Fractal Approaches in Materials: Structure and Mechanics

Dear Colleagues,

The fractal approach to material mechanics with a multiscale microstructure gives an efficient and concise explanation of the size effects on cohesive crack model parameters. In every physical theory, the topic of scaling is of prime significance. Because of the requirement for accurate prediction of mechanical characteristics in large-scale structures, the study of scale effects is becoming increasingly important in structural mechanics. The development of high-performance materials, along with more stringent safety regulations, necessitates a greater understanding of the structural behavior of large-scale materials. If the length scales are decoupled and the material microstructure possesses appropriate translational symmetry, traditional homogenization methods provide some effective way to describe the mechanical characteristics of heterogeneous materials. Real heterogeneous materials, on the other hand, typically have a formidably intricate design that exhibits statistical scale invariance over a wide range of length scales. Gels, polymers, and biological materials are examples; as are rocks, soils, and carbonate reservoirs. Because heterogeneities play an essential role at practically all scales, traditional homogenization procedures are ineffective for such materials. As a result, the mechanics of scale-invariant materials is crucial for both basic and technical reasons. Fractal geometry can be used to define scale-invariant structures in heterogeneous materials by using scaling ideas. Scale-invariant spatial and size distributions of solid phases and/or defects (e.g., pores or fractures); long-range correlations in the mass (or pore) density distribution; and the fractal geometry of fracture, pore, and crumpling networks are only a few examples. The ability to store data relevant to all scales of observation using a relatively small number of parameters that construct a structure of greater complexity and rich geometry is a significant benefit of the fractal approach.

Fractional calculus is a field of mathematics that investigates the various techniques to determine real number powers and complex number powers of the differentiation operator. The theory and applications of differential equations have played an essential role both in the development of mathematics and in exploring new horizons in science. From a theoretical viewpoint, the qualitative theory of differential equations, as well as investigative methods, have contributed to the development of many new mathematical ideas and methodologies for solving systems of differential equations. Mathematical modeling plays a critical role in engineering and science. The main importance of mathematical techniques and modeling appears in almost all fields of science, technology, finance and social science, imparting numerical simulations that simulate different phenomena and behaviors of all forms. Numerical models and modeling are accepted by engineers and scientists to represent the nature of a wide variety of phenomena and processes in engineering.

The main aim of this Special Issue is to focus on the study of different transformations and operators of mathematical modeling of complex systems in real-world problems, as well as on numerical and analytical methods. The articles appearing in this Special Issue will be of great interest to researchers working in scientific and engineering fields. This Special Issue will provide a deep understanding of the most important hot problems in the field of mathematics. From the fractional ordered systems perspective, manuscripts in dynamical systems, nonlinearity, chaos, fractional differential equation, and fractional dynamics are also encouraged.

Prof. Dr. Farooq Ahmad
Dr. Yeliz Karaca
Dr. Naveed Iqbal
Guest Editors

Manuscript Submission Information

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Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2700 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

• structure analysis (micro/macro)
• fractal approach
• fractional calculus
• fractional differential equations
• delay differential equations
• different fractional transformation and operators
• dynamical systems
• pattern formation
• stability analysis
• bifurcation theory
• analytical/numerical techniques for fractional differential equations