Two-dimensional Theory Of Elasticity Research Articles

A generalized thermoelastic dual-phase-lagging response of thick beams subjected to harmonically varying heat and pressure

The generalized thermoelastic problem of a thermo-mechanically loaded beam is studied. The upper surface of the beam is thermally isolated and subjected to a mechanical load while the bottom surface is traction free and subjected to a heating source. Based on the heat conduction equation containing the thermoelastic coupling term and the two-dimensional elasticity theory, thermoelastic coupling differential equations of motion are established. The generalized thermoelasticity theory with the dual-phase-laggings (DPLs) model is used to solve this problem. A closed-form analytical technique is used to calculate vibration of displacements and temperature. The effects of the phase-laggings (PLs), the intensity of the applied load and heat parameters on the field quantities of the beam are discussed. The variation along the axial direction and through-the-thickness distributions of all fields are investigated. Some comparisons have been also shown graphically to estimate the effects of the time on all the studied fields.

DYNAMIC RESPONSE OF FUNCTIONALLY GRADED NANOCOMPOSITE BEAMS SUBJECTED TO A MOVING LOAD USING A MESH-FREE METHOD

In this paper, the dynamic response of functionally graded nanocomposite beams under the action of a moving load are investigated. Three different types of Carbon NanoTubes (CNT’s) distributions in a polymer matrix material are studied; Uniform Distribution (UD), Symmetrically Functionally Graded (SFG) distribution and Unsymmetrically Functionally Graded (USFG) distribution. The analysis is carried out by a mesh-free method using the two-dimensional theory of elasticity. After validation, the effects of different design parameters such as CNT’s distribution, the velocity and position of the moving load on the dynamic behavior of the beam are examined. The results also highlight the importance of the reinforcement distribution type from a design perspective. The current approach can serve as a benchmark against which other semi-analytical and numerical methods based on classical beam theories can be compared.

Evaluation of support loss in micro-beam resonators: A revisit

This paper presents an analytical study on evaluation of support loss in micromechanical resonators undergoing in-plane flexural vibrations. Two-dimensional elastic wave theory is used to determine the energy transmission from the vibrating resonator to the support. Fourier transform and Green's function technique are adopted to solve the problem of wave motions on the surface of the support excited by the forces transmitted by the resonator onto the support. Analytical expressions of support loss in terms of quality factor, taking into account distributed normal stress and shear stress in the attachment region, and coupling between the normal stress and shear stress as well as material disparity between the support and the resonator, have been derived. Effects of geometry of micro-beam resonators, and material dissimilarity between support and resonator on support loss are examined. Numerical results show that ‘harder resonator’ and ‘softer support’ combination leads to larger support loss. In addition, the Perfectly Matched Layer (PML) numerical simulation technique is employed for validation of the proposed analytical model. Comparing with results of quality factor obtained by PML technique, we find that the present model agrees well with the results of PML technique and the pure-shear model overestimates support loss noticeably, especially for resonators with small aspect ratio and large material dissimilarity between the support and resonator.

Mechanics of fiber composites with fibers resistant to extension and flexure

A model of elastic solids reinforced with fibers resistant to extension and bending is formulated in finite-plane elastostatics. The linear theory of the proposed model is also derived through which a complete analytical solution is obtained. The presented model can serve as an alternative two-dimensional Cosserat theory of non-linear elasticity.

Time-dependent behavior of layered arches with viscoelastic interlayers

This work studies the time-dependent behavior of a layered arch adhesively bonded by viscoelastic interlayers. The deformation of the viscoelastic interlayer is represented by the Maxwell–Wiechert model. The constitutive relation in an interlayer is simplified through the quasi-elastic approximation approach. The mechanical property of an arch layer is described by the exact two-dimensional (2-D) elasticity theory in polar coordinates. The stress and displacement components in an arch layer, which strictly satisfy the simply supported boundary conditions, have been analytically derived out. The stresses and displacements are efficiently obtained by means of the recursive matrix method for the arch with any number of layers. The comparison study shows that the 2-D finite element solution has good agreement with the present one, while the solution based on the one-dimensional (1-D) Euler–Bernoulli theory has considerable error, especially for thick arches. The influences of geometrical and material parameters on the time-dependent behavior of the layered arch are analyzed in detail.

Free vibration of soft-core sandwich panels with general boundary conditions by harmonic quadrature element method

In this paper, the harmonic quadrature element method is presented for free vibration analysis of soft-core sandwich panels with general boundary conditions. To remove limitations existing in various beam theories, the thin faces are modeled by Euler-Bernoulli beam theory but the soft-core is directly modeled by two-dimensional elasticity theory. A novel harmonic quadrature sandwich panel element with arbitrary number of nodes is developed and the equation of motion is established utilizing Hamilton principle. Explicit formulations are worked out to ease the implementation of the element. To show the applicability of the method, several examples, including two types of core material and five combinations of boundary conditions, are investigated. Results are compared to existing data as well as finite element data of ABAQUS. Comparisons show that the proposed method exhibits a high convergence rate and can yield accurate results in all cases.

Solution of Multipoint Boundary Problem of Two-Dimensional Theory of Elasticity Based on Combined Application of Finite Element Method and Discrete-Continual Finite Element Method

The distinctive paper is devoted to solution of multipoint boundary problem of two-dimensional theory of elasticity (static analysis of two-dimensional structure) based on combined application of finite element method (FEM) and discrete-continual finite element method (DCFEM). The given domain, occupied by considering structure, is embordered by extended one. The field of application of DCFEM comprises fragments of structure (subdomains) with regular (constant or piecewise constant) physical and geometrical parameters in some dimension (“basic” dimension). DCFEM presupposes finite element mesh approximation for non-basic dimension of extended domain while in the basic dimension problem remains continual. Analytical solution for such typical subdomain is apparently preferable in all aspects for qualitative analysis of calculation data. It allows investigator to consider boundary effects when some components of solution are rapidly varying functions. Due to the abrupt decrease inside of mesh elements in many cases their rate of change can’t be adequately considered by conventional numerical methods while analytics enables study. FEM is used for approximation of all other subdomains. Discrete (within FEM) and discrete-continual (within DCFEM) approximation models for subdomains and coupled multilevel approximation model for extended domain are under consideration. Generally, discrete-continual formulations are contemporary mathematical models which currently becoming available for computer realization. Brief information about software systems and verification samples are presented as well.

Harmonic Differential Quadrature Analysis of Soft-Core Sandwich Panels under Locally Distributed Loads

Sandwich structures are widely used in practice and thus various engineering theories adopting simplifying assumptions are available. However, most engineering theories of beams, plates and shells cannot recover all stresses accurately through their constitutive equations. Therefore, the soft-core is directly modeled by two-dimensional (2D) elasticity theory without any pre-assumption on the displacement field. The top and bottom faces act like the elastic supports on the top and bottom edges of the core. The differential equations of the 2D core are then solved by the harmonic differential quadrature method (HDQM). To circumvent the difficulties in dealing with the locally distributed load by point discrete methods such as the HDQM, a general and rigorous way is proposed to treat the locally distributed load. Detailed formulations are provided. The static behavior of sandwich panels under different locally distributed loads is investigated. For verification, results are compared with data obtained by ABAQUS with very fine meshes. A high degree of accuracy on both displacement and stress has been observed.

Deformation of a rectangular plate with an arbitrarily located circular hole under in-plane pure shear loading

Exact solutions for stresses, strains, displacements, and the stress concentration factors of a rectangular plate perforated by an arbitrarily located circular hole subjected to in-plane pure shear loading are investigated by two-dimensional theory of elasticity using the Airy stress function. The hoop stresses, strains, and displacements occurring at the edge of the circular hole are computed and plotted. Comparisons are made for the hoop stresses and the stress concentration factors from the present study and those from a rectangular plate with a circular hole under uni-axial and bi-axial uniform tensions and in-plane pure bending moments on two opposite edges.

A simple single variable shear deformation theory for a rectangular beam

This paper proposes a simple single variable shear deformation theory for an isotropic beam of rectangular cross-section. The theory involves only one fourth-order governing differential equation. For beam bending problems, the governing equation and the expressions for the bending moment and shear force of the theory are strikingly similar to those of Euler–Bernoulli beam theory. For vibration and buckling problems, the Euler–Bernoulli beam theory governing equation comes out as a special case when terms pertaining to the effects of shear deformation are ignored from the governing equation of present theory. The chosen displacement functions of the theory give rise to a realistic parabolic distribution of transverse shear stress across the beam cross-section. The theory does not require a shear correction factor. Efficacy of the proposed theory is demonstrated through illustrative examples for bending, free vibrations and buckling of isotropic beams of rectangular cross-section. The numerical results obtained are compared with those of exact theory (two-dimensional theory of elasticity) and other first-order and higher-order shear deformation beam theory results. The results obtained are found to be accurate.

Forced vibration analysis of functionally graded beams by the meshfree boundary-domain integral equation method

Forced vibration of two-dimensional functionally graded beams is studied in this paper by a developed meshfree boundary-domain integral equation method. Material properties of the functionally graded beams are assumed varying continuously either in longitudinal or transvers direction following the exponential function. The boundary-domain integral equations are derived by using the elastostatic fundamental solutions based on the two-dimensional elastic theory. Radial integral method (RIM) is employed to transform the domain integrals into boundary integrals. A meshfree scheme is achieved through assuming the displacements and accelerations in the domain integrals by a combination of the radial basis function and polynomials with time dependent coefficients. Wilson-θ, Houbolt as well as two kinds of damped Newmark's algorithms are applied to accomplish the time integration. The forced vibration of the functionally graded beam subjected by the harmonic loading and transient loading are investigated in detail. Numerical examples demonstrate that the four mentioned time integral schemes are all adapted well to the developed meshfree boundary-domain integral equation method for analyzing the forced vibration of homogeneous structures. For the analysis of FG structures, it is shown that the damped Newmark's algorithm can achieve more stable and accurate results.

Two-dimensional linear elasticity theory of magneto-electro-elastic plates considering surface and nonlocal effects for nanoscale device applications

A novel two-dimensional linear elastic theory of magneto-electro-elastic (MEE) plates, considering both surface and nonlocal effects, is established for the first time based on Hamilton’s principle and the Lee plate theory. The equations derived are more general, suitable for static and dynamic analyses, and can also be reduced to the piezoelectric, piezomagnetic, and elastic cases. As a specific application example, the influences of the surface and nonlocal effects, poling directions, piezoelectric phase materials, volume fraction, damping, and applied magnetic field (i.e., constant applied magnetic field and time-harmonic applied magnetic field) on the magnetoelectric (ME) coupling effects are first investigated based on the established two-dimensional plate theory. The results show that the ME coupling coefficient has an obvious size-dependent characteristic owing to the surface effects, and the surface effects increase the ME coupling effects significantly when the plate thickness decreases to its critical thickness. Below this critical thickness, the size-dependent effect is obvious and must be considered. In addition, the output power density of a magnetic energy nanoharvester is also evaluated using the two-dimensional plate theory obtained, with the results showing that a relatively larger output power density can be achieved at the nanoscale. This study provides a mathematical tool which can be used to analyze the mechanical properties of nanostructures theoretically and numerically, as well as evaluating the size effect qualitatively and quantitatively.

Elasticity solution for vibration of generally laminated beams by a modified Fourier expansion-based sampling surface method

A modified Fourier expansion-based sampling surface (MFE-SaS) method for the vibration analysis of generally laminated beams with arbitrary boundary conditions is presented. The method combines the advantages of the sampling surface (SaS) method with those of the modified Fourier expansion (MFE). The formulation is based on the two-dimensional theory of elasticity. In this method, a certain number of non-equally spaced sampling surfaces which parallel to the beam midsurface are primarily collocated in each layer of the beam and, the displacements of these surfaces are introduced as fundamental beam unknowns. This fact gives the opportunity to derive the elasticity solutions for a thick beam with a prescribed accuracy by utilizing a sufficiently large number of sampling surfaces. The desired solutions are obtained by the variational operation. Several examples are given to demonstrate the convergency, accuracy and flexibility of the present method.

Buckling and Vibration Analysis of Functionally Graded Carbon Nanotube-Reinforced Beam Under Axial Load

Buckling and vibration analysis of cantilever functionally graded (FG) beam that reinforced with carbon nanotube (CNT) is the purpose of this paper. The beam is graded in the thickness direction, and compressive axial force impressed the beam. The volume fractions of randomly oriented agglomerated single-walled CNTs (SWCNTs) are assumed to be graded in the thickness direction. To determine the effect of CNT agglomeration on the elastic properties of CNT-reinforced FG-beam, a two-parameter micromechanics model of agglomeration is employed. In this paper, an equivalent continuum model based on the Eshelby–Mori–Tanaka approach is obtained. The stability and motion equations are based on the two-dimensional elasticity theory and Hamilton’s principle. The generalized differential quadrature method (GDQM) that has high accuracy is used to discretize the equations of stability and motion and to implement the boundary conditions. To study the accuracy of the present analysis, a compression is carried out with a known data. Convergence rate, the influence of graded agglomerated CNTs, and the effect of axial forces exerted on the beam, on the natural frequencies of reinforced beam by randomly oriented agglomerated CNTs are investigated.

Semianalytical Structural Analysis Based on Combined Application of Finite Element Method and Discrete-continual Finite Element Method Part 1: Two-Dimensional Theory of Elasticity

This paper is devoted to so-called semianalytical structural analysis, based on combined application of finite element method (FEM) [1] and discrete-continual finite element method (DCFEM) [2–7]. Boundary problems of two-dimensional theory of elasticity (static analysis of deep beam [1]) are under consideration. In accordance with the method of extended domain, the given domain is embordered by extended one. The field of application of DCFEM comprises structures with regular (constant or piecewise constant) physical and geometrical parameters in some dimension (“basic” dimension). DCFEM presupposes finite element mesh approximation for non-basic dimension of extended domain while in the basic dimension problem remains continual. Corresponding discrete and discrete-continual approximation models for subdomains and coupled multilevel approximation model for extended domain are under consideration. Brief information about software and verification samples are presented as well.

Young’s Modulus and Loss Factor Estimation of Sandwich Beam with Three Optimization Methods by FEM

physical properties identification of sandwich beams by measured flexural resonance frequencies. The physical parameters are the Young‟s modulus and the loss factor. They are estimated for each one of materials that constitute the structure of the sandwich beam which is made with the association of Hot-rolled steel, Polyurethane Rigid Foam and High Impact Polystyrene. This kind of the sandwich beam are widely used for the assembly of household refrigerators and food freezers. The solutions are obtained with parametric optimization of physical parameters of the materials that forming the sandwich beam with three methods: Genetic Algorithms (GA), Differential Evolution (DE), and Particle Swarm Optimization (PSO). Furthermore, this work intend verify the quality of the solutions obtained with parametric optimization. The parameters are estimated using measured and numeric frequency response functions (FRFs). The mathematical model to verify numeric FRF is obtained using the Finite Element Method and the two-dimensional elasticity theory coupled to three optimization methods. The results of the optimizations show that it is possible to determine effectively the physical parameters of a sandwich beam with this methodology.

2-D elasticity solution of layered composite beams with viscoelastic interlayers

This paper focuses on the mechanical properties of layered composite beams with viscoelastic interlayers. The exact two-dimensional elasticity theory is used to represent the deformation of each beam layer. The viscoelastic interlayer is described by the Maxwell–Wiechert model through the quasi-elastic approximation, which greatly simplifies the analytical process. The stress function with a series of undetermined coefficients depending on the time variable is derived for each beam layer. No matter how many layers the beam includes, the total solution can be obtained rapidly and efficiently by using the recursive matrix technique. The present method can give the exact stress and deformation distributions in the beam, which cannot be predicted by the approximate theories such as the one-dimensional Euler–Bernoulli theory. The convergence of the solution is numerically verified. A comparison study indicates that the present results are in agreement with those obtained from the finite element method; however, they have obvious differences from the results based on the Euler–Bernoulli theory for thick beams. Finally, the variations of stresses and displacements with respect to time in a five-layer beam are discussed in detail.

Construction of fundamental solution of static equations of medium-thickness isotropic plates

A fundamental solution of the elasticity theory equations for isotropic plates was obtained. To construct the two-dimensional elasticity theory equations, the approximation method of displacements, stresses and strains using Fourier series by Legendre polynomials on the transverse coordinate was used. This approach has allowed to take into account the transverse shear and normal stresses. Since the classical Kirchhoff-Love theory does not consider these stresses, the research based on the refined theories of the stress-strain state of isotropic plates under concentrated force actions is an urgent scientific and technical problem. The fundamental solution of these equations was found using the two-dimensional Fourier integral transform and the generalization method, built with a special G-function. The method allows to reduce the system of resolvent differential static equations of flat plates and shells to a system of algebraic equations. Then the inverse Fourier transform restores fundamental solution. Numerical studies that demonstrate behavior patterns of the stress-strain state components depending on the elastic constants of isotropic material were performed. The approach demonstrates the development of the refined theory of plates and shells based on the three-dimensional elasticity theory.

Collocation–quadrature methods and fast summation for Cauchy singular integral equations with fixed singularities

Basing on recent results on the stability of collocation methods applied to Cauchy singular integral equations with additional fixed singularities we give necessary and sufficient conditions for the stability of collocation–quadrature methods for such equations. These methods have the advantage that the respective system of equations has a very simple structure and allows to apply fast summation methods which results in a fast algorithm with O(nlog⁡n) complexity. We present numerical results of the application of the proposed collocation–quadrature methods to the notched half plane problem of two-dimensional elasticity theory.

Mechanical response of fabric sheets to three-dimensional bending, twisting, and stretching

A model for the mechanics of woven fabrics is developed in the framework of two-dimensional elastic surface theory. Thickness effects are modeled indirectly in terms of appropriate constitutive equations. The model accounts for the strain of the fabric and additional effects associated with the normal bending, geodesic bending, and twisting of the constituent fibers.