scholarly journals Nonlinear electromechanical analysis of axisymmetric thin circular plate based on flexoelectric theory

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Xue Ji
Keyword(s):  
Circular Plate ◽  
Electromechanical Coupling ◽  
Plate Theory ◽  
Equilibrium Equations ◽  
Nonlinear Free Vibration ◽  
Flexoelectric Effect ◽  
Nonlinear Static ◽  
The Galerkin Method ◽  
Stiffening Effect ◽  
Plate Type

AbstractFlexoelectricity will dominate the electromechanical coupling of intelligent components in MEMS/NEMS due to its size-dependency. This paper focuses on investigating the flexoelectric responses of intelligent components of the circular plate type, which are commonly used in MEMS/NEMS. Utilizing Hamilton’s principle, the nonlinear flexoelectric circular plate model is presented by combining von Kármán plate theory and flexoelectric theory. The equilibrium equations and all boundary conditions are obtained and then discretized. The nonlinear static bending of the simply supported axisymmetric flexoelectric circular plate is investigated by combining DQM and iteration method. The distributions of dimensionless bending deflection and electric potential are analyzed under different loads. Moreover, the nonlinear free vibration behaviors are also investigated by combining the Galerkin method and Lindstedt–Poincaré Method. The flexoelectric effect and stiffening effect of strain gradient are revealed. This paper will be helpful to promote the application of flexoelectric intelligent components of the circular plate type, which are encountered commonly in engineering.

Elastic Foundation Analysis of Uniformly Loaded Functionally Graded Viscoelastic Sandwich Plates

2012 ◽  
Vol 28 (3) ◽  
pp. 439-452 ◽  
Author(s):  
A. M. Zenkour ◽  
M. Sobhy
Keyword(s):  
Power Law ◽  
Sandwich Plate ◽  
Plate Theory ◽  
Functionally Graded ◽  
Sandwich Plates ◽  
Equilibrium Equations ◽  
Graded Material ◽  
Plate Aspect Ratio ◽  
Type V ◽  
Power Law Index

AbstractThis paper deals with the static response of simply supported functionally graded material (FGM) viscoelastic sandwich plates subjected to transverse uniform loads. The FG sandwich plates are considered to be resting on Pasternak's elastic foundations. The sandwich plate is assumed to consist of a fully elastic core sandwiched by elastic-viscoelastic FGM layers. Material properties are graded according to a power-law variation from the interfaces to the faces of the plate. The equilibrium equations of the FG sandwich plate are given based on a trigonometric shear deformation plate theory. Using Illyushin's method, the governing equations of the viscoelastic sandwich plate can be solved. Parametric study on the bending analysis of FG sandwich plates is being investigated. These parameters include (i) power-law index, (ii) plate aspect ratio, (iii) side-to-thickness ratio, (iv) loading type, (v) foundation stiffnesses, and (vi) time parameter.

scholarly journals Semi-analytical static analysis of nonlocal strain gradient laminated composite nanoplates in hygrothermal environment

2021 ◽  
Vol 43 (5) ◽  
Author(s):  
Giovanni Tocci Monaco ◽  
Nicholas Fantuzzi ◽  
Francesco Fabbrocino ◽  
Raimondo Luciano
Keyword(s):  
Thin Plate ◽  
Strain Gradient ◽  
Plate Theory ◽  
Laminated Composite ◽  
Equilibrium Equations ◽  
Thin Plate Theory ◽  
Bending Behavior ◽  
Hygrothermal Environment ◽  
Load Types ◽  
Novel Applications

AbstractIn this work, the bending behavior of nanoplates subjected to both sinusoidal and uniform loads in hygrothermal environment is investigated. The present plate theory is based on the classical laminated thin plate theory with strain gradient effect to take into account the nonlocality present in the nanostructures. The equilibrium equations have been carried out by using the principle of virtual works and a system of partial differential equations of the sixth order has been carried out, in contrast to the classical thin plate theory system of the fourth order. The solution has been obtained using a trigonometric expansion (e.g., Navier method) which is applicable to simply supported boundary conditions and limited lamination schemes. The solution is exact for sinusoidal loads; nevertheless, convergence has to be proved for other load types such as the uniform one. Both the effect of the hygrothermal loads and lamination schemes (cross-ply and angle-ply nanoplates) on the bending behavior of thin nanoplates are studied. Results are reported in dimensionless form and validity of the present methodology has been proven, when possible, by comparing the results to the ones from the literature (available only for cross-ply laminates). Novel applications are shown both for cross- and angle-ply laminated which can be considered for further developments in the same topic.

scholarly journals Conservation Integrals in Nonhomogeneous Materials with Flexoelectricity

2021 ◽  
Vol 11 (2) ◽  
pp. 681
Author(s):  
Pengfei Yu ◽  
Weifeng Leng ◽  
Yaohong Suo
Keyword(s):  
Electromechanical Coupling ◽  
Energy Momentum Tensor ◽  
Great Influence ◽  
Operator Method ◽  
Electric Polarization ◽  
Momentum Tensor ◽  
Noether Theorem ◽  
Strain Gradients ◽  
Flexoelectric Effect ◽  
Nonhomogeneous Materials

The flexoelectricity, which is a new electromechanical coupling phenomenon between strain gradients and electric polarization, has a great influence on the fracture analysis of flexoelectric solids due to the large gradients near the cracks. On the other hand, although the flexoelectricity has been extensively investigated in recent decades, the study on flexoelectricity in nonhomogeneous materials is still rare, especially the fracture problems. Therefore, in this manuscript, the conservation integrals for nonhomogeneous flexoelectric materials are obtained to solve the fracture problem. Application of operators such as grad, div, and curl to electric Gibbs free energy and internal energy, the energy-momentum tensor, angular momentum tensor, and dilatation flux can also be derived. We examine the correctness of the conservation integrals by comparing with the previous work and discuss the operator method here and Noether theorem in the previous work. Finally, considering the flexoelectric effect, a nonhomogeneous beam problem with crack is solved to show the application of the conservation integrals.

scholarly journals The Theoretical and experimental study of the bending of a thin substrate during electrolytic deposition

2020 ◽  
pp. 17-31
Author(s):  
D K Bout ◽  
P S Bychkov ◽  
S A Lychev
Keyword(s):  
Experimental Study ◽  
Plate Theory ◽  
Substrate Surface ◽  
Deposition Process ◽  
Material Composition ◽  
Electrolytic Deposition ◽  
Continuous Family ◽  
Equilibrium Equations ◽  
Shape Distortion ◽  
Local Deformations

The present paper is aimed at the theoretical and experimental study of the shape distortion of thin substrates during electrolytic deposition and gaccumulation of residual stresses in them. The theoretical modeling is provided in the framework of the theory of solids with variable material composition. The result of the deposition process is modeled with a continuous family of elastic bodies, which local deformations are incompatible. These deformations act as internal sources for stresses. Formally they are equivalent to the field of distributed defects. Unlike the classical approach adopted in nonlinear elasticity, the elements of the family which present a body with a variable material composition don’t have a global reference natural (free of stresses) form. Instead we used the continuous family being only locally free from stresses. To formulate the boundary value problem, continuous families of reference, intermediate and actual forms and corresponding families of deformations are defined. The deformations, belonging to these families, locally represent implants (local deformations of reference forms into intermediate ones) and deformations that bring intermediate forms into actual ones. Relations for stresses and strains in such bodies are obtained under the assumption that the displacement gradients are small with respect to unity and satisfy the kinematic hypothesis of the technical plate theory. Under these assumptions the equilibrium equations are derived. They include specific terms which determine formal loading that is caused by incompatible deformations. Axisymmetric problems for a circular substrate under various types of fixing and tension on the boundary, which characterize the conditions of the experiment, are obtained. The theoretical distribution for displacements of the substrate surface is formulated upon the obtained solution. They are intended to identify incompatible deformations that cause bending during the deposition process. The experimental measuring setup is constructed according to a holographic scheme of displacement measurements in real time. The deposition process is carried out in a cylindrical chamber with flange fastening of the cathode. The electrochemical process is implemented in sulphate electrolyte. As a result of comparing the theoretically obtained relations for bending surfaces of the substrate with the experimental results, the parameters that characterize the substrate shrinkage and tension are estimated.

A closed form solution for bending/stretching analysis of functionally graded circular plates under asymmetric loading using the Green function

2010 ◽  
Vol 224 (6) ◽  
pp. 1153-1163 ◽  
Author(s):  
A R Saidi ◽  
A Naderi ◽  
E Jomehzadeh
Keyword(s):  
Green Function ◽  
Closed Form ◽  
Volume Fraction ◽  
Plate Theory ◽  
Closed Form Solution ◽  
Form Solution ◽  
Functionally Graded ◽  
Transverse Displacement ◽  
Circular Plates ◽  
Equilibrium Equations

In this article, a closed-form solution for bending/stretching analysis of functionally graded (FG) circular plates under asymmetric loads is presented. It is assumed that the material properties of the FG plate are described by a power function of the thickness variable. The equilibrium equations are derived according to the classical plate theory using the principle of total potential energy. Two new functions are introduced to decouple the governing equilibrium equations. The three highly coupled partial differential equations are then converted into an independent equation in terms of transverse displacement. A closed-form solution for deflection of FG circular plates under arbitrary lateral eccentric concentrated force is obtained by defining a new coordinate system. This solution can be used as a Green function to obtain the closed-form solution of the FG plate under arbitrary loadings. Also, the solution is employed to solve some different asymmetric problems. Finally, the stress and displacement components are obtained exactly for each problem and the effect of volume fraction is also studied.

Thermal Buckling of a Circular Plate

1967 ◽  
Vol 71 (683) ◽  
pp. 798-798 ◽  
Author(s):  
E. H. Mansfield
Keyword(s):  
Circular Plate ◽  
Thermal Buckling ◽  
The Novel ◽  
Stiffening Effect

SummaryThis note considers the buckling of a circular lenticular plate subjected to a temperature which varies parabolically in its plane. The novel feature in the analysis is the determination of the stiffening effect due to the prevention of deflections, but not slopes, along a diameter.

A simplified plate theory for vibration analysis of composite laminated sector, annular and circular plate

2019 ◽  
Vol 143 ◽  
pp. 106252 ◽  
Author(s):  
Hong Zhang ◽  
Rupeng Zhu ◽  
Dongyan Shi ◽  
Qingshan Wang
Keyword(s):  
Circular Plate ◽  
Vibration Analysis ◽  
Plate Theory

Flutter instabilities of cantilevered piezoelectric pipe conveying fluid

2019 ◽  
Vol 30 (4) ◽  
pp. 606-617 ◽  
Author(s):  
Gang Wang ◽  
Jinwei Shen
Keyword(s):  
Electromechanical Coupling ◽  
Piezoelectric Materials ◽  
Fluid Velocity ◽  
Pipe Conveying Fluid ◽  
Performance Predictions ◽  
Parametric Studies ◽  
Cycle Oscillation ◽  
And Performance ◽  
The Galerkin Method ◽  
Conveying Fluid

In this article, a nonlinear model was developed for a cantilevered piezoelectric pipe conveying fluid that included geometric nonlinearity and electromechanical coupling. The Galerkin method discretized the system in order to characterize its behavior. Critical flutter velocity and its associated unstable mode can be determined based on linear analysis. Due to the presence of piezoelectric materials, the critical flutter velocity depends on the resistive piezoelectric damping and electromechanical coupling. This added resistive piezoelectric damping tends to decrease the flutter velocity. Comprehensive simulations were also conducted to characterize the post-flutter behaviors. System parameters including amplitude, deformed pipe shape, and collected voltage in piezoelectric materials were calculated. The system will undergo limited cycle oscillation when the fluid velocity passes the flutter velocity. Parametric studies were conducted as well to investigate the system responses under different flow velocities. Physical insights can be collected from these simulation results to conduct piezoelectric pipe design and performance predictions for future pipe vibration control and energy harvesting applications.

On the Existence of a Solution for a Solid Circular Plate Bilaterally Supported Along Two Antipodal Boundary Arcs and Loaded by a Central Transverse Concentrated Force

2000 ◽  
Vol 68 (5) ◽  
pp. 809-812 ◽  
Author(s):  
G. Monegato ◽  
A. Strozzi
Keyword(s):  
Circular Plate ◽  
Plate Theory ◽  
Reaction Force ◽  
Mechanical Analysis ◽  
Square Root ◽  
Existence Of A Solution ◽  
Concentrated Force ◽  
Boundary Constraints ◽  
Title Problem ◽  
The One

A purely flexural mechanical analysis is presented for a thin, solid, circular plate, deflected by a central transverse concentrated force, and bilaterally supported along two antipodal periphery arcs, the remaining part of the boundary being free. This problem is modeled in terms of a singular integral equation of the Prandtl type, which possesses a unique solution expressed in terms of a reaction force containing a factor exhibiting square root endpoint singularities. This solution is then shown not to respect the requested boundary constraints. It is therefore concluded that, within the framework of the purely flexural plate theory, the title problem cannot admit the weighted L2 solution here examined. It cannot, however, be excluded that a solution to the title problem exists, which possesses stronger endpoint singularities than those examined in this paper, or is of a more general form than the one considered here.

Nonlinear free vibration of orthotropic graphene sheets using nonlocal Mindlin plate theory

2011 ◽  
Vol 226 (7) ◽  
pp. 1896-1906 ◽  
Author(s):  
AR Setoodeh ◽  
P Malekzadeh ◽  
AR Vosoughi
Keyword(s):  
Free Vibration ◽  
Virtual Work ◽  
Plate Theory ◽  
Differential Quadrature Method ◽  
Nonlinear Partial Differential Equations ◽  
Mindlin Plate ◽  
Small Scale ◽  
Graphene Sheets ◽  
Nonlinear Free Vibration ◽  
Mindlin Plate Theory

This article deals with the small-scale effect on the nonlinear free vibration of orthotropic single-layered graphene sheets using the nonlocal elasticity plate theory. The formulations are based on the Mindlin plate theory, and von Karman-type nonlinearity is considered in strain displacement relations. Virtual work principle is used to derive the nonlinear nonlocal plate equations in which the effects of rotary inertia and transverse shear are included. The differential quadrature method is employed to reduce the governing nonlinear partial differential equations to a system of nonlinear algebraic eigenvalue equations. The efficiency and accuracy of the method are demonstrated by comparing the developed result with those available in literature. The methodology is capable of studying large-amplitude vibration characteristics of nanoplates with different sets of boundary conditions. The effects of various parameters on the nonlinear vibrations of nanoplates are presented.

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