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A single Gauss point continuum finite element formulation for gradient-extended damage at large deformations. (English) Zbl 1506.74339

Summary: Some years ago, a family of large deformation continuum finite elements based on reduced integration was investigated by the last author [Comput. Methods Appl. Mech. Eng. 194, No. 45–47, 4685–4715 (2005; Zbl 1221.74075)], M. Schwarze and the last author [Int. J. Numer. Methods Eng. 85, No. 3, 289–329 (2011; Zbl 1217.74135)] and J. Frischkorn and the last author [Comput. Methods Appl. Mech. Eng. 291, 42–63 (2015; Zbl 1423.74478)]. Many structural components with different kinds of elastic and inelastic material behavior were considered and these elements showed accurate results while being more efficient than similar three-dimensional formulations based on full integration. The objective of the present contribution is to extend the analysis to non-local damage and fracture. To this end, the incorporation of the gradient-extended damage plasticity model for large deformations of H. Holthusen et al. [“An anisotropic constitutive model for fiber-reinforced materials including gradient-extended damage and plasticity at finite strains”, Theor. Appl. Fracture Mech. 108, Article ID 102642 (2020; doi:10.1016/j.tafmec.2020.102642)] into the framework of reduced integration-based continuum elements is presented. For the sake of brevity, the present work is restricted to solids with only one integration (Gauss) point in the center of the element. The weak form of the formulation, which is based on a two-field variational functional closely related to the enhanced-assumed strain (EAS) method is extended by the weak form of the micromorphic balance equation. The steps required in order to transform the extended formulation into a stable, robust and efficient single Gauss point concept are described in detail. Due to the analogy to fully-coupled thermomechanical problems, the derivation of a novel micromorphic hourglass stabilization is based on earlier contributions of the research group. Therein, the Taylor series expansion of all constitutively dependent quantities plays a crucial role. Representative numerical examples of quasi-brittle and ductile fracture reveal the accuracy and efficiency of the proposed approach. Besides the ability to deliver mesh-independent results, the framework is especially suitable for constrained situations in which conventional low-order finite elements suffer from well-known locking phenomena.

MSC:

74R05 Brittle damage
74S05 Finite element methods applied to problems in solid mechanics
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
74R10 Brittle fracture

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[1] Kachanov, L. M., Time of the rupture process under creep conditions, Izv. Akad. Nauk SSSR, 8, 26-31 (1958) · Zbl 0107.18501
[2] Rabotnov, Y. N., Creep rupture, (Hetényi, M.; Vincenti, W. G., Applied Mechanics (1969), Springer Berlin Heidelberg: Springer Berlin Heidelberg Berlin, Heidelberg), 342-349
[3] Chaboche, J.-L., Description thermodynamique et phenomenologique de la viscoelasticite cyclique avec endommagement (1978)
[4] Lemaitre, J., A three-dimensional ductile damage model applied to deep-drawing forming limits, (Mechanical Behaviour of Materials (1984), Elsevier), 1047-1053
[5] Lemaitre, J., Coupled elasto-plasticity and damage constitutive equations, Comput. Methods Appl. Mech. Engrg., 51, 1-3, 31-49 (1985) · Zbl 0546.73085
[6] Marquis, D.; Lemaitre, J., Constitutive equations for the coupling between elasto-plasticity damage and aging, Rev. Phys. Appl., 23, 4, 615-624 (1988)
[7] Lemaitre, J.; Desmorat, R.; Sauzay, M., Anisotropic damage law of evolution, Eur. J. Mech. A Solids, 19, 2, 187-208 (2000) · Zbl 0986.74007
[8] Pires, F. A.; de Sá, J. C.; Sousa, L. C.; Jorge, R. N., Numerical modelling of ductile plastic damage in bulk metal forming, Int. J. Mech. Sci., 45, 2, 273-294 (2003) · Zbl 1171.74422
[9] Desmorat, R.; Cantournet, S., Modeling microdefects closure effect with isotropic/anisotropic damage, Int. J. Damage Mech., 17, 1, 65-96 (2008)
[10] Saanouni, K.; Lestriez, P., Modelling and numerical simulation of ductile damage in bulk metal forming, Steel Res. Int., 80, 9, 645-657 (2009)
[11] Badreddine, H.; Yue, Z.; Saanouni, K., Modeling of the induced plastic anisotropy fully coupled with ductile damage under finite strains, Int. J. Solids Struct., 108, 49-62 (2017)
[12] Simo, J. C.; Ju, J., Strain-and stress-based continuum damage models - I. Formulation, Int. J. Solids Struct., 23, 7, 821-840 (1987) · Zbl 0634.73106
[13] Voyiadjis, G. Z.; Shojaei, A.; Li, G., A thermodynamic consistent damage and healing model for self healing materials, Int. J. Plast., 27, 7, 1025-1044 (2011) · Zbl 1454.74136
[14] Zhu, Q.; Zhao, L.; Shao, J., Analytical and numerical analysis of frictional damage in quasi brittle materials, J. Mech. Phys. Solids, 92, 137-163 (2016)
[15] Voyiadjis, G. Z.; Kattan, P. I., A plasticity-damage theory for large deformation of solids—I. Theoretical formulation, Internat. J. Engrg. Sci., 30, 9, 1089-1108 (1992) · Zbl 0756.73038
[16] Vignjevic, R.; Djordjevic, N.; Panov, V., Modelling of dynamic behaviour of orthotropic metals including damage and failure, Int. J. Plast., 38, 47-85 (2012)
[17] Brünig, M.; Gerke, S.; Schmidt, M., Damage and failure at negative stress triaxialities: Experiments, modeling and numerical simulations, Int. J. Plast., 102, 70-82 (2018)
[18] Bažant, Z. P.; Belytschko, T. B.; Chang, T.-P., Continuum theory for strain-softening, J. Eng. Mech., 110, 12, 1666-1692 (1984)
[19] de Borst, R.; Sluys, L. J.; Muhlhaus, H.-B.; Pamin, J., Fundamental issues in finite element analyses of localization of deformation, Eng. Comput. Int. J. Comput. Aided Eng., 10, 2, 99-121 (1993)
[20] Jirásek, M.; Grassl, P., Evaluation of directional mesh bias in concrete fracture simulations using continuum damage models, Eng. Fract. Mech., 75, 8, 1921-1943 (2008)
[21] Brepols, T.; Wulfinghoff, S.; Reese, S., Gradient-extended two-surface damage-plasticity: micromorphic formulation and numerical aspects, Int. J. Plast., 97, 64-106 (2017)
[22] Needleman, A., Material rate dependence and mesh sensitivity in localization problems, Comput. Methods Appl. Mech. Engrg., 67, 1, 69-85 (1988) · Zbl 0618.73054
[23] Langenfeld, K.; Junker, P.; Mosler, J., Quasi-brittle damage modeling based on incremental energy relaxation combined with a viscous-type regularization, Contin. Mech. Thermodyn., 30, 5, 1125-1144 (2018) · Zbl 1396.74091
[24] Pijaudier-Cabot, G.; Bažant, Z. P., Nonlocal damage theory, J. Eng. Mech., 113, 10, 1512-1533 (1987)
[25] Bažant, Z. P.; Jirásek, M., Nonlocal integral formulations of plasticity and damage: survey of progress, J. Eng. Mech., 128, 11, 1119-1149 (2002)
[26] Peerlings, R. H.; de Borst, R.; Brekelmans, W. M.; De Vree, J., Gradient enhanced damage for quasi-brittle materials, Internat. J. Numer. Methods Engrg., 39, 19, 3391-3403 (1996) · Zbl 0882.73057
[27] Geers, M., Finite strain logarithmic hyperelasto-plasticity with softening: a strongly non-local implicit gradient framework, Comput. Methods Appl. Mech. Engrg., 193, 30-32, 3377-3401 (2004) · Zbl 1060.74503
[28] Forest, S., Micromorphic approach for gradient elasticity, viscoplasticity, and damage, J. Eng. Mech., 135, 3, 117-131 (2009)
[29] Forest, S., Nonlinear regularization operators as derived from the micromorphic approach to gradient elasticity, viscoplasticity and damage, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 472, 2188, Article 20150755 pp. (2016) · Zbl 1371.74226
[30] Brepols, T.; Wulfinghoff, S.; Reese, S., A micromorphic damage-plasticity model to counteract mesh dependence in finite element simulations involving material softening, (Multiscale Modeling of Heterogeneous Structures (2018), Springer), 235-255
[31] Brepols, T.; Wulfinghoff, S.; Reese, S., A gradient-extended two-surface damage-plasticity model for large deformations, Int. J. Plast., 129, Article 102635 pp. (2020)
[32] Fassin, M.; Eggersmann, R.; Wulfinghoff, S.; Reese, S., Efficient algorithmic incorporation of tension compression asymmetry into an anisotropic damage model, Comput. Methods Appl. Mech. Engrg., 354, 932-962 (2019) · Zbl 1441.74194
[33] Fassin, M.; Eggersmann, R.; Wulfinghoff, S.; Reese, S., Gradient-extended anisotropic brittle damage modeling using a second order damage tensor-theory, implementation and numerical examples, Int. J. Solids Struct., 167, 93-126 (2019)
[34] 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
[35] Kuhn, C.; Müller, R., A continuum phase field model for fracture, Eng. Fract. Mech., 77, 18, 3625-3634 (2010)
[36] Ambati, M.; Gerasimov, T.; De Lorenzis, L., Phase-field modeling of ductile fracture, Comput. Mech., 55, 5, 1017-1040 (2015) · Zbl 1329.74018
[37] Zienkiewicz, O. C.; Taylor, R. L.; Taylor, R. L.; Taylor, R. L., The finite element method: solid mechanics, Vol. 2 (2000), Butterworth-heinemann · Zbl 0991.74003
[38] Babuška, I.; Suri, M., Locking effects in the finite element approximation of elasticity problems, Numer. Math., 62, 1, 439-463 (1992) · Zbl 0762.65057
[39] de Borst, R.; Groen, A. E., Some observations on element performance in isochoric and dilatant plastic flow, Internat. J. Numer. Methods Engrg., 38, 17, 2887-2906 (1995) · Zbl 0836.73065
[40] de Borst, R.; Pamin, J., Some novel developments in finite element procedures for gradient-dependent plasticity, Internat. J. Numer. Methods Engrg., 39, 14, 2477-2505 (1996) · Zbl 0885.73074
[41] Miehe, C.; Aldakheel, F.; Mauthe, S., Mixed variational principles and robust finite element implementations of gradient plasticity at small strains, Internat. J. Numer. Methods Engrg., 94, 11, 1037-1074 (2013) · Zbl 1352.74408
[42] Arnold, D. N.; Brezzi, F.; Fortin, M., A stable finite element for the Stokes equations, Calcolo, 21, 4, 337-344 (1984) · Zbl 0593.76039
[43] Simo, J. C.; Hughes, T. J., Computational Inelasticity, Vol. 7 (2006), Springer Science & Business Media
[44] Duda, F. P.; Ciarbonetti, A.; Sánchez, P. J.; Huespe, A. E., A phase-field/gradient damage model for brittle fracture in elastic-plastic solids, Int. J. Plast., 65, 269-296 (2015)
[45] Simo, J. C.; Rifai, M., A class of mixed assumed strain methods and the method of incompatible modes, Internat. J. Numer. Methods Engrg., 29, 8, 1595-1638 (1990) · Zbl 0724.73222
[46] Simo, J.-C.; Armero, F., Geometrically non-linear enhanced strain mixed methods and the method of incompatible modes, Internat. J. Numer. Methods Engrg., 33, 7, 1413-1449 (1992) · Zbl 0768.73082
[47] Neuner, M.; Schreter, M.; Hofstetter, G., Enhanced assumed strain methods for implicit gradient-enhanced damage-plasticity, PAMM, 18, 1, Article e201800054 pp. (2018)
[48] Kienle, D.; Aldakheel, F.; Keip, M.-A., A finite-strain phase-field approach to ductile failure of frictional materials, Int. J. Solids Struct., 172, 147-162 (2019)
[49] Korelc, J.; Šolinc, U.; Wriggers, P., An improved EAS brick element for finite deformation, Comput. Mech., 46, 4, 641-659 (2010) · Zbl 1358.74059
[50] Wriggers, P.; Reese, S., A note on enhanced strain methods for large deformations, Comput. Methods Appl. Mech. Engrg., 135, 3-4, 201-209 (1996) · Zbl 0893.73072
[51] de Souza Neto, E.; Perić, D.; Huang, G.; Owen, D., Remarks on the stability of enhanced strain elements in finite elasticity and elastoplasticity, Commun. Numer. Methods. Eng., 11, 11, 951-961 (1995) · Zbl 0836.73066
[52] Reinoso, J.; Paggi, M.; Linder, C., Phase field modeling of brittle fracture for enhanced assumed strain shells at large deformations: formulation and finite element implementation, Comput. Mech., 59, 6, 981-1001 (2017) · Zbl 1398.74394
[53] Hortig, C., Local and Non-Local Thermomechanical Modeling and Finite-Element Simulation of High-Speed Cutting (2010), Institute of Mechanics, TU Dortmund University, (Ph.D. thesis)
[54] Seupel, A.; Hütter, G.; Kuna, M., An efficient FE-implementation of implicit gradient-enhanced damage models to simulate ductile failure, Eng. Fract. Mech., 199, 41-60 (2018)
[55] Ostwald, R.; Kuhl, E.; Menzel, A., On the implementation of finite deformation gradient-enhanced damage models, Comput. Mech., 64, 3, 847-877 (2019) · Zbl 1465.74142
[56] Verhoosel, C. V.; Scott, M. A.; Hughes, T. J.; De Borst, R., An isogeometric analysis approach to gradient damage models, Internat. J. Numer. Methods Engrg., 86, 1, 115-134 (2011) · Zbl 1235.74320
[57] Borden, M. J.; Hughes, T. J.; Landis, C. M.; Verhoosel, C. V., A higher-order phase-field model for brittle fracture: Formulation and analysis within the isogeometric analysis framework, Comput. Methods Appl. Mech. Engrg., 273, 100-118 (2014) · Zbl 1296.74098
[58] Thai, T. Q.; Rabczuk, T.; Bazilevs, Y.; Meschke, G., A higher-order stress-based gradient-enhanced damage model based on isogeometric analysis, Comput. Methods Appl. Mech. Engrg., 304, 584-604 (2016) · Zbl 1425.74047
[59] Beirão da Veiga, L.; Brezzi, F.; Cangiani, A.; Manzini, G.; Marini, L. D.; Russo, A., Basic principles of virtual element methods, Math. Models Methods Appl. Sci., 23, 01, 199-214 (2013) · Zbl 1416.65433
[60] de Bellis, M. L.; Wriggers, P.; Hudobivnik, B.; Zavarise, G., Virtual element formulation for isotropic damage, Finite Elem. Anal. Des., 144, 38-48 (2018)
[61] Aldakheel, F.; Hudobivnik, B.; Hussein, A.; Wriggers, P., Phase-field modeling of brittle fracture using an efficient virtual element scheme, Comput. Methods Appl. Mech. Engrg., 341, 443-466 (2018) · Zbl 1440.74352
[62] Aldakheel, F.; Hudobivnik, B.; Wriggers, P., Virtual element formulation for phase-field modeling of ductile fracture, Int. J. Multiscale Comput. Eng., 17, 2 (2019)
[63] Cangiani, A.; Manzini, G.; Russo, A.; Sukumar, N., Hourglass stabilization and the virtual element method, Internat. J. Numer. Methods Engrg., 102, 3-4, 404-436 (2015) · Zbl 1352.65475
[64] Malkus, D. S.; Hughes, T. J., Mixed finite element methods - reduced and selective integration techniques: a unification of concepts, Comput. Methods Appl. Mech. Engrg., 15, 1, 63-81 (1978) · Zbl 0381.73075
[65] Kosloff, D.; Frazier, G. A., Treatment of hourglass patterns in low order finite element codes, Int. J. Numer. Anal. Methods Geomech., 2, 1, 57-72 (1978)
[66] Belytschko, T.; Ong, J. S.-J.; Liu, W. K.; Kennedy, J. M., Hourglass control in linear and nonlinear problems, Comput. Methods Appl. Mech. Engrg., 43, 3, 251-276 (1984) · Zbl 0522.73063
[67] Belytschko, T.; Bindeman, L. P., Assumed strain stabilization of the eight node hexahedral element, Comput. Methods Appl. Mech. Engrg., 105, 2, 225-260 (1993) · Zbl 0781.73061
[68] Puso, M. A., A highly efficient enhanced assumed strain physically stabilized hexahedral element, Internat. J. Numer. Methods Engrg., 49, 8, 1029-1064 (2000) · Zbl 0994.74075
[69] Reese, S.; Wriggers, P.; Reddy, B., A new locking-free brick element technique for large deformation problems in elasticity, Comput. Struct., 75, 3, 291-304 (2000)
[70] Reese, S., On a physically stabilized one point finite element formulation for three-dimensional finite elasto-plasticity, Comput. Methods Appl. Mech. Engrg., 194, 45-47, 4685-4715 (2005) · Zbl 1221.74075
[71] Reese, S., A large deformation solid-shell concept based on reduced integration with hourglass stabilization, Internat. J. Numer. Methods Engrg., 69, 8, 1671-1716 (2007) · Zbl 1194.74469
[72] Schwarze, M.; Reese, S., A reduced integration solid-shell finite element based on the EAS and the ANS concept-geometrically linear problems, Internat. J. Numer. Methods Engrg., 80, 10, 1322-1355 (2009) · Zbl 1183.74315
[73] Schwarze, M.; Reese, S., A reduced integration solid-shell finite element based on the EAS and the ANS concept-Large deformation problems, Internat. J. Numer. Methods Engrg., 85, 3, 289-329 (2011) · Zbl 1217.74135
[74] Frischkorn, J.; Reese, S., A solid-beam finite element and non-linear constitutive modelling, Comput. Methods Appl. Mech. Engrg., 265, 195-212 (2013) · Zbl 1286.74101
[75] Frischkorn, J.; Reese, S., Solid-beam finite element analysis of nitinol stents, Comput. Methods Appl. Mech. Engrg., 291, 42-63 (2015) · Zbl 1423.74478
[76] Daniel, W. J.T.; Belytschko, T., Suppression of spurious intermediate frequency modes in under-integrated elements by combined stiffness/viscous stabilization, Internat. J. Numer. Methods Engrg., 64, 3, 335-353 (2005) · Zbl 1181.74134
[77] Krysl, P., Mean-strain eight-node hexahedron with optimized energy-sampling stabilization for large-strain deformation, Internat. J. Numer. Methods Engrg., 103, 9, 650-670 (2015) · Zbl 1352.74386
[78] Krysl, P., Mean-strain eight-node hexahedron with stabilization by energy sampling, Internat. J. Numer. Methods Engrg., 102, 3-4, 437-449 (2015) · Zbl 1352.74077
[79] Krysl, P., Mean-strain 8-node hexahedron with optimized energy-sampling stabilization, Finite Elem. Anal. Des., 108, 41-53 (2016)
[80] Adam, L.; Ponthot, J.-P., Thermomechanical modeling of metals at finite strains: First and mixed order finite elements, Int. J. Solids Struct., 42, 21-22, 5615-5655 (2005) · Zbl 1113.74418
[81] Reese, S., On a consistent hourglass stabilization technique to treat large inelastic deformations and thermo-mechanical coupling in plane strain problems, Internat. J. Numer. Methods Engrg., 57, 8, 1095-1127 (2003) · Zbl 1062.74626
[82] Juhre, D.; Reese, S., A reduced integration finite element technology based on a thermomechanically consistent stabilisation for 3D problems, Comput. Methods Appl. Mech. Engrg., 199, 29-32, 2050-2058 (2010) · Zbl 1231.74421
[83] Holthusen, H.; Brepols, T.; Reese, S.; Simon, J.-W., An anisotropic constitutive model for fiber-reinforced materials including gradient-extended damage and plasticity at finite strains, Theor. Appl. Fract. Mech., Article 102642 pp. (2020)
[84] Wulfinghoff, S.; Forest, S.; Böhlke, T., Strain gradient plasticity modeling of the cyclic behavior of laminate microstructures, J. Mech. Phys. Solids, 79, 1-20 (2015) · Zbl 1349.74071
[85] 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
[86] De Sousa, R. A.; Jorge, R. N.; Valente, R. F.; de Sá, J. C., A new volumetric and shear locking-free 3D enhanced strain element, Eng. Comput. (2003) · Zbl 1063.74537
[87] Armero, F., On the locking and stability of finite elements in finite deformation plane strain problems, Comput. Struct., 75, 3, 261-290 (2000)
[88] Taylor, R.; Simo, J.; Zienkiewicz, O.; Chan, A., The patch test - a condition for assessing FEM convergence, Internat. J. Numer. Methods Engrg., 22, 1, 39-62 (1986) · Zbl 0593.73072
[89] Taylor, R., FEAP-A Finite Element Analysis Program, Version 8.4, User Manual (2013), Department of Civil and Environmental Engineering, University of California at Berkeley: Department of Civil and Environmental Engineering, University of California at Berkeley Berkeley
[90] Brepols, T., Theory and Numerics of Gradient-Extended Damage Coupled with Plasticity (2019), Institute of Applied Mechanics, RWTH Aachen University, (Ph.D. thesis)
[91] Hughes, T. J., The Finite Element Method: Linear Static and Dynamic Finite Element Analysis (2012), Courier Corporation
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