Microscopic Theory Of Elasticity Research Articles

Deformation-induced nontetragonality of martensite in carbon steels

With supersaturated carbon atoms ordering in the preferential octahedral sites of α-Fe lattice, the martensite formed in medium and high carbon steels exhibit a unique structural feature, tetragonality. In this letter, we reported anomalous nontetragonality of martensite in carbon steels by deforming quenched martensite, indicating that the disordering of interstitial carbon atoms in α-Fe lattice occurs under mechanical deformation. This order-disorder transition of interstitial carbon atoms in carbon steels provides solid experimental evidences of the prediction based on microscopic elasticity theory.

Orientational ordering of interstitial atoms and martensite formation in dilute Fe-based solid solutions

We study the thermodynamic stability of dilute C, N, O, B, and C$+$N interstitial distributions in bcc iron, combining parameter-free density functional theory (DFT) in the generalized gradient approximation and microscopic elasticity theory. This scheme allows us to fully capture the long-range elastic impurity-impurity interactions using moderately sized DFT calculations. Employing this approach we compute temperature-concentration phase diagrams including the effects of external pressure, and provide direct insight into the formation mechanisms of martensite. For all investigated impurities, except for B, tetragonal states are predicted to be preferred even at low impurity concentrations. The preference is shown to originate from a thermodynamically driven orientational ordering of the interstitials.

Calculation of the elastic energy of a solid solution

A corrected version of the microscopic elasticity (ME) theory is proposed. Previous versions of this theory, like that of Khachaturyan, contained a discrepancy that caused severe problems in certain applications. One such case is the simulation of precipitation of secondary phases with long-ranged elastic interactions: Monte Carlo simulations using the published ME theories to account for the elastic stresses profoundly fail to predict the precipitation of plate-like particles, such as Guinier–Preston zones in Al–Cu alloys. The quoted discrepancy is resolved, so that Monte Carlo simulations using the new ME theory comply with all theoretical expectations. The revised theory also introduces new parameters such that real materials can be considered more accurately.

Long-Wavelength Elastic Interactions in Complex Crystals

The long-wavelength (LWL) limit of the elastic interactions in complex non-Bravais lattices is investigated on the basis of microscopic elasticity theory. The conceptual simplicity of our approach enables large-scale simulations in materials with complex crystalline structures. We demonstrate the method by calculating the LWL elastic energy of hcp-based Mg binary alloys for a variety of impurities. Our results show that for large coherent precipitates, the strain-induced interactions control the shape along the hexagonal axis, whereas the surface energy dictates the basal growth. The present formalism enables a straightforward treatment of the long-range elastic interactions in the cluster expansion method for complex crystals.

Interplay between long-range elastic and short-range chemical interactions in Fe-C martensite formation

The thermodynamic stability of tetragonal and cubic states of dilute Fe-C interstitial solid solutions is studied combining atomistic simulations with the microscopic elasticity theory. This approach allows us to accurately describe the effect of long-ranged elastic C-C interactions on the martensite formation. The predicted compositional dependence of the critical temperature at which the cubic phase transforms into tetragonal states is in good agreement with available experimental data and thus removes previous controversies regarding the relative importance of strain-induced and chemical interactions.

Lattice study of ω phase formation involving volume change

The incommensurate structure of the ω phase, which has been observed as the pre-transition phenomena of ω phase transformation, was studied by focusing on the elastic strain energy in a domain model using the microscopic theory of elasticity. The convolution theorem was applied to simplify the mathematics of describing an incommensurate ellipsoidal droplet with an internal structure consisting of alternate lamellae of ω and β phases. It was clear that the decomposition into the two-phase structure yielded much less elastic strain energy than the single ω phase. The interface of the ω and β lamellae in the incommensurate structure should be (1 1 1) β when formed by the simple ω wave k ( q ω) of Burgers displacement, but the deviation of the interface orientation from the (1 1 1) β plane occurs due to the additional volume change by Bain distortion. It was also found that there is no interaction in the elastic strain energy originated by the Burgers displacement and the Bain distortion.

Elastic Energy Analysis of Carbide and Nitride-Type Precipitates in an Fe–Mn–Si–Cr Shape Memory Alloy

The application of the microscopic theory of elasticity in a discrete lattice model is made on the transitional metal carbide and nitride precipitates formed in the Fe–Mn–Si–Cr shape memory alloy in conjunction with the improvement of shape memory effect (strain) and the improvement of strength. Two distinguishable methods of analysis have been established using the microscopic theory of elasticity in a discrete lattice model: One is the establishment of the description of precipitate and misfit dislocations in Fourier space. The second is the rigorous estimation of interaction energies among precipitate and misfit dislocations. The results could successfully describe the shape of the precipitate observed in the experimental investigation. It was also concluded that the elastic strain energy increases with the lattice parameter of the precipitate. Among the transition carbides and nitrides under investigations, VN, which revealed the minimum value of the elastic energy, is manifested to be the most favored one for the precipitation enhanced Fe–Mn–Si–Cr shape memory alloy. Homogeneously precipitated VN containing materials could show large deformability, higher strength by precipitation hardening and the higher shape recovery strain due to the nucleation sites of the precipitates in its reverse phase transformation.

Crystallography and elastic energy analysis of VN precipitates in Fe–Mn–Si–Cr shape memory alloys

High-resolution electron microscopy investigations are carried out to describe the morphology and crystallography of VN precipitates which are formed in an Fe–28Mn–6Si–5Cr (mass%) shape memory alloy. It is revealed that the shape change from a cube with {1 0 0} interfaces to an octahedral shape with {1 1 1} interfaces occurs on aging in the precipitate. In order to identify the equilibrium shape of the VN precipitate, elastic strain energy of the precipitate has been estimated on the basis of microscopic theory of elasticity. It is found that a coherent precipitate (∼4 nm in edge width) in a cube shape with {1 0 0} surfaces can be formed, at an early stage of precipitation, with no misfit dislocations existing at the interface of the precipitate. It is also shown that the octahedral-shaped precipitate (∼15 nm in edge width) has a minimum elastic energy, among the cube-shaped, sphere-shaped and octahedron-shaped precipitates, only when the misfit dislocations are introduced at the interfaces. The elastic interaction energy between the misfit dislocations and the precipitate–misfit dislocations is estimated for the first time using the Fourier transformed microscopic theory of elasticity.

Heterophase fluctuation of omega phase and X-ray diffuse scattering from dual phase structure

Abstract Heterophase fluctuation of athermal omega embryos has been analyzed by assuming a dual phase structure of omega embryos composed of omega and bcc matrix phase. The two-dimensional modulation of dual phase was suggested from the quantitative estimation of coherent free energy of omega embryos using microscopic theory of elasticity and the Landau anharmonic theory for phase transformation. The X-ray diffraction theory was developed in connection to the formation of omega embryos having the dual phase structure. The offset of the diffuse peak position from the ideal omega point in the X-ray diffraction pattern is attributed to the dual phase (incommensurate phase) of omega embryos. It was also shown that the ellipsoidal shape of the diffuse intensity tailing toward the fundamental spot of the matrix phase is originated from the equilibrium shape of the omega embryo. The quantitative estimation of elastic energy modulus (EEM) in the disordered bcc matrix and in the ordered bcc matrix indicates a difference in the deviation amount of the minimum point k ( q m ) from the ideal omega point k ( q ω ) and a difference in the elliptical shape of embryos.

Microscopic elasticity theory of ionic crystal and its application to cubic ZrO 2–5.2 mol% Y 2O 3 system

An analytical expression of the strain energy in an ionic crystalline solid solution was obtained by modifying the microscopic theory of elasticity in metal alloys. The Lorentz field originating in the short-range interaction of dipoles, along with a non-Coulomb force field produces specifically the elastic strain. On the other hand, the macroscopic electric field induced by the long-range interaction of dipoles works on each dipole lattice to produce the piezoelectric strain. The theory was applied to a cubic ZrO 2–Y 2O 3 system of ionic crystalline solid solution in order to estimate the microscopic strain energy of the system. The trace of the minimum points in the energy surface indicates possible spinodal decomposition along [1 1 1] direction. It is also revealed that local electrical neutrality can mostly be preserved during the decomposition due to the macroscopic electric field in the ionic crystal.

Microscopic theory of elasticity of the crystalline solid solution based on pseudopotential method

A theoretical method is developed using the pseudopotential method of electron band theory for calculating the strain energy posed by a crystalline solid solution having solute and solvent atoms that differ in size. In the harmonic approximation, the expression of the strain energy obtained from the pseudopotential method is formally identical to that of the phenomenological theory of elasticity. However, it clarifies the physical significance of the phenomenological parameters (solute, solute-lattice and lattice coupling parameters), so that important characteristics of coupling parameters such as the concentration dependence, the effect of the Fermi surface and the effect of the phonon dispersion curves have been made clear. The analysis reveals the direct effect of the Kohn anomaly on the relaxation of the strain energy of a crystalline solid solution. By applying the result to β-brass, the occurence of spinodal decomposition is consistently explained in the scheme of elastic energy. The pair of concentration (deviation) waves, each of which arises in the α- and β-sublattice, respectively, is theoretically described as an acoustic mode of waves with amplitude ratio β Q( q) Q ( α q ) = 999 .

Theoretical Analysis of Solute-Lattice Coupling Parameters in the Microscopic Elasticity Theory and Application to Elastic Energy in β-Brass

The microscopic theory of elasticity was reconsidered in detail in an ordered alloy system following the line of reasoning pursued by the present author

Instrument-independent dynamic mechanical analysis of polymeric systems

Large discrepancies are often observed among data obtained by different dynamic mechanical instruments, different analysis methods, or even different laboratories using identical instruments. Four commonly used dynamic mechanical instruments, the Rheovibron, the DMA 982 and 983, the Dynastat, and the PPM-R5 sonic tester, were methodically analysed for type of deformation as well as frequency-temperature relationships that may affect comparison of their data. Such discrepancies were shown to arise from the instruments' limited ranges of test frequencies, types of deformation, types of material tested, and the material morphology, orientation and conformation as well as limitations of theoretical methods of analysis and overall test sensitivity to external factors. Through the use of macro- and microscopic theory of elasticity as well as the time-temperature superposition principle, along with careful experimental set-ups, the dynamic mechanical responses of a set of well characterized polyethylene samples were successfully compared with one another across all instruments.

Invariance conditions of the microscopic theory of elasticity

The Born-Huang theory for calculation of the elastic constants of crystalline solids is assessed in the light of criticism directed at the validity or completeness of its invariance conditions. It is shown that the original formulation and invariance conditions are correct and complete. It is observed that a majority of the misunderstanding of the Born-Huang theory stem from the diversity of force constant sets that can be produced in the modeling process.

Theoretical and experimental investigations on elastic interactions between ??-precipitates in a Ni-Al alloy

Elastic interaction energies between two identical inhomogeneous plates in an anisotropic matrix were calculated with respect to particle separation distance and alignment orientation on the basis of the microscopic theory of elasticity developed by Yamauchi and de Fontaine. Four types of plate-arrangement along the cube directions; two plates whose faces were mutually parallel, two plates which were arranged on a cube plane as a raft, two plates which were perpendicularly arranged with face—edge configuration and two plates which were perpendicularly aligned with edge—edge configuration were energetically stable, while other arrangements were not stable. Compared with experimental observations on the local arrangement of γ′plates in a Ni-Al single crystal, the calculated results were confirmed to be qualitatively correct.

On the elastic free energy of solid solutions—II. Influence of the effective modulus on precipitation from solution and the order-disorder reaction

The reciprocal space representation of the microscopic elasticity theory described in part I is used in reconsidering the role that lattice strain energy plays in precipitation and order-disorder transformations. The results are stated in terms of the effective elastic modulus M( h) where h denotes the reciprocal space variable. The elastic energy of a single defect is shown to be proportional to the average value of the effective modulus, M̄, and it follows that the energy of a random distribution of solute atoms is also proportional to M̄. Neither of these energies are found to differ significantly from those obtained from continuum elasticity theory. The change in elastic energy in going from a random solute distribution to an ordered structure is shown to be proportional to the difference, M(h′) − M ̄ , where ( h′) denotes the position of the superlattice reflection. Only a slight change in elastic energy occurs on forming the ordered Cu 3Au structure contrasting with the result, presented in I, of a complete removal of elastic energy on forming the ordered β-brass structure. The method of Cahn and Hilliard for calculating the “chemical” gradient energy coefficients is extended to solid solutions in which there is a significant contribution to the heat of mixing from the elastic energy. The “elastic” gradient energy coefficient, however, is found to contribute significantly to the total gradient energy only for large size differences, greater than 10 per cent. The microscopic theory is also used to explore the paradoxical role that the elastic energy has in precipitation in which it contributes to the formation of a miscibility gap but inhibits coherent phase separation by lowering the spinodal.