Review about the Application of Fractal Theory in the Research of Packaging Materials

Abstract

The work is intended to summarize the recent progress in the work of fractal theory in packaging material to provide important insights into applied research on fractal in packaging materials. The fractal analysis methods employed for inorganic materials such as metal alloys and ceramics, polymers, and their composites are reviewed from the aspects of fractal feature extraction and fractal dimension calculation methods. Through the fractal dimension of packaging materials and the fractal in their preparation process, the relationship between the fractal characteristic parameters and the properties of packaging materials is discussed. The fractal analysis method can qualitatively and quantitatively characterize the fractal characteristics, microstructure, and properties of a large number of various types of packaging materials. The method of using fractal theory to probe the preparation and properties of packaging materials is universal; the relationship between the properties of packaging materials and fractal dimension will be a critical trend of fractal theory in the research on properties of packaging materials.

Recent strategy to study fractal-order viscoelastic polymer materials using an ancient Chinese algorithm and He’s formulation

“…5 Considering the relevance that polymers have in engineering applications, there is a need for a better understanding of their network structure behavior that affects physical and mechanical properties; therefore, a mathematical model that takes into account fractal patterns that are formed in the preparation of polymers and their composites is highly desirable. 58 In this article, we use a dynamics mathematical model that can be obtained from the fractal Lagrange equation…”

“…Considering the relevance that polymers have in engineering applications, there is a need for a better understanding of their network structure behavior that affects physical and mechanical properties; therefore, a mathematical model that takes into account fractal patterns that are formed in the preparation of polymers and their composites is highly desirable. 58 In this article, we use a dynamics mathematical model that can be obtained from the fractal Lagrange equationthat considers the Langevin polymer chain’s viscoelastic effects. In equation (1), the kinetic and Langevin potential energies T and V can be computed from the following expressionsHere, normalℒ−1(x) is known as the inverse Langevin function defined as x=ℒ(β)=coth(β)−1/β with β = normalℒ−1(x), 59,60 q i are the generalized coordinates, the overdot denotes derivative with respect to time, Q i are the generalized nonconservative forces, M is the mass, and α is the fractal dimension.…”

A Recurrent Neural Network-Based Method for Dynamic Load Identification of Beam Structures

“…The error introduced in any segment will greatly affect the ultimate identification result, which is similar to other fuzzy fields [50][51][52]. Moreover, different material properties will affect the solution for the problem relating to structural dynamic characteristics [53][54][55][56], which makes identification difficult. Scholars usually use the metaphor of the "black box" to describe the problem of dynamic load identification and neural networks.…”

Power spectral fractalysis: a surrogate method for laser-induced plasma temperature analysis

“…Fractal analysis is one such mathematical technique, introduced by the famous fractalist Benoit Mandelbrot, that comes under the branch of measure theory to describe the complex structures, their roughness, and self-similarities [1][2][3]. The omnipresence of fractal objects in nonlinear dynamical systems enabled its widespread application not only in the arts but also in physical, chemical, biological, and technological fields [1,[4][5][6]. In fractal geometry, a fractal is defined as an object that appears similar at different scales, and it is measured in terms of a metric called fractal dimension (D).…”

“…In all these examples, fractal dimension is adopted as a tool to unwrap the hidden complex dynamics of spatio-temporal systems through various a e-mail: drssraman@gmail.com (corresponding author) methods. The diverse methods for fractalysis are the main reason for its application in various fields of science and technology [1,2,6,8,12]. The walking divider method is common for finding the fractal nature of linear features, the box-counting method for objects in any dimension, and the variogram method for area-based applications [13].…”