Advanced materials modelling via fractional calculus: challenges and perspectives

Failla, Giuseppe; Zingales, Massimiliano

doi:10.1098/rsta.2020.0050

Cited by 87 publications

(37 citation statements)

References 116 publications

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“…They can be treated as viscoelastic, provided nonlinear theories are used for any appreciable level of strain. Fractional calculus has been used to extend the linear theory (e.g., [ 36 ]) and also in the nonlinear case [ 37 ]. However, most polymers, including the subject of our study, show yield phenomena.…”

Section: Modellingmentioning

confidence: 99%

Constitutive Modelling of Polylactic Acid at Large Deformation Using Multiaxial Strains

et al. 2021

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Sheet specimens of a PLLA-based polymer have been extended at a temperature near to the glass transition in both uniaxial and planar tension, with stress relaxation observed for some time after reaching the final strain. Both axial and transverse stresses were recorded in the planar experiments. In all cases during loading, yielding at small strain was followed by a drop in true stress and then strain hardening. This was followed by stress relaxation at constant strain, during which stress dropped to reach an effectively constant level. Stresses were modelled as steady state and transient components. Steady-state components were identified with the long-term stress in stress relaxation and associated with an elastic component of the model. Transient stresses were modelled using Eyring mechanisms. The greater part of the stress during strain hardening was associated with dissipative Eyring processes. The model was successful in predicting stresses in both uniaxial and planar extension over a limited range of strain rate.

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Section: Modellingmentioning

confidence: 99%

Constitutive Modelling of Polylactic Acid at Large Deformation Using Multiaxial Strains

et al. 2021

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“…Unlike integer-order operators, the intrinsic multiscale nature of fractional operators enabled a very unique and effective approach to model historically challenging physical processes involving, as an example, nonlocality or memory effects. Indeed, many of the early applications of FC to physical modeling included viscoelastic effects [ 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 ], nonlocal behavior [ 8 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 ], anomalous and hybrid transport [ 9 , 10 , 11 , 24 , 25 , 26 , 27 , 28 , 29 , 30 ], fractal media [ 12 , 31 , 32 , 33 , 34 , 35 ], and even control theory [ 36 , 37 , 38 , 39 ]. The interested reader is referred to the work in [ 40 ] for a detailed account of the birth and evolution of fractional calculus.…”

Section: Introductionmentioning

confidence: 99%

Applications of Distributed-Order Fractional Operators: A Review

Ding

Patnaik

Sidhardh

et al. 2021

Entropy

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Distributed-order fractional calculus (DOFC) is a rapidly emerging branch of the broader area of fractional calculus that has important and far-reaching applications for the modeling of complex systems. DOFC generalizes the intrinsic multiscale nature of constant and variable-order fractional operators opening significant opportunities to model systems whose behavior stems from the complex interplay and superposition of nonlocal and memory effects occurring over a multitude of scales. In recent years, a significant amount of studies focusing on mathematical aspects and real-world applications of DOFC have been produced. However, a systematic review of the available literature and of the state-of-the-art of DOFC as it pertains, specifically, to real-world applications is still lacking. This review article is intended to provide the reader a road map to understand the early development of DOFC and the progressive evolution and application to the modeling of complex real-world problems. The review starts by offering a brief introduction to the mathematics of DOFC, including analytical and numerical methods, and it continues providing an extensive overview of the applications of DOFC to fields like viscoelasticity, transport processes, and control theory that have seen most of the research activity to date.

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“…According to this hypothesis, mechanical analogues were built to bridge the gap between a fractal micro–nano structure and its fractional-order hereditary properties [14]. In the 1990s, Blumen & Schiessel [15] proposed a ladder-like spring–dashpot network to present fractional-order viscoelastic behaviours, and Paola & Zingales [16] further developed its continuous version by introducing a shear layer [17,18].…”

Section: Introductionmentioning

confidence: 99%

Fractional-order viscoelastic model of musculoskeletal tissues: correlation with fractals

2021

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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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Advanced materials modelling via fractional calculus: challenges and perspectives

Cited by 87 publications

References 116 publications

Constitutive Modelling of Polylactic Acid at Large Deformation Using Multiaxial Strains

Constitutive Modelling of Polylactic Acid at Large Deformation Using Multiaxial Strains

Applications of Distributed-Order Fractional Operators: A Review

Fractional-order viscoelastic model of musculoskeletal tissues: correlation with fractals

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