Materials

FACE-CENTRED CUBIC (FCC) CRYSTAL STRUCTURE:

In the face-centred cubic (FCC) crystal structure, atoms are located at all corner and face-centred positions. The atoms touch across a face diagonal. The FCC structure is found in some common metals such as gold, silver, aluminium and copper. The cube length a and the atomic radius R are related through:

a=4 R√2

=2√2R

The coordination number is 12.

The atomics packing factor (APF) is 0.74 – this is the maximum packing possible for spheres having the same radius. (insert diagram here ; insert calculation and 2-D for APF calculation)

Metals typically have relatively large atomic packing factors to maximise the shielding provided by the sea of delocalised electrons.

BODY-CENTRED CUBIC CRYSTAL STRUCTURE:

In the body-centred cubic (BCC) crystal structure, atoms are located at corner and cell centre positions. Centre and corner atoms touch one another along cube diagonals. Chromium, tungsten, iron (alpha), as well as several other metals exhibit a BCC structure. The cube length a and the atomic radius R are related through:

a=4 R√3

The coordination number is 8.

The atomics packing factor (APF) is 0.74 – this is the maximum packing possible for spheres having the same radius. (insert diagram here ; insert calculation and 2-D for APF calculation)

Metals typically have relatively large atomic packing factors to maximise the shielding provided by the sea of delocalised electrons.

Lecture 2 – Mechanical Properties of Materials

Read pages 234 – 236 on ‘Terminology for Mechanical Properties’ in the 4th ed. Askeland book!!!Stress, in general, is defined as the force per unit area

Strain, in general, is defined as the change in length per unit length

True stress uses ‘instantaneous’ area

True strain uses the integral of the reciprocal of the length

Tension Test:

In a tensile test, the strain rate (e dot) is controlled because mechanical properties of many materials vary with the strain rate. Strain rate is the rate of change in strain (deformation) of a material with respect to time.

Elastic strain is defined as fully recoverable strain resulting from an applied stress. The strain is ‘‘elastic’’ if it develops instantaneously (i.e., the strain occurs as soon as the force is applied), remains as long as the stress is applied, and disappears as soon as the force is withdrawn. A material subjected to an elastic strain does not show any permanent deformation (i.e., it returns to its original shape after the force or stress is removed).

Permanent or plastic deformation in a material is known as the plastic strain. In this case, when the stress is removed, the material does not go back to its original shape.

Stiffness is a qualitative measure of the elastic deformation produced in a material. A stiff material has a high modulus of elasticity and maintains its size and shape even under an elastic load. Stiffness also depends upon geometry.

The Young's modulus (E) describes tensile elasticity, or the tendency of an object to deform along an axis when opposing forces are applied along that axis; it is defined as the ratio of tensile stress to tensile strain. It is often referred to simply as the elastic modulus or modulus of elasticity.

Poisson's ratio is the negative ratio of transverse to axial strain. When a material is stretched in one direction, it usually tends to contract in the other two directions perpendicular to the direction of stretching. Conversely, if the material is compressed rather than stretched, it usually tends to expand in the directions transverse to the direction of compression. The Poisson’s ratio is about 0.3 for metals.

A tensile stress is to be applied along the long axis of a cylindrical brass rod of Young’s modulus 97 MPa and diameter of 10mm. Determine the magnitude of the load require to produce a 2.5x10-3 mm change in diameter if the deformation is entirely elastic. (Ans: 71.3MPa)

An aluminium rod, with a Young’s modulus of 68.0 MPa is to withstand an applied force of 200 kN. To assure sufficient safety, the maximum allowable stress on the rod is limited to 170 MPa. The rod must be at least 3.8m long but must deform elastically no more than 6mm when the force is applied. Design an appropriate rod. (Ans: the cross sectional area of the rod must be at least 1820mm2)

Brittle materials fail without significant plastic strain (plastic deformation). Examples include ceramics and glasses.

Ductility measures the amount of plastic deformation that a material can withstand before fracture. It can either be quantified by % elongation, %EL, (i.e. the engineering strain at the point of fracture in a tensile test) or the % reduction in area, %RA, (i.e. the amount of thinning undergone by the specimen during a tensile test). Ductility is important because a ductile material will show obvious deformation before fracture (this acts as a warning that the applied stress is too high), and it will usually absorb more energy than a brittle material. Fabricators of engineering components (metallic and polymers) want a ductile material in order to form complicated shapes without breaking the material in the process. Ductility depends on temperature and strain rate.

Difference between elastic limit and proportional limit?

The critical stress value needed to initiate plastic deformation is defined as the elastic limit of the material.

The proportional limit is defined as the level of stress above which the relationship between stress and strain in not linear.

In most materials the elastic limit (minimum stress required to initiate plastic deformation, i.e. permanent, non-recoverable deformation) and the proportional limit (stress above which the stress and strain relationship is no longer linear) are quite close and cannot be determined precisely; and so the yield strength is not well defined. We, therefore, use proof strength (or offset yield strength) in place of yield strength in such cases.

The proof strength is determined by drawing a line parallel to the linear region on the stress-strain plot. This line is offset from the linear region by a specific value of strain (usually 0.002 or 0.2%). The stress value corresponding to the intersection of this line and the engineering stress-strain curve is defined as the proof strength.

Mechanical properties of materials depend on temperature (Figure 6-11). Yield strength, tensile strength, and modulus of elasticity usually decrease at higher temperatures, whereas ductility commonly increases.

Necking – local deformation in a tensile specimen. Necking begins at the ultimate tensile stress in an engineering stress-strain curve.

This is because beyond maximum load the distribution of strain along gage length is not uniform. True stress continues to increase after necking because, although the load required decreases, the area decreases even more, so the stress is increasing up to the fracture.

For structural applications we often do not require true stress and true strain. When we exceed the yield strength, the material deforms. The component would fail because it can no longer support the applied stress. Furthermore, a significant difference develops between the two curves only when necking begins But when necking begins, our component is grossly deformed and no longer satisfies its intended use.

The ultimate tensile strength (UTS) is the maximum recorded tensile stress on the engineering stress-strain curve. In many ductile materials deformation does nor uniform. At some point, one region deforms more than others and a large local decrease in the cross-sectional area occurs. The region where this localised deformation occurs is called a neck. The tensile strength is the stress at which necking begins in ductile materials. The stress in a tensile test (calculated using the original area A0) decreases once necking occurs, since the cross-sectional area in the necked region decreases (less force required for smaller area to cause same deformation).

Toughness – a qualitative measure of the impact properties of a material. A tough material resists failure by impact.

Toughness refers to the ability of a material to absorb energy before fracturing. The area under a stress strain curve is a measure of toughness.

A material with a high toughness will have a better resistance to sudden fracture, particularly if the material contains a crack or flaw.

Specimen geometry as well as the manner of load application are important in toughness determinations. For dynamic (high strain rate) loading conditions and when a notch (or point of stress concentration) is present, notch toughness is assessed by using an impact test. Furthermore, fracture toughness is a property indicative of a material’s resistance to fracture when a crack is present. For the static (low strain rate) situation, toughness may be ascertained from the results of a tensile stress–strain test. It is the area under the stress-strain curve up to the point of fracture. The units for toughness are the same as for resilience (i.e., energy per unit volume of material). For a material to be tough, it must display both strength and ductility; often, ductile materials are tougher than brittle ones. This is demonstrated in Figure 6.13, in which the stress–strain curves are plotted for both material types. Hence, even though the brittle material has higher yield and tensile strengths, it has a lower toughness than the ductile one, by virtue of lack of ductility;