Nichtlineare Theorie und adaptive FEM martensitischer Phasentransformationen (PT) mit technischen Anwendungen

• Applicant: Professor Dr.-Ing. Erwin Stein (†)
• Subject Area: Mechanics
• Term: from 2005 to 2010
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 5445478

• Final Report Year: 2010

Final Report Abstract

In this report new contributions and results to the theory and computation of cyclic martensitic phase transformations in mono- and poly- crystalline metallic shape memory alloys are presented. Both micro-macro material models are based on the Cauchy-Born hypothesis and Bain’s principle. A C1 -continuous thermo-mechanical constitutive macro model for metallic monocrystals at finite strains, based on crystal theory of metals and assuming a hyperelastic free energy function is developed. This is represented, together with the phase transformation constraints, by a unified global Lagrangian variational functional, including the constraints of phase evolution equations with mass conservation and quasi-convexification during phase transformation processes. The quasi-convexification problem for finite strain is solved here for Neo-Hookean (polyconvex) elastic material by the incremental numerical calculation of the lower Reuß bound of the energy of mixing for all combinations of active phases. The reason for extending theory and algorithms from linear to finite strains is the fact that the transformation strains of metallic SMAs can reach up to 12% in case of mono crystals which is far beyond the validity of small strain theory. However, transformation strains of polycrystalline SMAs only reach about 6%, which also requires nonlinear kinematics. The unified setting presented here includes polycrystalline shape memory alloys whose microstructure is modeled using lattice variants. A pre-averaging scheme for randomly distributed polycrystalline variants (in grains) of transformation strains within a representative volume element (RVE) is used to transform them into those of a fictitious monocrystal. Thus, the parametric integration in process time and the spatial integration algorithms of the discrete variational problems for both mono- and poly- crystalline phase transformations can be done in same way. Furthermore, an error-controlled adaptive 3D finite element method in space is presented for phase transformation problems. Developed models are implemented in Abaqus via UMAT-interface which requires Jaumann rate of Kirchhoff stresses. Convergence checks of 3D finite element types available in Abaqus are made for Neo-Hookean material with finite strains. Error control and adaptivity of process time is realized by precalculations with error control and then choosing constant time steps for certain time intervals of PT processes. Computations of interesting examples with convergence studies, and comparisons with published experimental results, showing good agreements, are presented using 3D finite elements. So far the developed monocrystalline model at finite strains has been only validated for quasiplastic phase transformation (QP). It was also used for superelastic phase transformations (SE), but further comparisons with experimentally measured data are necessary. One of the deficiencies of the polycrystalline model is that it does not include surface energies of the grains and interactions between them which might be significant in some polycrystalline shape memory alloys. The effect of elastic anisotropy on the presented polycrystalline model was not implemented due to lack of material parameters from experiments. Further studies on this are recommendable to capture the influence on the macroscopic behavior. In order to detect the influence of stochastic grain distribution in polycrystals further experimental, theoretical and numerical studies are required. A further interesting aspect for future research work lies on implementation of mathematically sound error estimators for quantities of interest, especially for the driving forces which are essential for local phase transformations. Similar to plasticity, an error in time should be implemented besides the discretization error in space to verify the algorithm.

Publications

• E. Onate and D. R. J. Owen (eds.), Theory and computation of quasiplastic and superelastic martensitic phase transformations. Computational Plasticity VIII, CIMNE, Barcelona, 4-7 Sep., 2005
  E. Stein, O. Zwickert
  
• Cyclic martensitic phase transformations in monocrystals - Computational and experimental results for Shape memory and superelastic effects. Proceedings in 3rd ECCM Conference on Solids and Structure, Lisbon, Portugal, 5-8 Jun., 2006
  E. Stein, G. Sagar, O. Zwickert
  
• Implementation of a non-linear elastic material model with a free energy function in Abaqus 6.4 via UMAT. Proceedings in 18. Deutschsprachige ABAQUS Benutzerkonferenz, Erfurt, 18-19 Sept., Abaqus Inc., 2006, 2.12:15
  G. Sagar, E. Stein
  
• Shape memory and superelastic effects of cyclic martensitic phase transformations in monocrystals - Computational and experimental results. 5th GAMM Seminar on Microstructures, CIMNE, Essen, 13-14 Jan., 2006
  E. Stein, O. Zwickert
  
• Cyclic martensitic phase transformation in multi-variant monocrystals at finite strains using UMAT. Proceedings in 20th annual international ABAQUS User’s Conference, Paris, France, 22-24 May, Abaqus Inc., 2007, 571-582
  E. Stein, G. Sagar
  
• Cyclic martensitic phase transformation of monocrystals at finite strains. Proceedings in 9th United States National Congress on Computational Mechanics, San Francisco, USA, 23-26 Jul., 2007
  E. Stein, G. Sagar
  
• Cyclic martensitic phase transformation of monocrystals with finite strains. Proceedings in Applied Mathematics and Mechanics, Proc. Appl. Math. Mech. 7, 4060049-4060050 (2007)
  G. Sagar, E. Stein
  (See online at https://doi.org/10.1002/pamm.200700797/)
• E. Onate, D. R. J. Owen and B. Suárez (eds.) Shape memory effects at finite strains of cyclic martensitic phase transformations in monocrystals. Computational Plasticity IX, Fundamentals and Applications, Part1, CIMNE, Barcelona, 2007, 68-71
  E. Stein, G. Sagar
  
• J. Schröder, D. Lupascu and D. Balzani (eds.)Theory and computation of a micro-macro monocrystalline model for martensitic phase pransformation at finite strains. Proceedings in First Seminar on Mechanics of multifunctional Materials, Bad Honnef, Germany, 7-10 May, 2007, addendum 1-6
  E. Stein, G. Sagar
  
• Shape memory effect of cyclic martensitic phase transformation in monocrystals with finite strains - Theoretical and computational aspects. Proceedings in 6th GAMM Seminar on Microstructure, WIAS, Berlin, 12-13 Jan., 2007
  E. Stein, G. Sagar
  
• Theory and finite element computations of a unified cyclic phase transformation model for monocrystalline materials at small strains. Comp. Mech., 2007, 40, 429-445
  E. Stein, O. Zwickert
  
• An integrated methodology for analyzing mono- and poly- crystalline martensitic phase transformation . Proceedings in 7 th GAMM Seminar on Microstructure, Ruhr-Universität Bochum, 25-26 Jan., 2008
  G. Sagar, E. Stein
  
• Convergence behavior of 3-D finite elements for Neo-Hookean material models using Abaqus. Int. J. for Compputer-Aided Engng. and Software, 2008, 25, 220-232
  E. Stein, G. Sagar
  
• Theory and finite element computation of cyclic martensitic phase transformation at finite strain. Int. J. Numer. Meth. Engng., 2008, 74, 1-31
  E. Stein, G. Sagar
  
• Unified micro-macro models and computation of mono- and polycrystalline cyclic martensitic phase transformatios. Proceedings in 9th GAMM Seminar on Microstructure, Universität Stuttgart, 21-23 Jan., 2010
  E. Stein, G. Sagar