Discontinuous modelling of shear bands using adaptive meshfree methods. (English) Zbl 1169.74655

Summary: A simple methodology to model shear bands as strong displacement discontinuities in an adaptive meshfree method is presented. The shear band is represented by a displacement jump at discrete particle positions. The displacement jump in normal direction is suppressed with penalty method. Loss of material stability is used as transition criterion from continuum to discontinuum. The method is two- and three-dimensional. Examples of complicated shear banding including transition from brittle-to-ductile failure are studied and compared to experimental data and other examples from the literature.


74S30 Other numerical methods in solid mechanics (MSC2010)
74R20 Anelastic fracture and damage
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[3] Areias, P. M.A.; Cesar de Sa, J. M.A.; Conceicao Antonio, C. A.; Carneiro, J. A.S. A.O.; Teixeira, V. M.P., Strong displacement discontinuity and lagrange multipliers in the analysis of finite displacement fracture problems, Comput. Mech., 35, 1, 54-71 (2004) · Zbl 1109.74350
[4] Armero, F., On the characterization of localized solutions in inelastic solids: an analysis of wave propagation in a softening bar, Comput. Methods Appl. Mech. Engrg., 191, 181-213 (2001) · Zbl 1037.74027
[5] Armero, F.; Garikipati, K., An analysis of strong discontinuities in multiplicative finite strain plasticity and their relation with the numerical simulation of strain localization in solids, Int. J. Solids Struct., 33, 20-22, 2863-2885 (1996) · Zbl 0924.73084
[6] Batra, R. C.; Gummalla, R. R., Effect on material and geometric parameters on deformations near the notch-tip of a dynamically loaded prenotched plate, Int. J. Fracture, 101, 99-140 (2000)
[7] Batra, R. C.; Jaber, N. A., Failure mode transition in an impact loaded pre-notched plate with four thermoviscoplastic relations, Int. J. Fracture, 110, 47-71 (2001)
[8] Batra, R. C.; Jaber, N. A.; Malsbury, M. E., Analysis of failure modes in an impact loaded thermoviscoplastic prenotched plate, Int. J. Plasticity, 19, 139-196 (2003) · Zbl 1032.74647
[9] Batra, R. C.; Ravisankar, M. V.S., Three-dimensional numerical simulation of the kalthoff experiment, Int. J. Fracture, 105, 161-186 (2000)
[10] Bazant, Z. P.; Belytschko, T., Wave propagation in a strain-softening bar: Exact solution, J. Engrg. Mech. - ASCE, 111, 3, 381-389 (1985)
[11] Belytschko, T.; Bindemann, L. P., Assumed strain stabilization of the 4-node quadrilateral with 1-point quadrature for nonlinear problems, Comput. Methods Appl. Mech. Engrg., 88, 311-340 (1991) · Zbl 0742.73019
[12] Belytschko, T.; Black, T., Elastic crack growth in finite elements with minimal remeshing, Int. J. Numer. Methods Engrg., 45, 5, 601-620 (1999) · Zbl 0943.74061
[13] Belytschko, T.; Chen, H.; Xu, J.; Zi, G., Dynamic crack propagation based on loss of hyperbolicity and a new discontinuous enrichment, Int. J. Numer. Methods Engrg., 58, 12, 1873-1905 (2003) · Zbl 1032.74662
[14] Belytschko, T.; Chiang, H-Y; Plaskacz, E., High resolution two-dimensional shear band computations: imperfections and mesh dependence, Comput. Methods Appl. Mech. Engrg., 119, 1-15 (1994) · Zbl 0849.73064
[15] Belytschko, T.; Fish, J.; Englemann, B., A finite element method with embedded localization zones, Comput. Methods Appl. Mech. Engrg., 70, 59-89 (1988) · Zbl 0653.73032
[16] Belytschko, T.; Liu, W. K.; Moran, B., Nonlinear Finite Elements for Continua and Structures (2000), Wiley: Wiley Chichester · Zbl 0959.74001
[17] Belytschko, T.; Lu, Y. Y., Element-free Galerkin methods for static and dynamic fracture, Int. J. Solids Struct., 32, 2547-2570 (1995) · Zbl 0918.73268
[18] Belytschko, T.; Lu, Y. Y.; Gu, L., Element-free Galerkin methods, Int. J. Numer. Methods Engrg., 37, 229-256 (1994) · Zbl 0796.73077
[19] Belytschko, T.; Ong, J. S-J., Hourglass control in linear and nonlinear problems, Comput. Methods Appl. Mech. Engrg., 43, 251-276 (1984) · Zbl 0522.73063
[20] Belytschko, T.; Xiao, S. P., Stability analysis of particle methods with corrected derivatives, Comput. Math. Appl., 43, 329-350 (2002) · Zbl 1073.76619
[21] Bonet, J.; Bhargava, P., A uniform deformation gradient hexadron element with artificial hourglass control, Int. J. Numer. Methods Engrg., 38, 2809-2828 (1995) · Zbl 0835.73069
[22] Bruhns, O. T.; Xiao, H.; Meyers, A., Constitutive inequalities for an isotropic elastic strain-energy function based on Hencky’s logarithmic strain tensor, Proc. Royal Soc. A, London, 457, 2207-2226 (2001) · Zbl 1048.74505
[23] Camacho, G. T.; Ortiz, M., Computational modeling of impact damage in brittle materials, Int. J. Solids Struct., 33, 2899-2938 (1996) · Zbl 0929.74101
[24] Cesar de Sa, J. M.A.; Areias, P. M.A.; Jorge, R. M.N., Quadrilateral elements for the solution of elasto-plastic finite strain problems, Int. J. Numer. Methods Engrg., 51, 883-917 (2001) · Zbl 1023.74045
[25] Chen, J. S.; Yoon, S.; Wang, H.; Liu, W. K., An improved reproducing kernel method for nearly incompressible finite elasticity, Comput. Methods Appl. Mech. Engrg., 181, 1-3, 117-145 (2000) · Zbl 0973.74088
[26] Clifton, R. J.; Duffy, J.; Hartley, K. A., On critical conditions for shear band formation at high strain rates, Scripta Metall., 18, 5, 443-448 (1984)
[27] Diez, P.; Arroyo, M.; Huerta, A., Adaptivity based on error estimation for viscoplastic softening materials, Mech. Cohesive-Frictional Mater., 5, 87-112 (2000)
[28] Dolbow, J.; Belytschko, T., Volumetric locking in the element free Galerkin method, Int. J. Numer. Methods Engrg., 46, 925-942 (1999) · Zbl 0967.74079
[29] Dolbow, J.; Moes, N.; Belytschko, T., Discontinuous enrichment in finite elements with a partition of unity method, Finite Elem. Anal. Des., 36, 3, 235-260 (2000) · Zbl 0981.74057
[30] Evans, L.; Gariepy, R., Measure Theory and Fine Properties of Functions (1992), CRC Press: CRC Press New York · Zbl 0804.28001
[31] Ewing, D. J.F.; Hill, R., The plastic constraint of v-notched tension bars, J. Mech. Phys. Solids (1967)
[32] Falk, M. L.; Needleman, A.; Rice, J. R., A critical evaluation of cohesive zone models of dynamic fracture, J. Phys. IV, 11, PR5, 43-50 (2001)
[33] Flory, R. J., Thermodynamic relations for highly elastic materials, Trans. Faraday Soc., 57, 829-838 (1961)
[34] Gasser, T. C.; Holzapfel, G. A., Modeling 3d crack propagation in unreinforced concrete using pufem, Comput. Methods Appl. Mech. Engrg. (2005) · Zbl 1176.74180
[35] Grady, D. E., Dissipation in adiabatic shear bands, Mech. Mater., 17, 289-293 (1994)
[36] Gravouil, A.; Moes, N.; Belytschko, T., Non-planar 3D crack growth by the extended finite element and level sets – part ii: level set update, Int. J. Numer. Methods Engrg., 53, 2569-2586 (2002) · Zbl 1169.74621
[38] Hao, S.; Comec, A.; Schwalbe, K. H., Plastic stress-strain fields and limit loads of a plane strain cracked tensile panel with a mismatched welded joint, Int. J. Solids Struct., 34, 297-311 (1997) · Zbl 0946.74562
[39] Hao, S.; Liu, W. K.; Chang, C. T., Computer implementation of damage models by finite element and meshfree methods, Comput. Methods Appl. Mech. Engrg., 187, 3-4, 401-440 (2000) · Zbl 0980.74063
[40] Hao, S.; Moran, B.; Liu, W. K.; Olson, B., A hierarchical multi-physics model for design of high toughness steels, J. Comput.-Aided Mater. Des., 10, 2, 99-142 (2003)
[41] Huerta, A.; Fernandez-Mendez, S., Locking in the incompressible limit for the element-free Galerkin method, Int. J. Numer. Methods Engrg., 51, 1361-1383 (2001) · Zbl 1065.74635
[42] Huerta, A.; Vidal, Y.; Vilon, P., Pseudo-divergence-free element free Galerkin method for incompressible fluid flow, Comput. Methods Appl. Mech. Engrg., 193, 12-14, 1119-1136 (2004) · Zbl 1060.76626
[43] Hughes, T. J.R., Generalization of selective integration procedures to anisotropic and nonlinear media, Int. J. Numer. Methods Engrg., 15, 9, 1413-1418 (1980) · Zbl 0437.73053
[44] Jirasek, M., Comparative study on finite elements with embedded discontinuities, Comput. Methods Appl. Mech. Engrg., 188, 307-330 (2000) · Zbl 1166.74427
[46] Kalthoff, J. F.; Winkler, S., Failure mode transition at high rates of shear loading, Int. Conf. Impact Loading Dynam. Behav. Mater., 1, 185-195 (1987)
[47] Kalthoff, J. F., Modes of dynamic shear failure in solids, Int. J. Fracture, 101, 1-31 (2000)
[48] Krysl, P.; Belytschko, T., The element free Galerkin method for dynamic propagation of arbitrary 3-D cracks, Int. J. Numer. Methods Engrg., 44, 6, 767-800 (1999) · Zbl 0953.74078
[49] Lasry, D.; Belytschko, T., Localization limiters in transient problems, Int. J. Solids Struct., 24, 6, 581-597 (1988) · Zbl 0636.73021
[50] Lee, Y. W.; Woertz, J. C.; Wierzbicki, T., Fracture prediction of thin plates under hemi-spherical punch with calibration and experimental verification, Int. J. Mech. Sci. (2004)
[51] Lemonds, J.; Needleman, A., Finite element analysis of shear localization in rate and temperature dependent solids, Mech. Mater., 5, 339-361 (1986)
[52] Li, S.; Hao, W.; Liu, W. K., Mesh-free simulations of shear banding in large deformation, Int. J. Solids Struct., 37, 7185-7206 (2000) · Zbl 0995.74082
[53] Li, S.; Liu, W. K.; Rosakis, A. J.; Belytschko, T.; Hao, W., Mesh free Galerkin simulations of dynamic shear band propagation and failure mode transition, Int. J. Solids Struct., 39, 1213-1240 (2002) · Zbl 1090.74698
[54] Lin, R., Numerical study of consistency of rate constitutive equations with elasticity at finite deformations, Int. J. Numer. Methods Engrg., 55, 1053-1077 (2002) · Zbl 1037.74004
[55] Lu, Y. Y.; Belytschko, T.; Tabbara, M., Element-free Galerkin method for wave-propagation and dynamic fracture, Comput. Methods Appl. Mech. Engrg., 126, 1-2, 131-153 (1995) · Zbl 1067.74599
[56] Meyers, M. A.; Nesternko, V. F.; LaSalvia, J. C.; Xue, Q., Shear localization in dynamic deformation of materials: microstructural evolution and self-organization, Mater. Sci. Eng. (2001)
[57] Minnaar, K.; Zhou, M., An analysis of the dynamic failure resistance of structural metals, J. Mech. Phys. Solids, 46, 10, 2155-2170 (1998)
[58] Moes, N.; Belytschko, T., Extended finite element method for cohesive crack growth, Eng. Fracture Mech., 69, 813-834 (2002)
[59] Moes, N.; Dolbow, J.; Belytschko, T., A finite element method for crack growth without remeshing, Int. J. Numer. Methods Engrg., 46, 1, 133-150 (1999) · Zbl 0955.74066
[60] Nagtegaal, J. C.; Parks, D. M.; Rice, J. R., On numerically accurate finite element solutions in the fully plastic range, Comput. Methods Appl. Mech. Engrg., 4, 153-177 (1974) · Zbl 0284.73048
[61] Needleman, A., Dynamics shear band development in plane strain, J. Appl. Mech., 56, 1-9 (1989)
[62] Needleman, A.; Tvergaard, V., Analysis of brittle-ductile transition under dynamic shear loading, Int. J. Solids Struct., 32, 2571-2590 (1995) · Zbl 0919.73222
[63] Nesternko, V. F.; Meyers, M. A.; Wright, T. W., Self-organization in the initiation of adiabatic shear bands, Acta Mater. (1998)
[64] Oliver, J.; Cervera, M.; Manzoli, O., Strong discontinuities and continuum plasticity models: the strong discontinuity approach, Int. J. Plasticity, 15, 319-351 (1999) · Zbl 1057.74512
[65] Oliver, J.; Huespe, A. E.; Samaniego, E., A study on finite elements for capturing strong discontinuities, Int. J. Numer. Methods Engrg., 56, 1291-1305 (2003) · Zbl 1038.74645
[66] Olmstead, W. E.; Nemat-Nasser, S.; Ni, L., Shear bands as surfaces of discontinuity, J. Mech. Phys. Solids, 42, 697-709 (1994) · Zbl 0803.73034
[67] Organ, D.; Fleming, M.; Terry, T.; Belytschko, T., Continuous meshless approximations for nonconvex bodies by diffraction and transparency, Comput. Mech., 18, 225-235 (1996) · Zbl 0864.73076
[68] Ortiz, M.; Leroy, Y.; Needleman, A., Finite element method for localized failure analysis, Comput. Methods Appl. Mech. Engrg., 61, 2, 189-214 (1987) · Zbl 0597.73105
[69] Perzyna, P., Fundamental problems in viscoplasticity, (Recent Advances in Applied Mechanics (1966), Academic Press: Academic Press New York) · Zbl 1207.74025
[70] Piltner, R.; Taylor, R. L., A symmetric construction of b-bar functions for linear and non-linear mixed-enhanced finite elements for plane elasticity problems, Int. J. Numer. Methods Engrg., 44, 615-639 (1999) · Zbl 0947.74067
[71] Rabczuk, T.; Belytschko, T., Cracking particles: A simplified meshfree method for arbitrary evolving cracks, Int. J. Numer. Methods Engrg., 61, 13, 2316-2343 (2004) · Zbl 1075.74703
[72] Rabczuk, T.; Belytschko, T., Adaptivity for structured meshfree particle methods in 2D and 3D, Int. J. Numer. Methods Engrg., 63, 11, 1559-1582 (2005) · Zbl 1145.74041
[73] Rabczuk, T.; Belytschko, T., A three dimensional large deformation meshfree method for arbitrary evolving cracks, Comput. Methods Appl. Mech. Engrg., 196, 29-30, 2777-2799 (2007) · Zbl 1128.74051
[74] Rabczuk, T.; Belytschko, T.; Xiao, S. P., Stable particle methods based on lagrangian kernels, Comput. Methods Appl. Mech. Engrg., 193, 1035-1063 (2004) · Zbl 1060.74672
[75] Reinhardt, W. D.; Dubey, R. N., Application of objective rates in mechanical modelling of solids, ASME J. Appl. Mech., 118, 692-698 (1996) · Zbl 0893.73004
[76] Samaniego, E.; Belytschko, T., Continuum-discontinuum modelling of shear bands, Int. J. Numer. Methods Engrg., 62, 1857-1872 (2005) · Zbl 1121.74476
[77] Shih, C. J.; Meyers, M. A.; Nesterenko, V. F., High-strain-rate deformation of granular silicon carbide, Acta Mater. (1998)
[78] Simo, J. C.; Armero, F., Geometrically non-linear enhanced strain mixed methods and the method of incompatible modes, Int. J. Numer. Methods Engrg., 33, 1413-1449 (1992) · Zbl 0768.73082
[79] Ventura, G.; Moran, B.; Belytschko, T., Dislocations by partition of unity, Int. J. Numer. Methods Engrg., 62, 11, 1463-1487 (2005) · Zbl 1078.74665
[80] Vidal, Y.; Villon, P.; Huerta, A., Locking in the incompressible limit: pseudo-divergence-free element-free Galerkin, Revue Europeene des Elements Finis, 11, 7-8, 869-892 (2002) · Zbl 1120.74837
[81] Wang, D.; Chen, J-S., Locking-free stabilized conforming nodal integration for meshfree mindlind-reissner plate formulation, Comput. Methods Appl. Mech. Engrg., 193, 1065-1083 (2004) · Zbl 1060.74675
[82] Wang, W. M.; SLuys, L. J.; deBorst, R., Viscoplasticity for instabilities due to strain softening and strain-rate softening, Int. J. Numer. Methods Engrg., 40, 20, 3839-3864 (1997) · Zbl 0974.74511
[83] Wright, T. W.; Walter, J. W., The asymptotic structure of an adiabatic shear band in antiplane motion, J. Mech. Phys. Solids, 44, 1, 77-97 (1996) · Zbl 1054.74510
[84] Xiao, S. P.; Belytschko, T., Material stability analysis of particle methods, Adv. Comput. Math. (2005) · Zbl 1060.74070
[85] Xu, X.-P.; Needleman, A., Numerical simulations of fast crack growth in brittle solids, J. Mech. Phys. Solids, 42, 1397-1434 (1994) · Zbl 0825.73579
[86] Xue, Q.; Nesterenko, V. F.; Meyers, M. A., Evaluation of the collapsing thick-walled cylinder technique for shear-band spacing, Int. J. Impact Engrg. (2003)
[87] Yang, Q.; Mota, A.; Ortiz, M., A class of variational strain-localization finite elements, Int. J. Numer. Methods Engrg., 62, 8, 1013-1037 (2005) · Zbl 1081.74045
[88] Zhou, F.; Molinari, J. F., Dynamic crack propagation with cohesive elements: a methodology to address mesh dependence, Int. J. Numer. Methods Engrg., 59, 1, 1-24 (2004) · Zbl 1047.74074
[89] Zhou, M.; Ravichandran, G.; Rosakis, A., Dynamically propagating shear bands in impact-loaded prenotched plates-ii, J. Mech. Phys. Solids, 44, 1007-1032 (1996)
[90] Zhu, T.; Atluri, S. N., A modified collocation method and a penalty formulation for enforcing the essential boundary conditions in the element free Galerkin method, Comput. Mech., 21, 3, 211-222 (1998) · Zbl 0947.74080
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