© AES-Advanced Engineering Solutions (Ottawa, Canada) All rights are reserved Kh. Farhangdoost, Ehsan Pooladi B. Determine of residual stresses around cold-worked hole based on measured residual strains Kh. Farhangdoost 1*, Ehsan Pooladi B 2 1 Assoc. Prof., Dept. of Mechanical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran (Email: farhang @um.ac.ir) 2 PhD student, Dept. of Mechanical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran (Email: ehsanpb @ yahoo.com) *Corresponding Author Abstract Compressive residual stress due to cold working decreases tendency of mechanical parts to initiate and growing fatigue crack and increases fatigue life. So determining residual stress is one of the important subjects in the last decades. There are two categories for measuring residual stress as destructive and nondestructive tests. In this paper distribution of residual stress around a hole which cold worked by expansion due to interface of a shaft into the hole and then removing it, is determined by measuring residual strains. There is a good agreement between residual strain measurements and residual stress in which is a good characteristic feature of residual stress. Fourier series for relating measured strains to residual strains in each position, Hook's law in elastic region and power law in plastic region have been used for derive the residual stresses. This analytical technique is nondestructive and simple compared to other methods. Keywords Residual stress, Cold working, cold-worked hole, residual strain, total strain theory, Henkey 2 © AES-Advanced Engineering Solutions (Ottawa, Canada) All rights are reserved total strain theory, thick-walled cylinder, plasticity 1 Introduction Residual stresses can be introduced into parts during manufacturing, e.g., cold-reducing operations, autofrettage process,..., or can produced by the next process after manufacturing for being them as beneficial effect. Some complementary process after manufacturing produce compressive residual stress which is useful for increasing fatigue life, e.g., shot penning, cold working,.... The fatigue life improvement of cold expanded fasten holes is attributed to the presence of compressive residual stress induced by cold expansion. The fatigue fracture of fasten holes account for 50-90% of structure fracture of aircrafts. Over the last 40 years, because of its simple realization and remarkable enhancement of the fatigue life of holes (usually 3-5 times than that of holes without cold expansion), the cold expansion process has been widely used to improve the fatigue life of components with fasten holes [1]. The cold-expansion, which is developed by the "Fatigue Technology Inc. (FTI, 1994), is obtained by using increased pressure to Kh. Farhangdoost, Ehsan Pooladi B. plasticize an annular zone around the hole. The pressure on the surrounding material is realized by interference generated between the drilled plate and the pressuring element, i.e. the mandrel. When the mandrel is removed and the superficial pressure on the hole is erased, a residual stress field is created due to action of the elastic deformed material on that under plastic condition [2]. In the past, analytical models, experimental techniques and numerical simulations have been developed to predict the residual stress field induced by a cold-expansion process of a hole. Analytical studies have determined the closed-form solution of the residual stress for considering the material's yield limit on unloading step (Guo, 1993; Nadai, 1943; Hsu and Forman, 1975; Rich and Impellizzeri, 1977) [2]. H. Jahed et al. [3] presented elasticplastic boundaries and residual stress fields induced by cold expansion of fastener holes using variable material properties. Generally there are two ways to determine residual stress field as destructive, e.g. hole drilling method, Sach's boring technique and non-destructive test as X-ray and neutron diffraction, ultrasonic, Barekhausen parazit and.... These methods have been presented by many previous investigators. A comparison between the mentioned techniques have been considered and concluded by P.J. Withers et al. [4]. An analytical solution using strain gradient plasticity theory is presented for the borehole problem of an elasto-plastic plane strain body containing a traction-free circular hole and subjected to uniform far field stress, by X-L Gao [5,6]. In this paper, an analytical solution based on classic plasticity theory using Henkey total strain theory for plastic region is presented. The main objective of this study is to solve analytical equations in order to derive residual stress field based on measured strains. The material behaviour considered in this method is linear work-hardening with power law relation in plastic region and also Baushinger effect has been neglected. The condition of problem is plane strain. Of course according to A.Stacy [7], neglecting of the Baushinger effect causes the residual compression at the bore to be overestimate. But in this paper we have not reverse loading and there is only elastic unloading during removing the mandrel, so Bushinger effect can be neglected. According to D.J. Smith et al.[8,9], in this paper ,the strains are represented by Fourier series. In fact Fourier series have been used to relate strain in each position to measured strains. Also because of existing symmetry, all of shear stress components are zero. The conditions of entrance mandrel into the hole such as velocity, friction coefficients,... affect the rate of plasticity around the hole, but latest state which mandrel fully entered has been considered and also for removing the mandrel out. At final state of enter mandrel, a radial pressure, named shrunk pressure, applied to internal surface of hole. So for simplicity, this problem has been simulated with a thick-walled cylinder with an internal pressure. This pressure dependent on interface value which is the difference between mandrel and hole diameter. If this value is small enough, the parts will be in elastic region and the shrunk pressure is calculated by Lame's equation. But if it's larger than special value, some part of hub and shaft will be in plastic zone and Lame's equation won't be valid, as seen in this paper. 2 Theory 2 © AES-Advanced Engineering Solutions (Ottawa, Canada) All rights are reserved Kh. Farhangdoost, Ehsan Pooladi B. As stated before, the effect of the velocity of entrance (and removing) the mandrel, contact and Baushinger effect have been neglected. So with omitting the contact effects, formulation concentrated on final state of entrance, i.e., creation a radial pressure from the mandrel to internal surface of hole and also for final state of removing the mandrel or fully unloading. So for simplicity, let us simulate a hole at the end of loading (expanding) by a thick-walled cylinder with internal pressure. Similarly the mandrel can be simulated by the mentioned cylinder with external pressure only. We assume that the unloading process is elastic, and plane strain condition is dominant. Also for plastic region, power law relation and total strain theory (Henkey strain theory) are used, as seen later. Analytical solution for plane stress problem has been presented by other investigators before, e.g. Nadai, of course the previous researches need some modifications which authors will do it soon. At first let take a thick-walled cylinder with internal pressure only, into account for loading process as seen in Fig.1. ro -Strain-displacement relations: u du ε = , ε rr = θθ r dr -Compatibility equation: dε r θθ = ε rr - ε θθ dr (2) (3) -Power law relation for plastic region: rpo rso Where, rso is inner radius of the hole, Ps internal (shrunk) pressure, Ppo the radial pressure created at plastic radius (elasto-plastic limit). When the interface is more than a special value, some material around the hole, up to plastic radius (rpo), pass the elastic limit and will be in the plastic region meanwhile the outer ring, with radius bigger than the plastic radius, is in elastic region yet. Removing the mandrel leads to create tendency for spring back the elastic zone to previous (initial) position. This tendency is precluded by the inner plastic zone. So a compressive force is applied from material in outer ring to that of inner part. This compression enhances the fatigue life and fatigue cracks can't initiate or grow in compressed region which is immediately around the hole. Now let solve thick-walled cylinder with an internal pressure in both plastic and elastic states. -Equilibrium equation in cylinder coordinates: dσrr σθθ - σrr = (1) r dr Ps σe = kεem = (σ1y mEm )εem (4) Where σe and εe are Von-Mises effective stress and strain, also σy , E, m are yield stress, elastic modulus Figure 1. Thick-walled cylinder with internal pressure. 3 © AES-Advanced Engineering Solutions (Ottawa, Canada) All rights are reserved and strain respectively. -Von-Mises effective stress: hardening power Kh. Farhangdoost, Ehsan Pooladi B. 3 (σ - σ ) (5) 2 θθ rr -(Henkey) Total strain theory for plastic Where, D is effective strain at inner surface of the hole. Boundary conditions of the above problem are as follows: region: σ σe = 3 εe ε p = λσ ' = λ(σ - σ m ) where λ = ij ij ij 2σ rr r = r so (6) e (6 -1) 2 1 ε p = λ(σ - (σ rr + σ zz )) θθ 3 θθ 2 (6 - 2) ε yields (12) =D σ ε (6 - 3) (7) y E (14) Substitution (14) into (10): σ r2 y po D= E r2 so (15) Now put (15) & (10) into (4): r po 2m σ =σ ( ) e y r 2.1 Plastic solution (rso≤ r ≤rpo) With substitution equation (5) into (6), constitutive equation is derived as: (8) Using equation (5) in ( 8) leads to: 3 3 ε rr = εe , ε = ε θθ 2 2 e Satisfying compatibility equation causes: = er = r po Where, υ is Poision ratio. 3 εe (σ σ ) ε θθ 4 σ e rr θθ (13) er = r so → 1 σ zz = (σ rr + σ ) θθ 2 Hook's law for elastic region: E σ rr = [(1 υ)ε rr + υε ] θθ 1 υ 2υ2 E σ = [(1 υ)ε + υε rr ] θθ 1 υ 2υ2 θθ σ zz = υ(σ rr + σ ) θθ 2 r so ε =D e 2 r = P po rr r = r po 2 1 εp rr = 3 λ(σ rr - 2 (σθθ + σ zz )) ε rr (11) σ Or 2 1 εp zz = 3 λ(σ zz - 2 (σ rr + σθθ )) = 0 = P s (9) (16) Using latest relation and equation (5), solve equation (1); with considering (16) , (5) and integrating over the radius, a statement for radial stress in plastic zone is derived which satisfying boundary conditions (11) , (12) leads to: σ r 2m y po 1 1 p σ = [ ]-P rr s 2 m 2 m 3 r r m so (17) Tangential and axial plastic stresses are derived as: (10) σ dσ θθ = rr +σ rr dr leads to → 4 © AES-Advanced Engineering Solutions (Ottawa, Canada) All rights are reserved Kh. Farhangdoost, Ehsan Pooladi B. σ σ r 2m y po 1 2m 1 p = [ ]-P θθ s m 3 r 2m r 2m so σ r 2m y po 1 m 1 p σ = [ ]- P zz s 2 m m 3 r r 2m so (18) P r2 po po σe = [1 + θθ r 2 - r 2 o po r2 o ] r2 2 υP r 2 po po e σ = zz 2 r - r2 o po (20 - 1) (22 - 1) P r2 r2 s o un σ = [1 + o ] θθ r 2 - r 2 r2 o so (22 - 2) 2 υP r 2 s so un σ = zz 2 r -r2 o so (22 - 3) Radial and tangential strain during unloading are as follows: P rs 2 (1 - υ - 2υ2 ) r2 s o un ε = [1 - 2υ - o ] 2 2 rr E(r - r )(1 - 2υ) r2 o so (23 - 1) P rs 2 (1 - υ - 2υ2 ) r2 s o un ε = [1 - 2υ + o ] θθ E(r 2 - r 2 )(1 - 2υ) r2 o so (23 - 2) (20 - 2) (20 - 3) 2.3 For calculating radial pressure at elasto-plastic limit, according to validation of both elastic and plastic relations at exactly elastic boundary (rpo), equations (20-1) should be substituted into (5) and finally satisfying σe(rpo)=σy leads to: σ (r 2 _ r 2 ) σ r y o po y po 2 P = = [1 - ( ) ] po 2 r 3 3r o o P r2 r2 s o o un σ = [1 - ] rr 2 2 r -r r2 o so (19) 2.2 Elastic solution (rpo≤ r ≤ro) There are many papers which introduced elastic solution of cold-worked holes. So we use only the results as follows: P r2 r2 po po o e σ = [1 - ] rr r 2 - r 2 r2 o po relations are derived for elastic unloading; similar to elastic loading: (21) 2.3 Unloading (rso≤ r ≤ro) During unloading which assumed to be elastic, some stress created around the hole. The below 5 © AES-Advanced Engineering Solutions (Ottawa, Canada) All rights are reserved Residual stress field Residual stress field is obtained by subtracting the unloading stresses from loading parts. So in plastic region (rso≤ r ≤ rpo), they are as: σ r 2m 1 1 y po σ res = [ ]- P rr s m 3 r 2m r 2m so P r2 r2 s so (1 - o ) r2 - r2 r2 o so (24 - 1) Kh. Farhangdoost, Ehsan Pooladi B. series as follows: σ r 2m y po 1 2m - 1 σ res = [ ]- P θθ s m 3 r 2m r 2m so P r2 r2 s so o (1 + ) 2 2 r -r r2 o so (24 - 2) σ r 2m 1 m -1 y po res σ = [ ]- P zz s m 3 r 2m r 2m so 2 υP r 2 s so 2 r - r2 o so (24 - 3) And in elastic region (rpo≤ r ≤ ro), we have: P r2 r2 P r2 r2 po po o s so o res σ = [1 - ] [1 - ] (25 - 1) rr r2 -r2 r2 r2 -r2 r2 o po o so P r2 r2 P r2 r2 po po o s so res σ = [1 + ] [1 + o ] (25 - 2) 2 2 2 θθ r 2 - r 2 r r -r r2 o po o so 2 υ P r 2 2 υP r 2 po po s so σ res = θθ 2 2 2 r -r r -r2 o po o so (25 - 3) It should be noted that there are three unknown parameters, rpo, Ppo, Ps which should be determined. 2.4 Strain series expansion with Fourier If some strain gages are erected on some special points around the hole, the measured strains recorded by the mentioned gages, can be related to strain profile using Fourier series. In fact according to [8, 9], strains (similarly to each function) can be expanded by the Fourier ε (r, θ) = ε 0 (r, θ) + ε1A (r, θ) cos θ + ij ij ij ∞ ε1B (r, θ) sin θ + ∑ ε nA (r, θ) cos nθ + ij ij n=2 ∞ ∑ ε nB (r, θ) sin nθ (26) ij n=2 Which (r, θ) is the position of point in cylindrical coordinates and n is the amount of gauges; each at angle θ. With neglecting of high order coefficients and due to symmetry we have: 2B 4B ε 1θθA = ε 1θθB = ε θθ = ε 3θθA = ε 3θθB = ε θθ =0 3 New analytical procedure When entire the mandrel moves into the hole, the strain gauges record loading strains and when entire the mandrel removes out the measured value introduces residual strain. Strain due to unloading can derived as subtracting loading strain and residual strain, as: ε res = ε load - ε unload → ε unload = ε load - ε res (27) For determining residual stress field here, 3 pieces strain gages used at angles 0º, 120º, 270º. For simplicity, it's recommended that the gages erected in plastic zone. A criterion to find out a point is in plastic zone or not, according to equations (8), (9), is equality and opposite sign of radial and tangential strains. Because of symmetry, the strains recorded by gages at angle 90º and 270º should be the same. So at first a gage used at angle 90º and a known radius in radial direction and another at angle 270º and at same radius. After loading, if the values showed by the used gages are equal and 6 © AES-Advanced Engineering Solutions (Ottawa, Canada) All rights are reserved Kh. Farhangdoost, Ehsan Pooladi B. in opposite sign, then that radius would be in plastic zone, otherwise is in elastic zone. Now three gages are erected at the mentioned angles and in plastic zone tangentially. If the measured values by each gage are named εθθI, εθθII, εθθIII respectively, then Fourier coefficients of strain series will be derived as: ε 0 I II = ε + 2ε ε 2A ε 4A = 1 I 1 III ε - ε 2 2 1 I II 1 III ε - 2ε + ε 2 2 =- ( 28 ) So Fourier series for relating strain at each point to measured strains is rewritten as follows: 1 1 ( r , θ) = (ε I + 2ε II ) + ( ε I - ε II ) cos 2θ + θθ 2 2 1 1 (- ε I - 2ε II + ε III ) cos 4θ ( 29) 2 2 ε After fully loading, the tangential strain of a desired point in the same radius of strain gages position is calculated by using equation (29). Because the point is in plastic zone, the effective strain is derived through second equation of (9). One of unknown parameters is obtained by equations (10) & (15), i.e. plastic radius: r e D r po 2 2 r so DEr 2 so y 7 © AES-Advanced Engineering Solutions (Ottawa, Canada) All rights are reserved Where, r is the point radius. And the radial pressure at elasto-plastic limit (Ppo) is calculated by equation (21). Also by measuring residual strain and using equation (27), unloading strain will be derived. By second equation of (23) the shrunk pressure can be calculated. Finally with knowing plastic radius (rpo), shrunk pressure (Ps) and elasto-plastic pressure (Ppo) and using equations (24) , (25) residual stress field is determined. 4 Conclusions As stated before there are two ways for determining residual stress field as destructive and non-destructive tests which have some advantages or disadvantages. Approximately there is not an analytical background in form of existed paper in previous researches. In this paper an analytical formulation which can verifies finite element models, have been presented. Simplicity, take reality material behaviour into consideration and accuracy compared with numerical solution are some of benefits of these relations. Also the suggested technique for measuring strains is non-destructive, commodious, and fast. Determining residual stresses through measuring residual strains is a new method which can performed quickly and online with doing test, i.e., there is no need to waste time for measuring strains and no complex relations for estimating strain by e.g., micro structural changes such as X-ray diffraction and ....Of course in this method just surface strains are measured. Take stochastic properties of residual stresses and affected parameters into consideration is the next authors' research. Kh. Farhangdoost, Ehsan Pooladi B. References [1] Liu Yongshou,Shao Xiaojun,Liu Jun,Yue Zhufeng (2009), Finite element method and experimental investigation on the residual stress fields and fatigue performance of cold expansion hole, Materials and Design, Article In Press. [2] V.Nigrelli, S.Pasta (2008), Finite-element simulation of residual stress induced by split-sleeve cold-expansion process of holes, Journal of Material Processing Technology, 205, 290-296 [3] H.Jahed, S.B.Lambert, RN.Dubey (2000), Variable material property method in the analysis of cold-worked fastener holes, Journal of Strain Analysis,35(2),137-142. [4] P.J.Withers, M.Turski, L.Edwards, P.J.Bouchard, D.J.Buttle (2008), Recent advances in residual stress measurement, International Journal of Pressure Vessels and Piping, 85,118-127. [5] X.-L.Gao (2002), Analytical solution of a borehole problem using strain gradient plasticity, Journal of Engineering Materials and Technology, 124,365-370. [6] X.-L.Gao (2003), Elasto-plastic analysis of an internally pressurized thick-walled cylinder using a strain gradient plasticity theory, International Journal of Solids and Structures, 40, 6445-6455. [7] A.Stacy, G.A.Webster (1988), Determination of residual stress distribution in autofrettaged tubing, International Journal of Pressure Vessels and Piping, 31, 205-220. [8] A.A.Garcia-Granada, D.J.Smith, M.J.Pavier (2000), A new procedure based on Sach's boring for measuring nonaxisymmetric residual stresses, International Journal of Mechanical Science, 42, 1027-1047. [9] A.A.Garcia-Granada, V.D.Lacarac, D.J.Smith, M.J.Pavier (2001), A new procedure based on Sach's boring for measuring non-axisymmetric residual stresses: experimental application, International Journal of Mechanical Science, 43, 2753-2768. [10] M.Su, A.Amrouche, G.Mesmacque, N.Beneseddiq (2008), Numerical study of double cold expansion of the hole at the crack tip and the influence of the residual stresses field, Computational Materials Science, 41, 350-355. 8 © AES-Advanced Engineering Solutions (Ottawa, Canada) All rights are reserved

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