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Note on the Notion of Incompressibility in Thermodynamic Theories of Porous and Granular Materials
Author(s) -
Wilmański K.
Publication year - 2001
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/1521-4001(200101)81:1<37::aid-zamm37>3.0.co;2-h
Subject(s) - overdetermined system , lagrange multiplier , second law of thermodynamics , constraint (computer aided design) , mathematics , work (physics) , extension (predicate logic) , simple (philosophy) , thermodynamics , physics , mathematical optimization , computer science , geometry , philosophy , epistemology , programming language
We present a simple two‐component model of a porous material based on the constraint assumption that the so‐called true components are incompressible. In my previous work on this subject [1] I pointed out that many such models are not thermodynamically admissible. Namely the second law of thermodynamics led to the conclusion that an additional field of reaction force on the constraint cannot be introduced, and, consequently, the set of field equations was overdetermined. However I speculated as well that an extension of the set of variables may lead to thermodynamic admissibility. Indeed an example presented in this paper supports this speculation. According to results of this work it seems to be necessary to introduce higher gradients to multicomponent models with constraints in order to satisfy the second law of thermodynamics. The latter is exploited by means of Lagrange multipliers eliminating constraints from the entropy inequality. This procedure is known in the literature as the method of Lagrange‐Liu multipliers (see the original work of I‐Shih Liu [2]).