Three-dimensional Theory Of Elasticity Research Articles

Legendre–Ritz solutions for vibration characteristics of three-dimensional double-layer lattice truss sandwich plates

In this paper, Legendre–Ritz solutions for vibration characteristics of double-layer lattice truss sandwich plates are proposed based on three-dimensional (3-D) elasticity theory. Double-layer lattice truss sandwich plates consist of three face sheets and two cores. The face sheets are composed of fiber-reinforced composites and the cores are composed of pyramidal lattice trusses made of isotropic materials. The pyramidal lattice core is equated to a homogeneous plate according to equivalence principle. Based on 3-D elasticity theory and pseudo excitation method (PEM), each layer's energy expressions of double-layer lattice truss sandwich plates are constructed. After that, Legendre polynomials are adopted to describe displacement components in length, width and thickness directions. On this basis, the Rayleigh-Ritz method is used to solve energy functions to obtain motion equations of double-layer lattice truss sandwich plates. The accuracy and validity of the present method are corroborated by comparisons with references and finite element method (FEM). Additionally, in order to find out effects of relevant parameters on dynamical characteristics of double-layer lattice truss sandwich plates, parametric studies are carried out by means of some numerical examples.

Free vibration analysis and multi-objective robust optimization of three-dimensional pyramidal truss core sandwich plates with interval uncertain parameters

In this study, the free vibration analysis and multi-objective robust optimization of three-dimensional pyramidal truss core sandwich plates with interval uncertain parameters are fulfilled. The numerical model for free vibration of the plate is derived by combining the three-dimensional elasticity theory and Rayleigh-Ritz method, and the validity of the model is illustrated by numerical results. On this basis, considering various uncertainties within the plate, a new uncertainty-propagation analysis method is constructed by integrating the numerical model, interval-analysis model, kriging model and optimization. The one-dimensional and multi-source uncertainty-propagation analysis of the fundamental frequency is finished using this method, and the accuracy is proved by comparing with Monte Carlo simulation results. Meanwhile, to minimize the effects of uncertainties on the performance of plates at the design stage, an objective multi-objective robust optimization model with the objectives of maximizing the fundamental frequency and minimizing the robustness factor is established. Finally, an improved pelican optimization algorithm is proposed by introducing the improved opposition-based learning strategy, tracking strategy with step control and convergence strategy. And the Pareto front and corresponding design schemes applicable to different working environments are obtained without repeating the optimization.

Exact three-dimensional elasticity analysis for buckling of composite laminated plates resting on viscoelastic foundation

This paper aims to present novel exact solutions for the buckling of a laminated plate resting on the viscoelastic foundation with both normal and shear viscoelastic layers. The governing equations of plate buckling are derived using three-dimensional elasticity theory and state-space formulation. The normal and shear layers of the viscoelastic foundations are modeled using the generalized Maxwell model to represent both the elastic and viscose properties of the foundation. To couple the viscoelastic foundation equation with the buckling equation, Boltzmann’s superposition principle along with the Laplace transform is utilized. Then, the effects of geometry, relaxation modulus of normal and shear layers, viscosity, and time are investigated on the buckling load. The results reveal that the higher viscosity coefficient leads to a slower rate of change in the buckling loads. In addition, the viscoelastic properties have a significant impact on the buckling behavior of the plate. In this regard, the results show that instead of the expected second mode at a constant aspect ratio, the plate experiences the first mode as time passes. The computed results also show that there is a critical threshold. When the foundation stiffness exceeds this threshold, the conventional method of reducing the aspect ratio to prevent buckling not only proves ineffective in reducing the probability of buckling but also, in fact, leads to an increase in buckling occurrences. In addition to the analytical investigation, a finite element (FE) analysis is carried out to study the buckling response of the composite plate. The finite element results also show a reasonably good agreement with those of the analytical method.

Parametric Analysis of Free Vibration of Functionally Graded Porous Sandwich Rectangular Plates Resting on Elastic Foundation.

Based on the three-dimensional elasticity theory, the free vibration of functionally graded porous (FGP) sandwich rectangular plates is studied, and a unified solution for free vibration of the plates is proposed in this study. The arbitrary boundary conditions of FGP sandwich rectangular plates are simulated by using the Rayleigh-Ritz method combined with artificial spring theory. The calculation performances of the unified solution for FGP sandwich rectangular plates such as convergence speed and computational efficiency are compared extensively under different displacement functions. In addition, three kinds of elastic foundation (Winkler/Pasternak/Kerr foundations) and three porosity distributions are considered. Some benchmark results and accurate values for the free vibration of FGP sandwich rectangular plates resting on elastic foundations are given. Finally, the effects of diverse structural parameters, elastic foundations with different parameters, and boundary conditions on the free vibration of the FGP sandwich rectangular plates are analyzed.

Free vibration analysis of fiber-reinforced composite multilayer cylindrical shells under hydrostatic pressure

With the development and application of composite materials in deepwater submersibles, the influence of high external pressure from deepwater environments on the vibration characteristics of composite pressure-resistant cylindrical shells cannot be ignored. In this paper, the free vibration characteristics of prestressed composite multilayer shells are methodically examined by establishing partial differential equations based on the three-dimensional elasticity theory. To solve these equations, the state-space technique and double Fourier series expansion are employed to transform them into state-space ordinary differential equations. These equations are utilized to analyze vibration problems of simply supported composite multilayer cylindrical shells with arbitrary layered angles. The initial stress equilibrium equation is solved to determine the prestress induced by the hydrostatic pressure. This overcomes the limitation of membrane stress and considers non-uniform distributions of radial and interlaminar prestresses. The proposed theoretical model and solution strategies are validated by comparing with theoretical and experimental results from the existing literature and the finite element method. Additionally, the present investigation reveals that the natural frequencies of thick shells can be calculated more accurately using the prestress derived by the proposed approach, especially subjected to noticeable hydrostatic pressure. Finally, the performed investigations indicate that the circumferential prestressing has a more pronounced effect on the shell's natural frequencies than axial prestressing.

Coupled machine-learning approach and thermoelasticity for analyzing thermal stability of the nanocomposite sandwich system

To offer actionable strategies for mitigating the negative impacts of thermal shock loading and enhancing structural design, this study emphasizes the analysis of the transient coupled thermo-elastic behavior of a sandwich cylindrical panel. This panel is subjected to thermal and heat-flux shocks and features face-layers reinforced by functionally graded graphene platelets (FG-GPLRC) alongside a polymeric core. Leveraging compatibility conditions, a three-layered sandwich structure model is developed. Ensuring calculation precision, the governing equations are formulated based on the comprehensive three-dimensional elasticity theory. A numerical approach is adopted for the analytical solution, targeting the response of both fully simple and clamped-supported setups. The Dubner and Abate method aids in inverting the Laplace transform to ascertain the system’s time evolution. In a subsequent section, the derived results are integrated into a novel machine learning algorithm, the Forest Regression for LAyer-wise Structures (FRAS), to anticipate deflection and stress within the layers. The machine learning outcomes align reasonably well with the multi-layered nature of the structure and diverse regression across layers. This suggests that, for swift preliminary insights in engineering applications, machine learning techniques might be a more efficient alternative to traditional numerical computations.

Asymptotically accurate and locking-free finite element implementation of first order shear deformation theory for plates

A formulation of the asymptotically exact first-order shear deformation theory for linear-elastic homogeneous plates in the rescaled coordinates and rotation angles is considered. This allows the development of its asymptotically accurate and shear-locking-free finite element implementation. As applications, numerical simulations are performed for circular and rectangular plates, showing complete agreement between the analytical solution and the numerical solutions based on two-dimensional theory and three-dimensional elasticity theory.

Application of Transition Finite Elements in hpq-Adaptive Modeling and Analysis of Machine Elements

This paper concerns modeling and analysis of machine elements using the adaptive finite element method. The adaptation used is of hpq type, which means that the finite element dimension h and element transverse q and longitudinal p approximation order may be different in each element. These parameters are determined automatically by the program so as to obtain modelling and approximation error levels not higher than the assumed admissible level of the errors. The presented paper focuses on the use of transition elements between basic elements corresponding to the three-dimensional theory of elasticity and the first-order shell model. Three applied transition elements differ in their assumptions regarding the continuity of the displacement, strain and stress fields between the basic models. The effectiveness of the application of transition elements was assessed in terms of: removal of the internal boundary layer at the boundary between the models and convergence of adaptive solutions taking into account these models.

Study of nanotube waviness influence on the behaviors of spherical nanocomposites

Considering the prominent properties of nanocomposites, the closer their modeling is to reality, the more suitable it is for engineering applications. One of the steps that can be taken for this purpose is to include the waviness of the reinforcements in the study of nanocomposites. In this article, for the first time, the effects of the waviness of reinforcing nanotubes in spherical nanocomposites on the behavior of mechanical waves have been studied. Several different ideas, such as adopting random contact model, the excluded volume of two spherocylinders, microscopic images and experimental results, are used for modeling wavy nanotubes in spherical nanocomposites, each of which has its own characteristics. For modeling the spherical nanocomposite itself, three-dimensional elasticity theory in spherical coordinates is used. For several different cases, the results of the present models are compared and calibrated with the results of experimental tests, which adds to the attractiveness of the work. The influences of various parameters such as radius of spherical nanocomposite, waviness factor, nanotubes volume fraction and wave number on the results are also investigated. From the results of this article, the idea can come to mind that the effects of the waviness of the nanotubes cannot be ignored in some cases and should be included in the modeling, otherwise it will produce significant errors in the results.

Vibration Characteristics of FML Cylindrical Shell Bonded by Thin Piezoelectric Actuator and Sensor Layer with and Without Fluid–Structure Interaction Resting on Pasternak Elastic Foundation

The present study investigates the vibration analysis of cylindrical shells composed of fiber metal laminate (FML) with embedded piezoelectric layers, undergoing fluid-structure interaction (FSI) and resting on a Pasternak elastic foundation based on the principles of three-dimensional elasticity theory. Using the state space approach, the equations of motion were derived under simply supported boundary conditions. The natural frequencies of the FML cylindrical shell, accounting for the presence of a moving fluid, were computed by solving the eigenfrequency equations. The study examined the influence of various parameters, including boundary conditions, length-to-radius ratio, fluid type, fluid velocity, circumferential wave number, and radius-to-thickness ratio, on glass-reinforced aluminum laminate (GLARE), aramid-reinforced aluminum laminate (ARALL), and carbon-reinforced aluminum laminate (CARALL). A constant composite/metal volume ratio was assumed. The results obtained were validated by comparing with natural frequency values from the existing literature, confirming the agreement and convergence with previous studies. The results confirm that the highest natural frequency values are assigned to the CARALL, ARALL and GRALE structures in descending order. Furthermore, an increase in fluid flow velocity through the cylindrical shell correlates with a reduction in natural frequency.

ANALYTICAL AND NUMERICAL ANALYSIS OF SOLID CYLINDER MADE OF RADIALLY NONUNIFORM MATERIAL UNDER EXTENSION

Functionally graded materials are inhomogeneous composite materials, composed of two or more constituents selected to achieve desirable properties for specific applications. In this work, a solid cylindrical tube made of inhomogeneous composite materials under tensile action was analyzed within the context of three-dimensional elasticity theory. An analytical solution was obtained for computing the displacement and stress fields. It has been assumed that the elastic stiffness is varying through the functionally graded material according to radial variation laws: linear, power, and exponential laws, while Poisson's ratio is considered as constant. In order to check the relevance of the analytical solution, a finite element model of the cylindrical tube was constructed, taking into account variations in Young's modulus. Very good agreement has been found between the numerical results and the predictions of the analytical solution, which confirms the accuracy of our model. Numerous curves were plotted by adjusting the inhomogeneity parameter and the elongation value, revealing a significant effect. Thus, the inhomogeneity in material properties can be exploited to optimize stress distribution. Indeed, by tailoring the material properties to match the stress distribution in a specific load scenario, stress concentrations can be minimized in high-stress areas.

Electro-mechanical coupling model and interlaminar stress analysis of laminated plates containing GSR actuator

Due to its remarkable physical features, graphene nanosheets (GPN) are one of the most appealing reinforcing materials for composites. For polyvinylidene fluoride (PVDF), GPN reinforced composites can dramatically increase its piezoelectric and mechanical characteristics. If the interlaminar shear deformation of laminated plates containing uniform graphene sheets reinforced (GSR) smart piezoelectric layer, which material properties vary widely from layer to layer and subjected to electromechanical loading cannot be accurately predicted, the interlaminar stresses may be very high, eventually leading to interlaminar failure. In light of this, an effective mechanoelectrical coupling model for the accurate prediction of interlaminar stress for composite plates contains GSR actuators is developed in present study. Meanwhile, the finite element formulation (FEF) can be substantially simplified due to the expression of transverse shear stress components becoming more succinct. Therefore, by using the suggested electro-mechanical coupling theory, a three-node FEF is easily constructed. The refinement of transverse shear stress prediction in the context of electromechanical coupling can be accomplished through the application of the Reissner mixed variation theory (RMVT). The performance of the recommended plate model will be evaluated using the results derived from three-dimensional (3D) elastic theory and the selected model. By employing the RMVT method, we improve predictions of transverse shear stresses while considering the electromechanical coupling effect. The results from our model are compared with alternative models and 3D elasticity theory, demonstrating its superiority in satisfying the continuity requirements of transverse shear stresses and exhibiting excellent agreement with exact solutions. This validates the accuracy and applicability of our proposed model. Further to that, the prediction of mechanical characteristics for laminated plates with GSR actuators were systematically studied from the thoroughly perspectives of electromechanical load, piezoelectric layer thickness, graphene volume fraction, and some other parameters.

Analysis of three-dimensional thermal vibration coupling characteristics of composite shaft under axial load

The Carrera unified formulation (CUF) is employed to simplify the complete three-dimensional dynamic model based on the three-dimensional elasticity theory into a hierarchical one-dimensional dynamic model that maintains three-dimensional solution accuracy. The displacement function is formulated using Taylor polynomials and the improved Fourier series method (IFSM). Subsequently, the study solves the thermal vibration characteristics of the composite shaft under axial load through the application of energy functional and Hamilton's principle. The accuracy and stability of the present method are verified by comparing the present results with simulation. This study investigates the influence of thermal effects, structural parameters, and axial load on the vibrational characteristics of the composite shaft. The analysis results indicate that temperature, structural parameters, and axial load all significantly affect the fundamental properties of the composite shaft. As temperature and length-radius ratio increase, the natural frequency of the composite shaft gradually decreases. Regarding axial load, axial tension exhibits a positive correlation with the structure's natural frequency, whereas axial pressure shows a negative correlation. These findings hold significant engineering implications for utilizing composite shaft in the marine engineering sector.

Thin shells reinforced by fibers with intrinsic flexural and torsional elasticity

A two-dimensional thin shell model, derived from a three-dimensional Cosserat elasticity theory for fiber-reinforced solids, is presented. Models for single laminae and two-ply laminates are derived. The novelty of these models lies in the assignment of intrinsic flexural and torsional elasticity to the embedded fibers, regarded as continuously distributed material curves.

Levy-type analytical series solution for the three-dimensional free vibrations of functionally graded material rectangular plates with piezoelectric layers

Analytical solutions founded on three-dimensional theories play a crucial role in evaluating the credibility and precision of different plate theories and numerical methodologies. While Levy-type analytical solutions are widely recognized, they have been primarily confined to purely elastic plates. This study introduces a Levy-type analytical series solution for three-dimensional vibrations in a sandwich rectangular plate featuring a functionally graded material (FGM) core, along with piezoelectric material (PM) layers on the top and bottom surfaces. The behaviors of the FGM and PM layers were described using three-dimensional elasticity and piezoelasticity theories, respectively. In this study, the displacement functions and electric potential of each layer were expanded by Fourier series and polynomial auxiliary functions. An analytical series solution was then established by satisfying the governing equations of each layer, the mechanical and electric boundary conditions on the six faces of the plate, and the continuity conditions on the interfaces between the PM and FGM layers. To validate the proposed solutions, in-depth convergence studies were conducted for the vibration frequencies of the first six modes of sandwich square plates with various boundary conditions on the other pair of side faces. The well-converged results were then compared with published data based on various plate theories to verify the accuracy of these published data. Finally, accurate nondimensional frequencies were tabulated for the first six modes of sandwich rectangular plates with various aspect ratios, thickness-to-width ratios, PM-to-FGM layer thickness ratios, power law indices for the FGM layer, and six combinations of boundary conditions. These new numerical results when piezoelectric coupling is considered should be very useful to future analytical and numerical studies.

Vibration characteristic analysis of three-dimensional sandwich cylindrical shell based on the Spectro-Geometric method

This paper presents the numerical investigations on the free and random vibration properties of thick cylindrical shells under the three-dimensional (3-D) elasticity theory. The analysis of these properties is conducted using the Rayleigh-Ritz method in conjunction with the Spectro-Geometric method (SGM). The cylindrical shell considered in this study is composed of fiber-reinforced composite and functionally graded porous graphene platelet reinforced composites (FGP-GPLRC). To simulate different boundary conditions, an artificial spring technique is utilized. The displacement components, as determined by SGM, are incorporated into the energy expression of the cylindrical shell. By applying the Rayleigh-Ritz method, the intrinsic frequencies and vibration modes of the cylindrical shell are obtained. Additionally, the power spectral density (PSD) function of the cylindrical shell under random excitation is obtained by the Pseudo-Excitation method (PEM). The accuracy and effectiveness of the present method are verified through comparative studies with existing literature and finite element method (FEM). Furthermore, an investigation on the effects of various parameters on the intrinsic frequency and random response of the cylindrical shell is conducted.

A method to analyze the axisymmetric problem of FRP laminated tubes based on singularity correction

In order to solve the problem of singularity in the stress and strain calculation process of FRP laminated tubes with arbitrary layer angles under axisymmetric thermomechanical loading, a three-dimensional elasticity theory analysis method based on singularity correction by modifying the stiffness matrix coefficients is proposed. Firstly, the basic three-dimensional elasticity theory model for FRP laminated tubes under axisymmetric thermomechanical loading is established through the displacement method. Then, the singularity plies in the laminate are determined, and the stiffness matrix coefficients of the singularity plies are modified respectively. Modifications to the displacement function coefficients, stiffness coefficients, boundary condition, and interlayer continuous condition equilibrium equations are also conducted subsequently. Verifications of the present approach are conducted by comparing the stress distributions calculated by the singularity correction method with those extracted by traditional isotropic elasticity models and then with numerical results provided by finite element analysis. Good consistency was obtained; thus, the correctness of the theory is proved from both theoretical and numerical aspects.

Application of the optimized deep networks in the prediction of the coupled thermoelasticity response of the axially graded composite system subjected to thermal shock loading

This research is done to examine the coupled thermoelastic response of composite axially graded beams. Thermal shock is applied axially to the beam. Three-dimensional elasticity theory is used to generate the governing equations of motion. While the Fourier heat conduction relationship and the Lord–Shulman heat transfer theorem are used to construct the time-varying heat conduction connection. Using the discrete singular convolution approach and the Laplace transform, the time-varying equations of motion are solved at design locations. A modified version of the Dubner and Abate method is used to translate the system response from the Laplace domain to the time domain. The Whale Optimization Algorithm and the optimized deep neural network are employed as a prediction mechanism to estimate the system reaction under various scenarios in the final stage. By comparing the outcomes of the procedures utilized with those found in published resources, the correctness of the approaches was confirmed. Results on the doubly thermoelastic response of axially graded composite beams exposed to thermal shock loading are important and are presented in this work. As applicable results for related industries can be observed that by changing the boundary conditions from SS to CC, the location of maximum normal strain tends to be at the center of the axially graded beam

LOCALIZATION OF SOLUTION OF THE PROBLEM OF THREE-DIMENSIONAL THEORY OF ELASTICITY WITH THE USE OF B-SPLINE DISCRETE-CONTINUAL FINITE ELEMENT METHOD

Localization of solution of the problem of three-dimensional theory of elasticity with the use of B-spline discrete-continual finite element method (specific version of wavelet-based discrete-continual finite ele-ment method) is under consideration in the distinctive paper. The original operational continual and discrete-continual formulations of the problem are given, some actual aspects of construction of normalized basis func-tions of a B-spline are considered, the corresponding local constructions for an arbitrary discrete-continual finite element are described, some information about the numerical implementation and an example of analysis are presented.

Wave propagation analysis in pre-stressed incompressible hyperelastic multi-layered plates using a plate theory

Wave propagation analysis of pre-stressed incompressible hyperelastic thin multi-layered plates is developed based on the first-order shear deformation plate theory (FSDT) with second order thickness extensibility. Variables describing the thickness-direction deformation are obtained from the incompressibility condition. The incremental theory of nonlinear elasticity is employed in conjunction with a plate theory. The multi-layered plate is assumed to be symmetric in the thickness direction. Since the plate theory has not been used for the wave propagation analysis of hyperelastic plates, the results of flexural wave modes are compared with results of the straight-crested flexural waves based on the three-dimensional (3D) elasticity theory available in the literature and the results for in-plane wave modes are compared with those developed in this study using the plane-stress elasticity theory. The dispersion curves of both extensional and flexural waves for single-layer and three-layer plates are presented and the effects of different parameters including the homogenous pre-stretch, the wave propagation direction and the plate initial thickness on the dispersion curves are discussed in detail. The results indicate that the phase speed of the extensional waves depends on the wave propagation angle due to the induced anisotropy. As the overall shear modulus of the three-layer plate increases, the phase velocity increases. Also, the effect of pre-stretch and thickness on phase speed of flexural waves is not pronounced for high frequencies.