Third-order Theory Research Articles

Analysis of bi-directional functionally graded plates by FEM and a new third-order shear deformation plate theory

Modern structures and components may require advanced materials whose properties vary continuously not only in one specified direction, but also different other directions. In particular, the bi-directional functionally graded materials (2D-FGMs) introduced are expected to have more effective properties, consequently eliminating commonly awkward problems such as local stress concentrations and delamination. In this paper, buckling and bending behaviors of 2D-FGM plates, which are of great importance in the design and development of engineering applications, are numerically analyzed by a finite element model. The plate kinematics are described using a new third-order shear deformation plate theory (TSDT), without the need for special treatment of shear-locking effect and shear correction factors. The present TSDT theory based on rigorous kinematic of displacements, which is shown to be dominated over other preceding theories, is derived from an elasticity formulation, rather by the hypothesis of displacements. The materials are assumed to be graded in two directions and their effective properties are computed through the rule of mixture. The accuracy of the proposed approach assessed on numerical results is confirmed by comparing the obtained results with respect to reference published solutions. The effects of some numerical aspect ratios such as volume fraction, boundary conditions, thickness to length ratio, etc. on static deflections and critical buckling are numerically studied. The investigation of results confirms that such aforementioned aspect ratios have significant effects on the mechanical behaviors of plates.

Linking the Character of the Metal–Ligand Bond to the Ligand NMR Shielding in Transition-Metal Complexes: NMR Contributions from Spin–Orbit Coupling

Relativistic effects significantly affect various spectroscopic properties of compounds containing heavy elements. Particularly in Nuclear Magnetic Resonance (NMR) spectroscopy, the heavy atoms strongly influence the NMR shielding constants of neighboring light atoms. In this account we analyze paramagnetic contributions to NMR shielding constants and their modulation by relativistic spin-orbit effects in a series of transition-metal complexes of Pt(II), Au(I), Au(III), and Hg(II). We show how the paramagnetic NMR shielding and spin-orbit effects relate to the character of the metal-ligand (M-L) bond. A correlation between the (back)-donation character of the M-L bond in d10 Au(I) complexes and the propagation of the spin-orbit (SO) effects from M to L through the M-L bond influencing the ligand NMR shielding via the Fermi-contact mechanism is found and rationalized by using third-order perturbation theory. The SO effects on the ligand NMR shielding are demonstrated to be driven by both the electronic structure of M and the nature of the trans ligand, sharing the σ-bonding metal orbital with the NMR spectator atom L. The deshielding paramagnetic contribution is linked to the σ-type M-L bonding orbitals, which are notably affected by the trans ligand. The SO deshielding role of σ-type orbitals is enhanced in d10 Hg(II) complexes with the Hg 6p atomic orbital involved in the M-L bonding. In contrast, in d8 Pt(II) complexes, occupied π-type orbitals play a dominant role in the SO-altered magnetic couplings due to the accessibility of vacant antibonding σ-type MOs in formally open 5d-shell (d8). This results in a significant SO shielding at the light atom. The energy- and composition-modulation of σ- vs π-type orbitals by spin-orbit coupling is rationalized and supported by visualizing the SO-induced changes in the electron density around the metal and light atoms (spin-orbit electron deformation density, SO-EDD).

On natural frequencies of Levy-type thick porous-cellular plates surrounded by piezoelectric layers

In this paper, an analytical solution for free vibration of rectangular porous-cellular plates enclosed by piezoelectric layers is presented by using third-order shear deformation plate theory. Using Hamilton’s principle and Maxwell equation, the governing equations of the system are obtained for both closed and open circuit conditions. Due to the coordinate dependency of mechanical properties of porous materials, the governing equations of motion are highly coupled. By using four auxiliary functions, these equations convert into two independent partial differential equations. The decoupled equations are solved analytically by employing Levy-type boundary conditions for the plate. Finally, after validation of the obtained results, the effects of various parameters such as porosity and geometrical dimensions on the natural frequencies of plate are investigated for different electrical and mechanical boundary conditions. It is found that the natural frequencies of the plate decrease as the coefficient of plate porosity increases. Also, the piezoelectric layers cause the natural frequency of the plate to increase in various vibrating modes.

Vibration analysis of parabolic shear-deformable piezoelectrically actuated nanoscale beams incorporating thermal effects

ABSTRACTThermoelectric-mechanical vibration behavior of functionally graded piezoelectric (FGP) nanobeams is first investigated in this article, based on the nonlocal theory and third-order parabolic beam theory by presenting a Navier-type solution. Electro-thermo-mechanical properties of a nanobeam are supposed to change continuously throughout the thickness based on the power-law model. To capture the small-size effects, Eringen's nonlocal elasticity theory is adopted. Using Hamilton's principle, the nonlocal governing equations for the third-order, shear deformable, piezoelectric, FG nanobeams are obtained and they are solved applying an analytical solution. By presenting some numerical results, it is demonstrated that the suggested model presents accurate frequency results of FGP nanobeams. The influences of several parameters, including external electric voltage, power-law exponent, nonlocal parameter, and mode number on the natural frequencies of the size-dependent FGP nanobeams are discussed in detail. The results should be relevant to the design and application of the piezoelectric nanodevices.

Nonlinear dynamic response and vibration of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) shear deformable plates with temperature-dependent material properties and surrounded on elastic foundations

ABSTRACTBased on Reddy’s third-order shear deformation plate theory, the nonlinear dynamic response and vibration of imperfect functionally graded carbon nanotube-reinforced composite (FG-CNTRC) plates on elastic foundations subjected to dynamic loads and temperature are presented. The plates are reinforced by single-walled carbon nanotubes which vary according to the linear functions of the plate thickness. The plate’s effective material properties are assumed to depend on temperature and estimated through the rule of mixture. By applying the Airy stress function, Galerkin method and fourth-order Runge–Kutta method, nonlinear dynamic response and natural frequency for imperfect FG-CNTRC plates are determined. In numerical results, the influences of geometrical parameters, elastic foundations, initial imperfection, dynamic loads, temperature increment, and nanotube volume fraction on the nonlinear vibration of FG-CNTRC plates are investigated. The obtained results are validated by comparing with those of other authors.

Analytical solution of some delamination scenarios in thick structural sandwich plates

The first-, second- and third-order shear deformation plate theories are applied in this work to model thick rectangular sandwich plates with through-width delamination. The models are based on the concept of the four equivalent single layers and the system of exact kinematic conditions. Three different scenarios are considered: the failure of the core, the delamination between the top facesheet and the core, and finally, the case when the delamination takes place in the local midplane of the top facesheet. A general model is derived and applied to sandwich plates with Lévy type boundary conditions. The governing equations are summarized and the state-space model of the system is created. The mechanical fields are calculated and compared to finite element results. The comparison shows that the first-order sandwich plate model is inaccurate, on the other hand, the second- and third-order theories capture very well the mechanical fields compared to finite element results. The J-integral distribution is also calculated along the delamination front and it is concluded that the third- and second-order models give very good approximations of the results by finite element analysis and the virtual crack closure technique.

Analytical investigation on buckling and postbuckling of FGM toroidal shell segment surrounded by elastic foundation in thermal environment and under external pressure using TSDT

In this paper, FGM toroidal shell segments surrounded by elastic foundation and subjected to uniform external pressure are investigated by analytical method. A novelty of the study is that the Reddy’s third-order shear deformation shell theory (TSDT) with von Karman geometrical nonlinearity combined with deflection function selected with three terms is used to investigate the nonlinear stability of thicker FGM toroidal shell segments. In addition, the thermal element in the shell is also taken into account. The FGM shell is a convex and concave toroidal shell segment. It is a general form for a circular cylindrical shell. Closed-form expressions for determining the static critical external pressure load and postbuckling load–deflection curves are obtained. Effects of temperature field, foundations, material and dimensional parameters on the stability of shells are considered. This paper also shows that the use of TSDT to analyze the nonlinear stability of thicker toroidal shell segments is necessary and more suitable.

Third-order perturbative lattice and complex Langevin analyses of the finite-temperature equation of state of nonrelativistic fermions in one dimension

We analyze the pressure and density equations of state of unpolarized non-relativistic fermions at finite temperature in one spatial dimension. For attractively interacting regimes, we perform a third-order lattice perturbation theory calculation, assess its convergence properties by comparing with hybrid Monte Carlo results (there is no sign problem in this regime), and demonstrate agreement with real Langevin calculations. For repulsive interactions, we present lattice perturbation theory results as well as complex Langevin calculations, with a modified action to prevent uncontrolled excursions in the complex plane. Although perturbation theory is a common tool, our implementation of it is unconventional; we use a Hubbard-Stratonovich transformation to decouple the system and automate the application of Wick's theorem, thus generating the diagrammatic expansion, including symmetry factors, at any desired order. We also present an efficient technique to tackle nested Matsubara frequency sums without relying on contour integration, which is independent of dimension and applies to both relativistic and non-relativistic systems, as well as all energy-independent interactions. We find exceptional agreement between perturbative and non-perturbative results at weak couplings, and furnish predictions based on complex Langevin at strong couplings. We additionally present perturbative calculations of up to the fifth-order virial coefficient for repulsive and attractive couplings. Both the lattice perturbation theory and complex Langevin formalisms can easily be extended to a variety of situations including polarized systems, bosons, and higher dimension.

Nonlinear stability analysis of stiffened functionally graded material sandwich cylindrical shells with general Sigmoid law and power law in thermal environment using third-order shear deformation theory

This paper investigates analytically nonlinear buckling and postbuckling of functionally graded sandwich circular thick cylindrical shells filled inside by Pasternak two-parameter elastic foundations under thermal loads and axial compression loads. Shells are reinforced by closely spaced functionally graded material (FGM) rings and stringers. The temperature field is taken into account. Two general Sigmoid law and general power law, with four models of stiffened FGM sandwich cylindrical shell, are proposed. Using the Reddy’s third-order shear deformation shell theory (TSDT), stress function, and Lekhnitsky’s smeared stiffeners technique, the governing equations are derived. The closed form to determine critical axial load and postbuckling load-deflection curves are obtained by the Galerkin method. The effects of the face sheet thickness to total thickness ratio, stiffener, foundation, material, and dimensional parameters on critical thermal loads, critical mechanical loads and postbuckling behavior of shells are analyzed. In addition, this paper shows that for thin shells we can use the classical shell theory to investigate stability behavior of shell, but for thicker shells the use of TSDT for analyzing nonlinear stability of shell is necessary and suitable.

A new nonlocal elasticity theory with graded nonlocality for thermo-mechanical vibration of FG nanobeams via a nonlocal third-order shear deformation theory

ABSTRACTIn the present research, free vibration study of functionally graded (FG) nanobeams with graded nonlocality in thermal environments is performed according to the third-order shear deformation beam theory. The present nanobeam is subjected to uniform and nonlinear temperature distributions. Thermo-elastic coefficients and nonlocal parameter of the FG nanobeam are graded in the thickness direction according to power-law form. The scale coefficient is taken into consideration implementing nonlocal elasticity of Eringen. The governing equations are derived through Hamilton's principle and are solved analytically. The frequency response is compared with those of nonlocal Euler–Bernoulli and Timoshenko beam models, and it is revealed that the proposed modeling can accurately predict the vibration frequencies of the FG nanobeams. The obtained results are presented for the thermo-mechanical vibrations of the FG nanobeams to investigate the effects of material graduation, nonlocal parameter, mode number, slenderness ratio, and thermal loading in detail. The present study is associated to aerospace, mechanical, and nuclear engineering structures that are under thermal loads.

Ratcheting caused by moving loads

Progressive deformation (ratcheting) can occur as a response to variable loads as soon as the elastic limit is exceeded. If this is the case, strains and displacements accumulate in the event of cyclic loading in each load cycle. Widely known as triggers for ratcheting and already being considered in some design codes are configurations, in which a structure is subjected to at least two different types of load, namely a constant load (the primary load) and a superimposed cyclic load. In this paper, another mechanism that generates ratcheting is introduced. It can be attributed solely to the effect of a single load. In the simplest case, this can be explained by the successive activation of (an infinite number of) plastic hinges if a load of constant magnitude is moved in space. The increments of strains and displacements can decrease or increase from cycle to cycle, when the material is hardening, or if elastic foundation is present, or if the equilibrium condition is formulated for the deformed system (second-order theory) or if “large” rotations are taken into account (third-order theory).

Analysis of nonlinear aeroelastic characteristics of a trapezoidal wing in hypersonic flow

Nonlinear aeroelastic behavior of a trapezoidal wing in hypersonic flow is investigated. The aeroelastic governing equations are built by von Karman large deformation theory and the third-order piston theory. The Rayleigh–Ritz approach combined with the affine transformation is formulated and employed to transform the equations of a trapezoidal wing structure, modeled as a cantilevered wing-like plate, into modal coordinates. And then the modal equations are solved by numerical integrations. Several typical cases are studied to validate the capability of the proposed method for linear and nonlinear aeroelastic analysis of trapezoidal cantilever plate in hypersonic flow. The effects of Rayleigh–Ritz mode truncation for various wing-plate geometrical characteristics, i.e., sweep angle of leading edge, taper ratio and span, are examined to determine the appropriate mode number for accurate modeling and fast calculation. Meanwhile, the effects of various geometries of trapezoidal cantilever plates on the flutter stability are investigated. The nonlinear dynamic behaviors of the model with three typical geometries, namely, the rectangular, parallelogram and trapezoidal wing-like plate, are simulated numerically. Furthermore, complex dynamic behaviors are observed and identified via the phase plot, the Poincare map and the largest Lyapunov exponent. The results demonstrate that geometrical parameters of trapezoidal wing have significant effects on the nonlinear aeroelastic behaviors of wing structure. In particular, the evolution processes of chaos exhibit remarkable difference for these three wing configurations.

A dynamic stiffness method for analysis of thermal effect on vibration and buckling of a laminated composite beam

Thermal effects on vibration and buckling behaviors of generally layered composite beams with arbitrary boundary conditions are dealt with in this paper. The composite beam is modeled using third-order shear deformation beam theory in which the Poisson effect is incorporated. A constant temperature change through the beam thickness is assumed. An exact dynamic stiffness matrix is formulated by directly solving the differential equations of motion governing the natural vibration of the composite beams subjected to uniform temperature changes along the beam thickness. Application of the derived dynamic stiffness matrix together with the Wittrick–Williams algorithm to compute the natural frequencies and buckling temperature changes of two particular composite beams is discussed. The correctness and accuracy of the derived dynamic stiffness matrix is evaluated by comparing the present results with the available solutions in literature. The influences of Poisson effect, boundary condition, temperature change, thermal expansion coefficient and material anisotropy on the natural frequencies of the composite beams are studied.

Driven similarity renormalization group: Third-order multireference perturbation theory.

A third-order multireference perturbation theory based on the driven similarity renormalization group (DSRG-MRPT3) approach is presented. The DSRG-MRPT3 method has several appealing features: (a) it is intruder free, (b) it is size consistent, (c) it leads to a non-iterative algorithm with O(N6) scaling, and (d) it includes reference relaxation effects. The DSRG-MRPT3 scheme is benchmarked on the potential energy curves of F2, H2O2, C2H6, and N2 along the F-F, O-O, C-C, and N-N bond dissociation coordinates, respectively. The nonparallelism errors of DSRG-MRPT3 are consistent with those of complete active space third-order perturbation theory and multireference configuration interaction with singles and doubles and show significant improvements over those obtained from DSRG second-order multireference perturbation theory. Our efficient implementation of the DSRG-MRPT3 based on factorized electron repulsion integrals enables studies of medium-sized open-shell organic compounds. This point is demonstrated with computations of the singlet-triplet splitting (ΔST=ET-ES) of 9,10-anthracyne. At the DSRG-MRPT3 level of theory, our best estimate of the adiabatic ΔST is 3.9 kcal mol-1, a value that is within 0.1 kcal mol-1 from multireference coupled cluster results.

Vibration analysis of embedded size dependent FG nanobeams based on third-order shear deformation beam theory

In this paper, free vibration characteristics of functionally graded (FG) nanobeams embedded on elastic medium are investigated based on third order shear deformation (Reddy) beam theory by presenting a Navier type solution for the first time. The material properties of FG nanobeam are assumed to vary gradually along the thickness and are estimated through the power-law and Mori-Tanaka models. A two parameters elastic foundation including the linear Winkler springs along with the Pasternak shear layer is in contact with beam. The small scale effect is taken into consideration based on nonlocal elasticity theory of Eringen. The nonlocal equations of motion are derived based on third order shear deformation beam theory through Hamilton's principle and they are solved applying analytical solution. According to the numerical results, it is revealed that the proposed modeling can provide accurate frequency results of the FG nanobeams as compared to some cases in the literature. The obtained results are presented for the vibration analysis of the FG nanobeams such as the influences of foundation parameters, gradient index, nonlocal parameter and slenderness ratio in detail.

Generalized James' effective Hamiltonian method

James' effective Hamiltonian method has been extensively adopted to investigate largely detuned interacting quantum systems. This method is just corresponding to the second-order perturbation theory, and cannot be exploited to treat the problems which should be solved by using the third or higher-order perturbation theory. In this paper, we generalize James' effective Hamiltonian method to the higher-order case. Using the method developed here, we reexamine two examples published recently [Phys. Rev.Lett. 117, 043601 (2016), Phys. Rev A 92, 023842 (2015)], our results turn out to be the same as the original ones derived from the third-order perturbation theory and adiabatic elimination method respectively. For some specific problems, this method can simplify the calculating procedure, and the resultant effective Hamiltonian is more general.

Electro-magnetic effects on nonlocal dynamic behavior of embedded piezoelectric nanoscale beams

This article investigates vibration behavior of magneto-electro-elastic functionally graded nanobeams embedded in two-parameter elastic foundation using a third-order parabolic shear deformation beam theory. Material properties of magneto-electro-elastic functionally graded nanobeam are supposed to be variable throughout the thickness based on power-law model. Based on Eringen’s nonlocal elasticity theory which captures the small size effects and using Hamilton’s principle, the nonlocal governing equations of motions are derived and then solved analytically. Then, the influences of elastic foundation, magnetic potential, external electric voltage, nonlocal parameter, power-law index, and slenderness ratio on the frequencies of the embedded magneto-electro-elastic functionally graded nanobeams are studied.

On the 3-dimensional Hopf bifurcation via averaging theory of third order

We apply the averaging theory of third order to polynomial quadratic vector fields in $\mathbb{R}^3$ to study the Hopf bifurcation occurring in that polynomial. Our main result shows that at most $10$ limit cycles can bifurcate from a singular point having eigenvalues of the form $\pm bi$ and $0$. We provide an example of a quadratic polynomial differential system for which exactly $10$ limit cycles bifurcate from a such singular point.

Vibration analysis of FG nanobeams based on third-order shear deformation theory under various boundary conditions

In this study, free vibration of functionally graded (FG) micro/nanobeams based on nonlocal third-order shear deformation theory and under different boundary conditions is investigated by applying the differential quadrature method. Third-order shear deformation theory can consider the both small-scale effects and quadratic variation of shear strain and hence shear stress along the FG nanobeam thickness. The governing equations are obtained by using the Hamilton's principle, based on third-order shear deformation beam theory. The differential quadrature (DQ) method is used to discretize the model and attain the natural frequencies and mode shapes. The properties of FG micro/nanobeam are assumed to be chanfged along the thickness direction based on the simple power law distribution. The effects of various parameters such as the nonlocal parameter, gradient index, boundary conditions and mode number on the vibration characteristics of FG micro/nanobeams are discussed in detail.

Elastic Analysis of Carbon Nanotube-Reinforced Composite Plates with Piezoelectric Layers Using Shear Deformation Theory

This paper investigates the deflection and stress behavior of composite plates reinforced by single-walled carbon nanotubes (SWCNTs) with piezoelectric layers which are under transverse mechanical load. Two kinds of carbon nanotube-reinforced composite (CNTRC) plates, namely uniformly distributed (UD) and functionally graded (FG) along the plate thickness, are considered. The extended rule of mixture approach is used to estimate the effective material properties. The governing equations are derived using the Hamilton approach based on the first-order shear deformation plate theory (FSDT) and third-order shear deformation plate theory (TSDT). In addition, the Navier technique is employed to obtain the deflection and stress response of the nanocomposite plates. The results of present work are also compared with those available in the literature and show good agreement. The effects of applied force, volume fraction of CNT, distribution of CNT, thickness of piezoelectric layer, thickness to width ratio and aspect ratio on the static behavior are studied. In previous studies, deflection and stress analysis of nanocomposite plate with piezoelectric layers based on third-order shear deformation plate theory has not investigated.