Virtual elements for finite thermo-plasticity problems. (English) Zbl 1464.74382
Summary: The paper outlines a multi-dimensional virtual element scheme for the coupled thermo-mechanical response of finite strain plasticity problems. The virtual element method (VEM) has been developed over the last decade and applied to problems in elasticity for small strains and other areas in the linear range. Enlargements of VEM to problems of compressible and incompressible nonlinear elasticity and finite plasticity have been reported in the last years. This work is further extending VEM to problems of 2D and 3D finite strain thermo-plasticity. Several numerical results substantiate our developments. For comparison purposes, results of different finite element discretization schemes are demonstrated as well.
MSC:
| 74S99 | Numerical and other methods in solid mechanics |
| 74C15 | Large-strain, rate-independent theories of plasticity (including nonlinear plasticity) |
| 74F05 | Thermal effects in solid mechanics |
Software:
AceFEMReferences:
| [1] | Argyris J, Doltsinis J (1981) On the natural formulation and analysis of large deformation coupled thermomechanical problems. Comput Methods Appl Mech Eng 25:195-253 · Zbl 0489.73126 |
| [2] | Argyris J, Doltsinis J (1982) Thermomechanical response of solids at high strains—natural approach. Comput Methods Appl Mech Eng 32:3-57 · Zbl 0505.73062 |
| [3] | Simó J, Miehe C (1992) Associative coupled thermoplasticity at finite strains: formulation, numerical analysis and implementation. Comput Methods Appl Mech Eng 98:41-104 · Zbl 0764.73088 |
| [4] | Wriggers P, Miehe C, Kleiber M, Simó J (1992) A thermomechanical approach to the necking problem. Int J Numer Methods Eng 33:869-883 · Zbl 0760.73070 |
| [5] | Stainier L, Cuitino A, Ortiz M (2002) A micromechanical model of hardening, rate sensitivity and thermal softening in bcc single crystals. J Mech Phys Solids 50(7):1511-1545 · Zbl 1071.74553 |
| [6] | Chapuis A, Driver JH (2011) Temperature dependency of slip and twinning in plane strain compressed magnesium single crystals. Acta Materialia 59(5):1986-1994 |
| [7] | Cereceda D, Diehl M, Roters F, Raabe D, Perlado JM, Marian J (2016) Unraveling the temperature dependence of the yield strength in single-crystal tungsten using atomistically-informed crystal plasticity calculations. Int J Plasticity 78:242-265 |
| [8] | Lion A (2000) Constitutive modelling in finite thermoviscoplasticity: a physical approach based on nonlinear rheological models. Int J Plasticity 16(5):469-494 · Zbl 0996.74022 |
| [9] | Anand L, Ames NM, Srivastava V, Chester SA (2009) A thermo-mechanically coupled theory for large deformations of amorphous polymers. Part i: formulation. Int J Plasticity 25:1474-1494 · Zbl 1165.74011 |
| [10] | Miehe C, Méndez Diez J, Göktepe S, Schänzel L (2011) Coupled thermoviscoplasticity of glassy polymers in the logarithmic strain space based on the free volume theory. Int J Solids Struct 48:1799-1817 |
| [11] | Canadija M, Mosler J (2011) On the thermomechanical coupling in finite strain plasticity theory with non-linear kinematic hardening by means of incremental energy minimization. Int J Solids Struct 48:1120-1129 · Zbl 1236.74032 |
| [12] | Bartels A, Bartel T, Canadija M, Mosler J (2015) On the thermomechanical coupling in dissipative materials: a variational approach for generalized standard materials. J Mech Phys Solids 82:218-234 |
| [13] | Yang Q, Stainier L, Ortiz M (2006) A variational formulation of the coupled thermo-mechanical boundary-value problem for general dissipative solids. J Mech Phys Solids 54:401-424 · Zbl 1120.74367 |
| [14] | Stainier L, Ortiz M (2010) Study and validation of thermomechanical coupling in finite strain visco-plasticity. Int J Solids Struct 47:704-715 · Zbl 1183.74037 |
| [15] | Wriggers P (2008) Nonlinear finite elements. Springer, Berlin · Zbl 1153.74001 |
| [16] | Zienkiewicz OC, Taylor RL (2000) The finite element method, vol 1, 5th edn. Butterworth-Heinemann, Oxford · Zbl 0991.74002 |
| [17] | Hughes T, Cottrell J, Bazilevs Y (2005) Isogeometric analysis: cad, finite elements, nurbs, exact geometry and mesh refinement. Comput Methods Appl Mech Eng 194(39):4135-4195 · Zbl 1151.74419 |
| [18] | Cottrell JA, Hughes TJR, Bazilevs Y (2009) Isogeometric analysis: toward integration of CAD and FEA. Wiley, Hoboken · Zbl 1378.65009 |
| [19] | Beirão Da Veiga L, Brezzi F, Cangiani A, Manzini G, Marini LD, Russo A (2013) Basic principles of virtual element methods. Math Models Methods Appl Sci 23(1):199-214 · Zbl 1416.65433 |
| [20] | Beirão Da Veiga L, Brezzi F, Marini LD (2013) Virtual elements for linear elasticity problems. SIAM J Numer Anal 51(2):794-812 · Zbl 1268.74010 |
| [21] | Gain AL, Talischi C, Paulino GH (2014) On the virtual element method for three-dimensional linear elasticity problems on arbitrary polyhedral meshes. Comput Methods Appl Mech Eng 282:132-160 · Zbl 1423.74095 |
| [22] | Artioli E, Beirão da Veiga L, Lovadina C, Sacco E (2017) Arbitrary order 2D virtual elements for polygonal meshes: part I, elastic problem. Comput Mech 60:355-377 · Zbl 1386.74132 |
| [23] | Wriggers P, Rust WT, Reddy BD (2016) A virtual element method for contact. Comput Mech 58:1039-1050 · Zbl 1398.74420 |
| [24] | Hudobivnik B, Aldakheel F, Wriggers P (2018) A low order 3d virtual element formulation for finite elasto-plastic deformations. Comput Mech. https://doi.org/10.1007/s00466-018-1593-6 · Zbl 1468.74085 |
| [25] | Wriggers P, Hudobivnik B (2017) A low order virtual element formulation for finite elasto-plastic deformations. Comput Methods Appl Mech Eng. https://doi.org/10.1016/j.cma.2017.08.053 · Zbl 1468.74085 |
| [26] | Wriggers P, Hudobivnik B, Korelc J (2018) Efficient low order virtual elements for anisotropic materials at finite strains. Springer, Cham, pp 417-434 · Zbl 1493.74112 |
| [27] | Wriggers P, Hudobivnik B, Schröder J (2018) Finite and virtual element formulations for large strain anisotropic material with inextensive fibers. Springer, Cham, pp 205-231 |
| [28] | Reddy BD, van Huyssteen D (2019) A virtual element method for transversely isotropic elasticity. Comput Mech. https://doi.org/10.1007/s00466-019-01690-7 · Zbl 1462.74159 |
| [29] | Bellis MLD, Wriggers P, Hudobivnik B, Zavarise G (2018) Virtual element formulation for isotropic damage. Finite Elem Anal Des 144:38-48 |
| [30] | Taylor RL, Artioli E (2018) VEM for inelastic solids. Springer, Cham, pp 381-394 · Zbl 1493.74111 |
| [31] | Chi H, Beirão da Veiga L, Paulino G (2016) Some basic formulations of the virtual element method (VEM) for finite deformations. Comput Methods Appl Mech Eng. https://doi.org/10.1016/j.cma.2016.12.020 · Zbl 1439.74397 |
| [32] | Wriggers P, Reddy BD, Rust W, Hudobivnik B (2017) Efficient virtual element formulations for compressible and incompressible finite deformations. Comput Mech 60(2):253-268 · Zbl 1386.74146 |
| [33] | Hussein A, Aldakheel F, Hudobivnik B, Wriggers P, Guidault P-A, Allix O (2019) A computational framework for brittle crack-propagation based on efficient virtual element method. Finite Elem Anal Des 159:15-32 |
| [34] | Aldakheel F, Hudobivnik B, Hussein A, Wriggers P (2018) Phase-field modeling of brittle fracture using an efficient virtual element scheme. Comput Methods Appl Mech Eng 341:443-466 · Zbl 1440.74352 |
| [35] | Aldakheel F, Hudobivnik B, Wriggers P (2018) Virtual element formulation for phase-field modeling of ductile fracture. Int J Multiscale Comput Eng. https://doi.org/10.1615/IntJMultCompEng.2018026804 · Zbl 1468.74085 |
| [36] | Aldakheel F (2016) Mechanics of nonlocal dissipative solids: gradient plasticity and phase field modeling of ductile fracture. PhD thesis, Institute of Applied Mechanics (CE), Chair I, University of Stuttgart. https://doi.org/10.18419/opus-8803 |
| [37] | Korelc J, Wriggers P (2016) Automation of finite element methods. Springer, Berlin · Zbl 1367.74001 |
| [38] | Aldakheel F, Miehe C (2017) Coupled thermomechanical response of gradient plasticity. Int J Plasticity 91:1-24 |
| [39] | Aldakheel F (2017) Micromorphic approach for gradient-extended thermo-elastic-plastic solids in the logarithmic strain space. Contin Mech Thermodyn 29(6):1207-1217 · Zbl 1392.74069 |
| [40] | Korelc J, Stupkiewicz S (2014) Closed-form matrix exponential and its application in finite-strain plasticity. Int J Numer Methods Eng 98:960-987 · Zbl 1352.74482 |
| [41] | Wriggers P, Reddy B, Rust W, Hudobivnik B (2017) Efficient virtual element formulations for compressible and incompressible finite deformations. Comput Mech 60:253-268 · Zbl 1386.74146 |
| [42] | Wriggers, P.; Hudobivnik, B.; Schröder, J.; Soric, J. (ed.); Wriggers, P. (ed.), Finite and virtual element formulations for large strain anisotropic material with inextensive fibers (2017), Cham |
| [43] | Simó JC (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 68:1-31 · Zbl 0644.73043 |
| [44] | Hallquist JO (1984) Nike 2D: An implicit, finite deformation, finite element code for analyzing the static and dynamic response of two-dimensional solids. Rept. UCRL-52678, Lawrence Livermore National Laboratory, University of California, Livermore, CA |
| [45] | Simó JC, Taylor RL, Pister KS (1985) Variational and projection methods for the volume constraint in finite deformation elasto-plasticity. Comput Methods Appl Mech Eng 51:177-208 · Zbl 0554.73036 |
| [46] | Miehe C, Aldakheel F, Mauthe S (2013) Mixed variational principles and robust finite element implementations of gradient plasticity at small strains. Int J Numer Methods Eng 94:1037-1074 · Zbl 1352.74408 |
| [47] | Kienle D, Aldakheel F, Keip M-A (2019) A finite-strain phase-field approach to ductile failure of frictional materials. Int J Solids Struct. https://doi.org/10.1016/j.ijsolstr.2019.02.006 |
| [48] | Wriggers P, Korelc J (1996) On enhanced strain methods for small and finite deformations of solids. Comput Mech 18:413-428 · Zbl 0894.73179 |
| [49] | Korelc J, Solinc U, Wriggers P (2010) An improved EAS brick element for finite deformation. Comput Mech 46:641-659 · Zbl 1358.74059 |
| [50] | Wriggers P, Reese S (1996) A note on enhanced strain methods for large deformations. Comput Methods Appl Mech Eng 135:201-209 · Zbl 0893.73072 |
| [51] | Aldakheel F, Mauthe S, Miehe C (2014) Towards phase field modeling of ductile fracture in gradient-extended elastic-plastic solids. Proc Appl Math Mech 14:411-412 |
| [52] | Miehe C, Welschinger F, Aldakheel F (2014) Variational gradient plasticity at finite strains. Part II: local-global updates and mixed finite elements for additive plasticity in the logarithmic strain space. Comput Methods Appl Mech Eng 268:704-734 · Zbl 1295.74014 |
| [53] | Dittmann M, Krüger M, Schmidt F, Schuß S, Hesch C (2018) Variational modeling of thermomechanical fracture and anisotropic frictional mortar contact problems with adhesion. Comput Mech. https://doi.org/10.1007/s00466-018-1610-9 · Zbl 1469.74097 |
| [54] | Soyarslan C, Bargmann S (2016) Thermomechanical formulation of ductile damage coupled to nonlinear isotropic hardening and multiplicative viscoplasticity. J Mech Phys Solids 91:334-358 |
| [55] | Aldakheel F, Wriggers P, Miehe C (2018) A modified gurson-type plasticity model at finite strains: formulation, numerical analysis and phase-field coupling. Comput Mech 62(4):815-833 · Zbl 1459.74024 |
| [56] | Voyiadjis Z, Faghihi D (2012) Thermo-mechanical strain gradient plasticity with energetic and dissipative length scales. Int J Plasticity 30-31:218-247 |
| [57] | Fohrmeister V, Bartels A, Mosler J (2018) Variational updates for thermomechanically coupled gradient-enhanced elastoplasticity—implementation based on hyper-dual numbers. Comput Methods Appl Mech Eng 339:239-261 · Zbl 1440.74082 |
| [58] | Voyiadjis Z, Song Y (2019) Strain gradient continuum plasticity theories: theoretical, numerical and experimental investigations. Int J Plasticity. https://doi.org/10.1016/j.ijplas.2019.03.002 |
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