ASME2015 IDETC/CIE, 11th International Conference on Multibody Systems, Nonlinear Dynamics, and Control (MSNDC) MSNDC-10 Vehicle Dynamics

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1 UNCLASSIFIED: Dstrbuton Statement A. Approved for publc release. #26261 ASME215 IDETC/CIE, 11th Internatonal Conference on Multbody Systems, Nonlnear Dynamcs, and Control (MSNDC) MSNDC-1 Vehcle Dynamcs Techncal Publcaton DEVELOPMENT OF SHEAR DEFORMABLE LAMINATED SHELL ELEMENT AND ITS APPLICATION TO ANCF TIRE MODEL Hrok Yamashta Department of Mechancal and Industral Engneerng The Unversty of Iowa 2312 Seamans Center Iowa Cty, IA Paramsothy Jayakumar US Army TARDEC 651 E. 11 Mle Road Warren, MI Hroyuk Sugyama Department of Mechancal and Industral Engneerng The Unversty of Iowa 2416C Seamans Center Iowa Cty, IA Ths materal s declared a work of the U.S. Government and s not subject to copyrght protecton n the Unted States. Approved for publc release; dstrbuton s unlmted.

2 Report Documentaton Page Form Approved OMB No Publc reportng burden for the collecton of nformaton s estmated to average 1 hour per response, ncludng the tme for revewng nstructons, searchng exstng data sources, gatherng and mantanng the data needed, and completng and revewng the collecton of nformaton. Send comments regardng ths burden estmate or any other aspect of ths collecton of nformaton, ncludng suggestons for reducng ths burden, to Washngton Headquarters Servces, Drectorate for Informaton Operatons and Reports, 1215 Jefferson Davs Hghway, Sute 124, Arlngton VA Respondents should be aware that notwthstandng any other provson of law, no person shall be subject to a penalty for falng to comply wth a collecton of nformaton f t does not dsplay a currently vald OMB control number. 1. REPORT DATE 24 APR REPORT TYPE 3. DATES COVERED to TITLE AND SUBTITLE Development of Shear Deformable Lamnated Shell Element and ts Applcaton to ANCF Tre Model 5a. CONTRACT NUMBER 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) 5d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) US Army RDECOM-TARDEC,651 E. 11 Mle Road,Warren,MI, PERFORMING ORGANIZATION REPORT NUMBER 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 1. SPONSOR/MONITOR S ACRONYM(S) 12. DISTRIBUTION/AVAILABILITY STATEMENT Approved for publc release; dstrbuton unlmted 11. SPONSOR/MONITOR S REPORT NUMBER(S) 13. SUPPLEMENTARY NOTES ASME 215 IDETC/CIE, 11th Internatonal Conference on Multbody Systems, Nonlnear Dynamcs, and Control (MSNDC) 14. ABSTRACT See Report 15. SUBJECT TERMS 16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF ABSTRACT a. REPORT unclassfed b. ABSTRACT unclassfed c. THIS PAGE unclassfed Same as Report (SAR) 18. NUMBER OF PAGES 29 19a. NAME OF RESPONSIBLE PERSON Standard Form 298 (Rev. 8-98) Prescrbed by ANSI Std Z39-18

3 UNCLASSIFIED: Dstrbuton Statement A. Approved for publc release. #26261 ABSTRACT In ths nvestgaton, a contnuum mechancs based shear deformable shell element of the absolute nodal coordnate formulaton (ANCF) s generalzed to a lamnated shell element for applcaton to the modelng of fber-renforced rubber (FRR) structure of the physcs-based ANCF tre model. The complex deformaton couplng exhbted n fber-renforced composte materals can be automatcally consdered n the shear deformable lamnated shell element usng the contnuum mechancs approach, and the element lockngs are systematcally elmnated by the assumed natural stran and enhanced stran approaches, thereby leadng to a lockng-free shear deformaton ANCF lamnated shell element. Furthermore, varous nonlnear materal models can be consdered for each layer n a way same as sold elements. Usng the ANCF lamnated shell element developed, a physcs-based ANCF tre model s developed by consderng the detaled tre geometry and materal propertes. The expermental valdaton of the tre model s conducted for the load-deflecton curve to ensure that the fundamental structural tre propertes can be correctly captured n the ANCF tre model. Ths materal s declared a work of the U.S. Government and s not subject to copyrght protecton n the Unted States. Approved for publc release; dstrbuton s unlmted.

4 UNCLASSIFIED: Dstrbuton Statement A. Approved for publc release. # INTRODUCTION An accurate modelng of the complex tre geometry and the ansotropc materal propertes of tres s essental to the tre performance evaluaton ncludng the tre contact pressure and the brakng/tracton and cornerng forces. Snce a tre conssts of layers of ples and steel belts embedded n rubber, the tre structure needs to be modeled by cord-rubber composte materals and varous fber-renforced rubber materal models are proposed for use n detaled fnte element tre models. Snce Young s modulus of the steel cord s sgnfcantly hgher than that of the rubber matrx, mechancal property of the fber-renforced rubber (FRR) s hghly nonlnear [1]. In partcular, the tre cross-secton property that ncludes the geometrc and materal propertes s of sgnfcant mportance n characterzng the normal contact pressure dstrbuton. Furthermore, the n-plane shear deformaton of the carcass contrbutes to the cornerng characterstcs of tres. For ths reason, large-dmensonal hgh-fdelty fnte element tre models that account for the tre geometrc and materal nonlneartes are developed and used for the tre performance evaluaton. However, exstng fnte element tre models cannot be ntegrated nto the vehcle dynamcs smulaton due to the essental dfference n formulatons and soluton procedures used n multbody dynamcs and nonlnear fnte element codes. Ths prevents an ntegraton of the hgh-fdelty tre model nto the multbody vehcle dynamcs smulaton [2] and, therefore, the structural characterstcs of tres and transent tre dynamcs are, n general, evaluated usng dfferent computatonal models and dfferent smulaton approaches. To overcome ths fundamental and essental problem n the tre dynamcs smulaton, a tre model based on the flexble multbody dynamcs approach [2-4] s developed usng the absolute nodal coordnate formulaton (ANCF [6, 7]). The n-plane ANCF-LuGre tre model developed for the transent brakng analyss allows for consderng the nonlnear couplng between the dynamc Ths materal s declared a work of the U.S. Government and s not subject to copyrght protecton n the Unted States. Approved for publc release; dstrbuton s unlmted. 1

5 UNCLASSIFIED: Dstrbuton Statement A. Approved for publc release. #26261 structural deformaton of the tre and ts transent tre force n the contact patch usng general multbody dynamcs computer algorthms [4]. The generalzaton of the n-plane ANCF tre model to the three-dmensonal model requres the development of the new ANCF shell element suted for the tre model, whch allows for modelng the nonlnear fber-renforced rubber materals and the accurate three-dmensonal stresses under varous maneuverng scenaros. To ths end, the contnuum mechancs based shear deformable shell element of the absolute nodal coordnate formulaton (ANCF) [5] s generalzed to a lamnated shell element n ths study for applcaton to the modelng of fber-renforced rubber (FRR) structure of the physcs-based ANCF tre model. Furthermore, a physcs-based ANCF tre model s developed usng the shear deformable lamnated shell elements such that the hgh-fdelty tre model can be ntegrated nto general multbody dynamcs computer algorthms for ground moblty smulaton. 2. CONTINUUM MECHANICS BASED SHEAR DEFORMABLE ANCF SHELL ELEMENT 2.1 Knematcs of ANCF Shell Element As shown n Fg. 1, the global poston vector element s defned as [5] r of a materal pont T x [ x y z ] n shell r r r m( x, y ) z ( x, y ) (1) z where r ( x, y ) s the global poston vector n the mddle surface and r ( x, y ) m z s the transverse gradent vector used to descrbe the orentaton and deformaton of the nfntesmal volume n the element. Usng the b-lnear polynomals, the poston vector n the mddle surface and the transverse gradent vector are approxmated as follows: Ths materal s declared a work of the U.S. Government and s not subject to copyrght protecton n the Unted States. Approved for publc release; dstrbuton s unlmted. 2

6 UNCLASSIFIED: Dstrbuton Statement A. Approved for publc release. #26261 r rm ( x, y ) Sm( x, y ) e p, ( x, y ) Sm( x, y ) e g (2) z where Sm S1I S2I S3I S 4I and S1 1 1, S2 1 1, S3 1 1, S4 1 1 (3) where 2 x / and 2 y / w. and w are lengths along the element x and y axes, respectvely. In Eq. 2, the vectors e p and e g represent the element nodal coordnates assocated wth the global poston vector n the mddle surface and the transverse gradent vector. That s, for node k of element, one has e k k p r and k k g z e r. In the contnuum mechancs approach, the elastc forces of the shell element are evaluated as a contnuum volume and the Green-Lagrange stran tensor E at an arbtrary materal pont n element s defned as follows: where 1 ( ) T E F F I (4) 2 F s the global poston vector gradent tensor. The precedng equaton can be expressed n terms of the covarant stran tensor E as T 1 E ( J ) E ( J ) (5) where J X x and X represents the global poston vector of element at an arbtrary reference confguraton. The covarant stran tensor s defned as 1 ( ) T ( ) T E J J J J (6) 2 where J r x. Usng Eq. 5, the stran vector [ ] T xx yy xy zz xz yz ε s defned as Ths materal s declared a work of the U.S. Government and s not subject to copyrght protecton n the Unted States. Approved for publc release; dstrbuton s unlmted. 3

7 UNCLASSIFIED: Dstrbuton Statement A. Approved for publc release. #26261 where ε ( T ) T ε (7) ε s the covarant stran vector obtaned by Eq. 6, and the transformaton matrx gven n lterature [5]. T s as 2.2 Generalzed Elastc Forces In the contnuum mechancs based shear deformable ANCF shell element, element lockngs occur due to the use of low-order polynomals, resultng n erroneously stffer bendng behavor. The lockng n the contnuum mechancs based shear deformable ANCF shell element ncludes the transverse shear lockng; Posson s thckness lockng; curvature thckness lockng; n-plane shear lockng [5]. These lockngs are systematcally allevated by applyng the assumed natural stran [8, 9] and the enhanced assumed stran approaches [1, 11]. The modfed stran feld for the contnuum mechancs-based shear deformable ANCF shell element can then be defned as follows [5]: T EAS εˆ T ε ε (8) where the covarant transverse normal and shear strans are evaluated by the assumed stran approach, whle the other covarant strans are evaluated as compatble strans obtaned drectly from the assumed global dsplacement feld. Ths leads to the followng covarant stran vector: ANS ANS ANS ε xx yy xy zz xz yz (9) T EAS EAS EAS EAS EAS The enhanced assumed stran vector ε xx yy xy zz T n Eq. 8 s defned as [1, 11] J EAS T ε T N ξ α J() ξ () (1) where J (ξ) and J are the global poston vector gradent matrces at the reference Ths materal s declared a work of the U.S. Government and s not subject to copyrght protecton n the Unted States. Approved for publc release; dstrbuton s unlmted. 4

8 UNCLASSIFIED: Dstrbuton Statement A. Approved for publc release. #26261 confguraton evaluated at the Gaussan ntegraton pont ξ and at the center of element ( ξ ), respectvely. ξ s a vector of the element coordnates n the parametrc doman and T s the constant transformaton matrx evaluated at the center of element. The matrx N() ξ defnes polynomals for the enhancement of the stran feld n the parametrc doman and α s a vector of nternal parameters assocated wth the nterpolatng polynomals N() ξ of the enhanced stran feld. The generalzed elastc forces of the shell element are obtaned as a contnuum sold usng the vrtual work as follows: Q T ε W εˆ s V e ( ) dv ε (11) where dv s the nfntesmal volume at the reference confguraton of element, and W s an elastc energy densty functon. The contnuum mechancs based shear deformable ANCF shell element allows for consderng general hyperelastcty materal models n a way same as exstng sold elements. 3. LAMINATED SHELL FORMULATION 3.1 Classcal Lamnaton Theory In fber-renforced composte materals that are wdely used n many engneerng applcatons, lamnae havng dfferent fber angles are bonded together to produce desred materal propertes. Snce many lamnae are stacked at dfferent fber angles, the complex deformaton couplng between the extenson, shearng, bendng and twstng occurs, and such a deformaton couplng characterzes the mechancal behavor of fber-renforced composte materals [12]. In the frst part of ths secton, the macro-mechancal behavor of fber-renforced composte materals s overvewed usng the classcal lamnaton theory. Ths materal s declared a work of the U.S. Government and s not subject to copyrght protecton n the Unted States. Approved for publc release; dstrbuton s unlmted. 5

9 UNCLASSIFIED: Dstrbuton Statement A. Approved for publc release. #26261 In the Krchhoff plate theory, the plane stress s assumed and the n-plane strans of a plate ε [ ] T are defned as p xx yy xy ε ε κ (12) p p z where ε [ ] T s a vector of the n-plane strans n the mddle plane, and p xx yy xy κ [ ] T s a curvature vector assocated wth bendng and twstng. Usng a lnear x y xy orthotropc consttutve law, the n-plane stress vector s related to ts stran vector by σ p Cp ε p, and the materal modul s gven as [12] C R C R (13) p 1 T p In the precedng equaton, the transformaton matrx R s a functon of the fber angle that defnes the orentaton of the fber coordnate system o-12 wth respect to the materal frame o-xy of the plate as shown n Fg. 1. Ths matrx s defned by 2 2 cos sn sn 2 sn 2 sn 2 cos R sn cos sn 2 (14) and C p s the materal modul of an orthotropc materal n the fber coordnate system as C C C p C C (15) 1212 C where 1111 C E (1 ), 2222 C E (1 ), 1122 C 21E 1( ), and 1212 C G 12. Whle the couplng terms between the normal and shear strans n the fber coordnate system are zero as observed n Eq. 15, the extenson and shear couplng occurs for the stress and stran feld defned n the materal frame and the couplng terms n the materal modul matrx C p of Eq. 13 Ths materal s declared a work of the U.S. Government and s not subject to copyrght protecton n the Unted States. Approved for publc release; dstrbuton s unlmted. 6

10 UNCLASSIFIED: Dstrbuton Statement A. Approved for publc release. #26261 are not zero. That s, a plate subjected to a unaxal load produces n-plane shear deformaton due to the extenson and shear couplng [12]. Usng the lnear consttutve law for an orthotropc materal, the n-plane stresses of a lamna are defned as σ C ε zc κ (16) p p p p The force and moment resultants of a fber-renforced composte consstng of N orthotropc lamnae can then be defned as N z N k z k k k p p z p k1 zk1 k1 k1 N C ε dz C z κ dz (17) and z N k z k k 2 z dz z z dz z N k p p p k1 k1 k1 k1 M C ε C κ (18) where N [ N N N ] T, M [ M M M ] T, x y xy x y xy k C p s the materal modul matrx for the k-th layer, and z k s the thckness coordnate at the upper surface of the k-th layer. It s mportant to notce here that the force and moment resultants are defned as forces and moments per unt length [12]. Snce the stran and curvature are not functon of the thckness coordnate z n the Krchhoff plate theory, Eqs. 17 and 18 are wrtten n a matrx form as [12] N A B ε p M B D κ (19) N k where Aj ( Cp ) j ( zk zk 1), k1 N k 2 2 j p j k k 1 k1 B ( C ) ( z z ) / 2, and N k 3 3 j p j k k 1 k1 D ( C ) ( z z ) / 3. The presence of the matrx B mples the extenson and bendng/twstng couplng of a lamnate. However, f each lamna above the md-plane s dentcal to that below the md-plane n both geometry and materal propertes (.e., the lamnate s md-plane symmetrc), the matrx B becomes dentcally zero (.e., Bj ) and the extenson and bendng/twstng couplng vanshes. Ths materal s declared a work of the U.S. Government and s not subject to copyrght protecton n the Unted States. Approved for publc release; dstrbuton s unlmted. 7

11 UNCLASSIFIED: Dstrbuton Statement A. Approved for publc release. #26261 It s mportant to notce here that the extenson and shear couplng stll exsts. Another mportant case s a balanced lamnate, n whch a lamnate conssts of a par of lamnae that have opposte fber angles ( and ) above and below the md-plane, regardless of the stackng sequence. Ths elmnates the extenson and shear couplng. That s, couplng terms of n-plane normal and n-plane shear strans n matrx A are dentcally zero. That s, for a two-layer lamnate wth the same layer thckness and opposte fber angle ( / ) about the md-plane, the extenson and shear couplng vanshes, but the extenson and twstng couplng n B matrx exsts. Ths causes twstng deformaton of a lamnated plate subjected to a unaxal tensle loadng [12]. The presence of such a complex couplng s dscussed usng the ANCF lamnated shell element n the last part of ths Secton. 3.2 Generalzaton to Shear Deformable ANCF Lamnated Shear Element As dscussed n Secton 2, the contnuum mechancs based ANCF shell element s formulated as a contnuum sold that accounts for the three-dmensonal stress state, thus the complex deformaton couplng exhbted n lamnates can be automatcally consdered wthout specal elastc force formulatons. That s, the generalzed elastc force of the lamnated shell element that conssts of N layers can be defned as follows: Q T N k k k ε W εˆ s k V k k1 e ε ( ) dv k (2) In the precedng equaton, the ntegraton nterval for the k-th layer n the thckness drecton s from zk 1 to z k. In other words, the element generalzed elastc forces are evaluated layer by layer and the resultng generalzed elastc forces of each layer are smply added together to defne the elastc force vector of the lamnated shell. It s mportant to notce here that there s no restrcton n materal models consdered n each lama, despte the fact the orthotropc materal Ths materal s declared a work of the U.S. Government and s not subject to copyrght protecton n the Unted States. Approved for publc release; dstrbuton s unlmted. 8

12 UNCLASSIFIED: Dstrbuton Statement A. Approved for publc release. #26261 law s the most popular materal model used for renforced composte materals. Two Gaussan ntegraton ponts are used along the thckness when the elastc forces of each layer are evaluated. Orthotropc Sant-Venant-Krchhoff Materal Model For an orthotropc Sant-Venant- Krchhoff materal, the materal modul jkl 2 C W / j kl of an orthotropc lamna n the materal frame s defned as [13] C ( b a )( b a )( b a )( b a ) C (21) jkl j k l abcd a b c d where ( ) [ ] J b b b and abcd C s the tangent materal modul defned usng 9 materal parameters n the fber coordnate system [ a1 a2 a 3] as shown n Fg. 1, where the drecton of fber s defned along the coordnate 1. The materal modul are gven as follows [13]: abcd C n the fber coordnate system C C C C C C 1212 jkl C [ C ] C C C 2323 C C 1313 (22) where C E (1 ) / C E (1 ) / C E (1 ) / C E ( ) / C E ( ) / C E ( ) / C G C G C G (23) 1 2. and Mooney-Rvln Materal Model For modelng ncompressble materals such as rubbers, Mooney-Rvln materal model s wdely used. The energy densty functon s defned as [14] K W C I C I J 2 2 1( 1 3) 2( 2 3) ( 1) (24) Ths materal s declared a work of the U.S. Government and s not subject to copyrght protecton n the Unted States. Approved for publc release; dstrbuton s unlmted. 9

13 UNCLASSIFIED: Dstrbuton Statement A. Approved for publc release. #26261 where C 1 and C 2 are materal constants, I I I / ( 3), I I I / ( 3) and J 12 ( I3), where I 1, I 2 and I 3 are nvarants of rght Cauchy-Green tensor [14]. K s a bulk modulus. The second Pola Krchhoff stress tensor S s obtaned by dfferentatng the energy densty functon W wth respect to Green-Lagrange stran tensor E as S W C I I 2 C I I I KJ (J1) E I 3 C I C 3 C C (25) Equatons of Moton Usng the prncple of vrtual work n dynamcs, the equatons of moton of the shear deformable lamnated shell element can be expressed as where vectors the matrx Q s and M e Q ( e, α ) Q ( e, e, ) (26) s e t Q e are, respectvely, the element elastc and external force vectors; and M s the constant element mass matrx defned by N k T k ( ) dv k V k1 M S S (27) k where s the materal densty of k-th layer at the reference confguraton. The nternal parameters α ntroduced for the enhanced assumed strans are determned by solvng the followng equatons [1, 11]: V ε α EAS T W ( εˆ ) dv ε (28) It s mportant to notce here that the precedng equatons can be solved at element level for the unknown nternal parameters α usng the procedure presented n the lterature [5]. 3.3 Comparson wth Analytcal Solutons To valdate the ANCF lamnated shell element presented n ths Secton, a unaxal tensle test of Ths materal s declared a work of the U.S. Government and s not subject to copyrght protecton n the Unted States. Approved for publc release; dstrbuton s unlmted. 1

14 UNCLASSIFIED: Dstrbuton Statement A. Approved for publc release. #26261 a two-layer lamnate at / fber angles s consdered as a benchmark problem as shown n Fg. 2 [12]. The length, wdth and thckness of the lamnated plate are 2. m, 1. m, and.1 m, respectvely. The thcknesses of both layers are same (.e.,.5 m), but the sgn of the fber angles of the upper and lower layers are opposte as shown n Fg. 2. The tensle dstrbuted unaxal load of 5 N/m s appled n the X drecton. As dscussed n Secton 3.1, the extenson and bendng/twstng couplng descrbed by B matrx n Eq. 19 are not zero n ths lamnate, causng the warpage (twstng) under the unaxal tensle loadng [12]. It s also mportant to notce n ths example that the extenson and shear couplng exsts n each lamna. However, shear deformaton of the upper and lower layers developed by the unaxal tensle load are same n magntude, but opposte n drecton. For ths reason, the shear deformatons of both layers are canceled out and no n-plane shearng occurs at the lamnate level. That s, the extenson and shear couplng term of A matrx of the lamnate n Eq. 19 s dentcally zero n ths problem. To demonstrate ths fundamental couplng behavor of the lamnated composte materal usng the ANCF lamnated shell element, twstng angles of the two-layer lamnated plate are presented n Fg. 3 as a functon of the fber angle. In ths fgure, results obtaned by the ANCF lamnated shell element, the lamnated sold shell element n ANSYS (SOLSH19), and the analytcal model dscussed n Secton 3.1 are compared. In the analytcal model, the warpage s defned by [12] w xx y y xy xy (29) 2 and the curvature vector κ [ ] T s determned by solvng the followng equaton: x y xy 1 ε p A B N κ B D M (3) Ths materal s declared a work of the U.S. Government and s not subject to copyrght protecton n the Unted States. Approved for publc release; dstrbuton s unlmted. 11

15 UNCLASSIFIED: Dstrbuton Statement A. Approved for publc release. #26261 for N [ ] T and M [ ] T N x. It s observed from Fg. 3 that the twstng deformaton obtaned by the three dfferent models agree well, and the warpage developed by the unaxal tensle load appled to the two-layer lamnated plate s well predcted by the ANCF lamnated shell element developed n ths study. It s mportant to notce here that the sgn of the twstng angle of the composte plate changes at the fber angle of 54.7 degree, and ths mportant fber angle s also correctly predcted wth the ANCF lamnated shell element. 4. APPLICATION TO PHYSICS-BASED ANCF TIRE MODEL A tre has a complex structure that conssts of layers of ples and steel belts that are embedded n rubber, thus an accurate modelng of the complex tre geometry and the ansotropc materal propertes s essental to the tre performance evaluaton ncludng the tre contact pressure and the brakng/tracton and cornerng forces. Whle the n-plane tre belt deformaton can be modeled by an equvalent materal model [3,4], such a smplfed materal model cannot be used for predctng the overall tre structural deformaton n the three-dmensonal analyss. Ths s attrbuted to the fact that the tre secton property n both geometry and materal s of crucal mportance n characterzng the contact pressure dstrbuton. Ths necesstates varous fberrenforced rubber materal models that can be ntegrated nto hgh-fdelty fnte element tre models [15,16]. However, despte the fact that accurate solutons can be obtaned usng exstng fnte element tre models, dffcultes arse when they are ntegrated nto the vehcle dynamcs smulaton due to the essental dfference n formulatons and soluton procedures used n multbody dynamcs and nonlnear fnte element codes. Ths prevents an ntegraton of the hghfdelty tre model nto the multbody vehcle dynamcs smulaton. That s, the structural characterstcs of tres and transent tre forces are, n many cases, evaluated usng dfferent computatonal models and dfferent smulaton approaches. For ths reason, a physcs-based Ths materal s declared a work of the U.S. Government and s not subject to copyrght protecton n the Unted States. Approved for publc release; dstrbuton s unlmted. 12

16 UNCLASSIFIED: Dstrbuton Statement A. Approved for publc release. #26261 absolute nodal coordnate formulaton tre model usng the shear deformable lamnated shell elements s developed such that the hgh-fdelty ANCF tre model can be ntegrated nto general multbody dynamcs computer algorthms for ground moblty smulaton. To ths end, the tre cross secton geometry s mported from the tre cut secton and data ponts are nterpolated by a cubc smoothng splne for extractng the nodal poston and slope coordnates. As shown n Fg. 4, the tre cross-secton s dvded nto the tread, sdewall, and bead sectons. The number of layers, cord angles of layers, materal propertes are provded n each secton to create the tre model data as shown n Fg. 5. The tread secton conssts of a carcass ply, two steel belts, a belt cover, and tread blocks. The carcass ply and steel belt are modeled as an orthotropc materal wth nylon and steel cords embedded n rubber, respectvely. A rubber layer s consdered between the upper and lower steel belts and between the carcass ply and the lower steel belt. The sdewall secton s modeled by two carcass ples and a rubber that les n between. The bead secton s modeled by two carcass ples, a steel belt, and a rubber as shown n Fg. 5. Havng determned the cross-secton property, the three-dmensonal tre geometry s generated by rotatng the tre secton model and the nodal coordnates of the ANCF tre model are created as summarzed n Fg. 4. The tre ar pressure s 22 kpa that s consdered by the normal dstrbuted load appled to the nner surface of the tre. The penalty approach s used for modelng the normal contact force at each node n contact. The load-deflecton curve s mportant for characterzng the fundamental structural propertes of tres. The lateral and vertcal deflectons of the ANCF tre model shown n Fg. 6 are compared wth the measurement results n Fgs. 7 and 8. It s observed from these fgures that the local tre deflectons are well predcated n both lateral and vertcal drectons for the varous wheel loads. Furthermore, the lengths of the contact patch n the longtudnal and Ths materal s declared a work of the U.S. Government and s not subject to copyrght protecton n the Unted States. Approved for publc release; dstrbuton s unlmted. 13

17 UNCLASSIFIED: Dstrbuton Statement A. Approved for publc release. #26261 lateral drectons also agree well wth those of the measurement results as shown n Fg. 9 and SUMMARY AND CONCLUSIONS In ths study, a contnuum mechancs based shear deformable shell element of the absolute nodal coordnate formulaton (ANCF) s generalzed to a lamnated shell element for applcaton to the modelng of fber-renforced rubber (FRR) structure of the physcs-based ANCF tre model. It s shown that the complex deformaton couplng exhbted n fber-renforced composte materals can be automatcally consdered n the shear deformable lamnated shell element usng the contnuum mechancs approach. Furthermore, the element lockngs are systematcally elmnated by the assumed natural stran and enhanced stran approaches, thereby leadng to a lockng-free shear deformaton ANCF lamnated shell element. The benchmark problem of the extenson and twstng couplng of two-layer lamnated plate s used to valdate the ANCF lamnated shell element. The numercal results are n good agreement wth those predcated by the analytcal model based on the classcal lamnaton theory and the lamnated sold shell element n ANSYS. Furthermore, usng the ANCF lamnated shell element developed, a physcs-based ANCF tre model s developed by consderng the detaled tre geometry and materal propertes. The fberrenforced rubber s consdered for modelng the carcass ples and steel belts usng the multlayered lamnated shell elements. The load-deflecton curves predcted by the physcs-based ANCF tre model are n good agreement wth the measurement results. In the future work, the tangental tre force model based on the dstrbuted LuGre frcton model s ntegrated nto the ANCF tre model (ANCF-LuGre tre model) for use n the transent brakng/tracton and cornerng force evaluaton of tres n the context of multbody vehcle dynamcs smulaton. Ths materal s declared a work of the U.S. Government and s not subject to copyrght protecton n the Unted States. Approved for publc release; dstrbuton s unlmted. 14

18 UNCLASSIFIED: Dstrbuton Statement A. Approved for publc release. #26261 ACKNOWLEDGEMENTS Ths research s supported by the Automotve Research Center (ARC) n accordance wth Cooperatve Agreement W56HZV U.S. Army Tank Automotve Research, Development and Engneerng Center (TARDEC). Ths materal s declared a work of the U.S. Government and s not subject to copyrght protecton n the Unted States. Approved for publc release; dstrbuton s unlmted. 15

19 UNCLASSIFIED: Dstrbuton Statement A. Approved for publc release. #26261 REFERENCES [1] Clark, S. K. (ed.), 1981, Mechancs of Pneumatc Tres, US DOT HS NHTSA. [2] Sugyama, H., Yamashta, H. and Jayakumar, P., 214, "Rght on Tracks - An Integrated Tre Model for Ground Vehcle Smulaton from the Unversty of Iowa", Tre Technology Internatonal, vol. 67, pp [3] Sugyama, H. and Suda, Y., 29, "Nonlnear Elastc Rng Tre Model Usng the Absolute Nodal Coordnate Formulaton", IMechE Journal of Mult-Body Dynamcs, vol. 223, pp [4] Yamashta, H., Matsutan, Y. and Sugyama, H., "Longtudnal Tre Dynamcs Model for Transent Brakng Analyss: ANCF-LuGre Tre Model", ASME Journal of Computatonal and Nonlnear Dynamcs, n press. [5] Yamashta, H., Valkeapää, A., Jayakumar, P. and Sugyama, H., "Contnuum Mechancs Based B-Lnear Shear Deformable Shell Element usng Absolute Nodal Coordnate Formulaton", ASME Journal of Computatonal and Nonlnear Dynamcs, n press. [6] Shabana, A. A., Dynamcs of Multbody Systems, 25, Cambrdge Unversty Press, New York. [7] Gerstmayr, J., Sugyama, H., and Mkkola, A., 213, Revew on the Absolute Nodal Coordnate Formulaton for Large Deformaton Analyss of Multbody Systems, ASME Journal of Computatonal and Nonlnear Dynamcs, vol. 8, pp [8] Dvorkn, E. N., and Bathe, K. J., 1984, A Contnuum Mechancs Based Four-Node Shell Element for General Non-Lnear Analyss, Engneerng Computatons, vol. 1, pp Ths materal s declared a work of the U.S. Government and s not subject to copyrght protecton n the Unted States. Approved for publc release; dstrbuton s unlmted. 16

20 UNCLASSIFIED: Dstrbuton Statement A. Approved for publc release. #26261 [9] Betsch, P., and Sten, E., 1995, An Assumed Stran Approach Avodng Artfcal Thckness Stranng for A Non-Lnear 4-Node Shell Element, Communcatons n Numercal Methods n Engneerng, vol. 11, pp [1] Smo, J. C., and Rfa, M. S., 199, A Class of Mxed Assumed Stran Methods and The Method of Incompatble Modes, Internatonal Journal for Numercal Methods n Engneerng, vol. 29, pp [11] Andelfnger, U., and Ramm, E., 1993, EAS-Elements for Two-Dmensonal, Three- Dmensonal, Plate and Shell Structures and Ther Equvalence to HR-Elements, Internatonal Journal for Numercal Methods n Engneerng, vol. 36, pp [12] Jones, R. M., 1999, Mechancs of Composte Materals, Taylor and Francs. [13] Vu-Quoc, L., and Tan, X. G., 23, Optmal Sold Shells for Non-Lnear Analyses of Multlayer Compostes: I Statcs, Computer Methods n Appled Mechancs and Engneerng, vol. 192, pp [14] Bathe, K.J., Fnte Element Procedures, 1996, Prentce Hall. [15] Lee, C.R., Km, J.W., Hallqust, J.O., Zhang, Y. and Farahan, A.D., 1997, "Valdaton of a FEA Tre Model for Vehcle Dynamc Analyss and Full Vehcle Real Tme Provng Ground Smulatons", SAE Techncal Paper [16] Gruber, P., Sharp, R. S. and Crocombe, A. D., 212, "Normal and Shear Forces n the Contact Patch of a Braked Racng Tyre. Part 2: Development of a Physcal Tyre Model", Vehcle System Dynamcs, vol. 5, pp Ths materal s declared a work of the U.S. Government and s not subject to copyrght protecton n the Unted States. Approved for publc release; dstrbuton s unlmted. 17

21 UNCLASSIFIED: Dstrbuton Statement A. Approved for publc release. #26261 Fgure 1. Knematcs of shear deformable ANCF lamnated shell element Ths materal s declared a work of the U.S. Government and s not subject to copyrght protecton n the Unted States. Approved for publc release; dstrbuton s unlmted. 18

22 UNCLASSIFIED: Dstrbuton Statement A. Approved for publc release. #26261 Fgure 2. Unaxal tensle test of two-layer lamnate Ths materal s declared a work of the U.S. Government and s not subject to copyrght protecton n the Unted States. Approved for publc release; dstrbuton s unlmted. 19

23 Twstng angle (deg) UNCLASSIFIED: Dstrbuton Statement A. Approved for publc release. #26261 : ANCF lamnated shell : ANSYS lamnated sold shell : Analytcal soluton Fber angle (deg) Fgure 3. Twstng of two-layer lamnate subjected to unaxal tensle load Ths materal s declared a work of the U.S. Government and s not subject to copyrght protecton n the Unted States. Approved for publc release; dstrbuton s unlmted. 2

24 UNCLASSIFIED: Dstrbuton Statement A. Approved for publc release. #26261 Fgure 4. ANCF tre model creaton procedure Ths materal s declared a work of the U.S. Government and s not subject to copyrght protecton n the Unted States. Approved for publc release; dstrbuton s unlmted. 21

25 UNCLASSIFIED: Dstrbuton Statement A. Approved for publc release. #26261 Fgure 5. Physcs-based ANCF tre model usng mult-layered lamnated shell element Ths materal s declared a work of the U.S. Government and s not subject to copyrght protecton n the Unted States. Approved for publc release; dstrbuton s unlmted. 22

26 UNCLASSIFIED: Dstrbuton Statement A. Approved for publc release. #26261 Fgure 6. Deformed shape of tre cross secton Ths materal s declared a work of the U.S. Government and s not subject to copyrght protecton n the Unted States. Approved for publc release; dstrbuton s unlmted. 23

27 Lateral deflecton (mm) UNCLASSIFIED: Dstrbuton Statement A. Approved for publc release. #26261 Load (kn) Fgure 7. Lateral deflecton of tre for varous wheel loads Ths materal s declared a work of the U.S. Government and s not subject to copyrght protecton n the Unted States. Approved for publc release; dstrbuton s unlmted. 24

28 Vertcal deflecton (mm) UNCLASSIFIED: Dstrbuton Statement A. Approved for publc release. #26261 Load (kn) Fgure 8. Vertcal deflecton of tre for varous wheel loads Ths materal s declared a work of the U.S. Government and s not subject to copyrght protecton n the Unted States. Approved for publc release; dstrbuton s unlmted. 25

29 Longtudnal contact patch length (mm) UNCLASSIFIED: Dstrbuton Statement A. Approved for publc release. #26261 Load (kn) Fgure 9. Longtudnal contact patch length for varous wheel loads Ths materal s declared a work of the U.S. Government and s not subject to copyrght protecton n the Unted States. Approved for publc release; dstrbuton s unlmted. 26

30 Lateral contact patch length (mm) UNCLASSIFIED: Dstrbuton Statement A. Approved for publc release. #26261 Load (kn) Fgure 1. Lateral contact patch length for varous wheel loads Ths materal s declared a work of the U.S. Government and s not subject to copyrght protecton n the Unted States. Approved for publc release; dstrbuton s unlmted. 27