Higher-order Boundary Conditions Research Articles

Size-dependent axial dynamics of magnetically-sensitive strain gradient microbars with end attachments

The multi-parameter Sturm–Liouville eigenvalue problems, associated with the size-dependent longitudinal vibration of a finite micro-scale bar embedded in orthogonal transverse magnetic fields, are addressed in this study. Derived as a fourth-order partial differential equation, and augmented by enriched higher-order boundary conditions, the mathematical model of the micro-scale bar is predicated on the duo of the strain gradient theory of elasticity and the extended Hamilton׳s principle. The derived model is tackled with the computation scheme of the power series method. A thorough validation of the simplified form of the model is presented with benchmark results published in the literature. Presented along with the validation study is a comprehensive parametric examination of the influence of the aspect ratio, the material length scale, the mass and stiffness ratios of attachments and the magnetic field strength. Results from the analyses affirm that in the case of a lightweight mass attached to the end of the microbar, the axial resonant frequencies approach that of a microbar with a fixed–free boundary condition. However, in the case of an attachment with a heavy mass the fundamental resonant frequency of the microbar tends to zero. A Pareto analysis of the order of influence of the model׳s variables unmasks the ratio of the stiffness of the microbar and an attached elastic spring as having the most significant effect on the fundamental axial natural frequency, in the absence of the magnetic field. In the presence of the magnetic field, however, the effects of the end attachments are totally overshadowed by the influence of the magnetic field strength.

High Order Boundary Conditions for High Order Finite Difference Schemes on Curvilinear Coordinates Solving Compressible Flows

In the present paper, high order boundary conditions for high order finite difference schemes on curvilinear coordinates are implemented and analyzed. The finite difference scheme employed in the present paper is a conservative scheme on a finite volume type grid arrangement for which the implementation of the boundary conditions is characterized by a proper evaluation of the numerical fluxes at the boundaries. Simple extrapolation procedures dealing with both the inviscid and viscous fluxes at the boundary are proposed. Specifically, for the inviscid fluxes, a characteristic-variable-based extrapolation procedure is adopted, which effectively reduces the numerical reflection at the far-field boundaries. The emphasis of the present paper is to design straightforward and efficient numerical boundary closure procedures which are capable of achieving high order of accuracy near the boundaries. Special attention is paid to the influence of the extrapolation accuracy on the overall accuracy of the scheme. Numerical results show that the finite difference scheme can achieve its nominal order accuracy when the extrapolation accuracy is sufficiently high.

A transmission problem for the Helmholtz equation with higher order boundary conditions

A transmission problem for the Helmholtz equation with higher order boundary conditions

Two-scale nonlocal shear rate formulation of Bingham plastic fluid

This paper deals with a phenomenological model based on a nonlocal Bingham constitutive law. This nonlocal viscoplastic law accounts for possible microstructured effects of a heterogeneous fluid. Thermodynamic arguments are presented for the justification of this two-scale nonlocal shear rate law, which can be specialized either as a purely nonlocal or a gradient-type Bingham law. Plane shearing and uniaxial flow assumptions between parallel walls are basic and tractable hypotheses for a first investigation of nonlocal and gradient shear rate law. Analytical solutions are presented for the plane Poiseuille stationary flow inside a straight smooth channel of infinite width. Two higher boundary conditions at the walls are assumed for this so-called micromorphic medium, a static higher-order boundary condition and a kinematic one. A boundary layer phenomenon can be observed for such nonlocal problems. Parametric studies show the key role of the characteristic lengths of the nonlocal model on the shear flow phenomena. Consequently, the heterogeneous nature of microstructured yield-stress fluids may significantly affect velocity profiles, maximum velocity and total flow rate, which have been recently reported in literature.

Prediction of inverted velocity profile for gas flow in nanochannel

Velocity inversion is an interesting phenomenon of nanoscale which means that the velocity near the wall is greater than that of center. To solve this problem, fluid flow in nanochannel attracts more attention in recent years. The physical model of gas flow in two-dimensional nanochannel was established here. To describe the process with conventional control equations, Navier–Stokes equations combined with high-order accurate slip boundary conditions was used as mathematical model. With the introduction of new dimensionless variables, the problem was reduced to an ordinary differential equation. Then it was analytically solved and investigated using homotopy analysis method (HAM). The results were verified by comparing with other available experiment data. Result shows that the proposed method could predict velocity phenomenon.

On the numerical implementation of the higher-order strain gradient-dependent plasticity theory and its non-classical boundary conditions

The higher-order gradient plasticity theory is successful in explaining the size effects encountered at the micron and submicron length scale. Due to the incorporation of spatial gradients of one or more internal variables in these theories and the associated non-classical boundary conditions, special types of elements in the finite element method maybe necessary. This makes the numerical implementation of this higher-order theory not straightforward. In this paper, a robust but straightforward numerical implementation of higher-order gradient-dependent plasticity theories is presented. The novelty of this paper is in (1) the application of the meshless methods, particularly the moving weighted least square method, combined with the finite element method for the numerical computation of plastic strain gradients, and (2) the numerical implementation of different types of higher-order microscopic boundary conditions at internal/external surfaces, interfaces, and moving elastic–plastic boundaries. The proposed numerical implementation algorithms can be easily adapted in the implementation of any form of higher-order gradient-dependent constitutive models. Examples of applying the current numerical approach is demonstrated for capturing mesh-objective shear band formation and size effect and boundary layer formation in thin films on elastic substrates and metal matrix composites with embedded elastic inclusions through the consideration of stiff, intermediate, and soft interfaces.

Validation of Higher-Order Approximations and Boundary Conditions for Lossy Conducting Bodies

Validation of Higher-Order Approximations and Boundary Conditions for Lossy Conducting Bodies

A nonlocal analytical approach for torsion of cylindrical nanostructures and the existence of higher-order stress and geometric boundaries

A new analytical approach considering the effective nonlocal shear stress field is proposed and subsequently the torsion of cylindrical nanostructures, such as nanorods, nanoshafts, and/or nanotubes is investigated. The analytical solutions and torsional characteristics of such nanostructures subjected to an external end torque or distributed torques are established based on the Eringen’s gradient type of nonlocal elasticity theory. For bending, buckling and vibration, it has been proved that the classical equations of equilibrium and/or equations of motion could not be applied to nonlocal nanostructures directly unless the stress and moment quantities in classical equations are replaced by the corresponding effective nonlocal quantities. The work is extended to derive the governing equations here via the variational principle and effective nonlocal stress theory for torsion of cylindrical nanostructures for the first time. A new strain energy density expression with higher-order nonlocal shear strain gradients and a nonlocal nanoscale parameter is established in detail, alongside the corresponding generalized non-classical stress and geometric boundary conditions. The effective nonlocal twisting moment is proposed for torsion deformation for the first time. A few benchmarked examples are solved and the analytical solutions for specific cases of interest demonstrate that the maximum angular deformation for static twist decreases with an increase in the nonlocal nanoscale parameter, or the torsion rigidity of nonlocal nanostructures tends to harden when the effective nonlocal effects are considered. The effective nonlocal torsional solutions obtained are compared with the classical and previous softening nonlocal solutions to reveal the significant effects of additional higher-order nonlocal boundary conditions. The nanostructural enhancement effect derived here is consistent with some other existing approaches including the strain gradient elasticity theory and the modified couple stress theory.

Buckling and post-buckling of gradient and nonlocal plasticity columns experiencing softening

The buckling and the post-buckling behaviors of a perfect axially loaded column are analytically investigated through a global bilinear moment–curvature elastoplastic constitutive law. Three plasticity cases are studied, namely the linear hardening plasticity law, the perfect elastoplastic case and the softening case. The applications of such a study can be found in various structural engineering problems, including reinforced concrete, steel, timber or composite structures. It is analytically shown that for all kinds of elastoplastic behaviors, the plasticity phenomena lead to a global softening branch in the load–deflection diagram. The propagation of the plasticity zone during the post-buckling process is analytically characterized in case of linear hardening or softening plasticity laws. However, it is shown that the unphysical elastic unloading solution necessarily occurs in presence of local softening moment–curvature constitutive law. A nonlocal plasticity moment–curvature softening law is then used to control the localization branch in the post-buckling stage. This nonlocal plasticity law includes the explicit and the implicit gradient plasticity law. Higher-order plasticity boundary conditions are derived from an extended variational principle. Some parametric studies finally illustrate the main findings of this paper, including the plasticity modulus effect on the post-buckling behavior of these plasticity structural systems.

Vibration analysis of single-walled carbon nanotubes conveying nanoflow embedded in a viscoelastic medium using modified nonlocal beam model

In this study, the vibration and stability analysis of a single-walled carbon nanotube (SWCNT) coveying nanoflow embedded in biological soft tissue are performed. The effects of nano-size of both fluid flow and nanotube are considered, simultaneously. Nonlocal beam model is used to investigate flow-induced vibration of the SWCNT while the small-size effects on the flow field are formulated through a Knudsen number ( Kn ), as a discriminant parameter. Pursuant to the viscoelastic behavior of biological soft tissues, the SWCNT is assumed to be embedded in a Kelvin–Voigt foundation. Hamilton’s principle is applied to the energy expressions to obtain the higher-order governing differential equations of motion and the corresponding higher-order boundary conditions. The differential transformation method (DTM) is employed to solve the differential equations of motion. The effects of main parameters including Kn , nonlocal parameter and mechanical behaviors of the surrounding biological medium on the vibrational properties of the SWCNT are examined.

Three-dimensional isogeometric solutions to general boundary value problems of Toupin’s gradient elasticity theory at finite strains

We present, to the best of our knowledge, the first complete three-dimensional numerical solutions to a broad range of boundary value problems for a general theory of finite strain gradient elasticity. We have chosen for our work, Toupin’s theory (Toupin, 1962)–one of the more general formulations of strain gradient elasticity. Our framework has three crucial ingredients: The first is isogeometric analysis (Hughes et al., 2005), which we have adopted for its straightforward and robust representation of C1-continuity. The second is a weak treatment of the higher-order Dirichlet boundary conditions in the formulation, which control the development of strain gradients in the solution. The third ingredient is algorithmic (automatic) differentiation, which eliminates the need for linearization “by hand” of the rather complicated geometric and material nonlinearities in gradient elasticity at finite strains. We present a number of numerical solutions to demonstrate that the framework is applicable to arbitrary boundary value problems in three dimensions. We discuss length scale effects, the role of higher-order boundary conditions, and perhaps most importantly, the relevance of the framework to problems with elastic free energy density functions that are non-convex in strain space.

Local artificial boundary conditions for Schrödinger and heat equations by using high-order azimuth derivatives on circular artificial boundary

The aim of the paper is to design high-order artificial boundary conditions for the Schrödinger equation on unbounded domains in parallel with a treatment of the heat equation. We first introduce a circular artificial boundary to divide the unbounded definition domain into a bounded computational domain and an unbounded exterior domain. On the exterior domain, the Laplace transformation in time and Fourier series in space are applied to achieve the relation of special functions. Then the rational functions are used to approximate the relation of the special functions. Applying the inverse Laplace transformation to a series of simple rational function, we finally obtain the corresponding high-order artificial boundary conditions, where a sequence of auxiliary variables are utilized to avoid the high-order derivatives in respect to time and space. Furthermore, the finite difference method is formulated to discretize the reduced initial–boundary value problem with high-order artificial boundary conditions on a bounded computational domain. Numerical experiments are presented to illustrate the performance of our method.

Solving the discrete S-model kinetic equations with arbitrary order polynomial approximations

A numerical method for solving the model Boltzmann equation using arbitrary order polynomials is presented. The S-model is solved by a discrete ordinate method with velocity space discretized with a truncated Hermite polynomial expansion. Physical space is discretized according to the Conservative Flux Approximation scheme with extension to allow non-uniform grid spacing. This approach, which utilizes Legendre polynomials, allows the spatial representation and flux calculation to be of arbitrary order. High order boundary conditions are implemented. Various results are shown to demonstrate the utility and limitations of the method with comparison to solutions of the Euler and Navier–Stokes equations and from the Direct Simulation Monte Carlo and Unified Gas Kinetic schemes. The effect of both the velocity space discretization and Knudsen number on the convergence properties of the scheme are also investigated.

Size-dependent thermal behaviors of axially traveling nanobeams based on a strain gradient theory

This work is concerned with transverse vibrations of axially traveling nanobeams including strain gradient and thermal effects. The strain gradient elasticity theory and the temperature field are taken into consideration. A new higher-order differential equation of motion is derived from the variational principle and the corresponding higher-order non-classical boundary conditions including simple, clamped, cantilevered supports and their higher-order "offspring" are established. Effects of strain gradient nanoscale parameter, temperature change, shape parameter and axial traction on the natural frequencies are presented and discussed through some numerical examples. It is concluded that the factors mentioned above significantly influence the dynamic behaviors of an axially traveling nanobeam. In particular, the strain gradient effect tends to induce higher vibration frequencies as compared to an axially traveling macro beams based on the classical vibration theory without strain gradient effect.

Micromechanical analysis of heterogeneous materials subjected to overall Cosserat strains

In the framework of the computational homogenization procedures, the problem of coupling a Cosserat continuum at the macroscopic level and a Cauchy medium at the microscopic level, where a heterogeneous periodic material is considered, is addressed. In particular, non-homogeneous higher-order boundary conditions are defined on the basis of a kinematic map, properly formulated for taking into account all the Cosserat deformation components and for satisfying all the governing equations at the micro-level in the case of a homogenized elastic material. Furthermore, the distribution of the perturbation fields, arising when the actual heterogeneous nature of the material is taken into account, is investigated. Contrary to the case of the first-order homogenization where periodic fluctuations arise, in the analyzed problem more complex distributions emerge.

On the higher-order boundary conditions for incompressible nonlinear bipolar fluid flow

On the higher-order boundary conditions for incompressible nonlinear bipolar fluid flow

Torsional Vibration Analysis of Carbon Nanotubes Based on the Strain Gradient Theory and Molecular Dynamic Simulations

In the current study, the torsional vibration of carbon nanotubes is examined using the strain gradient theory and molecular dynamic simulations. The model developed based on this gradient theory enables us to interpret size effect through introducing material length scale parameters. The model accommodates the modified couple stress and classical models when two or all material length scale parameters are set to zero, respectively. Using Hamilton's principle, the governing equation and higher-order boundary conditions of carbon nanotubes are obtained. The generalized differential quadrature method is utilized to discretize the governing differential equation of the present model along with two boundary conditions. Then, molecular dynamic simulations are performed for a series of carbon nanotubes with different aspect ratios and boundary conditions, the results of which are matched with those of the present strain gradient model to extract the appropriate value of the length scale parameter. It is found that the present model with properly calibrated value of length scale parameter has a good capability to predict the torsional vibration behavior of carbon nanotubes.

An energy residual-based approach to gradient effects within the mechanics of generalized continua

AbstractGeneralized continua exhibiting gradient effects are addressed through a method grounded on the energy residual (ER)-based gradient theory by the first author and coworkers. A main tool of this theory is the Clausius-Duhem inequality cast in a form differing from the classical one only by a nonstandard extra term, the (nonlocality) ER, required to satisfy the insulation condition (its global value has to vanish or to take a known value). The ER carries in the nonlocality features of the mechanical problem through a strain-like rate field, being the specific nonlocality source, and a concomitant higher-order long-range stress (or microstress) field. The thermodynamic restrictions on the constitutive equations are determined by the latter inequality with no need for microstress equilibrium equations, whereas the principle of virtual power (PVP) is left in a standard format. The derived state equations include a set of partial differential equations involving the nonlocality-source strain-like quantity and the related long-range stress, as well as the associated higher-order boundary conditions determined by the insulation condition. Second-grade materials within gradient elasticity, gradient plasticity and crystal plasticity, as well as materials with microstructure (micromorphic and Cosserat materials) are considered to derive the pertinent constitutive equations. The proposed ER-based approach to gradient effects is shown to constitute a more straightforward and “economic” way to formulate the relevant constitutive equations than the PVP-based one.

Out-of-Plane Buckling of Microstructured Beams: Gradient Elasticity Approach

AbstractThis paper deals with the lateral-torsional buckling of elastic microstructured beams. A linear gradient elasticity approach is presented. The elastic constitutive law is linear for the generalized beam model, expressed with first- and second-order moment variables in a sixth-order dimensional stress-strain space. The boundary conditions (including the higher-order boundary conditions) are derived from application of the variational principle applied to second-grade elastic models. The effect of prebuckling deformation is taken into consideration based on the Kirchhoff-Clebsch theory. Some analytical solutions are presented for a hinged-hinged strip beam. The lateral-torsional buckling moment is sensitive to small length terms inherent in the microstructured constitutive law. It is shown that the gradient parameter tends to increase the critical lateral-torsional buckling moment, a conclusion different from the one obtained with an Eringen-based nonlocal model. This tendency is consistent with tha...

Micromechanical modeling of damage in periodic composites using strain gradient plasticity

Damage evolution at the fiber matrix interface in Metal Matrix Composites (MMCs) is studied using strain gradient theory of plasticity. The study includes the rate independent formulation of energetic strain gradient plasticity for the matrix, purely elastic model for the fiber and cohesive zone model for the fiber–matrix interface. For the micro structure, free energy holds both elastic strains and plastic strain gradients. Due to the gradient theory, higher order boundary conditions must be considered. A unit cell with a circular elastic fiber is studied by the numerical finite element cell model under simple shear and transverse uniaxial tension using plane strain and periodic boundary conditions. The result of the overall response curve, effective plastic strain, effective stress and higher order stress distributions are shown. The effect of the material length scale, maximum stress carried by the interface and the work of separation per unit interface area on the composites overall behavior are investigated. The results are compared with those for strong interface.