Fractional calculus in biomechanics: A 3D viscoelastic model using regularized fractional derivative kernels with application to the human calcaneal fat pad

Biomechanics and Modeling in Mechanobiology

Volumn 5, Issue 4, 2006, Pages 203-215

  Freed, A D ^a,b   Diethelm, K ^c,d  

^a NASA GLENN RESEARCH CENTER   (United States)

^b LERNER RESEARCH INSTITUTE   (United States)

^c GNS Systems GmbH   (Germany)

^d TECHNICAL UNIVERSITY OF BRAUNSCHWEIG   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

ENERGY EFFICIENCY; HEALTH CARE; THREE DIMENSIONAL; TISSUE; VISCOELASTICITY;

ELASTIC SOLIDS; MECHANOBIOLOGY; MULTI-MODEL INFERENCE; VISCOELASTIC MODEL;

BIOMECHANICS;

ADIPOSE TISSUE; BIOLOGICAL MODEL; BIOMECHANICS; CALCANEUS; ELASTICITY; HUMAN; MATHEMATICS; NONLINEAR SYSTEM; PHYSIOLOGY; REVIEW; TIME; VISCOSITY;

ADIPOSE TISSUE; BIOMECHANICS; CALCANEUS; ELASTICITY; HUMANS; MATHEMATICS; MODELS, BIOLOGICAL; NONLINEAR DYNAMICS; TIME FACTORS; VISCOSITY;

EID: 33750340240     PISSN: 16177959     EISSN: 16822978     Source Type: Journal    
DOI: 10.1007/s10237-005-0011-0     Document Type: Article

References (64)

1. Adolfsson K (2003) Models and numerical procedures for fractional order viscoelasticity. PhD thesis, Chalmers University Tech., Göteborg, Sweden
2. Bagley RL (1987) AIAA J 27:1412-1417
3. Baker CTH, Bocharov GA, Paul CAH, Rihan FA (2005) Appl Numer Math 53:107-129
4. Bernstein B, Kearsley EA, Zapas LJ (1963) Trans Soc Rheol 7:391-410
5. Bernstein B, Kearsley EA, Zapas LJ (1964) J Res NBS B Math Phys 68:103-113
6. Boltzmann L (1874) Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften: Mathematisch-naturwissenschaftlichen Klasse, Wien 70:275-300
7. Burnham KP, Anderson DR (2002) Model, selection and multimodel inference: a practical information-theoretic apporach, 2nd edn. Springer, Berlin Heidelerg New York
8. Caputo M (1967) Geophys J Roy Astr Soc 13:529-539
9. Caputo M, Mainardi F (1971a) Riv Nuovo Cimento 1:161-198
10. Caputo M, Mainardi F (1971b) Pure Appl. Geophys 91:134-147
11. Carew EO, Doehring TC, Barber JE, Freed AD, Vesely I (2003) In: L. J. Soslowsky, T. C. Skalak, J. S. Wayne, G. A. Livesay, (eds) Proceedings of the 2003 summer bioengineering conference, ASME, New York, pp 721-722
12. Cavanagh PR, Licata AA, Rice AJ (2005) Gravit Space Biol Bull 18(2):39-58
13. Cavanagh PR, Valiant GA, Misevich KW (1984) In: E. C. Frederick, (ed.) Sport shoes and playing surfaces, Human Kinetics Publishers, Champaign, pp 24-46
14. Chen Q, Suki B, An K-N (2004) J Biomech Eng 126:666-671
15. Coleman BD, Gurtin ME (1967) J Chem Phys 47:597-613
16. Coleman BD, Mizel VJ (1968) Arch Ration Mech Anal 29:18-31
17. Diethelm K, Ford NJ, Freed AD (2002) Nonlinear Dynam 29:3-22
18. Diethelm K, Ford NJ, Freed AD (2004) Numer Algorithms 36:31-52
19. Diethelm K, Ford NJ, Freed AD, Luchko Yu (2005) Comp Meth Appl Mech Eng 194:743-773
20. Diethelm K, Freed AD (2006) Comput Math Appl 51:51-72
21. Doehring TC, Carew EO, Vesely I (2004) Ann Biomed Eng 32:223-232
22. Douglas JF (2000) In: R. Hilfer, (ed) Applications of fractional, calculus in physics. World Scientific, Singapore, pp 241-330
23. Finger J (1894) Sitzungsberichte der Akademie der Wissenschaften, Wien, 103:1073-1100
24. Flory PJ (1961) Trans Faraday Soc 57:829-838
25. Freed AD (2004) J Eng Mater Technol 126:38-44.
26. Fulchiron R, Verney V, Cassagnau P, Michael A, Levoir P, Aubard J (1993) J Rheol 37:17-34
27. Fung Y-C (1967) Am J Physiol 213:1532-1544
28. Fung Y-C. (1971) In: Y.-C. Fung, N. Perrone, and M. Anliker, (eds) Biomechanics: Its foundations and objectives Prentice-Hall, Englewood Cliffs, pp 181-208
29. Fung Y-C (1993) Biomechanics: Mechanical properties of living tissues, 2nd edn Springer, Berlin Heidelberg New York
30. Gimbel JA, Sarver JJ, Soslowsky LJ (2004) J Biomech Eng 126:844-848
31. Glöckle WG, Nonnenmacher TF (1995) Biophys J 68:46-53
32. Gorenflo R, Rubin B (1994) Inverse Probl 10:881-893
33. Green G (1841) Trans Cambridge Phil Soc 7:121-140
34. Gross B (1947) J Appl Phys 18:212-221
35. Hencky H (1928) Z tech Phys 9:215-220
36. Holzapfel GA (2000) Nonlinear solid mechanics: a continuum approach for engineering, Wiley, Chichester
37. Johnson GA, Livesay GA, Woo SL-Y, Rajagopal KR (1996) J Biomech Eng 118:221-226
38. Kaye A (1962) Non-Newtonian flow in incompressible fluids, Technical Report 134, College of Aeronautics, Cranfield
39. Kohlrausch R (1847) Ann Phys Chem 72:353-405
40. Lang T, LeBlanc A. Evans H, Lu Y, Genant H, Yu A (2004) J Bone Miner Res 19:1006-1012
41. Ledoux WR, Meaney DR Hillstrom HJ (2004) J Biomech Eng 126:831-837
42. Leonov AI (1996) Macromolecules 29:8383-8386
43. Lodge AS (1964) Elastic liquids: an introductory vector treatment of finite-strain polymer rheology. Academic, London
44. Mainardi F (2002) Acoustic Interactions with submerged elastic structures Part IV In: A. Guran, A. Boström, O. Leroy, G. Maze B: series (eds) on stability, vibration and control of systems, World Scientific, New Jersey, pp 97-126
45. Miller-Young JE (2003) Factors affecting human heel pad mechanics: A finite element study. PhD thesis, University of Calgary, Calgary, Alberta
46. Miller-Young JE, Duncan NA, Baroud G (2002) J Biomech 35:1523-1531
47. Mitton RG (1945) J Soc Leath Trades Ch 29:169-194
48. Oldham KB, Spanier J (1974) The fractional calculus. In: Mathematics in science and engineering, vol. 3. Academic, New York
49. Podlubny I (1999) Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. In: Mathematics in science and engineering. Academic, San Diego
50. Puso MA, Weiss JA (1998) J Biomech Eng 120:62-70
51. Renardy M (1985) Arch Ration Mech Anal 88:83-94
52. Rivlin RS (1948) Phil Trans R Soc London A 241:379-397
53. Rubin B (1996) Fractional integrals and potentials. Addison Wesley Longman, Harlow
54. Simhambhatla M, Leonov AI (1993) Rheol Acta 32:589-600
55. Simo JC, Hughes TJR (1998) Computational inelasticity. In: Interdisciplinary applied mathematics, vol. 7. Springer-Verlag, Berlin Heidelberg New York
56. Suki B, Barabási A-L, Lutchen KR (1994) J Appl Physiol 76:2749-2759
57. Tuan VK, Gorenflo R (1994) Z Anal Anwend 13:537-545
58. Tuan VK, Gorenflo R (1994b) Numer Func Anal Opt 15:695-711
59. Venzon DJ, Moolgavkar SH (1988) Appl Statistician 37:87-94
60. Williams G, Watts DC (1970) Trans Faraday Soc 66:80-85
61. Williams ML (1964) AIAA J 2:785-808
62. Yuan H, Ingenito EP, Suki B (1997) J Appl Physiol 83:1420-1431
63. Yuan H, Kononov S, Cavalcante FA, Lutchen KR, Ingenito EP, Suki B (2000) J Appl Physiol 89:3-14
64. Zener C (1948) Elasticity and anelasticity of metals. University of Chicago, Chicago