Classical Boundary Conditions Research Articles

Bending behavior of squared cutout nanobeams incorporating surface stress effects

In nanosized structures as the surface area to the bulk volume ratio increases the classical continuum mechanics approaches fails to investigate the mechanical behavior of such structures. In perforated nanobeam structures, more decrease in the bulk volume is obtained due to perforation process thus nonclassical continuum approaches should be employed for reliable investigation of the mechanical behavior these structures. This article introduces an analytical methodology to investigate the size dependent, surface energy, and perforation impacts on the nonclassical bending behavior of regularly squared cutout nanobeam structures for the first time. To do this, geometrical model for both bulk and surface characteristics is developed for regularly squared perforated nanobeams. Based on the proposed geometrical model, the nonclassical Gurtin-Murdoch surface elasticity model is adopted and modified to incorporate the surface energy effects in perforated nanobeams. To investigate the effect of shear deformation associated with cutout process, both Euler-Bernoulli and Timoshenko beams theories are developed. Mathematical model for perforated nanobeam structure including surface energy effects are derived in comprehensive procedure and nonclassical boundary conditions are presented. Closed forms for the nonclassical bending and rotational displacements are derived for both theories considering all classical and nonclassical kinematics and kinetics boundary conditions. Additionally, both uniformly distributed and concentrated loads are considered. The developed methodology is verified and compared with the available results and an excellent agreement is noticed. Both classical and nonclassical bending profiles for both thin and thick perforated nanobeams are investigated. Numerical results are obtained to illustrate effects of beam filling ratio, the number of hole rows through the cross section, surface material characteristics, beam slenderness ratio as well as the boundary and loading conditions on the non-classical bending behavior of perforated nanobeams in the presence of surface effects. It is found that, the surface residual stress has more significant effect on the bending deflection compared with the corresponding effect of the surface elasticity, Es. The obtained results are supportive for the design, analysis and manufacturing of perforated nanobeams.

A deterministic game interpretation for fully nonlinear parabolic equations with dynamic boundary conditions

This paper is devoted to deterministic discrete game-theoretic interpretations for fully nonlinear parabolic and elliptic equations with nonlinear dynamic boundary conditions. It is known that the classical Neumann boundary condition for general parabolic or elliptic equations can be generated by including reflections on the boundary to the interior optimal control or game interpretations. We study a dynamic version of such type of boundary problems, generalizing the discrete game-theoretic approach proposed by Kohn-Serfaty (2006, 2010) for Cauchy problems and later developed by Giga-Liu (2009) and Daniel (2013) for Neumann type boundary problems.

Задача граничного управления колебаниями струны смещением левого конца при закрепленном правом конце с заданными значениями функции прогибав промежуточные моменты времени

We consider the boundary control problem for the homogeneous string vibrationequation with given the classical boundary (initial and final) conditions and with given valuesof the deflection function at intermediate times. The control is performed by displacementof the left end of the string when the right end is fixed. The problem is reduced to thecontrol problem with zero boundary conditions. We propose the constructive method forconstructing the boundary control of the process of string vibrations with given values ofthe deflection function at intermediate times.We present the results of numerical experimentsand the corresponding graphs confirm the validity of the results.

Vibration optimization of laminated composite plates using genetic algorithm with various discrete fiber angles

The mechanical characteristics of a laminated plate can be tailored by adjusting the fiber orientation angle of each orthotropic lamina. To find the fiber orientation angle of each lamina that maximizes the fundamental frequency, previous works analyses using various optimization algorithms have been performed so far. However, the solution space becomes significantly larger as the number of combinations in orientation angles increases. Thus the optimization process takes much more time. This paper sets up the orientation angles to discrete design variables, such as 15°, 30°, etc. Genetic algorithm is employed to find the optimal solutions. The CFRP rectangular composite plates with 8, 12 and 16 layers are considered as an example using the Ritz method for finding the eigenvalues. Two sets of classical boundary conditions are used as edge conditions. It was revealed that the fundamental frequency makes about 5% difference from the optimal value if it is in 30° steps and about 24% in 90° steps.

A unified modified couple stress model for size-dependent free vibrations of FG cylindrical microshells based on high-order shear deformation theory

The aim of present work is to propose a unified modified couple stress-based shear deformable shell model that has capability of analyzing size-dependent free vibrations of functionally graded (FG) cylindrical microshells. In the current model, the classical strain tensors and curvature tensor components are given within the framework of a unified high-order shear deformation theory. The trapezoidal shape factor is taken into account for achieving reasonable stress resultants over thickness and makes the proposed model more accurate for thick shell structures. Various classical boundary conditions of the cylindrical microshells are considered using the admissible functions of the displacements containing a beam modal function. The governing equations associated with free vibrations of microshells are derived using a standard Lagrange method. A few comparisons are conducted and verify that the results from the current model coincide favorably to the ones from three-dimensional elasticity solutions. The comparisons also prove that the proposed model has a sufficient capability of dealing with free vibration problems of the thick functionally graded cylindrical microshells. To the end, the size-dependent free vibrations of an FG cylindrical microshell are investigated. As new results, natural frequencies of the FG microshells with various boundary conditions are tabulated, and the effects of some important parameters on vibration characteristics of FG microshells are discussed.

Existence of solution for 2-D time-fractional differential equations with a boundary integral condition

In this article, we prove the existence and uniqueness of a solution for 2-dimensional time-fractional differential equations with classical and integral boundary conditions. We start by writing this problem in the operator form and we choose suitable spaces and norms. Then we establish prior estimates from which we deduce the uniqueness of the strong solution. For the existence of solution for the fractional problem, we prove that the range of the operator generated by the considered problem is dense.

A New Conformal FDTD for Lossy Thin Panels

A novel conformal technique for the finite-difference time-domain (FDTD) method is proposed to deal with curved and arbitrarily oriented lossy thin panels. This is formulated as an extension of the existing conformal relaxed Dey–Mittra (CRDM) method combined with subcell algorithms to handle wave propagation across the panel. Two alternatives are presented for this: the classical impedance network boundary condition (INBC) and a recent subgridding boundary condition (SGBC) developed by the authors. Several test cases are employed to demonstrate the capability of the proposed method to remove errors associated with the usual staircased FDTD method.

Scattering of acoustic waves from a point source over an impedance wedge

In this work we study diffraction of a spherical acoustic wave due to a point source, by an impedance wedge In the exterior of the wedge the acoustic pressure satisfies the stationary wave (Helmholtz) equation and classical impedance boundary conditions on two faces of the wedge, as well as Meixner’s condition at the edge and the radiation conditions at infinity. Solution of the boundary value problem is represented by a Weyl type integral and its asymptotic behavior is discussed. On this way, we derive various components in the far field interpreting them accordingly and discussing their physical meaning.

Calculation of tensorial flow properties on pore level: Exploring the influence of boundary conditions on the permeability of three-dimensional stochastic reconstructions.

While it is well known that permeability is a tensorial property, it is usually reported as a scalar property or only diagonal values are reported. However, experimental evaluation of tensorial flow properties is problematic. Pore-scale modeling using three-dimensional (3D) images of porous media with subsequent upscaling to a continuum scale (homogenization) is a valuable alternative. In this study, we explore the influence of different types of boundary conditions on the external walls of the representative modeling domain along the applied pressure gradient on the magnitude and orientation of the computed permeability tensor. To implement periodic flow boundary conditions, we utilized stochastic reconstruction methodology to create statistically similar (to real porous media structures) geometrically periodic 3D structures. Stochastic reconstructions are similar to encapsulation of the porous media into statistically similar geometrically periodic one with the same permeability tensor. Seven main boundary conditions (BC) were implemented: closed walls, periodic flow, slip on the walls, linear pressure, translation, symmetry, and immersion. The different combinations of BCs amounted to a total number of 15 BC variations. All these BCs significantly influenced the resulting tensorial permeabilities, including both magnitude and orientation. Periodic boundary conditions produced the most physical flow patterns, while other classical BCs either suppressed crucial transversal flows or resulted in unphysical currents. Our results are crucial to performing flow properties upscaling and will be relevant to computing not only single-phase but also multiphase flow properties. Moreover, other calculation of physical properties such as some mechanical, transport, or heat conduction properties may benefit from the technique described in this study.

The BEG Model in the Disordered Region and at the Antiquadrupolar-Disordered Line of Parameters

We analyse the d-dimensional BEG model with $$d\ge 2$$ and parameters in the disordered region and at the antiquadrupolar-disordered line. We obtain a subset of these regions for which the odd correlation functions are zero and the even ones decay exponentially fast, at all non-zero temperatures, for a wide class of boundary conditions.

Boundary Characteristic Bernstein Polynomials Based Solution for Free Vibration of Euler Nanobeams

This paper is concerned with the free vibration problem of nanobeams based on Euler–Bernoulli beam theory. The governing equations for the vibration of Euler nanobeams are considered based on Eringen’s nonlocal elasticity theory. In this investigation, computationally efficient Bernstein polynomials have been used as shape functions in the Rayleigh-Ritz method. It is worth mentioning that Bernstein polynomials make the computation efficient to obtain the frequency parameters. Different classical boundary conditions are considered to address the titled problem. Convergence of frequency parameters is also tested by increasing the number of Bernstein polynomials in the simulation. Further, comparison studies of the results with existing literature are done after fixing the number of polynomials required from the said convergence study. This shows the efficacy and powerfulness of the method. The novelty of using the Bernstein polynomials is addressed in detail and solutions obtained by this method provide a better representation of the vibration behavior of Euler nanobeams.

Free In-Plane Vibrations of Orthotropic Rectangular Plates by Using an Accurate Solution

Many numerical methods have been developed for in-plane vibration of orthotropic rectangular plates with various boundary conditions; however, the exact results for such structures with elastic boundary conditions are very scarce. Therefore, the object of this paper is to present an accurate solution for free in-plane vibration of orthotropic rectangular plates with various boundary conditions by the method of reverberation ray matrix (MRRM) and improved golden section search (IGSS) algorithm. The boundary condition studied in this paper is defined as that a set of opposite edges is with one kind of simply supported boundary conditions, while the other set is with any kind of classical and general elastic boundary conditions or their combination. Its accuracy, reliability, and efficiency are verified by some numerical examples where the results are compared with other exact solutions in the published literature and the FEA results based on the ABAQUS software. Finally, some new accurate results for free in-plane vibration of orthotropic rectangular plates with elastic boundary conditions are examined and further can be treated as the reference data for other approximate methods or accurate solutions.

DLR TAU-Code uRANS Turbofan Modeling for Aircraft Aerodynamics Investigations

In the context of an increased focus on fuel efficiency and environmental impact, turbofan engine developments continue towards larger bypass ratio engine designs, with Ultra-High Bypass Ratio (UHBR) engines becoming a likely power plant option for future commercial transport aircraft. These engines promise low specific fuel consumption at the engine level, but the resulting size of the nacelle poses challenges in terms of the installation on the airframe. Thus, their integration on an aircraft requires careful consideration of complex engine–airframe interactions impacting performance, aeroelastics and aeroacoustics on both the airframe and the engine sides. As a partner in the EU funded Clean Sky 2 project ASPIRE, the DLR Institute of Aerodynamics and Flow Technology is contributing to an investigation of numerical analysis approaches, which draws on a generic representative UHBR engine configuration specifically designed in the frame of the project. In the present paper, project results are discussed, which aimed at demonstrating the suitability and accuracy of an unsteady RANS-based engine modeling approach in the context of external aerodynamics focused CFD simulations with the DLR TAU-Code. For this high-fidelity approach with a geometrically fully modeled fan stage, an in-depth study on spatial and temporal resolution requirements was performed, and the results were compared with simpler methods using classical engine boundary conditions. The primary aim is to identify the capabilities and shortcomings of these modeling approaches, and to develop a best-practice for the uRANS simulations as well as determine the best application scenarios.

Natural Frequencies of Laminated CFRP Square Plates with Slightly Different Material Properties

The influence of using different material properties is studied on the natural frequencies of laminated composite square plates composed of recently developed carbon fiber reinforced plastic (CFRP) materials. The plate is symmetrically laminated by thin layers with each layer reinforced by parallel straight carbon fibers. Using the Ritz method in conjunction with the thin plate theory and the lamination theory, the displacements are assumed as finite polynomial functions with the boundary index to accommodate any combination of classical boundary conditions. The material properties are expressed by a set of four elastic constants, and some typical constants are referred from the recent literature. In addition to these sets of constants, a new standard set of discretized constants is proposed to consider the underlying characteristics of the existing constants. In numerical experiment, the convergence study is carried out, and the lowest five natural frequencies are presented in frequency parameters for five sets of classical boundary conditions. Next, for better insight, a new definition of frequency parameters is introduced to promote more physically meaningful comparison, and the influence of using slightly different constants in frequency calculation is clarified for the purpose of deriving approximate frequency formula.

A semi-analytical solution for in-plane free waves analysis of rectangular thin plates with general elastic support boundary conditions

A semi-analytical solution for in-plane free waves analysis of rectangular thin plates with general elastic support boundary conditions (BCs) is developed. Based on the classical thin plate theory, the governing differential equations of the rectangular thin plate are introduced. The semi-analytical solution is obtained by superposition of in-plane waves of different patterns and different propagation directions. And it is validated by comparisons of the present results with those given in literatures. Eigensolutions for general BCs and the specific expressions of characteristic equations for various classical BCs are demonstrated. Characteristic curves of the dimensionless wavenumbers of the in-plane longitudinal wave and the in-plane shear wave of the rectangular thin plate with various BCs are presented and discussed. Perfect orthogonality of the in-plane longitudinal wave and the in-plane shear wave of different mode numbers is presented in the characteristic curves of the rectangular thin plate with two opposite sides simply supported. The characteristic curves corresponding to the rest of the classical BCs and all of the non-classical BCs show non-orthogonality of different extents, which reveals the formation mechanism of the variety of the mode shapes of the rectangular thin plate with various BCs. The strengthening and weakening process of the orthogonality and the formation of the intertwinement or independence of the characteristic curves of different mode numbers are elaborated through analysis on elastic support BCs intermediate between two classical BCs, which shows the influence of the BCs on the intrinsic characteristics of in-plane waves of the rectangular thin plate.

Benchmark Vibration Frequencies of Square Thin Plates with all Possible Combinations of Classical Boundary Conditions

In this work, a new method is used for the exact vibration analysis of plates with classical boundary conditions. Four classical edge conditions are included: C — clamped, S — Simply supported, F — free, and G — guided. For square plates, all the possibilities add up to 55 cases. The solutions for the natural frequencies of the plates are found in this paper using static analysis. Starting from the equations of motion of an isotropic rectangular thin plate supported on Winkler elastic foundation, with a positive or negative value, the solution for the vibration frequencies of the plate is equivalent to finding the values of the negative elastic foundation that will yield infinite deflection under a point load on the plate. The solution is composed of three parts, the sum of which satisfies exactly both the field equation and the boundary conditions. For zero force, the vibration frequencies are found up to the desired accuracy. Benchmark results of the first six normalized natural frequencies, of isotropic square plates, for all possible 55 combinations of classical boundary conditions are given, many for the first time.

Analytical approach to coupled bending-torsional vibrations of cracked Timoshenko beam

The explicit expressions of steady-state responses of a coupled bending-torsional Timoshenko beam with cracks and damping effect subjected to external harmonic loadings are presented in this paper. The Green's functions method is employed to obtain the analytical solutions. The beam is split to several segments due to the existence of cracks. General Green's functions of each segment with unknown boundary constants are given by using Laplace transform method. A mixture line-spring is introduced to obtain the compatibility conditions of the cracked cross-section between the segments. Based on the transfer matrix method, those constants in the Green's functions for each segment are determined with those matching conditions, as well as boundary relationships and end boundary conditions. Then, the solutions of the whole cracked beam with one or multiple cracks can be expressed in terms of the piecewise functions. The solutions of the multi-cracked beam cover those of the beam with only one crack and the uncracked beam. The theory developed can be used for the beam with classical boundary conditions, but for illustration, results for a cantilever cracked beam with T shape cross-section are illustrated, emphasizing the effect of cracks on the solutions. Comparisons between results in the published literature and those obtained from the developed method show a good agreement, which confirms the validity and accuracy of the proposed approach. The influences of crack depth, location and number on the natural frequencies are discussed. The changes of the steady-state responses of beam are investigated due to the existence of crack. Moreover, the symmetric property of the Green's functions and damping effect on the amplitude of steady-state responses of the cracked beam are studied particularly.

Well-Posed Solvability of the Neumann Problem for a Generalized Mangeron Equation with Nonsmooth Coefficients

For a fourth-order generalized Mangeron equation with nonsmooth coefficients defined on a rectangular domain, we consider the Neumann problem with nonclassical conditions that do not require matching conditions. We justify the equivalence of these conditions to classical boundary conditions for the case in which the solution to the problem is sought in an isotropic Sobolev space. The problem is solved by reduction to a system of integral equations whose well-posed solvability is established based on the method of integral representations. The well-posed solvability of the Neumann problem for the generalized Mangeron equation is proved by the method of operator equations.

Analytical Buckling Loads for Rectangular Orthotropic and Symmetrically Laminated Plates

In this research, a new analytical solution for finding the buckling loads of thin orthotropic rectangular plates with different boundary conditions is derived. The methods currently known in the literature for finding the buckling loads of plates are mainly numerical. Although some plates with specific boundary conditions have analytical solutions, a comprehensive analytical method providing analytical solutions that fit all possible combinations of boundary conditions is lacking. The solution method in this study is based on the development of a static solution for a plate. The physical meaning of buckling is the loss of stiffness, and it is found as the value of the in-plane loading intensity at which a zero force on the plate surface will generate infinite displacement. The solution is obtained in series form, and the coefficients are solved to match the edge conditions. The solution presented for this plate will fit different boundary conditions of the deflection, slope, shear force, and bending moment along the common edges of neighboring plates. Using this new method, exact buckling loads and the buckling modes of many new cases of classical boundary conditions are found (cases involving clamped, simply supported, guided, and free boundaries). Results are given for several stiffness ratios in both directions of the plate, as well as for unidirectional and several cases of bidirectional loading.

Analytical–Numerical Solution for the Skin and Proximity Effects in Two Parallel Round Conductors

This paper describes an analytical-numerical method for the skin and proximity effects in a system of two parallel conductors of circular cross section—a system very frequently encountered in various applications. The magnetic field generated by the current applied on each conductor is expressed by means of vector magnetic potential and expanded into Fourier series. Using the Laplace and Helmholtz equations, as well as the classical boundary conditions, the current density induced due to the proximity and skin effect is determined in each conductor. The resulting current density is expressed as a series of successive reactions. The results obtained are compared with those obtained via finite elements. Although the paper is theoretical, the considered problem has a practical significance, because transmission lines with round conductors are universally used. Besides, the results can be used to estimate errors when only the first reaction is taken into account, which gives relatively simple formulas.