Extended Stroh Formalism Research Articles

Study on the attributes of transverse surface waves in a layered piezoelectric semi-space with surface elasticity theory and extended stroh formalism

Previous works on surface acoustic wave sensors have shown great limitations in selecting piezoelectric materials and the wave propagation direction. To eliminate such limitations in the technological revolution of SAW sensors, the current paper's main purpose is to explore how wave propagation orientation affects the performance of SAW macro- and nano-sensors. Based on Extended Stroh formalism, the theoretical forms are derived and exploited to present the wavenumber of transverse waves in an arbitrary direction of the piezoelectric medium. Furthermore, we consider surface elasticity theory to acquire the phase velocity equation on the basis of the expression of the wavenumber. More intuitively, a physical model is set up to obtain the horizontal shear stiffness of the surface and bulk layers. Then, the numerical case is carried out to determine the relationship between phase velocity and wave propagation orientation. By comparing the numerical study and the physical model, it can be found that the empirical formula of shear stiffness for bulk and surface layers offers a helpful route to precisely predict the mechanical attributes of SAW macro- and nano-sensors, respectively. The summaries of the current theoretical work benefit the manufacturing of surface acoustic wave sensors with improved performance.

Theoretical Analysis of Piezoelectric Semiconductor Thick Plates with Periodic Boundary Conditions.

Piezoelectric semiconductors, being materials with both piezoelectric and semiconducting properties, are of particular interest for use in multi-functional devices and naturally result in multi-physics analysis. This study provides analytical solutions for thick piezoelectric semiconductor plates with periodic boundary conditions and includes an investigation of electromechanical coupling effects. Using the linearization of the drift-diffusion equations for both electrons and holes for small carrier concentration perturbations, the governing equations are solved by the extended Stroh formalism, which is a method for solving the eigenvalues and eigenvectors of a problem. The solution, obtained in the form of a series expansion with an unknown coefficient, is solved by matching Fourier series expansions of the boundary conditions. The distributions of electromechanical fields and the concentrations of electrons and holes under four-point bending and three-point bending loads are calculated theoretically. The effects of changing the period length and steady-state carrier concentrations are covered in the discussion, which also reflects the extent of coupling in multi-physics interactions. The results provide a theoretical method for understanding and designing with piezoelectric semiconductor materials.

Theoretical analysis of surface waves in piezoelectric medium with periodic shunting circuits

The investigations of surface waves in the piezoelectric medium bring out great possibility in designing smart surface acoustic wave (SAW) devices. It is important to study the dispersion properties and manipulation mechanism of surface waves in the semi-infinite piezoelectric medium connected with periodic arrangement of shunting circuits. In this study, the extended Stroh formalism is developed to theoretically analyze the dispersion relations of surface waves under different external circuits. The band structures of both the Rayleigh wave and the Bleustein-Gulyaev (BG) wave can be determined and manipulated with proper electrical boundary conditions. Furthermore, the electromechanical coupling effects on the band structures of surface waves are discussed to figure out the manipulation mechanism of adjusting electric circuit. The results indicate that the proposed method can explain the propagation behaviors of surface waves under the periodic electrical boundary conditions, and can provide an important theoretical guidance for designing novel SAW devices and exploring extensive applications in practice.

Extended Stroh formalism for plane problems of thermoelasticity of quasicrystal media

This study provides a mathematical method for the analysis of plane phonon-phason thermomechanical fields in quasi-crystalline bodies, which was developed based on the complex variable calculus and the Stroh formalism. Constitutive and balance equations of thermoelasticity of quasicrystal solids are written in a generalized form using extended phonon-phason vectors and tensors. Adapting the methods of the Stroh formalism, the formulated general problem is reduced to the search for 7 analytical functions (1 for temperature and 6 for phonon-phason fields). The obtained approach was used to construct the integral equations of the corresponding two-dimensional problems of the thermoelasticity of quasicrystal solids, and on its basis a specific problem for a quasi-crystalline medium with a crack was solved. Distributions of temperature and mechanical fields in the body and functions of asymptotic distributions of phonon-phason fields near the crack tip using intensity coefficients of phonon and phason stresses were obtained.

Theoretical analysis of guided waves propagation in periodic piezoelectric plates with shunting circuits

The tunable manipulation of guided waves in plates brings out great potential applications in engineering practices, and the electromechanical coupling effects of piezoelectric material with shunting circuits have exhibited powerful tunability and flexibility for guided wave propagation. In this paper, a theoretical model is established to analyze the guided wave propagation in one-dimensional periodic piezoelectric plate constructed from a periodic array of anisotropic piezoelectric materials under periodic electrical boundary conditions. The extended Stroh formalism incorporating with the plane wave expansion method is developed to transform the wave motion equations of the periodic piezoelectric plate into a linear eigenvalue system, and a more concise and elegant solution of generalized displacement and generalized stress can be derived. There are various dispersion relations in terms of the altering electrical boundary conditions to be acquired, if the thin electrodes with shunting circuits are attached periodically to both surfaces of the piezoelectric plate. Analytical results show that the coupling of the local electric resonant mode and propagating elastic wave modes can induce hybridization bandgaps, and the bandgaps of Lamb waves and SH waves in the piezoelectric plate can be tuned by designing appropriate material polarization orientations and shunting circuits. In addition, the Bragg bandgaps can also be influenced by the external circuits. Results indicate that the proposed theoretical model can effectively analyze the performances of guided waves in periodic piezoelectric plate and provide useful theoretical guidance for designing smart wave control devices.

Influence of Imperfect Interface of Anisotropic Thermomagnetoelectroelastic Bimaterial Solids on Interaction of Thin Deformable Inclusions

Abstract This work studies the problem of thermomagnetoelectroelastic anisotropic bimaterial with imperfect high-temperature conducting coherent interface, whose components contain thin inclusions. Using the extended Stroh formalism and complex variable calculus, the Somigliana-type integral formulae and the corresponding boundary integral equations for the anisotropic thermomagnetoelectroelastic bimaterial with high-temperature conducting coherent interface are obtained. These integral equations are introduced into the modified boundary element approach. The numerical analysis of new problems is held and results are presented for single and multiple inclusions.

Riemann–Hilbert approach-based analytical solutions for strip saturated two unequal collinear cracks in piezoelectric media

The objective of the work is to derive analytical solutions based on the Riemann–Hilbert (R–H) approach for semipermeable strip saturated two unequal collinear cracks in arbitrary polarized piezoelectric media. We particularly consider the influence of far field electromechanical loadings, poling direction and different crack-face boundary conditions. The problem is mathematically formulated into a set of non-homogeneous R–H problems in terms of complex potential functions (related to field components) using complex variable and extended Stroh formalism approach. After solving these equations, explicit solutions are obtained for the involved unknown complex potential functions and hence, the stress and electric displacement components at any point within the domain. Furthermore, after employing standard limiting conditions, explicit expressions for some conventional fracture parameters such as saturated zone lengths (in terms of nonlinear equations), local stress intensity factors and crack opening displacement are obtained. Numerical studies are presented for the PZT-4H material to analyze the effects of prescribed electromechanical loadings, inter-cracks distance, crack-face conditions and poling direction on the defined fracture parameters.

Complex variable solution for mode-III quadratically varying PS model in piezoelectric media

Complex variable technique and extended Stroh formalism approach are applied to develop the closed-form solutions for mode-III quadratically varying polarization saturation (PS) model in an infinite piezoelectric media. A semipermeable centre crack problem is considered for study under the influence of out of plane far-field mechanical loading and in-plane electrical displacement loading. Similar to PS model, the electrical yielding zones or saturated zones are considered just infront of the crack-tips in the form of a strip. But in place of constant PS condition imposed on yielding zones, the quadratically varying PS condition is considered for study. Mathematically, this problem reduces into simultaneous non-homogeneous Riemann-Hilbert problems which are solved by finding the solutions of involved singular integrals. Hence, some standard fracture parameters are derived in explicit forms such as saturated zone lengths, crack sliding displacement, crack potential drop and local stress intensity factor. Also, the comparative studies for quadratically varying PS model and PS model are presented graphically.

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.

Free vibration of fully coupled thermoelastic multilayered composites with imperfect interfaces

A fully coupled thermoelastic framework is formulated to cope with the free vibration response of anisotropic multilayered plates in three dimensions. The laminated structure consists of homogeneous laminae of arbitrary thickness and width under simply supported edge conditions in thermal environment. The general and exact field expressions of the temperature, heat flux, displacement and stress components are expressed in terms of double Fourier series expansions in any rectangular plate, which lead to the extended Stroh formalism with thermomechanical coupling effects in a concise and compact matrix form. Different imperfect interface conditions are introduced to characterize specific structural and thermal contact properties at the bounding interfaces, and further to determine the finite complex valued coefficients in the suitable series relations. The complete time-harmonic solutions in the laminated composites in the presence of perfect/imperfect interfaces are recursively obtained by means of the modified dual variable and position technique with explicit layer-to-layer transfer matrices. Results are obtained for different layups, length-to-thickness ratios and interfacial boundary conditions for two application examples, namely the graphite/epoxy cross-ply composites and the thermal barrier coatings on superalloys, without suffering from numerical exponential instability. These investigations reveal that the natural frequencies and first and higher vibration mode shapes of the multilayered structures can be considerably affected by increasing the environmental temperature and the severity of the interfacial imperfections. Since the through-thickness stress distribution in 2, 5, and 10 layered composites appears to be strongly correlated to the layups, such modal stress analysis could be exploited to locate the fatigue hotspots operated in dynamic structures and to guide the structural design of aircraft and spacecraft composite laminates subjected to residual vibrations.

SH waves in multilayered piezoelectric semiconductor plates with imperfect interfaces

In this paper, analytical solutions for SH waves in transversely isotropic multilayered piezoelectric semiconductor (PSC) plates with imperfect interfaces are obtained. The extended displacements and stresses are expressed in terms of the eigenvalues and eigenvectors by introducing the extended Stroh formalism. Making use of the dual variable and position (DVP) method and the imperfect interface conditions, the transfer matrix which relates the extended displacement and traction on the lower and upper interfaces of the multilayered plates is derived. Then the dispersion relation is obtained by using the boundary conditions on the top and bottom surfaces of the multilayered plates. Effect of the steady-state carrier density in single-layer ZnO plate, and effect of stacking sequences and imperfect interfaces in sandwich plates are discussed via numerical examples. Particularly, the critical elastic (E), piezoelectric (PE), and PSC wave domains are identified for the given carrier density, plate thickness and frequency, which could be very helpful as theoretical guidance for the design of PSC devices.

Mixed Boundary Value Problem for an Anisotropic Thermoelastic Half-Space Containing Thin Inhomogeneities

Abstract The paper presents a rigorous and straightforward approach for obtaining the 2D boundary integral equations for a thermoelastic half-space containing holes, cracks and thin foreign inclusions. It starts from the Cauchy integral formula and the extended Stroh formalism which allows writing the general solution of thermoelastic problems in terms of certain analytic functions. In addition, with the help of it, it is possible to convert the volume integrals included in the equation into contour integrals, which, in turn, will allow the use of the method of boundary elements. For modelling of solids with thin inhomogeneities, a coupling principle for continua of different dimensions is used. Applying the theory of complex variable functions, in particular, Cauchy integral formula and Sokhotski–Plemelj formula, the Somigliana type boundary integral equations are constructed for thermoelastic anisotropic half-space. The obtained integral equations are introduced into the modified boundary element method. A numerical analysis of the influence of boundary conditions on the half-space boundary and relative rigidity of the thin inhomogeneity on the intensity of stresses at the inclusions is carried out.

New analytical solutions for modified polarization saturation models in piezoelectric materials

In this paper, we present new analytical solutions for modified polarization saturation (PS) models for arbitrary polarized and semipermeable 2-D piezoelectric media. The PS model is modified to various other non-linear fracture models by varying the normal electric displacement saturated conditions in place of a constant saturated value. These variations are hereby defined as interpolating linear, quadratic and cubic type’s polynomials into saturated value interpolated on the basis of possible electric displacement values at the centre of the crack, actual crack tip and saturated zone tip. To present the analytical study, a centre crack problem in 2-D semipermeable piezoelectric media under arbitrary poling direction and combined in-plane mechanical and electric displacement loadings is considered. Using extended Stroh formalism and complex variable approach, these models are mathematically reduced into generalized non-homogeneous Riemann–Hilbert problems. Applying Riemann–Hilbert technique, the solution of these problems is obtained in terms of generalized complex potential functions representing traction forces and electric displacement components. These lead to obtain the explicit expressions for saturated zone length, crack opening displacement, crack opening potential and local intensity factor in each case. Also, the results of saturation zones and local intensity factor are obtained in each case and compared with the established numerical results. The studies show that saturated zone length increases with increasing electric loading for each case and also depends on the polynomial varying saturation condition. For a particular electric loading applied within the range of critical value, it increases with increasing degree of polynomial type saturated condition i.e. from constant to cubic type’s normal electric displacement saturated condition.

Some characteristics of elastic waves in a piezoelectric semiconductor plate

Devices based on piezoelectric semiconductors (PSCs) have recently received particular attention due to their wide bandgap where strain energy band engineering under both static and time-harmonic deformations is the key. In this paper, we investigate and characterize the elastic waves propagating in an anisotropic n-type PSC plate. To achieve our goals, we first introduce the new notations for the extended displacements, stresses, strains, and modulus to arrive at a mathematically elegant extended Stroh formalism. Then, the elastic wave problem is converted into a linear eigenvalue system from which the extended displacements and stresses are expressed in terms of the eigenvalues and eigenvectors. Finally, making use of the boundary conditions on the top and bottom surfaces of the plate, wave dispersion and attenuation are derived analytically. Numerical examples are presented to systematically study the effect of the surface boundary condition, steady-state carrier density, plate thickness, and biasing electric field on the wave speed and attenuation of both shear horizontal and Lamb waves in the transversely isotropic ZnO PSC plate. Some interesting characteristics of the elastic waves observed in this paper could be helpful as theoretical guidance when designing PSC-based devices.

Analytical solution for two equal collinear modified strip saturated cracks in 2‐D semipermeable piezoelectric media

AbstractAn analytical solution and study is presented for two equal collinear cracks in 2‐D semipermeable piezoelectric media based on modified strip saturation models. The constant saturated condition defined in strip saturation model is modified here by considering the symmetric and polynomial varying saturation conditions. These proposed saturated conditions are multiplicative of saturated electric displacement value and polynomial (constant to cubic order) of a variable defined as ratio of distance of a point on saturated zone length to the extended half‐crack length. Moreover, these are interpolated on the basis of possible saturated values near the crack‐tips and zone tips. To present the analytical study, a problem of two cracks of equal length situated in series is considered in 2‐D arbitrary polarized infinite piezoelectric domain subjected to semipermeable crack‐face and electromechanical loading conditions. Applying the extended Stroh formalism and complex variable approach, these problems are mathematically modeled into non‐homogeneous Riemann Hilbert problems in terms of unknown complex functions representing stress and electric displacement components at any point within the domain. Using Muskhelishvili [1] and Collin's [2] mathematical techniques, these complex functions are obtained from the developed non‐homogeneous Riemann Hilbert problems. Hence, the explicit expressions for saturated zone lengths (inner and outer), crack opening potentials (COPs), crack opening displacements (CODs) and local stress intensity factors (LSIFs) are obtained. Further, the impact of polynomial varying saturated conditions is highlighted on these fracture parameters with respect to inter crack space distance, crack‐face conditions, polarization angle by presenting the numerical applications in infinite 2‐D PZT‐4 media.

Semicoherent heterophase interfaces with core-spreading dislocation structures in magneto-electro-elastic multilayers under external surface loads

Using the extended Stroh formalism combining with the Fourier-transform and dual variable and position techniques, semicoherent interfaces are introduced to simulate the appropriate structural interface conditions in three-dimensional magneto-electro-elastic (MEE) multilayered composites under external loads. The field solutions caused by both internal and external actions are expressed in terms of biperiodic Fourier series expansions in multilayered systems that are made of dissimilar, linear and anisotropic plates with planar lattice mismatches. For non-perfectly bonded internal interfaces, the semicoherent heterophase interfaces are atomistically described by infinitely-long, parallel, periodically-spaced, and core-spreading intrinsic dislocation structures in free-standing multilayers. On the other hand, subsequent solutions for general loads on the external surfaces with coherent interfacial conditions are consistently derived and superposed to the former dislocation-induced fields.Three application examples in bi-, tri-, and six-layered MEE systems are investigated. The first bilayered case illustrates the significant effects of the semicoherent interface with dislocation core-spreading widths on the elastic, electric, and magnetic fields in the CoFe2O4/BaTiO3 composite materials. The second example is a trilayered system made of A/B/A with two discrete sets of intrinsic dislocation dipoles on their semicoherent interfaces, analyzed by proposing a novel energy based criterion. Critical dislocation spacings and thicknesses of capped interlayers are quantified in nanoscale MEE heterostructures with different stacking sequences, providing possible guides for stable growth of layered nanostructures. Finally, the complex distribution of field solutions in a six-layered MEE system with a prescribed externally mechanical load shows the crucial role played by the topological defects in localizing the coupled electric and magnetic components at the semicoherent interfaces, which indicates the important interplay between the internal and external actions. These resulting features open up novel opportunities for designing atomic-scale heterostructures with unprecedented MEE properties by engineering the interfacial dislocation patterns when the multilayered structures are free or under external action.

Study on a straight dislocation in an icosahedral quasicrystal with piezoelectric effects

An electro-elastic analysis is performed on an icosahedral quasicrystal with piezoelectric effects containing a straight dislocation. The closed-form expressions for the elastic and electric fields are obtained using the extended Stroh formalism. The effects of piezoelectric constant on the phonon displacement, phason displacement, and electric potential are discussed in detail.

Effect of Maxwell stress on a moving crack with polarization saturation region in ferroelectric solid

The focus of this work is on a generalized two-dimensional problem of a crack moving in a piezoelectric solid subjected to uniform electrical load at infinity. The novel point includes that the electric field inside the crack is taken into account when polarization saturation region exists. Based on the extended Stroh formalism and complex function method, explicit expressions of both the stress fields in the solid and electric fields inside the crack are derived by using semi-permeable crack model, respectively. Effect of Maxwell stress along the crack surface is investigated and the results are illustrated graphically. It is shown that the moving speed of the crack cannot exceed the lowest bulk wave speed. It is also found that the medium properties inside the crack and surrounding the ferroelectric solid at infinity directly affect the Maxwell stress, and as a result the Maxwell stresses are remarkable and cannot be ignored under different electric load.

Fundamental elastic field in an infinite plane of two-dimensional piezoelectric quasicrystal subjected to multi-physics loads

• Green's functions of 2D decagonal piezoelectric QCs in an infinite plane are obtained. • The effects of the phason field on the generalized displacement and stress are analyzed. • The piezoelectric effect of 2D decagonal piezoelectric QCs is shown. • Mechanical change regulations under different material constants are discussed. By utilizing the extended Stroh formalism, the Green's function of infinite plane is obtained for the problem of two-dimensional decagonal quasicrystals with the piezoelectric effect subjected to multi-physics loads. By numerical computations, the piezoelectric effect of the two-dimensional decagonal quasicrystals is revealed; the changes of the stress and displacement fields with multi-physics loads are discussed. The variation laws of material constants in stress and displacement fields are investigated. The results show that the effect of the phason field on the generalized displacement is larger than that on the generalized stress; and the effects of material parameters are different in diverse field.

Interfacial cracks in piezoelectric bimaterials: an approach based on weight functions and boundary integral equations

The focus of this paper is on the analysis of a semi-infinite crack lying along a perfect interface in a piezoelectric bimaterial with arbitrary loading on the crack faces. Making use of the extended Stroh formalism for piezoelectric materials combined with Riemann–Hilbert formulation, general expressions are obtained for both symmetric and skew-symmetric weight functions associate with plane crack problems at the interface between dissimilar anisotropic piezoelectric media. The effect of the coupled electrical fields is incorporated in the derived original expressions for the weight function matrices. These matrices are used together with Betti’s reciprocity identity in order to obtain singular integral equations relating the extended displacement and traction fields to the loading acting on the crack faces. In order to study the variation of the piezoelectric effect, two different poling directions are considered. Examples are shown for both poling directions with a number of mechanical and electrical loadings applied to the crack faces.