THE ANALYSIS OF STRESS DISTRIBUTION BASED ON MEASUREMENTS USING TWO METHODS OF X-RAY APPLICATION

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1 182 THE ANALYSIS OF STRESS DISTRIBUTION BASED ON MEASUREMENTS USING TWO METHODS OF X-RAY APPLICATION ABSTRACT Barbara Kucharska Institute of Materials Engineering, Czestochowa University of Technology, Poland The results of measurements of the state of stress in the sub-surface layer of steel C45 are presented. The state of stress was examined in a standard steel tensile test specimen that had been previously ground and then statically loaded with an effective strain of 0.45% and 1%, respectively. Stress measurements were carried out by the sin 2 ψ method with a ψ-diffractometer and using K α Co radiation. The classical version of this method and a version with a fixed angle of incidence of X-ray radiation applied onto the steel surface were used, alternatively. Based on the reflection from plane (211), the stresses in steel sub-surface layers of different thickness, as determined from the depth of X-ray penetration (with the classical θ-θ method and the fixed incidence angle method at α=30, combined with the material layer stripping method), were calculated. It has been demonstrated that the determination of the stress distribution over the specimen depth based on a single method of radiation exposure providing the identical depth of penetration into the material gives as a result a distribution curve that is shifted toward either the lower or higher stresses. The use of two (or more) methods of radiation exposure from two (or more) effective depths of penetration into the material from the same testing surface enables the gradients of stresses in the material sub-surface layers to be determined and, what is the most important, their averaged values will define a stress distribution in the specimen, which will be closer to the actual stress distribution. INTRODUCTION Mechanical surface treatments, to which metal products are subjected, induce a certain state of stress within the surface region. Thus produced state of stress can exist in the material without any interaction of external forces, therefore those stresses are commonly referred to as internal stresses irrespective of whether they are the result of heat treatment carried out, or have been introduced mechanically, e.g. with the aim of surface hardening. Normally, the zone of the produced state of stress reaches a depth ranging from a few microns to several millimeters, i.e. it covers the so called top layer of the item. One of the basic mechanical treatments most commonly used for machine parts is grinding. The examination of surface stresses introduced by this treatment are widely conducted using various measuring techniques, of which the X-ray method plays the most important role [1-4]. The particular importance of this method results from the fact that it enables the determination of stress variations with the depth from the material surface. The stress distribution over the material depth, σ(z) (1), is obtained by the analysis of diffractions originating from sub-surface layers of a different effective depth, τ (2) [3, 5]: σ ( τ ) e τ dz σ ( z) = 0 τ z (1)

2 This document was presented at the Denver X-ray Conference (DXC) on Applications of X-ray Analysis. Sponsored by the International Centre for Diffraction Data (ICDD). This document is provided by ICDD in cooperation with the authors and presenters of the DXC for the express purpose of educating the scientific community. All copyrights for the document are retained by ICDD. Usage is restricted for the purposes of education and scientific research. DXC Website ICDD Website -

3 183 τ = sin θ cos ψ 1 ln (2) 2 µ 1 G τ where: θ - diffraction half-angle, ψ - angle of rotation of the normal to the diffracting lattice plane, µ - linear absorption coefficient, G τ (0,1) ratio of the intensity of radiation diffracted on a specimen of the thickness τ to the intensity of radiation diffracted on a specimen of infinite thickness. The stress distribution can also be obtained by taking measurements on the surfaces, from which material layers of the specified thickness z have been previously stripped off. Conventional diffractometers most commonly use the sin 2 ψ method for stress measurement [6]. This method measures the deformation of the crystal lattice, ε, as a function of the angles ϕ and ψ defining the position of the X-ray diffracting plane. Due to the relatively little depth of X- radiation penetration into the material it is assumed that a flat state of stress occurs in the surface layers of materials examined, which results in the following notation of the sin 2 ψ method (2): dφψ d0 1+ ν 2 ν ε φ,ψ = = σ sin ψ (σ1 + σ2 ) d0 E E ϕ (3) where: E- Young modulus, ν - Poisson coefficient, σ ϕ = σ 1 cos 2 ϕ + σ 2 sin 2 ϕ - stress component in the specimen plane, which can be obtained from the measurement of the slopes of the curve of the relationship of crystal lattice deformation, ε ϕ,ψ, as a function of sin 2 ψ [2,7,8]. The drawbacks of the conventional sin 2 ψ method include the variable depth of radiation penetration into the material during measurement, even when using a fixed angle of radiation incidence [8]. The present study compares the stresses determined by the conventional sin 2 ψ method and by the fixed angle of incidence version this method on the example of steel C45 after mechanical working and tensile deformation. MATERIAL The C45 low-alloy toughening steel was examined, which had an approximate chemical composition (in wt%) as follows: C ; Mn ; Si Round cross-section specimens with a measurement-base diameter of 12 mm were prepared from the steel, as for a tensile test. The specimen dimensions and surface preparation (by grinding) complied with typical standards applicable to the tensile test. The specimens were ground circumferentially. In the microscopic image taken from the specimen surface and cross-section (in the measurement part), numerous scratches typical of surface mechanical Fig.1. Shape of testing specimens points of X-ray measurements working were observed - Fig. 1. In observations made on an optical microscope at a magnification of 800x, no evidence of plastic deformation in the sub-surface ferritic-pearlitic steel structure was found on the specimen cross-sections. Two of the specimens were subjected to tensile test on a testing machine, with a deformation of 0.45% and 1.0%, respectively.

4 184 EXPERIMENTAL a) Steel specimens were subjected to X-ray examination within the measurement-base region. The measurement base was not cut out, but the whole specimen was fixed on the goniometer table. The examination was performed on a Seifert 3003TT diffractometer using a K α Co radiation (λ=0.179 nm). Reflections (211) corresponding to the diffraction angle of 99.8 were recorded. The measurements were taken for three azimuth angles of ϕ=0, 45 and 90, respectively, with the azimuth of ϕ=0 corresponding to the measuring direction parallel to the specimen axis. The diffraction vector rotation angles, ψ, varied, taking on the following positions: 0, ±7.5, ±15, ±22.5 and ±30. For each azimuth specimen positioning measurements were taken in the symmetrical θ-θ option and with a fixed beam incidence angle [8]. A fixed beam incidence angle of α=30 was applied, which was still a safe angle, i.e. the one that allowed a nonconflicting measurement with specimen positioning at the azimuth of ϕ=0 (due to the size of specimen Fig.2. Methods of recording reflections from plane 211 in steel: (a) the θ-θ method, (b) the method with a fixed incidence angle of 30 heads). Both options of radiation exposure onto the steel surface allowed the measurement of stresses in material layers at different depths. During measurements, the thickness of steel layers covered with measurements (G τ =0.9 was assumed in Eq.(2)) varied in accordance with the value of ψ ( ψ max 0) in the following range: µm ( τ=2.72 µm) in the symmetrical measurement, - 13,89 16,04 µm ( τ=2.15 µm) in the measurement with α=30 Fig. 2. The average measurement depth, allowing for the weight of the number of diffractions (and thus the measurement time) from a given depth, was as follows: µm in the symmetrical measurement, µm in the measurement with fixed angle of incidence α=30. On the specimen subjected to a deformation of 1.0%, measurements were additionally taken after twice stripping off 0.2 mm-thick steel surface layers. The steel layers were stripped off by means of manual polishing using cotton cloth soaked with diamond paste. All measurements were conducted under the same current and voltage conditions and with the same X-ray beam optics. The diameter of the radiation beam was 2 mm. Each measurement was taken at three points on the specimen perimeter. For this purpose, the specimen was rotated twice by 45 around its axis. Stresses given for one azimuth, e.g. σ 90, are the average of three measurements taken on the specimen perimeter. The computation of stresses were made by the sin 2 ψ method employing the Rayflex\Analyze program, on assumption of existing a flat state of stress and using the mechanical elasticity constants E=220 GPa and ν=0.2. Diffraction reflections were fitted with a pseudo-voigt function and PL, Bkg and smooth-corrected. The diffraction angles, 2θ Bragg, were determined from the reflection maximum. b) mean

5 185 RESULTS a) τ 1 =15.14 µm τ 2 =17.74 µm The stress results obtained in measurements directly on the steel specimen surface before and after the tensile test, respectively, are shown in Fig. 3. In measurements taken from layers of an average thickness of µm using a fixed angle of beam incidence, larger stress values compared to symmetrical method measurements (from µm-thick layers) were always obtained. In measurements taken on starting specimens it was demonstrated that compressive stresses of different magnitudes depending on the specimen azimuth, σ ϕ, and the measurement depth, τ, had been introduced in the process of their standard surface preparation. Those stresses were contained in the range MPa (av MPa) for measurements taken by the symmetrical θ-θ method (from the µm-thick layer) and σϕ, MPa b) σ, MPa ϕ=0 ϕ=45 ϕ=90 ϕ=0 ϕ=45 ϕ=90 τ 1 =15.14 µm τ 2 =17.74 µm initial sample tensile 0.45% tensile 1% Fig.3. Surface stresses in the initial steel, as tensile deformed: (a) azimuth stress, σ ϕ ; (b) stresses averaged out of three azimuths, σ MPa (av MPa) for measurements taken with a fixed angle of radiation incidence of α=30 (from the 15.14µm-thick layer). Stresses determined for the direction perpendicular to the grinding direction, ϕ=0, were the lowest in both measuring methods. Introducing stresses in the tensile test resulted in an increase in compressive stresses for a majority of the azimuth specimen positions, ϕ. This increase is due to summing up of stresses resulting from grinding and tension. The most uniform increase in stresses, caused by the tensile test, was noted for the azimuths ϕ=0 and ϕ=45 in measurements from the µm layer and for the azimuth ϕ=90 in measurements from the µm layer. Figure 4b shows the results of measurements of the stress σ, averaged out of three azimuths: Table 1. Relative increments in surface stress in steel due to the tensile strain, ε Depth of X-ray penetration in steel, τ σ ε ε = 0.45% ε = 1.0% µm 5.7% 28.3% µm 35.1% 68.4% ϕ=0, 45 and 90 (from Fig. 4a). The averaged values of the stress σ increase with increasing tensile stress, regardless the measuring method. The relative compressive stress increments are larger for the measurements from the µm layer (Table 1). Compressive stresses measured on a specimen tensioned by 1%, from which a 0.2 mm-thick layer had been stripped off twice, increased with increasing depth from the specimen surface - Fig. 4.

6 186 It was assumed that the difference in the values of stresses determined in both radiation exposure methods used (θ-θ and α=30 ) was a gradient of stresses in the layer of a thickness of 2.6 µm being the difference of the effective radiation penetration depths. Thus, it can be stated that, in the initial specimen, the stress gradient (calculated from three azimuth positions) in the 2.6 µm-thick layer situated at a depth of µm amounted to σ 2.6/15.14 = MPa. σϕ, MPa stresses in layer (z, z+17.74)µm: initial specimen, specimen as tensile deformed by 1% stresses in layer (z, z+15.14)µm: initial specimen, specimen as tensile deformed by 1% ϕ= ,2 0,4 0 0, , , ,4 z, mm ϕ= ϕ=90 Fig. 4. Distribution of azimuth stresses over the specimen depth, as obtained by the method of stripping off 0.2 mm-thick layers Table 2. Gradients of stresses in 2.6 µm-thick layers situated at different depth under the steel surface, z σ 2.6/z Specimen z=15.14 µm z= µm z= µm initial ε=0.45% ε=1.0% It follows from Table 2 that the gradients of stresses in such a layer decrease with increasing tensile strain and with increasing distance from the specimen surface. The latter relationship, noted in the method of stripping off 200 µm-thick layers, leads to the conclusion that in the deeper parts of the material the distribution of stresses is more uniform, and thus the gradient values decrease. SUMMARY As a summary of the obtained investigation results, a schematic diagram of the distribution of stresses in the initial specimen and in the specimen tensioned by 1.0% is proposed Fig. 5. The analysis of this schematic diagram shows that determining the stress distribution over the specimen depth based on a single method of radiation exposure giving the identical effective penetration into the material results in obtaining a distribution curve which is shifted towards either the lower stresses (with τ 1, the curve ) or the higher stresses (with τ 2, the curve ). Using two (or more) methods of radiation exposure from two (or more) effective depth of penetration into the material, from the identical testing surface, enables the gradients of stresses in the material sub-surface layers to be determined and, what is the most important, their averaged values will define a stress distribution in the specimen, which will be closer to the actual stress distribution (the curve ).

7 / / τ τ 2.6/ Fig. 5. Graphical scheme of stress distribution in steel REFERENCES 1. L. Suominen, D.Carr, Advances in X-ray Analysis, Vol.43, (2000) Y. Calik, J.T. Evans, B.A.Shaw (1996), Scr. Mater., Vol.34, No.7, M.J.Marques, A.M.Dias, P.Gergaud, J.L.Lebrun (2000), Mater. Sc. and Eng., A287, B.D.Cullity, Elements of X-ray Diffraction. Reading, Mass: Addison-Wesley X.Chen, W.B.Rowe, D.F. McCormack (2000), J. Mater. Proc. Techn., 107, , 6. S.J.Skrzypek, A.Baczmański (2001), Advances in X-ray Analysis, Vol.44, H.Behnken, V.Hauk (2000), Mat. Sc. and Eng., A300, D.Senczyk, X-ray diffractometry in the examination of the state of stress and elastic properties of polycrystalline materials (in Polish), The Publishers of the Poznan University of Technology, B.Kämpfe, Mat. Sc. and Eng., A288, (2000) B.Kucharska, G.Starzyński (2004), Internal subsurface stresses in steel 45 cyclically deformed to a steady state (in Polish), The Publishers of the Częstochowa University of Technology, Metallurgy Series No. 39,