Static–kinematic duality and the principle of virtual work in the mechanics of fractal media

Abstract The framework for the mechanics of solids, deformable over fractal subsets, is outlined. While displacements and total energy maintain their canonical physical dimensions, renormalization group theory permits to define anomalous mechanical quantities with fractal dimensions, i.e., the fractal stress [ σ * ] and the fractal strain [ e * ]. A fundamental relation among the dimensions of these quantities and the Hausdorff dimension of the deformable subset is obtained. New mathematical operators are introduced to handle these quantities. In particular, classical fractional calculus fails to this purpose, whereas the recently proposed local fractional operators appear particularly suitable. The static and kinematic equations for fractal bodies are obtained, and the duality principle is shown to hold. Finally, an extension of the Gauss–Green theorem to fractional operators is proposed, which permits to demonstrate the Principle of Virtual Work for fractal media.

[1]  I. Podlubny Fractional differential equations , 1998 .

[2]  E. S. Mistakidis,et al.  Fractal geometry and fractal material behaviour in solids and structures , 1993 .

[3]  B. Sprušil,et al.  Fractal character of slip lines of Cd single crystals , 1985 .

[4]  C. Tricot Dimension fractale et spectre , 1988 .

[5]  Hans Wallin,et al.  The dual of Besov spaces on fractals , 1995 .

[6]  Alberto Carpinteri,et al.  Structural Mechanics: A unified approach , 1997 .

[7]  Pietro Cornetti,et al.  A scale-invariant cohesive crack model for quasi-brittle materials , 2002 .

[8]  P. Panagiotopoulos,et al.  Fractal Geometry in Contact Mechanics and Numerical Applications , 1997 .

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

[10]  Francesco Mainardi,et al.  Linear models of dissipation in anelastic solids , 1971 .

[11]  F. Borodich Parametric homogeneity and non-classical self-similarity. I. Mathematical background , 1998 .

[12]  Alberto Carpinteri,et al.  Multifractal scaling laws in the breaking behaviour of disordered materials , 1997 .

[13]  K. M. Kolwankar Studies of fractal structures and processes using methods of fractional calculus , 1998, chao-dyn/9811008.

[14]  G. W. Blair The role of psychophysics in rheology , 1947 .

[15]  H. Eugene Stanley,et al.  Fractal Concepts for Disordered Systems: The Interplay of Physics and Geometry , 1991 .

[16]  Kiran M. Kolwankar,et al.  Local Fractional Fokker-Planck Equation , 1998 .

[17]  L. C. Young,et al.  On Fractional Integration by Parts , 1938 .

[18]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

[19]  Alberto Carpinteri,et al.  Three-dimensional fractal analysis of concrete fracture at the meso-level , 1999 .

[20]  G. I. Barenblatt Scaling: Self-similarity and intermediate asymptotics , 1996 .

[21]  R. Bagley,et al.  A Theoretical Basis for the Application of Fractional Calculus to Viscoelasticity , 1983 .

[22]  G. I. Barenblatt,et al.  Similarity, Self-Similarity and Intermediate Asymptotics , 1979 .

[23]  A. Carpinteri Scaling laws and renormalization groups for strength and toughness of disordered materials , 1994 .

[24]  Kenneth Falconer,et al.  Fractal Geometry: Mathematical Foundations and Applications , 1990 .

[25]  Yuri Y. Podladchikov,et al.  FRACTAL PLASTIC SHEAR BANDS , 1994 .

[26]  J. Harrison,et al.  The Gauss-Green theorem for fractal boundaries , 1992 .

[27]  Alberto Carpinteri,et al.  Fractal nature of material microstructure and size effects on apparent mechanical properties , 1994 .

[28]  F. Borodich Parametric homogeneity and non-classical self-similarity. II. Some applications , 1998 .

[29]  Fractional differentiation of devil's staircases , 1992 .

[30]  Kiran M. Kolwankar,et al.  Fractional differentiability of nowhere differentiable functions and dimensions. , 1996, Chaos.

[31]  F. Tatom THE RELATIONSHIP BETWEEN FRACTIONAL CALCULUS AND FRACTALS , 1995 .

[32]  K. Miller,et al.  An Introduction to the Fractional Calculus and Fractional Differential Equations , 1993 .