Integral quation Method for Ring-Core Residual Stress Measurement Adam Civín, Miloš Vlk & Petr Navrátil 3 Astract: The ring-core method is the semi-destructive experimental method used for evaluation of the homogeneous and non-homogeneous residual stresses. In this paper, the integral equation method to quantify non-uniform residual stress fields is discussed. Therefore, correctly determined and properly used caliration factors a i and i, necessary for the residual stress measurement y the ring-core method, are essential. The finite element method is used for the simulation of residual states of stress and to calculate relieved strains on the top face of the core, which are necessary for the susequent caliration factors determination. Keywords: Caliration factor, Relaxation factor, Integral equation method, Residual stress, Ring-core method, Strain gauge rosette. Introduction The ring-core method (RCM) is a semi-destructive experimental method used for the evaluation of homogeneous and non-homogeneous residual stresses, acting over depth of drilled core. Therefore, the specimen is not totally destroyed during measurement and it could e used for further application in many cases. In this paper, the integral equation method (IM) to quantify non-uniform residual stress fields, acting over depth of drilled core, is discussed. This method is descried y a physical theory with a proper mathematical model. The IM overcomes typical drawacks of the incremental strain method (ISM), which leads to incorrect results, where a steep gradient of residual state of stress occurs. The incremental strain method assumes, that the measured deformations d a, d, and d c are functions only of the residual stresses, acting in the current depth z of the drilled groove and they do not depend on the previous increments dz, including another residual stresses. More information aout the ISM could e found in papers [, ]. In fact, relieved strains on the top face of the core do not depend only on the stress acting within the drilled layer and its position, ut also on the geometric changes of the ring groove during deepening. These two factors are taken into account y the integral equation method, which has een particularly developed for the Hole-drilling method in 988 [3, 4]. Anyway, the integral equation method assumes, that strain relaxation on the top face of the core, for the particular depth of the drilled groove, is superposition of all deformations. These deformations are caused y a partial residual stresses, acting within every drilled layer, of all depth increments (see Figs. and ). Ing. Adam Civín; Brno University of Technology; Institute of Solid Mechanics, Mechatronics and Biomechanics, Technická 896/; 66 69 Brno; Czech Repulic, civin.adam@seznam.cz doc. Ing. Miloš Vlk, CSc.; Brno University of Technology; Institute of Solid Mechanics, Mechatronics and Biomechanics, Technická 896/; 66 69 Brno; Czech Repulic, vlk@fme.vutr.cz 3 Ing. Petr Navrátil; Brno University of Technology; Institute of Solid Mechanics, Mechatronics and Biomechanics, Technická 896/; 66 69 Brno; Czech Repulic, ynavra6@stud.fme.vutr.cz xperimental Stress Analysis 0, June 6-9, 0 Znomo. Czech Repulic. 3
Aovalasit et al. [5] and Zuccarello [6] extended theory of the IM into the ring-core application. They generally descrie the IM like a method, with a high sensitivity to the measurement errors, due to the numerical ill-conditioning of the equation set. This is caused y the position of the strain gauge on the top of the core, which is not enough sensitive to strains, relieved in the deeper layers of the drilled groove. Fig.. The ring-core method: geometry and general notation Fig.. Loading cases ased on theory of the integral equation method This paper descries how, application of the ring-core method with theory of the IM and the finite element method (FM) could e used for a numerical simulation and for determination of the uniform or non-uniform residual state of stress. The numerical simulation is used for the measurement of relieved strains on the top face of the model s core, at real positions of the strain gauge rosette s measuring grids. Caliration factors matrices a and, which are lower triangular, need to e calculated first in order to quantify the uniform or non-uniform residual state of stress, y the integral equation method. 3
. Integral quation Method Like each method, the integral equation method has its own theoretical ackground to define certain relations etween known and unknown parameters. Generally, total deformation measured on the top face of the core, for the ring-groove having depth H, is the integral of an infinitesimal strain relaxation components, caused y the residual stresses, acting at all depths in the range of 0 z H: k(h) H {[ A ( σ + σ ) + B ( σ σ )] cosα } dz k a,, c = (H,z) (z) (z) (H,z) (z) (z) k(z) = () 0 where a(h), (H), c(h) are the strains, measured y the strain gauge rosette on the top of the core s surface after a milling a groove having depth H, σ (z) and σ (z) are the unknown residual stresses acting at current depth z, α k(z) is the angle etween the maximum principal stress σ (z) and the direction of the strain gauge s measuring grid k = a and A (H,z), B (H,z) are the caliration functions, dependent on the shape and geometry of the ring-groove (Fig.). Using a three-grid rosette, q. () leads to a three linear equation set, from which the principal residual stresses and their orientation can e evaluated too. For i =,..., n finite depth increments q. () can e written as: i ki = ki k = a,, c () where the strain ki depends only on the stresses existing in the th layer y means of q. (3). Consequently, it is necessary to divide the maximum depth H into i intervals with the depth increment of Δz i and to approximate the function of the principal residual stresses σ (z) and σ (z) in each interval y the proper uniform distriution (Fig.3). Therefore, considering i finite depth increments, q. () can e written as: ai i = σ + σ + σ σ cos α (3) ki ( ) ( ) k in which ki is the strain component, relaxed on the surface solely due to the stress acting in the th layer, when i th depth increments have een achieved, σ, σ are the stresses within the th layer and a i, i are caliration factors. Knowledge of their dependence on the geometric changes of the ring-groove and on the disposition of the residual state of stress cross the depth of metallic material is essential and correspond of theirs appropriate application. Fig. 3. Approximation of non-homogenous residual stress field y the integral equation method Caliration factors a i and i of the lower triangular matrices a and cannot e determined y caliration coefficients K and K used for the incremental strain method [, ]. They can e possily otained y the finite element simulation. a a = a a ; = (4) a 3 a3 a33 3 3 33 33
.. Caliration factors determination For the correct determination of the depth-varying principal residual stresses σ, σ y the IM, it is necessary to determine caliration factors a i, i, respective generalized caliration factors A i, B i for each type of used depth increment distriution. Generalized caliration factors A i and B i are not dependent on the material properties. q. (3), followed from the asic equation of the integral equation method q. (), can e rewritten in order to determine the principal strains i and i as follows [5]: ( σ + σ ) + Bi ( σ ) ( σ + σ ) B ( σ ) i Ai σ = (5) i Ai i σ = (6) In order to determine factors a i, A i it is necessary to consider a iaxial state of uniform stress with σ = σ = MPa and the position of the strain gauge rosette placed centric on the top face of the core. Then, y considering q. (3) and relieved strains ai = i = ci = i : i ai i ai = ; Ai = = (7, 8) σ σ To evaluate factors i, B i, it is necessary to consider a pure shear state of uniform stress with σ = -σ = MPa. Then, considering q. (3), for α = 0 are relieved strains ai = - ci : ai i = ; σ B i ai i = = (9, 0) σ Finally, particular principal residual stresses σ,σ (α = 0 ), acting in th layer of drilled groove (Fig. and Fig.3), made y i =,..., n depth increments, can e determined as follows: ai + ci ai ci ai + ci ai ci σ = + ; σ = (, ) 4 Ai Bi 4 Ai Bi.. Residual stress determination The strain components e i, d i, m i [6], calculated from strains ai, i and ci, measured on the top face of the core y the three-element strain gauge rosette (Fig. ), after milling the i th step of the groove, are superposition of all strains, relieved in every th layer (see q. ()): ei = ai i s ci + = ai (3) di = i i p ci = ai (4) mi = i i q ci + ai i = The stress components s, p and q, represented y qs. (6 8), are calculated from stresses σ a, σ and σ c, acting in every th layer, when i =,..., n depth increments have een reached: s σ c + σ a = ; p σ c σ a = ; q c (5) σ σ a σ = (6, 7, 8) 34
Then, desired residual stresses s i, p i and q i, calculated from the strain components e i, d i, m i and stress components s, p and q, after every drilled groove s depth, can e found y following equations: i = s i ei ai s (9) a ii i = p i di i p (0) ii i = q i mi i q () ii The asis of the integral equation method is descried y qs. (9 ). These equations consider stresses, acting in every th layer, when i =,, n depth increments have een reached, and therefore strains, measured on the top face of core after milling another dept increment. Furthermore, correctly determined and properly used caliration factors a i, i, are required too. Finally, the principal residual stresses σ i and σ i, corresponding to the i th layer of drilled groove, can e re-calculated y q. (): ( p q ) σ σ = s ± + () i, i i i i.3. Step optimization Papers [5, 6] made y Aovalasit et al. and Zuccarello are focused on the step optimization, which is important for the IM too. The strain measurement errors influence, which occurs in practice, affects accuracy of the residual state of stress determination. Influence of the strain measurement errors depends particularly on the numer and magnitude of the depth increment distriutions Δz i and consequently on the maximum depth H of the drilled groove. Study of the step distriution s influence on the determination of caliration factors a i, i and on the susequent residual stress state determination is not in the scope of this paper. Therefore, total depth of drilled groove H = 5 mm has een made y n = 8 optimized depth increments (see Tale ), as recommended in [5, 6]. Optimum depth increment distriution minimizes error sensitivity of the experimental measurement and consideraly improves the numerical conditioning. Tale. Distriution of the depth n = 8 increments Δz i for a total depth of H = 5 mm Depth increment Δz i [mm]: Δz Δz Δz 3 Δz 4 Δz 5 Δz 6 Δz 7 Δz 8 0.6 0.45 0.4 0.4 0.45 0.5 0.7.5 3. F-simulation A prerequisite for correct and accurate measurement of residual strains on the top of the core is to use the finite element simulation. Therefore, the ANSYS analysis system is used for the susequent F-simulation. F-analysis is ased on a specimen volume with dimensions of a x a = 50 mm and thickness of t = 50 mm. Due to symmetry, only a quarter of the model has een modelled with centre of the core on the surface as the origin. The shape of the model is simply represented y 35
a lock with planar faces with a quarter of the annular groove drilled away (Figs. 4 and 5). The annular groove has een made y n = 8 increments with the different step s size Δz i (Tale ). The maximum depth of drilled groove is H = 5 mm. Dimension of the outer diameter is D = r i = 8 mm and groove width is h = mm. Fig. 4. Quarter of gloal solid model Fig. 5. Detail of the core with finite element mesh Linear, elastic and isotropic material model is used with material properties of Young s modulus = 0 GPa and Poisson s ratio μ = 0.3. Length and width of each measuring of grid of FR-5--3LT strain gauge rosette is l = 5 mm and w =.9 mm respectively [7]. In case of known directions of principal residual stresses, placing of the three-element strain gauge rosette on the top of the ring-core is shown in Fig.. Strain measurement on the top of the core is made y the integration across rosettes measuring grid surface. In case of the iaxial state of uniform stress simulation, applied surface s pressure p = p r = MPa is demonstrated in Fig. 6a and pressure p r = p cos(α), p t = p sin(α) (load s magnitude is varying with the angle 0 α 90 ) in order to simulate the pure shear state of uniform stress is demonstrated in Fig. 6. All loading conditions are applied to the oth inner sides of the ring-groove, in the range of the each depth increment. Fig. 6a. Loading conditions for the iaxial state of uniform stress simulation Fig. 6. Loading conditions for the pure shear state of uniform stress simulation 4. Results Process of the ring-core method s F-simulation has een carried out. Strains, relieved on the top face of the core, have een measured for two types of loading oundary conditions, in order to determine caliration factors. Tale contains magnitudes of strains, measured in 36
case of the iaxial state of uniform stress ( ai = i = ci = i ), see Fig. 6a. Appropriate caliration factors a i, calculated from measured strains y the q. (7), are shown in Tale 3a. The last column of the Tale represents the sum of all strain increments, caused y the unit stress σ = MPa, acting in every drilled layer (see Fig. ). This sum i = e i is a necessary part of residual stress determination y the q. (9). Tale. Relieved strains measured y the F-simulation of the iaxial state of uniform residual stress i [] 3 4 5 6 7 8 e i = i i= 3.03 07 3.03 07 4.94 07.44 07 7.39 07 3 6.8 07 3.66 07. 07. 06 4 7.33 07 4.5 07 3.3 07.99 07.70 06 5 8.3 07 5. 07 3.84 07.97 07.03 07.3 06 6 8.9 07 5.76 07 4.36 07 3.58 07 3.0 07.90 07.75 06 7 9.50 07 6. 07 4.79 07 4.06 07 3.66 07.95 07. 07 3.33 06 8 9.93 07 6.55 07 5. 07 4.4 07 4.0 07 3.54 07 3.35 07.48 07 3.95 06 Tale of strains, measured y the simulation of pure shear state of uniform stress is not included in this scope. Corresponding caliration factors i, calculated y the q. (9) and necessary for solving qs. (0, ), are shown in Tale 3. Represented results are valid only for dimension of the groove s outer diameter D = 8 mm, groove s width h = mm and total depth of drilled ring-groove H = 5 mm. Furthermore, results are valid for the groove made y n = 8 increments, with the depth distriution according to Tale and for the FR-5--3LT strain gauge rosette. Tale 3a. Caliration factors a i for optimized depth increment a i [] 3 4 5 6 7 8 i= 0.038 0.059 0.057 3 0.0659 0.0384 0.0 4 0.0770 0.0474 0.039 0.008 5 0.0864 0.0548 0.0403 0.03 0.03 6 0.0937 0.0605 0.0458 0.0376 0.037 0.000 7 0.0998 0.065 0.0503 0.046 0.0384 0.030 0.03 8 0.04 0.0688 0.0537 0.0463 0.043 0.037 0.035 0.06 Tale 3. Caliration factors i for optimized depth increment i [] 3 4 5 6 7 8 i= 0.0300 0.050 0.063 3 0.0670 0.040 0.044 4 0.0793 0.055 0.0374 0.047 5 0.0903 0.0605 0.0469 0.0383 0.080 6 0.0998 0.068 0.0544 0.0473 0.049 0.030 7 0.09 0.0756 0.066 0.0554 0.0539 0.048 0.044 8 0.98 0.0839 0.0693 0.0636 0.064 0.067 0.0696 0.0830 The residual stress magnitudes s i = σ i = σ i, corresponding to the every drilled i th layer, in case of considered iaxial state of uniform stress, are shown in Tale 4. They have een calculated according to q. (9), y using sum of strains i = e i from Tale and y using caliration factors a i shown in Tale 3a. 37
Tale 4. Re-calculated stress components for the iaxial state of uniform residual stress s i [MPa] 3 4 5 6 7 8 i=.00.49.00 3.9.58.00 4.3.3.63.00 5.7.69.30.68.00 6 3.09 3.3.95.37.63.00 7 3.50 3.83 3.67 3.5.38.7.00 8 3.98 4.5 4.50 4.06 3.9.65.74.00 The residual stresses results, located on the diagonal in Tale 4, proove the correctness of used equations. In case of non-uniform residual state of stress distriution, qs. (9 ) in order to determine the principal residual stresses σ i and σ i y q. (), must e calculated all together. 5. Conclusion This paper descried how application of the ring-core method with theory of the integral equation method and the finite element method could e used for the numerical simulation and susequent determination of uniform or non-uniform residual state of stress. In order to descrie the non-uniform residual state of stress y the integral equation method a numerical caliration factor s matrices a and, which are lower triangular, have een determined. For this reason, two types of loading conditions have een applied to the F-model. Appropriate equations to evaluate particular principal residual stresses σ,σ (α = 0 ), acting in th layer, when groove has i =,, n depth increments and the principal residual stresses σ i and σ i, corresponding to the i th layer of drilled groove, have een demonstrated for optimized step distriution. Acknowledgement This work was supported y the specific research FSI-S--. References [] Civín A. and Vlk M.: Assessment of Incremental Strain Method Used for Residual Stress Measurement y Ring-Core Method, In 48 th International Scientific Conference xperimental Stress Analysis 00, Velké Losiny, 00, pp. 7-34, ISBN 078-80-44-533-7 [] Civín A. and Vlk M.: Ring-Core Residual Stress Measurement: Analysis of Caliration Coefficients for Incremental Strain Method, Bulletin of Applied Mechanics, Vol. 6, No.4, pp.77-8, ISSN 80-7 [3] Schaer G. S.: Measurement of Non-Uniform Residual Stresses Using the Hole-Drilling method, Part I - Stress Calculation Procedures, Transactions of the ASM, ser. H, Journal of ngineering Materials and Technology, Octoer 988, vol. 0, no. 4, pp. 338-343 [4] Schaer G. S.: Measurement of Non-Uniform Residual-Stresses Using The Hole-Drilling Method, Part II - Practical Application of the Integral Method, Transactions of the ASM, ser. H, Journal of ngineering Materials and Technology, Octoer 988, vol. 0, no.4, pp. 344-349 [5] Aovalasit A., Petrucci G. and Zuccarello B.: Determination of Nonuniform Residual Stresses Using the Ring-Core Method Transactions of the ASM Journal of ngineering Materials and Technology, April 996, vol. 8, no., pp. 4-8 [6] Zuccarello B.: Optimization of Depth Increment Distriution in the Ring-Core Method, Journal of Strain Analysis for ngineering Design, 996, vol. 3, no. 4, pp. 5-58 [7] Preusser Messtechnik GmH [online] URL: <http://www.dms-technik.de/> [cit. 00-0-8] 38