Effects of deformation mode on surface roughening of austenitic stainless steels

Introduction

Free surfaces of metals become rougher during plastic deformation. This roughness is undesired for many applications because it can influence important properties such as reflectivity, frictional behaviour and corrosion resistance. In addition, it provides initial sites for strain localisation, which may degrade the formability of a material. Hence, extensive investigations on the roughening of free surfaces have been reported. These studies show the influence of strain, grain size, stress mode and texture on surface roughness characteristics. The surface roughness is found to increase linearly with strain and depends linearly on grain size.1^–5 The roughness increment was irrespective of the stress state,6 whereas texture was found to have a great influence on the roughness.7^–9 Most of these studies are aimed at finding the mechanisms of surface roughening during deformation. A number of studies have shown that plastic deformation roughens a surface by introducing slip bands within grains and strain incompatibilities between grains.3,8,10 The latter is the major factor for the measured surface roughness parameters.

Recently, fractal theory was introduced to study the surface roughness during deformation. Zaiser et al.11 found that copper samples deformed in tension develop self-affine roughness over a large range in length scales. Wouters et al.1,7 did further investigations of the surface roughening process using a height–height correlation technique, and found the statistical parameters α, ξ and ω to be a good approach to obtaining valuable information about the nature of the roughness and the underlying physical mechanisms. However, these experiments were only conducted in tensile deformation mode.

In the present study, the authors aimed at gaining more knowledge of the nature and development of roughness of polycrystalline metal surfaces under different deformation modes: tensile deformation and deep drawing. In this paper, fractal analytical method is applied to give descriptions of the rough surface, and surface roughness are compared between two deformation modes, which is useful to predict the roughness by deep drawing deformation. And it gives a direct insight into the typical length scales that play a role as the roughness develops.

Experimental procedure

Results

Fractal character of tensile surface

The results of the deformed tensile sample surface measured in the direction parallel to the tensile axis and the direction orthogonal to the former are shown in Fig. 1. Typical plots exhibit linear fit to the first section of the structure function, that is, the curve indicates a self-affine scaling behaviour on a certain length scales. Hurst exponents deduced from the slope of the roughness plots (and the corresponding fractal dimension D_F = 2−H) are shown in Fig. 2a. The error bars reflect the scatter of exponents obtained from different profiles taken at the same strain. In the longitudinal direction of the tensile sample, the value of H is small for small strains, but increases with increasing strain. Above a strain of 12·3, the value of H is almost the same and is 0·88. However, in the transverse direction, the value of H stabilised to ∼0·88.

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Figure 1

Roughness plots for profiles obtained after tensile deformation to certain strains

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Figure 2

Fractal dimension and Hurst exponent as function of strain

Fractal character of deep drawing surface

The log–log plots of the function of the deep drawing surface profile are illustrated in Fig. 3. As the tensile sample surface, these plots exhibited a linear relationship for small distances. However, in the direction orthogonal to the cup axis, the plots exhibited two sections of straight lines, which mean that the surface was bifractal. The Hurst exponent H determined from the roughness plots is shown in Fig. 2b. The value of H increases initially with increasing strain, but reaches a constant value of 0·89 for the two directions of the deep drawn sample surface.

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Figure 3

Roughness plots for profiles obtained after deep drawing at different strains

The correlation length ξ of the two sample surfaces in different directions were plotted in Fig. 4. There were gaps in the different directions. For the transverse direction, the ξ stabilised to a value of ∼30 μm, which is the same order of magnitude as the grain size (21 μm). In the longitudinal direction, however the ξ is ∼60 μm for the two samples.

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Figure 4

Correlation length ξ

The rms surface roughness of the two samples was shown in Fig. 5. The values of rms appear to lie on a straight line for each direction of the samples, and there was a small difference for the two samples in the different directions. For the tensile sample, the rms in the longitudinal direction is a little larger than the one in the transverse direction. However, the rms in the longitudinal direction is a little smaller than the one in the transverse direction for the deep drawn sample.

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Figure 5

Root mean square roughness as function of strain in two directions

The average values for rms as the function of true strain in the two different deformation modes are shown in Fig. 6. It can be seen that the trends of plots can be represented by a linear fit for each sample with different slopes (5469·4 in tension test and 3515·7 in deep drawing).

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Figure 6

Average rms roughness in tension test and deep drawing deformation

Discussion

The surface structure of plastically deformed metal samples in terms of self-affine properties was reported by Zaiser et al.11 for the first time. The authors analysed the rough surface profiles in terms of self-affine roughness. From Figs. 1 and 3, it is clear that the height difference correlation functions are scaling invariant. To demonstrate self-affine roughness more convincingly, the authors performed a check by multiscaling analysis where the authors evaluate the nth order structure function12 (3) where Δy_L = y(x)−y(x+L), and the average runs over all pairs (x+L) within the profile. Self-affine behaviour is characterised by a power–law relationship (4) where H_n = H does not depend on the order of the structure function. One reliable way to estimate what is the highest moment that the data can faithfully reproduce is to plot the integrand in the definition of the nth moment13,14 (5) where P(Δy_L) is the probability distribution function, and the integrand has to be sufficiently smooth and decrease to small values at large Δy_L. Accordingly, the integrand was tested for assessing the statistical reliability of the numerically determined structure functions which led us to discard structure functions of order n>4 in our case.

Figure 7 shows that power–law relationship is indeed observed in certain scale. Within the scaling regimes, there is a weak n-dependence of the scaling exponent H_n = 0·70−0·77 for 1⩽n⩽4, thus the available data are consistent with self-affine scaling. So the rough surface profiles caused by deformation indeed develop self-affine roughness.

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Figure 7

Multiscaling analysis of surface profile taken at ϵ = 6·14 during tensile deformation in longitudinal direction; dotted line indicate limits of scaling regimes used in fitting scaling exponents H_n

Wouters et al.^1,7 did studies on the roughness of deformed metal surface and got many valuable results by height–height correlation functions. In the authors’ experiments, the H value is higher than the values obtained by Zaiser et al.11 (H = 0·75), and is close to the value obtained by Wouters et al.1 (H = 0·88±0·05). The explanation for the difference can be that the measurement is conducted with different scan sizes between the two groups. The scan sizes performed by Zaiser et al. is smaller, or on the order of the grain size, which excludes the influence of the roughening mechanism that acts on the grain size scale. This is different from the scan sizes adopted by Wouters et al. which is 10 times larger than the grain size. In the authors’ experiments, the scan size is almost 20 times larger than the grain size, which enables the roughening mechanism that acts on the subgrain and grain size scales to be included. Furthermore, the H value of the two deformation modes are the same, which means that the value of H≈0·88 for deformed metal surfaces may be universal. The correlation lengths are different in the two directions of the surface profile for the two deformation modes, but the values are on the same order of magnitude as the grain size. The ξ in the longitudinal direction is approximately one time larger than the value in the transverse direction, which can be related to different stains in the two directions.

Figure 5 clearly shows the roughness to increase linearly with increasing strain, but the slopes are different for the two directions for both deformation modes. For the tensile testing sample, the roughness in the two directions is initially the same, and the roughness in the longitudinal direction increases faster than the one in the transverse direction as the strain increases. It may be related to the strain anisotropy. Grains at the free surface have less restrain than grains under the surface and the deformation of the grains become different as the strain increases. However, for deep drawing sample, the roughness are different at small equivalent strains (the roughness in the transverse direction is larger than the one in the longitudinal direction), but the gap decreases with strain increasing. It can be linked with the deep drawing deformation where radial shrinkage can create destabilising effect of the inner surface, which induces a larger roughness in the transverse direction.

In Fig. 6, the average roughness is plotted as a function of strain for the two deformation modes, which is useful for engineering application. It is clear that the average roughness increase linearly. Assuming that the linear relationship between rms roughness and grain size, the rms roughness equation can be written as (6) This yields c = 0·26 for the tension test and c = 0·17 for deep drawing. The roughening rate of the tension test sample is higher than the one obtained by deep drawing. In deep drawing, the inner surface is under circumferential compression stress and axial tensile stress, and compression is larger than in the tensile stress. The roughness caused by the circumferential strain may be reduced by the tensile strain. While the tensile testing sample is only under axial stress, this could explain the higher surface roughness in this case.

Conclusion

The value of statistical parameters Hurst exponent H, correlation length ξ and rms roughness ω were gained to obtain information of the roughening mechanisms by different deformation modes.

It is shown that the surface profile obeys self-affine structure on the magnitude of grain size and the H values are close under the two deformation modes. The correlation length is directional and the value in the longitudinal direction is about two times larger than the one in the transverse direction. The roughening mechanism is different in the two deformation modes. During deep drawing, the inner surface may be unstable because of shrinkage in the transverse direction. As a result, the surface profile of deep drawing sample is bifractal. The roughening process is correlated to the strain state. In the tension test, the roughness rms in the longitudinal direction is larger than the one in the transverse direction, while the result is reverse during deep drawing deformation. The average rms increases linearly with strain increasing for the two deformation modes, and the roughening rate during tensile deformation is larger.