Towards thermomechanics of fractal media

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Hans Ziegler's thermomechanics [1,2,3], established half a century ago, is extended to fractal media on the basis of a recently introduced continuum mechanics due to Tarasov [14,15]. Employing the concept of internal (kinematic) variables and internal stresses, as well as the quasiconservative and dissipative stresses, a field form of the second law of thermodynamics is derived. In contradistinction to the conventional Clausius-Duhem inequality, it involves generalized rates of strain and internal variables. Upon introducing a dissipation function and postulating the thermodynamic orthogonality on any lengthscale, constitutive laws of elastic-dissipative fractal media naturally involving generalized derivatives of strain and stress can then be derived. This is illustrated on a model viscoelastic material. Also generalized to fractal bodies is the Hill condition necessary for homogenization of their constitutive responses.

Original languageEnglish (US)
Pages (from-to)1085-1096
Number of pages12
JournalZeitschrift fur Angewandte Mathematik und Physik
Issue number6
StatePublished - Nov 2007


  • Fractional calculus
  • Random media
  • Viscoelastic material

ASJC Scopus subject areas

  • General Mathematics
  • General Physics and Astronomy
  • Applied Mathematics


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