Physical Meaning and Experimental Check of a Variational Principle for Macro-to-micro Transition

Maximum entropy principle and texture formation

“…[3] that the macro-to-micro transition, i.e., determining the distribution (D k ) k = 1, ..., n from data D (plus necessary additional data: either h or U(D), in the present case), is very close in that model and in the above MAXENT model based on constraints (9), (10) and (17). But it is simpler to impose directly the average heterogeneity h, because it is computable from the unknown (p j k ) of the MAXENT procedure.Indeed, using eq.…”

“…It would be easy to give many more examples.) The ideal goal of the macro-to-micro transition is to determine the micro-fields d(x), σ(x) from the mere data of the macro-stimulus, say D. This seems to be an unattainable goal, the more so if the microscopic constitutive relation is non-linear [3]. In order that the macro-to-micro transition in the polycrystal (for example) may fit with the MAXENT procedure, the first step is to define the micro-state in the heterogeneous medium.…”

“…In order that the macro-to-micro transition in the polycrystal (for example) may fit with the MAXENT procedure, the first step is to define the micro-state in the heterogeneous medium. This we define as the joint data of the microscopic stimulus [thus here the value of d(x)] and the local state X(x) in the heterogeneous medium [3], with X(x) being the set of the internal and/or geometrical variables that make the microscopic constitutive law depend on the micro position x [4]. For a polycrystal, the heterogeneity is mainly due to the anisotropy of the constitutive crystals.…”

“…Many procedures exist to discretize the orientation distribution in a real polycrystal, i.e., to approximate the real orientation field by a piecewise constant field satisfying (3).] The volume fractions f k (k = 1, ..., n) of the different orientations are given, with f 1 + ... + f n = 1.…”

“…add constraints in MAXENT. One possible new constraint is to impose that the average potential is known (assuming therefore that the micro-law expressing σ(x) as a function of d(x) does derive from a potential, say u k in the orientation R k ) [3] :…”