Introduction of pseudo-stress for local residual and algebraic derivation of consistent tangent in elastoplasticity. (English) Zbl 1519.74074
Summary: In this article, an introduction of pseudo-stress for local residual and an algebraic derivation of consistent tangent are presented. The authors define a coupled problem of the equilibrium equation for the overall structure and the constrained equations for stress state at every material point, and the pseudo-stress and the derived consistent tangent can be implemented easily to finite element analysis. In the proposed block Newton method, the internal variables are also updated algebraically without any local iterative calculations. In addition, the authors demonstrate the performance of the proposed approach for both \(J_2\) plasticity and \(J_2\) plasticity under plane stress state.
MSC:
| 74S05 | Finite element methods applied to problems in solid mechanics |
| 74C15 | Large-strain, rate-independent theories of plasticity (including nonlinear plasticity) |
Keywords:
block Newton method; plane stress; finite element analysis; large strain; updated internal variableSoftware:
HYPLASReferences:
| [1] | Yamamoto, T.; Yamada, T.; Matsui, K., Simultaneously iterative procedure based on block newton method for elastoplastic problems, Int J Numer Methods Eng, 122, 9, 2145-2178 (2021) · Zbl 07863747 · doi:10.1002/nme.6613 |
| [2] | Kulkarni, DV; Tortorelli, DA; Barth, TJ; Griebel, M.; Keyes, DE, A domain decomposition based two-level newton scheme for nonlinear problems, Domain decomposition methods in science and engineering, 615-622 (2005), Berlin: Springer, Berlin · Zbl 1066.65056 · doi:10.1007/3-540-26825-1_65 |
| [3] | Michaleris, P.; Tortorelli, D.; Vidal, C., Tangent operators and design sensitivity formulations for transient non-linear coupled problems with applications to elastoplasticity, Int J Numer Methods Eng, 37, 14, 2471-2499 (1994) · Zbl 0808.73057 · doi:10.1002/nme.1620371408 |
| [4] | Michaleris, P.; Tortorelli, D.; Vidal, C., Analysis and optimization of weakly coupled thermoelastoplastic systems with applications to weldment design, Int J Numer Methods Eng, 38, 8, 1259-1285 (1995) · Zbl 0823.73047 · doi:10.1002/nme.1620380803 |
| [5] | Wisniewski K, Kowalczyk P, Turska E (2003) On the computation of design derivatives for Huber-Mises plasticity with non-linear hardening. Int J Numer Methods Eng 57(2):271-300 · Zbl 1062.74631 |
| [6] | Wallin, M.; Jönsson, V.; Wingren, E., Topology optimization based on finite strain plasticity, Struct Multidiscip Optim, 54, 783-793 (2016) · doi:10.1007/s00158-016-1435-0 |
| [7] | Okada, J.; Washio, T.; Hisada, T., Study of efficient homogenization algorithms for nonlinear problems: approximation of a homogenized tangent stiffness to reduce computational cost, Comput Mech, 46, 2, 247-258 (2010) · Zbl 1398.74261 · doi:10.1007/s00466-009-0432-1 |
| [8] | Lange, N.; Hütter, G.; Kiefer, B., An efficient monolithic solution scheme for FE2 problems, Comput Methods Appl Mech Eng, 382, 113, 886 (2021) · Zbl 1506.74414 · doi:10.1016/j.cma.2021.113886 |
| [9] | Fritzen, F.; Hassani, MR, Space-time model order reduction for nonlinear viscoelastic systems subjected to long-term loading, Meccanica, 53, 1333-1355 (2018) · doi:10.1007/s11012-017-0734-x |
| [10] | Matthies, HG; Steindorf, J., Partitioned strong coupling algorithms for fluid-structure interaction, Comput Struct, 81, 8, 805-812 (2003) · doi:10.1016/S0045-7949(02)00409-1 |
| [11] | Matthies, HG; Niekamp, R.; Steindorf, J., Algorithms for strong coupling procedures, Comput Methods Appl Mech Eng, 195, 17, 2028-2049 (2006) · Zbl 1142.74050 · doi:10.1016/j.cma.2004.11.032 |
| [12] | Ellsiepen, P.; Hartmann, S., Remarks on the interpretation of current non-linear finite element analyses as differential-algebraic equations, Int J Numer Methods Eng, 51, 6, 679-707 (2001) · Zbl 1014.74068 · doi:10.1002/nme.179.abs |
| [13] | Kulkarni, DV; Tortorelli, DA; Wallin, M., A Newton-Schur alternative to the consistent tangent approach in computational plasticity, Comput Methods Appl Mech Eng, 196, 7, 1169-1177 (2007) · Zbl 1173.74474 · doi:10.1016/j.cma.2006.06.013 |
| [14] | Simo, J.; Taylor, R., Consistent tangent operators for rate-independent elastoplasticity, Comput Methods Appl Mech Eng, 48, 1, 101-118 (1985) · Zbl 0535.73025 · doi:10.1016/0045-7825(85)90070-2 |
| [15] | Hartmann S, Quint KJ, Arnold M (2008) On plastic incompressibility within time-adaptive finite elements combined with projection techniques. Comput Methods Appl Mech Eng 198(2):178-193 · Zbl 1194.74404 |
| [16] | Nakshatrala, P.; Tortorelli, D., Topology optimization for effective energy propagation in rate-independent elastoplastic material systems, Comput Methods Appl Mech Eng, 295, 305-326 (2015) · Zbl 1423.74755 · doi:10.1016/j.cma.2015.05.004 |
| [17] | Owen, D.; Hinton, E., Finite elements in plasticity: theory and practice (1980), Swansea: Pineridge Press, Swansea · Zbl 0482.73051 |
| [18] | Hartmann, S., A remark on the application of the Newton-Raphson method in non-linear finite element analysis, Comput Mech, 36, 2, 100-116 (2005) · Zbl 1102.74040 · doi:10.1007/s00466-004-0630-9 |
| [19] | Rempler, HU; Wieners, C.; Ehlers, W., Efficiency comparison of an augmented finite element formulation with standard return mapping algorithms for elastic-inelastic materials, Comput Mech, 48, 5, 551-562 (2011) · Zbl 1384.74044 · doi:10.1007/s00466-011-0602-9 |
| [20] | Braudel, H.; Abouaf, M.; Chenot, J., An implicit and incremental formulation for the solution of elastoplastic problems by the finite element method, Comput Struct, 22, 5, 801-814 (1986) · Zbl 0578.73066 · doi:10.1016/0045-7949(86)90269-5 |
| [21] | Braudel, H.; Abouaf, M.; Chenot, J., An implicit incrementally objective formulation for the solution of elastoplastic problems at finite strain by the F.E.M, Comput Struct, 24, 6, 825-843 (1986) · Zbl 0604.73045 · doi:10.1016/0045-7949(86)90292-0 |
| [22] | de Souza, Neto E.; Peric, D.; Owen, D., Computational methods for plasticity: theory and applications (2008), New York: Wiley, New York |
| [23] | Simo, J., Numerical analysis and simulation of plasticity, Numerical methods for solids (part 3) numerical methods for fluids (part 1), handbook of numerical analysis, 183-499 (1998), B.V.: Elsevier, B.V. · Zbl 0930.74001 · doi:10.1016/S1570-8659(98)80009-4 |
| [24] | Simo, J.; Hughes, T., Computational inelasticity (1998), New York: Springer, New York · Zbl 0934.74003 |
| [25] | Voce, E., A practical strain hardening function, Metallurgia, 51, 219-226 (1955) |
| [26] | Cook RD, Malkus DS, Plesha ME et al (2001) Concepts and applications of finite element analysis, 4th edn. Wiley, New York |
| [27] | Simo, J., A framework for finite strain elastoplasticity based on maximum plastic dissipation and the multiplicative decomposition: Part I. Continuum formulation, Comput Methods Appl Mech Eng, 66, 2, 199-219 (1988) · Zbl 0611.73057 · doi:10.1016/0045-7825(88)90076-X |
| [28] | Simo J (1988) A framework for finite strain elastoplasticity based on maximum plastic dissipation and the multiplicative decomposition. Part II: computational aspects. Comput Methods Appl Mech Eng 68(1):1-31 · Zbl 0644.73043 |
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