Full Article - Open Access.

Idioma principal


Rodrıguez, A. Campos; Pozo, L. Pérez;

Full Article:

In this paper, the meshless Finite Points Method (FPM) is used for numerical simulation of one and two-dimensional elastoplastic problems, which develop a softening response after to exceed the elastic range. In such problems, a pathological behaviour of the solutions is induced by the ill-conditioning of the partial differential equations after reaching the yield limit, producing the localization of deformations. Specifically, this work presents the implementation of a regularization technique through the enrichment of the constitutive equations using strain gradients, with a characteristic length based on a non-local plastic strain formulation in order to maintain the ellipticity of the differential equations. The strain localization phenomenon is replicated objectively in the computational simulation considering an isotropic Von Mises (J2) softening model. This localization phenomenon is induced weakening a region of the material and also developing problems with geometrical discontinuities. The FPM performs the domain’s discretization in a finite number of points in each of which the punctual collocation of the equations is performed. For these reasons the FPM applies the strong formulation, allowing the use of high-order differentiability shape functions (C2 class or higher). This features can improve the computational cost because of the same shape functions are used in order to approximate non-local strain and the displacement nodal fields. Both fields are obtained by using the iterative Newton-Raphson method. To ensure convergence of the iterative method, the algorithmic tangent operator is obtained by Perturbation Method. The theory is developed for one-dimensional geometries, being extended to two and three-dimensional problems. In order to validate the proposal, a benchmarking is developed with typical problems extracted from literature.

Full Article:

Palavras-chave: Meshless, Localization, Non-local plasticity, Strain-gradient plasticity,


DOI: 10.5151/meceng-wccm2012-19755

Referências bibliográficas
  • [1] Ba?zant Z., Lin F. Non-local yield limit degradation. International Journal for Numerical Methods in Engineering, 26:1805–1823, 1988.
  • [2] Chen J., Wu C., Belytschko T. Regularization of material instabilities by meshfree approximations with intrinsic length scales. International Journal for Numerical Methods in Engineering, 47:1303–1322, 2000.
  • [3] Comi C., Perego U. A generalized variable formulation for gradient dependent softening plasticity. International Journal for Numerical Methods in Engineering, 39:3731–3755, 1996.
  • [4] De Borst R., Mühlhaus H. Gradient-dependent plasticity: Formulation and algorithmic aspects. International Journal for Numerical Methods in Engineering, 35:521–539, 1992.
  • [5] Dunne F., Petrinic N. Introduction to Computational Plasticity. Oxford University Press Inc., New York, 200
  • [6] Etse G., Vrech S. Teor´ia constitutiva de gradientes para modelos materiales elastoplásticos. Mecánica Computacional, 20:155–162, 2001.
  • [7] Fleck N.A., Hutchinson J.W. Advances in Applied Mechanics. Academic Press, 199
  • [8] Jirásek M., Rolshoven S. Comparison of integral-integral type nonlocal plasticity models for strain-softening materials. International Journal of Engineering Science, 41:1553–1602, 2003.
  • [9] Liebe T., Menzel A., Steinmann P. Theory and numerics of geometrically non-linear gradient plasticity. International Journal of Engineering Science. 41, (2003), 1603–162
  • [10] Neto E., Peri’ c D., D. O. Computational Methods for Plasticity. Theory and Applications. Wiley, 2008.
  • [11] Oller S. Fractura Mecánica. Un enfoque global. CIMNE, 2001.
  • [12] Oller S., Mart´inez X., Barbat A., Rastellini F. Advanced composite material simulation. Ciència e tecnologia des materiais.20:1-14. 2008.
  • [13] Oñate E, Idelsohn S, Zienkiewicz O., Taylor R. A finite point methods in computational mechanics, application to convective transport and fluid flow. Int J Numer Methods Eng (1996);39:3839-66.
  • [14] Oñate E, Idelsohn S, Zienkiewicz O, Taylor R., Sacco C. A stabilized finite point method for analysis of fluid mechanics problems. Comput Methods Appl Mech Eng (1996); 139:315-46.
  • [15] Oñate E., Perazzo F., J.Miquel. A finite point method for elasticity problems. Computer and Structures, 79:2151–2163, 2001.
  • [16] Owen D., Hinton E. Finite Elements in Plasticity: Theory and Practice. Pineridge Press Limited, 1980.
  • [17] Peerlings R., De Borst R., Brekelmans W., De Vree J. Gradient enhanced damage for quasibrittle materials. International Journal for Numerical Methods in Engineering, 39:3391– 3403, 1996.
  • [18] Perazzo F. Una metodolog´ia numerica sin malla para la resolución de las ecuaciones de elasticidad mediante el m´etodo de puntos finites. Univeritat Polit´ecnica de Cataluña, Barcelona España. Tesis doctoral; 2002.
  • [19] Perazzo F., Oller S., Miquel J., Oñate E. Avances en el m´etodo de puntos finitos para la mecánica de sólidos. Revista Internacional de M´etodos Num´ericos en Ingenier´ia, 22:153–168, 2006.
  • [20] P´erez-Foguet A., Rodr´iguez-Ferran A., Huerta A. Numerical differentiation for nontrivial consistent tangent matrices: an application to the MRS-Lade Model. Department de Matem`atica Aplicada III, Universitat Politècnica de Catalunya.
  • [21] P´erez L. Simulación num´erica del comportamiento no-lineal de materiales utilizando aproximaciones elásticas y el m´etodo sin malla de puntos finitos. Universidad T´ecnica Federico Santa Mar´ia,Valparaiso, Chile. Tesis doctoral; 2008.
  • [22] P´erez L., Campos A. Regularización de localización por medio de gradientes de deformaci ón plástica no local y el m´etodo sin malla de puntos finitos. Mecánica Computacional, Vol XXX: 739-753, 2011.
  • [23] P´erez L., Chacana F., Quel´in J. Regularización de la energ´ia de fractura para el análisis de daño isotrópico mediante el m´etodo sin malla de puntos finitos. Mecánica Computacional, Vol XXX: 755-772, 2011.
  • [24] P´erez-Pozo L., Perazzo F., Angulo A. A meshless FPM model for solving nonlinear material problems with proportional loading based on deformation theory. Advances in Engineering Software, 40:1148–1154, 2009.
  • [25] P´erez-Pozo L., Perazzo F. Non-linear material behaviour analysis using meshless finite point method. In 2nd ECCOMAS Thematic Conference on Meshless Methods, 251–268. Porto, Portugal, 2007.
  • [26] Polizzotto C. Gradient elasticity and nonstandard boundary conditions. International Journal of Solids and Structures, 40:7399–7423, 2003.
  • [27] Rodr´iguez-Ferran A., Bennet T., Askes H., Tamayo-Mas E. A general framework for softening regularization based on gradient elasticity. International Journal of Solids and Structures. 48:1382-1394. 2011.
  • [28] Rodr´iguez-Ferran A., Morata I., Huerta A. A new damage model based on non-local displacements. International Journal for Numerical and Analytical Methods in Geomechanics, 29:473–493, 2005.
  • [29] Rolshoven S., Jirásek M. Nonlocal formulations of softening plasticity. Fifth World Congress on Computational Mechanics, 2002.
  • [30] Taylor R, Idelsohn S, Zienkiewics O., Oñate E. Moving least square approximations for solution of differential equations. CIMNE research report, 1995; 74.
  • [31] Vrech S., Etse G. Análisis geom´etrico de localización para plasticidad regularizada mediante teor´ia de gradientes. Mecánica Computacional, 23, 2004.
  • [32] Zienkiewicz O. Andamp; Taylor R. El m´etodo de los elementos finitos, vol. 1. Centro internacional de m´etodos num´ericos en ingenier´ia, Barcelona España; 2000.
Como citar:

Rodrıguez, A. Campos; Pozo, L. Pérez; "APPLICATIONS OF THE FINITE POINTS METHOD IN THE STRAIN-GRADIENT PLASTICITY", p. 4176-4192 . In: In Proceedings of the 10th World Congress on Computational Mechanics [= Blucher Mechanical Engineering Proceedings, v. 1, n. 1]. São Paulo: Blucher, 2014.
ISSN 2358-0828, DOI 10.5151/meceng-wccm2012-19755

últimos 30 dias | último ano | desde a publicação