/tvsb Transactions of the VŠB Technical University of Ostrava Civil Engineering Series, Vol. 17, No.

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1 /tvsb Transatins f the VŠB Tehnial University f Ostrava Civil Engineering Series, Vl. 17, N., 017 paper #4 Ivan SHATSKYI 1, Mykla MAKOVIICHUK, Lubmyr ROPYAK ANALYTICAL ESTIMATIONS OF STRENGTH OF TWO LAYER COATING LOADED ALONG A LINE Abstrat The engineering alulatin predure f the stress-strain state f the tw-layer ating under the lad lalized alng a line is prpsed. The plane upled prblem f the plate equilibrium has been frmulated. The analytial slutins f the prblem have been btained and mpared with previus results fr the unupled prblem. The analytial estimatins f strength f the mpsite ating have been given. Keywrds Tw-layer ating, Winkler fundatin, lal lad, stress distributin, strength. 1 INTRODUCTION Wrkability f mahines, mehanisms and equipment subjeted t ntat lads and t interatin with aggressive r abrasive media is mainly ntrlled by the state and struture f the surfae layers f their ritial mpnents. Cnventinal predure f surfae hardening f mahine parts implies depsitin f funtinally gradient atings whse integral parts have different funtins. Suh hetergeneus strutures inlude, in partiular, tw-layered atings f the eramisaluminum system, whih ver the surfae f steel [1] r aluminium [] parts. The external (xide) layer exhibits a high hardness and ensures the prtetin frm mehanial wear, while the subsurfae (aluminum) layer prvides the neessary bnd with the base metal (steel) and prtets the latter frm hydrgen saturatin in aggressive media. Similar prblems rise als in ivil engineering within interatin f building walls and base plates n sil bdies. The available literature in this dmain [ 7] prvides insuffiient insight int the effet f struture and mehanial prperties f the tw-layer ating mpnents n the limit lad values. Espeially, this is vital fr lalized spatial lads. Study [6] desribes the apprah t analytial alulatin f lal stresses in a tw-layer ating treated as a plate lying n the Winkler bed and prvides the assessment f the limit lad transferred frm the tl applied during the finishing mehanial treatment f the ated part. The predure fr design ptimizatin f a tw-layer ating fr partiular lal lading nditins is nsidered in the artile [7]. The aim f the present study is elabratin f the tehnique [6, 7] in the ase f taking int nsideratin internnetin f the membrane and bending fields in ating. 1 Prf., Ivan Shatskyi, DS., Department f mdelling f damping systems, Ivan-Frankivsk Branh f Pidstryhah Institute fr Applied Prblems f Mehanis and Mathematis, NAS f Ukraine, Mykytunetska str., ; 7600, Ivan-Frankivsk, Ukraine, phne: (+80) , ipshatsky@gmail.m Ass. Prf., Mykla Makviihuk, PhD., Department f mdelling f damping systems, Ivan-Frankivsk Branh f Pidstryhah Institute fr Applied Prblems f Mehanis and Mathematis, NAS f Ukraine, Mykytunetska str., ; 7600, Ivan-Frankivsk, Ukraine, phne: (+80) , makviy@ua.fm Ass. Prf., Lubmyr Rpyak, PhD, Department f Cmputer Engineering Manufaturing, Ivan-Frankivsk Natinal Tehnial University f Oil and Gas, Karpatska Str., 15, Ivan-Frankivsk, 76019, Ukraine, phne: (+80) , l_rpjak@ukr.net 109 Dwnlad Date 1/9/18 10:7 PM

2 PROBLEM FORMULATION AND STRENGTH CRITERION Let s nsider a flat rigid fundatin with depsited tw-layer ating nsisting f a quite stiff surfae layer f thikness h (external ating layer) and f a quite mpliant subsurfae layer f thikness h (substrate), whih is illustrated by Fig. 1. The external (upper) layer is mdeled by a plate subjeted t tensile (mpressive) and bending lads, while the lwer (substrate) layer is treated as an interlayer whse behavir is ntrlled by the Winkler hyptheses n the prprtinality f stresses and elasti displaements. Fr the sake f the prblem simplifiatin, the base is assumed t be perfetly rigid, while the mehanial ntat between the ating mpnents is treated as a perfet ne. The mpsite is laded by distributed fre whih is unifrmly distributed alng the straight line (nrmal t the plane in Fig. 1) and is represented as tw mpnents X and Y (N/m). Anther assumptin is the plane strain state f the mpsite under study ( ε z = 0 ). The stress distributin in the upled tw-layer mpsite and the strength values are t be derived. y Y h E, ν, [ σ] O X x h E, ν, [ σ] z Fig. 1: Sheme f a tw-layer ating and lad Based n the abve assumptins, the equilibrium equatin fr the ating n an elasti bed takes the fllwing frm [8, 9]: d u du x y B k u h X ( x) x x = δ + 4 d u y d du y d D + k u h k u h Y ( x) h X ( x) 4 y y x x = δ + δ +, x R, (1) where: u x, u y are mpnents f the elasti displaement f the plate midsetin [m], δ (x) is the Dira funtin, B = E h /(1 ν ) are stiffness values fr tensin [Pa m], D = E h /((1 ν )) are stiffness values fr bending, [Pa m ], k x = G / h = E /((1 + ν ) h ), k y = E / h are the Winkler bed effiients, [Pa/m], 110 Dwnlad Date 1/9/18 10:7 PM

3 E, E are elasti mduli f ating and substrate, respetively [Pa], ν, ν are Pissn s ratis. At infinity the lads and bending mments turn t zer: du B x d u y d u y ( ± ) = 0, D ( ± ) = 0, D ( ± ) = 0. () Obtained displaements rrespnd t stress in the ating, in partiular in the external ating midsetin ( y = h / ) have the fllwing frm: d du y σ x = E u x + h. () Using the Winkler hyptheses, the stresses in the substrate are assessed as fllws: du y τ xy = kx ux + h, σ y = k yu y. (4) Fr the strength assessment f eah layer, the Mises riterin is used, whih yields the fllwing equatins fr the plane-strained ating σ eq, с ( x) (1 ν + ν )( σ x + σ y ) (1 + ν ν ) σ xσ y + τ xy [ σ ] and fr the Winkler bed-type substrate where [σ ], ], о ( ) σ y + τ xy [ σ 111, (5) σ eq x ], (6) [σ are maximal permissible stresses fr the ating and substrate materials [Pa]. Тhe underlined members in equatins (1) and frmulae (), (4) haraterize internnetin f membrane and bending fields and distinguish this statement frm the previusly nsidered unupled prblem [6, 7]. STRESSED STATE ANALYSIS Using the dimensinless values u 1 = u x / h, u = u y / h, ξ = x / h the prblem (1), () was presented in the frm: L 11 u 1 + L1 u = B1, L1 u 1 L u = B +, ξ R. (7) du 1 ( ± ) = d u 0, d u ( ± ) = 0, ( ± ) = 0. (8) The elements f symmetrial matrix f differential peratrs and the lads vetr are the fllwing: L 11 d = ε, Here k xh Eh 1 ν ε = =. B E h 4(1 + ν ) d 1 d d L 1 = L 1 = ε, L = ε + (1 + ν ) ε d 4, ξ X Y X B1 = δ( ξ), B = δ ( ξ ) + δ ( ξ ). B B B 4 Dwnlad Date 1/9/18 10:7 PM

4 The analytial slutin f bundary prblem (7), (8) has been btained using the Furier transfrmatin. The result in image spae parametrised by ω is the fllwing: u F k ( ω) = Δk ( ω) / Δ( ω), k = 1, ; (9) X 1 4 Y X Δ1 ( ω) = ω + (1 + ν ) ε εiω, Y Δ ( ω) = iω + ( ω + ε ), B B B B ) = ( ω + ε ) ω + (1 + ν ) ε. Δ ( ω + εω The prblem f Furier inversin f the expressin (9) is redued t finding zers f the denminatr (the rts f the biubi equatin Δ ( ω ) = 0 ). As we see, rejetin f underlined member whih is respnsible fr nnetivity f prblem signifiantly simplifies the searhing f rts. We nsidered the example f the alulatin with parameters: X = 0, Y / B = 1, ε = 1, ν = 0.5, ν = 0.. The results f simulatin f dimensinless values f the displaement and vn Mises stresses are represented n Fig. and Fig.. Fig. : The hrizntal and vertial displaements f upper layer ating: dashed lines rrespnd t unupled prblem [6, 7], slid lines rrespnd t nsider upled prblem 11 Dwnlad Date 1/9/18 10:7 PM

5 Fig. : The equivalent stresses in the ating and in the substrate: dashed lines rrespnd t unupled prblem [6, 7], slid lines rrespnd t nsider upled prblem 11 Dwnlad Date 1/9/18 10:7 PM

6 6 CONCLUSIONS The prpsed alulatin predure makes it pssible t prvide analytial preditin f the stressed state and limiting equilibrium f tw-layer ating fr varius ratis between gemetrial and mehanial harateristis f the ating mpnents. The prinipal issue f the prpsed alulatin sheme is appliatin f the strength riteria t bth mpnents f the pieewise-unifrm bdy. The nlusin abut the real strength f the mpsitin shuld be dne by mparing the value f relatinship σ eq, /[ σ] and σ eq, /[ σ]. The mparisn f presented mdel that takes int aunt the nnetivity f membrane and bending fields and previusly develped mdel [6, 7] shws that at small ε the simplified thery is wrth using. LITERATURE [1] SAAKIYAN, L. S., EFREMOV A. P., ROPYAK L. Ya. & EPELFELD A. V. Appliatin f surfae strengthening f aluminum allys and atings fr imprving rrsin and mehanial resistane f il-and-gas industrial equipment mpnents. Msw: VNIIOENG, pp. (in Russian). [] STUDENT, М. М., DOVHUNYK V. М., KLAPKIV М. D., POSUVAILO V. М., SHMYRKO V. V. & KYTSYA А. Р. Triblgial prperties f mbined metal-оxide сerami layers n light allys. Materials Siene. 01, XLVIII. Nr., pp ISSN X. DOI: /s [] DOLGOV, N. A. Determinatin f stresses in a tw-layer ating. Strength f materials. 005, XXXVII. Nr. 4, pp ISSN DOI: /s [4] DOLGOV, N. A. Analytial methds t determine the stress state in the substrate ating system under mehanial lads. Strength f materials. 016, XLVIII. Nr. 5, pp ISSN DOI: /s [5] GORYACHEVA, I. G., TORSKAYA E. V., KORNEV Yu. V., KOVALEVA I. N. & MYSHKIN N. K. Theretial and experimental study f the mehanial prperties f bimpnent metal vapr depsited atings. Jurnal f Fritin and Wear. 015, XXXVI. Nr., pp ISSN DOI: 10.10/S [6] ROPYAK, L. Ya., SHATSKYI I. P. & MAKOVIICHUK M. V. Strength assessment f twlayer ating subjeted t the lal lading. Sientifi ntes. Lutsk. 015, Nr. 50, pp ISSN (in Ukrainian). [7] SHATSKYI, I. P., ROPYAK L. Ya. & MAKOVIICHUK M. V. Strength ptimizatin f a tw-layer ating fr the partiular lal lading nditins. Strength f Materials. 016, XLVIII. Nr. 5, pp ISSN DOI: /s [8] VLASOV, V. Z. & LEONT EV N. N. Beams, plates, and shells n the elasti bed. Msw: Fizmatgiz, (in Russian). [9] GRIGOLYUK, E. I. & TOLKACHEV V. M. Cntat prblems f plates and shells thery. Msw: Mashinstrenie, pp. (in Russian). 114 Dwnlad Date 1/9/18 10:7 PM