Engineering Transactions, 65, 1, pp. 209–215, 2017

Dynamical Systems Approach of Internal Length in Fractional Calculus

Peter Balazs BEDA
Budapest University of Technology and Economics
Hungary

Conventionally, non-local properties are included in the constitutive equations in the form of strain gradient-dependent terms. In case of the second gradient dependence an internal material length can be obtained from the critical eigenmodes in instability problems. When non-locality is included by using fractional calculus, a generalized strain can be defined. Stability investigation can be also performed and internal length effects can be studied by analysing the critical eigenspace. Such an approach leads to classical results for second gradient, but new phenomena appear in the first gradient case
Keywords: rate and gradient dependence; fractional calculus; static and dynamic internal length
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References

Mindlin R.D., Second gradient of strain and surface tension in linear elasticity, Intl. Journal of Solids and Structures, 1: 417–438, 1965.

Aifantis E.C., On the role of gradients in the localization of deformation and fracture. International Journal of Engineering Science, 30(10): 1279–1299, 1992.

Lazopoulos K.A., Lazopoulos A.K., Fractional derivatives and strain gradient elasticity, Acta Mechanica, 227(3): 823–835, 2016.

Drapaca C.S., Sivaloganathan S., A fractional model of continuum mechanics, Journal of Elasticity, 107(2):105–123, 2012.

Tarasov V.E., Aifantis, E.C., Non-standard extensions of gradient elasticity: Fractional non-locality, memory and fractality, Communications in Nonlinear Science and Numerical Simulation, 22(1–3): 197–227, 2015.

Sumelka W., Thermoelasticity in the framework of the fractional continuum mechanics, Journal of Thermal Stresses, 37(6): 678–706, 2014.

Atanackovic T.M., Stankovic. B., Generalized wave equation in non-local elasticity, Acta Mech., 208(1): 1–10, 2009.




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