Exact Solutions For Displacements Research Articles

Axisymmetric Wave Propagation in Functionally Grade Cylinder with Isotropic Concentric Layers

The axisymmetric wave propagation is investigated in coaxially layered isotropic functionally graded cylinders with different mass and stiffness properties of the layers. Exact solutions of the governing equations of the wave propagation in the cylinders exist only for isotropic, transversely isotropic and piezoelectric transversely isotropic cylinders. In the case of the functionally graded cylinders exact solutions are not known. The authors developed a method of exact solution of the problem, which is based on matching of continuity and boundary conditions on junctions of the layers. The continuity and boundary conditions are formulated in terms of exact solutions for displacements and stresses in the layers obtained in Bessel functions. The spectral diagrams of the dispersion curves of frequencies versus wave numbers are plotted for both propagating and evanescent waves and graphs of the phase and group velocities are obtained from these diagrams.

Analytical and Numerical Analyses for a Variety of Submerged Hemi-Ellipsoidal Shells

AbstractThis paper concerned an analytical method to obtain exact solutions for displacements of submerged hemi-ellipsoidal shells. Based on the linear membrane theory, the membrane forces and stra...

Three-dimensional free vibration analysis of thick cylindrical shells with general end conditions and resting on elastic foundations

Abstract A unified method based on the three-dimensional theory of elasticity is developed for the free vibration analysis of thick cylindrical shells with general end conditions and resting on elastic foundations. Each shell displacements, regardless of boundary conditions, is expanded as a standard Fourier cosine series supplemented with four auxiliary functions introduced to eliminate any possible discontinuities of the original displacement and its derivatives throughout the entire shell space including the boundaries and then to effectively enhance the convergence of the results. Mathematically, such series expansions are capable of representing any functions (including the exact displacement solutions). Since the displacement field is constructed adequately smooth throughout the whole solution domain, an accurate solution can obtained by using Rayleigh–Ritz procedure based on the energy functions of the shell. The current method can be universally apply to a variety of end conditions including all the classical cases and their combinations and arbitrary elastic foundations. The excellent accuracy and reliability of current solutions are demonstrated by numerical examples and comparisons with the results available in the literature. Effects of the boundary restraining parameters and foundation coefficients on frequency parameters are investigated as well. New results for thick cylindrical shells with various end conditions and resting on elastic foundations are presented, which may serve as benchmark solutions for validating new computational techniques in future.

Analysis of Functionally Graded Piezoelectric Cylinders in a Hygrothermal Environment

AbstractThis paper presents an analytical solution for the interaction of electric potentials, electric displacement, elastic deformations, and describes hygrothermal effect responses in hollow and solid cylinders, subjected to mechanical load and electric potential. Exact solutions for displacement, stresses and electric potentials in functionally graded piezoelectric material are determined using the infinitesimal theory. The material properties coefficients of the present cylinder are assumed to be graded in the radial direction by a power law distribution. Numerical examples display the significant of influence of material inhomogeneity. It is interesting to note that selecting a specific value of inhomogeneity parameter can optimize the piezoelectric hollow and solid cylinders responses, which will be of particular importance in modern engineering designs.

Analytical study on elastic transition in short-fiber composites for plane strain case

An analytical approach for short-fiber-reinforced composites is developed for three-dimensional (3D) elastic stress field distribution subjected to an applied axial load. Two sets of exact displacement solutions for matrix and fiber, which are respectively called far-field and transient solutions, are derived based on the theory of elasticity. The superposition state of these solutions are then used to obtain the analytical expressions for the 3D stress field components over the entire composite system, including the fiber end region, through the adding imaginary fiber technique. The fiber/matrix 3D stress field components fully satisfy the equilibrium and compatibility conditions in the theory of elasticity. The stress field components also satisfy the overall boundary, interface continuity, and axial force equilibrium conditions. The analytical results obtained are then validated by finite element method modeling.

Free vibration analysis of laminated composite shallow shells with general elastic boundaries

A unified and exact solution method is developed for the free vibration analysis of composite laminated shallow shells with general elastic boundary conditions, a class of problem which is of practical interest and fundamental importance but rarely attempted in the literature. Under the current framework, each of the shell displacements, regardless of boundary conditions, is expanded as a standard Fourier cosine series supplemented with closed-form auxiliary functions introduced to eliminate all the relevant discontinuities with the displacement and its derivatives at the edges and accelerate the convergence of series representations. Mathematically, such series expansions are capable of representing any functions (including the exact displacement solutions). Rayleigh–Ritz procedure is employed to obtain the exact solution based on the energy functions of the shell. The current method can be universally applicable to a variety of boundary conditions including all the classical cases, elastic restraints and their combinations. The excellent accuracy and reliability of the current solutions are validated by numerical examples, and the effects of boundary restraining stiffnesses and lamination schemes on frequency parameters are illustrated. New results for various shell curvatures including spherical, cylindrical and hyperbolical paraboloidal shells with elastically restrained edges are presented, which may serve as benchmark solutions.

An exact solution for the free vibration analysis of laminated composite cylindrical shells with general elastic boundary conditions

A unified and exact solution method is developed for the free vibration analysis of composite laminated cylindrical shells with general elastically restrained boundaries and arbitrary lamination schemes. Each of the shell displacements, regardless of boundary conditions, is constructed as a standard Fourier cosine series supplemented with auxiliary functions introduced to eliminate all the relevant discontinuities with the displacement and its derivatives at the edges and accelerate the convergence of series representations. Mathematically, such series expansions are capable of representing any functions (including the exact displacement solutions). Rayleigh–Ritz procedure is employed to obtain the exact solution based on the energy functions of the shell. The current method can be universally applicable to a variety of boundary conditions including all the classical cases, elastic restraints and their combinations. Several numerical examples are presented to validate the excellent accuracy and reliability of current solutions, and the effects of boundary restraining stiffnesses and lamination schemes on frequency parameters are illustrated. New results for different lamination schemes with elastically restrained edges are presented, which may serve as benchmark solutions.

Basic modifications in 3D micromechanical modeling of short fiber composites with bonded and debonded fiber end

A modified analytical model is developed for analysis of 3D elastic stress fields in short fiber composites subjected to an applied axial load. Two sets of exact displacement solutions for the matrix and fiber are derived based on the theory of elasticity. The superposition state of these solutions are then used to obtain the analytical expressions for the 3D stress field components over the entire composite system including the fiber end region, which is modeled by the use of imaginary fiber technique. The main difference with the previous works here is that the stress field is considered to be a function of both radial and axial directions. Such an assumption made it possible to calculate the effects of radial position on the stress and strain distributions in the fiber and imaginary fiber regions, the fact that has been ignored by previous works. The analytical results obtained here are then validated by the FEM modeling. Interestingly, good agreements are found between the analytical and numerical predictions for all the stress, strain and displacement components. Another superiority of the proposed analytical model is in its capability of solving problems with fiber/matrix debonding defect.

Exact solutions for functionally graded pressure vessels in a uniform magnetic field

Analytical studies for magnetoelastic behavior of functionally graded material (FGM) cylindrical and spherical vessels placed in a uniform magnetic field, subjected to internal pressure are presented. Exact solutions for displacement, stress and perturbation of magnetic field vector in FGM cylindrical and spherical vessels are determined by using the infinitesimal theory of magnetoelasticity. The material stiffness and magnetic permeability obeying a simple power law are assumed to vary through the wall thickness and Poisson’s ratio is assumed constant. Stresses and perturbation of magnetic field vector distributions depending on an inhomogeneous constant are compared with those of the homogeneous case and presented in the form of graphs. The inhomogeneous constant, which includes continuously varying volume fraction of the constituents, is empirically determined. The values used in this study are arbitrary chosen to demonstrate the effect of inhomogeneity on stresses and perturbation of magnetic field vector distributions.

Vibration analysis of beams treated with passive constrained layer damping using spectral element method

This paper introduces a spectrally formulated finite element method (SFEM) for the beams treated with passive constrained layer damping (PCLD). The viscoelastic core of the beams has a complex modulus that varies with frequency. The SFEM is formulated in the frequency domain using dynamic shape functions based on the exact displacement solutions from progressive wave methods, which implicitly account for the frequency-dependent complex modulus of the viscoelastic core. The dynamic responses obtained by the spectral element method and the conventional finite element method (CFEM) are compared to evaluate the validity and accuracy of the present spectral PCLD beam element model. The spectral PCLD beam element model is found to provide very reliable results when compared with the conventional finite element model.

Spectral Finite Element Analysis of Sandwich Beams With Passive Constrained Layer Damping1

We present a spectral finite element model (SFEM) for sandwich beams with passive constrained layer damping (PCLD) treatments. The viscoelastic core has a complex modulus that varies with frequency. The SFEM is formulated in the frequency domain using dynamic shape functions based on the exact displacement solutions from progressive wave methods, where we implicitly account for the frequency dependent complex modulus of the viscoelastic core. The SFEM results of natural frequencies and frequency response functions are compared to those calculated using conventional finite element (CFEM), where the Golla-Hughes-McTavish method is used to account for the frequency dependent complex modulus of a viscoelastic core. Also experimental data are used to validate both analyses using frequency response functions measured for two cantilevered sandwich beams with PCLD treatments having 50% and 75% coverage of the beam length. SFEM shows improved computational efficiency and accuracy, because many more elements must be incorporated into the CFEM for comparable accuracy.

Exact solutions for axisymmetric bending of continuous annular plates

This paper is concerned with obtaining exact solutions for the axisymmetric bending problem of elastic annular plates with internal concentric supports. The standard stiffness method of analysis is employed to solve the problems. The solutions can be obtained either numerically or in closed-form. Both thin and thick plates can be handled. For the thick plate analysis, the Mindlin plate theory has been adopted to cater for the effect of transverse shear deformation. Relationships between Kirchhoff and Mindlin plate solutions have been used to derive the stiffness matrix of thick plates. Present solutions are compared with published results for some representative annular plate bending problems to demonstrate their validity. Exact solutions for displacement, shear force and bending moment for an annular plate with an internal concentric support are presented. Numerical results showing the influence of transverse shear deformation on edge displacement under a concentric ring load applied at the free inner edge of a free/continuous support/fixed plate are also presented.

A creeping and surface breaking long strike-slipfault inclined to the vertical in a viscoelastic half space

An asoismically creeping surface-breaking strike-slip fault inclined to the vertical at an arbitrary angle, situated in a simple model of the lithosphere-asthenosphere system consisting of a visoelastic half space is considered. The exact solutions for displacements, stresses and strains In the model are obtained. Computed results show that the inclination of the fault has a significant influence on the values of the displacements, stresses and strains. The rate of accumulation of shear stress tending to cause strike-slip movement has been found to be greatest for vertical strike-slip fault, while for faults inclined at smaller angles to the horizontal, this rate is significantly smaller. The uses of such theoretical models in obtaining greater insight into the earthquake processes in seismically active regions and their relations to the dynamics of the lithosphere-asthenosphere system are examined.

The convergence of finite element method in solving linear elastic problems

This paper presents a theoretical development to show the sufficient conditions that will insure a finite element displacement analysis to converge to the exact displacement solutions when the size of the elements are progressively reduced. The order of such convergence is also estimated. The development is in connection with the three dimensional elasticity problem and the plate bending problem. A study is made to determine the possible means of evaluating the merits of different stiffness matrices to be used in the finite element analysis.