Quantitative microscopy of strain-hardening. For simple comparisons of rela-

5.2.2.5 Quantitative microscopy of strain-hardening. For simple comparisons of rela-

tive microhardness, hardness values at a fixed load can Important standard methods for measuring grain size

be compared without allowance for load-dependence. and contents of inclusions and phases have evolved On the other hand, if absolute values of hardness are

in metallurgy and mineralogy. Grain size in metal- required from low-load tests, it is advisable to deter-

lic structures is commonly assessed by the ASTM mine the Meyer line for the particular material over

(McQuaid-Ehn) method in which etched microstruc-

a comparatively small load range by plotting P val- tures are compared with diagrams or standard sets of ues against the corresponding d values, using log-log

photographs at a standard magnification, say 100ð. graph paper. (Extrapolations beyond the chosen load

Numerous manual methods have been developed to range are unwise because the profile of the Meyer

assess the cleanliness of steels. They range from line may change.) Figure 5.5 shows the Meyer line,

direct methods, in which particles beneath linearly- slope n, for a material giving load-dependent micro-

traversed crosswires are individually counted, to com- hardness values. The slope n is less than 2, which

parative methods based on photomicrographs or charts is usual. The H m curve has a negative slope and

(e.g. Fox, Jernkontoret (JK), SAE-ASTM, Diergarten). microhardness values increase as the load increases.

Sometimes these methods attempt to identify inclu- One way of reporting load-dependent microhardness

sions according to type and form. Nowadays, auto- results is to state three hardness numbers in terms of

matic systems are available to convert microscopical information, from a light or electron microscope, to electronic signals that can be rapidly processed and evaluated. Although more objective than former meth- ods, they must be set up correctly and their limitations appreciated. For instance, the plane of sectioning is critical when highly-directional features are present, such as filaments in composites and slag stringers in metals. Quantitative methods have made it possible to relate many microstructural characteristics, such as volume fractions, interparticle distances, grain bound- ary area per unit volume, interlamellar spacing, etc., to mechanical properties. In 1848, the French petro- grapher Delesse established mathematically that, in a fully random cross-section of a uniform aggregate, the area fraction of a given microconstituent (phase) in

a field of view is equal to the corresponding volume fraction in the three-dimensional aggregate. There are three basic methods for measuring the area fraction, as the following exercise demonstrates.

Suppose that a certain field of view contains a dark phase and a light-coloured matrix, as shown in Figure 5.6a. Using the systematic notation for stereol- ogy given in Table 5.1, the total area occupied by the

dark phase in the test area A T ⊲D L 2 ⊳ is A; it is the sum of i areas, each of area a. This areal fraction is A A . Figure 5.5 Meyer line for material with load-independent

Alternatively, the field of view may be systematically hardness (by courtesy of Carl Zeiss, Germany) .

traversed with a random test line, length L T , and a

132 Modern Physical Metallurgy and Materials Engineering

Figure 5.6 Comparison of methods for measuring areal fraction .

Table 5.1 Stereological notation (after Underwood)

Symbol Dimensions P

Points of intersection

mm 0

N Number of objects

mm 0

L Lines

mm 1

A Flat surfaces

mm 2

S Curved surfaces

mm 2

V Volumes

mm 3

length L derived from the sum of j intercepts, each of

length l. The lineal fraction L L D L/L T (Figure 5.6b).

In the third method (Figure 5.6c), a regular grid of points is laid over the field and all coincidences of grid intersections with the dark phase counted to give

a point count fraction P/P T which is P P . Only one field of view is shown in Figure 5.6; in practice, numerous different fields are tested in order to give statistically-

significant average values, i.e. A A ,L L ,P P . The areal

fraction method is chiefly of historical interest. The lin- Figure 5.7 Relationships between stereological quantities . eal method was used in the original Quantimet instru- ments which scanned 1 the images electronically with a raster. The point count method is the basis of modern

in unit volume S V of a single-phase alloy is poten- instruments in which electronically-scanned fields are

tially more significant than average grain size. The composed of thousands of pixels.

diagram of Figure 5.7 shows some of the stereolog- Certain three-dimensional properties of a structure

ical equations that permit non-measurable quantities to be calculated. Measurable quantities are enclosed

have a great influence on a material’s behaviour in circles, calculable quantities are enclosed in boxes but cannot be measured directly by microscope. For

and arrows show the possible routes in the triangu- instance, the amount of curved grain boundary surface

lar matrix. Important stereological equations provid- ing the necessary links are shown below the matrix.

1 Contrary to popular belief, the term ‘scan’ implies Thus, the above-mentioned quantity S V may be derived accuracy and fine resolution.

from intersections on test lines or traverses ⊲P L ⊳ since

The characterization of materials 133 2P L D S V . Similarly, length of dislocation lines, acic-

ular precipitates or slag stringers per unit volume L V

may be obtained from point counts over an area, P A .