Discontinuities and Plasticity
The guide line of these lectures is that Plasticity and spatial discontinuities are two companion phenomena:
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P.M. Suquet Universite des Sciences et Techniques du Languedoc:, Montepellier, France
J. J. Moreau et al. (eds.), Nonsmooth Mechanics and Applications © Springer-Verlag Wien 1988
280
P.M. Suquet
1.
INTRODUCTION
The guide line of these lectures is that Plasticity and spatial discontinuities are two companion phenomena : a) on the one hand displacements (or displacement rates) solutions of Plasticity problems are naturally discontinuous. This conclusion is easily drawn from the asymptotically linear growth of the functionals arising in the variational formulation of Plasticity problems : Inf J(u) = u
J
n
j(E(u))dx- L(u)
(I. 1)
where E stands for the usual deformation operator (infinitesimal strains), L is a linear form and j satisfies : j(e) ~ k 0 iel- k 1
(1.2)
Accounting for (1.2) the natural space of definition for J contains discontinuous vector fields. But this is only on~ of the difficulties arising from the variational problems (1.1) . Another one is the non lower semi continuity of functionals, which has for main consequence a ~elaxation p~oe~dun~ necessary to obtain well posed variational problems. Our presentation of this subject follows and extends TEMAM & STRANG's [1] original work. b) on the other hand it is known from experimental observation that Plasticity takes its source in microscopic slips along specific cristallographic planes. To give a more mathematical basis to this assertion we examine a model of rigid blocks slipping ones over others. The blocks size is assumed to be a small parameter 1/n, and the problem can be described by minimizing a functional Jn . In order to derive the homog~niz~d ~obl~m (when 1/n tends to 0), we use the theory of ~p~ eonv~g~ne~ which allows to compute the limits of variational problems. We shall point out a few singular aspects of epi-convergence in Plasticity mainly arising from (roughly speaking) the non eommutativ~ty o6 ~e la~on
and
~p~ eonv~genee.
c) located midway between these two aspects of "Plasticity and discontinuities" the problems of ~n p!Mile ta.yeJL6 will be studied in these notes. Our aim is to bring ~ention 06 ~ea.deJL6 on recent mathematical developments which could open significantly wide fields of investigation for researchers interested in mathematical aspects of Mechanics. Therefore
Discontinuities and Plasticity
281
detailled proofs are often omitted, but conjectures and theorems are, as far as possible, distinguished. The author's personal contribution in theoretical developments is most of the time rather limited, but the goal is to attract readers to a m~cha~L[cal app~oach of TEMAM's or ATTOUCH' s books [2] [3] , to BOUCHITTE' s thesis [4l , or to recent works of the Italian School. The background necessary to follow the text includes a basic knowledge of convex analysis, of measure theory and of classical functional analysis, and can be found, together with numerous other subjects, in PANAGIOTOPOULOS [9] .
ACKNOWLEDGMENTS Several discussions with Guy BOUCHITTE and numerous uses of his results are gratefully acknowledg
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