EEE-112Mechanics of Materials

InstructorYasser M Siddiqi

Materials CharacterizationThe American Society for Testing and

Materials (ASTM) specifies test procedures for determining the various properties of a material.

These descriptions are guidelines used by experimentalists to obtain reproducible results for material properties.

Tension TestIn the tension test, standard specimen are

placed in a tension-test machine, where they are gripped at each end and pulled in the axial direction.

Figure 1 shows two types of standard geometry: a specimen with a rectangular cross-section and specimen with a circular cross-section.

Tension test machine and specimens. Fig 1

Two marks are made in the central region, separated by a gage length Lo.

The deformation δ is movement of the two marks.

For metals, such as aluminum or steel, ASTM recommends a gage length Lo = 2 in. and diameter do = 0.5 in.

The normal strain ε is the deformation δ divided by Lo.

The tightness of the grip, the symmetry of the grip, friction, and other local effects are assumed and are observed to die out rapidly with the increase in distance from the ends.

This dissipation of local effects is further facilitated by the gradual change in the cross-section.

The specimen is designed so that its central region is in a uniform state of axial stress.

The normal stress is calculated by dividing the applied force P by the area of cross section Ao, which can be found from the specimen’s width or diameter.

The tension test may be conducted by controlling the force P and measuring the corresponding deformation δ.

Alternatively, we may control the deformation δ by the movement of the grips and measuring the corresponding force P.

The values of force P and deformation δ are recorded, from which normal stress σ and normal strain ε are calculated.

As the force is applied, initially a straight line (OA) is obtained.

The end of this linear region is called the proportional limit.

For some metals, the stress may then decrease slightly (the region AB), before increasing once again.

Stress-Strain CurveFig. 2.

The largest stress (point D on the curve) is called the ultimate stress.

In a force-controlled experiment, the specimen will suddenly break at the ultimate stress.

In a displacement-controlled experiment we will see a decrease in stress (region DE).

The stress at breaking point E is called fracture or rupture stress.

Elastic and Plastic regionsIf we load the specimen up to any point along

line OA- or even a bit beyond- and then start unloading, we find that we retrace the stress-strain curve and return to point O.

In this elastic region, the material regains its original shape when the applied force is removed.

If we start unloading only after reaching point C, however, then we will come down the straight line FC, which will be parallel to line OA.

At point F, the stress is zero, but the strain is nonzero. C this lies in the plastic region of the stress-strain curve, in which the material is deformed permanently, and the permanent strain at point F is the plastic strain.

The region in which the material deforms permanently is called plastic region.

The total strain at point C is sum of the plastic strain (OF) and an additional elastic strain (FG).

The point demarcating the elastic from the plastic region is called the yield point.

The stress at yield point is called the yield stress.

In practice, the yield point may lie anywhere in the region AB, although for most metals it is close to the proportional limit. For many materials it may not even be clearly defined.

It should be emphasized that elastic and linear are two distinct material descriptions.

Figure 3 shows the stress-strain curve for a soft rubber that can stretch several times its original length and still return to its original geometry.

Soft rubber is thus elastic but nonlinear material.

Examples of nonlinear and brittle materials. (a) Soft rubber. (b) Glass. Fig. 3.

Ductile and brittle materialsDuctile materials, such as aluminum and

copper, can undergo large plastic deformations before fracture.

Soft rubber can undergo large deformations but it is not a ductile material.

Glass, on the other hand, is brittle: it exhibits little or no plastic deformation.

A material’s ductility is usually described as percent elongation before rupture.

The elongation values of 17% for aluminum and 35% for copper before rupture reflect the large plastic strains these materials undergo before rupture, although they show small elastic deformation as well.

A ductile material usually yields when the maximum shear stress exceeds the yield shear stress.

A brittle material usually ruptures when the maximum tensile normal stress exceeds the ultimate tensile stress.

Hard and Soft materialsA material hardness is its resistance to

scratches and indentation.In Rockwell test, the most common hardness

test, a hard indenter of standard shape is pressed into the material using a specified load.

The depth of indentation is measured and assigned a numerical scale for comparing hardness of different materials.

A soft material can be made harder by gradually increasing its yield point by strain hardening.

As we have seen, at point C in Figure 2 the material has a permanent deformation even after unloading.

If the material now is reloaded, point C becomes the new yield point, as additional plastic strain will be observed only after stress exceeds this point.

Strain hardening is used, for example, to make aluminum pots and pans more durable.

In the manufacturing process, known as deep drawing, the aluminum undergoes large plastic deformation.

Of course, as the yield point increases, the remaining plastic deformation before fracture decreases, so the material becomes more brittle.

True stress and true strainWe noted that stress decreases with

increasing strain between the ultimate stress and rupture (region DE in Figure 2).

However, this decrease is seen only if we plot Cauchy’s stress versus engineering strain.

Cauchy’s stress is the load P divided by the original undeformed cross-sectional area.

An alternative is to plot true stress versus true strain, calculated using the actual, deformed cross-section and length.

In such a plot, the stress in region DE continues to increase with increasing strain and just as in region BD.

Past ultimate stress a specimen also undergoes a sudden decrease in cross-sectional area called necking.

Fig. 4.

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