Elastic-plastic stress analysis of a thermoplastic composite disc under parabolic temperature distribution

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1 Indian Jounal of Engineeing & Mateials Sciences Vol. 4, August 007, Elastic-lastic stess analysis of a themolastic comosite disc unde aabolic temeatue distibution Fauk Sen a *, Yeliz Pekbey b, Onu Sayman a a Dokuz Eylul Univesity, Deatment of Mechanical Engineeing, 3500, Bonova, Izmi, Tukey b Ege Univesity, Deatment of Mechanical Engineeing, 3500, Bonova, Izmi, Tukey Received Febuay 006; acceted 6 Mach 007 In this study, an elastic-lastic themal stess analysis was caied out on a steel fibe einfoced themolastic comosite disc unde a aabolic temeatue distibution. The themal elastic and elastic-lastic s wee obtained both analytically and numeically. Theefoe, FORTRAN and ANSYS ogams wee used to calculate of themal stesses fo comaison. Tsai-Hill theoy was used as a yield citeion duing the. Themal stesses occued in the disc fo the aabolic temeatue distibution due to the diffeent themal exansions in adial and tangential diections. The magnitudes of the adial stess comonents wee obtained to be small in comaison with the tangential stess comonents. Additionally, the tangential stess comonent was found to be comessive and the highest aound the inne suface. Accoding to obtained esults these two s to be in good ageement. The intensities of the esidual stess comonent of the tangential stess and lastic flow wee the highest at the inne suface. IPC Code: C08B, G0L/00 Themolastic comosites offe many advantages in industial alications due to such as high imoved factue toughness, secific stiffness, and inceased imact esistance. They can be emelted, efomed and efomed. Themal stess analysis in comosite discs is a consideable subject due to the wide ange of alications in actice. Plastic yielding may occu at high temeatues. Consequently, esidual stesses and emanent defomation may occu. Otimum design of inhomogeneous non-unifom otating disc was investigated unde themal loading. A themo-elasto-lastic stess analysis in functionally gaded mateials unde a themal loading was efomed Shabana and Noda. Lanin and Deyavko 3 studied the influence of esidual stesses on the themal stess esistance in a ZC disc unde a themal load. Guven and Altay 4 calculated themal stesses in a linea hadening solid disc with igid casing subjected to a unifom heat souce. Themal stesses as a esult of cacking in the disc bakes wee studied by Mackin 5 An elastic-lastic stess analysis on a steel fibeeinfoced themolastic comosite disc was studied unde unifom themal distibution by Sayman 6. A numeical method was develoed fo the analysis of defomations and stesses in the elastic-lastic *Fo coesondence ( fauk.sen@deu.edu.t) otating discs with abitay coss-sections of continuously vaying thicknesses and abitay vaiable density made of nonlinea stain-hadening mateials by using Von Mises yield citeion 7. Jahed and Shiazi 8 examined loading and esidual stesses, and associated stains and dislacements in themolastic otating discs at elevated temeatues. Elastic-lastic stesses in thick-walled cylindes made of non-hadening Tesca mateial was investigated 9. Huang 0 esented a stength fomula fo the unidiectional comosites including themal esidual stesses. Rossit and Laua examined unsteady themal stesses caused by the esence of a hot, cental nucleus, in disc o cylindes. Tende et al. studied esidual stesses and dimensional changes in the comession moulded glass-mat einfoced themolastic ats. Bektas and Sayman 3 investigated elastic-lastic and esidual stesses in themolastic comosite laminated lates unde linea themal loading. In the esent study, an elastic-lastic themal stess analysis was caied out on a steel fibe einfoced themolastic comosite disc unde a aabolic temeatue distibution. The elastic and elastic-lastic themal stesses wee obtained both analytically and numeically ste by ste at diffeent aabolic temeatues. Futhemoe, esidual stesses

2 SEN et al.: ELASTIC-PLASTIC STRESS ANALYSIS 83 Table Mechanical oeties of the comosite disc E E G ν Y S K α (/ C) α (/ C) wee calculated via sueosition method using esults of these two s. Mateials and Method Mechanical oeties of the comosite disc ae shown in Table. Accoding to Sayman et al. 6 the comosite disc consists of low-density olyethylene as a themolastic matix and steel fibes as cuvilinea. The yield oint, the modulus of elasticity and the themal exansion coefficient of the low density themolastic ae 9.3 MPa, 70 MPa and (/ºC), esectively. The aw ganules of the olyethylene wee laced into the mold heated by an electical esistance and the temeatue was inceased u to 60ºC fo eaing olyethylene layes. The aw mateial was held fo 5 min unde.5 MPa essue at this temeatue. And then the temeatue was deceased to 30ºC in 3 min unde 5 MPa essue. Thus, a olyethylene laye with thickness of mm was oduced. The fibe volume faction was measued as 6%. The dimensions of a cuvilinea othotoic disc that is used duing the ae illustated in Fig.. It can be seen in this figue inne and oute adii of comosite disc ae selected as 30 and 70 mm, esectively Elastic and elastic-lastic stess analyses ae caied out both analytically and numeically by comaing with obtained esults. The finite element method (FEM) is used fo numeical. Fo this uose, ANSYS that is known geneal uose finite element softwae is used. Meanwhile, analytical is efomed via FORTRAN ogam with a eaed ogam by the authos. Additionally, the finite element model ceated in ANSYS is illustated with bounday conditions in Fig.. It is seen in this figue that a maed mesh is intoduced to disc model using PLANE 4 element tye that is used fo -D modeling of solid stuctues. In addition, the element is defined by fou nodes having two degees of feedom at each node: tanslations in the nodal x and y diections. The element has lasticity, cee, swelling, stess stiffening, lage deflection, and lage stain caabilities 4,5. Afte the mesh geneation ocess, the model was divided into 4400 elements Fig. Cuvilinea othotoic disc Fig. Finite element model and bounday conditions in ANSYS

3 84 INDIAN J. ENG. MATER. SCI., AUGUST 007 and 455 nodes. Because of the symmety, the quate at of the whole disc is modeled. Elastic Solution In this study, a comosite disc is assumed that it is vey thin and as a esult of this, the temeatue distibution does not change acoss the thickness of the disc. The stess stain elations in a comosite disc einfoced cuvilinea ae witten as, du ε aσ a σ d u ε a σ aσ () whee u is the dislacement in the adial diection and a, a, a ae the comonents of the comliance matix, α and α ae the themal exansion coefficients in the adial and tangential diections, esectively. The equation of equilibium in the lane stess case fo a cuvilinea othotoic laye can be educed to the following fom, σ σ σ 0 () The comability equation is witten fom the elation between the dislacement and stains. ε ε ε (3) The stess function F satisfies the equation of equilibium as, F σ df σ d (4) The govening stess function is witten by using Eqs ()-(4), df df a a a F d d dt α αt αt d (5) A aabolic temeatue distibution can be witten as, T i ( b ) T b a (6) whee a, b ae the inne and oute adii of the disc. It is consideed that the temeatue is T i and zeo at the inne and oute sufaces, esectively. The of the equations is given as, k k 3 F C C A C (7) The stess comonents ae found as, F k k C C σ A C df kc k k σ C k 3 A C d whee A and C ae found as, 3α α A λ λ b, a (9 k ) T i a, k a a ( α α ) b C λ a, ( k ) (8) (9) The integation constants C and C ae obtained fom the bounday conditions. They ae inne and oute sufaces of the disc and ae fee. Consequently, σ 0 at the a and b. The integation constants ae found as, k 3 k k 3 k Aa Ca Ab Cb k k C b a C Ca Aa Ca k k 3 k (0) The dislacement comonent in the adial diection is obtained easily, fom Eq. () u a σ a σ α T () Elastic-Plastic Solution The stain incements in the adial and tangential diections ae witten as, dε a dσ a dσ dε α dt dε a dσ a dσ dε α dt () The stain incements fo anisotoic mateials ae obtained fom the lastic otential f, f dεij dλ (3) σ ij

4 SEN et al.: ELASTIC-PLASTIC STRESS ANALYSIS 85 whee f is the otential function which is equal to the equivalent stess (σ ), λ is a scala quantity and it is equal to the equivalent lastic stain incement dε. The equivalent stess fo the Tsai-Hill citeion is witten as, σ σ σ σσ (4) Y whee, and Y ae the yield stengths of the disc in the fibe and tansvese diections, esectively. In this, it is assumed that the mateial hadens linealy. σ σ o Kε whee σ o is equal to and K is the lasticity constant. If the tangential stess comonent is witten in tems of the adial stess comonent and then is substituted in Eq. The Tsai-Hill citeion, d σ the adial stess incement is obtained as, σ± σ 4 σ σ Y dσ d (5) The lastic stain incements in the fibe and tansvese diections ae obtained fom the lastic otential, accoding to the Pandtl-Reuss equations, σ f σ dε Y dλ dε σ σ f σ σ dε dλ dε σ σ (6) The total stain incements ae obtained fom the integation of Eq. () as, du ε a σ a σ d ε α dt B d u a a d T B ε σ σ ε α (7) whee B and B ae the integation constants. B and B ae found to be equal to zeo by using the elation that the elastic and lastic stain incements ae equal at the beginning of lastic yielding. When the stain comonents ae substituted in the comability equation the following elation is found, a σ a σ ε α T a σ a σ ε α T dσ dσ dε dσ a a α d d d d (8) o d aσ aσε αt aσ aσ ε αt adσ adσ dε α dt 0 (9) The temeatue incement can be witten as, dti Ti dt ( b ) ( d) b a b a whee T i (0) is the temeatue at the inne suface. The yield citeion can be witten in the diffeential fom as, σ dσ σdσ σdσ σ dσ σkdε Y () If the tangential stess incement dσ is taken out fom the Eq. () and then the tangential stess and temeatue incement Eq. (0) ae substituted in Eq. (9), the following equation is obtained, σ K σ σ a ε σ σ σ ( a a) σ( a a) ε ε T ( α α ) σ Σ Σ σ σ / Y a σ a σ σ Ti Ti σ α ( b ) α b a b a () whee ε, σ, σ, and Ti ae used instead of dε, dσ, d σ, d and dti. ε equivalent lastic stain is calculated fom the evious equation. When the lastic yielding occus, T is inceased 500 stes fom the yield oint to the given temeatue. The stess comonents at this ste can be calculated as, ( ) ( ) ( ) ( ) σ σ σ i i σ σ σ i i ( ) ( ) ε ε ε i i (3)

5 86 INDIAN J. ENG. MATER. SCI., AUGUST 007 whee i is the ste numbe. If the lastic egion exands fom a to any adius c, the integation comonents ae found fom the bounday conditions at the elastic-lastic bounday, since the adial stess comonent is equal at the bounday of the elastic and lastic egions. If the lastic egion exands u to c the integation constants in the elastic egion fo the stess function ae obtained as, C C C b k 3 k 3 k k ( b ) C( c b ) A c k Ab k 3 k b c Cb k k c k (4) whee is equal to the σ at the bounday of the lastic egion. Results and Discussion Plastic yielding begins at 6.3 o C. Fo this eason, the elastic is shown at 60 o C in Fig. 3. As seen in this figue, the magnitudes of the tangential stess comonents ae bigge than the adial stess comonent. Esecially, the magnitude of the tangential stess comonent is the highest at the inne suface of the disc. It is comessive and tensile at the inne and oute sufaces, esectively. The distibutions of the themal elastic, lastic and esidual stess comonents fo the adial diection at 80 and 00 o C ae shown in Figs 4 and 5, esectively. As seen in these figues, all the stess comonents fo adial diection ae calculated vey small. The distibution of the tangential stess comonents fo the elastic and lastic cases at 70 o C is shown in Fig. 6. As seen in this figue, it is comessive and tensile aound the inne and oute sufaces, esectively, since the disc wants to exand aound the inne suface due to its high temeatue the aticles aound the oute suface aly a comessive Fig. 3 Distibution of the elastic stess comonents at 60 o C Fig. 5 Distibution of the adial stess comonents at 00 o C Fig. 4 Distibution of the adial stess comonents at 80 o C Fig. 6 Distibution of the tangential stess comonents at 70 o C

6 SEN et al.: ELASTIC-PLASTIC STRESS ANALYSIS 87 foces to the aound the inne sufaces. Theefoe, the diffeent temeatue distibution oduces the comessive and tensile stesses aound the inne and oute sufaces, esectively. The magnitudes of the esidual stess comonents fo the tangential diection ae the highest and tensile at the inne suface, while it is comessive at the oute suface. The distibutions of the tangential stess comonents fo the elastic and lastic cases at 80, 90 and 00 o C ae shown in Figs 7-9. They ae simila to that in Fig. 6. Accoding to these figues, the magnitudes of the themal and esidual stess comonents inceases by inceasing alied aabolic temeatue loadings. Consequently, the maximum values of themal and esidual stesses ae obtained on the inne suface at 00 o C. The distibution of equivalent lastic stess in the disc at 00 o C is illustated with colo contous in Fig. 0 as examle. Besides, the distibution of the equivalent lastic stess fom 70 to 00 o C is shown in Fig. 9 Distibution of the tangential stess comonents at 00 o C Fig. 7 Distibution of the tangential stess comonents at 80 o C Fig. 0 Equivalent lastic stess in the disc at 00 o C Fig. 8 Distibution of the tangential stess comonents at 90 o C Fig. Distibution of the equivalent lastic stess

7 88 INDIAN J. ENG. MATER. SCI., AUGUST 007 Table A comaison of analytical and numeical esults of tangential stess comonent (σ ) at the inne and oute sufaces Temeatue ( C) Sufaces Analytical Elastic stess Numeical Analytical Plastic stess Numeical Analytical Residual stess Numeical Inne Oute Inne Oute Inne Oute Inne Oute ,0-0.0 Fig., because of the fact that lastic yielding is stated 6.3 o C. It can be seen in this figue, the magnitude of equivalent lastic stess is inceased deending on incease of aabolic temeatue loads fo inne suface, although it has got a constant value that is 4 MPa on the oute suface. It is mentioned befoe the is comleted in two ats which ae analytical and numeical using FORTRAN and ANSYS ogams, esectively. Theefoe, comaisons of analytical and numeical esults of tangential stess comonents (σ ) at the inne and oute sufaces ae esented in Table. As seen in this table, these two methods oduce simila esults. Esecially, the elastic-lastic esults obtained fom these methods ae simila than those obtained fom the elastic fo inne suface. In geneal, analytical esults ae highe than numeical esults fo both inne and oute sufaces, but the diffeence is vey small. In othe wods, the diffeences between analytical and numeical esults ae negligible levels. Futhemoe, themal stesses ae calculated as comessive and tensile foms at the inne and oute sufaces, esectively. The magnitude of comessive stesses is highe than tensile stesses. Accoding to elastic and elastic-lastic esults, esidual stesses at the inne suface ae bigge than oute sufaces. Additionally, the maximum value of esidual stesses is comuted as 7.86 MPa on the inne suface at 00 o C. (iii) The tangential stess comonent is comessive and tensile aound the inne and oute sufaces, esectively. (iv) The magnitudes of the esidual stesses fo the tangential diection ae the geatest at the inne suface. (v) The esults of both analytical and numeical methods ae in good ageement. (vi) The magnitudes of all themal and esidual stess comonents incease with incease in alied aabolic temeatue loadings. Refeences Fashi B, Jahed H & Mehabian A, Comut & Stuct, 8 (004) Shabana Y M & Noda N, Comos Pat B 3 (00), -. 3 Lanin A G & Deyavko I I, J Euo Ceam Soc, 0 (000) Guven U & Altay O, Mec Res Commun, 5 (998) Mackin J T, Noe SC & Ball K J, Eng Failue Anal, 9() (00) Sayman O, Yangıncı S & Saye M, J Reinfoced Plast Comos, 4 (005) You L H, Tang Y Y, Zhang J J & Zheng C Y, Int J Solids Stuct, 37 (000) Jahed H & Shiazi R, Int J Pes Ves & Piing, 78 (00) Jahed, H, Sethuaman H R & Dubey R N, Int J Pes Ves & Piing, 7 (997) Huang Z M, Mate Lett, 43 (000) Rossit C A, Laua P A A, Ocean Eng, 7 (000) Tende A, Astom B T & Nilsson G, Comos. Pat A, 3 (000) Conclusions 3 Bektas N B & Sayman O, J Reinfoced Plast Comos, The following conclusions have been dawn fom this study: (i) The magnitude of the tangential stess comonent is highe than that of the adial stess comonent (ii) The lastic defomation begins at the inne suface, fistly. (003) ANSYS Documentation, The geneal-uose finite element softwae, ANSYS, Inc. 5 Moaveni S, Finite element analysis: Theoy and alication with ANSYS, (Peason Education. Inc., New Jesey, USA), 003.