Transformation Stretch Tensor Research Articles

Electronic Mechanism of Martensitic Transformation in Nb-doped NiTi Alloys: A First-Principles Investigation.

The effect of Nb on the crystal structures and electronic mechanism of martensitic transformation in Ni50Ti50–xNbx alloys is investigated by first principles. The lattice parameters, the formation energy, the middle eigenvalue of the transformation stretch tensor (λ2), and the energy difference between the parent and martensite (ΔE) as a function of Nb content x (x = 0, 2.08, 6.25, 8.33, 10.42, 12.5, 18.75) are calculated. Lattice parameters increase with the increase of Nb content. The formation energies of the parent B2 phase, martensite orthorhombic B19, and monoclinic B19′ increase with the increase of Nb content. It is also found that at ≤10.42 at. % Nb, the martensite stable phase is monoclinic structure B19′; at >10.42 at. % Nb, the orthorhombic crystal structure B19 is formed. The energy difference between the parent and martensite means that the transformation temperature decreases with increasing Nb concentration at Nb ≤ 10.42 at. % and increases at >10.42 at. % Nb. The λ2 of the NiTiNb alloys have the same value of about 0.95 with low Nb content. Furthermore, the electronic structure mechanisms behind the martensitic transformations are discussed in detail based on the density of states.

Experimentally correlating thermal hysteresis and phase compatibility in multifunctional Heusler alloys

Thermal hysteresis is recognized as one of the main drawbacks for cyclical applications of magnetocaloric and ferromagnetic shape memory materials with first-order transformations. As such, the challenge is to develop strategies that improve the compatibility between the phases involved in the transitions and study its influence on thermal hysteresis. With this purpose, we explore the thermal, structural, and magnetic properties of the ${\mathrm{Ni}}_{2}{\mathrm{Mn}}_{1\ensuremath{-}x}{\mathrm{Cu}}_{x}{\mathrm{Ga}}_{0.84}{\mathrm{Al}}_{0.16}$ Heusler alloys. The alloys present a thermal hysteresis reduction of \ensuremath{\sim}60% when the Cu content in the compound varies from $x=0.10$ to $x=0.25$, with a minimum hysteresis width of 6 K being achieved. We applied the geometric nonlinear theory of martensite to address the phase compatibility, quantified by the parameter ${\ensuremath{\lambda}}_{2}$, the middle eigenvalue of the transformation stretch tensor, and found that the minimum of hysteresis is associated with a better crystallographic compatibility (${\ensuremath{\lambda}}_{2}$ closer to 1) between the austenite and martensite phases. In addition, we show that the valleylike properties of hysteresis found in the ${\mathrm{Ni}}_{2}{\mathrm{Mn}}_{1\ensuremath{-}x}{\mathrm{Cu}}_{x}{\mathrm{Ga}}_{0.84}{\mathrm{Al}}_{0.16}$ compounds is present in several other alloys in the literature. These results provide pathways to understand as well as to master the phase compatibility and ultimately achieve a low thermal hysteresis in multifunctional Heusler alloys.

Martensitic transformation in stoichiometric NiMn and Ni–Mn-X alloys: A first principles study

The martensitic phase transformation in stoichiometric NiMn alloy and its shape memory effect is investigated using first principles method based on Density Functional Theory (DFT). The elastic properties of the B2 austenite and the L10 martensite phases were compared and the results indicated that the B2 phase has a low elastic shear modulus along the [110] direction giving rise to the formation of L10. Electronic density of states (DOS) calculations showed that the existence of the pseudo-gap appearing at the Fermi level which stabilizes the martensite phase against the austenite depends on both magnetism and the tetragonality ratio. Using geometrically nonlinear theory of martensite it was found that large volume change and twinning shear, and the low value of middle eigenvalue of the transformation stretch tensor (λ2 = 0.8766) are some of the properties of the martensitic transformation in NiMn which inhibit the shape memory effect in this system despite being structurally similar to other shape memory alloys such as NiAl. The effect of the addition of 8 alloying elements on the martensitic transformation and shape memory effect is investigated. The results showed, in agreement with the experimental findings, that addition of Ti and Al can improve the shape memory effect. Also, it was found that among the 8 elements considered here, Al, Ti, and Ge are the most effective alloying elements in enhancing the shape memory properties of NiMn by lowering the transformation temperature, decreasing the volume change, and increasing the value of λ2..

An exact formulation for exponential-logarithmic transformation stretches in a multiphase phase field approach tomartensitic transformations

A general theoretical and computational procedure for dealing with an exponential-logarithmic kinematic model for transformation stretch tensor in a multiphase phase field approach to stress- and temperature-induced martensitic transformations with N martensitic variants is developed for transformations between all possible crystal lattices. This kinematic model, where the natural logarithm of transformation stretch tensor is a linear combination of natural logarithm of the Bain tensors, yields isochoric variant–variant transformations for the entire transformation path. Such a condition is plausible and cannot be satisfied by the widely used kinematic model where the transformation stretch tensor is linear in Bain tensors. Earlier general multiphase phase field studies can handle commutative Bain tensors only. In the present treatment, the exact expressions for the first and second derivatives of the transformation stretch tensor with respect to the order parameters are obtained. Using these relations, the transformation work for austenite ↔ martensite and variant ↔ variant transformations is analyzed and the thermodynamic instability criteria for all homogeneous phases are expressed explicitly. The finite element procedure with an emphasis on the derivation of the tangent matrix for the phase field equations, which involves second derivatives of the transformation deformation gradients with respect to the order parameters, is developed. Change in anisotropic elastic properties during austenite–martensitic variants and variant–variant transformations is taken into account. The numerical results exhibiting twinned microstructures for cubic to orthorhombic and cubic to monoclinic-I transformations are presented.

Reversible low-field magnetocaloric effect in Ni-Mn-In-based Heusler alloys

Ni-Mn-$X$ ($X=\mathrm{In}$, Sn, and Sb) based Heusler alloys show a strong potential for magnetic refrigeration owing to their large magnetocaloric effect (MCE) associated with first-order magnetostructural transition. However, the irreversibility of the MCE under low field change of 0--1 T directly hinders its application as an efficient magnetic coolant. In this work, we systematically investigate thermal and magnetic properties, crystalline structure and magnetocaloric performance in $\mathrm{N}{\mathrm{i}}_{51\ensuremath{-}x}\mathrm{M}{\mathrm{n}}_{33.4}\mathrm{I}{\mathrm{n}}_{15.6}{\mathrm{V}}_{x}$ alloys. With the introduction of V, a stable magnetostructural transition near room temperature is observed between martensite and austenite. An extremely small hysteresis of 2.3 K is achieved for the composition $x=0.3$. Due to this optimization, the magnetic-field induced structural transition is partially reversible under 0--1 T cycles, resulting in a reversible MCE. Both magnetic and calorimetric measurements consistently show that the largest value for the reversible magnetic entropy change can reach about $5.1\phantom{\rule{0.16em}{0ex}}\mathrm{J}\phantom{\rule{0.16em}{0ex}}\mathrm{k}{\mathrm{g}}^{\ensuremath{-}1}\phantom{\rule{0.16em}{0ex}}{\mathrm{K}}^{\ensuremath{-}1}$ in a field change of 0--1 T. A considerable and reversible adiabatic temperature change of \ensuremath{-}1.2 K by the direct measurement is also observed under a field change of 0--1.1 T. Furthermore, the origin of this small hysteresis is discussed. Based on the lattice parameters, the transformation stretch tensor is calculated, which indicates an improved geometric compatibility between the two phases. Our work greatly improves the MCE performance of Ni-Mn-$X$-based alloys and make them suitable as realistic magnetic refrigeration materials.

Interfacial stresses within boundary between martensitic variants: Analytical and numerical finite strain solutions for three phase field models

The origin of a large elastic stress within an interface between martensitic variants (twins) within a finite strain phase field approach has been determined. Notably, for a sharp interface this stress is absent. Three different constitutive relations for the transformation stretch tensor versus order parameters have been considered: a linear combination of the Bain tensors (kinematic model-I, KM-I), an exponential-logarithmic combination (KM-II) of the Bain tensors, and a stretch tensor corresponding to simple shear (KM-III). An analytical finite-strain solution has been found for an infinite sample for tetragonal martensite under plane stress condition. In particular, explicit expression for the stresses have been obtained. The maximum interfacial stress for KM-II is more than twice that which corresponds to KM-I. Stresses are absent for KM-III, but it is unclear how to generalize this model for multivariant martensitic transformation. An approximate analytical solution for a finite sample has been found as well. It shows good correspondence with numerical results obtained using the finite element method. The obtain results are important for developing phase field approaches for multivariant martensitic transformations coupled to mechanics, especially at the nanoscale.

Determination of the stretch tensor for structural transformations

Structural transformations in crystalline solids are increasingly the basis of the functional behavior of materials. Recently, in diverse alloy systems, both low hysteresis and reversibility of phase transformations have been linked to the satisfaction of the nongeneric conditions of compatibility between phases. According to the Cauchy–Born rule, these conditions are expressed as properties of transformation stretch tensor. The transformation stretch tensor is difficult to measure directly due to the lack of knowledge about the exact transforming pathway during the structural change, and the complicating effects of microstructure. In this paper we give a rigorous algorithmic approach for determining the transformation stretch tensor from X-ray measurements of structure and lattice parameters. For some traditional and emerging phase transformations, the results given by the algorithm suggest unexpected transformation mechanisms.

Study of the cofactor conditions: Conditions of supercompatibility between phases

The cofactor conditions, introduced in James and Zhang(2005), are conditions of compatibility between phases in martensitic materials. They consist of three subconditions: (i) the condition that the middle principal stretch of the transformation stretch tensor U is unity (λ2=1), (ii) the condition a·Ucof(U2−I)n=0, where the vectors a and n are certain vectors arising in the specification of the twin system, and (iii) the inequality trU2+detU2−(1/4)|a|2|n|2≥2. Together, these conditions are necessary and sufficient for the equations of the crystallographic theory of martensite to be satisfied for the given twin system but for any volume fraction f of the twins, 0≤f≤1. This contrasts sharply with the generic solutions of the crystallographic theory which have at most two such volume fractions for a given twin system of the form f⁎ and 1−f⁎. In this paper we simplify the form of the cofactor conditions, we give their specific forms for various symmetries and twin types, we clarify the extent to which the satisfaction of the cofactor conditions for one twin system implies its satisfaction for other twin systems. In particular, we prove that the satisfaction of the cofactor conditions for either Type I or Type II twins implies that there are solutions of the crystallographic theory using these twins that have no elastic transition layer. We show that the latter further implies macroscopically curved, transition-layer-free austenite/martensite interfaces for Type I twins, and planar transition-layer-free interfaces for Type II twins which nevertheless permit significant flexibility (many deformations) of the martensite. We identify some real material systems nearly satisfying the cofactor conditions. Overall, the cofactor conditions are shown to dramatically increase the number of deformations possible in austenite/martensite mixtures without the presence of elastic energy needed for coexistence. In the context of earlier work that links the special case λ2=1 to reversibility (Cui et al., 2006; Zhang et al., 2009; Zarnetta et al., 2010), it is expected that satisfaction of the cofactor conditions for Type I or Type II twins will lead to further lowered hysteresis and improved resistance to transformational fatigue in alloys whose composition has been tuned to satisfy these conditions.

Influence of crystallographic compatibility on residual strain of TiNi based shape memory alloys during thermo-mechanical cycling

The primary focus of this study was to determine the influence of the inherent microstructure, specifically the crystallographic compatibility between martensite and austenite during the transformation process, on residual strain and strain evolution during isobaric thermal cycling. A series of predominantly single-phase Ti-rich TiNi based shape memory alloys (SMAs), including binary Ti50.1Ni49.9, two ternary TiNiPd25 compositions with slightly different Ni contents, Ti50.5Ni28.5Pt21, and a quaternary Ti50Ni24.5Pd25Sc0.5 alloy, were thermally cycled nominally 100 times at different stress levels. In all cases, the amount of residual strain per cycle was found to decrease asymptotically with increasing number of cycles to some finite value that was related to the compatibility of the transforming phases as determined by the middle eigenvalue of the transformation stretch tensor. In addition, the amount of residual strain generated during the early thermal cycles was also dependent on the resistance of the SMAs against defect propagation, which can be controlled by initial dislocation density and grain size among other factors. In light of these findings, it was concluded that a dimensionally stable SMA actuator for multiple cycle operation should have compatible transforming phases to reduce the generation of defects and a high resistance against defect propagation.

A weak compatibility condition for precipitation with application to the microstructure of PbTe–Sb 2Te 3 thermoelectrics

We propose a weak condition of compatibility between phases applicable to cases exhibiting full or partial coherence and Widmanstätten microstructure. The condition is applied to the study of Sb 2Te 3 precipitates in a PbTe matrix in a thermoelectric alloy. The weak condition of compatibility predicts elongated precipitates lying on a cone determined by a transformation stretch tensor. Comparison of this cone with the long directions of precipitates determined by a slice-and-view method of scanning electron microscopy combined with focused ion beam sectioning shows good agreement between theory and experiment. A further study of the morphology of precipitates by the Eshelby method suggests that interfacial energy also plays a role and gives an approximate value of interfacial energy per unit area of 250 dyn cm −1.

Combinatorial search of thermoelastic shape-memory alloys with extremely small hysteresis width

Reversibility of structural phase transformations has profound technological implications in a wide range of applications from fatigue life in shape-memory alloys (SMAs) to magnetism in multiferroic oxides. The geometric nonlinear theory of martensite universally applicable to all structural transitions has been developed. It predicts the reversibility of the transitions as manifested in the hysteresis behaviour based solely on crystal symmetry and geometric compatibilities between phases. In this article, we report on the verification of the theory using the high-throughput approach. The thin-film composition-spread technique was devised to rapidly map the lattice parameters and the thermal hysteresis of ternary alloy systems. A clear relationship between the hysteresis and the middle eigenvalue of the transformation stretch tensor as predicted by the theory was observed for the first time. We have also identified a new composition region of titanium-rich SMAs with potential for improved control of SMA properties.