Abstract
This paper is devoted to the mechanics of fractally heterogeneous media. A model of fractal continuum with a fractional number of spatial degrees of freedom and a fractal metric is suggested. The Jacobian matrix of the fractal continuum deformation is defined and the kinematics of deformations is elucidated. The symmetry of the Cauchy stress tensor for continua with the fractal metric is established. A homogenization framework accounting for the connectivity, topological, and metric properties of fractal domains in heterogeneous materials is developed. The mapping of mechanical problems for fractal media into the corresponding problems for the fractal continuum is discussed. Stress and strain distributions in elastic fractal bars are analyzed. An approach to fractal bar optimization is proposed. Some features of acoustic wave propagation and localization in fractal media are briefly highlighted.
Similar content being viewed by others
References
M. Sahimi, in Heterogeneous Materials (Springer, New York, 2003), Vol. II
D. Cioranescu, P. Donato, An Introduction to Homogenization (Oxford University Press, Oxford, 1999)
B.B. Mandelbrot, The Fractal Geometry of Nature (Freeman, San Francisco, 1999)
G. Korvin, Fractal Models in the Earth Sciences (Elsevier, New York, 1992)
K.J. Falconer, Fractal Geometry – Mathematical Foundations and Applications (Wiley, New York, 2003)
K. Oleschko, G. Korvin, A.S. Balankin, R.V. Khachaturov, L. Flores, B. Figueroa, J. Urrutia, F. Brambila, Phys. Rev. Lett. 89, 188501 (2002)
A.S. Balankin, T. López, R. Alexander-Katz, A. Córdova, O. Susarrey, R. Montiel, Langmuir 19, 3628 (2003)
A. Carpinteri, P. Cornetti, N. Pugno, A. Sapora, Adv. Sci. Technol. 58, 54 (2008)
A.S. Balankin, A. Horta, G. García, F. Gayosso, H. Sanchez, C.L. Martínez-González, Phys. Rev. E 87, 052806 (2013)
F. Yang, Fuel 115, 378 (2013)
J.-F. Gouyet, Physics and Fractal Structures (Springer, Paris, 1996)
D. Ben-Avraham, S. Havlin, Diffusion and Reactions in Fractal and Disordered Systems (Cambridge University Press, Cambridge, 2002)
F.H. Stillinger, J. Math. Phys. 18, 1224 (1977)
B. O’Shaughnessy, I. Procaccia, Phys. Rev A 32, 3073 (1985)
P.D. Panagiotopoulos, O. Panagouli, Chaos Solitons Fractals 8, 253 (1997)
A.S. Balankin, Eng. Fract. Mech. 57, 135 (1997)
A. Carpinteri, P. Cornetti, Chaos Solitons Fractals 13, 85 (2002)
V.E. Tarasov, Phys. Lett. A 336, 167 (2005)
J. Li, M. Ostoja-Starzewski, Proc. Royal Soc. A 465, 2521 (2009)
S.I. Muslih, O.P. Agrawal, J. Math. Phys. 50, 123501 (2009)
S.I. Muslih, D. Baleanu, Romanian Rep. Phys. 62, 689 (2010)
V.E. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media (Springer, New York, 2011)
X.-J. Yang, Advanced Local Fractional Calculus and Its Applications (World Science Publisher, New York, 2012)
M. Zubair, M.J. Mughal, Q.A. Naqvi, Electromagnetic fields and waves in fractional dimensional space (Springer, New York, 2012)
C.S. Drapaca, S. Sivaloganathan, J. Elasticity 107, 105 (2012)
A.S. Balankin, B. Espinoza, Phys. Rev. E 85, 056314 (2012)
A.S. Balankin, B. Mena, J. Patiño, D. Morales, Phys. Lett. A 377, 783 (2013)
A.S. Balankin, Phys. Lett. A 377, 2535 (2013)
A.S. Balankin, Phys. Lett. A 377, 1606 (2013)
A.S. Balankin, B. Espinoza, Phys. Rev. E 88, 057002 (2013)
M. Schmutz, Europhys. Lett. 2, 897 (1986)
A. Dathea, M. Thullner, Geoderma 129, 279 (2005)
A. Neimark, Physica A 191, 258 (1992)
M. Ciccotti, F. Mulargia, Phys. Rev. E 65, 037201 (2002)
F.M. Borodich, Z. Feng, Z. Angew. Math. Phys. 61, 21 (2010)
F. Bianco, S. Chibbaro, D. Vergni, A. Vulpiani, Phys. Rev. E 87, 062811 (2013)
A.S. Balankin, B. Mena, C.L. Martínez-González, D. Morales, Phys. Rev. E 86, 052101 (2012)
U. Mosco, Phys. Rev. Lett. 79, 4067 (1997)
G. Calcagni, Phys. Rev. E 87, 012123 (2013)
L.V. Meisel, Phys. Rev. A 45, 654 (1992)
S. Miyazima, H.E. Stanley, Phys. Rev. B 35, 8898 (1987)
J. Feder, Fractals (Plenum Press, New York, 1988)
G. Calcagni, J. High Energy Phys. 2012, 65 (2012)
W. Chen, Chaos Solitons Fractals 28, 923 (2006)
A.S. Balankin, B. Espinoza, Phys. Rev. E 83, 025302(R) (2012)
R. Abreu-Blaya, J. Bory-Reyes, T. Moreno-García, D. Peña-Peña, Math. Meth. Appl. Sci. 31, 849 (2008)
P. Moon, D.E. Spencer, J. Franklin Institute 256, 551 (1953)
C. Palmer, P.N. Stavrinou, J. Phys. A 37, 6987 (2004)
X.F. He Phys. Rev. B 43, 2063 (1991)
P. Christol, P. Lefebvre, H. Mathieu, J. Appl. Phys. 74, 5626 (1993)
A. Thilagam, A. Matos-Abiague, J. Phys.: Condens. Matter 16, 3981 (2004)
A. Yavari, J.E. Marsden, Rep. Math. Phys. 63, 1 (2009)
A. Yavari, A. Goriely, Arch. Rational Mech. Anal. 205, 59 (2012)
P. Topping, Lectures on the Ricci Flow (Cambridge University Press, New York, 2006)
A. Yavari, J. Nonlinear Sci. 20, 781 (2010)
A. Ozakin, A. Yavari, J. Math. Phys. 51, 032902 (2010)
B.N. Obyfrlea, Advanced Particle Physics, V.I: Particles, Fields, and Quantum Electrodynamics (Taylor & Francis, New York, 2011)
A. Riotto, Lecture Notes on Cosmology (Université de Genève, Genève, 2013)
D. Aubram, Differential Geometry Applied to Continuum Mechanics (Shaker Verlag, Berlin, 2009)
A.C. Eringen, Microcontinuum field theories I: Foundations and solids (Springer, Berlin, 2009)
G.T. Mase, G.E. Mase, Continuum Mechanics for Engineers, 2nd edn. (CRC Press LLC, New York, 1999)
D. Rayneau-Kirkhope, Y. Mao, R. Farr, Phys. Rev. E 87, 063204 (2013)
A. Carpinteri, B. Chiaia, P. Cornetti, Mater. Sci. Eng. A 365, 235 (2004)
H. Khezrzadeh, M. Mofid, Theor. Appl. Fracture Mech. 46, 46 (2006)
A. Carpinteri, P. Cornetti, A. Sapora, Z. Angew. Math. Mech. 89, 207 (2009)
E. Larose, L. Margerin, B.A. Van Tiggelen, M. Campillo, Phys. Rev. Lett. 93, 048501 (2004)
J.F. Kelly, R.J. McGough, J. Acoust. Soc. Am. 126, 2072 (2009)
A.M. García-García, E. Cuevas, Phys. Rev. B 82, 033412 (2010)
T.A. Tafti, M. Sahimi, F. Aminzadeh, C.G. Sammis, Phys. Rev. E 87, 032152 (2013)
F. Shahbazi, A. Bahraminasab, S.M.V. Allaei, M. Sahimi, M.R.R. Tabar, Phys. Rev. Lett. 94, 165505 (2005)
A. Bahraminasab, S.M.V. Allaei, F. Shahbazi, M. Sahimi, M.D. Niry, M.R.R. Tabar, Phys. Rev. B 75, 064301 (2007)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Balankin, A.S. A continuum framework for mechanics of fractal materials I: from fractional space to continuum with fractal metric. Eur. Phys. J. B 88, 90 (2015). https://doi.org/10.1140/epjb/e2015-60189-y
Received:
Revised:
Published:
DOI: https://doi.org/10.1140/epjb/e2015-60189-y