Exact Three-dimensional Solution Research Articles

Three-dimensional exact solutions of transversely isotropic coated structures under tilted circular flat punch contact

Because of the large stress gradient in the coated structure and the more and more thin thickness of coating, it is too hard to guarantee the calculation accuracy by the commercial software in the design of the coated structure. In this case, a research work for an accurate and efficient calculation method for the coated structure is necessary and will contribute to the accurate design and quality promotion of the coated structure. In this paper, the three-dimensional solutions of transversely isotropic coated structure under tilted circular flat punch contact are derived based on the three-dimensional general solutions of the transversely isotropic material. The solutions are obtained in two cases including the smooth contact and the frictional contact. The numerical results show that this method has the good convergence, high accuracy, high efficiency and high stability. Based on this method, the optimal thickness of the coating and the interfacial stress between the coating and the substrate can be obtained accurately. The corresponding analysis will contribute to the strengthening design of the coated structure.

Fully coupled thermo-mechanical analysis of multilayered plates with embedded FGM skins or core layers using a layerwise mixed model

This work presents a new layerwise mixed model for the fully coupled thermo-mechanical static analysis of multilayered plates with embedded functionally graded material (FGM) layers, either skins or core layers, under thermal and/or mechanical loads. This model is able to fully describe a two-constituent metal-ceramic FGM layer z-continuous effective properties through-thickness, using any given homogenization method, and is here extended to the fully coupled thermo-mechanical analysis. It is based on a mixed least-squares formulation with a layerwise variable description for displacements, transverse stresses and in-plane strains, along with temperature, transverse heat flux and in-plane components of the thermal gradient, taken as independent variables. This mixed formulation ensures that the interlaminar C0 continuity requirements, where the material properties may actually change, are fully fulfilled a priori by all chosen independent variables. The numerical results consider single-layer and multilayered plates with different side-to-thickness ratios, under thermal or mechanical loads, using mainly Mori-Tanaka estimate for the FGM effective properties with different material gradation profiles. The results are assessed by comparison with three-dimensional (3D) exact solutions, and demonstrate the model capability to predict a highly accurate quasi-3D thermo-mechanical description of the through-thickness distributions of displacements and stresses, as well as temperature and heat flux.

Three-dimensional exact solutions of homogeneous transversely isotropic coated structures under spherical contact

Because of the large stress gradient in the coated structure and the more and more thin thickness of coating, it is too hard to guarantee the calculation accuracy by the commercial software in the design of the coated structure. In this case, a research work for an accurate and efficient calculation method for the coated structure is necessary and will contribute to the accurate design and quality promotion of the coated structure. In this paper, the three-dimensional solutions of homogeneous transversely isotropic coated structure under spherical contact loadings are derived based on the three-dimensional general solutions of the transversely isotropic material. The solutions are obtained in two cases including the smooth contact and the frictional contact. The numerical results show that this method has the good convergence, high accuracy, high efficiency and high stability. Based on this method, the optimal thickness of the coating and the interfacial stress between the coating and the substrate can be obtained accurately. Because the interfacial stress can result to the debonding failure of coated structures, a method for predicting interface security state with evaluation criteria for interfacial failure is proposed. The corresponding analysis will contribute to the strengthening design of the coated structure.

Static response of functionally graded multilayered one-dimensional quasicrystal cylindrical shells

Functionally graded materials have shown better mechanical behavior in multilayered structures, which have attracted much attention. A three-dimensional exact elastic solution for layered cylindrical shells made of functionally graded one-dimensional orthorhombic quasicrystals is carried out in this paper. The two edges of the cylindrical shells are simply supported, and functionally graded materials of the cylindrical shells are assumed to obey a power law along the radial direction. The exact solution for a single-layer functionally graded one-dimensional quasicrystal cylindrical shell is derived in terms of the pseudo-Stroh formalism. After introduction of the propagator matrix method, the exact solution for the corresponding layered case is obtained. Comprehensive numerical investigation, encompassing different exponential factors, different inner surface boundary conditions as well as different stacking sequences, has been conducted.

Making shapes of glassy nematic sheets with three-dimensional director fields

Programmable shaping of glassy nematic sheets with three-dimensional director fields is explored. In contrast to planar inscriptions, the out-of-plane director orientations in these nematic sheets cause transverse shear strains upon heating or illumination. We quantify such an effect by extending the Reissner-Mindlin's nonlinear theory of elastic plates. The validity and necessity are examined via a special problem which has exact three-dimensional solution. Then we show, through a few typical examples, that the three-dimensional director fields can be exploited to create rich deformed shapes of the nematic sheets. Especially, some complicated morphologies can be programmed without producing resultant forces and moments.

Three-dimensional exact solution for the free vibration of thick functionally graded annular sector plates with arbitrary boundary conditions

A new three-dimensional exact solution for the free vibrations of arbitrary thick functionally graded annular sector plates with arbitrary boundary conditions is presented. The three-dimensional elasticity theory is employed to formulate the theoretical model. According to a power law distribution of the volume of the constituents, the material properties change continuously through the thickness of the functionally graded annular sector plates. Each of displacements of the annular sector plates, regardless of boundary conditions, is expanded as a three-dimensional (3-D) Fourier cosine series supplemented with closed-form auxiliary functions introduced to eliminate all the relevant discontinuities with the displacements and its derivatives at the edges. Since the displacement fields are constructed adequately smooth throughout the entire solution domain, an exact solution is obtained based on the Ritz procedure by the energy functions of plate. The excellent accuracy and reliability of the current solutions are demonstrated by numerical examples and comparison of the present results with those available in the literature, and numerous new results for thick FG annular sector plates with elastic boundary conditions are presented. The effects of gradient indexes are also illustrated.

Three-dimensional exact solution for vibration analysis of thick functionally graded porous (FGP) rectangular plates with arbitrary boundary conditions

In this paper, a novel three-dimensional exact solution is performed for vibration analysis of thick functionally graded porous rectangular plates subject to arbitrary boundary conditions. The theoretical model is based on the three-dimensional elastic theory. Three kinds of porosity distributions including even, uneven and the logarithmic-uneven are performed. Non consideration boundary conditions, all displacements of the rectangular plate are unified expanded as a improve Fourier series, which consists of standard three-dimensional (3-D) Fourier cosine series supplemented with closed-form auxiliary functions introduced to eliminate all the relevant discontinuities with the displacements and its derivatives at the edges. As the displacement field is sufficiently smooth in the entire solution domain, the exact solution of the energy function of the plate is obtained through the Rayleigh Ritz process. The numerical examples are given to verify the accuracy and reliability of the present method, in which both the present results and those reported in the literature are provided. Besides, several numerous new results for thick FGP plates with elastic boundary conditions are also presented.

Deformations and stresses of multilayered plates with embedded functionally graded material layers using a layerwise mixed model

This work presents a new layerwise mixed model for the static analysis of multilayered plates with embedded functionally graded material (FGM) layers subjected to transverse mechanical loads. This model is capable to fully describe a two-constituent metal-ceramic FGM layer continuous variation of material properties in the thickness direction, using any given homogenization method to estimate its effective properties. The present model is based on a mixed least-squares formulation with a layerwise variable description for displacements, transverse stresses and in-plane strains, chosen as independent variables. This mixed formulation ensures that the interlaminar C0 continuity requirements at the layers interfaces, where the material properties actually change, are fully fulfilled a priori for all independent variables. The order of the in-plane two-dimensional finite element approximations and the order of the z-expansion through each layer thickness, as well as the number of layers, whether FGM layers or not, are considered free parameters. The full description of the FGM effective properties is achieved by applying to the z-continuous elastic coefficients a z-expansion through the layer thickness of a given order, set as an added free parameter, in a similar approach to finite element approximations. The numerical results consider both single-layer and multilayered functionally graded plates with different side-to-thickness ratios, using either Mori-Tanaka or the rule of mixtures estimates for the FGM effective properties with different material gradation profiles. The present model results are assessed by comparison with three-dimensional (3D) exact solutions and closed form solutions, which demonstrate its capability to predict a highly accurate quasi-3D description of the displacements and stresses distributions altogether.

Rigorous formulation of space-charge wake function and impedance by solving the three-dimensional Poisson equation

In typical numerical simulations, the space-charge force is calculated by slicing a beam into many longitudinal segments and by solving the two-dimensional Poisson equation in each segment. This method neglects longitudinal leakage of the space-charge force to nearby segments owing to its longitudinal spread over 1/γ. By contrast, the space-charge impedance, which is the Fourier transform of the wake function, is typically calculated directly in the frequency-domain. So long as we follow these approaches, the longitudinal leakage effect of the wake function will remain to be unclear. In the present report, the space-charge wake function is calculated directly in the time domain by solving the three-dimensional Poisson equation for a longitudinally Gaussian beam. We find that the leakage effect is insignificant for a bunch that is considerably longer than the chamber radius so long as the segment length satisfies a certain condition. We present a criterion for how finely a bunch should be sliced so that the two-dimensional slicing approach can provide a good approximation of the three-dimensional exact solution.

Solution of the Cauchy Problem for the Three-Dimensional Telegraph Equation and Exact Solutions of Maxwell’s Equations in a Homogeneous Isotropic Conductor with a Given Exterior Current Source

For the solution of the Cauchy problem for the linear telegraph equation in three-dimensional space, we derive a formula similar to the Kirchhoff one for the linear wave equation (and turning into the latter at zero conductivity). Additionally, the problem of determining the field of a given exterior current source in an infinite homogeneous isotropic conductor is reduced to a generalized Cauchy problem for the three-dimensional telegraph equation. The derived formula enables us to reduce this problem to quadratures and, in some cases, to obtain exact three-dimensional solutions with a propagating front, which are of great applied importance for testing numerical methods for solving Maxwell’s equations. As an example, we construct the exact solution of the field from a Hertzian dipole with an arbitrary time dependence of the current in an infinite homogeneous isotropic conductor.

Mutual effect of Coriolis acceleration and temperature gradient on the stress and strain field of a glass/epoxy composite-pipe

• Three-dimensional thermoelastic solution for a composite pipe using the heat transfer equation along thickness. • Effect of body forces created by rotation and the Coriolis acceleration. • Approximation of Bessel equation as an Euler type differential equation while assuming a thin wall composite pipe. • Evaluation of angular velocity and the composite pipe wall thickness as two important parameters in making the Coriolis effect. Rotating composite pipes due to their light-weight, high specific strength and stiffness and excellent fatigue resistance are a good substitute for most common metallic alloys. This paper presents the exact three-dimensional thermoelastic solution for a composite pipe using the heat transfer equation along thickness, by taking into account the body forces created by rotation and the Coriolis effect. A Bessel nonhomogeneous differential equation is derived for displacement along the radial direction, which can be approximated by a Euler equation using the pipe wall thinness restriction. The composite pipe rotational speed and wall thickness are two key parameters which are used to illustrate the Coriolis effect on the pipe stress and strain filed. Results indicated that for the wall thickness to inner radius ratios less than 0.1, all stresses, strains, and displacements were experienced 4% increase, except for radial stress, upon which Coriolis acceleration had no effect . In the case of thermomechanical loading, increasing the pipe wall thickness resulted in variation of the thermal stress and the centrifugal force values. Numerical verifications showed that for thickness to radius ratios less than 0.05 the amount of deviation between Bessel (numerical) and Euler (analytical) formulation is less than 5%.

Benchmark exact free vibration solutions for multilayered piezoelectric composite plates

The development of advanced finite element models suitable for analysis of smart structures depends on a comprehensive understanding of the high layerwise inhomogeneity in mechanical and electric properties present in multilayered piezoelectric composite structures. The assessment of advanced models still continues to rely only on a few well-known benchmark three-dimensional (3D) exact solutions provided by pioneer works, namely, in the 1970s by Pagano as well as Srinivas for composite plates and later in the 1990s by Heyliger and Saravanos for piezoelectric composite plates. To overcome the limited number of test cases whose 3D exact solutions have been published, the method introduced by Heyliger to derive the 3D exact solutions has been successfully implemented using symbolic computing. New benchmark 3D exact free vibration solutions are provided for piezoelectric composite plates that complete a previous work with the static solutions. Two multilayered plates using PVDF piezoelectric material are chosen as test cases under three sets of electrical boundary conditions and considering three plate aspect ratios (from thick to thin plate), in a total of 18 test cases. For each case, the frequencies of the first six thickness modes are presented, along with some representative through-thickness mode shapes.

Exact Three-Dimensional Elasticity Solution for Buckling of Beam of a Circular Cross-Section

Exact Three-Dimensional Elasticity Solution for Buckling of Beam of a Circular Cross-Section

Static analysis of doubly curved film-substrate shells with thickness-dependent material properties

Based on Reissner’s mixed variational theorem (RMVT), we develop a weak-form formulation of finite doubly curved layer (FDCL) methods for the static analysis of simply-supported, multilayered functionally graded material (FGM) doubly curved shells (DCSs) under mechanical loads. This formulation is used to examine the spatial distributions of interlaminar stress and displacement components induced in a pressure-loaded FGM film-substrate DCS, and these solutions are compared with those obtained by using the principle of virtual displacements. The material properties of the multilayered (or film-substrate) shell are thickness-dependent, and the effective material properties of the FGM layer are estimated using the rule of mixtures and Mori–Tanaka scheme. The trigonometric functions and Lagrange polynomials are used to interpolate the in-surface and thickness variations of the primary variables of each individual layer, respectively. The orders used to expand the primary variables through the thickness direction are taken to be the same as one another, which are linear, quadratic and cubic variations. The accuracy and convergence rate of these RMVT-based FDCL methods are validated by comparing their solutions with the exact three-dimensional and accurate two-dimensional solutions available in the literature.

Harbingers and latecomers – the order of appearance of exact coherent structures in plane Poiseuille flow

ABSTRACTThe transition to turbulence in plane Poiseuille flow (PPF) is connected with the presence of exact coherent structures. We here discuss a variety of different structures that are relevant for the transition, compare the critical Reynolds numbers and optimal wavelengths for their appearance, and explore the differences between flows operating at constant mass flux or at constant pressure drop. The Reynolds numbers quoted here are based on the mean flow velocity and refer to constant mass flux. Reynolds numbers based on constant pressure drop are always higher. The Tollmien–Schlichting (TS) waves bifurcate subcritically from the laminar profile at Re = 5772 at wavelength 6.16 and reach down to Re = 2610 at a different optimal wave length of 4.65. Their streamwise localised counter part bifurcates at the even lower value Re = 2334. Three-dimensional exact solutions appear at much lower Reynolds numbers. We describe one exact solutions that has a critical Reynolds number of 316. Streamwise localised versions of this state require higher Reynolds numbers, with the lowest bifurcation occurring near Re = 1018. The analysis shows that the various branches of TS-waves cannot be connected with transition observed near Re ≈ 1000 and that the exact coherent structures related to downstream vortices come in at lower Reynolds numbers and prepare for the transition.

Simulating Stable Isotope Ratios in Plumes of Groundwater Pollutants with BIOSCREEN-AT-ISO.

BIOSCREEN is a well-known simple tool for evaluating the transport of dissolved contaminants in groundwater, ideal for rapid screening and teaching. This work extends the BIOSCREEN model for the calculation of stable isotope ratios in contaminants. A three-dimensional exact solution of the reactive transport from a patch source, accounting for fractionation by first-order decay and/or sorption, is used. The results match those from a previously published isotope model but are much simpler to obtain. Two different isotopes may be computed, and dual isotope plots can be viewed. The dual isotope assessment is a rapidly emerging new approach for identifying process mechanisms in aquifers. Furthermore, deviations of isotope ratios at specific reactive positions with respect to "bulk" ratios in the whole compound can be simulated. This model is named BIOSCREEN-AT-ISO and will be downloadable from the journal homepage.

Three-dimensional nonlinear ideal MHD equilibria with field-aligned incompressible and compressible flows

The equilibrium properties of three-dimensional ideal magnetohydrodynamics (MHD) are investigated. Incompressible and compressible flows are considered. The governing equations are taken in a steady state such that the magnetic field is parallel to the plasma flow. Equations of stationary equilibrium for both of incompressible and compressible MHD flows are derived and described in a mathematical mode. For incompressible MHD flows, Alfvénic and non-Alfvénic flows with constant and variable magnetofluid density are investigated. For Alfvénic incompressible flows, the general three-dimensional solutions are determined with the aid of two potential functions of the velocity field. For non-Alfvénic incompressible flows, the stationary equilibrium equations are reduced to two differential constraints on the potential functions, flow velocity, magnetofluid density, and the static pressure. Some examples which may be of some relevance to axisymmetric confinement systems are presented. For compressible MHD flows, equations of the stationary equilibrium are derived with the aid of a single potential function of the velocity field. The existence of three-dimensional solutions for these MHD flows is investigated. Several classes of three-dimensional exact solutions for several cases of nonlinear equilibrium equations are presented.

An analytical approach to three-dimensional coupled thermoelectroelastic analysis of functionally graded piezoelectric plates

This article deals with the sampling surfaces method developed recently by the authors and its implementation for the three-dimensional coupled steady-state thermoelectroelastic analysis of functionally graded piezoelectric laminated plates subjected to thermal loading. The sampling surfaces formulation is based on choosing inside the nth layer [Formula: see text] not equally spaced sampling surfaces parallel to the middle surface of the plate in order to introduce temperatures, electric potentials, and displacements of these surfaces as basic plate variables. Such choice of unknowns with the consequent use of Lagrange polynomials of degree [Formula: see text] in the thickness direction for each layer permits the presentation of the proposed functionally graded piezoelectric plate formulation in a very compact form. The sampling surfaces are located inside each layer at Chebyshev polynomial nodes that allow one to minimize uniformly the error due to the Lagrange interpolation. As a result, the sampling surfaces method can be applied efficiently to analytical solutions for functionally graded piezoelectric laminated plates, which asymptotically approach the three-dimensional exact solutions of thermoelectroelasticity as [Formula: see text].

Benchmark solution for free vibration of thick open cylindrical shells on Pasternak foundation with general boundary conditions

In the present article, a new three-dimensional exact solution for free vibration of thick open cylindrical shells on Pasternak foundation with general boundary conditions is presented. The three-dimensional elasticity theory is employed to formulate the theoretical model. The admissible functions of the thick shells are described as a combination of a three-dimensional (3-D) Fourier cosine series and auxiliary functions. Compared with the traditional Fourier series, the improved Fourier series can eliminate all the relevant discontinuities of the displacements and their derivatives at the edges regardless of boundary conditions. The excellent accuracy and reliability of the current solutions are demonstrated by numerical examples and comparison of the present results with those available in the literature and obtained by using ABAQUS which is based on the finite element method. Numerous new results for thick open cylindrical shells on Pasternak foundation with elastic boundary conditions are presented. In addition, comprehensive studies on the effects of the elastic restraint parameters, geometric parameters and elastic foundation coefficients are also reported.

On a Consistent Dynamic Finite-Strain Plate Theory and Its Linearization

This paper develops a dynamic finite-strain plate theory consistent with three-dimensional Hamilton’s principle under general loadings with a fourth-order error. Starting from the three-dimensional field equations for a compressible hyperelastic material and by a series expansion about the bottom surface, we deduce a vector dynamic plate equation with three unknowns, which exhibits the local momentum-balance structure. Associated weak formulations are considered, in connection with various boundary conditions. Then, by linearization, we provide a novel linear plate theory for orthotropic materials, which takes into account both stretching and bending effects. For isotropic materials, it is further modified to a refined linear plate theory, leading to higher-order results under certain circumstances. To verify the present plate theory, an exhaustive study of the free vibration and static bending problems is carried out. Comparing with the available exact three-dimensional solutions for these problems, it is shown that the plate theory indeed provides correct asymptotic results for displacements and stresses distributions. The advantages of this linear plate theory include (i) its simplicity (as simple as the Kirchhoff-Love theory), (ii) high accuracy for frequencies and deflections, as well as capability of capturing the distributions of all concerned quantities, and (iii) applicability for general loadings.