On formulating continuum mechanics of fractal media

Martin Ostoja-Starzewski, Department of Mechanical Science & Engineering, UIUC

Continuum thermomechanics of elastic-dissipative media is extended to fractal media on the basis of a recently introduced continuum mechanics using dimensional regularization [Tarasov, Ann. Phys., 2005]. Next, employing the concept of internal (kinematic) variables and internal stresses, as well as the quasi-conservative and dissipative stresses, a field form of the second law of thermodynamics is derived. In conradistinction 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. Finally, focusing on thermoelasticity, a new form of Duhamelís differential equation of heat conduction coupled to deformation is derived. When the fractal dimension of the medium becomes integer, conventional balance laws of continuum thermomechanics are smoothly recovered.