PHYSICAL INTERPRETATION OF THE REFERENCE FEATURES IN TEXTURAL FRACTOGRAPHY OF FATIGUE FRACTURES

Reference features of a fatigue fracture surface ar e the reference texture and reference crack growth rate which are unambiguously mutually relate d. The reference texture is a subset of the image texture in SEM fractographs. It is expected t o be common to all fractures caused by loadings in which significant events occur sufficie ntly regularly and frequently. The ratio of the reference and the conventional crack growth rate ca ll d reference factor is a characteristic of a particular loading. Its value may be related to the sequence of successive sizes of the cyclic plastic zone, while the mechanism of the effect of overloads follows the models of Wheeler and Willenborg. Application to a set of nine test speci mens from aluminium alloy loaded by three different loading regimes is shown.


Introduction 1.Morphology of fracture surface in mezoscopical dimensional range
The morphology of a crack surface reflects the interaction of the material, including its microstructure, and the process of crack growth.The coincidences of the morphological structure of crack surface and material microstructure were proved for example in [1,2].The topic of this paper is an investigation of the relation between morphology of a fatigue crack surface and parameters derived from fracture mechanics.Typical features of fatigue fractures were discovered a long time ago.In the field of microfractography, striations were found, related to the mechanism of crack growth [3], and widely used for quantitative analyses [4][5][6][7].In the dimensional range of macrofractography, beach lines were joined with significant changes in parameters of loading, especially overloads.In cases when they could be related to a known time, the reconstitution of the history of crack growth was possible [8][9][10][11][12].In contrast to this, the mezoscopical range between micro-and macrofractography holds its secrets up to now.The mezoscopical fractographic range is characterized by SEM magnifications from about 100 to 500x.These magnifications were rarely used in quantitative fractography, because images of fracture surfaces taken under them contain a complicated random structure without any distinct borders.Therefore, no objects can be simply extracted to be counted and measured.Since 1989, textural fractography is developed to utilize the information contained in fracture morphology in the mezoscopical dimensional scale.Images of fracture surface are understood as random fields -image textures, and analysed by means of the image textural analysis.Global characteristics -image features -are estimated for the whole image of fracture surfaces.

Expected relation to cyclic plasticity
The main mechanism of fatigue crack growth is plastic deformation.In microvolume, the primary process is reflected -creation of plastic striations by single crack increments.Also beach lines visible by macro-observation are traces of plastic deformation.Therefore, it may be expected the same also in the mezoscopical range: the morphological structure of fracture surface should be related first of all to characteristics of plastic deformation.Within this dimensional range, the relevant characteristic of plastic processes is the size of plastic zone at the crack tip.However, there are two plastic zones accompanying fatigue crack growth: the static and the plastic one.Static plastic zone, governed by K max , is responsible for a shift of stress/strain loop at the crack tip towards a position symmetrical around zero, as well as for the effects of retardation or acceleration of crack growth after overloads.Under a constant cycle loading, it develops slowly and quasi continuously.Under a variable cycle loading, it significantly changes only in overload cycles.On the contrary, the cyclic plastic zone, governed by ∆K ef , "breathes" with single loading cycles and is closely related to individual crack increments.So, just the cyclic plastic zone should be expected to be "inscribed" in the morphology of fracture surfaces.Therefore, we looked for the "trace" of plasticity in mezoscopical fracture morphology as some coincidence with the size of the cyclic plastic zone.

Textural fractography
Various alternatives of the textural method were proposed during the last 12 years [13][14][15].Among them, the direct fractographic solution of the reference concept brought the best results.Its application is limited to loadings satisfying the condition of stationarity over short distances.
It means that all significant events, especially overloads, occur sufficiently regularly and frequently.
The Reference concept [14,15] is based on a discovery of the reference texture -a textural component in fatigue fractographs which is common to fracture under various loading regimes.The reference texture is unambiguously related with the reference crack growth rate v ref , a product of the conventional crack rate v and the reference factor B, . .
where h q denotes a set of selected functions of image features, e.g.h = {f, log(f), f 1/2 , f 2 , 1/f, etc}.Free coefficients c 0 and c u,q (common to all images) and reference parameters B k (common to images of specimens loaded by the same loading regime) are estimated by the least squares method.Statistical significance of particular terms of the sum in eq. ( 2) is tested by a t-test and non-significant terms are excluded [16].The set of image features composing the final model defines the reference texture.

Physical explanation of the reference features
By reference features we mean the reference texture, crack rate v ref , and factor B. For reasons which were discussed in Introduction, we will look for a relation between reference features and the size of the cyclic plastic zone w * [17].Under a constant cycle loading, both the crack growth rate (CGR) v and the size of cyclic plastic zone w * are functions of ∆K ef : Simultaneously, the morphology of crack surface is strictly related to the CGR, and, therefore, it is also governed by ∆K ef .Due to the equality v ref = v and the relationship between v ref and the reference texture, the reference texture and the reference crack rate are also controlled by ∆K ef .
In case of various variable cycle loadings, the same reference texture corresponds to different conventional crack growth rates, and, consequently, also to different mean values of ∆K ef and to different mean sizes of the cyclic plastic zone.A seemly contradiction to the case of constant cycle loading may be overcome by assuming that not all but only major cycles dominate in the process of creating the morphological structure of fracture surface, in particular the reference texture.
The algorithm of crack growth models of Wheeler and Willenborg was found to fit well experimental results.As illustrated in Fig. 1, a new major plastic zone arises when the front of the theoretical cyclic plastic zone in a given cycle exceeds the front of the foregoing major plastic zone: where 2K is the length of the moving average.For periodical loadings by repeating a block of cycles, w M is computed for particular blocks and assessed to corresponding middle crack length.
Let the symbols ( )

M v w and
( ) v w denote crack growth rate and reference crack growth rate related to the given value of w M .The assumption that w M , i.e. the mean local size of the residual major cyclic plastic zone, controls the reference texture and crack rate, implies following expectations: a) For a given variable cycle loading, the ratio should be approximately constant (independent of w M ) and similar to the corresponding value of the parameter B, estimated within the fractographic reference solution (eq. ( 1), ( 2)).
b) The dependence v w should be independent of the type of loading.

Experimental materials and methods
CT specimens (Fig. 2) from aluminum alloy 2024 were loaded at 20°C in air by various loading regimes.Crack growth was regularly measured and recorded.
For the present study, nine specimens were selected, loaded in groups of three by a constant cycle, regime 199+1 (constant cycle with a periodical overload after each 199 cycles), and a block of 1000 cycles with random characteristics [17].Markers represent single images of the fracture surface.
The final model contains 16 most significant features.Resulting estimates of reference parameters B are presented in Table 1 [17].The quality of the model is documented in Fig. 4.
Examples of reference textures are shown in Fig.
The sizes of residual major cyclic plastic zones * m w ∆ and their moving averages w M were computed from equations ( 3) - (7).Dependences between w M and the conventional crack growth rate v are shown in Fig. 5.The data characterizing each test body follow a linear trend, and hence they may be represented by linear regression.On a logarithmic scale, the ratios B' (eq.( 8)) are represented by distances between graphs for constant and variable cycle loadings.These distances are almost constant, exact values for the middle of the range are presented in Table 1.
For the loading regime 199+1 the ratio B' is almost equal to the reference parameter B. In the case of random loading, a discrepancy of about 20% has been obtained.Up to this degree, expectation a) was verified.Dependences between the magnitude of w M and the reference crack growth rate v ref are shown in Fig. 6.Also the expectation b) may be said to be approximately valid.

Conclusion
Results obtained allow to argue that reference features are governed by cyclic plasticity corresponding to the major values of the effective SIF range ∆K ef .This fact opens a way for a detailed structural investigation of the mezoscopical dimensional component of the morphology of fracture surface in relation to plastic processes at the crack tip.

Fig. 5 Fig. 6
Fig.5 The dependence between w M 1/ and the mean conventional crack growth rate v.2/ -test specimens, j -images of the fracture surface, k -the applied loading regime, u -image features, q -selected functions.Each image is characterized by a set of numerical textural characteristics -image features f uij , and assigned a mean local macroscopic crack growth rate v ij estimated from experimental records of crack growth.The relation between the crack growth rate v and image features f u may be expressed via a multilinear model (equation for j-th image of i-th specimen) .) Factor B is a characteristic of the type of loading.In case of constant cycle loading, B = 1, i.e., v ref is equal to the conventional crack growth rate: v ref = v.For variable cycle loadings with dominating effects of tensile overload, B > 1, i.e., v ref > v.By means of introducing of the factor B, an unambiguous relation between the reference texture and reference crack growth rate v ref was achieved (in contrast to the relation between the morphology of crack surface and the conventional crack growth rate v).Examples of reference textures corresponding with the same v ref are shown in Fig. 7. DOI 10.12776/ams.v19i2.98p-ISSN 1335-1532 e-ISSN 1338-1156 Let following indices denote: i 7. Original images assigned various crack rates Reference textures are similar -visually as well as analytically in the sense of a random field.In the next step, cycle-by-cycle crack growth predictions were computed.The generalized Paris and Erdogan equation was used for crack growth under a constant cycle, and Wheeler's model for variable cycle loading.The variability of crack growth rates was respected by means of a multiplicative parameter β for each individual specimen.The value of ∆K ef in each loading cycle was computed from model crack increment ∆a according to the equation