Journal of the Mechanical Behaviour of Materials, 2013
The modest aim of this short article is to provide some new results for a screw dislocation in a ... more The modest aim of this short article is to provide some new results for a screw dislocation in a functionally graded material within the theory of gradient elasticity. These results, based on a displacement formulation and the Fourier transform technique, complete earlier findings obtained with the stress function method and extends them to the case of the second strain gradient elasticity. Rigorous and easy-to-use analytical expressions for the displacements, strains, and stresses are obtained, which are free from singularities at the dislocation line.
Journal of the Mechanical Behavior of Materials, 2004
Stress time series from the PLC effect typically exhibit stick-slips of upload and download type.... more Stress time series from the PLC effect typically exhibit stick-slips of upload and download type. These data contain strong short-term correlations of a nonlinear type. We investigate whether there are also long term correlations, i.e. the successive updown patterns are generated by a deterministic mechanism. A statistical test is conducted for the null hypothesis that the sequence of the up-down patterns is totally random. The test is constructed by means of surrogate data, suitably generated to represent the null hypothesis. Linear and nonlinear estimates are used as test statistics, namely autocorrelation, mutual information and Lyapunov exponents, which are found to have proper performance for the test. The test is then applied to three stress 1 time series under different experimental conditions. Rejections are obtained for one of them and not with all statistics. From the overall results we cannot conclude that the underlying mechanism to the PLC effect has long memory.
The relations between mesoscopic plastic strain gradients, Ôgeometrically necessaryÕ dislocations... more The relations between mesoscopic plastic strain gradients, Ôgeometrically necessaryÕ dislocations (GND), and dislocation dynamics are discussed. It is argued that the connection between GND and size effects in crystal plasticity should be established on the basis of dislocation dynamics, taking into account the specific deformation conditions. It is demonstrated that dislocation dynamics based models for size effects lead to different phenomenological forms of gradient plasticity ÔlawsÕ proposed in the literature.
A non-singular gradient elasticity solution for the dislocation self-energy is employed and compa... more A non-singular gradient elasticity solution for the dislocation self-energy is employed and compared to previous results for 1/6〈2023〉 partial dislocations fault in wurtzite GaN obtained by empirical potential calculations. Twenty-four previously obtained stable dislocation cores, twelve for each GaN polarity, are considered. It is shown that the gradient elasticity solution can be fitted to the atomistic simulations with good agreement and can be employed to trace the energetically favourable cores without the requirement of defining a core radius. As a byproduct of this continuum approach, a dislocation self-energy quantity is derived as a constant material property in terms of a gradient coefficient.
The term ''gradient nanomechanics'' is used here to designate a generalized continuum mechanics f... more The term ''gradient nanomechanics'' is used here to designate a generalized continuum mechanics framework accounting for ''bulk-surface'' interactions in the form of extra gradient terms that enter in the balance laws or the evolution equations of the relevant constitutive variables that govern behavior at the nanoscale. In the case of nanopolycrystals, the grain boundaries may be viewed either as sources/sinks of ''effective'' mass and internal force or as a separate phase, interacting with the bulk phase that it surrounds, and supporting its own fields, balance laws, and constitutive equations reflecting this interaction. In either view, a further common assumption introduced is that the constitutive interaction between bulk and ''interface'' phases enters in the form of higher order gradient terms, independently of the details of the underlying physical mechanisms that bring these terms about. The effectiveness of the approach is shown by considering certain benchmark problems for nanoelasticity, nanoplasticity, and nanodiffusion for which standard continuum mechanics theory fails to model the observed behavior. Its implications to interpreting size-dependent stress-strain curves for nanopolycrystals with varying grain size are also discussed.
By utilizing methods recently developed in the theory of fluid interfaces, we provide a new frame... more By utilizing methods recently developed in the theory of fluid interfaces, we provide a new framework for considering the localization of deformation and illustrate it for the case of hyperelastic materials. The approach overcomes one of the major shortcomings in constitutive equations for solids admitting localization of deformation at finite strains, i.e. their inability to provide physically acceptable solutions to boundary value problems in the post-localization range due to loss of ellipticity of the governing equations. Specifically, strain-induced localized deformation patterns are accounted for by adding a second deformation gradient-dependent term to the expression for the strain energy density. The modified strain energy function leads to equilibrium equations which remain always elliptic. Explicit solutions of these equations can be found for certain classes of deformations. They suggest not only the direction but also the width of the deformation bands providing for the first time a predictive unifying method for the study of pre-and post-localization behavior. The results derived here are a three-dimensional extension of certain one-dimensional findings reported earlier by the second author for the problem of simple shear.
IOP Conference Series: Materials Science and Engineering, 2009
Non-singular solutions for dislocation and disclination fields have recently been obtained by the... more Non-singular solutions for dislocation and disclination fields have recently been obtained by the author and his co-workers by using a robust model of gradient elasticity theory. These solutions, whose form is simple and easy to implement, are obtained by reducing the gradient elasticity problem to a corresponding linear elasticity boundary value problem through the solutions of an inhomogeneous Helmholtz equation where the source term is the classical singular solution. The Laplacian in the Helmholtz equation, involving the extra gradient coefficient, produces a new term in the gradient solution which asymptotically approaches the negative of the classical elasticity solution on the dislocation line. Thus, the singularity is eliminated and an arbitrary estimate of the dislocation core size introduced in classical theory, is not required. These predictions are tested against atomistic calculations and their implications to various dislocation related configurations are discussed. Due to the simple and elegant form of these solutions, it is hoped that they will be useful in discrete dislocation dynamics simulations.
The applicability of fractal analysis to medical sciences has been well-known for almost thirty y... more The applicability of fractal analysis to medical sciences has been well-known for almost thirty years now. However, the sheer volume of data produced by most medical imaging apparatuses, and the extreme inefficiency of most methods of fractal analysis, has presented a roadblock in the width of their application. To remediate that, very fast methods of fractal analysis are required. In a previous work of ours, the Box Merging method was introduced, which implements Box Counting by counting nonempty boxes directly from the coordinates of each element of a set. Its chief drawback is that it needs to sort a large array several times, slowing it down. This paper proposes another method that only requires one sorting. Afterwards, it offers a way to also calculate the lacunarity with negligible changes in algorithm structure and running time. To our knowledge, this marks the first time that the lacunarity can be computed in an acceptable time-frame. Finally, after offering some example app...
The authors would like to make some corrections/revisions to their previous article cited above. ... more The authors would like to make some corrections/revisions to their previous article cited above. These pertain mainly to the way the introduction to the Quantized Fracture Mechanics method, which was initially proposed by the fi rst author, should have appeared in order to be more clear and transparent for the benefi t of the readers. In addition, an error in the form of the parameter α is corrected, thus leading to slightly different results. It turns out that this correction establishes closer connection between the two methods being compared, i.e., gradient theory and quantized fracture mechanics. This Erratum was due much earlier but due to misunderstandings with the previous JMBM publisher its appearance has been delayed. Below, detailed accounts of the revisions needed are indicated in bullets.
The double diffusion model was proposed in the late 1970s and extended in 1980s and 1990s by Aifa... more The double diffusion model was proposed in the late 1970s and extended in 1980s and 1990s by Aifantis and co-workers. It is a continuum model that assumes two local non-equilibrium concentration fields abiding separate mass–momentum equations mediated by a linear mass exchange term. The present work deals with stochastic double diffusion of two competing phase densities to study transport in nanocrystals. We show that the presence of surface inhomogeneities and temporal delay, represented by stochastic dynamics, could have substantial impact at the interface of the two phases. The effect is most pronounced at a critical point (stochastic resonance), where maximum energy transfer occurs, thereby quantifying the impact of surface imperfections in nanocrystal dynamics. The results have been favourably compared with experimental data.
The double diffusivity model proposed earlier by Aifantis and co-workers was applied in this work... more The double diffusivity model proposed earlier by Aifantis and co-workers was applied in this work for modelling the diffusion of metals in sandy aquifers, as well as chloride diffusion in concrete specimens. The theoretical predictions are in very good agreement with the measured concentrations in all cases, showing that the model is capable of dealing with a large variety of double diffusivity problems.
The "double diffusivity" model was proposed in the late 1970s, and reworked in the early 1980s, a... more The "double diffusivity" model was proposed in the late 1970s, and reworked in the early 1980s, as a continuum counterpart to existing discrete models of diffusion corresponding to high diffusivity paths, such as grain boundaries and dislocation lines. It was later rejuvenated in the 1990s to interpret experimental results on diffusion in polycrystalline and nanocrystalline specimens where grain boundaries and triple grain boundary junctions act as high diffusivity paths. Technically, the model pans out as a system of coupled Fick type diffusion equations to represent "regular" and "high" diffusivity paths with "source terms" accounting for the mass exchange between the two paths. The model remit was extended by analogy to describe flow in porous media with double porosity, as well as to model heat conduction in media with two non-equilibrium local temperature baths e.g. ion and electron baths. Uncoupling of the two partial differential equations leads to a higher-ordered diffusion equation, solutions of which could be obtained in terms of clasical diffusion equation solutions. Similar equations could also be derived within an "internal length" gradient (ILG) mechanics formulation applied to diffusion problems, i.e., by introducing nonlocal effects, together with inertia and viscosity, in a mechanics based formulation of diffusion theory. While being remarkably successful in studies related to various aspects of transport in inhomogeneous media with deterministic microstructures and nanostructures, its implications in the presence of stochasticity have not yet been considered. This issue becomes particularly important in the case of diffusion in nanopolycrystals whose deterministic ILG based theoretical calculations predict a relaxation time that is only about one-tenth of the actual experimentally verified timescale. This article provides the "missing link" in this estimation by adding a vital element in the ILG structure, that of stochasticity, that takes into account all boundary layer fluctuations. Our stochastic-ILG diffusion calculation confirms rapprochement between theory and experiment, thereby benchmarking a new generation of gradient-based continuum models that conform closer to real life fluctuating environments.
Journal of the Mechanical Behaviour of Materials, 2013
The modest aim of this short article is to provide some new results for a screw dislocation in a ... more The modest aim of this short article is to provide some new results for a screw dislocation in a functionally graded material within the theory of gradient elasticity. These results, based on a displacement formulation and the Fourier transform technique, complete earlier findings obtained with the stress function method and extends them to the case of the second strain gradient elasticity. Rigorous and easy-to-use analytical expressions for the displacements, strains, and stresses are obtained, which are free from singularities at the dislocation line.
Journal of the Mechanical Behavior of Materials, 2004
Stress time series from the PLC effect typically exhibit stick-slips of upload and download type.... more Stress time series from the PLC effect typically exhibit stick-slips of upload and download type. These data contain strong short-term correlations of a nonlinear type. We investigate whether there are also long term correlations, i.e. the successive updown patterns are generated by a deterministic mechanism. A statistical test is conducted for the null hypothesis that the sequence of the up-down patterns is totally random. The test is constructed by means of surrogate data, suitably generated to represent the null hypothesis. Linear and nonlinear estimates are used as test statistics, namely autocorrelation, mutual information and Lyapunov exponents, which are found to have proper performance for the test. The test is then applied to three stress 1 time series under different experimental conditions. Rejections are obtained for one of them and not with all statistics. From the overall results we cannot conclude that the underlying mechanism to the PLC effect has long memory.
The relations between mesoscopic plastic strain gradients, Ôgeometrically necessaryÕ dislocations... more The relations between mesoscopic plastic strain gradients, Ôgeometrically necessaryÕ dislocations (GND), and dislocation dynamics are discussed. It is argued that the connection between GND and size effects in crystal plasticity should be established on the basis of dislocation dynamics, taking into account the specific deformation conditions. It is demonstrated that dislocation dynamics based models for size effects lead to different phenomenological forms of gradient plasticity ÔlawsÕ proposed in the literature.
A non-singular gradient elasticity solution for the dislocation self-energy is employed and compa... more A non-singular gradient elasticity solution for the dislocation self-energy is employed and compared to previous results for 1/6〈2023〉 partial dislocations fault in wurtzite GaN obtained by empirical potential calculations. Twenty-four previously obtained stable dislocation cores, twelve for each GaN polarity, are considered. It is shown that the gradient elasticity solution can be fitted to the atomistic simulations with good agreement and can be employed to trace the energetically favourable cores without the requirement of defining a core radius. As a byproduct of this continuum approach, a dislocation self-energy quantity is derived as a constant material property in terms of a gradient coefficient.
The term ''gradient nanomechanics'' is used here to designate a generalized continuum mechanics f... more The term ''gradient nanomechanics'' is used here to designate a generalized continuum mechanics framework accounting for ''bulk-surface'' interactions in the form of extra gradient terms that enter in the balance laws or the evolution equations of the relevant constitutive variables that govern behavior at the nanoscale. In the case of nanopolycrystals, the grain boundaries may be viewed either as sources/sinks of ''effective'' mass and internal force or as a separate phase, interacting with the bulk phase that it surrounds, and supporting its own fields, balance laws, and constitutive equations reflecting this interaction. In either view, a further common assumption introduced is that the constitutive interaction between bulk and ''interface'' phases enters in the form of higher order gradient terms, independently of the details of the underlying physical mechanisms that bring these terms about. The effectiveness of the approach is shown by considering certain benchmark problems for nanoelasticity, nanoplasticity, and nanodiffusion for which standard continuum mechanics theory fails to model the observed behavior. Its implications to interpreting size-dependent stress-strain curves for nanopolycrystals with varying grain size are also discussed.
By utilizing methods recently developed in the theory of fluid interfaces, we provide a new frame... more By utilizing methods recently developed in the theory of fluid interfaces, we provide a new framework for considering the localization of deformation and illustrate it for the case of hyperelastic materials. The approach overcomes one of the major shortcomings in constitutive equations for solids admitting localization of deformation at finite strains, i.e. their inability to provide physically acceptable solutions to boundary value problems in the post-localization range due to loss of ellipticity of the governing equations. Specifically, strain-induced localized deformation patterns are accounted for by adding a second deformation gradient-dependent term to the expression for the strain energy density. The modified strain energy function leads to equilibrium equations which remain always elliptic. Explicit solutions of these equations can be found for certain classes of deformations. They suggest not only the direction but also the width of the deformation bands providing for the first time a predictive unifying method for the study of pre-and post-localization behavior. The results derived here are a three-dimensional extension of certain one-dimensional findings reported earlier by the second author for the problem of simple shear.
IOP Conference Series: Materials Science and Engineering, 2009
Non-singular solutions for dislocation and disclination fields have recently been obtained by the... more Non-singular solutions for dislocation and disclination fields have recently been obtained by the author and his co-workers by using a robust model of gradient elasticity theory. These solutions, whose form is simple and easy to implement, are obtained by reducing the gradient elasticity problem to a corresponding linear elasticity boundary value problem through the solutions of an inhomogeneous Helmholtz equation where the source term is the classical singular solution. The Laplacian in the Helmholtz equation, involving the extra gradient coefficient, produces a new term in the gradient solution which asymptotically approaches the negative of the classical elasticity solution on the dislocation line. Thus, the singularity is eliminated and an arbitrary estimate of the dislocation core size introduced in classical theory, is not required. These predictions are tested against atomistic calculations and their implications to various dislocation related configurations are discussed. Due to the simple and elegant form of these solutions, it is hoped that they will be useful in discrete dislocation dynamics simulations.
The applicability of fractal analysis to medical sciences has been well-known for almost thirty y... more The applicability of fractal analysis to medical sciences has been well-known for almost thirty years now. However, the sheer volume of data produced by most medical imaging apparatuses, and the extreme inefficiency of most methods of fractal analysis, has presented a roadblock in the width of their application. To remediate that, very fast methods of fractal analysis are required. In a previous work of ours, the Box Merging method was introduced, which implements Box Counting by counting nonempty boxes directly from the coordinates of each element of a set. Its chief drawback is that it needs to sort a large array several times, slowing it down. This paper proposes another method that only requires one sorting. Afterwards, it offers a way to also calculate the lacunarity with negligible changes in algorithm structure and running time. To our knowledge, this marks the first time that the lacunarity can be computed in an acceptable time-frame. Finally, after offering some example app...
The authors would like to make some corrections/revisions to their previous article cited above. ... more The authors would like to make some corrections/revisions to their previous article cited above. These pertain mainly to the way the introduction to the Quantized Fracture Mechanics method, which was initially proposed by the fi rst author, should have appeared in order to be more clear and transparent for the benefi t of the readers. In addition, an error in the form of the parameter α is corrected, thus leading to slightly different results. It turns out that this correction establishes closer connection between the two methods being compared, i.e., gradient theory and quantized fracture mechanics. This Erratum was due much earlier but due to misunderstandings with the previous JMBM publisher its appearance has been delayed. Below, detailed accounts of the revisions needed are indicated in bullets.
The double diffusion model was proposed in the late 1970s and extended in 1980s and 1990s by Aifa... more The double diffusion model was proposed in the late 1970s and extended in 1980s and 1990s by Aifantis and co-workers. It is a continuum model that assumes two local non-equilibrium concentration fields abiding separate mass–momentum equations mediated by a linear mass exchange term. The present work deals with stochastic double diffusion of two competing phase densities to study transport in nanocrystals. We show that the presence of surface inhomogeneities and temporal delay, represented by stochastic dynamics, could have substantial impact at the interface of the two phases. The effect is most pronounced at a critical point (stochastic resonance), where maximum energy transfer occurs, thereby quantifying the impact of surface imperfections in nanocrystal dynamics. The results have been favourably compared with experimental data.
The double diffusivity model proposed earlier by Aifantis and co-workers was applied in this work... more The double diffusivity model proposed earlier by Aifantis and co-workers was applied in this work for modelling the diffusion of metals in sandy aquifers, as well as chloride diffusion in concrete specimens. The theoretical predictions are in very good agreement with the measured concentrations in all cases, showing that the model is capable of dealing with a large variety of double diffusivity problems.
The "double diffusivity" model was proposed in the late 1970s, and reworked in the early 1980s, a... more The "double diffusivity" model was proposed in the late 1970s, and reworked in the early 1980s, as a continuum counterpart to existing discrete models of diffusion corresponding to high diffusivity paths, such as grain boundaries and dislocation lines. It was later rejuvenated in the 1990s to interpret experimental results on diffusion in polycrystalline and nanocrystalline specimens where grain boundaries and triple grain boundary junctions act as high diffusivity paths. Technically, the model pans out as a system of coupled Fick type diffusion equations to represent "regular" and "high" diffusivity paths with "source terms" accounting for the mass exchange between the two paths. The model remit was extended by analogy to describe flow in porous media with double porosity, as well as to model heat conduction in media with two non-equilibrium local temperature baths e.g. ion and electron baths. Uncoupling of the two partial differential equations leads to a higher-ordered diffusion equation, solutions of which could be obtained in terms of clasical diffusion equation solutions. Similar equations could also be derived within an "internal length" gradient (ILG) mechanics formulation applied to diffusion problems, i.e., by introducing nonlocal effects, together with inertia and viscosity, in a mechanics based formulation of diffusion theory. While being remarkably successful in studies related to various aspects of transport in inhomogeneous media with deterministic microstructures and nanostructures, its implications in the presence of stochasticity have not yet been considered. This issue becomes particularly important in the case of diffusion in nanopolycrystals whose deterministic ILG based theoretical calculations predict a relaxation time that is only about one-tenth of the actual experimentally verified timescale. This article provides the "missing link" in this estimation by adding a vital element in the ILG structure, that of stochasticity, that takes into account all boundary layer fluctuations. Our stochastic-ILG diffusion calculation confirms rapprochement between theory and experiment, thereby benchmarking a new generation of gradient-based continuum models that conform closer to real life fluctuating environments.
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