轉(zhuǎn)向節(jié)文獻(xiàn)翻譯--一種無摩擦接觸問題的有限元方法(參考版)
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【正文】 ( 2)2(1)1( ??????? CgCgCg tt (3) Consequently, imperability condition (1) can be rewritten in terms of the above gap functions as 0,0 )2()1( ?? gg At the absence of inertial effects, the local form of the equations governing the motion of each body αare given by div 0?? ??? ? bt in ?t? (4a) ?? ??uu on ?u? (4b) ??? ?ntnt ? on ?q? (4c) 0)( ??g on ?t? (4d) Where ?t is the Cauchy stress tensor, ?? the mass density, ?b the body force per unit mass, ?u the prescribed boundary displacement, and ??nt the prescribed traction vector on ?q? . Application of the standard weightedresidual method,in conjunction with the introduction of Lagrange multiplier p≥0 for the imperability constraint,results in the weak form of the equations of motion, which states that the displacement solution ?u of equations (4) and the Lagrange multiplier field p satisfy ? ? ? ? ?? ? ? ? ????????2 1 221 0)(}:{? ???????? ? ? ? ???t t q Cn dnwwpdtwdvbwdvtdi v w (5a) 11 ??? ??? ??t dgpq 0)( )( (5b) For all admissible function ?w and q. Without loss of generality )2(g is used in equation (5a) for the definition of gap function on C. Displacement fields ?u belong to spaces ?? with ?????? ??????? ???uxxx onuuHu |)(1 and weight functions ?w belong to spaces x? defined as }0|)}({ 1 ??? uxx onwHw ?????? The admissible functions q≥0 are piecewise continuous. To prove that equation (5a) holds, note that the work done by contact forces along C on admissible functions 1w and 2w is given by ? ? ??????? C Cc dnwwpdnwwpW ?? 221121 )()( (6) At the absence of friction and recalling that 21 nn ?? on C, Cauchy’s lemma on the stress vector implies that 12 )(1 )( :21 pntt nn ???? (7) With the aid of equation (7), equation (6) is written as ?? ??????? CCc dynwwpdynwwpW 221121 )()( , Which shows that the Lagrange multiplier field is naturally identified with normal traction (pressure) along the contact region. The pressure field p is generally assumed to be only piecewise smooth, thus allowing each body to feature material interfaces in the neighbourhood of C. Moreover, since 0)( ??pg on ?t?? (8) inequalities (5b) follow from (4d) and the assumed nonnegativeness of q. In order to further clarify the role of the Lagrange multipliers in the twobody contact problem and provide some motivation for the ensuing numerical approximations, use (2b) and (8) to rewrite the integral in (5b) on surface C as 12 ?? ???? CC dynxxqgdypq ])[()( 221 ?? ?????)2()1( 2211 CC dyxqxdynqx (9) The notation )(?C is employed to merely emphasize that the contact surface C can be selectively viewed as part of ?t?? ,as indicated in (3).The integral expression (9) suggests that, given a properly defined Lagrange multiplier field )(?p on the boundary of each body, integration of (5b) can be performed separately on each of the contacting surfaces. However, it is clear that fields )(?p should satisfy balance of linear momentum on the (mon) contact surface ?C , namely that )2()1( pp ? The above observation will be exploited in the approximate solution of (5a) and (5b). Folowing the procedure used in the derivation of (9), equations (5a) and (5b) are rewritten as ???? ? ??????? ? ??? ? 0}:{ )()(2 1 dynwpdtwdvbwpdvtdi v w xxnxxx q xtt ????? ? ???? (10a) ? ?? ??2 1 )( 0)(? ? dynXq xxC x (10b) for all admissible ?w and )(?q ,where )2()1( qq ? . The classical penalty regularization of (5a) and (5b) is obtained by setting ???? gp ? in the last integral term of (5a),where ( 2122 nxxx ? satisfies 0)( 221 ??? nxx .Defining equations (2a) and (2b) imply that gap functions )1(g and )2(g are identically equal to zero on C, namely that 0)(:)。 finite elements 1. INTRODUCTION Finite element methods are
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