Elastic Softening Research Articles

Tuning the glass forming ability and mechanical properties of Ti-based bulk metallic glasses by Ga additions

A new series of Ti40Zr10Cu36−xPd14Gax (x = 2, 4, 8 and 10) alloys with improved plasticity and glass-forming ability (GFA) are produced. The strategy is to create local structural heterogeneity in Ti-based bulk metallic glasses (BMGs) by alloying with a soft element which has high negative enthalpy of mixing with Ti-atoms. The GFA of Ti40Zr10Cu36−xPd14Gax rods increases with increasing Ga content up to 4 at.%, and for more Ga it decreases. Minor addition of Ga (x = 2) is beneficial to improve the strength and ductility of the TiZrCuPd-based BMG due to the formation of local structural heterogeneities which facilitate the formation of a highly organized pattern of multiple shear bands under quasi-static compression. Ti40Zr10Cu34Pd14Ga2 BMG shows the best combination of compressive global strain and maximum compressive strength of ∼4.5% and ∼2000 MPa, respectively. Comparing the Young's modulus of Ga-containing alloys in amorphous and crystalline state we find that the amorphous state shows significant elastic softening (40–48% smaller) which is useful to reduce stress shielding effect for biomedical applications. The alloys show a remarkable thermoplastic formability as the viscosity decreases sharply by an order of magnitude in the supercooled liquid region.

A nano-embryonic mechanism for superelasticity, elastic softening, invar and elinvar effects in defected pre-transitional materials

A vast amount of physically-different materials near their structural phase transitions have been widely used due to their rich and extraordinary properties, e.g., superelasticity, elastic softening and invar/elinvar effects. These pre-transitional materials are known to have complex microstructures consisting of stress-generating defects such as dislocations and coherent nano-precipitates. However, effects of such defects on properties have not been well understood, which hinders fully exploiting the potential applications of these materials. In this paper we investigated a nano-embryonic mechanism in a generic case of pre-transitional materials with stress-generating defects at temperatures close but above the starting temperature of phase transformation, Ms. We demonstrated that the stress concentration generated by defects could induce localized displacive phase transformation near defects, producing equilibrium nano-size embryos of orientation variants of the product phase. The obtained mixed state consisting of nano embryos is in a thermoelastic equilibrium in which the total volume and sizes of embryos are equilibrium internal thermodynamic parameters. The subsequent imposition of an applied stress causes these embryos to grow, generating superelastic responses with an increasing applied field. If the defects are stationary the growth maintains thermoelastic equilibrium, and is, hence, fully reversible and anhysteretic. Moreover, cooling toward the Ms also causes embryo growth resulting in a diffuse phase transformation, which increases the volume and softens the modulus. These effects counteract the thermal contraction and modulus increase in the untransformed matrix, and may explain the invar and elinvar affects in alloys with low-temperature displacive transformations.

A NURBS-based inverse analysis of thermal expansion induced morphing of thin shells

Soft, active materials have been widely studied due to their ability to undergo large, complex shape changes in response to both mechanical and non-mechanical external stimuli. However, the vast majority of such studies has focused on investigating the forward problem, i.e. determining the shape changes that result from the applied stimuli. In contrast, very little work has been done to solve the inverse problem, i.e. that of identifying the external loads and stimuli that are needed to generate desired shapes and morphological changes. In this work, we present a new inverse methodology to study residual thermal expansion induced morphological changes in geometric composites made of soft, thin shells. In particular, the method presented in this work aims to determine the prescribed external stimuli needed to reconstruct a specific target shape, with a specific focus and interest in morphological changes from two-dimensional (2D) to three-dimensional (3D) shapes by considering the external stimuli within a thermohyperelastic framework. To do so, we utilize a geometrically exact, rotation-free Kirchhoff–Love shell formulation discretized by NURBS-based shape functions. We show that the proposed method is capable of identifying the stimuli, including cases where thermal expansion induced shape changes involving elastic softening occur in morphing from the initially flat 2D to non-planar 3D shapes. Validation indicates that the reconstructed shapes are in good agreement with the target shape.

Magnetoelastic and magnetoelectric couplings across the antiferromagnetic transition in multiferroic BiFeO3

Clear anomalies in the lattice thermal expansion (deviation from linear variation) and elastic properties (softening of the sound velocity) at the antiferromagnetic-to-paramagnetic transition are observed in the prototypical multiferroic BiFeO3 using a combination of picosecond acoustic pump-probe and high-temperature X-ray diffraction experiments. Similar anomalies are also evidenced using first-principles calculations supporting our experimental findings. Those calculations in addition to a simple Landau-like model we also developed allow to understand the elastic softening and lattice change at T_N as a result of magnetostriction combined with electrostrictive and magnetoelectric couplings which renormalize the elastic constants of the high-temperature reference phase when the critical T_N temperature is reached.

Magnetic phase diagram and crystal-field effects in the kagome-lattice antiferromagnet U3Ru4Al12

We report on the magnetic phase diagram of the distorted kagome-lattice antiferromagnet ${\mathrm{U}}_{3}{\mathrm{Ru}}_{4}{\mathrm{Al}}_{12}$ determined through measurements of magnetic and elastic properties. For field applied along the [100] and [120] axes of the hexagonal crystal structure, we find pronounced anomalies in the magnetization and elastic moduli that signal the existence of unknown magnetic phases. Our crystal-electric-field (CEF) analysis evidences interlevel quadrupolar interactions between the ground-state singlet and the first excited doublet. These interactions lead to a large softening of the shear elastic modulus ${C}_{44}$. The large number of phases and pronounced elastic softening suggest that geometric frustrations and CEF effects play an important role in the physical properties of ${\mathrm{U}}_{3}{\mathrm{Ru}}_{4}{\mathrm{Al}}_{12}$.

Strain relaxation behaviour of vortices in a multiferroic superconductor

The elastic and anelastic properties of a single crystal of Co-doped pnictide Ba(Fe0.957Co0.043)2As2 have been determined by resonant ultrasound spectroscopy in the frequency range 10–500 kHz, both as a function of temperature through the normal-superconducting transition (Tc ≈ 12.5 K) and as a function of applied magnetic field up to 12.5 T. Correlation with thermal expansion, electrical resistivity, heat capacity, DC and AC magnetic data from crystals taken from the same synthetic batch has revealed the permeating influence of strain on coupling between order parameters for the ferroelastic (QE) and superconducting (QSC) transitions and on the freezing/relaxation behaviour of vortices. Elastic softening through Tc in zero field can be understood in terms of classical coupling of the order parameter with the shear strain e6, λe6, which means that there must be a common strain mechanism for coupling of the form λQE. At fields of ~5 T and above, this softening is masked by Debye-like stiffening and acoustic loss processes due to vortex freezing. The first loss peak may be associated with the establishment of superconductivity on ferroelastic twin walls ahead of the matrix and the second is due to the vortex liquid–vortex glass transition. Strain contrast between vortex cores and the superconducting matrix will contribute significantly to interactions of vortices both with each other and with the underlying crystal structure. These interactions imply that iron-pnictides represent a class of multiferroic superconductors in which strain-mediated coupling occurs between the multiferroic properties (ferroelasticity, antiferromagnetism) and superconductivity.

On the mode I toughness of adhesive bonds exhibiting strain-softening and re-hardening

The mode I fracture toughness of an interface between an adhesive material exhibiting strain-softening behavior and an elastic substrate is investigated by finite element modeling. The model is based on an asymptotic K-field formulation relying on cohesive zone elements to simulate the fracture process. A parametric phenomenological hardening law is used to cover the plastic response behavior of a wide range of amorphous glassy polymers, involving a first hardening peak followed by a softening and a re-hardening response. The effect of the peak hardening stress and of the softening stress on the joint toughness is investigated for different elastic mismatch and interface strength. The joint toughness is found to significantly increase with increasing amplitude of the softening, and, this, at constant yield stress and peak hardening stress. This is related to larger plastic zone sizes and larger plastic strains, hence to more energy dissipation. Increasing the interface work of fracture magnifies this effect. There is an interesting coupling between the elastic mismatch and softening, related to their effect on the fracture process zone length. One outcome of these findings is the potential impact of ageing which is known to modify the amplitude of the peak hardening and softening, hence leading to an evolution of the joint toughness with time.

Understanding size-dependent migration of a two-phase lithiation front coupled to stress

Recent in-situ experiments show that stress-driven migration of the phase interface during two-phase lithiation in nanosized particles exhibits self-limiting and size-dependent behaviors wherein the mechanism remains unclear. In the reaction-limited regime, we develop a mechano-kinetic coupling model with a nonlinear kinetic law to study the size effect of such phase boundary movement accounting for possible sources of stresses by chemical lithiation, concurrent plasticity, surface/interface elasticity, and elastic softening of the lithiated phase. We show that both hydrostatic and non-hydrostatic stresses contribute to the driving force for the phase interface movement and result in the size-dependent slowing down behavior of the phase interface. The obtained results reveal why the interface movement slows down more dramatically in the smaller particle, and there are similar lithiation time scales in nanoparticles of different sizes observed in experiments.

Engineering rheological properties of edible oleogels with ethylcellulose and lecithin

Addition of 1% (w/w) soy lecithin increased the shear moduli 10-fold and gel hardness 20-fold for 10% ethylcellulose (EC) oleogels. Higher lecithin addition levels or addition to gels with a higher EC concentration caused smaller increases. Similar trends were observed in the penetration force of the gels. Gels displayed thermal reversibility and a high temperature plateau at T≈120–130 °C. Large amplitude oscillatory shear rheology demonstrated similar solid-to-fluid transitions indicating that the polymer drives elastic softening and failure of the network. However, EC oleogels differed in their resistance to flow: the addition of unsaturated lecithin promoted a more gradual thickening response compared to gels containing saturated lecithin or only EC (the last two types of gels display strong intra-cycle thickening and thinning, more indicative of brittle failure). The thickening response of EC oleogels containing unsaturated lecithin, resembles more closely that of a model edible fat (lard).

Elastic properties of SmPt2Cd20 probed by ultrasound measurements

Ultrasound measurements have been carried out to investigate the elastic and low-temperature 4f-electronic properties of the Sm-based 1-2-20 system SmPt2Cd20. It was found that SmPt2Cd20 shows a pronounced elastic softening toward the magnetic transition temperature Tc = 0.64 K in the temperature dependence of the transverse elastic constant CE = (C11 -C12)/2. The clear elastic softening behavior is suppressed gradually with increasing a magnetic field. The obtained results can be explained reasonably by a deformation potential approximation in which a coupling between conduction electrons and elastic strains caused by sound waves leads to a temperature dependence of elastic constants. This suggests possibly that the 4f electrons would become itinerant in the ground state in SmPt2Cd20. The nature of the low-temperature elastic and electronic states involving the ground state multiplets of Sm ions are discussed.

Elastic and Dielectric Evaluation of the Piezoelectric Response of Ferroelectrics Using Unpoled Ceramics

The evaluation of the piezoelectric properties of ferroelectric ceramics generally has a high level of uncertainty, due to incomplete poling, porosity, domain wall clamping and other effects. In addition, the poling process is often difficult and dangerous, due to the risk of breaking or damaging the sample. A method is described for the evaluation of the potential intrinsic piezoelectric response that a ceramic would have after full poling, without poling it. The method relies on the fact that any material undergoes an elastic softening below the ferroelectric transition temperature, whose magnitude can be expressed in terms of the intrinsic piezoelectric and dielectric coefficients of the material. Such a softening is equivalent to an electromechanical coupling factor averaged over all the components, due to the unpoled state of the sample, and can be deduced from a single temperature scan of an elastic modulus of a ceramic sample, spanning the ferroelectric and paraelectric states. The strengths, limits and possible applications of the method are discussed.

A Generalized Structure-Dependent Semi-Explicit Method for Structural Dynamics

In this paper, a novel generalized structure-dependent semi-explicit method is presented for solving dynamical problems. Some existing algorithms with the same displacement and velocity update formulas are included as the special cases, such as three Chang algorithms. In general, the proposed method is shown to be second-order accurate and unconditionally stable for linear elastic and stiffness softening systems. The comprehensive stability and accuracy analysis, including numerical dispersion, energy dissipation, and the overshoot behavior, are carried out in order to gain insight into the numerical characteristics of the proposed method. Some numerical examples are presented to show the suitable capability and efficiency of the proposed method by comparing with other existing algorithms, including three Chang algorithms and Newmark explicit method (NEM). The unconditional stability and second-order accuracy make the novel methods take a larger time-step, and the explicitness of displacement at each time-step succeeds in avoiding nonlinear iterations for solving nonlinear stiffness systems.

Geometric micromechanical modeling of structure changes, fracture and grain boundary layers in polycrystals

A constitutive framework based on concepts from phase field theory and pseudo-Finsler geometry is exercised in numerical simulations of deformation and fracture of ceramic polycrystals. The material system of interest is boron carbide, a hard but brittle ceramic. Some microstructures are enabled with thin layers of a secondary amorphous phase of boron nitride between grains of boron carbide. The constitutive theory accounts for physical mechanisms of twinning, crystal-to-glass phase transformations, cleavage fracture within grains and separation and cavitation at grain boundaries (GBs). According to the generalized Finsler approach, geometric quantities such as the metric tensor and connection coefficients can depend on one or more director vectors, also called internal state vectors, that enter the energy potential in a manner similar to order parameters of phase field models. A partially linearized version of the theory is invoked in finite element simulations of polycrystals, with and without GB layers, subjected to pure shear loading. Effects of grain size and layer properties — thickness, shear modulus and surface energy — are studied parametrically. Results demonstrate that twinning and amorphization occur prominently in nanocrystals but less so in aggregates with larger grains that tend to fail earlier by fracture. Structural changes occur readily in the latter at smaller applied strains only in conjunction with elastic shear softening in localized degraded or damaged regions. Hall–Petch scaling of peak shear strength with grain size is observed. Strength is increased via addition of amorphous layers that shift the failure mode from transgranular to intergranular and further by cavity expansion in layers that induces local elastic compression and suppresses crack extension. Stiff layers provide the largest peak strength enhancement, while elastically compliant layers may improve toughness and strength in the softening regime.

Superconductivity and Elastic Properties of La1.85-1.5xSr0.15+1.5xCu1-xMnxO4 with x = 0, 0.02 and 0.04

The effects of mixed-valence Mn substitution at the Cu site on the superconducting and elastic properties of La1.85-1.5xSr0.15+1.5xCu1-xMnxO4 (x = 0, 0.02 and 0.04) were studied. The samples were characterized by X-ray diffraction (XRD) method, temperature-dependent resistance measurements, scanning electron microscopy (SEM) and sound velocity measurements at room temperature. An elastic softening and a decrease in the critical temperature Tc was observed as Mn was substituted for Cu. The Debye temperature θD for x = 0 and 0.02 void free samples were calculated to be 459 K and 430 K, respectively. Internal lattice strain which is related to lattice parameter ratio a/c did not correlate with Tc. The electron-phonon coupling estimated using the conventional Bardeen-Cooper-Schrieffer (BCS) theory was λ = 0.367 (x = 0) and 0.368 (x = 0.02). The electron-phonon coupling in two-dimensional van Hove scenario was calculated to be λvH = 0.0148 (x = 0) and 0.0149 (x = 0.02). These results were compared with other cuprates and related superconductors. The comparison showed that other than electron-phonon coupling, the elastic moduli might play an important role in the mechanism of superconductivity in these materials.

Structural and mechanical characterization of heterogeneities in a CuZr-based bulk metallic glass processed by high pressure torsion

Cu45Zr45Al5Ag5 bulk metallic glass samples, processed by high pressure torsion (HPT) under various conditions, were characterized using synchrotron X-ray diffraction, nanoindentation, differential scanning calorimetry, atomic force and transmission electron microscopy. The experimental results clearly show that HPT modifies the amorphous structure by increasing the mean atomic volume. The level of rejuvenation, correlated with the excess mean atomic volume, is enhanced at higher shear strains as inferred from relaxation enthalpies. By mapping of structural and mechanical quantities, the strain-induced rejuvenated state is characterized on cross-sectional HPT samples on a local scale. A clear correlation both between elastic and plastic softening and between softening and excess mean atomic volume is obtained. But also the heterogeneity of the HPT induced rejuvenation is revealed, resulting in the formation of highly strain-softened regions next to less-deformed ones. A hardness drop of up to 20% is associated with an estimated increase of the mean atomic volume of up to 0.75%. Based on synchrotron X-ray diffraction and nanoindentation measurements it is concluded that elastic fluctuations are enhanced in the rejuvenated material on different length scales down to atomic scale. Furthermore, the calculated flexibility volume and the corresponding average mean square atomic displacement is increased. The plastic response during nanoindentation indicates that HPT processing promotes a more homogeneous-like deformation.

Anisotropic Grüneisen Parameter and Diverse Order Parameter Fluctuations in Iron-Based Superconductor Ba(Fe1−xCox)2As2

In this study, the temperature dependence of elastic constants $C_{11}$, $C_{33}$, $C_{\rm E} = (C_{11}-C_{12})/2$, $C_{66}$ and $C_{44}$ of the iron-based superconductor Ba(Fe$_{1-x}$Co$_{x}$)$_{2}$As$_{2}$ (0 $\leqq x \leqq$ 0.245) have been measured. This system shows a large elastic softening in $C_{66}$ towards low temperatures. In addition to $C_{66}$, which originates from orthorhombic structural fluctuation, the samples near the optimal concentration show remarkable structural fluctuation in $C_{11}$ and $C_{33}$ elastic modes, which correspond to $\Gamma_{1}$ (C4) symmetry. It suggests the existence of diverse fluctuations in this system. Gr{\" u}neisen parameters were analyzed under some assumptions for structural and magnetic transition temperatures. Results showed that the Gr{\" u}neisen parameters for the inter-plane strain are remarkably enhanced toward the QCP, while those for the in-plane stress tend to turn down near the QCP. Gr{\" u}neisen parameters for the superconducting transition are anisotropic and shows remarkable Co-concentration dependence, suggesting that the in-plane isotropic compression and inter-layer elongation enhance the superconductivity. The correlation of Gr{\" u}neisen parameters between $T_{\rm S}$, $T_{\rm N}$ and $T_{\rm sc}$ shows $c$-axis elongation and its relevant role in the emergence of superconductivity in this system.

Magnetic-field-induced quadrupolar ordering and the crystal electric field effect in the distorted kagome lattice antiferromagnet Dy3Ru4Al12

To investigate the $4f$-electronic states under a crystal electric field (CEF) and the phase transition in ${\mathrm{Dy}}_{3}{\mathrm{Ru}}_{4}{\mathrm{Al}}_{12}$ with the antiferromagnetic transition temperature ${T}_{\mathrm{N}}=7$ K, we performed ultrasonic measurements on a single-crystalline sample at zero magnetic field and under fields. The transverse elastic modulus ${C}_{44}$ shows a characteristic elastic softening. The CEF analyses indicate that the softening is caused by an interlevel quadrupole interaction between the ground and excited Kramers doublets. Under fields, we found a magnetic-field-induced phase transition along both the [100] and [001] directions in addition to the antiferromagnetic ordering. A plausible origin of the field-induced transition is quadrupolar ordering, which is estimated from our CEF calculation. These results and the negative sign of a quadrupole-quadrupole coupling constant suggest that the effect of geometrical frustration alignment due to the kagome lattice also appears on the electric quadrupoles of the Dy ions with the antiferroquadrupolar-type interaction.

Nonlinear elastic response of single crystal Cu under uniaxial loading by molecular dynamics study

Molecular dynamics simulations were carried out to study the mechanical response of single crystal Cu with various orientations under uniaxial loading. Cu shows a high degree of elastic anisotropy as a function of the crystal orientations, and the yield stress varies significantly with the crystal orientations and the loading directions. An obvious nonlinear stress-strain response was observed at the elastic deformation stage under applied loading, namely elastic hardening and elastic softening. This nonlinear elastic behavior is derived from the potential energy between atoms and can play a significant role in yield stress and yield asymmetry of crystals under tension and compression.

Search for multipolar instability in URu2Si2 studied by ultrasonic measurements under pulsed magnetic field

The elastic properties of URu$_2$Si$_2$ in the high-magnetic field region above 40 T, over a wide temperature range from 1.5 to 120 K, were systematically investigated by means of high-frequency ultrasonic measurements. The investigation was performed at high magnetic fields to better investigate the innate bare 5$f$-electron properties, since the unidentified electronic thermodynamic phase of unknown origin, so called `hidden order'(HO) and associated hybridization of conduction and $f$-electron ($c$-$f$ hybridization) are suppressed at high magnetic fields. From the three different transverse modes we find contrasting results; both the $\Gamma_4$(B$_{\rm 2g}$) and $\Gamma_5$(E$_{\rm g}$) symmetry modes $C_{66}$ and $C_{44}$ show elastic softening that is enhanced above 30 T, while the characteristic softening of the $\Gamma_3$(B$_{\rm 1g}$) symmetry mode $(C_{11}-C_{12})/2$ is suppressed in high magnetic fields. These results underscore the presence of a hybridization-driven $\Gamma_3$(B$_{\rm 1g}$) lattice instability in URu$_2$Si$_2$. However, the results from this work cannot be explained by using existing crystalline-electric field (CEF) schemes applied to the quadrupolar susceptibility in a local $5f^2$ configuration. Instead, we present an analysis based on a band Jahn-Teller effect.

Elastic, mechanical, and thermodynamic properties of Bi-Sb binaries: Effect of spin-orbit coupling

Using first principles calculations, we systematically study the elastic stiffness constants, mechanical properties, elastic wave velocities, Debye temperature, melting temperature, and specific heat of several thermodynamically stable crystal structures of Bi$_{x}$Sb$_{1-x}$ ($0 < x < 1$) binaries, which are of great interest due to their numerous inherent rich properties, such as thermoelectricity, thermomagnetic cooling, strong spin-orbit coupling (SOC) effects, and topological features in the electronic bandstructure. We analyze the bulk modulus ($B$), Young's modulus ($E$), shear modulus ($G$), $B/G$ ratio, and Poisson's ratio ($\nu$) as a function of the Bi concentration in Bi$_{x}$Sb$_{1-x}$. The effect of SOC on above mentioned properties is further investigated. In general, we observe that the SOC effects cause elastic softening in most of the studied structures. Three monoclinic structures of Bi-Sb binaries are found to exhibit significantly large auxeticity. The Debye temperature and the magnitude of the elastic wave velocities monotonically decrease with increasing Bi-concentration. We also discuss the specific heat capacity versus temperature data for all studied binaries. Our theoretical results are in excellent agreement with the existing experimental and theoretical data. The comprehensive understanding of the material properties such as hardness, mechanical strength, melting temperature, propagation of the elastic waves, auxeticity, and heat capacity is vital for practical applications of the studied binaries.