Stresses and strains in a deformable fractal medium and in its fractal continuum model

Abstract The model of fractal continuum accounting the topological, metric, and dynamic properties of deformable physical fractal medium is suggested. The kinematics of fractal continuum deformation is developed. The corresponding geometric interpretations are provided. The concept of stresses in the fractal continuum is defined. The conservation of linear and angular momentums is established. The mapping of mechanical problems for physical fractal media into the corresponding problems for fractal continuum is discussed.

[1]  A. Yavari.  Generalization of Barenblatt's cohesive fracture theory for fractal cracks , 2002 .

[2]  A. Balankin,et al.  Hydrodynamics of fractal continuum flow. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[3]  A. Balankin,et al.  Map of fluid flow in fractal porous medium into fractal continuum flow. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[4]  Y. Xiaojun,et al.  Advanced Local Fractional Calculus and Its Applications , 2012 .

[5]  Alain Goriely,et al.  Riemann–Cartan Geometry of Nonlinear Dislocation Mechanics , 2012 .

[6]  C. L. Martínez-González,et al.  Fractal features of a crumpling network in randomly folded thin matter and mechanics of sheet crushing. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[7]  A. Balankin,et al.  Electromagnetic fields in fractal continua , 2013 .

[8]  The concept of multifractal elasticity , 2005, cond-mat/0503694.

[9]  A. Balankin.  Models of self-affine cracks in brittle and ductile materials , 1996 .

[10]  C. Drapaca,et al.  A Fractional Model of Continuum Mechanics , 2012 .

[11]  M. Sahimi,et al.  Numerical simulation of the localization of elastic waves in two- and three-dimensional heterogeneous media , 2008, 1308.1775.

[12]  J. Bory‐Reyes,et al.  Laplacian decomposition of vector fields on fractal surfaces , 2008, Journal of Mathematical Analysis and Applications.

[13]  M. Ostoja-Starzewski.  Continuum mechanics models of fractal porous media: Integral relations and extremum principles , 2009 .

[14]  Alberto Carpinteri,et al.  Are scaling laws on strength of solids related to mechanics or to geometry? , 2005, Nature materials.

[15]  M. Ostoja-Starzewski,et al.  Fractal solids, product measures and fractional wave equations , 2009, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[16]  V. E. Tarasov.  Fractional hydrodynamic equations for fractal media , 2005, physics/0602096.

[17]  W. Chen.  Time-space fabric underlying anomalous diffusion , 2005, math-ph/0505023.

[18]  A. Milovanov.  Topological proof for the Alexander-Orbach conjecture , 1997 .

[19]  Vladimir V. Konotop,et al.  Wave interaction with a random fat fractal: dimension of the reflection coefficient , 1995 .

[20]  J. Suck,et al.  Do high-frequency acoustic vibrations propagate in structurally disordered solids? , 1998 .

[22]  Q. Naqvi,et al.  Electromagnetic Fields and Waves in Fractional Dimensional Space , 2012 .

[23]  O. K. Panagouli,et al.  Mechanics on fractal bodies. Data compression using fractals , 1997 .

[24]  A. Balankin.  Physics of fracture and mechanics of self-affine cracks , 1997 .

[25]  Jerrold E. Marsden,et al.  On Spatial and Material Covariant Balance Laws in Elasticity , 2006 .

[26]  A. Balankin,et al.  Self-affine nature of the stress-strain behavior of an elastic fractal network , 2002 .

[27]  Zdeněk P. Bažant,et al.  Is the cause of size effect on structural strength fractal or energetic-statistical? , 2004 .

[28]  Electromagnetism on anisotropic fractal media , 2011, 1106.1491.

[29]  Jiaxin Hu,et al.  Laplace operators related to self-similar measures on R d , 2006 .

[30]  A. Caetano,et al.  The fractal Dirichlet Laplacian , 2011 .

[31]  Gianluca Calcagni,et al.  Geometry and field theory in multi-fractional spacetime , 2011, 1107.5041.

[32]  Benoit B. Mandelbrot,et al.  Fractal Geometry of Nature , 1984 .

[33]  Mathieu Desbrun,et al.  On the geometric character of stress in continuum mechanics , 2007 .

[34]  P. Panagiotopoulos.  Fractal geometry in solids and structures , 1992 .

[35]  Kinetic roughening and pinning of two coupled interfaces in disordered media. , 2006, Physical review letters.

[36]  C. L. Martínez-González,et al.  Random walk in chemical space of Cantor dust as a paradigm of superdiffusion. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[37]  M Kardar,et al.  Fractional Laplacian in bounded domains. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[38]  M. Sahimi,et al.  Self-affine fractal distributions of the bulk density, elastic moduli, and seismic wave velocities of rock. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[39]  M. Schmutz.  The Hausdorff dimension as an intrinsic metric property of fractals , 1986 .

[40]  A. Balankin,et al.  Fractal topology of hand-crumpled paper. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[41]  M. Ostoja-Starzewski,et al.  Micropolar continuum mechanics of fractal media , 2011 .

[42]  Gabor Korvin,et al.  Fractal radar scattering from soil. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[43]  Scaling properties of pinned interfaces in fractal media. , 2003, Physical review letters.

[44]  Arash Yavari,et al.  A geometric theory of thermal stresses , 2009, 0912.1298.

[45]  M. Ostoja-Starzewski.  Towards thermomechanics of fractal media , 2007 .

[46]  Stefano Zapperi,et al.  Role of disorder in the size scaling of material strength. , 2007, Physical review letters.

[47]  H. Sagan.  Space-filling curves , 1994 .

[48]  S. Havlin,et al.  Diffusion in disordered media , 2002 .

[49]  V. E. Tarasov.  Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media , 2011 .

[50]  Alexander S. Balankin,et al.  Physics in space–time with scale-dependent metrics , 2013 .

[51]  V. E. Tarasov.  Continuous Medium Model for Fractal Media , 2005, cond-mat/0506137.

[52]  Feodor M. Borodich,et al.  Fractals and fractal scaling in fracture mechanics , 1999 .

[53]  Frank H. Stillinger,et al.  Axiomatic basis for spaces with noninteger dimension , 1977 .

[54]  Paul N. Stavrinou,et al.  Equations of motion in a non-integer-dimensional space , 2004 .

[55]  F. Borodich,et al.  Scaling of mathematical fractals and box-counting quasi-measure , 2010 .

[56]  A. Yavari,et al.  A correspondence principle for fractal and classical cracks , 2005 .

[57]  Charles Thompson,et al.  Ultrasonic characterization of fractal models , 1993 .

[58]  Peter Haupt,et al.  Continuum Mechanics and Theory of Materials , 1999 .

[59]  Arash Yavari,et al.  A Geometric Theory of Growth Mechanics , 2009, J. Nonlinear Sci..

[60]  Robert S. Strichartz,et al.  Function spaces on fractals , 2003 .

[61]  J. Marsden,et al.  COVARIANT BALANCE LAWS IN CONTINUA WITH MICROSTRUCTURE , 2008, 0811.2234.

[62]  I. Procaccia,et al.  Diffusion on fractals. , 1985, Physical review. A, General physics.

[63]  A. Balankin,et al.  Stress concentration and size effect in fracture of notched heterogeneous material. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[64]  A. Balankin,et al.  Entropic rigidity of a crumpling network in a randomly folded thin sheet. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[65]  A. Balankin,et al.  Phosphate Alumina Process by Sol−Gel: Textural and Fractal Properties , 2003 .

[66]  A S Balankin,et al.  Self-affine nature of the stress-strain behavior of thin fiber networks. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[67]  P. Topping.  Lectures on the Ricci Flow , 2006 .