AN EXAMINATION OF THE RELEVANCE OF CO-INDENTATION STUDIES TO INCOMPLETE COVERAGE IN SHOT-PEENING USING THE FINITE-ELEMENT METHOD

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1 Journal of Mechanical Working Technology, 11 (1985) Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands AN EXAMINATION OF THE RELEVANCE OF CO-INDENTATION STUDIES TO INCOMPLETE COVERAGE IN SHOT-PEENING USING THE FINITE-ELEMENT METHOD S.A. MEGUID and M.S. KLAIR Applied Mechanics Group, School of Mechanical Engineering, Cranfield Institute of Technology, Cranfield, Bedford MK43 OAL (England) (Received February 24, 1984; accepted October 5, 1984) Industrial Summary The achievement of complete coverage during the application of shot-peening to critical components represents a major concern in the aircraft and nuclear industries. It has been accepted, for many years, that complete coverage of the component is needed to attain the beneficial effects of the treatment upon the fatigue life of such components. It has also been proposed, by a number of industrialists, that incomplete coverage may apparently shorten the fatigue life of the component because of the presumed presence of tensile residual stresses in the uncovered areas. The complete and accurate solution of this problem is very complex. However, in order to simplify the situation, an elasto-plastic finite-element analysis of simultaneous indentations of a bounded solid, by two smooth, flat, rigid punches under plane-strain conditions was cnnsiderecl. TWG aspects of the co-indentation process were accordingly examined in the initial analysis of the results presented: (i) the influence of punch separation upon the resulting residual-stress field, and (ii) the successive development of the plastic zone for different incremental-indentation pressures. The results of the work were also extended to take into account the relationship of separation between the punches to incomplete coverage in the shot-peening process. In particular, it was desired to determine the critical separation beyond which the relevant residual stresses change from compressive to tensile, The present work highlights the fact that compressive residual stresses resulting from incomplete coverage were attained even where the separation between the punches was four times the punch width. Notation total surface area of target projected area produced by plastic indentation due to a single shot surface area covered by plastic indentations due to shot-peening coverage height of the bounded solid width of the bounded solid thickness of the bounded solid width of each of the punches 1985 Elsevier Science Publishers B.V

2 1. Introduction separation of the two punches - interference ratio elastic modulus of the material plastic modulus of the material yield stress equivalent stress equivalent plastic strain deviatoric stress tensor incremental plastic strain tensor non-negative constant strain-hardening coefficient indentation pressure cartesian co-ordinates Poisson's ratio punch displacement normal stresses in x- and y directions, respectively shear stress residual normal stresses in x- and y directions respectively residual shear stress maximum compressive residual stress along x direction Shot-peening is a cold-working process used mainly to improve the fatigue life and corrosion resistance of metallic components. (For a detailed desrrlptim of the prcccss see, for exliaii~ple [i-31 j. The resuits are accomplished by bombarding the surface of the component with small spherical shots made of hardened castcsteel or conditioned cut-wire or glass beads at a relatively high velocity. After contact between target and shot has ceased, a small plastic indentation will have been made, causing stretching of the top layers of the exposed surface. Upon unloading, the elastically stressed subsurface layers tend to recover their original dimensions, but the continuity of the material in both zones, the elastic and the plastic, does not allow this to occur. Consequently, a compressive residual stress system followed by tensile will be trapped in the bounded solid. These compressive residual stresses are highly effective in preventing premature failure under conditions of cyclic loading, since fatigue failure generally propagates from the free surface of the component, and starts in a region which is subjected to tensile stresses. Suppose in a peening process a target surface area A is exposed to the jet stream. Let the projected area produced by plastic indentation due to each shot striking the surface be b. Thus, even if the number of shots impinging on the target surface is (Alb), the whole of the target surface area would not necessarily be covered by indentations: some parts of the target would be struck by a shot more than once, and the others would escape. The ratio

3 RIGID SMOOTH PUNCH BOUNDED - SOLID I ///fl//////////////////// Y RIGID SMOOTH FOUNDATION Fig. 1. A schematic showing the notation used in the study. between the area covered by the plastic indentations S and the total surface area of the component A is the coverage C. Ccmplete or hll coverage means that S = A. It is worth noting that in some circumstances, for example in the peening of very hard metals, and very complex geometries, it is difficult not only to achieve but also to monitor full coverage. The presence of unpeened areas in a peened component could represent a problem to the designer, especially if the implications of incomplete coverage are not well understood. Ultimately, it is the compressive residual-stress field that controls the initiation and the early stage propagation of fatigue damage. It is therefore of paramount importance to examine the effect of incomplete coverage upon the resulting residual-stress field. The complete and accurate solution of this problem is very complex. However, in order to simplify the situation, an elasto-plastic finite-element analysis of the simultaneous indentations of a bounded solid of height h, width w and thickness t by two smooth, flat, rigid punches under planestrain conditions has been considered. The punch width is assumed to be 2a and the separation distance 2c; see Fig. 1 for details of nomenclature. The bounded solid is assumed to be made from an elastic linearly strainhardening material, which could be identified by the elastic modulus E, the

4 plastic modulus Ep and the yield stress oy. The analysis pertains to quasistatic indentations, and strain-rate and inertia effects have been ignored. It is worth mentioning that problems concerned with the elasto-plastic indentation behaviour of metallic components have received considerable attention since the pioneering investigation of Prandtl in 1920 [4]. None of these investigations, except the work by Meguid [2] and Meguid, Collins and Johnson [5], have dealt with the problem of co-indentation of a bounded solid by two smooth flat rigid punches. In their work, the co-indentation problem was investigated experimentally using etching techniques, and theoretically using an upper-bound solution; no calculations were made of the unloading residual stresses, and for this reason the present investigation was undertaken. In this study, particular attention has been devoted to two related questions: (i) the influence of punch separation upon the resulting residual-stress field, and (ii) the successive development of the plastic zone for different incremental-indentation pressures. The results of the work have also been extended to take into account the relevance of separation between the punches to incomplete coverage in the shot-peening process. In particular, it was desired to determine the critical separation ratio (cla) beyond which the relevant residual stresses change from compressive to tensile. 2. Theoretical considerations The determination of the equivalent stress Z as defined by: and the equivalent plastic strain as defined by: was based upon the von Mises' yield criterion and the following Prandtl- Reuss relations: The above definitions were used in the elasto-plastic quasi-static solution for the stresses and strains resulting from the present co-indentation studies. The present solutions were obtained by utilising the standard facility of the "PAFEC 75" suite of finite-element programs, detailed in [6]. These solutions were developed for different interference ratios (cla), specimen heights h and strain-hardening coefficients H'. In order to obtain good approximations of the residual stress field, it was thought desirable to use the finite-element idealisation shown in Fig. 2,

5 PUNCH \ RELATIVELY FINE MESH TRANSITION ELEMENTS RELATIVELY COARSE MESH Fig. 2. One-half of the bounded solid with the finite-element mesh showing the restraints and the punch. u hich includes three distinct regions, as follows: a relatively fine mesh and a relatively coarse mesh, joined together by a transition region. The 8-noded isoparametric element was used to describe the bounded solid. Due to the symmetrical nature of the problem and consequently of the solution, only one-half of the bounded solid was examined. It is also worth mentioning that slight modifications were introduced for other geometrical and loading conditions considered. The assumed rigidity of the punch meant that all nodes under it must experience the same y displacement. This has been allowed for in the study by the use of "repeated freedom" for these nodes. The current elasto-plastic solutions were obtained using incremental punch pressure and method of successive elastic solutions. The appropriate solution for a pre-determined increment of indentation pressure was obtained iteratively using a compromise method of solution between initial strain and initial stress approaches. For the solutions reported here, the accuracy criterion adopted was that the discrepancy resulting from successive estimates of the calculated equivalent stress at each gauss point

6 92 RESIDUAL STRESS, nxrx MPa RESIDUAL STRESS, ojii MPa Fig. 3. Effect of interference ratio upon residual normal stress distributions (G,) for H' = 0.2 and h/a = 20: (a) beneath the centre-line of the punch; (b) at the centre-line of the bounded solid.

7 should not exceed 1% of the provided equivalent stress (material property) at a given value of an equivalent plastic strain. The material behaviour was assumed to be elastic-linear strain-hardening. The relevant properties of the model material (mild steel) used in this study are as follows. Young's modulus E: 209 GN/m2; Poisson's ratio v: 0.30; yield stress oy : 240 MN/m2. The finite element results were also utilised using a specially developed computer program (containing the subroutines POLY, MATINV, COEFF, H PLOT AXISMO, CHAWRT, AXIPLO) to obtain the best fit for the relevant quantities; the results are discussed in the following section. 3. Analysis of results and discussions The present investigation into the incomplete coverage of shot-peened components is simplified by co-indentation of a bounded solid by two rigid smooth, flat, punches. In spite of the fact that in the present study no account has been taken of the inertia, strain rate and multiple-indentation effects, the residual-stress patterns illustrated in Figs. 3, 4 and 5 closely resemble those generated by shot-peening. This resemblance could be attributed to the hypothesis that the interaction between the plastic zones and recovery of both elastic and plastic regions is similar for both the static and dynamic problems. It is anticipated however, that the magnitudes of the resulting plastic zone and compressive residual stresses will be different. In view of the importance of the normal unloading residual stress $, to the beneficial effects of shot-peening on cornp~nent fztigue life, it was thought desirable to devote the discussion to these stresses. It was also thought reasonable to concentrate upon the case where (c/a) = 4 as this is an INTERFERENCE RATIO (c/o ) INTERFERENCE RATIO Icla) Fig. 4. (a) relationship between maximum residual stress (g:,) and interference ratio (cla); (b) relationship between depth of maximum residual stress and interference ratio (cla).

8 94 RESIDUAL STRESS, ax[ Mpa RESIDUAL STRESS, d Mpa Fig. 5. Effect of height upon residual normal stress (6,) distribution for H' = 0.2 and c/o = 4: (a) beneath the centre-line of the punch; (b) at the centre-line of the bounded solid.

9 extreme case and would highlight the effects of incomplete coverage. Two positions were analysed; beneath the centre line of the punch and at the centre line of the bounded solid. It is worth noting that the results obtained at the centre line of the bounded solid will be of interest to incomplete coverage. Figures 3(a) and 3(b) illustrate the effect of the interference ratio (cla) upon the resulting residual-stress pattern under the centre line of the punch and at the centre line of the bounded solid, respectively. It is interesting to note that the maximum residual stress occurs at a definite distance below the surface in both cases. Figure 3(b) also demonstrates that increasing the interference ratio (cla) from 0 to 4 results in a decrease in the maximum compressive residual stress E:, from MN/m2 to MN/m2 and a decrease in depth of the maximum compressive residual stress from 6.8 mm to 1.2 mm. This is clearly demonstrated in Fi$s. 4(a) and 4(b). The best fit of the decay of maximum residual stress of o:, with the increased interference ratio (cla) can be given by the following: where 0 < cla < 4, while the straight line relationship given in Fig. 4(b) correlates the depth of maximum compressive residual-stress to the interference ratio, as follows: The study was extended to include the effect of varying the height h and the strain-hardening characteristics of the bounded solid upon the resulting residual stresses for c/a = 4. It appears from Figs. 5(a) and 5(b) that the residual-stress profile changes from compressive near the surface to tensile near the middle and back again to compressive at the bottom of the bounded solid for hla = 20. For hla < 10, the profile changes sign only once and thereafter remains unaltered. Figures 6(a) and 6(b) demonstrate that an increase in the strain-hardening coefficient H' results in an increase in the magnitude of the maximum compressive residual stresses for the same indentation pressures. This is to be expected, since the increased H' results in a decrease in the plastic zone developed by the indenters. It is also interesting to note that the maximum residual stress occurs quite close to the surface at the centre line of the bounded solid (see Fig. 6(b)), while it occurs at a definite distance below the surface beneath the centre line of the punch (Fig. 6(a)). The development of the plastic zone and the manner in which the zone spreads are important aspects of the finite-element solution to the present problem. Figures 7(a), 7(b) and 7(c) show a sequential illustration of the effect of the interference ratio (cla), normalised height (hla) and strainhardening coefficient (H') upon the growth of the plastic zone for different punch loads. These figures confirm that there exists interference between the

10 RESIDUAL STRESS, u,: MPa RESIDUAL STRESS, 20 T MPa Fig. 6. Effect of strain-hardening upon normal residual-stress distributions (4,) for c/a = 4 and hla = 20: (a) beneath the centre-line of the punch; (b) at the centre-line of tha bounded solid.

11 C 1 I 22-4 rnm Fig. 7. Plastic zone development during various incremental loadings: (a) effect of interference ratio (cla) for H' = 0.2 and hla = 20,

12 Fig. 7(b). Effect of strain hardening (H' ) for C/a = 1 and h/a = 20.

13 Fig. 7(c). Effect of hight (hla) for H' = 0.2 and C/a = 1.

14 Fig. 8. Etching results demonstrating the effect of interference ratio upon the development, interaction and spread of plastic zone for h/a = 10 and p = 618 MN/m2 : (a) c/a = 3; (b) c/a = 2; (c) cia = 1 (after [2] and [5]).

15 evel- = 2;

16 plastic zones for 0 < c/a < 1 for a maximum pressure of 680 MN/m2 and strain-hardening coefficient H' < 0.1. The figures also indicate that an increase in the strain-hardening coefficient H' (> 0.1) decreases the spread of the plastic zone, resulting in separate plastic zones for the punches for thc same co-indentation conditions. Figure 7(c) demonstrates the effect of height upon both the spread of the plastic zone to the foundation for 0 < hla < 10, and the interference between the punches for c/a = 1. The above findings confirm the overall trend of the experimental results obtained earlier by Meguid [2] and Meguid et al. [5] on the same topic, in which rimming steel was loaded by two flat, plane indenters and etched by immersion in a solution of Fry's reagent. Typical etching results are shown in Figs. 8 and 9. Those shown in Fig. 8 are for hla = 10. In Fig. 8(a) where c/a = 3 and p = 618 MN/m2, the two deformation zones are separate and localised under the surface. However, with h/a = 10, c/a = 1 and p = 618 MN/m2, Fig. 8(c) indicates that the plastic zones resulting from each punch interact with one another and that the deformation spreads through to the foundation of the bounded solid: this is in accord with the finiteelement predictions of Fig. 7(c). The etching in Fig. 8(b) shows the intermediate situation, with c/a = 2, where the two deformation zones are beginning to interact and to spread to the base of the bounded solid. Similar general agreement between the finite-element predictions and the earlier experimental results [2,5] is borne out by the etchings of Fig. 9, where hla = 18, and p = 618 MN/m2. For this depth, the finite-element predictions (Fig. 7(a)) are that the plastic deformation will be localised under the two punches. In fact, both the finite-element predictions and the etching results indicate that, the p!astic deformztion wi! be lccafised underneath the two punches irrespective of the punch interference ratio and will never spread to the foundation. 4. Conclusions The main aim of the present investigation was to study the effect of large separation ratios upon the resulting residual stresses beneath the uncovered area. In particular, it was desired to examine whether for large separation ratios (cla = 4), the residual stress pattern changes to tensile near the surface. The predictions of the present finite-element analysis, applicable to smooth, flat, rigid punches, can be summarised as follows: (i) The compressive residual stresses do not reach their maximum value at the surface, but rather at a definite distance below the surface for both the centre line of the bounded solid and underneath the centre line of the punch. (ii) The depth of the maximum compressive residual stress increases with the decrease in the interference ratio (cia) for both the centre line of the bounded solid and underneath the centre line of the punch. (iii) The maximum compressive residual stress QiX increases with the

17 decrease in the interference ratio (cla) at the centre line of the bounded solid, while values of :ix are almost the same underneath the centre line of the punch. (iv) The residual-stress profile changes from compressive near the surface to tensile near the middle and back to compressive at the bottom of the bounded solid for hla = 20. For (hla) < 10, the profile changes sign only once and thereafter remains unchanged. (v) An increase in the strain-hardening coefficient H', results in an increase in maximum compressive residual stress both beneath the centre line of the punch and at the centre line of the bounded solid. Finally, it is worth noting that for all the separation ratios considered in this study, the residual stresses near the surface, both beneath the centre line of the punch and at the centre line of the bounded solid, were compressive. It must be appreciated, however, that the inclximum separation ratio used in the present study (cla = 4) is unrealistic in actual peening applications. However, it represents an extreme value in order to enable the authors to assess its effect upon the residual-stress pattern. It is clearly possible to strengthen the relevance of the study to incomplete coverage in shot-peening, for example, by considering the inertia, friction, statistical nature, and strain-rate effects. Further, it is appreciated that the present work deals only with the plane-strain situation, as opposed to the axially-symmetric nature of indentations arising from the actual shotpeening process. It is therefore reasonable to assume that the shot-peening results of the study must be viewed with some caution. It is also worth mentioning that the improvement in component fatigue life is controlled by the percentage coverage; full coverage is obviously recommended for best fatigue-life improvement. Acknowledgements The authors gratefully acknowledge the financial support given by the Science and Engineering Research Council (S.E.R.C.) under Research Grant GR/B/ The support given by lmpact Finishers Limited and Vacu-Blast Limited is also acknowledged. The advice provided by Dr. A. Stafford of PAFEC Limited and Mr. A. Lawrance of the Cranfield Computing Centre on the use of the PAFEC suite of programs is gratefully acknowledged. References 1 S.A. Meguid, Shot-Peening and Peen-Forming: Theory and Applications, Short Course organised by Cranfield Institute of Technology, S.A. Meguid, The Mechanics of the Shot-Peening Process. Ph.D. Thesis, U.M.I.S.T., S.A. Meguid, Troubles caused by metal fatigue and engineering interest in compressive residual stresses. Paper presented at the First International Conference on Impact Treatment Processes, Cranfield Institute of Technology, Sept. 5-8th, 1983.

18 4 W. Johnson, R. Sowerby and R.D. Venter, Plane-Strain Slip-Line Field for Metal- Deformation Processes, Pergamon Press, Oxford, S.A. Meguid, I.F. Collins and W. Johnson, Int. J. Mech. Sci., 19 (1977) 1. 6 R.D. Henshell (Ed.), "PAFEC 75" Manual, PAFEC Limited, Nottingliam, 1980.