THE SMOOTH INDENTATION OF A CYLINDRICAL INDENTOR AND ANGLE-PLY LAMINATES W. C. Lao Department of Cvl Engneerng, Feng Cha Unverst 00 Wen Hwa Rd, Tachung, Tawan SUMMARY: The ndentaton etween clndrcal ndentor and angle-pl compostes s nvestgated. An exact Green s functon for the surface dsplacement of an angle-pl eam s derved usng Jones s dsplacement formulaton. B matchng the dsplacements of the top surface of the angle-pl lamnate and those of the ndentor wthn the contact regon, we can fnd the contact stress dstruton, the magntude of contact force and the ndentaton. In order to speed up the formulaton of fundamental deflecton mode shape of Green s functon, method of ntal functon (MIF) s adopted to fnd the dsplacements n the top and ottom surface of an angle-pl lamnate. In the MIF, the order of sstem equaton s onl 3 3, whch wll reduce the computaton tme. Numercal results for angle-pl lamnates sujected to clndrcal ndentaton are evaluated to verf the applcalt of the proposed method. KEYWORDS: angle-pl lamnates, ndentaton, pont matchng method, method of ntal functons INTRODUCTION The contact prolems of composte lamnate are mportant n the analss of foregn oject mpact and flexural measurement of composte materals. Several authors have studed the contact ehavors etween a clndrcal ndentor and orthotropc eams[,,3]. Sanar solved the
contact prolem of orthotropc eam the superposton of half space soluton and eam soluton[]. B applng the exact dsplacement soluton for clndrcal endng of orthotropc eams, the exact Green s functon s formulated n solvng the contact prolems[3]. However the ndentaton etween clndrcal ndentor and angle-pl compostes s seldom nvestgated. In ths stud, an exact Green s functon for the surface dsplacement of an angle-pl eam s derved usng Jones s[4] dsplacement formulaton. Ths Green s functon satsfes the dsplacement and stress contnut condton at the nterface, and the prescred stress condton at the top and ottom surfaces. B matchng the dsplacements of the top surface of the angle-pl lamnate and those of the ndentor wthn the contact regon, we can fnd the contact stress dstruton, the magntude of contact force and the ndentaton. In constructng the exact Green s functon, we have to solve a matrx of order 6N 6N, wth M tmes, where N s the pl numer and M s the numer of asc flexural modes. If the pl numer ncreases, the computaton tme wll ncrease drastcall. In order to speed up the calculaton, method of ntal functon (MIF)[5] s adopted to fnd the dsplacements n the top and ottom surface of an angle-pl lamnate. In the MIF, the order of sstem equaton s onl 3 3, whch wll reduce the computaton tme. Numercal examples are llustrated to demonstrate the contact ehavors of angle-pl lamnates. CYLINDRICAL BENDING OF ANGLE-PLY LAMINATES In order to stud the contact ehavor of angle-pl lamnates, the clndrcal endng prolems of an ansotropc lamnate s nvestgated frst whch wll provde the Green s functon for contact prolem. Assumng plane stan condton n the x-drecton, an ansotropc eam has wdth and thcness h n the -drecton s shown n Fgure. The stran feld s defned as u v w ε x 0, ε, ε x v w u u γ +, γ x, γ x () The consttutve relaton for a generall ansotropc materal s xx c c c c 0 3 0 0 6 c c c c ε 3 0 0 6 c c c c ε 3 3 33 0 0 36 () 0 0 0 c c γ 44 45 0 x 0 0 0 c c 0 γ 45 55 x x c c c 0 0 c γ 6 6 36 66 x
In whch c j s the elastc constant for each laer. The eam s smpl supported along 0 and, wth the followng oundar condtons w(0,)w(,)0 (3a) ( 0, ) (, ) 0 (3) x ( 0, ) (, ) 0 (3c) x (, 0) (, 0) (, 0) 0 (3d) x ( h, ) q ( ), ( h, ) ( h, ) 0 (3e) x If the nerta force s neglected, the equlrum equaton can e wrtten as + 0 (4a) x, x, + 0 (4),, + 0 (4c),, The Dsplacement felds for ths prolem can e expressed as nπ u (, ) Un ( ) cos nπ v(, ) Vn ( ) cos nπ w (, ) Wn ( ) sn The aove canddates satsf Eqns. (3a), (3) and (3c). When susttute Eqn. (5) nto Eqn.(4) and Eqn.(), the followng equatons can e derved. '' '' pc66 U( ) + c55 U ( ) pc6 V( ) + c45v ( ) + pc ( 36 + c45) W' 0 (6) '' '' pc6 U( ) + c45 U ( ) pc V( ) + c44 V ( ) + pc ( 3 + c44 ) W' 0 (7) '' pc ( 45 + c36) U'( ) pc ( 44 + c3) V'( ) + c33 W( ) pc44 W ( ) 0 (8) In whch pnπ /, the suscrpt n of U, V and W are omtted for smplct. In order to solve the U (), V () and W (), we let [4] U ( ) U V ( ) V 0 0 W ( ) W e 0 e λ λ e λ a c55, a c44, a3 c33 p c66, p c, 3 p c44 (0) c c45, d pc6, e pc ( 36 + c45), e pc ( 3 + c44 ) B susttutng Eqns. (9) and (0) nto Eqns. (6)-(8), the nontrval solutons exst f the followng relaton holds λ a + λ c d λe det λ c d λ a + λe 0 () λe λe λ a + 3 3 (5) (9)
After solvng the 6th order characterstc equaton of λ, the fnal form of U n (), V n () and W n () can e elded as 6 * λn n U ne 6 * λn n ( ) V ne () 6 * λn n ( ) W ne U ( ) V W * * * In whch U, V and W are the correspondng egenvectors when solvng Eqn.(). The onl unnowns are n. For a fxed p, there are sx n s n each laer. These constants can e solved from the nterface contnut and the tracton condtons at the top and ottom surfaces. If the lamnate s made of N laers, the contnut of dsplacements and stresses at the nterface requres that + u (, h ) u (, 0) + v ( h, ) v (, 0) + w (, h ) w (, 0) + ( h, ) (, 0 ) (3) + ( h, ) (, 0) + x ( h, ) x (, 0),,,..., N In whch h s the thcness of the -th laer. The tracton condtons on the top and ottom surface mpose that N N (, 0) (, 0) x (, 0) ( h, N) x( h, N ) 0 N ( h, N ) q( ) (4) If the normal stress on the top surface s q( ) o sn p (5) then the coeffcents n can e solved through a set of matrx of order 6N 6N. The vertcal dsplacements w(,h) and w(,0) are of great mportance n the contact prolem. CYLINDRICAL INDENTATION OF SIMPLY SUPPORTED ANGLE-PLY LAMINATES Consder a smpl supported angle-pl lamnate sujected to the ndentaton of a rgd clnder as shown n Fg.. The radus of the ndentor s R. The frcton etween the ndentor and lamnate s neglected. The ndentaton s assumed to e smmetrc aout the mdspan of the eam. The contact length s c. In the contact prolems the dstruton of contact pressure has to e determned. The pont matchng method s adopted to fnd the contact stress dstrutons.
From prevous secton, the dsplacement at the top surface of the eam due to a normal stress o sn( nπ/ ) appled at h s shown to e nπ wh (, ) Wn ( h) sn (6) For an artrar normal stress q() at the top surface, the dsplacement of the eam wll e nπ wh (, ) qw n n( h) sn (7) In whch nπ qn q d ( )sn 0 (8) + c nπ q d ( )sn c From Eqns. (7) and (8), we have nπξ nπ w(, h) Wn ( h) q( ξ )sn dξsn Ω Ω G( ξ) q( ξ ) dξ Where Ω s contact regon, and the Green s functon s defned as[3] nπ nπξ G( ξ ) Wn ( h) sn sn (0) The relatve vertcal dsplacements of an pont w(,h) nsde the contact regon and the center pont w(0.5,h) can e expressed as[] 0.5 ( 0.5) w(, h) w(0.5, h) R R R () [ G( ξ) G(0.5 ξ )] q( ξ ) dξ Ω The contact regon s dscreted nto several segments, wthn each segment the contact stress s assumed to e constant. B the least square method and pont matchng method[,3], we can fnd the contact stress dstrutons. (9) METHOD OF INITIAL FUNCTIONS In constructng the Green s functon for the contact prolems, the vertcal deflecton at top surface W n (h) due to a snusodal stress s needed. If the laer numer ncreases, t wll e tme consumng to fnd the asc deflecton mode. The method of ntal functon[5] s appled to smplf the calculaton of (h). From the stran defntons, the stress-stran relaton and W n equlrum equatons, the followng equaton can e derved[5]:
U V Z 0 A X B 0 Y W In whch c44 / d c45 / d 0 A c / / 45 d c55 d 0 0 U V Z X Y W () (3a) * c66 * B c6 c36 c33 c c * 6 * c c 3 33 c36 c 33 c3 c 33 / c33 (3) d c44c55 c45,,, c c * 3 j3 cj cj,, j,, 3, 6 (4) c33 X, Y, Z x The materal constants for the -th laer should e susttuted n Equatons () and (3). The soluton of Eqn. () s shown to e R( ) e Where 0 B A 0 R(0) R( ) T { U( ), V( ), Z( ), X( ), Y( ), W( )} (6) The state vector R() n the -th laer at artrar heght can e expressed n terms of the state vector at the ntal reference heght of that laer. B a successve multplcaton of the state vector at each laer, W n (h) can e found solvng a set of 6 6 smultaneous equaton. (5)
NUMERICAL EXAMPLES A [± 45] S composte lamnate made of Hexcel graphte/epox s analed as an llustratve example. The eam has a wdth of 0 cm and thcness of cm. Each lamna assumes the followng propertes, E 05.7GPa, E E3 8.57GPa, υ υ3 0.37, υ3 0.306 G G 4.39GPa, G 3.05GPa 3 3 The radus R of the rgd ndentor s 7.6cm. The through-the-thcness stress dstruton s calculated for contact length cmm. Fgure 3 shows the thcness-wse dstruton of the normal stress. The results at three dfferent postons (the contact center, the contact edge and end of eam) are compared. It seems that the normal stress wll have maxmum value at the top surface of the contact center. The nterlamnar stress dstrutons of and x are demonstrated n Fgs. 4 and 5. The solutons satsf the tracton free condtons at oth top and ottom surfaces. Orthotropc solutons are also compared, n whch the effectve elastc constants of ths angle-pl lamnate s calculated usng a 3-D effectve modul theor[6]. The orthotropc results are comparale to the laer-wse solutons. Fgure 6 shows the n-plane stress dstruton of. The relaton of ndentaton and contact force s evaluated n Fg. 7. The ndentaton s xx defned as the dfference of vertcal dsplacement of w(0.5, h) and w(0.5, 0). Fgure 8 s the contact pressure dstruton for dfferent contact length. In order to verf the applcalt of method of ntal functon, results of W n (h) for dfferent n are revealed n Fg. 9. It seems that oth methods agree well for ths case. In some other case, f n s large, the successve multplcaton of state vectors wll cause some numercal nstalt. If ths nstalt can e overcome, the method of ntal functon wll e an effcent tool for contact analss. CONCLUSIONS A method for the analss of ndentaton of angle-pl lamnates sujected to rgd clndrcal ndentor s proposed. Numercal results show that ths method satsfes the gven oundar condtons and tracton condtons. The contact pressure dstruton, magntude of contact force and ndentaton can e evaluated. Ths analtcal results wll compare to the expermental results n the near future.
REFERENCES. Sanar, B. V., Smooth Indentaton of Orthotropc Beams, Compostes Scence and Technolog, 34, (989) 95-.. Sun, C. T. and Sanar, B. V., Smooth Indentaton of an ntall stressed orthotropc eam, Int. J. Solds and Structures,, (985), 6-76. 3. Wu, E., Chao, J. C. and Yen, C. S., Smooth Contact of Orthotropc Lamnates Rgd Clnders, AIAA Journal, 3, 0, (993), 96-9. 4. Jones, A. T., Exact Natural Frequences and Modal Functons for a Thc Off-Axs Lamna, J. of Composte Materals, 5, Oct. (97), 504-50. 5. Sundara Raja Iengar, K. T. and Panda, S. K., Analss of Orthotropc Rectangular Thc Plates, Fre Scence and Technolog, 8 (983), 9-36. 6. Sun, C. T. and Lee, S., Three-dmensonal Effectve Elastc Constants for Thc Lamnates, J. of Composte Materals, 988,, 69-636.