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A Micromorphic Damage-Plasticity Model to Counteract Mesh Dependence in Finite Element Simulations Involving Material Softening

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Multiscale Modeling of Heterogeneous Structures

Part of the book series: Lecture Notes in Applied and Computational Mechanics ((LNACM,volume 86))

Abstract

A gradient-extended damage-plasticity material model is presented which belongs to the class of micromophic media as proposed by Forest (J Eng Mech 135:117–131, 2009) [17]. A ‘two-surface’ formulation is utilized in which damage and plasticity are treated as independent but strongly coupled dissipative phenomena. To this end, separate yield and damage criteria as well as loading/unloading conditions are introduced. The model is thermodynamically consistent and accounts for both nonlinear kinematic and isotropic hardening as well as damage hardening. Various theoretical and numerical aspects of the formulation are discussed. Emphasis is also put on a procedure to enforce stress constraints at the local integration point level which provides, for instance, the basis for a straightforward integration of 3D gradient-extended material models into beam or shell elements or for their usage in 2D plane stress computations. A structural example problem illustrates the merits of the model and its ability to deliver mesh-independent results in coupled damage-plasticity finite element simulations.

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Notes

  1. 1.

    \(\nabla ^{\text {s}}\left( \bullet \right) := \frac{1}{2} \left[ \left( \bullet \right) + \left( \bullet \right) ^{\text {T}} \right] \) denotes the symmetric part of a quantity \(\left( \bullet \right) \).

  2. 2.

    \(\left( \bullet \right) ^{\prime } := \left( \bullet \right) - \frac{1}{3}\,\text {trace}\left( \bullet \right) \,\mathbf {I}\) denotes the deviatoric part of a second-order tensor \(\left( \bullet \right) \).

  3. 3.

    In the following, if not explicitly denoted otherwise, any quantity is referred to time \(t_{n+1}=t_{n} + \varDelta t\).

  4. 4.

    The corresponding residuals in case of a purely elastoplastic step or elastic step with concurrently evolving damage are obtained by deleting either \(r_{3}\) or \(\mathbf {r}_{1}\) and \(r_{2}\) in (29), respectively.

  5. 5.

    Remember that, during the considered time interval, \(\varvec{\sigma }\) and D do additionally depend on the history variables at time \(t_{n}\) which are, however, constant within \([t_{n},\,t_{n+1}]\).

  6. 6.

    No extra symbols are introduced to avoid an excessive proliferation of notation. The change in meaning of the quantities is implicitly understood.

  7. 7.

    Quadrilateral elements are preferred in the meshing process, triangular ones are only used rarely as transitional elements.

  8. 8.

    The top and bottom values in the legends of the contour plots always indicate the corresponding minimum and maximum values attained in the computations.

References

  1. Aravas, N.: On the numerical integration of a class of pressure-dependent plasticity models. Int. J. Numer. Meth. Eng. 24(7), 1395–1416 (1987)

    Article  MATH  Google Scholar 

  2. Auricchio, F., Bonetti, E., Scalet, G., Ubertini, F.: Theoretical and numerical modeling of shape memory alloys accounting for multiple phase transformations and martensite reorientation. Int. J. Plast. 59, 30–54 (2014)

    Article  Google Scholar 

  3. Bartel, T., Hackl, K.: A micromechanical model for martensitic phase-transformations in shape-memory alloys based on energy-relaxation. ZAMM J. Appl. Math. Mech. 89(10), 792–809 (2009)

    Article  MATH  Google Scholar 

  4. Bažant, Z.P., Belytschko, T., Chang, T.-P.: Continuum theory for strain-softening. J. Eng. Mech. 110(12), 1666–1692 (1984)

    Article  Google Scholar 

  5. Bayerschen, E., Stricker, M., Wulfinghoff, S., Weygand, D., Böhlke, T.: Equivalent plastic strain gradient plasticity with grain boundary hardening and comparison to discrete dislocation dynamics. Proc. R. Soc. A 471(2184) (2015)

    Google Scholar 

  6. Brepols, T., Wulfinghoff, S., Reese, S.: Gradient-extended two-surface damage-plasticity: micromorphic formulation and numerical aspects. Int. J. Plast. 97, 64–106 (2017)

    Google Scholar 

  7. Cervera, M., Chiumenti, M.: Mesh objective tensile cracking via a local continuum damage model and a crack tracking technique. Comput. Meth. Appl. Mech. Eng. 196(1–3), 304–320 (2006)

    Article  MATH  Google Scholar 

  8. Chaboche, J.-L.: Thermodynamics of local state: overall aspects and micromechanics based constitutive relations. Tech. Mech. 23(2–4), 113–119 (2003)

    Google Scholar 

  9. Chow, C.L., Wang, J.: An anisotropic theory of continuum damage mechanics for ductile fracture. Eng. Fract. Mech 27(5), 547–558 (1987)

    Article  Google Scholar 

  10. de Borst, R.: The zero-normal-stress condition in plane-stress and shell elastoplasticity. Commun. Appl. Numer. Meth. 7(1), 29–33 (1991)

    Article  MATH  Google Scholar 

  11. de Borst, R., Sluys, L.J., Mühlhaus, H.-B., Pamin, J.: Fundamental issues in finite element analyses of localization of deformation. Eng. Comput. 10(2), 99–121 (1993)

    Article  Google Scholar 

  12. Dimitrijevic, B.J., Hackl, K.: A method for gradient enhancement of continuum damage models. Tech. Mech. 28(1), 43–52 (2008)

    Google Scholar 

  13. Dimitrijevic, B.J., Hackl, K.: A regularization framework for damage-plasticity models via gradient enhancement of the free energy. Int. J. Numer. Method. Biomed. Eng. 27(8), 1199–1210 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Dodds, R.H.: Numerical techniques for plasticity computations in finite element analysis. Comput. Struct. 26(5), 767–779 (1987)

    Article  MATH  Google Scholar 

  15. Duda, F.P., Ciarbonetti, A., Sánchez, P.J., Huespe, A.E.: A phase-field/gradient damage model for brittle fracture in elastic-plastic solids. Int. J. Plast. 65, 269–296 (2015)

    Article  Google Scholar 

  16. Dvorkin, E.N., Pantuso, D., Repetto, E.A.: A formulation of the MITC4 shell element for finite strain elasto-plastic analysis. Comput. Meth. Appl. Mech. Eng. 125(1), 17–40 (1995)

    Article  Google Scholar 

  17. Forest, S.: Micromorphic approach for gradient elasticity, viscoplasticity, and damage. J. Eng. Mech. 135(3), 117–131 (2009)

    Article  Google Scholar 

  18. Forest, S.: Nonlinear regularization operators as derived from the micromorphic approach to gradient elasticity, viscoplasticity and damage. Proc. R. Soc. A 472(2188), 20150755 (27 pages) (2016)

    Google Scholar 

  19. Germain, P., Nguyen, Q.S., Suquet, P.: Continuum thermodynamics. J. Appl. Mech. 50(4b), 1010–1020 (1983)

    Article  MATH  Google Scholar 

  20. Grassl, P., Jirásek, M.: On mesh bias of local damage models for concrete. In: Proceedings of FraMCoS-5, pp. 252–262. Vail, USA (2004)

    Google Scholar 

  21. Heinrich, C., Aldridge, M., Wineman, A.S., Kieffer, J., Waas, A.M., Shahwan, K.W.: The role of curing stresses in subsequent response, damage and failure of textile polymer composites. J. Mech. Phys. Solid. 61(5), 1241–1264 (2013)

    Article  MathSciNet  Google Scholar 

  22. Hütter, G., Linse, T., Mühlich, U., Kuna, M.: Simulation of ductile crack initiation and propagation by means of a non-local gurson-model. Int. J. Solid. Struct. 50(5), 662–671 (2013)

    Article  Google Scholar 

  23. Hütter, G., Roth, T.L.S., Mühlich, U., Kuna, M.: A modeling approach for the complete ductile-brittle transition region: cohesive zone in combination with a non-local gurson-model. Int. J. Fract. 185(1), 129–153 (2014)

    Article  Google Scholar 

  24. Jetteur, P.: Implicit integration algorithm for elastoplasticity in plane stress analysis. Eng. Comput. 3(3), 251–253 (1986)

    Article  Google Scholar 

  25. Jirásek, M., Grassl, P.: Evaluation of directional mesh bias in concrete fracture simulations using continuum damage models. Eng. Fract. Mech 75(8), 1921–1943 (2008)

    Article  Google Scholar 

  26. Ju, J.W.: On energy-based coupled elastoplastic damage theories: constitutive modeling and computational aspects. Int. J. Solid. Struct. 25(7), 803–833 (1989)

    Article  MATH  Google Scholar 

  27. Junker, P., Schwarz, S., Makowsk, J., Hackl, K.: A relaxation-based approach to damage modeling. Continuum Mech. Therm. 29(1), 291–310 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  28. Kachanov, L.M.: ‘Time of the rupture process under creep conditions’, Izvestiya Akademii Nauk SSSR. Otdelenie Tekhnicheskikh Nauk 8, 26–31 (1958)

    Google Scholar 

  29. Kachanov, M.: Elastic solids with many cracks and related problems. In: Hutchinson, J. W., Wu, T. Y. (eds.) Advances in Applied Mechanics, vol. 30, pp. 259–445. Elsevier (1993)

    Google Scholar 

  30. Kiefer, B., Bartel, T., Menzel, A.: Implementation of numerical integration schemes for the simulation of magnetic sma constitutive response. Smart Mater. Struct. 21(9), 094007 (8 pages) (2012)

    Google Scholar 

  31. Kiefer, B., Waffenschmidt, T., Sprave, L., Menzel, A.: A gradient-enhanced damage model coupled to plasticity—multi-surface formulation and algorithmic concepts. Int. J. Damage Mech. 1–43 (2017)

    Google Scholar 

  32. Kirchner, E., Reese, S., Wriggers, P.: A finite element method for plane stress problems with large elastic and plastic deformations. Commun. Numer. Meth. Eng. 13(12), 963–976 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  33. Klinkel, S., Govindjee, S.: Using finite strain 3D-material models in beam and shell elements. Eng. Comput. 19(8), 902–921 (2002)

    Article  MATH  Google Scholar 

  34. Lemaitre, J.: Evaluation of dissipation and damage in metals submitted to dynamic loading. In: Proceedings International Conference Mechanical Behavior of Materials, vol. 1. Kyoto, Japan (1971)

    Google Scholar 

  35. Lemaitre, J., Chaboche, J.-L.: Mechanics of Solid Materials, 1st edn. Cambridge University Press (1990)

    Google Scholar 

  36. Lion, A.: Constitutive modelling in finite thermoviscoplasticity: a physical approach based on nonlinear rheological models. Int. J. Plast. 16(5), 469–494 (2000)

    Article  MATH  Google Scholar 

  37. Miehe, C.: Variational gradient plasticity at finite strains. Part I: mixed potentials for the evolution and update problems of gradient-extended dissipative solids. Comput. Meth. Appl. Mech. Eng. 268, 677–703 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  38. Miehe, C., Aldakheel, F., Raina, A.: Phase field modeling of ductile fracture at finite strains: a variational gradient-extended plasticity-damage theory. Int. J. Plast. 84, 1–32 (2016)

    Article  Google Scholar 

  39. 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. Meth. Appl. Mech. Eng. 268, 704–734 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  40. Mozaffari, N., Voyiadjis, G.Z.: Coupled gradient damage—viscoplasticty model for ductile materials: phase field approach. Int. J. Plast. 83, 55–73 (2016)

    Article  Google Scholar 

  41. Naderi, M., Jung, J., Yang, Q.D.: A three dimensional augmented finite element for modeling arbitrary cracking in solids. Int. J. Fract. 197(2), 147–168 (2016)

    Article  Google Scholar 

  42. Pietryga, M.P., Vladimirov, I.N., Reese, S.: A finite deformation model for evolving flow anisotropy with distortional hardening including experimental validation. Mech. Mater. 44, 163–173 (2012)

    Article  Google Scholar 

  43. Rabotnov, Y. N.: Paper 68: on the equation of state of creep. Proc. Inst. Mech. Eng. (Conf. Proc.) 178(1), 117–122 (1963)

    Google Scholar 

  44. Reese, S.: On the equivalent of mixed element formulations and the concept of reduced integration in large deformation problems. Int. J. Nonlinear Sci. Numer. Simul. 3(1), 1–34 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  45. Reese, S.: On a consistent hourglass stabilization technique to treat large inelastic deformations and thermo-mechanical coupling in plane strain problems. Int. J. Numer. Meth. Eng. 57(8), 1095–1127 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  46. Reese, S.: On a physically stabilized one point finite element formulation for three-dimensional finite elasto-plasticity. Comput. Meth. Appl. Mech. Eng. 194(45), 4685–4715 (2005)

    Article  MATH  Google Scholar 

  47. Saanouni, K., Forster, C., Ben Hatira, F.: On the anelastic flow with damage. Int. J. Damage Mech. 3(2), 140–169 (1994)

    Article  MATH  Google Scholar 

  48. Saanouni, K., Hamed, M.: Micromorphic approach for finite gradient-elastoplasticity fully coupled with ductile damage: Formulation and computational aspects. Int. J. Solid. Struct. 50(14–15), 2289–2309 (2013)

    Article  Google Scholar 

  49. Seabra, M.R.R., Šuštarič, P., Cesar de Sa, J.M.A., Rodič, T.: Damage driven crack initiation and propagation in ductile metals using XFEM. Comput. Mech. 52(1), 161–179 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  50. Simo, J.C., Kennedy, J.G., Govindjee, S.: Non-smooth multisurface plasticity and viscoplasticity. Loading/unloading conditions and numerical algorithms. Int. J. Numer. Meth. Eng. 26(10), 2161–2185 (1988)

    Article  MATH  Google Scholar 

  51. Simo, J.C., Taylor, R.L.: A return mapping algorithm for plane stress elastoplasticity. Int. J. Numer. Meth. Eng. 22(3), 649–670 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  52. Truesdell, C., Noll, W.: The non-linear field theories of mechanics. In: Encyclopedia of Physics, vol. III/3. Springer, Berlin, Heidelberg (1965)

    Google Scholar 

  53. Vladimirov, I.N., Pietryga, M.P., Kiliclar, Y., Tini, V., Reese, S.: Failure modelling in metal forming by means of an anisotropic hyperelastic-plasticity model with damage. Int. J. Damage Mech. 23(8), 1096–1132 (2014)

    Article  Google Scholar 

  54. Vladimirov, I.N., Pietryga, M.P., Reese, S.: On the modelling of non-linear kinematic hardening at finite strains with application to springback—comparison of time integration algorithms. Int. J. Numer. Meth. Eng. 75(1), 1–28 (2008)

    Article  MATH  Google Scholar 

  55. Vladimirov, I.N., Pietryga, M.P., Reese, S.: Prediction of springback in sheet forming by a new finite strain model with nonlinear kinematic and isotropic hardening. J. Mater. Process. Tech. 209(8), 4062–4075 (2009)

    Article  Google Scholar 

  56. Vladimirov, I.N., Pietryga, M.P., Reese, S.: Anisotropic finite elastoplasticity with nonlinear kinematic and isotropic hardening and application to sheet metal forming. Int. J. Plast. 26(5), 659–687 (2010)

    Article  MATH  Google Scholar 

  57. Waffenschmidt, T., Polindara, C., Menzel, A., Blanco, S.: A gradient-enhanced large-deformation continuum damage model for fibre-reinforced materials. Comput. Meth. Appl. Mech. Eng. 268, 801–842 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  58. Wulfinghoff, S., Bayerschen, E., Böhlke, T.: A gradient plasticity grain boundary yield theory. Int. J. Plast. 51, 33–46 (2013)

    Article  Google Scholar 

  59. Wulfinghoff, S., Böhlke, T.: Equivalent plastic strain gradient enhancement of single crystal plasticity: theory and numerics. Proc. R. Soc. A 468(2145), 2682–2703 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  60. Ziemann, M., Chen, Y., Kraft, O., Bayerschen, E., Wulfinghoff, S., Kirchlechner, C., Tamura, N., Böhlke, T., Walter, M., Gruber, P.A.: Deformation patterns in cross-sections of twisted bamboo-structured Au microwires. Acta Mater. 97, 216–222 (2015)

    Article  Google Scholar 

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Authors and Affiliations

  1. Institute of Applied Mechanics, RWTH Aachen University, 52074, Aachen, Germany

    Tim Brepols, Stephan Wulfinghoff & Stefanie Reese

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  1. Tim Brepols
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  2. Stephan Wulfinghoff
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Correspondence to Tim Brepols .

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Editors and Affiliations

  1. University of Zagreb, Zagreb, Croatia

    Jurica Sorić

  2. Institute of Continuum Mechanics, Leibniz Universität Hannover, Hannover, Germany

    Peter Wriggers

  3. LMT Cachan, Cachan, France

    Olivier Allix

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Brepols, T., Wulfinghoff, S., Reese, S. (2018). A Micromorphic Damage-Plasticity Model to Counteract Mesh Dependence in Finite Element Simulations Involving Material Softening. In: Sorić, J., Wriggers, P., Allix, O. (eds) Multiscale Modeling of Heterogeneous Structures. Lecture Notes in Applied and Computational Mechanics, vol 86. Springer, Cham. https://doi.org/10.1007/978-3-319-65463-8_12

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  • DOI: https://doi.org/10.1007/978-3-319-65463-8_12

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