This paper continues the extension of continuum mechanics and thermodynamics to
fractal porous media which are specified by a mass (or spatial) fractal dimension
, a surface fractal
dimension
, and a
resolution length-scale
.
The focus is on a theory based on dimensional regularization, in which
is
also the order of fractional integrals employed to state global balance laws. Thus, we
first generalize the main integral theorems of continuum mechanics to fractal
media: Stokes, Reynolds, and Helmholtz–Żórawski. Then, we review balance
equations and recently obtained extensions of several subfields of continuum
mechanics to fractal media. This is followed by derivations of extremum and
variational principles of elasticity and Hamilton’s principle for fractal porous
materials. In all the cases, we derive relations which depend explicitly on
,
and
, and which,
upon setting
and
,
reduce to the conventional forms of governing equations for continuous media with
Euclidean geometries.
Department of Mechanical Science and
Engineering and Institute for Condensed Matter Theory
University of Illinois at Urbana–Champaign
1206 W. Green Street
Urbana, IL 61801-2906
United States
