A mesoscopical model of shape memory alloys/kujumaluga sulamite mesoskoopiline mudel.

1. INTRODUCTION, STORED ENERGY, MICROSTRUCTURE, DISSIPATION ENERGY

Shape memory alloys (SMAs) belong to the so-called smart materials which enjoy important applications. These exhibit specific, hysteretic stress/strain/ temperature response and a so-called shape memory effect. The mechanism behind it is quite simple: atoms tend to be arranged in several crystallographical configurations having different symmetry groups: higher symmetrical one (referred to as the austenite phase, typically cubic) has higher thermal capacity while lower symmetrical one (called the martensite phase, typically tetragonal, orthorhombic, or monoclinic) has lower thermal capacity and may exist, by symmetry, in several variants (typically 3, 6, or 12, respectively). We refer to [1-7] for a thorough survey. Here we consider only isothermal stress-strain response modelling.

We consider a bounded Lipschitz domain [OMEGA] [subset] [R.sup.3] as a reference configuration (canonically the stress-free austenite). Standardly, the displacement u : [OMEGA] [right arrow] [R.sup.3] and the deformation y : [OMEGA][right arrow][R.sup.3] are related by y(x) = x + u(x), x[member of][OMEGA]. Hence the deformation gradient is F = [nabla]y = [??] + [nabla]u, where [??] [member of][R.sup.3x3] denotes the identity matrix. Mechanical response is phenomenologically described by a specific stored energy [??] = [??](F), assumed to have a p-polynomial growth/coercivity structure. The frame-indifference, i.e. [??](F) = [??](RF) for any R [member of] SO(3), the group of orientation-preserving rotations, requires that [??](*) in fact depends only on the (right) Cauchy-Green stretch tensor C := [F.sup.T]F. We abbreviate

[psi](*) := [??]([??] + *) : (1)

The overall free energy related to a displacement profile u is [PHI](u):=[[integral].sub.[OMEGA]] [psi]([nabla]u)dx. Considering a (time-varying) elastic support w(t, x) on a part [GAMMA] of the boundary [partial derivative][OMEGA], we expand it to the stored energy G(t, u) = [PHI](u)+ 1/2 [[integral].sub.[GAMMA]][(u - w(t, *)).sup.[perpendicular to]]B(u - w(t, *))dS with [B.sup.[perpendicular to]] = B. Due to the multiwell character of [psi], the deformation gradient usually tends to develop fast spatial oscillations if it tends to minimize the overall stored energy under given boundary conditions, see [1,4,8,9], resulting in a microstructure that can effectively be described by so-called gradient Young measures, which are measurably parameterized probability measures x [??] [v.sub.x] on [R.sup.3x3] that can be attained by gradients in the sense [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for some sequence [{[u.sub.k]}.sub.k[member of]N] [subset] [W.sup.1,p]([OMEGA];[R.sup.3]) and all g [member of] [L.sup.[infinity]]([OMEGA]) and v [member of] [C.sub.0]([R.sup.3x3]), see [9]; the notation [C.sub.0], [L.sup.p], [W.sup.1,p] for function spaces is standard. Let us denote the set of all such parameterized measures by [G.sup.p]([OMEGA];[R.sup.3x3]). The continuously extended (so-called relaxed) stored energy is then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

The pair of "macroscopical" displacement u and the gradient Young measures v represents a quite natural mesoscopical description of the state of the body. The "kinematically" admissible pairs (u, v) are in

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Within microstructure evolution due to time-varying loading w, SMAs dissipate energy. For sufficiently slow loading, these processes are activated and quite rate-independent, leading to a hysteretic stress-strain response. We assume dissipative forces having a (pseudo)potential, say R, and that the energy dissipated during the phase-transformation process depends (counting phenomenologically, beside possible rank-one connections, with various impurities) on the starting and final (phase) variants, only; this (simplifying) concept has been adopted also in [10-14]. We implement this philosophy with the help of a frame-invariant "phase indicator" being a smooth bounded function [??] : [R.sup.3x3] [right arrow] [R.sup.L], with L denoting the number of (phase) variants. Then, with L(A) := [??]([??] + A) like (1), the dissipation potential is postulated as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

with a convex compact K [subset] [R.sup.L] determining the activation stresses, [[delta].sub.K] being its indicator function, and [[delta].sup.*.sub.K] its conjugate which is, of course, homogeneous degree-1. The quantity [lambda] plays the role of a macroscopic volume fraction assigned through (3) to the microstructure described by v.

2. ENERGETIC SOLUTION, LAMINATES, NUMERICAL APPROXIMATION

With neglecting kinetic energy and based on the minimum-stored-energy principle competing with the maximum-dissipation (or rather realizability [15]) principle, in the scalar (hence convex) case, the desired evolution (u, v) = (u(t), v(t)) : [0, T] [right arrow] Q would be governed by the doubly-nonlinear evolution inclusion

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

considered completed by an initial condition, here on [lambda]. In the convex case, it is equivalent (see [16,17]) to the energetic formulation, i.e. stability

[for all]([??], [??]) [member of] Q : [bar.G](t, u(t), v(t)) [less than or equal to] [bar.G](t, [??],[??]) + R(v(t) - [??]), (5)

together with the energy equality

[??](t) + [Var.sub.R](v, s, t) = [??](s) - [[integral].sub.(s,t)x[GAMMA]] [(u - w).sup.[perpendicular to]] B [partial derivative]w/[partial derivative]t dSdt (6)

to be satisfied for any 0 [less than or equal to] s < t [less than or equal to] T, where [??](t) := [bar.G](t, u(t), v(t)) is the Gibbs energy and [Var.sub.R](v, s, t) denotes the total variation over [s, t] of v(*) with respect to R from (3). The particular terms in (6) represent the stored energy at time t, the energy dissipated by changes of the internal structure during the time interval [s, t], the stored energy at the initial time s, and work done by external loadings during the time interval [s, t]. In our vectorial case, the set of admissible configurations Q is no longer convex, hence (4) has no longer a good sense and we must rely on the energetic formulation (5)-(6) as a natural generalization.

Mathematical advantage of the energetic formulation (5)-(6) by Mielke and Theil [16-18] is that it is free of time derivatives. The existence of thus defined energetic solution (u, v) : [0; T] [right arrow] Q has been shown in [19], provided [bar.G] is still regularized by counting energy of possible spatial jumps in [lambda], as proposed in [20], p. 364.

For computational implementation, additional discretization of the set Q is necessary. The canonical approach is to apply P1-finite elements on a triangulation (with a discretization parameter h) of a polyhedral domain [OMEGA] for discretization uh of u and elementwise constant (= homogeneous) so-called laminates (see [9]) to discretize v. We implemented the second-order laminate, which leads to the four-atomic Young measure [v.sub.h], where

[v.sub.h] = [[xi].sub.0h][[xi].sub.1h][[delta].sub.F1h] + [[xi].sub.0h](1-[[xi].sub.1h])[[delta].sub.F2h] + (1-[[xi].sub.0h])[[xi].sub.2h][[delta].sub.F3h] + (1-[[xi].sub.0h])(1-[[xi].sub.2h])[[delta].sub.F4h]

with

[F.sub.1h] = [nabla][u.sub.h] - (1-[[xi].sub.0h])[a.sub.h] [cross product] [n.sub.h] - (1-[[xi].sub.1h])[a.sub.1h] [cross product] [n.sub.1h],

[F.sub.2h] = [nabla][u.sub.h] - (1-[[xi].sub.0h])[a.sub.h] [cross product] [n.sub.h] + [[xi].sub.1h][a.sub.1h] [cross product] [n.sub.1h],

[F.sub.3h] = [nabla][u.sub.h] + [[xi].sub.0h][a.sub.h] [cross product] [n.sub.h] - (1-[[xi].sub.2h])[a.sub.2h] [cross product] [n.sub.2h],

[F.sub.4h] = [nabla][u.sub.h] + [[xi].sub.0h][a.sub.h] [cross product] [n.sub.h] + [[xi].sub.2h][a.sub.2h] [cross product] [n.sub.2h].

Here 0 [less than or equal to] [[xi].sub.ih] [less than or equal to] 1, i = 0, 1, 2, are elementwise constant. The vectors [a.sub.ih] [member of] [R.sup.3] and [n.sub.ih] [member of] [R.sup.3] are elementwise constant as well and, moreover, we may choose |[n.sub.ih]| = 1. Hence, the whole Young measure [v.sub.h] is identified by means of [nabla][u.sub.h] and {[[xi].sub.ih], [a.sub.ih], [n.sub.ih]}. This ensures that ([u.sub.h], [v.sub.h]) [member of] Q. The same approximation was used, for instance, in [21;22].

In order to find an approximate energetic solution, we consider a fully-implicit time discretization based on the following incremental problem: take a time step [tau] > 0 and let [v.sup.0.sub.h] be a given initial condition (we do not prescribe an initial condition for [u.sub.h] because R depends only on v), and, for k = 1, ..., T/[tau] member of] N we define recursively [([u.sup.k.sub.h], [v.sup.k.sub.h]).sub.k=1,...,T/[tau]] as a solution to the minimization problems

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

3. COMPUTATIONAL EXPERIMENTS WITH CuAlNi

The orthorhombic martensite has 6 variants, i.e., counting also austenite, L = 7. The frame-indifferent stored energy composed of St. Venant-Kirchhoff-type materials for each (phase) variant is postulated as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

where [C.sup.l] = {[C.sup.l.sub.ijkl]} is the 4th-order tensor of elastic moduli, [R.sub.l] are rotation matrices relating the martensitic coordinates to the reference austenite, [d.sub.l] are some offsets, and [U.sub.l] the distortion matrices: [U.sub.0] = [??] corresponds to austenite while

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

while the other three, i.e. [U.sub.4], ...,[U.sub.6], take -[[eta].sub.3] in place of [[eta].sub.3]. An example of Cu-14.0wt%Al-4.2wt%Ni counts with [[eta].sub.1] = 1.04245, [[eta].sub.2] = 0.9178, and [[eta].sub.3] = 0.01945. The specific values of elastic moduli are determined from experiments; we refer to Sedlak et al. [23]. We also use the usual Voigt's notation, which (in a one-to-one way) replaces [C.sup.l] by [{[C.sup.l.sub.ij}.sup.6.sub.i,j=1]. For l = 0, i.e. for austenite, by symmetry there are only 3 nonvanishing elastic moduli, i.e. here [C.sup.0.sub.11] = [C.sup.0.sub.22] = [C.sup.0.sub.33] = 142.8 GPa, [C.sup.0.sub.44] = [C.sup.0.sub.55] = [C.sup.0.sub.66] = 93.5 GPa, [C.sup.0.sub.12] = [C.sup.0.sub.23] = [C.sup.0.sub.13] = 129.7 GPa. The specific values for martensite (in the basis of a particular variant) are [C.sub.11] = 189 GPa, [C.sub.22] = 141 GPa, [C.sub.33] = 205 GPa, [C.sub.44] = 54.9 GPa, [C.sub.55] = 19.7 GPa, [C.sub.66] = 62.6 GPa, [C.sub.12] = 124 GPa, [C.sub.13] = 45.5 GPa, [C.sub.23] = 115 GPa. Matrices [R.sub.l] in (8) are proper rotations transforming C to the basis of austenite and can be found in [21]. The offset [d.sub.l] in (8) has been chosen as 3 MPa, which corresponds to the process temperature of 312 K.

As to the construction of the phase-indicator function L : [R.sup.3x3] [right arrow] [R.sup.7], we take some [delta] > 0 small and a smooth function d : R [right arrow] R such that d = 1 in a neighbourhood of 0 and d = [delta] far from that neighbourhood, and put

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

The set K in (3) is chosen as a simplex in [R.sup.7] and specific dissipation energies (or, equally, activation stresses) are set to be 2 MJ/[m.sup.3] (= 2 MPa) for transformations between austenite and martensite and 1 Pa for transformations between various variants of martensite, which makes the so-called re-orientation of martensite almost nondissipative. It is an unfortunate reality that the data for the phenomenological dissipation model are very difficult to obtain. Moreover, dissipation mechanisms are often not fully autonomous and, e.g., may vary within the number of cycles in cyclical loadings. Here, the concrete value 2 MJ/[m.sup.3] is approximately fitted with experiments reported in [24], fig. 1 or [25] fig. 4, while the value 1 Pa is to reflect that the reorientation of two martensite variants, which are rank-one connected, is nearly nondissipative at least if there are not much impurities in the material so that pinning effects are small (cf. also [26] for the case of austenite/martensite transformation).

4. RESULTS OF COMPRESSION TESTS

Our specimen is a block with dimensions 4mm x 9mm x 4 mm, referring to the stress-free austenite [OMEGA]. Its bottom is fixed by the zero-displacement Dirichlet boundary condition, while on its top we apply varying stress, ranging the interval 0 - 300 MPa in the vertical direction (cf. Fig. 1). The initial condition is [v.sup.0.sub.h] = [[delta].sub.0], i.e., the whole specimen is in the austenite. The form of stored energy (8) together with variants (9) reflect the case when the crystal lattice of austenite has the orientation (001). In many applications, however, the specimen is oriented differently, see e.g. [27]. Various material orientations can be easily implemented by using the specific stored energy [??](F) = [??](F[R.sub.A]), where [R.sub.A] is a rotation of the austenite from (001). Four compression tests were performed for (0,tan[alpha],1)-oriented single crystal with [alpha] = 0, 10, 20, and 30 degrees (cf. Fig. 2).

It should be remarked that, in real CuAlNi single crystals, the 2H ([[gamma]'.sub.1]) orthorhombic martensite, considered in the above text, occurs in compression tests near the (001) directions, while in directions closer to (011) or (111) another type of martensite, namely 18R ([[beta]'.sub.1]) which is monoclinic, may be observed, too. To model it, other 12 wells would have to be included into the stored energy and other dissipation energies would have to be specified. Beside such expansion of the energies in the model, the simulations would expectedly be more difficult because the optimization algorithms are computationally less efficient if the landscape of the minimized energy in (7) has more local valleys. In the compression test presented here, the monoclinic martensite seems, indeed, relatively negligible, as documented in [25], fig. 5, and therefore we dared neglect it. Also, our aim has been rather to present the modelling aspects and the ability of the model itself.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

ACKNOWLEDGEMENTS

This study was supported by grants 201/06/0352 (GA CR), A107 5402 (GA AV CR), LC 06052, MSM21620839 and VZ6840770021 (MSMT CR), and MRTN-CT-2004-505226 (EU).

Received 15 January 2007

REFERENCES

[1.] Bhattacharya, K. Microstructure of Martensite. Why it Forms and How it Gives Rise to the Shape-Memory Effect. Oxford University Press, New York, 2003.

[2.] Fremond, M. and Miyazaki, S. Shape Memory Alloys. Springer, Wien, 1996.

[3.] James, R. D. and Hane, K. F. Martensitic transformations and shape-memory materials. Acta Mater., 2000, 48, 197-222.

[4.] Muller, S. Variational models for microstructure and phase transitions. In Calculus of Variations and Geometric Evolution Problems (Hildebrandt, S. et al., eds). Lect. Notes in Math., 1999, 1713, 85-210.

[5.] Otsuka, K. and Shimizu, K. Morphology and crystallography of thermoelastic Cu-Al-Ni martensite analyzed by the phenomenological theory. Trans. Japan Inst. Metals, 1974, 15, 103-113.

[6.] Pitteri, M. and Zanzotto, G. Continuum Models for Phase Transitions and Twinning in Crystals. Chapman & Hall, Boca Raton, 2003.

[7.] Roubicek, T. Models of microstructure evolution in shape memory materials. In Nonlinear Homogenization and its Application to Composites, Polycrystals and Smart Materials (Ponte Castaneda, P., Telega, J. J. and Gambin, B., eds). Kluwer, Dordrecht, 2004, 269-304.

[8.] Ball, J. M. and James, R. D. Fine phase mixtures as minimizers of energy. Archive Rat. Mech. Anal., 1988, 100, 13-52.

[9.] Pedregal, P. Parametrized Measures and Variational Principles. Birkauser, Basel, 1997.

[10.] Huo, Y. and Muller, I. Nonequilibrium thermodynamics of pseudoelasticity. Continuum Mech. Thermodyn., 1993, 5, 163-204.

[11.] Petryk, H. Thermodynamic conditions for stability in materials with rate-independent dissipation. Phil. Trans. Roy. Soc. A, 2005, 363, 2479-2515.

[12.] Stupkiewicz, S. and Petryk, H. Modelling of laminated microstructures in stress-induced martensitic transformations. J. Mech. Phys. Solids, 2002, 50, 2303-2331.

[13.] Thamburaja, P. and Anand, L. Polycrystalline shape-memory materials: effect of crystallographic texture. J. Mech. Physics Solids, 2001, 49, 709-737.

[14.] Vivet, A. and Lexcelent, C. Micromechanical modelling for tension-compression pseudoelastic behaviour of AuCd single crystals. Euro Phys. J. A. P., 1998, 4, 125-132.

[15.] Levitas, V. I. The postulate of realizibility. Int. J. Eng. Sci., 1995, 33, 921-971.

[16.] Mielke, A. Evolution of rate-independent systems. In Handbook of Differential Equations (Dafermos, C. and Feireisl, E., eds). Elsevier, Amsterdam, 2005, 461-559.

[17.] Mielke, A. and Theil, F. On rate-independent hysteresis models. Nonlin. Diff. Eq. Appl., 2004, 11, 151-189.

[18.] Mielke, A., Theil, F. and Levitas, V. I. A variational formulation of rate-independent phase transform using an extremum principle. Arch. Rat. Mech. Anal., 2002, 162, 137-177.

[19.] Mielke, A. and Roubicek, T. Rate-independent model of inelastic behaviour of shape-memory alloys. Multiscale Modeling Simul., 2003, 1, 571-597.

[20.] Fremond, M. Non-Smooth Thermomechanics. Springer, Berlin, 2002.

[21.] Kruzik, M., Mielke, A. and Roubicek, T. Modelling of microstructure and its evolution in SMA single-crystals, in particular in CuAlNi. Meccanica, 2005, 40, 389-418.

[22.] Roubicek, T. and Kruzik, M. Mesoscopic model of microstructure evolution in shape memory alloys, its numerical analysis and computer implementation. GAMM Mitteilungen, 2006, 29, 192-214.

[23.] Sedlak, P., Seiner, H., Landa, M., Novak, V., Sittner, P. and Manosa, L. l. Elastic constants of bcc austenite and 2H orthorhombic martensite in CuAlNi shape memory alloy. Acta Mater., 2005, 53, 3643-3661.

[24.] Novak, V., Sittner, P. and Zarubova, N. Anisotropy of transformation characteristics of Cu-base alloys. Mater. Sci. Eng. A, 1997, 234-236, 414-417.

[25.] Novak, V., Sittner, P., Vokoun, D. and Zarubova, N. On the anisotropy of martensitic transformation in Cu-based alloys. Mater. Sci. Eng. A, 1999, 273-275, 280-285.

[26.] James, R. D. and Zhang, Z. A way to search for multiferroic materials with "unlikely" combination of physical properties. In Magnetism and Structure in Functional Materials, Ch. 9 (Planes, A., Manoza, L. and Saxena, A., eds). Springer, 2005, 159-176.

[27.] Novak, V., Sittner, P., Ignacova, S. and Cernoch, T. Transformation behavior of prism shaped shape memory alloy single crystals. Mater. Sci. Eng. A, 2006, 438-440, 755-762.

Tomas Roubicek (a,b), Martin Kruzik (b,c), and Jan Koutny (b)

(a) Mathematical Institute, Charles University, Sokolovska 83, CZ-186 75 Praha 8, Czech Republic; [e-mail]

(b) Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, Pod vodarenskou vezi 4, CZ-182 08 Praha 8, Czech Republic; [e-mail], [e-mail]

(c) Department of Physics, Faculty of Civil Engineering, Czech Technical University, Thakurova 7, CZ-16629 Praha 6, Czech Republic