Length Scales Interaction in Nonlocal Plastic Strain Localization of Bars of Varying Section
Publication: Journal of Engineering Mechanics
Volume 136, Issue 8
Abstract
In computational mechanics, strain softening generates ill-posed boundary value problems, which cannot be solved without being regularized, e.g., through the introduction of an internal length scale. This paper investigates how the internal length scale that is introduced by regularization may interact with the external length scale arising from boundary conditions in the particular case of a strain-softening bar of varying cross section and a nonlocal averaging regularization. The interaction of internal and external length scales is examined using an analytical closed-form solution for overnonlocal softening plasticity that derives from a Fredholm equation of the second kind. In the absence of external length (bars of constant section), the analysis shows that the overnonlocal averaging confines and smoothly distributes plastic strain into a localized band. The localization width, plastic strain distribution inside the band, and load-displacement response are controlled by the internal length of the averaging function and the overnonlocal weighting factor. In the presence of external length (bar of varying section), the analytical solution shows that the localization width is controlled by the interaction of external and internal length scales. This interaction is significant when the external and internal lengths are of comparable magnitude, and decreases when the external length becomes large compared to the internal length. The bandwidth is found to depend on the internal and external lengths and stress level while strain localizes, and to relate only to the internal length when the bar collapses.
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Acknowledgments
This work was financially supported by National Science Foundation of China (NSFC) (through Grant Nos. UNSPECIFIED50908171 and UNSPECIFIED50825803). These supports are gratefully acknowledged. We are grateful to the anonymous reviewers for their helpful comments and suggestions.
References
Abu Al-Rub, R. K., and Voyiadjis, G. Z. (2005). “A direct finite element implementation of the gradient-dependent theory.” Int. J. Numer. Methods Eng., 63(4), 603–629.
Aifantis, E. C. (1992). “On the role of gradients in the localization of deformation and fracture.” Int. J. Eng. Sci., 30(10), 1279–1299.
Alsaleh, M. I., Voyiadjis, G. Z., and Alshibli, K. A. (2006). “Modelling strain localization in granular materials using micropolar theory: Mathematical formulations.” Int. J. Numer. Analyt. Meth. Geomech., 30(15), 1501–1524.
Alshibli, K. A., Alsaleh, M. I., and Voyiadjis, G. Z. (2006). “Modelling strain localization in granular materials using micropolar theory: Numerical implementation and verification.” Int. J. Numer. Analyt. Meth. Geomech., 30(15), 1525–1544.
Bardet, J. P., and Proubet, J. (1991). “A numerical investigation of the structure of persistent shear bands in granular media.” Geotechnique, 41(4), 599–613.
Bažant, Z. P. (1976). “Instability, ductility, and size effect in strain softening concrete.” J. Eng. Mech., 102(2), 331–344.
Bažant, Z. P., Belytschko, T., and Chang, T. P. (1984). “Continuum theory for strain softening.” J. Eng. Mech., 110(12), 1666–1692.
Bažant, Z. P., and Jirásek, M. (2002). “Nonlocal integral formulations of plasticity and damage: Survey of progress.” J. Eng. Mech., 128(11), 1119–1149.
Bažant, Z. P., and Lin, F. B. (1988). “Nonlocal yield-limit degradation.” Int. J. Numer. Methods Eng., 26(8), 1805–1823.
Belytschko, T., Chiang, H. -Y., and Plaskacz, E. (1994). “High resolution two-dimensional shear band computations: Imperfections and mesh dependence.” Comput. Methods Appl. Mech. Eng., 119(1–2), 1–15.
Borja, R. I. (2000). “A finite element model for strain localization analysis of strongly discontinuous fields based on standard Galerkin approximation.” Comput. Methods Appl. Mech. Eng., 190(11–12), 1529–1549.
Comi, C. (2001). “A non-local model with tension and compression damage mechanisms.” Eur. J. Mech. A/Solids, 20(1), 1–22.
de Borst, R. (2001). “Some recent issues in computational failure mechanics.” Int. J. Numer. Methods Eng., 52(1–2), 63–95.
Jackiewicz, J., and Kuna, M. (2003). “Non-local regularization for FE simulation of damage in ductile materials.” Comput. Mater. Sci., 28(3–4), 684–695.
Jirásek, M. (1998). “Nonlocal models for damage and fracture: Comparison of approaches.” Int. J. Solids Struct., 35(31–32), 4133–4145.
Jirásek, M., and Rolshoven, S. (2003). “Comparison of integral-type nonlocal plasticity models for strain-softening materials.” Int. J. Eng. Sci., 41(13–14), 1553–1602.
Lasry, D., and Belytschko, T. (1988). “Localization limiters in transient problems.” Int. J. Solids Struct., 24(6), 581–597.
Lu, X. L., Bardet, J. P., and Huang, M. S. (2009). “Numerical solutions of strain localization with nonlocal softening plasticity.” Comput. Methods Appl. Mech. Eng., 198(47–48), 3702–3711.
Luzio, G. D., and Bažant, Z. P. (2005). “Spectral analysis of localization in nonlocal and over-nonlocal materials with softening plasticity or damage.” Int. J. Solids Struct., 42(23), 6071–6100.
Needleman, A. (1988). “Material rate dependence and mesh sensitivity in localization problems.” Comput. Methods Appl. Mech. Eng., 67(1), 69–85.
Pietruszczak, S., and Niu, X. (1993). “On the description of localized deformation.” Int. J. Numer. Analyt. Meth. Geomech., 17(11), 791–805.
Pijaudier-Cabot, G., and Bažant, Z. P. (1987). “Nonlocal damage theory.” J. Eng. Mech., 113(10), 1512–1533.
Polyanim, A. D., and Manzhirov, A. V. (2008). Handbook of integral equations, CRC, Boca Raton, Fla.
Strömberg, L., and Ristinmaa, M. (1996). “FE-formulation of a nonlocal plasticity theory.” Comput. Methods Appl. Mech. Eng., 136(1–2), 127–144.
Vardoulakis, I., and Aifantis, E. C. (1991). “A gradient flow theory of plasticity for granular materials.” Acta Mech., 87(3–4), 197–217.
Vermeer, P. A., and Brinkgreve, R. B. J. (1994). “A new effective nonlocal strain measure for softening plasticity.” Localization and bifurcation theory for soil and rocks, Balkema, Rotterdam, The Netherlands, 89–100.
Wells, G. N., Sluys, L. J., and de Borst, R. (2002). “Simulating the propagation of displacement discontinuities in a regularized strain-softening medium.” Int. J. Numer. Methods Eng., 53(5), 1235–1256.
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© 2010 ASCE.
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Received: Mar 4, 2009
Accepted: Feb 3, 2010
Published online: Feb 8, 2010
Published in print: Aug 2010
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