×

Application of a variational iteration method to linear and nonlinear viscoelastic models with fractional derivatives. (English) Zbl 1112.74009

Summary: We present two nonlinear anelastic models with fractional derivatives, describing the properties of polymers and polycrystalline materials. These models are studied analytically using a variational iteration method. The paper clarifies different ways in which the fractional differentiation operator can be defined. A Volterra series method of model parameter identification from experimental data is also presented.

MSC:

74D10 Nonlinear constitutive equations for materials with memory
74D05 Linear constitutive equations for materials with memory
26A33 Fractional derivatives and integrals
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Oldham K. B., Fractional Calculus (1974) · Zbl 0292.26011
[2] DOI: 10.1007/BF01134604 · doi:10.1007/BF01134604
[3] DOI: 10.1088/0305-4470/26/19/034 · doi:10.1088/0305-4470/26/19/034
[4] DOI: 10.1088/0305-4470/26/19/034 · doi:10.1088/0305-4470/26/19/034
[5] DOI: 10.1023/A:1016552503411 · Zbl 1028.74013 · doi:10.1023/A:1016552503411
[6] DOI: 10.1016/S0022-460X(02)01530-4 · doi:10.1016/S0022-460X(02)01530-4
[7] DOI: 10.1016/S0022-460X(02)01530-4 · doi:10.1016/S0022-460X(02)01530-4
[8] DOI: 10.1177/0583102404039131 · doi:10.1177/0583102404039131
[9] DOI: 10.1177/0583102404039131 · doi:10.1177/0583102404039131
[10] DOI: 10.1016/S0165-1684(99)00118-8 · Zbl 1037.94524 · doi:10.1016/S0165-1684(99)00118-8
[11] DOI: 10.1007/s11071-004-3744-x · Zbl 1142.34385 · doi:10.1007/s11071-004-3744-x
[12] DOI: 10.1063/1.533231 · Zbl 0986.34055 · doi:10.1063/1.533231
[13] DOI: 10.1063/1.533231 · Zbl 0986.34055 · doi:10.1063/1.533231
[14] DOI: 10.1063/1.533231 · Zbl 0986.34055 · doi:10.1063/1.533231
[15] DOI: 10.1007/s11071-004-3750-z · Zbl 1134.65300 · doi:10.1007/s11071-004-3750-z
[16] DOI: 10.1016/S0020-7462(98)00048-1 · Zbl 1342.34005 · doi:10.1016/S0020-7462(98)00048-1
[17] DOI: 10.1016/S0020-7462(98)00085-7 · Zbl 1068.74618 · doi:10.1016/S0020-7462(98)00085-7
[18] DOI: 10.1016/S0045-7825(98)00108-X · Zbl 0942.76077 · doi:10.1016/S0045-7825(98)00108-X
[19] DOI: 10.1016/S0020-7462(00)00117-7 · Zbl 1116.34321 · doi:10.1016/S0020-7462(00)00117-7
[20] DOI: 10.1016/S0020-7462(00)00117-7 · Zbl 1116.34321 · doi:10.1016/S0020-7462(00)00117-7
[21] DOI: 10.1016/S0020-7462(00)00117-7 · Zbl 1116.34321 · doi:10.1016/S0020-7462(00)00117-7
[22] DOI: 10.1142/S0217979206033796 · Zbl 1102.34039 · doi:10.1142/S0217979206033796
[23] DOI: 10.1016/j.chaos.2005.10.100 · Zbl 1147.35338 · doi:10.1016/j.chaos.2005.10.100
[24] DOI: 10.1515/IJNSNS.2006.7.1.27 · Zbl 1401.65087 · doi:10.1515/IJNSNS.2006.7.1.27
[25] DOI: 10.1515/IJNSNS.2006.7.1.27 · Zbl 1401.65087 · doi:10.1515/IJNSNS.2006.7.1.27
[26] DOI: 10.1515/IJNSNS.2006.7.1.27 · Zbl 1401.65087 · doi:10.1515/IJNSNS.2006.7.1.27
[27] He J. H., Appl. Math. Comput. 114 pp 115– (2000) · Zbl 1027.34009 · doi:10.1016/S0096-3003(99)00104-6
[28] J. H. He, dissertation. de-Verlag im Internet GmbH, 2006.
[29] DOI: 10.1007/978-3-7091-2664-6_7 · doi:10.1007/978-3-7091-2664-6_7
[30] DOI: 10.1137/1018042 · Zbl 0324.44002 · doi:10.1137/1018042
[31] M. M. Benghorbal, Ph.D. thesis, University of Western Ontario, London, 2004.
[32] DOI: 10.1007/s11071-004-3747-7 · Zbl 1095.70010 · doi:10.1007/s11071-004-3747-7
[33] DOI: 10.1515/IJNSNS.2003.4.3.219 · doi:10.1515/IJNSNS.2003.4.3.219
[34] DOI: 10.1515/IJNSNS.2003.4.3.219 · doi:10.1515/IJNSNS.2003.4.3.219
[35] Drăgănescu G. E., J. Optoelectron. Adv. Mater. 7 pp 877– (2004)
[36] Abramowits M., Handbook of Mathematical Functions · doi:10.1119/1.1972842
[37] Erdely A., Tables of Integral Transforms 2 (1954)
[38] Schetzen M., The Volterra and Weiner Theories of Non-linear Systems (1980) · Zbl 0501.93002
[39] Bendat J. S., Non-linear System Analysis and Identification from Random Data (1990) · Zbl 0715.93063
[40] DOI: 10.1109/PROC.1971.8525 · doi:10.1109/PROC.1971.8525
[41] DOI: 10.1109/PROC.1971.8525 · doi:10.1109/PROC.1971.8525
[42] DOI: 10.1109/PROC.1971.8525 · doi:10.1109/PROC.1971.8525
[43] DOI: 10.1063/1.2189199 · Zbl 1111.81071 · doi:10.1063/1.2189199
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.