3D Exact Solutions Research Articles

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.

A refined two‐dimensional theory for thermal bending of functionally graded plates with thickness normal strains

AbstractA refined theory is presented for the thermal response of composite functionally graded plates under thermo‐mechanical loads. The theory includes shear deformation and normal strain effects by assuming through‐the‐thickness linear in‐plane displacements and parabolic transverse displacement. A mixed variational approach is used to introduce stresses satisfying the loading conditions on the plate surfaces without employing any shear correction factor used in other theories of the same order. A thermal bending problem is solved for metal/ceramic rectangle plates. Some numerical results are compared with numerical results of 3D exact solutions. The importance of the normal strain effect for such problems is studied.

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.

Thermo-elastic analysis of multilayered plates and shells based on 1D and 3D heat conduction problems

Abstract The present work shows a generic 3D exact shell solution for the thermo-mechanical analysis of a heterogeneous group of one- and multi-layered isotropic, composite and sandwich structures. Plates, cylinders, cylindrical and spherical shells can be investigated using orthogonal mixed curvilinear coordinates. The 3D equilibrium equations for spherical shells automatically degenerate in those for simpler geometries. The elastic part of the proposed 3D model is based on a consolidated layer-wise exact solution which uses the exponential matrix method to solve the equilibrium differential equations through the thickness direction. The closed-form solution is obtained assuming simply-supported boundary conditions and harmonic forms for displacement and temperature fields. The temperature amplitudes are imposed at the top and bottom external surfaces in steady-state conditions. Therefore, the temperature profile can be evaluated through the thickness direction in three different ways: – calculation of the temperature profile via the steady-state 3D Fourier heat conduction equation; – evaluation of the temperature profile using the steady-state simplified 1D version of the Fourier heat conduction equation; – a priori assumed linear temperature profile through the entire thickness direction ranging from the bottom temperature value to the top temperature value. Once the temperature profile is defined at each thickness coordinate, it is considered as a known term in the 3D differential equilibrium equations. The obtained system consists in a set of non-homogeneous second order differential equilibrium equations which can be solved introducing appropriate mathematical layers. After a reduction to a first order differential equation system, the exponential matrix method is used to calculate both the general and the particular solutions. The effects of the temperature field on the static response of plates and shells are evaluated in terms of displacements and stresses. The proposed solution will be validated using reference results available in the literature. Then, new analyses will be presented for different thickness ratios, geometries, lamination schemes, materials and temperature values at the external surfaces. Results will demonstrate the importance in the 3D shell model of both the correct definition of the elastic part and the appropriate evaluation of the temperature profile through the thickness of the structure. .

Modeling of three dimensional thermocapillary flows with evaporation at the interface based on the solutions of a special type of the convection equations

• New 3D exact solution of the problem of evaporative convection is constructed. • Numerical algorithm for finding basic characteristics is proposed. • Basic mechanisms defining the structure of flows are described. • Influence of thermal load and gravity is studied. • Applicability of the different types of boundary conditions is discussed. Theoretical and numerical study of the convection processes, which are accompanied by evaporation/condensation, in the framework of new non-standard problem is largely motivated by new physical experiments. One of the principal questions is to understand the character and to evaluate the degree of influence of particular factors or their combined action on the structure of the joint flows of liquid and gas-vapor mixture. The flow topology is determined by four main mechanisms: natural and thermocapillary convection, tangential stresses and mass transfer due to evaporation at the interface. The mathematical modeling of the fluid flows in an infinite channel with a rectangular cross section is carried out on the basis of the solution of a special type of the convection equations. The effects of thermodiffusion and diffusive thermal conductivity in the gas phase and evaporation at the thermocapillary interface are taken into consideration. Numerical investigations are performed for the liquid – gas (ethanol – nitrogen) system under normal and low gravity. The fluid flows are characterized as translational and progressively rotational motions and can be realized in various forms.

A 3D layer-wise model for the correct imposition of transverse shear/normal load conditions in FGM shells

A general elasticity 3D layer-wise shell model is proposed for the static investigation of plates and shells including functionally graded material (FGM) layers. A closed-form solution is used considering simply-supported sides and harmonic forms for displacements and loads. The partial differential equations obtained from the 3D equilibrium relations developed in curvilinear orthogonal coordinates are solved using the exponential matrix methodology. These equations have constant coefficients because an opportune number of mathematical layers has been introduced in order to calculate the parametric coefficients including the radii of curvature for the shells and the elastic coefficients that are variable through the thickness direction in the case of functionally graded materials. The main aim of the present work is to fill the gap found in the literature where the 3D elasticity theories always give solutions for FGM plates or shells in the only case of a transverse normal load positioned at the top or at the bottom surfaces. The present work proposes an exhaustive static analysis where the load boundary conditions have been appropriately rewritten in order to allow the use of several transverse normal and transverse shear loads separately or simultaneously positioned at top and/or bottom surfaces. One-layered and sandwich FGM plates, cylinders, cylindrical shells and spherical shells are analyzed changing the material laws and properties, the applied loads and the thickness ratios. The importance of the zigzag features, the interlaminar continuity in terms of compatibility and equilibrium requirements, the boundary load requirements, the considerations about the symmetry, the thickness ratio effect and the three-dimensional behavior have been opportunely discussed. Advantages connected with the use of FGM layers have also been analyzed. These new 3D exact results will allow the validation of recent advanced 2D shell models in the literature for the static investigation of FGM structures subjected to different load conditions. The proposed 3D model is general for several geometries (plates and shells) and materials (classical ones, composites and FGMs) and it allows a unique 3D exact solution for a large variety of structures.

Dynamic equations for an orthotropic cylindrical shell

A hierarchy of dynamic shell equations is derived for an orthotropic cylindrical shell. The displacement components are expanded into power series in the thickness coordinate and the three-dimensional elastodynamic equations then yield a set of recursion relations among the expansion functions that can be used to eliminate all but the six lowest-order functions. Applying the boundary conditions on the surfaces of the shell and eliminating all but the six lowest-order expansion functions give the shell equations as a power series in the shell thickness. In principle, these six differential equations can be truncated to any order. Numerical examples showing eigenfrequencies for a ring and for a simply supported shell show the convergence of the method to the 3D solution, and a comparison with previous investigations is also made. Finally, the exact 3D solution is given for a simply supported transversely isotropic shell of arbitrary thickness.

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.

Singular optics of spin waves in a two-sublattice antiferromagnet with uniaxial magnetic anisotropy

Based on exact 3D solutions of the Landau-Lifshitz equations in a two-sublattice antiferromagnet with uniaxial magnetic anisotropy, the existence of nonlinear spin waves with singular points on the wavefront is predicted. These waves are spin-wave analogs of optical singularities.

Strong sampling surfaces formulation for laminated composite plates

The paper focuses on the application of the sampling surfaces (SaS) method to exact solutions of elasticity for laminated plates. The SaS method is based on choosing the arbitrary number of SaS parallel to the middle surface in order to introduce the displacements of these surfaces as basic plate unknowns. Such choice of unknowns with the use of the Lagrange polynomials in assumed approximations of displacements through the layer thicknesses leads to a compact laminated plate formulation. The feature of the proposed approach is that all SaS are located inside the layers at Chebyshev polynomial nodes. The use of outer surfaces and interfaces is avoided that makes possible to minimize uniformly the error due to the Lagrange interpolation. Therefore, the strong SaS formulation based on direct integration of the equilibrium equations of elasticity can be applied efficiently to the obtaining of 3D exact solutions for laminated plates.

A robust, four-node, quadrilateral element for stress analysis of functionally graded plates through higher-order theories

ABSTRACTA hybrid-mixed, four-node, quadrilateral element for the three-dimensional (3D) stress analysis of functionally graded (FG) plates using the method of sampling surfaces (SaS) is developed. The SaS formulation is based on choosing an inside the plate body N, not equally spaced SaS parallel to the middle surface, in order to introduce the displacements of these surfaces as basic plate variables. Such a choice of unknowns, with the consequent use of Lagrange polynomials of the degree N − 1 in the assumed distributions of displacements, strains, and mechanical properties through the thickness leads to a robust FG plate formulation. All SaS are located at Chebyshev polynomial nodes that permit one to minimize uniformly the error due to the Lagrange interpolation. To avoid shear locking and spurious zero-energy modes, the assumed natural strain method is employed. The proposed four-node quadrilateral element passes 3D patch tests for FG plates and exhibits a superior performance in the case of coarse distorted meshes. It can be useful for the 3D stress analysis of thin and thick metal/ceramic plates because the SaS formulation gives an opportunity to obtain the solutions with a prescribed accuracy, which asymptotically approach the 3D exact solutions of elasticity as the number of SaS tends to infinity.

Electric Current Filamentation Induced by 3D Plasma Flows in the Solar Corona

Abstract Many magnetic structures in the solar atmosphere evolve rather slowly, so they can be assumed as (quasi-)static or (quasi-)stationary and represented via magnetohydrostatic (MHS) or stationary magnetohydrodynamic (MHD) equilibria, respectively. While exact 3D solutions would be desired, they are extremely difficult to find in stationary MHD. We construct solutions with magnetic and flow vector fields that have three components depending on all three coordinates. We show that the noncanonical transformation method produces quasi-3D solutions of stationary MHD by mapping 2D or 2.5D MHS equilibria to corresponding stationary MHD states, that is, states that display the same field-line structure as the original MHS equilibria. These stationary MHD states exist on magnetic flux surfaces of the original 2D MHS states. Although the flux surfaces and therefore also the equilibria have a 2D character, these stationary MHD states depend on all three coordinates and display highly complex currents. The existence of geometrically complex 3D currents within symmetric field-line structures provides the basis for efficient dissipation of the magnetic energy in the solar corona by ohmic heating. We also discuss the possibility of maintaining an important subset of nonlinear MHS states, namely force-free fields, by stationary flows. We find that force-free fields with nonlinear flows only arise under severe restrictions of the field-line geometry and of the magnetic flux density distribution.

Three-dimensional free vibrations analysis of functionally graded rectangular plates by the meshless local Petrov–Galerkin (MLPG) method

In this paper, the meshless local Petrov–Galerkin (MLPG) method is used for three-dimensional (3D) elastodynamic analysis of functionally graded (FG) isotropic plates. The 3D linear elasticity theory and infinitesimal strains are the basic assumptions in this analysis. Unlike 2-D theories, in 3D theory, no restricting assumptions are made about the kinematics of deformation in plates. Therefore, a higher accuracy can be achieved by using 3D theory. Heaviside step functions are used as test functions on the local sub-domain of each node and the field variables are interpolated using the 3D moving least squares (MLS) approximation. To impose the essential boundary conditions, the penalty method is adopted and Young's modulus is assumed to be graded through the thickness of plates by the Mori–Tanaka estimation and Poisson's ratio is assumed to be constant. The natural frequencies of thick rectangular plates are obtained for different boundary conditions. Also, the effects of different parameters such as functionally graded power law index, thickness-to-length ratio, and the aspect ratio on the natural frequencies of plates are also studied in details. In order to validate this method, the results are compared with the available exact 3D solutions. The results show the accuracy and effectiveness of the present method.

Mechanical behaviour of FGM sandwich plates using a quasi-3D higher order shear and normal deformation theory

This paper presents an original hyperbolic (first present model) and parabolic (second present model) shear and normal deformation theory for the bending analysis to account for the effect of thickness stretching in functionally graded sandwich plates. Indeed, the number of unknown functions involved in these presents theories is only five, as opposed to six or even greater numbers in the case of other shear and normal deformation theories. The present theory accounts for both shear deformation and thickness stretching effects by a hyperbolic variation of ail displacements across the thickness and satisfies the stress-free boundary conditions on the upper and lower surfaces of the plate without requiring any shear correction factor. It is evident from the present analyses; the thickness stretching effect is more pronounced for thick plates and it needs to be taken into consideration in more physically realistic simulations. The numerical results are compared with 3D exact solution, quasi-3-dimensional solutions and with other higher-order shear deformation theories, and the superiority of the present theory can be noticed.

Assessment of the sampling surfaces formulation for thermoelectroelastic analysis of layered and functionally graded piezoelectric shells

ABSTRACTThe article focuses on the implementation of the sampling surfaces (SaS) concept for the three-dimensional (3D) coupled steady-state thermoelectroelastic analysis of layered and functionally graded (FG) piezoelectric shells subjected to thermal loading. The SaS formulation is based on choosing inside the nth layer In, not equally spaced SaS parallel to the middle surface, in order to introduce the temperatures, electric potentials, and displacements of these surfaces as basic shell variables. Such choice of unknowns with the consequent use of Lagrange polynomials of degree In − 1 in the assumed distributions of the temperature, electric potential, displacements, and mechanical properties through the thickness of the layer leads to the robust FG piezoelectric shell formulation. The SaS are located inside each layer at Chebyshev polynomial nodes, which permits one to minimize uniformly the error due to the Lagrange interpolation. As a result, the SaS formulation can be applied efficiently to deriving the analytical solutions for FG piezoelectric shells, which asymptotically approach the 3D exact solutions of thermoelectroelasticity as the number of SaS tends to infinity.

An assessment of a non-polynomial based higher order shear and normal deformation theory for vibration response of gradient plates with initial geometric imperfections

In the present study, a new non-polynomial based higher order shear and normal deformation theory is proposed and implemented for the vibration response of geometrically imperfect functionally graded material (FGM) plates. The present theory accounts for nonlinear variation in the in-plane and transverse displacements, respectively in the thickness coordinate. This theory contains the trigonometric shear strain shape functions and having only four unknowns in the displacement field. It is variationally consistent and accommodates thickness stretching effects without employing shear correction factor. Two micromechanics models (Mori-Tanaka and Voigt) have been employed to determine the effective material properties of the plate, and are graded continuously through the thickness direction according to a simple power law and exponential law. The accuracy and efficiency of the proposed theory have been conferred by comparing the results with an existing 3D exact solution and various higher order theories. Frequency parameters with various side-to-thickness ratios, boundary conditions, imperfection sizes, volume fraction indexes and exponential indexes have been computed for perfect and imperfect FGM plate. It is found that the proposed theory is not only accurate but also simple in predicting the free vibration responses of functionally graded ceramic-metal plates.

2D and 3D shell models for the free vibration investigation of functionally graded cylindrical and spherical panels

The paper shows the free vibration investigation of simply supported functionally graded material (FGM) shells. Spherical and cylindrical shell geometries are investigated for two different material configurations which are one-layered FGM structures and sandwich structures embedding an internal FGM core. A three-dimensional (3D) exact shell model and different two-dimensional (2D) computational models are compared in terms of frequencies and vibration modes. The proposed numerical solutions are typical 2D finite elements (FEs), and classical and advanced generalized 2D differential quadrature (GDQ) solutions. High and low frequency orders are investigated for thin and thick simply supported shells. Vibration modes are fundamental to compare the 3D exact shell model and 2D numerical solutions. The 2D finite element results based on the classical Reissner–Mindlin theory are calculated using a typical commercial FE software. Classical and advanced GDQ 2D models use the generalized unified formulation. The 3D exact shell model uses the differential equations of equilibrium written in general orthogonal curvilinear coordinates and developed in layer-wise (LW) form. The differences between 2D numerical results and 3D exact results depend on the thickness ratio and geometry of the structure, the considered mode and the frequency order, the lamination sequence and materials.

Bending analysis of composite laminated and sandwich structures using sublaminate variable-kinematic Ritz models

This paper presents a novel numerical tool for the bending analysis of thin and thick composite plates, including monolithic and sandwich structures. The formulation is developed within a displacement-based approach, where the Principle of Virtual Displacements (PVD) and the method of Ritz are adopted to derive the governing equations. The approach relies upon the Sublaminate Generalized Unified Formulation (S-GUF) as underlying kinematic theory describing the behavior across the plate thickness. Main idea of the S-GUF is to group the plies into a number of smaller units called sublaminates, each of them characterized by an independent, variable-kinematic theory. Continuity conditions between the sublaminates are enforced in strong form during the assembly procedure of the governing equations. The S-GUF appears particularly useful when theories of different order are needed to approximate the displacement field of different portions of the structure, such as in the case of sandwich panels. A number of test cases from the literature is discussed, and results are validated against exact 3D solutions. The results demonstrate the ability of the approach to obtain accurate results, both in terms of deformed shapes, and intra- and inter-laminar stress distributions. A set of novel results is also presented for future benchmarking purposes.

A high-performance parametrized mixed finite element model for bending and vibration analyses of thick plates

Based on the parametrized mixed variational principle, a high-performance finite element model has been developed for bending and vibration analysis of thick plates. The proposed mixed variational formulation combines the concept of Reissner’s mixed variational theorem and Rong’s generalized mixed variational theorem. In addition to unknown parameters of the displacement fields, the transverse normal component of the stress tensor is also assumed as the independent field variable in the present variational formulation. The latter allows the fulfillment of the boundary conditions of the transverse normal stress at the top and bottom surfaces of the plate in a natural way. The variational constraint of the transverse normal strain–displacement relations is relaxed via the parametrized Lagrange multipliers method. The functional of the proposed variational formulation contains an arbitrary free parameter, called the splitting factor. Simple formulations have been proposed for selecting the splitting factor which leads to the results of higher precision. The numerical results obtained from the proposed mixed formulation have been compared with the results of three-dimensional (3D) theory of elasticity and other plate theories available in the literature. The comparison study shows that the proposed parametrized mixed plate theory is capable of achieving the accuracy of nearly exact 3D solutions.

A statically compatible layerwise stress model for the analysis of multilayered plates

This paper proposes a new layerwise model for multilayered plates. The model, called SCLS1 for Statically Compatible Layerwise Stresses with first-order membrane stress approximations per layer in thickness direction, complies exactly with the 3D equilibrium equations and with the free-edge boundary conditions. As in the LS1 model initially proposed by Naciri et al. (1998), the laminated plate is considered as a superposition of Reissner plates coupled by interfacial stresses which are considered as generalized stresses. However, the divergences of the interlaminar transverse shears are introduced as additional generalized stresses in the SCLS1 model. Also, a refined version of the new model is obtained by introducing several mathematical layers per physical layer. Contrary to the LS1 model which is derived by means of the Hellinger–Reissner principle, the new model is derived by means of the minimum of the complementary energy principle. This ensures the convergence of the refined SCLS1 solution to the exact 3D solution as the number of mathematical layers per physical layer increases. The new model has been implemented in a new version of the in-house finite element code MPFEAP . Several comparisons are made between LS1, SCLS1 and full 3D FE models in order to assess the performances of the new model which shows itself very effective.