Softening Of Elastic Constant Research Articles

On the habit planes between elastically distorted austenite and martensite in NiTi

The cubic B2 to monoclinic B19’ martensitic transformation in NiTi shape memory alloys proceeds by the nucleation and propagation of habit plane interfaces that remain undistorted and unrotated during the transformation. Due to incompatibility of austenite and martensite lattices in NiTi, habit plane interfaces are assumed to form between austenite and twinned martensite as described by Phenomenological Theory of Martensite Crystallography (PTMC). This view is currently widely spread in the literature in spite of the fact that it contradicts experimental observations of interfaces between austenite and single martensite lattice by TEM and EBSD methods frequently reported in the literature. On this account, we propose a different solution for the formation of strain compatible habit planes propagating during stress induced martensitic transformation by considering the effect of elastic deformation of both lattices caused by the external stress. Using modified PTMC theory, we evaluate the magnitudes of the uniaxial stress in tension and compression along a general austenite crystal direction, for which strain compatible habit plane interfaces can be established between austenite and single variant of martensite. The stress calculated in such a way is, however, too high. Nevertheless, considering the pre-transformation softening of austenite, the critical stress decreases to 500 MPa commonly observed in experiments. Three scenarios are simulated assuming softening of elastic constants of austenite. Orientation dependence and tension-compression asymmetry of predicted habit planes of stress induced martensitic transformation is compared with available experimental results.

Pathways to Titanium Martensite

The structural relationship between the parent and product phases in the martensitic transformation from the parent β phase is described. The atomic movements leading to the martensite are accomplished by a long-range shear {112} < 111 > that transforms the parent to the product lattice and a short-wavelength displacement or shuffle {110} < 110 > that induces the correct stacking. The microstructures arising out of these paths depend on which of these modes initiates the transformation. When the shear precedes the shuffle, conventional martensite forms as dislocated laths or internally twinned plates. A signature of the {110} < 110 > shuffle that follows or accompanies the shear is present in both cases as stacking fault-related domains. The shuffle displacement precedes the shear with increasing β stabilizer addition, resulting in a nanodispersion of a structure with orthorhombic symmetry, which we have designated as O′. Al, Sn, Zr and O additions promote the shuffle. The O′ dispersion acts as embryos for the formation of nanomartensite on cooling or with the application of stress. The resulting continuous and controlled strain incorporation into the lattice within the constrained nanoembryos results in nonlinear superelasticity or the invar and elinvar effects. The stability of the bcc parent is discussed in terms of phonon mode or related elastic constant softening.

High-pressure monoclinic phase of MoAlB

The pressure-induced phase transitions of MoAlB have been fully studied to provide insights into the typical MAB phases under high pressure. A novel high-pressure monoclinic structure (C2/m, Z = 4) was firstly identified for the MoAlB above 234 GPa, using the first principles crystal structure prediction approach. First-order structure transition character of MoAlB from the ambient-pressure orthorhombic Cmcm phase to this high-pressure monoclinic C2/m phase was demonstrated by a volume drop of 1.28%. Inspections of crystal structures showed that the occurrence of C2/m phase follows the slip in the metallic Al-Al layers and the fracture of linear zigzag B-B chains in the Cmcm phase under compression, confirming the earlier reports. Our argument has been also supported by the softening of elastic constant under high pressure. Phonon curves calculations suggest the high-pressure C2/m phase can be quenchable to ambient pressure. The electronic and mechanical properties of this novel C2/m phase have thus been systematically investigated.

Ultrasound Response in Quantum Critical β-YbAlB4 and α-YbAl0.986Fe0.014B4

We analyze the key origin of quantum valence criticality in the heavy electron metal $\beta$-YbAlB$_4$ evidenced in the sister compound $\alpha$-YbAl$_{0.986}$Fe$_{0.014}$B$_4$. By constructing a realistic canonical model for $\beta$-YbAlB$_4$, we evaluate Coulomb repulsion between the 4f and 5d electrons at Yb $U_{\rm fd}\approx 6.2$ eV realizing the quantum critical point (QCP) of the Yb-valence transition. To reveal the Yb 5d contribution to the quantum critical state, we propose ultrasound measurement. We find that softening of elastic constants of not only the bulk modulus but also the shear moduli is caused by electric quadrupole fluctuations enhanced by critical 4f and 5d charge fluctuations for low temperatures at the valence QCP. Possible relevance of these results to $\beta$-YbAlB$_4$ and also $\alpha$-YbAl$_{1-x}$Fe$_x$B$_4$ is discussed.

Strength and Brittleness of Interfaces in Fe-Al Superalloy Nanocomposites under Multiaxial Loading: An ab initio and Atomistic Study.

We present an ab initio and atomistic study of the stress-strain response and elastic stability of the ordered FeAl compound with the D0 structure and a disordered Fe-Al solid solution with 18.75 at.% Al as well as of a nanocomposite consisting of an equal molar amount of both phases under uniaxial loading along the [001] direction. The tensile tests were performed under complex conditions including the effect of the lateral stress on the tensile strength and temperature effect. By comparing the behavior of individual phases with that of the nanocomposite we find that the disordered Fe-Al phase represents the weakest point of the studied nanocomposite in terms of tensile loading. The cleavage plane of the whole nanocomposite is identical to that identified when loading is applied solely to the disordered Fe-Al phase. It also turns out that the mechanical stability is strongly affected by softening of elastic constants and/or and by corresponding elastic instabilities. Interestingly, we found that uniaxial straining of the ordered FeAl with the D0 structure leads almost to hydrostatic loading. Furthermore, increasing lateral stress linearly increases the tensile strength. This was also confirmed by molecular dynamics simulations employing Embedded Atom Method (EAM) potential. The molecular dynamics simulations also revealed that the thermal vibrations significantly decrease the tensile strength.

Effect of ripples on the finite temperature elastic properties of hexagonal boron nitride using strain-fluctuation method

This work intents to put forth the results of a classical molecular dynamics study to investigate the temperature dependent elastic constants of monolayer hexagonal boron nitride (h-BN) between 100 and 1000 K for the first time using strain fluctuation method. The temperature dependence of out-of-plane fluctuations (ripples) is quantified and is explained using continuum theory of membranes. At low temperatures, negative in-plane thermal expansion is observed and at high temperatures, a transition to positive thermal expansion has been observed due to the presence of thermally excited ripples. The decrease of Young's modulus, bulk modulus, shear modulus and Poisson's ratio with increase in temperature has been analyzed. The thermal rippling in h-BN leads to strong anharmonic behaviour that causes large deviation from the isotropic elasticity. A detailed study shows that the strong thermal rippling in large systems is also responsible for the softening of elastic constants in h-BN. From the determined values of elastic constants and elastic moduli, it has been elucidated that 2D h-BN sheets meet the Born's mechanical stability criterion in the investigated temperature range. The variation of longitudinal and shear velocities with temperature is also calculated from the computed values of elastic constants and elastic moduli.

Lattice Softening in Fe3Pt Exhibiting Three Types of Martensitic Transformations

We have investigated the relation between the softening of elastic constants and martensitic transformation in Fe3Pt, which exhibits various kinds of martensitic transformation depending on its long-range order parameter S. The martensite phases of the examined alloys are BCT (S = 0.57), FCT1 (S = 0.75, c/a &lt; 1) and FCT2 (S = 0.88, c/a &gt; 1). The elastic constants C′ and C44 of these alloys decrease almost linearly with decreasing temperature. Although the temperature coefficient of C′ decreases as S increases, C′ at the transformation temperature is the smallest in the alloy with S = 0.75, which transforms to FCT1. This result implies that softening is most strongly related to the formation of the FCT1 martensite with tetragonality c/a &lt; 1 among the three martensites.

Origin of enhanced piezoelectric constants of MgNbAlN studied by first-principles calculations

We theoretically investigate the piezoelectric properties of Mg–Nb codoped wurtzite AlN by first-principles calculations and find the following two points. First, doping induces a remarkable softening of elastic constants, especially the diagonal component along the c-axis. This results in the increased piezoelectric strain constant d33, which is about 4.3 times larger than that of wurtzite AlN. Second, the c-axis lattice length increases up to the dopant concentration of around 60 at. %, and then decreases. These features are in excellent agreement with the experimental results previously reported.

Surface acoustic wave diagnosis of vacancy orbital with electric quadrupoles in silicon

We demonstrate ultrasonic diagnosis of vacancies in boron-doped silicon wafers currently used in device manufacturing. The low-temperature softening of elastic constants measured by surface acoustic waves (SAW) as well as bulk ultrasonic waves is caused by a coupling of elastic strains to electric quadrupoles of the vacancy orbital in silicon wafers. Using interdigital transducers with a comb gap of 2.5 μm on a piezoelectric ZnO film deposited on the (001) surface of the wafer, we observed the softening of 1.9×10−4 in relative amount of the elastic constant Cs below 2 K down to 23 mK. Taking account of the strong quadrupole-strain interaction, we deduced a small vacancy concentration 3.1×1012 cm−3 in the surface layer of the wafer within a penetration depth 3.5 μm of the SAW.

Influence of carbon vacancy formation on the elastic constants and hardening mechanisms in transition metal carbides

For group VB transition metal carbides, as compared to group IVB carbides, an anomalous rise in hardness occurs for substoichiometric carbon concentrations as compared to the stoichiometric monocarbides. Here we report the computationally derived elastic constants, electronic density of states, and activation energies for carbon vacancy migration as a function of carbon content to elucidate their effect on the hardening responses. The changes in elastic properties with respect to carbon vacancy concentration show similar behaviors of elastic softening and decreasing hardness for all of the cubic carbides. The consistent trends of vacancy diffusion energy barriers between all of the group IVB and VB transition metal carbides also suggests that carbon diffusion may not be a significant contributor to the reported hardness differences. Consequently, we propose that the anomalous hardening for substoichiometric behavior is a competition between elastic constant softening and a microstructural-based effect, i.e. domain hardening, that is present in group VB carbides but not in group IVB carbides.

Quadrupole Susceptibility of Gd-Based Filled Skutterudite Compounds

It is shown that quadrupole susceptibility can be detected in Gd compounds contrary to our textbook knowledge that Gd$^{3+}$ ion induces pure spin moment due to the Hund's rules in an $LS$ coupling scheme. The ground-state multiplet of Gd$^{3+}$ is always characterized by $J$=7/2, where $J$ denotes total angular momentum, but in a $j$-$j$ coupling scheme, one $f$ electron in $j$=7/2 octet carries quadrupole moment, while other six electrons fully occupy $j$=5/2 sextet, where $j$ denotes one-electron total angular momentum. For realistic values of Coulomb interaction and spin-orbit coupling, the ground-state wavefunction is found to contain significant amount of the $j$-$j$ coupling component. From the evaluation of quadrupole susceptibility in a simple mean-field approximation, we point out a possibility to detect the softening of elastic constant in Gd-based filled skutterudites.

Melting curve of the cI16 sodium at high pressure from ab initio calculations

The melting curve of cI16 sodium at high pressure is investigated extensively by means of Lindemann melting criterion and ab initio molecular dynamics simulations. Different from previous theoretical results, our results show that the melting point of cI16 sodium decreases with increasing pressure. It is found that the elastic constant softening plays an important role and results in a negative-slope Lindemann melting curve of cI16 sodium. The melting mechanism is different from that of f.c.c. sodium reported in previous work, that the Peierls-like liquid transition is the origin of negative-slope melting curve. Electronic structure analysis shows that more electrons in liquid sodium localize in the voids of lattice just like in its solid state, the exclusionary effect of atomic cores on valence electrons in the liquid still takes effect.

Ultrasonic study of vacancy in single crystal silicon at low temperatures

We have performed ultrasonic measurements at low temperatures in order to investigate vacancy in single crystal silicon. The longitudinal elastic constants of non-doped and boron-doped silicon grown by a floating zone method exhibit appreciable softening with decreasing temperature down to 20 mK. The softening of boron-doped silicon is easily suppressed in applied magnetic field up to 2 T, while the softening of non-doped silicon is robust in fields even up to 16 T. The softening of elastic constants in high-purity crystalline silicon is certainly caused by the coupling of elastic strains of the ultrasonic waves to electric quadrupoles of the vacancy orbital.

Quadrupole Ordering in Clathrate Compound Ce3Pd20Si6

Low-temperature elastic properties of the clathrate compound Ce 3 Pd 20 Si 6 were investigated using ultrasonic measurements. Pronounced softening of elastic constants ( C 11 - C 12 )/2 and C 44 with decreasing temperature indicates a precursor of electric quadrupole ordering due to a Γ 8 quartet ground sate with an orbital degeneracy for 8c site of a Ce ion. A minimum of ( C 11 - C 12 )/2 at T C =170 mK indicates a transition from paramagnetic phase I to an ordered phase III. Applied magnetic fields induce an antiferro-quadrupole (AFQ) ordered phase II with highly anisotropic character depending on the field direction, which resembles the AFQ phase with O x y -type order parameter in CeB 6 with the Γ 8 ground sate. The order paramater of the phase III may be characterized by either magnetic dipole or magnetic octupole. The Γ 7 ground state at 4a site may fall into a singlet screened by conduction electrons at low temperatures without long-range magnetic ordering.

Experimental study of elastic constant softening prior to stress-induced martensitic transformation

There has been a large body of experimental evidence for the existence of precursor lattice softening prior to temperature-induced martensitic transformation, but little is known about whether such softening effect also exists prior to stress-induced martensitic transformation. Recent molecular dynamics simulations suggested such a possibility, but direct experimental evidence remains unclear. In the present study, we established a simple experimental method that can measure elastic constant ${C}^{\ensuremath{'}}$ from zero stress up to the critical stress prior to stress-induced martensitic transformation. By applying this method to a Cu-Al-Ni single crystal, we found ${C}^{\ensuremath{'}}$ of $(101)[\overline{1}01]$ shear mode does soften with increasing applied stress along the [001] direction, until a stress-induced martensitic transformation occurs at a critical stress. This is direct evidence for the existence of lattice softening prior to a stress-induced martensitic transformation. Our result completes an important conclusion that the softening of elastic constant ${C}^{\ensuremath{'}}$ is a common feature prior to both temperature- and stress-induced martensitic transformations. Furthermore, we also analyzed the dependence of ${C}^{\ensuremath{'}}$ on uniaxial tensile stress through a theoretical model and obtained similar result as that of our experiment.

Theory on Softening of Elastic Constants in Non-Doped and B-Doped Si with a Vacancy

Softening of elastic constants in non-doped and B-doped Si with a vacancy is discussed theoretically on the basis of a numerical diagonalization study. To explain the difference of the magnetic field dependence of the elastic constant between B-doped Si and non-doped Si, it is crucial to take the spin–orbit coupling into account. As a result, taking also the many body effect, we explain the experiments on the softening of elastic constants so far, and predict that the softening ceases at the temperarture of the order of mK in non-doped Si with the high magnetic field of 10 T.

Bond stiffening in small nanoclusters and its consequences for mechanical and thermal properties

We have used density functional perturbation theory to investigate the stiffness of interatomic bonds in small clusters of Si, Sn, and Pb. As the number of atoms in a cluster is decreased, there is a marked shortening and stiffening of bonds. The competing factors of fewer but stiffer bonds in clusters result in softer elastic moduli but higher (average) frequencies as size is decreased, with clear signatures of universal scaling relationships. The stiffness of bonds is found to scale as the inverse tenth power of length. A significant role in understanding trends is played by the coordination number of the bulk structure: The higher this is, the lesser is the relative softening of elastic constants and the greater the relative damping of vibrational amplitudes for clusters compared to the bulk. Our results could provide a framework for understanding recent reports that some clusters remain solid above the bulk melting temperature. Our results suggest that Sn and Pb clusters (but not Si clusters) are more thermally stable than the bulk.

Observation of Vacancy in High Purity Silicon Crystal Using Low-Temperature Ultrasonic Measurements

We have succeeded in direct observation of isolated vacancies in high purity silicon crystals grown by a floating zone (FZ) method using low-temperature ultrasonic measurements. The softening of elastic constants below 20 K down to 20 mK is observed in non-doped FZ silicon and B-doped FZ silicon. This softening is caused by an interaction of electric quadrupoles of triply degenerate vacancy orbital to elastic strains of ultrasonic waves. The low-temperature elastic softening measured by ultrasonic methods verifies existence of the isolated vacancies in Pv-region distributed in pure crystal of a Czockralski ingot. The ultrasonic measurement of the low- temperature softening is a faithful probe for vacancy evaluation in high purity silicon crystals in commercial base.

Study on elastic constant softening in stress-induced martensitic transformation by molecular dynamics simulation

Precursor phenomena are critical issues for martensitic transformation, as they provide important clues for understanding the origin of the transformation and the structure of transformation products. Prior to temperature-induced martensitic transformation, it has been widely recognized for a long time that the basal plane shear modulus C′ (=( C 11 − C 12)/2) of the parent phase decreases with approaching transformation temperature. On the other hand, martensitic transformation can also be induced by stress. But little is known about whether similar precursor phenomenon also exists prior to such a stress-induced martensitic transformation (SIM). In the present study, we used molecular dynamics method to simulate the precursory stage of a stress-induced martensitic transformation. A new method for calculating the elastic constants C′ prior to SIM was proposed. The relationship between C′ and applied stress at different temperatures was calculated. Our results showed that the softening of elastic constant C′ is a common feature for both temperature and stress-induced martensitic transformation. For stress-induced martensitic transformation, the critical stress to induce the transformation increases with increasing temperature and obey Clausius–Clapyeyron relationship; at a given temperature, C′ decreases with the applied stress approaching the critical stress.

Quadrupole and lattice effects of orbitally degenerate 4f-electron systems

The quadrupole orderings and lattice effects in 4f-electron systems withorbital degeneracy in rare earth compounds have been investigated byultrasonic and thermal expansion measurements. A ferroquadrupole (FQ)Oxy-typeordering associated with a trigonal distortion in HoB6 witha Γ5ground state is presented. The trigonal distortion in orderedphase IV CexLa1−xB6 (x = 0.75 and 0.70) with aΓ8 ground stateindicates Oxy-typeFQ ordering in phase IV. In both systems, HoB6and CexLa1−xB6,C44anomalies are observed. In tetragonal HoB2C2,an ordered phase IV appears near a tetra-critical point, where the antiferroquadrupolephase II and antiferromagnetic phase III compete with each other. Noticeableultrasonic attenuation and softening of elastic constants in phase IV of HoB2C2 dueto quadrupole fluctuation with a slow relaxation time τ ∼ 7 × 10−9 sare found.