pseudo-Stroh Formalism Research Articles

Time-dependent behavior of layered magneto-electro-elastic cylindrical shell with viscoelastic interlayer

In this work, an analytical solution for layered magneto-electro-elastic (MEE) cylindrical shell adhesively bonded by viscoelastic interlayer is developed to predict its time-dependent mechanical, electric and magnetic behaviors. The viscoelastic characteristic of the interlayer is modelled by the standard linear solid model. Each MEE layer is governed by the equations of magneto-electro-elasticity. The imperfect electric conditions between adjacent MEE layers are also considered. Using the Pseudo-Stroh formalism, a general solution with unknown coefficients is derived for each MEE layer. The Laplace transformation is applied to the constitutive equations of the viscoelastic interlayer. The coefficients are determined by the surface conditions as well as the interface conditions. The present solution can be used as the benchmark to assess results from numerical approaches. It is shown that the finite element solution converges to the present one as the mesh density increases; however, the finite element method is time-consuming in mesh division and calculation. Finally, the effects of time, shell angle, interlayer thickness and imperfect electric coefficient on the mechanical, electric and magnetic behaviors are investigated.

Cylindrical bending analysis of a layered two-dimensional piezoelectric quasicrystal nanoplate

The piezoelectric effect is a significant property of quasicrystal. In this article, the exact solution is derived for a layered piezoelectric quasicrystal nanoplate with nonlocal effect in cylindrical bending. Based on the nonlocal theory and the pseudo-Stroh formalism, the exact solution for a homogeneous simply supported nanoplate is obtained. With the aid of the propagator matrix, the exact solution for a multilayered nanoplate is achieved. Numerical examples are carried out to reveal the influences of span-to-thickness ratio, nonlocal parameter, and stacking sequence on piezoelectric quasicrystal nanoplates with their top surface subjected to two loadings. One is a z-direction mechanical loading and the other is an electric potential loading. These results can be served as benchmarks for the design, numerical modeling, and simulation of layered two-dimensional piezoelectric quasicrystal nanoplates under cylindrical bending.

An exact solution for a functionally graded multilayered one-dimensional orthorhombic quasicrystal plate

This paper investigates the problem of a functionally graded multilayered one-dimensional orthorhombic quasicrystal plate with simply supported edge conditions. Assuming that the functionally graded material properties vary exponentially along the thickness direction, a solution for a functionally graded plate subjected to top surface loading is obtained by using the pseudo-Stroh formalism. The propagator matrix method is utilized to get the solution for the corresponding multilayered case. The exact solution is applied to a multilayered plate made of quasicrystals and crystals. The influences of the exponential factor, load form, and stacking sequence on physical quantities are studied in numerical examples. The exact solution can be used to design a functionally graded multilayered plate composed of one-dimensional quasicrystals and crystals. The numerical results can also serve as a basis for other numerical methods, such as finite element and boundary element methods.

Bending deformation of multilayered one-dimensional hexagonal piezoelectric quasicrystal nanoplates with nonlocal effect

Based on the nonlocal elasticity theory, the static bending deformation of one-dimensional (1D) hexagonal piezoelectric quasicrystal (PQC) nanoplates is investigated under surface electroelastic loadings. The general solutions for the extended displacement and traction vectors of a simply supported and homogeneous PQC nanoplate are derived by solving an eigenvalue problem reduced from the pseudo-Stroh formalism. By utilizing the propagator matrix method, exact closed-form solutions of multilayered 1D hexagonal PQC nanoplates are then obtained by assuming that the layer interfaces are perfectly contacted. Numerical examples for six kinds of sandwich nanoplates made up of piezoelectric crystals (PE), quasicrystal (QC) and PQC are presented to illustrate the effect of the nonlocal parameter and stacking sequence of the nanoplates on the phonon, phason and electric fields, which play an important role in designing new composite structures in engineering.

Three-dimensional exact electric-elastic analysis of a multilayered two-dimensional decagonal quasicrystal plate subjected to patch loading

An exact electric-elastic analysis of a multilayered two-dimensional decagonal quasicrystal plate subjected to patch loading with simply supported boundary conditions is presented. The pseudo-Stroh formalism and propagator matrix method are used to obtain the exact three-dimensional mechanical behaviors of the plate. By expressing the patch loading in the form of a double Fourier series expansion an exact closed-form solution with a concise and elegant expression is deduced. Three different kinds of patch surface loadings are applied to the surface of the plate and the response of the plate is investigated. Comprehensive numerical results are shown for a sandwich plate subjected to the three patch loadings with two different stacking sequences. The results show that the stacking sequences, patch loading areas, and patch loading types have a great influence on the stress, displacement and electric components of the plate. Also, different coupling constants between the phonon and phason fields will influence the physical quantities. The useful features observed from numerical results can be used in the design of composite laminates made of two-dimensional piezoelectric quasicrystals. The numerical results can also serve as a reference in convergence studies of other numerical methods and for verification of existing or future plate theories.

Deformation of a layered magnetoelectroelastic simply-supported plate with nonlocal effect, an analytical three-dimensional solution

In this paper, we present an exact closed-form solution for the three-dimensional deformation of a layered magnetoelectroelastic simply-supported plate with the nonlocal effect. The solution is achieved by making use of the pseudo-Stroh formalism and propagator matrix method. Our solution shows, for the first time, that for a homogeneous plate with traction boundary condition applied on its top or bottom surface, the induced stresses are independent of the nonlocal length whilst the displacements increase with increasing nonlocal length. Under displacement boundary condition over a homogeneous or layered plate, all the induced displacements and stresses are functions of the nonlocal length. Our solution further shows that regardless of the Kirchoff or Mindlin plate model, the error of the transverse displacements between the thin plate theory and the three-dimensional solution increases with increasing nonlocal length revealing an important feature for careful application of the thin plate theories towards the problem with nonlocal effect. Various other numerical examples are presented for the extended displacements and stresses in homogeneous elastic plate, piezoelectric plate, magnetostrictive plate, and in sandwich plates made of piezoelectric and magnetostrictive materials. These results should be very useful as benchmarks for future development of approximation plate theories and numerical modeling and simulation with nonlocal effect.

An exact closed-form solution for a multilayered one-dimensional orthorhombic quasicrystal plate

By extending the pseudo-Stroh formalism to multilayered one-dimensional orthorhombic quasicrystal plates, we derive an exact closed-form solution for simply supported plates under surface loadings. The propagator matrix method is introduced to efficiently and accurately treat the multilayered cases. As a numerical example, a sandwich plate made of quasicrystals and crystals with two different stacking sequences is investigated. The displacement and stress fields for these two stacking sequences are presented, which clearly demonstrate the importance of the stacking sequences on the induced physical quantities. Our exact closed-form solution should be of particular interest to the design of one-dimensional quasicrystal laminated plates. The numerical results can be further used as benchmarks to various numerical methods, such as the finite element and finite difference methods, on the analysis of laminated composites made of one-dimensional quasicrystals.

An exact solution for a multilayered two-dimensional decagonal quasicrystal plate

By extending the pseudo-Stroh formalism to two-dimensional decagonal quasicrystals, an exact closed-form solution for a simply supported and multilayered two-dimensional decagonal quasicrystal plate is derived in this paper. Based on the different relations between the periodic direction and the coordinate system of the plate, three internal structure cases for the two-dimensional quasicrystal layer are considered. The propagator matrix method is also introduced in order to treat efficiently and accurately the multilayered cases. The obtained exact closed-form solution has a concise and elegant expression. Two homogeneous quasicrystal plates and a sandwich plate made of a two-dimensional quasicrystal and a crystal with two stacking sequences are investigated using the derived solution. Numerical results show that the differences of the periodic direction have strong influences on the stress and displacement components in the phonon and phason fields; different coupling constants between the phonon and phason fields will also cause differences in physical quantities; the stacking sequences of the multilayer plates can substantially influence all physical quantities. The exact closed-form solution should be of interest to the design of the two-dimensional quasicrystal homogeneous and laminated plates. The numerical results can also be employed to verify the accuracy of the solution by numerical methods, such as the finite element and difference methods, when analyzing laminated composites made of quasicrystals.

Pattern instability of functionally graded and layered elastic films under van der Waals forces

This paper investigates surface instability of a functionally graded and layered elastic film interacting with another flat rigid body [or interacting with another functionally graded and layered elastic film (or simply-supported elastic plate)] through surface van der Waals forces under plane strain conditions. The shear modulus in each functionally graded layer is assumed to be exponentially varied in the thickness direction. A homogeneous elastic layer, which is the focus of this research, can be considered as a special case of the functionally graded layer by taking the magnitude of the gradient parameter to be very small. The solution for any functionally graded layer is obtained in terms of the pseudo-Stroh formalism; then the solution for the multilayered system is derived based on the transfer-matrix method. As a result the displacement and traction vectors at the top surface of the layered film (or plate) can be expressed in terms of those at the bottom surface of the layered film (or plate). We can thus obtain simple relationships between the surface normal traction and surface deflection. Expressions for the interaction coefficient as a function of the wave number of the instability mode are therefore obtained. The critical value of the interaction coefficient for surface instability and the associated instability mode can be determined easily by identifying the minimum of the interaction coefficient. An advantage of the present method lies in that it is very convenient to address a film (or a plate) with an arbitrary number of layers. The correctness of the present approach is verified by comparison with known results. The results show that it is possible to find N distinct surface instability modes for an N-layered elastic film interacting with another flat rigid body; and that it is also possible to find that there are at most N1+N2 distinct surface instability modes for an N1-layered elastic film interacting with another N2-layered elastic film. When a multilayered elastic film interacts with a simply-supported multilayered elastic plate, the film-plate system will exhibit the instability mode of the film or that of the plate depending on the stability strength of the plate versus that of the film.

Exact Solutions for Simply Supported and Multilayered Piezothermoelastic Plates with Imperfect Interfaces

Exact solutions are derived for three-dimensional, orthotropic, linearly piezothermoelastic, simply-supported, and multilayered rectangular plates with imperfect interfaces under static thermo-electro-mechanical loadings. In this re- search the imperfect interface is described as thermally weakly (or highly) conducting, mechanically compliant and di- electrically weakly (or highly) conducting. While the homogeneous solutions for one layer are obtained in terms of the so- called pseudo-Stroh formalism, solutions for multilayered plates are expressed in terms of the transfer matrices for both the layer and the imperfect interface. Due to the fact that the thermal effect is incorporated, we adopt a special form of the transfer matrix, resulting in a very concise solution structure for piezothermoelastic multilayered plates. Numerical results are presented to validate the developed formulas and to demonstrate the influence of the interface imperfection on the dis- tributions of the field variables.

Three-dimensional analysis of multi-layered functionally graded anisotropic cylindrical panel under thermomechanical loading

A general method is presented for the analysis of a thermoelastic multi-layered cylindrical panel made of an oblique pile of functionally graded layers having orthotropic material properties. The two edges of the panel are under simply-supported and zero temperature conditions. Steady-state thermal and mechanical boundary conditions can be applied on the innermost and outermost surfaces of the cylindrical panel. In addition, the functionally graded material is assumed to be of a power law type in the radial direction and the homogeneous solution in each layer is obtained in terms of the pseudo-Stroh formalism. The solution for the multi-layered cylindrical panel is derived based on the transfer matrix method. Novel to this work, is an alternative but equivalent formulation for the transfer matrix method which is more concise in expression. The derived solution is subsequently applied to investigate a five-layered cylindrical panel. The influence of the gradation of the material and ply angle on the distribution of temperature, displacements and stresses are investigated. The results clearly show that material gradation and ply angle significantly influence the elastic fields and temperature.

Exact solution for functionally graded and layered magneto-electro-elastic plates

In this paper, an exact solution is presented for the multilayered rectangular plate made of functionally graded, anisotropic, and linear magneto-electro-elastic materials. While the edges of the plate are under simply supported conditions, general mechanical, electric and magnetic boundary conditions can be applied on both the top and bottom surfaces of the plate. The functionally graded material is assumed to be exponential in the thickness direction and the homogeneous solution in each layer is obtained based on the pseudo-Stroh formalism. For multilayered plate structure, the propagator matrix method is employed so that only a 5 × 5 system of linear algebraic equations needs to be solved. The exact solution is then applied to two functionally graded (exponential) sandwich plates made of piezoelectric BaTiO 3 and magnetostrictive CoFe 2O 4, under mechanical and electric loads on the top surface. While the numerical results clearly show the influence of the exponential factor, magneto-electro-elastic properties, and loading types on induced magneto-electric-elastic fields, they can also serve as benchmarks to numerical methods such as the finite and boundary element methods.

Three-dimensional solution of smart laminated anisotropic circular cylindrical shells with imperfect bonding

This paper presents an exact solution for a simply-supported and laminated anisotropic cylindrical shell strip with imperfect bonding at the off-axis elastic layer interfaces and with attached anisotropic piezoelectric actuator and sensor subjected to transverse loading. In this research, the imperfect interface conditions are described in terms of linear relations between the interface tractions in the normal and tangential directions, and the respective discontinuities in displacements. The solution for an elastic (or piezoelectric) layer of the smart laminated cylindrical shell strip is obtained in terms of the six-dimensional (or eight-dimensional) pseudo-Stroh formalism, solution for multilayered system is then derived based on the transfer matrix method. Finally, a numerical example is presented to demonstrate the effect of imperfect interface on the static response of the smart laminated cylindrical shell. The derived solutions can serve as benchmark results to assess various approximate shell theories and numerical methods.