Shear Modulus C44 Research Articles

Single crystal shear moduli of β-phase stabilized by thermomechanical treatment in TiNbSn alloys with ultralow elastic modulus

The single crystal shear moduli is crucial to explore the origin of low Young’s modulus for β-type titanium alloys. In this study, the single crystal elastic constants of two TiNbSn alloys with Nb concentration below a critical quantity (βc) were extracted from their polycrystalline specimens using an in-situ synchrotron X-ray diffraction technique, although it was impossible to fabricate their single crystals due to the ultralow β-phase stability. The results show that the shear modulus C’ can be further reduced by decreasing the Nb concentration to below βc. Meanwhile, strong correlation between the shear modulus C44 and the Nb concentration was observed in TiNbSn alloys, although the C44 is independent on β-stabilizing elements in binary alloys. It is proposed that ultralow Young’s modulus can be realized in the alloy composition below βc by concurrently reducing C’ and C44.

Microstructure-elastic property relationships in carbon fibers: A nanoindentation study

Carbon fibers with high axial stiffness and strength have been used to reinforce polymer-matrix materials in advanced composites. However, these fibers exhibit large anisotropies in the radial and axial directions. In addition, limited data has been reported on the elastic properties of fibers in other directions than longitudinal and transversal directions, and on the relationship between elastic properties and fiber structures. In this study, we prepared five epoxy composites each containing a different type of carbon fiber and used a nanoindentation method in combination with finite element analysis to investigate five elastic constants of the carbon fibers. The results demonstrate that the crystallite size and the orientation of its graphite-crystal inclusions affect the elastic constants c11, c12, c13 and c33, while the carbon fibers with well-aligned large-sized crystallite possess a large mechanical anisotropy. In contrast, the shear modulus (c44) is less affected by the alignment of the crystallite. Instead, the fibers with larger interlayer spacing of (002) basal planes and large crystallite size exhibit a smaller c44.

Experimental and Theoretical Study of Water Retention Effects on Elastic Properties of Opalinus Shale

Understanding of shales elastic properties behaviour with saturation changes is important for geological storage of nuclear waste, CO2 sequestration as well as for development of conventional and unconventional shale oil and gas reservoirs. Existing data describing effects of saturation on elastic properties of shales are sparse and contradictory. To improve understanding of the effects of changing water content on elastic properties in shales, we conduct an experimental study on Opalinus shale samples. We measure vertical and horizontal ultrasonic P- and S-wave velocities of the same set of samples with controlled water content. The measured velocities are used to calculate components of elastic stiffness tensor in the shale at different saturations assuming its vertical transverse isotropy. Obtained results show increasing C11 and decreasing C33 with drying of the samples. Moreover, we observe 80% and 60% increase of shear moduli C44 and C66, respectively, with reduction of the water content from 5.5% weight in the preserved state to 0.3%. Conventional rock physics models are not designed to explain the observed dynamics. Here we perform a theoretical investigation of the influence on the shale moduli of following factors: (1) mechanical softening of the rock with the decrease of water saturation; (2) shrinkage of clay leading to reduction of porosity; (3) chemical hardening of clay particles; and (4) enhancing stiffness of contacts due to removing of water between clay particles.

Design of low modulus β-type titanium alloys by tuning shear modulus C44

In the present paper, a Ti-39Nb (wt.%) was employed to illustrate the feasibility of designing β-type titanium alloys with low Young's modulus through concurrently reducing shear moduli C44 and C'. The solution treated and quenched alloy exhibits a dual (β+α") phase due to the insufficient Nb content, implying the intrinsically low β-phase stability. After severe cold rolling and annealing treatment at 523 K for 30 min, the martensitic transformation was suppressed by high density of defects. As a result, a nearly single β-phase alloy with a low Young's modulus of 39 GPa and a high ultimate tensile strength of 745 MPa was successfully fabricated. The single-crystal elastic constants of the cold rolled plus annealed Ti-39Nb alloy were extracted from its polycrystalline specimen by using an in-situ synchrotron X-ray technique. It is indicated that the alloy has an anomalously low C44 of 21.4 GPa as well as a low C′ of 17 GPa, both of which contribute to the low Young's modulus. This suggests that it is a feasible approach to concurrently reduce C44 and C′ through composition design and microstructure optimization. Our findings provide a new strategy of designing β-type titanium alloys with ultralow Young's modulus.

Stability of rhombohedral phases in vanadium at high-pressure and high-temperature: first-principles investigations.

The pressure-induced transition of vanadium from BCC to rhombohedral structures is unique and intriguing among transition metals. In this work, the stability of these phases is revisited by using density functional theory. At finite temperatures, a novel transition of rhombohedral phases back to BCC phase induced by thermal electrons is discovered. This reentrant transition is found not driven by phonons, instead it is the electronic entropy that stabilizes the latter phase, which is totally out of expectation. Parallel to this transition, we find a peculiar and strong increase of the shear modulus C44 with increasing temperature. It is counter-intuitive in the sense that it suggests an unusual harding mechanism of vanadium by temperature. With these stability analyses, the high-pressure and finite-temperature phase diagram of vanadium is proposed. Furthermore, the dependence of the stability of RH phases on the Fermi energy and chemical environment is investigated. The results demonstrate that the position of the Fermi level has a significant impact on the phase stability, and follows the band-filling argument. Besides the Fermi surface nesting, we find that the localization/delocalization of the d orbitals also contributes to the instability of rhombohedral distortions in vanadium.

Origin of ultralow Young׳s modulus in a metastable β-type Ti–33Nb–4Sn alloy

Although there is difficulty in growing a Ti–33Nb–4Sn single crystal due to its ultralow β-phase stability, the single-crystal elastic constants of metastable β-type Ti–33Nb–4Sn (wt%) alloy were extracted successfully from its polycrystal by in-situ synchrotron X-ray diffraction technique, to clarify the origin of the ultralow Young’s modulus in its polycrystal. It is indicated that compared to binary TiCr, TiV and TiNb alloys, the Ti–33Nb–4Sn alloy possesses slightly lower β-phase stability with respect to {110}<110>-shear (i.e., C׳) but much lower β-phase stability regarding to {001}〈100〉 shear (i.e., C44). An analysis by the Hill approximation suggests that the ultralow isotropic polycrystalline Young׳s modulus (EH) of Ti–33Nb–4Sn alloy originates from the extremely low shear modulus C44 as well as the relatively low C׳. This indicates that in addition to C׳, C44 has a significant contribution to the Young’s modulus of polycrystal, which challenges a conventional understanding that the Young’s modulus of β-type Ti alloys is predominantly determined by C׳.

Micromechanism of Cu and Fe alloying process on the martensitic phase transformation of NiTi-based alloys: First-principles calculation

Using first-principles pseudo-potential plane wave method, the formation enthalpy ΔH, binding energy ΔE, elastic constants, and electronic structure were calculated and analyzed carefully for NiTiX (X = Cu, Fe) shape memory alloy. The results show that the Cu or Fe element prefers to occupy the Ni site in the NiTi matrix phase respectively. Compared with the NiTi matrix phase, the ΔH, ΔE, c44 and c′ of NiTi (Cu) are similar to each other. However, the structural stability of the NiTi phase is improved obviously by the Fe alloying process. Simultaneously, the shear modulus c44 and c′ of NiTi (Fe) are larger than those of the NiTi matrix phase. Furthermore, Milliken population results indicate that QCu–Ti is smaller than QNi–Ti after the Cu alloying process, but QFe–Ti is larger than QNi–Ti. The electron density difference shows that some covalent bonding exists between Fe and Ti elements. Based on the upward analysis, the difference in the phase stability and elastic constants of NiTiX (X = Cu, Fe) is the substantial mechanism for the different Ms of NiTiX (X = Cu, Fe) although Cu or Fe substitutes for the same atom Ni elements in the NiTi matrix phase.

Universal elastic-hardening-driven mechanical instability in α-quartz and quartz homeotypes under pressure.

As a fundamental property of pressure-induced amorphization (PIA) in ice and ice-like materials (notably α-quartz), the occurrence of mechanical instability can be related to violation of Born criteria for elasticity. The most outstanding elastic feature of α-quartz before PIA has been experimentally reported to be the linear softening of shear modulus C44, which was proposed to trigger the transition through Born criteria B3. However, by using density-functional theory, we surprisingly found that both C44 and C66 in α-quartz exhibit strong nonlinearity under compression and the Born criteria B3 vanishes dominated by stiffening of C14, instead of by decreasing of C44. Further studies of archetypal quartz homeotypes (GeO2 and AlPO4) repeatedly reproduced the same elastic-hardening-driven mechanical instability, suggesting a universal feature of this family of crystals and challenging the long-standing idea that negative pressure derivatives of individual elastic moduli can be interpreted as the precursor effect to an intrinsic structural instability preceding PIA. The implications of this elastic anomaly in relation to the dispersive softening of the lowest acoustic branch and the possible transformation mechanism were also discussed.

Effect of thermo-mechanical treatment on mechanical and elastic properties of Ti–36Nb–5Zr alloy

The evolutions of phase constitutions and mechanical properties of a β-phaseTi–36Nb–5Zr (wt%) alloy during thermo-mechanical treatment were investigated. The alloy consisted of dual (β+α″) phase and exhibited a double yielding phenomenon in solution treated state. After cold rolling and subsequent annealing at 698K for 20min, an excellent combination of high strength (833MPa) and low modulus (46GPa) was obtained. The high strength can be attributed to high density of dislocations, nanosized α phase and grain refinement. On the other hand, the low Young׳s modulus originates from the suppression of chemical stabilization of β phase during annealing, which guarantees the low β-phase stability. Furthermore, the single-crystal elastic constants of the annealed Ti–36Nb–5Zr alloy were extracted from polycrystalline alloy using an in-situ synchrotron X-ray technique. The results indicated that the low shear modulus C44 contributes to the low Young׳s modulus for the Ti–36Nb–5Zr alloy, suggesting that reducing C44 through thermo-mechanical treatment might be an efficient approach to realize low Young׳s modulus in β-phase Ti alloys. The results achieved in this study could be helpful to elucidate the origin of low modulus and sheds light on developing novel biomedical Ti alloys with both low modulus and high strength.

Possible contribution of low shear modulus C44 to the low Young's modulus of Ti-36Nb-5Zr alloy

Despite the importance of single-crystal elastic constants of β-phase titanium alloys in understanding their low Young's modulus—a property crucial for many applications, such data are often difficult to obtain when the alloy composition is close to the instability limit of the β phase, where the presence of α" martensite precludes the fabrication of β-phase single crystal. In the present study, we extracted the single-crystal elastic constants of such a β-phase titanium alloy with low Young's modulus, Ti-36Nb-5Zr (wt. %), from polycrystalline specimens by using an in-situ synchrotron X-ray diffraction technique. It is indicated that the low Young's modulus of the alloy originates from the anomalously low shear modulus C44 as well as the low shear modulus C′, which is different from a common viewpoint that the Young's modulus of β-phase titanium alloys is dominantly controlled by the C′. This suggests that low C44 is an important contributor to low Young's modulus for instable β-phase titanium alloys.

Single crystal elastic constants of high-manganese transformation- and twinning-induced plasticity steels determined by a new method utilizing nanoindentation

A better understanding of single crystal elastic anisotropy in high-manganese austenitic transformation- and twinning-induced plasticity (TRIP/TWIP) steels is necessary primarily to improve the accuracy of stacking-fault energy measurements and validate elastic constants determined by ab initio simulations. In the present work, a method utilizing nanoindentation in combination with orientation imaging microscopy is developed for the purpose of calculating single crystal elastic constants from cubic polycrystalline specimens. Applying the method on two Fe–(22/25)Mn–3Al–3Siwt% alloys yielded single crystal elastic constants C11, C12 and C44 of 175/83/97 and 174/85/99GPa respectively. Indentation moduli found in the literature for a third TWIP steel, with composition Fe–18Mn–1.5Al–0.6C, produced elastic constants of 169, 82 and 96GPa via the proposed method. Anisotropy ratios of the three alloys ranged from 2.11 to 2.22, considerably lower than values of ∼3.5–3.9 measured for binary austenitic Fe–Mn alloys. The discrepancy is attributed to alloying additions of Al, Si and C, which cause the Néel transition to occur below room temperature in the TRIP/TWIP alloys, thereby reducing suppression of the tetragonal shear modulus (C11–C12)/2 typically associated with antiferromagnetic ordering. In addition, decreases in Mn content reduced the normal shear modulus C44. In a test, the model calculated single crystal elastic constants to be within ∼4% of established values for a series of materials with a wide range of elastic anisotropy.

Band structure, phase transition, phonon and elastic instabilities in calcium polonide under pressure: A first-principles study

Ab initio self-consistent calculations were performed to study the structure, electronic, elastic and lattice dynamical properties of calcium polonide under hydrostatic pressure. Enthalpy calculations show that the material undergoes a first-order phase transition from rocksalt (RS) to CsCl structure at 16.7GPa with 5.0% volume collapse. A soft transverse acoustic phonon mode at the zone boundary X point is identified at 21.1GPa for RS phase, signifying a pressure-induced dynamical instability. Moreover, the shear modulus C44 in RS phase softens with increasing pressure. Our results imply that the underlying physical origin of pressure-induced RS→CsCl phase transition might be in close relation with the coupling between unstable X-point phonon mode and shear modulus C44. The RS phase under pressure undergoes a direct→indirect bandgap transition. The anomalous behavior of band structure originates from the fact that the top valence and bottom conduction states at different reciprocal lattice points show the different dependence on hydrostatic pressure.

The ground-state structure and physical properties of RuC: first-principles calculations

We have extensively explored the ground-state structure of RuC using the particle swarm optimization algorithm for crystal structural prediction. A hexagonal R-3m structure has been proposed as the best candidate, which is energetically more favorable than the previously proposed zinc blend structure. The R-3m-RuC possesses alternative stacking of double hexagonal close-packed Ru atom layers and C atom layers, and it is dynamically stable evidenced by the calculation of phonon dispersion. The calculated large bulk modulus, shear modulus, and elastic constant C44 reveal that it is an ultra-incompressible and hard material. The evidence of strong covalent bonding of Ru—C, which plays an important role to form a hard material, is manifested by the partial densities of states analysis.

Simulation of borehole sonic waveforms in dipping, anisotropic, and invaded formations

A numerical simulation study has been made of borehole sonic measurements that examined shoulder-bed, anisotropy, and mud-filtrate invasion effects on frequency-dispersion curves of flexural and Stoneley waves. Numerical simulations were considered for a range of models for fast and slow formations. Computations are performed with a Cartesian 3D finite-difference time-domain code. Simulations show that presence of transverse isotropy (TI) alters the dispersion of flexural and Stoneley waves. In slow formations, the flexural wave becomes less dispersive when the shear modulus (c44) governing wave propagation parallel to the TI symmetry axis is lower than the shear modulus (c66) governing wave propagation normal to the TI symmetry axis; conversely, the flexural wave becomes more dispersive when c44 &gt; c66. Dispersion decreases by as much as 30% at higher frequencies for the considered case where c44 &lt; c66. Dispersion of Stoneley waves, on the other hand, increases in TI formations when c44 &gt; c66 and decreases when c44 &lt; c66. Dispersion increases by more than a factor of 2.5 at higher frequencies for the considered case where c44 &lt; c66. Simulations also indicate that the impact of invasion on flexural and Stoneley dispersions can be altered by the presence of TI. For the case of a slow formation and TI, where c44 decreases from the isotropic value, separation between dispersion curves for cases with and without the presence of a fast invasion zone increases by as much as 33% for the flexural wave and by as much as a factor of 1.4 for the Stoneley wave. Lastly, presence of a shoulder bed intersecting the sonic tool at high dip angles can alter flexural dispersion significantly at low frequencies. For the considered case of a shoulder bed dipping at 80°, ambiguity in the flexural cutoff frequency might lead to shear-wave velocity errors of 8%–10%.

Finnis–Sinclair potentials for fcc Au–Pd and Ag–Pt alloys

Abstract The thermal and mechanical properties of Au, Pd, Ag, Pt pure metals and their alloys Au–Pd and Ag–Pt are studied using molecular dynamics simulations. The Finnis–Sinclair potential is used to describe the atomic interactions. Lattice constants and elastic constants of Au3Pd, AuPd, AuPd3, Ag3Pt, AgPt and AgPt3 at different temperatures are predicted for the first time. The melting temperatures of these alloys are also calculated. Furthermore, the values of lattice constants, elastic constants and melting temperatures of pure metals Au, Pd, Ag, and Pt are calculated for comparison. By comparing the simulation results of pure metals and alloys, it is found that for Au3Pd, AuPd, AuPd3 and Ag3Pt, lattice constants, elastic constants and melting temperatures lie between those of their components. For AgPt and AgPt3, the values of shear modulus C44 and melting temperatures are higher than those for both Ag and Pt pure metals.

Elastic Properties of Polycrystalline Solid Helium

Recent experiments have shown that the shear modulus of solid helium undergoes a large temperature variation in the range 20 to 200 mK, possibly due to changes in the pinning of dislocations. In this note we report on computer simulations of the elastic properties of polycrystalline solid helium. We calculate how the elastic coefficients of a sample made up of a large number of randomly oriented grains are affected by the changes in the shear modulus c44 of the individual grains.

First-principles study of elastic and stability properties of ZrC–ZrN and ZrC–TiCalloys

Ab initio calculations of the elastic constants for several cubic ordered structures of zirconium carbonitride(ZrCxN1−x) and zirconium–titaniumcarbide (ZrxTi1−xC) alloys were carried out. The calculations of total and formation energies, bulk modulus andelastic constants as functions of composition were performed with an ab initio pseudo-potentialmethod. The predicted equilibrium lattice parameters are slightly higher than those foundexperimentally (on average by 0.2–0.4%). The predicted formation energies indicate that theZrCxN1−x alloys are stable even at 0 K in the whole concentration range, while the homogeneousZrxTi1−xC alloys can be stabilized only at high temperatures. Spinodal decompositionof the latter alloys into cubic domains takes place over a wide range ofcompositions and temperatures. For the carbonitrides, the shear modulusG, the Young’smodulus E and thePoisson ratio σ reach an extremum for carbon-rich alloys, and this is attributed to a maximum value of the shear modulusC44 that corresponds to a valence-electron concentration in the range of 8.2–8.3. Thisextremal behavior finds its origin in the response of the band structure ofZrCxN1−x alloysfor 0≤x≤1, caused by the monoclinic strain that determines this shear modulus. In contrast, the othershear modulus does not exhibit any extremum over the whole composition range. These results are incontrast with those for Zr–Ti carbides for which the elastic properties gradually increasefrom ZrC to TiC.

Calculated structural, electronic and elastic properties of M2GeC (M=Ti, V, Cr, Zr, Nb, Mo, Hf, Ta and W)

Using ab initio calculations, we have studied the structural, electronic and elastic properties of M2GeC, with M=Ti, V, Cr, Zr, Nb, Mo, Hf, Ta and W. Geometrical optimizations of the unit cell are in agreement with the available experimental data. The band structures show that all studied materials are electrical conductors. The analysis of the site and momentum projected densities shows that bonding is due to M d-C p and M d-Ge p hybridizations. The elastic constants are calculated using the static finite strain technique. The shear modulus C 44, which is directly related to the hardness, reaches its maximum when the valence electron concentration is in the range 8.41–8.50. We derived the bulk and shear moduli, Young’s moduli and Poisson’s ratio for ideal polycrystalline M2GeC aggregates. We estimated the Debye temperature of M2GeC from the average sound velocity. This is the first quantitative theoretical prediction of the elastic constants of Ti2GeC, V2GeC, Cr2GeC, Zr2GeC, Nb2GeC, Mo2GeC, Hf2GeC, Ta2GeC and W2GeC compounds, and it still awaits experimental confirmation.

Effect of hydrogen on electronic structure of fcc iron in relation to hydrogen embrittlement of austenitic steels

AbstractThe total structural energy per primitive lattice cell, density of electron states, spatial distribution of electrons and elastic modulus in fcc Fe–H solid solutions are studied using the density functional theory and Wien2k program package. It is shown that hydrogen increases the density of electron states at the Fermi level. The density of conduction electrons is increased in the vicinity of hydrogen atoms, which suggests that the latter migrate over the crystal lattice surrounded by the clouds of conduction electrons. Calculations of elastic modulus show that hydrogen decreases the shear modulus c44. The consequences for mechanical properties of hydrogen‐charged austenitic steels are analysed taking into account the stress for activation of dislocation sources, the line tension of dislocations and the distance between dislocations in pile‐ups. It is concluded that hydrogen‐induced brittleness of austenitic steels can be satisfactorily interpreted in terms of the hydrogen effect on the electronic structure. (© 2008 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Influence of pressure and impurity hydrogen on the elastic property of metal lithium

The pressure effects on the elastic properties of lithium doped by hydrogen are studied using first-principles methods based on density functional theory. A supercell including 16 Li atoms in hcp lattice is constructed with a hydrogen atom interstitially doped in. The supercell is fully relaxed at 0K. The elastic properties of the stable lattice are studied. The result shows that the bulk modulus increases after hydrogen atom is added. The elastic constants C11, C33, C66 and C12 are larger than those of pure Li，while the shear modulus C44 is smaller. The elastic constants increase with external pressure, with C11, C33 and C66 larger than those of pure Li while C12 being smaller. C13 is abnormally lower than that of pure Li within the pressure range from 2GPa to 4GPa, while C44 and C13 are greater at higher pressure area. The discrepancy of shear modulus between the doped and the pure system is enhanced by external pressure. The analysis of anisotropy ΔW shows that the doped system maintains isotropic under external pressure as does pure Li.