Deformed Shape Memory Alloys (SMAs) can recover their original shape upon transformation of their crystallographic structure from a low symmetry (martensite) to a higher symmetry (austenite) phase [1][2][3]. SMAs are, thus, desirable in engineering applications such as vascular stents and monolithic, frictionless, compact, lightweight, solid-state actuators [4][5][6][7][8]. The crystallographic phase transformation is non-diffusive, reversible, triggered by thermal and/or mechanical loading. The martensite phase is formed as thin platelets, needles, or laths within the austenite parent phase, resulting in crystallographic slip, termed TRansformation-Induced Plasticity (TRIP), as a mechanism to accommodate the deformation incompatibility at the austenite-martensite interfaces [9,10]. TRIP accumulates with transformation cycling, degrading the desired functionality of SMAs and is further responsible for a reduction in transformation stress, strain, and hysteresis [11][12][13][14][15][16][17][18].
Numerous constitutive equations of the deformation response of SMAs at the single-crystal level have been developed based on continuum mechanics [19][20][21][22][23][24][25][26][27][28][29][30][31]. These models, which are formulated by incorporating crystallographic information into a local continuum formulation, allow continuum stresses to be resolved onto individual planes responsible for mediating inelastic deformation. The scale in these models is smeared out and, thus, the spatial arrangement of the crystallographic details is not captured. However, they are (i) easily implementable in numerical methods for the full-field solution of boundary value problems in polycrystalline settings where misorientation across grains, grain boundaries, and triple joints can give rise to complex stress states, and (ii) are amenable to meanfield scale translation rules (e.g., Mori-Tanaka and selfconsistent approaches). Due to limitations in simultaneous measuring infinitesimal stress and strain increments at the mesoscale, the constitutive equations are in general derived and calibrated through a top-down approach by fitting material parameters to macroscopic experiments.
Given that material models are abstractions of reality, necessarily simplifying or omitting physical phenomena, assessing the credibility of performance predictions is of critical importance, particularly when these models are used in engineering design. Traditional engineering design is based on empirical safety factors. To allow for more efficient designs, empirical safety factors should be replaced by confidence bounds that account for (i) measurement errors, (ii) deficiencies of the material models used in the material characterization and performance predictions, and (iii) material variability. The growing importance of the related field of Uncertainty Quantification (UQ) is highlighted by the extensive guidelines and standards by the American Society of Mechanical Engineers [32,33]. UQ methods provide a robust framework for (i) the quantification of uncertainties by handling multiple error sources, (ii) the forward propagation of uncertainty, (iii) model selection in terms of data fit and model simplicity, and (iv) revealing model sensitivities and correlations among model parameters [34][35][36][37][38][39][40].
Instead of the classical, deterministic approach to model calibration that yields the single set of material parameter values that best matches the observed data in some appropriate sense, Bayesian inference determines a probability density function (pdf) for the model parameters, i.e., accounts for uncertainty in the estimated parameters. In turn, the model predictions are descriptors of the random field that emerges as a solution of the underlying stochastic model [36,[41][42][43]. Bayesian inference, rooted in Bayes' theorem, leads to an optimal update on prior knowledge conditional on the observational data. In Bayes' theorem, the likelihood pdf plays a role analogous to the cost/objective function in traditional fitting/optimization describing the error between model predictions and observational data. The error is formulated in terms of either the traditional additive error models [44][45][46] or embedded error models [47][48][49][50]; the latter assigning statistical bias correction terms to the model parameters directly. In the case of nonlinearity and high-dimensionality, the posterior distribution is evaluated as a stationary distribution of a Markov chain generated by a Markov Chain Monte Carlo (MCMC) method [51,52], which can be based on adaptive Metropolis-Hastings algorithms [53][54][55]. Ranking of the influence of the model parameters and their interdependencies on a model's predictions for the sake of model refinement, e.g., reduction of cost and complexity in model optimization problems, or gaining an insight into the physical process described by the model (conditional to the model) can be provided by ANalysis Of VAriance (ANOVA), which is a global variance-based sensitivity analysis [56][57][58][59]. Furthermore, Bayesian inference is a powerful approach for model selection among competing models emulating the response of a physical system [60][61][62], balancing ability to reproduce data with predictiveness and model simplicity [63][64][65].
In this paper, the Bayesian approach is adopted for the characterization of the inelastic deformation response of NiTi single crystals in a top-down approach from macroscopic experimental data. Uniaxial compressive loading experiments at six different crystallographic directions are utilized for this purpose [66]. The constitutive model accounts for phase transformation and plastic deformation based on micromechanics to accurately reflect the internal stress states that contribute to TRIP. Due to the complexity of the deformation response several simplifying assumptions are necessarily adopted. The presented analysis provides (i) material parameter values within confidence intervals; (ii) ranking the relative influence of the various material parameters on the deformation response; and (iii) a quantitative evaluation of the importance of accounting for the internal stress rise due to the deformation incompatibility among the phases in the overall deformation response. The obtained results offer an insight into the importance of the various deformation mechanisms in the overall deformation response conditional to the adopted model, and allow for a propagation of the quantified uncertainty onto performance predictions, which can be used toward a more efficient design methodology in which empirical safety factors can be replaced with confidence bounds.
The paper is organized as follows. In ''Deformation Response of Single NiTi Crystals'' section, the NiTi singlecrystal deformation mechanisms are briefly reviewed. The constitutive model accounting for those is presented in
In NiTi, austenite transforms from a cubic B2 crystallographic structure into 12 monoclinic (B19 0 ) martensite variants by mechanical loading and/or cooling. The martensite phase forms within the austenite parent phase with the two phases fitting together along planes, called invariant or habit planes, that remain unchanged, i.e., neither deform nor rotate. These planes/interfaces are between austenite and twins of martensite variants, called Habit Plane Variants (HPVs), that comprise of two Lattice Correspondent Variants (LCVs). Crystallographic theory predicts 192 HPVs, i.e., 192 possible distinct interfaces between austenite and martensite [67]. Movement of interfaces between HPVs is referred to as HPV reorientation, and movement of interfaces between LCVs as detwinning.
Plastic deformation in austenite is strongly influenced by h100i 011 f g and h101i 001 f g slip modes [68] and, as recently observed, by h111i 110 f g [69]. TRIP in austenite is observed during phase transformation as a mechanism to accommodate the deformation incompatibility at the austenite-martensite interfaces [9,10]. The phase transformation-plasticity coupling detrimentally affects performance, reflected in a reduced work output (functional fatigue) and early fatigue failure (structural fatigue) during repeated thermomechanical cycling [9,[70][71][72][73][74][75][76][77][78][79][80][81][82][83][84][85][86]. The plastic deformation in martensite is mainly due to twin activity, 11 possible twinning systems were pointed out by [87], while only one slip system (001) [100] exists due to the low symmetry of the martensite monoclinic crystal structure.
The adopted single-crystal model accounts for reversible phase transformation from austenite to HPVs, dislocation slipping in the austenite state, and anisotropy in the elastic properties of the two phases, neglecting reorientation of HPVs, detwinning of LCVs, and deformation twinning in martensite [88]. Thus, the model targets pseudoelasticity and shape memory effect for nearly proportional loading within a range that does not allow for considerable martensite plastic deformation or formation of self-accommodated martensite. The interaction between the two phases is described through the Eshelby tensor by regarding the HPVs as ellipsoidal inclusions embedded in the austenite matrix in order to reflect the internal stress states that can activate dislocation slipping, i.e., TRIP, even for applied load levels that wouldn't otherwise [89,90].
The inelastic deformation of an SMA crystal is defined as an average over a Representative Volume Element (RVE), which should be large enough to include a sizable set of martensite HPVs and slip systems within a single-crystal austenite. It is further assumed that the austenite-martensite formed interfaces are coherent and their motion along with the dislocation motion is rate-independent.
For later use, the volume fraction of martensite corresponding to the a th -HPV system in an RVE is denoted as n a ; restricted by 0 n a 1: The total volume fraction of martensite in a crystal, n ¼ P a n a ; must lie in the range 0 n a 1:
Assuming infinitesimal strains, additive decomposition of the total macroscopic strain tensor reads as
where e e ; e t ; and e p stand for the elastic, transformation, and plastic strain tensors, respectively. Thermal strain is an order of magnitude smaller than the transformation strain and is thus not included for simplicity. The thermal expansion of the monoclinic martensite variants is highly anisotropic [91] and its proper implementation in a singlecrystal model is not trivial [92].
By the rule of mixtures, the transformation strain can be written as
where b e a t ¼ 1 2 g t l a d a þ d a l a ð Þ ; l a ; d a ; and g t are the stress free transformation strain, the habit plane normal, the transformation direction, and the magnitude of transformation, respectively, for each of the N t martensite HPVs, given by crystallography considerations.
The rate of e t thus reads as
The overall plastic strain tensor can be written as
where e p A and e p M a stand for the plastic strain tensors in the regions occupied by austenite and a th -martensite HPV, respectively.
The rate of e p is thus given as
The rate of e p can moreover be described by crystallographic slip mechanisms in the austenite phase
is the orientation tensor of the l th -slipping system of austenite, q l A ; r l A ; _ c l A are the respective shear direction, slip plane normal, and average shearing rate, respectively, and N A denotes the number of slip systems.
Combining ( 5) and ( 6), which hold for every n a 2 ½0; 1; a ¼ 1; . . .; N t ; yields
Thus, the rates of plastic strain in the austenite and a thmartensite HPV are dependent on the dislocation slip rates on austenite's slip systems and on the rates of expansion/ shrinkage of the HPVs, thus, the model accounts for the inheritance of plastic strain from one phase to another.
Following the choice of the applied strain tensor e and absolute temperature T as external state variables, the Helmholtz free energy per unit reference volume is taken to be
where the interaction energy is defined through its rate as in [89] foot_0
and
The model parameters C and c denote the effective stiffness tensor and specific heat at the reference state, respectively. The effective stiffness tensor, C; can be evaluated in terms of the martensite volume fraction, n; by the rule of mixtures, i.e., CðnÞ
where the subscripts A and M denote austenite and martensite, respectively. Here, the assumption C ¼ C a M ¼ C A is adopted for simplicity since first principal calculations show that the elastic properties of the B2 and B19 0 phases are similar [104]. Moreover, c is assumed to be phase-independent, which is a common engineering assumption. The parameter T T is the phase equilibrium temperature and k is the latent heat of transformation at temperature T T : r A and r M a stand for the average stress values in the austenite and the a th -martensite HPV [89], S a stands for the Eshelby's tensor of the a th -martensite HPV, which depends on the elastic constants and shape of the variant, and I is the fourth-order unit tensor.
Using the above expression of the Helmholtz free energy, the standard thermodynamical procedure, commonly referred to as the Coleman-Noll procedure [93], applied to the dissipation inequality
where s is the entropy, yields the constitutive relationships
and reduces the dissipation expression to
Driving Forces
From the above dissipation expression, ( 14), the driving forces for phase transformation and plastic deformation can be invoked as the quasi-conservative thermodynamic forces conjugate to the respective internal variables. Martensitic transformation For transformation of austenite to a particular martensite HPV, a; the driving force, F t a ; for this HPV should satisfy the following nonequilibrium condition
where f f a t [ 0 is the HPV hardness, and r : b e a t is the resolved stress on the a th -transformation system, but not in the classical Schmid sense since l a is typically not perpendicular to d a .
For this particular martensite HPV to transform back to austenite, the following condition must be met
where r f a t [ 0. Plastic deformation of austenite For plastic deformation of austenite, the driving force for dislocation slip of the l thslip system, A F l p , should satisfy the following condition
where A f l p [ 0 is the respective slip system hardness.
The evolution laws of martensitic transformation and plastic deformation are given by the following power-law relations, in which the exponents are chosen sufficiently large to approximate rate-independent conditions. Martensitic transformation The evolution law for the volume fraction of the a th -martensite HPV follows the power-law relation
where _ n 0 is a reference transformation strain rate, H ab t h i is the interaction energy (constant) matrix between the different martensite HPVs, the scalar J t [ 0 describes the transformation hardening due to plastic deformation, and
is the accumulated total slip. c f f a t and c r f a t are positive scalars.
Plastic deformation of austenite The slip rate in the l thslip system of austenite is given as
with the evolution law of the hardness, A f l p ; reading as
where _ c 0 is reference plastic strain rate, the matrix
describes the history-dependent rate of increase of the deformation resistance on slip system l due to shearing on slip system r, given in terms of the accumulated total slip, q l stands for a constant latent-hardening parameter that ranges between 1 and 1.04, the positive scalar A H p is the initial slip system hardening rate, m p is the strain hardening exponent, and
The numerical procedure adopted for the integration of the constitutive law falls into the class of forward gradient methods, presented in detail in [88] and discussed in [94]. This method leads to improved numerical stability by resulting in a tangent stiffness expression which is considerably reduced from the elastic stiffness; in explicit integration, the maximum allowable time step is inversely related to the relevant material stiffness [95]. The Eshelby tensor is assumed identical for all HPVs and corresponds to oblate spheroids-the martensite phase is formed as thin platelets, needles, or laths within the austenite parent phase-in an isotropic matrix obtained by the isotropization of the stiffness tensor, C iso C ::
First, the relevant notation and terminology is introduced. Let Mðx; hÞ denote the numerical model of an SMA structure with the constitutive response presented in the previous section, which depends on a set of control variables x (e.g., displacement, temperature), and a set of material parameters h to be identified (characterized by a pdf) from a set of reference data d: The reference data comprises of N data points d
The set of parameters characterizing the error terms-i.e., hyperparameters-to be defined below, is denoted by /: pðÁjÁÞ and pðÁÞ denote conditional and marginal pdfs, respectively.
The relation between the measured values d i and the true process Tðx i Þ is represented as
where g i is the measurement error for the i th -observation and is modeled as independent and identically distributed (iid) normal random variable, g i ¼ N ð0; r 2 Þ; where r is a hyperparameter in the set of inferred parameters /: A strategy to model the relationship between the model output and experimental data is the additive error model [44][45][46]
where d is the model discrepancy function that accounts for missing physics in the model (and numerical approximation errors). d can modeled as a Gaussian process,
is the Mahalanobis distance between
x i and x i 0 with roughness parameters x d k [96]. The integer part of m d determines the mean square differentiability of the underlying process. The rest of the parameters
k  à T k¼1...q x ; are to be inferred from the measurement data. Note that the discrepancy term d is not explicitly considered in this paper. As a result, the assumed level of confidence in the simulations (''model = reality'') may result in a non-conservative uncertainty reduction.
The Bayes's theorem updates any prior information regarding the parameters h by incorporating new information obtained from the reference data d; i.e., combines information from the prior and likelihood pdfs to give a pdf for the parameters
where -pðhjMÞ is the prior pdf of the parameters; -pðdjh; MÞ is the likelihood pdf that plays an analogous role to the cost/objective function in traditional fitting/ optimization in the sense that it describes the discrepancy between model predictions and observational data;
pðhjd; MÞ is the posterior pdf that contains all the information on the parameters h; conditional on d and M; -pðdjMÞ is the evidence pdf, typically ignored when sampling from the posterior since it is a normalizing factor that ensures that the posterior pdf integrates to unity; however, this term plays a central role in model selection, as will be described later.
For simpler notation, M is kept implicit in the following.
The prior distribution indicates the initial degree of belief in the parameters' values by either prior quantitative knowledge or subjective expert opinion. It is common practice to assign uninformative prior pdfs to parameters for which little knowledge is available (for example, a uniform distribution or normal distribution with large variances) and assume statistical independence among them
Conjugate priors such that the prior and posterior distributions of the parameter belong to the same family are recommended for practical and computational purposes. Further discussion on the choice of priors may be found in, e.g., [39,97].
From the statistical models and the above definitions and assumptions, the likelihood function is expressed as a multivariate normal distribution, assuming statistical independence of reference data, as follows
where l ¼ d i ½ i¼1...N ; C ¼ r 2 I; and I is the N Â N identity matrix.
Given that the model M is nonlinear and multi-dimensional, the posterior pdf pðhjdÞ is evaluated by a Markov Chain Monte Carlo (MCMC) method [51], specifically Gibbs sampler and Metropolis-Hastings (MH) algorithm [53,54]. The key idea behind MCMC is to generate a Markov chain whose stationary distribution corresponds to the posterior distribution [52]. An adaptive MH algorithm proposed in [98] is used in this work in order to enhance the rates at which the chains generated by the algorithm converge to the posterior distribution. In each iteration of this algorithm, a parameter candidate is sampled from an adaptive proposal distribution, i.e., a multivariate normal distribution (q), centered at the previous parameter sample in the chain (or the initial guess for the first iteration) with a variance-covariance matrix that is adapted using the variance-covariance of all previous parameter samples in the chain as described by [98]. Then, the parameter candidate is accepted or rejected in a probabilistic manner based on the Metropolis-Hastings ratio
where the joint densities, i.e., prior  likelihood, for the sampled candidate, h cand , and the previous sample in the chain, h iÀ1 , which are proportional to their posterior densities, are compared using the first ratio, known as the Metropolis ratio. The probabilities of the forward and backward moves from h iÀ1 to h cand are also compared using the second ratio (Hastings ratio) in order to account for the asymmetry in the proposal distribution. At the end, the burn-in period, which that includes the samples before the parameter convergence, is discarded from the sample chain fh 1 . . .h n g. The mean of remaining samples and the square root of diagonal terms in their variance-covariance matrix (standard deviations) indicate the most plausible values and uncertainties of model parameters.
Model selection is performed based on the Bayesian hypothesis testing [99]. Assuming all models have equal prior probabilities ahead of analysis, the Bayes factor, i.e., the ratio of marginal likelihoods (or model evidences), is considered as a metric to identify to what extent a model (a null hypothesis) is favored by evidence (data)
where
is the model evidence. Here, the model selection process involves the pairwise comparison of models, where BðM i ; M j Þ [ 1 indicates optimality of M i over M j and BðM i ; M j Þ\1 the other way around.
Among the different methods proposed to approximate (30) [100], the MCMC sampling from the posterior in an importance sampling-based integration scheme is adopted, according to which, the model evidence equals a harmonic average of likelihood values
) À1
; ð31Þ associated with the MCMC sampled parameters [60,100].
ANOVA by Design Of Experiment (DOE) is a rigorous technique for global sensitivity analysis that performs well in high-dimensional problems. This approach relies on a set of frequentist hypothesis testing, where the insensitivities of model outcomes to the individual parameters and their interactions are assumed to be the null hypotheses. The model outcome variation (uncertainty) is decomposed into a sum of contributions due to the input factors and their interactions (according to a chosen DoE and appropriate levels for the parameters) to calculate the corresponding F values, i.e., ratios between variances, for ranking the parameters based on their influence on the model outcomes [101,102]. Herein, a two-level Full-Factorial Design (FFD) is applied to take into account all the level combinations for the parameters, which is equivalent to 2 S level-parameter combinations, where S is the number of parameters.
In this section, a summary of the model parameters is given. Those related to the inelastic response are subjected to Bayesian calibration from the experimental data reported in [66] for Ni 50:9 Ti (at.%) single-crystal isothermal uniaxial loading; deterministic values are adopted for the rest of the parameters. The model parameters are also listed in Table 1a-c. Three dots ''…'' in the parameter values column indicate that the respective parameters are subjected to probabilistic calibration. Elastic parameters The B2 structure belongs to the cubic crystal system and thus the elastic tensor of austenite phase in NiTi SMA single crystals possesses three independent constants, i.e., C A 11 ¼ 130 GPa, C A 22 ¼ 98 GPa, and C A 44 ¼ 34 GPa [103]. As already mentioned, the assumption that C a M ¼ C A is adopted for simplicity [104]. Transformation parameters Of the 192 possible HPVs predicted by the crystallographic theory of martensite only the 24 Type II-1 HPVs frequently observed in experiments are considered. The components of the vectors l a and d a are given in [105,106] and g t ¼ 0:1308. The interaction matrix H ab t h i , given in [107], is not accounted for since simulations showed that its inclusion overestimates the transformation hardening observed in the experimental data. The ''viscous'' parameter n is set to a high value, n ¼ 50, to approximate the rate-independent response of NiTi. The reference transformation rate value, _ n 0 , is representative of the applied loading rate, determined using the method suggested in [108].
A deterministic value is adopted for the latent heat, k ¼ 154 MJ/m 3 , calibrated from differential scanning calorimetry [30].
The initial critical forces for forward phase transformation are assumed identical for all martensite HPVs,
r f t , and J t are subject to Bayesian calibration.
Parameters related to dislocation slipping-Experimentally, only slip in the system families h100i 001 f g, h100i 011 f g, h110i 111 f g has been observed [69], and, thus, only these slip families are included in the simulations. The initial critical forces for slip in these systems, c A f l p , are assumed identical for each family, and are thus reduced to c A f r p (r ¼ 1; 2; 3). c A f r p (r ¼ 1; 2; 3) and A H p are Ratio of self to latent-hardening, q l 1.4
Critical force for slip in h110i
The crystallographic data for the 24 martensite HPVs in NiTi is given [105,106]. The strain rate exponent n is set to n ¼ 50
Shap. Mem. Superelasticity evaluated by Bayesian calibration, while the reference plastic strain rate, _ c 0 , is determined as in [108]. Note that c 0 p and m p in ( 22) cannot be reliably calibrated from the experiments reported in [66] and are thus assumed null; experiments at a temperature above M d would be required for their calibration, i.e., a temperature at which stressinduced phase transformation is suppressed.
The Bayesian inference methodology described in Section 4 is used to perform the probabilistic calibration of the model for the seven parameters, listed in Table 1 with three dots ''…'' as their values, from the experimental stressstrain curves reported in [66] for Ni 50:9 Ti (at.%) single crystals uniaxially loaded in six crystallographic directions at room temperature. Since there is no prior knowledge available for these parameters, the prior pdfs are assumed to be uniform over the parameter ranges specified in Table 2. The level of discrepancy between the simulations and experimental data is measured by minimizing the squared Euclidean distance between predicted and experimental stress values, R i;j pred and R i;j exp ; respectively, at specified strain values (denoted by index i) for each loading direction (denoted by index j)
i.e., by minimizing the differences between the predicted and experimental stress-strain curves.
The model calibration is performed by generating 30,000 parameter samples during the MCMC process. After the elimination of the burn-in period, the mean value and standard deviation of the remaining samples for each parameter are calculated and listed in Table 2 as their calibrated value and uncertainty.
Uncertainty propagation from the probabilistically calibrated parameters to the model outputs, i.e., the uniaxial stress-strain responses in different loading directions, is performed by running the model for the mean parameter values and each sample in the convergence region, sorting the resulting output samples, and then discarding 2.5% of these samples from the upper and lower bounds to estimate 95% credible intervals. The most plausible model predictions (red lines) and 95% credible intervals (green shaded areas) in addition to the corresponding experimental data (blue lines) are plotted in Fig. 1. Depending on the loading orientation the experimental stress-strain responses differ in terms of the required load level for initiation of forward/ reverse phase transformation, strain hardening, and amount of residual deformation with the simulations to quantitatively reproduce the experimental data in good agreement. The discrepancy between the most plausible model predictions and the experimental data should be attributed to the constitutive model's assumptions/simplifications, mostly to those related to self-and latent-hardening laws due to both phase transformation and plastic deformation and to a lesser extend to others, such as the equal hardness assumption for all HPVs.
Correlations among the parameters is examined via the Pearson correlation coefficient
where r X , r Y , and r X;Y denote the standard deviation of parameter X, the standard deviation of parameter Y, and the covariance between X and Y, respectively, quantifies the linear correlations for each pair of model parameters. The coefficient varies between À 1 and 1, where the sign demonstrates the correlation direction. Values closer to À 1 and 1 imply higher linear correlations between pair parameters, while values closer to 0 correspond to lower correlations. In Table 3, the Pearson correlation coefficients for all pairs of calibrated parameters are tabulated. All coefficient values indicate almost no or weak linear correlations, except for the two hardening parameters, J t and A H p , that show some degree of correlation (q % 0:4). The marginal distributions of J t and A H p and their joint (pair) distribution are shown in Fig. 2. Figure 2a andb shows a distinct convergence peak in their marginal frequency distributions, while Fig. 2c andd indicate the convergence region in the joint pdf of these two parameters in 3D and 2D, respectively. Some degree of linearity of color features in Fig. 2d may qualitatively provide the insight indicated from the Pearson correlation coefficient into the linear correlation between the two parameters. Table 3 Pearson correlation coefficients between the pair model parameters
As already mentioned, the model accounts for reversible phase transformation from austenite to HPVs and dislocation slipping in the austenite state. In contrast to most existing micromechanics-based single-crystal models, with the exception of [89,90], a mean-field contribution of the deformation incompatibility at the interphases on the internal stress states that drive dislocation slipping during phase transformation, i.e., TRIP, is introduced. In an effort to elucidate the relative contribution of the inelastic deformation mechanisms and of the aforementioned micromechanical description of the internal stress rise due to the deformation incompatibility among the phases in the overall deformation response, two extra models are constructed and compared pairwise with each other and the adopted model M using the Bayesian model selection outlined in ''4.3'' section. Model M A does not account for dislocation slipping and model M B does account for dislocation slipping but without the introduced contribution of the deformation incompatibility among the phases into the driving forces for dislocation slipping, i.e., r A ¼ r ð¼ r M a Þ in (10). The probabilistic calibration of M A and M B follows the same procedure adopted for M; note that just 3 parameters, c f f t , c r f t , and J t , need to be calibrated for M A . The models' outputs for uniaxial loading in the [1 1 1]-direction are compared in Fig. 3. The expected high importance of dislocation slipping in the deformation response is obvious. However, model selection between M B and M in terms of model's ability to reproduce data is not possible from a visual comparison alone. The Bayes factor calculated for all pairs of models (Table 4) based on (29) suggest dramatic evidence in favor of models M B and M over model M A , in accordance with the visual comparison of the subfigures in Fig. 3, while the evidence in favor of model M over M B is rather weak. M is the most likely model (based on model evidence), with M B being 10% less likely. Thus, the incorporation of the mean-field evaluation of the internal stress that contributes to TRIP does improve the model's ability to reproduce the experimental data and should result in parameter values that are more representative of reality. Note that the calibrated values of the initial critical forces for slip are substantially lower in model M B and the calibrated values of the hardening parameters higher (Table 5).
A two-level FFD-based seven-way ANOVA by DoE is utilized to identify the sensitivity of model outputs to the variations of model parameters listed in Table 2. To perform this analysis, two levels are considered for each parameter based on its 95% credible interval bounds obtained from Table 2, i.e., hhi AE 2 Â r h : The l 2 -norm of squared Euclidean distances (in the form of (32)) between the stress-strain results for the parameter mean values and their counterparts for each level-parameter combination is obtained and input into the ANOVA analysis.
The ANOVA results are shown in Table 6 in descending order for F values that corresponds to a reduction in the parameter influence. Unsurprisingly, the most influential parameters are the ones related to the phase transformation, namely, the critical force for forward phase transformation, the transformation hardening due to plastic deformation, and the critical force for reverse phase transformation in a decreasing order of the level of influence on the model outputs.
Based on energetic grounds [69], h100i 011 f g has the lowest energy barrier (the unstable peak value) and hence is most likely to be activated, h110i 111 f g is a harder slip system, and h100i 001 f g is the hardest slip system to activate and it has hardly been observed. h100i 011 f g permits glide only in three independent slip systems out of the six physically possible (or geometric) slip systems, thus h100i 011 f g cannot produce five independent slip systems to satisfy the Mises criteria for arbitrary deformations, same as h100i 001 f g. h110i 111 f g can produce five independent slip systems (out of physically possible 12 slip systems). According to the ANOVA by DoE analysis, the ranking of the critical forces for dislocation slipping from the most influential to the least is as follows: h110i 111 f g; h100i 011 f g; and h100i 001 f g: The difference in F values between h110i 111 f g; h100i 011 f g is quite small and both values are orders of magnitude greater than the F value corresponding to h100i 001 f g; which is a small value (( 1), i.e., the influence of the h100i 001 f g on the overall deformation response is negligible. (ii) a quantitative evaluation of the importance of the nucleation and build-up of slip in austenite to accommodate the high transformation strains upon traversing austenite-martensite interphases, are provided, conditional on the adopted constitutive response.
The derivation of the interaction term in[89] is based on the Mori-Tanaka and Kro ¨ner micromechanical assumptions and the instantaneous growth hypothesis according to which the martensitic domains form instantaneously. Shap. Mem. Superelasticity
Shap. Mem. Superelasticity