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A semi-smooth Newton method for orthotropic plasticity and frictional contact at finite strains. (English) Zbl 1423.74159

Summary: A new approach for the unified treatment of frictional contact and orthotropic plasticity at finite strains using semi-smooth Newton methods is presented. The contact discretization is based on the well-known mortar finite element method using dual Lagrange multipliers to facilitate the handling of the additional Lagrange multiplier degrees of freedom. Exploiting the similarity of the typical inequality constraints of plasticity and friction, all involved discrete inequalities are reformulated as nonlinear non-smooth equations using complementarity functions. The resulting set of discrete semi-smooth equations can be solved efficiently by a variant of Newton’s method, where all additionally introduced variables are condensed from the global system so that a linear system only consisting of the displacement degrees of freedom has to be solved in each iteration step. In contrast to classical radial return mapping methods for computational plasticity, the plastic constraints are not required to hold at every iterate in the nonlinear solution procedure, but only at convergence. This relaxation in the pre-asymptotic behavior results in an increased flexibility regarding algorithm design and a potentially higher robustness compared to radial return mapping algorithms. The presented elasto-plasticity algorithm includes arbitrary isotropic hyperelasticity, an anisotropic Hill-type yield function with isotropic and kinematic hardening, plastic spin and appropriate finite element technology for nearly incompressible materials. Therefore, it is well suited for the modeling of sheet metal forming and similar processes. Several numerical examples underline the robustness of the proposed plasticity algorithm and the efficient treatment of elasto-plastic contact problems.

MSC:

74C05 Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials)
74M10 Friction in solid mechanics
74M15 Contact in solid mechanics

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[1] Puso, M. A.; Laursen, T. A., A mortar segment-to-segment contact method for large deformation solid mechanics, Comput. Methods Appl. Mech. Engrg., 193, 601-629 (2004) · Zbl 1060.74636
[2] Puso, M. A.; Laursen, T. A., A mortar segment-to-segment frictional contact method for large deformations, Comput. Methods Appl. Mech. Engrg., 193, 4891-4913 (2004) · Zbl 1112.74535
[3] Fischer, K. A.; Wriggers, P., Frictionless 2D contact formulations for finite deformations based on the mortar method, Comput. Mech., 36, 226-244 (2005) · Zbl 1102.74033
[4] Tur, M.; Fuenmayor, F. J.; Wriggers, P., A mortar-based frictional contact formulation for large deformations using Lagrange multipliers, Comput. Methods Appl. Mech. Engrg., 198, 2860-2873 (2009) · Zbl 1229.74141
[5] Hesch, C.; Betsch, P., A mortar method for energy-momentum conserving schemes in frictionless dynamic contact problems, Internat. J. Numer. Methods Engrg., 77, 1468-1500 (2009) · Zbl 1156.74378
[6] Wohlmuth, B. I., A mortar finite element method using dual spaces for the Lagrange multiplier, SIAM J. Numer. Anal., 38, 989-1012 (2000) · Zbl 0974.65105
[7] Wohlmuth, B. I., Discretization Methods and Iterative Solvers Based on Domain Decomposition (2001), Springer-Verlag: Springer-Verlag Berlin, Heidelberg · Zbl 0966.65097
[8] Puso, M. A., A 3D mortar method for solid mechanics, Internat. J. Numer. Methods Engrg., 59, 315-336 (2004) · Zbl 1047.74065
[9] Lamichhane, B. P.; Stevenson, R. P.; Wohlmuth, B. I., Higher order mortar finite element methods in 3D with dual Lagrange multiplier bases, Numer. Math., 102, 93-121 (2005) · Zbl 1082.65120
[10] Flemisch, B.; Wohlmuth, B. I., Stable Lagrange multipliers for quadrilateral meshes of curved interfaces in 3D, Comput. Methods Appl. Mech. Engrg., 196, 1589-1602 (2007) · Zbl 1173.74416
[11] Hintermüller, M.; Ito, K.; Kunisch, K., The primal-dual active set strategy as a semi-smooth Newton method, SIAM J. Optim., 13, 865-888 (2002) · Zbl 1080.90074
[12] Hüeber, S.; Wohlmuth, B. I., A primal-dual active set strategy for non-linear multibody contact problems, Comput. Methods Appl. Mech. Engrg., 194, 3147-3166 (2005) · Zbl 1093.74056
[13] Hüeber, S.; Stadler, G.; Wohlmuth, B. I., A primal-dual active set algorithm for three-dimensional contact problems with Coulomb friction, SIAM J. Sci. Comput., 30, 572-596 (2008) · Zbl 1158.74045
[14] Hartmann, S.; Brunssen, S.; Ramm, E.; Wohlmuth, B. I., Unilateral non-linear dynamic contact of thin-walled structures using a primal-dual active set strategy, Internat. J. Numer. Methods Engrg., 70, 883-912 (2007) · Zbl 1194.74218
[15] Popp, A.; Gee, M. W.; Wall, W. A., A finite deformation mortar contact formulation using a primal-dual active set strategy, Internat. J. Numer. Methods Engrg., 79, 1354-1391 (2009) · Zbl 1176.74133
[16] Popp, A.; Gitterle, M.; Gee, M. W.; Wall, W. A., A dual mortar approach for 3D finite deformation contact with consistent linearization, Internat. J. Numer. Methods Engrg., 83, 1428-1465 (2010) · Zbl 1202.74183
[17] Gitterle, M.; Popp, A.; Gee, M. W.; Wall, W. A., Finite deformation frictional mortar contact using a semi-smooth Newton method with consistent linearization, Internat. J. Numer. Methods Engrg., 84, 543-571 (2010) · Zbl 1202.74121
[18] Popp, A.; Wohlmuth, B. I.; Gee, M. W.; Wall, W. A., Dual quadratic mortar finite element methods for 3D finite deformation contact, SIAM J. Sci. Comput., B421-B446 (2012) · Zbl 1250.74020
[19] Popp, A.; Seitz, A.; Gee, M. W.; Wall, W. A., Improved robustness and consistency of 3D contact algorithms based on a dual mortar approach, Comput. Methods Appl. Mech. Engrg., 264, 67-80 (2013) · Zbl 1286.74106
[20] Alart, P.; Curnier, A., A mixed formulation for frictional contact problems prone to Newton like solution methods, Comput. Methods Appl. Mech. Engrg., 92, 353-375 (1991) · Zbl 0825.76353
[21] Qi, L.; Sun, J., A nonsmooth version of Newton’s method, Math. Program., 58, 353-367 (1993) · Zbl 0780.90090
[22] Wohlmuth, B. I., Variationally consistent discretization schemes and numerical algorithms for contact problems, Acta Numer., 20, 569-734 (2011) · Zbl 1432.74176
[23] Simo, J. C., A framework for finite strain elastoplasticity based on maximum plastic dissipation and the multiplicative decomposition: Part I. Continuum formulation, Comput. Methods Appl. Mech. Engrg., 66, 199-219 (1988) · Zbl 0611.73057
[24] Simo, J. C., A framework for finite strain elastoplasticity based on maximum plastic dissipation and the multiplicative decomposition. Part II: Computational aspects, Comput. Methods Appl. Mech. Engrg., 68, 1-31 (1988) · Zbl 0644.73043
[25] Simo, J. C.; Hughes, T. J.R., Computational Inelasticity (1998), Springer: Springer New York · Zbl 0934.74003
[26] de Souza Neto, E. A.; Perić, D.; Owen, D. R.J., Computational Methods for Plasticity: Theory and Applications (2008), Wiley: Wiley Chichester
[27] Han, W.; Reddy, B., Plasticity: Mathematical Theory and Numerical Analysis (1999), Springer: Springer New York · Zbl 0926.74001
[28] Wieners, C., Nonlinear solution methods for infinitesimal perfect plasticity, Z. Angew. Math. Mech., 87, 643-660 (2007) · Zbl 1128.74008
[29] Krabbenhoft, K.; Lyamin, A. V.; Sloan, S. W.; Wriggers, P., An interior-point algorithm for elastoplasticity, Internat. J. Numer. Methods Engrg., 69, 592-626 (2007) · Zbl 1194.74422
[30] Christensen, P. W., A nonsmooth Newton method for elastoplastic problems, Comput. Method. Appl. Mech. Engrg., 191, 11-12, 1189-1219 (2002) · Zbl 1021.74049
[31] Hager, C.; Wohlmuth, B., Nonlinear complementarity functions for plasticity problems with frictional contact, Comput. Methods Appl. Mech. Engrg., 198, 3411-3427 (2009) · Zbl 1230.74225
[32] Ortiz, M.; Stainier, L., The variational formulation of viscoplastic constitutive updates, Comput. Methods Appl. Mech. Engrg., 171, 419-444 (1999) · Zbl 0938.74016
[33] Fancello, E.; Vassoler, J. M.; Stainier, L., A variational constitutive update algorithm for a set of isotropic hyperelastic-viscoplastic material models, Comput. Methods Appl. Mech. Engrg., 197, 4132-4148 (2008) · Zbl 1194.74519
[34] Mosler, J., Variationally consistent modeling of finite strain plasticity theory with non-linear kinematic hardening, Comput. Methods Appl. Mech. Engrg., 199, 2753-2764 (2010) · Zbl 1231.74059
[35] Mosler, J.; Bruhns, O., On the implementation of rate-independent standard dissipative solids at finite strain—variational constitutive updates, Comput. Methods Appl. Mech. Engrg., 199, 417-429 (2010) · Zbl 1227.74014
[36] Bleier, N.; Mosler, J., Efficient variational constitutive updates by means of a novel parameterization of the flow rule, Internat. J. Numer. Methods Engrg., 89, 1120-1143 (2012) · Zbl 1242.74202
[37] Dafalias, Y. F., Plastic spin: necessity or redundancy?, Int. J. Plast., 14, 909-931 (1998) · Zbl 0947.74008
[38] Lee, E. H., Finite-strain elastic-plastic theory with application to plane-wave analysis, J. Appl. Phys., 38, 19 (1967)
[39] Mandel, J., Plasticité Classique et Viscoplasticité: Course held at the Department of Mechanics of Solids at Udine, September-October 1971 (1972), Springer: Springer Wien [etc.] · Zbl 0298.73048
[40] Hill, R., A theory of the yielding and plastic flow of anisotropic metals, Proc. R. Soc. A, 193, 281-297 (1948) · Zbl 0032.08805
[41] Dafalias, Y. F., The plastic spin, J. Appl. Mech., 52, 865-871 (1985) · Zbl 0587.73052
[42] Zbib, H.; Aifantis, E., On the concept of relative and plastic spins and its implications to large deformation theories. Part I: Hypoelasticity and vertex-type plasticity, Acta Mech., 75, 15-33 (1988) · Zbl 0667.73031
[43] Ulz, M. H., A finite isoclinic elasto-plasticity model with orthotropic yield function and notion of plastic spin, Comput. Methods Appl. Mech. Engrg., 200, 1822-1832 (2011) · Zbl 1228.74014
[44] Carstensen, C.; Hackl, K.; Mielke, A., Non-convex potentials and microstructures in finite-strain plasticity, Proc. R. Soc. A, 458, 299-317 (2002) · Zbl 1008.74016
[45] Mielke, A., Existence of minimizers in incremental elasto-plasticity with finite strains, SIAM J. Math. Anal., 36, 384-404 (2004) · Zbl 1076.74012
[46] Wohlmuth, B. I.; Popp, A.; Gee, M. W.; Wall, W. A., An abstract framework for a priori estimates for contact problems in 3D with quadratic finite elements, Comput. Mech., 49, 735-747 (2012) · Zbl 1312.74047
[47] Reddy, B.; Martin, J., Algorithms for the solution of internal variable problems in plasticity, Comput. Methods Appl. Mech. Engrg., 93, 253-273 (1991) · Zbl 0744.73023
[48] Hager, C., Robust numerical algorithms for dynamic frictional contact problems with different time and space scales (2010), Universität Stuttgart, (Ph.D. thesis)
[49] Ristinmaa, M.; Ottosen, N. S., Consequences of dynamic yield surface in viscoplasticity, Int. J. Solids Struct., 37, 4601-4622 (2000) · Zbl 0982.74014
[50] de Souza Neto, E. A.; Perić, D.; Dutko, M.; Owen, D. R.J., Design of simple low order finite elements for large strain analysis of nearly incompressible solids, Int. J. Solids Struct., 33, 3277-3296 (1996) · Zbl 0929.74102
[51] Wall, W. A.; Gee, M. W., BACI: a multiphysics simulation environment, Technical Report (2014), Technische Universität München
[52] Fuschi, P.; Perić, D.; Owen, D., Studies on generalized midpoint integration in rate-independent plasticity with reference to plane stress \(J_2\)-flow theory, Comput. Struct., 43, 1117-1133 (1992) · Zbl 0771.73023
[53] Simo, J. C.; Taylor, R. L., Consistent tangent operators for rate-independent elastoplasticity, Comput. Methods Appl. Mech. Engrg., 48, 101-118 (1985) · Zbl 0535.73025
[54] Simo, J. C.; Taylor, R. L., A return mapping algorithm for plane stress elastoplasticity, Internat. J. Numer. Methods Engrg., 22, 649-670 (1986) · Zbl 0585.73059
[55] Simo, J. C., Algorithms for static and dynamic multiplicative plasticity that preserve the classical return mapping schemes of the infinitesimal theory, Comput. Methods Appl. Mech. Engrg., 99, 61-112 (1992) · Zbl 0764.73089
[56] Simo, J. C.; Armero, F., Geometrically non-linear enhanced strain mixed methods and the method of incompatible modes, Internat. J. Numer. Methods Engrg., 33, 1413-1449 (1992) · Zbl 0768.73082
[57] de Souza Neto, E. A.; Pires, F.; Owen, D., F-bar-based linear triangles and tetrahedra for finite strain analysis of nearly incompressible solids. Part I: formulation and benchmarking, Internat. J. Numer. Methods Engrg., 62, 353-383 (2005) · Zbl 1179.74159
[58] Andrade Pires, F. M.; de Souza Neto, E. A.; de la Cuesta Padilla, J. L., An assessment of the average nodal volume formulation for the analysis of nearly incompressible solids under finite strains, Comm. Numer. Methods Engrg., 20, 569-583 (2004) · Zbl 1302.74173
[60] Doll, S.; Schweizerhof, K.; Hauptmann, R.; Freischläger, C., On volumetric locking of low-order solid and solid-shell elements for finite elastoviscoplastic deformations and selective reduced integration, Eng. Comput., 17, 874-902 (2000) · Zbl 1012.74069
[61] Holzapfel, G. A., Nonlinear Solid Mechanics: A Continuum Approach for Engineering (2000), Wiley · Zbl 0980.74001
[62] Miehe, C.; Apel, N.; Lambrecht, M., Anisotropic additive plasticity in the logarithmic strain space: modular kinematic formulation and implementation based on incremental minimization principles for standard materials, Comput. Methods Appl. Mech. Engrg., 191, 5383-5425 (2002) · Zbl 1083.74518
[63] Papadopoulos, P.; Lu, J., On the formulation and numerical solution of problems in anisotropic finite plasticity, Comput. Methods Appl. Mech. Engrg., 190, 4889-4910 (2001) · Zbl 1001.74020
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