Piecewise Homogeneous Body Model Research Articles

Effect of interaction between polarization direction and inclined force on the dynamic stability of a pre-stressed piezoelectric plate

The current study deals with forced vibration analysis of a pre-stressed piezoelectric plate with finite lengths resting on a rigid foundation, exposed to an inclined time-harmonic force. The present investigation is modeled according to a new theoretical consideration in the context of the three-dimensional linearized theory of elasticity for solids under initial stress (TLTESIS). It is assumed that the plate exhibits a polarization effect in both vertical and horizontal directions to the top surface and there is an exact clamping condition at the interface between the plate and the rigid foundation, that is, no jumping in the quantities. For obtaining the governing and boundary condition equations of our problem, the virtual work principle is used according to Hamilton's principle and the piecewise homogeneous body model. The three-dimensional displacement-based finite element model is developed for solving the system of partial differential equations and analyzing the dynamic behavior of the plate. The reliability of the solution procedure is proved by convergence analysis and comparing the results to the current literature. The target of the paper for surveying the numerical results and findings is divided into two sections. The first one presents the influence of the inclination of the dynamic external force on the forced vibration characteristics of the current model. In the second one, the influence of inclination angle, thickness ratio, aspect ratio, and initial stress parameter on the frequency response of the harmonic force are investigated in detail. As a desirable result for the designer, while a decrease in the angle of the force with the vertical axis exceeds the resonant mode of the stress, increasing values of the initial stress prevents that mode.

Vibration of a Two-Layer “Metal+PZT” Plate Contacting with Viscous Fluid

The present work investigates the mechanically forced vibration of the hydro-elasto-piezoelectric system consisting of a two-layer plate “elastic+PZT”, a compressible viscous fluid, and a rigid wall. It is assumed that the PZT (piezoelectric) layer of the plate is in contact with the fluid and time-harmonic linear forces act on the free surface of the elastic-metallic layer. This study is valuable because it considers for the first time the mechanical vibration of the metal+piezoelectric bilayer plate in contact with a fluid. It is also the first time that the influence of the volumetric concentration of the constituents on the vibration of the hydro-elasto-piezoelectric system is studied. Another value of the present work is the use of the exact equations and relations of elasto-electrodynamics for elastic and piezoelectric materials to describe the motion of the plate layers within the framework of the piecewise homogeneous body model and the use of the linearized Navier-Stokes equations to describe the flow of the compressible viscous fluid. The plane-strain state in the plate and the plane flow in the fluid take place. For the solution of the corresponding boundary-value problem, the Fourier transform is used with respect to the spatial coordinate on the axis along the laying direction of the plate. The analytical expressions of the Fourier transform of all the sought values of each component of the system are determined. The origins of the searched values are determined numerically, after which numerical results on the stress on the fluid and plate interface planes are presented and discussed. These results are obtained for the case where PZT-2 is chosen as the piezoelectric material, steel and aluminum as the elastic metal materials, and Glycerin as the fluid. Analysis of these results allows conclusions to be drawn about the character of the problem parameters on the frequency response of the interfacial stress. In particular, it was found that after a certain value of the vibration frequency, the presence of the metal layer in the two-layer plate led to an increase in the absolute values of the above interfacial stress.

Dispersion of torsional surface waves in a threefold concentric compounded cylinder with imperfect interface

The present paper investigates the torsional wave propagation in a threefold concentric pre-stressed compounded cylinder with imperfect contact conditions. The three-dimensional linearized theory of elastic waves and the piecewise homogeneous body model has been employed to formulate the problem. The mathematical modeling has been carried out in two independent cases. In the first case, a solid cylinder encased in a hollow cylinder embedded in an infinite elastic medium has been considered. Whereas the second case comprises a hollow cylinder of finite thickness in place of a solid cylinder. By means of Murnaghan potential, the mechanical characteristics of the three materials have been used. Further, the dispersion relations for both the cases have been obtained in terms of the Bessel and modified Bessel functions. In order to validate the present findings, two particular cases have been derived that matches with the previous works. The first case is obtained by removing the outermost cylinder, while the second case has been derived by removing the imperfection in addition to that. To summarize the computations, a complete numerical simulation has been carried out, and graphical illustrations have been shown to aid the mathematical analyses.

The stability loss of a locally curved double-walled carbon nanotube in visco-elastic matrix

In the current study, stability loss investigation is carried out for a viscoelastic medium containing a double-walled carbon nanotube with local curvature. This research has been carried out using the three dimensional linearized stability theory (TDLTS) within the framework of the piecewise-homogeneous body model. In this case, it is supposed that the carbon nanotube (CNT) has an insignificant initial local defect, and the increase of this defect with the flow of time is examined. Moreover, it is accepted as the criterion for determining the loss of stability that local curvature of the CNT starts to grow indefinitely. The values of the critical time and critical force are obtained from this criterion. In addition, van der Waals forces between the inner surface of the outer tube of the CNT and the outer surface of the inner tube of the CNT were considered. The fractional-exponential Rabotnov operator defines the composite material’s viscoelasticity properties. It is expected that the results obtained from this paper may be a useful guide in applications related to exhibiting the mechanical behavior of the composite material under consideration. Therefore, the effects of viscoelasticity towards composite materials containing a locally curved DWCNT and the limits to be considered in the production of this material are obtained.

Frequency Response of an Initially Stressed Slab Made from Three Compressible Materials

In this study, the frequency response of a pre-stressed slab, which stands on a rigid foundation, subject to a timely harmonic loading was considered. The investigation is implemented according to the piecewise homogeneous body model utilizing the three-dimensional linearized theory of elastic waves in initially stressed bodies (TLTEWISB). The considered body was designed joining to three discrete slabs side-by-side. It was assumed that there exists a rigidly clamping state at all interface planes on the system. A mathematical model of the problem is constructed and the system related to equations of motion is numerically solved using the finite element method (FEM). Particularly, the effect the ratio of the layer length has on the frequency response of the slab was presented.

Stress distribution in an elastic body with a locally curved double-walled carbon nanotube

Nanotubes are the most important structural elements of nanomaterials used in nanotechnological applications and are also used in the production of nanocomposite materials. In the reinforcement (in the fibers) of the structure of composite materials, the initial curvature occurring due to structural reasons or technological processes causes the appearance of self-balancing stresses. The magnitudes of these stresses can lead to exceeding of the safety limits of the material. Therefore, theoretical investigation of the mechanical behaviors of the material, under tension and compression in the direction of the strengthening (fibers), is important for engineering. In this study, composite materials containing the double-walled nanotube are investigated in the scope of the piecewise homogeneous body model with the use of the geometric nonlinear exact equations of the three-dimensional theory of elasticity. The investigation involves stress analysis on the outer surface of the outer layer of the carbon nanotube (CNT) and on the matrix intersection, and non-ideal contact conditions between CNT and the matrix are used. Van der Waals forces which exist between the CNT walls are considered. The results include the impact of the geometric linearity and geometric nonlinearity on the parameters of the problem at the interface of the study. Numerical results on the influence of the problem parameters on the stress distribution are presented and discussed. The effect of van der Waals force on the stresses is shown by comparison with the case where the interlayer gap is disregarded and full contact between the CNT is assumed. It is seen that the maximum absolute values of the stress increase significantly when there is no van der Waals force. In addition, it has been observed that the values of the stress change monotonically with the increase of the thickness of the nanotube. Moreover, it is established that the increase of elasticity constants significantly affects the stresses.

Normal stress distribution in infinite elastic matrix with a locally curved triple-walled carbon nanotube

Nanocomposite materials are produced by using of nanotubes, the most significant structural elements of nanomaterials used in nanotechnologic applications. In the reinforcement (in the fibers) of the structure of composite materials, the appearance of the self-balancing stresses results from the initial curvature, caused by either structural reasons or technological processes. Because of exceeding safety limits of material caused by high magnitude self-balancing stresses, investigating the mechanical behaviors of the material theoretically, both under tensile and compression in the direction of strengthening (fiber) is essential for the engineering. Unlike the literature, in this study, composite materials containing triple-walled nanotube are investigated in the scope of the piecewise homogeneous body model by using of geometric non-linear exact equations of the 3-D theory of elasticity. The normal stress analysis, on the outermost surface of the carbon nanotube and the matrix intersection, is investigated under various external effects. Nanotube is first formed as having a small local curvature. Van der Waals forces existing between the carbon nanotube walls are taken into consideration.

Stress Distribution in Composites with Co-Phase Periodically Curved Two Neighboring Hollow Fibers

In this paper, stress distribution is examined in the case where infinite length co-phase periodically curved two neighboring hollow fibers are contained by an infinite elastic body. The midline of the fibers is assumed to be in the same plane. Using the three-dimensional geometric linear exact equations of the elasticity theory, research is carried out by use of the piecewise homogeneous body model. Moreover, the body is assumed to be loaded at infinity by uniformly distributed normal forces along the hollow fibers. On the inter-medium between the hollow fibers and matrix surfaces, complete cohesion conditions are satisfied. The boundary form perturbation method is used to solve the boundary value problem. In this investigation, numerical results are obtained by considering the zeroth and first approximations to calculate the self-equilibrium shear stresses and normal stress at the contact surfaces between the hollow fibers and matrix. Numerous numerical results have been obtained and interpreted about the effects of the interactions between the hollow fibers on this distribution.

The Influence of Imperfect Contact Conditions between Layers of a Hollow Sandwich Sphere on its Natural Frequencies

The influence of imperfect contact conditions between layers of a hollow sandwich sphere on its natural frequencies is studied. It is assumed that the materials of face layers are the same, but different from the core layer material. Each layer is made of a homogeneous isotropic material and has a constant thickness. Investigations are performed using a piecewise homogeneous body model with the use of exact three-dimensional equations and relations of elastodynamics. The solution of the corresponding equations of motion of the hollow sandwich sphere is presented through the Helmholtz potentials in spherical coordinates. Numerical results for the influence of imperfect contact conditions between layers on the natural frequencies of the sphere are presented and discussed. It is established that an imperfect contact between sphere layers can decrease its natural frequencies significantly.

3D FEM Analysis of Buckling Delamination of a Piezoelectric Sandwich Rectangular Plate with Interface Edge Cracks

The problem of local buckling delamination around interface edge cracks in a sandwich PZT/Metal/PZT rectangular thick plate is studied within the scope of the 3D linearized buckling instability theory. Crack faces have an insignificant initial imperfection, and the plate is compressed by uniformly distributed normal forces acting on two opposite ends of the plate. The evolution of the initial imperfection is investigated, and the critical buckling forces for the plate are found. A mathematical modeling of the problem considered is performed using the 3D exact geometrically nonlinear electroelasticity theory within the framework of a piecewise homogeneous body model. The corresponding boundary-value problems are solved numerically by employing a 3D finiteelement method. Numerical results for the critical forces and the effect problem parameters on these forces are presented and discussed.

Dynamic response of a bi-axially pre-stressed bi-layered plate resting on a rigid foundation under a harmonic force

This study aims to investigate the forced vibrations caused by a time-harmonic force from a pre-stressed bi-layered plate resting on a rigid foundation under the action of a time-harmonic pointwise loading. Our investigation was conducted according to a piecewise homogeneous body model utilizing the three-dimensional linearized theory of elastic waves in initially stressed bodies. Throughout this study, we assumed that there is complete contact between the plate and the rigid foundation. The purpose of this study is threefold: the development of a mathematical model to investigate the dynamic response of the pre-stressed bi-layered plate, the analysis of the frequency response of the plate under consideration, and finally, demonstrating the relationship between the initial stress and the dimensionless frequency of the plate. We solved the mathematical model by employing the finite element method. We present our numerical results on the dynamic behavior of the plate. In particular, we have shown that an increase in the values of the aspect ratio of a plate under fixed thickness leads to a decrease in the normal stress resonance values.

Stress distribution caused by co-phase locally spatially curved layers in an infinite elastic body under bi-axial compression

In the present paper, the stress distribution is studied in an infinite elastic body, reinforced by an arbitrary number of non-intersecting co-phase locally spatially curved filler layers under bi-axial compression is studied. It is assumed that this system is loaded at infinity with uniformly distributed normal forces with intensity p1(p3) acting in the direction which is parallel to the layers? location planes. It is required to determine the self-equilibrated stresses within, caused by the spa?tially local curving of the layers. The corresponding boundary and contact value problem is formulated within the scope of geometrically non-linear exact 3-D equations of the theory of elasticity by utilizing of the piece-wise homogeneous body model. The solution the formulated problem is represented with the series form of the small parameter which characterizes the degree of the aforementioned local curving. The boundary-value problems for the zeroth and the first approximations of these series are determined with the use of the exponential double Fourier transform. The original of the sought values is determined numerically. Consequently, in the present investigation, the effect of the local curving on the considered interface stress distribution is taken into account within the framework of the geometrical non-linear statement. The numerical results related to the considered interface stress distribution and to the influence of the problem parameters on this distribution are given and discussed.

On internal stability loss of a row unidirected periodically located fibers in the visco-elastic matrix

In the present paper, the microbuckling or internal stability loss in the viscoelastic composites containing unidirected fibers under compression along the fibers is studied by use of piecewise homogeneous body model. In this model, it is used the 3-D geometrically non-linear exact equations of viscoelasticity theory. The composite material was considered as an infinite viscoelastic body with a row unidirected periodically located elastic fibers that have an initial infinitesimal imperfection. When the initial imperfection starts to increase and becomes indefinitely, this is taken as a stability loss criterion and co-phase microbuckling mode out of plane are taken into account. The numerical results about the influence of the interaction between the fibers on the values of the critical time are obtained and presented.

Forced Vibrations of Pre-Stressed Sandwich Plate-Strip with Elastic Layers and Piezoelectric Core*

The dynamical stress field in a sandwich plate-strip is mathematically modeled using the piecewise-homogeneous body model and the three-dimensional linearized theory of electro-elastic waves in initially stressed bodies. The plate consists of a piezoelectric core perfectly bonded to elastic layers with initial stress, rests on a rigid foundation, and is subject to harmonic force. It is assumed that the piezoelectric material is poled perpendicularly to the free surface of the body. The governing system of partial differential equations of motion is solved by the finite-element method. The numerical results are presented, illustrating the effect of certain relationships of the problem of the propagation of stresses and electric displacements at the interfaces between the elastic layers and the piezoelectric core and between the plate-strip and the rigid foundation. The effect of the initial stress parameter and a change in the thickness of the piezoelectric core on the frequency response of the plate-strip is studied.

Effect of imperfect bonding on the dynamic response of a pre-stressed sandwich plate-strip with elastic layers and a piezoelectric core

In this paper, we consider the dynamic response of a pre-stressed sandwich plate-strip with a piezoelectric core and elastic layers under the action of a time-harmonic force resting on a rigid foundation. The investigation is carried out within the framework of the piecewise homogeneous body model by utilizing the exact equations of motion and relations of the linear theory of electro-elasticity. It is assumed that there is a shear-spring-type imperfect contact between the layers, but a complete contact between the plate-strip and the rigid foundation. A mathematical model of the problem is constructed, and the governing equations of motion are solved by employing the finite element method (FEM). Numerical results illustrating the influence of a change in the value of the shear-spring parameter on the dynamic response of the plate-strip are then presented.

Buckling delamination of a PZT/Metal/PZT sandwich rectangular thick plate containing interface inner band cracks

This work studies the 3D buckling delamination problem around the interface band cracks which are in the PZT/Metal/PZT sandwich rectangular thick plate within the scope of the piecewise homogeneous body model by utilizing the so-called 3D linearized theory of stability loss for piezoelectric materials. It is assumed that the plate is simply supported with respect to mechanical quantities and short-circuit conditions with respect to the electric potential along all its lateral edge-surfaces. On the face planes of the PZT layers, the open-circuit condition with respect to the electrical displacement is satisfied. It is also assumed that on the opposite two ends of the plate which are parallel to the band cracks’ front, uniformly distributed normal compressive mechanical forces act. The critical values of these forces under are determined by employing an initial imperfection criterion. For solution of the corresponding boundary value problems the 3D FEM is utilized and numerical results on the critical forces and on the influence of the problem parameters on these forces are presented and discussed. In particular, it is established that the piezoelectricity of the face layers and the three dimensionality of the problem cause to increase the values of the critical forces.

Effect of Initial Stress on the Dynamic Response of a Multi-Layered Plate-Strip Subjected to an Arbitrary Inclined Time-Harmonic Force

The forced vibration of a multi-layered plate-strip with initial stress under the action of an arbitrary inclined time-harmonic force resting on a rigid foundation is considered. Within the framework of the piecewise homogeneous body model with the use of the three-dimensional linearized theory of elastic waves in initially stressed bodies (TLTEWISB), a mathematical modelling is presented in plane strain state. It is assumed that there exists the complete contact interaction at the interface between the layers and the materials of the layer are linearly elastic, homogeneous and isotropic. The governing system of the partial differential equations of motion for the considered problem is solved approximately by employing the Finite Element Method (FEM). Further, the influence of the initial stress parameter on the dynamic response of the plate-strip is presented.

Dynamic response of a pre-stressed bi-layered plate-strip subjected to an arbitrary inclined time-harmonic force

Within the scope of the piecewise homogeneous body model with utilizing of the three dimensional linearized theory of elastic waves in initially stressed bodies the dynamical stress field problem in a bi-layered plate-strip with initial stress under the action of an arbitrary inclined timeharmonic force resting on a rigid foundation is investigated. The concrete materials such as a pair of Aluminum and Steel are selected. It is assumed that there exists a complete contact interaction between the layers. The mathematical modeling of the problem under consideration is carved out, and the governing system of the partial differential equations of motion is approximately solved by employing Finite Element Method. The numerical results related to the influence of certain parameters on the dynamic response of the plate-strip are presented.

Influence of the Interaction Between Fibers Periodically Located in a Composite Material on the Distribution of Stresses in It

Within the framework of a piecewise homogeneous body model, with the use of exact equations of the threedimensional geometrically nonlinear elasticity theory, the stress distributions in a unidirectional fibrous composite containing a row of periodically curved fibers is investigated. The midlines of the fibers are located in different parallel planes, and the curvatures of these lines are periodic and cophasal. The body is loaded at infinity by uniformly distributed normal forces in the direction of fiber location, and the stress distribution in it is investigated. The self-balanced normal and shear stresses arising as a result of fiber curvature are analyzed, and the effects of interaction between the fibers and of geometrical nonlinearity on their distribution are studied. The corresponding numerical results are presented and discussed.

On the total electro-mechanical potential energy and energy release rate at the interface crack tips in an initially stressed sandwich plate-strip with piezoelectric face and elastic core layers

An investigation into the values of the total electro-mechanical potential energy and energy release rate (ERR) at the interface crack tips in an initially stressed sandwich plate-strip is carried out for the opening crack mode. It is assumed that the face layers of the considered sandwich plate are made from piezoelectric materials but that the core is made of elastic material, and that this sandwich plate has initial stresses caused by the uniformly distributed normal forces acting on the planes of the two parallel grounded ends of the plate. The plate has two parallel interface cracks symmetrically located with respect to the middle planes and additional uniformly distributed normal opening mechanical forces acting on the cracks’ edges. Moreover, the face planes of the face layers and the cracks’ edges are unelectroded with vanishing normal electric displacement and are free of electrical charge. The investigations are made by utilizing the three-dimensional linearized theory of electro-elasticity in a plane-strain state within the framework of the piecewise homogeneous body model. The solution to the corresponding boundary value problems are found numerically by employing the finite element method. Numerical results, which are presented and analyzed, illustrate the influences of the geometrical parameters and the initial stresses, as well as the coupling effects between the electro-mechanical fields, on the values of the total electro-mechanical potential energy and of the ERR.