Elastic Boundary Research Articles

Droplets Capped with an Elastic Film Can Be Round, Elliptical, or Nearly Square.

We present experiments that show that the partial wetting of droplets capped by taut elastic films is highly tunable. Adjusting the tension allows the contact angle and droplet morphology to be controlled. By exploiting these elastic boundaries, droplets can be made elliptical, with an adjustable aspect ratio, and can even be transformed into a nearly square shape. This system can be used to create tunable liquid lenses and, moreover, presents a unique approach to liquid patterning.

Free vibrations of functionally graded porous rectangular plate with uniform elastic boundary conditions

In this paper, the free vibrations of functionally graded porous (FGP) rectangular plate with uniform elastic boundary conditions is investigated by means of an improved Fourier series method (IFSM). It is assumed that the distributions of porosity are uniform or non-uniformly along a certain direction and three types of the porosity distribution are considered, among which material property of two non-uniform porous distributions was expressed as the simple cosine. The size of the pore in a rectangular plate is determined by the porosity coefficients. Using the first-order shear deformation theory(FSDT), the energy expression of FGP rectangular plate is created. In order to obtain the admissible function of displacement for functionally graded porous rectangular plate, the IFSM is employed. Then, the Rayleigh-Ritz method is used to solve coefficients in the Fourier series which determine natural frequencies and modal shapes. Convergence and comparative research are performed to prove the convergence, reliability and accuracy of the current method. On this foundation, some new results covering the influence of the geometrical parameters subject to classical and elastic boundary condition are presented, and the parametric studies are also investigated in detail, which can provide a reference for future research by other researchers.

Torsion of bi-directional functionally graded truncated conical cylinders

Analytical solutions are presented for the torsion of bi-directional functionally graded (FG) linearly elastic truncated conical cylinders for six functional forms of the shear modulus varying in both the radial and the axial directions. Furthermore, for an arbitrary variation of the shear modulus along the two directions, we employ a weighted residual approach (WRA) to obtain accurate numerical solutions for the linear elastic boundary value problem. The influence of the variation of the shear modulus and of the cone angle on the stress distribution in the cylinder is discussed through numerical examples. The analytical solutions presented herein can serve as benchmarks for ascertaining the accuracy of approximate or numerical solutions. The WRA combined with an optimization algorithm can be used to find the shear modulus variation to maximize the torsional stiffness.

Nonlinear bending analysis of FG-CNTRC annular plates with variable thickness on elastic foundation

As a first endeavor, the nonlinear bending behavior of functionally graded carbon nanotube reinforced composite (FG-CNTRC) annular plates with variable thickness on an elastic foundation is scrutinized in this research. The nonlinear governing equations are established according to the third-order shear deformation theory (TSDT) in conjunction with the nonlinear von Karman strain field. A Pasternak elastic foundation is assumed to be in contact with the annular plate during deformation. To discretize the nonlinear governing equations and boundary conditions, the generalized differential quadrature method (GDQM) is adopted and the nonlinear system of equations is solved via the Newton-Raphson iterative method. Through the numerical results, effects of the elastic foundation, geometrical parameters, boundary conditions, CNT dispersion and volume fraction, thickness-to-radius ratio and thickness profile are investigated in detail. It is shown that CNT volume fraction and dispersion have a remarkable effect on improving bending behavior of the annular plate. It is found that thickness profile plays an essential role in the bending response of the annular plate.

Letter: Counter-jet formation of an expanding bubble near a curved elastic boundary

A high-speed liquid jet is commonly observed during the collapse phase of a cavitation bubble. In this letter, we present a surprising counter-jet (directed away from the nearby boundary) that forms in the early expansion phase of the bubble near a curved elastic boundary, which has not been reported before. The boundary integral simulation reveals the most possible mechanism as follows: First, the expansion of the bubble causes a local concave deformation on the boundary. Thereafter, the bubble partly envelops the curved elastic boundary, leading to the confinement of some liquid in a local dent. During the deformation recovery, a localised high pressure region is built up within the narrow dent, which drives a thin liquid jet away from the boundary.

The use of FWI in coal exploration

Seismic surveys are routinely used for building precise structural images of the coal bearing formations in Australia but coal production related hazards such as weak strata and zones with an increased gas content remain to be fully resolved by seismic measurements. One way of investigating these issues is through the application of Full Waveform Inversion (FWI) methods. To utilise the full power of these methods a high quality seismic dataset is needed. Such conditions are often met by 2D and 3D reflection seismic data acquired over coal seams in Bowen and Sydney basins. FWI can be used for a high resolution estimate of P- and S-wave velocities and the density that can also be translated into geotechnical parameters of interest to mining operations. In this study, we evaluate the applicability of FWI methods for estimating elastic parameters (P and S wave velocities and density) from the inversion of a synthetic seismic dataset that was recorded over the surface of a 2D earth model that represents subsurface geology in Goonyella coal mine in Queensland, Australia. We generated elastic synthetic shot records using finite difference algorithm and inverted these data back for model parameters to assess the potential of the FWI algorithm. Using only a short array of surface receivers (cheaper option), we show that the application of FWI method can still improve the original earth models towards the true solutions. We were also able to reconstruct the elastic boundaries for a major part of the subsurface models within the seismic bandwidth. Interpretation of the estimated parameters for coal mining objectives is then straightforward

Vibration analysis of nanorods by the Rayleigh-Ritz method and truncated Fourier series

Abstract A solution procedure applicable for analyzing the free longitudinal vibration of nanorods is presented in this study. The longitudinal displacement of the nanorod is sought as a linear combination of a Fourier series and auxiliary trigonometric functions. The auxiliary functions are introduced to account for all the relevant discontinuities in the boundary displacement and its derivatives. The improved Fourier series represents a residual or conditioned displacement that is of at least C1 class. The basic theory of the nanorod is shown, which included the total potential energy and the kinematic energy of the nanorod system. The Rayleigh-Ritz method is applied to determine the coefficients of the improved Fourier series expansion for the displacements. The solution approach here is referred to as the improved Fourier-Ritz method. The generality, accuracy and efficiency of the presented approach are fully demonstrated and verified through benchmark examples involving classical and elastic boundary conditions.

Geometrically nonlinear vibration of laminated composite cylindrical thin shells with non-continuous elastic boundary conditions

The geometrically nonlinear forced vibration response of non-continuous elastic-supported laminated composite thin cylindrical shells is investigated in this paper. Two kinds of non-continuous elastic supports are simulated by using artificial springs, which are point and arc constraints, respectively. By using a set of Chebyshev polynomials as the admissible displacement function, the nonlinear differential equation of motion of the shell subjected to periodic radial point loading is obtained through the Lagrange equations, in which the geometric nonlinearity is considered by using Donnell’s nonlinear shell theory. Then, these equations are solved by using the numerical method to obtain nonlinear amplitude–frequency response curves. The numerical results illustrate the effects of spring stiffness and constraint range on the nonlinear forced vibration of points-supported and arcs-supported laminated composite cylindrical shells. The results reveal that the geometric nonlinearity of the shell can be changed by adjusting the values of support stiffness and distribution areas of support, and the values of circumferential and radial stiffness have a more significant influence on amplitude–frequency response than the axial and torsional stiffness.

A simplified calculation model for the sliding contact boundaries in a hydrostatic piston mechanism

The objective of this study was to discuss simplified calculation models for the piston/cylinder sliding mechanism in which boundary contact partly occurs invariably. An efficient prediction of the boundary leakage and friction is often needed, such as in a swash-plate axial piston machine whose lubrication test is hard to perform due to the mechanism complexity. In order to model this physically uncertain lubrication regime, two calculation models were compared to compute the lubrication behaviors: “rigid boundary model”, whose theoretical concept was previously reported in the literature, and “elastic boundary model”, newly proposed in this study. Developed numerical algorithms commonly facilitated the simultaneous calculation of body motion and fluid film pressure to observe piston motion, reaction forces, and power loss. The results showed that simulations using the elastic boundary model should be more helpful for the prediction in the earlier development stage than the previous model since the methodology provides much less simulation time than full-order calculation, higher accuracy than the rigid model, and useful engineering parameters such as surface stress. The proposed calculation model can be extended to various asymmetrically loaded reciprocating piston mechanisms for efficiently predicting the lubrication behavior.

On the electro-thermo-mechanical vibration characteristics of elastically restrained functionally graded nanobeams using differential transformation method

In this investigation, free vibration of piezoelectric functionally graded nanobeams in the presence of thermal field effects for various elastic boundary conditions is studied. Formulations are established in the framework of nonlocal elasticity theory of Eringen accompanied by Euler–Bernoulli beam theory. Further, the material characteristics are supposed to change through the thickness based on the power law. Discretization of the equations of motion and the relative boundary conditions are carried out by utilizing the differential transformation method. The presented model is validated through obtaining numerical results for conventional boundary conditions in comparison with the corresponding benchmark results. Finally, numerical results are given in details to demonstrate the impact of material gradient, nonlocal effect, piezoelectric voltage, along with temperature changes on vibration characteristic of piezoelectric functionally graded nanobeams, with various elastic boundary conditions.

Experimental analysis of the elastic boundary conditions of press machines for modelling the deep-drawing process

The elastic deflection of the working areas of the active tool parts has a direct effect on the workpiece contact and thus on the component quality. Knowledge regarding the elastic behavior for typical process loads is of great importance for simulation of manufacturing processes as well as for the process layout, for example to maintain high predictive accuracy of the simulation methods or to reduce costs of the tool tryout process. With the aid of an experimental analysis, the deflection of the working areas for centric and eccentric loads is investigated using the example of a servo-screw press. The analyses focus on the evaluation of the machine’s symmetry properties as well as the stroke position-dependent stiffness behavior. Based on the results of the experimental analysis, the quantification of the stiffness properties of the press slide for the process typical working space is carried out and boundary conditions for modeling of machine and process interactions are evaluated.

Impedance modeling of elastic boundary support in the vibration field of a thin plate

Impedance modeling of elastic boundary support in the vibration field of a thin plate

IB-WENO method for incompressible flow with elastic boundaries

Abstract We present a new approach to compute the interaction between an elastic boundary and an incompressible viscous fluid, by integrating the immersed boundary method and a finite difference WENO scheme. The hybrid algorithm maintains simple and efficient treatment of the moving interface based on the immersed boundary formulation, and achieves high-order accuracy, with the WENO method and the Hodge decomposition technique employed, in the majority of the fluid domain that does not contain the interface. We demonstrate the performance of the proposed algorithm and its various numerical components through a number of computational examples.

Free in-plane vibration analysis of rotating annular panels with elastic boundary restraints

In this paper, an accurate solution for the in-plane vibration analysis of rotating circular panels with general edge restraints is established by solving the governing equations and boundary conditions, simultaneously. Two coupled in-plane displacement fields are constructed as Fourier series with smoothed supplementary polynomials to remove the relevant differential discontinuities associated with the original radial component at inner and/or outer edges. Any type of boundary conditions can be easily achieved by setting the boundary restraining stiffness accordingly. System characteristic matrix is obtained through the comparison of all expanded Fourier series coefficients. Numerical examples are then presented to validate the correctness and reliability of the proposed solution by comparing them with those solved from other approaches. Based on the model established, significant influence of elastic boundary condition on the in-plane modal characteristics of rotating annular panel is investigated and addressed. This work represents the first time that analytical solution for the in-plane vibration of rotating circular panel with general boundary conditions is derived, and can also shed some new light on the understanding of complex dynamic behavior of such structural system.

On the linear elastic responses of the 2D bonded discrete element model

SummaryThe bonded discrete element model (DEM) is a numerical tool that is becoming widely used when studying fracturing, fragmentation, and failure of solids in various disciplines. However, its abilities to solve elastic problems are usually overlooked. In this work, the main features of the 2D bonded DEM which influence Poisson's ratio and Young's modulus, and accuracy when solving elastic boundary value problems, are investigated. Outputs of numerical simulations using the 2D bonded DEM, the finite element method, a hyper elasticity analysis, and the distinct lattice spring model (DLSM) are compared in the investigation. It is shown that a shear interaction (local) factor and a geometric (global) factor are two essential elements for the 2D bonded DEM to reproduce a full range of Poisson's ratios. It is also found that the 2D bonded DEM might be unable to reproduce the correct displacements for elastic boundary value problems when the represented Poisson's ratio is close to 0.5 or the long‐range interaction is considered. In addition, an analytical relationship between the shear stiffness ratio and the Poisson's ratio, derived from a hyper elasticity analysis and applicable to discontinuum‐based models, provides good agreement with outputs from the 2D bonded DEM and DLSM. Finally, it is shown that the selection of elastic parameters used the 2D bonded DEM has a significant effect on fracturing and fragment patterns of solids.

Investigations on the aerothermoelastic properties of composite laminated cylindrical shells with elastic boundaries in supersonic airflow based on the Rayleigh–Ritz method

This paper is devoted to investigate the aerothermoelastic flutter and thermal buckling characteristics of composite laminated cylindrical shells with elastic boundary conditions. In the structural modeling, Donnell's shell theory is employed, and the supersonic piston theory is applied to evaluate the aerodynamic pressure. The elastic boundary constraints for the cylindrical shell are simulated by a series of distributed artificial springs. The shape functions of the laminated cylindrical shell with elastic boundary conditions are composed of characteristic orthogonal polynomials which are derived by the Rayleigh–Ritz method. The equation of motion of the structural system is established using Hamilton's principle. The high accuracy of the present method is verified by comparing the natural frequencies and flutter bounds with published results. Based on the frequency domain method, the aerothermoelastic properties of the laminated cylindrical shell with elastic boundary conditions are analyzed. The influences of different types of spring stiffnesses on the flutter and thermal buckling bounds are studied in detail. In addition, the influences of several parameters including the length to radius ratio, ply angle and boundary condition on the aerothermoelastic behaviors of the laminated cylindrical shell with elastic boundary conditions are researched.

Three-dimensional exact solution for the free vibration of thick functionally graded annular sector plates with arbitrary boundary conditions

A new three-dimensional exact solution for the free vibrations of arbitrary thick functionally graded annular sector plates with arbitrary boundary conditions is presented. The three-dimensional elasticity theory is employed to formulate the theoretical model. According to a power law distribution of the volume of the constituents, the material properties change continuously through the thickness of the functionally graded annular sector plates. Each of displacements of the annular sector plates, regardless of boundary conditions, is expanded as a three-dimensional (3-D) Fourier cosine series supplemented with closed-form auxiliary functions introduced to eliminate all the relevant discontinuities with the displacements and its derivatives at the edges. Since the displacement fields are constructed adequately smooth throughout the entire solution domain, an exact solution is obtained based on the Ritz procedure by the energy functions of plate. The excellent accuracy and reliability of the current solutions are demonstrated by numerical examples and comparison of the present results with those available in the literature, and numerous new results for thick FG annular sector plates with elastic boundary conditions are presented. The effects of gradient indexes are also illustrated.

Free vibration analysis of laminated composite elliptic cylinders with general boundary conditions

In this paper, a unified analysis model is proposed for the first time to study the free vibration of laminated composite elliptic cylinders with general boundary conditions including the classical boundary, elastic boundary and their combinations. The theoretical model is established by means of the modified variational principle and multilevel partition technique based on the first-order shear deformation theory. The interface continuity and boundary constraints are enforced by using the coupling and boundary spring technique. On the basis of that, the displacement components of each shell domain are expanded in the form of double Jacobi polynomials along the meridional and circumferential direction. The convergence and comparison analysis for laminated composite elliptic cylinders subject to different classical boundary conditions is conducted to show the reliability and accuracy of the present method. To make the research topic understood better, some mode shapes are also depicted. The present solutions show stable and rapid convergence characteristics, and the natural frequencies and mode shapes agree well with the Finite Element Analysis results. Some new vibration results and parameterized results are presented and may be as the reference data by other researchers in the future.

Free vibration of arbitrary-shaped laminated triangular thin plates with elastic boundary conditions

Abstract This paper studies the free vibration of arbitrary shaped laminated triangular thin plates based on a modified Fourier series method. An arbitrary shaped triangular plate is mapped into a right-angled isosceles triangular plate with unit length by the coordinate transformation. By padding another right–angled isosceles triangular plate which is near zero thickness on the plate after transformation. The displacement functions are then generally expressed as the combinations of Fourier cosine series and supplementary functions introduced to eliminate the discontinuous or jumping phenomenon in the boundaries. The classical thin plate elasticity theory is employed to construct the energy expressions of the plate. The Rayleigh-Ritz method is used in the present method to obtain all the unknown series expansion coefficients. A number of results are presented to verify the convergence and accuracy of current solution method to laminated triangular thin plates with arbitrary shapes, different material parameters and different boundary conditions. Additionally, some new results are given as the benchmark for future research, which are based on the various boundary conditions and geometric dimensions.

Three-dimensional exact solution for vibration analysis of thick functionally graded porous (FGP) rectangular plates with arbitrary boundary conditions

In this paper, a novel three-dimensional exact solution is performed for vibration analysis of thick functionally graded porous rectangular plates subject to arbitrary boundary conditions. The theoretical model is based on the three-dimensional elastic theory. Three kinds of porosity distributions including even, uneven and the logarithmic-uneven are performed. Non consideration boundary conditions, all displacements of the rectangular plate are unified expanded as a improve Fourier series, which consists of standard three-dimensional (3-D) Fourier cosine series supplemented with closed-form auxiliary functions introduced to eliminate all the relevant discontinuities with the displacements and its derivatives at the edges. As the displacement field is sufficiently smooth in the entire solution domain, the exact solution of the energy function of the plate is obtained through the Rayleigh Ritz process. The numerical examples are given to verify the accuracy and reliability of the present method, in which both the present results and those reported in the literature are provided. Besides, several numerous new results for thick FGP plates with elastic boundary conditions are also presented.