Fractal Mechanics of Engineering Materials

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Engineering".

Deadline for manuscript submissions: closed (30 April 2024) | Viewed by 11724

Special Issue Editors


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Guest Editor
School of Mechanical Engineering, Faculty of Engineering, Universiti Teknologi Malaysia, Johor Bahru 81310, Malaysia
Interests: materials characterization; constitutive modelling; fatigue and fracture mechanics; damage mechanics; experimental methods (DIC, fractal, hybrid expt-FEA)

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Guest Editor
Faculty of Mechanical Engineering, Universität der Bundeswehr München, Werner-Heisenberg-Weg 39, 85577 Neubiberg, Germany
Interests: advanced materials; finite element analysis; stress analysis; finite element modeling; mechanics of materials; mechanical behavior of materials; material characterization; polymers
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Special Issue Information

Dear Colleagues,

Numerous engineering materials are inhomogeneous and inherit complex structures that exhibit statistical scale invariance over several length scales. These materials and structures include metals and alloys, geomaterials (rocks and aggregates), nanocomposites, engineered coatings and interfaces. The straining of these materials is quantified with appropriate theories within the domains of continuum, fracture, and damage mechanics. The fractality of geometrical features such as microstructures, granular aggregates, surface morphological irregularities and fatigue cracks can be quantified in terms of their fractal dimensions. The fractal analysis of these self-similar and multi-scale features aims to correlate the fractal geometry with meaningful mechanical quantities at micro-to-macro scale. These include mechanical properties, fracture toughness, crack-tip driving force and fracture potentials. Several methods including digital image correlation (DIC) as well as optical and scanning electron microscopy (SEM) have been employed in acquiring microscopic images of the material regions representing the phenomenon of interest. These images are then analyzed to establish relevant fractal parameters including fractal dimension and fractional fractal dimension. Several algorithms for greyscale thresholding, image partitioning and pixel counting are available and continuously being improved. The resulting fractal parameters could serve as material variables in the classical and new/unconventional approaches to continuum mechanics, LEFM, fatigue crack growth analysis, and damage mechanics.

This Special Issue on "Fractal Mechanics of Engineering Materials" aims to present the state-of-the-art research methodologies and outcomes in fractal approaches for engineering materials. Therefore, we invite authors to submit quality review papers, research articles, and technical notes addressing the fractal aspects of engineering materials and related issues. Research topics are described by, but not limited to, the following keywords:

  • Materials: metals and alloys, geomaterials (rocks, aggregates), polymers, composites, nanomaterials, ceramics, functionally graded materials (FGMs), interfaces, modified surfaces, etc
  • Continuum mechanics
  • Fracture mechanics: LEFM, EPFM
  • Fatigue and fatigue crack growth
  • Damage mechanics
  • Experimental mechanics: DIC, SEM, optical microscopy
  • Fractal analysis: theoretical development, image analysis, algorithms
  • Fractal dimension
  • Fractional fractal dimension
  • Computational method: FEA, BEM, numerical prediction

Prof. Dr. Mohd Nasir Tamin
Dr. S. S. R. Koloor
Guest Editors

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Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Fractal and Fractional is an international peer-reviewed open access monthly journal published by MDPI.

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.

Published Papers (7 papers)

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Research

14 pages, 5758 KiB  
Article
Glass Beads Test with True Triaxial Stress Path Achieved by Conventional Triaxial Apparatus
by Xuefeng Li, Jiahui Ma and Longlong Lv
Fractal Fract. 2024, 8(4), 193; https://doi.org/10.3390/fractalfract8040193 - 28 Mar 2024
Viewed by 715
Abstract
The impact of fabric anisotropy, fractal dimension, and breakage on the strength and deformation of granular materials were diminished by uniform-size spherical glass beads. Triaxial drained and undrained tests were performed on glass beads based on a novel method to substitute true triaxial [...] Read more.
The impact of fabric anisotropy, fractal dimension, and breakage on the strength and deformation of granular materials were diminished by uniform-size spherical glass beads. Triaxial drained and undrained tests were performed on glass beads based on a novel method to substitute true triaxial stress paths with conventional triaxial apparatus equivalents with varying intermediate principal stress coefficients (b-values). The result indicates that all specimens manifested a noticeable strain-softening phenomenon. The peak strength decreased with increasing b-value, and the specimens showed more pronounced dilatancy. This pattern is similar to the results of the true triaxial test in current research. Compared to the undrained test, the peak friction angle in the drained test displayed a greater variation with varying b-values, which indicated that the mechanical response of glass beads is sensitive to water. This difference provides experimental evidence for comprehending effective stress in granular materials with constant friction coefficients. The experiments reflect the effect of b-value changes on the p-q stress path, as well as on the peak stress ratio, the state transition stress ratio, and the critical state stress ratio. The specimens exhibited a distinct shear band at different b-values ranging from 0.2 to 0.6, which is different from observations in conventional triaxial tests for granular materials. Full article
(This article belongs to the Special Issue Fractal Mechanics of Engineering Materials)
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12 pages, 4859 KiB  
Article
Preparation of Steel Slag Foam Concrete and Fractal Model for Their Thermal Conductivity
by Guosheng Xiang, Danqing Song, Huajian Li, Yinkang Zhou, Hao Wang, Guodong Shen and Zhifeng Zhang
Fractal Fract. 2023, 7(8), 585; https://doi.org/10.3390/fractalfract7080585 - 29 Jul 2023
Cited by 1 | Viewed by 1154
Abstract
The innovation of structural forms and the increase in the energy-saving requirements of buildings have led to higher requirements regarding the application conditions of steel slag foam concrete (SSFC) to ensure that the SSFC has a lower thermal conductivity and sufficient compressive strength, [...] Read more.
The innovation of structural forms and the increase in the energy-saving requirements of buildings have led to higher requirements regarding the application conditions of steel slag foam concrete (SSFC) to ensure that the SSFC has a lower thermal conductivity and sufficient compressive strength, which has become the primary research object. Through a comprehensive consideration of 7 d compressive strength and thermal conductivity, the recommend mix ratio of SSFC was as follows: maximum SS size = 1.18 mm, water–cement ratio = 0.45, replacement rate of SS = 20–30%. Moreover, a theoretical formula was derived to determine thermal conductivity versus porosity based on fractal theory. The measured values of the foam concrete found elsewhere corroborate the fractal relationship regarding thermal conductivity versus porosity. This fractal relationship offers a straightforward and scientifically sound way to forecast the thermal conductivity of SSFC. Full article
(This article belongs to the Special Issue Fractal Mechanics of Engineering Materials)
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13 pages, 8503 KiB  
Article
Relationship between Remanence and Micromorphology of Nd-Fe-B Permanent Magnets Revealed by Fractal Theory and EBSD Data
by Qisong Sun, Minggang Zhu, Jiaming Bai and Qiang Wang
Fractal Fract. 2023, 7(5), 393; https://doi.org/10.3390/fractalfract7050393 - 9 May 2023
Cited by 1 | Viewed by 1328
Abstract
Remanence is an important parameter of magnetic property for Nd-Fe-B magnets, and high remanent magnetization is a prerequisite for high-performance magnets. In this paper, the surface morphology perpendicular to the texture orientation direction and parallel to the texture orientation direction is analyzed by [...] Read more.
Remanence is an important parameter of magnetic property for Nd-Fe-B magnets, and high remanent magnetization is a prerequisite for high-performance magnets. In this paper, the surface morphology perpendicular to the texture orientation direction and parallel to the texture orientation direction is analyzed by Nd-Fe-B permanent magnets with different compositions. For the first time, the relationship between the remanence of a magnet and the degree of texture orientation is explained in depth using the fractal dimension. The fractal dimension of surface morphology combined with the remanence equation yields the degree of texture orientation of the magnet, which is in agreement with the trend of the squareness factor of the demagnetization curves. Among the three samples, the Nd-Fe-B sample has the highest degree of texture orientation, the Pr-Nd-Fe-B sample has the lowest degree of texture orientation, and the Nd-Ce-Fe-B sample is in between the first two. The multiples of uniform (pole) density obtained by EBSD further prove the correctness of the degree of texture orientation calculated by the fractal dimension. The combination of EBSD morphology and fractal dimension to obtain novel insights into the correlation between remanence and the degree of texture orientation will contribute to the development of high-performance Nd-Fe-B with high remanence. Full article
(This article belongs to the Special Issue Fractal Mechanics of Engineering Materials)
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15 pages, 6426 KiB  
Article
Estimation Method of Ideal Fractal Parameters for Multi-Scale Measurement of Polished Surface Topography
by Hongju Li, Ying Liu, Yuechang Wang and Haoran Liao
Fractal Fract. 2023, 7(1), 17; https://doi.org/10.3390/fractalfract7010017 - 25 Dec 2022
Cited by 1 | Viewed by 1274
Abstract
A surface topography characterization parameter system based on fractal parameters has been established, and several estimation methods for these fractal parameters have been suggested accordingly. Since scale dependence exists in these conventional methods, it is necessary to find an estimation method for characterization [...] Read more.
A surface topography characterization parameter system based on fractal parameters has been established, and several estimation methods for these fractal parameters have been suggested accordingly. Since scale dependence exists in these conventional methods, it is necessary to find an estimation method for characterization parameters with uniqueness. An estimation method for ideal fractal parameters for multi-scale measurement of polished surface topography is proposed in this study. Polished surfaces of two materials, WC-Ni and 9Cr18Mo, are measured under multi-scale for frequency component analysis. This study proposes an estimation method for ideal fractal parameters based on a modified determination method for the scale-free region and the decomposition of frequency components into three classifications. The reasonable results verify the existence of ideal fractal parameters: for the WC-Ni surface, ideal fractal dimension D = 1.3 and scale coefficient G = 2.23×1020 μm; for the 9Cr18Mo surface, ideal fractal dimension D = 1.2 and scale coefficient G = 3.33×1033 μm. Additionally, it is revealed that the scale-dependent components conform to the same regulation on the same instrument by comparing the results of two materials. The conclusions of this study are expected to support tribology research and mechanical engineering related to surface topography. Full article
(This article belongs to the Special Issue Fractal Mechanics of Engineering Materials)
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12 pages, 3413 KiB  
Article
Fractal Analysis for Fatigue Crack Growth Rate Response of Engineering Structures with Complex Geometry
by Mudassar Hussain Hashmi, Seyed Saeid Rahimian Koloor, Mohd Foad Abdul-Hamid and Mohd Nasir Tamin
Fractal Fract. 2022, 6(11), 635; https://doi.org/10.3390/fractalfract6110635 - 1 Nov 2022
Cited by 2 | Viewed by 1461
Abstract
A growing fatigue crack in metallic materials and structures exhibits multifractal features that inherit signatures of the crack growth rate behavior of the material. This study exploits the recently established multifractal fatigue crack growth model to quantify the characteristic fatigue crack growth rate [...] Read more.
A growing fatigue crack in metallic materials and structures exhibits multifractal features that inherit signatures of the crack growth rate behavior of the material. This study exploits the recently established multifractal fatigue crack growth model to quantify the characteristic fatigue crack growth rate response of the AISI 410 martensitic stainless steel using an L-shaped bell crank structure. The objective is to demonstrate that the fatigue crack growth rate response of the material could be established by quantifying the fractality of the growing crack. The fractal approach avoids the need of the crack geometry factor when calculating the crack tip driving force. The fractal analysis of the crack image employs the box-counting algorithm to determine the fractal dimension along the edge of the crack length. The analysis is confined to the power law crack growth rate stage (Paris crack growth regime). Results show that the fatigue crack growth path in the bell crank structure is dictated by the Mode I (opening) component of the crack loading. The distribution of fractal-based fatigue crack growth rate data is within the 99% confidence limit of the median crack growth response by the Paris equation. Thus, the model could be employed for prediction of the fatigue crack growth response of engineering structures where the crack geometry factor is not readily available. Full article
(This article belongs to the Special Issue Fractal Mechanics of Engineering Materials)
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17 pages, 9285 KiB  
Article
Application of Asymmetric Notched Semi-Circular Bending Specimen to Evaluate Mixed-Mode I-II Fracture Behaviors of Sandstone
by Gang Ma, Jiangteng Li, Xiang Zhou, Lianying Zhang, Peitao Qiu and Yang Yu
Fractal Fract. 2022, 6(6), 336; https://doi.org/10.3390/fractalfract6060336 - 17 Jun 2022
Cited by 13 | Viewed by 2264
Abstract
In this paper, to investigate mixed-mode I-II fracture behaviors, three different asymmetric notched semi-circular bending specimens (ANSCB) were designed by adjusting the angle and the distance between supporting rollers to conduct asymmetric three-point bending tests. Several aid technologies, including acoustic emission (AE), digital [...] Read more.
In this paper, to investigate mixed-mode I-II fracture behaviors, three different asymmetric notched semi-circular bending specimens (ANSCB) were designed by adjusting the angle and the distance between supporting rollers to conduct asymmetric three-point bending tests. Several aid technologies, including acoustic emission (AE), digital image correlation (DIC), crack propagation gauge (CPG), and scanning electron microscopy (SEM), was utilized to monitor and assess the fracture characteristic. Meanwhile, the fractal dimension of the fracture surface was assessed based on the reconstructed digital fracture surface. The results show that mixed-mode I-II ANSCB three-point bending fracture is a brittle failure with the characteristics of the main crack being rapidly transfixed and the bearing capacity decreasing sharply. Based on the DIC method, the whole fracture process consists of a nonlinear elastic stage, fracture process zone, crack initiation stage and crack propagation stage. The crack initiation is mainly caused by the tension-shear strain concentration at the pre-existing crack tip. At the microscale, the crack propagation path is always along the grain boundary where the resultant stress is weakest. According to the monitoring of the AE, it can be found that micro-tensile cracks are mainly responsible for the asymmetric three-point bending fracture. The data obtained by CPG suggest that the subcritical crack growth rate is positively correlated to the ultimate load. In addition, asymmetric loading leads to a coarser fracture surface, and thus a higher fractal dimension of the fracture surface. The current study can provide a better understanding of the mixed-mode I-II fracture behaviors of rock. Full article
(This article belongs to the Special Issue Fractal Mechanics of Engineering Materials)
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13 pages, 530 KiB  
Article
Map of a Bending Problem for Self-Similar Beams into the Fractal Continuum Using the Euler–Bernoulli Principle
by Didier Samayoa Ochoa, Lucero Damián Adame and Andriy Kryvko
Fractal Fract. 2022, 6(5), 230; https://doi.org/10.3390/fractalfract6050230 - 22 Apr 2022
Cited by 11 | Viewed by 2138
Abstract
The bending of self-similar beams applying the Euler–Bernoulli principle is studied in this paper. A generalization of the standard Euler–Bernoulli beam equation in the FdH3 continuum using local fractional differential operators is obtained. The mapping of a bending problem [...] Read more.
The bending of self-similar beams applying the Euler–Bernoulli principle is studied in this paper. A generalization of the standard Euler–Bernoulli beam equation in the FdH3 continuum using local fractional differential operators is obtained. The mapping of a bending problem for a self-similar beam into the corresponding problem for a fractal continuum is defined. Displacements, rotations, bending moments and shear forces as functions of fractal parameters of the beam are estimated, allowing the mechanical response for self-similar beams to be established. An example of the structural behavior of a cantilever beam with a load at the free end is considered to study the influence of fractality on the mechanical properties of beams. Full article
(This article belongs to the Special Issue Fractal Mechanics of Engineering Materials)
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