Gradient Theory Research Articles

Couple-stress–based gradient theory of poroelasticity

In this research, we present a gradient theory of poroelasticity based on the couple-stress. Within the context of finite deformations and in a thermodynamically consistent manner, the constitutive equations for the porous solid are derived by including the solid vorticity gradient and its power-conjugate counterpart, namely, the couple-stress. Subsequently, a linearized theory for an isotropic porous solid is developed in which two microstructure-dependent constitutive moduli (or equivalently, two material length-scale parameters) are introduced. To investigate the gradient effects on the responses of the material, the problem of wave propagation in fluid-saturated porous solids is formulated and solved based on the proposed theory. For comparison, the wave dispersion and attenuation curves are compared with those obtained from classical theory of poroelasticity.

Wave propagation in periodic nano structures through second strain gradient elasticity

The study of wave propagation in the periodic nano-waveguides is of importance to advance the development of energy and wave controlling systems. In this paper, firstly, the element discretization of nano structures is introduced based on the second strain gradient theory. The displacement field is determined through the utilization of six quintic Hermite polynomial interpolating functions. By applying Hamilton’s principle, the weak form, which incorporates the element matrices, is determined and the global dynamic stiffness matrix of a unit cell is assembled. Then, the wave finite element method, based on the inverse form, direct form and contour integral solution, is used to analyze the two-dimensional free wave propagation properties of complex nano structures by solving the eigenvalue problems. The interpretation of the frequency spectrum is confined to the irreducible first Brillouin zone, encompassing both the dispersion relation and band structure. Furthermore, the slowness surfaces and energy flow vector fields are discussed via the second strain gradient theory. The novel work, which utilizes the second strain gradient theory equipped with the wave finite element method to investigate two-dimensional complex nano structures, has revealed significant potential in elucidating the two-dimensional wave propagation characteristics of the complex nano-waveguides.

Local gradient theory of dielectrics incorporating polarization inertia and flexodynamic effect

A higher-grade theory of non-ferromagnetic thermo-elastic dielectrics which incorporates the local mass displacement, the heat flux gradient, polarization inertia, and flexodynamic effects is developed. The process of local mass displacement is associated with changes in material microstructure. Using the fundamental principles of continuum mechanics, electrodynamics, and non-equilibrium thermodynamics, the gradient-type constitutive equations are derived. Due to accounting for the polarization inertia, the rheological constitutive equation for the polarization vector is obtained. In the balance equation of linear momentum, an additional term with the second time derivative of the polarization vector appears in comparison with the classical theory. This term controls the influence of the dynamic flexoelectric effect on the mechanical motion of dielectric solids. The propagation of a plane harmonic wave is analyzed within the context of the developed theory. It is shown that the theory allows for capturing the experimentally observed phenomenon of high-frequency dispersion of a longitudinal elastic wave. The theory may be useful for modeling coupled processes in nanodielectrics and heterogeneous polarized systems.

Isogeometric analysis of shear-deformable, in-plane functionally graded microshells by Mindlin’s strain gradient theory

Isogeometric analysis of shear-deformable, in-plane functionally graded microshells by Mindlin’s strain gradient theory

Magnetic field and thermal environment effects on sound transmission loss of double-walled cracked piezoelectric cylindrical nanoshells

Based on a non-local strain gradient theory (NSGT), this article analytically studies the sound transmission loss (STL) performance of a special type of double-walled partially cracked piezoelectric nanoshell structures of cylindrical shape undergoing initial electric potential and longitudinal magnetic field. The structure, which includes a specific number of axial and circumferential surface cracks, is struck by a plane wave in a thermal setting. Such cracks have minimal size compared to the nano-shell structure as a whole, minimizing the effects of curvature for this problem. The first-order shear deformation theory (FSDT) is utilized to explain the displacement components, and Hamilton’s concept is applied to obtain the coupled vibroacoustic equations that are used to generate the governing equations of the system for the two crack types, giving us an idea of the resulting structural stiffness in the presence of cracks using coefficients of crack compliance that are derived from the line spring model (LSM). The mechanical and acoustic properties are then checked to see their effect on the system response followed by a series of validation case studies. This evaluation also allows one to optimize a similar structure by selecting appropriate external parameters such as electric voltage, magnetic field intensity, different crack types, nonlocal and strain gradient parameters, variations of temperature, and incident angle of sound waves.

Temperature-dependent vibrational behavior of bilayer doubly curved micro-nano liposome shell: Simulation of drug delivery mechanism

This article analyzes the vibrational behavior of the double-layered micro-nanosphere to simulate the drug delivery mechanism in the biological structure under the influence of the temperature environment and the viscoelastic substrate for the polar lipid fraction E (PLFE) liposome isotropic material model. In order to obtain the micro-nano structural equations of the double-layer spherical shell, the displacement-strain relations of the shell have been expressed according to the first-order shear deformation theory and the non-local strain gradient theory. The partial differential equations of motion have been obtained by applying Hamilton’s principle. The system of linear couple equations have been solved using the Galerkin method. After validating the model with the results of the articles available in the literature to determine the accuracy of the presented model, numerical results are presented to investigate the effects of various parameters such as the radius of curvature ratio to length, damping coefficient, Kelvin-Voight damping coefficient, boundary conditions and temperature on vibration frequency response are provided and discussed. Results of the current research indicates that natural frequency decreases as the damping coefficient related to the viscous effect between the liposome bilayers increase. Results of the current research can be used as a benchmark for drug delivery applications.

Geometrically nonlinear dynamic analysis of a damped porous microplate resting on elastic foundations under in-plane nonuniform excitation

This article uses the semi-analytical approach to study the combined nonlinear vibration and nonlinear response of a damped porous microplate under nonuniform periodic parametric excitation to understand the complete nonlinear dynamic behavior of the plate. The plate is supported by a Winkler-Pasternak elastic foundation and modeled using modified strain gradient and third-order shear deformation theories to simulate the small-scale effects and shear deformation, respectively. Using Hamilton’s principle, the governing partial differential equations of motion are derived and solved using Galerkin’s method to convert them into ordinary differential equations (ODEs). These ODEs are solved using a combined incremental harmonic balance (IHB) and arc-length continuation approaches to get the nonlinear vibration (frequency–amplitude curves). The same ODEs are solved using the Newmark-β technique to obtain the nonlinear response (time–amplitude curves). The effect of elastic foundation parameters and aspect ratio on mode shape is presented. The effect of parameters such as the porosity coefficient, type of porosity, Winkler-Pasternak elastic foundation parameters, different size-dependent theories, plate thickness, size of plate, damping coefficient, different loading profiles, and loading concentrations on the nonlinear vibration and nonlinear response is examined. Also, the dependence of initial displacements on the frequency–amplitude curves with respect to the excitation frequency is demonstrated with the help of time-amplitude curves.

A size-dependent meshfree approach for magneto-electro-elastic functionally graded nanoplates based on nonlocal strain gradient theory

In this paper, we introduce a novel size-dependent meshfree approach for analyzing free vibrations of magneto-electro-elastic (MEE) functionally graded (FG) nanoplates. This approach combines the nonlocal strain gradient theory (NSGT), the higher-order shear deformation theory (HSDT), and meshfree method for the first time. MEE materials are essentially mixed by the barium titanate and cobalt ferrite which are definitely coupled by both piezoelectric and piezomagnetic effects. Our approach captures both the nonlocal and strain gradient effects in nanoplates, which result in the reduced stiffness and increased stiffness, respectively, by adjusting two parameters. The kind of effective material properties of MEE-FG nanoplates are expressed using a power-law scheme. The coupled governing equations, based on the principle of extended virtual displacement, are solved using the moving Kriging (MK) meshfree method. Numerical examples are given to investigate the effect of geometrical parameters, initial electric voltage, power index, initial magnetic potential, strain gradient parameter, and nonlocal parameter on the natural frequency of MEE-FG nanoplates.

Interfacial tension and phase equilibria for binary systems containing (CH4-CO2)+(n-dodecane; n-butanol; water)

This study aims to investigate and predict the interfacial tension at elevated pressure for binary systems, that are essential for the ongoing energy transition. A comprehensive review is carried out on saturated mixture densities as well as interfacial tension in binary, ternary and quaternary systems comprising nonpolar and polar compounds at elevated pressures. In view of the research gap especially with regard to quaternary model systems that contain H2O, a hydrocarbon, a gas and a surface-active compound, new experimental and theoretical data on binary systems are presented that are required for systematically approaching the respective quaternary system. Mixture densities are determined for the nonpolar systems CO2+n-dodecane and CH4+n-dodecane, as well as systems comprising n-butanol with CO2 or CH4 at pressures up to 30 MPa in a range of temperatures between 313.15-353.15 K as well as its interfacial tension plus the systems CO2+H2O and CO2+CH4. In order to theoretically determine these properties, the Perturbed Chain Polar Statistical Associating Fluid Theory is applied for describing the bulk phase equilibrium, subsequently being combined with the Density Gradient Theory for calculating the concentrations across the interface from which the interfacial tension is derived. Experimental results are in very good agreement with the simulation as well as previously available data ranging from an absolute average deviation of 0.44 mN/m for CO2+n-butanol to 2.58 mN/m for CO2+water. Furthermore, the simulation is able to reflect the characteristic crossover with respect to pressure and temperature.

Dynamic modelling of the interfacial concentration profiles of fast chemical reactions in reactive liquid–liquid systems

A theoretical framework regarding the interfacial properties of reactive liquid–liquid systems has been applied to specific model examples. Based on the kinetic rate equations, which rely on activities rather than concentrations, and the incompressible density gradient theory, which provides an expression for the Gibbs energy of a nonuniform system, the new thermodynamic treatment of inhomogeneous reactive liquid mixtures is used to study the impact of the various thermodynamic and kinetic parameters of the theoretical framework on the dynamics of chemical reactions taking place in the coexisting liquid bulk phases as well as in the interface between them. More precisely, the equations are used to describe the spatial and temporal evolution of the mole fractions in the interface and the temporal evolution of the mole fractions in the bulk phases for a ternary mixture and a simple chemical reaction. Predictions and calculations were compared to a standard model example. It was found that the studied examples follow the expected reactive and phase behaviour when varying the applied thermodynamic and kinetic parameters.

Thermomechanical Response of Functionally Graded Ti-6Al-4V and Zirconia Biomaterial Plates

This article studies the free vibration responses of functionally graded material (FGM) porous nanoplates exposed to thermal load. The developed mathematical model includes a shear deformation, size-scale, and microstructure influence by a high-order shear deformation (HSDT) and nonlocal strain gradient (NGST) theories. The study considers four different porosity patterns across the thickness: uniform, symmetrical, asymmetric bottom, and asymmetric top distributions. The equation of motion of the FGM porous nanoplate, including the effects of thermal load, is derived with Hamilton's principle, and then solved analytically by employing the Navier method. For the free vibration responses of the nanoplate, the effects of nonlocal and strain gradient elasticities, temperature rise, porosity volume fraction and its distribution are analyzed.

On the thermal stresses in chiral porous elastic beams

This paper is concerned with the strain gradient theory of porous thermoelastic solids. We study the deformation of isotropic chiral cylinders subjected to a temperature field that is linear in the axial coordinate. It is shown that the solution can be reduced to the study of two-dimensional problems. The results are used to investigate the deformation of a circular cylinder subjected to a uniform temperature variation. In contrast to the case of achiral materials, the thermal field in chiral cylinders produces torsional effects.

Bending Responses of Bi-Directional Advanced Composite Nanobeams Using Higher Order Nonlocal Strain Gradient Theory

The bending response of two-dimensional (2D) functionally graded (FG) nonlocal strain gradient nanobeams is explored analytically in this work. The longitudinal and transverse orientations vary in material gradation and material characteristics. Kinematic relations of nanobeams are proposed according to hybrid hyperbolic-parabolic functions. The virtual work principle obtains the equilibrium equations, which are then solved using Navier's method. The accuracy and dependability of the suggested analytical model are demonstrated by comparing the results to predictions made in the literature. A thorough parametric study also determines how sensitive the material distribution, the nonlocal length-scale parameter, the strain gradient microstructure-scale parameter, and the geometry are to how the bending response and stresses of 2D FG nanobeams. The results obtained provide benchmark results, which can be used in the design of composite structures.

Molecular perspectives of interfacial properties of the hydrogen+water mixture in contact with silica or kerogen

Interfacial behaviours in multiphase systems containing H2 are crucial to underground H2 storage but are not well understood. Molecular dynamics simulations were carried out to investigate interfacial properties of the H2+H2O and H2+H2O+silica/kerogen systems over a broad range of temperatures (298 - 523 K) and pressures (1 - 160 MPa). The combination of the H2 model with the INTERFACE force field and TIP4P/2005 H2O model can accurately predict the experimental interfacial tensions (IFTs) of the H2+H2O mixture. The IFTs from simulations are also in accordance with predictions from density gradient theory combined with the PC-SAFT equation of state. In general, the IFT decreases with pressure and temperature. However, at relatively high temperatures and pressures, the IFTs increase with pressure. The opposite pressure influence on IFTs can be explained by the inversion of the sign of the relative adsorption of H2. The H2 enrichment in the interfacial regions was identified in density distributions. Meanwhile, the behaviours of contact angles (CAs) in the H2+H2O+silica system are noticeably different from those in the H2+H2O+kerogen system. The H2O CAs for the H2+H2O+silica and H2+H2O+kerogen systems increase with pressure and decrease with temperature. However, the influence of temperature and pressure on these CAs is less pronounced for the H2+H2O+silica system at low temperatures. The behaviours of CAs were understood based on the variations of IFTs in the H2+H2O system (fluid-fluid interaction) and adhesion tensions (fluid-solid interaction). Furthermore, the analysis of the atomic density distributions indicates that the presence of H2 in between the H2O and the silica/kerogen is almost negligible. Nevertheless, the adsorption of H2O on the silica outside the H2O is strong, while less H2O adsorption is seen on the kerogen.

Aeroelastic flutter analysis of tilted curved nanopipe in supersonic flow

One of the novel applications of nano-scale structure is in the field of sensors in supersonic flow conditions. In this regard, a cylindrical nano-scale structure made of two-dimensional functionally graded (2D-FG) material in two-phase supersonic flow condition is considered in this study to explore the state of instability and flutter. It is obvious from the problem definition that several aspects from different fields have to be incorporated into the formulation of the problem. First, the small scale of the structure enforced the utilization of size-dependent theories of elasticity. Here, the nonlocal strain gradient theory is utilized to relate the stress and strain fields together. Moreover, the strain field of the cylindrical structure is computed from assumed displacement fields from higher-order shear deformation theory (HSDT). On the other hand, the equations of motion and boundary conditions are obtained using Hamilton’s principle considering the variational of all the work and energy components. The external energy of the system comes from the pressure loading from the aerodynamic supersonic flow. The validation and convergence examination of the presented method is conducted and a detailed parametric study is performed to find the most important parameter affecting the stability of the nanopipe structure. It is revealed that the Mach number and gas volume fraction of two-phase supersonic flow is extremely influential in determining flutter in the nanopipes.

Thermal vibration of perforated nanobeams with deformable boundary conditions via nonlocal strain gradient theory

Abstract Due to nonlocal and strain gradient effects with rigid and deformable boundary conditions, the thermal vibration behavior of perforated nanobeams resting on a Winkler elastic foundation (WEF) is examined in this paper. The Stokes transformation and Fourier series have been used to achieve this goal and to determine the thermal vibration behavior under various boundary conditions, including deformable and non-deformable ones. The perforated nanobeams’ boundary conditions are considered deformable, and the nonlocal strain gradient theory accounts for the size dependency. The problem is modeled as an eigenvalue problem. The effect of parameters such as the number of holes, elastic foundation, nonlocal and strain gradient, deformable boundaries and size on the solution is considered. The effects of various parameters, such as the length of the perforated beam, number of holes, filling ratio, thermal effect parameter, small-scale parameters and foundation parameter, on the thermal vibration behavior of the perforated nanobeam, are then illustrated using a set of numerical examples. As a result of the analysis, it was determined that the vibration frequency of the nanobeam was most affected by the changes in the dimensionless WEF parameter in the first mode and the changes in the internal length parameter when all modes were considered.

Effect of flexoelectricity on the Pull-in instability of beam-type NEMS

Considering the increasing number of smart materials in advanced industries and technologies, the need to investigate the use of these materials is felt more and more. The flexoelectric effect is one of the most important phenomena created in micro-scale beams, which will have a significant effect on the results and should be investigated. Therefore, in this article, the Pull-in instability in the beam-type microelectromechanical systems has been investigated under the flexoelectric effect. The governing equations of the system have been derived based on the theory that is founded on the strain gradient theory for the Euler-Bernoulli beam model. This beam is under the effect of intermolecular Casimir force and electrostatic force. The equations and boundary conditions related to them have been non-dimensionalized first in direct and indirect flexoelectric effect. With the help of the differential quadrature method, the Pull-in voltage's critical parameters and the system's free-standing behavior have been studied. Among the significant results obtained from this research, it is possible to mention the Pull-in voltage increase due to the flexoelectric effect reduction. Also, in the indirect state of the flexoelectric effect, the increase in the initial applied voltage causes a decrease in the Pull-in voltage.

Thermomechanical vibration and buckling response of magneto-electro-elastic higher order laminated nanoplates

This study investigated the free vibration and buckling behavior of a magneto-electro-elastic (MEE) sandwich porous nanoplate under nonlinear thermal loads and electric and magnetic potentials. The equation of motion of the nanoplate consisting of barium-titanate and cobalt-ferrite components, with a functionally graded (FG) porous core and two face plates, was modeled by a sinusoidal higher-order shear (SHSDT) and the nonlocal strain gradient (NSGT) theories. Due to the significant effect of the temperature in the dynamics of the magneto-electro-elastic nanoplate, the temperature-dependent material properties of barium-titanate (BaTiO3) and cobalt ferrite (CoFe2O4) were defined and applied to the modeling for the first time in the literature. The laminated plate consists of upper and lower surface auxiliary plates and a main FG core plate. The core plate consists of functionally grading cobalt-ferrite and barium-titanate, while face plates can be in three different material compositions consisting of pure cobalt-ferrite, pure barium-titanate, or a combination of the two. It has been shown in this study that the dynamic behavior of the core plate, exposed to high temperatures, can be controlled through auxiliary surface plates. Especially in nanosensor applications, the disadvantage of thermal load can be reduced by the electric and magnetic field's strength applied to the surface plates. This study will make an essential contribution to the design and application of nanosensor or nanoelectromechanical systems, especially by controlling nanosensor responses in high-temperature applications.

Seismic wave propagation analysis in the framework of generalized continuum mechanics theory

For the problems of seismic wave propagation, scale effects will appear while considering the microstructures of medium. Scale effects mean that new components of wave fields appear in the seismograms due to the microstructure interactions in the medium and vary with the change of the characteristic length scale parameters. The generalized continuum mechanics (GCM) theory which introduces the intrinsic length scales of the microstructures is suitable for analyzing the scale effects of seismic wave propagation. We adopt the concept of multi-scale microstructure interactions, that is, the modified couple stress theory, and the one-parameter second strain gradient theory, as two branches of the GCM theory, have the same definition characteristic length scale parameters of medium, to describe the microstructure interactions at different scales. Through the quantitative relationship between the characteristic length scale parameters of the medium and the micropore, the layered model of the characteristic length scale parameters of the micropore is constructed for numerical modeling in a unified framework, and the scale effects of seismic wave propagation is analyzed. More specifically, the asymmetric elastic wave equations under a unified framework are derived, and the physical definition of the characteristic length scale parameters of medium is given. And the influence of the characteristic length scale parameter of the micropore on the seismic wave propagation is focused on. Finally, the scale effects of seismic waves excited by high-speed trains under the viaduct system are analyzed, and some conclusions are drawn.

Geometrically nonlinear deformation reconstruction of based on Euler–Bernoulli beam theory using a nonlinear iFEM algorithm

Deformation reconstruction plays a vital role in the structural health monitoring systems. The inverse finite element method (iFEM) has been demonstrated to be an accurate and robust method of deformation reconstruction. Current iFEM formulations have been applied to the linear deformation of structures based on small-displacement assumption. However, the assumption is inapplicable to some structures with large displacements in practical engineering. Therefore, the geometric nonlinearity needs to be considered in deformation reconstruction model. In this paper, a novel nonlinear iFEM algorithm is proposed based on strain gradient theory. The advantage of the proposed iFEM is that the nonlinear responses does not need to be linearized, which eliminates the influence of the improper strain linearization on stability of reconstruction displacements. The simulation analyses and experimental tests are used to verify the proposed nonlinear iFEM method. Numerical results show that the large displacements can be accurately predicted and the nonlinear iFEM algorithm can improve the reconstruction accuracy by 9% as compared to the linear iFEM strategy. Hence, the proposed nonlinear iFEM approach can be used as a viable tool to accurately reconstruct geometrically nonlinear deformations of structures in real-time applications.