Crystalline Plasticity and Structured Deformations
The connection between the geometrical changes at small length scales in single crystals and their macroscopic response has been the subject of extensive experimental studies. In (Deseri and Owen (2000)), experimental evidence that points to a connection
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Series Editors: The Rectors Manuel Garcia Velarde - Madrid Jean Salen 0 if B, C are not compatible, note first that the above construction of Uj is no longer possible. But this of course does not mean that there
M. Silhavy
8
is not some other sequence satisfying (13). The proof that there is no such a sequence is obtained by lifting the segment C into the space M 2 x2 x IR of pairs (A,8) where A E M2X2, 8 E IR and employing the convexity notions in the space of lifted objects. Thus let C C M2x2 X IR be line segment with endpoints (B, det B) and (C, det C), Le.,
C=
{t(B,detB) + (1- t)(C,det C) : 0
< t < I}.
Let dist(·, C) : M 2X2 X IR -> IR be the distance from C and note that since C is convex, so also is the function dist(·, C). Independently of whether or not are B, C compatible, one can prove that there exists a strictly increasing convex function w : [0,(0) -> [0,(0) such that f(A) ~ w(dist(A, det A; C)) for every A E M2x2. Then the function g : M 2x2 X IR -> IR defined by g(A,8) = w(dist(A, 8; C)), (A, 8) E M 2x2 X IR, is convex. If u : n -> IR 2 satisfies u(x) = Ax on an, then we have (10). Hence Jensen's inequality says that
In f(Du) dx ~ In g(Du, det Du) dx ~ g(A, det A). The proof will now be completed by showing that if B, C are not compatible, then g(A, det A) > o. Indeed, the equality g(A, det A) = 0 means that (A, det A) E C, which in turn means that (A,detA) = s(B,detB) + (1- s)(C,detC)
(15)
for some s, 0< s < 1. However, we have A = tB + (1 - t)C; thus the first component of (15) implies s = t and then the second reads det(tB + (1 - t)C) = t det B + (1 - t) det C.
(16)
Using the formula det(P+Q) = det P+ P·cof Q+det Q twice, Equation (16) is rearranged to the form t(l - t) det(B - C) = 0 and thus 0 < t < 1 implies det(B - C) = O. Since m = n = 2, the last equation implies that B - C is a rank 1 matrix, in contradittion with the assumption that B, C are incompatible. 0
Remarks 1.7. • The energy (12) admits an explicit relaxation, see Lurie and Cherkaev (1987), Pipkin (1991), Kohn (1991) for a more general dass. • The construction of Uj is frequently used; the procedure is called lamination and the deformations Uj (simple) laminates. If is often necessary to refine the construction to form layers within the layers of an already constructed laminate. Such deformations are called laminates of higher order. • If Fj is the mean deformation
Energy Minimization for Isotropie Nonlinear Elastic Bodies
9
then
Fj
-+
A,
and
Uj(x) -+ Ax uniformlyon 0,
but the energy I(uj) does not converge to that of the homogeneous deformation of gradient A. Thus if one views A as the macroscopic deformation, the correct macroscopic energy is not f(A) but rC(A). • In crystal structures, laminates of orders 2 (layers within layers) are frequently observed, but rarely laminates of order ~ 3.
1.2
Minima of Effective Energy
The calculation of r c is generally difficult; an easier problem is to find the set L of all A for which rC(A) takes the minimum value. It turns out that L is completely determined by th
