Calculation of the tensile and flexural strength of disordered materials using fractional calculus

Carpinteri, Alberto; Cornetti, Pietro; Kolwankar, Kiran M.

doi:10.1016/j.chaos.2003.12.081

Cited by 81 publications

(28 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…[1][2][3][4][5][6][7][8] The theory of the local fractional derivative (LFD) is a mathematical tool for describing fractals, that was used to model the fractal complexity in shallow water surfaces, 14 LC-electric circuit, 15 traveling-wave solution of the Burgerstype equation, 16 PDEs, [17][18][19][20] ODEs, 21 and inequalities. 22,23 The useful models for the LFD were considered [24][25][26][27][28][29] and discussed. 30 However, the nonlinear local fractional Boussinesq equations and their non-differentiable-type traveling-wave solutions have not yet been tackled.…”

Section: Introductionmentioning

confidence: 99%

Exact Traveling-Wave Solution for Local Fractional Boussinesq Equation in Fractal Domain

2017

View full text Add to dashboard Cite

The new Boussinesq-type model in a fractal domain is derived based on the formulation of the local fractional derivative. The novel traveling wave transform of the non-differentiable type is adopted to convert the local fractional Boussinesq equation into a nonlinear local fractional ODE. The exact traveling wave solution is also obtained with aid of the non-differentiable ¶ Corresponding author. This is an Open Access article published by World Scientific Publishing Company. It is distributed under the terms of the Creative Commons Attribution 4.0 (CC-BY) License. Further distribution of this work is permitted, provided the original work is properly cited. X. -J. Yang, J. A. T. Machado & D. Baleanu graph. The proposed method, involving the fractal special functions, is efficient for finding the exact solutions of the nonlinear PDEs in fractal domains. 1740006-1

show abstract

Section: Introductionmentioning

confidence: 99%

Exact Traveling-Wave Solution for Local Fractional Boussinesq Equation in Fractal Domain

2017

View full text Add to dashboard Cite

show abstract

“…The scaling behavior of the kinematical parameters shows that the critical nominal strain ε c decreases as the bar length increases. As shown experimentally (and also analytically by Carpinteri et al 2004), by increasing the size, the bar progressively loses its deformation capacity (or ductility) and tends to a more brittle behavior. For a given material, the fractal exponent d ε increases with the size of the bar, from the value d ε = 0 (homogeneous deformation) to d ε = 1 (highly localized deformation).…”

Section: The Multifractal Scaling Lawsmentioning

confidence: 88%

Self-similarity in concrete fracture: size-scale effects and transition between different collapse mechanisms

Carpinteri

Puzzi

2008

Int J Fract

Self Cite

View full text Add to dashboard Cite

Since the pioneering paper by Mandelbrot (Nature, 308:721-722, 1984) on the fractal character of the fracture surfaces in metals, the fractal aspects in the deformation and failure of materials have been investigated by several Researchers (see the reviews by Bouchaud (J Phys Condens Matter, 9:4319-4344) and Carpinteri et al. (Appl Mech Rev, 59:283-305, 2006)) and the attempts to apply fractals to fracture have grown exponentially. Aim of this paper is 2-fold: on one hand, it summarizes in a detailed yet concise fashion the major results of the fractal approach to the scaling of mechanical properties in solid mechanics; on the other hand, it reports some recent results concerning the size effect in the failure of reinforced concrete (RC) beams. These recent findings clearly show that the picture of the size-scale effects is much more complex when interaction among different collapse mechanisms occurs. The consequences on the size-scale effects are discussed in detail.

show abstract

“…The above formula is a direct generalization of Cauchy's formula for repeated integration When p is not an integer, the fractional derivative (20) is a nonlocal operator since it depends on the lower integration limit a. In addition, when a = 0, the following scaling property exists providing a link between fractional calculus and fractals and p being the fractal dimension [31], since the fractional operator shows the same scaling laws as the a-dimensional Hausdorff measure of a fractal set V, for e.g. .…”

Section: Reaction-diffusion Equationsmentioning

confidence: 99%

Complex Systems: Phenomenology, Modeling, Analysis

Iliopoulos¹

2016

Int J Appl Exp Math

View full text Add to dashboard Cite

In this paper a short overview on complex systems and their basic features, as well as the models and mathematical tools developed for their analysis, is given. This review is formed according to the related experience, activity and scientific interests of the author, namely focused on mainly in nonlinear time series analysis and statistics and their applications on different physical systems. In particular, rough outlines are given concerning the phenomenology of complex systems, e.g. far from equilibrium thermodynamics and Tsallis statistics, power law scaling, multi-fractality, low dimensional chaos, SOC, strange kinetics and anomalous diffusion and turbulent intermittency. In addition, a non-complete list of models, based on equations or agent based, is briefly described such as Kuramoto -Sivashinsky equation, cubic complex Ginzburg-Landau Equation, reaction-diffusion Equation, fractional equations, cellular automata, complex networks and artificial neural networks. A more extended review is provided concerning the nonlinear time series analysis complex systems, describing tools like mutual information, flatness coefficient, structure functions, Tsallis q-triplet, correlation dimension and Lyapunov exponents in the reconstructed phase space, which can provide valuable information for the complex system's dynamics. Finally, applications of nonlinear time series analysis on various physical systems, such as earthquakes, Earth's magnetosphere, solar plasma and solar wind, plastic deformation of materials, epilepsy, economical indices and DNA structure, are provided. Mini ReviewOpen Access Complex SystemsIn general, even though there is no precise definition of complex systems, a complex system can be thought of as a collection of nonlinearly interacting elements summing up as a whole which is characterized by novel large scale effects. These effects arise as emergent properties related with nonlinear-complex behavior, which is the most distinguishing feature of complex systems [1]. A nonexhaustive list of complex systems contains among others (in fact most systems can be considered as complex systems): geophysical processes (earthquakes), biophysics (brain dynamics, DNA), space plasmas (solar wind, solar flares, Earth's magnetosphere), plastic deformation of materials, economy (stock indices), socio-technical systems and many others. The aforementioned complex systems, even though they are completely different in many aspects (e.g. different elements), share common characteristics such as:1. They consist of a large number of interacting elements. This fact allows the description of these systems in two different hierarchical levels, namely microscopic and macroscopic. 2. Their subsystems and their interactions are nonlinear. 3. They exhibit emergent behavior, namely a self-organizing collective behavior, which is not predictable from the knowledge of the element's behavior. 4. These systems are open, namely they interact with their environment. This allows these systems to be driven to metastable state...

show abstract

Calculation of the tensile and flexural strength of disordered materials using fractional calculus

Cited by 81 publications

References 19 publications

Exact Traveling-Wave Solution for Local Fractional Boussinesq Equation in Fractal Domain

Exact Traveling-Wave Solution for Local Fractional Boussinesq Equation in Fractal Domain

Self-similarity in concrete fracture: size-scale effects and transition between different collapse mechanisms

Complex Systems: Phenomenology, Modeling, Analysis

Contact Info

Product

Resources

About