WIAS Preprint No. 1286, (2007)

A higher gradient theory of mixtures for multi-component materials with numerical examples for binary alloys


  • Böhme, Thomas
  • Dreyer, Wolfgang
  • Duderstadt, Frank
  • Müller, Wolfgang H.

2010 Mathematics Subject Classification

  • 74A15 74A50 74N15 74N25 80A17 80A20 82D35


  • Thermodynamics, structured surfaces and interfaces, coexistent phases, analysis of microstructure, transformations involving diffusion


A theory of mixture for multi-component materials is presented based on a novel, straightforward method for the exploitation of the Second Law of thermodynamics. In particular the constitutive equations for entropy, heat and diffusion flux as well as the stress tensor are formulated as a consequence of the non-negative entropy production. Furthermore we derive the established Gibbs equation as well as the Gibbs Duhem relation which also follow from the formalism. Moreover, it is illustrated, how local mechanical strains due to eigenstrains or external loadings, modify the free energy and, consequently, change the chemical potentials of the components. All consecutive steps are illustrated, first, for simple mixtures and, second, for a system containing two different phases. So-called higher gradients of the concentrations are considered, which take the nonuniform composition into account. It will also become apparent that more/other variables of modified/different physical pr oblems beyond the illustrated ones can be easily treated within the presented framework. This work ends with the specification to binary alloys and with the presentation of various numerical simulations.

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