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A gradient theory is applied to the mechanical constitutive equations for piezoelectric semiconductor nanostructures. This is achieved by considering the strain gradients in the constitutive equation with high-order stresses and electric displacements in advanced continuum model. The C1 continuous interpolations of displacements or a mixed formulation is required in the finite element method (FEM) due to the presence of the second-order derivative on the elastic displacements. A mixed FEM is then developed from the principle of virtual work. Numerical examples clearly show the significant effect of flexoelectricity on the induced electric potential and electric current in the piezoelectric semiconductor nanostructures.

Thermal transport cannot be well described by classical Fourier’s law in nanosized structures. A novel gradient theory is developed in such structures adopting the size effect of heat conduction. This is achieved by considering the second derivatives of temperature in the constitutive equation for the high-order heat flux in an advanced continuum model. The variational principle is applied to derive the governing equations. The general two-dimensional boundary-value problem for heat conduction is analyzed by the finite element method (FEM). A mixed FEM with two independent C0 continuous interpolations for temperature and temperature gradients is developed. The constraints between temperature and its gradients are performed by the collocation approach. Parametric studies are given to evaluate the influence of the internal size on the temperature distribution.

A simplified two length-scale model is applied in direct flexoelectricity for two-dimensional problems. Numerical experiments are performed to obtain the two unknown flexoelectric parameters that a...

Size effects are important in nano-energy applications. In this paper, a novel gradient theory is introduced to describe nonlocal heat transport in nano-scale structures. This is achieved by considering the second derivatives of temperature in the constitutive equation for the high-order heat flux in an advanced continuum model. The variational principle is then applied to derive the governing equations for the coupled phonons and electrons in thermoelectric materials. The general two-dimensional boundary-value problem in the thermoelectric solid is then analyzed by the finite element method (FEM). The mixed FEM with the C0 continuous interpolation for temperature and temperature gradients is developed. A collocation approach for constraints between temperature and its gradients is applied. Numerical examples are presented to study the influence of the internal size on the temperature and electric potential, demonstrating clearly significant enhancement of thermoelectric properties via the internal material size.

In this article, coupled photo-thermoelasticity analysis is carried out using an analytical method in a semiconducting micro/nano beam resonator, considering Green – Naghdi theory (with energy dissipation) and small scale effects. The governing equations for temperature and displacement fields are derived using Eringen nonlocal theory combined with Rayleigh beam theory. One end of the assumed semiconducting MEMS/NEMS is excited by three types of suddenly increasing carrier density and temperature as the plasma and thermal shock loading. The transient behaviours of carrier density field are studied and the effects of disturbances in plasma field on other fields including temperature and deflection are obtained using the proposed analytical solution. The presented analytical solution is based on Laplace transform. To find the dynamic and transient behaviours of fields’ variables in time domain, an inversion Laplace technique is utilized, which is called Talbot method. The effects of small scale parameter and dimensions of the semiconducting micro/nano beam on the dynamic behaviours of fields’ variables are discussed in detail. The axial wave propagation and the distribution of fields’ variables along axial direction are studied at various times.

The non-classical linear governing equations of strain gradient piezoelectricity with micro-inertia effect are used to investigate Love wave propagation in a layered piezoelectric structure. The influence of flexoelectricity and micro-inertia effect on the phase wave velocity in a thin homogeneous flexoelectric layer deposited on a piezoelectric substrate is investigated. The dispersion relation for Love waves is obtained. The phase velocity is numerically calculated and graphically illustrated for the electric open-circuit and short-circuit conditions and for distinct material properties of the layer and substrate. The influence of direct flexoelectricity, micro-inertia effect, as well as the layer thickness on Love wave propagation is studied individually. It is found that flexoelectricity increases the Love-wave phase velocity, while the micro-inertia effect reduces its value. These effects become more significant for Love waves with shorter wavelengths and small guiding layer thicknesses.

The finite element method (FEM) is developed to analyse the size effect (flexoeletricity) for 2-D crack problems in thermo-piezoelectricity. Flexoelectricity is observed in micro/nanoelectronic structures, where large strain gradients destroy the symmetric structure of atoms in crystals and thereby causing polarization, even in dielectric materials. In contrast to using classical Fourier heat conduction theory, a finite speed of the thermal wave is considered in the higher order transport equation. The variational principle is applied to derive the FEM equations and C1-continuous elements are employed in the implementation of the FEM. An example is presented to demonstrate the effect of the characteristic time parameter on the crack opening displacement and temperature distribution.

Modeling the elastic behavior of solids in energy conversion and storage devices such as fuel cells and lithium-ion batteries is usually difficult because of the nonlinear characteristics and the coupled chemo-mechanical behavior of these solids. In this work, a perturbation finite element (FE) formulation is developed to analyze chemo-elastic boundary value problems (BVPs) under chemical equilibrium. The perturbation method is applied to the FE equations because of the nonlinearity in the chemical potential expression as a function of solute concentration. The compositional expansion coefficient is used as the perturbation parameter. After the perturbation expansion, a system of partial differential equations for the displacement and dimensionless solute concentration functions is obtained and solved in consecutive steps. The presence of a numerical solution enables modeling 3D chemo-elastic solids, such as battery electrodes or ionic gels, of any geometric shape with defects of different shapes. The proposed method is tested in several numerical examples such as plates with circular or elliptical holes, and cracks. The numerical examples showed how the shape of the defect can change the distribution of solute concentration around the defect. Cracks in chemo-elastic solids create sharp peaks in solute concentration around crack tips, and the intensity of these peaks increases as the far field solute concentration decreases.

Given that the thermal stresses mainly at the interface between two different materials are the significant factors of failure of the laminated composite structures, there is an enhanced request to replace laminated composite structures by structures made of micro-composite materials exhibiting the properties of continuously nonhomogeneous continua. In addition to elimination of interface discontinuities, the functional gradation (FG) of material coefficients brings new phenomena in bending of FGM plates as compared with homogeneous ones. There are known coupling effects between the in-plane deformations and bending modes even in plates subjected to stationary mechanical loadings. Another coupling can arise between thermal and mechanical fields in thermoelasticity. In the classical thermoelasticty, the heat conduction is described by the parabolic PDE when the temperature change propagates with infinite velocity. Much more realistic is the non-classical treatment with wave propagation of heat. In this paper, the unified formulation for bending of FGM plates under transient thermal loads is developed within the generalized thermoelasticity with taking into account the assumptions of the Lord-Shulman theory of thermoelasticty and the assumptions of the Kirchhoff-Love theory as well as the 1st and 3rd order shear deformation plate bending theories. Moreover, we can study various coupling effects by changing the parameters of functional gradations of particular material coefficients in numerical simulations. For numerical solution, the strong formulation is developed with meshless approximation of spatial variations of field variables. The time integration is carried out by the Wilson time stepping technique.

In this paper we investigate the vibration response of thin elastic FGM plates with combination of transversal and/or in-plane gradation of various material parameters subjected to transient tension loading. The equations of motion and initial-boundary conditions for transient problems are derived within Kirchhoff-Love plate theory. For numerical modelling of plates with dynamic multi-gradation coupling effects, it is developed the strong formulation with using the meshless approximation of field variables by the Moving Least Square (MLS) approximation scheme. Several numerical examples are presented for illustration of the multi-gradation coupling effects in vibration response of elastic FGM plates.

In this paper, we present the development of completely 2D formulation for bending of functionally graded plates subjected to stationary thermal loading. Consistently with the assumptions made in plate bending theories, the temperature field in 3D domain is expanded into low order power-law series with respect to transversal coordinate and the original 3D formulation (governing equation and boundary conditions) is recast according to physical principles into 2D formulation. A unified formulation is developed using the assumptions of three plate bending theories such the Kirchhoff-Love theory for bending of thin plates (KLT) and the shear deformation plate theories of the 1st and 3rd order (FSDPT, TSDPT) valid for thick plates too. The derivation is performed with assuming the power-law gradation of material coefficients (such as the Young modulus, linear thermal expansion coefficient, and heat conduction coefficient) across the plate thickness. For variation of material coefficients and plate thickness within the mid-plane, arbitrary continuous function is allowed. In numerical simulations, the derived formulation is implemented numerically by using the strong formulation and mesh-free Moving Least Square-approximation. The original PDE with high order derivatives are decomposed into a system of 2nd order PDE.

The size effect observed in nano-sized structures is considered in the proposed advanced continuum model for heat transfer. It is important for structures, where characteristic microstructural length is comparable with the phonon mean free-path. This feature can be captured by higher-grade continuum models and/or nonlocal modelling of constitutive laws in continuum theories. Both these approaches can be shown equivalent under certain assumptions. The governing equations are given by the PDE with higher-order derivatives than in classical continuum models, with the response of physically conjugated field being proportional to the gradients of primary fields. The variational principle is applied to derive the finite-element formulation for the solution of a thermoelectric 2-d boundary-value problem. Due to higherorder derivatives in gradient theory, it is necessary to use C1-continuous elements to guarantee the continuity of the derivatives at the element interfaces. Since it is not an easy task, a mixed FEM formulation is developed here.

In this paper we derived a formulation for bending of thermo-piezoelectric FGM plates with taking into account the assumptions of the Kirchhoff–Love plate bending theory within the classical theory of thermoelasticity. Uncoupled thermoelasticity is considered; thus, the heat conduction problem is analyzed independently of the mechanical fields. The functional gradation of material coefficients complies the rule of mixture. For numerical solution of complex boundary value problems a strong form meshless method is developed with using the Moving Least Square approximation for spatial variations of field variables. The inaccuracy of solution due to the high order derivatives of field variables is overcome by decomposing the original governing partial differential equations (PDE) into the system of PDEs with lower order derivatives. Several numerical experiments are presented for illustration of coupling effects in bending of FGPM plates under stationary thermal loading.

This paper presents a computational method to analyze 2D crack problems in magneto-electro-elastic solid described by the gradient theory with the direct flexoelectric effect. The finite-element me...

In this work, the finite element method (FEM) is used to characterize the effective thermo-electro-mechanical material properties of cement-based porous piezoelectric ceramic composites used in structural health monitoring (SHM) systems. The micromechanics representative volume element (RVE) approach is used with a realistic distribution of piezoelectric particles in the porous cement matrix. The effects of the piezoelectric particle volume fraction and pore volume fraction on the effective composite properties are determined using sets of different boundary conditions. Results show very good agreement with published experimental and computational results.

The influence of microstructure on macro-structural plate bending is incorporated into plate bending theories by replacing the classical elasticity theory by strain gradient elasticity. The microstructure is represented by gradients of macro-strains and one new material coefficient which is called the micro-scale length parameter. The strain gradients can be introduced either in the non-symmetrized or in the symmetrized formulation. Though the former is physically inconsistent, both these formulations are considered and compared in numerical simulations. The governing equations as well as boundary conditions are derived from the principle of variations applied to unified formulation of plate bending with assuming the assumptions of the classical thin plate bending theory (Kirchhoff-Love theory) and/or the shear deformation plate theory of the 1st and 3rd order (FSDPT and TSDPT). The derived formulation is applicable to FGM (Functionally Graded Material) plates, since the gradation of Young’s modulus is allowed in both the transversal and in-plane directions. Some numerical examples are considered in numerical simulations, in order to illustrate the influence of microstructure on the behavior of macro-structural plate subjected to static transversal loading.

In this paper, we present a systematic development of unified formulation for bending of elastic plates with micro-voids. The homogenization concept is applied with replacing real medium with micro-voids by a homogeneous medium with effective material coefficients. The effective material coefficients are calculated in advance by solving properly selected micro-structural 3D boundary value problems in representative volume elements with specified shape and volume content of micro-voids. The unified formulation for plate bending problems is developed from 3D elasticity with taking into account the assumptions of three plate bending theories such as Kirchhoff-Love theory for bending of thin plates, 1st and 3rd order shear deformation theories for thick plates. Switching among these theories is allowed by proper selection of two key factors introduced in the unified formulation. The influence of the porosity on bending of porous plates subjected to transversal loading is studied via the numerical simulations.

The finite element method (FEM) is developed to analyse 2-D crack problems where the electric field and displacement gradients exhibit a size effect penomenon. This phenomenon in micro/nanoelectronic structures is described by the strain-and electric field-gradients in constitutive equations. The governing equations are derived using variational principles with the corresponding boundary conditions. The FEM formulation with C1-continuous elements is subsequently developed and implemented. An example is presented and discussed to demonstrate the effects of the strain-and electric intensity-gradients on the electro-mechanical behavior of cracked solids.

Thermal stresses, especially at the interface between two different materials, often play an important role in the failure of laminated composite structures. Thus, it is a strong motivation to replace laminated plate structures by FGM ones if possible. The functional gradation of material coefficients, however, yields new coupling effects between the in-plane deformation and bending modes. Therefore the study of behaviour of FGM plates under thermal loadings has become important. In this paper, the bending of thin and/or thick FGM plates under thermal load is considered within the classical theory of thermoelasticity. The variable material properties of plate (such as the Young's modulus, thermal expansion coefficient, etc.) are allowed to be continuous functions of the position. The governing equations which are given by the 4th order partial differential equations (PDE) are decomposed into the 2nd order PDEs in order to overcome the inaccuracy of approximation of high order derivatives of field variables. The strong form meshless formulations for solution of thin plate bending problem is developed in combination with Moving Least Squares (MLS) approximation scheme. The attention is paid to the study of the influence of various parameters of gradations of material coefficients on bending of plates.

The production methodology of alloyed quantum-dot (QD) structures introduced a new design degree of freedom for QD arrays which is the grading of the material composition in the QD growth direction. This enables QDs of same size to generate different colors when exposed to blue light based on the grading of each QD. The grading of the material composition affects the material properties as well as the lattice mismatch strain between the QDs and the host matrix. Previous studies modeled graded QDs by just considering graded lattice mismatch strain while the material properties were kept uniform. Because these previous studies were seeking analytical solutions, including a graded material property model would have complicated the solutions. In this paper, a fully-coupled thermo-electro-mechanical finite element model of a cylindrical functionally graded QD (FGQD) in a host piezoelectric matrix is developed with both graded material properties and graded lattice mismatch strain. Different cases are considered corresponding to separately increasing and decreasing the strength of the lattice mismatch strain and the material properties in the QD thickness direction. The grading function is expressed using the power law that enables fractional exponents. The results show the effect of grading on the electromechanical quantities and demonstrate the flexibility that grading can add to the design of QD arrays. This work contributes to the development of quantum dots with "grading-dependent color" rather than the traditional "size-dependent color." The model can be easily extended to other cases such as different shapes of QDs, addition of wetting layer, and any applied thermo-electro-mechanical loads.

The flexoelectric effect on elastic waves is investigated in nano-sized cracked structures. The strain gradients are considered in the constitutive equations of a piezoelectric solid for electric displacements and the higher-order stress tensor. The governing equations with the corresponding boundary conditions are derived from the variational principle. The finite element method (FEM) is developed from the principle of virtual work. It is equivalent to the weak-form of derived governing equations in gradient elasticity. The computational method can be applied to analyze general 2D boundary value problems in size-dependent piezoelectric elastic solids with cracks under a dynamic load. The FEM formulation is implemented for strain-gradient piezoelectricity under a dynamic load.

A fully coupled thermo-electro-mechanical models of cylindrical and truncated conical GaN/AlN Functionally Graded Quantum Dot (FGQD) systems with and without WL are analyzed in this study to determine the effect of lattice mismatch strain grading on the electromechanical behavior of the FGQD system. This has a technological and fundamental importance because the production methodology adopted for manufacturing QDs enables the composition of the QD material to be graded in the growth direction, so the material properties as well as the induced mismatch strain between the QD and the carrier matrix are accordingly graded. The power law is used to describe the grading function. Based on the obtained results, grading of material properties and lattice mismatch strain have significant effect on the distribution of the electromechanical quantities inside the QD and can be used as another tuning parameter in the design of QD systems.

This paper presents the dynamic crack analysis in two-dimensional (2D) and linear piezoelectric composites. Stationary cracks in infinite and finite piezoelectric solids subjected to different time-harmonic loadings are investigated. A symmetric Galerkin boundary element method (SGBEM) is developed for this purpose. The frequency-domain fundamental solutions for linear piezoelectric solids are utilized. The double boundary integrals are computed analytically for straight elements to enhance the efficiency of the present SGBEM. Non-linear electrical crack-face boundary conditions are applied to take the limited electrical permeability of the crack into account. Numerical examples are presented and discussed to show the accuracy of the presented frequency-domain SGBEM and the influences of the coupled fields on the dynamic field intensity factors.

The local radial basis function collocation method (LRBFCM) is proposed for plate bending analysis in orthorhombic quasicrystals (QCs) under static and transient dynamic loads. Three common types of the plate bending problems are considered: (1) QC plates resting on Winkler foundation (2) QC plates with variable thickness and (3) QC plates under a transient dynamic load. According to the Reissner–Mindlin plate bending theory, there is allowed to simulate the behavior of the two excitations in QC plates, phonon and phason, by 2D strong formulations for the system of governing equations. The governing equations, which describe the phason displacements, are based on Agiasofitou and Lazar elastodynamic model. Numerical results demonstrate the effect of the elastic foundation, as well as plate thickness on the phonon and phason characteristics in this paper. For the transient dynamic analysis, the influence of the phason friction coefficients on the responses of QC plate to transient dynamic loads is also studied.

General 2D boundary value problems of piezoelectric nanosized structures with cracks under a thermal load are analyzed by the finite element method (FEM). The size-effect phenomenon observed in nanosized structures is described by the strain-gradient effect. The strain gradients are considered in the constitutive equations for electric displacement and the high-order stress tensor. For this model, the governing equations are derived with the corresponding boundary conditions using the variational principle. Uncoupled thermoelasticity is considered, thus, the heat conduction problem is analyzed independently of the mechanical fields in the first step. A numerical example is presented and discussed to demonstrate the effects of the strain-gradient.

General 2D boundary value problems of piezoelectric nano-sized structures with cracks under a thermal load are analyzed by the finite element method (FEM). The size-effect phenomenon observed in nano-sized structures is described by the strain-gradient effect. The strain gradients are considered in the constitutive equations for electric displacement and the high-order stress tensor. For this model, the governing equations and the corresponding boundary conditions are derived using the variational principle. Uncoupled thermoelasticity is considered; thus, the heat conduction problem is analyzed independently of the mechanical fields in the first step. The veracity of the derived formulations and their implementation into the finite element scheme is demonstrated by some numerical examples.

Time-harmonic crack analysis in two-dimensional piezoelectric functionally graded materials (FGMs) is presented in this paper. A frequency-domain boundary element method (BEM) is developed for this purpose. Since fundamental solutions for piezoelectric FGMs are not available, a boundary-domain integral formulation is derived. This requires only the frequency-domain fundamental solutions for homogeneous piezoelectric materials. The radial integration method is adopted to compute the resulting domain integrals. The collocation method is used for the spatial discretization of the frequency-domain boundary integral equations. Adjacent the crack-tips square-root elements are implemented to capture the local square-root-behavior of the generalized crack-opening-displacements properly. Special regularization techniques based on a suitable change of variables are used to deal with the singular boundary integrals. Numerical examples will be presented and discussed to show the influences of the material gradation and the dynamic loading on the intensity factors.

New coupling effects are revealed and described in plates with multi-gradation of material coefficients. Three variants of plate bending theory (such as the Kirchhoff-Love theory, the 1st and 3rd order shear deformation plate theory) are considered within unified formulation. It is known that transversal gradation of Young’s modulus gives rise to coupling between the bending and in-plane deformation modes in plates under transversal loading. In this paper it is shown that combination of transversal gradation of Young’s modulus with in-plane gradation of Young’s modulus and/or variation of plate thickness leads to deflections of such plates subject to in-plane loading. The governing equations and boundary conditions for static problems are derived in the unified formulation using the variational principle. For numerical simulations of multi-gradation coupling effects, it is developed the strong formulation with using the meshless approximation of field variables by the Moving Least Square (MLS) approximation. Several numerical examples are presented for illustration of the multi-gradation coupling effects in bending of elastic FGM (Functionally Graded Material) plates. The role of the boundary conditions as well as the thickness and shear deformations is studied via numerical simulations and comparisons of the plate responses obtained in three plate bending theories.

The finite element method (FEM) is developed to analyse general 2D boundary value problems in size-dependent piezoelectric, elastic solids with cracks. The size-effect phenomenon in micro/nano electronic structures is described by the strain-gradient effect. The electric field-strain gradient coupling is considered in the constitutive equations of the material and the governing equations are derived with the corresponding boundary conditions using the variational principle. The FEM formulation is subsequently developed and implemented for strain-gradient piezoelectricity. The path-independent J -integral is also derived for fracture mechanics analysis of such solids. Numerical examples are presented to demonstrate the veracity of the formulations.

The finite element method is developed for 3-D general boundary value problems for a piezoelectric semiconductor with functionally graded material properties. The electron density and electric current are additionally considered in the constitutive equations for piezoelectric semiconductors. A general variation of material properties with Cartesian coordinates is treated in the numerical analyses. The influence of material parameter gradation and initial electron density is investigated in the static case. Numerical results are presented for a 3-D beam under a static and impact mechanical load. Transversely isotropic material properties are considered in this study.

The behaviour of small size dielectric elastic beams is described within higher-grade theory with including electric polarization. The coupling between strain gradients and polarization is incorporated into the constitutive laws in the form of flexoelectricity, while piezoelectricity is involve in the classical form. Both the governing equations and boundary conditions are derived using variational formulation for electro-elastic continuous media and deformation assumptions employed in three various beam bending theories such as the classical theory (Euler-Bernoulli theory), the 1st order shear deformation theory (Timoshenko theory) and 3rd order shear deformation theory. The unified formulation allows switching between theories with various bending assumptions by a proper selection of two key factors.

The field of protein residue network (PRN) research has brought several useful methods and techniques for structural analysis of proteins and protein complexes. Many of these are ripe and ready to be used by the proteomics community outside of the PRN specialists. In this paper we present software which collects an ensemble of (network) methods tailored towards the analysis of protein-protein interactions (PPI) and/or interactions of proteins with ligands of other type, e.g. nucleic acids, oligosaccharides etc. In parallel, we propose the use of the network differential analysis as a method to identify residues mediating key interactions between proteins. We use a model system, to show that in combination with other, already published methods, also included in pyProGA, it can be used to make such predictions. Such extended repertoire of methods allows to cross-check predictions with other methods as well, as we show here. In addition, the possibility to construct PRN models from various kinds of input is so far a unique asset of our code. One can use structural data as defined in PDB files and/or from data on residue pair interaction energies, either from force-field parameters or fragment molecular orbital (FMO) calculations. pyProGA is a free open-source software available from https://gitlab.com/Vlado_S/pyproga.

The recent progress in the development and research of functionally graded materials (FGMs) has prompted the development of numerical methods in engineering analysis for continuously non-homogeneous media. This paper presents the hybrid Meshless Displacement Discontinuity Method (MDDM) for a cracked structure with functionally graded materials. The fundamental solutions of displacement discontinuity for an isotropic homogenous body form a part of the general solutions for non-homogenous materials, in order to create the gap on the crack surface. The governing equations are satisfied by using the Meshless Local Petrov-Galerkin method (MLPG) at the scattered nodes in the domain. The stress intensity factors (SIFs) are evaluated by either crack opening displacement (COD) method or equivalent stress technique. The crack growth paths with different FGMs gradient parameters are investigated and compared with the solutions given by the finite element method (FEM). The efficiency and accuracy of hybrid MDDM are demonstrated by four examples and comparisons have been e with other analytical and numerical methods.

Health monitoring of structures can be improved if a more universal two-way electromechanical coupling, the flexoelectric effect, is utilized. It is a coupling between the electric polarization and the strain gradients. In the direct flexoelectricity, the electric polarization is induced by strain gradients, which yield also a finite higher-order stress tensor in the considered phenomenological theory. Because of the size-effect in higher-grade theories of continua, the polarization in piezoelectric solids under a non-uniform strains in nano-sized structures is significantly influenced by flexoelectricity. This effect is substantially enhanced near the crack defects, in regions with large strain gradients. The mixed finite element method (FEM) is developed from the variational formulation of flexoelectric boundary value problems. The C0 continuous approximation is applied independently for both the displacements and displacement gradients. The kinematic constraints between them are satisfied by collocation at some internal points of elements. The developed computational method is applied to general 2D boundary value problems with cracks under a dynamic load. The influence of flexoelectricity on the induced electric potential and the crack opening displacement is investigated.

Although micro-dilatation theory is very suitable and effective in modeling elastic porous materials, the absence of any guidance to evaluate or characterize its porosity-related parameters in the literature limited its use and applicability. This paper is proposing a methodology to characterize two of such important parameters, namely porosity change stress parameter and void stiffness coefficient, in terms of microstructural details such as average void radius and density of voids. Two numerical experiments were used to characterize these parameters, and results were validated by a comparison with a high-resolution finite element model of the microstructure with voids explicitly considered.

In this paper, the bending of thin functionally graded plates under thermal load is considered within the classical theory of thermoelasticity. The variable material properties of plate (such as the Young's modulus, thermal expansion coefficient, etc.) are allowed to be continuous functions of the position. The governing equations which are given by the 4th order partial differential equations are decomposed into the 2nd order partial differential equations in order to overcome the inaccuracy of approximation of high order derivatives of field variables. The strong form meshless formulations for solution of thin plate bending problem is developed in combination with Moving Least Squares approximation scheme. The attention is paid to the numerical investigation of the influence of several parameters of gradations of material coefficients on the deflection of the plate.

In this paper, the effective material parameters of magnetoelectroelastic composites are computed on the base of homogenization techniques performed on the representative volume element (RVE). For this purpose, the resulting boundary value problem is formulated as boundary integral equations (BIEs). The Galerkin method is applied for spatial discretization. The required average stresses and strains are obtained directly from the primary values of the BIEs and therefore additional numerical differentiations as necessary in other methods are avoided. Numerical examples will be presented and discussed to show the efficiency of the present boundary element method (BEM) and the influences of the voids on the effective material properties.

Numerical computational techniques can be conveniently utilized for the analysis of porous structural materials with vacuous pores. The presence of voids or pores in these materials affects their material properties and overall performance therefore it is of great importance to study the structural integrity of such materials. The micro-dilatation theory firstly developed by Cowin-Nunziato (C-N) has been applied in this study to describe the porous materials with vacuous pores. This theory belongs to the general continuum theories and uses linear elastic theory with additional kinematic variable called porosity change function. Thus, the deformation and porosity are coupled fields giving the response of the body under external loads. However in spite of the number of papers concerning the numerical analysis of C-N theory, this theory still remains unresolved and physically unclear. It is the aim of this paper to provide some numerical examples in order to contribute to better understanding of the theory. The derived FEM formulation has been developed and incorporated into the commercial software COMSOL via weak form module. The numerical studies are verified with analytical solution.

Using the quarter-point crack tip element, a simple formula is derived which provides a quick and expedient means to evaluate the elastic T-stress in the finite element method (FEM) fracture mechanics analysis of functionally graded materials (FGMs). The formula is derived by comparing the variation of displacements in the quarter-point element with the corresponding asymptotic expression for the displacement field in the vicinity of a crack-tip. The asymptotic fields in the crack-tip vicinity in a FGM with continuously varying material properties are the same as in a homogeneous one. The simplicity of the use of this formula for T-stress evaluation in FEM fracture mechanics analysis is demonstrated by its application to some transient elastodynamic and thermoelastic problems.

The finite element method (FEM) is developed to analyze general two-dimensional boundary value problems in size-dependent magnetoelectroelastic solids. The size-effect phenomena in micro/nano electronic structures are described by the strain gradient effect. The electric and magnetic field-strain gradient coupling is considered in the constitutive equations of the material and the governing equations are derived with the corresponding boundary conditions by virtue of the variational principle. The FEM formulation is subsequently developed and implemented for strain-gradient magnetoelectroelasticity and a couple of numerical examples are presented to illustrate the strain gradient effect on the fields.

The size-dependent features concerning the mechanical behavior of the micro/nanoelectronic structures are well known from experiments. They are described by the strain-gradient effect in this paper since the classical elasticity theory fails to capture the size effect of the nanostructures. The electric field-strain gradient coupling is considered in the constitutive equations of the material and the governing equations are derived with the corresponding boundary conditions using the variational principle. The path independent J-integral is derived for fracture mechanics analysis of piezoelectric solids described by the gradient theory.

The finite element method is proposed to analyze coated fiber composites with piezoelectric and piezomagnetic phases. The computational homogenization technique is applied for fiber composites with magnetoelectroelastic properties to determine effective material parameters. The evolution of the magnetoelectroelastic fields at the macroscopic level is resolved through the incorporation of the microstructural response. The microstructural analyses are performed on the representative volume element, where essential physical geometrical information about the microstructural components is included. Circular cross section of fibers is considered in numerical analyses. A thin coating layer is considered on the surface of the piezoelectric fiber which is embedded in the piezomagnetic matrix. Influence of the coating layer on the effective material properties is analyzed.

The scaled boundary-finite element method (SBFEM) is employed to analyze cracks in porous piezoelectric solids. A large (magistral) crack and a short crack emanating from a single pore are analyzed. These cracks with their tips in the solid skeleton have the weakening effects on the fracture strength. The crack tip region is considered as a subdomain where both the crack tip and pores are modeled inside the piezoelectric skeleton. The remaining part of the analyzed domain is modeled with effective material properties obtained from the analysis on the representative volume element (RVE). A regular distribution of circular voids is considered in the numerical analyses. The scaled boundary-finite element method (SBFEM) is applied to solve all the boundary value problems.

The scaled boundary finite element method (SBFEM) is proposed to analyze porous magnetoelectroelastic solids. The presence of voids affects material properties and functionality of these elements in smart structures. For determination of influence of voids on material properties in a simplified case with uniform distribution of voids, it is efficient to analyze the representative volume element (RVE). The SBFEM combines the main advantages of the finite element method and the boundary element method (BEM). In this method, only the boundary is discretized with elements leading to a reduction of spatial dimension by one and sparing the human and computational efforts in mesh generation. In contrast to the boundary element method, no fundamental solution is required, which permits to analyze general boundary value problems, where the conventional BEM cannot be applied due to missing fundamental solution.

The paper deals with transient analysis of homogeneous as well as FGM (functionally graded material) thin and/or thick plates subjected to transversal dynamic loading. The Young modulus and mass density are allowed to be continuously variable through the thickness of the plate. The unified formulation is developed for plate bending problems including three various theories such as the classical Kirchhoff–Love theory for bending of thin elastic plates, the 1st and 3rd order shear deformations plate theories. Switching among these three theories is controlled by two key factors. From the derived equations of motion, the coupling between the bending and in-plane deformation modes is discovered in FGM plates with specification of the necessary conditions. For the numerical implementation, the strong formulation is proposed with meshless approximation of spatial variations of field variables. The semi-discretized equations of motion yield a system the ordinary differential equations which can be solved by standard time stepping techniques. The great attention is paid to the numerical study of coupling effects in FGM plates subjected to transversal Heaviside impact loading and/or Heaviside pulse loading. The achieved numerical results are thoroughly discussed and interpreted.