Isotropic Elasticity Theory Research Articles

Analysis of local strain fields around individual threading dislocations in GaN substrates by nanobeam x-ray diffraction

Position-dependent three-dimensional reciprocal space mapping (RSM) by nanobeam x-ray diffraction (nanoXRD) was performed to reveal the strain fields produced around individual threading dislocations (TDs) in GaN substrates. The distribution and Burgers vector of TDs for the nanoXRD measurements were confirmed by prerequisite analysis of multi-photon excited photoluminescence and etch pit methods. The present results demonstrated that the nanoXRD can identify change in the lattice plane structure for all types of TDs, i.e., edge-, screw-, and mixed TDs with the Burgers vector of b = 1a, 1c and 1m + 1c. Strain tensor components related to edge and/or screw components of the TDs analyzed from the three-dimensional RSM data showed a nearly symmetrical strained region centered on the TD positions, which were in good agreements with simulation results based on the isotropic elastic theory using a particular Burgers vector. The present method is beneficial in that it allows non-destructive analysis of screw components of TDs that tend to contribute to leakage characteristics and are not routinely accessible by conventional structural analysis. These results indicate that nanoXRD could be a powerful way to reveal three-dimensional strain fields associated with arbitrary types of TDs in semiconductor materials, such as GaN and SiC.

Comprehensive structural analysis, magnetic, elastic, and optical properties of Cu–Ni–Co spinel and its potential applications

This work investigates the Cu0·6Ni0·2Co0·2FeAlO4 (CNCFAO) spinel nanoparticle, focusing on its structural, magnetic, elastic, and optical properties. The structural analysis provided various parameters, such as lattice parameter, crystallite size, and bond lengths, which were compared to theoretical values. The force constants of the sample were used to calculate stiffness constants (C11, C12, and C44), which in turn were used to determine elastic moduli such as the bulk modulus (B), longitudinal modulus (L), and rigidity modulus (G). The calculated Pugh ratio (B/G) indicated the brittle mechanical nature of the sample, while the Poisson ratio (σ) was estimated as σ = 0.25, consistent with isotropic elasticity theory. The sample demonstrated a low coercive field, indicating its potential for soft magnetic devices. Optical characteristics were evaluated using the Tauc method, revealing a direct optical transition with estimated band-gap (Eg) and Urbach (Eu) energies of 2.38 eV and 0.59 eV, respectively. Additional optical properties, including the penetration depth, refractive index, and extinction coefficient, were thoroughly examined. These results provide valuable insights into the potential optoelectronic applications of the synthesized sample.

Refined linearly anisotropic couple-stress elasticity

A recently developed, refined version of the conventional linear couple-stress theory of isotropic elasticity is extended to include the influence of anisotropic material effects. With this development, the implied refined theory (1) retains ability to determine the spherical part of the couple-stress and (2) is further furnished with constitutive ability to embrace modelling of linearly elastic solids that exhibit inherent polar material anisotropy of advanced levels that reach the class of locally monoclinic materials. This type of anisotropy embraces most of the structural material subclasses met in practice, such as those of general and special orthotropy, as well as the subclass of transverse isotropy. The thus obtained, enhanced version of the refined theory is furnished with ability to also handle structural analysis problems of polar fibrous composites reinforced by families of perfectly flexible fibres or, more generally, polar anisotropic solids possessing one or more material preference directions that do not possess bending resistance. A relevant example application considers and studies in detail the subclass of polar transverse isotropy caused by the presence of a single family of perfectly flexible fibres. By developing the relevant constitutive equation, and explicitly presenting it in a suitable matrix rather than indicial notation form, that application also exemplifies the way that the spherical part of the couple-stress is determined when the fibres are straight. It further enables this communication to initiate a discussion of further important issues stemming from (1) the positive definiteness of the full, polar form of the relevant strain energy function and (2) the lack of ellipticity of the final form attained by the governing differential equations.

Review of the Basic Elastic Mechanical Properties and Their Realignment to Establish Ductile Versus Brittle Failure Behaviors

AbstractThe underlying formalism of isotropic elasticity theory is shown to benefit from a review and re-examination of its structure thereby yielding a re-alignment of the basic moduli type properties. When this is accomplished the pathway to understanding ductile versus brittle failure behaviors becomes much more accessible. The ductile/brittle transition in uni-axial tension then admits a simple and direct specification in terms of the two elastic moduli 2 μ and k. The consequences of these results are discussed in the context of general failure theory.

Dislocations interacting with a pore in an elastically anisotropic single crystal nickel-base superalloy during hot isostatic pressing

The formation of pores in CMSX-4 nickel based superalloys is detrimental to the service life of the material. A way to avoid the problem is to treat the superalloys under Hot Isostatic Pressing (HIP), which enables a large volume fraction of pores to be annihilated. This paper aims to understand the contribution of plastic activity related to the gliding of dislocations on the pore annihilation. Simulations based on a phase-field model of dislocation are performed and make it possible to consider the strong anisotropy of the CMSX-4 under HIP conditions in conjunction to the strong elastic heterogeneity introduced by the pore. For pores with a radius of few micrometers, it is shown that edge parts of dislocation lines that present an extra half atomic plane oriented towards the pore are stacked above and under it in the direction which is perpendicular to their slip-planes, causing an increase of the number of dislocation along the four octahedral directions of the FCC single crystal which intersect the pore center. Results are streamlined within the isotropic elastic theory of dislocations. Effects of elastic anisotropy and dislocation reactions are also investigated in order to specify what would be the dislocation configuration around a pore in CMSX-4 under HIP conditions. Notably, the elastic anisotropy is shown to significantly modify the arrangement of dislocations close to the pore equator. Simulations also allow for the characterization of pore/dislocation interactions when dislocations are involved in Low Angle Boundaries as experimentally observed.

Some Remarks on the Weakly Nonlinear Theory of Isotropic Elasticity

Some Remarks on the Weakly Nonlinear Theory of Isotropic Elasticity

Analytical Solution of Stress in a Transversely Isotropic Floor Rock Mass under Distributed Loading in an Arbitrary Direction

Rock masses with a distinct structure may present a transversely isotropic character; thus, the stress state in a transversely isotropic elastic half-plane surface is an important way to assess the behavior of the interaction between the distributed loading and the surroundings. Most previous theoretical analyses have considered a loading direction that is either vertical or horizontal, and the stress distribution that results from the effect of different loading directions remains unclear. In this paper, based on the transversely isotropic elastic half-plane surface theory, a stress solution that is applicable to distributed loading in any direction is proposed to further examine the loading effect. The consistency between the analytical solution and numerical simulations showed the effectiveness of the proposal that was introduced. Then, it was utilized to analyze the stress distribution rule by changing the Poisson’s ratio and Young’s modulus of the model. The effects of the formation dip angle on the stress state are also examined. The stress distribution, depending on the physical property parameters and relative angle, is predicted using an analytical solution, and the mechanisms associated with the transversely isotropic elastic half-plane surface subjected to the loading in any direction are clarified. Additionally, extensive analyses regarding this case study, with respect to the mechanical behavior associated with changes in the stress boundary, is available. Hence, the proposed analytical solution can more realistically account for the loading problem in many engineering practices.

Mathematical modeling of the elastic properties of cubic crystals at small scales based on the Toupin\u2013Mindlin anisotropic first strain gradient elasticity

In this work, a mathematical modeling of the elastic properties of cubic crystals with centrosymmetry at small scales by means of the Toupin–Mindlin anisotropic first strain gradient elasticity theory is presented. In this framework, two constitutive tensors are involved, a constitutive tensor of fourth-rank of the elastic constants and a constitutive tensor of sixth-rank of the gradient-elastic constants. First, 3+11 material parameters (3 elastic and 11 gradient-elastic constants), 3 characteristic lengths and 1+6 isotropy conditions are derived. The 11 gradient-elastic constants are given in terms of the 11 gradient-elastic constants in Voigt notation. Second, the numerical values of the obtained quantities are computed for four representative cubic materials, namely aluminum (Al), copper (Cu), iron (Fe) and tungsten (W) using an interatomic potential (MEAM). The positive definiteness of the strain energy density is examined leading to 3 necessary and sufficient conditions for the elastic constants and 7 ones for the gradient-elastic constants in Voigt notation. Moreover, 5 lattice relations as well as 8 generalized Cauchy relations for the gradient-elastic constants are derived. Furthermore, using the normalized Voigt notation, a tensor equivalent matrix representation of the two constitutive tensors is given. A generalization of the Voigt average toward the sixth-rank constitutive tensor of the gradient-elastic constants is given in order to determine isotropic gradient-elastic constants. In addition, Mindlin’s isotropic first strain gradient elasticity theory is also considered offering through comparisons a deeper understanding of the influence of the anisotropy in a crystal as well as the increased complexity of the mathematical modeling.

Incompatible strain gradient elasticity of Mindlin type: screw and edge dislocations

The fundamental problem of dislocations in incompatible isotropic strain gradient elasticity theory of Mindlin type, unsolved for more than half a century, is solved in this work. Incompatible strain gradient elasticity of Mindlin type is the generalization of Mindlin’s compatible strain gradient elasticity including plastic fields providing in this way a proper eigenstrain framework for the study of defects like dislocations. Exact analytical solutions for the displacement fields, elastic distortions, Cauchy stresses, plastic distortions and dislocation densities of screw and edge dislocations are derived. For the numerical analysis of the dislocation fields, elastic constants and gradient elastic constants have been used taken from ab initio DFT calculations. The displacement, elastic distortion, plastic distortion and Cauchy stress fields of screw and edge dislocations are non-singular, finite, and smooth. The dislocation fields of a screw dislocation depend on one characteristic length, whereas the dislocation fields of an edge dislocation depend on up to three characteristic lengths. For a screw dislocation, the dislocation fields obtained in incompatible strain gradient elasticity of Mindlin type agree with the corresponding ones in simplified incompatible strain gradient elasticity. In the case of an edge dislocation, the dislocation fields obtained in incompatible strain gradient elasticity of Mindlin type are depicted more realistic than the corresponding ones in simplified incompatible strain gradient elasticity. Among others, the Cauchy stress of an edge dislocation obtained in incompatible isotropic strain gradient elasticity of Mindlin type looks more physical in the dislocation core region than the Cauchy stress obtained in simplified incompatible strain gradient elasticity and is in good agreement with the stress fields of an edge dislocation computed in atomistic simulations. Moreover, it is shown that the shape of the dislocation core of an edge dislocation has a more realistic asymmetric form due to its inherent asymmetry in incompatible isotropic strain gradient elasticity of Mindlin type than the dislocation core possessing a cylindrical symmetry in simplified incompatible strain gradient elasticity. It is revealed that the considered theory with the incorporation of three characteristic lengths offers a more realistic description of an edge dislocation than the simplified incompatible strain gradient elasticity with only one characteristic length.

Propagation of threading dislocations and effects of Burgers vectors in HVPE-grown GaN bulk crystals on Na-flux-grown GaN substrates

The propagation behavior of threading dislocations (TDs) and the effects of Burgers vectors in hydride vapor phase epitaxy (HVPE) GaN bulk crystals generated on Na-flux-grown GaN and in a commercially available HVPE-grown GaN bulk crystal were investigated. Analyses based on chemical etching and transmission electron microscopy (TEM) revealed a close correlation between the etch pit sizes and the Burgers vector of these TDs. The existence of TDs with the unique Burgers vector b = 1m + 1c was observed for the first time ever using a large-angle convergent-beam electron diffraction technique and plan-view bright-field scanning TEM. Multi-photon excitation photoluminescence microscopy observations showed that TDs with b = 1c had a meandering morphology in contrast to the linear morphology of TDs with b = 1a, 1a + 1c, or 1m + 1c in both types of HVPE-grown GaN crystals. The inclinations of TDs with b = 1a and 1a + 1c in HVPE-grown GaN on Na-flux-grown GaN were greatly affected by large symmetrical hexagonal hillocks. The TDs with b = 1a were inclined in the slope directions of the hillock planes, while those with b = 1a + 1c were inclined in the a directions parallel or antiparallel to the a component in their Burgers vector. These inclinations were readily explained by the isotropic elastic theory of an individual TD. The hillocks were produced around pairs of parallel mixed TDs for which the a components were opposite to one another. This phenomenon was attributed to inclusions at the interface between the Na-flux-grown GaN substrate and the HVPE-grown layer. The origins of other TDs including unusual ones having b = 1m + 1c are also discussed herein.

Shock Loading of Heteromodular Elastic Materials under Plane-Strain Condition

The paper discusses the results of mathematical modeling the two-dimensional nonlinear dynamics of heteromodular elastic materials. The resistance of these materials under tension and compression is various. The deformation properties of the heteromodular medium are described within the framework of the isotropic elasticity theory with stress-dependent elastic moduli. In the plane strain case, it is shown that only two types of the nonlinear deformation waves can appear in the heteromodular elastic materials: a plane-polarized quasi-longitudinal wave and a plane-polarized quasi-transverse wave. Basing on obtained properties of the plane shock waves, two plane self-similar boundary value problems are formulated and solved.

Implementation of double scalar elastic damage constitutive model in UMAT interface

This paper aims to simulate the isotropic elastic damage theory of Liu Jun (2012) using the self-programmed UMAT subroutine in the interface of ABAQUS. Liu Jun (2012)'s method based on the mechanic theory can not be used interactively with the currently commonly used finite element software ABAQUS. The advantage of this method in the paper is that it can interact with ABAQUS and provide a constitutive program framework that can be modified according to user need. The model retains the two scalar damage variables and the corresponding two energy dissipation mechanisms and damage criteria for considering the tensile and compressive asymmetry of concrete. Taking C45 concrete as an example, the relevant damage evolution parameters of its tensile and compressive constitutive model are given. The study demonstrates that the uniaxial tensile stress calculated by the subroutine is almost the same as the Chinese Concrete Design Specification (GB50010) before the peak stress, but ends soon after the peak stress. The stress-strain curve of uniaxial compression calculated by the subroutine is in good agreement with the peak stress in Chinese Concrete Design Specification (GB50010), but there is a certain deviation in the descending stage. In addition, this paper uses the newly compiled subroutine to simulate the shear bearing capacity of the shear key in a new structural system, namely the open-web sandwich slab. The results show that the damage constitutive subroutine has certain reliability.

Residual Stress Monitoring for ITER Diagnostic Windows

This article presents the measurement of residual stress distribution in ITER (“The Way” in Latin) diagnostics windows using a circular polariscope. The method being used is based on the measurement of the change in the state of polarization of a circularly polarized light after transmission/reflection from the stressed window under test. The theory based on Mueller calculus for the experimental setup has been presented. The measurement technique is validated for transmission mode by comparing the stress values measured for borosilicate glass (BK7) sample disks under diametrical compression with the theoretical values obtained using linear isotropic theory of elasticity. The method is, then, applied to an undamaged and a damaged ITER trial window, both in transmission and reflection modes, to observe the stress patterns inside the glass material and on the glass to metal seal. The major observations are that the undamaged window shows higher stress in majority of the cross section with lower stress values at localized regions. The glass near the seal shows high values of stress all along the circumference in case of the undamaged window. The damaged window that has got detached from the metal seal shows much lower stress values in majority of the cross section of the window with higher stress values in some regions. The region where the seal seems to have got detached from the glass shows lower values of stress at the seal location, and also the stress values in the glass near those locations are low. In conclusion, by observing the stress patterns inside the window, this technique helps determine the health of vacuum seal, and hence can be used in taking preventive measures.

Stress Concentration Near Stiff Cylindrical Inclusions under Anti-Plane Shear Loading

Analytical solutions for the problems of stress concentration near cylindrical inclusion with circular or elliptical cross section under the anti-plane shear loading are presented. The solutions are obtained in the framework of the isotropic strain gradient elasticity theory with the assumption of high stiffness of the inclusions as compared to the matrix, which corresponds to the typical properties of the fiber-reinforced composite materials. It is shown that near the thin fibers, diameter of which is comparable to the characteristic size of the matrix microstructure, the stress concentration can decrease in comparison with conventional estimates known in the theory of elasticity. For circular cylindrical inclusions, the closed-form solutions are obtained for composites with low volume fraction of inclusions and can be used for the strength prediction of composites under the longitudinal shear and for the identification of additional parameters of the strain gradient theory of elasticity.

The reformulated micro-beam models by incorporating the general strain gradient elasticity theory (GSGET)

Two independent material parameters (Lamé constants) are incorporated in classical isotropic elasticity theory, while how many independent material parameters are incorporated in the higher-order isotropic elasticity theory? It is answered by Zhou's general strain gradient elasticity theory (GSGET), which firstly proved that only three material parameters are independent. In this paper, the static and dynamic analyses of micro-beam models are proposed based on the GSGET, which can capture the size effects of micro-structures. Then the variational formulations of the micro-beams including Bernoulli–Euler and Timoshenko beams are deduced by taking advantage of Hamilton's principle. Governing equations, initial conditions, and all possible boundary conditions are derived subsequently. What's more, the static bending and free vibration problems for a simply supported beam have been analytically solved, and the effects of normalized maximum deflection, stiffness, and natural frequency are studied by comparing the present results with the classical and couple stress results respectively. The results indicate that the GSGET should be used to characterize the size effects of mechanical properties for micro-structures.

Combining the method of boundary states and the Lindstedt–Poincaré method in geometrically nonlinear elastostatics

The study considers a statement of geometrically non-linear problems of isotropic theory of elasticity for a simply connected bounded body. The Lindstedt–Poincaré method reduces the problem to a weakly non-linear variation containing a small parameter. The solution is then effected at each step of the perturbations method through the means of the method of the boundary states with perturbations (MBSP). Each iteration is accompanied by a generation of fictitious volume forces of a polynomial nature. The study proposes a strict particular solution and a solution of an elasticity problem for iteration. The approach is also applicable to thermoelasticity problems. The method is tested on classical axially symmetrical bodies, i.e., cylinders subjected to uniaxial tensioning or twisting. The study confirms the applicability of flat section hypotheses to linear engineering calculations. Nonlinear distortions mostly happen at the first iteration, followed by a weak correction. We also provide a calculation for and an illustration of a stress-strain state (SSS) of a semicylinder under unbalanced axial loads and a concentration of compression stresses in extended bodies’ middle areas that are remote from load-free surfaces.

Do all screw dislocations cause leakage in GaN-based devices?

Screw dislocations are generally considered to be one of the main causes of GaN-based device leakage, but so far, nearly no reports have focused on the effects of open-core screw dislocations on device leakage currents experimentally. In this paper, we use a conductive atomic force microscope to characterize the electronic properties of threading dislocations (TDs) in the GaN layer. The full-core screw dislocations and mixed dislocations are found to provide conductive paths for device leakage currents. In terms of the contribution to device leakage currents, the edge and open-core screw dislocations have smaller effects than the full-core screw dislocations and mixed dislocations. We use isotropic linear elasticity theory and density functional theory calculations to model the core atomic structures of TDs and calculate the corresponding electronic structures. The results show that screw dislocations with full-core structures are found to introduce both deep and shallow energy states within the energy gap dispersedly, while the open-core screw dislocations and the most edge dislocations introduce only shallow energy states. The calculated electronic structures of each type of dislocation are systematically compared and correlated with experimental observations. Our findings demonstrate that full-core screw dislocations and mixed dislocations in the GaN layer have a far more detrimental impact on device leakage than edge and open-core screw dislocations.

Phase diagram and phonon-induced backscattering in topological insulator nanowires

We present an effective low-energy theory of electron-phonon coupling effects for clean cylindrical topological insulator nanowires. Acoustic phonons are modelled by isotropic elastic continuum theory with stress-free boundary conditions. We take into account the deformation potential coupling between phonons and helical surface Dirac fermions, and also include electron-electron interactions within the bosonization approach. For half-integer values of the magnetic flux $\Phi_B$ along the wire, the low-energy theory admits an exact solution since a topological protection mechanism then rules out phonon-induced $2k_F$-backscattering processes. We determine the zero-temperature phase diagram and identify a regime dominated by superconducting pairing of surface states. As example, we consider the phase diagram of HgTe nanowires. We also determine the phonon-induced electrical resistivity, where we find a quadratic dependence on the flux deviation $\delta\Phi_B$ from the nearest half-integer value.

Thermodynamic analysis of dissociation of periodic dislocation dipoles in isotropic crystals.

In the past, experimentally observed dislocations were often interpreted using an isolated dislocation assumption because the effect of background dislocation density was difficult to evaluate. Contrarily, dislocations caused by atomistic simulations under periodic boundary conditions can be better interpreted because linear elastic theory has been developed to address the effect of periodic dislocation array in the literature. However, this elastic theory has been developed only for perfect dislocations, but not for dissociated dislocations. The periodic boundary conditions may significantly change the dissociation energy of dislocations and stacking fault width, which in turn, change the deformation phenomena observed in simulations. To enable materials scientists to understand the dislocation behavior under the periodic boundary conditions, we use isotropic elastic theory to analyze the thermodynamics of dissociated periodic dislocations with an arbitrary dislocation character angle. Analytical expressions for force, stacking fault width, and energies are presented in the study. Results obtained from the periodic dislocation array were compared with those obtained from isolated dislocations to shed light on the interpretation of experimentally observed and simulated dislocations.

Augmented Lévy–Michell equations for flexural plates

A new sixth order, isotropic elastic and flexural plate theory is presented based on the fourth order equations of Lévy (1877) and Michell (1900). The lack of surface loading in these Lévy–Michell equations is overcome by including the dominant flexural component of Dougall’s (1904) solution for a point load on the surface of a three dimensional isotropic, elastic layer. The augmented plate equations are arranged to fit the form of a consistent sixth order system of isotropic plate equations which include the well known work of Reissner and static equations of Mindlin. Numerical applications to two three dimensional problems with analytical solutions show good comparative results.