Energy Approach-Based Simulation of Structural Materials High-Cycle Fatigue

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1 IOP Conference Serie: Materia Science and Engineering PAPER OPEN ACCESS Energy Approach-Baed Simuation of Structura Materia High-Cyce Fatigue To cite thi artice: A F Baayev et a 016 IOP Conf. Ser.: Mater. Sci. Eng View the artice onine for update and enhancement. Thi content wa downoaded from IP addre on 14/05/018 at 0:0

2 AMNT 015 IOP Pubihing IOP Conf. Serie: Materia Science and Engineering 116 (016) doi: / x/116/1/01039 Energy Approach-Baed Simuation of Structura Materia High-Cyce Fatigue A F Baayev¹, A V Koroev¹, A V Kochetkov, A I Skyarova¹, O V Zakharov¹ ¹ Yuri Gagarin State Technica Univerity, Saratov, Ruia, Perm Nationa Reearch Poytechnic Univerity, Perm, Ruia E-mai: zov17@mai.ru Abtract. The paper decribe the mechanim of micro-crack deveopment in oid tructura materia baed on the theory of britte fracture. A probabiity function of materia crack energy ditribution i obtained uing a probabiitic approach. The paper tate energy condition for crack growth at materia high-cyce oading. A formua aowing to cacuate the amount of energy aborbed during the crack growth i given. The paper propoe a highcyce fatigue evauation criterion aowing to determine the maximum permiibe number of oid body oading cyce, at which micro-crack tart growing rapidy up to detruction. 1. Introduction Durabiity i one of the mot important product performance criterion. That i why the crucia tak i to evauate durabiity under high-cyce oading ubject to tructura materia high-cyce fatigue. Highcyce fatigue wa tudied in many paper [1-8] which dea with detruction of oid bodie due to deveopment of micro-crack. Thee tudie propoed durabiity evauation criteria baed on determinitic approach to decription of detruction proce conidering random ditribution of microcrack. In fact, not ony ditribution of micro-crack in materia but the proce of their growth and ditribution of interna energy i random a we. In our opinion, in order to determine the condition of micro-crack occurrence and deveopment and to chooe a criterion of evauation of reidua ife of a rea oid body in high-cycic oading it i neceary to ue a probabiitic approach to decribe energy proce of crack deveopment.. Probem Statement Let' conider a pate of voume V tretched out and contracted aternatey under a oad P (Figure 1). The pate incude mutipe crack ditributed at random. When approximating the crack for the firt time, one can take it a fat, eiptica. Let' take the theory of A. Griffith a a bai for energy criterion of britte body fracture. According to the paper [1] the energy of a crack can be determined uing the foowing formua: w (1), where i the crack materia pecific urface energy; i the crack urface area; i the crack equivaent ength equa to the average D. The materia incude a arge number of random micro-crack of different ize. Therefore, the Rayeigh aw can decribe the ditribution of micro-crack ength, and the exponentia aw can decribe the denity of the crack energy ditribution: Content from thi work may be ued under the term of the Creative Common Attribution 3.0 icence. Any further ditribution of thi work mut maintain attribution to the author() and the tite of the work, journa citation and DOI. Pubihed under icence by IOP Pubihing Ltd 1

3 AMNT 015 IOP Pubihing IOP Conf. Serie: Materia Science and Engineering 116 (016) doi: / x/116/1/01039 where o i the expected vaue. f ( w ) e w w /(8 o ) e 8 o, () Figure 1. P pate cycic oading diagram. The conidered voume V of the materia contain a random integra nonnegative n number of crack, which ditribution i decribed by the Poion aw. Then, a probabe occurrence of microcrack with tota energy W n in the voume V of the materia i decribed with the probabiity function: n k wk V o k 1 k1 V ( W ) fw ( Wn ) H n e n1 n0 n! f n w n V o, (3) where H n i the poibe preence of n number of micro-crack (n = 1,, 3,...) with the tota energy W n in the voume V of the materia; o i the average denity of the crack ditribution in the voume V of the materia; w k i the energy of the k-th crack. Let' conider the pate tretched out and contracted aternatey under the ymmetrica cycic oad P (Figure 1). The pate tretching-out require the amount of work determined with the foowing formua: U V, (4) E where i the tretching-out tre occurring in the pate under the P force. The energy gained by the pate i concentrated in the materia defect and i ditributed among them. If the energy gain portion i proportiona to the area of the crack, then the energy gain of the crack of ize i equa to: u, (5) where i the pecific portion of the externa energy U faing to a unit of free micro-crack urface area. Simiary with the equation (1) and () for the crack energy, ditribution of the crack energy gain i expreed with the foowing formua: u /(8 o ) u e f ( u ) e. (6) 8 Auming that the energy required for tretching-out (4) wi be aborbed by the micro-crack, the expected vaue of the probabiity function of the tota energy gain i a foow: o

4 AMNT 015 IOP Pubihing IOP Conf. Serie: Materia Science and Engineering 116 (016) doi: / x/116/1/ E o o. (7) A crack can grow under the impact of the energy u gain if energy i ufficient for thi. The crack may remain unchanged if the energy gain i inufficient to form a new free urface. Hence, if, a a reut of externa impact on the materia, the energy u gained by the crack i e than the tandardized vaue u g (u < u g ), the crack doe not grow. If the energy gained by the crack i arger than thi vaue (u u g ), the crack tart growing. The vaue of thi energy u g doe not depend on the ize of the crack. In thi cae, externa force impact on the materia i a foow. Ony a portion of a the microcrack being under the externa oad in the voume V can grow. To enure the growth thi portion of the crack wi aborb the additiona energy u attributabe to them. The remaining part of the crack wi not grow, and, conequenty, after unoading and eatic recovery of the materia the part of the additiona energy attributabe to them diipate into the urrounding pace. 3. Simuation reut Baed on the equation (6) the part of the externa energy aborbed by the micro-crack can be decribed a foow: u g Pg f ( u ) du e. (8) u g According to the formua (1) upon receiving additiona energy u u g (equation (5)) the microcrack gain vaue with initia energy w, wi be equa to: w u (9) w During cycic deformation a tretching-out haf-cyce i foowed by a contraction haf-cyce, which energy i cacuated in the ame way. Occurrence and deveopment of the crack in tructura materia are accompanied by crack etting under the oad oppoite to thoe favoring crack growth. Thi iue wa tudied in detai in the foowing paper [, 3]. According to thee paper, the crack etting require much more energy than it deveopment, therefore crack deveopment proce wi outpace their etting. Theoretica and experimenta tudie [4, 5] have hown that high hydrotatic preure or ubtantia biaxia contraction in the eiptic pore caue reduction of a minor axi, whie the major axi tay unchanged in term of dimenion. Thu, the micro-crack etting require certain condition, which are difficut to enure under a impe cycic oad. Therefore, we can aume that the micro-crack can grow in the cycic aternating contraction and tretching-out, if the energy concentrated in them exceed a certain threhod, whie in the micro-crack contraction haf-cyce nothing happen. If in one oading cyce the micro-crack increaed in ize by a vaue determined with the equation (9), then in the i-th oading cyce the micro-crack ength gain wi be: If microcrack at one oading cyce ha increaed in ize by an amount determined by the equation (9), then the i-th cyce of oading ength of microcrack increment wi be: i i 1 при u ug (10), whie the micro-crack ength wi increae -fod. Since the externa deformation energy i aborbed ony by the crack, which energy exceed the u g, then the crack, which did not tart growing in the firt oading cyce, woud not grow in the ubequent cyce too. Therefore, the vaue of the externa energy aborbed by the micro-crack i the um of the energy gain of a the micro-crack in the i-th cyce which origina energy exceeded the u g : i 3

5 AMNT 015 IOP Pubihing IOP Conf. Serie: Materia Science and Engineering 116 (016) doi: / x/116/1/01039 i ui i. (11) It appear from the equation (11) that the gain of the energy aborbed with the micro-crack i proportiona to the quare in parenthee. Initia energy conumption to deform the tretching-out form the U vaue from the formua (4). Then the tota energy o to deform the micro-crack in the k cyce i defined with the foowing formua: k Ui U P g i1 i. (1) According to the formua (1) the reative energy o in the i-th cyce wi be equa to: i i. (13) If we know the permiibe ength of the micro-crack G, which dramaticay increae probabe detruction of an item, then uing the formua (10) and (13) we can determine the number of maximum permiibe cyce of eatic oading: n G n i (n G n ) k. (14) n n i The reuting formua (14) can be ued a a criterion for the reidua ife of the item: k-i. 4. Experimenta tudie We carried out experimenta tudie of vibromechanic reaxation of reidua tree of the item haped ike a rod with dia. = 4 mm and ength of L = 10 mm made of tructura tee 70 GOST The rod wa bent under the concentrated oad, which reuted in generation of interna tree added to the initia reidua tree. The cycic change of the interna tree were obtained by uniform rotation of the rod, which reuted in harmonic ociation of the rod bent axi reative to it rotation axi. Rotation peed, treatment time and bending moment determining the interna tree were taken a variabe parameter. During the experiment, it wa found that if the number of oading cyce i e than the vaue determined with the formua (14), the tre i reieved from the item ony partiay. The number of the oading cyce wa defined by the duration of treatment and rotation peed. If the vibromechanic treatment time exceed the pecified vaue, new tree can appear in the item and materia cycic trength decreae. The reut obtained are hown in Figure. The figure how dependence of the reidua tree in the item (k, t) at the item rotation peed of t =160 min -1 on the treatment time t for three vaue of reative tre k t m (0.6; 0.8; 0.9), where m i the maximum tre of the item eatic deformation, t i the yied trength of the item materia. The Figure how that during the vibromechanic treatment the reidua tree firt decreae down to zero, and then increae. The decreae of the reidua tree i due to the fact that during the vibromechanic tabiization the energy generated in the treatment proce graduay aborb the potentia energy generating reidua tree in the item. A oon a the potentia energy of the item materia become equa to zero, further vibromechanic treatment caue increae in the potentia energy of the item materia and generate interna tree. 5. Finding The mathematica imuation of crack occurrence mechanim with the ue of probabiitic and energy approache reuted in the foowing finding: britte fracture energy i exponentiay dependent on a oid body detroyed voume; increae in the voume of a body increae the probabiity of it detruction; crack ength gain per one oading cyce i characterized by power-aw dependence on the vaue of the energy gained during uch cyce; 4

6 AMNT 015 IOP Pubihing IOP Conf. Serie: Materia Science and Engineering 116 (016) doi: / x/116/1/01039 increae in the energy aborbed by the micro-crack i proportiona to the quare of the microcrack ize gain. Figure. Reidua tree dependence on vibromechanic treatment time. According to the reut of the experimenta tudie, it wa found that increae in the number of oading cyce caue reaxation of interna tree that characterize accumuation of interna energy in the crack. After compete reief of the reidua tree, whie continuing the cycic oading proce, interna tree tart to increae. It characterize croing of critica energy threhod and ubequent rapid growth of the micro-crack foowed by the detruction of the materia. The experimenta reut confirm the adequacy of the mathematica mode obtained and criteria of evauation of the materia reidua ife. 6. Concuion The above method aow to evauate high-cyce fatigue and reidua ife of tructura materia baed on the energy approach to decription of the crack occurrence mechanim. The reut can be ued in deigning of tee tructure and in deveopment of new technoogica method of reidua tree reaxation. Acknowedgment Thi paper ha been deveoped under the tate order of the Minitry of Education and Science of the Ruian Federation (No /K). Reference [1] Gumbe E I 1958 Statitick of extreme (New Jork: Coumbia Univerity Pre) p 375 [] Bezobrazov Yu A, Kobanikov N G and Naumov A A 014 Tenion-compreion method in the imuation of mutitage patic deformation Stee in Tranation 44 pp [3] Betekhtin V I, Veekov S Yu, Da Yu M, Kadomtev A G and Amoova O V 003 Theoretica and Experimenta Invetigation of the Effect of an Appied Load on Pore in Soid Phyic of the Soid State [4] Betekhtin V I and Kadomtev A G 005 Evoution of microcopic crack and pore in oid under oading Phyic of the Soid Stat [5] Petukhov A N 005 High-Cyce Fatigue of Materia and Part of Ga-Turbine Engine Strength of Materia [6] Petukhov A N 010 Probem of the high-cyce fatigue of the materia intended for the part of modern ga-turbine engine and power pant Ruian Metaurgy [7] Xiaojing Cai and Jinquan Xu 015 Effective Fatigue Stre and Criterion for High-Cyce Mutiaxia Fatigue Journa of Materia Engineering and Performance

7 AMNT 015 IOP Pubihing IOP Conf. Serie: Materia Science and Engineering 116 (016) doi: / x/116/1/01039 [8] McDiarmid D L 1991 A Genera Criterion for High Cyce Mutiaxia Fatigue Faiure Fatigue Fract. Eng. Mater. Struct

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