The exponentiated Hencky-logarithmic strain energy. III: Coupling with idealized multiplicative isotropic finite strain plasticity. (English) Zbl 1348.74053
Summary: We investigate an immediate application in finite strain multiplicative plasticity of the family of isotropic volumetric-isochoric decoupled strain energies
\[
F\mapsto W_{\mathrm{eH}}(F):=\widehat W_{\mathrm{eH}}(U):=\begin{cases} \frac{\mu}{k}e^{k\|\mathrm{dev}_n\log U\|^2}+\frac{\kappa}{2\widehat k}^{e^{\widehat k[\mathrm{tr}(\log U)]^2}}\quad &\text{if }\det F>0,\\ +\infty\quad & \text{if }\det F\leq 0,\end{cases}
\]
based on the Hencky-logarithmic (true, natural) strain tensor \(\log U\) . Here, \(\mu>0\) is the infinitesimal shear modulus, \(\kappa=\frac{2\mu+3\lambda}{3}>0\) is the infinitesimal bulk modulus with \(\lambda\) the first Lamé constant, \(k,\widehat k\) are additional dimensionless material parameters, \(F=\nabla\varphi\) is the gradient of deformation, \(U=\sqrt{F^TF}\) is the right stretch tensor, and \(\mathrm{dev}_n\log U=\log U-\frac{1}{n}\) \(\mathrm{tr}(\log U)\cdot\mathbf 1\) is the deviatoric part of the strain tensor \(\log U\) . Based on the multiplicative decomposition \(F=F_eF_p\), we couple these energies with some isotropic elasto-plastic flow rules \(F_p\frac{\mathrm d}{\mathrm{dt}}[F_p^{-1}]\in-\partial\chi(\mathrm{dev}_3\Sigma_e)\) defined in the plastic distortion \(F_p\), where \(\partial\chi\) is the subdifferential of the indicator function \(\chi\) of the convex elastic domain \(\mathcal E_{\mathrm{e}}(\Sigma_e,\frac{1}{3}\sigma_{\mathbf y}^2)\) in the mixed-variant \(\Sigma_e\)-stress space, \(\Sigma_e=F_e^TD_{F_e}W_{\mathrm{iso}}(F_e)\), and \(W_{\mathrm{iso}}(F_e)\) represents the isochoric part of the energy. While \(W_{\mathrm{eH}}\) may loose ellipticity, we show that loss of ellipticity is effectively prevented by the coupling with plasticity, since the ellipticity domain of \(W_{\mathrm{eH}}\) on the one hand and the elastic domain in \(\Sigma_e\)-stress space on the other hand are closely related. Thus, the new formulation remains elliptic in elastic unloading at any given plastic predeformation. In addition, in this domain, the true stress-true strain relation remains monotone, as observed in experiments.
For part I, II see [P. Neff et al., J. Elasticity 121, No. 2, 143–234 (2015; Zbl 1325.74028); Z. Angew. Math. Phys. 66, No. 4, 1671–1693 (2015; Zbl 1320.74022)].
For part I, II see [P. Neff et al., J. Elasticity 121, No. 2, 143–234 (2015; Zbl 1325.74028); Z. Angew. Math. Phys. 66, No. 4, 1671–1693 (2015; Zbl 1320.74022)].
Keywords:
logarithmic strain; Hencky energy; multiplicative decomposition; elasto-plasticity; bounded elastic distortions; nine-dimensional flow rule; associated plasticity; convex elastic domainReferences:
| [1] | Armero F., Simo J.C.: A priori stability estimates and unconditionally stable product formula algorithms for nonlinear coupled thermoplasticity. Int. J. Plast. 9(6), 749-782 (1993) · Zbl 0791.73026 · doi:10.1016/0749-6419(93)90036-P |
| [2] | Balzani, D.; Schröder, J.; Gross, D.; Neff, P.; Owen, D. R.J. (ed.); Onate, E. (ed.); Suarez, B. (ed.), Modeling of anisotropic damage in arterial walls based on polyconvex stored energy functions, 802-805 (2005), Barcelona |
| [3] | Bertram A.: An alternative approach to finite plasticity based on material isomorphisms. Int. J. Plast. 15(3), 353-374 (1999) · Zbl 1016.74009 · doi:10.1016/S0749-6419(98)00074-6 |
| [4] | Bertram, A.: Elasticity and Plasticity of Large Deformations: An Introduction. Springer, Berlin (2012) · Zbl 1231.74003 |
| [5] | Bruhns O.T., Xiao H., Mayers A.: Constitutive inequalities for an isotropic elastic strain energy function based on Hencky’s logarithmic strain tensor. Proc. R. Soc. Lond. A 457, 2207-2226 (2001) · Zbl 1048.74505 · doi:10.1098/rspa.2001.0818 |
| [6] | Bruhns O.T., Xiao H., Mayers A.: Finite bending of a rectangular block of an elastic Hencky material. J. Elast. 66(3), 237-256 (2002) · Zbl 1078.74517 · doi:10.1023/A:1021959329598 |
| [7] | Bruhns O.T., Xiao H., Meyers A.: Self-consistent Eulerian rate type elasto-plasticity models based upon the logarithmic stress rate. Int. J. Plast. 15(5), 479-520 (1999) · Zbl 1036.74010 · doi:10.1016/S0749-6419(99)00003-0 |
| [8] | Caminero M.A., Montáns F.J., Bathe K.J.: Modeling large strain anisotropic elasto-plasticity with logarithmic strain and stress measures. Comput. Struct. 89(11), 826-843 (2011) · doi:10.1016/j.compstruc.2011.02.011 |
| [9] | Carstensen C., Hackl K., Mielke A.: Non-convex potentials and microstructures in finite-strain plasticity. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 458, 299-317 (2002) · Zbl 1008.74016 · doi:10.1098/rspa.2001.0864 |
| [10] | Cleja-Ţigoiu S.: Consequences of the dissipative restrictions in finite anisotropic elasto-plasticity. Int. J. Plast. 19(11), 1917-1964 (2003) · Zbl 1059.74014 · doi:10.1016/S0749-6419(03)00045-7 |
| [11] | Cleja-Ţigoiu S., Iancu L.: Orientational anisotropy and strength-differential effect in orthotropic elasto-plastic materials. Int. J. Plast. 47, 80-110 (2013) · doi:10.1016/j.ijplas.2013.01.005 |
| [12] | Cleja-Ţigoiu S., Maugin G.A.: Eshelby’s stress tensors in finite elastoplasticity. Acta Mech. 139(1-4), 231-249 (2000) · Zbl 0951.74009 · doi:10.1007/BF01170191 |
| [13] | Dettmer W., Reese S.: On the theoretical and numerical modelling of Armstrong-Frederick kinematic hardening in the finite strain regime. Comput. Methods Appl. Mech. Eng. 193(1), 87-116 (2004) · Zbl 1063.74020 · doi:10.1016/j.cma.2003.09.005 |
| [14] | Dvorkin E.N., Pantuso D., Repetto E.A.: A finite element formulation for finite strain elasto-plastic analysis based on mixed interpolation of tensorial components. Comput. Methods Appl. Mech. Eng. 114(1), 35-54 (1994) · doi:10.1016/0045-7825(94)90161-9 |
| [15] | Ebobisse F., Neff P.: Existence and uniqueness for rate-independent infinitesimal gradient plasticity with isotropic hardening and plastic spin. Math. Mech. Solids 15(6), 691-703 (2010) · Zbl 1257.74023 · doi:10.1177/1081286509342269 |
| [16] | Eterovic A.L., Bathe K.-J.: A hyperelastic-based large strain elasto-plastic constitutive formulation with combined isotropic-kinematic hardening using the logarithmic stress and strain measures. Int. J. Numer. Methods Eng. 30, 1099-1114 (1990) · Zbl 0714.73035 · doi:10.1002/nme.1620300602 |
| [17] | Gabriel G., Bathe K.J.: Some computational issues in large strain elasto-plastic analysis. Comput. Struct. 56(2), 249-267 (1995) · Zbl 0918.73148 · doi:10.1016/0045-7949(95)00019-D |
| [18] | Ghiba I.D., Neff P., Martin R.J.: On the rank-one convexity domain for isochoric, isotropic and quadratic energies defined in logarithmic strain tensor (in preparation) (2015) · Zbl 1194.74026 |
| [19] | Ghiba, I.D., Neff, P., Šilhavý, M.: The exponentiated Hencky-logarithmic strain energy. Improvement of planar polyconvexity. Int. J. Non-Linear Mech. 71, 48-51 (2015) · Zbl 1320.74022 |
| [20] | Geers M.G.D.: Finite strain logarithmic hyperelasto-plasticity with softening: a strongly non-local implicit gradient framework. Comput. Methods Appl. Mech. Eng. 193(30), 3377-3401 (2004) · Zbl 1060.74503 · doi:10.1016/j.cma.2003.07.014 |
| [21] | Glüge R., Kalisch J.: Graphical representations of the regions of rank-one-convexity of some strain energies. Tech. Mech. 32, 227-237 (2012) |
| [22] | Gupta A., Steigmann D.J., Stölken J.S.: On the evolution of plasticity and incompatibility. Math. Mech. Solids 12(6), 583-610 (2007) · Zbl 1133.74009 · doi:10.1177/1081286506064721 |
| [23] | Gupta A., Steigmann D.J., Stölken J.S.: Aspects of the phenomenological theory of elastic-plastic deformation. J. Elast. 104(1-2), 249-266 (2011) · Zbl 1320.74027 · doi:10.1007/s10659-010-9288-z |
| [24] | Gurtin M.E., Spear K.: On the relationship between the logarithmic strain rate and the stretching tensor. Int. J. Solids Struct. 19(5), 437-444 (1983) · Zbl 0525.73048 · doi:10.1016/0020-7683(83)90054-9 |
| [25] | Hanin M., Reiner M.: On isotropic tensor-functions and the measure of deformation. Z. Angew. Math. Phys. 7(5), 377-393 (1956) · Zbl 0073.18802 · doi:10.1007/BF01606325 |
| [26] | Heiduschke K.: The logarithmic strain space description. Int. J. Solids Struct. 32(8), 1047-1062 (1995) · Zbl 0865.73016 · doi:10.1016/0020-7683(94)00186-Z |
| [27] | Heiduschke K.: Computational aspects of the logarithmic strain space description. Int. J. Solids Struct. 33(5), 747-760 (1996) · Zbl 0901.73073 · doi:10.1016/0020-7683(95)00058-I |
| [28] | Henann D., Anand L.: A large deformation theory for rate-dependent elastic-plastic materials with combined isotropic and kinematic hardening. Int. J. Plast. 25(10), 1833-1878 (2009) · doi:10.1016/j.ijplas.2008.11.008 |
| [29] | Horák, M., Jirásek, M.: An extension of small-strain models to the large-strain range based on an additive decomposition of a logarithmic strain. In: Chleboun, J., Segeth, K., Šıstek, J., Vejchodskỳ, T. (eds.) Programs and Algorithms of Numerical Mathematics, vol. 16, pp. 88-93 (2013) · Zbl 1340.74012 |
| [30] | Hutchinson, J.W., Neale, K.W.: Finite strain J2-deformation theory. In: Carlson, D.E., Shield, R.T. (eds.) Proceedings of the IUTAM Symposium on Finite Elasticity, pp. 237-247. Martinus Nijhoff (1982) https://www.uni-due.de/imperia/md/content/mathematik/ag_neff/hutchinson_ellipticity80. · Zbl 1342.74028 |
| [31] | Jog C.S.: Foundations and Applications of Mechanics: Continuum Mechanics, vol. 1. CRC Press, Boca Raton (2002) |
| [32] | Jog C.S.: On the explicit determination of the polar decomposition in n-dimensional vector spaces. J. Elast. 66(2), 159-169 (2002) · Zbl 1026.74014 · doi:10.1023/A:1021253906202 |
| [33] | Jog C.S., Patil K.D.: Conditions for the onset of elastic and material instabilities in hyperelastic materials. Arch. Appl. Mech. 83, 1-24 (2013) · Zbl 1293.74121 · doi:10.1007/s00419-012-0711-8 |
| [34] | Krishnan J., Steigmann D.: A polyconvex formulation of isotropic elastoplasticity theory. IMA J. Appl. Math. 79, 722-738 (2014) · Zbl 1305.74026 · doi:10.1093/imamat/hxt049 |
| [35] | Kröner E.: Der fundamentale Zusammenhang zwischen Versetzungsdichte und Spannungsfunktionen. Z. Angew. Math. Phys. 142(4), 463-475 (1955) · Zbl 0068.40803 |
| [36] | Kröner E.: Kontinuumstheorie der Versetzungen und Eigenspannungen. Springer, Berlin (1958) · Zbl 0084.40003 · doi:10.1007/978-3-642-94719-3 |
| [37] | Kröner, E.: Allgemeine Kontinuumstheorie der Versetzungen und Eigenspannungen. Arch. Rat. Mech. Anal. 4(1), 273-334 (1959) · Zbl 0090.17601 |
| [38] | Lankeit J., , Neff P., Nakatsukasa Y.: The minimization of matrix logarithms: on a fundamental property of the unitary polar factor. Linear Algebra Appl. 449(0), 28-42 (2014) · Zbl 1302.15009 · doi:10.1016/j.laa.2014.02.012 |
| [39] | Lankeit J., Neff P., Pauly D.: Uniqueness of integrable solutions to \[{\bigtriangledown \zeta=G \zeta,\ \zeta|_{\Gamma}=0}\]▽ζ=Gζ,ζ|Γ=0 for integrable tensor coefficients G and applications to elasticity. Z. Angew. Math. Phys. 64, 1679-1688 (2013) · Zbl 1414.35046 · doi:10.1007/s00033-013-0314-4 |
| [40] | Lee, E.H.: Elastic-plastic deformation at finite strains. J. Appl. Mech. 36(1), 1-6 (1969) · Zbl 0179.55603 |
| [41] | Martin, R.J., Neff, P.: Minimal geodesics on GL(n) for left-invariant, right-O(n)-invariant Riemannian metrics. Preprint arXiv:1409.7849 (2014) · Zbl 1366.53028 |
| [42] | Masud A., Panahandeh M., Aurrichio F.: A finite-strain finite element model for the pseudoelastic behavior of shape memory alloys. Comput. Methods Appl. Mech. Eng. 148(1), 23-37 (1997) · Zbl 0924.73215 · doi:10.1016/S0045-7825(97)00080-7 |
| [43] | Maugin G.: Eshelby stress in elastoplasticity and ductile fracture. Int. J. Plast. 10(4), 393-408 (1994) · Zbl 0808.73026 · doi:10.1016/0749-6419(94)90040-X |
| [44] | Meyers A.: On the consistency of some Eulerian strain rates. Z. Angew. Math. Mech. 79(3), 171-177 (1999) · Zbl 0983.37111 · doi:10.1002/(SICI)1521-4001(199903)79:3<171::AID-ZAMM171>3.0.CO;2-6 |
| [45] | Meyers A., Xiao H., Bruhns O.T.: Choice of objective rate in single parameter hypoelastic deformation cycles. Comput. Struct. 84(17), 1134-1140 (2006) · doi:10.1016/j.compstruc.2006.01.012 |
| [46] | Miehe, C.: Kanonische Modelle multiplikativer Elasto-Plastizität. Thermodynamische Formulierung und numerische Implementation. Habilitationsschrift, Universität Hannover, Germany (1992) · Zbl 1068.74522 |
| [47] | Miehe C.: On the representation of Prandtl-Reuss tensors within the framework of multiplicative elastoplasticity. Int. J. Plast. 10(6), 609-621 (1994) · Zbl 0810.73016 · doi:10.1016/0749-6419(94)90025-6 |
| [48] | Miehe C.: Variational gradient plasticity at finite strains. Part I: mixed potentials for the evolution and update problems of gradient-extended dissipative solids. Comput. Methods Appl. Mech. Eng. 268, 677-703 (2014) · Zbl 1295.74013 · doi:10.1016/j.cma.2013.03.014 |
| [49] | 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. Eng. 191(47), 5383-5425 (2002) · Zbl 1083.74518 · doi:10.1016/S0045-7825(02)00438-3 |
| [50] | Miehe C., Welschinger F., Aldakheel F.: 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 (2014) · Zbl 1295.74014 · doi:10.1016/j.cma.2013.07.015 |
| [51] | Mielke A.: Finite elastoplasticity Lie groups and geodesics on SL (d). In: Newton, P., Holmes, P., Weinstein, A. (eds) Geometry, Mechanics, and Dynamics, pp. 61-90. Springer, Berlin (2002) · Zbl 1146.74309 |
| [52] | Mielke A.: Energetic formulation of multiplicative elasto-plasticity using dissipation distances. Contin. Mech. Thermodyn. 15(4), 351-382 (2003) · Zbl 1068.74522 · doi:10.1007/s00161-003-0120-x |
| [53] | Montella, G., Govindjee, S., Neff, P.: The exponentiated-Hencky strain energy in modeling tire derived material for moderately large deformation (submitted) (2015) · Zbl 1295.74014 |
| [54] | Mosler J., Ortiz M.: Variational h-adaption in finite deformation elasticity and plasticity. Int. J. Numer. Methods Eng. 72(5), 505-523 (2007) · Zbl 1194.74451 · doi:10.1002/nme.2011 |
| [55] | Müller Ch., Bruhns O.T.: A thermodynamic finite-strain model for pseudoelastic shape memory alloys. Int. J. Plast. 22(9), 1658-1682 (2006) · Zbl 1099.74050 · doi:10.1016/j.ijplas.2006.02.010 |
| [56] | Naghdabadi R., Baghani M., Arghavani J.: A viscoelastic constitutive model for compressible polymers based on logarithmic strain and its finite element implementation. Finite Elem. Anal. Des. 62, 18-27 (2012) · doi:10.1016/j.finel.2012.05.001 |
| [57] | Neff, P.: Mathematische analyse multiplikativer Viskoplastizität. Ph.D. thesis, Technische Universität Darmstadt. Shaker Verlag. ISBN:3-8265-7560-1. https://www.uni-due.de/ hm0014/Download_files/cism_convexity08. Aachen (2000) · Zbl 0970.74002 |
| [58] | Neff P.: On Korn’s first inequality with nonconstant coefficients. Proc. Roy. Soc. Edinb. A 132, 221-243 (2002) · Zbl 1143.74311 · doi:10.1017/S0308210500001591 |
| [59] | Neff P.: Finite multiplicative plasticity for small elastic strains with linear balance equations and grain boundary relaxation. Contin. Mech. Thermodyn. 15(2), 161-195 (2003) · Zbl 1035.74015 · doi:10.1007/s00161-002-0109-x |
| [60] | Neff, P.: Some results concerning the mathematical treatment of finite plasticity. In: Deformation and Failure in Metallic Materials, pp. 251-274. Springer (2003) · Zbl 1263.74017 |
| [61] | Neff P.: Local existence and uniqueness for quasistatic finite plasticity with grain boundary relaxation. Q. Appl. Math. 63, 88-116 (2005) · Zbl 1072.74013 · doi:10.1090/S0033-569X-05-00953-9 |
| [62] | Neff P.: A finite-strain elastic-plastic Cosserat theory for polycrystals with grain rotations. Int. J. Eng. Sci. 44, 574-594 (2006) · Zbl 1213.74032 · doi:10.1016/j.ijengsci.2006.04.002 |
| [63] | Neff P., Chełmiński K., Alber H.D.: Notes on strain gradient plasticity: finite strain covariant modelling and global existence in the infinitesimal rate-independent case. Math. Mod. Methods Appl. Sci. 19, 307-346 (2009) · Zbl 1160.74009 · doi:10.1142/S0218202509003449 |
| [64] | Neff P., Chełmiński K., Müller W., Wieners C.: A numerical solution method for an infinitesimal elasto-plastic Cosserat model. Math. Mod. Methods Appl. Sci. 17(08), 1211-1239 (2007) · Zbl 1137.74012 · doi:10.1142/S021820250700225X |
| [65] | Neff, P., Eidel, B., Martin, R.J: Geometry of logarithmic strain measures in solid mechanics (submitted) (2015) · Zbl 1348.74039 |
| [66] | Neff P., Eidel B., Osterbrink F., Martin R.J.: The Hencky strain energy \[{\|\log{U}\|^2} \]‖logU‖2 measures the geodesic distance of the deformation gradient to SO(3) in the canonical left-invariant Riemannian metric on GL(3). PAMM 13(1), 369-370 (2013) · doi:10.1002/pamm.201310180 |
| [67] | Neff, P., Ghiba, I.D.: Loss of ellipticity in additive logarithmic finite strain plasticity. arXiv:1410.2819 (submitted) (2014) |
| [68] | Neff, P., Ghiba, I.D., Lankeit, J.: The exponentiated Hencky-logarithmic strain energy. Part I: constitutive issues and rank-one convexity. J. Elast. (2015). doi:10.1007/s10659-015-9524-7 · Zbl 1325.74028 |
| [69] | Neff, P., Ghiba, I.D., Lankeit, J., Martin, R.J., Steigmann, D.: The exponentiated Hencky-logarithmic strain energy. Part II: coercivity, planar polyconvexity and existence of minimizers. Z. Angew. Math. Phys. arXiv:1408.4430 (2015). doi:10.1007/s00033-015-0495-0 · Zbl 1320.74022 |
| [70] | Neff P., Knees D.: Regularity up to the boundary for nonlinear elliptic systems arising in time-incremental infinitesimal elasto-plasticity. SIAM J. Math. Anal. 40(1), 21-43 (2008) · Zbl 1193.35224 · doi:10.1137/070695824 |
| [71] | Neff P., Müller W., Wieners C.: Parallel simulation of an infinitesimal elasto-plastic Cosserat model. GAMM-Mitteilungen 33(1), 79-94 (2010) · Zbl 1342.74028 · doi:10.1002/gamm.201010007 |
| [72] | Neff P., Nakatsukasa Y., Fischle A.: A logarithmic minimization property of the unitary polar factor in the spectral norm and the Frobenius matrix norm. SIAM J. Matrix Anal. 35, 1132-1154 (2014) · Zbl 1309.15016 · doi:10.1137/130909949 |
| [73] | Neff P., Sydow A., Wieners C.: Numerical approximation of incremental infinitesimal gradient plasticity. Int. J. Numer. Methods Eng. 77(3), 414-436 (2009) · Zbl 1155.74316 · doi:10.1002/nme.2420 |
| [74] | Neff P., Wieners C.: Comparison of models for finite plasticity. A numerical study. Comput. Vis. Sci. 6, 23-35 (2003) · Zbl 1129.74303 · doi:10.1007/s00791-003-0104-1 |
| [75] | Ogden R.W.: Compressible isotropic elastic solids under finite strain-constitutive inequalities. Q. J. Mech. Appl. Math. 23(4), 457-468 (1970) · Zbl 0215.57401 · doi:10.1093/qjmam/23.4.457 |
| [76] | Ogden, R.W.: Non-linear Elastic Deformations. Mathematics and its Applications, vol. 1. Ellis Horwood, Chichester (1983) · Zbl 1060.74503 |
| [77] | Papadopoulos P., Lu J.: A general framework for the numerical solution of problems in finite elasto-plasticity. Comput. Methods Appl. Mech. Eng. 159(1), 1-18 (1998) · Zbl 0954.74068 · doi:10.1016/S0045-7825(98)80101-1 |
| [78] | Perić D., Souza Neto E.A.: A new computational model for Tresca plasticity at finite strains with an optimal parametrization in the principal space. Comput. Methods Appl. Mech. Eng. 171(3), 463-489 (1999) · Zbl 0958.74070 |
| [79] | Perić D., Owen D.R.J., Honnor M.E.: A model for finite strain elasto-plasticity based on logarithmic strains: computational issues. Comput. Methods Appl. Mech. Eng. 94(1), 35-61 (1992) · Zbl 0747.73020 · doi:10.1016/0045-7825(92)90156-E |
| [80] | Raoult A.: Non-polyconvexity of the stored energy function of a St.Venant-Kirchhoff material. Aplikace Matematiky 6, 417-419 (1986) · Zbl 0608.73023 |
| [81] | Reese S., Christ D.: Finite deformation pseudo-elasticity of shape memory alloys—constitutive modelling and finite element implementation. Int. J. Plast. 24(3), 455-482 (2008) · Zbl 1145.74005 · doi:10.1016/j.ijplas.2007.05.005 |
| [82] | Reese S., Wriggers P.: A material model for rubber-like polymers exhibiting plastic deformation: computational aspects and a comparison with experimental results. Comput. Methods Appl. Mech. Eng. 148, 279-298 (1997) · Zbl 0921.73155 · doi:10.1016/S0045-7825(97)00034-0 |
| [83] | Sansour C.: On the dual variable of the logarithmic strain tensor, the dual variable of the Cauchy stress tensor, and related issues. Int. J. Solids Struct. 38(50), 9221-9232 (2001) · Zbl 0999.74006 · doi:10.1016/S0020-7683(01)00073-7 |
| [84] | Sansour C.: On the physical assumptions underlying the volumetric-isochoric split and the case of anisotropy. Eur. J. Mech. A Solids 27(1), 28-39 (2008) · Zbl 1129.74009 · doi:10.1016/j.euromechsol.2007.04.001 |
| [85] | Sansour C., Wagner W.: Viscoplasticity based on additive decomposition of logarithmic strain and unified constitutive equations: theoretical and computational considerations with reference to shell applications. Comput. Struct. 81(15), 1583-1594 (2003) · doi:10.1016/S0045-7949(03)00149-4 |
| [86] | Schröder, J., Neff, P.: Poly, quasi and rank-one convexity in mechanics. CISM-Course Udine. Springer (2009) · Zbl 0791.73026 |
| [87] | Shutov A.V., Ihlemann J.: Analysis of some basic approaches to finite strain elasto-plasticity in view of reference change. Int. J. Plast 63, 183-197 (2014) · doi:10.1016/j.ijplas.2014.07.004 |
| [88] | Shutov A.V., Kreißig R.: Finite strain viscoplasticity with nonlinear kinematic hardening: phenomenological modeling and time integration. Comput. Methods Appl. Mech. Eng. 197(21), 2015-2029 (2008) · Zbl 1194.74026 · doi:10.1016/j.cma.2007.12.017 |
| [89] | 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. Eng. 99(1), 61-112 (1992) · Zbl 0764.73089 · doi:10.1016/0045-7825(92)90123-2 |
| [90] | Simo, J. C.; Stein, E. (ed.), Recent developments in the numerical analysis of plasticity, 115-173 (1993), New York · doi:10.1007/978-3-7091-2626-4_3 |
| [91] | Simo, J. C.; Ciarlet, P. G. (ed.); Lions, J. L. (ed.), Numerical analysis and simulation of plasticity (1998), Amsterdam · Zbl 0930.74001 |
| [92] | Simo, J.C., Hughes, J.R.: Computational Inelasticity, vol. 7, Interdisciplinary Applied Mathematics. Springer, Berlin (1998) · Zbl 0934.74003 |
| [93] | Simo J.C., Ortiz M.: A unified approach to finite deformation elastoplastic analysis based on the use of hyperelastic constitutive equations. Comput. Methods Appl. Mech. Eng. 49, 221-245 (1985) · Zbl 0566.73035 · doi:10.1016/0045-7825(85)90061-1 |
| [94] | Steigmann D.J., Gupta A.: Mechanically equivalent elastic-plastic deformations and the problem of plastic spin. Theor. Appl. Mech. 38(4), 397-417 (2011) · Zbl 1299.74030 · doi:10.2298/TAM1104397S |
| [95] | Tanaka E.: Finite element investigation of the problem of large strains, formulated in terms of true stress and logarithmic strain. Acta Mech. 34(1-2), 129-141 (1979) · Zbl 0426.73072 · doi:10.1007/BF01176262 |
| [96] | Vallée C.: Lois de comportement élastique isotropes en grandes déformations. Int. J. Eng. Sci. 16(7), 451-457 (1978) · Zbl 0381.73021 · doi:10.1016/0020-7225(78)90078-2 |
| [97] | Vallée C., Fortuné D., Lerintiu C.: On the dual variable of the Cauchy stress tensor in isotropic finite hyperelasticity. Comptes Rendus Mecanique 336(11), 851-855 (2008) · Zbl 1154.74303 · doi:10.1016/j.crme.2008.10.003 |
| [98] | Xiao H., Bruhns O., Meyers A.: A consistent finite elastoplasticity theory combining additive and multiplicative decomposition of the stretching and the deformation gradient. Int. J. Plast. 16(2), 143-177 (2000) · Zbl 1005.74015 · doi:10.1016/S0749-6419(99)00045-5 |
| [99] | Xiao H., Bruhns O.T., Meyers A.: Existence and uniqueness of the integrable-exactly hypoelastic equation \[{{\tau}^{\circ}=\lambda ({\rm tr} D){\rm{I}}+ 2\mu{D}}\] τ∘=λ(trD)I+2μD and its significance to finite inelasticity. Acta Mech. 138(1-2), 31-50 (1999) · Zbl 0978.74011 · doi:10.1007/BF01179540 |
| [100] | Xiao H., Bruhns O.T., Meyers A.: Elastoplasticity beyond small deformations. Acta Mech. 182(1-2), 31-111 (2006) · Zbl 1116.74005 · doi:10.1007/s00707-005-0282-7 |
| [101] | Zhu Y., Kang G., Kan Q., Bruhns O.: Logarithmic stress rate based constitutive model for cyclic loading in finite plasticity. Int. J. Plast. 54, 34-55 (2014) · doi:10.1016/j.ijplas.2013.08.004 |
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