Internal Strain Tensor Research Articles

Ab initio calculations of second-, third-, and fourth-order partial and inner elastic constants of diamond

By means of ab initio calculations, a unified framework is presented to investigate the effect of internal displacement on the linear and nonlinear elasticity of single diamond crystals. The calculated linear and nonlinear elastic constants, internal strain tensor and internal displacement in single diamond crystals are compatible with the available experimental data and other theoretical calculations. The complete set of second-, third- and fourth-order elastic constants and internal strain tensor not only offer a better insight into the nonlinear and anisotropic elasticity behaviors, but also shows us the basic internal mechanical response of diamond. This study provides a route to calculate the nonlinear internal and external elasticity response in a nonprimitive lattice.

Theory-Guided Exploration of the Sr2Nb2O7 System for Increased Dielectric and Piezoelectric Properties and Synthesis of Vanadium-Alloyed Sr2Nb2O7

Ab initio methods provide a powerful tool in the search for novel polar materials. In particular, there has been a surge to identify lead-free piezoelectric materials to replace PbZr0.52Ti0.48O3. This study examines a computational strategy to identify increased piezoelectric and dielectric responses of alloy systems based on the linear interpolation of force constants, Born effective charges, and internal strain tensors from their end-point compounds. We choose the ferroelectric layered perovskite Sr2Nb2O7 as a parent structure and employ this alloying strategy for 19 potential cation substitutions, targeting thermodynamically metastable alloys with high piezoelectric response. From this screening, we identify Sr2Nb2–2xV2xO7 as a promising polar system. We conduct large-unit-cell calculations of Sr2Nb2–2xV2xO7 at x = 0.0625, 0.125 for multiple cation orderings and find a significant 184% enhanced piezoelectric response. The solid solution system is synthesized as single-crystalline thin-film heterostructures using pulsed-laser deposition, and an enhanced dielectric response is observed at x = 0.05 and at x = 0.1. We present the Sr2Nb2–2xV2xO7 alloy system designed through high-throughput computational screening methods with a large calculated piezoelectric response and experimentally verified increased dielectric response. Our methodology is provided as a high-throughput screening tool for novel materials with enhanced polarizability and alloy systems with potential morphotropic phase boundaries.

Finite-temperature atomic relaxations: Effect on the temperature-dependent C44 elastic constants of Si and BAs.

The effect of atomic relaxations on the temperature-dependent elastic constants (TDECs) is usually taken into account at zero temperature by the minimization of the total energy at each strain. In this paper, we investigate the order of magnitude of this approximation on a paradigmatic example: the C44 elastic constant of diamond and zincblende materials. We estimate the effect of finite-temperature atomic relaxations within the quasi-harmonic approximation by computing ab initio the internal strain tensor from the second derivatives of the Helmholtz free-energy with respect to strain and atomic displacements. We apply our approach to Si and BAs and find a visible difference between the softening of the TDECs computed with the zero-temperature and finite-temperature atomic relaxations. In Si, the softening of C44 passes from 8.6% to 4.5%, between T = 0 K and T = 1200 K. In BAs, it passes from 8% to 7%, in the same range of temperatures. Finally, from the computed elastic constant corrections, we derive the temperature-dependent Kleinman parameter, which is usually measured in experiments.

Digital volume correlation analysis of polylactic acid based fused filament fabrication printed composites.

Fused filament fabrication (FFF) has rapidly begun to see implementation in industrial fields as a method of rapid manufacturing. Traditional FFF parts are made from a single thermoplastic polymer. The polymer is heated to its melting point and deposited on a work bed where a model is gradually built from the base up. While traditional FFF parts have low mechanical properties, a reinforcing phase allows for improved mechanical properties. The addition of a reinforcing material to the base polymer and complex internal microstructure of the 3 D printed party leads to anisotropic mechanical properties. Thus, these materials’ mechanical properties become challenging to characterize using traditional measurement techniques due to the previously mentioned factors. Therefore, it is essential to develop a method in which mechanical properties can be measured and analyzed. This study aims to characterize the mechanical behaviour under a uniaxial tensile load of an FFF produced polylactic acid (PLA)-copper particulate composite. The internal response of the FFF sample was imaged using micro-computed tomography at predetermined loads. The μ-CT images were input into an open-source digital volume correlation (DVC) software to measure the internal displacements and strain tensor fields. The study results show the development of different strain fields and interior features of the FFF parts.

Effect of Earth Models on Coulomb Stress Change Caused by Surface Load

Based on the elastic spherical Earth models, this study presents the radial Love numbers and Green’s functions of the surface load problem, then evaluates the effect of different Earth models on the Coulomb stress changes due to elastic deformation by surface load. First of all, the radial load Love numbers are introduced and verified by comparing them with the results of an analytical method. Second, the formulae of the radial Green’s functions of the internal displacements and strain tensors are obtained by summing the Love numbers. The results are compared with Green’s function of half-space to verify the correctness. Then, the stress tensors are obtained by the constitutive relation. Finally, the Coulomb stress changes due to elastic deformation around the Wenchuan Mw7.9 earthquake by the impoundment load of the Zipingpu reservoir are calculated based on five Earth models. The results show that the elastic deformations caused by the impoundment load do not increase the seismicity rate in the study area. At the epicenter of the great event, the maximum effect due to the Earth models is about 38%. The stratification of the Earth is the main effect factor on the Coulomb stress change, and it should not be ignored in actual studies. The biggest difference of Coulomb stress changes based on the different stratified Earth models is close to 15%, which also cannot be ignored.

Volumetric and Shear Strain Localization in Mt. Etna Basalt

AbstractTo examine the impact of preexisting weaknesses on fracture coalescence during volcanic edifice deformation, we triaxially compressed Mount Etna basalt while acquiring in situ dynamic X‐ray microtomograms and calculated the internal strain tensor fields using image correlation. Contraction localization preceded dilation and shear strain localization into the protofault zone. This onset of strain localization preceded macroscopic yielding and coincided with increases in the magnitude and volume of rock experiencing dilation, and spatial clustering of the strain populations. The exploitation of weaknesses by propagating fractures enabled the dominant shear strain to switch senses as propagating fractures lengthened along 30–60° from σ1. Scanning electron microscopy images reveal pore‐emanated fractures, and fractures linking pores. These experiments provide evidence of internal contraction preceding dilation and shear, consistent with inferences from field and laboratory observations. The transition from contraction to dilation may provide a precursory signal of volcanic flank eruption.

Hybrid functional study of nonlinear elasticity and internal strain in zinc-blende III-V materials

We investigate the elastic properties of selected zinc-blende III-V semiconductors. Using hybrid functional density functional theory, we calculate the second- and third-order elastic constants and first- and second-order internal strain tensor components for Ga, In, and Al containing III-V compounds. For many of these parameters, there are no available experimental measurements, and this work is the first to predict their values. The stricter convergence criteria for the calculation of higher-order elastic constants are demonstrated, and arguments are made based on this for extracting these constants via the calculated stresses, rather than the energies, in the context of plane-wave-based calculations. The calculated elastic properties are used to determine the strain regime at which higher-order elasticity becomes important by comparing the stresses predicted by a lower- and a higher-order elasticity theory. Finally, the results are compared with available experimental literature data and previous theory.

Synchrotron X-ray microbeam diffraction measurements of full elastic long range internal strain and stress tensors in commercial-purity aluminum processed by multiple passes of equal-channel angular pressing

Synchrotron X-ray microbeam diffraction was used to measure the full elastic long range internal strain and stress tensors of low dislocation density regions within the submicrometer grain/subgrain structure of equal-channel angular pressed (ECAP) aluminum alloy AA1050 after 1, 2, and 8 passes using route BC. This is the first time that full tensors were measured in plastically deformed metals at this length scale. The maximum (most tensile or least compressive) principal elastic strain directions for the unloaded 1 pass sample for the grain/subgrain interiors align well with the pressing direction, and are more random for the 2 and 8 pass samples. The measurements reported here indicate that the local stresses and strains become increasingly isotropic (homogenized) with increasing ECAP passes using route BC. The average maximum (in magnitude) LRISs are −0.43 σa for 1 pass, −0.44 σa for 2 pass, and 0.14 σa for the 8 pass sample. These LRISs are larger than those reported previously because those earlier measurements were unable to measure the full stress tensor. Significantly, the measured stresses are inconsistent with the two-component composite model.

Poroelasticity-I: Governing Equations of the Mechanics of Fluid-Saturated Porous Materials

We present here the approach to the theory of fluid-filled poroelastics based on consideration of poroelastics as a continuum of “macropoints” (representative elementary volumes), which “internal” states can be described by as a set of internal parameters, such as local relative velocity of fluid and solid, density of fluid, internal strain tensor, specific area, and position of the center of mass of porous space. We use the generalized Cauchy–Born hypothesis and suggest that there is a system of (structural) relationships between external parameters, describing the deformation of the continuum and internal parameters, characterizing the state of representative elementary volumes. We show that in nonhomogenous (and, particularly, nonlinear) poroelastics, an interaction force between solid and fluid appears. Because this force is proportional to the gradient of porosity, absent in homogeneous poroelastics, and one can neglect with dynamics of internal degrees of freedom, this force is equivalent to the interaction force, introduced earlier by Nikolaevskiy from phenomenological reasons. At last, we show that developed theory naturally incorporates three mechanisms of energy absorption: visco-inertial Darcy mechanism, “squirt flow” attenuation, and skeleton attenuation.

An Internal Variable Approach to Inelastic Deformation Including Structural Superplasticity

An internal variable theory has been proposed to account for the essential microstructures during inelastic deformation. The framework of the theory is built on the basis of well known dislocation dynamics to provide the concept of an internal strain tensor as the most fundamental deformation state variable. The plastic and inelastic strain rate tensors are then naturally defined and also a kinematics relation among them can further be derived from the time rate of change of this internal strain tensor, which in fact accounts for the evolution of microstructures during inelastic deformation. To complete the theory, the constitutive relations between the various stress variables and their conjugate deformation rate variables are then derived based on the dislocation kinetics. The theory is then further extended to describe the structural superplasticity, taking this slip zone model with dislocation pile-ups as the major accommodation mechanism for grain boundary sliding. The experimental results obtained from the various crystalline materials are then presented and compared with each other in relation to the internal variable theory for inelastic deformation.

Elasticity of carbon allotropes. III. Hexagonal graphite: Review of data, previous calculations, and a fit to a modified anharmonic Keating model

The experimental data relating to the second- and third-order elasticity and the zone-center optic modes of hexagonal graphite are reviewed and some amendments proposed. A modified Keating model involving three sets of interactions, one planar and two interlayer, has been developed. The harmonic parameters, four planar and seven interlayer, have been fitted by least-squares procedures to five second-order elastic constants, five zone-center optic-mode frequencies and two assumptions relating to internal strain. The anharmonic parameters comprise three planar and three interlayer ones. They have been fitted to the pressure derivatives of the five second-order constants and of three of the optic-mode frequencies. The full spectrum of inner elastic constants and internal strain tensors is given, the composition of the second- and third-order elastic constants is exposed, and the corresponding elastic compliances calculated. A pressure-induced phase transition is correctly predicted at around 16 GPa.

Formal elasticity offour carbon allotropes: I. The inner elastic constants, internalstrain tensors, and zone-centre optic mode frequencies and theirpressure dependences

The symmetry structure of the six families of inner elasticconstants (derivatives of the free energy with respect to one,two, or three components of the relative sublattice displacement)that enter the elasticity, through the third order, of the cubic andhexagonal diamond allotropes and of the hexagonal andrhombohedral graphite allotropes is analysed in detail. This isfollowed by derivations (i) of the forms of the linear andquadratic internal strain tensors, (ii) of the expressions forthe zone-centre optic mode frequencies and eigenvectors, and(iii) of the effective inner elastic constants that determinethese frequencies in arbitrarily strained crystals together with(iv) the derived pressure dependences of those frequencies.

An internal variable theory of structural superplasticity

A new approach for structural superplasticity (SSP) is attempted in this study based on the internal deformation variables. The basic mechanisms of SSP are thought here to consist of interface sliding (IS), i.e. grain or phase boundary sliding (GBS/PBS), and a dislocation glide process accommodating the incompatibilities due to IS. For this purpose, a new constitutive framework for inelastic deformation is first developed by a simple consideration of dislocation kinematics to reveal the existence of “internal strain” and “internal spin” tensors. The evolution relationship of internal strain tensor is then shown to lead into a kinematic relationship between the observable deformation variables. To complete the constitutive structures, the constitutive relationships for each deformation variable including the strain rate due to IS are also prescribed by kinetics consideration of each process. The theoretical results are then applied to the experimental results on a fine-grained 7475 Al and a Pb–Sn eutectic alloy obtained from load relaxation and tensile tests.

The relation of inner compressibilities to the pressure dependence of atomic position parameters in simple crystal structures

The application of hydrostatic pressure to crystals in which some atoms occupy sites with polar point-group symmetry causes the position coordinates of these atoms to change. This phenomenon is here related to inner compressibility tensors. These arise naturally in the theory of inner elasticity, where they are linear combinations of the macroscopic linear compressibilities weighted by components of the internal strain tensors, and they indicate the ease with which the separation between pairs of sublattices may change under pressure. The form of the inner compressibility tensors is presented for eleven simple crystal structures involving up to four atoms in the basis. Finally, the inner compressibilities and con- straining equations for components of the internal strain are obtained from the pressure dependence of the structure of the elements As, Sb, Bi, Se, Te and I.

Inner displacement and the internal strain tensor in the fluorite structure

The inner displacements in a stressed crystal of the fluorite structure are calculated from the parameters of both rigid-ion and shell lattice dynamic models by considering the response of the lattice to long-wavelength phonons. The general results are applied to the three crystals CaF2, SrF2 and BaF2. The magnitudes of the inner displacements are found to be model dependent and are significant compared with the uniform strain displacements.

The relation of the stress dependence ofF(H)2to inner compliance and internal strain tensors in 20 simple crystal structures

Internal strain tensors describe the relative displacements of sublattices in stressed crystals in which some atoms occupy sites lacking inversion symmetry. The variation of |F(H)|2 in such crystals is shown to depend on inner compliance tensors which are essentially products of internal strain tensors with the macroscopic elastic compliance tensor. The forms of the inner compliance tensors and the uniaxial stress derivatives of |F(H)|2 are tabulated for 20 simple crystal structures in which the number of atoms per lattice point is 2, 3 or 4.

Determination of the internal strain tensor of germanium

The single independent component A of the internal strain tensor of germanium has been determined by observing the 006 and 0,0,10 forbidden X-ray reflections whilst applying a stress along the (110) axis. The authors find A=-0.18+or-0.01. The corresponding bond-bending parameter is zeta identical to -4A=0.72+or-0.04, a value somewhat larger than the one currently accepted.

Internal Strain of Silicon Studied by X-Ray Energy-Dispersive Diffraction

The single independent component Ā of the internal strain tensor of silicon has been determined by observing the 006 forbidden reflection whilst applying a stress along the [1̄10] axis. A value Ā = -0.18 ± 0.01 has been found. Advantages of the energy dispersive method are indicated.

Determination of internal strain tensors by energy-dispersive X-ray diffraction: Results for Si using the 006 forbidden reflection

The internal strain tensor of crystals having the diamond structure has a single independent component {\bar A}. A value {\bar A} = −0.18±0.01 for silicon has been found by observing the 006 forbidden reflection whilst applying a stress along the [1{\bar 1}0] axis. An energy-dispersive method was used so that advantage could be taken of the simultaneous presence of strong reflections (004 and 008) to align the sample with precision and to normalize the intensity measurements. In the course of investigation it was found that the strong reflections increased in intensity as a function of stress, suggesting that the crystal was undergoing a reversible change from the ideally perfect to the ideally imperfect state.

The corundum structure: internal strain tensors and cluster configurations under stress

The corundum structure ( alpha -R2O3 where R=Al, Ti, V, Cr, Fe) has been chosen for the first detailed study of inner displacement and internal strain in a moderately complex system. It is shown that the inner displacements of sublattices of ions are governed by two independent internal strain tensors: these relate to O-O and R-R displacements and have eight and six independent components respectively. These results are used to give a proper description of the deformation of octahedral clusters of O2- ions surrounding an R3+ cation when the crystal is subject to uniaxial stress. Such clusters and their deformations are important in the understanding of a variety of recent experiments but have been the subject of controversy and erroneous description. The major components of the internal strain tensors have been estimated using a pseudo-HCP model of the anion structure.