Growth control of cracks under contact conditions based on the topological derivative of the Rice’s integral

• Autorzy: da Silva Xavier Marcel Duarte , Novotny Antonio André , Sokołowski Jan
• Języki publikacji: EN

Abstrakty

EN

In the present paper we propose a simple method for dealing with growth control of cracks under contact type boundary conditions on their lips. The aim is to find a mechanism for decreasing the energy release rate of cracked components, which means increasing their fracture toughness. The method consists in minimizing a shape functional defined in terms of the Rice’s integral, with respect to the nucleation of hard and/or soft inclusions, according to the information provided by the associated topological derivative. Based on Griffith’s energy criterion, this simple strategy allows for an increase in fracture toughness of the cracked component. Since the problem is non-linear, the domain decomposition technique, combined with the Steklov-Poincaré pseudo-differential boundary operator, is used to obtain the sensitivity of the associated shape functional with respect to the nucleation of a small circular inclusion with different material property from the background. Then, the obtained topological derivatives are used to indicate the regions, where the controls should be positioned in order to solve the minimization problem we are dealing with. Finally, a numerical example is presented showing the applicability of the proposed methodology.

Słowa kluczowe

EN

Rice’s integral; Griffith’s criterion; Eshelby’s tensor; topological derivative

• Wydawca: Systems Research Institute, Polish Academy of Sciences
• Czasopismo: Control and Cybernetics
• Rocznik: 2019
• Tom: Vol. 48, No. 2
• Strony: 307--323
• Opis fizyczny: Bibliogr. 29 poz., rys., tab.

Twórcy

autor

da Silva Xavier Marcel Duarte

• Universidade Federal Fluminense UFF, TEM - Departamento de Engenharia Mecânica, Rua passo da Pátria 156, 24210-240 Niterói - RJ, Brasil

autor

Novotny Antonio André

• Laboratório Nacional de Computa，cão Cientìfica LNCC/MCTIC, Coordena，cão de Matemática Aplicada e Computacional, Av. Getúlio Vargas 333, 25651-075 Petrópolis - RJ, Brasil

autor

Sokołowski Jan

• Departamento de Computa，cão Cientìfica, Centro de Informática, Universidade Federal da Paraìba, Rua dos Escoteiros s/n, Mangabeira, 58058-600, João Pessoa, PB, Brasil
• Systems Research Institute of the Polish Academy of Sciences, ul Newelska 6, 01-447 Warsaw, Poland

Bibliografia

• Amigo, R. C. R., Giusti, S. M., Novotny, A. A., Silva, E. C. N. and Sokołowski, J. (2016) Optimum design of extensional piezoelectric actuators into two spatial dimensions. SIAM Journal on Control and Optimization, 52(2), 760–789.
• Ammari, H., Kang, H., Kim, K. and Lee, H. (2013) Strong convergence of the solutions of the linear elasticity and uniformity of asymptotic expansions in the presence of small inclusions. Journal of Differential Equations, 254(12), 4446–4464.
• Ammari, H., Kang, H., Nakamura, G. and Tanuma, K. (2002) Complete asymptotic expansions of solutions of the system of elastostatics in the presence of inhomogeneities of small diameter. Journal of Elasticity, 67, 97–129.
• Amstutz, S. (2006) Sensitivity analysis with respect to a local perturbation of the material property. Asymptotic Analysis, 49(1-2), 87–108.
• Bittencourt, T. N., Wawrzynek, P. A., Ingratea, A. R. and Sousa, J. L. (1996) Quasi-automatic simulation of crack propagation for 2d LEFM problems. Engineering Fracture Mechanics, 55(2), 321–334.
• Destuynder, P. (1989) Remarques sur le contrˆole de la propagation des fissures en régime stationnaire. Comptes Rendus de l’Academie des Sciences de Paris Serie II, 308(8), 697–701.
• Eshelby, J. D. (1975) The elastic energy-momentum tensor. Journal of Elasticity, 5(3-4), 321–335.
• Fancello, E. A., Taroco, E. O. and Feij´oo, R. A. (1993) Shape sensitivity analysis in fractura mechanics. In: Structural Optimization – The World Congress on Optimal Design of Structural System, 2, 239–248.
• Feijóo, R. A., Padra, C., Saliba, R., Taroco, E., Venere, M. And Marcelo, J. (2000) Shape sensitivity analysis for energy release rate evaluation and its application to the study of three-dimensional cracked bodies. Computer Methods in Applied Mechanics and Engineering, 188(4), 649–664.
• Griffth, A. A. (1921) The phenomena of rupture and flow in solids. Philosophical Transactions of the Royal Society, 221, 163–198.
• Hild, P., Munch, A. and Ousset, Y. (2008) On the active control of crack growth in elastic media. Computer Methods in Applied Mechanics and Engineering, 198(3-4), 407–419.
• Hömberg, D. and Khludnev, A. M. (2002) On safe crack shapes in elastic bodies. European Journal of Mechanics, A/Solids, 21, 991–998.
• Ingraffea, A. R. and Grigoriu, M. (1990) Probabilistic fracture mechanics: A validation of predictive capability. Technical report, Cornell University, Ithaca, New York.
• Khludnev, A., Leugering, G. and Specovius-Neugebauer, M. (2012) Optimal control of inclusion and crack shapes in elastic bodies. Journal of Optimization Theory and Applications, 155(1), 54–78.
• Kovtunenko, V. A. and Leugering, G. (2016) A shape-topological control problem for nonlinear crack-defect interaction: the antiplane variational model. SIAM Journal on Control and Optimization, 54(3), 1329–1351.
• Leugering, G., Sokołowski, J. and Żochowski, A. (2015) Control of crack propagation by shape-topological optimization. Discrete and Continuous Dynamical Systems. Series A, 35(6), 2625–2657.
• Lopes, C. G., Santos, R. B., Novotny, A. A. and Sokołowski, J. (2017) Asymptotic analysis of variational inequalities with applications to optimum design in elasticity. Asymptotic Analysis, 102, 227–242.
• Münch, A. and Pedregal, P. (2010) Relaxation of an optimal design problem in fracture mechanic: the anti-plane case. ESAIM: Control, Optimization and Calculus of Variations, 16(3), 719–743.
• Nazarov, S. A., Sokołowski, J. and Specovius-Neugebauer, M. (2010) Polarization matrices in anisotropic heterogeneous elasticity. Asymptotic Analysis, 68(4), 189–21.
• Novotny A. A. and Soko lowski, J. (2013) Topological Derivatives in Shape Optimization. Interaction of Mechanics and Mathematics. Springer-Verlag, Berlin, Heidelberg.
• Rice, J. R. (1968) A path independent integral and the approximate analysis of strain concentration by notches and cracks. Journal of Applied Mechanics, 35, 379–386.
• Saliba, R., Padra, C., Venere, M., Marcelo, J., Taroco, E. And Feijóo, R. A. (2005) Adaptivity in linear elastic fracture mechanics based on shape sensitivity analysis. Computer Methods in Applied Mechanics and Engineering, 194(34-35), 3582–3606.
• Saurin, V. V. (2000) Shape design sensitivity analysis for fracture conditions. Computers and Structures, 76, 399–405.
• Sokołowski, J., Leugering, G. and Żochowski, A. (2014) Shapetopological differentiability of energy functionals for unilateral problems in domains with cracks and applications. In: R. Hoppe, ed., Optimization with PDE Constraints. Lecture Notes in Computational Science and Engineering, 101. Springer, Cham, 243–284.
• Sokołowski, J., Leugering, G. and Żochowski, A. (2016) Passive control of singularities by topological optimization: The second-order mixed shape derivatives of energy functionals for variational inequalities. In: J.B. Hiriart-Urruty, A. Korytowski, H. Maurer, and M. Szymkat, eds., Advances in Mathematical Modeling, Optimization and Optimal Control. Springer Optimization and Its Applications, 109, Springer, Cham. 65–102.
• Sokołowski, J. and Żochowski, A. (1999) On the topological derivative in shape optimization. SIAM Journal on Control and Optimization, 37(4), 1251–1272.
• Sokołowski, J. and Żochowski, A. (2005) Modelling of topological derivatives for contact problems. Numerische Mathematik, 102(1), 145–179.
• Van Goethem, N. and Novotny, A. A. (2010) Crack nucleation sensitivity analysis. Mathematical Methods in the Applied Sciences, 33(16), 1978–1994.
• Xavier, M., Novotny, A. A. and Sokołowski, J. (2018) Crack growth control based on the topological derivative of the Rice’s integral. Journal of Elasticity, 134(2), 175–191.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2021).