Review of fractal analysis of fracture surfaces in various materials using three-dimensional images reconstructed by stereo matching method

Abstract

The fractal analysis of fracture surfaces using the three-dimensional images reconstructed by the computer-aided stereo matching method was reviewed in various materials. The fractal dimension of the fracture surface was estimated on 15 kinds of fracture surfaces such as fatigue fracture surfaces, creep fracture surfaces and impact fracture surfaces of metallic materials or ceramics, with fracture surface patterns including dimples, striations or grain-boundary facets. The fractal dimension of the fracture surface was compared to that of the fracture surface profile or the fracture surface contour on the reconstructed fracture surface images. The results of the fractal analysis seemed to indicate that the reconstructed material fracture surface images are generally not completely isotropic. When estimated in the length scale range that is associated with the size ranges of the principal fracture surface patterns on 10 kinds of fracture surfaces, the results of the three-dimensional fractal analysis were well correlated to those of the two-dimensional fractal analysis of the actual fracture surface profile of metallic materials or the indentation crack of ceramics. It was confirmed that the computer-aided stereo matching method can reproduce the principal fracture surface topography to give important information for investigation of material fracture, except subsurface cracks and overlaps of the fracture surface. For detailed geometrical analysis of material fracture surfaces, it is suitable to conduct the fractal analysis in correlation with the size range of the principal fracture surface patterns.

Keywords

Fractal dimension fracture surfaces fracture surface topography stereo matching method metallic materials ceramics

Nomenclature

The box size (the length scale for two-dimensional or three-dimensional fractal analysis)

The number of boxes covering a fracture surface

The number of boxes covering a fracture surface profile or a fracture surface contour

The fractal dimension of the fracture surface (in the three-dimensional space) estimated on the reconstructed fracture surface image by the stereo matching method

D ^′

The value of D estimated in the length scale (r) range that is associated with the size range of the principal fracture surface patterns

D _p

The fractal dimension of a fracture surface profile estimated on the reconstructed fracture surface image by the stereo matching method

D _p

The averaged value of D_p

D _p ^′

The value of D_p estimated in the length scale (r) range that is associated with the size range of the principal fracture surface patterns

D _p ^′

The averaged value of D_p^′

D _c

The fractal dimension of a fracture surface contour estimated on the reconstructed fracture surface image by the stereo matching method

D _c

The averaged value of D_c

D _c ^′

The value of D_c estimated in the length scale (r) range that is associated with the size range of the principal fracture surface patterns

D _f

The fractal dimension of an actual fracture surface profile estimated in the length scale (r) range that is associated with the size range of the principal fracture surface patterns

D _f

The averaged value of D_f or the fractal dimension of the indentation crack estimated in the length scale (r) range that is associated with the size range of the principal fracture surface patterns

Δ𝜀_t

The maximum total strain range at the specimen surface in fatigue by repeated bending

Averaged grain diameter

d ^′

Grain diameter

d _max

The maximum grain diameter

1 GBL

One grain-boundary length

1. Introduction

First of all, the authors would like to express special thanks to the respected scientist, Dr. Takeo Yokobori, Professor Emeritus of Tohoku University, for his leadership and outstanding contribution in the academic field of strength and fracture of materials. His famous monograph entitled “Strength of Materials (Zairyo-kyodo-gaku)” (Iwanami Zensho, 1964, Tokyo) has become the bible for research scientists in this field, and has been inspiring one of the authors (M. T.) from days of graduate school. He has also let the authors turn their eyes to the field of materials fracture, and led them to another respected scientist, B.B. Mandelbrot, the founder of fractal geometry.

B.B. Mandelbrot et al. [11] first applied fractal geometry to the quantitative description of impact fracture surfaces in steels. From a phenomenological point of view, the fracture of materials is considered to be a kind of self-organized pattern formation processes. In a statistical sense, the fractal dimension represents self-similarity and complex nature of fracture surface patterns such as dimples, striations and grain-boundary facets which are formed by different fracture mechanisms, and depends on the length scale range of the fractal analysis [2,5,19,20]. Experimental methods such as the vertical section method (VSM) [3,4,14,32] and the slit island method (SIM) [11,13,14] were proposed for the estimation of the fractal dimension of microstructures and fracture surfaces in materials. However, fracture surface samples are damaged by cutting and polishing in these methods. As a non-destructive method for preservation of fracture surface samples, a computer-aided stereo matching method using scanning electron micrographs (a stereo pair) has been developed for the reconstruction and analysis of three-dimensional fracture surfaces in materials [6,7,15,16,25,26]. The stereo matching method generates the height data of fracture surfaces such as the elevation maps and fracture surface profiles necessary for the fracture surface topography analysis (FRASTA) [6,15,16] and the fractal dimension map (FDM) [22,27,29] for investigation of fracture process or fracture mechanism in materials.

The fractal dimension of a surface of an object in the three-dimensional space can be estimated by adding unity to the fractal dimension of its cross-section from general properties of fractal set [18]. However, there is no systematic study on the relationship between the fractal dimension of the fracture surface in the three-dimensional space and the fractal dimension of the fracture surface profile or the fracture surface contour in materials. It is also important to confirm the reproducibility of fracture surface topography by the computer-aided stereo matching method. The fractal dimension is considered to be a suitable parameter to examine quantitatively the results of the three-dimensional image reconstruction of fracture surfaces by the computer-aided stereo matching method, because the fractal dimension has been used as an important descriptor for characterizing various kinds of geometrical features of fracture surfaces in materials [2–5,8,11,13,14,18–20,22,26,27,29,32]. Further, the box-counting method seems to be advantageous for the fractal analysis using a computer [9,17].

In this review, the fractal dimension of the fracture surface in the three-dimensional space (the fractal dimension of the fracture surface) was estimated on 15 kinds of fracture surfaces such as fatigue fracture surfaces, creep fracture surfaces and impact fracture surfaces of metallic materials or ceramics, with fracture surface patterns including dimples, striations or grain-boundary facets. The computer-aided stereo matching method that was developed in the previous study [25] was used for the reconstruction of the three-dimensional fracture surface images for fractal analysis. Computer-aided box-counting methods were also employed for the two-dimensional and three-dimensional fractal analyses of the fracture surfaces [23,24]. The fractal dimension of the fracture surface was compared to that of the fracture surface profile or the fracture surface contour on the reconstructed material fracture surface images. When estimated in the length scale range that was associated with the size range of the principal fracture surface patterns on 10 kinds of the reconstructed fracture surface images, the fractal dimension of the fracture surface was compared to that of the actual fracture surface profile in the separate specimens, the published data in metallic materials [19,20,26,28] or the fractal dimension of the indentation crack in ceramics [21,26]. Then, discussion was made on the reproducibility of the fracture surface topography by the computer-aided stereo matching method on various fracture surfaces of materials.

Table 1
Test condition, grain size and principal fracture surface patterns on the analyzed fracture surfaces

Material (type of fracture surface) Test condition Averaged grain diameter (d), 1 GBL, grain diameter (d^′) or maximum grain diameter (d_max) Principal fracture surface patterns (size range)

Cu–Be alloy (stage I Slip steps and dimples

fatigue fracture surface) Fatigued by repeated bending d = 2.4 × 10⁻⁵ m (r ≤ 1 GBL)

Cu–Be alloy (stage II (Δ𝜀_t = 0.0171) 1 GBL = 1.4 × 10⁻⁵ m GB facets and fine dimples

fatigue fracture surface) (r ≤ 1 GBL)

SUS316 steel (stage I Slip steps and ledges

fatigue fracture surface) Fatigued by repeated bending d = 1.3 × 10⁻⁵ m (r ≤ 1 GBL)

SUS316 steel (stage II (Δ𝜀_t = 0.0169) 1 GBL = 7.5 × 10⁻⁶ m Striations and slip steps

fatigue fracture surface) (r ≤ 1 GBL)

SUS631 steel (fatigue Fatigued by repeated bending d = 2.50 × 10⁻⁵ m Small steps and microcracks

fracture surface) (Δ𝜀_t = 0.0162) 1 GBL = 1.44 × 10⁻⁵ m (r ≤ 1 GBL)

21Cr steel (fatigue Fatigued by repeated bending d = 1.00 × 10⁻⁵ m Large shear steps and

fracture surface) (Δ𝜀_t = 0.0127) 1 GBL = 1. 74 × 10⁻⁵ m striations (r > 1 GBL)

Pure Zn (creep Creep-ruptured under a stress d = 1.2 × 10⁻⁵ m Dimples (r ≤ 1 GBL)

fracture surface) of 14.7 MPa at 373 K 1 GBL = 6.9 × 10⁻⁶

SUS430 steel (creep Creep-ruptured under a stress d = 2.5 × 10⁻⁵ m Dimples and GB facets

fracture surface) of 24.5 MPa at 973 K 1 GBL = 1.4 × 10⁻⁵ m (r ≤ 1 GBL)

HS–21 alloy (creep Microcracks on straight GBs

fracture surface) Creep-ruptured under a stress d = 1.30 × 10⁻⁴ m (r > 1 GBL)

HS–21 alloy (creep of 108 MPa at 1089 K 1 GBL = 7.73 × 10⁻⁵ m Microcracks on serrated GBs

fracture surface) (r > 1 GBL)

SS400 steel (impact Impact-loaded and fractured d = 2.48 × 10⁻⁵ m River patterns and steps

fracture surface) at 194 K (Charpy test) 1 GBL = 1.43 × 10⁻⁵ m (r ≤ 1 GBL)

SS400 steel (impact Impact-loaded and fractured Dimples (r ≤ 1 GBL)

fracture surface) at 273 K (Charpy test)

Silicon carbide (impact Impact-loaded and fractured 3.7 × 10⁻⁷ m ≤ d^′ ≤ 8.6 Brittle–ductile mixed-type

fracture surface) at RT × 10⁻⁵ m fracture surface (r ≤ d_max)

d_max = 8.6 × 10⁻⁵ m

Alumina (impact Impact-loaded and fractured 6.4 × 10⁻⁷ m ≤ d^′ ≤ 9.2 Brittle fracture surface with

fracture surface) at RT × 10⁻⁶ m micropores (r ≤ d_max)

d_max = 9.2 × 10⁻⁶ m

Mullite (impact Impact-loaded and fractured 2.73 × 10⁻⁶ m ≤ Brittle fracture surface with

fracture surface) at RT d^′ ≤ 2.35 × 10⁻⁴ m micropores (r ≤ d_max)

d_max = 2.35 × 10⁻⁴ m

Material (type of fracture surface)	Test condition	Averaged grain diameter (d), 1 GBL, grain diameter (d^′) or maximum grain diameter (d_max)	Principal fracture surface patterns (size range)
Cu–Be alloy (stage I			Slip steps and dimples
fatigue fracture surface)	Fatigued by repeated bending	d = 2.4 × 10⁻⁵ m	(r ≤ 1 GBL)
Cu–Be alloy (stage II	(Δ𝜀_t = 0.0171)	1 GBL = 1.4 × 10⁻⁵ m	GB facets and fine dimples
fatigue fracture surface)			(r ≤ 1 GBL)
SUS316 steel (stage I			Slip steps and ledges
fatigue fracture surface)	Fatigued by repeated bending	d = 1.3 × 10⁻⁵ m	(r ≤ 1 GBL)
SUS316 steel (stage II	(Δ𝜀_t = 0.0169)	1 GBL = 7.5 × 10⁻⁶ m	Striations and slip steps
fatigue fracture surface)			(r ≤ 1 GBL)
SUS631 steel (fatigue	Fatigued by repeated bending	d = 2.50 × 10⁻⁵ m	Small steps and microcracks
fracture surface)	(Δ𝜀_t = 0.0162)	1 GBL = 1.44 × 10⁻⁵ m	(r ≤ 1 GBL)
21Cr steel (fatigue	Fatigued by repeated bending	d = 1.00 × 10⁻⁵ m	Large shear steps and
fracture surface)	(Δ𝜀_t = 0.0127)	1 GBL = 1. 74 × 10⁻⁵ m	striations (r > 1 GBL)
Pure Zn (creep	Creep-ruptured under a stress	d = 1.2 × 10⁻⁵ m	Dimples (r ≤ 1 GBL)
fracture surface)	of 14.7 MPa at 373 K	1 GBL = 6.9 × 10⁻⁶
SUS430 steel (creep	Creep-ruptured under a stress	d = 2.5 × 10⁻⁵ m	Dimples and GB facets
fracture surface)	of 24.5 MPa at 973 K	1 GBL = 1.4 × 10⁻⁵ m	(r ≤ 1 GBL)
HS–21 alloy (creep			Microcracks on straight GBs
fracture surface)	Creep-ruptured under a stress	d = 1.30 × 10⁻⁴ m	(r > 1 GBL)
HS–21 alloy (creep	of 108 MPa at 1089 K	1 GBL = 7.73 × 10⁻⁵ m	Microcracks on serrated GBs
fracture surface)			(r > 1 GBL)
SS400 steel (impact	Impact-loaded and fractured	d = 2.48 × 10⁻⁵ m	River patterns and steps
fracture surface)	at 194 K (Charpy test)	1 GBL = 1.43 × 10⁻⁵ m	(r ≤ 1 GBL)
SS400 steel (impact	Impact-loaded and fractured		Dimples (r ≤ 1 GBL)
fracture surface)	at 273 K (Charpy test)
Silicon carbide (impact	Impact-loaded and fractured	3.7 × 10⁻⁷ m ≤ d^′ ≤ 8.6	Brittle–ductile mixed-type
fracture surface)	at RT	× 10⁻⁵ m	fracture surface (r ≤ d_max)
		d_max = 8.6 × 10⁻⁵ m
Alumina (impact	Impact-loaded and fractured	6.4 × 10⁻⁷ m ≤ d^′ ≤ 9.2	Brittle fracture surface with
fracture surface)	at RT	× 10⁻⁶ m	micropores (r ≤ d_max)
		d_max = 9.2 × 10⁻⁶ m
Mullite (impact	Impact-loaded and fractured	2.73 × 10⁻⁶ m ≤	Brittle fracture surface with
fracture surface)	at RT	d^′ ≤ 2.35 × 10⁻⁴ m	micropores (r ≤ d_max)
		d_max = 2.35 × 10⁻⁴ m

r: the length scale of the fractal analysis; 1 GBL: one grain-boundary length (≈0.577d); GB: grain boundary; Δ𝜀_t: the maximum total strain range at the specimen surface; RT: room temperature.

2. Materials and experimental methods

The fractal dimension of fracture surfaces was estimated on 15 kinds of fracture surfaces (8 kinds of metallic materials and 3 kinds of ceramics) in this study. Table 1 lists the test condition, grain size and principal fracture surface patterns on the analyzed fracture surfaces. Fracture surfaces chosen for fractal analysis were fatigue fracture surfaces of a Cu–Be alloy [26], the SUS 316 (AISI 316) steel [28], the SUS 631 (17–7PH) steel [22] and a 21Cr steel [22], creep fracture surface of a pure Zn (99.99 mass %Zn) [26], the SUS 430 (AISI 430) steel and the cobalt-base HS–21 alloy [19,20], and impact-loaded and fractured surfaces of the SS400 structural steel [30], a silicon carbide (SiC, Norton NC–430) [21,26], an alumina (Al₂O₃, SSA–H) [21,26] and a mullite [21]. The details of the materials, specimen preparations and test conditions are described in the references. The principal fracture surface patterns have a certain size range in the fracture surfaces of materials, which were formed by different fracture mechanisms (Table 1). For example, the principal fracture surface patterns are slip steps and dimples within the grain (r ≤ 1 GBL, where 1 GBL is one grain-boundary length) on the stage I fatigue fracture surface in the specimen of the Cu–Be alloy fatigued by bending [26], while the principal fracture surface patterns are microcracks with the size of more than one grain-boundary length (r > 1 GBL) on the creep-fracture surfaces of the HS–21 alloy [19,20].

Photographs of stereo pairs (the basic image and the tilted image by 10 degrees) for the computer-aided stereo matching method were taken on the fracture surfaces of materials using a scanning electron microscope (SEM) at the magnifications of 30 to 3000 times. For additional two-dimensional fractal analysis, some fracture surface profiles were obtained by the VSM for separate specimens on the stage I fatigue fracture surface of the Cu–Be alloy, and the creep fracture surfaces of a pure Zn and the SUS430 steel [3,4,14,32]. Photographs were taken using an optical microscope on the fracture surface profiles of these specimens at the magnification of 400 or 200 times.

3. Analytical methods

Photographs of stereo pairs on the fracture surfaces were then taken into a computer and were converted to the digital images of 256 grey scale levels. The three-dimensional image reconstruction was carried out using stereo pair images (SEM photographs) to obtain the elevation map (height data), fracture surface profiles and fracture surface contours by the computer-aided stereo matching method of coarse-to-fine scheme (pyramid scheme) that had been developed in the previous study [25]. Table 2 lists the size of the area reconstructed by stereo matching method and the size of the central square region used for the fractal analysis of fracture surfaces in materials. The real length corresponding to 1 pixel is also shown for each fracture surface in the table. The fractal dimension of the fracture surface in the three-dimensional space (the fractal dimension of the fracture surface), D, was estimated for the central square region with the size of 400 × 400 to 720 × 720 in pixel of the elevation map obtained by the computer-aided box-counting method [24]. For the three-dimensional fractal analysis, the number of boxes, N, covering a fracture surface can be related to the box size (the length scale of the fractal analysis), r, through the value of D by the following power law relationship: $\begin{eqnarray}\displaystyle N=kr^{-D}\quad \text{or}\quad \log N=\log k-D\log r & & \displaystyle\end{eqnarray}$ (1) where k is a constant. For the two-dimensional fractal analysis using another computer program of the box-counting method [23], the number of boxes covering a fracture surface profile or a fracture surface contours, n, is correlated to the box size, r, through the fractal dimensions, D_i (i = p for the fractal dimension of a fracture surface profile and i = c for that of a fracture surface contour) by similar power law relationship: $\begin{eqnarray}\displaystyle n=k^{\prime }r^{-Di}\quad \text{or}\quad \log n=\log k^{\prime }-D_{i}\log r & & \displaystyle\end{eqnarray}$ (2) where k^′ is a constant. The values of the fractal dimension were calculated from Eq. (1) or Eq. (2) by the regression analysis using the datum sets of N and r, or n and r. The minimum value of r was set to be 2 pixels in the present fractal analysis.

Table 2

Size of the area reconstructed by the stereo matching method and the size of the central square region used for the fractal analysis of fracture surfaces in materials

Material (type of fracture surface)	Size of reconstructed area (pixel)	Size of analyzed region (pixel)	Real length corresponding to 1 pixel
Cu–Be alloy (stage I fatigue fracture surface)	403 × 467	400 × 400	1.81 × 10⁻⁷ m
Cu–Be alloy (stage II fatigue fracture surface)	663 × 601	600 × 600	1.17 × 10⁻⁷ m
SUS316 steel (stage I fatigue fracture surface)	765 × 615	600 × 600	1.15 × 10⁻⁷ m
SUS316 steel (stage II fatigue fracture surface)
SUS631 steel (fatigue fracture surface)	775 × 681	660 × 660	7.81 × 10⁻⁷ m
21Cr steel (fatigue fracture surface)	775 × 681	660 × 660	1.18 × 10⁻⁶ m
Pure Zn (creep fracture surface)	817 × 739	720 × 720	1.46 × 10⁻⁷ m
SUS430 steel (creep fracture surface)	775 × 681	660 × 660	3.81 × 10⁻⁸ m
HS–21 alloy with straight GBs (creep fracture surface)	779 × 675	660 × 660	4.22 × 10⁻⁶ m
HS–21 alloy with serrated GBs (creep fracture surface)			4.34 × 10⁻⁶ m
SS400 steel (inpact fracture surface at 194 K)	817 × 739	720 × 720	3.95 × 10⁻⁸ m
SS400 steel (inpact fracture surface at 273 K)			1.17 × 10⁻⁷ m
Silicon carbide (inpact fracture surface at RT)	663 × 601	600 × 600	1.12 × 10⁻⁷ m
Alumina (inpact fracture surface at RT)	575 × 513	480 × 480	1.48 × 10⁻⁷ m
Mullite (inpact fracture surface at RT)	637 × 607	600 × 600	4.66 × 10⁻⁷ m

GBs: grain boundaries; Δ𝜀_t: the maximum total strain range at the specimen surface; RT: room temperature.

Totally 14 to 20 fracture surface profiles in the horizontal or vertical direction were analyzed for each fracture surface in this study. The values of the fractal dimension of a fracture surface profile, D_p, were estimated in the horizontal and vertical directions, and were averaged for each direction. The mean value of these two averaged values was defined as the fractal dimension of the fracture surface profile and was denoted by D _p , because there was almost no difference in the averaged values of the fractal dimension between horizontal and vertical directions in the fatigue fracture surfaces of the Cu–Be alloy [26] and the SUS316 steel [28]. Further, totally 14 to 20 fracture surface contours were analyzed for each fracture surface. The averaged value of the fractal dimension of a fracture surface contour (D_c) was defined as the fractal dimension of the fracture surface contour and was denoted by D _c .

The principal fracture surface patterns that were formed by different fracture mechanisms have a certain size range (Table 1). Therefore, the value of the fractal dimension was also estimated in the length scale (r) range of the fractal analysis that was associated with the size range of the principal fracture surface patterns. When estimated in this length scale range, the fractal dimension of the fracture surface was shown as D^′, and the fractal dimensions of a fracture surface profile and that of a fracture surface contour were denoted by $D_{p}^{\prime }$ and $D_{c}^{\prime }$ , respectively. The averaged value of $D_{p}^{\prime }$ was defined as the fractal dimension of the fracture surface profile, D _p ^′. The fractal dimension of an actual fracture surface profile estimated in this length scale range was denoted by D_f, and the averaged value of D_f or that of the indentation crack that was estimated in this length scale range was denoted by D _f in this study. The values of D_f and D _f were also estimated by the box-counting method [9,17,23].

4. Results and discussion

4.1. Reconstructed fracture surface images for fractal analysis

As a result of the image reconstruction of material fracture surfaces by the computer-aided stereo matching method, Fig. 1 shows the stage I fatigue fracture surface of the Cu–Be alloy analyzed in this study (the maximum total strain range at the specimen surface, Δ𝜀_t, is 0.0171). The fatigue crack has grown from right to left in the figure (i.e., the crack growth direction is horizontal). Figure 1(a) is the basic image with the reconstructed area (enclosed by white lines, 403 × 467 in pixel). The fracture surface is covered with slip steps and dimples (Table 1), while flat grain-boundary surface is partly observed in the central region. Figure 1(b) is the elevation map obtained by stereo matching method (the central region chosen for three-dimensional fractal analysis is enclosed by white lines and the size is 400 × 400 in pixel in this case). The higher region in the fracture surface is shown brighter in the elevation map (Fig. 1(b)). The fractal dimension of the fracture surface estimated in this review is not always strictly the same as that obtained in the previous study [26,28,30] even on the identical fracture surface, because the size or location of the analyzed fracture surface region is slightly different. The horizontal white lines in Fig. 1(c) and vertical ones in Fig. 1(d) indicate the locations of fracture surface profiles selected for fractal analysis. The projected length of the fracture surface profile is the same as the side length of the elevation map (the reconstructed area) in horizontal or vertical direction (Table 2).

Fig. 1.

Stage I fatigue fracture surface of the Cu–Be alloy analyzed in this study (the maximum total strain range at the specimen surface, Δ𝜀_t, is 0.0171). The fatigue crack has grown from right to left in this figure. (a) is the basic image with the reconstructed area (enclosed by white lines, 403 × 467 in pixel) and (b) is the elevation map obtained by the stereo matching method (the central square region chosen for three-dimensional fractal analysis is enclosed by white lines and the size is 400 × 400 in pixel in this case). Horizontal white lines in (c) and vertical ones in (d) indicate the locations of the reconstructed fracture surface profiles selected for two-dimensional fractal analysis.

Figure 2 shows the representative fracture surface profiles and fracture surface contours reconstructed by the stereo matching method on the stage I fatigue fracture surface of the Cu–Be alloy (Fig. 1). Figures 2(a) and 2(b) are examples of the fracture surface profiles in the horizontal (No. 4) and vertical (No. 3) directions, respectively. The fracture surface contours (indicated by white lines in Fig. 2(c)) used for the estimation of the fractal dimension of a fracture surface contour, D_c. The height of a fracture surface contour decreases with increasing number from 1 to 14 in Fig. 2(c). There are “small islands” adjacent to a fracture surface contour that are at the same level (the same altitude). These small islands are also included in the fractal analysis in this study.

Fig. 2.

Representative fracture surface profiles and fracture surface contours reconstructed by the stereo matching method on the stage I fatigue fracture surface of the Cu-Be alloy (Fig. 1). (a) is a fracture surface profile in the horizontal direction (No. 4 in Fig. 1(c)), (b) is that in the vertical direction (No. 3 in Fig. 1(d)) and (c) is the fracture surface contours related to Fig. 1(b). Small islands adjacent to a fracture surface contour are also visible in (c).

4.2. Fractal analysis of reconstructed fracture surface images

Figure 3 shows the fractal dimension of the fracture surface (D or D^′) estimated on the elevation map of the stage I fatigue fracture surface image of the Cu–Be alloy reconstructed by the stereo matching method (the analyzed region is shown in Fig. 1(b)). The value of D was estimated in the length scale (r) range from 2 to 400 pixels, while the value of D^′ was calculated in the length scale range (2 to 50 pixels) that is associated with the size range of the principal fracture surface patterns (slip steps and dimples) on the fracture surface (Table 1). The value of D^′ (2.198) is almost the same as the value of D (2.184) in this case. Figure 4 shows the examples of the fractal dimension (D_p or $D_{p}^{\prime }$ ) estimated on the reconstructed fracture surface profiles shown in Figs 2(a) and 2(b). The fractal dimension of the fracture surface profile (D_p) is slightly smaller for the No. 4 profile (1.140) in the plane in parallel with horizontal direction compared to the No. 3 profile (1.196) in the plane in parallel with vertical direction in this case when estimated in the length scale range from 2 to 134 pixels. However, the values of $D_{p}^{\prime }$ are similar (1.163 and 1.188) for both fracture surface profiles when estimated in the length scale range that is associated with the size range of the principal fracture surface patterns (from 2 to 68 pixels). Figure 5 shows the examples of the fractal dimension (D_c or $D_{c}^{\prime }$ ) estimated on the reconstructed fracture surface contours shown in Fig. 2(c). When estimated in the length scale range from 2 to 118 pixels, the values of D_c for the Nos. 3, 6 and 12 fracture surface contours (from 1.190 to 1.270) are slightly larger the values of D_p. The values of $D_{c}^{\prime }$ (from 1.207 to 1.238), which are estimated in the length scale range associated with the size range of the principal fracture surface patterns (from 2 to 68 pixels), are also slightly larger than the value of $D_{p}^{\prime }$ .

Fig. 3.

The fractal dimension of the fracture surface (D or D^′) estimated on the elevation map of the stage I fatigue fracture surface image (Fig. 1(b)) of the Cu–Be alloy reconstructed by the stereo matching method (the analyzed region is shown in Fig. 1(b)).

Fig. 4.

Examples of the fractal dimension (D_p or $D_{p}^{\prime }$ ) estimated on the reconstructed fracture surface profiles shown in Figs 2(a) and 2(b).

Fig. 5.

Examples of the fractal dimension (D_c or $D_{c}^{\prime }$ ) estimated on the reconstructed fracture surface contours shown in Fig. 2(c).

As already known, the fractal dimension of a surface of an object in the three-dimensional space can be obtained by adding unity to the fractal dimension of its cross-section from general properties of fractal sets [18]. Therefore, the fractal dimension of the fracture surface, D, was compared to the averaged value of the fractal dimension of a fracture surface profile, D _p , or the averaged value of the fractal dimension of a fracture surface contour, D _c , on 15 kinds of reconstructed fracture surface images. Figure 6 shows the relationship between the fractal dimension of the fracture surface (D) and the averaged value of the fractal dimension of the fracture surface profile ( D _p ) on the reconstructed fracture surface images of materials. The value of D lies in the range from 2.123 to 2.299 in these materials. The datum points are close to the relationship of D = D _p + 1, while the fractal dimension of the fracture surface calculated from this relationship with the value of D _p seems to be slightly smaller than the actual value of D on some fracture surfaces. Figure 7 shows the relationship between the fractal dimension of the fracture surface (D) and the averaged value of the fractal dimension of the fracture surface contour ( D _c ) on the reconstructed fracture surface images of materials. The datum points are close to similar relationship of D = D _c + 1, although the value of ( D _c + 1) gives slightly larger value compared to the actual D value in many fracture surfaces, and the value of D _c is slightly larger than that of D _p on all the reconstructed fracture surface images. As described in the Section 4.1, the present fractal analysis includes “small islands” adjacent to a fracture surface contour. B.B. Mandelbrot [10] pointed out that the fractal dimension of a coastline with small islands is larger than that of a coastline alone. This may be one of the reasons that the value of D _c is slightly larger than that of D _p in this study. These results of the fractal analysis seem to indicate that the reconstructed material fracture surface images are generally not completely isotropic.

Fig. 6.

Relationship between the fractal dimension of the fracture surface (D) and the averaged value of the fractal dimension of the fracture surface profile ( D _p ) on the reconstructed fracture surfaces of materials.

Fig. 7.

Relationship between the fractal dimension of the fracture surface (D) and the averaged value of the fractal dimension of the fracture surface contour ( D _c ) on the reconstructed fracture surfaces of materials.

4.3. Fractal dimensions of reconstructed fracture surfaces and actual fracture surface profiles

The fractal dimension represents the characteristic fracture surface patterns that have a certain size range [2,5,19,20]. Therefore, in order to compare the reconstructed fracture surfaces with the actual fracture surface profiles or the morphology of indentation cracks, it is appropriate to evaluate the fractal dimension in the length scale range of the fractal analysis that is associated with the size range of the principal fracture surface patterns such as dimples, striations and grain-boundary facets (Table 1). The fractal dimension of the fracture surface and that of the fracture surface profile, which were estimated on 10 kinds of fracture surface images reconstructed by the computer-aided stereo matching method, were compared to the fractal dimension of the fracture surface profile estimated on the actual fracture surfaces or that of the indentation crack in materials. Table 3 lists the comparison of the fractal dimension of the fracture surface (D^′) and that of the fracture surface profile ( D _p ^′) on the reconstructed fracture surface images with the fractal dimension of the actual fracture surface profile or the indentation crack ( D _f ) when estimated in the length scale range that is associated with the size range of the principal fracture surface patterns (Table 1). The value of D _f is the averaged value of the fractal dimension of the actual fracture surface profile (D_f) for metallic materials obtained in this study or the data cited from the previous study [19,20,26,28], and is the fractal dimension of the indentation crack for ceramics [21,26]. The value of D^′ lies in the range from 2.137 to 2.296 in these materials.

Table 3
Comparison of the fractal dimension of the fracture surface (D^′) and that of the fracture surface profile ( D _p ^′) on the reconstructed fracture surface images with the fractal dimension of the actual fracture surface profile or the indentation crack ( D _f ) when estimated in the length scale range that is associated with the size range of the principal fracture surface patterns (Table 1)

Material (type of fracture surface) Fracture surfaces reconstructed by stereo matching method Actual fracture surface profiles, D _f (length scale range of fractal analysis)

D^′ (length scale range of fractal analysis) Fracture surface profiles, D _p ^′ (length scale range of fractal analysis)

Cu–Be alloy (stage I fatigue fracture surface) 2.198 (3.62 × 10⁻⁷ m to 9.05 × 10⁻⁶ m) 1.171 (3.62 × 10⁻⁷ m to 1.23 × 10⁻⁵ m) 1.200^(a) (6.7 × 10⁻⁷ m to 1.7 × 10⁻⁵ m)

Cu–Be alloy (stage II fatigue fracture surface) 2.167 (2.34 × 10⁻⁷ m to 1.17 × 10⁻⁵ m) 1.114 (2.34 × 10⁻⁷ m to 1.17 × 10⁻⁵ m) 1.168^(a) (6.7 × 10⁻⁷ m to 1.7 × 10⁻⁵ m)

SUS316 steel (stage I fatigue fracture surface) 2.286 (2.31 × 10⁻⁷ m to 6.94 × 10⁻⁶ m) 1.218 (2.31 × 10⁻⁷ m to 6.48 × 10⁻⁶ m) 1.230^(b) (6.4 × 10⁻⁷ m to 8.0 × 10⁻⁶ m)

SUS316 steel (stage II fatigue fracture surface) 2.249 (2.31 × 10⁻⁷ m to 6.94 × 10⁻⁶ m) 1.199 (2.31 × 10⁻⁷ m to 6.48 × 10⁻⁶ m) 1.240^(b) (6.4 × 10⁻⁷ m to 8.0 × 10⁻⁶ m)

Pure Zn (creep fracture surface) 2.162 (2.92 × 10⁻⁷ m to 6.58 × 10⁻⁶ m) 1.135 (2.92 × 10⁻⁷ m to 6.43 × 10⁻⁶ m) 1.179 (4.5 × 10⁻⁷ m to 6.8 × 10⁻⁶ m)

SUS430 steel (creep fracture surface) 2.196 (7.62 × 10⁻⁸ m to 1.25 × 10⁻⁵ m) 1.172 (7.62 × 10⁻⁸ m to 6.47 × 10⁻⁶ m) 1.201 (8.2 × 10⁻⁷ m to 1.06 × 10⁻⁵ m)

HS–21 alloy with straight GBs (creep fracture surface) 2.173 (8.44 × 10⁻⁵ m to 2.78 × 10⁻³ m) 1.232 (9.29 × 10⁻⁵ m to 8.19 × 10⁻⁴ m) 1.209^(c) (8.90 × 10⁻⁵ m to 4.45 × 10⁻³ m)

HS–21 alloy with serrated GBs (creep fracture surface) 2.296 (8.68 × 10⁻⁵ m to 2.86 × 10⁻³ m) 1.279 (9.54 × 10⁻⁵ m to 8.42 × 10⁻⁴ m) 1.245^(c) (8.90 × 10⁻⁵ m to 4.45 × 10⁻³ m)

Silicon carbide (inpact fracture surface at RT) 2.141 (2.25 × 10⁻⁷ m to 6.77 × 10⁻⁵ m) 1.142 (2.25 × 10⁻⁷ m to 1.87 × 10⁻⁵ m) 1.16^(d) (2.7 × 10⁻⁷ m to 5.5 × 10⁻⁶ m)

Alumina (inpact fracture surface at RT) 2.137 (2.97 × 10⁻⁷ m to 8.92 × 10⁻⁶ m) 1.127 (2.97 × 10⁻⁷ m to 7.44 × 10⁻⁶ m) 1.15–1.19^(e) (2.7 × 10⁻⁸ m to 1.1 × 10⁻⁵ m)

Material (type of fracture surface)	Fracture surfaces reconstructed by stereo matching method	Actual fracture surface profiles, D _f (length scale range of fractal analysis)
Cu–Be alloy (stage I fatigue fracture surface)	2.198 (3.62 × 10⁻⁷ m to 9.05 × 10⁻⁶ m)	1.171 (3.62 × 10⁻⁷ m to 1.23 × 10⁻⁵ m)	1.200^(a) (6.7 × 10⁻⁷ m to 1.7 × 10⁻⁵ m)
Cu–Be alloy (stage II fatigue fracture surface)	2.167 (2.34 × 10⁻⁷ m to 1.17 × 10⁻⁵ m)	1.114 (2.34 × 10⁻⁷ m to 1.17 × 10⁻⁵ m)	1.168^(a) (6.7 × 10⁻⁷ m to 1.7 × 10⁻⁵ m)
SUS316 steel (stage I fatigue fracture surface)	2.286 (2.31 × 10⁻⁷ m to 6.94 × 10⁻⁶ m)	1.218 (2.31 × 10⁻⁷ m to 6.48 × 10⁻⁶ m)	1.230^(b) (6.4 × 10⁻⁷ m to 8.0 × 10⁻⁶ m)
SUS316 steel (stage II fatigue fracture surface)	2.249 (2.31 × 10⁻⁷ m to 6.94 × 10⁻⁶ m)	1.199 (2.31 × 10⁻⁷ m to 6.48 × 10⁻⁶ m)	1.240^(b) (6.4 × 10⁻⁷ m to 8.0 × 10⁻⁶ m)
Pure Zn (creep fracture surface)	2.162 (2.92 × 10⁻⁷ m to 6.58 × 10⁻⁶ m)	1.135 (2.92 × 10⁻⁷ m to 6.43 × 10⁻⁶ m)	1.179 (4.5 × 10⁻⁷ m to 6.8 × 10⁻⁶ m)
SUS430 steel (creep fracture surface)	2.196 (7.62 × 10⁻⁸ m to 1.25 × 10⁻⁵ m)	1.172 (7.62 × 10⁻⁸ m to 6.47 × 10⁻⁶ m)	1.201 (8.2 × 10⁻⁷ m to 1.06 × 10⁻⁵ m)
HS–21 alloy with straight GBs (creep fracture surface)	2.173 (8.44 × 10⁻⁵ m to 2.78 × 10⁻³ m)	1.232 (9.29 × 10⁻⁵ m to 8.19 × 10⁻⁴ m)	1.209^(c) (8.90 × 10⁻⁵ m to 4.45 × 10⁻³ m)
HS–21 alloy with serrated GBs (creep fracture surface)	2.296 (8.68 × 10⁻⁵ m to 2.86 × 10⁻³ m)	1.279 (9.54 × 10⁻⁵ m to 8.42 × 10⁻⁴ m)	1.245^(c) (8.90 × 10⁻⁵ m to 4.45 × 10⁻³ m)
Silicon carbide (inpact fracture surface at RT)	2.141 (2.25 × 10⁻⁷ m to 6.77 × 10⁻⁵ m)	1.142 (2.25 × 10⁻⁷ m to 1.87 × 10⁻⁵ m)	1.16^(d) (2.7 × 10⁻⁷ m to 5.5 × 10⁻⁶ m)
Alumina (inpact fracture surface at RT)	2.137 (2.97 × 10⁻⁷ m to 8.92 × 10⁻⁶ m)	1.127 (2.97 × 10⁻⁷ m to 7.44 × 10⁻⁶ m)	1.15–1.19^(e) (2.7 × 10⁻⁸ m to 1.1 × 10⁻⁵ m)

GBs: grain boundaries; RT: room temperature; ^(a): the value cited from reference [26]; ^(b): the value cited from reference [28]; ^(c): the value cited from references [19,20]; ^(d): the value of the indentation crack [21,26]; ^(e): the value of the indentation crack in similar alumina (SSA–999W, SSA–999H and SSA–S) [21,26].

Figure 8 shows the examples of actual fracture surface profiles in materials. Rugged parts on the fracture surface profile seem to correspond to slip steps or dimples on the stage I fatigue fracture surface of the Cu–Be alloy in Figs 8(a) and 8(b), while the straight part of the fracture surface profile that is observed on the left side of Fig. 8(b) may be a grain-boundary facet. Convexes and concaves on the fracture surface profile seem to represent dimple walls on the creep fracture surface of a pure Zn in Fig. 8(c). Similar features are visible on the creep fracture surface profile of the SUS430 steel in Fig. 8(d), while grain-boundary facets are not clear in the figure. There are subsurface cracks (pointed by arrows) and overlaps of the fracture surface with overhangs (enclosed by circles) on the actual fracture surface in Fig. 8. As pointed out in the previous study [4,4,25], subsurface cracks and overlaps of the fracture surface are only partly reproducible by the stereo matching method, and cannot be fully detected by any other observations using a scanning probe microscope (SPM) [1], an atom force microscope (AFM) [12] or a scanning laser microscope [31]. Figure 9 shows the fractal dimension of an actual fracture surface profile (D_f) in Fig. 8 when estimated in the length scale range that is associated with the size range of the principal fracture surface patterns. In the stage I fatigue fracture surface of the Cu–Be alloy, the fractal dimension of an actual fracture surface profile (D_f) is almost the same in the plane in parallel with the crack growth direction (1.197) and in that normal to the crack growth direction (1.181). These values of D_f are similar to the values of $D_{p}^{\prime }$ estimated on the reconstructed fracture surface image (1.163 and 1.188 in Fig. 4) or the value of D _p ^′ (1.171, the averaged value of $D_{p}^{\prime }$ ) (Table 3). However, the values of D_f (1.184 for a pure Zn and 1.206 for the SUS430 steel) are slightly larger than the corresponding values of D _p ^′ estimated on the reconstructed fracture surface images (1.135 for a pure Zn and 1.172 for the SUS430 steel) on the creep fracture surfaces.

Fig. 8.

Examples of the actual fracture surface profiles in materials. (a) is a fracture surface profile in the plane in parallel with and (b) is that in the plane normal to the fatigue crack growth direction on the stage I fatigue fracture surface in the Cu–Be alloy, (c) is a fracture surface profile in the creep-ruptured specimen of a pure Zn and (d) is a fracture surface profile in the creep-ruptured specimen of the SUS430 steel. Overhangs on the fracture surface are enclosed by circles and arrows indicate subsurface cracks.

Fig. 9.

The fractal dimension of an actual fracture surface profile (D_f) in Fig. 8 when estimated in the length scale range that is associated with the size range of the principal fracture surface patterns.

Figure 10 shows the relationship between the fractal dimension of the reconstructed fracture surface profile ( D _p ^′) and that of the actual fracture surface profile or the indentation crack ( D _f ) in materials when estimated in the length scale range that is associated with the size range of the principal fracture surface patterns. The value of D _f is in the range from 1.15 to 1.245 in these materials, while the value of D _p ^′ lies in the range from 1.114 to 1.279 (Table 3). The value of D _f (or D _p ^′) is slightly larger in the creep fracture surface of the specimen with serrated grain boundaries than in the one with straight grain boundaries in the HS–21 alloy, because the former specimen exhibited more complicated fracture surface appearance with more grain-boundary microcracks [19,20]. Similarly, the stage I fatigue fracture surface with slip steps and dimples shows a slightly larger value of D _f (or D _p ^′) compared to the stage II fatigue fracture surface with grain-boundary facets and fine dimples in the Cu–Be alloy [26]. However, the value of D _f (or D _p ^′) are much the same in the stage I fatigue fracture surface with slip steps and ledges and the stage II fatigue fracture surface with striations and slip steps in the SUS316 steel, because these fracture surface patterns are formed by the common mechanism (plastic deformation) [28]. The value of D _p ^′ is slightly smaller than the value of D _f on many material fracture surfaces except the creep fracture surfaces of the HS–21 alloy. Nevertheless, the value of D _p ^′ seems to be well correlated to the value of D _f in these materials. When estimated in the corresponding length scale range, a relationship similar to Fig. 10 was found between the averaged value of the fractal dimension of the fracture surface contour and the value of D _f , while the former value was slightly larger than the value of D _p ^′.

Fig. 10.

Relationship between the fractal dimension of the reconstructed fracture surface profile ( D _p ^′) and that of the actual fracture surface profile or the indentation crack ( D _f ) in materials when estimated in the length scale range that is associated with the size range of the principal fracture surface patterns.

Figure 11 shows the relationship between the fractal dimension of the fracture surface (D^′) (estimated on the reconstructed fracture surface images) and that of the actual fracture surface profile or the indentation crack ( D _f ) in materials when estimated in the length scale range that is associated with the size range of the principal fracture surface patterns. The datum points of the stage I fatigue fracture surface with slip steps and dimples (D^′ = 2.198) and the stage II fatigue fracture surface with grain-boundary facets and fine dimples (D^′ = 2.167) of the Cu–Be alloy, and the creep fracture surfaces with dimples and grain-boundary facets of the SUS430 steel (D^′ = 2.196) are very close to the relationship of D^′ = D _f + 1. The actual value of D^′ is slightly smaller than the value of ( D _f + 1) for the brittle fracture surface with micropores of an alumina (D^′ = 2.137) (Table 3), while the value of D^′ is slightly larger for the creep fracture surface with microcracks in the specimen with serrated grain boundaries of the HS–21 alloy (D^′ = 2.296) and the stage I fatigue fracture surface with slip steps and ledges of the SUS316 steel (D^′ = 2.286). As a whole, the values of D _f are well correlated to those of D^′ through a relationship of D^′ = D _f + 1 on these fracture surfaces.

Fig. 11.

Relationship between the fractal dimension of the fracture surface (D^′) (estimated on the reconstructed fracture surface images) and that of the actual fracture surface profile or the indentation crack ( D _f ) in materials when estimated in the length scale range that is associated with the size range of the principal fracture surface patterns.

Subsurface cracks and overlaps of the fracture surface (Fig. 8) may be one of the reasons that the fractal dimension of the actual fracture surface profile, D _f , was slightly larger than the fractal dimension of the reconstructed fracture surface profile, D _p ^′, on many material fracture surfaces (Fig. 10). The fractal dimension of the fracture surface (D^′) can reasonably be estimated using a relationship of D^′ = D _f + 1 with the averaged values of the fractal dimension of the actual fracture surface profile in metallic materials or that of the indentation crack in ceramics ( D _f ) (Fig. 11), when estimated in the length scale range that is associated with the size range of the principal fracture surface patterns on 10 kinds of fracture surfaces. Thus, it was confirmed that the computer-aided stereo matching method can reproduce the principal fracture surface topography on the actual fracture surfaces necessary for the investigation of material fracture using the FRASTA [6,15,16] or the FDM [22,27,29], except subsurface cracks and overlaps of the fracture surface. Further, the present results suggest that for detailed morphological analysis of material fracture surfaces it is suitable to conduct the fractal analysis in the length scale range that is associated with the size range of the principal fracture surface patterns.

5. Conclusions

The fractal analysis of fracture surfaces using the three-dimensional images, which were reconstructed by the computer-aided stereo matching method, was reviewed in various materials. The fractal dimension of the fracture surface was estimated on 15 kinds of fracture surfaces such as fatigue fracture surfaces, creep fracture surfaces and impact fracture surfaces of metallic materials or ceramics with fracture surface patterns including dimples, striations or grain-boundary facets. The two-dimensional and three-dimensional fractal analyses of the fracture surfaces were carried out by the computer-aided box-counting methods. The fractal dimension of the fracture surface was compared to the fractal dimension of the fracture surface profile or the fracture surface contour on the reconstructed fracture surface images. The results of the fractal analysis seemed to indicate that the reconstructed material fracture surface images are generally not completely isotropic. The results of the three-dimensional fractal analysis, which were carried out on 10 kinds of reconstructed fracture surface images, were compared to those of the two-dimensional fractal analysis of the actual fracture surface profiles of metallic materials or on the indentation cracks of ceramics. When estimated in the length scale range that was associated with the size range of the principal fracture surface patterns, both fractal dimensions of the fracture surface and of the fracture surface profile were well correlated to the fractal dimension of the actual fracture surface profile of metallic materials or the indentation crack. It was confirmed that the computer-aided stereo matching method can reproduce the principal fracture surface topography to give important information for investigation of material fracture, except subsurface cracks and overlaps of the fracture surface. Further, for detailed morphological analysis of material fracture surfaces, it is suitable to conduct the fractal analysis in correlation with the size range of the principal fracture surface patterns.

Footnotes

Acknowledgements

The authors thank Dr. Toshimitsu Yokobori, Prof. Emeritus of Tohoku University, for giving them an opportunity for publishing this research article.

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