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Guo J, Yin Y, Peng G. Fractional-order viscoelastic model of musculoskeletal tissues: correlation with fractals. Proc Math Phys Eng Sci 2021. [DOI: 10.1098/rspa.2020.0990] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/24/2022] Open
Abstract
Self-similar fractals are widely obtained from biomaterials within the human musculoskeletal system, and their viscoelastic behaviours can be described by fractional-order derivatives. However, existing viscoelastic models neglect the internal correlation between the fractal structure of biomaterials and their fractional-order temporal responses. We further expanded the fractal hyper-cell (FHC) viscoelasticity theory to investigate this spatio-temporal correlation. The FHC element was first compared with other material elements and spring–dashpot viscoelastic models, thereby highlighting its discrete and fractal nature. To demonstrate the utility of an FHC, tree-like, ladder-like and triangle-like FHCs were abstracted from human cartilage, tendons and muscle cross-sections, respectively. The duality and symmetry of the FHC element were further discussed, where operating the duality transformation generated new types of FHC elements, and the symmetry breaking of an FHC altered its fractional-order viscoelastic responses. Thus, the correlations between the staggering patterns of FHCs and their rheological power-law orders were established, and the viscoelastic behaviour of the multi-level FHC elements fitted well in stress relaxation experiments at both the macro- and nano-levels of the tendon hierarchy. The FHC element provides a theoretical basis for understanding the connections between structural degeneration of bio-tissues during ageing or disease and their functional changes.
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Affiliation(s)
- Jianqiao Guo
- MOE Key Laboratory of Dynamics and Control of Flight Vehicle, School of Aerospace Engineering, Beijing Institute of Technology, Beijing, People’s Republic of China
| | - Yajun Yin
- Department of Engineering Mechanics, Tsinghua University, Beijing, People’s Republic of China
| | - Gang Peng
- Department of Engineering Mechanics, Tsinghua University, Beijing, People’s Republic of China
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Bologna E, Di Paola M, Dayal K, Deseri L, Zingales M. Fractional-order nonlinear hereditariness of tendons and ligaments of the human knee. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2020; 378:20190294. [PMID: 32389091 DOI: 10.1098/rsta.2019.0294] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 03/31/2020] [Indexed: 06/11/2023]
Abstract
In this paper the authors introduce a nonlinear model of fractional-order hereditariness used to capture experimental data obtained on human tendons of the knee. Creep and relaxation data on fibrous tissues have been obtained and fitted with logarithmic relations that correspond to power-laws with nonlinear dependence of the coefficients. The use of a proper nonlinear transform allows one to use Boltzmann superposition in the transformed variables yielding a fractional-order model for the nonlinear material hereditariness. The fundamental relations among the nonlinear creep and relaxation functions have been established, and the results from the equivalence relations have been contrasted with measures obtained from the experimental data. Numerical experiments introducing polynomial and harmonic stress and strain histories have been reported to assess the provided equivalence relations. This article is part of the theme issue 'Advanced materials modelling via fractional calculus: challenges and perspectives'.
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Affiliation(s)
- E Bologna
- Dipartimento di Ingegneria, Viale delle Scienze ed.8, 90128 Palermo, Italy
- Bio/NanoMechanics for Medical Sciences Laboratory, Viale delle Scienze ed.8, 90128 Palermo, Italy
| | - M Di Paola
- Dipartimento di Ingegneria, Viale delle Scienze ed.8, 90128 Palermo, Italy
| | - K Dayal
- Department of Civil and Environmental Engineering, Carnegie Mellon University Pittsburgh, Pittsburgh, PA 15213, USA
- Center for Nonlinear Analysis, Carnegie Mellon University Pittsburgh, Pittsburgh, PA 15213, USA
- Department of Materials Science and Engineering, Carnegie Mellon University Pittsburgh, Pittsburgh, PA 15213, USA
| | - L Deseri
- Department of Mechanical Engineering, University of Pittsburgh, Benedum Hall, Pittsburgh, PA, USA
- Dipartimento Civile, ambientale e meccanica, Università degli Studi di Trento, Via Mesiano, 77 - 38123 Trento, Italy
- Department of Mechanical Engineering, Department of Civil & Env. Engineering, Carnegie Mellon University, Pittsburgh, PA 15213-3890, USA
- The Methodist Hospital Research Institute, Department of Nanomedicine, 6565 Fannin St., MS B-490, Houston, TX 77030, USA
| | - M Zingales
- Dipartimento di Ingegneria, Viale delle Scienze ed.8, 90128 Palermo, Italy
- Bio/NanoMechanics for Medical Sciences Laboratory, Viale delle Scienze ed.8, 90128 Palermo, Italy
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Failla G, Zingales M. Advanced materials modelling via fractional calculus: challenges and perspectives. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2020; 378:20200050. [PMID: 32389077 PMCID: PMC7287319 DOI: 10.1098/rsta.2020.0050] [Citation(s) in RCA: 11] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 04/03/2020] [Indexed: 06/11/2023]
Abstract
Fractional calculus is now a well-established tool in engineering science, with very promising applications in materials modelling. Indeed, several studies have shown that fractional operators can successfully describe complex long-memory and multiscale phenomena in materials, which can hardly be captured by standard mathematical approaches as, for instance, classical differential calculus. Furthermore, fractional calculus has recently proved to be an excellent framework for modelling non-conventional fractal and non-local media, opening valuable prospects on future engineered materials. The theme issue gathers cutting-edge theoretical, computational and experimental studies on advanced materials modelling via fractional calculus, with a focus on complex phenomena and non-conventional media. This article is part of the theme issue 'Advanced materials modelling via fractional calculus: challenges and perspectives'.
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Affiliation(s)
- Giuseppe Failla
- Department of Civil, Environmental, Energy and Materials Engineering (DICEAM), University of Reggio Calabria, Via Graziella, Località Feo di Vito, 89124 Reggio Calabria, Italy
| | - Massimiliano Zingales
- Department of Engineering, University of Palermo, Viale delle Scienze ed. 8, 90128, Palermo, Italy
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Abstract
Non-local time evolution of material stress/strain is often referred to as material hereditariness. In this paper, the widely used non-linear approach to single integral time non-local mechanics named quasi-linear approach is proposed in the context of fractional differential calculus. The non-linear model of the springpot is defined in terms of a single integral with separable kernel endowed with a non-linear transform of the state variable that allows for the use of Boltzmann superposition. The model represents a self-similar hierarchy that allows for a time-invariance as the result of the application of the conservation laws at any resolution scale. It is shown that the non-linear springpot possess an equivalent mechanical hierarchy in terms of a functionally-graded elastic column resting on viscous dashpots with power-law decay of the material properties. Some numerical applications are reported to show the capabilities of the proposed model.
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A Lower Limb-Pelvis Finite Element Model with 3D Active Muscles. Ann Biomed Eng 2017; 46:86-96. [PMID: 29038943 DOI: 10.1007/s10439-017-1942-1] [Citation(s) in RCA: 19] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/28/2017] [Accepted: 10/07/2017] [Indexed: 10/18/2022]
Abstract
A lower limb-pelvis finite element (FE) model with active three-dimensional (3D) muscles was developed in this study for biomechanical analysis of human body. The model geometry was mainly reconstructed from a male volunteer close to the anthropometry of a 50th percentile Chinese male. Tissue materials and structural features were established based on the literature and new implemented experimental tests. In particular, the muscle was modeled with a combination of truss and hexahedral elements to define its passive and active properties as well as to follow the detailed anatomy structure. Both passive and active properties of the model were validated against the experiments of Post-Mortem Human Surrogate (PMHS) and volunteers, respectively. The model was then used to simulate driver's emergency braking during frontal crashes and investigate Knee-Thigh-Hip (KTH) injury mechanisms and tolerances of the human body. A significant force and bending moment variance was noted for the driver's femur due to the effects of active muscle forces during emergency braking. In summary, the present lower limb-pelvis model can be applied in various research fields to support expensive and complex physical tests or corresponding device design.
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Baró J, Shyu P, Pang S, Jasiuk IM, Vives E, Salje EKH, Planes A. Avalanche criticality during compression of porcine cortical bone of different ages. Phys Rev E 2016; 93:053001. [PMID: 27300967 DOI: 10.1103/physreve.93.053001] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/18/2015] [Indexed: 06/06/2023]
Abstract
Crack events developed during uniaxial compression of cortical bones cut from femurs of developing pigs of several ages (4, 12, and 20 weeks) generate avalanches. These avalanches have been investigated by acoustic emission analysis techniques. The avalanche energies are power-law distributed over more than four decades. Such behavior indicates the absence of characteristic scales and suggests avalanche criticality. The statistical distributions of energies and waiting times depend on the pig age and indicate that bones become stronger, but less ductile, with increasing age. Crack propagation is equally age-dependent. Older pigs show, on average, larger cracks with a time distribution similar to those of aftershocks in earthquakes, while younger pigs show only statistically independent failure events.
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Affiliation(s)
- Jordi Baró
- Departament d'Estructura i Constituents de la Matèria, Facultat de Física. Universitat de Barcelona, Diagonal, 647, E-08028 Barcelona, Catalonia
| | - Peter Shyu
- Department of Bioengineering, University of Illinois, 1270 Digital Computer Laboratory, Urbana, Illinois 61801, USA
| | - Siyuan Pang
- Department of Mechanical Science and Engineering, University of Illinois, 1206 West Green Street, Urbana, Illinois 61801, USA
| | - Iwona M Jasiuk
- Department of Bioengineering, University of Illinois, 1270 Digital Computer Laboratory, Urbana, Illinois 61801, USA
- Department of Mechanical Science and Engineering, University of Illinois, 1206 West Green Street, Urbana, Illinois 61801, USA
| | - Eduard Vives
- Departament d'Estructura i Constituents de la Matèria, Facultat de Física. Universitat de Barcelona, Diagonal, 647, E-08028 Barcelona, Catalonia
| | - Ekhard K H Salje
- Department of Earth Sciences, University of Cambridge, Downing Street, Cambridge CB2 3EQ, United Kingdom
- State Key Laboratory for Mechanical Behavior of Materials, Xi'an Jiaotong University, Xi'an 710049, China
| | - Antoni Planes
- Departament d'Estructura i Constituents de la Matèria, Facultat de Física. Universitat de Barcelona, Diagonal, 647, E-08028 Barcelona, Catalonia
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Fraldi M, Cugno A, Deseri L, Dayal K, Pugno NM. A frequency-based hypothesis for mechanically targeting and selectively attacking cancer cells. J R Soc Interface 2015; 12:20150656. [PMID: 26378121 PMCID: PMC4614503 DOI: 10.1098/rsif.2015.0656] [Citation(s) in RCA: 37] [Impact Index Per Article: 4.1] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/22/2015] [Accepted: 08/27/2015] [Indexed: 12/27/2022] Open
Abstract
Experimental studies recently performed on single cancer and healthy cells have demonstrated that the former are about 70% softer than the latter, regardless of the cell lines and the measurement technique used for determining the mechanical properties. At least in principle, the difference in cell stiffness might thus be exploited to create mechanical-based targeting strategies for discriminating neoplastic transformations within human cell populations and for designing innovative complementary tools to cell-specific molecular tumour markers, leading to possible applications in the diagnosis and treatment of cancer diseases. With the aim of characterizing and gaining insight into the overall frequency response of single-cell systems to mechanical stimuli (typically low-intensity therapeutic ultrasound), a generalized viscoelastic paradigm, combining classical and spring-pot-based models, is introduced for modelling this problem by neglecting the cascade of mechanobiological events involving the cell nucleus, cytoskeleton, elastic membrane and cytosol. Theoretical results show that differences in stiffness, experimentally observed ex vivo and in vitro, allow healthy and cancer cells to be discriminated, by highlighting frequencies (from tens to hundreds of kilohertz) associated with resonance-like phenomena—prevailing on thermal fluctuations—that could be helpful in targeting and selectively attacking tumour cells.
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Affiliation(s)
- M Fraldi
- Department of Structures for Engineering and Architecture and Interdisciplinary Research Center for Biomaterials, Polytechnic School, College of Engineering, University of Napoli Federico, II via Claudio 21, 80125 Napoli, Italy
| | - A Cugno
- Department of Structures for Engineering and Architecture and Interdisciplinary Research Center for Biomaterials, Polytechnic School, College of Engineering, University of Napoli Federico, II via Claudio 21, 80125 Napoli, Italy Department of Civil, Environmental and Mechanical Engineering, University of Trento, via Mesiano 77, 38123 Trento, Italy
| | - L Deseri
- Department of Civil, Environmental and Mechanical Engineering, University of Trento, via Mesiano 77, 38123 Trento, Italy Department of Civil and Environmental Engineering, Carnegie Mellon University, Pittsburgh, PA 15213-3890, USA Department of Mechanical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213-3890, USA TMHRI-Department of Nanomedicine, The Methodist Hospital Research Institute, 6565 Fannin Street, MS B-490 Houston, TX 77030, USA
| | - K Dayal
- Department of Civil and Environmental Engineering, Carnegie Mellon University, Pittsburgh, PA 15213-3890, USA
| | - N M Pugno
- Department of Civil, Environmental and Mechanical Engineering, University of Trento, via Mesiano 77, 38123 Trento, Italy Laboratory of Bio-inspired and Graphene Nanomechanics, Department of Civil, Environmental and Mechanical Engineering, University of Trento, Via Mesiano 77, 38123 Trento, Italy Centre of Materials and Microsystems, Bruno Kessler Foundation, Via Santa Croce 77, 38122 Trento, Italy School of Engineering and Materials Science, Queen Mary University, Mile End Road, London E1 4NS, UK
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