Full-Field Micromechanics of Precipitated Shape Memory Alloys

Abstract

A full-field micromechanics approach is developed to predict the effective thermomechanical response of precipitation-hardened near-equiatomic Ni-rich NiTi alloys on the basis of composition and heat treatment. The microscale-informed model takes into account the structural effects of the precipitates (precipitate volume fraction, elastic properties, and coherency stresses due to the lattice mismatch between the precipitates and the matrix) on the reversible martensitic transformation under load as well as the chemical effects resulting from the Ni depletion of the matrix during precipitate growth. The post-aging thermomechanical response is predicted based on finite element simulations on representative microstructures, using the response of the solutionized material and time–temperature–martensitic transformation temperature maps. The predictions are compared with experiments for materials of different initial compositions and heat treatments and reasonably good agreement is demonstrated. The proposed methodology can be in principle extended to predict the post-aging thermomechanical response of other shape memory alloy systems as well.

Appendix: Constitutive Law for Polycrystalline SMAs

The model is developed within the framework of continuum thermodynamics and adopts the classical rate-independent small-strain flow theory for the evolution equations of the transformation strains (Lagoudas et al., 1996, 2012; Lagoudas, 2008). This model is used to describe the response of both precipitated and unprecipitated polycrystalline SMAs. By utilizing infinitesimal strains and assuming an additive strain rate decomposition, the increments of the total strain tensor components, dɛ _ij , are given as

$$\displaystyle{ d\varepsilon _{ij} = S_{ijkl}d\sigma _{kl} + dS_{ijkl}\sigma _{kl} + d\varepsilon _{ij}^{t}, }$$

(8.8)

where σ _ij and ɛ _ij ^t are the Cartesian components of the stress and transformation strain tensor, respectively, and S _ijkl represents the “current” compliance tensor. Standard Einstein notation is used with summation over repeated indices assumed. The “current” compliance tensor varies with the martensite volume fraction, ξ, as S _ijkl = (1 −ξ)S _ijkl ^A + ξS _ijkl ^M, where S _ijkl ^A and S _ijkl ^M are the components of the compliance tensor of the polycrystalline SMA material in the austenitic and martensitic phase, respectively. The assumption of elastic isotropy for both the austenitic and martensitic phases results in $S_{ijkl}^{\alpha } = \frac{1+\nu _{\alpha }} {2E_{\alpha }}(\delta _{il}\delta _{jk} +\delta _{ik}\delta _{jl}) - \frac{\nu _{\alpha }} {E_{\alpha }}\delta _{ij}\delta _{kl}$, where the index α stands for A and M in the cases of pure austenite and martensite, respectively. The Young’s modulus and Poisson’s ratios of the SMA’s two phases are denoted E _α and ν _α , respectively, and δ _ij is Kronecker’s delta.

An evolution equation of the transformation strain is defined so that it is related to the evolution of martensite volume fraction ξ,

$$\displaystyle{ d\varepsilon _{ij}^{t} = \Lambda _{ ij}d\xi,\ \ \Lambda _{ij} = \left \{\begin{array}{@{}l@{\quad }l@{}} \Lambda _{ij}^{fwd},\;d\xi> 0,\quad \\ \Lambda _{ij}^{rev},\;d\xi <0, \quad \end{array} \right. }$$

(8.9)

where, $\Lambda _{ij}$, the components of the direction tensor, are defined as

$$\displaystyle{ \Lambda _{ij}^{fwd} = \frac{3} {2} \frac{H^{cur}} {\sigma } \sigma _{ij}^{{\prime}},\ \ \Lambda _{ ij}^{rev} = \frac{\varepsilon _{ij}^{t}} {\xi }. }$$

(8.10)

Here, H ^cur is the uniaxial transformation strain magnitude for complete transformation, $\sigma = \sqrt{\frac{3} {2}\sigma _{ij}^{{\prime}}\sigma _{ij}^{{\prime}}}$ is the Mises equivalent effective stress, and σ _ij ^′ = σ _ij −σ _kk δ _ij ∕3 are the stress deviator components. Forward transformation generates transformation strain in the direction of the deviatoric stress, which motivates the selected J ₂ form of the direction tensor. During reverse phase transformation, it is assumed that the direction and magnitude of the transformation strain recovery is governed by the average orientation of martensite at transformation reversal (the cessation of forward transformation, be it partial or full). This definition ensures a zero transformation strain for every state with a null martensite volume fraction.

During transformation, the stress tensor components should remain on the transformation surface:

$$\displaystyle{ \Phi = 0,\;\;\Phi = \left \{\begin{array}{@{}l@{\quad }l@{}} \Phi ^{fwd} =\pi ^{fwd} - Y _{ 0},\;d\xi> 0, \quad \\ \Phi ^{rev} = -\pi ^{rev} - Y _{0},\;d\xi <0,\quad \end{array} \right. }$$

(8.11)

with π ^fwd, π ^rev being the thermodynamic driving forces for forward and reverse transformation, respectively, and Y ₀ being the critical value of the thermodynamic force to both initiate and sustain forward and reverse phase transformation.

The thermodynamic driving force for forward transformation is written as

$$\displaystyle{ \pi ^{fwd} =\sigma _{ ij}\Lambda _{ij}^{fwd} + \frac{1} {2}\Delta S_{ijkl}\sigma _{ij}\sigma _{kl} + \rho \Delta s_{0}T -\rho \Delta u_{0} - f^{fwd}, }$$

(8.12)

and for reverse transformation

$$\displaystyle{ \pi ^{rev} =\sigma _{ ij}\Lambda _{ij}^{rev} + \frac{1} {2}\Delta S_{ijkl}\sigma _{ij}\sigma _{kl} + \rho \Delta s_{0}T -\rho \Delta u_{0} - f^{rev}. }$$

(8.13)

f ^fwd and f ^rev are functions describing the transformation hardening behavior during forward and reverse phase transformation, respectively. s ₀ and u ₀ are the specific entropy and internal energy, respectively, ρ is the density, and $\Delta$ denotes the difference in property between the martensitic and the austenitic states.

8.1.1 Variation of the Transformation Strain Magnitude

The transformation strain magnitude, H ^cur, is a function of the stress state since precipitated polycrystalline SMA materials do not exhibit a constant maximum attainable transformation strain at all stress levels (Bo and Lagoudas, 1999c; Bo et al., 1999). A saturated value of maximum attainable transformation strain, H _sat , is reached at a high-stress level, which is dependent on the SMA material and the processing conditions. Following this observation, H ^cur is represented by the following decaying exponential function:

$$\displaystyle{ H^{cur}(\sigma ) = H_{ sat}\left (1 - e^{-k\sigma }\right ). }$$

(8.14)

The parameter k controls the rate at which H ^cur exponentially evolves from 0 to H _sat .

8.1.2 Description of a “Smooth” Thermomechanical Response

In precipitated polycrystalline SMAs, local transformation initiates in a non-uniform manner, resulting in an experimentally observed gradual transition from the elastic to transformation response and vice versa. To capture the gradual transformation initiation and completion response, the hardening functions are given as general power laws in terms of ξ with real components:

$$\displaystyle{ f^{fwd}(\xi ) = \frac{1} {2}\alpha _{1}\left [1 +\xi ^{n_{1} } - (1-\xi )^{n_{2} }\right ] + \alpha _{3}, }$$

(8.15)

$$\displaystyle{ f^{rev}(\xi ) = \frac{1} {2}\alpha _{2}\left [1 +\xi ^{n_{3} } - (1-\xi )^{n_{4} }\right ] -\alpha _{3}, }$$

(8.16)

where, α _i  (i = 1, 2, 3) are coefficients that assume real number values and n _i  (i = 1, 2, 3) are exponents that assume real numbers in the interval (0, 1]. If n ₁ and/or n ₃ take values less than 1, the forward and/or reverse phase transformations, respectively, are initiated in a “smooth” gradual fashion. Similarly, if n ₂ and/or n ₄ take values less than 1, the forward and/or reverse phase transformations, respectively, are completed in a “smooth” gradual fashion.

8.1.3 Calibration of the Model

Given the above constitutive relations, the following model parameters must be calibrated: (1) the elastic parameters of the precipitated polycrystalline SMA in the austenitic and martensitic states, (2) parameters contained in the functional form of the maximum transformation strain H ^cur(σ), and (3) six model parameters ($\rho \Delta s_{0}$, $\rho \Delta u_{0}$, α ₁, α ₂, α ₃, Y ₀) that are characteristic of the martensitic transformation. The common material properties that are used to calibrate the model are E _A , E _M , ν _A , ν _M , H _sat , M _s , M _f , A _s , A _f , C _M , and C _A . M _s , M _f , A _s , and A _f are the martensitic-start, martensitic-finish, austenitic-start, and austenitic-finish temperatures at zero load, respectively, and C _M and C _A are the forward and reverse transformation slopes in the stress-temperature phase diagram, respectively (Fig. 8.2). The elastic constants can be calculated directly from nominally isothermal stress–strain curves where loads are applied at temperatures outside the transformation regions. The parameters for H ^cur(σ) can be calibrated directly from material testing under thermal variations at a constant applied load, where the value of k in particular is chosen to best fit the experimental trend. The remaining six parameters are calibrated by considering the conditions under which forward transformation begins and ends in the stress-temperature space (Lagoudas, 2008). The hardening coefficients n ₁ − n ₄ do not have an associated material property but are directly chosen to best fit the four corners of the transformation hysteresis plots.