Chapter 2

MATERIALS AND PROCESSES

41

 

2.2

THE STATISTICAL NATURE OF MATERIAL PROPERTIES

 

f(x)

Some published data for material properties represent average values of many samples Sd Sd Sd Sd Sd Sd

2

tested. (Other data are stated as minimum values.) The range of variation of the published test data is sometimes stated, sometimes not. Most material properties will vary about the average or mean value according to some statistical distribution such as the Gaussian or normal distribution shown in Figure 2-11. This curve is defined in terms of two parameters, the arithmetic mean  and the standard deviation Sd. The equation of the Gaussian distribution curve is

x

1

( x − )2 

F I G U R E 2 - 11

 

f ( x) =

exp−

,

−  x  

(2.9a)

 

2 S

The Gaussian (Normal)

d

 2 S 2 d  

Distribution

where x represents some material parameter, f(x) is the frequency with which that value of x occurs in the population, and  and Sd are defined as n

1

 =  x

(2.9b)

n

i

i =1

n

 

1

2

Sd =

( x

(2.9c)

i )

n - 1 i =1

 

The mean  defines the most frequently occurring value of x at the peak of the curve, and the standard deviation Sd is a measure of the “spread” of the curve about the mean. A small value of Sd relative to  means that the entire population is clustered closely about the mean. A large Sd indicates that the population is widely dispersed about the mean. We can expect to find 68% of the population within   1 Sd, 95%

within   2 Sd, and 99% within   3 Sd

 

 

There is considerable scatter in multiple tests of the same material under the same test conditions. Note that there is a 50% chance that the samples of any material that you buy will have a strength less than that material’s published mean value. Thus, you may not want to use the mean value alone as a predictor of the strength of a randomly chosen sample of that material. If the standard deviation of the test data is published Table 2-2

along with the mean, we can “factor it down” to a lower value that is predictive of some Reliability Factors

larger percentage of the population based on the ratios listed above. For example, if you for Sd = 0.08 

want to have a 99% probability that all possible samples of material are stronger than your assumed material strength, you will subtract 3 Sd from  to get an allowable value Reliability %

Factor

for your design. This assumes that the material property’s distribution is Gaussian and not skewed toward one end or the other of the spectrum. If a minimum value of the 50

1.000

material property is given (and used), then its statistical distribution is not of concern.

90

0.897

Usually, no data are available on the standard deviation of the material samples 95

0.868

tested. But you can still choose to reduce the published mean strength by a reliability 99

0.814

factor based on an assumed Sd. One such approach assumes Sd to be some percentage 99.9

0.753

of  based on experience. Haugen and Wirsching[1] report that the standard deviations 99.99

0.702

of strengths of steels seldom exceed 8% of their mean values. Table 2-2 shows reliability 99.999

0.659

reduction factors based on an assumption of Sd = 0.08  for various reliabilities. Note 99.9999

0.620

that a 50% reliability has a factor of 1 and the factor reduces as you choose higher reli-

 

 

Image 86

 

42

MACHINE DESIGN -

An Integrated Approach

 

 

ability. The reduction factor is multiplied by the mean value of the relevant material property. For example, if you wish 99.99% of your samples to meet or exceed the assumed strength, multiply the mean strength value by 0.702.

2

In summary, the safest approach is to develop your own material-property data for the particular materials and loading conditions relevant to your design. Since this approach is usually prohibitively expensive in both time and money, the engineer often must rely on published material-property data. Some published strength data are expressed as the minimum strength to be expected in a statistical sample, but other data may be given as the average value for the samples tested. In that case, some of the tested material samples failed at stresses lower than the average value, and your design strength may need to be reduced accordingly.

 

 

2.3

HOMOGENEITY AND ISOTROPY

All discussion of material properties so far has assumed that the material is homogeneous and isotropic. Homogeneous means that the material properties are uniform throughout its continuum, e.g., they are not a function of position. This ideal state is seldom attained in real materials, many of which are subject to the inclusion of discontinuities, precipitates, voids, or bits of foreign matter from their manufacturing process.

However, most metals and some nonmetals can be considered, for engineering purposes, to be macroscopically homogeneous despite their microscopic deviations from this ideal.

An isotropic material is one whose mechanical properties are independent of orientation or direction. That is, the strengths across the width and thickness are the same as along the length of the part, for example. Most metals and some nonmetals can be considered to be macroscopically isotropic. Other materials are anisotropic, meaning that there is no plane of material-property symmetry. Orthotropic materials have three mutually perpendicular planes of property symmetry and can have different material properties along each axis. Wood, plywood, fiberglass, and some cold-rolled sheet metals are orthotropic.

One large class of materials that is distinctly nonhomogeneous (i.e., heterogeneous) and nonisotropic is that of composites (also see below). Most composites are man-made, but some, such as wood, occur naturally. Wood is a composite of long fibers held together in a resinous matrix of lignin. You know from experience that it is easy to split wood along the grain (fiber) lines and nearly impossible to do so across the grain. Its strength is a function of both orientation and position. The matrix is weaker than the fibers, and it always splits between fibers.

 

 

2.4

HARDNESS

The hardness of a material can be an indicator of its resistance to wear (but is not a guarantee of wear resistance). The strengths of some materials such as steels are also closely correlated to their hardness. Various treatments are applied to steels and other metals to increase hardness and strength. These are discussed below.

 

Image 87