Abstract
Thermoelastic martensitic transformations in shape memory alloys can be modeled on the basis of nonlinear elastic theory. Microstructures of fine phase mixtures are local energy minimizers of the total energy. Using a one-dimensional effective model, we have shown that such microstructures are inhomogeneous solutions of the nonlinear Euler–Lagrange equation and can appear upon loading or unloading to certain critical conditions, the bifurcation conditions. A hybrid numerical method is utilized to calculate the inhomogeneous solutions with a large number of interfaces. The characteristics of the solutions are clarified by three parameters: the number of interfaces, the interface thickness, and the oscillating amplitude. Approximated analytical expressions are obtained for the interface and inhomogeneity energies through the numerical solutions.
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References
Kaufman, L., Cohen, M.: Martensitic transformations. In: Chalmers, B., King, R. (eds.) Progress in Metal Physics, vol. 7, 165–246. Pergamon Press, Oxford (1958)
Nishiyama, M.: Transformations. Academic Press, San Diego (1978)
Olson, G.B., Cohen, M.: Thermoelastic behavior in martensitic transformations. Scripta Met. 9, 1247–1254 (1975)
Duerig, T.W., Melton, K.N., Stöckel, D.: Engineering Aspects of Shape Memory Alloys. Butterworth-Heinemann, London (1990)
Otsuka, K., Wayman, C.M.: Shape Memory Materials. Cambridge University Press, Cambridge (1999)
Wu, Z.Q., Zhang, Z.H.: Force-displacement characteristics of simply supported beam laminated with shape memory alloys. Acta Mechanica Sinica 27, 1065–1070 (2011)
Yang, S.B., Xu, M.: Finite element analysis of 2D SMA beam bending. Acta Mechanica Sinica 27, 738–748 (2011)
Rong, Q.Q., Cui, Y.H., Shimada, T., et al.: Self-shaping of bioinspired chiral composites. Acta Mechanica Sinica 30, 533–539 (2014)
Bhattacharya, K.: Microstructure of Martensite. Oxford University Press, Oxford (2003)
Cheng, P., Xingyao, W., Yongzhong, H.: Characteristics of stress-induced transformation and microstructure evolution in Cu-based SMA. Acta Mechanica Solida Sinica 21, 1–8 (2008)
Roubicek, T.: Models of microstructure evolution in shape memory alloys. In: Ponte Castaneda, P., et al. (eds.) Nonlinear Homogenisation and Its Applications to Composites, Polycrystals and Smart Materials, 269–304. Kluwer Academic publishers, Dordrecht (2004)
Patoor, E., Lagoudas, D.C., Entchev, P.B., et al.: Shape memory alloys, Part I: general properties and modeling of single crystals. Mech. Mater. 38, 391–429 (2006)
Lagoudas, D.C., Entchev, P.B., Popov, P., Patoor, E., Brinson, L.Catherine, Gao, Xiujie., et al.: Shape memory alloys, Part II: modeling of polycrystals. Mech. Mater. 38, 430–462 (2006)
Porta, M., Lookman, T.: Heterogeneity and phase transformation in materials : energy minimization, iterative methods and geometric nonlinearity. Acta Materialia 61, 5311–5340 (2013)
Cesana, P., Porta, M., Lookman, T.: Asymptotic analysis of hierarchical martensitic microstructure. J. Mech. Phys. Solids 72, 174–192 (2014)
Guquan, S., Qingping, S., Kehchih, H.: Effect of microstructure on the hardening and softening behavIors of polycrystalline shape memory alloys Part I: Micromechanics constitutive modeling. Acta Mechanica Sinica 16, 309–324 (2000)
Xiangyang, Z., Qingping, S., Shouwen, Y.: On the strain jump in shape memory alloys–a crystallographic-based mechanics analysis. Acta Mechanica Sinica 15, 134–144 (1999)
Li, J.Y., Lei, C.H., Li, L.J., et al.: Unconventional phase field simulations of transforming materials with evolving microstructures. Acta Mechanica Sinica 28, 915–927 (2012)
Maxwell, J.C., Lord, Rayleigh: Encyclopedia Britannica (1876)
Christian, J.W.: The Theory of Transformations in Metals and Alloys. Pergamon Press, New York (2002)
Khachaturyan, A.G.: Theory of Structural Transformations in Solids. John Wiley & Sons, New York (1983)
Mueller, I., Seelecke, S.: Thermodynamic aspects of shape memory alloys. Math. Comput. Model 34, 1307–1355 (2001)
Yan, Y., Yin, H., Sun, Q.P., et al.: Rate dependence of temperature fields and energy dissipations in non-static pseudoelasticity. Contin. Mech. Thermodyn. 24, 675–695 (2012)
Yin, H., Yan, Y., Huo, Y.Z., et al.: Rate dependent damping of single crystal CuAlNi shape memory alloy. Mater. Lett. 109, 287–290 (2013)
Hao, Yin, Yongjun, He, Qingping, Sun: Effect of deformation frequency on temperature and stress oscillations in cyclic phase transition of NiTi shape memory alloy. J. Mech. Phys. Solids 67, 100–128 (2014)
Shield, T.W.: Orientation dependence of the pseudoelastic behaviour of single crystals of Cu–Al–Ni in tension. J. Mech. Phys. Solids 43, 869–895 (1995)
Waitz, T., Antretterb, T., Fischerb, F.D., et al.: Size effects on the martensitic phase transformation of NiTi nanograins. J. Mech. Phys. Solids 55, 419–444 (2007)
Ueland, S.M., Schuh, C.A.: Transition from many domain to single domain martensite morphology in small-scale shape memory alloys. Acta Materialia 61, 5618–5625 (2013)
Sun, Q.P., Aslan, A., Li, M.P., et al.: Effects of grain size on phase transition behavior of nanocrystalline shape memory alloys. Sci. China Technol. Sci. 57, 671–679 (2014)
Ericksen, J.L.: Equilibrium of bars. J. Elast. 5, 191–202 (1975)
Carr, J., Gurtin, M.E., Slemrod, M.: Structured phase transition on a finite interval. Arch. Rat. Mech. Anal. 86, 317–351 (1984)
Müller, S.: Singular perturbations as a selection criterion for periodic minimizing sequences. Cal. Var. Partial Diff. Equ. 1, 169–204 (1993)
Truskinovsky, L., Zanzotto, G.: Ericksen’s bar revisited: energy wiggles. J. Mech. Phys. Solids 44, 1371–1408 (1996)
Vainchtein, A., Healey, T., Rosakis, P., et al.: The role of the spinodal in one dimensional phase transitions microstructures. Phys. Rev. D. 115, 29–48 (1998)
Anna, V., Healey, T.J., Rosakis, P.: Bifurcation and metastability in a new one-dimensional model for martensitic phase transitions. Comput. Methods Appl. Mech. Eng. 170, 407–421 (1999)
Ren, X., Truskinovsky, L.: Finite scale microstructures in nonlocal elasticity. J. Elast. 59, 319–355 (2000)
Vainchtein, A.: Dynamics of phase transitions and hysteresis in a viscoelastic Ericksen’s bar on an elastic foundation. J. Elast. 57, 243–280 (1999)
Vainchtein, A.: Hysteresis and stick-slip motion of phase boundaries in dynamic models of phase transitions. J. Nonlinear Sci. 9, 697–719 (1999)
Xuan, C., Peng, C., Huo, Y.: One dimensional model of Martensitic transformation solved by Homotopy analysis method. Z. Naturforsch 67a, 230–238 (2012)
Liao, S.J.: Beyond Perturbation: Introduction to the Homotopy Analysis Method. Chapman & Hall/ CRC, Boca Raton (2003)
Liao, S.: Homotopy Analysis Method in Nonlinear Differential Equations. Higher education press, Beijing (2011)
Wechsler, M., Lieberman, D., Read, T.: On the theory of the formation of martensite. Trans. AIME J. Metals 179, 1503–1515 (1953)
Bowles, J., MacKenzie, J.: The crystallography of martensitic transformations I and II. Acta Metal. Mater. 2, 129–147 (1954)
Santman, S., Guo, Z.: Large shearing oscillations of incompressible nonlinear elastic. J. Elast. 14, 249–262 (1984)
Ball, J.M., James, R.D.: Fine phase mixtures as minimizers of energy. Arch. Rat. Mech. Anal. 100, 13–52 (1987)
Kohn, R., Müller, S.: Surface energy and microstructure in coherent phase transitions. Comm. Pure. Appl. Math. 47, 405–435 (1994)
Huo, Y., Muller, I.: Interfacial and inhomogeneity penalties in phase transitions. Contin. Mech. Thermodyn. 15, 395–407 (2003)
Hui-Hui, D., Zongxi, C.: An analytical study on the instability phenomena during the phase transitions in a thin strip under uniaxial tension. J. Mech. Phys. Solids 60, 691–710 (2012)
Kalies, W.: Regularized models of phase transformation in one-dimensional nonlinear elasticity. [Ph. D. Thesis], Cornell University (1994)
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This project was supported by the National Natural Science Foundation of China (Grants 11461161008 and 11272092)..
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Dedicated to Professor Zhongheng Guo.
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Xuan, C., Ding, S. & Huo, Y. Multiple bifurcations and local energy minimizers in thermoelastic martensitic transformations. Acta Mech. Sin. 31, 660–671 (2015). https://doi.org/10.1007/s10409-015-0491-9
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DOI: https://doi.org/10.1007/s10409-015-0491-9