Non-classical Boundary Condition Research Articles

Surface effect on the partial-slip contact of a nano-sized flat indenter

The partial-slip contact at nano-scale is always influenced by a surface effect. A new model based on the Gurtin-Murdoch (G-M) surface elasticity theory is proposed in this paper to study the partial-slip contact behavior of a rigid flat nano-indenter on an elastic half-space. The surface effect in the partial-slip contact is characterized via a non-classical boundary condition involving a surface-induced normal traction related to the residual surface stress and a surface-induced tangential traction related to the surface elasticity, the influences of which on stress and displacement fields and the stick-slip state at the contact surface are investigated. It is found that, with the increase of residual surface stress, both the normal pressure and vertical and horizontal displacements in the contact zone decrease, the relative slip between indenter and substrate reduces and the stick region is enlarged. Such effects are basically attributed to the action of the surface-induced normal traction opposite to the externally applied compression. However, the increase of surface elastic constants is conductive to the relative slip and results in a shrinkage of the stick region, which is attributed to the action of the surface-induced tangential traction opposite to the frictional stress. An interesting phenomenon is further unveiled that, when the frictional coefficient increases, the dominant role in affecting the stick-slip state changes from the surface-induced tangential traction to the normal one, thus inspiring a feasible route to manipulate the surface effect by tuning the frictional coefficient of substrate. The present research enables one to better understand the partial-slip contact behavior of nano-indenters, which is of guiding value for anti-wear designs in nano-mechanical devices.

On modelling strain gradient viscoelasticity of polymer nanocomposites

Theories of generalized continuum mechanics have found great success in the analysis of nanostructures. However, there exists no work on analysing composites whose constituents are generalized continua. The present work fills this gap and studies the strain gradient viscoelasticity of polymeric nanocomposites. The key problem is to assign the nonclassical boundary condition of the representative volume element (RVE). To resolve it, a perturbation field is superposed on the homogeneous displacement boundary condition. The wavelength of perturbation is comparable to the strain gradient characteristic length. Simulations to obtain the macroscopic effective mechanical properties are performed, which agree well with the experimental data. The frequency dependence of the perturbation field is revealed, and it has a clear physical interpretation in terms of the segmental motions of polymer chains.

An Analytical Solution for the Bending of Anisotropic Rectangular Thin Plates with Elastic Rotation Supports

The mechanical analysis of thin-plate structures is a major challenge in the field of structural engineering, especially when they have nonclassical boundary conditions, such as those encountered in cement concrete road slabs connected by transfer bars. Conventional analytical solutions are usually limited to classical boundary conditions—clamped support, simple support, and free edges—and cannot adequately describe many engineering scenarios. In this study, an analytical solution to the bending problem of an anisotropic thin plate subjected to a pair of edges with free opposing elastic rotational constraints is found using a two-dimensional augmented Fourier series solution method. In the derivation process, the thin-plate problem can be transformed into a problem of solving a system of linear algebraic equations by applying Stoke’s transform method, which greatly reduces the mathematical difficulty of solving the problem. Complex boundary conditions can be optimally handled without the need for large computational resources. The paper addresses the exact analytical solutions for bending problems with multiple combinations of boundary conditions, such as contralateral free–contralateral simple support (SFSF), contralateral free–contralateral solid support–simple support (CFSF), and contralateral free–contralateral clamped support (CFCF). These solutions are realized by employing the Stoke transformation and adjusting the spring parameters in the analyzed solutions. The results of this method are also compared with the finite element method and analytical solutions from the literature, and good agreement is obtained, demonstrating the effectiveness of the method. The significance of the study findings lies in the simplification of complex nonclassical boundary condition problems using a simple and reliable analytical method applicable to a wide range of engineering thin-plate structures.

Direct and Inverse Initial Boundary Value Problems for Heat Equation with Non-Classical Boundary Condition

Direct and Inverse Initial Boundary Value Problems for Heat Equation with Non-Classical Boundary Condition

Reconstruction of Two Functions in the Model of Vibrations of a String One End of Which Is Placed in a Moving Medium

The paper considers an inverse problem of determining the coefficients in the model of small transverse vibrations of a homogeneous finite string one end of which is placed in a moving medium and the other is free. The vibrations are simulated by a hyperbolic equation on an interval. One boundary condition has a nonclassical form. Additional data for solving the inverse problem are the values of the solution of the forward problem with a known fixed value of the spatial argument. In the inverse problem, it is required to determine the function in the nonclassical boundary condition and a functional factor on the right-hand side of the equation. Uniqueness and existence theorems for the inverse problem are proved. For the forward problem, conditions for unique solvability are established in a form that simplifies the analysis of the inverse problem. For the numerical solution of the inverse problem, an algorithm is proposed for the stage-by-stage separate reconstruction of the sought-for functions using the method of successive approximations for integral equations.

A Study of a Thermally-Vigorous Rayleigh-Bénard Convection in a Hybrid Nanofluid Using Non-Classical Boundary Condition

A linear stability analysis of a thermally-vigorous Rayleigh-Bénard convection in a mono and a hybrid nanofluid is carried out using the single-phase model. We make use of non-classical boundary condition on velocity (rough boundary condition) and temperature (third-type boundary condition) instead of a specific boundary condition (free-free isothermal, rigid-rigid isothermal, etc.) which is traditionally considered. The thermophysical properties are calculated using phenomenological laws and mixture theory. A unique and novel combination of a single-term Galerkin technique and the Maclaurin series expansion is used to solve the boundary-eigen-value problem obtained in the problem. The critical value of the wave number and the Rayleigh number that are calculated using the procedure are quite accurate up to 5 and 3 decimal places respectively. A comparative study on the instability in two types of nanofluids is carried out so as to infer which type of nanofluid suits best for thermally vigorous systems. This paper aims to provide a theoretical basis to ease the selection process pertaining to the making of a choice from different types of nanofluids. To validate the study sixteen limiting cases have been obtained. To visualize the flow behavior physically, streamlines have been plotted.

On simplified deformation gradient theory of modified gradient elastic Kirchhoff–Love plate

A concise modified gradient elastic Kirchhoff–Love plate model with two length-scale parameters is proposed based on simplified deformation gradient theory. A sixth-order basic differential equation and boundary conditions applicable to arbitrary shapes are obtained by applying the principle of minimum potential energy and combining with the generalized strain energy related to classical strain, strain gradient and rotation gradient. By introducing couple stress, the classical bending stiffness corresponding to the fourth-order terms and force-related boundary conditions in the typical gradient elastic Kirchhoff–Love plate model are modified, and the bending deformation of thin plates can be described more flexibly. Under certain conditions, the modified gradient elastic Kirchhoff–Love plate model can be reduced to typical gradient elastic thin plate model, couple stress thin plate model and classical Kirchhoff–Love plate model. The bending boundary value problems of gradient elastic thin plates are further studied, and the specific modified classical boundary conditions and non-classical higher-order boundary condition acceptable to the sixth-order basic equation in Cartesian coordinates are derived. The analytical and numerical bending solutions to gradient Navier-type and Levy-type thin plates with various boundary conditions, including simply supported, clamped and free boundaries are presented, and the size-dependent bending stiffness that is jointly determined by geometric dimensions and length-scale parameters is defined to describe the size effect of thin plates comprehensively.

Reconstruction of a potential coefficient in the Rayleigh–Love equation with non-classical boundary condition

PurposeThe purpose of this paper is to reconstruct the potential numerically in the fourth-order Rayleigh–Love equation with boundary and nonclassical boundary conditions, from additional measurement.Design/methodology/approachAlthough, the aforesaid inverse identification problem is ill-posed but has a unique solution. The authors use the cubic B-spline (CBS) collocation and Tikhonov regularization techniques to discretize the direct problem and to obtain stable as well as accurate solutions, respectively. The stability, for the discretized system of the direct problem, is also carried out by means of the von Neumann method.FindingsThe acquired results demonstrate that accurate as well as stable solutions for the a(t) are accessed for {10–8, 10–7, 10–6, 10–5}, when p {0.01%, 0.1%} for both linear and nonlinear potential coefficient a(t). The stability analysis shows that the discretized system of the direct problem is unconditionally stable.Research limitations/implicationsSince the noisy data are introduced, the investigation and analysis model real circumstances where the practical quantities are naturally infested with noise.Practical implicationsThe acquired results demonstrate that accurate as well as stable solutions for the a(t) are accessed for {10–8, 10–7, 10–6, 10–5}, when p {0.01%, 0.1%} for both linear and nonlinear potential coefficient a(t). The stability analysis shows that the discretized system of the direct problem is unconditionally stable.Originality/valueThe potential term in the fourth-order Rayleigh–Love equation from additional measurement is reconstructed numerically, for the first time. The technique establishes that accurate, as well as stable solutions are obtained.

Quasinormal Mode Expansion Theory for Mesoscale Plasmonic Nanoresonators: An Analytical Treatment of Nonclassical Electromagnetic Boundary Condition

Nonclassical quantum effects will significantly affect the optical response of plasmonic nanoresonators with mesoscale feature sizes between about 2 and 20nm, and can be fully described by the nonclassical electromagnetic boundary condition (NEBC) expressed with the surface-response Feibelman d parameters. In this Letter, a quasinormal mode (QNM) expansion theory under the NEBC is proposed. By adopting the easily solved classical QNMs under the classical electromagnetic boundary condition as a complete set of basis functions, rigorous expansions of the nonclassical source-free QNMs and source-excited electromagnetic field under the nonperturbative NEBC are provided. With the obtained nonclassical QNMs as basis functions, expansions of the nonclassical source-excited field and Green's function tensor are further obtained. These expansions have a fully analytical dependence on the NEBC and classical QNMs, thus transparently unveiling their impact on the nonclassical QNMs and source-excited electromagnetic field. For instance, a new expression of mode volume is proposed for analyzing the nonclassically corrected Purcell factor. The proposed theory is physically intuitive and computationally efficient which is enabled by the dominance of a small set of classical QNMs, thus providing an efficient tool for understanding and designing mesoscale plasmonic nanoresonators.

Rothe–Legendre pseudospectral method for a semilinear pseudoparabolic equation with nonclassical boundary condition

A semilinear pseudoparabolic equation with nonlocal integral boundary conditions is studied in the present paper. Using Rothe method, which is based on backward Euler finitedifference schema, we designed a suitable semidiscretization in time to approximate the original problem by a sequence of standard elliptic problems. The questions of convergence of the approximation scheme as well as the existence and uniqueness of the solution are investigated. Moreover, the Legendre pseudospectral method is employed to discretize the time-discrete approximation scheme in the space direction. The main advantage of the proposed approach lies in the fact that the full-discretization schema leads to a symmetric linear algebraic system, which may be useful for theoretical and practical reasons. Finally, numerical experiments are included to illustrate the effectiveness and robustness of the presented algorithm.

Determination of a time-dependent potential in a Rayleigh-Love equation with non-classical boundary condition

Mathematical model of the longitudinal vibration of bars includes higher-order derivatives in the equation of motion under considering the effect of the lateral motion of a relatively thick bar. This paper considers such an inverse coefficient problem of determining time-dependent potential of a linear source together with the unknown longitudinal displacement from a Rayleigh-Love equation (containing the fourth-order space derivative) by using an additional measurement. Existence and uniqueness theorem of the considered inverse coefficient problem is proved for small times by using contraction principle.

MODELING PRESSURE FIELDS IN A PETROLEUM RESERVOIR TAKING INTO ACCOUNT THE CHANGE OF LIQUID LEVEL IN THE WELL

This article presents a numerical model of pressure fields during filtration of reservoir fluid, which accounts for the effect of the well. The mathematical formulation of the problem under consideration includes the equation of piezoconductivity in a cylindrical coordinate system and differs in a non-classical boundary condition obtained from the ratio of the balance of mass and momentum. The authors compare the numerical calculations and the analytical solution constructed in the Laplace — Carson image space. Den Iseger’s numerical algorithm is used to convert it to the original space. As a result of the comparison, the area of applicability of the computational experiment was determined. It is shown that in the case of a limited reservoir, the exact solution of the problem coincides with a numerical experiment over the entire domain of definition. In the case of an infinite reservoir, the numerical model is applicable only in the region of small times, the dimensions of which are determined by the values of the right boundary and the radial coordinate.

Wave Scattering by Cracks at Macro- and Nano-Scale in Anisotropic Plane by Boundary Integral Equation Method

AbstractElastic wave scattering by cracks at macro- and nano-scale in anisotropic plane under conditions of plane strain is studied in this work. Furthermore, time-harmonic loads due to incident plane longitudinal P- or shear SV- wave are assumed to hold. In a subsequent step, the elastodynamic fundamental solution for general anisotropic continua derived in closed-form via the Radon transform is implemented in a numerical scheme based on the traction boundary integral equation method (BIEM). The surface elasticity effect in the case of nano-crack is taken into consideration via non-classical boundary condition along the crack surface proposed by Gurtin and Murdoch [1]. The numerical results obtained herein reveal substantial differences between anisotropic materials containing a macro- and a nano-crack in terms of their dynamic stress response, where the latter case demonstrates clearly the strong influence of the size-effects. Finally, these types of examples serve to illustrate the present approach and to show its potential for evaluating the stress concentration fields (SCF) inside cracked nanocomposites. The obtained results concern the reliability and safety of the advancing nanomaterials.

Permeability evolution and water transfer in the excavation damaged zone of a ventilated gallery

The fluid transfers occurring around underground galleries are of paramount importance when envisaging the long-term sustainability of underground structures for nuclear waste disposal. These transfers are mainly conditioned by the behaviour of the surrounding material and by its interaction with the gallery air. The hydro-mechanical behaviour of the excavation damaged zone, which develops around galleries due to the drilling process, is thenceforward critical because it is composed of fractures having a significant irreversible impact on flow characteristics and transfer kinetics. Besides, the material interaction with the gallery air may engender water drainage and desaturation. Thus, a gallery air ventilation experiment, preceded by its excavation, is numerically modelled in an unsaturated argillaceous rock to study its influence on hydraulic transfers. The fractures are numerically represented with shear strain localisation bands by means of a microstructure enriched model including a regularisation method. The impact of fracturing on the transport properties is addressed by associating the intrinsic permeability increase with mechanical deformation which is amplified in the strain localisation discontinuities. Such dependence permits us to reproduce a significant permeability increase of several orders of magnitude in the excavation damaged zone, in agreement with available experimental measurements. After the excavation, the hydraulic transfers are studied through the reproduction of a gallery air ventilation experiment that implies drainage and desaturation of the surrounding rock. These transfers depend on liquid water and water vapour exchanges at gallery wall that are introduced through a non-classical boundary condition. The model prediction successfully captures the drainage and desaturation kinetics of undisturbed and damaged rock.

An inverse time-dependent source problem for the heat equation with a non-classical boundary condition

This paper investigates the inverse problem of determining the time-dependent heat source and the temperature for the heat equation with a non-classical boundary and an integral over-determination conditions. The existence, uniqueness and continuous dependence upon the data of the classical solution of the inverse problem is shown by using the generalised Fourier method. Furthermore in the numerical process, the boundary element method (BEM) together with the second-order Tikhonov regularization is employed with the choice of regularization parameter based on the generalised cross-validation (GCV) criterion. Numerical results are presented and discussed.

Comparison of meshless local weak and strong forms based on particular solutions for a non-classical 2-D diffusion model

Abstract In the current work, a new aspect of the weak form meshless local Petrov–Galerkin method (MLPG), which is based on the particular solution is presented and well-used to numerical investigation of the two-dimensional diffusion equation with non-classical boundary condition. Two-dimensional diffusion equation with non-classical boundary condition is a challenged and complicated model in science and engineering. Also the method of approximate particular solutions (MAPS), which is based on the strong formulation is employed and performed to deal with the given non-classical problem. In both techniques an efficient technique based on the Tikhonov regularization technique with GCV function method is employed to solve the resulting ill-conditioned linear system. The obtained numerical results are presented and compared together through the tables and figures to demonstrate the validity and efficiency of the presented methods. Moreover the accuracy of the results is compared with the results reported in the literature.

Thermostressed state of a cylinder with thin near-surface layer having time-dependent thermophysical properties

We develop a method for the solution of boundary-value problems of the thermostressed state of cylindrical bodies with thin near-surface layers having time-dependent thermophysical properties. The presence of a thin near-surface layer in a long solid cylinder is taken into account in the formulated boundary-value problem by means of a nonclassical nonstationary boundary condition with variable coefficients. The replacement of this condition by the classical first-kind condition enables one to reduce the original problem to finding the solution of an integro-differential equation with variable coefficients. This equation has a Volterra-type integral operator and is solved by using spline approximations. The efficiency of this method has been partially verified on the well-known problem of heating of a thin homogeneous plate with variable Biot number by environment. We investigate the thermostressed state of a cylinder for both linear and exponential normalized surface parameters of heat transfer and heat capacity over time. For different stages of heating, we analyze at what constant values of parameters the thermal regime of the cylinder is closest to the mode with variable parameters. The obtained solutions and data on their properties can be used in solving the problems of boundary parametric identification.

MLPG method for two-dimensional diffusion equation with Neumann's and non-classical boundary conditions

In this paper, a meshless local Petrov–Galerkin (MLPG) method is presented to treat parabolic partial differential equations with Neumann's and non-classical boundary conditions. A difficulty in implementing the MLPG method is imposing boundary conditions. To overcome this difficulty, two new techniques are presented to use on square domains. These techniques are based on the finite differences and the Moving Least Squares (MLS) approximations. Non-classical integral boundary condition is approximated using Simpson's composite numerical integration rule and the MLS approximation. Two test problems are presented to verify the efficiency and accuracy of the method.

Influence of evaporation and seepage on the convergence of a ventilated cavity

In the design of nuclear waste disposals, an important topic concerns the development of an excavated damage zone, where permeability increases. A correct numerical prediction of the coupled processes occurring during excavation is therefore needed. In this paper, a nonclassical hydraulic boundary condition is described. It mixes two modes of water exchanges in partial saturation: seepage and evaporation flows. Seepage flow avoids unphysical water inflow into the rock mass, which might be obtained with usual boundary condition in dilatant geomaterial. Evaporation flows enable the thermodynamical equilibrium between air relative humidity in the cavity and in the rock mass. The numerical modeling of a cavity excavation in dilatant geomaterial is carried out. The results show the influence of the hydraulic boundary condition on the convergence of the cavity. Depending on the value of the transfer coefficient, the proposed numerical model can recover the results obtained with an imposed atmospheric pressure or suction at the cavity wall. The determination of the mass transfer coefficient is thus needed and can be achieved through drying experiments or can be estimated thanks to in situ measurements.

Coupled FEM-BEM Approach for Axisymetrical Heat Transfer Problems

This work deals with a stationary axisymmetrical heat transfer problem in a combined domain. This domain consists of half-space joined with a bounded cylinder. An important feature of the problem is the possible flux singularity along the edge points of the transmission surface. Domain decomposition is used to separate the subdomains. The solution for an auxiliary mixed boundary value problem in the half space is found analytically by means of Hankel integral transform. This allows us to reduce the main problem in the infinite domain to another problem defined in the bounded subdomain. In turn, the new problem contains a nonlocal boundary conditions along the transmission surface. These conditions incorporate all basic information about the infinite sub-domain (material properties, internal sources etc.). The problem is solved then by means of the Finite Element Method. In fact it might be considered as a coupled FEM-BEM approach. We use standard MATLAB PDE toolbox for the FEM analysis. As it is not possible for this package to introduce directly a non-classical boundary condition, we construct an appropriate iterative procedure and show the fast convergence of the main problem solution. The possible solution singularity is taken into account and the corresponding intensity coefficient of the heat flux is computed with a high accuracy. Numerical examples dealing with heat transfer between closed reservoir (filled with some substance) and the infinite foundation are discussed.