THE MECHANICAL BEHAVIOR OF MATERIALS

Forging, rolling, extrusion, rod and wire drawing, and all sheet metal forming are plastic deformation processes. This discussion explores the fundamental aspects of mechanical behavior of metals during deformation.

For example, when stretching a piece of metal to create an aircraft or automobile component, the material is subjected to tension. By the same token, when a solid cylindrical piece of metal is forged to make a gear disk, the material is subjected to compression. Sheet metal, too, undergoes shearing stress when a hole is punched in it.

Strength, hardness, toughness, elasticity, plasticity, brittleness, ductility and malleability are mechanical properties used to measure how metals behave under a load. These properties are described in terms of the types of force or stress that the metal must withstand and how these forces are resisted. Common types of loading are compression, tension, shear, torsion, or a combination of these stresses, such as fatigue. Figure 1 shows the three most common types of stress.

Compression stresses develop within a sample material when forces compress or crush it. The material that supports an overhead beam, for instance, is in compression and the internal stresses that develop within the supporting column are compressive.

Tension (or tensile) stresses develop when a material is subject to a pulling load. An example is when a wire rope is used to lift a load or as a guy to anchor an antenna. Tensile strength is the resistance to longitudinal stress or pull and is measured in pounds per inch of cross section.

Shearing stresses occur within a material when external forces are applied along parallel lines in opposite directions. Shearing forces can separate a material by sliding part of it in one direction and the remainder in the opposite direction.

The mechanical properties of materials are revealed by testing them. Results from the tests depend on the size and shape of the material being tested (the specimen), how it is held, and how the test is performed. To make the results comparable, common procedures (or standards) are used which are published by the ASTM.

To compare specimens of different sizes, the load is calculated per unit area. The force divided by the areas is called stress. In tension and compression tests, the relevant area is that perpendicular to the force. In shear tests, the area of interest is perpendicular to the axis of rotation. The engineering stress is defined as the ratio of the applied force F to the original cross-sectional area A0 of the specimen. That is:

σ = F/A₀

where:

F = tensile strength or compressive force

A₀ = original cross-sectional area of the specimen.

The nominal strain or engineering strain can be defined in three ways. As the relative elongation, given by:

e = (l – l₀) / l₀ = Δl / l₀ = l / l – 1

where:

l₀ = the original gauge length

l = the instantaneous length

As the reduction of the cross-section area, given by:

Ψ = (A – A₀) / A₀ = ΔA / A₀ = 1 – A / A₀

Or as the logarithmic strain, given by:

φ = 1n x A₀ / A

where:

A₀ = the original cross-section area

A = the instantaneous cross-section area.

These definitions of stress and strain allow test results for specimens of different cross-sectional area A0 and of different length l0 to be compared. It is generally accepted that tension is positive and compression is negative. Shear stress is defined as:

τ = F/A₀

where:

F = force is applied parallel to the upper and lower faces, each of which has an area A₀.

Shear strain is defined as:

y = a/b = tg θ

Shear stresses produce strains according to:

τ = G ? y

where:

G = shear modulus.

Torsion is a variation of pure shear. Shear stress then is a function of applied torque shear strains related to the angle of twist. Materials subject to uniaxial tension shrink in the lateral direction. The ratio of lateral strain to axial strain is called Poisson's ratio:

ν = e_x/e_y

where:

e_x = laterial strains

e_y = axial strains.

The theory of isotropic elasticity defines Poisson's ratio in the next relationship:

-1 < ν ≤ 0.5

The Poisson ratio for most metals is between 0.25 and 0.35. Rubbery materials have Poisson's ratios very close to 0.5 and are therefore almost incompressible. Theoretical materials with a Poisson's ratio of exactly 0.5 are truly incompressible, because the sum of all their strains leads to a zero volume change. Cork, on the other hand, has a Poisson's ratio close to zero, which makes cork function well as a bottle stopper. The cork must be easily inserted and removed, yet it also must withstand the pressure from within the bottle.

The elastic modulus, shear modulus, and Poisson's ratio are related in the following way:

E = 2G (1+ &#957;)

STRESS/STRAIN CURVES
The relationship between the stress and strain that a material displays is known as a stress/strain curve. It is unique for each material. Figure 2 shows the characteristic stress/strain curves for three common materials.

Note that the concrete curve is almost a straight line. There is an abrupt end to the curve. This abrupt end, combined with the fact that the line is very steep, indicate that the material is brittle. The curve for cast iron is slightly curved. Cast iron is also a brittle material. Notice that the curve for mild steel seems to have a long, gently curving "tail", which indicates the high ductility of the material. Typical results from a tension test on mild-steel are shown in Figure 3.

Several significant points on a stress/strain curve help one understand and predict the way any building material will behave. Point A is known as the proportional limit. Up to this point, the relationship between stress and strain is exactly proportional. The linear relationship between stress and strain is called Hooke's law:

&#963; = E ? e

If the load is removed, the specimen returns to its original length and shape, which is known as elastic behavior. Strain increases faster than stress at all points on the curve beyond point A. Point B is known as the elastic limit; after this point, any continued stress results in permanent, or inelastic, deformation.

The stress resistance of the material decreases after the peak of the curve, and this point is also known as the yield point Y of the material. For soft and ductile materials, the exact position of the stress/strain curve where yielding occurs may not be easily determined because the slope of the straight portion of the curve decreases slowly. Therefore, Y is usually determined as the point on the stress/strain curve that is offset by a strain of 0.002 or 0.2 percent elongation.

If the specimen continues to elongate further under an increasing load beyond point Y, a domain curve begins in which the growth of strain is faster than that of stress. Plastic forming of metal is performed in this domain. If the specimen is released from stress between point Y and point M, the curve follows a straight line downward and parallel to the original slope (see Figure 4).

As the load and engineering stress increase further, the curve eventually reaches a maximum point M and then begins to decrease. The maximum engineering stress is called tensile strength or ultimate tensile strength (UTS) of the material:

&#963; = UTS = F_max/A₀

If the specimen is loaded beyond its ultimate tensile strength, it begins to "neck", or "neck down". The cross-sectional area of the specimen is no longer uniform along a gauge length, but is smaller in the necking region. As the test progresses, the engineering stress drops further and the specimen finally fractures at the point F. The engineering stress at fracture is known as the breaking or fracture stress.

The ratio of stress to strain in the elastic region is known as the modulus of elasticity (E) or Young's modulus and is expressed by:

E = &#963;/e

The modulus of elasticity is essentially a measure of the stiffness of the material.

DUCTILITY
Ductility is an important mechanical property because it is a measure of the degree of plastic deformation that can be sustained before fracture. Ductility may be expressed as either percent elongation or percent reduction in area.

Elongation can be defined as:

&#948; = (l_f – l₀)/ l₀ x 100

Reduction can be defined as:

&#936; = (A₀ – A_f) / A₀ x 100

l_f = length at the fracture. This length is measured between original gauge marks after the pieces of the broken specimen are placed together

l₀ = the original sample gauge length

A_f = cross-sectional area the fracture

A₀ = original sample gauge cross-sectional area.

Knowledge of the ductility of a particular material is important because it specifies the degree of allowable deformation during forming operations. Gauge length is usually determined by inscribing gauge marks on the sample prior to testing and measuring the distance between them, before and after elongation has occurred. Because elongation is always declared as a percentage, the original gauge must be recorded. Fifty millimeters (two inches) is the standard gauge length for strip tensile specimens and this is how the data are generally recorded.

The reduction in area is declared as a percentage decrease in the original cross-sectional area and, like percentage elongation, it is measured after the sample fractures. The percentage elongation is more a measure of the strain leading to the onset of necking than a measure of the strain at final fracture in a uniaxial tensile specimen. A better measure of the strain at final fracture is the percentage reduction in area.

The relationship between the elongation and reduction of area is different for some groups of metals, as shown in Figure 5.

Elongation ranges approximately between 10 and percent for most materials, and values between 20 and 90 percent are typical for reduction of area. Thermoplastics and super-plastic materials, of course, exhibit much higher ductility, and brittle materials have little or no ductility.

TRUE STRESS AND TRUE STRAIN
In the solution of technical problems in the processes of sheet metal forming, theoretical stress and strain do not have as crucial a significance as do true stress and true strain. True stress and true strain are much more important.

It is apparent that, since stress is defined as the ratio of force to area, true stress may be defined as:

k = F/A

where:

A = the instantaneous cross-section area.

As long as there is uniform elongation, true stress (k) can be expressed using the value for engineering stress. Assuming that volume at plastic deformation is constant: (this equation is in effect only to point M), the relationship between true and nominal stress may be defined as follows:

k = F/A = &#963; x A₀ / A = &#963; (1-e) = &#963; / 1-&#936; = &#963;&#949;ⁿ

Figure 6 shows a nominal (engineering) curve and the true stress and strain for medium carbon steel. Because the strains at the yield point Y are very small, the difference between the true and engineering yield stress is negligible for metals. This is because the difference in the cross-sectional areas A₀ and A, (A < A₀) above point Y is always greater, so the difference between the true and nominal stress is significant (k > &#963;). Because of the relationship, the given curve showing the true stress is known as the "hardening curve" of a metal.

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Vukota Boljanovic is a PhD in mechanical engineering who retired from the American automotive industry. He is the author of Sheet Metal Forming Processes and Die Design. This article is an excerpt from that book. For more information, contact Industrial Press Inc., 200 Madison Avenue, New York, NY 10016, 212-889-6330, Fax: 212-545-8327, www.industrialpress.com, induspress@aol.com.