Acta Mechanica

, Volume 230, Issue 5, pp 1565–1606

# Micropolar plasticity—Part I: modeling based on curvature tensors related by mixed transformations

• Daniel Johannsen
• Charalampos Tsakmakis
Original Paper

## Abstract

When formulating finite deformation micropolar plasticity, the structure of the theory strongly depends on the choice of the variables describing the deformation kinematics. This holds true even for classical plasticity. However, in contrast to classical plasticity, the set of kinematical variables in micropolar plasticity includes, besides strain tensors, so-called micropolar curvature tensors. There are only a few investigations addressing such aspects, so the aim of the paper is to highlight the effect of a specific micropolar curvature kinematics on the structure of a micropolar plasticity. We do this by developing a general finite deformation micropolar plasticity, which relies upon a class of micropolar curvature tensors related to each other by mixed transformations. That means, the pull-back and push-forward transformations characterizing the class involve both deformation gradient and micropolar rotation tensors. The curvature kinematics is discussed by using geometrical methods developed previously. The plasticity theory is based on the assumption that the yield function and the flow rules are functions of specific micropolar Mandel’s stress tensors. The definition of the Mandel’s stress tensor is suggested by the adopted curvature kinematics and reveals a characteristic feature of the resulting plasticity. Moreover, the presence of curvature variables in plastic arc length gives reason to introduce a characteristic internal material length, which in turn seems to urge the form of the formulation of the theory. A specific version of von Mises micropolar plasticity with kinematic and isotropic hardening, derived in the theoretical context of the present paper, is elaborated in Part II.

## References

1. 1.
Ask, A., Forest, S., Appolaire, B., Ammar, K., Salman, O.U.: A Cosserat crystal plasticity and phase field theory for grain boundary migration. J. Mech. Phys. Solids 115, 167–194 (2018).
2. 2.
Bilby, B.A., Bullough, R., Smith, E.: Continuous distributions of dislocations: a new application of the methods of non-Riemannian geometry. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 231(1185), 263–273 (1955). . ISSN 0080-4630
3. 3.
Broese, C., Sfyris, D., Tsakmakis, Ch.: Isoclinic versus arbitrary rotated intermediate configuration for gradient plasticity applications. Compos. Part B Eng. 43(6), 2633–2645 (2012). . ISSN 1359-8368. Homogenization and micromechanics of smart and multifunctional materials
4. 4.
Chaboche, J.-L.: Cyclic viscoplastic constitutive equations, part I: a thermodynamically consistent formulation. J. Appl. Mech. 60(4), 813–821 (1993). . ISSN 0021-8936
5. 5.
Chaboche, J.-L.: Cyclic viscoplastic constitutive equations, part II: stored energy—comparison between models and experiments. J. Appl. Mech. 60(4), 822–828 (1993). . ISSN 0021-8936
6. 6.
Châu Le, K., Stumpf, H.: Finite Elastoplasticity with Microstructure. Mitteilungen aus dem Institut für Mechanik // Ruhr-Universität Bochum. Ruhr-Univ. (1994). https://books.google.de/books?id=BdjLtgAACAAJ
7. 7.
Cho, H.W., Dafalias, Y.F.: Distortional and orientational hardening at large viscoplastic deformations. Int. J. Plast. 12(7), 903–925 (1996). . ISSN 0749-6419
8. 8.
Cleja-Ţigoiu, S.: Couple stresses and non-Riemannian plastic connection in finite elasto-plasticity. Zeitschrift für angewandte Mathematik und Physik ZAMP 53(6), 996–1013 (2002). . ISSN 1420-9039
9. 9.
Coleman, B.D., Noll, W.: The thermodynamics of elastic materials with heat conduction and viscosity. Arch. Rational Mech. Anal. 13(1), 167–178 (1963). . ISSN 0003-9527
10. 10.
Dafalias, Y .F.: On multiple spins and texture development. Case study: kinematic and orthotropic hardening. Acta Mech. 100(3), 171–194 (1993). . ISSN 1619-6937
11. 11.
de Borst, R.: A generalisation of J2-flow theory for polar continua. Eng. Comput. 10(2), 99–121 (1993)
12. 12.
deWit, R.: Relation between dislocations and disclinations. J. Appl. Phys. 42(9), 3304–3308 (1971).
13. 13.
Dłużewski, P.H.: Finite elastic-plastic deformations of oriented media. In: Benellal, A., Billardon, R. (eds.) Proceedings of international symposium on multiaxial plasticity, MECAMAT’92, Cachan, France. Labolatoire de Mecanique et Technologie (1992)Google Scholar
14. 14.
Eckart, C.: The thermodynamics of irreversible processes. IV. The theory of elasticity and anelasticity. Phys. Rev. 73, 373–382 (1948).
15. 15.
Eringen, A.C.: Microcontinuum Field Theories: Volume 1, Foundations and Solids. Microcontinuum Field Theories: Foundations and Solids. Springer, Berlin (1999). ISBN 9780387986203Google Scholar
16. 16.
Eringen, A .C.: Theory of micropolar elasticity. In: Liebowitz, H. (ed.) Fracture: An Advanced Treatise, vol. 2, pp. 621–729. Academic Press, Cambridge (1968)Google Scholar
17. 17.
Eringen, A.C., Kafadar, C.B.: Polar field theories. In: Eringen, A.C. (ed.) Continuum Physics, vol. 4, pp. 1–73. Academic Press, New York (1976)Google Scholar
18. 18.
Fleck, N.A., Hutchinson, J.W.: A reformulation of strain gradient plasticity. J. Mech. Phys. Solids 49(10), 2245–2271 (2001). . ISSN 0022-5096
19. 19.
Forest, S.: Micromorphic media. In: Altenbach, H., Eremeyev, V.A. (eds.) Generalized Continua from the Theory to Engineering Applications, vol. 541 of CISM International Centre for Mechanical Sciences, pp. 249–300. Springer, Vienna (2013). . ISBN 978-3-7091-1370-7
20. 20.
Forest, S., Sievert, R.: Elastoviscoplastic constitutive frameworks for generalized continua. Acta Mech. 160(1), 71–111 (2003). . ISSN 1619-6937
21. 21.
Forest, S., Cailletaud, G., Sievert, R.: A Cosserat theory for elastoviscoplastic single crystals at finite deformation. Arch. Mech. 49(4), 705–736 (1997)
22. 22.
Grammenoudis, P., Tsakmakis, Ch.: Hardening rules for finite deformation micropolar plasticity: restrictions imposed by the second law of thermodynamics and the postulate of Il’iushin. Contin. Mech. Thermodyn. 13(5), 325–363 (2001). . ISSN 0935-1175
23. 23.
Grammenoudis, P., Tsakmakis, Ch.: Finite element implementation of large deformation micropolar plasticity exhibiting isotropic and kinematic hardening effects. Int. J. Numer. Methods Eng. 62, 1691–1720 (2005)
24. 24.
Grammenoudis, P., Tsakmakis, Ch.: Predictions of microtorsional experiments by micropolar plasticity. Proc. R. Soc. A Math. Phys. Eng. Sci. 461(2053), 189–205 (2005).
25. 25.
Grammenoudis, P., Tsakmakis, Ch.: Incompatible deformations—plastic intermediate configuration. ZAMM J. Appl. Math. Mech. Zeitschrift für Angewandte Mathematik und Mechanik 88(5), 403–432 (2008). . ISSN 1521-4001
26. 26.
Grammenoudis, P., Tsakmakis, Ch.: Plastic intermediate configuration and related spatial differential operators in micromorphic plasticity. Math. Mech. Solids 15(5), 515–538 (2010).
27. 27.
Grammenoudis, P., Tsakmakis, Ch., Hofer, D.: Micromorphic continuum. Part II: finite deformation plasticity coupled with damage. Int. J. Non Linear Mech. 44(9), 957–974 (2009). . ISSN 0020-7462
28. 28.
Gurtin, M.E., Fried, E., Anand, L.: The Mechanics and Thermodynamics of Continua. Cambridge University Press, Cambridge (2010). ISBN 9781139482158Google Scholar
29. 29.
Gurtin, M.E.: On the plasticity of single crystals: free energy, microforces, plastic-strain gradients. J. Mech. Phys. Solids 48(5), 989–1036 (2000). . ISSN 0022-5096
30. 30.
Gurtin, M.E.: A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations. J. Mech. Phys. Solids 50(1), 5–32 (2002). . ISSN 0022-5096
31. 31.
Haupt, P., Tsakmakis, Ch.: Stress tensors associated with deformation tensors via duality. Arch. Mech. 48(2), 347–384 (1996)
32. 32.
Kondo, K. (ed.): Non-Riemannian geometry of the imperfect crystal from a macroscopic viewpoint. In: RAAG Memoirs of the Unifying Study of Basic Problems in Engineering Sciences by Means of Geometry, vol. 1, Division D, pp. 458–469. Gakuyusty Bunken Fukin-Day, Tokyo (1955)Google Scholar
33. 33.
Kondo, K., Shimbo, M., Amari, S.: On the standpoint of non-Riemannian plasticity theory. In: Kondo, K. (ed.) RAAG Memoirs of the Unifying Study of Basic Problems in Engineering Sciences by Means of Geometry, vol. 4, pp. 205–224. Gakujutsu Bunken Fukyu-kai, Tokyo (1968)Google Scholar
34. 34.
Kröner, E.: Kontinuumstheorie der Versetzungen und Eigenspannungen. Springer, Berlin (1958)
35. 35.
Kröner, E.: Continuum theory of defects. In: Balian, R., Kleman, M., Poirier, J.-P. (eds.) Physics of Defects. Les Houches Session XXXV, vol. 35, pp. 217–315. North-Holland, Amsterdam (1981)Google Scholar
36. 36.
Lee, E.H.: Elastic-plastic deformation at finite strains. J. Appl. Mech. 36, 1–6 (1969).
37. 37.
Lee, E.H., Liu, D.T.: Finite-strain elastic-plastic theory with application to plane-wave analysis. J. Appl. Phys. 38(1), 19–27 (1967).
38. 38.
Lubliner, J.: Normality rules in large-deformation plasticity. Mech. Mater. 5(1), 29–34 (1986). . ISSN 0167-6636
39. 39.
Materially Uniform Simple Bodies with Inhomogeneities. Department of Mathematics, Carnegie Institute of Technology, Pittsburgh (1967). http://books.google.de/books?id=vb3ztgAACAAJ
40. 40.
Minagawa, S.: Elastic fields of dislocations and disclinations in an isotropic micropolar continuum. Lett. Appl. Eng. Sci 5, 85–94 (1977)
41. 41.
Ogden, R.W.: Non-linear Elastic Deformations. Dover Civil and Mechanical Engineering, Dover Publications, New York (1997). ISBN 9780486696485Google Scholar
42. 42.
Regueiro, R.A.: On finite strain micromorphic elastoplasticity. Int. J. Solids Struct. 47(6), 786–800 (2010). . ISSN 0020-7683
43. 43.
Sansour, C.: A theory of the elastic-viscoplastic Cosserat continuum. Arch. Mech. 50(3), 577–597 (1998)
44. 44.
Steinmann, P.: A micropolar theory of finite deformation and finite rotation multiplicative elastoplasticity. Int. J. Solids Struct. 31(8), 1063–1084 (1994). . ISSN 0020-7683
45. 45.
Svendsen, B.: Objective frame derivatives for the hyperstress and couple stress. Arch. Mech. 46, 669–683 (1994). 01
46. 46.
Svendsen, B.: A thermodynamic formulation of finite-deformation elastoplasticity with hardening based on the concept of material isomorphism. Int. J. Plast. 14(6), 473–488 (1998). . ISSN 0749-6419
47. 47.
Svendsen, B.: On the modelling of anisotropic elastic and inelastic material behaviour at large deformation. Int. J. Solids Struct. 38(52), 9579–9599 (2001). . ISSN 0020-7683
48. 48.
Svendsen, B., Arndt, S., Klingbeil, D., Sievert, R.: Hyperelastic models for elastoplasticity with non-linear isotropic and kinematic hardening at large deformation. Int. J. Solids Struct. 35(25), 3363–3389 (1998). . ISSN 0020-7683
49. 49.
Tsakmakis, Ch.: On the loading conditions and the decomposition of deformation. In: Boehler, J.-P., Khan, A.S. (eds.) Anisotropy and Localization of Plastic Deformation, pp. 353–356. Springer, Dordrecht (1991). . ISBN 978-1-85166-688-1
50. 50.
Tsakmakis, Ch.: Description of plastic anisotropy effects at large deformations-part I: restrictions imposed by the second law and the postulate of Il’iushin. Int. J. Plast. 20(2), 167–198 (2004). . ISSN 0749-6419
51. 51.
Volk, W.: Untersuchung des Lokalisierungsverhaltens mikropolarer poröser Medien mit Hilfe der Cosserat-Theorie. Bericht Nr. II-2. Universität Stuttgart Inst. f. Mechanik (Bauwesen), Stuttgart (1999)Google Scholar

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

## Authors and Affiliations

• Daniel Johannsen
• 1
• Charalampos Tsakmakis
• 1