TY - JOUR
T1 - Waves in fractal media
AU - Demmie, Paul N.
AU - Ostoja-Starzewski, Martin
N1 - Funding Information:
Sandia National Laboratories is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under Contract DE-AC04-94AL85000.
Funding Information:
Acknowledgements Comments of an anonymous referee helped improve this paper. This research was made possible by the support from Sandia-DTRA (grant HDTRA1-08-10-BRCWMD) and the NSF (grant CMMI-1030940).
PY - 2011/8
Y1 - 2011/8
N2 - The term fractal was coined by BenoÎt Mandelbrot to denote an object that is broken or fractured in space or time. Fractals provide appropriate models for many media for some finite range of length scales with lower and upper cutoffs. Fractal geometric structures with cutoffs are called pre-fractals. By fractal media, we mean media with prefractal geometric structures. The basis of this study is the recently formulated extension of continuum thermomechanics to such media. The continuum theory is based on dimensional regularization, in which we employ fractional integrals to state global balance laws. The global forms of governing equations are cast in forms involving conventional (integerorder) integrals, while the local forms are expressed through partial differential equations with derivatives of integer order. Using Hamilton's principle, we derive the equations of motion of a fractal elastic solid under finite strains. Next, we consider one-dimensional models and obtain equations governing nonlinear waves in such a solid. Finally, we study shock fronts in linear viscoelastic solids under small strains. In all the cases, the derived equations for fractal media depend explicitly on fractal dimensions and reduce to conventional forms for continuous media with Euclidean geometries upon setting the dimensions to integers.
AB - The term fractal was coined by BenoÎt Mandelbrot to denote an object that is broken or fractured in space or time. Fractals provide appropriate models for many media for some finite range of length scales with lower and upper cutoffs. Fractal geometric structures with cutoffs are called pre-fractals. By fractal media, we mean media with prefractal geometric structures. The basis of this study is the recently formulated extension of continuum thermomechanics to such media. The continuum theory is based on dimensional regularization, in which we employ fractional integrals to state global balance laws. The global forms of governing equations are cast in forms involving conventional (integerorder) integrals, while the local forms are expressed through partial differential equations with derivatives of integer order. Using Hamilton's principle, we derive the equations of motion of a fractal elastic solid under finite strains. Next, we consider one-dimensional models and obtain equations governing nonlinear waves in such a solid. Finally, we study shock fronts in linear viscoelastic solids under small strains. In all the cases, the derived equations for fractal media depend explicitly on fractal dimensions and reduce to conventional forms for continuous media with Euclidean geometries upon setting the dimensions to integers.
KW - Elastic materials
KW - Fractals
KW - Micromechanical theories
KW - Waves
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U2 - 10.1007/s10659-011-9333-6
DO - 10.1007/s10659-011-9333-6
M3 - Article
AN - SCOPUS:80051671935
SN - 0374-3535
VL - 104
SP - 187
EP - 204
JO - Journal of Elasticity
JF - Journal of Elasticity
IS - 1-2
ER -