×

An algorithm based on incompatible modes for the global tracking of strong discontinuities in shear localization analyses. (English) Zbl 1439.74382

Summary: Numerical methods for predicting localized shear failure in elasto-plastic solids have experienced considerable advancements in the last decades. Among these approaches, the so-called “Embedded Strong Discontinuity (ESD)” method is often successfully used to accurately simulate the post-localization response with negligible dependence on the finite element discretization. However, it was observed that the employed discontinuity tracking strategy plays a crucial role in the successful localization analysis. In this contribution, we propose a novel strategy for the global tracking of discontinuity surfaces. It is based on exploiting information obtained from the enhanced parameters employed in Enhanced Assumed Strain (EAS) formulations. It is well known, that enhanced strain element formulations are able to better capture localized shear deformations as compared to standard finite elements. This can be explained as a consequence of the improved performance in bending. We observed, that the approximation of the strain jumps delimiting the shear band is connected with a deformation field characterized by opposite bending curvatures across these two discontinuities. Hence, in view of the relations existing between the kinematics of strong and weak discontinuities, we formulate a proper scalar function of the enhanced parameters to identify potential strong discontinuity surfaces, which are evaluated in each step of the analysis with negligible computational cost. This proposed approach has a global character, as it is based upon evaluating discontinuity surfaces defined in the complete analysis domain that are, by construction, continuous across elements. We demonstrate that the tracking algorithm correctly identifies the potential strong discontinuity surface already in early loading stages, even before a localization condition is fulfilled. In those elements which are crossed by the potential failure surface and which also satisfy the localization condition, the kinematics of embedded strong discontinuities is activated to capture the shear failure surface. The performance of the new tracking algorithm is demonstrated by means of several numerical shear localization analyses using associative and non-associative Drucker-Prager elastoplastic models to simulate 2-D and 3-D benchmarkanalyses.

MSC:

74S05 Finite element methods applied to problems in solid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
74C05 Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials)

Software:

COM-MAT-FAIL
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Simo, J.; Oliver, J.; Armero, F., An analysis of strong discontinuities induced by strain-softening in rate-independent inelastic solids, Comput. Mech., 12, 5, 277-296 (1993) · Zbl 0783.73024
[2] Oliver, J.; Huespe, A.; Pulido, M.; Chavez, E., From continuum mechanics to fracture mechanics: The strong discontinuity approach, Eng. Fract. Mech., 69, 113-136 (2002)
[3] Sulem, J.; Vardoulakis, I., Bifurcation Analysis in Geomechanics (2004), CRC Press: CRC Press London
[4] Bažant, Z.; Jirásek, M., Nonlocal integral formulations of plasticity and damage: survey of progress, J. Eng. Mech., 128, 1119-1149 (2002)
[5] Jirásek, M.; Bažant, Z., Inelastic Analysis of Structures (2002), John Wiley & Sons
[6] de Borst, R.; Mühlhaus, H., Gradient-dependent plasticity: Formulation and algorithmic aspects, Internat. J. Numer. Methods Engrg., 35, 521-539 (1992) · Zbl 0768.73019
[7] Mang, H.; Meschke, G.; Lackner, R.; Mosler, J., Computational modelling of concrete structures, (Milne, I.; Ritchie, R.; Karihaloo, B., Comprehensive Structural Integrity, Vol. 3 (2003), Elsevier), 1-67
[8] Bourdin, B.; Francfort, G. A.; Marigo, J.-J., The variational approach to fracture, J. Elasticity, 91, 1-3, 5-148 (2008) · Zbl 1176.74018
[9] Miehe, C.; Welschinger, F.; Hofacker, M., Thermodynamically consistent phase-field models of fracture: Variational principles and multi-field FE implementations, Internat. J. Numer. Methods Engrg., 83, 10, 1273-1311 (2010) · Zbl 1202.74014
[10] McAuliffe, C.; Waisman, H., A unified model for metal failure capturing shear banding and fracture, Int. J. Plast., 65, 131-151 (2015)
[11] Pandolfi, A.; Conti, S.; Ortiz, M., A recursive-faulting model of distributed damage in confined brittle materials, J. Mech. Phys. Solids, 54, 9, 1972-2003 (2006) · Zbl 1120.74735
[12] Dvorkin, E.; Cuitino, A.; Gioia, G., Finite elements with displacement interpolated embedded localization lines insensitive to mesh size and mesh distortions, Internat. J. Numer. Methods Engrg., 30, 541-564 (1990) · Zbl 0729.73209
[13] Simo, J. C.; Rifai, M. S., A class of mixed assumed strain methods and the method of incompatible modes, Internat. J. Numer. Methods Engrg., 29, 1595-1638 (1990) · Zbl 0724.73222
[14] F. Armero, K. Garikipati, (1995) Recent advances in the analyses and numerical simulation of strain localization in inelastic solids, in: Owen, D., Oñate, E., Hinton, E. (Eds.), Computational Plasticity, Proceedings of 4th Int. Conf., Barcelona, Spain, 1995, pp. 547-561.; F. Armero, K. Garikipati, (1995) Recent advances in the analyses and numerical simulation of strain localization in inelastic solids, in: Owen, D., Oñate, E., Hinton, E. (Eds.), Computational Plasticity, Proceedings of 4th Int. Conf., Barcelona, Spain, 1995, pp. 547-561.
[15] Oliver, J., Modelling strong discontinuities in solid mechanics via strain softening constitutive equations. Part 1: Fundamentals, Part 2: Numerical simulation, Internat. J. Numer. Methods Engrg., 39, 3575-3623 (1996) · Zbl 0888.73018
[16] Mosler, J.; Meschke, G., 3D modeling of strong discontinuities in elastoplastic solids: Fixed and rotating localization formulations, Internat. J. Numer. Methods Engrg., 57, 1553-1576 (2003) · Zbl 1062.74623
[17] Oliver, J.; Huespe, A., Continuum approach to material failure in strong discontinuity settings, Comput. Methods Appl. Mech. Engrg., 193, 3195-3220 (2004) · Zbl 1060.74507
[18] Borja, R. I.; Regueiro, R. A., Strain localization in frictional materials exhibiting displacement jumps, Comput. Methods Appl. Mech. Engrg., 190, 20, 2555-2580 (2001) · Zbl 0997.74009
[19] Jirásek, M., Comparative study on finite elements with embedded discontinuities, Comput. Methods Appl. Mech. Engrg., 188, 307-330 (2000) · Zbl 1166.74427
[20] Cazes, F.; Meschke, G.; Zhou, M., Strong discontinuity approaches: An algorithm for robust performance and comparative assessment of accuracy, Int. J. Solids Struct., 96, 355-379 (2016)
[21] Armero, F.; Callari, C., An analysis of strong discontinuities in a saturated poro-plastic solid, Internat. J. Numer. Methods Engrg., 46, 10, 1673-1698 (1999) · Zbl 0971.74029
[22] Callari, C.; Armero, F., Finite element methods for the analysis of strong discontinuities in coupled poro-plastic media, Comput. Methods Appl. Mech. Engrg., 191, 4371-4400 (2002) · Zbl 1124.74324
[23] Callari, C.; Armero, F., Analysis and numerical simulation of strong discontinuities in finite strain poroplasticity, Comput. Methods Appl. Mech. Engrg., 193, 27-29, 2941-2986 (2004) · Zbl 1067.74519
[24] Callari, C.; Armero, F.; Abati, A., Strong discontinuities in partially saturated poroplastic solids, Comput. Methods Appl. Mech. Engrg., 199, 23, 1513-1535 (2010) · Zbl 1231.74105
[25] Oliver, J.; Huespe, A.; Sánchez, P., A comparative study on finite elements for capturing strong discontinuities: E-FEM vs X-FEM, Comput. Methods Appl. Mech. Engrg., 195, 4732-4752 (2006) · Zbl 1144.74043
[26] Babuška, I.; Melenk, J., The partition of unity method, Internat. J. Numer. Methods Engrg., 40, 727-758 (1997) · Zbl 0949.65117
[27] Belytschko, T.; Moës, N.; Usui, S.; Parimi, C., Arbitrary discontinuities in finite elements, Internat. J. Numer. Methods Engrg., 50, 993-1013 (2001) · Zbl 0981.74062
[28] Wells, G.; Sluys, L., A new method for modelling cohesive cracks using finite elements, Internat. J. Numer. Methods Engrg., 50, 2667-2682 (2001) · Zbl 1013.74074
[29] Areias, P.; Belytschko, T., Two-scale shear band evolution by local partition of unity, Internat. J. Numer. Methods Fluids, 66, 878-910 (2006) · Zbl 1110.74841
[30] Alfaiate, J.; Wells, G.; Sluys, L., On the use of embedded discontinuity elements with crack path continuity for mode-I and mixed-mode fracture, Eng. Fract. Mech., 69, 661-686 (2002)
[31] Callari, C., Coupled numerical analysis of strain localization induced by shallow tunnels in saturated soils, Comput. Geotech., 31, 3, 193-207 (2004)
[32] Stolarska, M.; Chopp, D.; Moes, N.; Belytschko, T., Modelling crack growth by level sets in the extended finite element method, Internat. J. Numer. Methods Engrg., 51, 943-960 (2000) · Zbl 1022.74049
[33] Gravouil, A.; Moës, N.; Belytschko, T., Non-planar 3D crack growth by the extended finite element and level sets - Part II: Level set update, Internat. J. Numer. Methods Engrg., 53, 2569-2586 (2002) · Zbl 1169.74621
[34] J. Oliver, A. Huespe, On strategies for tracking strong discontinuities in computational failure mechanics, in :Online Proceedings of the Fifth World Congress on Computational Mechanics (WCCM V), 2002.; J. Oliver, A. Huespe, On strategies for tracking strong discontinuities in computational failure mechanics, in :Online Proceedings of the Fifth World Congress on Computational Mechanics (WCCM V), 2002.
[35] Oliver, J.; Huespe, A.; Samaniego, E.; Chaves, E., Continuum approach to the numerical simulation of material failure in concrete, Int. J. Numer. Anal. Methods Geomech., 28, 609-632 (2004) · Zbl 1112.74493
[36] Oliver, J.; Dias, I.; Huespe, A. E., Crack-path field and strain-injection techniques in computational modeling of propagating material failure, Comput. Methods Appl. Mech. Engrg., 274, 289-348 (2014) · Zbl 1296.74102
[37] Taylor, R. L.; Beresford, P. J.; Wilson, E. L., A non-conforming element for stress analysis, Internat. J. Numer. Methods Engrg., 10, 6, 1211-1219 (1976) · Zbl 0338.73041
[38] Ortiz, M.; Leroy, Y.; Needleman, A., A finite element method for localized failure analysis, Comput. Methods Appl. Mech. Engrg., 61, 189-214 (1987) · Zbl 0597.73105
[39] Steinmann, P.; Willam, K., Performance of enhanced finite element formulations in localized failure computations, Comput. Methods Appl. Mech. Engrg., 90, 1-3, 845-867 (1991)
[40] Steinmann, P.; Willam, K., Finite elements for capturing localized failure, Arch. Appl. Mech., 61, 259-275 (1991) · Zbl 0729.73904
[41] Callari, C.; Lupoi, A., Localization analysis in dilatant elasto-plastic solids by a strong-discontinuity method, (Novel Approaches in Civil Engineering (2004), Springer), 121-132 · Zbl 1130.74350
[42] Oliver, J.; Cervera, M.; Manzoli, O., Strong discontinuities and continuum plasticity models: the strong discontinuity approach, Int. J. Plast., 15, 319-351 (1999) · Zbl 1057.74512
[43] Wilson, E. L.; Taylor, R. L.; Doherty, W. P.; Ghaboussi, J., Incompatible displacement models, Numer. Comput. Methods Struct. Mech., 43 (1973)
[44] Wriggers, P.; Korelc, J., On enhanced strain methods for small and finite deformations of solids, Comput. Mech., 18, 6, 413-428 (1996) · Zbl 0894.73179
[45] LI, X.; Crook, A.; Lyons, L., Mixed strain elements for non-linear analysis, Eng. Comput., 10, 3, 223-242 (1993)
[46] Andelfinger, U.; Ramm, E., EAS-elements for two-dimensional, three-dimensional, plate and shell structures and their equivalence to HR-elements, Internat. J. Numer. Methods Engrg., 36, 8, 1311-1337 (1993) · Zbl 0772.73071
[47] Wriggers, P.; Reese, S., A note on enhanced strain methods for large deformations, Comput. Methods Appl. Mech. Engrg., 135, 201-209 (1996) · Zbl 0893.73072
[48] Korelc, J.; Wriggers, P., Improved enhanced strain four-node element with Taylor expansion of the shape functions, Internat. J. Numer. Methods Engrg., 40, 3, 407-421 (1997)
[49] Reddy, B.; Simo, J., Stability and convergence of a class of enhanced strain methods, SIAM J. Numer. Anal., 32, 6, 1705-1728 (1995) · Zbl 0855.73073
[50] Oliver, J., Continuum modelling of strong discontinuities in solid mechanics using damage models, Comput. Mech., 17, 49-61 (1995) · Zbl 0840.73051
[51] Gajo, A.; Wood, D. M.; Bigoni, D., On certain critical material and testing characteristics affecting shear band development in sand, Geotechnique, 57, 5, 449-461 (2007)
[52] Roscoe, K. H., The influence of strains in soil mechanics, Geotechnique, 20, 2, 129-170 (1970)
[53] Atkinson, J., An Introduction to the Mechanics of Soils and Foundations: Through Critical State Soil Mechanics (1993), McGraw-Hill Book Company (UK) Ltd
[54] Arthur, J. R.F.; Dunstan, T.; Al-Ani, Q. A.J.; Assadi, A., Plastic deformation and failure of granular media, Geotechnique, 27, 53-74 (1977)
[55] Vardoulakis, I., Shear band inclination and shear modulus of sand in biaxial tests, Int. J. Numer. Anal. Methods Geomech., 4, 103-119 (1980) · Zbl 0443.73092
[56] Linder, C.; Armero, F., Finite elements with embedded strong discontinuities for the modeling of failure in solids, Internat. J. Numer. Methods Engrg., 72, 1391-1433 (2007) · Zbl 1194.74431
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.