Elastic Boundary Research Articles

MAGPIE: a web-based, open-source framework for plate vibration analysis

The analysis and simulation of plate vibrations are central to many branches of science and engineering, specifically acoustics and musical acoustics. Well-developed plate theories exist, along with numerous simulation software packages, but they are seldom available in open-source form and are not easily accessible to a greater audience. MAGPIE is a framework for interacting with a novel finite difference scheme for plates under general elastic boundary conditions. Free, clamped, and simply supported boundary conditions are obtainable by setting stiffness coefficients to very small or large values. Boundary conditions are parameterised which allow for intermediate possibilities. A web browser based implementation of the framework includes interactive elements. Users begin by providing parameters for an isotropic plate. These parameters are then used to generate visualisations of the mode shapes from the finite difference scheme simulation. Elastic boundary conditions can then be changed on a continuous scale to see their effect on mode shapes. Results are then available for export in common image and data formats. MAGPIE aims to make it easier to explain concepts such as mode shapes to a wider audience and also provide a tool for research that is accessible to students, scientists and musical instrument builders.

Analysis of dynamic modeling and power flow of ABH laminated composite thin beam with elastic boundaries

To investigate the energy distribution characteristics of a laminated structure with an acoustic black hole (ABH), this study introduces the dynamic modeling and power flow analysis of an ABH laminated thin beam (ABH laminated beam) with elastic boundary conditions for the first time. ABH profile is defined by a power function in the dynamic modeling process. Utilizing the Euler-Bernoulli beam theory, the constitutive equation of ABH laminated beams is formulated by using the isogeometric method. Three sets of artificial springs are employed to replicate the elastic boundary conditions, incorporating the potential energy of the springs. Subsequently, the dynamic model of the ABH laminated beam is developed by solving the Lagrange equation concerning the total energy within the uniform region, ABH region, and boundary. Through numerical examples, the convergence and accuracy of the proposed modeling approach are validated by comparing the results with those obtained from COMSOL Multiphysics and experimental data. The analysis of power flow and structural intensity elucidates the energy propagation behavior in ABH laminated beams and the underlying mechanism of the ABH effect. The results demonstrate that the ABH laminated beam displays remarkable absorption and dissipation characteristics, introducing a new perspective for the design and application of ABH structures.

Accurate and efficient analytical simulation of free vibration for embedded nonlocal CNTRC beams with general boundary conditions

This study aims to evaluate the free vibrational response of embedded restrained nanobeams enriched by nanocomposites based on an exact Fourier series approach. In order to capture the small-scale effects on the dynamical response, Eringen's differential form of nonlocal elasticity is used which employs a scale (nonlocal) parameter. Within the framework of Rayleigh and Bernoulli-Euler beam theories, including the effect of nonlocality and employing the Fourier sine series together with Stokes' transformation, systems of linear equations are obtained and solved using the coefficient matrices. The combined effects of elastic boundary conditions, elastic foundation, dispersion patterns and volume fractions of carbon nanotubes, and nonlocal parameter are examined by solving eigenvalue problems constructed with Fourier infinite series. Free vibration frequencies are calculated for carbon nanotube-reinforced nanobeams under different rigid or restrained boundary conditions, including Winkler-Pasternak type elastic foundation. A comprehensive parametric study is performed, focusing on various effects for the free vibrational response of the composite nanobeam reinforced with carbon nanotubes. It is concluded that adding a small amount of carbon nanotube material can reinforce the stiffness of the composite nanobeam, and its free vibration performance is significantly affected by the distribution patterns, elastic medium, and boundary conditions.

Dynamics of a single cavitation bubble near an oscillating boundary

Cavitation and its effects are well investigated, especially single bubble cavitation and its collapse near rigid and elastic boundaries. In our current article, we investigated novel experiments of a single cavitation bubble near an oscillatory boundary. We generated the cavitation bubble by laser focusing in water. A flat glass plate was fixed to the shaft of the magnetostriction oscillator coil. We investigated the dynamics of bubbles at two relative wall distances (ratio of the distance between the bubble center and plate surface to the maximum radius of the bubble) of the bubble from the glass plate in combination with four modes of oscillation. Each mode has specific frequency and amplitude of oscillation. The high-speed camera captured the dynamics of the bubble using the back-illumination method with a framing rate of 120Kfps and simultaneously we used an optical CMOS sensor to measure the oscillation of the glass plate. We presented a clear comparison among the bubble dynamics near stationary and oscillating plates with parameters such as oscillating modes and direction. We correlated the dynamics of the bubble with the motion of the plate. In addition, we highlighted the differences including the characteristics of bubble shape and jetting that occurred during the collapse phase. The comparison of the time histories of the bubble’s equivalent size postulated that the bubble’s collapse times vary significantly in some cases compared to the bubble’s dynamics near the stationary plate. In all cases, we noticed the shortening of the bubble’s collapsing time, i.e. accelerated collapses. In our findings, we noticed a collapse times reduction of about 4–15%. Our finding signifies the importance of introducing the oscillation of the boundaries to obtain effective energy concentration over the time during the collapse. Our study also suggests that forced oscillation of boundaries is undesirable for destructive cavitation effects. The method we suggested for the manipulation of bubble dynamics holds potential for enhancing the efficiency of applications such as lithotripsy in biomedical devices, actuation and micro pumping in microfluidic devices, and effective semiconductor surface cleaning. Not but least, obtained results can be used as benchmark in future for validating numerical methods.

Dynamic response of sandwich functionally graded nanoplate under thermal environments and elastic foundations using dynamic stiffness method

The present paper introduces the development of dynamic stiffness method for analyzing small-scale sandwich functionally graded nanoplates resting on elastic foundation in thermal environments. The mathematical formulation is based on classical plate theory in conjunction with nonlocal elasticity theory. The governing equation is derived using Hamilton’s principle. The dynamic stiffness matrix is obtained through the application of the Levy displacement approach and assembled to form the global stiffness matrix. The final matrix is solved for natural frequency of the plates using the Wittrick–Williams algorithm. The proposed methodology is validated against existing literature, demonstrating a strong agreement. Various parametric studies explore the effects of thermal environments, volume fraction index, sandwich configurations, elastic foundation characteristics, nonlocal parameter and boundary conditions. The results show the versatility of the proposed approach in addressing small scaled complex engineering structures. This research significantly contributes to the understanding and analysis of sandwich functionally graded nanoplates, providing valuable insights for applications in aerospace, structural systems, sensors, actuators, and energy harvesting devices.

Homogenization of quasi-periodic conformal architectured materials and applications to chiral lattices

In this study, we propose to extend asymptotic periodic homogenization for non-periodic continuous microstructured media, assuming that the non-periodic geometry (called quasi-periodic) can be designed by a conformal planar transformation of a periodic parent domain architectured media with periodically disposed unit cells. Conformal transformations are shown to play a privileged role in the design of circular macroscopic heterogeneous domains tessellated with non-periodic unit cells, obtained from a periodic parent domain architectured with these unit cells. The conditions for conformal invariance are established, leading to the general form of conformal transformation in their dependencies upon the periodic coordinates. It is shown that any conformal map can be decomposed into the product of an isotropic dilatation function of the first periodic spatial position of decreasing exponential type and a rotation characterized by an angular function linear in the second periodic position. A general theory of quasi-periodic homogenization in the framework of conformal transformations is established for the first time, leading to an expression of the tensor of quasi-periodic moduli which is fully evaluated from the solution of the elasticity boundary value problem posed over the periodic unit cell. The influence of microcurvature distortion of individual unit cells on their effective properties is evaluated. Closed-form solutions are confronted to numerical examples issued from the implementation of circular periodicity in a finite element solver, showing overall a good agreement with the identified homogenized moduli.

Vibration of composite open shell of hydrogen-electric fuselage with rectangular cutout in hygrothermal circumstances: theoretical and experimental research

An effective formulation is performed to approximate calculate the vibration of the typical composite fuselage battery compartment structures of the hydrogen-electric aircraft, which are highly sensitive to hygrothermal circumstances. The strain energy, kinetic energy and hygrothermal potential energy of CLOCSRC are deduced in frame of Donnell's shell assumption and a series of hypothetical elastic springs operating on the edges of divided components are introduced to simulate coupling connection and the elastic supports of the open shell, which contribute to total system energy in the form of elastic spring potential energy. The equations of motion for CLOCSRC subjected to distinct hygrothermal circumstances and arbitrary boundaries are eventually obtained in frame of the Rayleigh-Ritz method by means of the modified orthogonal polynomial as the displacement admissible function. After confirming the convergence and accuracy of the presented formulation by experiment and finite element method, several numerical examples are presented to investigative the effects of geometric parameters, elastic boundaries and hygrothermal circumstances on the vibration of CLOCSRC. Differ from traditional investigation, the present paper offers an effective approach to insight the hygrothermal mechanism of open shell with rectangular cutouts creatively. The relevant investigation demonstrated that the current strategy is beneficial for further research on the vibration characteristics of CLOCSRC.

A semi-analytical method using auxiliary sine series for vibration and sound radiation of a rectangular plate with elastic edges

This paper proposes an efficient semi-analytical method using auxiliary sine series for transverse vibration and sound radiation of a thin rectangular plate with edges elastically restrained against translation and rotation. The formulation, constructed by two-dimensional sine and/or cosine series, can approximately express the bending displacement, and calculate vibration and sound radiation under excitation of point force, arbitrary-angle plane wave, or diffuse acoustic field with acceptable accuracy. It is also applied for baffled or unbaffled conditions. A post-process program is developed to predict vibrating frequencies and modes, mean square velocity spectrum, and sound transmission loss via reduced-order integrals of radiation impedances. The method is validated by experiment and simulation results, demonstrating accurate and efficient computation using a single program for transverse vibration and sound radiation of a plate under different elastic boundary conditions and different excitations. Formulas given in this paper provide a basis for the code development on transverse vibration and sound radiation analysis of thin plates.

Facteurs pronostiques : degré d’invasion de la sous-séreuse, distance tumeur-séreuse et invasion de la limitante élastique de la sous-séreuse dans l’adénocarcinome colique

The aim was to study the prognostic impact of tumor infiltration of the subserosa in colonic adenocarcinoma, by evaluating the degree of tumor infiltration in the subserosa (DISS), tumor-serosa distance (DTS), and invasion of the elastic boundary of the subserosa (ILE) after elastic fiber staining.Material and methods: All patients operated on for colonic adenocarcinoma classified as pT3 without lymph node or visceral metastasis operated on at CHU-Amiens between 2004 and 2017 were included. All slides were reviewed by 2 pathologists. Bivariate and subgroup analyses were performed according to the presence of a DISS ≤5 mm or >5mm, a DTS ≤1mm or >1mm and the presence or absence of an ILE. These statistical analyses were then correlated with 5-year survival.Results: 101 patients were included in the study. We performed elastic fiber staining on an average of 2 tumor blocks per case, and 39.6% of patients had invasion of the elastic boundary. However, bivariate and subgroup analyses showed no statistically significant association between DISS, DTS or ILE and 5-year survival.Conclusion: None of these three histological criteria proved to have prognostic value in our series, contrary to some results in the literature. However, as these data are subject to a number of confounding factors, we do not recommend that pathologists specify these different criteria in their reports.

Traveling wave vibration characteristic analysis of FRC sandwich cylindrical shells with FGP-GPLRC core under discontinuous elastic boundary conditions

In this study, a fiber-reinforced composite (FRC) sandwich cylindrical shell with a functionally graded porous graphene platelet reinforced composite (FGP-GPLRC) core is investigated, and the traveling wave vibration characteristics of this new sandwich cylindrical shell under discontinuous elastic boundary conditions are analyzed for the first time. The influences of centrifugal and Coriolis forces are introduced into the model. By employing the first-order shear deformation theory (FSDT) and the continuity assumption of the layerwise theory, the governing equations of the rotating sandwich shell are obtained. The artificial springs are implemented at the ends of the sandwich shell to represent the discontinuous elastic boundary conditions. The natural frequency of the shell is solved via the Rayleigh-Ritz method and the state space method. Then, the approach proposed is validated by comparing the present results with those reported in the literature and the finite element method. Finally, the effects of various parameters on the traveling wave frequency of the shell are evaluated. It was found that in practical engineering, the appropriate thickness of the FGP-GPLRC core should be selected according to the boundary conditions and other factors, and the ply angle should be adjusted to obtain more stability of vibration.

Semi-analytical solutions of kinked edge cracks

A surface crack may advance either by further penetrating inward or by kinking out in response to possible variations in driving force or material properties. This paper provides a general approach to solve the planar elastic boundary value problem (BVP) of a kinked edge crack in a half-space. Semi-analytical solutions to BVPs with far-field uniform tension and concentrated loads on the free surface are given. The solution enable us to derive related information for reliability analysis, including stress fields, stress intensity factors (SIFs), and crack opening displacement (COD). An extension to the Cotterell–Rice formula (Cotterell and Rice, 1980) on SIFs for infinitesimal kinks is supplied, which is applicable to cracks with kinked segments of arbitrary length. Zooming in on a broadly concerned BVP of a cross-coating crack in a coating–substrate with a kink in the interface, we show the SIFs of the kink tip following an empirical formula K∗=αexp(−βah)+K∞∗ as a/h>0.2, where a is the length of the interface segment and h the coating thickness, and K∞∗ is the asymptotic solution by Thouless et al. (1987, 1989). The formula is in accord with existing solutions on the two asymptotic limits of the kinked part.

Liquid-Crystal-Like Dynamic Transition in Ferroelectric-Dielectric Superlattices.

Nanostructured ferroelectrics display exotic multidomain configurations resulting from the electrostatic and elastic boundary conditions they are subject to. While the ferroelectric domains appear frozen in experimental images, atomistic second-principles studies suggest that they may become spontaneously mobile upon heating, with the polar order melting in a liquidlike fashion. Here, we run molecular dynamics simulations of model systems (PbTiO_{3}/SrTiO_{3} superlattices) to study the unique features of this transformation. Most notably, we find that the multidomain state loses its translational and orientational orders at different temperatures, resembling the behavior of liquid crystals and yielding an intermediate hexaticlike phase. Our simulations reveal the mechanism responsible for the melting and allow us to characterize the stochastic dynamics in the hexaticlike phase: we find evidence that it is thermally activated, with domain reorientation rates that grow from tens of gigahertzs to terahertzs in a narrow temperature window.

Theoretical and experimental study of the high-frequency nonlinear dynamic response of a 10 MW semi-submersible floating offshore wind turbine

To further investigate the hydrodynamic performance of a novel 10 MW semi-submersible floating offshore wind turbine (FOWT) OUCwind and the dynamic response of the FOWT, especially the high-frequency dynamic response induced by the high-order high-frequency hydrodynamic loads, an experimental study for OUCwind is carried out. A novel regular wave condition design method is proposed, i.e., making incidence wave periods that exactly meet an integer multiple of the natural period of the wind turbine, to stimulate the structural resonance of the tower. To determine the natural period of the wind turbine with an elastic boundary, an irregular wave test is carried out and the natural period of the wind turbine with the elastic boundary is proved to be 3 s. Different parts of the FOWT model are verified to satisfy the corresponding scale similarity through a series of validation tests. The hydrodynamic performance of OUCwind is investigated. Linear response is obtained by applying the band-pass filter to the total response. The pitch motions have apparent high-frequency components aligned with the natural frequency of the wind turbine under the environmental conditions with a wave periods of 6 s. For surge and heave motion, the linear component dominates these two responses and the effect of the high-frequency component on these two responses is negligible. The second-order doubling and third-order tripling wave effects have a tremendous impact on the tower-top shear force and mooring line tension. The experimental data providing high-frequency dynamic responses up to the quadruple order can be used for the validation and correction of mid-fidelity and high-fidelity numerical models. Finally, the mechanisms of the generation of the high-frequency dynamic response of the FOWT are also concluded in an illustrative form.

Single domain Chebyshev spectral method for analyses of the vibroacoustic characteristics of baffled irregularly shaped plates

In this study a Chebyshev spectral method is introduced to analyse the vibroacoustic characteristics of plates with arbitrary shape. This approach involves a unique expansion of the governing equations for plate displacements using Chebyshev polynomials, coupled with Gauss-Chebyshev-Lobatto sampling. A significant advancement in computational efficiency is the use of a tensor product configuration. To manage irregular shapes, the elements of the inner product matrix corresponding to the open regions were set to zero. For precise shape representation, a substantial number of Chebyshev-Gauss-Lobatto points, surpassing the polynomial truncation count, are carefully selected. the elastic boundary conditions of the plate were simulated by applying artificial spring techniques at different points. The prediction accuracy of the vibroacoustic phenomena, including those of the mean square velocity and sound power, were rigorously confirmed through comprehensive comparisons with results from previous studies and those obtained using the finite element method. Furthermore, the accuracy and versatility of the method are demonstrated through a variety of numerical examples encompassing plates with geometries shaped like rectangules, triangules, clouds, and pigs geometries, and different boundary conditions.

Warren truss inspired hierarchical beams for three dimensional hierarchical truss lattice materials

Networks of beams are a subject of increasing interest to create architected materials with exceptional mechanical properties and low density. This paper investigates the mechanical properties of one dimensional (1D) hierarchical beams for the development of three dimensional (3D) truss lattice materials. These 1D hierarchical beams are constructed in two configurations by placing axial and inclined struts in single and double laced Warren truss patterns in each side of a beam with polygon cross section. Analytical and numerical analyses have been used to characterize their mechanical properties, including the elastic modulus, second moment of area, and shear stiffness of hierarchical beams drawn from a broad design space. Also, the failure limits of the beams with respect to parent material failure and various buckling modes are probed. Finally, the hierarchical beams have been implemented as the constituent members of Kelvin and octet lattices, and the elastic modulus and failure boundaries of the second-order hierarchical lattices are evaluated. The investigation reveals the competition between the elastic properties in the individual hierarchical beams based on different combinations of the design variables. The stiffness of the designs under compression and bending is found to be a function of the axial member size and cross sectional shape of the hierarchical beam. On the other hand, the shear stiffness of hierarchical beam designs is a function of the inclined member size and their inclination angle. It is demonstrated that incorporating hierarchy in the Kelvin and octet truss lattices can enhance the load bearing capacity of designs at low relative densities when compared to their hollow counterparts. Also, it is shown that second-order hierarchical stretching and bending-dominated lattices incorporating first-order hierarchical beams, can not only achieve but also surpass the strength and stiffness scaling relations established for first-order lattices. This becomes particularly noteworthy when considering bending-dominated lattices, as the hierarchy can drive their stiffness toward the boundaries, enabling them to outperform their equivalent stretching-dominated rivals.

Analytical solutions for the free vibration of cross-ply composite laminated plates with arbitrary biaxial symmetric geometry

This paper presents an analytical model for the free vibration of composite laminated plates with arbitrary biaxial symmetric geometry under elastic boundary conditions, applicable to cross-ply fiber-reinforced composites. In this model, the domain segmentation integral method proposed by the authors is extended to the first-order shear deformation theory (FSDT) and the laminated plate theory. The energy integrals pertaining to the FSDT are analytically simplified by segmenting the integration domains and synthesizing orthogonal polynomial displacement functions over the plate domain coordinate intervals. This synthesis is achieved through the amalgamation of the penalty function method and the Gram-Schmidt orthogonalization process. Then, using the Rayleigh-Ritz procedure, a series solution for the free vibration problem of cross-ply fiber-reinforced composite laminated plates is obtained. The model provides an analytical mathematical relationship between the profile curve of composite laminated plates with arbitrary biaxial symmetric geometry and the strain energy, kinetic energy, and boundary potential energy during vibrational processes. The correctness and applicability of the method are validated by comparing the obtained results with those from the literatures. New results for laminated plates, such as hexagonal and astroid-shaped plates, were presented as references for future research. Taking a track shaped plate as an example, the effect of the number of layers on the vibration characteristics of cross-ply fiber-reinforced composite laminated plates with a certain thickness is studied.

Comparative analysis of heat transfer performance in cavities with varied wall conditions and nano encapsulated phase change material–water

This numerical study investigates convective flow within a square cavity subjected to a temperature gradient established between hot and cold walls. The cold wall is subjected to three different conditions: standard, elastic, and open boundary. The cavity is filled with an advanced nanofluid comprising water and nano encapsulated phase change material (NEPCM). The NEPCM feature a polyurethane shell encapsulating a nonadecane core, known for its phase transition capabilities and efficient storage or release of substantial latent heat. Numerical solutions for the momentum and energy equations are obtained for the three wall types, considering variations in the volume fraction of NEPCM (0.0≤ϕ≤0.05), Rayleigh number (105≤RaH≤106), and the fusion temperature of the core (0.1≤θF≤0.3). The results reveal that the type of wall in the cavity (standard, elastic, or open) has a significant impact on the flow patterns and heat transfer characteristics. The open wall, in particular, showed a distinct behavior with a lower mid-region temperature and a higher heat transfer rate at lower NEPCM concentrations compared to the standard and elastic walls. Adjusting the concentration from 1% to 5% increases the heat transfer rates by 44, 49, and 37 percent for standard, elastic, and open wall conditions, respectively.

The modeling method for vibration characteristics analysis of composite laminated rotationally stiffened shell.

The composite laminated rotationally stiffened shell is widely applied in aviation, aerospace, ship, machinery and other fields. To investigate the vibration characteristics of composite laminated rotationally stiffened shells with varying elastic boundary conditions, a modeling method of composite laminated rotationally stiffened shells is established. Firstly, the first-order shear deformation theory (FSDT) and the modified Fourier series method are effectively applied to establish the allowable displacement function of the composite laminated rotationally stiffened shell. Secondly, the energy function of composite laminated rotationally stiffened shell is established, and the simulation of complex elastic boundary and coupling boundary is realized by using artificial virtual spring technology. Thirdly, the Rayleigh-Ritz method is used to solve the energy function. Finally, the vibration characteristics of composite laminated rotationally stiffened shells are obtained and analyzed. In the analysis of numerical results, the fast and uniform convergence of analysis modeling and the accuracy of the calculated results are verified. On this basis, the effect of some important parameters such as thickness-to-radius ratio and length-to-radius ratio of shell, boundary spring stiffness values, cone apex angle, thickness and width of laminated beams, number of stiffeners on the vibration characteristics of composite laminated rotationally stiffened shell is studied. In theory, it makes up for the vibration characteristics analysis of composite laminated rotationally stiffened shells. In practical application, it guides the noise reduction design of related structures.

A study on attenuation patterns of acoustic waves in waveguide structures with flexural boundaries

This research article investigates the attenuation patterns of acoustic waves within waveguide structures featuring flexural boundaries. By employing advanced methods in vibrations and controls, the study aims to enhance our understanding of waveguide behavior and offer valuable insights for optimizing acoustic wave propagation in diverse applications. The entire waveguide structure is tuned through cooperative interactions involving elastic plates, membranes, and rigid boundary surfaces. The lower boundary of the expansion chamber consists of a rigid plate, with elastic membranes covering the upper surfaces of the cavity. The problem is formulated in terms of transmission and reflection coefficients of the modes, enabling the solution of the scattering problem. Matching acoustic pressure and velocity at the waveguide junction yields the scattering coefficients. The results reveal that the scattering coefficients are intricately influenced by both the geometric characteristics of the waveguide and the frequency of the incident waves. Notably, the first higher-order mode of the cut-off frequency of the waveguide structure corresponds to the frequency at which the dissipation coefficient reaches its maximum value. This research provides valuable insights into the scattering behavior of acoustic waves in waveguide configurations, contributing to the design of acoustic devices and systems. Rigorous support and justification for all aspects of the mode-matching method, including pressure and velocity matching, eigen properties, physical edge conditions, convergence criteria, and energy conservation, are provided through thorough algebraic and numerical analyses, confirming the validity of the solution methodology.

Elastic Solids with Strain-Gradient Elastic Boundary Surfaces

Elastic Solids with Strain-Gradient Elastic Boundary Surfaces