Micro-inertia Effects Research Articles

High Strain Rate Hardening of Metallic Cellular Metamaterials

Strain rate hardening caused by the changed deformation mode is a fascinating phenomenon in cellular metamaterials where the material’s stiffness and energy absorption capabilities increase as the strain rate increases. This unique behaviour is attributed to a combination of micro-inertia effects, base material’s strain rate hardening and inertia effects. At high strain rates, the metamaterial’s inertia influences its deformation response, which changes to shock mode. This work briefly presents the geometry and fabrication of different metallic metamaterials. Then, it evaluates their mechanical response at different strain rates, ranging from quasi-static to intermediate dynamic and shock, determined by experimental and computational investigation. The three deformation modes can be separated into two critical loading velocities, unique for each metamaterial, which are also presented and compared in this work for various metamaterials. The investigations show that the deformation mode change in metallic metamaterials depends on their porosity. The critical velocities separating the deformation modes decrease with increasing porosity, i.e., decreased density of the metamaterial results in reduced critical loading velocities. The shock deformation mode in cellular metamaterials is thus attainable at much lower loading velocities than in homogeneous (nonporous) materials.

Rayleigh wave propagation in centrosymmetric materials with micro-stiffness, flexoelectric and micro-inertia effects

A theoretical investigation of Rayleigh waves propagation in polarized media has been carried out using a reformulated flexoelectric theory for isotropic dielectrics with micro-inertia effect. Within this non-classical theory, the internal energy density is the functional of the strain tensor, dilatation gradient, deviatoric part of stretch gradient and rotation gradient tensors, polarization vector, and polarization gradient. The obtained system of governing equations additionally contains three material length-scale parameters to account the micro-stiffness effect, one material constant to capture the micro-inertia effect, two flexoelectric constants to describe the flexoelectric effect and three length scale parameters related to the polarization gradient. To solve the coupled governing equations, the method of Lamé-type potentials for mechanical displacement and electric polarization vectors is used. The influences of various factors such as micro-stiffness, flexoelectricity, electric quadrupoles and micro-inertia effects on the phase velocity of the Rayleigh waves in a homogeneous isotropic half-space are studied. It is found that above effects become significant with the increase of the wavenumber. This study can be important for the investigation of high frequency surface acoustic waves in dielectric materials.

Size-dependent and microinertia effects on statics and dynamics based on the modified couple stress theory with arbitrary boundary conditions

Size-dependent and microinertia effects on statics and dynamics based on the modified couple stress theory with arbitrary boundary conditions

Micropolar flow and heat transfer within a permeable channel using the successive linearization method

Abstract This research investigates the flow of micropolar fluid and heat transfer through a permeable channel using the successive linearization method (SLM). The study considers parameters such as coupling, spin-gradient viscosity, and micro-inertia density. The partial differential equations involved are transformed into a system of ordinary differential equations using similarity variables. The resulting nonlinear equations are solved using the SLM technique, and their accuracy and computational efficiency are validated through comparative analysis with previous results. The study shows that increasing the parameters of coupling and spin-gradient viscosity has a positive impact on fluid flow, microrotation, heat transfer, and mass transport, as demonstrated by the increased dimensionless profiles. Conversely, an increase in the micro-inertia density parameter leads to a reduction in these profiles. This decrease can be attributed to the increase in the micro-inertia effect of fluid flow and heat transfer, resulting in a decrease in convection and a change in the flow pattern in the channel. Additionally, higher Reynolds numbers are associated with decreases in velocity, microrotation, temperature, and concentration distribution. This implies a reduction in fluid flow intensity, weaker heat transfer, and decreased mass transport. However, an increased Peclet number results in increased fluid temperature and concentration profiles, indicating enhanced thermal convection and mass transport. These findings have significant implications for applications involving micropolar fluids, such as lubrication systems, blood flow, microchannels, and filtration systems.

Plane wave reflection in micro-structural piezomagnetic-flexomagnetic solid with impedance boundary conditions

The second-order gradient theory is reliable to predict the behavior of mechanical phenomena such as wave scattering, vibration, and buckling in nano/micro-structures. This theory incorporates size-dependent effects including strain-gradient elasticity and micro-inertia, crucial for constitutive modeling of piezomagnetic-flexomagnetic crystals. These small-scale size effects can have significant impact on the sensing mechanism of nanoscale transducers used in structural health monitoring (SHM) and nondestructive testing (NDT). In this article, we address the scattering of a plane wave (P-type longitudinal wave) in a piezomagnetic-flexomagnetic solid halfspace after it reflects from an open surface. The secular equation of quasi plane ( qP ) wave and multiple wave modes generated after reflection have been obtained analytically using non-classical governing equations. Impedance boundary conditions have been considered and conventional stress-free boundary conditions have been obtained as its case. Unlike a conventional piezomagnetic medium, five different wave modes have been traced after reflection (two different surface wave modes are present due to the gradient elasticity and micro-inertia effect). Closed expressions of amplitude ratios and energy flux ratios of reflected waves have been formulated from the system of linear equations. By taking a suitable numerical example, amplitude ratios and energy flux ratios for reflected waves have been plotted with respect to incidence angle. Detailed discussions have been made to understand the influence of impedance boundary and flexomagnetic effect on those ratios. The numerical results have been validated by energy conservation law. This investigation may provide necessary guidelines to understand the plane wave characteristics in micro-structural smart solid structures.

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.

Strain-gradient solution to elastodynamic scattering from a cylindrical inhomogeneity

In this paper, Mindlin’s first strain-gradient theory is adopted for the first time to come up with a solution to the problem of the elastic scattering of an anti-plane time-harmonic wave incident upon an isolated circular cylindrical inhomogeneity embedded in an infinite body. The effect of micro-inertia is also captured in this analysis, which alongside the strain-gradient effect enhances the adopted theory to account for different aspects of the size effect on the elastodynamic fields developed in the medium. The general case of an elastic inhomogeneity and two special cases of a rigid immovable obstacle and a cavity are addressed and the analytical expressions for the corresponding displacement fields are obtained. These solutions reveal the existence of some kind of decaying standing waves in the medium, which cannot be predicted by the classical elastodynamic theory. Moreover, the analytical expressions for the differential and total scattering cross-sections of the cylindrical inhomogeneity are determined. The results demonstrate that, if the cross-sectional radius of the inhomogeneity or the wave-length of the incident wave is comparable with the characteristic lengths of the medium, then the size effect will be significant and the discrepancies between the solutions obtained by the adopted theory and their counterparts in the classical elastodynamic theory will be pronounced.

On Flamant–Boussinesq problem with dynamical flexoelectric effect and micro-inertia effect in dielectrics subjected to dynamical wave loading

This work investigates the Flamant–Boussinesq problem for a half-space made of a homogeneous and isotropic dielectric material. The dynamical flexoelectric effect and the dynamical flexocoupling between displacement and polarization, due to mechanical and electrical states, are taken in consideration. The mechanical loading is taken as a wave of a decaying behavior in time at the surface of a half-space, while the electric potential is considered in an open circuit with no charge on the terminals. The first strain gradient theory of elasticity is used as a mathematical frame in the problem formulation. The equation of motion for the representative volume element additionally accounts for the micro-inertia effect because of the intrinsic structure of the dielectrics at the nanoscale. The governing equations and the boundary conditions for homogeneous, isotropic dielectric material are presented with reference to previous work, using a variational technique for internal energies and external forces. An analytical harmonic wave solution is obtained for the problem under consideration, involving different coupling parameters arising from the mechanical and electrical loadings. The results are analyzed and discussed. The solutions for the quantities of practical interest are represented graphically with different choices of material parameters and flexocoupling parameters. The solution is finite everywhere. The existing damping phenomenon arises, not only from the various physical parameters in the governing field equations as shown in the figures, but also through the boundary conditions.

Torsional vibration of a flexoelectric nanotube with micro-inertia effect

The modified strain gradient theory (MSGT) with flexoelectric and micro-inertia effects is developed to study the free vibration of tube made of elastic centrosymmetric materials. Within this theory, the internal energy density is the function of the strain tensor, deviatoric part of stretch gradient and rotation gradient tensors, dilatation gradient, polarization vector, and electric quadrupoles. The linear coupled governing equations and boundary conditions are derived from the Hamilton variation principle. The torsional vibration problem is analytically solved for non-piezoelectric nanotube with fixed edges. The effect of the nanotube geometry, flexoelectric coefficient, micro-inertia, and micro-stiffness parameters on the natural frequency is studied.

Nonlinear finite element free and forced vibrations of cellular plates having lattice-type metamaterial cores: a strain gradient plate model approach

The main objective of this paper is to develop a theoretically and numerically reliable and efficient methodology based on combining a finite element method and a strain gradient shear deformation plate model accounting for the nonlinear free and forced vibrations of cellular plates having lattice-type metamaterial cores. The proposed model provides a modelling framework computationally more efficient than framework of the conventional three-dimensional elasticity for the simulation of the underlying nonlinear dynamics of cellular plates with microarchitectures. The corresponding governing equations follow Mindlin’s strain gradient elasticity theory including the micro-inertia effect applied to the first-order shear deformation plate theory along with the nonlinear von Kármán kinematics. Standard and higher-order computational homogenization methods determine the classical and strain gradient material constants, respectively. A higher-order C1-continuous 6-node finite element is adopted for the discretization in the spatial domain, and an arc-length continuation technique along with time-periodic discretization is implemented to solve the resulting nonlinear time-dependent problem. Through a set of comparative studies with 3D full-field models as reliable validation references for triangularly prismatic example cores, the accuracy and efficacy of the proposed dimension reduction methodology are demonstrated for a diverse range of problem parameters for analyzing the large-amplitude dynamic structural response. The results show that the proposed strain gradient plate model can accurately capture the microarchitecture effects of cellular plates with low computational costs, which signifies a relevant contribution particularly to the computational analysis of nonlinear dynamics.

QSH WAVE PROPAGATION STUDY IN INFINITE FLEXOELECTRIC PIEZOELECTRIC MEDIUM

The quasi-SH (QSH) wave propagating through a flexoelectric piezoelectric infinite space is investigated in this paper. With the consideration of the microstructure effects (i.e., the flexoelectric effects, strain gradient effects, and micro-inertial effects), the frequency dispersive equations of the QSH wave propagation in infinite flexoelectric piezoelectric materials are derived. The impacts of the flexoelectric coefficient, static characteristic length, and inertial characteristic length of the microstructure upon the dispersion curves of the QSH wave are discussed numerically. We obtained that these microstructure effects have distinct impacts upon the QSH wave velocity. In particular, the micro-inertia effects have different impacts on the dispersion characteristics of the QSH wave. The results can provide very promising theoretical support for future applications of flexoelectric composites.

Self-consistent homogenization-based parametrically upscaled continuum damage mechanics model for composites subjected to high strain-rate loading

This paper develops a Parametrically Upscaled Continuum Damage Mechanics (PUCDM) model for carbon fiber/epoxy matrix composites subjected to high strain-rate loading. The PUCDM model for predicting high strain-rate induced deformation and damage evolution at the macroscopic scale, explicitly incorporates microstructural morphology and micro-inertia in its constitutive coefficients. It is developed using self-consistent homogenization that uses a concurrent multiscale model to account for the effect of micro-inertia and stress wave interaction with the microstructure. The concurrent model embeds a statistically equivalent representative volume element (SERVE) of the microstructure in an exterior domain, whose constitutive and damage response are described by the PUCDM model. Micromechanical modeling for the composite microstructure involves a unified cohesive zone enhanced phase-field damage model for fiber-matrix interface debonding, fiber breakage, and matrix cracking. The PUCDM model parameters for the upscaled domain are determined through micro-macro-scale energy equivalence, along with traction reciprocity and displacement continuity at the SERVE-exterior domain interface that are enforced using a Lagrange multiplier-based approach. Simulations using the concurrent model analyze stress wave propagation and damage evolution in composite microstructures with three different fiber volume fractions at multiple strain rates in the range of 102 − 105 s−1. The fiber volume fraction and spatial distributions affect the overall stiffness and damage evolution in the PUCDM model. Results of the analysis demonstrate the need for representation of micro-inertia, strain rates and microstructure morphology in the PUCDM-based stiffness and damage model parameters for high strain-rates above [Formula: see text].

Investigation of penetration behavior of combined geometry shells at quasi-static and intermediate strain rates: An experimental and numerical study

In this study, the penetration/perforation behavior of a core material with previously determined static and dynamic crushing characteristics was investigated both experimentally and numerically. Penetration/perforation problems occur due to shrapnel effect when sandwich structures containing energy-absorbing core materials by crushing are exposed to blast loads. The penetration behavior of combined geometry shells consisting of a hemispherical cap and a cylindrical segment was investigated experimentally using blunt, conical and hemispherical penetrator tips. The quasi-static penetration tests were performed in a universal test machine, and the intermediate strain rate penetration tests were performed in a drop weight test device. The numerical models of penetration tests were implemented in LS-DYNA at the test strain rates as well as at the higher strain rates. Results showed that different penetrator geometries induced damage forms of symmetrical tearing, petaling, plugging and inversely formed hemispherical domed cone. The increase in the thickness of core geometry resulted in a decent increase in force–displacement curves, as average of force levels increased around 140%, 200% and 220% for blunt, conical and hemispherical tip penetrators, respectively. Numerical results indicated very good correlation with experimental work and enabled to investigate effect of strain rate and micro-inertia over numerical models at elevated penetrator velocities. Penetration behavior was found to be affected from micro-inertia effects up to a threshold displacement of 4 mm for thicker and 5 mm for thinner core units and strain rate effects were found to be dominant beyond that point.

Acoustic surface wave dispersion with strain gradient elasticity and micro-inertia effect in lossy polymer-coated piezoelectric structure

This research paper utilizes second-order gradient electroelasticity theory to develop a theoretical model of Love-type surface wave transference considering strain and inertia gradient effects in a Kelvin–Voigt type viscoelastic lossy polymer-coated piezoelectric substrate. Non-classical constitutive equations derived from variational principles are used to obtain electromechanically coupled field equations. Using proper analytical techniques and non-classical boundary conditions, dispersion relations for electrically open and short cases are derived. Numerical example has been provided and graphs have been plotted to visualize the dependency of velocity and attenuation traits of Love-type waves on considered strain gradient, inertia gradient, and other relevant parameters like viscosity and layer width. It has been observed that the phase velocity of the fundamental mode of surface wave is highly affected by strain gradient elasticity and micro-inertia and high gradient elasticity helps to achieve high phase velocity. The investigation can provide proper guidelines to optimize biological or chemical surface acoustic wave (SAW) sensors with nanocoating. Highlights Nanoscale strain gradient elasticity and micro-inertia effect are considered in the shear wave propagation model. Non-conductive Kelvin–Voigt model-based viscoelasticity is assumed for coating polymer. Attenuation is also characterized along with phase velocity on a logarithmic scale. Dispersion characteristics are investigated for electrically open and short boundary types.

An electro-mechanical dynamic model for flexoelectric energy harvesters

Flexoelectricity is a universal electro-mechanical coupling effect that occurs in dielectrics of all symmetric groups and becomes dominant at the micro- and nano-scales. It plays an important role in evaluating micro-electro-mechanical systems (MEMS) such as energy harvesters which convert vibrational energy to electric energy. At finer length scales, micro-inertia effects significantly contribute to the behavior of flexoelectric materials due to the mechanical dispersion. Hence, to properly characterize the vibrational behavior of MEMS, a reliable theoretical approach is required accounting for all possible phenomena that affect the output of the system such as voltage or power density. In this work, we present a consistent (dynamic) model and associated computational framework for flexoelectric structures to study the characteristics of the vibrational behavior of energy harvesters showing the dominance of the flexoelectric effect at micro- and nano-scales. In this context, we quantify the impact of the micro-inertia length scale and the flexoelectric dynamic parameter on both frequency and time responses of energy harvesters.

Further study on the dynamic loading transmission in cellular solids based on one-dimensional mass-spring model

The characteristics of dynamic loading transmission in cellular solids are studied in the present paper using a one-dimensional (1D) deformation-contact mass-spring model. From the perspective of deformation process, the collisions among adjacent cell walls happen due to the shock-induced localized large deformation and the cellular structure of the material. Two springs representing deformation stress and contact stress, respectively, are introduced to the mass-spring model to represent different deformation and loading transmission mechanisms. Combined with shock-wave model, the contact stress in the densification stage is determined. Both quasi-static and dynamic compression of cellular solids can be described by the 1D mass-spring model, in which the critical impact velocity of the compaction shock is used to distinguish two different compression states. Based on the 1D deformation-contact mass-spring model, several important issues in the dynamic loading transmission in cellular solids, i.e. the deformation-contact process, unloading and reversed loading issue, deformation modes under different loading conditions, stress effectiveness and micro-inertia effect, are examined. The present paper clarifies previous concerns on the mass-spring model and demonstrates its enhanced capability to simulate the dynamic loading transmission in cellular solids.

Love waves propagation in layered waveguide structures including flexomagneticity/flexoelectricity and micro-inertia effects

A gradient-type theory with flexomagnetic/flexoelectric and micro-inertia effects is used to study Love wave propagation in the layered magneto-electro-elastic structures. Two waveguide structures are considered: flexo-piezoelectric layer deposited on the piezomagnetic half-space and flexo-piezomagnetic layer resting on the piezoelectric half-space. The dispersion equations are numerically investigated for the magneto-electrically open and short circuit conditions. It is concluded that the profile of dispersion curves depends on the material composition of the waveguide structure, the guiding layer thickness, and the relative values of flexomagnetic/flexoelectric coefficients and the micro-inertia length-scale parameter. The obtained results are helpful for mathematical modeling of new small-sized acoustic devices.

Effect of pore density on the compressive response of open-cell Al foams

The effects of pore density and matrix on the quasi-static and dynamic compressive behaviours of the high porosity open-cell aluminium foams were examined. The results show that pore density has little effect on the collapse strength of the samples under quasi-static conditions. However, as the strain rate increases, the collapse strength decreases with increasing pore density. The different matrix material also shows distinct strain rate response characteristics, which is related to their respective microstructures. The strain-rate sensitivity is mainly attributed to the micro-inertia effect caused by the larger aspect ratio of the cell edges, which was verified by the more transverse elongation of the dynamic compression samples. Moreover, the energy absorption capacity is also discussed in the present paper.

Waves Propagating in Nano-Layered Phononic Crystals with Flexoelectricity, Microstructure, and Micro-Inertia Effects.

The miniaturization of electronic devices is an important trend in the development of modern microelectronics information technology. However, when the size of the component or the material is reduced to the micro/nano scale, some size-dependent effects have to be taken into account. In this paper, the wave propagation in nano phononic crystals is investigated, which may have a potential application in the development of acoustic wave devices in the nanoscale. Based on the electric Gibbs free energy variational principle for nanosized dielectrics, a theoretical framework describing the size-dependent phenomenon was built, and the governing equation as well as the dispersion relation derived; the flexoelectric effect, microstructure, and micro-inertia effects are taken into consideration. To uncover the influence of these three size-dependent effects on the width and midfrequency of the band gaps of the waves propagating in periodically layered structures, some related numerical examples were shown. Comparing the present results with the results obtained with the classical elastic theory, we find that the coupled effects of flexoelectricity, microstructure, and micro-inertia have a significant or even dominant influence on the waves propagating in phononic crystals in the nanoscale. With increase in the size of the phononic crystal, the size effects gradually disappear and the corresponding dispersion curves approach the dispersion curves obtained with the conventional elastic theory, which verify the results obtained in this paper. Thus, when we study the waves propagating in phononic crystals in the micro/nano scale, the flexoelectric, microstructure, and micro-inertia effects should be considered.

Strain-gradient theory for shear deformation free-form microshells: Governing equations of motion and general boundary conditions

A first-order shear deformation (FOSD) free-form microshell model described in general curvilinear coordinates was developed within the complete framework of Mindlin’s form II linear isotropic strain-gradient theory (SGT), considering both strain-gradient and micro-inertia effects. The high-order governing equations of motion and consistent boundary conditions were simultaneously obtained through a variational formulation based on Hamilton’s principle. The established microshell model contains five strain-gradient material constants and one length parameter resulting from micro-inertia effects in addition to two classical Lamé constants, thereby capturing microstructure-induced size-dependent phenomena in both static and dynamic analyses. The constructed model within the general SGT can be flexibly reduced to those based on the modified strain-gradient, modified couple stress, and simplified strain-gradient theories. Moreover, by enforcing geometric restrictions, the general formulation for free-form microshells can be specialized for doubly curved microshells. Finally, the forced vibration results of a simply supported closed cylindrical microshell confirmed the reliability and accuracy of the proposed model and strain-gradient and high-order inertia effects on the size-dependent dynamic characteristics of microshells. Altogether, the presented mathematical formulations can provide a foundation for the in-depth understanding of microscale shell structures.