Training-induced criticality in martensites
We propose an explanation for the self-organization towards criticality observed in martensites during the cyclic process known as ‘training’. The scale-free behavior originates from the interplay between the reversible phase transformation and the concurrent activity of lattice defects. The basis of the model is a continuous dynamical system on a rugged energy landscape, which in the quasi-static limit reduces to a sandpile automaton. We reproduce all the principal observations in thermally driven martensites, including power-law statistics, hysteresis shakedown, asymmetric signal shapes, and correlated disorder.
Experiments in martensites reveal intermittent behavior with power-law statistics Vives et al. (1994); Carrillo et al. (1998); Pérez-Reche et al. (2004a, b), showing an intrinsic complexity comparable to that of turbulence, earthquakes, internet networks, and financial markets. Criticality is known to be an issue of great significance in contemporary science, giving a framework for understanding the emergence of complexity in a variety of natural systems Bak et al. (1987); Sornette (2000). Within materials science, criticality has been recognized in the last years as a key factor in crystal plasticity, brittle fracture and damage Alava et al. (2006). In the present paper we develop a model explaining why similar behavior is also observed in martensitic transformations.
Reversible martensitic transformations involve a coordinated distortion of the crystal lattice and belong to the class of ferroelastic first-order phase changes with athermal character Olson and Owen (1992); Pérez-Reche et al. (2004). In such systems the macroscopic strain discontinuity typically splits into a set of bursts (avalanches) corresponding to transitions between neighboring metastable states. The individual avalanches can be detected through the measurement of the intermittent acoustic or calorimetric signals. The size distribution observed in shape-memory alloys (Cu-Al-Ni, Cu-Zn-Al, Cu-Al-Mn, Ni-Mn-Ga) was shown to be scale free Vives et al. (1994); Carrillo et al. (1998); Pérez-Reche et al. (2004a, b, 2004). Despite the apparent similarity with driven ferromagnetic systems, where the scale-free Barkhausen noise has been known for a long time, the experiments on memory alloys show features not observed in magnets, and which are instead reminiscent of plastic shakedown. In particular, the critical character of the avalanches Carrillo et al. (1998); Pérez-Reche et al. (2004a) and the smoothing of the hysteresis profile Miyazaki and Ishida (1999); Pérez-Reche et al. (2004a) emerge only after multiple thermal cycling through the transition.
The mechanism leading to training-induced critical behavior in martensites strongly resembles the phenomenology associated with self-organized criticality (SOC) Bak et al. (1987). The SOC paradigm in the form of a sandpile automaton has been applied to martensitic transformations in Goicoechea and Ortín, 1994; which however, lacked a connection to the physics of martensitic transformations. A different set of models exploited the similarity between martensites and magnetics by interpreting both in an Ising-type framework, with zero temperature and quenched disorder Sethna et al. (2006). In this context the power-law over a few decades of avalanche sizes is viewed as a sign of proximity of the system to a classical critical point. Criticality then emerges only as a result of the external tuning of the disorder. Furthermore, the symmetric avalanche shapes with scaling collapse which are expected in these models Sethna et al. (2006), contrast those experimentally recorded Pérez-Reche et al. (2004). More recent modeling has focused on the direct simulation of martensitic transformations within the framework of elasticity theory Sreekala et al. (2004). While the corresponding numerical tests show some scaling in avalanche sizes, the system is unable to memorize its state of disorder upon unloading, failing therefore to exhibit the effects of training.
A key experimental observation left aside in the preceding theoretical work is the dislocational activity assisting the development of the phase transformation in these materials Ríos-Jara and Guénin (1987); Pons et al. (1990); Lovey and Torra (1999); Miyazaki and Ishida (1999); Cuniberti and Romero (2004). It has been repeatedly observed that dislocations are indeed introduced in shape memory alloys during training. In particular, under periodic driving the degree of defectiveness first increases monotonically and then saturates Pons et al. (1990); Lovey and Torra (1999). Our model shows that this dislocational activity is highly correlated and is ultimately responsible for the scale-free character of the reversible behavior of martensites. More precisely, the attainment of criticality is due to the ability of the crystal to develop an optimal amount of disorder.
To describe in an unified way the processes involving both dislocations and phase boundaries, we consider a prototypical 2D system of kinematically compatible elastic units resulting from a suitable triangulation of a square lattice (see an example of such procedure in Conti and Zanzotto, 2004). To each unit with index we assign a multi-well strain energy function depending on a single scalar order parameter which, in turn, is a combination of the components of the discrete strain tensor. The adiabatic elimination of the harmonic non-order-parameter strain variables by means of the equilibrium equations and the kinematic compatibility constraints leads to a non-local elastic energy of the form
where is the kernel of the long-range elastic interactions, and is a periodic function as in Fig. 1. In each period we use the three-parabolic approximation , where in the high symmetry phase (austenite) and in the two variants of the low symmetry phase (martensite). The parameter , where is the transformation strain, defines the location of the bottoms of the energy wells; the integer-valued parameter specifies the period of . We emphasize that no randomness have been assumed in the model.
The global periodicity of the energy takes large shearing distortions into account Ericksen (1980), so that both the phase change and dislocation formation can be handled simultaneously. When the transformation strain is small and the energy barriers for slip are much higher than the barriers for the phase transition and twinning (‘weak transitions’), no lattice-invariant shears occur (as with the R-phase of NiTi Miyazaki and Ishida (1999), or the pre-martensitic transformation in Ni-Mn-Ga Pérez-Reche et al. (2004b)) and modeling can proceed according to the Landau theory. Such phase changes do not generate significant dislocational activity and are largely reversible, as is assumed, for instance, in Sreekala et al. (2004); Shenoy et al. (2006). On the contrary, in ‘reconstructive transformations’ the transformation strain is large, as it lays at the boundary of the periodicity domain (e.g. the ideal Bain transformation from bcc to fcc). In those cases the energy barriers to slip are only as high as the transformation barriers, and the phase change advances ‘half way’ towards the formation of dislocations Conti and Zanzotto (2004). As a result, defects proliferate making the transition irreversible Bhattacharya et al. (2004). In between these two extremes a range of possibilities exists, where defect formation plays an increasing role as the transformation strain gets closer to the boundary of the maximal periodicity domain. In particular, all the martensitic transformations considered in the experiments that we are concerned with Vives et al. (1994); Carrillo et al. (1998); Pérez-Reche et al. (2004a, b, 2004), involve a transformation strain which is very close to the ideal Bain strain Balandraud and Zanzotto (2007). When slowly driven, these systems are expected to exhibit cell deformations not confined to one periodicity domain, but rather extending to an unbounded portion of the periodic energy landscape. This involves formation of dislocations, which our kinematic compatibility assumption does not exclude, as is exemplified in Fig. 2, where we compare the standard representation of a dislocation (b) with the one adopted in this paper (a).
We drive the system quasi-statically trough the function , by changing the temperature . Since the transformations are typically athermal Pérez-Reche et al. (2004), we consider a purely mechanical setting with overdamped dynamics. By making this assumption we treat the acoustic emission as a part of dissipation. In the limit of infinitely slow driving, the dynamics projects on the local minima of the total energy , which form a discrete set of branches , with , where the extremes correspond to marginally stable configurations Puglisi and Truskinovsky (2005). For piece-wise parabolic as in Fig. 1, the limits of each branch can be explicitly written as a function of and . When such limits are reached, the instability resolves through a fast event (avalanche) which brings the system to another equilibrium branch. In this way the dynamics becomes piece-wise continuous (see Truskinovsky and Vainchtein, 2004 for a study of the 1D case).
We proceed by eliminating through minimization the linear elastic strain at given and obtaining . The relaxed energy is of the Ising type
where the two discrete spin variables describe the phase transformation () and the plastic slip (), and . Since the energy (2) is supposed to penalize the inhomogeneity of the field induced by either phase boundaries or dislocations, we assume that the corresponding term has the particular form so that . We furthermore assume the kernel to be of the ANNNI type, to account for the competing interactions driving both the coarsening and the refinement of the microstructure Ren and Truskinovsky (2000); Shenoy et al. (2006). Specifically, we consider for nearest neighbors, and for next-to-nearest neighbors; the ensuing diagonal-dominated structure of the matrix does not prevent the original matrix in (1) from being dense. Suitable inequalities ensure the metastability of the individual equilibrium branches Truskinovsky and Vainchtein (2004); in particular, the condition is sufficient for our automaton to reach a steady state (we use = 0.062, = 0.03 in the simulations presented below).
Under the above hypotheses, the piece-wise continuous dynamics becomes a sandpile automaton, whose main variable is the elastic strain representing the ‘local height’. Once a cell becomes unstable (the condition is violated), is updated as
where for phase transitions, and for slips. Since each update may make new sites unstable, the updates continue at constant until the system is fully equilibrated. For , , and the automaton (3) reduces to the standard BTW sandpile Bak et al. (1987).
We implement the model numerically on a grid for an almost reconstructive transformation with and open boundary conditions. The initially homogeneous austenite is chosen to contain only a minimal dislocation loop. The crystal is then thermally cycled through the complete transformation, being a periodic triangular function of computational time. The intensity of the acoustic bursts registered experimentally is linked to the size of the avalanches (total number of updates before stabilization). Fig. 3a shows the development of the phase microstructure during the training period. The level of plastic deformation is monitored through the density of nearest neighbors with differing values of . Fig. 3b shows the formation of dislocations induced by training and marked by the steep initial increase in the variable (Fig. 4a). The creation of correlated dislocation microstructure quickly saturates, in accordance with the experiments Pons et al. (1990); Lovey and Torra (1999). In Fig. 4b, we observe the smoothing effect of the self-organized defects on the cooling curves (and hence on the hysteresis cycle). The dislocational activity leads to the increase of the martensite starting temperature, similarly to what is reported experimentally Pérez-Reche et al. (2004a). The parallel development of criticality is indicated by the emergence of the power-law statistics for the avalanche sizes (Fig. 4c). At the beginning of the training period the avalanche distribution is supercritical with a peak at large sizes evident from the sharp initial cooling curves in Fig. 4b. The peak eventually vanishes (Fig. 4c), as in the experiments Pérez-Reche et al. (2004a). Two further predictions of the model matching experimental data Pérez-Reche et al. (2004) concern the strong asymmetry of the avalanche shapes, and the absence of their scaling collapse (see Fig. 4d). Similar effects are also observed in Barkhausen noise, earthquakes, and dynamic fracture (see the discussion in Zapperi et al., 2005). Fig. 4c, shows the distribution of avalanche durations (number of simultaneous updates in an avalanche) predicted by the model, which deviates from a power-law. This indicates that in the present framework a scale-free size distribution does not always imply scaling in time. While this prediction for an idealized system (with neither inertia nor fixed pinning sites) disagrees with the scaling for durations reported experimentally Vives et al. (1994), our model does generate an almost power-law distribution of durations after the introduction of a small amount of quenched disorder represented by a Gaussian distribution of with zero average in the initial configuration. Such modification, however, does not influence the power-law distribution of avalanches, nor does it affect the pulse asymmetry.
When in the original setting with two variables and the phase
transformation is suppressed (no variable ), the model describes
the micromechanics of stress-driven intermittent plastic flow in
crystals Dimiduk et al. (2006). In this case the system is defined by the
single integer-valued order parameter as in the phase-field
description of plasticity Koslowski et al. (2002), to which our model
offers an analytically accessible automaton alternative
In summary, the proposed model accounts for all the main observed phenomena accompanying the training process in martensites leading to criticality. The agreement with experiment clearly indicates that SOC originates in these systems as a result of the interplay between the reversible phase change and the irreversible development of an optimal amount of plastic deformation.
We thank J. Aizenberg, P. Collet, Ll. Mañosa, A. Planes, S. Roux, G. Tarjus, and E. Vives for helpful comments. FJPR was supported by the European contract MRTN-CT-2004-505226 and by Spanish grant MEC EX2005-0792. G.Z. was partially supported by grants from PRIN2005 and INdAM, Italy.
- Models with one variable and random thresholds have been explored in the literature (e.g. Bak et al. (1987)). In our model such randomness is not postulated a priori but is instead developing due to the complex coevolution of the two variables and . A two-variable model with well-separated relaxation times has been proposed in Coolen et al. (1991).
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