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MODIFIED LAGRANGE FUNCTIONAL FOR SOLVING ELASTIC PROBLEM WITH A CRACK IN CONTINUUM MECHANICS

  • Namm, Robert V. (Computing Center of Far Eastern Branch Russian Academy of Sciences) ;
  • Tsoy, Georgiy I. (Computing Center of Far Eastern Branch Russian Academy of Sciences) ;
  • Woo, Gyungsoo (Department of Mathematics Changwon National University)
  • Received : 2018.08.15
  • Accepted : 2018.09.17
  • Published : 2019.10.31
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Abstract

Modified Lagrange functional for solving an elastic problem with a crack is considered. Two formulations of a crack problem are investigated. The first formulation concerns a problem where a crack extending to the outer boundary of the domain. In the second formulation, we consider a problem with an internal crack. Duality ratio is established for initial and dual problem in both cases.

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References

  1. I. Hlavachek, J. Haslinger, J. Necas, and J. Lovishek, Solving Variational Inequalities in Mechanics, translated from the Slovak by Yu. A. Kuznetsov and A. V. Lapin, translation edited and with a preface by N. S. Bakhvalov, "Mir", Moscow, 1986.
  2. A. M. Khludnev, Problems of Elasticity Theory in Nonsmooth Domains, Fizmatlit, Moscow, 2010.
  3. V. A. Kovtunenko, Numerical simulation of the non-linear crack problem with non-penetration, Math. Methods Appl. Sci. 27 (2004), no. 2, 163-179. https://doi.org/10.1002/mma.449
  4. A. Kufner and S. Fuchik, Nonlinear Dierential Equations, translated from the English by A. F. Zhukov, translation edited and with a foreword by S. I. Pokhozhaev, "Nauka", Moscow, 1988.
  5. O. L. Mangasarian, A generalized Newton method for absolute value equations, Optim. Lett. 3 (2009), no. 1, 101-108. https://doi.org/10.1007/s11590-008-0094-5
  6. N. F. Morozov, Mathematical Problems of Crack Theory, "Nauka", Moscow, 1984.
  7. R. V. Namm and G. I. Tsoy, A modied dual scheme for solving an elastic crack problem, Numer. Anal. Appl. 10 (2017), no. 1, 37-46; translated from Sib. Zh. Vychisl. Mat. 20 (2017), no. 1, 47-58. https://doi.org/10.15372/SJNM20170105
  8. E. M. Rudoy, Domain decomposition method for a model crack problem with a possible contact of crack edges, Comput. Math. Math. Phys. 55 (2015), no. 2, 305-316. https://doi.org/10.1134/S0965542515020165
  9. E. M. Rudoy, Numerical solution of an equilibrium problem for an elastic body with a thin delaminated rigid inclusion, J. Appl. Ind. Math. 10 (2016), no. 2, 264-276; translated from Sib. Zh. Ind. Mat. 19 (2016), no. 2, 74-87. https://doi.org/10.17377/sibjim.2016.19.207
  10. F. P. Vasiliev, Methods for Solving Extremal Problems, "Nauka", Moscow, 1981.
  11. E. M. Vikhtenko and R. V. Namm, A duality scheme for solving the semi-coercive Signorini problem with friction, Comput. Math. Math. Phys. 47 (2007), no. 12, 1938-1951; translated from Zh. Vychisl. Mat. Mat. Fiz. 47 (2007), no. 12, 2023-2036. https://doi.org/10.1134/S0965542507120068
  12. E. M. Vikhtenko, G. Woo, and R. V. Namm, Sensitivity functionals in contact problems of elasticity theory, Comput. Math. Math. Phys. 54 (2014), no. 7, 1190-1200. https://doi.org/10.1134/S0965542514070112
  13. E. M. Vikhtenko, G. Woo, and R. V. Namm, The methods for solving semicoercive variational inequalities of mechanics on the basis of modified Lagrangian functionals, Dal'nevost. Mat. Zh. 14 (2014), no. 1, 6-17.
  14. E. M. Vikhtenko, G. Woo, and R. V. Namm, A modified duality scheme for finite- and infinite-dimensional convex programming problems, Dal'nevost. Mat. Zh. 17 (2017), no. 2, 158-169.