FRICTIONLESS CONTACT OF ESTIC BODIES BY FINITE ELEMENT METHOD AND MATHEMATIÊL PROGRAMMING TECHNIQUE NGUYEN DANG HUNGt and GERY de xc£: partment of Suctural Mhanics and Stability of Consons, University of Lie, lum (Received 13 Ju 1979) A-The work pnts an analysis of the conta problems tw elastic di without friction. Extended variaonal principles in the contact probls constitute the theoretil foundations of the analysis. In fa, kinetilly admissible displament fields a subt to the non-interפttion nditions so that the cont probm ing uivalent to a tion of the with ntraints on the displament& may du to a partilar of the mathetil programming thniqu. Applications limit plane ste of sss or strain s whe l defoaons conde. The ometril spa dire into finite ement nely the 12 de e of om hybr tri or 16 d of from isoparam?ric hi-linear qdrgle elements. A s prdure of lintion is adopt to u the quadratic prming algorithm already prepar for lge ale probms. In th conditions, a mput prom has en written in FORTRAN H and the rults obtain om this pret go agrment comparing to analytil solutions or other numerical solution in the literature. Finly, theotil eorts have פrfoed to gerali the method by introdunR a conpt tio lk matial whh represents the assum ntact a ten the two bi. A geral and n condition of non-interפnetration is propos for gal of two elastic ds in nt. NOMENE x1 Lagranan coordinates , Eulerian ordinates V, v_., v. geometric domains of elastic bodi r., r,., r8 rtion of bodary of V,, V8 wה e p q, '1 1 noal vtor defin on the boundaries 0 rid domain ꜵ boundary of n s, t strain tr, strain field U(s11) density of the sain ener a,1, u stress tensor, sess field r11 stress tensor (Piola) (Euler-uchy) h body fo tractions presid on the boundaries q nodal displaent vtor g nodal galed for vtor K obal stiness matrix total potential strain ener * modifi total potential energy , contact ttial , diotion potential + ntact constraint function ; grange's pareter w slack variable A constraint matrix D., V Lagrangian adient oפrator ,1 displacement gradient [J] Jacobian matrix J Jacobian, deteinant of [!1 1 J 111 minor of element J11 in [J11] tMaitre de Conferens. tEngineer Physician. I. UON e problem of nta of elastic bodies en in small defoations is a nonlinear problem of the complicating factor: the pren of the unknown surface contt. One may find in the literature two distinct ways of numerical approach for this problem; the iterative meth; and the direct method. The first method consists of calculating the increment of loading and veriing the contact condition at each step. Following authors may cited among the adepts of this procure. Chan and Tuba[I], Gaertner [2], Zienkiewicz and Franca- villa [3 ], Frriksson[ 4 ]. e second method consists of reducing the elastic contact problem to a sפcial case of the mathematil programming techniques. By that way, Feng and Huang[5] have examin the contact problem of an inflated plane membrane. The variational forms of the contact problem is studi by Fremond[6] who has presented some numerical examples of contact between elastic bodies and rigid foundation. Panagio- topoulos [7] has generaliz this approach to the inelastic foundation and present some dual forms of the variational inequalities for the contact problem. The present work longs to the sond type of formulation where appropriate linearization of con- tact condition is adopt. It is assum that no friction between solid bodies exists so that the contact condition may expressed uniquely in terms of displacements. It is supsed also that the deformation is small and the material obeys the linear elastic constitutive equations. In the numerical examples, we will be dealing only with plane strain or stress two dimensions problems. The generalization of the formulation to the thr dimensions bodies would 1
Frictionless contact of elastic bodies by finite element method and mathematical programming technique
Jun 12, 2023
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