×

Analytical solutions for vibrating fractal composite rods and beams. (English) Zbl 1211.74115

Summary: Fractals have the potential to describe complex microstructures but presently no solution methodologies exist for the prediction of deformation on transiently deforming fractal structures. This is achieved in this paper with the development of analytical solutions on vibrating composite rods and beams. The fractals considered are necessarily deterministic and relatively simple in form to demonstrate the solution methodology. The solutions are limited to beams and rods constructed from an idealised-composite material consisting of relatively large rigid particles embedded in an infinitely thin pliable matrix. Although, as a result, the fractal composite system is not representative of a realistic physical system the methodologies presented do serve to highlight the practical difficulties in using fractals in structural dynamics. Static loading is restricted to spatially invariant axial forces and bending moments as solutions on a unified state of continuum stress are sought which then serve as initial conditions for the vibratory problem. It is demonstrated that measurable displacement is possible on a fractal structure and that finite measures of total, kinetic and strain energy are simultaneously achievable. The approach involves the use of modal analysis to determine modes at natural frequencies that satisfy boundary conditions. These are combined to provide a free vibration solution on a fractal that satisfies the initial conditions in the form of a fractal displacement field.

MSC:

74H45 Vibrations in dynamical problems in solid mechanics
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74E30 Composite and mixture properties
37N15 Dynamical systems in solid mechanics
74H10 Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) of dynamical problems in solid mechanics
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Mandelbrot, B.B.; Passoja, D.E.; Paullay, A.J., Fractal character of fracture surfaces of metals, Nature, 308, 721-722, (1984)
[2] Shek, C.H.; Lin, G.M.; Lai, J.K.L.; Tang, Z.F., Fractal fracture and transformation toughening in cunial single crystal, Metall. mater. trans. A, 28A, 1337-1340, (1997)
[3] Carpinteri, A.; Chiaia, B., Crack-resistant behaviour as a consequence of self-similar fracture topologies, Int. J. fract., 76, 4, 327-340, (1996), 20
[4] Mecholsky, J.J., Estimating theoretical strength of brittle materials using fractal geometry, Mater. lett., 60, 20, 2485-2488, (2006)
[5] Epstein, M.; Śniatycki, J., Fractal mechanics, Phys D nonlinear phenom., 220, 1, 54-68, (2006) · Zbl 1098.74008
[6] Capitanelli, R.; Lancia, M.R., Nonlinear energy forms and Lipschitz spaces on the Koch curve, J. convex anal., 9, 1, 245-257, (2002) · Zbl 1021.46030
[7] Carpinteri, A.; Chiaia, B.; Cornetti, P., Static-kinematic duality and the principle of virtual work in the mechanics of fractal media, Comput. methods appl. mech. eng., 191, 1-2, 3-19, (2001) · Zbl 0991.74013
[8] Carpinteri, A.; Cornetti, P., A fractional calculus approach to the description of stress and strain localization in fractal media, Chaos solitons fractals, 13, 1, 85-94, (2001) · Zbl 1030.74045
[9] Mosco, U., Energy functionals on certain fractal structures, J. convex anal., 9, 581-600, (2002) · Zbl 1018.28005
[10] Borodich, F.M., Fractals and fractal scaling in fracture mechanics, Int. J. fract., 95, 239-259, (1999) · Zbl 0969.74572
[11] Borodich, F.M.; Onishchenko, D.A., Similarity and fractality in the modelling of roughness by a multilevel profile with hierarchical structure, Int. J. solids struct., 36, 2585-2612, (1999) · Zbl 0939.74004
[12] Riewe, F., Nonconservative Lagrangian and Hamiltonian mechanics, Phys. rev. E, 53, 2, 1890-1897, (1996)
[13] Agrawal, Om.P., Formulation of euler – lagrange equations for fractional variational problems, Math. anal. appl., 272, 368-379, (2002) · Zbl 1070.49013
[14] Panagiotopoulos, P.D.; Panagouli, O.K., Mechanics on fractal bodies. data compression using fractals, Chaos solitons fractals, 8, 253-267, (1997) · Zbl 0924.58092
[15] Falconer, K., Fractal geometry, (2003), John Wiley & Sons
[16] Russel, L.; Hall, F.R.; Davey, K., (), 525-533
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.