Stress-strain diagram (W1)

• Stress is force per unit area (MPa). Strain is the deformation it causes, given as a change in length relative to the original length. The stress-strain diagram plots stress on the vertical axis against strain on the horizontal axis for a gradual static load.
• Four points are read along the curve, in order: proportional limit (σp), elastic limit (σe), yield strength (σy) and maximum strength (σmax).
• Up to the elastic limit the deformation is elastic and disappears when the load is removed. Beyond it the deformation is permanent, which is plastic deformation.
• Six properties are read from one curve: elasticity and rigidity (the slope, given by the modulus of elasticity E), brittleness (elastic limit close to maximum strength), toughness (energy absorbed, the area under the curve up to maximum strength), ductility (plastic elongation at fracture), resilience (energy absorbed with no permanent deformation) and, from a separate dynamic test, impact strength.
• Permanent or elastic deformation of a crown, bridge or denture damages the periodontium, the adhesion to the teeth and the contact with the antagonists, so the curve guides the choice of material and its thickness.
• A high modulus of elasticity allows thinner, finer prosthetic constructions without elastic deformation during chewing.

This topic answers written item 1.

Strength is the ability of a material to resist deformation and failure. The strength of a material is measured under a static load, a force that is increased gradually, in a testing machine that applies forces in different directions. Strength depends on the strain rate, the shape of the test specimen, the surface finish and the environment in which the material is tested.

Two other load types are named in the textbooks. An impact load is a sudden stroke (see section 7). A fatigue load is a periodically and repeatedly changing load; the material can fail under loads that would not break it on a single application, through a progressive reduction of the adhesion forces, the formation of microcracks and the breakdown of the material.

Types of static strength

Tensile strength is measured under the action of axial pulling forces. Approximate tensile values quoted for tooth tissues are about 10 MPa for enamel and 100 MPa for dentin.

When the load is applied perpendicularly, the material first bends, then cracks and then fractures. This is why porcelain is tested in flexure.

Shear strength

The shear strength is calculated by a formula, because the value depends on the force and on the contact surface between the two materials. It is used to investigate the adhesion between a restorative material and a tooth tissue, and the shear strength between some materials is given in the table below.

Masticatory loads

The mechanical properties of a dental material have to withstand masticatory forces, which are normally 10 to 20 kg and exceptionally up to 800 kg.

The deformation changes of materials are studied in the strength investigation under a static load. The results are presented on the stress-strain diagram, with the stress in MPa on the vertical axis and the deformation on the horizontal axis. The deformation is given in percentages, as the linear deformation caused by a prolongation in a tensile test or by a reduction in a compressive test.

The study of deformation before destruction matters more to dental practice than the load that causes it. Although prosthetic constructions in the mouth appear undamaged, invisible changes in shape and size during function can cause unfavourable effects on the living tissues. Elastic deformation in the pontics of a fixed bridge damages the periodontium of the abutments, and elastic and plastic deformation of crowns and restorations destroys their adhesion to the tooth and changes their relationship with the antagonists.

Points on the curve

The curve is read from left to right as the load increases. Each point marks a change in the behaviour of the material.

A labelled version of the same diagram marks the four points with the letters A to D: proportional limit (A), elastic limit (B), yield strength (C) and maximum strength (D).

Beyond the elastic limit, the material changes its shape and size permanently. Further increases in stress cause a greater increase in strain until the material is destroyed at σmax. The elastic limit lies at the start of the plastic deformation. If the stresses and strains in a restoration exceed the maximum value the material can withstand, fracture is the most likely outcome. To avoid deformation of prosthetic constructions, appropriate materials should be selected, with an adequate thickness and cross-section.

Tensile test

The tensile test is a relatively simple test to understand and interpret, and is possibly also the most useful. A sample is stretched in one direction in a tensile tester at a constant strain rate, that is a constant rate of extension. The load is measured by a load cell and the elongation that corresponds to each load is recorded.

Stress and strain

Stress (σ) is the force per unit area, in N/m² or pascals (Pa), and is quoted in MPa for dental materials. Strain is the change in length of a specimen relative to its original length. In a tensile test a rod is stretched in one direction, and the elongation is measured against its initial length.

A ductile metal curve

A typical stress-strain curve for a ductile metal, such as a brass alloy, shows a linear elastic region followed by plastic flow. The region where the curve is linear is the elastic region, and removing the stress in it returns the material to its original shape. Where the curve departs from its straight path the material has exceeded its elastic limit and will deform permanently. Removing the stress after this point does not return the material to its original shape; this permanent change is called plastic flow and is shown as the plastic region of the graph.

The stress at which plastic deformation begins is the yield stress, σy. In practice this point is hard to detect, because the change in slope is gradual rather than sudden. The proof stress is therefore used instead: the stress needed to produce a set amount of plastic strain, usually 0.2 %. The ultimate tensile strength (σUTS) is the maximum stress the specimen withstands; it is often different from the fracture strength, which is the stress at which the specimen breaks.

The elasticity and rigidity of a material indicate the degree of elastic deformation under a load in the region of proportionality. They are opposite qualities: the more rigid the material, the less elastic it is, and the harder it is to deform. A more elastic material deforms more easily.

Graphical representation

Elasticity is shown graphically by the angle that the straight line of the proportional region makes with the strain axis. A small angle means the material is more elastic; a greater angle means it is more rigid. Enamel is more rigid and dentin is more elastic, as shown in the diagram below.

Modulus of elasticity

The elasticity is represented mathematically by the modulus of elasticity, also called Young's modulus and written E. It is the ratio of stress to strain in the region of proportionality:

E = P (stress) / L (strain), in MPa.

When strain is expressed as a percentage, the formula becomes E = P / L x 100 MPa. For example, if a material loaded to 500 MPa in a tensile test elongates by 2 %, its modulus is E = 500 / 2 x 100 = 25 000 MPa = 25 GPa.

Every point of stress up to the proportional limit, divided by its corresponding strain, gives the same modulus, so E can be read from any point of the straight region of the curve.

The Young's modulus is also called the elastic modulus or the modulus of elasticity. It is often very large for real materials, so it is usually expressed in gigapascals (GPa), where 1 GPa = 10⁹ N/m². The stiffness of a structure depends on the modulus of the material it is made from and on its shape and dimensions.

Materials with a high modulus are rigid or inelastic, and materials with a low modulus are elastic. The moduli of some dental tissues and materials are given in the table below.

Why the modulus matters in the mouth

The modulus of elasticity guides the choice of restorative material and its thickness.

• A removable denture base of acrylic resin must be at least 3 to 5 mm thick. The same base made from cobalt-chromium alloy may be 0.5 mm thick, and the denture is more comfortable in the mouth.
• An artificial crown of base dental alloy may be 0.4 mm thick. The same crown of feldspar porcelain must be at least 1.2 mm thick, which means removing more tooth tissue with a potential risk to the vital pulp.
• The modulus is important for adhesion. While chewing, the more elastic material deforms more than the rigid one. This strains and can rupture the adhesive bonds between the restoration and the dental tissue.

Brittleness is the mechanical quality that makes a material break without any deformation, or with only a minimal one. On the stress-strain diagram the elastic limit and the maximum strength are very close together, so the curve rises almost straight to fracture.

Brittle materials are dental enamel, dental ceramics, zinc phosphate cement, acrylic resin and some composite materials. Brittle materials generally have lower toughness than tough materials, and a brittle material can fail suddenly and unexpectedly. This sudden failure is called fast fracture: a crack becomes unstable and grows very rapidly. It causes planes to fall out of the sky, ships to break in half and bridges to collapse. It is associated with glasses, ceramics and hard brittle resins, and with metals that are not ductile, such as dental amalgam, solders and welds.

Exam point: brittle means the elastic limit and the maximum strength lie almost on top of each other.

Toughness is the ability of a material to absorb energy under stress. It is the energy needed to destroy the material. Tough materials show great elastic and plastic deformation, so a considerable amount of energy is required to break them. On the stress-strain diagram, toughness is the area between the strain axis and the curve, up to the maximum strength.

Fracture toughness

Toughness is also measured through the fracture toughness, which is the resistance of a material to any destruction in the presence of one or several cracks. Destruction of a material begins at a crack, which then destroys its continuity and integrity. The fracture toughness is written K_IC, in MPa·m½, and quantifies the stress under which a crack in a given material propagates.

The fracture toughness is measured by introducing a crack of known size and shape, and then measuring the stress needed for the crack to grow.

Brittle materials usually have a lower toughness. Dentin is tougher than enamel, and dental alloys are tougher than ceramic materials. The energy absorbed is also expressed as energy per unit volume, in J/m³.

Ductility

Ductility is the degree of plastic deformation a material undergoes under stress beyond its proportional limit. In a tensile test, ductility is assessed by the elongation of the test body after rupture. It is given as the relative elongation, a percentage of the length of the test body before loading:

Relative elongation = (L1 - L0) / L0 x 100 %

where L1 is the length of the test body immediately after rupture and L0 is its initial length.

The amount of plastic strain at fracture is the ductility; on the curve it is found by drawing a line from the fracture point parallel to the elastic region until it meets the strain axis. The value is frequently quoted as the percentage elongation.

The relative elongation of some dental alloys is given in the table below. Values show how well a metal can be cold worked, by cutting, drilling, bending, hammering, rolling, drawing or stamping.

Resilience

Resilience describes a material with a large elastic deformation and no ductility. Such a material is destroyed with a minimal plastic deformation, or none at all, and is described as flexible or resilient. On the stress-strain diagram the maximum strength coincides with the elastic limit.

Resilience is also the energy a material absorbs without any permanent deformation. It is measured as the area under the linear elastic portion of the curve, and gives an indication of the potential springiness of the material. When a wire is bent and released it springs back to its original shape, as long as the stress does not exceed the elastic limit. The energy stored in the wire is recoverable when the stress is released. Resilience is expressed in J/m³.

The impact strength is the stress needed to fracture a material under a sudden maximum impact. It is a dynamic test, and is measured in a Charpy-type impact tester.

The pendulum is held in the upper position and the test bar, with a rectangular or round cross-section, is fixed at both ends. When the pendulum is released its potential energy becomes kinetic energy, it swings round its axis, breaks the test bar, and carries on with the rest of its kinetic energy. The less the pendulum swings after the break, the stronger the material is.

The impact strength test gives an indicative estimate of toughness, especially for brittle materials.

The six properties from one curve

Four curve shapes

Points on the stress-strain curve

Remember the order of the four points as P, E, Y, M: proportional, elastic, yield, maximum.

Exam tip: Be ready to define the four points in order, give E = stress / strain with the example of 500 MPa and 2 % giving 25 GPa, and state that brittle means the elastic limit is close to the maximum strength.