ISSN:
1572-9125
Keywords:
Non-quasiconvex variational problem
;
finite order rank-one convex envelope
;
laminates in laminates
;
high precision numerical solutions
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract Finite order rank-one convex envelopes are introduced and it is shown that the i-th order laminated microstructures, or laminates in laminates, can be solved by any of the k-th order rank-one convex envelopes with k ≥ i. It is also shown that in finite element approximations of microstructures, replacing the non-quasiconvex potential energy density by its k-th order rank-one convex envelope, one can generally obtain sharper numerical results. Especially, for crystalline microstructures with laminates in laminates of order no greater than k + 1, numerical results with up to the computer precision can be obtained. Numerical examples on the first and second order rank-one convex envelopes for the Ericksen-James two-dimensional model for elastic crystals are given. A numerical example on finite element approximations of a crystalline microstructure by using the first order rank-one convex envelope and the periodic relaxation method is also presented. The methods turn out to be very successful for microstructures with laminates in laminates.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1023/A:1022348603727
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