pseudo-Stroh Formalism Research Articles

Three-dimensional unsteady state solution of multilayered saturated anisotropic finite media subjected to point source

The three-dimensional (3D) unsteady state solution of a saturated anisotropic finite porous medium due to a point source is presented in this paper. First, Laplace transform is introduced to address the governing equations of anisotropic poroelasticity. Next, the eigen equation method and the modified pseudo-Stroh formalism are used to obtain the transient solution for homogeneous saturated finite media. Then, the propagation matrix method is used to address the corresponding multilayered porous elastic media, and the 3D unsteady state solution of the layered porous medium subjected to a point source is obtained. The solution for a finite porous foundation is presented to demonstrate the use of the closed form solution. The numerical results of a homogeneous foundation due to an internal time-dependent point source are given. Two different point sources are used to show the transient responses of the proposed method. By comparing with those obtained by the finite element method, the accuracy of the proposed method has been proved. Meanwhile, the influence of the depth of the point source on the hydromechanical responses of the homogeneous foundation is discussed. Finally, the transient response of an anisotropic sandwich foundation due to a point source at different depths is investigated. These numerical examples prove the feasibility of the transient 3D solution. The closed form solution presented in this paper can provide help for 3D transient fluid-deforming coupling analysis of laminated porous media.

Three-dimensional thermoelastic analysis of orthotropic laminated cylindrical shells subjected to uniformly localized thermal boundary conditions

Accurate evaluation of three-dimensional (3D) thermal-elastic analysis of laminated cylindrical shells under different loads has a significant impact on their safety design. In this study, we present a 3D thermoelastic solution of laminated orthotropic cylindrical shells subjected to uniformly localized thermal boundary conditions. For a homogeneous orthotropic layer, a thinning layer approach is introduced to deal with the heat conduction equation and the elastic equation. Based on the eigenequation method and pseudo-Stroh formalism, general solutions for physical quantities of a homogeneous layer are derived. The propagator matrix method is used to deal with multilayered media. A closed form solution for the laminated cylindrical shells is obtained. A numerical example of a 3-layered cylindrical shell subjected to convective heat transfer boundary conditions is presented to verify the solutions’ correctness. An excellent convergence of the current approach is found and the physical quantities predicted by the current method agree well with the numerical results by the finite element method. Finally, the numerical solutions of a 3-layered orthotropic cylindrical shell subjected to uniformly localized thermal boundary conditions are obtained by the current method. The closed form solution presented in this article can be used for 3D thermoelastic analysis of laminated cylindrical shells under complex boundary conditions.

Wave motion of magneto-electro-elastic laminated plates with membrane-type interfacial imperfections

A theoretical analysis for the free vibration of magneto-electro-elastic (MEE) laminated plates with generalized membrane-type interfacial imperfections is proposed. The wave motion characteristics, such as the dispersion curves and mode shapes, are investigated. The pseudo-Stroh formalism and the propagator matrix method are employed to derive the analytical solutions for an orthotropic and multilayered MEE plate with imperfect interfaces. The effect of generalized membrane-type interfacial imperfections on the dispersion curve, mode shape, and magnetoelectric coupling factor for piezoelectric BaTiO 3 and piezomagnetic CoFe 2O4 sandwich plates are studied in detail. Numerical calculations show that the membrane-type interfacial imperfection has a significant effect on the dispersion curve, mode shape, and magnetoelectric coupling factor in layered MEE composites. This exact solution provides important insight to assess the wave characteristics of the MEE laminated composites with membrane-type interfacial imperfections.

Multiferroic laminated composites with interfacial imperfections and the nonlocal effect

In this study, we investigate multiferroic magneto-electro-elastic (MEE) laminated plates with interfacial imperfections by using nonlocal theory. The multilayered MEE plate is considered a simply supported rectangular plate subjected to appropriate boundary conditions. Generalized membrane-type and generalized linear-spring-type imperfect interfaces are considered. We extend the pseudo-Stroh formalism and propagator matrix method to derive analytical solutions for an orthotropic and multilayered MEE plate with interfacial imperfections and the nonlocal effect. The key step is first to observe that the equilibrium equations and interface conditions are equivalent to those in the classical MEE multilayered problem. Next, we derive the relationship between nonlocal and classical tractions. The derived solutions are then applied to BaTiO3−CoFe2O4 sandwich plates. Numerical calculations show that interfacial imperfections and the nonlocal effect can be used to tune the field distribution and ME effect. The findings can provide insights for the application and design of various nanodevices.

Semi-analytical solution for static and free vibration of multilayered functionally graded elastic plates with imperfect interfaces

In this paper, the three-dimensional static behavior and the free vibration of simply supported multilayered functionally graded elastic plates with bonding imperfections is derived. The imperfect interfaces between the adjacent layers are modeled by the spring-type layer model. In each layer, a well-adapted semi-analytical solution coupling the state-space approach with the fourth-order Runge-Kutta numerical procedure is elaborated. In consequence, the intricate three-dimensional problem has been reduced to a one-dimensional recursive problemwith which arbitrary functionally graded material’s model and number of layers can be easily handled. The predicted solution in each layer has been propagated from the bottom to the top layers of the plate using the propagator matrix method and taking into account the transfer matrix at the imperfect interfaces. The transfer relationship linking the top and the bottom surfaces of the multilayered functionally graded plate is therefore obtained. The accuracy and reliability of the proposed methodological approach have been clearly demonstrated by several numerical tests. The predicted numerical results have been well compared with the available ones obtained by the pseudo-Stroh formalism, discrete layer approach, finite elements method, Layerwise method, variable kinematic model, sinusoidal and hyperbolic shears deformations theories, respectively. These results showed that the imperfect interfaces have a strong effect on the static behavior and the natural frequencies of multilayered functionally graded plates. In addition, the obtained results showed that when the dimensionless interface parameter increase the natural frequencies decrease quickly.

Static response and free vibration analysis for cubic quasicrystal laminates with imperfect interfaces

Quasicrystals have attracted the tremendous attention of researchers for their unusual properties. In this paper, an analytical treatment is presented for the static response and free vibration of a multilayered three-dimensional, cubic quasicrystal rectangular simply supported plate with bonding imperfections. Based on the basic elasticity equation of three-dimensional quasicrystals, we construct the linear eigenvalue system in terms of the pseudo-Stroh formalism, from which the general solutions of the extended displacements and stresses in any homogeneous layer can be obtained. Furthermore, these solutions along the thickness direction can be utilized to solve any physical variables under given boundary conditions. For multilayered plates, the propagator matrices are employed to connect the field variables at the upper interface to those at the lower interface of each layer. The special spring model, which describes the discontinuity of the physical quantities across the interface, is introduced into the overall propagator relationship of the structure. Compared with the conventional propagator matrix method, a new propagator relation is established to resolve numerical instabilities of the case of large aspect ratio and high-order frequencies for QC laminates with imperfect interface. In addition, the traction-free boundary condition on the top and bottom surfaces of the layered plate is considered to investigate the free vibration characteristics of the laminates. Finally, typical numerical examples are presented to illustrate the influence of imperfect interfaces on static response and free vibration of cubic quasicrystal plates.

A virtual layers-state-space method for 3D responses of arbitrary functionally graded magnetoelectroelastic plates

In this paper, a methodological approach for the three-dimensional static behavior of functionally graded magnetoelectroelastic rectangular plates is elaborated. The state-space method is used to establish the set of state-space equations for arbitrary graded functionalities throughout the thickness direction of multilayered composite plates. To solve the resulting state-space equations, a numerical scheme that consists of dividing the heterogeneous plate into a number of virtual layers having homogeneous properties calculated in the thickness average sense within the corresponding layer is elaborated. In each virtual homogeneous layer the propagator matrix method is used and a recursive relationship is resulted. The proposed mathematical modeling has been validated and the convergence according to the number of virtual layers has been analyzed. The obtained results are in agreement with the available ones given by the pseudo-Stroh formalism, the finite elements method, the modified Pagano method and the three-dimensional asymptotic approach. Various types of graded properties can be easily considered. It has been revealed that the materials properties gradient index and the loading conditions remarkably influence the static response of functionally graded composites plates.

Three-dimensional modeling of static deformation of arbitrary functionally graded multilayered multiferroic composites plates with weakly and highly conducting imperfect interfaces

In this paper, a three-dimensional static deformation of arbitrary functionally graded multilayered multiferroic composites plates with weakly and highly conducting imperfect interfaces is derived. The magnetoelectroelastic properties of each layer of the composite plates have been assumed varying throughout the thickness direction. The imperfect interfaces between the layers are assumed to be mechanically compliant, dielectrically and magnetically weakly or highly conducting. In each layer, the state-space approach is firstly applied leading to space variable. Cauchy’s problem and adapted Runge-Kutta numerical procedure is used to solve the established state-space equation. The elaborated semi-analytical solution has been propagated throughout the multilayered multiferroic composites plates using the propagator matrix method and accounting the transfers matrices at each imperfect interface. The developed formulas have been programmed and the numerical obtained results have been well compared with available ones. For the computation process, the piezoelectric material [Formula: see text] and piezomagnetic material [Formula: see text] are used. In addition, these numerical tests showed that the proposed solution is in good agreement with the available 3D asymptotic approach, the modified Pagano method, the pseudo-Stroh formalism, the finite elements method and the Peano series solution. Furthermore, the effects of mechanically compliant, dielectrically and magnetically weakly or highly conducting imperfect interface on the static response and the magnetoelectric coupling coefficient of the functionally graded multilayered multiferroic composites plates for various configurations have been also analyzed. It has been carried out that, the bending response and the magnetoelectric coupling coefficient of the multiferroic composites plates remarkably depend on the kind of imperfect interface, the used sequences as well as on the loadings conditions namely mechanical, electric or magnetic, respectively.

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.

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.

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.

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.

3D-dynamic modeling of cross-ply magneto-electro-elastic laminates based on the pseudo-Stroh formalism

This work presents methodological approaches to investigate 3D forced vibrations of simply supported multilayer magneto-electro-elastic plates. Multilayered multifunctional plates with orthotropic layers are considered. Harmonic transverse stress is used and the propagation matrix is presented at each layer. The dynamic responses of 3D displacements, stresses, and electric and magnetic displacements are analyzed with respect to the thickness direction and excitation frequency. This formalism is effective for thin and thick multilayered plates. Moreover, this study in forced vibrations makes it possible to overcome the 3D problems of computation time and storage commonly encountered in the solution of the eigenvalue problem.

Nonlocal free and forced vibration of multilayered two-dimensional quasicrystal nanoplates

A three-dimensional plate model capturing the size effect is developed for free and forced vibration of a layered two-dimensional quasicrystal nanoplate. The simply supported nanoplate subjected a top surface harmonic excitation is assumed for forced vibration analysis, and the traction free boundary conditions are considered for free vibration analysis. The exact solution for layered two-dimensional quasicrystal nanoplates is obtained by using the pseudo-Stroh formalism, Eringen’s nonlocal theory, and a dual variable and position method. Numerical examples show the effects of nonlocal parameter, stacking sequence and aspect ratio on natural frequency as well as forcing frequency on harmonic response.

Dynamic analysis of multilayered magnetoelectroelastic plates based on a pseudo-Stroh formalism and Lagrange polynomials

In this article, a semi-analytical three-dimensional modeling of dynamic behavior of the multilayered magnetoelectroelastic plates under simply supported edges boundary conditions is derived. A combination of pseudo-Stroh formalism and the Lagrange polynomials is elaborated for the space and time response. The time domain is subdivided into small intervals that are discretised using the associated Tchybechev points. The layer-time solution is elaborated in time-dependent matrix form. The propagator matrices are used for the laminated multifunctional plates with an arbitrary number of layers. Extended-traction vectors are obtained for mechanical, electrical, and magnetic excitations. To validate the elaborated numerical procedure, the dynamic behavior of the three layered plates made of piezoelectric material [Formula: see text] and piezomagnetic material [Formula: see text] is investigated. The lower surface of the plate is assumed to be traction free, whereas the upper surface is subjected to a dynamic sinusoidal loading. The obtained results are in good agreement with the available ones based on the Layer wise and the state-space approaches. These results demonstrated that a magnetoelectric coupling coefficient is time-independent but depends strongly on the kind of imperfect interfaces and the taking sequences of the multilayered plates. Furthermore, it is established that the imperfect interfaces have a strong influence on the dynamic behavior of the laminated structures.

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.

Exact thermoelectroelastic solution of layered one-dimensional quasicrystal cylindrical shells

Laminated shells have been widely used as intelligent structures in many industries due to their unique characteristics of high strength/weight and stiffness/weight ratios. This paper presents a thermoelectromechanical analysis of multilayered one-dimensional piezoelectric quasicrystal cylindrical shells under surface heating, which are simply supported at circumferential edges and infinitely at axial direction. The exact solution is obtained based on the extended pseudo-Stroh formalism and a dual variable and position method with good computational stability, and it is utilized to study the effects of shell curvature, loading condition and stacking sequence on phonon, phason, thermal and electric fields.

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.

Three-dimensional exact thermo-elastic analysis of multilayered two-dimensional quasicrystal nanoplates

• A 3D thermo-elastic analytical solution for 2D QC nanoplates is derived. • Pseudo-Stroh formalism and nonlocal theory is extended. • The effects of nonlocal parameters, stress-temperature coefficients, and stacking sequences are investigated. A three-dimensional thermo-elastic analytical solution for two-dimensional quasicrystal simply supported nanoplates subjected to a temperature change on their top surface is presented. The nonlocal theory and pseudo-Stroh formalism are used to obtain the exact solution for a homogeneous two-dimensional decagonal quasicrystal nanoplate with its thickness direction as a quasi-periodic direction. The propagator matrix method is introduced to deal with the corresponding multilayered nanoplates. Comprehensive numerical results show that nonlocal parameters, stress-temperature coefficients, stacking sequences have great influence on the stress, displacement components and heat fluxes of the nanoplates. In addition, the stacking sequences also influence the temperature and heat fluxes of the nanoplate. The exact thermo-elastic solution should be of interest to the design of the two-dimensional quasicrystal homogeneous and multilayered plates. The mechanical behaviors of the nanoplates in numerical results can also serve as benchmarks to verify various thin-plate theories or other numerical methods.

Size-dependent effect on functionally graded multilayered two-dimensional quasicrystal nanoplates under patch/uniform loading

The nonlocal effect on functionally graded multilayered quasicrystal nanoplates is investigated. The functionally graded quasicrystal is assumed to be exponentially distributed along the thickness direction of the simply supported nanoplates. The exact solution for functionally graded multilayered two-dimensional quasicrystal nanoplates subjected to a patch loading on their top surfaces is derived using the extended nonlocal elastic theory, pseudo-Stroh formalism, and propagator matrix method. The patch loading is indicated by the form of a double Fourier series expansion. Numerical examples are presented to reveal the influences of patch size, nonlocal parameter, and stacking sequence on the phonon, phason, and electric fields.