On the wave propagation in isotropic fractal media

2011

J Elast

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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.

Section: Closing Remarksmentioning

confidence: 99%

Waves in Fractal Media

Demmie

2011

J Elast

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“…Therefore, the Cauchy stress is generally asymmetric in fractal media, indicating that the micropolar effects should be accounted for and (38) should be augmented by the presence of couple stresses. It is important to note here that a material may have anisotropic fractal structure yet be isotropic in terms of its constitutive laws (Joumaa and Ostoja-Starzewski, 2011).…”

Section: Fractal Angular Momentum Equationmentioning

confidence: 99%

“…More work was done on waves in linear elastic fractal solids under small motions. Several cases of isotropic (Joumaa and Ostoja-Starzewski, 2011) or anisotropic (with micropolar effects) Joumaa and Ostoja-Starzewski, 2016) media have been considered through analytical and computational methods. It was found on the mathematical side that fractal versions of harmonic, Bessel, and Hankel functions.…”

Section: Elastodynamics Of a Fractal Timoshenko Beammentioning

confidence: 99%

See 1 more Smart Citation

Continuum Homogenization of Fractal Media

Handbook of Nonlocal Continuum Mechanics for Materials and Structures

Li²,

Demmie

2016

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This chapter reviews the modeling of fractal materials by homogenized continuum mechanics using calculus in non-integer dimensional spaces. The approach relies on expressing the global balance laws in terms of fractional integrals and, then, converting them to integer-order integrals in conventional (Euclidean) space. Via localization, this allows development of local balance laws of fractal media (continuity, linear and angular momenta, energy, and second law) and, in case of elastic responses, formulation of wave equations in several settings (1D and 3D wave motions, fractal Timoshenko beam, and elastodynamics under finite strains). Next, follows an account of extremum and variational principles, and fracture mechanics. 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. Sandia National Laboratories is a multimission laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under Contract DE-AC04-94AL85000.

“…Wave propagation in isotropic fractal media on spherical domains has been studied in [11,18], whereas a formulation of a three-dimensional (3d) elastodynamic model for anisotropic fractal solids was developed in [14] based on product measures. The latter work utilized two different approaches to derive the governing elastodynamic equations of this material model: the fractal mechanics approach and the variational energy approach with both approaches producing consistent results.…”

Section: Introductionmentioning

confidence: 99%

On the dilatational wave motion in anisotropic fractal solids

Joumaa

Mathematics and Computers in Simulation

2016

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This paper reports a study of wave motion in a generally anisotropic fractal medium (i.e. with different fractal dimensions in different directions), whose constitutive response is represented by an isotropic Hooke's law. First, the governing elastodynamic laws are formulated on the basis of dimensional regularization. It is discovered that the satisfaction of the angular momentum equation precludes the implementation of the classical elasticity theory which results in symmetry of the Cauchy stress tensor. Nevertheless, the classical elastic constitutive model can still be applied to explore dilatational wave propagation; in such a case, the angular momentum balance is "trivially" satisfied. The resulting problem, of eigenvalue type, is solved analytically.A computational finite element method solver is also developed to simulate the problem in its 1d form, it is validated by the reference solutions generated through the modal analysis. A 3d finite-difference-based solver is also developed and the obtained results match those of the 1d simulation.