ANALYSIS OF THERMOELASTIC STRESSES IN LAYERED PLATES

Transcription

1 6t Iteratioal DAAAM Baltic Coferece "INDUSTRIAL ENGINEERING" 4-6 April 8, Talli, Estoia ANALYSIS OF THERMOELASTIC STRESSES IN LAYERED PLATES Kõo, J. & Valgur, J. Abstract: Tis paper deals wit termoelastic effects i deformatio of plates wit arbitrarily cagig elastic parameters ad temperature troug tickess. Usig te semi-iverse metod, a simple aalytical solutio is obtaied for te termoelastic problem of a oomogeeous plate wit a arbitrary cotour. Plates wit free, slidig ad fixed edges are cosidered. Te results are applied to te layered plate. As a example of applicatio, termoelastic stresses ad deformatios are determied for a copper plate wit a steel coatig. Te edge effects for tis plate are examied by te fiite elemet metod. Key words: semi-iverse metod; termoelastic stresses; layered plates; edge effects; fiite elemet metod.. INTRODUCTION I moder egieerig desig, oomogeeous layered structures are widely used due to teir superiority i stiffess-toweigt ad stregt-to-weigt ratios, as well as logevity. Te study of mecaical, especially termoelastic, caracteristics of layered beams ad plates as bee of icreasig iterest for egieers. Termo-mecaical aalysis of layered beams (arrow strips) is preseted i papers [, ]. I te moograp [ 3 ] te termoelastic problem is solved for a omogeeous free plate wit temperature variatio troug tickess. I te paper [ 4 ] tis solutio is exteded to te plate wit variatio of termoelastic parameters troug tickess. I preset paper te solutio uder cosideratio will be applied to termoelastic aalysis of layered plates. I coectio wit tis, te plate wit free, slidig, ad fixed edges will be studied. As a example of suc applicatio, termoelastic stresses ad deformatios will be determied for a quadratic copper plate wit a steel coatig. Te edge effects for tis plate will be examied usig te fiite elemet program ANSYS.. PLATE WITH ARBITRARY VARIATION OF THERMOELASTIC PARAMETERS THROUGH THICKNESS.. Plate wit free edges Cosider a plate (Fig. ) of a arbitrary form ad of costat tickess. Te plate is completely free of te surface load. Te rectagular coordiates x, y ad are used, were te lower surface of te plate is take as te referece surface (x, y) ad te coordiate is directed upwards. Fig.. Free plate wit a arbitrary sape

2 Te plate is fixed at te origi of te coordiates, were deflectio ad its derivatios are ero, i.e. u u = = =. x y Te temperature T, te coefficiet of liear termal expasio α, te modulus of elasticity (Youg s modulus) E ad te Poisso s ratio μ vary troug tickess oly, = E = E ad i.e. T T( ), α=α( ), μ = μ ( ). By aalogy wit te omogeeous plate [ 3 ], it is reasoable to assume tat te stress compoets will be of te followig form: x = y = ( ). () = τxy = τ y = τx = Tese stresses satisfy idetically te followig equilibrium equatios of te teory of elasticity [ 3, 5 ]: τ x yx τx + + = (x, y, ). () x y Here ad below te symbol (x, y, ) deotes permutatio by meas of wic we ca write two more equatios correspodig to te oter two axes, cagig x for y, y for, ad for x. From te termoelastic stress-strai relatios [ 3, 5 ] ε = ( + ) μ + α T (3) x E x y (x, y, ) we ave εx = εy = ε( ) = + αt E, (4) μ ε = + αt E were E E = (5) μ is te biaxial elastic modulus. For strais (4) te compatibility equatios [ 3, 5 ] ε γ y x x y are reduced to ε x y xy + = (x, y, ) (6) + T =. E α (7) Terefore = E α T + a+ b, (8) were te costats a ad b are to be determied from te boudary coditios of ero resultat force ad momet o te edges of te plate. Tese coditios are reduced to: d = d =. (9) Substitutio of expressio (8) yields te two equatios for te ukow costats a ad b. From tese equatios we fid: CNT BMT a = C BD, () CMT DNT b = C BD were te quatities N T, M T, B, C ad D are defied as follows: NT = E αtd ; () M = T E αtd B= E d C = E d. () D= E d Te displacemets u x, u y ad u i te x, y ad directios, respectively, are obtaied by usig equatios [ 3, 5 ] ux ε x = (x, y, ) (3) x ad (4) as well as (8). Te solutio is te foud to be: ux = x( a+ b) ; (4) uy = y( a+ b)

3 a u = ( x + y ) + (5) μ + μ + ( a + b) + α T. d μ μ For completeess of te solutio, we determie te deformatio caracteristics of te referece surface ( = ). Usig equatios (4) ad (3) we ave for strai o te referece surface: ε ε =b. (6) I te case of small deflectios we ca use for curvatures te approximatios: u u æx = ; æ y =. x y Te, usig equatio (5) we fid te curvatures of te referece surface æx = æy æ = a. (7) Tus, te plate subjected to a uiform temperature cage is bet to a sperical surface... Plate wit slidig edges For a plate wit slidig edges (Fig. ), we ca suppose tat u =, = = x y for every poit of te referece surface. Fig.. Plate wit slidig edges Te a = ad from equatios () it follows tat N b = T (8) B ad expressio (8) for stresses is reduced to = E α T + b. (9).3. Plate wit fixed edges For a plate wit fixed edges (Fig. 3), we ca start from te assumptios tat ux = uy = u =, = = x y o te referece surface. Fig. 3. Plate wit fixed edges Te a = b = ad equatio (8) for stresses is reduced to = E α T. () 3. LAYERED PLATE SUBJECTED TO A UNIFORM TEMPERATURE CHANGE 3.. Plate wit free edges Cosider a layered plate (Fig. 4) cosistig of a arbitrary umber of layers wit differet costat tickesses i. Fig. 4. Layered plate Te locatio of a arbitrary layer i of te plate is specified by te coordiate i, wic is te distace from te bottom plae of te plate to te top plae of te it layer. Assume tat te coefficiet of termal expasio α i, te modulus of elasticity E i ad te Poisso s ratio μ i do ot cage troug te tickess of te it layer. Te, assumig tat te plate is subjected to a uiform temperature cage ΔT, ad usig piecewise itegratio, we ca write te parameters N T, M T, B, C ad D i equatios () ad () as NT =ΔT Eiαi( i i ) =ΔT Eiαii i= i= ;() ΔT M = ( ) T Eiα i i i i=

4 B E E = i ( i i ) = i i i= i= = i ( i i ) i= 3 3 = i ( i i ) 3 i= C E D E. () For te costats () we ave a = aδt ; b= bδt, were CNT BMT a = C BD. (3) CMT DNT b = C BD Here NT = Eiαi( i i ) = Eiαii i= i=. (4) M = ( ) T Eiαi i i i= I accordace wit equatio (8) stresses i te it layer are = E ΔT α + a+ b. (5) i i i Deformatio parameters are ε = bδt. (6) æ = aδt 3.. Plate wit slidig edges I tis case a =, ad N b = T B. (7) Stresses i te it layer are = E ΔT α + b. (8) i i i Strai o te referece surface ε = bδt (9) Plate wit fixed edges For tis case a = b =, ad stresses i te it layer are = E α ΔT. (3) i i i 4. APPLICATIONS: THERMAL STRESSES DUE TO UNIFORM TEMPERATURE CHANGE IN A COPPER PLATE WITH A GALVANIC STEEL COATING Usually, te coefficiets of termal expasio of te substrate ad coatig are differet ad te temperature of te coatig process differs from room temperature (+ C). Terefore, termal stresses are geerated i coated parts. I experimetal aalysis of residual stresses it may tur out tat coatig temperature differs from te temperature at wic te deformatio parameters of te substrate are measured. I suc cases measuremet results ave to be corrected takig ito accout termal stresses. As a umerical example, termal stresses ad displacemets for a quadratic copper plate 3 3 mm wit galvaic steel coatig deposited at 95 C are calculated. Fig. 5. Copper plate wit a steel coatig I additio, te followig data is used: = =. mm; E = GPa; μ =.34; α = = / C for te substrate ad = =.6 mm; E = GPa; μ =.8; α = = / C for te coatig.

5 4.. Plate wit free edges For uiform temperature cage ΔT = = 95 = 75 C, te values defied by equatios (5), () ad (4) are: E = 66.7 GPa; E = 8.6 GPa; N = 396 N/m; M =.638 N; T B = N/m; C = N; D = 49. N m. By meas of equatios (3) we ave a = /m; b = For stresses i layers from equatio (5) it follows tat: 3 6 i = 75Ei ( αi ) (i =, ). (3) At typical poits of tickess stresses i MPa are: = 4.9 ( 3.5) 3 ( ) = 3.7 (3.6). (3) 3 ( ) = 8.5 ( 7.56) 3 (.6 ) =. (.3) From (6) for te strai ad curvature of te referece surface we fid 6 6 ε = 398 ( ) (33) æ = 6 (.53 ) m 4.. Plate wit slidig edges Accordig to (7), we ave b = ad for stresses i layers (8) we obtai 6 = 75 ( 6.58 i Ei α i ) (34) (i =, ). From ere for stresses i MPa we ave: =.5 (.6). (35) = ( 46.77) Strai o te referece surface (9) as te value 6 6 ε = 4 ( 39.6 ). (36) 4.3. Plate wit fixed edges By meas of equatio (3) we calculate stresses i te layers i MPa: = 8.79 (8.7). (37) = 3.94 (3.8) T 4.4. FEM aalysis A similar problem was aalysed by te fiite elemet program ANSYS. Te obtaied distributio of ormal stress x = = y = i te middle regio of te plate is preseted i Fig. 6. Te values of stresses at typical poits of tickess are give i te pareteses at te give data sets (3), (35) ad (37). Te values of te deformatio parameters ε ad æ are preseted i te same maer at te data sets (33) ad (36). Compariso of te results sows tat discrepacy betwee data is less ta 5%. Some results of stress aalysis i te edge regio (.5 mm x 5 mm, y =,.6 mm) are preseted i Fig. 7. As we ca see, te ormal stresses x = y = i te edge regio are reduced. Te ormal stress, kow as peelig stress, ad te sear stress τ x are igly localied ear te edge. Tese facts oce agai cofirm te validity of te well-kow Sait-Veat s priciple. Fig. 6. Normal stresses x = y = () i te middle regio of a free copper plate wit a steel coatig

6 (a) (c) (b) (d) Fig. 7. Stress distributios i te edge regio: (a) ormal stress x, (b) ormal stress y, (c) ormal stress (peelig stress), (d) sear stress τ x (MPa) 5. CONCLUSIONS. Usig te semi-iverse metod, a aalytical solutio is obtaied for te termoelastic problem of a plate wit a arbitrary cotour oomogeeous troug tickess. Te results are applied for aalysis of a layered plate subjected to a uiform temperature cage.. As a applicatio, stresses due to uiform temperature cage i a copper plate wit a galvaic steel coatig are determied. 3. Usig te fiite elemet metod, te termo-mecaical state of te same plate wit te coatig is studied. For te middle regio, good agreemet is acieved betwee te data of te aalytical ad umerical metods. Te data obtaied wit te use of te fiite elemet metod for stresses i te edge regio of te plate cofirm te validity of te well-kow Sait- Veat s priciple. 6. REFERENCES. Feg, Z. C. ad Liu, H. D. Geeralied formula for curvature radius ad layer stresses caused by termal strai i semicoductor multilayer structures. J. Appl. Pys., 983, 54 (), Hsue, C. H. Termal stresses i elastic multilayer systems. Ti Solid Films,, 48, Boley, B. A. ad Weier, J. H. Teory of Termal Stresses. Jo Wiley ad Sos, Ic., New York, Lodo, Kõo, J. P. Problem of termoelasticity for a free oomogeeous plate wit temperature variatio troug tickess. I Heat Stresses i Structural Members. Naukova Dumka, Kyiv, 979, 9, 89 9 (i Russia). 5. Timoseko, S. P., Goodier, J. N. Teory of Elasticity. Tird Editio. McGraw-Hill, Sigapore, CORRESPONDING AUTHOR Prof. emer. Jakub Kõo Estoia Uiversity of Life Scieces Istitute of Forestry ad Rural Egieerig Kreutwaldi 5, Tartu 54, Estoia, jakub.koo@emu.ee