Handbook Fractography (1998) WW Part 7 doc

measurements, which are defined at the bottom of the table. If image analysis equipment with operator-interactive capability is available, most of the quantities listed in Table 1 can be measured directly. Table 1 Stereological relationships for features in a projection plane Primes indicate projected quantities or measurements made on projection planes. Variable Explanation Equation Part I: Individual projected features (a) L' Length of a linear feature L' = (π/2) P ' L A' T L' p Perimeter length of a closed figure L' p = (π/2) P ' L A' T L ' 2 Mean intercept length of a closed figure L ' 2 = (2 P ' P / P ' L A' Area of a closed figure A' = P ' P A' T Part II: Systems of projected features (b) L ' Mean length of discrete linear features L ' = L' A /N' A L ' p Mean perimeter length of closed figures L ' p = L' A /N' A L ' 2 Mean intercept length of closed figures L ' 2 = L' L /N' L A ' Mean area of closed figures A ' = A' A /N' A A' A Area fraction of closed figures A' A = L' L = P' P L' A Line length per unit area L' A = (π/2) P ' L λ' Mean free distance λ' = (1-A' A )/N' L d ' Mean tangent diameter of convex figures d ' = N' L /N' A k ' L Mean curvature of convex figures k ' L = 2N' A /N' L (a) A' T : The selected test area in the plane; P ' L : the number of intersections of a linear feature per unit length of tests line, averaged over several directions of the test grid; P ' P : the number of points hits in areal features per grid test point, averaged over several angular placements of test grid. (b) N' A : the number of features per unit area of the test plane; N' L : the number of interceptions of a feature per unit length of test line; L' A : the length of linear features per unit area of the test plane Fig. 1 Basic quantities for a convex figure in a projection plane When many individual measurements are required on a system of features, such as for a size distribution, more labor is involved. If appropriate or available, semiautomatic image analysis equipment can be used in such cases. For example, the areal size distribution of the facets shown in Fig. 2 was determined by tracing the perimeter of each facet in the SEM photomicrograph with an electronic pencil (Ref 18). With a typical semiautomatic image analysis system, a printout can be produced with the data in the form of a histogram, along with other statistics. It should be emphasized that the equations listed in Table 1 describe only the images of features in the projection plane and do not give spatial information. Nevertheless, due to the lack of anything better, the two-dimensional results are valid for that particular projected surface and may even be compared with other fractographs from fracture surfaces of comparable roughness. Assumption of Randomness. It would be better if the true magnitudes of areas, lengths, sizes, distances, and so on, were obtainable instead of the projected quantities. If this is not possible, however, order of magnitude calculations can be made by assuming that the fracture surface is composed of randomly oriented elements. With this assumption, the stereological equations can be used with only one (or a limited number) of projection planes (Ref 33, 48). This is so because the stereological equations are valid for structures with any degree of randomness, provided randomness of sampling is achieved. If the structure (in this case, the fracture surface) happens to possess complete angular randomness, the randomness of sampling is not required. Instead, directed measurements are adequate because for a random structure the measured value should be the same in any direction. Accordingly, with the assumption of randomness, the standard equations of stereology, based only on vertical (directed) measurements from the SEM picture, can be used. Basic equations that are valid under these conditions are listed in Table 2. The quantities to the left-hand side of Table 2 pertain to the fracture surface. Thus, instead of L A , for example, L S is used, which represents the length of the line per unit area of the (curved) fracture surface. On the right-hand side of Table 2 the working equations are expressed in terms of projected quantities. Table 2 Stereological relationships between spatial features and their projected images Primes indicate projected quantities or measurements made on projection planes. Variable Explanation Equation Part I: Individual features in the fracture surface L φ Length of linear feature (fixed direction φ in the projection plane) L φ = (π/2) L ' φ L c Length of curved linear feature (variable directions in the projection plane) L c = (4/ π) L ' c L 3 Mean intercept length of a closed figure (averaged over all directions in the projection plane) L 3 = (π /2) L ' L p Perimeter length of a closed figure L p = (4/ π) L ' p S Area of a curved two-dimensional feature (no overlap) S = 2 A ' Part II: Systems of features in fracture surface (a) L φ Mean length of linear features (fixed direction φ in projection plane) L φ = (π /2)L' A /N' A L c Mean length of curved lines (variable directions in the projection plane) L c = (4/ π)L' A /N' A L 3 Mean intercept length of closed features (averaged over all directions in the projection plane) L 3 = (π /2)L' L /N' L L p Mean perimeter length of closed figures L p = (4/ π)L' A /N' A S Mean area of curved two-dimensional features (no overlap) S = 2A' A /N' A S s Area fraction of curved two-dimensional features (no overlap) S s = A' A = L' L = P' P L s Length of linear features per unit area of fracture surface (variable directions in projection plane) L s = (2/ π)L' A (a) L' A = (π /2) P ' L ( φ ); P ' L ( φ ) is the number of intersections of linear features per unit length of test line (variable directions in the projection plane). Figure 3 shows the differences between L φ and L c in the fractures surface. The value L φ corresponds to a straight line L' φ with fixed direction in the projection plane, while L c relates to a curved line L' c with variable direction in the projection plane. Statistically, the angular dependence of L φ is averaged in a vertical plane defined by L φ , while L c must be averaged in three dimensions, amounting to a difference of up to 23%. This elementary distinction has not been recognized in work purported as three-dimensional fractography (Ref 42). Other quantities in Table 2 can be used as they are or can be combined into more complicated functions as required. Fig. 2 SEM project ion of facets in fractured Al- 4Cu alloy Fig. 3 Relation of curved and straight lines in the projection plane to lines in the fracture surface. L c = (4/π) L ' c and L φ = (π /2) L ' φ . Correlation of Fracture Path and Microstructure. It is possible to relate bulk microstructural properties to the fracture path only if there is no correlation between the configuration of the surface and the underlying structure. However, in general, a correlation does exist. Frequently, the fracture path passes preferentially through particles, or some feature of the microstructure, rather than along an independent path through the material. This correlation results in a statistically higher concentration of particles in the fracture surface, N S , than the metallographic plane of polish, N A . Figure 4 shows the relationship of the fracture surface to a horizontal plane of polish and to the SEM projection plane. The interrelationship of particles and dimples among these three planes has been thoroughly investigated (Ref 49). Fig. 4 Correlation among plane of polish, fracture surface, and projection plane If it is permissible to assume an absence of correlation, then measurements of P' P , L' L , or A' A in the flat SEM fractograph should yield the same values as from the plane of polish (Ref 50). That is, for a two-phase structure: A' A = A A , L' L = L L , and P' P = P P (Eq 1) where A A , L L , and P P refer to quantities measured in the plane at polish. The three projected quantities in Eq 1 are dimensionless ratios and are therefore independent of magnification and distortion in the SEM image. If it can also be assumed that the surface elements of the fracture surface are randomly oriented, then the relationships given in Table 2 can also be used. For example: ' ' 2 2 A LIs N NNandN π  ==   (Eq 2) Under these circumstances, ratios from the fracture surface for example, N L or N S have the same values, respectively, as their counterparts (N L or N A ) in the plane of polish. It is evident from the above discussion that considerable quantification is possible in several ways from SEM photomicrographs. First, calculations can be made in the plane of projection alone, without any attempt to convert to three dimensions (Table 1). Second, spatial quantities in the fracture surface can be-calculated if random orientation of surface elements can be reasonably assumed (Table 2). Correlation effects will determine whether additional relationships are possible with the plane of polish (Ref 51). The assumption of angular randomness in the fracture surface is admittedly somewhat tenuous, especially considering the strongly oriented nature of a fracture surface. This problem will be addressed in the section "Analytical Procedures" in this article, in which the subject of partially oriented surfaces is treated in a more quantitative manner. Stereoscopic Methods In this section, conventional stereoscopic imaging and photogrammetric methods will be considered, as well as a geometric method that requires no instrumentation. Basically, these methods measure the locations of points. Stereoscopic Imaging. Stereoscopic pictures can be readily taken by SEM and TEM (Ref 52). In any SEM picture, there are two main types of distortion: perspective error due to tilt of the surface and magnification error arising from surface irregularities. The first type of error can be minimized by keeping the beam close to perpendicular to the fracture surface. The second error can be understood by reference to Fig. 5, which is the rectilinear optical equivalent of the SEM image (Ref 50). Fig. 5 Geometry of image formation in the scanning electron microscope The magnification is defined as the ratio of the image distance to the object distance. In the case of an irregular surface, the object distance is not constant. Consequently, high points have higher magnification than low points on the surface. For example, at point p in Fig. 5, the magnification is proportional to ss'/sm, but at point q, it is proportional to ss'/sn. The coordinates of the points in the fracture surface are usually measured by stereo SEM pairs, that is, two photographs of the same field taken at small tilt angles with respect to the normal. The geometry of this case is shown in Fig. 6, in which the points A, B appear at A',B' and A'',B'' in stereo pictures taken at tilt angles ±α. The lengths A'B' and A''B'' can be measured either from the two photos separately or from the stereo image directly (Ref 53). The difference A'B' - A''B'' is called the parallax, ∆x. Fig. 6 Determination of parallax ∆x from stereo imaging According to the geometry shown in Fig. 6: 1 2sin zx M α  ∆=∆   (Eq 3) where the height difference ∆z between the two points is proportional to the measured parallax ∆x, and M is the average magnification (Ref 30, 53). Because the magnification and tilt angle are fixed for one pair of photographs, the terms in the square brackets are constant. If α= 10°, for example, the value of ∆z should be about 2.88 times greater than the corresponding measurement along the x- or y-direction. The x- and y-coordinate points can be measured directly with a superimposed grid or can be obtained automatically with suitable equipment (Ref 54). Equation 3 is strictly correct for an orthogonal projection; that is, the point S in Fig. 5 is situated at infinity. This is a reasonable assumption at higher magnifications (>1000×); however, at lower magnifications, there are induced errors (Ref 50). Once the (x,y,z) coordinates have been obtained at selected points in the fracture surface, elementary calculations can be made, such as the equation of a straight line or a planar surface, the length of a linear segment between two points in space, and the angle between two lines or two surfaces (the dihedral angle) (Ref 55). Some of these basic equations are given in Ref 16. Also available is a computerized graphical method for analyzing stereophotomicrographs (Ref 56). Photogrammetric Methods. Another procedure for mapping fracture surfaces uses stereoscopic imaging with modified photogrammetry equipment (Ref 14, 42, 57). Several reports from the Max-Planck-Institut in Stuttgart have described the operation in detail (Ref 14, 54, 58, 59). Their instrument is a commercially available mirror stereometer with parallax-measuring capability (adjustable light point type). It is linked to an image analysis system and provides semiautomatic measurement of up to 500 (x,y,z) coordinate points over the fracture surface. The output data generate height profiles or contours at selected locations, as well as the angular distribution of profile elements (Fig. 7). The accuracy of the z-coordinates is better than 5% of the maximum height difference of a profile, and the measurement time is about 3 s per point. Fig. 7 Contour map and profiles obtained by stereophotogrammetry. Source: Ref 58 Stereo pair photographs with symmetrical tilts about the normal incident position are used to generate the stereoscopic image. The built-in floating marker (a point light source) of the stereometer is adjusted to lie at the level of the fracture surface at the chosen (x',y') position. Small changes in height are recognized when the marker appears to float out of contact with the fracture surface. The accuracy with which the operator can place the marker depends on his stereo acuity and amount of practice. When the floating marker is positioned in the surface, a foot switch sends all three spatial coordinates to the computer. Thus, the operator can take a sequence of observations without interrupting the stereo effect. A FORTRAN program then calculates the three-dimensional coordinates and produces the corresponding profile or contour map on the plotter or screen. In another research program having the objective of mapping fracture surfaces, a standard Hilger-Watts stereophotogrammetry viewer was modified so that the operator does not have to take his eyes from the viewer to record the micrometer readings (Ref 57). The graphic output is based on a matrix of 19 by 27 data points and is in the form of contour plots, profiles, or carpet plots. Figure 8 shows an example of the latter for a fractured Ti-10V-2Fe-3Al tensile specimen. Fig. 8 Fracture surface map (carpet plot) of a Ti-10V-2Fe-3Al specimen by stereophotogrammetry. Source: Ref 57 These photogrammetric procedures are nondestructive with regard to the fracture surface. They also allow detailed scrutiny of some areas and wider spacings in regions of less interest. It is also possible to obtain a roughness profile from a fracture surface along a curved or meandering path, a task that is very difficult to accomplish by sectioning methods. However, this is still a point-by-point method that relies heavily on operator skill and training. No information is provided on any subsurface cracking or possible interactions between microstructure and fracture path. Overlaps and complex fracture surfaces are difficult, if not impossible, to handle. Moreover, smooth or structureless areas cannot be measured unless a marker is added (Ref 59). This method is best suited to relatively large facets and to samples that must be preserved. Geometrical Methods. The stereoscopic and photogrammetric methods discussed above usually require special equipment and a visually generated stereo effect. Three-dimensional information can also be obtained from stereo pair photomicrographs by a computer graphical method (Ref 56) or a microcomputer-based system (Ref 60). A geometric method has also been described in which the three dimensional data are obtained by analytical geometry (Ref 61). Special stereoscopic or photogrammetric equipment or visual stereo effects are not required. The three-dimensional information is derived from three micrographs taken at different tilting angles, without regard to conditions for a good stereo effect. The x- and y-coordinates are measured from the micrographs, while the z-coordinate is calculated from a simple angle formula. A computer program in BASIC calculates the metric characteristics. In all these methods that utilize stereoscopic viewing or photography, the basic limitations are the inherent inefficiency of mapping areas with points, the inability to see into reentrances, and the difficulty of obtaining the fine detail required to characterize complex, irregular fracture surface. However, in some cases, the requirement of retaining the specimen in the original condition precludes all other considerations. Profile Generation The analysis of fracture surfaces by means of profiles appears more amenable to quantitative treatment than by other methods. Profiles are essentially linear in nature, as opposed to the two-dimensional sampling of SEM pictures and the point sampling using photogrammetry procedures. Several types of profiles can be generated, either directly or indirectly, but this article will discuss only three categories of profiles. Those selected are profiles obtained by metallographic sectioning, by nondestructive methods, and by sectioning of fracture surface replicas. Metallographic Sectioning Methods. Although many kinds of sections have been investigated for example, vertical (Ref 16, 62, 63), slanted (Ref 64, 65, 66), horizontal (Ref 22, 47, 49, 67), and conical (Ref 20, 64, 68) this discussion will be confined to planar sections. The major experimental advantages of planar sections are that they are obtained in ordinary matallographic mounts and that any degree of complexity or overlap of the fracture profile is accurately reproduced. Moreover, serial sectioning (Ref 18, 45, 47) is quite simple and direct (Fig. 9). In addition, planar sections reveal the underlying microstructure and its relation to the fracture surface (Ref 16, 62, 63), the standard equations of stereology are rigorously applicable on the flat section (Ref 33), and the angular characteristics of the fracture profile (Ref 36) can be mathematically related to those of the surface facets (Ref 39). Fig. 9 Profiles obtained by serial sectioning of a fractured Al-4Cu alloy Some objections to planar sectioning have been raised on the grounds that it is destructive of the fracture surface and sample. Also, the fracture surface must be coated with a protective layer before sectioning to preserve the trace. Moreover, in retrospect, additional detailed scrutiny of special areas on the fracture surface is not possible once it is coated and cut. However, bearing these objections in mind, if the fracture surface is carefully inspected by SEM in advance, areas of interest can be photographed before coating and cutting take place (Ref 69, 70). The preliminary experimental procedures are straightforward. When the specimen is fractured, two (ostensibly matching) nonplanar surfaces are produced. After inspection and photography by SEM, one or both surfaces can be electrolytically coated to preserve the edge upon subsequent sectioning (Ref 71). The coated specimens is then mounted metallographically and prepared according to conventional metallographic procedures (Ref 71). One fracture surface can be sectioned in one direction, and the other at 90 ° to the first direction, if desired. Once the profile is clearly revealed and the microstructure underlying the crack path properly polished and etched, the measurements can begin. Photographs of the trace or microstructure can be taken in the conventional manner for subsequent stereological measurements. 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