The B.E.M. in plane elastic bodies with cracks and/or boundaries of fractal geometry. (English) Zbl 0818.73073
Summary: The scope is to present a method which examines the influence of the fractal geometry on the stress and strains fields in cracked plane elastic bodies through a B.E. scheme combined with an iterative approximation procedure. The method proposed is based on the description of the fractal as the attractor of a deterministic or a random iterated function system. It is an iterative method which approximates the fractal boundary by classical \(C^ 1\)-curves in order to avoid additional singularities. The method proposed may be seen as an extension of the classical B.I.E.M. to the case of bodies having cracks and/or boundaries of fractal geometry.
MSC:
| 74S15 | Boundary element methods applied to problems in solid mechanics |
| 74R99 | Fracture and damage |
| 28A80 | Fractals |
References:
| [1] | Antes, H.; Panagiotopoulos, P. D. 1992: The boundary integral approach to static and dynamic contact problems. Equality and inequality methods. Basel: Birkhäuser Verlag · Zbl 0764.73002 |
| [2] | Barnsley, M. 1988: Fractals everywhere. Boston, New York: Academic Press · Zbl 0691.58001 |
| [3] | Blandford, G. E., Ingraffea, A. R.; Liggett, J. A. 1981 Two dimensional stress intensity factor computations using the boundary element method. Int. J. for Numerical Methods in Engineering, 17: 387-404 · Zbl 0463.73082 · doi:10.1002/nme.1620170308 |
| [4] | Cruse, T. A. 1969: Numerical solutions in three-dimensional elastostatics. Int. J. Solids and Struct. 5, 1259-1274 · Zbl 0181.52404 · doi:10.1016/0020-7683(69)90071-7 |
| [5] | Cruse, T. A. Van Buren, W. 1971: Three-dimensional elastic stress analysis of a fractured speciment with an edge crack. Intern. J. Fract. Mech. 7: 1-15 |
| [6] | Cruse, T. A. 1973: Applications of the boundary integral equation method to three-dimensional stress analysis.; Comp. and Struct. 3: 509-527 · doi:10.1016/0045-7949(73)90094-1 |
| [7] | Falconer, K. J. 1985: The geometry of fractal sets. Cambridge: Cambridge Univ. Press · Zbl 0587.28004 |
| [8] | Feder, J. 1988: Fractals. New York: Plenum Press · Zbl 0648.28006 |
| [9] | Gol’dshtein, R.; Mosolov, A. 1992: Fractal cracks. J. Appl. Maths. Mechs. 56(4): 563-571 · doi:10.1016/0021-8928(92)90012-W |
| [10] | Grisvard, P. 1985: Elliptic problems in nonsmooth domains. Pitman, Publ. Ltd., London · Zbl 0695.35060 |
| [11] | Hutchinson, J. F. 1981: Fractals and selfsimilarity. Indiana Univ. J. of Math. 30: 713-747 · Zbl 0598.28011 · doi:10.1512/iumj.1981.30.30055 |
| [12] | Jonson, A.; Wallin, H. 1984: Function spaces on subsets of ? n Math. Report. 2. London: Harwood Acad. Publ |
| [13] | Le Méhauté, A. 1990: Les géométries fractales. Lermes, Paris |
| [14] | Léné, F. 1974: Sur les matériaux élastiques énergie de déformation non quadratique. J. de Mécanique. 13: 499-534 |
| [15] | Mandelbrot, B. 1972. The fractal geometry of nature. New York: W. H. Freeman and Co. · Zbl 0227.76081 |
| [16] | Massopust, P. R. 1993: Smooth interpolating curves and surfaces generated by iterated function systems. Zeitschrift für Analysis und ihre Anwendungen, 12: 201-210 · Zbl 0777.41024 |
| [17] | Mistakidis, E. S., Panagiotopoulos, P. D.; Panagouli, O. K. 1992: Fractal surfaces and interfaces in structures. Methods and algorithms. Chaos Solitons and Fractals. 2(5): 551-574 · Zbl 0770.58022 · doi:10.1016/0960-0779(92)90030-Q |
| [18] | Moreau, J. J.; Panagiotopoulos, P. D. 1989: Nonsmooth Mechanics and Applications In Moreau, J. J. and Panagiotopoulos, P. D. (ed): CISM Lect. Notes, Vol. 302. New York: Springer-Verlag |
| [19] | Moreau, J. J., Panagiotopoulos, P. D.; Strang, G. 1988: topics in nonsmooth mechanics. Basel: Birkhäuser Verlag · Zbl 0646.00014 |
| [20] | Ne?as, J., Jarusek, J. and Haslinger, J.; 1980: On the solution of the variational inequality to the Signorini problem with small friction. Bulletino. U.M.I. 17B: 796-811 · Zbl 0445.49011 |
| [21] | Panagiotopoulos, P. D. 1975: A nonlinear programming approach to the unilateral contact- and friction- boundary value problem in the theory of elasticity. Ing. Archiv. 44: 421-432 · Zbl 0332.73018 · doi:10.1007/BF00534623 |
| [22] | Panagiotopoulos, P. D. 1978: A variational inequality approach to the friction problem of structures with convex energy density and application to the frictional unilateral contact problem. J. Struct. Mech. 6: 303-318 |
| [23] | Panagiotopoulos, P. D. 1983: Nonconvex energy functions Hemivariational inequalities and substationarity principles. Acta Mechanica. 42: 160-183 · Zbl 0538.73018 |
| [24] | Panagiotopoulos, P. D. 1985: Inequality problems in mechanics and applications. Convex and nonconvex energy functions. Basel: Birkhäuser Verlag. (Russian Translation 1985: Moscow MIr Publ). · Zbl 0579.73014 |
| [25] | Panagiotopoulos, P. D. 1990a: On the fractal nature of mechanical theories. ZAMM. 70: 258-260 |
| [26] | Panagiotopoulos, P. D. 1990b: Fractals in mechanics. In: Schneider, W.; Troger, H.; Ziegler, F. (ed): Trends in applications of pure mathematics to mechanics STAMM 8 London: Longman Scientific and Technical Press · Zbl 0817.73046 |
| [27] | Panagiotopoulos, P. D. 1991: Coercive and semicoercive hemivariational inequalities. Nonlinear Analysis T.M.A. 16: 209-231 · Zbl 0733.49012 · doi:10.1016/0362-546X(91)90224-O |
| [28] | Panagiotopoulos, P. D. 1992a: Fractal geometry in solids and structures. Int. J. Solids and Structures. 29(17): 2159-2175 · Zbl 0764.73012 · doi:10.1016/0020-7683(92)90063-Y |
| [29] | Panagiotopoulos, P. D. 1992b: Fractals and fractal approximation in structural mechanics. Meccanica. 27: 25-33 · Zbl 0771.73044 · doi:10.1007/BF00453000 |
| [30] | Panagiotopoulos, P. D.; Panagouli, O. K. 1992: Fractal interfaces in structures. Comp. and Struct. 452: 369-380 · Zbl 0771.73061 · doi:10.1016/0045-7949(92)90420-5 |
| [31] | Panagiotopoulos, P. D. 1993: Hemivariational inequalities and their applications in mechanics and engineering. Berlin: Springer Verlag · Zbl 0826.73002 |
| [32] | Panagiotopoulos, P. D., Mistakidis, E. S.; Panagouli, O. K. 1992: Fractal interfaces with unilateral contact and friction conditions. Computer Methods in Applied Mechanics and Engineering. 99: 395-412 · Zbl 0764.73076 · doi:10.1016/0045-7825(92)90043-J |
| [33] | Panagiotopoulos, P. D., Panagouli, O. K.; Mistakidis, E. S. 1993: Fractal geometry and fractal material behaviour in solids and structures. Archive of Applied Mechanics 63: 1-64 · Zbl 0767.73007 · doi:10.1007/BF00787906 |
| [34] | Panagiotopoulos, P. D., Panagouli, O. K.; Mistakidis, E. S. 1994: On the consideration of the geometric and physical fractality in solid Mechanics. I. Theoretical results. ZAMM, 74(3): 167-176 · Zbl 0816.73018 · doi:10.1002/zamm.19940740306 |
| [35] | Panagouli, O. K., Panagiotopoulos, P. D.; Mistakidis, E. S. 1992: On the numerical solution of structures with fractal geometry: The FE approach. Meccanica. 27: 263-274 · Zbl 0768.73078 · doi:10.1007/BF00424365 |
| [36] | Takayasu, H. 1990: Fractals in the physical sciences. Manchester: Manchester Univ. Press · Zbl 0689.58001 |
| [37] | Wallin, H. 1989a: Interpolating and orthogonal polynomials on fractals. Constr. Approx. 5: 137-150 · Zbl 0695.41014 · doi:10.1007/BF01889603 |
| [38] | Wallin, H. 1989b: The trace to the boundary of Sobolev spaces on a snowflake. Rep. Dep. of Math., Univ. of Umea, Sweden · Zbl 0793.46015 |
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.
