## Abstract

This work is concerned with different estimates of the quasiconvexification of multi-well energy landscapes of NiTi shape memory alloys, which models the overall behavior of the material. Within the setting of the geometrically linear theory of elasticity, we consider a formula of the quasiconvexification which involves the so-called energy of mixing.We are interested in lower and upper bounds on the energy of mixing in order to get a better understanding of the quasiconvexification. The lower bound on the energy of mixing is obtained by convexification; it is also called Sachs or Reuß lower bound. The upper bound on the energy of mixing is based on second-order lamination. In particular, we are interested in the difference between the lower and upper bounds. Our numerical simulations show that the difference is in fact of the order of 1% and less in martensitic NiTi, even though both bounds on the energy of mixing were rather expected to differ more significantly. Hence, in various circumstances it may be justified to simply work with the convexification of the multi-well energy, which is relatively easy to deal with, or with the lamination upper bound, which always corresponds to a physically realistic microstructure, as an estimate of the quasiconvexification. In order to obtain a potentially large difference between upper and lower bound, we consider the bounds along paths in strain space which involve incompatible strains. In monoclinic shape memory alloys, three-tuples of pairwise incompatible strains play a special role since they form so-called *T*
_{3}-configurations, originally discussed in a stress-free setting. In this work, we therefore consider in particular numerical simulations along paths in strain space which are related to these *T*
_{3}-configurations. Interestingly, we observe that the second-order lamination upper bound along such paths is related to the geometry of the *T*
_{3}-configurations. In addition to the purely martensitic regime, we also consider the influence of adding R-phase variants to the microstructure. Adding single variants of R-phase is shown to be energetically favorable in a compatible martensitic setting. However, the combination of several R-phase variants with compatible or incompatible martensite yields significant differences between the bounds considered.

This is a preview of subscription content,

to check access.## References

Arghavani J., Auricchio F., Naghdabadi R., Reali A., Sohrabpour S.: A 3-D phenomenological constitutive model for shape memory alloys under multiaxial loadings. Int. J. Plast.

**26**, 976–991 (2010)Avellaneda M., Milton G.: Bounds on the effective elasticity tensor of composites based on two-point correlations. In: Hui, D., Koszic, T. (eds.) Proceedings of the ASME Energy-Technology Conference and Exposition, ASME, New York (1989)

Ball J.M., James R.D.: Fine phase mixtures as minimizers of energy. Arch. Rat. Mech. Anal.

**100**, 13–52 (1987)Ball J.M., James R.D.: Proposed experimental tests of a theory of fine microstructure, and the two-well problem. Phil. Trans. R. Soc. Lond. A

**338**, 389–450 (1992)Bartel T., Hackl K.: A novel approach to the modelling of single-crystalline materials undergoing martensitic phase-transformations. Mater. Sci. Eng. A

**481**, 371–375 (2008)Bartel T., Menzel A., Svendsen B.: Thermodynamic and relaxation-based modeling of the interaction between martensitic phase transformations and plasticity. J. Mech. Phys. Solids

**59**, 1004–1019 (2011)Bhattacharya K.: Comparison of the geometrically nonlinear and linear theories of martensitic transformation. Cont. Mech. Thermodyn.

**5**, 205–242 (1993)Bhattacharya K.: Microstructure of Martensite—Why It Forms and How It Gives Rise to the Shape-Memory Effect. Oxford University Press, New York (2003)

Bhattacharya K., Schlömerkemper A.: Stress-induced phase transformations in shape-memory polycrystals. Arch. Rat. Mech. Anal.

**196**, 715–751 (2010)Bruno O.P., Reitich F., Leo P.H.: The overall elastic energy of polycrystalline martensitic solids. J. Mech. Phys. Solids

**44**, 1051–1101 (1996)Chenchiah I.V., Schlömerkemper A.: Non-laminate microstructures in monoclinic-I martensite. Arch. Rat. Mech. Anal.

**207**, 39–74 (2013)Dacorogna B.: Direct Methods in the Calculus of Variations. 2nd edn. Springer, Berlin (2008)

Govindjee S., Hackl K., Heinen R.: An upper bound to the free energy of mixing by twin-compatible lamination for

*n*-variant martensitic phase transformations. Cont. Mech. Thermodyn.**18**, 443–453 (2007)Govindjee S., Miehe C.: A multi-variant martensitic phase transformation model: formulation and numerical implementation. Comput. Methods Appl. Mech. Eng.

**191**, 215–238 (2001)Govindjee S., Mielke A., Hall G.J.: The free energy of mixing for

*n*-variant martensitic phase transformations using quasi-convex analysis. J. Mech. Phys. Solids**51**, 763+ (2003)Hackl K., Heinen R.: A micromechanical model for pretextured polycrystalline shape-memory alloys including elastic anisotropy. Cont. Mech. Thermodyn.

**19**, 499–510 (2008)Hackl K., Heinen R.: An upper bound to the free energy of

*n*-variant polycrystalline shape memory alloys. J. Mech. Phys. Solids**56**, 2832–2843 (2008)Hara T., Ohba T., Okunishi E., Otsuka K.: Structural study of R-phase in Ti—50.23 at% Ni and Ti—47.75 at% Ni—1.50 at% Fe alloys. Mater. Trans.

**38**, 11–17 (1997)Hall G.J., Govindjee S.: Application of a partially relaxed shape memory free energy function to estimate the phase diagram and predict global microstructure evolution. J. Mech. Phys. Solids

**50**, 501–530 (2001)Hane K.F., Shield T.W.: Microstructure in the cubic to monoclinic transition in titanium–nickel shape memory alloys. Acta Mater.

**47**, 2603–2617 (1999)Heinen R., Hackl K.: On the calculation of energy-minimizing phase fractions in shape memory alloys. Comput. Methods Appl. Mech. Eng.

**196**, 2401–2412 (2007)Helm D., Haupt P.: Shape memory behaviour: modelling within continuum thermomechanics. Int. J. Solids Struct.

**40**, 827–849 (2003)Junker P.: A novel approach to representative orientation distribution functions for modeling and simulation of polycrystalline shape memory alloys. Int. J. Numer. Meth. Eng.

**98**, 799–818 (2014)Knowles K.M., Smith D.A.: The crystallography of the martensitic transformation in equiatomic nickel–titanium. Acta Metal. Mater.

**29**, 101–110 (1981)Kochmann D., Hackl K.: The evolution of laminates in finite crystal plasticity: a variational approach. Cont. Mech. Thermodyn.

**23**, 63–85 (2011)Kohn R.: The relaxation of a double-well problem. Cont. Mech. Thermodyn.

**3**, 193–236 (1991)Lexcellent C., Boubakar M.L., Bouvet C., Calloch S.: About modelling the shape memory alloy behaviour based on the phase transformation surface identification under proportional loading and anisothermal conditions. Int. J. Solids Struct.

**43**, 613–626 (2006)Lexcellent C., Schlömerkemper A.: Comparison of several models for the determination of the phase transformation yield surface in shape-memory alloys with experimental data. Acta Mater.

**55**, 2995–3006 (2007)Mielke A.: Estimates on the mixture function for multiphase problems in elasticity. In: Sändig, A.M., Schiehlen, W., Wendland, W.L. Multifield Problems (State of the Art), pp. 96–103. Springer, Berlin (2000)

Mielke A., Roubíček T.: Rate-Independent Systems: Theory and Application. Springer, Berlin (2015)

Müller, S.: Variational models for microstructure and phase transitions. In: Bethuel, F., Huisken, G., Müller, S., Steffen, K. (Eds.) Calculus of Variations and Geometric Evolution Problems: Lectures Given at the 2nd Session of the Centro Internazionale Matematico Estivo (CIME) Held in Cetraro, Italy, June 15–22, 1996, pp. 85–210. Springer, Berlin (1999)

Müller Ch., Bruhns O.T.: A thermodynamic finite-strain model for pseudoelastic shape memory alloys. Int. J. Plast.

**22**, 1658–1682 (2006)Mercatoris B.C.N., Massart T.J.: A coupled two-scale computational scheme for the failure of periodic quasi-brittle thin planar shells and its application to masonry. Int. J. Numer. Meth. Eng.

**85**, 1177–1206 (2011)Otsuka K., Ren X.: Physical metallurgy of Ti–Ni-based shape memory alloys. Prog. Mater. Sci.

**50**, 511–678 (2005)Otsuka K., Sawamura T., Shimizu K.: Crystal structure and internal defects of equiatomic TiNi martensite. Phys. Status Solidi (A)

**5**, 457–470 (1971)Peigney M.: A non-convex lower bound on the effective energy of polycrystalline shape memory alloys. J. Mech. Phys. Solids

**57**, 970–986 (2009)Peigney M.: On the energy-minimizing strains in martensitic microstructures—part 1: geometrically nonlinear theory. J. Mech. Phys. Solids

**61**, 1489–1510 (2013)Peigney M.: On the energy-minimizing strains in martensitic microstructures—part 2: geometrically linear theory. J. Mech. Phys. Solids

**61**, 1511–1530 (2013)Pijaudiercabot G., Benallal A.: Strain localization and bifurcation in a nonlocal continuum. I. J. Solids Struct.

**30**, 1761–1775 (1993)Sagar G., Stein E.: Contributions on the theory and computation of mono- and poly-crystalline cyclic martensitic phase transformations. Z. Angew. Math. Mech.

**90**, 655–681 (2010)Saleeb A.F., Padula S.A., Kumar A.: A multi-axial, multimechanism based constitutive model for the comprehensive representation of the evolutionary response of SMAs under general thermomechanical loading conditions. Int. J. Plast.

**27**, 655–687 (2011)Schlömerkemper A., Chenchiah I.V., Fechte-Heinen R., Wachsmuth D.: Upper and lower bounds on the set of recoverable strains and on effective energies in cubic-to-monoclinic martensitic phase transformations. MATEC Web Conf.

**33**, 02011 (2015)Sedlák P., Frost M., Benešová B., Ben Zineb T., Seiner H.: Šittner, thermomechanical model for NiTi-based shape memory alloys including R-phase and material anisotropy under multi-axial loadings. Int. J. Plast.

**39**, 132–151 (2012)Shaw J.A., Kyriakides S.: Thermomechanical aspects of NiTi. J. Mech. Phys. Solids

**43**, 1243–1281 (1995)Smyshlyaev V., Willis J.: A ‘non-local’ variational approach to the elastic energy minimization of martensitic polycrystals. Proc. R. Soc. Lond. A

**454**, 1573–1613 (1998)Smyshlyaev V., Willis J.: On the relaxation of a three-well energy. Proc. R. Soc. Lond. A

**455**, 779–814 (1998)Tartar L.: H-measures, a new approach for studying homogenization, oscillation and concentration effects in partial differential equations. Proc. R. Soc. Edinb. A

**115**, 193–230 (1990)Wagner M.F.-X., Windl W.: Lattice stability, elastic constants and macroscopic moduli of NiTi martensites from first principles. Acta Mater.

**56**, 6232–6245 (2008)Yawny A., Sade M., Eggeler G.: Pseudoelastic cycling of ultra-fine-grained NiTi shape-memory wires. Z. Metallkunde

**96**, 608618 (2005)

## Author information

### Authors and Affiliations

### Corresponding author

## Additional information

Communicated by Andreas Öchsner.

## Rights and permissions

## About this article

### Cite this article

Fechte-Heinen, R., Schlömerkemper, A. About lamination upper and convexification lower bounds on the free energy of monoclinic shape memory alloys in the context of *T*
_{3}-configurations and R-phase formation.
*Continuum Mech. Thermodyn.* **28**, 1601–1621 (2016). https://doi.org/10.1007/s00161-016-0494-1

Received:

Accepted:

Published:

Issue Date:

DOI: https://doi.org/10.1007/s00161-016-0494-1