Internal Characteristic Length Research Articles

An asymptotic formulation for boundary value problems in a non-local elastic half-space

The primary purpose of this study is to develop an asymptotic formulation for boundary value problems in a non-local elastic half-space. For the sake of simplicity, the non-locality is limited to the vertical direction, which is represented by a one-dimensional exponential kernel, and the problem is formulated within the framework of Eringen’s theory. The proposed asymptotic approach is based on the assumption that the internal characteristic length is significantly smaller than a typical wavelength. This assumption allows for the development of an asymptotic formulation that expresses the considered boundary value problem in terms of local stresses. Additionally, the formulation includes explicit correction terms to the classical boundary conditions, which arise from the non-local effects. As an example application of the derived formulation, the Rayleigh surface waves in a plane strain problem are considered. Finally, numerical results are presented for certain specific values of elastic parameters to illustrate the effects of non-locality on the analyzed system.

Memory Effects in the Magnetoresistance of Two-Component Electron Systems

The theory of the magnetotransport in a two-component electron system with rare macroscopic defects has been developed. In such a system, the classical memory effects in the scattering of electrons by defects and a slow transfer of electrons between the components of the liquid occurring due to the electron–electron scattering play a decisive role. It has been shown that the flow regime depends on the ratio of the sample width to the characteristic internal length, which is determined by the rate of electron transfer between the components. In samples wider than the internal length, the flow of the two-component liquid as a whole is formed within the bulk of the sample and is described by the corresponding Drude formulas taking into account memory effects. In this case, the magnetoresistance is positive at low magnetic fields and negative at high fields. In samples narrower than the characteristic length, the transfers involving a change in the type of electrons do not provide enough time to form a unified liquid. As a result, the flows of different components are independent and described by their own conductivities, taking into account the memory effects, while the magnetoresistance is strictly negative.

Nonlocal elasticity of Klein‐Gordon type with internal length and time scales: Constitutive modelling and dispersion relations

AbstractIn this work, a nonlocal elasticity theory with nonlocality in space and time is presented by considering nonlocal constitutive equations with a dynamical scalar nonlocal kernel. Based on the proposed theory, we consider the isotropic nonlocal elasticity of Klein‐Gordon type including a characteristic internal time scale parameter in addition to the characteristic internal length scale parameter. The dispersion relations for homogeneous isotropic media in the framework of nonlocal elasticity of Klein‐Gordon type are analytically determined. The obtained results reveal the ability of the presented nonlocal elasticity model to predict, for the first time in the framework of nonlocal elasticity, in addition to the acoustic modes (low‐frequency modes), optic modes (high‐frequency modes) as well as frequency band‐gaps. The phase and group velocities for all four modes (acoustic and optic branches of longitudinal and transverse waves) are determined showing that all four modes exhibit normal dispersion with positive group velocity. The presented model allows for physically realistic dispersive wave propagation.

Elastic waves in fluid-saturated porous materials with a couple-stress solid phase

Fluid-saturated porous materials have attracted extensive attention in micro/nanoscale applications owing to their excellent mechanical properties. Couple-stress theory can be used to depict the size effect of micro/nanoelastic materials, which cannot be ignored when the internal characteristic length of materials is comparable with the material size or wavelength. However, it has not been extended to porous materials. In this paper, we investigate the size effect of elastic-wave propagation in micro/nano fluid-saturated porous materials by combining couple-stress theory with Biot theory and introducing the higher order micro-inertia effect, establish the motion equations for elastic waves in fluid-saturated porous materials containing a couple-stress solid phase. The reflection and transmission properties of elastic waves at the interface between an elastic solid and a couple-stress fluid-saturated porous material are numerically analyzed. The results suggest that an increase in characteristic length can lead to an increase in the phase velocity of shear wave and further influence the reflection and transmission of elastic waves. A comparison among the proposed theory, Biot theory and experimental data indicates that the introduction of couple-stress and higher order micro-inertia effects allows for a more accurate calculation of the propagation properties of elastic waves in fluid-saturated porous materials, and it can provide a theoretical basis for applications of elastic waves in micro/nano porous materials, such as nondestructive testing.

Nonlocal antiplane shear interfacial waves

In this paper, non-local shear interfacial waves between two non-local elastic half-spaces in perfect contact along a common boundary are investigated. For the sake of simplicity, an anti-plane problem formulated through the framework of Eringen’s theory is considered. The non-locality is taken only in the transverse coordinate making the analysis of the problem easier than that of non-local problems of 2D. Effective boundary conditions are obtained, with the help of a small internal characteristic length scale, along the interface utilizing an asymptotic approach enabling one to write the non-local stresses through their local counterparts. Antiplane non-local shear interfacial waves are analysed by the derived formulation. Numerical results are presented for certain characteristic values of the problem parameters.

Thermoelastic damping in circular microplate resonators based on fractional dual-phase-lag model and couple stress theory

In the present work, thermoelastic damping in a micro-resonator operating in vacuum as a vibrating size-dependent circular thin plate is studied. To better describe the heat conduction process, a generalized thermoelasticity theory of fractional dual-phase-lag (DPL) incorporating the couple stress theory is put forward to capture the experimentally observed small-scale effect of thermoelastic dissipation. The complex frequency of the micro-plate is obtained through Laplace transform technique. Finally, Lifshitz and Roukes’s complex frequency method is employed to derive an analytical expression for thermoelastic damping. The influences of the parameters including fractional order, phase-lag time, and internal characteristic length on the inverse quality factor Q−1 are analyzed. The obtained results show that the influences of these parameters on thermoelastic damping are pronounced. These results are helpful to the design of micro-/nano-resonators with high-performance.

Nonlocal elasticity of Klein–Gordon type: Fundamentals and wave propagation

A nonlocal elasticity theory with nonlocality in space and time is developed by considering nonlocal constitutive equations with a dynamical scalar nonlocal kernel function. The proposed theory is specified to isotropic nonlocal elasticity of Klein–Gordon type, which is an extension of nonlocal elasticity of Helmholtz type, and it possesses one characteristic internal length scale parameter as well as one characteristic internal time scale parameter in addition to the elastic material parameters. Dynamical nonlocal kernels are analytically derived as retarded Green functions of the Klein–Gordon operator. The dispersion relations of the considered isotropic nonlocal elasticity model of Klein–Gordon type are analytically determined. The obtained results reveal the advantage of the proposed nonlocal elasticity model, that is, its ability to predict for the first time in the framework of nonlocal elasticity in addition to the acoustic modes (low-frequency modes), optic modes (high-frequency modes) as well as frequency band-gaps between the acoustic and optic modes. A qualitative examination of the phase and group velocities for all four modes (acoustic and optic branches of longitudinal and transverse waves) shows that the proposed model allows for physically realistic dispersive wave propagation.

Geotechnical Strain Localization Analysis Based on Micropolar Continuum Theory Considering Evolution of Internal Characteristic Length

Within the framework of second-order cone programming optimized micropolar continuum finite-element method (CosFEM-SOCP), the geotechnical strain localization can be adequately modeled. In most existing literatures, however, the constant internal characteristic length lc has been adopted and less attention has been paid to the evolution of lc. To more accurately predict the strain localization, response, and stability of a geotechnical system, one relationship for evolving lv that relies on the equivalent plastic strain is implemented and investigated. Based on one homogeneous slope example and one rigid strip footing example, it was found that geotechnical stability may not be significantly affected by evolving lv, indicating that constant lc can be simply applied to geotechnical stability analysis. For the rigid strip footing problem, nevertheless, the effects of evolving lv on the pressure–displacement response curves should not be ignored, and the influence range of the shear band predicted by CosFEM-SOCP with evolving lv is generally smaller than that predicted by CosFEM-SOCP with constant lc. Consequently, in order to more accurately predict the pressure–displacement response curves and the failure zone, the evolving lv will be adequately assessed and modeled.

Space-fractional small-strain plasticity model for microbeams including grain size effect

Elastic–plastic bending of microstructure-dependent beams is analyzed, i.e., beams whose external dimensions and characteristic internal length are close. The formulations presented concern space-fractional Euler–Bernoulli (s-FEBB) and Timoshenko (s-FTB) beams, enriched by including the plastic properties of the material. The Huber–Mises–Hencky plasticity criterion and isotropic hardening are applied in the developed models. The incremental iterative numerical procedure is elaborated to solve the elastic–plastic bending problem. The analysis covers not only the scale effect but also the grain size-dependent strengthening described by the general Hall–Petch relation. Based on the experimental results of WC-Co micropillars compression available in the literature, model parameters are identified for specific microstructures. Significant results were found — it was revealed that the length scale could be directly correlated with the grain size. Finally, a numerical study of elastic–plastic bending of a fractional beam demonstrated how the microstructure affects the propagation of plastic hinges.

Band Structure Analysis of SH Wave Propagating in Nanoscale Layered Metamaterial Structures

This study is devoted to the analysis of the band structures of the anti-plane transverse wave (SH wave) in nanoscale layered metamaterial structures. Attention is restricted to normal incidence of waves. The localization factor is introduced to characterize the band structures. The general transfer matrix method based on the nonlocal elastic continuum theory is employed to calculate the localization factor. Based on the analysis of band structures, the influences of random disorder of the internal characteristic length and the external thickness of each sub-layer, the aperiodic arrangements, the location of different material components, the ratio of mass density, the ratio of the transverse wave velocity, the ratio of the internal characteristic length or the external thickness of each sub-layer on the band structures, the cut-off frequency, the peak points and the dense band zones are investigated and discussed in detail, which can provide some new thoughts for the designs and applications of the nanoscale wave devices.

Analysis of stress fields at the crack tip and fracture resistance parameters under conditions of gradient plasticity

This paper presents a numerical analysis based on previously obtained experimental data on crack growth for P2M and 34X steels, aluminum 7050 and titanium alloy Ti-6Al-4V compact samples (CTS) with a one-sided lateral incision, within the framework of the conventional theory of strain gradient plasticity (CMSG), based on the Taylor dislocation model. In this study, the initial points of the curved crack trajectory were considered for two classical states: plane strain and plane stress with normal separation and pure shear. The constitutional equations of material behavior for CMSGP theories were introduced into the finite element computational complex and new fields of stress-strain state parameters for the conditions of strain gradient plasticity were obtained. The results of FE calculations show a significant increase in the magnitude of the true stress fields at the crack tip, taking into account the plastic deformation gradients and the internal characteristic length of the material. The numerical results also show that the singularity in the crack tip region is different for the model of gradient plasticity of deformations and depends on the mode I/II. An important conclusion regarding the numerical results regarding the parameters of the material's fracture resistance is that the new plastic stress intensity coefficients for gradient plasticity differ for plane strain and plane stress, and also show significant sensitivity to the plastic properties of the material and to the scale parameter of the intrinsic material length, which is attractive from the point of view of practical application and further fundamental research.

Internal structure of ultralow-crosslinked microgels: From uniform deswelling to phase separation.

We perform small angle neutron scattering on ultralow-crosslinked microgels and find that while in certain conditions both the particle size and the characteristic internal length scale change in unison, in other instances this is not the case. We show that nonuniform deswelling depends not only on particle size, but also on the particular way the various contributions to the free energy combine to result in a given size. Only when polymer-solvent demixing strongly competes with ionic or electrostatic effects do we observe nonuniform behavior, reflecting internal microphase separation. The results do not appreciably depend on particle number density; even in concentrated suspensions, we find that at relatively low temperature, where demixing is not very strong, the deswelling behavior is uniform, and that only at sufficiently high temperature, where demixing is very strong, does the microgel structure change akin to internal microphase separation.

Geotechnical stability analysis considering strain softening using micro-polar continuum finite element method

Geotechnical stability analyses based on classical continuum may lead to remarkable underestimations on geotechnical safety. To attain better estimations on geotechnical stability, the micro-polar continuum is employed so that its internal characteristic length (4) can be utilized to model the shear band width. Based on two soil slope examples, the role of internal characteristic length in modeling the shear band width of geomaterial is investigated by the second-order cone programming optimized micro-polar continuum finite element method. It is recognized that the underestimation on factor of safety (FOS) calculated from the classical continuum tends to be more pronounced with the increase of 4. When the micro-polar continuum is applied, the shear band dominated by 4 is almost kept unaffected as long as the adopted meshes are fine enough, but it does not generally present a slip surface like in the cases from the classical continuum, indicating that the micro-polar continuum is capable of capturing the non-local geotechnical failure characteristic. Due to the coupling effects of 4 and strain softening, softening behavior of geomaterial tends to be postponed. Additionally, the bearing capacity of a geotechnical system may be significantly underestimated, if the effects of 4 are not modeled or considered in numerical analyses.

Size effects in nanostructured Li-ion battery cathode particles

Abstract Cathode materials for Li-ion batteries exhibit volume expansions on the order of 10% upon maximum lithium insertion. As a result internal stresses are produced and after continuous electrochemical cycling damage accumulates, which contributes to their failure. Battery developers resort to using smaller particle sizes in order to limit damage and some models have been developed to capture the effect of particle size on damage. In this paper, we present a gradient elasticity framework,which couples the mechanical equilibrium equations with Li-ion diffusion and allows the Young’s modulus to be a function of Li-ion concentration. As the constitutive equation involves higher order gradient terms, the conventional finite element method is not suitable, while, the two-way coupling necessitates the need for higher order shape functions. In this study, we employ B-spline functions with the framework of the iso-geometric analysis for the spatial discretization. The effect of the internal characteristic length on the concentration evolution and the hydrostatic stresses is studied. It is observed that the stress amplitude is significantly affected by the internal length, however, using either a constant Young’s modulus or a concentration dependent one yields similar results.

Analytical Solutions for Bending of Nanoscaled Bars Based on Eringen’s Nonlocal Differential Law

The one-dimensional nanoscaled bar is commonly seen in current nanodevices, and it plays a significant role in nanoelectromechanical system (NEMS) and related nanotechnology. Motivated by this, the paper is concerned with the bending and stability of a nanoscaled bar especially a flexible bar subjected to vertical concentrated, vertical linearly distributed, and horizontal concentrated forces simultaneously. The theoretical model is developed by employing Eringen’s nonlocal differential law. Hence, the nonlocal differential constitutions in terms of stress and bending moment at a nanoscale are applied to the classical equilibrium formulation in order to construct the nonlocal constitutive model and then determine the analytical solutions for bending of such a nanoscale bar. Subsequently, the effect of a nonlocal scale on the midpoint deflections and internal forces is shown numerically, and the internal and external characteristic length scales are also examined. It is revealed that the capacity of resisting compression decreases with increasing the nonlocal effect since the internal characteristic scale cannot be neglected by comparing with the external characteristic scale. There exists a threshold value for bending stiffness. Both the midpoint deflection and bending moment vary monotonously when the material bending stiffness exceeds the threshold value, while the mechanical quantities fluctuate when the bending stiffness is less than its threshold value, namely, for a flexible nanoscaled bar. Accordingly, two different kinds of nonlocal predictions and the related disputes are resolved in the context of Eringen’s nonlocal differential law. The work is expected to be useful for the design, application, and optimization of nanoscaled bars sustaining bending with linear vertical and horizontal forces.

Geotechnical localization analysis based on Cosserat continuum theory and second-order cone programming optimized finite element method

Based on the Hellinger-Reissner mixed variational principle, a second-order cone programming optimized finite element method for elastic and elastoplastic analyses of Cosserat continuum (named CosFEM-SOCP) is proposed. To examine the correctness and effectiveness of CosFEM-SOCP, three well-known numerical examples are revisited. In the plane-strain elastic problem of stress concentration around a circular hole, the stress concentration factors calculated by CosFEM-SOCP are in good agreement with the existing solutions, and the correctness of CosFEM-SOCP can be verified. In the analysis of load-displacement response of a flexible strip footing resting on the soil ground, it is found that load-displacement response may be affected by the internal characteristic length lc, and generally a larger lc may result in a wider and deeper slip surface as well as more elements being included in the shear bands. From the plane-strain soil strip with the Drucker-Prager (DP) failure criterion subjected to the prescribed incremental displacement, it can also be observed that the pathologically mesh-dependent problem can be partially or even completely removed by employing CosFEM-SOCP, as long as the mesh is fine enough so that the average element size in the mesh discretization is smaller than lc.

Development and implementation of new triangular finite element based on MGE theory for bi-material analysis

Abstract Compared with the classical continuum theory, the modified gradient elastic theory (MGE theory) better explains the size effect, strain and damage localization of materials. In this paper, we constructed a C1 planar triangular finite element based on the MGE theory. We also derived its finite element format and compiled the corresponding finite element calculation program by the MATLAB software. The program was used to analyze the bi-material shear boundary layer problem with unidirectional infinite length. The results show that the change of the internal characteristic length of the material in the infinite direction (the same direction as the shear stress) has little effect on calculation results of the shear strain, while the change of the internal characteristic length in the finite direction has a certain impact on calculation results of the shear strain, and its impact is even greater on the material with small elastic modulus in the bi-material; when the size of the finite element mesh does not exceed the internal characteristic length of the material, the mesh dependence of calculation results can be eliminated. Compared with the MGE rectangular element, the triangular element constructed in this paper has higher calculation accuracy and better adaptability.

On the nano-structural dependence of nonlocal dynamics and its relationship to the upper limit of nonlocal scale parameter

Nonlocal elasticity theory is one of the most popular theoretical approaches to investigate the intrinsic scale effect of nano-materials/structures. The coupling of an internal characteristic length and a material parameter can be regarded as a nonlocal scale parameter in nano-meters. The range of this non-dimensional scale parameter is from zero up to different values previously. There is no doubt that the zero nonlocal scale parameter corresponds to a situation without any nonlocal effect. However, the determination of a peak value for the scale parameter is still uncertain. In fact, we frequently ask a simple but unresolved question, i.e., how strong is the nonlocal scale effect? This question is equivalent to what the maximum value of the nonlocal scale parameter is, since it was introduced to characterize the scale effect theoretically. Until now, various maximum values have been selected without rigorous verifications. In this paper, the nano-structural dependence of nonlocal dynamical behavior is investigated to present the existence of an upper limit for the scale parameter. Through three typical examples, the size-dependent behavior of nonlocal dynamics for various nano-structures is analyzed. The upper limit of the scale parameter can be determined accordingly. It is shown that an interval for the scale parameter in the illustrative examples can be found on the basis of the nonlocal softening physical mechanism, in which the equivalent stiffness of nano-structures is weakened than that predicted by the classical continuum theory. The present study contributes to a fuzzy zone in nonlocal elasticity where people are puzzled over the question how to select the upper limit of the nonlocal scale parameter. It is not only beneficial to the refinement of the nonlocal theory of elasticity, and also useful for the exploration of similar theories in nano-mechanics.

Crack dynamics and crack tip shielding in a material containing pores analysed by a phase field method

Many naturally occurring materials, such as wood and bone, have intricate porous micro-structures and high stiffness and toughness to density ratios. Here, the influence of pores in a material on crack dynamics in brittle fracture is investigated. A dynamic phase field finite element model is used to study the effects of pores with respect to crack path, crack propagation velocity and energy release rate in a strip specimen geometry with circular pores. Four different ordered pore distributions are considered, as well as randomly distributed pores. The results show that the crack is attracted by the pores; this attraction is stronger when there is more energy available for crack growth. Crack propagation through pores also enables higher crack propagation velocities than are normally seen in strip specimens without pores (i.e. homogeneous material), without a corresponding increase in energy release rate. It is further noticed that as the porosity of an initially solid material increases, the crack tip is increasingly likely to become shielded or arrested, which may be a key to the high relative strength often exhibited by naturally occurring porous materials. We also find that when a pore is of the same size as the characteristic internal length then the pore does not localise damage. Since the characteristic internal length only regularises the damage field and not the strain end kinetic energy distributions, crack dynamics are still affected by small pores.

Semi-analytic solution of Eringen’s two-phase local/nonlocal model for Euler-Bernoulli beam with axial force

Eringen’s two-phase local/nonlocal model is applied to an Euler-Bernoulli nanobeam considering the bending-induced axial force, where the contribution of the axial force to bending moment is calculated on the deformed state. Basic equations for the corresponding one-dimensional beam problem are obtained by degenerating from the three-dimensional nonlocal elastic equations. Semi-analytic solutions are then presented for a clamped-clamped beam subject to a concentrated force and a uniformly distributed load, respectively. Except for the traditional essential boundary conditions and those required to be satisfied by transferring an integral equation to its equivalent differential form, additional boundary conditions are needed and should be chosen with great caution, since numerical results reveal that non-unique solutions might exist for a nonlinear problem if inappropriate boundary conditions are used. The validity of the solutions is examined by plotting both sides of the original integro-differential governing equation of deflection and studying the error between both sides. Besides, an increase in the internal characteristic length would cause an increase in the deflection and axial force of the beam.