•The Zak phase, a constant to determine the topological properties in 1D systems, is theoretically calculated by employing transfer matrix method.•The topological phase inversion is realized in a ...homogeneous piezoelectric system by varying the electrical boundary conditions.•The location of the Dirac point can be changed smoothly over a wide range by controlling the electrical boundary conditions.•Different arrays of electrically open and electrically closed boundary conditions can switch the topological phase.
By employing periodic electrical boundary conditions, an innovative method to generate actively tunable topologically protected interface mode in a “homogeneous” piezoelectric rod system is proposed. Made of homogeneous material and with uniform structure, each unit cell of the piezoelectric rod system consists of three sub-rods forming an A-B-A structure, where the two A sub-rods have the same geometry and electric boundary conditions. It is discovered that the switch of electrical boundary conditions from A-closed and B-open to A-open and B-closed will yield topological phase inversion, based on which topologically protected interface mode is realized. When capacitors CA and CB are connected to the electrodes of sub-rods A and B, respectively, a variety of physical phenomena is observed. On one hand, varying the capacitance in a certain path leads to topological phase transition. On the other hand, different variation paths of the capacitors give rise to different locations of topological phase transition points. This discovery allows the eigenfrequency of the topologically protected edge mode thus formed be actively controlled by appropriately varying the capacitance. The active topological protected interface mode may find wide engineering applications that require high sensitivity sensing, nondestructive testing, reinforcing energy harvesting, information processing, and others.
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This paper presents an investigation on the nonlinear flexural vibrations of carbon nanotube-reinforced composite (CNTRC) laminated cylindrical shells with negative Poisson’s ratios in thermal ...environments. The material properties of the CNTRCs are temperature-dependent and the functionally graded (FG) in a piece-wise pattern in the thickness direction of the shell. An extended Voigt (rule of mixture) model is employed to estimate the CNTRC material properties. The motion equations for the nonlinear flexural vibration of FG-CNTRC laminated cylindrical shells are based on the Reddy’s third order shear deformation theory and the von Kármán-type kinematic nonlinearity, and the effects of thermal environmental conditions are included. The nonlinear vibration solutions for the FG-CNTRC laminated cylindrical shells can be obtained by applying a singular perturbation technique along with a two-step perturbation approach. The effects of material property gradient, the temperature variation, shell geometric parameter, stacking sequence as well as the end conditions on the vibration characteristics of CNTRC laminated cylindrical shells are discussed in detail through a parametric study. The results show that negative Poisson’s ratio has a significant effect on the linear and nonlinear vibration characteristics of CNTRC laminated cylindrical shells.
•Both piece-wise FG configurations and NPRs are considered.•A multi-scale approach is proposed for nonlinear vibration of auxetic CNTRC shells.•NPR has a significant effect on the nonlinear vibration behavior of FG-CNTRC shells.
The Eringen nonlocal theory of elasticity formulated in differential form has been widely used to address problems in which size effect cannot be disregarded in micro- and nano-structured solids and ...nano-structures. However, this formulation shows some inconsistencies that are not completely understood. In this paper we formulate the problem of the static bending of Euler–Bernoulli beams using the Eringen integral constitutive equation. It is shown that, in general, the Eringen model in differential form is not equivalent to the Eringen model in integral form, and a general method to solve the problem rigorously in integral form is proposed. Beams with different boundary and load conditions are analyzed and the results are compared with those derived from the differential approach showing that they are different in general. With this integral formulation, the paradox that appears when solving the cantilever beam with the differential form of the Eringen model (increase in stiffness with the nonlocal parameter) is solved, which is one of the main contributions of the present work.
A method that employs a dual mesh, one for primary variables and another for dual variables, for the numerical analysis of functionally graded beams is presented. The formulation makes use of the ...traditional finite element interpolation of the primary variables (primal mesh) and the concept of the finite volume method to satisfy the integral form of the governing differential equations on a dual mesh. The method is used to analyze bending of straight, through-thickness functionally graded beams using the Euler–Bernoulli and the Timoshenko beam theories, in which the axial and bending deformations are coupled. Both the displacement and mixed models using the new method are developed accounting for the coupling. Numerical results are presented to illustrate the methodology and a comparison of the generalized displacements and forces/stresses computed with those of the corresponding finite element models. The influence of the coupling stiffness on the deflections is also brought out.
Finite element analysis of functionally graded plates based on a general third-order shear deformation plate theory with a modified couple stress effect and the von Kármán nonlinearity is carried out ...to bring out the effects of couple stress, geometric nonlinearity and power-law variation of the material composition through the plate thickness on the bending deflections of plates. The theory requires no shear correction factors. The principle of virtual displacements is utilized to develop a nonlinear finite element model. The finite element model requires
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continuity of all dependent variables. The microstructural effects are captured using a length scale parameter via the modified couple stress theory. The variation of two-constituent material is assumed through the thickness direction according to a power-law distribution. Numerical results are presented for static bending problems of rectangular plates with various boundary conditions to bring out the parametric effects of the power-law index and length scale parameter on the load–deflection characteristics of plates with various boundary conditions.
This work presents the static and dynamic analyses of laminated doubly-curved shells and panels of revolution resting on the Winkler–Pasternak elastic foundation using the generalized differential ...quadrature (GDQ) method. The analyses are worked out considering the first-order shear deformation theory (FSDT) for the aforementioned moderately thick structural elements. The solutions are given in terms of generalized displacement components of points lying on the middle surface of the shell. Several types of shell structures such as doubly-curved and revolution shells, singly-curved and degenerate shells are considered in this paper. The main novelty of this paper is the application of the differential geometry within GDQ method to solve doubly-curved shells resting on the Winkler–Pasternak elastic foundation. The discretization of the differential system by means of the GDQ technique leads to a standard linear problem for the static analysis and to a standard linear eigenvalue problem for the dynamic analysis. In order to show the accuracy of this methodology, numerical comparisons between the present formulation and finite element solutions are presented. Very good agreement is observed. Finally, new results are presented to show effects of the Winkler modulus, the Pasternak modulus, and the inertia of the elastic foundation on the behavior of laminated doubly-curved shells.
We look for an enhancement of the correspondence model of peridynamics, emphasizing the elimination of zero-energy deformation modes. We propose an approach based on the notion of sub-horizons. The ...most useful feature of this proposal is the setup which, whilst providing solutions with the necessary stability, deviates only marginally from the original correspondence formulation. A thorough analysis of the sub-horizon based method is furnished based on the well-posedness of integral equations and energy spectrum, which clearly demonstrate a removal of zero energy modes. We also show how other forms of unphysical deformation modes, e.g. material collapse within horizon, jump discontinuities and vanishing energy modes, can be prevented with the present proposal. Finally, a set of numerical simulations are undertaken that attest to the remarkable efficacy of the sub-horizon based approach.
•Computationally expedient Peridynamics constitutive correspondence method is proposed.•The formulation imparts the necessary stability to the solution.•Suppresses zero-energy deformation modes along with other instabilities.•Neither introduces penalizing energy nor leads to change in the original form of EOM.•The new strategy can seamlessly be incorporated within already existing PD codes.
In this work, nonlocal nonlinear analysis of functionally graded plates subjected to static loads is studied. The nonlocal nonlinear formulation is developed based on the third-order shear ...deformation theory (TSDT) of Reddy (1984, 2004). The von Kármán nonlinear strains are used and the governing equations of the TSDT are derived accounting for Eringen’s nonlocal stress-gradient model (Eringen, 1998). The nonlinear displacement finite element model of the resulting governing equations is developed, and Newton’s iterative procedure is used for the solution of nonlinear algebraic equations. The mechanical properties of functionally graded plate are assumed to vary continuously through the thickness and obey a power-law distribution of the volume fraction of the constituents. The variation of the volume fractions through the thickness have been computed using two different homogenization techniques, namely, the rule of mixtures and the Mori–Tanaka scheme. A detailed parametric study to show the effect of side-to-thickness ratio, power-law index, and nonlocal parameter on the load-deflection characteristics of plates have been presented. The stress results are compared with the first-order shear deformation theory (FSDT) to show the accuracy of nonlocal nonlinear TSDT formulation.
Finite element models of microstructure-dependent geometrically nonlinear theories for axisymmetric bending of circular plates, which accounts for through-thickness power-law variation of a ...two-constituent material, the von Kármán nonlinearity, and the strain gradient effects are developed for the classical and first-order plate theories. The strain gradient effects are included through the modified couple stress theory that contains a single material length scale parameter which can capture the size effect in a functionally graded material plate. The developed finite element models are used to determine the effect of the geometric nonlinearity, power-law index, and microstructure-dependent constitutive relations on the bending response of functionally graded circular plates with different boundary conditions.
•Models account for modified couple stress effects and von Karman nonlinearity.•The formulation is developed for power-law based functionally graded plates.•Novel finite element models of the theories are developed.•Parametric studies include power-law index, material length scale, and geometric nonlinearity.
A microstructure-dependent Timoshenko beam model is developed using a variational formulation. It is based on a modified couple stress theory and Hamilton's principle. The new model contains a ...material length scale parameter and can capture the size effect, unlike the classical Timoshenko beam theory. Moreover, both bending and axial deformations are considered, and the Poisson effect is incorporated in the current model, which differ from existing Timoshenko beam models. The newly developed non-classical beam model recovers the classical Timoshenko beam model when the material length scale parameter and Poisson's ratio are both set to be zero. In addition, the current Timoshenko beam model reduces to a microstructure-dependent Bernoulli–Euler beam model when the normality assumption is reinstated, which also incorporates the Poisson effect and can be further reduced to the classical Bernoulli–Euler beam model. To illustrate the new Timoshenko beam model, the static bending and free vibration problems of a simply supported beam are solved by directly applying the formulas derived. The numerical results for the static bending problem reveal that both the deflection and rotation of the simply supported beam predicted by the new model are smaller than those predicted by the classical Timoshenko beam model. Also, the differences in both the deflection and rotation predicted by the two models are very large when the beam thickness is small, but they are diminishing with the increase of the beam thickness. Similar trends are observed for the free vibration problem, where it is shown that the natural frequency predicted by the new model is higher than that by the classical model, with the difference between them being significantly large only for very thin beams. These predicted trends of the size effect in beam bending at the micron scale agree with those observed experimentally. Finally, the Poisson effect on the beam deflection, rotation and natural frequency is found to be significant, which is especially true when the classical Timoshenko beam model is used. This indicates that the assumption of Poisson's effect being negligible, which is commonly used in existing beam theories, is inadequate and should be individually verified or simply abandoned in order to obtain more accurate and reliable results.
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