SELFCONSISTENT EVALUATION OF RESIDUAL SHEAR STRESS PROFILES NEAR GROUND SURFACES

Transcription

1 SELFCONSISTENT EVALUATION OF RESIDUAL SHEAR STRESS PROFILES NEAR GROUND SURFACES H. Wern, Hochschule fir Technik und Wirtschafl des Saarlandes, Goebenstrane 40, D Saarbrticken, Germany ABSTRACT The presence of large residual shear stresses near ground surfaces are often revealed by x-ray diffraction analysis. With the X-Ray-Integral-AJethod(RIM)Z it is now possible to determine the depth profiles of strains and stresses as a function of the true depth below the surface. However, any shear stress component parallel to a free surface must be zero at the surface itself Therefore, the detection of shear stress implies the existence of steep gradients, but often they are at variance with the laws of static equilibrium at a boundary when the conventional Cauchy definition of stress is assumed. The observations can be reconciled with the laws of static equilibrium if a torque-stress theory is adopted3, but this requires the existence of a particular deformation structure in a thin region below the ground surface. Some measurements of residual strain patterns near ground surfaces are analyzed and interpreted according to the phase compensation hypothesis and according to the torque stress formulation as suggested by Gola and Coppa3. In this couple-stress hypothesis, the residual shear stress 013 (z) must have a sign inversion in depth. The results obtained with RIM seem to favor this hypothesis but also other explanations will be given. With the aid of a selfconsistency criterion it is possible to find a proper normalization of the measured strains which allows for the first time an independent calculation of the stress free lattice spacing which is often not known. NORMALIZATION OF STRAINS AND STRESSES Residual strain and stress profiles measured by X-rays in the near surface region of polycrystalline materials are always averaged quantities because the counted intensities are averages over the diffracted volume. Since the true z-profiles of residual stresses are generally of more interest in evaluation the effects of surface treatments, several analytical and numerical approaches have been employed to retrieve these z-profiles from the measured z-profiles 2P4-12. The author introduced a Fourier method in which it is assumed that the z-profile E(Z) can be represented by trigonometric basis functions. The mathematical details are described elsewhere2. One serious advantage of this procedure is the capability to find a proper and selfconsistent normalization of the measured strains which allows an in-situ evaluation of the stress free lattice spacing. The principles are summarized in the flow chart diagram of Fig. 1. From the measured d-values in a first step a do start value is evaluated. Because we deal with an iterative technique this step is called iter=o. With the use of orthonormal trigonometric basis functions as a representation of strain gradients, the true strain z-profiles are determined. Now, Hooke s law is used to convert strains to stresses with the known x-ray elastic constants. The general form of Hooke s law in terms of stress (oij), strain (aij), and the elastic stiffness (C&l) or compliance (Sijkl) tensors read as follows:

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 Selfconsistent normalization of strains [Eva, ~te Star/Fl iter = 0 ormulation of strain gradients with orthonormal basis, functions t / From a33 (z=o)=o t I x-ray elastic constants i*,1/2 s2 Boissons s ratio: c = - S&1/2 s2+ S,) vs= e33(0) / (&33(0)-e22( )-El I to)) if (iter == 0) VI = vs I optimum - found do such that 1 decrease/increase do if (v, d vc,(vs) N 0 iter ++ Fig. 1: Flow chart of normalization procedure Eij = ijkl Okl (14 oij = ijkl kl where summation over repeated indices is implied. The terms oij and EG are symmetric second rank tensors and can contain six independent unknown components in the general case. C+ and Sijki are fourth-rank tensors. The number of independent components in this case depends on the symmetry of the material in question. For isotropic and quasiisotropic materials, the number reduces to two, Young s modulus E and Poisson s ratio v. In polycrystalline materials these two calibration constants that link the peak shifts measured with x-rays to the (lb)

4 macroscopic stress are sometimes termed x-ray elastic constants. These terms contain both material parameters (such as elastic stiffness or compliance terms, eq. 2) and configurational parameters such as the Miller indices hkl. A detailed description can be found in reference13. In this sense the x-ray elastic constants are often abbreviated as E and vx. In the following these terms will be used without indexing. From the physical boundary condition that 033 has to be zero at a free surface, one can derive an equation for a surface x-ray Poisson s ratio vs which is given by eq. 2. &33 (0) vj = &33 (0) - % (0) - El1 (0) (2) The corresponding strain values at the surface are simply obtained from the calculated strain gradients. Because of the functional relationship in terms of sine and cosines the surface values need not to be extrapolated. If this surface Poisson ratio matches with Poisson s ratio from the x-ray elastic constants, the corresponding do is already found. However, in general, this will not be the case. Therefore a search direction for do is determined which is based on the idea that there may be a (nonlinear) relationship between Poisson s ratio and the stress free lattice spacing. The search direction is determined in a numerical way by using a linear Taylor series development where the first derivative of Poisson s ratio with respect to do is approximated by a symmetric difference formula. Depending on the sign of the first derivative, do is increased or decreased in small steps until the prescribed Poisson ratio is fixed within a certain interval (see eq. 3). Whenever one of the inequalities of eq. 3 holds, the Pegasus method is invoked to calculate the optimum stress free lattice value such that surface Poisson s ratio from eq. 2 is identical with Poisson s ratio from the x-ray elastic constants. The Pegasus method14 is a numerical root finding procedure which converges absolutely whenever one of the conditions of eq.3 is valid. < < v, ; vc ; v, (3) The present technique was first applied to experimental data from the book of Noyan and Cohen. The data were collected in R-geometry of a shot peened SAE52100 steel using CrKai radiation. The peening direction was 30 degrees with respect to the surface normal in order to produce a Y-splitting. The role of residual shear stress profiles will be discussed in the next chapter. For polycrystalline aggregates of cubic materials, the Voigt and Reuss averages of Poisson s ratio can be calculated from the single crystal data (see eq. 1) where the orientation term r is defined as: r = h2k2 + k212 + h212 (h2 + k2 + Z2)2 (4) Here h, k and 1 are the Miller indices of the diffracting plane and the compliances are referred to the crystal axes.

5 Voigt = - s,(s,lll ) + 10s1122s1212 s,(s,,,, + 2s,,,, ) + wl2,s**,2 + 10s,212(& ) (5) S w V Reuss = - S SJ (6) so = s,,,, - &,22-2&2,2 Hi1116 showed that, for a random polycrystal, the Voigt and Reuss averages are, respectively, the upper and lower bounds for the average x-ray elastic constants. Very often the mean of Voigt and Reuss values agree closely with experimental measured values. In order to get an information of the relationship between Poisson s ratio and the stress free lattice spacing, Poisson s ratio has been varied between 0.4 and 0.15 in steps of 0.01 and the algorithm of Fig. 1 was used to find the corresponding stress free lattice values. The numerical result in shown in Fig. 2. 0,45 I I I I 0,40 v= a+ b*do 0,35 0 z= F 0,30 rn -C s 0, a 0,20 a = b = R = ,15 0,lO I I I I I I 0, , , , , , stress free lattice spacing [nm] Fig.2: Poisson s ratio vs stress free lattice spacing of a shot peened steel specimen. It is interesting to note that a nearly linear relationship is observed. The values of a linear regression analysis are given in Fig.2. A physical explanation is still open. do is relatively insensitive to variances in Poisson s ratio. A standard deviation of 10% in Poisson s ratio results in an uncertainty of about 0.01% in do. With this relationship it is now possible to find a proper normalization of the measured d-values as long as the x-ray elastic constants are known( and here especially only Poisson s ratio). This advantage is not known by any other method. It is independent of any model and only based on the boundary condition that 033 has Copyright 0 JCPDS-international Centre for Diffraction Data 1997

6 to be zero at a free surface. It is important to note that the observed behaviour of a negative slope in Poisson s ratio vs stress free lattice spacing diagram was also obtained with other data sets from ground steel or ceramics specimens. The slopes are different and seem to depend on the surface treatment techniques used. In order to illustrate the normalization procedure, Fig. 4 shows the normal stress profiles of the steel specimen under the assumption of two arbitrarily chosen Poisson ratios. 400 I I I I I I 1 z g O- f! ii Q) L; U-J w $ GIL d0= v= 0.17 depth from surface [vm] do= v= I I I I I I depth from surface [pm] Fig.3: Normalized normal stresses for two different Poisson ratios.

7 Because of the different Poisson ratios there must of course be differences in the stress profiles. However, these differences are small as can be seen by inspection of Fig.3. Note that the absolute scale for the stresses is the same in both figures. RESIDUAL SHEAR STRESS PROFILES The presence of residual shear stress gradients is in general indicated by the observation of,,psi-splitting in one of the measuring directions. Of course, apparent sin*y splitting can be caused by misalignment of the goniometer, but this can easily be distinguished from the real effect because the former is invariant with respect to specimen rotation. A useful procedure to see whether gradients are present or not was first introduced by Dolle and Hauk. This method uses a linear combination of the measured strains by defining al vs sin*y and a2 as given in eq. 7. A curvature, or deviation from linearity, in the plots a2 vs. sinl2yi are due only to gradients in z. However, this procedure requires in any case the knowledge of the stress free lattice spacing which is often not known. With the normalization procedure described in the previous chapter, there is a powerful feature to calculate the corresponding do value. In so far Y-splitting is explained by the presence of term sin2y in eq. 7. However, the very presence of this term is not obvious, since both equilibrium equations and boundary conditions have to be satisfied in the subsurface layer. Recently, two hypotheses have been put forward to justify Y-splitting. They are as follows: -Phase Compensation Hypothesis18 1g Shear stresses take place in different phases of the material and have opposite signs, so that they compensate internally. As Noyan and Cohen16 state, these stresses are no longer macrostresses in the Cauchy sense, but are pseudomacrostresses being by definition zero at the surface. This hypothesis has gained more favor with the majority of people in the residual stress field. -Couple Stress Hypothesis3 Gola and Coppa proposed in or couple- or torque stress model. Within this model the existence of a deformation structure in a thin region below a ground surface is required. The hypothesis is based on the assumption that a surface separating neighboring elements in a continuum can transmit torque as well as force. Due to additional equilibrium equations for rotations, shear stresses are explained with torque-stress components. In terms of the equilibrium equations, the residual shear stress (~13 must have a sign inversion in depth. This sign inversion is necessary in the couple-stress hypothesis and unnecessary in the hypothesis of inner compensation. The result for the shear stress profiles from the shot-peened steel specimen is shown in Fig.4. Because the shot-peening direction was almost incident in the x-z plane of the reference coordinate system, the results seem to be quite reasonable with a small 023 component and a vanishing csr2 term. It is important to note that the shear profiles 013 and 023 are almost zero at the surface in accordance with the equilibrium condition of Newton s second law. This behaviour is not required by the computational procedure. Thus, this data set is an excellent example which confirms the hypothesis of inner compensation.

8 250 I I I I I I I do= Y = I 1 I I I I depth from surface [pm] Fig.4: Residual shear stresses of a shot peened steel specimen. On the other hand, the analysis of many data sets obtained from ground surfaces seem to contradict this hypothesis. In all cases which have been analyzed according to the normalization procedure described in this paper were successful in giving very reliable do values. However, the results obtained for the corresponding shear stress profiles are at variance with the boundary conditions at free surfaces. It is believed that grinding causes plastic deformation in the near surface region, but, even for plastic strains, the boundary conditions need to be satisfied Fig.5: Residual shear stress profiles of a ground steel specimen vs depth of a camshaft lobe after subsequent removal of 5pm layers.

9 In Fig. 5 a serie of shear stress profiles are shown. The measurements were performed on camshaft lobes which were induction hardened and ground. Camshaft steel was AISI (2 1 1 } lattice planes were investigated with CrKa radiation in modified Y-geometry. In this serie, measurements were repeated after a surface removal of about 5um using the technique of electropolishing. Whereas a proper normalization with respect to d0 was always possible, the corresponding shear stress profiles are nonzero at the surface. Here one could give many more examples from ground surfaces which all show the same tendency. Possible explanations may _ be given - as follows: 4 measurements from ground surfaces contradict the inner compensation hypothesis. b) stress profiles from ground surfaces are more complicated so that the Fourier approach which is used to evaluate the profiles may be insufficient. 4 surface treatments create surface roughness*. Shear stresses may be more sensitive to surface roughness than normal stresses and therefore more uncertain than predicted by the boundary condition of the hypothesis of inner compensation. d) surface treatment creates a deformation structure and the couple stress hypothesis may be valid3. The simplest way of accounting for the results is to drop the Cauchy definition of stress which assumes oji=oij. To sati@ equilibrium, the torque-stress theory as suggested by Gola and Coppa needs to be adopted (case d). Because in this theory it is assumed that the stress tensor is no longer symmetric, the equilibrium equations are splitted in a translational and rotational part. Whereas the translational part, involving force stress, is the same as that in the classical theory15, they are supplemented by additional equations involving torque stresses pji, i.e. (8) where &;jk is the permutation tensor and, j stands for the derivative with respect to the jth variable. Due to symmetry arguments given in ref. 3 and 1, the following equilibrium relation involving (313 is obtained from eq. 8. The existence of a stress ~713 at a free surface thus requires a gradient in torque-stress p32. Of course, ~32 itself must be zero at the surface. Once the corresponding gradients are determined, a numerical integration technique can be used to calculate the torque-stress ~32 in eq. 9. The result for the first profile in Fig.5 is shown in Fig.6. It is interesting to note that the obtained profile matches with the suggested form of Gola and Coppa. The required sign inversion in the shear stress profiles was observed in all those cases when ground surfaces were investigated and analyzed with RIM. Removal of a small depth of the surface layer causes only a minor redistribution of oil and (3z2 but a greater redistribution occurs in the 013 component. This arises because ~32 as well as 033 must be zero at a free surface. However, one should keep in mind that from equilibrium of an element, a torque-stress component ~32 should be produced by a (333 stress component that fluctuates from positive to negative values on a microscopic scale and being zero on a macroscopic scale. The obtained results with RIM, on the other hand, evaluate (~33 as a pseudomacrostress gradient being zero only at the surface. ISSN Copyright , 0 JCPDS-international Advances X-ray Centre Analysis, for Diffraction Volume 40 Data 1997

10 o,o depth from surface [pm] Fig.6: Possible torque stress component ~3~32 vs depth of a ground steel surface. All approaches which have been employed to retrieve the true z-profiles from the measured z-profiles suffer from the fact that inverse problems are extremely ill-conditioned. As a consequence the linear system of equations which has to be solved in any case is in general singular. Therefore, the uniqueness of the obtained solution in a physical sense is still open. Future progress will hopefully be achieved by applying wavelet theory to this problem21. The Fourier approach used in this paper, on the other hand, gives very reasonable results. Because the normalization procedure gives do values which are very close to literature values, it is believed that the representation of the stress profiles by trigonometric basis functions is sufficient even for the shear profiles. In a paper by Li et a1.20, the effects of surface roughness on stress determination by the x-ray diffraction method are discussed. In this investigation, however, the results are interpreted only in terms of average stresses. So, it would be interesting to quantify the effects of surface roughness on the stress gradients. A first insight in this problem can be checked by inspection of the corresponding al versus sin2y and a2 versus sinl2yi plots. The RlMsoftware also can provide the theoretical plots which are calculated for the whole available Y- ranges. These results are shown as solid lines in Fig. 7. It is interesting to note that al is always well defined with a very small experimental scatter2. This may be the reason that it was always possible to find a proper normalization of the data with respect to do. Fig.7 summarizes the results of a2 for the shot peened steel specimen and a ground Al203 specimen measured in Y- geometry with CrKa radiation. Whereas the former is well defined, much more experimental scatter is observed for the ground surface. This situation seems to be typical for ground or machined surfaces. Because the strain gradients are evaluated according to a least squares procedure, it is understandable that the obtained shear profiles are less reliable than the normal profiles which are involved in al. If there is less scatter in a2 as shown in Fig.7a, it turns out that the corresponding shear stress profile (here 013) is almost zero at the surface as shown in Fig. 4. For comparison the calculation for the ground surface has been repeated with a supplementary boundary condition which requires 013 and 023 to be zero at the surface. The result is shown in Fig.7b as a dashed line and also can fit the data quite well.

11 Fig.7a: a2 vs sinl2yl of a shot peened steel specimen. Fig.7h: a2 vs sinl2yl of a ground ceramics. CONCLUSIONS For XRD stress determination with RIM, it is now possible to find a selfconsistent normalization procedure as long as the x-ray elastic constants are known. This procedure is based on the assumption that the orthogonal stress 033 has to be zero at a free surface and allows for the evaluation of do which in many cases is often not known. It has been shown that an almost linear relationship between Poisson s ratio and the stress free lattice spacing exists. There is evidence that this relationship depends on the different surface treatment techniques used. The presence of residual shear stresses especially near ground surfaces are often revealed by x-ray diffraction experiments in terms of sin2y splitting. In the literature, one finds two hypotheses which just@ Y-splitting. In the so called phase compensation hypothesis, Noyan and Cohen state that ~33, 013 and 023 can exist only as gradients (pseudomacrostresses) in the near surface layers being by definition zero at the surface which mutually compensate between phases of the material. Based on a kinematical description of elastoplastic displacements which occur during grinding, the presence of shear stresses also can be explained in terms of a couple stress model. The hypothesis of inner compensation is confirmed in a number of cases and seems to be contradicted in an equivalent number of other cases because of the apparent violation of equilibrium. It is interesting to see that one also can find stress profiles which seem to confirm the suggestions made by Gola and Coppa within a couple- or torque stress model. The experimental technique of grazing incidence22 could give a good answer to distinguish between both hypotheses. However, the results obtained with RIM indicate that residual shear profiles are more sensitive to the grinding parameters and/or surface roughness. Therefore, the equilibrium conditions at the surface are not fulfilled automatically but must be required within the analysis. In this sense the hypothesis of inner compensation is favored.

12 REFERENCES 1. Calik, Y., Evans, J.T. and Shaw, B.A., ScriptaMaterialia, 34, (1996). 2. Wern, H. and Suominen, L., Advances in X-Ray Analysis, 39, (1996). 3. Gola, M.M and Coppa, P., Journal of Applied Mechanics, 55, (1988). 4. Zhu, X., Ballard, B. and Predecki, P., Advances in X-Ray Analysis, 38, (1995). 5. Zhu, X., Predecki, P. and Ballard, B., Advances in X-Ray Analysis, 38, (1995). 6. Predecki, P., Powder Diffraction, 8, (1993). 7. Wern, H., Journal of Strain, 27, (1991). 8. Zhu, X. and Predecki, P., Advances in X-Ray Analysis, 37, (1994). 9. Ruppersberg, H., Advances in X-Ray Analysis, 37, (1994). 10. Wern, H. and Suominen, L., Advances in X-Ray Analysis, 37, (1994). 11. Wern, H. and Suominen, L., Journal de Physique, (1997). 12. Eigenmann, B., Advances in X-Ray Analysis, 40 (1997). 13. Murray, C.E. and Noyan, I.C., Phil. Mag. (1996). 14. Engeln-Mullges, G., Reutter, F.,,,Formelsammlung zur Numerischen Mathematik mit Standard FORTRAN 77-Programmen, BI Wissenschaflsverlag (1988). 15. Noyan, I.C. and Cohen, J.B.,,,Residual Stress Measurement by Diffraction and Interpretation, Springer (1987). 16. Hill., R., Proc. Phys. Sot. London Sect. A, 65, 349 (1952). 17. Ddlle, H. and Hauk, V., Harterei-Techn. Mitt., 31, 83 (1976). 18. Hauk, V., Oudelhoven, R. and Vaessen, G., Harterei-Techn. Mitt., 36, (1981). 19. Noyan, I. C. and Cohen, J.B., Advances in X-Ray Analysis, 27, (1983). 20. Li, A., Ji, V., Lebrun, J. and Ingelbert, G., Exper. Tech., 19 (2), 9-l 1 (1995). 21. Wem, H. and Moryson R., to be presented at the ICRS-5 in Linkoping(l997). 22. Noyan, I.C., Huang, T.C. and York, B.R., Critical Reviews in Solid State and Materials Sciences, 20 (2), (1995).