Geodesic
Legendre spectral element for plastic localization problems at large scale strains
Čebyševskij sbornik, Tome 21 (2020) no. 3, pp. 306-316

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In paper the method of spectral elements based on the Legendre polynomial for time-independent elastic-plastic plane problems at large strains is proposed. The method of spectral elements is based on the variational principle (Galerkin's method). The solution of these problems has the phenomenon of localization of plastic deformations in narrow areas called slip-line or shear band. The possibility of using a spectral element for the numerical solution of these problems with discontinuous solutions is investigated. The yield condition of the material is the von Mises criterion. The stresses are integrated by the radial return method by backward implicit Euler scheme. The system of nonlinear algebraic equations is solved by the Newton's iterative method. A numerical solution is given of an example of stretching a strip weakened by cuts with a circular base in a plane stress and plane deformed state. Kinematic fields and limit load are obtained. Comparisons of numerical results with the analytical solution obtained for incompressible media constructed by the method of characteristics are presented.
Keywords: spectral method, localization phenomenon, plasticity, slip-line, finite strains, iterative Newton's method. 
@article{CHEB_2020_21_3_a24,
     author = {V. A. Levin and K. M. Zingerman and K. Yu. Krapivin and M. Ya. Yakovlev},
     title = {Legendre spectral element for plastic localization problems at large scale strains},
     journal = {\v{C}eby\v{s}evskij sbornik},
     pages = {306--316},
     publisher = {mathdoc},
     volume = {21},
     number = {3},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHEB_2020_21_3_a24/}
}
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V. A. Levin; K. M. Zingerman; K. Yu. Krapivin; M. Ya. Yakovlev. Legendre spectral element for plastic localization problems at large scale strains. Čebyševskij sbornik, Tome 21 (2020) no. 3, pp. 306-316. http://geodesic.mathdoc.fr/item/CHEB_2020_21_3_a24/