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The best material-property data will be obtained from destructive or nondestructive testing under actual service loadings of prototypes of your actual design, made from the actual materials by the actual manufacturing process. This is typically done only 2
when the economic and safety risks are high. Manufacturers of aircraft, automobiles, motorcycles, snowmobiles, farm equipment, and other products regularly instrument and test finished assemblies under real or simulated service conditions.
lo
In the absence of such specific test data, the engineer must adapt and apply published material-property data from standard tests to the particular situation. The American Society for Testing and Materials (ASTM) defines standards for test specimens and do
test procedures for a variety of material-property measurements.* The most common
material test used is the tensile test.
The Tensile Test
F I G U R E 2 - 1
A typical tensile test specimen is shown in Figure 2-1. This tensile bar is machined from A Tensile Test Specimen
the material to be tested in one of several standard diameters do and gage lengths lo. The gage length is an arbitrary length defined along the small-diameter portion of the specimen by two indentations so that its increase can be measured during the test. The larger-diameter ends of the bar are threaded for insertion into a tensile test machine which is capable of applying either controlled loads or controlled deflections to the ends of the bar, and the gage-length portion is mirror polished to eliminate stress concentrations from surface defects. The bar is stretched slowly in tension until it breaks, while the load and the distance across the gage length (or alternatively the strain) are continuously monitored. The result is a stress-strain plot of the material’s behavior under load as shown in Figure 2-2 a, which depicts a curve for a low-carbon or “mild” steel.
STRESS AND STRAIN Note that the parameters measured are load and deflection, but those plotted are stress and strain. Stress () is defined as load per unit area (or unit load) and for the tensile specimen is calculated from
P
=
(2.1 a)
Ao
where P is the applied load at any instant and Ao is the original cross-sectional area of the specimen. The stress is assumed to be uniformly distributed across the cross section. The stress units are psi or Pa.
Strain is the change in length per unit length and is calculated from
l − l
=
o
(2.1 b)
lo
where lo is the original gage length and l is the gage length at any load P. The strain is unitless, being length divided by length.
MODULUS OF ELASTICITY This tensile stress-strain curve provides us with a number of useful material parameters. Point pl in Figure 2-2 a is the proportional limit below which the stress is proportional to the strain, as expressed by the one-dimensional form of Hooke’s law:
* ASTM, 1994 Annual Book of
ASTM Standards, Vol. 03.01, Am.
E =
(2.2)
Soc. for Testing and Materials,
Philadelphia, PA.
32
MACHINE DESIGN -
An Integrated Approach
Stress
Stress
true
true
S
2
ut
f
u
f
u
eng'g
y
eng'g
y
Sy
el
pl el
pl
offset line
(a)
(b)
E
E
Elastic
Range
Plastic Range
Strain
Strain
0.002
F I G U R E 2 - 2
Engineering and True Stress-Strain Curves for Ductile Materials: ( a) Low-Carbon Steel ( b) Annealed High-Carbon Steel where E defines the slope of the stress-strain curve up to the proportional limit and is called Young’s modulus or the modulus of elasticity of the material. E is a measure of the stiffness of the material in its elastic range and has the units of stress. Most metals exhibit this linear stiffness behavior and also have elastic moduli that vary very little with heat treatment or with the addition of alloying elements. For example, the highest-strength steel has the same E as the lowest-strength steel at about 30 Mpsi (207 GPa).
For most ductile materials (defined below), the modulus of elasticity in compression is the same as in tension. This is not true for cast irons and other brittle materials (defined below) or for magnesium.
ELASTIC LIMIT The point labeled el in Figure 2-2 a is the elastic limit, or the point beyond which the material will take a permanent set, or plastic deformation. The elastic limit marks the boundary between the elastic-behavior and plastic-behavior regions of the material. Points el and pl are typically so close together that they are often considered to be the same.
YIELD STRENGTH At a point y slightly above the elastic limit, the material begins to yield more readily to the applied stress, and its rate of deformation increases (note the lower slope). This is called the yield point, and the value of stress at that point defines the yield strength Sy of the material.
Materials that are very ductile, such as low-carbon steels, will sometimes show an apparent drop in stress just beyond the yield point, as shown in Figure 2-2 a. Many less ductile materials, such as aluminum and medium- to high-carbon steels, will not exhibit this apparent drop in stress and will look more like Figure 2-2 b. The yield strength of a material that does not exhibit a clear yield point has to be defined with an offset line, drawn parallel to the elastic curve and offset some small percentage along the strain axis.
An offset of 0.2% strain is most often used. The yield strength is then taken at the intersection of the stress-strain curve and the offset line as shown in Figure 2-2 b.
ULTIMATE TENSILE STRENGTH The stress in the specimen continues to increase nonlinearly to a peak or ultimate tensile strength value Sut at point u. This is considered to be the largest tensile stress the material can sustain before breaking. However, for