# Fractional Differential Calculus and Continuum Mechanics

• K. A. Lazopoulos
• A. K. Lazopoulos
Reference work entry

## Abstract

The present essay is an attempt to present a meaningful continuum mechanics formulation into the context of fractional calculus. The task is not easy, since people working on various fields using fractional calculus take for granted that a fractional physical problem is set up by simple substitution of the conventional derivatives to any kind of the plethora of fractional derivatives. However, that procedure is meaningless, although popular, since laws in science are derived through differentials and not through derivatives. One source of that mistake is that the fractional derivative of a variable with respect to itself is different from one. The other source of the same mistake is that the well-known derivatives are not able to form differentials. This leads to erroneous and meaningless quantities like fractional velocity and fractional strain. In reality those quantities, that nobody understands what physically represent, alter the dimensions of the physical quantities. In fact the dimension of the fractional velocity is L/Tα, contrary to the conventional L/T. Likewise, the dimension of the fractional strain is L−α, contrary to the conventional L0. That fact cannot be justified. Imagine that even in relativity theory, where everything is changed, like time, lengths, velocities, momentums, etc., the dimensions remain constant. Fractional calculus is allowed up to now to change the dimensions and to accept derivatives that are not able to form differentials, according to differential topology laws. Those handicaps have been pointed out in two recent conferences dedicated to fractional calculus by the authors, (K.A. Lazopoulos, in Fractional Vector Calculus and Fractional Continuum Mechanics, Conference “Mechanics though Mathematical Modelling”, celebrating the 70th birthday of Prof. T. Atanackovic, Novi Sad, 6–11 Sept, Abstract, p. 40, 2015; K.A. Lazopoulos, A.K. Lazopoulos, Fractional vector calculus and fractional continuum mechanics. Prog. Fract. Diff. Appl. 2(1), 67–86, 2016a) and were accepted by the fractional calculus community. The authors in their lectures (K.A. Lazopoulos, in Fractional Vector Calculus and Fractional Continuum Mechanics, Conference “Mechanics though Mathematical Modelling”, celebrating the 70th birthday of Prof. T. Atanackovic, Novi Sad, 6–11 Sept, Abstract, p. 40, 2015; K.A. Lazopoulos, in Fractional Differential Geometry of Curves and Surfaces, International Conference on Fractional Differentiation and Its Applications (ICFDA 2016), Novi Sad, 2016b; A.K. Lazopoulos, On Fractional Peridynamic Deformations, International Conference on Fractional Differentiation and Its Applications, Proceedings ICFDA 2016, Novi Sad, 2016c) and in the two recently published papers concerning fractional differential geometry of curves and surfaces (K.A. Lazopoulos, A.K. Lazopoulos, On the fractional differential geometry of curves and surfaces. Prog. Fract. Diff. Appl., No 2(3), 169–186, 2016b) and fractional continuum mechanics (K.A. Lazopoulos, A.K. Lazopoulos, Fractional vector calculus and fractional continuum mechanics. Prog. Fract. Diff. Appl. 2(1), 67–86, 2016a) added in the plethora of fractional derivatives one more, that called Leibniz L-fractional derivative. That derivative is able to yield differential and formulate fractional differential geometry. Using that derivative the dimensions of the various quantities remain constant and are equal to the dimensions of the conventional quantities. Since the establishment of fractional differential geometry is necessary for dealing with continuum mechanics, fractional differential geometry of curves and surfaces with the fractional field theory will be discussed first. Then the quantities and principles concerning fractional continuum mechanics will be derived. Finally, fractional viscoelasticity Zener model will be presented as application of the proposed theory, since it is of first priority for the fractional calculus people. Hence the present essay will be divided into two major chapters, the chapter of fractional differential geometry, and the chapter of the fractional continuum mechanics. It is pointed out that the well-known historical events concerning the evolution of the fractional calculus will be circumvented, since the goal of the authors is the presentation of the fractional analysis with derivatives able to form differentials, formulating not only fractional differential geometry but also establishing the fractional continuum mechanics principles. For instance, following the concepts of fractional differential and Leibniz’s L-fractional derivatives, proposed by the author (K.A. Lazopoulos, A.K. Lazopoulos, Fractional vector calculus and fractional continuum mechanics. Prog. Fract. Diff. Appl. 2(1), 67–86, 2016a), the L-fractional chain rule is introduced. Furthermore, the theory of curves and surfaces is revisited, into the context of fractional calculus. The fractional tangents, normals, curvature vectors, and radii of curvature of curves are defined. Further, the Serret-Frenet equations are revisited, into the context of fractional calculus. The proposed theory is implemented into a parabola and the curve configured by the Weierstrass function as well. The fractional bending problem of an inhomogeneous beam is also presented, as implementation of the proposed theory. In addition, the theory is extended on manifolds, defining the fractional first differential (tangent) spaces, along with the revisiting first and second fundamental forms for the surfaces. Yet, revisited operators like fractional gradient, divergence, and rotation are introduced, outlining revision of the vector field theorems. Finally, the viscoelastic mechanical Zener system is modelled with the help of Leibniz fractional derivative. The compliance and relaxation behavior of the viscoelastic systems is revisited and comparison with the conventional systems and the existing fractional viscoelastic systems are presented.

## Keywords

Fractional Derivative Fractional Differential Fractional Stress Fractional Strain Fractional Principles in Mechanics Fractional Continuum Mechanics

## References

1. F.B. Adda, Interpretation geometrique de la differentiabilite et du gradient d’ordre reel. CR Acad. Sci. Paris 326(Serie I), 931–934 (1998)
2. F.B. Adda, The differentiability in the fractional calculus. Nonlinear Anal. 47, 5423–5428 (2001)
3. O.P. Agrawal, A general finite element formulation for fractional variational problems. J. Math. Anal. Appl. 337, 1–12 (2008)
4. E.C. Aifantis, Strain gradient interpretation of size effects. Int. J. Fract. 95, 299–314 (1999)
5. E.C. Aifantis, Update in a class of gradient theories. Mech. Mater. 35, 259–280 (2003)
6. E.C. Aifantis, On the gradient approach – relations to Eringen’s nonlocal theory. Int. J. Eng. Sci. 49, 1367–1377 (2011)
7. H. Askes, E.C. Aifantis, Gradient elasticity in statics and dynamics: an overview of formulations, length scale identification procedures, finite element implementations and new results. Int. J. Solids Struct. 48, 1962–1990 (2011)
8. T.M. Atanackovic, A generalized model for the uniaxial isothermal deformation of a viscoelastic body. Acta Mech. 159, 77–86 (2002)
9. T.M. Atanackovic, B. Stankovic, Dynamics of a viscoelastic rod of fractional derivative type. ZAMM 82(6), 377–386 (2002)
10. T.M. Atanackovic, S. Konjik, S. Philipovic, Variational problems with fractional derivatives. Euler-Lagrange equations. J. Phys. A Math. Theor. 41, 095201 (2008)
11. R.L. Bagley, P.J. Torvik, Fractional calculus model of viscoelastic behavior. J. Rheol. 30, 133–155 (1986)
12. N. Bakhvalov, G. Panasenko, Homogenisation: Averaging Processes in Periodic Media (Kluwer, London, 1989)
13. A. Balankin, B. Elizarrataz, Hydrodynamics of fractal continuum flow. Phys. Rev. E 85, 025302(R) (2012)
14. D. Baleanu, K. Golmankhaneh Ali, K. Golmankhaneh Alir, M.C. Baleanu, Fractional electromagnetic equations using fractional forms. Int. J. Theor. Phys. 48(11), 3114–3123 (2009)
16. H. Beyer, S. Kempfle, Definition of physically consistent damping laws with fractional derivatives. ZAMM 75(8), 623–635 (1995)
17. G. Calcani, Geometry of fractional spaces. Adv. Theor. Math. Phys. 16, 549–644 (2012)
18. A. Carpinteri, B. Chiaia, P. Cornetti, Static-kinematic duality and the principle of virtual work in the mechanics of fractal media. Comput. Methods Appl. Mech. Eng. 191, 3–19 (2001)
19. A. Carpinteri, P. Cornetti, A. Sapora, Static-kinematic fractional operator for fractal and non-local solids. ZAMM 89(3), 207–217 (2009)
20. A. Carpinteri, P. Cornetti, A. Sapora, A fractional calculus approach to non-local elasticity. Eur. Phys. J. Spec Top 193, 193–204 (2011)
21. D.R.J. Chillingworth, Differential Topology with a View to Applications (Pitman, London, 1976)
22. M. Di Paola, G. Failla, M. Zingales, Physically-based approach to the mechanics of strong non-local linear elasticity theory. J. Elast. 97(2), 103–130 (2009)
23. C.S. Drapaca, S. Sivaloganathan, A fractional model of continuum mechanics. J. Elast. 107, 107–123 (2012)
24. A.C. Eringen, Nonlocal Continuum Field Theories (Springer, New York, 2002)
25. G.I. Evangelatos, Non Local Mechanics in the Time and Space Domain-Fracture Propagation via a Peridynamics Formulation: A Stochastic\Deterministic Perspective. Thesis, Houston Texas, 2011Google Scholar
26. G.I. Evangelatos, P.D. Spanos, Estimating the ‘In-Service’ modulus of elasticity and length of polyester mooring lines via a non linear viscoelastic model governed by fractional derivatives. ASME 2012 Int. Mech. Eng. Congr. Expo. 8, 687–698 (2012)
27. J. Feder, Fractals (Plenum Press, New York, 1988)
28. A.K. Goldmankhaneh, A.K. Goldmankhaneh, D. Baleanu, Lagrangian and Hamiltonian mechanics. Int. J. Theor. Rhys. 52, 4210–4217 (2013)
29. K. Golmankhaneh Ali, K. Golmankhaneh Alir, D. Baleanu, About Schrodinger equation on fractals curves imbedding in R3. Int. J. Theor. Phys. 54(4), 1275–1282 (2015)
30. H. Guggenheimer, Differential Geometry (Dover, New York, 1977)
31. G. Jumarie, An approach to differential geometry of fractional order via modified Riemann-Liouville derivative. Acta Math. Sin. Engl. Ser. 28(9), 1741–1768 (2012)
32. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations (Elsevier, Amsterdam, 2006)
33. K.A. Lazopoulos, On the gradient strain elasticity theory of plates. Eur. J. Mech. A/Solids 23, 843–852 (2004)
34. K.A. Lazopoulos, Nonlocal continuum mechanics and fractional calculus. Mech. Res. Commun. 33, 753–757 (2006)
35. K.A. Lazopoulos, in Fractional Vector Calculus and Fractional Continuum Mechanics, Conference “Mechanics though Mathematical Modelling”, celebrating the 70th birthday of Prof. T. Atanackovic, Novi Sad, 6–11 Sept, Abstract, p. 40 (2015)Google Scholar
36. A.K. Lazopoulos, On fractional peridynamic deformations. Arch. Appl. Mech. 86(12), 1987–1994 (2016a)
37. K.A. Lazopoulos, in Fractional Differential Geometry of Curves and Surfaces, International Conference on Fractional Differentiation and Its Applications (ICFDA 2016), Novi Sad (2016b)Google Scholar
38. A.K. Lazopoulos, On Fractional Peridynamic Deformations, International Conference on Fractional Differentiation and Its Applications, Proceedings ICFDA 2016, Novi Sad (2016c)Google Scholar
39. K.A. Lazopoulos, A.K. Lazopoulos, Bending and buckling of strain gradient elastic beams. Eur. J. Mech. A/Solids 29(5), 837–843 (2010)
40. K.A. Lazopoulos, A.K. Lazopoulos, On fractional bending of beams. Arch. Appl. Mech. (2015).
41. K.A. Lazopoulos, A.K. Lazopoulos, Fractional vector calculus and fractional continuum mechanics. Prog. Fract. Diff. Appl. 2(1), 67–86 (2016a)
42. K.A. Lazopoulos, A.K. Lazopoulos, On the fractional differential geometry of curves and surfaces. Prog. Fract. Diff. Appl., No 2(3), 169–186 (2016b)Google Scholar
43. G.W. Leibnitz, Letter to G. A. L’Hospital. Leibnitzen Mathematishe Schriftenr. 2, 301–302 (1849)Google Scholar
44. Y. Liang, W. Su, Connection between the order of fractional calculus and fractional dimensions of a type of fractal functions. Anal. Theory Appl. 23(4), 354–362 (2007)
45. J. Liouville, Sur le calcul des differentielles a indices quelconques. J. Ec. Polytech. 13, 71–162 (1832)Google Scholar
46. H.-S. Ma, J.H. Prevost, G.W. Sherer, Elasticity of dlca model gels with loops. Int. J. Solids Struct. 39, 4605–4616 (2002)
47. F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity (Imperial College Press, London, 2010)
48. M. Meerschaert, J. Mortensen, S. Wheatcraft, Fractional vector calculus for fractional advection–dispersion. Physica A 367, 181–190 (2006)
49. R.D. Mindlin, Second gradient of strain and surface tension in linear elasticity. Int. Jnl. Solids & Struct. 1, 417–438 (1965)
50. W. Noll, A mathematical theory of the mechanical behavior of continuous media. Arch. Rational Mech. Anal. 2, 197–226 (1958/1959)
51. K.B. Oldham, J. Spanier, The Fractional Calculus (Academic, New York, 1974)
52. I. Podlubny, Fractional Differential Equations (An Introduction to Fractional Derivatives Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications) (Academic, San Diego, 1999)
53. I.R. Porteous, Geometric Differentiation (Cambridge University Press, Cambridge, 1994)
54. B. Riemann, Versuch einer allgemeinen Auffassung der Integration and Differentiation, in Gesammelte Werke, vol. 62 (1876)Google Scholar
55. F. Riewe, Nonconservative Lagrangian and Hamiltonian mechanics. Phys. Rev. E 53(2), 1890–1899 (1996)
56. F. Riewe, Mechanics with fractional derivatives. Phys. Rev. E 55(3), 3581–3592 (1997)
57. J. Sabatier, O.P. Agrawal, J.A. Machado, Advances in Fractional Calculus (Theoretical Developments and Applications in Physics and Engineering) (Springer, The Netherlands, 2007)
58. S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications (Gordon and Breach, Amsterdam, 1993)
59. S.A. Silling, Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solids 48, 175–209 (2000)
60. S.A. Silling, R.B. Lehoucq, Peridynamic theory of solid mechanics. Adv. Appl. Mech. 44, 175–209 (2000)Google Scholar
61. S.A. Silling, M. Zimmermann, R. Abeyaratne, Deformation of a peridynamic bar. J. Elast. 73, 173–190 (2003)
62. W. Sumelka, Non-local Kirchhoff-Love plates in terms of fractional calculus. Arch. Civil and Mech. Eng. 208 (2014).
63. V.E. Tarasov, Fractional vector calculus and fractional Maxwell’s equations. Ann. Phys. 323, 2756–2778 (2008)
64. V.E. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media (Springer, Berlin, 2010)
65. R.A. Toupin, Theories of elasticity with couple stress. Arch. Ration. Mech. Anal. 17, 85–112 (1965)
66. C. Truesdell, A First Course in Rational Continuum Mechanics, vol 1 (Academic, New York, 1977)
67. C. Truesdell, W. Noll, The non-linear field theories of mechanics, in Handbuch der Physik, vol. III/3, ed. by S. Fluegge (Springer, Berlin, 1965)
68. I. Vardoulakis, G. Exadactylos, S.K. Kourkoulis, Bending of a marble with intrinsic length scales: a gradient theory with surface energy and size effects. J. Phys. IV 8, 399–406 (1998)Google Scholar
69. H.M. Wyss, A.M. Deliormanli, E. Tervoort, L.J. Gauckler, Influence of microstructure on the rheological behaviour of dense particle gels. AIChE J. 51, 134–141 (2005)
70. K. Yao, W.Y. Su, S.P. Zhou, On the connection between the order of fractional calculus and the dimensions of a fractal function. Chaos, Solitons Fractals 23, 621–629 (2005)