Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 52, 2, pp. 359-372, Warsaw 2014 INQUIRE INTO THE MARVELLOUSNESS OF AUTOFRETTAGE FOR MONO-LAYERED CYLINDERS Ruilin Zhu, Quan Li College of Engineering and Design, Hunan Normal University, Changsha, China e-mail: zrl200701@sina.com With the help of the equation of optimum overstrain or depth of the plastic zone, a set of concise and accurate equations for residual stresses and their equivalent stress as well as the total stress and their equivalent stresses are obtained, and features of these stresses are discussed, thereupon the law of distribution and the varying tendency of these stresses become clearer. Safe and optimum load-bearing conditions for a cylinder are presented. Keywords: thick-wall cylinder, autofrettage, load-bearingcapacity,overstrain, residual stress 1. Introduction Much of mechanical problems is involved in the autofrettage of cylinders. Currently, researches on autofrettage have been concentratedmostly on specific engineering problems, while a general theoretical study is rare.Due to their structural geometric features and the load bearingpattern, we feel that theremustbeamathematicalmysterybehindautofrettage theoryof cylinders,which theoretically describes the physical meaning of autofrettaged cylinders. To discover the general law contained in autofrettage theory, the autofrettage of a cylinder is investigated based on the fourth strength theory by theoretical analysis and the image method. The autofrettage technology is a clever and effective measure to obtain a favorable stress pattern inside the wall of a cylinder and raise of load-bearing capacity for (ultra-)high pressure vessels. Much of mechanical problems is concerned in the autofrettage of cylinders. Lots of researches concentrated upon specific engineering problems in the autofrettage have been done (Gao et al., 2008; Hameed et al., 2004; Huang et al., 2009, 2011; Levy et al., 2003; Lin et al., 2009; Zheng et al., 2010; Zheng and Xuan, 2010; etc.), nevertheless, many theoretical problems in the autofrettage remain unsolved.Nodoubt itwould benecessary to solve specific engineering problems about the autofrettage, but theoretical studies are more penetrating and can probe deeply into the essence of things, thus have universality and generality. Moreover, because of their structural geometric features and load bearing pattern, we feel that there must be more to it than meets the eye in autofrettage theory of cylinders, which theoretically describes the physicalmeaning of autofrettaged cylinders. Therefore, we dismiss specific engineering problems and do general research about the autofrettage in this paper. For an autofrettaged cylinder, depth of the plastic zone kj or overstrain ε is key, which affects residual stresses and load-bearing capacity. For determination of kj, previous researchers presented various methods. A repeated trial calculation method to determine the radius of elastic-plastic juncture rj was presented by Yu (1990), which is too tedious and inaccurate, and this method is based on limiting only the hoop stress and is essentially based on the first strength theory which is in agreement with brittle materials, while pressure vessels are made usually fromductilematerials which are in excellent agreementwith the third or fourth strength theory (Yu, 1990). Another method for determination of rj to ensure the equivalent stress of total stress at elastic-plastic juncture σej to be minimum was also suggested by Yu (1990). 360 R. Zhu, Q. Li However, to ensure σej to be minimum is not ideal and optimal, for it cannot ensure that a cylinder is not yielded compressively (reversed yielding) when processed with autofrettage, and its load-bearing capacity cannot be raised as high as possible. Thus, Zhu (2008) advanced an expression to calculate the depth of the plastic zone kj for a cylinder with the radius ratio k not to be yielded compressively when it is autofrettaged in his previous research, which is k2 lnk2j −k2−k2j +2=0. If kj of a cylinder is determined by this equation, its ultimate load- bearing capacity can reach two times the initial yieldpressure (themaximumelastic load-bearing capability of an unautofrettaged cylinder), 2pe (Zhu, 2008). By use of k 2 lnk2j−k2−k2j+2=0, Zhu and Zhu (2013a) simplified the equations for the residual stresses and the total stress, thus the laws of distribution and varying tendency of these stresses were discovered and relations among various parameters were revealed. By limiting the hoop residual stress, Zhu and Zhu (2013b) studied load-bearing capacity anddepth of the plastic zone of an autofrettaged cylinder, where load-bearingcapacity anddepthof theplastic zonearebothfixed for acertain k, andthere is a sole corresponding depth of the plastic zone kjθ for a certain k.We found that the research should be carried on in a more extensive andmore general sense, and it is well known that the greater the kj, the greater the load-bearing capacity, while the more different to perform the autofrettage technology.Therefore, if it is notnecessary for a cylinder tobear 2pe or ( √ 3+2)pe/2 (Zhu and Zhu, 2013b), kj can be lowered to be beneficial to the performance of autofrettage technology. Then, how to determine kj of a cylinder for a certain load-bearing capacity and the radius ratio k? Or what is the relation between kj and k for a certain load-bearing capacity when p< 2pe?What results will be brought about by this relation? How to determine the load- bearing capacity of a cylinder for a certain kj and k? What is the characteristics of residual stresses and their equivalent stress as well as total stress and their equivalent stress under new conditions?Therefore, on thebasis of theauthor’spreviouswork, thispaper is intended to resolve more general theoretical problems in the autofrettage and bring to light essential relations and laws contained in the current theory on the autofrettage according to the fourth strength theory (Mises yield criterion). Because we deal with the ideal case, and problems about the autofrettage under specific engineering conditions can be resolved by reference to the results of this paper on the basis of the specific engineeringconditions,webypass specificengineering conditionswhichvary in thousands of ways and do our research based on the following ideal conditions as in our previous works: (1) the material of a cylinder is perfectly elastic-plastic and Bauschinger’s effect is neglected, the compressive yield limit is equal to the tensile one; (2) strain hardening is ignored; (3) there is not any defect in the material. It is hoped that the obtained theoretical results are of academic value and are referential as well as applicable to the design of (ultra-)high pressure apparatus. 2. Residual stresses under ordinary condition At a general location (relative location, r/ri) within the plastic zone, the residual stresses are as follows(ZHU, 2008) σ′z σy = 1√ 3 [k2j k2 +ln (r/ri) 2 k2j − ( 1− k2j k2 +lnk2j ) 1 k2−1 ] σ′r σy = 1√ 3 [k2j k2 −1+ln (r/ri) 2 k2j − ( 1− k2j k2 +lnk2j ) 1 k2−1 ( 1− k2 (r/ri)2 )] σ′θ σy = 1√ 3 [k2j k2 +1+ln (r/ri) 2 k2j − ( 1− k2j k2 +lnk2j ) 1 k2−1 ( 1+ k2 (r/ri)2 )] (2.1) Inquire into the marvellousness of autofrettage for mono-layered cylinders 361 Therefore, the equivalent residual stress at a general radius location within the plastic zone is (Yu, 1990) σ′e σy = √ 3 2 (σ′θ σy − σ′r σy ) =1− k2−k2j +k2 lnk2j (k2−1)(r/ri)2 (2.2) where σ′z,σ ′ r,σ ′ θ are axial, radial andhoopresidual stress, respectively; ri, rj, ro are inside radius, elastic-plastic juncture radius, outside radius, respectively; k is the radius ratio or ratio of the outside to inside radius, k= ro/ri; kj is depth of the plastic zone, or plastic depth, kj = rj/ri; σy is yield strength; σ ′ e is equivalent residual stress; σ ′ e/σy is relative equivalent residual stress; subscript i represents the internal surface, subscript j represents the elastic-plastic juncture. The residual stresses at a general location within the elastic zone are as follows (Zhu, 2008) σ′z σy = 1√ 3 [k2j k2 − ( 1− k2j k2 +lnk2j ) 1 k2−1 ] σ′r σy = 1√ 3 ( 1− k2 (r/ri)2 )[k2j k2 − ( 1− k2j k2 +lnk2j ) 1 k2−1 ] = ( 1− k2 (r/ri)2 )σ′z σy σ′θ σy = 1√ 3 ( 1+ k2 (r/ri)2 )[k2j k2 − ( 1− k2j k2 +lnk2j ) 1 k2−1 ] = ( 1+ k2 (r/ri)2 )σ′z σy (2.3) Therefore, the equivalent residual stress at a general radius location within the elastic zone is (Yu, 1990) σ′e σy = √ 3 2 (σ′θ σy − σ ′ r σy ) = k2(k2j −1− lnk2j) (k2−1)(r/ri)2 (2.4) 3. Discussion about plastic depth or overstrain Whentheequivalent stressof total stress (residual stressplus the stresses causedby theoperation pressure p) at the elastic-plastic juncture reaches the yield strength, or σej = σy, the relation for p, the pressure a cylinder can contain, σy, k and kj is as follows(Yu, 1990) p σy = k2−k2j +k2 lnk2j√ 3k2 (3.1) Zhu (2008) showed that when radius ratio is greater than critical radius ratio, or k > kc = 2.2184574899167 . . ., if kj ¬ kj∗, where kj∗ is determined by k2 lnk2j∗ −k2−k2j∗ +2=0, the absolute value of equivalent stress of residual stress at the internal surface |σ′ei| ¬ σy, when kj = kj∗, |σ′ei|=σy and p σy =2 k2−1√ 3k2 =2 pe σy when k kc, to reflect the level of increase in the load-bearing capacity, letting p=λpe, then p σy =λ k2−1√ 3k2 =λ pe σy (k>kc) (3.3) where λ is also called the reinforcing coefficient. SubstitutingEq. (3.3) intoEq. (3.1), oneobtains k2 lnk2jλ− (λ−1)k2−k2jλ+λ=0 (3.4) where kj is written as kjλ to indicate that the safe plastic depth kj is related with λ. The overstrain is defined as ε= rj −ri ro−ri = kj −1 k−1 (3.5) Substituting kj from Eq. (3.5) into Eq. (3.4), one obtains k2 ln[ελ(k−1)+1]2− (λ−1)k2− [ελ(k−1)+1]2+λ=0 (3.6) If ε and k meet Eq. (3.6), where ε is written as ελ, and a cylinder contains pressure determined byEq. (3.3), σej/σy =1, |σ′ei| ¬σy.When λ=1, σ′ei =−σy. ελ determined byEq. (3.6) is called the optimum overstrain, and kjλ determined by Eq. (3.4) is called the optimum plastic depth. They are plotted in Fig. 1. Fig. 1. The optimum plastic depth and optimum overstrain; (a) the optimum plastic depth, (b) the optimum overstrain It can be known that from Eq. (3.4), Eq. (3.6) and Fig. 1 that: (1) If λ¬ 1 (curves 1-3), k 1, this is meaningless in application; below the line, k > kj and ελ < 1. So, it is the part below the line k= kj or ελ =1 that is of significance. When kkcλ, kj should be determinedbyEq. (3.4).Equation (3.7) is the sameas theabove η in form.Theprerequisite to kj = k is k kc, the greatest reinforcing coefficient is λ ≡ 2. So, integrating Eq. (3.2), we obtain the greatest reinforcing coefficient for any k (1¬ k¬∞), as shown in Fig. 2. (3) kj and ελ decrease with k increasing on the right of the two branches. Thus, discussion about the autofrettage is not significant unless λ> 1. Since compressive yield occurs when λ > 2 (when λ > 2, ελ is higher than the value on curve 8 on which σ′ei =−σy), curve 9 in Fig. 1a ismeaningless. Besides, curves 1-3 and the left of curves 4-8 in Fig. 1 are meaningless. The abscissa can be taken as the curve with λ = 1 (ε = 0). Thus, significant and possible plastic depth lies in a trapezoid surrounded by the abscissa (λ=1), the slanting straight line k= kj and the curve k 2 lnk2−k2−k2jλ+2=0 (curve 8 inFig. 1a for λ=2).The coordinates of four vertexes of this quasi-infinite area (m,o,v,n) are shown in Fig. 1a. The significant and possible overstrain lies in a trapezoid surrounded by the horizontal line ε= 0 (λ = 1), the vertical line (k = 1), the horizontal line ε= 1 and the curve k2 ln[ελ(k−1)+1]2−k2− [ελ(k−1)+1]2+2=0 (curve 8 in Fig. 1b for λ=2). The coordinates of five vertexes of this quasi-infinite area (m,u,o,v,n) are shown in Fig. 1b.When k→∞, points n and v coincide. Fig. 2. The greatest reinforcing coefficient 4. Discussion about residual stresses and their equivalent stress under kjλ If kj is determined by Eq. (3.4) or kj = kjλ and k ­ kcλ, with the help of Eq. (3.4), Eqs. (2.1)-(2.4) become σ′z σy = 1√ 3 (lnx2−λ+1) σ ′ r σy = 1√ 3 ( lnx2+ λ x2 −λ ) σ′θ σy = 1√ 3 ( lnx2− λ x2 −λ+2 ) σ′e σy =1− λ x2 (4.1) where x= r/ri, the same below.When λ¬ 1+ √ 3/2, |σ′θi/σy| ¬ 1; when λ¬ 2, |σ′ei/σy| ¬ 1. σ′z,σ ′ r,σ ′ θ and σ ′ e havenothing to dowith kj and kwithin theplastic zone,whichmeans that for some λ, the curves of residual stress inwhichever direction (axial, radial andhoopdirection), 364 R. Zhu, Q. Li and the equivalent residual stress (i.e. σ′z, σ ′ r, σ ′ θ and σ ′ e) for various plastic depth kjλ and the radius ratio k coincide σ′z σy ≡ 1√ 3k2 (k2jλ−λ) σ′r σy = ( 1− k 2 x2 )σ′z σy σ′θ σy = ( 1+ k2 x2 )σ′z σy σ′e σy = 1 x2 (k2jλ−λ) (4.2) The curves of residual stresses at a general location for k=3,λ=1.2 and1.8when kj = kjλ are plotted in Fig. 3. In this three cases, |σ′ei|<σy for kj = kjλ and λ< 2. Fig. 3. Curves of residual stresses and their equivalent stress at a general location; (a) k=3, λ=1.2, kjλ =1.106693, (b) k=3, λ=1.8, kjλ =1.539944, (c) k=5, λ=1.8, kjλ =1.50584 The three curves of residual stress (σ′z/σy, σ ′ r/σy and σ ′ θ/σy) at a general location collect at a fixed point within plastic zone: [ √ λ,(lnλ+1−λ)/2] for any k, kj and λ, and the coordinate of the intersection is not related with k and kj but only with λ. If kj 6= kjλ, this situation does not happen, the coordinate of the intersection is related not only with λ but also with k and kj. The equivalent stress of residual stress at the internal surface is the most dangerous, and when λ ­ 1, σ′ei ¬ 0, which implies compressive stress; when λ ¬ 2, σ′ei ­ −σy. Since kj ­ √ λ (kj ­ e(λ−1)/2 at the same time), then,when λ¬ 1,k2j−λ> 0within the elastic zone, or equivalent stress of residual stress σ′e > 0 (tension) within the elastic zone. At the elastic-plastic juncture, where x = kjλ, the equivalent stress of residual stress is the maximum (algebraic value, not absolute value) within the whole elastic zone, or σ′ej/σy =(k 2 jλ−λ)/k2jλ =1−λ/k2jλ. Obviously, 0<σ′ej/σy < 1. From Eq. (3.4), when k=∞, kjλ =e(λ−1)/2 = k∞jλ, then from Eqs. (4.2), within the whole elastic zone σ′z σy ≡ eλ−1−λ√ 3k2 σ′r σy = ( 1− k2 x2 )σ′z σy σ′θ σy = ( 1+ k2 x2 )σ′z σy σ′e σy = eλ−1−λ x2 Inquire into the marvellousness of autofrettage for mono-layered cylinders 365 x=e(λ−1)/2 ∼∞within the elastic zone, therefore σ′z σy ≡ e λ−1−λ√ 3k2 σ′r σy = ( 1− k 2 eλ−1 )σ′z σy ∼ 0 σ′θ σy = ( 1+ k2 eλ−1 )σ′z σy ∼ eλ−1−2λ k2 σ′e σy =1− λ eλ−1 ∼ 0 Thedistribution of σ′e/σy within thewholewall for λ=1.8 anddifferent k and kjλ is shown in Fig. 4. Fig. 4. Distribution of σ′e/σy within the whole wall for λ=1.8 and various k and kjλ Figure 4 is explained as follows: • CurveBAA: k=1.93322 . . ., kjλ = k= kcλ =1.93322 . . ..Within the plastic zone or point B to A, x varies from 1 to kjλ, σ ′ e/σy varies from −0.8 to 0.51837 . . .; within elastic zone (no elastic zone) or point A toA, x varies from 1.93322 . . . (kjλ) to 1.93322 . . . (k), σ ′ e/σy varies from 0.51837 . . . to 0.51837 . . .. • Curve BCD: k=2, kjλ =1.736906 . . .. Within the plastic zone or point B to C, x varies from 1 to kjλ, σ ′ e/σy varies from −0.8 to 0.40335 . . .; within the elastic zone or point C to D, x varies from kjλ to k, σ ′ e/σy varies from 0.40335 . . . to 0.304211 . . .. • Curve BEF: k = kc or kcλ when λ = 2, kjλ = 1.624631 . . .. Within the plastic zone or point B to E, x varies from 1 to kjλ, σ ′ e/σy varies from −0.8 to 0.318043 . . .; within the elastic zone or point E to F, x varies from kjλ to k, σ ′ e/σy varies from 0.318043 . . . to 0.170561 . . .. • Curve BGH: k=3, kjλ =1.539944 . . .. Within the plastic zone or point B to G, x varies from 1 to kjλ, σ ′ e/σy varies from −0.8 to 0.240964 . . .; within the elastic zone or point G to H, x varies from kjλ to k=3, σ ′ e/σy varies from 0.240964 . . . to 0.063492 . . .. • Curve BMN: k=∞, kjλ =e0.4.Within the plastic zone or point B toM, x varies from 1 to kjλ =e 0.4 =1.491825 . . ., σ′e/σy varies from −0.8 to 1−λ/e0.8 =0.191208 . . .; within the elastic zone or pointM toN (far infinitely), x varies from kjλ =e 0.4 to k=∞, σ′e/σy varies from 1−λ/e0.8 =0.191208 . . . to 0. FromFig. 4 andEq. (4.1)4, it is known that all curves of equivalent residual stresses for any k and kjλ within the plastic zone are located on the identical curveABandpass through the same point (1.80.5,0), except that a different curve for different k and kjλ is located on a different section of curve AB. Saying, the above curves for the plastic zone, BA, BC, BE, BG, BM, are all on curve BA, or they coincide with each other. However, if kj 6= kjλ, or relation between kj and k does not satisfy Eq. (3.4), the above conclusion is untenable, even |σ′ei/σy|> 1. This case is illustrated in Fig. 5 (λ = 1.8), where curve 1 and 2 coincide with each other in the 366 R. Zhu, Q. Li plastic zone and both pass through the point (1.80.5,0) for kj = kjλ, but curve 3 and 4 do not coincide with each other in the plastic zone and neither pass through the point (1.80.5,0), and they do not coincide with curve 1 and 2 for kj 6= kjλ. When k = 3, kj = 1.9 > kjλ, σ′ei/σy = −1.11792 < −1; when k = 3, kj = 1.4 < kjλ, σ′ei/σy = −0.63706 > −1, but the equivalent stress of total stress σei may exceed σy. Fig. 5. A comparison of the equivalent residual stress To know these phenomena and laws about the autofrettage of cylinders well is beneficial to design, manufacturing and academic research on pressure vessels. When kj = kjλ = kcλ (entire yielded), distributions of σ ′ e/σy and ,σ ′ θ/σy within the plastic zone for various λ are shown in Figs. 6a and 6b, respectively. Fig. 6. Distribution of σ′e/σy (a) and σ ′ θ/σy (b) for various λ The top dash curve in Fig. 6b is themaximumhoop residual stress σ′θm/σy under the critical radius ratio kcλ and different λ, the equation of which is σ′θm σy = 2√ 3 ( 1− lnk2cλ k2cλ−1 ) (4.3) d(σ′ θm /σy) dkcλ = 4√ 3 λ−1 kcλ(k 2 cλ −1) ­ 0 for λ­ 1, and kcλ increases with λ increasing, when λ= 2, kcλ gets the maximum kc, thereby σ ′ θm/σy gets the maximum σ′θm σy = 2√ 3 ( 1− lnk 2 c k2c −1 ) (4.4) From Zhu (2008), it is known that k2c lnkc k2c−1 = 1, then, σ′ θm σy = 2√ 3 ( 1− 2 k2c ) . From Eq. (4.4), when kc < 2 √ 2+ √ 3= 3.86, σ′θm/σy < 1. So the hoop residual tension is safe. However, when Inquire into the marvellousness of autofrettage for mono-layered cylinders 367 λ > 1+ √ 3/2, the hoop residual compressive stress is not safe. When λ ¬ 2, the equivalent residual stress σ′e is invariably safe. σ′e = 0 at x= √ λ, which is just the abscissa of intersection of the three curves of residual stress at a general location. Generally, in Eq. (2.2), letting 1− k 2−k2 j +k2 lnk2 j (k2−1)(r/ri)2 =0, one obtains x= √ k2−k2j +k2 lnk2j k2−1 = √ σ′ei σy +1 1 when λ> 1 if a cylinder is not treated with the autofrettage and p>pe. Fig. 7. Stresses caused by p at a general location From Eqs (5.3) and Fig. 7, when x­ √√ 3−1k, σpe/σy ¬σ p θ/σy. The total stresses σ/σy include the residual stresses and the stresses caused by p, or σz =σ ′ z +σ p z σr =σ ′ r+σ p r σθ =σ ′ θ +σ p θ (5.4) 368 R. Zhu, Q. Li The equivalent stress of total stress is σe = √ 3 2 (σθ−σr)= √ 3 2 [(σ′θ+σ p θ)−(σ ′ r+σ p r)] = √ 3 2 (σ′θ−σ′r)+ √ 3 2 (σ p θ−σ p r)=σ ′ e+σ p e (5.5) For k = 3, λ = 1.8, p = λpe, kj = 1.4 < kjλ as mentioned above (see Fig. 5), σ′ei/σy = −0.63706, σ p ei/σy = 1.8, then the equivalent stress of total stress at the internal surface σei = σ ′ ei +σ p ei = 1.1629 > 1. So, for p = λpe, kj must be determined by Eq. (3.4), i.e. kj = kjλ (in this example, kjλ =1.539944 . . . > 1.4). If kj kjλ, the residual stresses will be dangerous. The greater the λ, the higher the load-bearing capacity, but the deeper the plastic zone, leading to a more difficult autofrettage treatment; conversely, the less the λ, the lower the load-bearing capacity, but the shallower the plastic zone, leading to an easier autofrettage treatment. This finding helps us to weigh the advantages and disadvantages in the design of pressure vessels. If p=λpe and kj = kjλ, the components of total stresses are: —Within the plastic zone σz σy = σ′z σy + σpz σy = 1√ 3 ( lnx2−λ+1+λ 1| k2 ) σr σy = σ′r σy + σpr σy = 1√ 3 ( lnx2−λ+λ 1 k2 ) σθ σy = σ′θ σy + σ p θ σy = 1√ 3 ( lnx2−λ+2+λ 1 k2 ) σe σy = σθ σy − σr σy ≡ 1 (5.6) Eqation (5.6)4 means that if a cylinder is subject to p=λpe and its plastic depth is determined byEq. (3.4), the equivalent stress of total stress everywherewithin the plastic zone is σe/σy ≡ 1. —Within elastic zone σz σy = σ′z σy + σpz σy = k2 jλ√ 3k2 σr σy = σ′r σy + σpr σy = k2jλ√ 3 ( 1 k2 − 1 x2 ) σθ σy = σ′θ σy + σ p θ σy = k2jλ√ 3 ( 1 k2 + 1 x2 ) σe σy = σθ σy − σr σy = k2jλ x2 (5.7) Equations (5.6) and (5.7) are plotted in Fig. 8 for k=3, λ=1.2 and 1.8, respectively. Fig. 8. Total stresses and their equivalent stress; (a) k=3, λ=1.2, (b)) k=3, λ=1.8 From Eqs. (5.7)3,4 and Fig. 8, when x­ √√ 3−1k, σe/σy ¬σθ/σy. When kj, k are related by Eq. (3.4), at r = rj, the stresses determined by Eqs. (5.6) are consistent with the corresponding stresses determined by Eqs. (5.7). This testifies reliability of this paper. According toEqs. (5.7), seemingly the total stresseswithin the elastic zone are not concerned with λ. Nevertheless, kjλ depends on λ as seen in Eq. (3.4). Inquire into the marvellousness of autofrettage for mono-layered cylinders 369 Actually, it is Eq. (3.1) that ensures σe/σy ≡ 1 everywhere within the plastic zone and σe/σy = k 2 j/x 2 within the elastic zone irrespective of kj and k. Substituting Eq. (3.1) into Eq. (5.2), results in σpe σy = k2−k2j +k2 lnk2j (k2−1)x2 (5.8) Substituting Eqs. (5.8) and (2.2) into Eq. (5.5), one just obtains σe/σy ≡ 1; substituting Eqs. (5.8) and (2.4) into Eq. (5.5), one just obtains σe/σy = k 2 j/x 2. Therefore, as long as p/σy =(k 2−k2j +k2 lnk2j)/( √ 3k2) (i.e. Eq. (3.1)), the results for σe/σy ≡ 1 within the plastic zone and for σe/σy = k 2 j/x 2 within the elastic zone have nothing to do with the magnitude of kj and k. In other words, providing that p/σy = (k 2 − k2j + k2 lnk2j)/( √ 3k2), for any kj and k, which are not needed to be related by Eq. (3.4), σe/σy ≡ 1 within the plastic zone and σe/σy = k 2 j/x 2 (0 kc, the load-bearing capacity is p/σy = λpe/σy = λ(k 2 − 1)/( √ 3k2) for a cylinder with k and kjλ; when k 1 never occurs irrespective of kj even if kj = k; when kj = k, p/σy = py/σy = lnk 2/ √ 3 (<λ(k2−1)/( √ 3k2)). When k­ 1.2071 . . ., if kj ¬ kjλ, |σ′eiσy|> 1 never occurs; if kj = kjλ, p/σy =λpe/σy =1.2pe/σy (< lnk2/ √ 3). If λ = 1.8, when k ¬ 1.93322 . . . (kcλ), |σ′eiσy| > 1 never occurs irrespective of kj even if kj = k; when kj = k, p/σy = py/σy = lnk 2/ √ 3 (<λ(k2 −1)/( √ 3k2)). When k ­ 1.93322 . . ., if kj ¬ kjλ, |σ′eiσy|> 1 never occurs; if kj = kjλ, p/σy =λpe/σy =1.8pe/σy (< lnk2/ √ 3). If λ = 2, when k ¬ kc, |σ′eiσy| > 1 never occurs irrespective of kj even if kj = k; when kj = k, p/σy = py/σy = lnk 2/ √ 3 (<λ(k2−1)/( √ 3k2)). When k­ kc, if kj ¬ kjλ, |σ′eiσy|> 1 Inquire into the marvellousness of autofrettage for mono-layered cylinders 371 Fig. 10. Load-bearing capacity of a cylinder never occurs, if kj = kjλ, p/σy = 2pe/σy (< lnk 2/ √ 3). When λ= 2, kjλ is marked as kj∗ by Zhu (2008). Substituting Eq. (3.7) into p/σy = λ(k 2 − 1)/( √ 3k2), one obtains p/σy = lnk 2 cλ/ √ 3. It is easy to prove that when k¬ kcλ, lnk¬ (k2−1)/k2. 8. Conclusions • The optimum operation conditions are: for any k, the plastic depth is determined by k2 lnk2jλ−(λ−1)k2−k2jλ+λ=0, and the load-bearing capacity is determined by p=λpe, where λ= η= k2 lnk2/(k2−1)when k¬ kc (λ= η=1∼ 2 calculated by k2 lnk2/(k2−1) for k=1∼ kc) and λ=1∼ 2 for choosing when k­ kc. λ¬ 2 is required for |σ′ei| ¬σy. • When k ¬ kcλ, |σ′eiσy|> 1 never occurs irrespective of kj even if kj = k, if kj = k, the ultimate load-bearing capacity p/σy = py/σy = lnk 2/ √ 3 (<λ(k2−1)/k2).When k­ kcλ, if kj ¬ kjλ, |σ′eiσy|> 1 never occurs, if kj = kjλ, the load-bearing capacity p/σy =λpe/σy (< lnk2/ √ 3).When λ=2, kjλ and p reach the maxima: k 2 lnk2j −k2−k2j2+2=0 and p/σy =2pe/σy. • Thepossible andoptimumplasticdepth kj is situated in thequasi-infinitearea constructed of the horizontal axis, the straight line kj = k and the curve k 2 lnk2jλ−k2−k2jλ+2=0. • If k ¬ kc (or kcλ for λ= 2), kjλmax = k = kcλ; if k ­ kc, kjλmax < k (kcλ), kjλmax is determined by k2 lnk2jλ−k2−k2jλ+2=0 (λ=2). The greater the λ, the greater the kjλ. • As long as p/σy =(k2−k2j+k2 lnk2j)/( √ 3k2), irrespective of kj, σe ≡σy within thewhole plastic zone, or σe is even, and the equivalent stress of total stress within the elastic zone is always lower than σy. However, if kj is outside the quasi-infinite area of the possible and optimum plastic depth, the compressive yield occurs. • Due to the equation k2 lnk2jλ − (λ− 1)k2 − k2jλ +λ = 0, the relations between various parameters and their varying tendency become concise and clearer, and the equations concerned with the autofrettage are simplified greatly. Acknowledgement This project is supported by Scientific Research Fund of Hunan Provincial Education Department (Grant No. 12A087) References 1. Gao J., Xie G., Zhou H.-J., et al., 2008, Research on autofrettage technology (in Chinese), Mechanical Research and Application, 21, 6, 21-23 372 R. Zhu, Q. Li 2. Hameed A., Brown R.D., Hetherington J., 2004, A study of the residual stress distribution in an autofrettaged, thick-walled cylinder with cross-bore, Journal of Pressure Vessel Technology, 126, 4, 497-503 3. HuangL., LiQ.,YangP., LiH., 2009, Self-strengthening processing of high- pressure fuel pipes (in Chinese),Diesel Engine, 31, 2, 45-54 4. Huang Y., Lu C., 2011, Study on pressure of high-pressure fuel pipes of the high-speed diesel engine burning heavy fuel oil (in Chinese), Ship and Ocean Engineering, 40, 2, 91-93 5. Levy C., Perl M., Ma Q., 2003, The influence of finite three-dimensional multiple axial ero- sions on the fatigue life of partially autofrettaged pressurized cylinders, Journal of Pressure Vessel Technology, 125, 4, 379-384 6. Lin Y., Dong Q., Jia J., 2009, Autofrettage damage residual stress of thick-walled cylinder (in Chinese), Science Technology and Engineering, 9, 24, 7306-7309 7. YuG., 1990,Chemical Pressure Vessels and Equipment (in Chinese), Beijing: Chemical industrial Press 8. Zheng G., Ma M., Wang D., 2010, Improvement of tightness test equipment for high-pressure fuel pipe (in Chinese),Mechanical Management and Development, 25, 6, 35-36 9. Zheng X., Xuan F., 2010, Investigation on autofrettage and safety of the thick-walled cylinder under thermo-mechanical loadings (inChinese),Journal ofMechanical Engineering,46, 16, 156-161 10. Zhu R., 2008, Ultimate load-bearing capacity of cylinder derived from autofrettage under ideal condition,Chinese Journal of Mechanical Engineering, 21, 5, 80-87 11. Zhu R., Zhu G., 2013a, Effect of optimum plastic depth on stresses and load-bearing capacity of autofrettaged cylinder,Chinese Journal of Mechanical Engineering, 26, 2, 365-370 12. Zhu R., Zhu G,, 2013b, On autofrettage of cylinders by limiting circumferential residual stress based onMises yield criterion, Journal of Theoretical and Applied Mechanics, 51, 3, 697-710 Manuscript received February 10, 2013; accepted for print October 21, 2013