ABSTRACT
The conservation of linearmomentum gives thenthe complementary compatibility equations :
[ do 0 n = p (-G+vn ) [day] (42) In the same manner the jump in the Jauman stress rate is given by :
[1]t 0] = ( vn - G) [ dn +(a. [ dn ) Os n - [ dn v]ODs (0. n ) (43) and the el astoplastic constitutive equation:
[ 0] = D: ([ dnv]®s n) - D: [ate] = D ([dny]g s n) - [(NAP] /7: `It (44)
where respectively r),. Ep, gyp, 'F stands for the elastic tensor, the plastic deformation and multiplier and the gradient of the plastic potential. The jump in the plastic multiplier may be computed when plastic loading occurs on each side of the singular surface :
[atXP] = ((4r D it).[dnv]) / A (45)
where the denominators is given by :
A = .49 h (46)
while 4:1) is the gradient of the yield function and h is the hardening modulus. The non associatedness of the constitutive equation is illustrated by the fact that IV is not assumed to be equal to T. By collecting the above equations an expression for the jump in the derivative of the stresses is obtained :
(vn-G) [dn a] = 11'11® sn) - ((4 : D a). [dnv] / A) D: - (0 .[dnv])0 sn + [diiv]t s(a .n) (47)
The eigenvalue problem for the discontinuity mode When the superficial constitutive equation is plugged into the superficial equilibrium equation the following eigen problem is obtained for [dny] :
n. D.n - 1/2 a - (- 1/2 Ora, + p (-G+vn )2) I } [ dor] =
a_ D: ((4): D n).[ dnv] / A) + 1/2 cridavia n + 1/2 [dnvA) O. n (48) Let us restrict the analysis to the stationnary case ( G=0 ) and neglect the influence of the rotation rates. Then the preceeding equation simplifies to :
a. D.n [ dav] = a. Di ((P: D a). [ dof] / A) (49) which gives the direction of the discontinuity mode :
[ (Inv] + (a. D. n)-1 a. D ` P (50)
AN OVERVIEW OF FINITE ELEMENT METHODS FOR THE ANALYSIS OF STRAIN LOCALIZATION
Y. Leroy, A. Needleman and M. Ortiz Division of Engineering
Brown University Providence RI 02912
ABSTRACT
An overview is given of finite element analyses of strain localization phenomena. Analyses based on 'crossed triangle' quadrilaterals and on the enhanced mode method of Ortiz et al. [1] are used to illustrate a range of localization phenomena. Mesh effects and convergence issues are discussed. Comparisons are made with other finite element treatments of localization problems.
