Three-dimensional Theory Of Elasticity Research Articles

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.

Nonlinear Oscillations and Buckling Prediction for Shallow Convex Shells

Since 2010, the testing of thin-walled shallow shells has revealed that the ultimate internal stresses, under which the shells lose their stability (collapse), appeared to be lower than those predicted by the buckling theory used in American and European standards. The existing norms are based on the static theory of shallow shells proposed in 1930 with stable solutions for thin-walled structures within the nonlinear theory. These stable solutions differ significantly from the forms of equilibrium common to small initial loads. The minimum load, under which an alternative form of equilibrium exists, was used as the ultimate load. In the 1970s, this approach was proved to be unacceptable for complex loadings that were not a part of the real world in the past but are now the case with thinner products operated under complex conditions. Therefore, the initial theories on bearing capacity evaluations must be revisited. The new theory could be built on the basis of recent mathematical results establishing that the estimates made per two schemes (three-dimensional dynamic theory of elasticity and dynamic theory of shallow convex shells) were asymptotically close. Green's special function is introduced to convert the Foppl-von Karman system into a resolving integro-differential equation. The obtained nonlinear equation allows for the separation of variables and has numerous time-periodic solutions that satisfy the Duffing equation with a "soft spring". Numerical analysis enables determining the amplitude and period of oscillations depending on the properties of Green's function. This paper reviews an experimental setup in which oscillations are generated with the probing load directed normally to the surface of the shell. The experimental measurements of the resonance displacements as a function of coordinates and time enable the calculation of the safety factor for the bearing capacity of the structure using the non-destructive method under operating conditions.

Optimization of multi-directional functionally graded plates in thermal environment based on 3D isogeometric analysis and adaptive-hybrid evolutionary firefly algorithm

In this study, a numerical model based on three-dimensional isogeometric analysis and adaptive hybrid evolutionary firefly algorithm is developed to concurrently optimize the shape and material distribution of multi-directional functionally graded plates. The optimization problems focus on maximizing the fundamental frequency of the plates subjected to thermal effects, while the volume fraction of ceramic constituent material and volume of the plates are considered as side constraints. Isogeometric multi-mesh design approach is employed to generate design and analysis domain. The free vibration analysis of multi-directional functionally graded plates with variable thickness is conducted within the framework of three-dimensional elasticity theory and isogeometric analysis. The generalized numerical framework developed in this study could accurately capture the thermal effects on the behavior of nonhomogeneous plates with variable thickness. Various numerical examples are conducted to illustrate the optimal shapes and material distributions within square, rectangular, and circular plates. Influences of temperature field, boundary conditions, and geometries of the plates on the optimal results are also addressed.

Asymptotic Study of Plate Bending for a Strongly Orthotropic Materia

The technique of asymptotic averaging has been developed for three-dimensional partial differential equations with rapidly oscillating coefficients. For example, for the equations of elasticity theory. Then it was modified and applied to thin bodies in the form of plates (homogeneous or inhomogeneous, with even front surfaces or not), described by the three-dimensional theory of elasticity. In these cases, asymptotic solutions were constructed with respect to one small parameter, which is usually the ratio of the plate thickness to the characteristic plan size. The averaging technique in this case also reduces the dimension of the problem, i.e. reduces the three-dimensional boundary value problem to some two-dimensional one. In this article, we substantiate the application of the method to a problem with two small parameters in the case of a homogeneous thin, highly orthotropic plate bent by a surface load without taking into account body forces. The second small parameter is the ratio of the transverse elastic moduli to the moduli in the plan of the plate. It is shown that strong orthotropy is equivalent to an increase in the thickness of the equivalent plate. A procedure for obtaining the stress distribution over the plate thickness is described for three approximations. The first approximation gives the classical Kirchhoff theory, also called the Kirchhoff-Love theory, and the third approximation coincides with the Ambartsumian theory and allows one to find transverse shear and normal stresses. The consideration of cylindrical bending makes it possible to find solutions in the framework of classical plate theories in the form of formulas, as well as three approximations of the asymptotic theory, which simplifies comparison. Examples are considered when the averaged orthotropic moduli are taken for a single-layer fibrous composite.

On the Equilibrium of Nonthin Cylindrical Shells with a Dent

On the basis of the linear three-dimensional theory of elasticity, we consider the problem of equilibrium of nonthin isotropic cylindrical shells with dents under certain boundary conditions imposed on the end faces. To describe the cross section of the reference surface, we use the Pascal snail equation in polar coordinates. By the method of separation of variables, with the help of the approximation of functions by discrete Fourier series, we reduce the three-dimensional boundary-value problem to a one-dimensional problem, which is solved by the stable numerical method of discrete orthogonalization. We estimate the accuracy of the obtained solutions. The results of solving the problems are presented in the form of plots and tables.

Improving soundproof characteristics of an FG-CNT–reinforced composite structure by adding a coating magneto-electro-elastic layer

This paper deals with sound propagation through a two-layer sandwich plate in which the carbon nanotubes with functionally graded (FG) distribution are coated by a thin layer of magneto-electro-elastic (MEE) materials. Firstly, the derivation of governing equations for each layer of the sandwich plate is followed according to the three-dimensional elasticity theory. In this regard, the relationship between the equations of motion for the layer, made of MEE materials, and the electric and magnetic potentials is established. Additionally, the effective properties of the nanocomposite layer are extracted by employing the rule of mixture. Afterward, the solution is finalized by employing the approximate layer and transfer matrix technique, in accordance with the definition of state space and sound transmission loss. In the next stage, to confirm the reliability of the derived relationships, three configurations of the panel are examined. Firstly, according to the obtained results, it is evident that adding a thin coating layer of triple-phase MEE materials effectively improves the sound insulation in the sandwich plate. Secondly, considering the electromagnetic boundary conditions can lead to a noticeable enhancement in the STL in the stiffness region. Moreover, the magnetic potential, compared to the electric one, is seen to be more effective on the acoustic radiation of the sandwich panel.

Three-Dimensional Vibration Model of Cylindrical Shells via Carrera Unified Formulation.

In this paper, we present a novel and unified model for studying the vibration of cylindrical shells based on the three-dimensional (3D) elastic theory and the Carrera Unified Formulation. Our approach represents a significant advancement in the field, as it enables us to accurately predict the vibrational behavior of cylindrical shells under arbitrary boundary conditions. To accomplish this, we expand the axial, circumferential, and radial displacements of the shell using Chebyshev polynomials and Taylor series, thereby reducing the dimensionality of the expansion and ensuring the precision and rigor of our results. In addition, we introduce three groups of artificial boundary surface springs to simulate the general end boundary conditions of the cylindrical shell and coupling springs to strongly couple the two surfaces of the cylindrical shell φ = 0 and φ = 2π to ensure continuity of displacements on these faces. Using the energy function of the entire cylindrical shell model, we obtain the characteristic equation of the system by finding the partial derivatives of the unknown coefficients of displacement in the energy function. By solving this equation, we can directly obtain the vibration characteristics of the cylindrical shell. We demonstrate the convergence, accuracy, and reliability of our approach by comparing our computational results with existing results in the literature and finite element results. Finally, we present simulation results of the frequency features of cylindrical shells with various geometrical and boundary parameters in the form of tables and figures. Overall, we believe that our novel approach has the potential to greatly enhance our understanding of cylindrical shells and pave the way for further advancements in the field of structural engineering. Our comprehensive model and simulation results contribute to the ongoing efforts to develop efficient and reliable techniques for analyzing the vibrational behavior of cylindrical shells.

Three-Dimensional Stress Fields in Thick Orthotropic Plates with Sharply Curved Notches under In-Plane and Out-of-Plane Shear.

In this paper, an analytical solution for the stress fields in the close neighbourhoods of radiused notches in thick orthotropic plates under shear loading and twisting is provided. In the first step, the equations of the three-dimensional theory of elasticity are successfully reduced to two uncoupled equations in two-dimensional space. Later, the 3D stress field solution for orthotropic plates with radiused notches is presented and its degree of accuracy is discussed by comparing theoretical results and numerical data from 3D FE analyses. The solution proposed can be satisfactorily used to characterise the stress field in plates made with polymeric composite materials, such as fibre-reinforced polymers and natural composites.

Inverse of initial stress in carbon fiber reinforced polymer laminates using lamb waves and deep neural network

The prediction of the initial stress in composites is essential for the non-destructive testing (NDT) and structural health monitoring (SHM) of carbon fibre reinforced polymer (CFRP). This paper examines the potential of Lamb waves in the inverse of initial stress by calculating the influence of initial stress on the dispersion characteristics of Lamb waves propagating in multilayered CFRP laminates. By introducing the mechanics of incremental deformation into the linear three-dimensional elasticity theory, the Legendre orthogonal polynomial expansion (LOPE) method is used to mathematically model the Lamb wave propagating in multilayered CFRP laminates subjected to horizontal and vertical homogeneous initial stresses. Then, a three-hidden-layers Feed Forward Deep Neural Network (DNN) with Back Propagation (BP) algorithm is constructed to invert the magnitude and direction of the initial stresses. The input features are the phase velocities of fundamental Lamb wave A0 mode at five different frequencies. Both training and testing samples are obtained by LOPE forward calculation. An ablation experiment is presented to compare the two different activation functions. Finally, the accuracy of the inverse is verified by comparing with the available outcomes of LOPE forward calculation.

A one-dimensional model for axisymmetric deformations of an inflated hyperelastic tube of finite wall thickness

We derive a one-dimensional (1d) model for the analysis of bulging or necking in an inflated hyperelastic tube of finite wall thickness from the three-dimensional (3d) finite elasticity theory by applying the dimension reduction methodology proposed by Audoly and Hutchinson (2016). The 1d model makes it much easier to characterize fully nonlinear axisymmetric deformations of a thick-walled tube using simple numerical schemes such as the finite difference method. The new model recovers the diffuse interface model for analyzing bulging in a membrane tube and the 1d model for investigating necking in a stretched solid cylinder as two limiting cases. It is consistent with, but significantly refines, the exact linear and weakly nonlinear bifurcation analyses. Comparisons with finite element simulations show that for the bulging problem, the 1d model is capable of describing the entire bulging process accurately, from initiation, growth, to propagation. The 1d model provides a stepping stone from which similar 1d models can be derived and used to study other effects such as anisotropy and electric loading, and other phenomena such as rupture.

MOMENT-MEMBRANE THEORY OF ELASTIC FLEXIBLE PLATES AS A CONTINUAL GEOMETRICALLY NONLINEAR THEORY OF A GRAPHENE SHEET

In the present work, under the assumption of smallness of deformations, bending-torsional characteristics and angles of rotation (including the angles of free rotation) of the elements of the plate, based on the three-dimensional geometrically-nonlinear moment theory of elasticity, preserving only those nonlinear terms, that come from normal displacement (deflection) and its derivatives, a geometrically nonlinear moment-membrane theory of elastic plates is constructed as a continual theory of deformations of a flexible graphene. For the indicated nonlinear theory of elastic plates, by introducing stress functions, the resolving equations are presented also in a mixed form: these are the system of equilibrium equations for transverse-bending deformation, compiled in the deformed state of the plate, and deformations continuity equations, expressed in stress functions and deflection functions. For the geometrically nonlinear moment-membrane theory of elastic plates Lagrange-type variational principle is established.

Stress analysis of rotating thick-walled nonhomogeneous sphere under thermomechanical loadings

In this work, a rotating thick-walled spherical vessel made of nonhomogeneous materials subjected to internal and/or external pressure under thermal loading was analyzed within the context of three-dimensional elasticity theory. An analytical formulation was established for computing the displacement and stress fields. It has been assumed that the mechanical and thermal properties are varying through thickness of the functionally graded material (FGM) according to a power-law nonlinear expression, while Poisson's ratio is considered as constant. Based on equilibrium equation, Hooke's law, stress-strain relationship in the spheres and other theories from mechanics a second-order ordinary differential equation well-known as Navier equation is obtained that represents the thermoelastic field in hollow FGM sphere. Navier equation derived from the mechanical equilibrium equation was solved to obtain the exact solution of the displacement and stress distributions. Different results of thermoelastic field are presented across the thickness of the sphere. The analysis of the different results reveals that stress and strain in the FGM sphere are significantly depend upon variation made in temperature profile, rotation and inhomogeneity parameter on the thermoelastic field. Thus, the inhomogeneity in material properties can be exploited to optimize the distribution of displacement and stress fields.

Efficient machine-learning algorithm applied to predict the transient shock reaction of the elastic structure partially rested on the viscoelastic substrate

Due to numerous applications of piezoelectric materials in the modern technologies, this study assesses the thermomechanical shock behavior of the functionally graded graphene platelets reinforced nanocomposite (FG-GPLRN) annular plate surrounded by two piezoelectric layers and partially rested on the viscoelastic substrate for different cases of boundary conditions by metaheuristic optimized machine-learning methods, for the first time. Thermal and mechanical shocks are simultaneously applied on the upper surface of the mentioned structure. Governing equations of the system are formulated in the background of three-dimensional (3D) elasticity theory. Energy balance of the system is considered based on the Lord-Shulman theorem. Differential quadrature method (DQM) is selected as the main solver to determine the spatial response of the system from the state-space form of the governing differential equations. Additionally, Laplace transform is collaborated with modified Dubner and Abate’s approach to predict the temporal response of the system. Accuracy of the applied methods is carefully examined and verified through comparative study performed between the current results and those determined in the published high-quality studies. Valuable outcomes of the current approach would be directly employed in designing process of similar structures interacting with probable thermomechanical shocks.

Three-dimensional bending and free vibration analyses of laminated cylindrical panel with/without elastic foundation using two-dimensional discrete method

In this paper, a novel isogeometric scaled boundary finite element method (IGSBFEM) is developed to provide a comprehensive investigation into bending and free vibration of laminated cylindrical panels resting on Pasternak foundation. The significant novelty of this method lies in that the proposed formulation is derived based on the three-dimensional (3D) theory of elasticity, while only two-dimensional (2D) reference surface associated with the curvilinear coordinate system needs to be discretized by using isogeometric elements constructed with the non-uniform rational B-splines (NURBS) basis functions, which allows to reduce the computation cost and maintain the exact geometry of problem with desired precision flexibly. The SBFEM governing equation of each single lamina is reduced to a matrix form of first-order ordinary differential equations by introducing a vector of dual variables made of nodal displacements and forces, and the solution can be expressed analytically in terms of the radial coordinate. Therefore, shear locking phenomenon can be overcome naturally without any shear correction factors or reduced integration. The global solving equations of laminated cylindrical panels resting on Pasternak foundation are assembled with a layer-wise method. In addition, the high-order continuous features of isogeometric elements are insensitive to distorted mesh. Therefore, it is compatible with the research of complex geometrical structures with holes. The superior performances of the current approach are testified in the analyses of cylindrical panels with or without elastic foundation, solutions obtained by many other researchers are used for comparison purpose, and effects of several parameters including length and thickness, fiber orientation, stacking sequence and foundation stiffnesses coefficient on the bending or free vibration characteristics are also discussed.

Numerical investigation of sound transmission through single and double walls with periodic arrays of resonators

Lightweight partition walls are widely used in buildings, especially nowadays when sustainability is of a major interest. At the same time noise pollution is becoming a growing problem across the globe. All this implies the need to find possible solutions for sound insulation improvement of the lightweight partitions. This paper is focused on exploring the possibilities of exploiting the dynamic effects of periodic subwavelength arrays of resonators for sound insulation enhancement. In order to improve the sound transmission loss in a specific frequency region, periodic resonant units are tuned and introduced to single and double panels. It can be shown that the resonance mode of the resonators couples with the host plate vibration in the way of breaking the mass law and overcoming some phenomena like coincidence effect and mass-air-mass resonance. The investigation of the considered panels under acoustic field excitation is conducted using different methods. Sound transmission loss curves and dispersion diagrams are calculated. Finite element method is employed for this purpose, but also the theory of three-dimensional elasticity and well-known transfer matrix method (TMM). Several designs of resonators are proposed.

Acoustic insulation characteristics improvement of a thick CNT-reinforced doubly-curved shell by using GPLRC and MEE composite layers

Noise reduction in structures and human living environments is one of the most important issues in engineering that is always given special attention. Sound insulation has always been improved using different methods, one of which is to use the properties of materials. Herewith, the aim of this paper is to take advantage of graphene-platelet reinforced composites and magneto-electro-elastic (MEE) material properties for sound attenuation. The present paper deals with the analysis of sound transmission loss (STL) through a three-layer sandwich doubly-curved shell where an MEE sheet is integrated with two nanocomposite sheets. In addition, these two nanocomposite sheets are reinforced by functionally graded (FG) distributions of CNT and graphene platelet (GPL)-reinforced composites, respectively. Firstly, the three-dimensional elasticity theory is employed to derive the governing equations of motion. Then, the vibroacoustic analysis for the resultant equations is completed according to the state space and transfer matrix method. Comparing the obtained results with the available literature discloses that the offered procedure has a high precision for structural acoustic problems. In the next step, in addition to inspecting two kinds of MEE composites, the effective parameters, such as layup configuration, FG distribution, volume fraction, weight fraction, radii of curvature, electromagnetic boundary conditions, and interphase thickness, are assessed on the STL. This assessment shows that the parameters involved in this paper are highly interdependent. Accordingly, the analysis of these parameters is done simultaneously with the aid of three- and four-dimensional plots in order that the optimal value for each parameter can be realized. As seen clearly in the outcomes, the electromagnetic boundary conditions parameters, compared to the other parameters, can much more alter the STL trend, so that a slight change in electric potential results in great change in the STL.

Estimation of the stress-strain state of orthotropic plates on elastic foundation using the bimoment theory

The study is devoted to the development of the bimoment theory and the method for calculating thick plates in the framework of the three-dimensional theory of elasticity. The basic relations and equations of motion of the plate presented are constructed with respect to the forces, moments and bimoments arising from the nonlinearity of the law of distribution of displacements and stresses over the plate thickness. Bending and vibrations of isotropic and orthotropic plates are considered an example of calculations. The resulting solution showed the efficiency and accuracy of the proposed bimoment theory in assessing the stress-strain state of thick plates.

Contact interaction of deformable continuous media of various physical nature

An approach is proposed for modeling the contact interaction processes of reinforcement bundles with concrete, as well as for underground elements of foundations and buildings reinforced foundations by compensatory injection with adjacent soil. A special contact finite element of a continuous medium with specific properties has been constructed which makes it possible to simulate the elastic interaction of two contacting surfaces their displacement relative to each other and separation. The developed algorithm allows within the framework of a single finite element method, to calculate the joint deformation of the elements of metal reinforcing elements, concrete and “weak” soils. The proposed calculation methods are implemented within the consistent equations framework of a geometrically nonlinear three-dimensional elasticity theory. On the basis of the proposed methods, the problem of "pulling out" a steel reinforcement bar from a concrete block was solved, a comparison was made with experimental data and with the results obtained in publicly available application software packages, a good agreement was noted.

To the calculation of orthotropic plates taking into account bimoments

The work is devoted to the development of the bimoment theory and the method for calculating thick plates in the framework of the three-dimensional theory of elasticity. The basic relations and equations of motion of the plate are given, constructed with respect to forces, moments and bimoments, arising from the nonlinearity of the law of distribution of displacements and stresses over its thickness. Bending and vibrations of isotropic and orthotropic plates are considered as an example of calculation. The resulting solution showed the efficiency and accuracy of the proposed bimoment theory in assessing the stress-strain state of thick plates. Figs.: Refs.: 11 titles.
 Keywords: thick orthotropic plate, three-dimensional theory of elasticity, bi-moment theory, bending and vibrations, stress-strain state.

Three-Dimensional Thermoelasticity Analysis of Viscoelastic FGM Plate Embedded in Piezoelectric Layers under Thermal Load

Due to the high importance of viscoelastic materials in modern industrial applications, besides the intensive popularity of piezoelectric smart structures, analyzing their thermoelastic response in extreme temperature conditions inevitably becomes very important. Accordingly, this research explores the thermoviscoelastic response of sandwich plates made of a functionally-graded Boltzmann viscoelastic core and two surrounding piezoelectric face-layers subjected to electrothermal load in the platform of three-dimensional elasticity theory. The relaxation modulus of the FG viscoelastic layer across the thickness follows the power law model. the plate’s governing equations are expressed in the Laplace domain to handle mathematical complications corresponding to the sandwich plate with a viscoelastic core. Then, the state-space method, combined with Fourier expansion, is utilized to extract the plate response precisely. Finally, the obtained solution is converted to the time domain using the inverse Laplace technique. Verification of the present formulation is compared with those reported in the published papers. Finally, the influences of plate dimension, temperature gradient, and relaxation time constant on the bending response of the above-mentioned sandwich plate are examined. As an interesting finding, it is revealed that increasing the length-to-thickness ratio leads to a decrease in deflections and an increase in stresses.