Two-dimensional Theory Of Elasticity Research Articles

Exact solutions for multilayer functionally graded beams bonded by viscoelastic interlayer considering memory effect

This work proposes exact solutions for simply supported multilayer functionally graded (FG) beams with viscoelastic interlayer to predict its time-dependent mechanical behavior. The stresses and deformations of FG layers are described by the two-dimensional (2-D) elasticity theory. The viscoelastic property of the interlayer is simulated by the standard linear solid model, and its constitutive relation is expressed in the convolution form which considers the strain memory effect. The general solutions of FG layers involving undetermined coefficients are obtained by virtue of the Fourier series expansion. By the use of the Laplace transform method, the solutions of stresses and deformations are determined analytically. The present solutions including Fourier series possess good convergence property. The comparison studies indicate that the finite element (FE) solutions approach the present ones when each FG layer is divided into many isotropic sub-layers in FE model and the gradient factor is taken as small value. The relative error of the one-dimensional (1-D) Euler-Bernoulli solutions is slight for slender beam with small gradient factor, but it significantly increases as the beam thickness or the gradient factor grows. Finally, the influences of the material constants on the time-dependent behavior in the beam are investigated in detail.

Inversion Symmetry of the Solutions of Basic Boundary-Value Problems of Two-Dimensional Elasticity Theory for a Wedge

By using the Mellin integral transformation, we obtain the solutions of basic two-dimensional problems for an elastic wedge. It is shown that, in the case of boundary conditions symmetric under the inversion transformation, some components of the solutions of plane strain problems and the solutions of antiplane strain problems for a wedge also have the inversion symmetry.

Elasticity Solutions for Sandwich Arches considering Permeation Effect of Adhesive

In this work, analytical solution of simply supported sandwich arches considering permeation effect of adhesives is presented. The permeation layer is described by the functionally graded material, exponentially graded in the radial direction. The stresses and deformations of each layer are based on the two-dimensional (2D) elasticity theory in the polar coordinate. The governing equations of the arch are solved by the layer-wise method, which turns the differential equations with variable coefficients into constant coefficients. The solution can be obtained efficiently by means of the recursive matrix method, especially for the arch with many layers. The present solution agrees well with the finite element solution with a fine mesh, while the finite element method is time consuming in mesh division and calculation. The one-dimensional (1D) solution based on the Euler–Bernoulli theory is close to the present one; however, the error increases as the arch becomes thick. The effect of permeation layer thickness on the stresses is studied. It is indicated that the stress distributions tend to be smooth along the radial direction as the permeation layer thickness increases.

Clamped-Free Non Homogeneous Magneto Electro Elastic Plate of Polygonal Cross-Sections with Hydrostatic Stress and Gravity

In this article, the influence of hydrostatic stress and gravity on a clamped- free non homogeneous magneto electro elastic plate of polygonal cross sections is studied using linear theory of elasticity. The equations of motion based on two-dimensional theory of elasticity are applied under the plane strain assumption of prestressed and gravitated magneto electro elastic plate of polygonal cross-sections composed of non homogeneous isotropic material. The frequency equations are obtained by satisfying the boundary conditions along the irregular surface of the polygonal plate using Fourier expansion collocation method. The complex roots of the frequency equations are obtained by secant method. The numerical computations are carried out for triangular, square, pentagon and hexagon cross sectional plates. Graphical representation is given for the various physical variables via gravity and different edge boundaries and its characteristics are discussed. This result can be applied for optimum design of concrete plates with polygonal cross sections.

A Novel Analytical Solution for the Brazilian Test with Loading Arcs

This research study presents a new theoretical model to calculate the indirect tensile strength for the Brazilian disk with loading arcs, based on numerical simulations, two-dimensional elasticity theory, and Griffith failure criterion. The new expression incorporates a no uniform contact pressure distribution determined by the results of the simulations with the finite element method. A computational experiment design has been developed to test the accuracy of the predictions made with the proposed model. This study demonstrates that the stresses predicted with the new model are closer to those determined by the finite element models than other theoretical solutions available in the literature. Additionally, a comparative analysis with experimental results obtained by other authors also indicates that the new model provides a more accurate magnitude of the indirect tensile strength.

Two-dimensional elastic theory methods for describing the stress state and the modes of elastic boring

Two-dimensional elastic theory methods for describing the stress state and the modes of elastic boring

Construction of the bending model of micropolar elastic thin beams with a circular axis and its implementation using finite element method

This paper considers the problem of transition from the system of two-dimensional equations of the micropolar (moment) theory of elasticity in a thin curved area to the one-dimensional system of equations describing deformation of the micropolar elastic thin beam with a circular axis. During the transition process, Timoshenko's hypotheses generalized to the micropolar case are applied. As a result, the applied model (with independent fields of displacements and rotation) of a micropolar elastic thin beam with a circular axis has been constructed. It is shown that the model includes the law of conservation of energy, energy theorems and variation principles. All main functionals for the model of the micropolar elastic thin beam with a circular axis are obtained from the functional of the two-dimensional micropolar theory of elasticity, containing only the first derivatives of displacements and rotations. The finite element method (FEM) is taken to study the boundary problems of statics and dynamics of applied model of the micropolar elastic thin beam with a circular axis. The basic concepts and stages of the FEM are formulated: the discretization, the selection of basic nodal unknowns, the approximation of the solution, and the construction of the basic FEM equations. The finite-element solutions of some problems of statics and problems on natural vibrations of beams with a circular axis are considered according to the micropolar theory of elasticity. A comparative analysis with similar problems of beams with a circular axis according to the classical theory of elasticity is carried out. Based on the results, some effective properties of beams with a circular axis have been established when considering their deformations in the context of the micropolar theory of elasticity.

Elasticity solution of functionally graded beams with consideration of the flexoelectric effect

This article studies the electroelastic response in bending behavior of a functionally graded elastic beam with consideration of the flexoelectric effect. The variational principle is employed to deduce the governing differential equations along with appropriate boundary conditions including nonclassic conditions based on a two-dimensional theory of elasticity and flexoelectricity. An exact elasticity solution is derived for a simply supported functionally graded beam subjected to mechanical loading and/or electric voltage between the top and bottom surfaces. Numerical computations are carried out for a SrTiO3 functionally graded beam of cubic symmetry subjected to typical sinusoidally mechanical and electric loading, respectively. Obtained results show the influence of the gradient parameter on the deflection, stress distribution, and electric polarization of the functionally graded beams. The suitable choice of the gradient parameter may decrease the stress or enhance the polarization, and improve the performance of the flexoelectricity of the beam.

Stress concentration around an elliptical hole in a large rectangular plate subjected to linearly varying in-plane loading on two opposite edges

Stress concentration in plates due to geometric irregularities such as holes and cracks is a crucial factor in design. The main objective of this paper is to derive the formula for tangential stress concentration factor around an elliptical hole in a large rectangular plate subjected to linearly varying in-plane loading on two opposite edges. The problem is investigated by the two-dimensional theory of elasticity using Muskhelishvili’s complex variable method. The stress functions are evaluated using the conformal mapping method and Cauchy’s integral formula. The stress functions thus obtained are coded to compute the non-dimensional stress components in the plate. The stress concentration factors at the edge of the elliptical hole are compared for different aspect ratios under various in-plane loading conditions. The results indicate that the formula obtained for tangential stress matches to the Kang’s [1] formula when elliptical hole is reduced to a circular hole, by taking its aspect ratio equal to 1. Furthermore, the classical solutions proposed by Inglis and Kirsch are easily obtained as special cases of the problem presented in this work, which confirms the efficacy of the solution. A plane stress finite element model is prepared in ABAQUS and results are compared with present method for a particular case of linearly varying in-plane loading.

A Novel Method for Assessing Material Heterogeneity-Induced Interfacial Stress Singularities of Dissimilar Joints Under Normal Loading

Interfacial stress singularities have a great effect on the strength, stiffness, and fracture behavior of dissimilar joints. Commonly used stress singularity assessment methods are based on the two-dimensional theory of elasticity, which is too complicated for design purposes. Three-dimensional (3-D) finite element analysis results show that the traditional method does not fully reflect the influence of material heterogeneity, especially Poisson’s ratio, on interfacial stress singularities. This paper presents an interfacial stress intensity factor based on the 3-D theory of elasticity, which characterizes the increasing rate of interfacial stress as the external force in dissimilar joints increases. Compared with traditional methods, the interfacial stress intensity factor proposed here provides a more accurate and efficient assessment of material heterogeneity-induced interfacial stress singularities and can be a valuable aid in determining the material properties in practice.

Two-dimensional elasticity solution for free vibration of simple-supported beams with arbitrarily and continuously varying thickness

This paper studies the analytical solution for the vibration of simply supported beams with arbitrarily and continuously varying thickness based on the two-dimensional elasticity theory. The general expression of stress function, which exactly satisfies the governing differential equations and the boundary conditions, is derived. Frequency equation governing the free vibration of beams with variable thickness can be obtained by using the Fourier sinusoidal series expansion on the upper and lower surfaces of the beam. The present solution method ensures a rapid convergence and meets the need of high accuracy in modern precise instruments. Several examples are provided to show the application of the proposed solution method which can be used to assess the validity of various approximate solutions and numerical methods for the beams with arbitrarily and continuously varying thickness.

A new mechanism of energy dissipation in nanomechanical resonators due to the Casimir force

In this study, we report a new energy dissipation mechanism of nanomechanical resonators due to the Casimir effect originating from quantum fluctuation of the vacuum electromagnetic field at the nanoscale. An analytical study on the evaluation of the Casimir effect-induced energy loss in nanobeam resonators undergoing in-plane flexural vibration is presented. Two-dimensional elastic wave theory is employed to determine the energy transmission from the vibrating resonator to the substrate. Fourier transform and Green's function technique are adopted to solve the problem of wave motions on the surface of the substrate excited by the Casimir force. Analytical expressions of the Casimir effect-induced energy loss in terms of the quality factor, taking into account both pressure wave propagation in the noncontact substrate and shear wave propagation in the supporting substrate, as well as linear and nonlinear terms of time-varying Casimir force, have been derived. Effects of beam geometry, initial separation gap, and structural boundary conditions on energy loss are examined. Results of the present study demonstrate that the Casimir effect-induced energy loss plays an important role in the dissipation of the nanobeam resonators, in which the influence of shear wave propagation is remarkable. Also, as reflected by our results, the influence of nonlinear terms of time-varying Casimir force on the energy dissipation cannot be neglected for large-amplitude vibration, which is obviously a feature of nonlinear damping. Furthermore, we propose a possible way to experimentally measure the Casimir force by using the energy dissipation mechanism due to the Casimir force.

In-Plane Free Vibration of Circular and Annular FG Disks

The analysis of the in-plane free vibration of the circular and annular FG disks by a meshfree boundary-domain integral equation method is presented in this paper. The material properties of the disks are assumed to vary in the radial direction obeying an exponential law. Based on the two-dimensional linear elastic theory, the motion equations of the FG disks are derived by using the static fundamental solutions. Radial integration method as an efficient tool is adopted to treat the domain integrals which are raised due to the material inhomogeneous and inertial effects. The natural frequencies and associate mode shapes are calculated for the FG disks with combinations of free and clamped boundary conditions. Parametric studies are also conducted to study the effects of the material gradients, radius ratios and boundary conditions on the frequency of the FG disks.

Bending analysis of functionally graded curved beams with different properties in tension and compression

In this study, we obtained analytical solutions for functionally graded curved beams with different properties in tension and compression, in which the moduli of elasticity in tension and compression are assumed as two different exponential functions. First, by determining the unknown neutral layer, we established a simplified mechanical model concerning tension and compression subzone and derived the one-dimensional solution (i.e., the solution in the scope of mechanics of materials). Given that the one-dimensional solution is a relatively simplified one, thus a comprehensive understanding of this problem is still needed. For this purpose, we established the consistency equation expressed in terms of stress function under two-dimensional theory of elasticity. Combining boundary conditions of inner and outer edges with continuity conditions of the neutral layer, we applied power series method for the solution of stress components under pure bending. The variations of radial and circumferential stresses in different cases of bimodular functionally graded parameters are comprehensively analyzed with numerical examples. Results indicate that the position of the neutral layer is generally related to the elastic modulus and the functionally graded coefficients of the materials. Moreover, the maximum tensile or compressive bending stress may not take place at the outer or inner edges of the curved beam but inside the beam, which should be given more attention in the analysis and design of functionally graded curved beams with different properties in tension and compression.

Dynamic behaviors of tapered bi-directional functionally graded beams with various boundary conditions under action of a moving harmonic load

In this paper, the dynamic response of beams made of bi-directional functionally graded materials (FGMs) with varied cross-sections subjected to a moving harmonic load has been studied by a developed boundary-domain integral equation method based on the two-dimensional linear elastic theory. It is assumed that the elastic modulus and mass density of the FG beams vary separately following each exponential distribution parameters through longitudinal or transvers direction. Then, the elastostatic homogeneous fundamental solution was substituted into the dynamic governing equation for deriving the vibration boundary-domain integral equation of the tapered FG beams. A meshfree scheme can be achieved by applying the radial integration technique to transform the domain integrals into boundary integrals. Houbolt's algorithm is employed to obtain the dynamic response of the tapered FG beams. The effectiveness of the present method applied to the non-uniformed FG beams is verified by comparing numerical results with those available for free vibration of tapered beams of linearly variable width or depth. A detailed parametric study is carried out to investigate the influences of the taper ratios, material gradients, gradation directions and boundary conditions as well as the moving load frequencies and velocities on the dynamic behaviors of tapered FG beams.

Dynamic stability of cylindrical nanoshells under combined static and periodic axial loads

The dynamic stability of cylindrical nanoshells subjected to combined static and time-dependent periodic axial forces is studied by employing the two-dimensional nonlocal elasticity theory together with the first-order shear deformation theory of shells. The differential quadrature method as an efficient and accurate numerical technique is applied to discretize the equations of motion under different boundary conditions in the spatial domain and transform them into a system of coupled Mathieu–Hill-type equations in time domain. Subsequently, the Bolotin’s first approximation method is employed to extract the dynamic instability regions of the cylindrical nanoshells. The approach is validated by showing its fast convergence rate and carrying out comparison studies with existing results in the limit cases. Afterward, the effects of the nonlocal parameter, length and thickness-to-mean radius ratios together with different boundary conditions on the principal dynamic instability regions of the cylindrical nanoshells are studied in detail.

Strain-Gradient Elasticity Theory for the Mechanics of Fiber Composites Subjected to Finite Plane Deformations: Comprehensive Analysis

A general model for the mechanics of fiber-reinforced composites is delivered in finite-plane elastostatics. Within the framework of the nonlinear strain-gradient theory, we obtain the Euler equilibrium equations and admissible boundary conditions. In particular, we present a third gradient continuum model and associated formulations. Soft composite materials are also considered by employing the Mooney-Rivlin energy potential. The presented model can be used as an alternative two-dimensional Cosserat theory of nonlinear elasticity.

Wave dispersion properties in imperfect sigmoid plates using various HSDTs

In this paper, wave propagations in sigmoid functionally graded (S-FG) plates are studied using new Higher Shear Deformation Theory (HSDT) based on two-dimensional (2D) elasticity theory. The current higher order theory has only four unknowns, which mean that few numbers of unknowns, compared with first shear deformations and others higher shear deformations theories and without needing shear corrector. The material properties of sigmoid functionally graded are assumed to vary through thickness according sigmoid model. The S-FG plates are supposed to be imperfect, which means that they have a porous distribution (even and uneven) through the thickness of these plates. The governing equations of S-FG plates are derived employed Hamilton's principle. Using technique of Navier, differential equations of S-FG in terms displacements are solved. Extensive results are presented to check the efficient of present methods to predict wave dispersion and velocity wave in S-FG plates.

A two-dimensional elasticity model for bending and free vibration analysis of laminated graphene-reinforced composite beams

By assuming the plane-stress state in each layer, a two-dimensional elasticity model is proposed for laminated graphene-reinforced composite (GRC) beams. It is assumed that the graphene disperses uniformly in each layer but the graphene volume fraction may vary from layer to layer. For an arbitrary individual layer, the governing partial differential equations and boundary conditions are given directly from the two-dimensional elasticity theory. Then, the multi-term Kantorovich-Galerkin method is employed to build a state-space equation for the layer, in which the axial and transverse displacements are expressed as products of trial function matrix and unknown function matrix. Eventually, a global equation for the laminated beam is established by virtue of the displacement and stress continuity conditions at the interfaces. Non-dimensional displacements, stresses and natural frequencies are obtained for laminated GRC beams with different boundary conditions. The effects of graphene distribution patterns, boundary conditions, length-to-thickness ratios, layer fraction increments and the number of layers are examined. It is found that the laminated GRC beam with graphene distribution pattern X has the smallest deflection and largest fundamental frequency at high length-to-thickness ratios, but it has largest deflection and smallest fundamental frequency at a very low length-to-thickness ratio due to its reduced transverse shear stiffness.

Fracture Cross Extension Arithmetic Research Based on Non-continuum Theory

The cross extension of hydraulic fracture and natural fracture is the key to realize the volume fracture in unconventional reservoir. Based on the non-continuum theory, with the help of numerical simulation algorithm, a mathematical model for in-situ stress about fracturing expansion when the hydraulic fracture intersects with a natural fracture is presented. Then, using two-dimensional linear elastic theory verify the validity of degenerate in-situ stress mode. So it makes the natural fissure error about natural fracture surface pressure less than 1%. As a result, the accuracy of the model was proved. On the basis of the above, considering fluid-structure coupling effect, the stress change rules on natural fracture surface are obtained during hydraulic fracture extension process. The calculation results are as follows: In the process of hydraulic fracture approaching natural fracture, the surface of natural fracture occurs shear deformation first, and then turns into open deformation. Under the influence of shear deformation, the direction of the maximum principal stress on natural fracture surface is the same as initial horizontal maximum principal geostress. Due to the expansion of hydraulic fracture and fracture tip concentration effect, the maximum principal stress appears in the hydraulic fracture tip, this two points may induce the natural fracture begin to extend. And for initial ground stress in a stratum, the smaller the difference, the greater the pressure required to open the natural fracture. At this time, the maximum principal stress on the surface of the natural fracture is less concentrated; multiple fracture initiations are easily formed. Under high pressure, when the approaching angle between natural fracture and hydraulic fracture closes to 45° hydraulic fracture is easier to activate natural fracture, the formed hydraulic fracture shape is more complicated.