Fractional Vector Calculus and Fractional Continuum Mechanics

2016 ◽  
Vol 2 (2) ◽  
pp. 85-104 ◽  
Author(s):  
Konstantinos A. Lazopoulos ◽  
Anastasios A. Lazopoulos
Keyword(s):  
Continuum Mechanics ◽  
Vector Calculus ◽  
Fractional Continuum

Fractional vector calculus and fluid mechanics

2017 ◽  
Vol 26 (1-2) ◽  
pp. 43-54 ◽  
Author(s):  
Konstantinos A. Lazopoulos ◽  
Anastasios K. Lazopoulos
Keyword(s):  
Differential Geometry ◽  
Porous Media ◽  
Fluid Mechanics ◽  
Research Field ◽  
Flow In Porous Media ◽  
Navier Stokes ◽  
Basic Fluid ◽  
Vector Calculus ◽  
Fractional Differential ◽  
Fractional Continuum

AbstractBasic fluid mechanics equations are studied and revised under the prism of fractional continuum mechanics (FCM), a very promising research field that satisfies both experimental and theoretical demands. The geometry of the fractional differential has been clarified corrected and the geometry of the fractional tangent spaces of a manifold has been studied in Lazopoulos and Lazopoulos (Lazopoulos KA, Lazopoulos AK. Progr. Fract. Differ. Appl. 2016, 2, 85–104), providing the bases of the missing fractional differential geometry. Therefore, a lot can be contributed to fractional hydrodynamics: the basic fractional fluid equations (Navier Stokes, Euler and Bernoulli) are derived and fractional Darcy’s flow in porous media is studied.

scholarly journals On selected aspects of space-fractional continuum mechanics model approximation

2020 ◽  
Vol 167 ◽  
pp. 105287 ◽  
Author(s):  
Krzysztof Szajek ◽  
Wojciech Sumelka ◽  
Tomasz Blaszczyk ◽  
Krzysztof Bekus
Keyword(s):  
Continuum Mechanics ◽  
Model Approximation ◽  
Mechanics Model ◽  
Fractional Continuum

scholarly journals Fractional calculus for continuum mechanics – anisotropic non-locality

2016 ◽  
Vol 64 (2) ◽  
pp. 361-372 ◽  
Author(s):  
W. Sumelka
Keyword(s):  
Fractional Calculus ◽  
Continuum Mechanics ◽  
General Framework ◽  
Deformation Gradient ◽  
Theoretical Discussion ◽  
Geometrical Meaning ◽  
Numerical Examples ◽  
Non Local ◽  
Fractional Continuum ◽  
Non Locality

Abstract In this paper, a generalisation of previous author’s formulation of fractional continuum mechanics for the case of anisotropic non-locality is presented. The discussion includes a review of competitive formulations available in literature. The overall concept is based on the fractional deformation gradient which is non-local due to fractional derivative definition. The main advantage of the proposed formulation is its structure, analogous to the general framework of classical continuum mechanics. In this sense, it allows to define similar physical and geometrical meaning of introduced objects. The theoretical discussion is illustrated by numerical examples assuming anisotropy limited to single direction.

scholarly journals Plane strain and plane stress elasticity under fractional continuum mechanics

2014 ◽  
Vol 85 (9-10) ◽  
pp. 1527-1544 ◽  
Author(s):  
Wojciech Sumelka ◽  
Krzysztof Szajek ◽  
Tomasz Łodygowski
Keyword(s):  
Plane Strain ◽  
Continuum Mechanics ◽  
Plane Stress ◽  
Fractional Continuum

On Modeling Flow in Fractal Media form Fractional Continuum Mechanics and Fractal Geometry

2013 ◽  
Vol 99 (1) ◽  
pp. 161-174 ◽  
Author(s):  
Miguel Angel Moreles ◽  
Joaquin Peña ◽  
Salvador Botello ◽  
Renato Iturriaga
Keyword(s):  
Continuum Mechanics ◽  
Fractal Geometry ◽  
Fractal Media ◽  
Media Form ◽  
Fractional Continuum ◽  
Modeling Flow

scholarly journals A primer on exterior differential calculus

2003 ◽  
pp. 85-162 ◽  
Author(s):  
D.A. Burton
Keyword(s):  
Continuum Mechanics ◽  
Differential Calculus ◽  
Differential Forms ◽  
Lie Derivative ◽  
Exterior Differential ◽  
Exterior Calculus ◽  
Vector Calculus ◽  
Exterior Forms ◽  
Pedagogical Application

A pedagogical application-oriented introduction to the cal?culus of exterior differential forms on differential manifolds is presented. Stokes' theorem, the Lie derivative, linear con?nections and their curvature, torsion and non-metricity are discussed. Numerous examples using differential calculus are given and some detailed comparisons are made with their tradi?tional vector counterparts. In particular, vector calculus on R3 is cast in terms of exterior calculus and the traditional Stokes' and divergence theorems replaced by the more powerful exterior expression of Stokes' theorem. Examples from classical continuum mechanics and spacetime physics are discussed and worked through using the language of exterior forms. The numerous advantages of this calculus, over more traditional ma?chinery, are stressed throughout the article. .

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