Propagator Matrix Method Research Articles

Effect of position-dependent effective mass on electron tunneling of InAs/GaSb type-II superlattice having triangular and parabolic geometries

In a recent paper, we presented transmission probability of a triple quantum well triple barrier structure having triangular and parabolic geometries in presence of finite thick contact barriers using propagation matrix method for InAs/GaSb type-II superlattice. In this letter, effect of different barriers and wells widths and wells geometry are independently studied on transmission coefficient, and also for a specified structure, material composition of barriers is varied to observe the tunnelling effect. We show that the number of peaks in transmission (-ln(Trans(E))) for 30 nm wells width and 10 nm barriers width is more than10nm wells width and 30 nm barriers width.

Bending Analysis of Functionally Graded One-Dimensional Hexagonal Piezoelectric Quasicrystal Multilayered Simply Supported Nanoplates Based on Nonlocal Strain Gradient Theory

In this study, the nonlocal strain gradient theory is adopted to investigate the static bending deformation of a functionally graded (FG) multilayered nanoplate made of one-dimensional hexagonal piezoelectric quasicrystal (PQC) materials subjected to mechanical and electrical surface loadings. The FG materials are assumed to be exponential distribution along the thickness direction. Exact closed-form solutions of an FG PQC nanoplate including nonlocality and strain gradient micro-size dependency are derived by utilizing the pseudo-Stroh formalism. The propagator matrix method is further used to solve the multilayered case by assuming that the layer interfaces are perfectly contacted. Numerical examples for two FG sandwich nanoplates made of piezoelectric crystals and PQC are provided to show the influences of nonlocal parameter, strain gradient parameter, exponential factor, length-to-width ratio, loading form, and stacking sequence on the static deformation of two FG sandwich nanoplates, which play an important role in designing new smart composite structures in engineering.

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.

Acoustic Vibration of Hexagonal Nanoparticles With Damping and Imperfect Interface Effects

Abstract In this paper, acoustic vibration of hexagonal nanoparticles is investigated. In terms of the spherical system of vector functions, the first-order differential equation with constant coefficients for a layered sphere is obtained via variable transformation and mass conservation. The propagation matrix method is then used to obtain the vibration equation in the multilayered system. Further utilizing a new root-searching algorithm, the present solution is first compared to the existing solution for a uniform and isotropic sphere. It is shown that, by increasing the sublayer number, the present solution approaches the exact one. After validating the formulation and program, we investigate the acoustic vibration characteristics in nanoparticles. These include the effects of material anisotropy, damping, and core–shell imperfect interface on the vibration frequency and modal shapes of the displacements and tractions.

Coupled state-space and Gauss integration procedures for static analysis of general functionally graded magnetoelectroelastic laminated plates

In this article, a 3D semi-analytical procedure is elaborated for static behavior of simply supported functionally graded magnetoelectroelastic laminated plates subjected to various types of loads. An arbitrary graded formulation is considered in the thickness direction. The laminated plates have been divided into some layers in which the materials properties vary through the thickness. In each layer, the solution has been established using the state-space approach coupled with the Gauss-Tchybechev integration. The predicted solution has been propagated from the bottom to the top layers of the plate using the propagator matrix method. The presented methodological procedure is general and allow considering various types of graded functionalities in the thickness direction. For the computation process, the piezoelectric material and the piezomagnetic material are used. Two kinds of functionally graded materials have been hindered. First, the polynomial variation of the materials properties through the thickness direction of the plate is considered. The effective properties of functionally graded materials are assumed to follow the law of mixture. Second, the exponential variation of the materials properties through the thickness of the plate was used. The obtained numerical results have been well compared with the available ones obtained by the 3D asymptotic approach, the modified Pagano method, the pseudo-Stroh formalism, and the finite elements method, respectively. The convergence study showed the accuracy, the efficiency, and the reliability of the presented methodological procedure.

Axisymmetric bending analysis of functionally graded one-dimensional hexagonal piezoelectric quasi-crystal circular plate.

Within a framework of the state space method, an axisymmetric solution for functionally graded one-dimensional hexagonal piezoelectric quasi-crystal circular plate is presented in this paper. Applying the finite Hankel transform onto the state space vector, an ordinary differential equation with constant coefficients is obtained for the circular plate provided that the free boundary terms are zero and an exponential function distribution of material properties is assumed. The ordinary differential equation is then used to obtain the stress, displacement and electric components in the physical domain of the elastic simply supported circular plate through the use of the propagator matrix method and the inverse Hankel transform. The numerical studies are carried out to show the validity of the present solution and reveal the influence of material inhomogeneity on the axisymmetric bending of the circular plate with different layers and loadings, which provides guidance for the design and manufacture of functionally graded one-dimensional hexagonal piezoelectric quasi-crystal circular plate.

Joint PP and PS Pre-stack Seismic Inversion for Stratified Models Based on the Propagator Matrix Forward Engine

Pre-stack seismic inversion of the P- and S-wave velocities and bulk density is important in seismic exploration for evaluating lithological units and fluid properties. Generally, this inversion is based on ray-tracing modeling, which introduces errors and requires substantial pre-processing for stratified models due to its oversimplified single-interface assumption. To overcome those problems, we propose a pre-stack inversion method, using wave-equation-based modeling as a forward engine. Most wave-equation-based pre-stack inversions are based on the reflectivity method and adopt nonlinear optimization algorithms, although accurate, but computationally expensive. Hence, we use a fast propagator matrix (PM) method valid for layered media. To improve the stability and accuracy, the PP data inversion is extended to joint PP and PS PM-based inversion (JPMI). A linear inversion scheme is adopted to reduce the computational cost, and the Frechet derivatives are computed accordingly. Moreover, to obtain an optimal model solution, the L-BFGS (Limited-memory Broyden–Fletcher–Goldfarb–Shanno) optimization algorithm and L-curve criterion, an adaptive regularization parameter acquisition method, are implemented. A posterior probability analysis shows that the method has a higher parameter sensitivity than the joint exact Zoeppritz-based inversion and gives better estimations than the single-data inversion. We discuss the effects of dataset weight, internal multi-reflections, time window setting, noise level and initial model by using model tests. Synthetic and real-data examples demonstrate that the algorithm is better than the single PP inversion in terms of stability and accuracy, especially for S-wave velocity and density estimations.

Stress Wave Propagation through Rock Joints Filled with Viscoelastic Medium Considering Different Water Contents

A rock mass often contains joints filled with a viscoelastic medium of which seismic response is significant to geophysical exploration and seismic engineering design. Using the propagator matrix method, an analytical model was established to characterize the seismic response of viscoelastic filled joints. Stress wave propagation through a single joint highly depended on the water content and thickness of the filling as well as the frequency and incident angle of the incident wave. The increase in the water content enhanced the viscosity (depicted by quality factor) of the filled joint, which could promote equivalent joint stiffness and energy dissipation with double effects on stress wave propagation. There existed multiple reflections when the stress wave propagated through a set of filled joints. The dimensionless joint spacing was the main controlling factor in the seismic response of the multiple filled joints. As it increased, the transmission coefficient first increased, then it decreased instead, and at last it basically kept invariant. The effect of multiple reflections was weakened by increasing the water content, which further influenced the variation of the transmission coefficient. The water content of the joint filling should be paid more attention in practical applications.

Dynamic analysis of a multilayered piezoelectric two-dimensional quasicrystal cylindrical shell filled with compressible fluid using the state-space approach

The state-space approach is developed to analyze the dynamic behaviors of a multilayered two-dimensional piezoelectric quasicrystal circular cylinder filled with the compressible fluid. With simple support at both ends, the hollow cylindrical shell has imperfect bonding between the layers. The analytical solution of a homogeneous cylindrical shell has been derived based on the state equations. The general solution for the corresponding multilayered case is also obtained by utilizing the propagator matrix method. The numerical results present the natural frequencies in free vibration with different length-to-radius and radius-to-thickness ratios. The critical load and dynamic behaviors of the model are exactly predicted in the axial buckling problem. For the impulse case, the influences of the density of the filled fluid and coefficients of interfacial imperfections on the dynamic responses are also discussed.

Enhancing magnetoelectric effect in magneto-electro-elastic laminated composites via interface modulus and stress

Magnetoelectric (ME) coupling is of interest for a variety of applications, but it is weak in a single-phase material. Strain-coupled laminates are an efficient way to achieve a high ME coupling. In this paper, we first derive an exact solution for a simply supported multilayered anisotropic magneto-electro-elastic plate with interfacial magneto-electro-elasticity. It is expressed in terms of the extended pseudo Stroh formalism along with the propagator matrix method. The interface modulus and stress are considered by constructing an interface matrix. Based on this solution, sandwich plates made of different layups of piezoelectric and piezomagnetic materials are numerically studied. Our results indicate that for the given layup, it is possible to tune the ME coupling by the interface modulus and stress. Considering interface effect is common in nanoplate, this study could provide an extra channel for turning the ME coupling in the layered magneto-electro-elastic plates with possible device applications.

Interaction between P, S1 and S2 waves in multilayered periodic monoclinic media at low frequencies

SUMMARYApplication of the Floquet theorem and the matrix propagator method reduces the problem of the plane wave propagation in a periodically layered anisotropic media, to analysis of the properties of stationary envelopes of different wave modes propagating up- and downwards. We analyse the interchanging of stop- and pass-bands and their structure at low frequencies for a periodically layered medium with monoclinic symmetry. The analysis shows the effect of interaction between P,S1 and S2 wave multipliers for stop- and pass-band structure and gives insight into the wave propagation in vertically heterogeneous anisotropic media which is important in modelling and interpretation of seismic data.

A propagation matrix method for the solution of the parabolic equation in ocean acoustics.

A propagation matrix method for the solution of the parabolic equation in ocean acoustics is presented, where the sound fields at an arbitrary distance are expressed as a product of the initial fields and a sequential multiplication of the propagation matrices. The method contains two main calculation steps, i.e., producing and multiplying the propagation matrices. The first step allows a two-level parallel algorithm. In the second step an eigenvalue decomposition technique is developed to improve the efficiency in the case of a range-independent waveguide. The efficiency of the parallelized algorithm shall be tested in the future.

Three-dimensional buckling of embedded multilayered magnetoelectroelastic nanoplates/graphene sheets with nonlocal effect

This article presents an exact analysis for the three-dimensional buckling problem of embedded multilayered magnetoelectroelastic and simply supported nanoplates/graphene sheets with nonlocal effect. The interaction between the multilayered nanoplates/graphene sheets and their surrounding medium is simulated by a Pasternak-type foundation. The critical loads for embedded multilayered magnetoelectroelastic nanoplates/graphene sheets under uniaxial and biaxial compression at small scale are then derived by solving the linear eigensystem and making use of the propagator matrix method. A comparison between the present anisotropic three-dimensional model and previous results (an asymptotic nonlocal elasticity theory for single elastic graphene sheet and classical orthotropic plate theories) is made to show the effectiveness and correctness of the present anisotropic three-dimensional model. Numerical examples are then presented for the variation of the dimensionless critical buckling loads for the homogeneous elastic graphene sheet with nonlocal effect, the homogeneous orthotropic thick plate without nonlocal effect, and the sandwich magnetoelectroelastic nanoplates made of piezoelectric and magnetostrictive materials with nonlocal effect. Furthermore, the effects of the thickness of nanoplates, nonlocal parameter, Winkler stiffness, and shear modulus of the elastic medium on the critical load of sandwich magnetoelectroelastic nanoplates/graphene sheets are demonstrated. These results should be very useful as benchmarks for the future development of approximate nanoplate/graphene sheet theories and numerical methods for modeling and simulation of multilayered nanoplates/graphene sheets with nonlocal effect.

Full-wave method for the analysis of the radiation characteristics of a VLF source in the atmosphere

The basis of the full-wave method is the propagation matrix method and a plane wave expansion, which is generally used for the analysis of radiation problems in layered media. The radiation field in the spatial domain is represented as an integral in the wavenumber domain. By integrating the vertical wavenumber, the radiation can be represented as a superposition of plane waves with different directions. If the atmosphere and ionosphere are regarded as layered media, the vertical wavenumber integral contains first-order singularities when the radiation source is located in the ionosphere, which are relatively easy to handle. When the radiation source is located in the air, the vertical wavenumber integral contains second-order singularities, which are more complicated. Therefore, there are few full-wave studies on radiation characteristics in the air and ionosphere when a very-low-frequency (VLF) source is located in the air. In this paper, air is considered as an isotropic medium with a micro-conductivity, the vertical wave number integral that has second-order singularities is solved, and the radiation fields in the air and the ionosphere are calculated. The numerical results show the correctness and effectiveness of the proposed algorithm.

Three-dimensional nonlocal buckling of composite nanoplates with coated one-dimensional quasicrystal in an elastic medium

Based on the nonlocal theory, three-dimensional (3D) buckling of composite nanoplates with coated one-dimensional (1D) quasicrystal (QC) is analyzed. The nanoplate is embedded in an elastic medium and is under uniaxial or biaxial compression. All edges of the QC nanoplate are simply supported and its interaction with the surrounding medium is simulated by the Pasternak-type model. In terms of the extended displacement and traction vectors, the eigensystem is first derived from the basic equations of nonlocal QC materials. Then 3D analytical solutions of the critical buckling load under compression are derived by using the propagator matrix method and the continuity condition on the interfaces of the nanoplate. The influence of the thickness and length-to-width ratio of the nanoplate, Winkler stiffness and shear modulus of the elastic medium, coating thickness and nonlocal parameter on the critical buckling load is analyzed. For a sandwich nanoplate made of QC and soft metallic aluminium, our numerical results indicate that QC coatings could offer an interesting alternative to surface reinforcement of soft metallic materials in industrial applications. The present 3D buckling model could further serve as a benchmark for various thin-nanoplate theories and for numerical methods in multilayered QC nanoplate modeling with nonlocal effect.

Three-dimensional static analysis of multilayered one-dimensional orthorhombic quasicrystal spherical shells with the piezoelectric effect

The static analysis of multilayered one-dimensional orthorhombic piezoelectric quasicrystals spherical shells is presented by using the state vector method. Based on two separated state equations, the solution of a homogenous spherical shell is derived, and the solution for the corresponding multilayered case is obtained via the propagator matrix method. Mechanical behaviors of spherical shells subjected to two kinds of symmetric loads are analyzed, respectively. Under uniformed loads, smaller phonon displacements are achieved in a quasicrystal-skin structure than those in crystal-skin one. Under the compression of rigid half-plane, the radius of the circular contact area increases with decreasing thickness of quasicrystal layers.

EM Waves Propagation Characteristics Based on Modified Dielectric Constant Model in Nonuniform Weakly Ionized Dusty Plasma

The influence of weakly ionized dusty plasma on electromagnetic (EM) wave propagation was investigated to understand blackouts in communications during the reentry of hypersonic vehicles. Adopting the Bhatnagar–Gross–Krook (BGK) collision model and charging theory, the total complex dielectric constant of weakly ionized dusty plasma is derived considering solely: 1) a nonzero minimum velocity of the electron; 2) a contribution of the second term of the collision cross-section; and finally, 3) both 1) and 2) considered. Using the propagation matrix method, we analyzed the dielectric constant and propagation properties from our modified models and those of the traditional model for weakly ionized plasma with and without dust. The results show that, regarding EM waves, the dust-free weakly ionized plasma has a larger coefficient of reflection than weakly ionized dusty plasma. Simultaneous considering nonzero velocity and contribution of the second term of the collision cross-section significantly changed the dielectric constant and propagation properties, indicating the nonnegligible role of these two factors. Furthermore, the influence of dust size and concentration on the propagation properties of the EM wave was discussed with considering the two factors. The simulated results show that the coefficient of absorption increases and the coefficient of transmission decreases with increase in size and concentration of dust particles.

Bending Deformation of Multilayered One-Dimensional Quasicrystal Nanoplates Based on the Modified Couple Stress Theory

The modified couple stress theory has been applied in many nanomaterials except for nano-quasicrystals. In this paper, the modified couple stress theory is firstly adopted to analyze the static bending deformation of multilayered one-dimensional (1D) hexagonal quasicrystal (QC) nanoplates under surface loadings. The general solutions for the extended displacement and traction vectors in a simply supported and homogeneous QC nanoplate are derived by solving an eigenvalue problem reduced from the governing equations. Utilizing the propagator matrix method, the analytical solutions of multilayered 1D QC nanoplates are then obtained by assuming that the layer interfaces are continuous. Numerical examples for some kinds of nanoplates made up of QC and crystal ($$\hbox {BaTiO}_{3}$$) are provided to illustrate the effects of the material length parameter, the number of layers and the stacking sequence of nanoplates on the phonon and phason fields, which is helpful for the application of QCs to surface coating and solar energy selective absorber.

Static response of functionally graded multilayered one-dimensional hexagonal piezoelectric quasicrystal plates using the state vector approach

The effect of the non-homogeneity of material properties has been considered the important variation mechanism in the static responses of quasicrystal structures, but the existing theoretical model for it is unable to simulate the material change format beyond the exponential function. In this paper, we create a new model of functionally graded multilayered 1D piezoelectric quasicrystal plates using the state vector approach, in which varying functionally graded electro-elastic properties can be extended from exponential to linear and higher order in the thickness direction. Based on the state equations, an analytical solution for a single plate has been derived, and the result for the corresponding multilayered case is obtained utilizing the propagator matrix method. The present study shows, in particular, that coefficient orders of two varying functions (the power function and the exponential function) of the material gradient provide the ability to tailor the mechanical behaviors in the system’s phonon, phason, and electric fields. Moreover, the insensitive points of phonon stress and electric potential under functionally graded effects in the quasicrystal layer are observed. In addition, the influences of stacking sequences and discontinuity of horizontal stress are explored in the simulation by the new model. The results are very useful for the design and understanding of the characterization of functionally graded piezoelectric quasicrystal materials in their applications to multilayered systems.

Nonlocal analytical solution of functionally graded multilayered one-dimensional hexagonal piezoelectric quasicrystal nanoplates

Based on the nonlocal elasticity theory, the static bending deformation of a functionally graded multilayered one-dimensional (1D) hexagonal piezoelectric quasicrystal (PQC) simply supported nanoplate is investigated under surface mechanical loadings. The functionally graded material is assumed to be exponential along the thickness direction. By utilizing the pseudo-Stroh formalism and propagator matrix method, exact closed-form solutions of functionally graded multilayered 1D hexagonal PQC nanoplates are then obtained by assuming that the layer interfaces are perfectly contacted. Numerical examples for six kinds of sandwich functionally graded nanoplates made up of piezoelectric crystals, quasicrystal and PQC are presented to illustrate the influence of the exponential factor, nonlocal parameter and stacking sequence on the phonon, phason and electric fields, which play an important role in designing new composite structures in engineering.