Exact Three-dimensional Solution Research Articles

Layerwise electro-elastic user-elements in Abaqus for static and free vibration analysis of piezoelectric composite plates

In light of the limited number of available literature on Abaqus user-elements implementation, this work presents two layerwise electro-elastic user-elements for static and free vibration analysis of piezoelectric composite plates. Three discrete layers are considered, each one described by a first-order shear deformation displacement field along with a linear or quadratic z-expansion of the electric potential. The models predictive capabilities are assessed by a comparison with three-dimensional exact solutions for simply supported plates. Numerical results demonstrate an advantage of the electric potential quadratic z-expansion for static analysis, with negligible improvements on the evaluation of the first 12 natural frequencies.

Zigzag Theory for Piezoelectric-Layer-Integrated Functionally Graded Material Plates

An electromechanically coupled zigzag theory is presented for static and free vibration responses in functionally graded material (FGM) plates bonded with piezoelectric layers. It has the capability to consider continuity as well as discontinuity in material properties at all interfaces. The displacement field approximation, which is initially layer based, has been made layer independent by using continuity conditions on displacements and transverse shear stresses as well as zero traction conditions on top and bottom surfaces. Material properties in FGM layers vary continuously in the thickness direction. Piezoelectric layers have constant property across their thickness. Extended Hamilton’s principle for piezoelectric continuum has been used in deriving governing equations of motion and boundary conditions. An analytical solution is obtained for elastic as well as piezoelectric layer bonded FGM plates. Static and free vibration responses have been obtained in a number of FGM plates, with each plate considered to contain a number of FGM layers with discontinuity or continuity in material properties at the interfaces. Wherever available, present responses have been compared with three-dimensional exact solutions and other shear deformation theories. The effects of the aspect ratio, length-to-width ratio, and different schemes for effective material property calculation have been illustrated.

Thermoelasticity of multilayered plates with imperfect interfaces

Three-dimensional exact solutions for temperature and thermoelastic stresses in multilayered anisotropic plates are derived for advanced boundary-value problems with general boundary conditions. The extended Stroh formalism is formulated to include the thermal coupling with the Eringen nonlocal elasticity theory that captures small scale effects. The simply supported structures are subjected to time-harmonic distributions of temperature, combined with tractions, which are represented by means of Fourier series expansions. In particular, the prescribed loads on both the bottom and top surfaces include the uniformly and heterogeneously distributed normal stresses, while imperfect thermal and mechanical contacts between constituents are incorporated at the internal interfaces. Recursive field relations for multilayered plates with imperfect interfaces are consistently articulated by virtue of the traditional propagation matrix method, which is further completed by the dual variable and position technique to overcome numerical instability issues. Three application examples are proposed to throw light on various effects of the externally applied loads and internal imperfections on the thermoelastic fields in multilayered structures. The residual stress fields in graphite fiber-reinforced epoxy matrix composites are shown to be drastically different from those predicted by the classical elasticity theory when nonlocal effects are significant. The stacking sequence and the number of copper and molybdenum in laminated anisotropic plates are of great importance in tailoring interfacial properties, especially if the thermally conducting boundaries are taken into account. The forced vibration analysis of thermal barrier coatings on nickel based superalloys is investigated, including interfacial bonding effects between adjoining layers. Depending on the input frequency amplitude, severe oscillating displacements and stresses take place in the single crystal superalloys that can endanger the safety-related stability and integrity of aircraft engines. Overall, the present formalism should be utilized in the optimal design of sophisticated multilayered structures with desired steady-state and time-harmonic thermoelastic responses.

On the mechanics of composite sandwich plates with three-dimensional stress recovery

By using the variational-asymptotic method, a universal asymptotic model for composite sandwich plates is established to achieve a great compromise between efficiency and accuracy through the synthesis of two competing theories: equivalent single-layer theories and layer-wise theories. When each layer of a sandwich structure can be individually considered as an elastic plate, and all material constitutive constants of such plates can be assumed to be of the same order, an equivalent plate model can be constructed from the equivalent single-layer perspective. It has an asymptotically correct energy functional, capable of capturing the transverse deformations, but still limited to the zeroth-order approximation. Then, by taking into account mismatched constituent material properties, a universal model for predicting mechanical behavior of composite sandwich plates is then systematically derived from the layer-wise perspective. It has another asymptotically correct energy functional with respect to the core-layer’s plate variables and implements into a single unified representation for subsequent application to sandwich plate problems with any face-to-core-stiffness and length-to-thickness ratios. In particular, to resolve theoretical shortcomings found in published works, complementary theoretical procedures are incorporated and used into the present approach by considering the interlaminar transverse stress continuity conditions as essential conditions in conventional layer-wise theories and by providing three-dimensional recovery relations as necessary ingredients for theories based on various dimensional reduction processes. Finally, as a preliminary validation, several examples available in the literature are presented and investigated. Together with critical comparisons of the three-dimensional exact solutions, the close agreement demonstrates the capability and accuracy of this present approach.

Exact axisymmetric adhesive contact analysis for a pre-deformed soft electroactive half-space

In this paper, three-dimensional exact solutions of adhesive contact between a pre-deformed compressible soft electroactive half-space and an axisymmetric rigid indenter are presented. The change of surface adhesion energy during the contact is examined by using the modified JKR model, which accounts for the real contact area instead of the projected area. With the help of new results in the potential theory method, all physical (field) variables are derived in terms of elementary functions for three common types of axisymmetric indenters (flat-ended, conical, and spherical). The analytical contact relations for different indenter geometries and material properties are provided and summarized in Tables 2 and 3 to serve a solid base for revealing the underlying electromechanical mechanism of soft electroactive materials. For numerical illustration, neo-Hookean isotropic electroactive material is considered. The simulation results clearly demonstrate that both the mechanical and electric biasing fields significantly affect the indentation measurement of the electroactive material. Moreover, at either micro- or nano-scale, adhesion plays a prominent role in the indentation responses. It is of interest that, even without adhesion, the normal stress somewhere in the contact region may become tensile under a prescribed pre-stretch when the biasing electric displacement exceeds a certain value. This abnormal phenomenon actually corresponds to surface instability of the half-space under the biasing field. In the case of adhesive contact, other than the surface instability, the value of surface adhesion energy between the probe and the sample will impose a constraint on the validity of indentation analysis within the linear elastic regime.

Generalized Grad–Shafranov equation for non-axisymmetric MHD equilibria

The structure of static MHD equilibria that admit continuous families of Euclidean symmetries is well understood. Such field configurations are governed by the classical Grad–Shafranov equation, which is a single elliptic partial differential equation in two space dimensions. By revealing a hidden symmetry, we show that in fact all smooth solutions of the equilibrium equations with non-vanishing pressure gradients away from the magnetic axis satisfy a generalization of the Grad–Shafranov equation. In contrast to solutions of the classical Grad–Shafranov equation, solutions of the generalized equation are not automatically equilibria, but instead only satisfy force balance averaged over the one-parameter hidden symmetry. We then explain how the generalized Grad–Shafranov equation can be used to reformulate the problem of finding exact three-dimensional smooth solutions of the equilibrium equations as finding an optimal volume-preserving symmetry.

Evaluation of exact electro-elastic static and free vibration solutions of multilayered plates for benchmarking: Piezoelectric composite laminates and soft core sandwich plates

This work provides a study on three-dimensional exact electro-elastic static and free vibration solutions of multilayered plates, focused on a comprehensive evaluation of well-known benchmarks for piezoelectric and/or composite laminates as well as soft core sandwich plates, adding much to thus far available in the literature. The exact solution method for simply supported multilayered plates is fully described in line with earlier leading works, compiled in a single study in a consistent form throughout. The layers involved can be either piezoelectric layers poled through-thickness (i.e. extension mode) or purely elastic layers, including composite layers. For each layer, the form of the through-thickness exact solution depends strongly on its material properties, thus each case arising from an orthotropic, transversely isotropic or simply isotropic layer is considered specifically. Within the free vibration solution, in agreement with an in-plane mode (nx, ny), the so-called special modes for either nx=0 or ny=0, though often overlooked, are purposely addressed, along with thickness modes for each pair (nx, ny). The benchmarks cover purely elastic solutions, as in composite laminates and soft core sandwich plates, as well as electro-elastic solutions, namely piezoelectric composite laminates, involving not only but especially thick plates. For each multilayered plate, the static solution considers either an applied load or an applied electric potential, providing a detailed through-thickness evaluation of displacements and stresses, and when present, the electric potential and electric displacements. The respective free vibration solution reveals the first twenty natural frequencies and associated modes, including all together special modes and thickness modes.

Topological characteristics of building blocks in the domain wall of an antiferromagnet with the Dzyaloshinskii–Moriya interaction

Topological charges are calculated for a number of exact three-dimensional analytical solutions to the Landau–Lifshitz equation, which describe the distributions of the vector fields for the vectors of antiferromagnetism and antiferromagnet magnetization. It is shown that in the case of samples with dimensions that are comparable to the characteristic scales of topological objects of the antiferromagnetism and magnetization vector fields, there are modified characteristics that depend not only on the topological properties of these objects, but also on the geometry of the sample. These modified characteristics in samples with finite dimensions may assume non-integer values.

Three-Dimensional Vibration Analysis of Laminated Composite Rectangular Plate with Cutouts.

The main purpose of this paper is to establish an analysis model of laminated composite rectangular plate with/without cutouts on the basis of three-dimensional elasticity theory and provide the exact three-dimensional solution. In the present work, the effect of the cutout is considered by subtracting the energies of cutouts from the total energy of the entire plate. The standard three-dimensional trigonometric cosine Fourier series and auxiliary Fourier series are chosen as admissible functions, and the Hamilton’s principle and Rayleigh-Ritz procedure are used to obtain the exact solution. In order to verify the effectiveness and reliability of the proposed method, some numerical results are obtained, and the results are compared with the ones available in the literature or finite element analysis. Finally, the effects of some key parameters which will affect the vibration characteristics are analyzed, and the non-dimensional frequency parameters are obtained, which can serve as benchmark data for the future research.

Three-dimensional exact solution of layered two-dimensional quasicrystal simply supported nanoplates with size-dependent effects

A size-dependent plate model is developed to investigate the elastic responses of the multilayered two-dimensional quasicrystal nanoplates based on the nonlocal strain gradient theory for the first time. A nonlocal stress field parameter and a length scale parameter are taken into account in the new model to capture both stiffness-softening and stiffness-hardening size effects. The exact solution for a single-layer two-dimensional quasicrystal simply supported nanoplate is derived by utilizing the pseudo-Stroh formalism in conjunction with the nonlocal strain gradient theory. Afterward, a dual variable and position method is used to deal with the multilayered case. Numerical examples are presented to study the dependence of size-dependent effect on nanoplate length and the influences of scale parameters on the quasicrystal nanoplate subjected to a z-direction mechanical load on its top surface. The proposed model should be useful to verify various nanoplate theories and other numerical methods.

Performance of combined spectral collocation method and artificial compressibility method for 3D incompressible fluid flow and heat transfer

Purpose The purpose of this paper is to develop a combined method for three-dimensional incompressible flow and heat transfer by the spectral collocation method (SCM) and the artificial compressibility method (ACM), and further to study the performance of the combined method SCM-ACM for three-dimensional incompressible flow and heat transfer. Design/methodology/approach The partial differentials in space are discretized by the SCM with Chebyshev polynomial and Chebyshev–Gauss–Lobbatto collocation points. The unsteady artificial compressibility equations are solved to obtain the steady results by the ACM. Three-dimensional exact solutions with trigonometric function form and exponential function form are constructed to test the accuracy of the combined method. Findings The SCM-ACM is developed successfully for three-dimensional incompressible flow and heat transfer with high accuracy that the minimum value of variance can reach. The accuracy increases exponentially along with time marching steps. The accuracy is also improved exponentially with the increasing of nodes before stable accuracy is achieved, while it keeps stably with the increasing of the time step. The central processing unit time increases exponentially with the increasing of nodes and decreasing of the time step. Research limitations/implications It is difficult for the implementation of the implicit scheme by the developed SCM-ACM. The SCM-ACM can be used for solving unsteady impressible fluid flow and heat transfer. Practical implications The SCM-ACM is applied for two classic cases of lid-driven cavity flow and natural convection in cubic cavities. The present results show good agreement with the published results with much fewer nodes. Originality/value The combined method SCM-ACM is developed, firstly, for solving three-dimensional incompressible fluid flow and heat transfer by the SCM and ACM. The performance of SCM-ACM is investigated. This combined method provides a new choice for solving three-dimensional fluid flow and heat transfer with high accuracy.

Layerwise mixed elements with node-dependent kinematics for global–local stress analysis of multilayered plates using high-order Legendre expansions

Carrera Unified Formulation (CUF) is taken a step further to render node-dependent kinematics (NDK) capabilities to new layerwise finite elements based on Reissner’s Mixed Variational Theorem (RMVT), especially suited for global–local stress analysis of multilayered plates, ensuring high numerical accuracy and computational efficiency, all together. In the framework of CUF, as introduced originally for multilayered structures, any degree of kinematic refinement can be considered in agreement with Equivalent Single-Layer (ESL) or Layer-Wise (LW) theories to develop advanced finite element models, whether based on the Principle of Virtual Displacements (PVD) or RMVT. The degree of kinematic refinement, which usually holds equally for the entire element, can be taken a step further, by being assigned locally to each of its nodes, making full use of CUF to render NDK capabilities to the elements. Besides, even though the elements can adopt any type of nodal shape functions, high p-order hierarchical Legendre expansions (HLE) can also be combined with NDK, achieving excellent convergence rates. These capabilities combined, explored first under the PVD, are for once integrated in the proposed elements under RMVT to further benefit accurate stress analysis. These elements can be applied throughout the entire mesh, adapting to local, transitional and global regions straightforwardly, providing high numerical accuracy, locally, with minimal computational efforts, globally. The numerical results focus on stress analysis of multilayered composite plates, including local effects, to demonstrate the predictive capabilities of the proposed RMVT-based LW elements with NDK and HLE combined, considering well-known benchmark three-dimensional exact solutions for assessment.

Solutions to Equations of a Three-Dimensional Electron Beam in Elementary Functions

Abstract—The exact three-dimensional solutions describing nonstationary nonrelativistic electron flows in a uniform magnetic field are given, with preference given to the study of oscillatory regimes.

Assessment of the consistent second-order plate theory for isotropic plates from the perspective of the three-dimensional theory of elasticity

In this paper, the consistent second-order plate theory for isotropic plates is validated against the three-dimensional elasticity theory using a well-known benchmark problem of a simply-supported rectangular plate subjected to symmetric transverse sinusoidal loading. The choice of the benchmark problem is based on the fact that it allows for an exact three-dimensional elasticity solution to be derived in closed-form. In the paper, two equivalent closed-form solutions are employed for validation purposes, one of which is specifically derived for this study. Once the equivalence of the two closed-form analytical solutions is established, they are expanded into a power-law series with respect to the non-dimensionalised plate thickness. This enables a direct term-by-term comparison with the consistent second-order plate theory solution and provides a valuable mechanism to validate the consistent plate theory in purely analytical form. The term-by term comparison reveals that the first terms of the above power-law series coincide exactly with the expressions of the consistent second-order plate theory. In addition to the analytical validation, a parametric study is carried out with a view to establish the range of applicability of the consistent second-order plate theory in terms of the thickness-to-length ratio. It is demonstrated that the consistent plate theory can predict displacements and stresses in thick plates with very high degree of accuracy, such that even for very thick plates with thickness-to-length ratio of 1/2, the deviation from the three-dimensional elasticity solution is less than 1%.

Hygro-thermo-mechanical modelling of multilayered plates: Hybrid composite laminates, fibre metal laminates and sandwich plates

A new layerwise mixed model is developed for the coupled hygro-thermo-mechanical static analysis of multilayered plates, addressing hybrid composite laminates, fibre metal laminates and sandwich plates under hygro-thermo-mechanical loadings. This model is based on a mixed least-squares formulation, i.e. a strong form model, 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, as well as moisture, transverse moisture flux and in-plane components of the moisture gradient, all taken as independent variables. This mixed formulation ensures that the interlaminar C0 continuity requirements, where the material properties change, are fully fulfilled a priori by all independent variables. The effects of hygrothermal environments in the behaviour of multilayered plates are here demonstrated considering hybrid composite laminates, fibre metal laminates and sandwich plates with different side-to-thickness ratios under a series of hygro-thermo-mechanical loadings. Three-dimensional (3D) exact solutions are used to assess the results by the present model and some further by an alternative weak form model in the framework of Carrera Unified Formulation (CUF). The present model is shown to predict a highly accurate quasi-3D hygro-thermo-mechanical description of the through-thickness distributions of displacements and stresses, temperature and heat flux, moisture and moisture flux, all together.

The electro-mechanics of a coating/substrate system under charged spherical contact

A general theory of the piezo-electric contact mechanics for a transversely isotropic piezo-electric elasticity coated system (TIPECS) under charged spherical contact is presented. Three-dimensional exact solutions for the stresses, displacements, electric displacements and electric potential are derived. The contact radius versus the total load relation is obtained to guide the indentation experiments. The relations of coating thickness versus voltage and coating thickness versus voltage are studied. The optimal thickness can be determined with an optimization procedure. The interfacial delamination area of the TIPECS can be predicted with the interfacial failure criterion. The theoretical results are helpful for the prediction of some transient electrical effects, which can be combined with experiments to estimate some elastic constants, dielectric constants and piezo-electric constants in the indentation process.

Two-dimensional linear models of multilayered anisotropic plates

A two-dimensional model describing the multilayered anisotropic plate deformations is proposed. The plate is assumed to consist of some orthotropic layers with arbitrary orientation of axes relative to the plate frame. The studied multilayered plate is replaced by the equivalent plate composed of a monoclinic material with piecewise elastic modules. An asymptotic solution is constructed for long-wave deformations. This problem was solved earlier in the first approximation; however, the obtained solution is not applicable for the case in which the stiffness of layers differs essentially from each other. The second asymptotic approximation is constructed in the present paper. It takes into account the effects of transversal shear and the normal fibers extension. Some special cases resulting in simple equations are studied in detail. The asymptotic solution error is estimated by comparison with the exact three-dimensional solutions for some test examples.

A unified modeling method for free vibration of open and closed functionally graded cylindrical shell and solid structures

In this research, a novel three-dimensional (3D) exact solution for the dynamic analyses of functionally graded (FG) cylindrical structures with arbitrary boundary conditions is put forward. In accordance with the power law distribution of constituents volume, the material properties vary consecutively along the length of the FG cylindrical structures. The theoretical model is derived based on the 3D elasticity theory. The displacements of the FG cylindrical structures are expanded as a 3D cosine series accompanied with auxiliary functions. In comparison with traditional Fourier series, these complementary functions can be helpful to remove the discontinuities of the displacements and the correlated derivatives at the edges. The advantage of the present method compared with others is that it is able to be extended to arbitrary boundary conditions without peculiar procedures. The present results are compared with those from other literature to validate its reliability. Several numerical examples are exhibited for FG cylindrical structures (open cylindrical shell, open cylindrical solid, cylindrical shell and cylindrical solid) with different types of boundary conditions (both classical and elastic boundary conditions). In addition, the effect of gradient index on the FG cylindrical structures is also reported.

Three-dimensional exact solutions of elastic transversely isotropic coated structures under conical contact

Due to the strong reduction of the coating thickness and to the large stress gradient that may occur in coated structures, it is hard to guarantee the calculation accuracy in the design of coated structures by the commercial software. In this case, an accurate and efficient calculation method for researching the coated structure is valuable. It will contribute to the accurate design and quality promotion of coated structures. In this paper, the 3D solutions of elastic transversely isotropic coated structure under conical contact are derived based on the general solutions of the elastic transversely isotropic material. The solutions are derived in two cases including the normal contact load and the tangential contact load. The numerical results present that this method presents good convergence, high efficiency and stability. Based on this method, the optimal thickness of the coating and the interfacial stress can be obtained accurately. Because the interfacial stress can lead to the debonding failure of coated structures, a method for predicting interface security state with evaluation criterion for interfacial failure is proposed. The corresponding analysis will contribute to the strengthening design of the coated structure.

The boundary layer flow induced above the torsional motion of a disk

The boundary layer flow above a disk in torsional motion with azimuthal velocity proportional to rm is investigated for all values of m ≥ 1; here, r is the radial coordinate measured from the center of the disk. The resulting flow is a fully three-dimensional exact solution to the steady, axisymmetric boundary layer equations in the form of similarity solutions. We compute wall shear stresses in the radial and azimuthal directions as a function of the torsional exponent m, as well as the flow induced in the far field. The large m asymptotics of the problem are computed and compared with numerical solutions. The induced radial velocity profiles have a “wall-jet” structure, and it is found that both the radial and azimuthal velocity components become confined close to the surface of the disk as m increases.