention is achieved through small-scale elastic deformation. This elastic or revers- ible deformation describes the function of elastic bands, clasps, o-rings, and implant screws.
Ultimate Strength
In Fig. 4.5 the test specimen exhibits a maximum stress at point C. The ultimate tensile strength or stress (UTS) is defined as the maximum stress that a mate- rial can withstand before failure in tension, whereas the ultimate compressive strength or stress (UCS) is the maximum stress a material can withstand in com- pression. The ultimate engineering stress is deter- mined 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 as used in den- tistry specifies the maximum load and minimum cross-sectional area when designing a 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. A safety margin should be incorporated into the design of the restoration and choice of material to ensure that the ultimate strength is not approached in normal function. The yield strength is often of greater importance than ultimate Shear
stress A
B Shear
force Shear
force d
A
Plastic shear strain A
B
B
FIG. 4.7 Sketch of an atomic model showing atoms in original position (A) and after plastic deformation (B). (Modified from Anusavice KJ. Phillips’ Science of Dental Materials. 11th ed. St. Louis: Saunders; 2003:79.)
strength in design and material selection because it is an estimate of when a material will start to deform permanently.
Fracture Strength
In Fig. 4.5 the test specimen fractured at point D. The stress at which a brittle material fractures is called the fracture strength or fracture stress (SF or σF). Note that a material does not necessarily fracture at the point at which the maximum stress occurs. After a maximum tensile force is applied to some materials, the speci- men begins to elongate excessively, resulting in “neck- ing” or a reduction of cross-sectional area (see Fig.
4.5). The stress calculated from the force and the origi- nal cross-sectional area may decrease dramatically before final fracture occurs because of the reduction in cross-sectional area. Accordingly, the stress at the end of the curve is less than that at some intermediate point on the curve. Therefore, in materials that exhibit necking, the ultimate and fracture strengths are dif- ferent. However, for the specific cases of many dental alloys and ceramics subjected to tension, the ultimate and fracture strengths are similar, as is shown later in this chapter. Note that the reduction in stress that is observed after the ultimate stress in materials that show necking is an artifact of using the original cross- sectional area in the calculation of stress. If the true cross-sectional area is used, the stress would increase.
Elongation
The deformation that results from the applica- tion of a tensile force is elongation. Elongation is extremely important because it gives an indica- tion of the possible manipulation 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; and (2) the elonga- tion beyond the proportional limit and up to the fracture strength (from A to D), which is perma- nent. The permanent deformation may be mea- sured with an extensometer while the material is being tested and calculated from the stress-strain curve. Total elongation is commonly expressed as a percentage. The percent elongation is calculated as follows:
Elongation=
Original length)×100 (Increase in length/
Total elongation includes both the elastic elonga- tion and the plastic elongation. Plastic elongation is usually the greatest of the two, except in materials that are quite brittle or those with very low stiffness.
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 per- manent 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.
An alloy that has a high value for total elongation can be bent permanently without danger of frac- ture. Clasps can be adjusted, orthodontic wires can be adapted, and crowns or inlays can be burnished if alloys with high values for elongation are used.
Elongation and yield strength are generally related in many materials, including dental gold alloys, where, generally, the higher the yield strength, the lower 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 word modulus means ratio and in this case, the ratio of stress to strain. The elastic modu- lus represents the stiffness of a material within the elastic range. The elastic modulus can be determined from a stress-strain curve (see Fig. 4.5) by calculating the ratio of stress to strain or the slope of the linear region of the curve. The modulus is calculated from the following equation:
Elastic modulus=Stress/Strain or
E= σ/ε
This equation is also known as Hooke’s law.
Because strain is unitless (length/length), the mod- ulus 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 inter- atomic 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 mate- rial 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:
σ1=150 MPa ε1=0.005 and
σ2=300 MPa ε2=0.010 The slope is therefore:
σ2− σ1/ε2− ε1 = 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 greater than for B. This difference can be demonstrated numeri- cally, 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 0.01 (1%) and the elastic modulus is as follows:
E=300 MPa/0 010=30 000 MPa=30 GPa 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 modu- lus for B is
E=300 MPa/0.020=15 000 MPa=15 GPa The fact that material A has a steeper slope in the elastic range than material B means that a larger force is required to deform material A to a given amount than is required 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 to which B is deformed by a stress of 150 MPa. Therefore A is stiffer or more rigid than B. Conversely, B is more flexible than A. Materials such as elastomers and other polymers have low values for elastic modulus, whereas many metals and ceramics have much higher values, as shown in Table 4.1.
Poisson’s Ratio
During axial loading in tension or compression there is a simultaneous strain in the axial and transverse, or lateral, directions. Under tensile loading, as a material elongates in the direction of load, there is a reduction in cross section, known as necking. 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 (ν). In tensile loading, the Poisson’s ratio indicates that the reduction in cross sec- tion is proportional to the elongation during the elastic deformation. The reduction in cross section continues until the material is fractured. Poisson’s ratio is a unit- less value because it is the ratio of two strains.
Most rigid materials, such as enamel, dentin, amalgam, and dental composite, exhibit a Poisson’s ratio of about 0.3. 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 higher degree of reduc- tion in cross-sectional area and higher Poisson’s ratios. Rubber has a Poisson’s ratio of nearly 0.5.
Cork exhibits little lateral expansion under compres- sion and has a Poisson’s ratio close to 0. This prop- erty has made cork a common material for sealing wine bottles.
Ductility and Malleability
Two significant properties of metals and alloys are ductility and malleability. The ductility of a material
60 55 50 45
B z'
40 35 30 25 20
A z
x x'
y' yN
y
15 10 5 00 100 200 300 400
Strain (103)
Stress (MPa)
FIG. 4.8 Stress-strain curves of two hypothetical mate- rials subjected to tensile stress.
TABLE 4.1 Elastic Modulus (GPa) of Selected dental Materials
Material Elastic Modulus (GPa)
Enamel 84
Dentin 17
Gold (type IV) alloy 90–95
Amalgam 28–59
Cobalt-chromium removable
partial denture alloy 218–224 Feldspathic porcelain 69–70 Resin composite with hybrid
filler 17–21
Poly (methyl methacrylate) 2.4 Silicone elastomer for
maxillofacial prosthesis 0.002–0.003