A new higher-order shear deformation theory and refined beam element of composite laminates

A new higher-order shear deformation theory based on global-local superposition technique is developed. The theory satisfies the free surface conditions and the geometric and stress continuity conditions at interfaces. The global displacement components are of the Reddy theory and local components are of the internal first to third-order terms in each layer. A two-node beam element based on this theory is proposed. The solutions are compared with 3D-elasticity solutions. Numerical results show that present beam element has higher computational efficiency and higher accuracy.

KRMN-TYPE EQUATIONS FOR A HIGHER-ORDER SHEARDEFORMATION PLATE THEORY AND ITS USE IN THE THERMAL POSTBUCKLING ANALYSIS

arman-type nonlinear large deflection equations are derived occnrding to theReddy's higher-order shear deformation plate theory and used in the thermal postbuckling analysis The effects of initial geometric imperfections of the plate areincluded in the present study which also includes th thermal effects.Simply supported,symmetric cross-ply laminated plates subjected to uniform or nomuniform parabolictemperature distribution are considered. The analysis uses a mixed GalerkinGolerkinperlurbation technique to determine thermal buckling louds and postbucklingequilibrium paths.The effects played by transverse shear deformation plate aspeclraio, total number of plies thermal load ratio and initial geometric imperfections arealso studied.

AN ANALYTICAL SOLUTION OF RECTANGULAR LAMINATEDPLATES BY HIGHER-ORDER THEORY

On the basis of the Reddy's higher-order theory of composites, this paper introduces a displacement function Phi into it and transforms its three differential equations for symmetric cross-ply composites into only one eight-order differential equation generated by the displacement-function. When a proper Phi is chosen, both solutions are obtained, namely, the Navier-type solution of simply supported rectangular laminated plates and the Levy-type solution with the boundary condition where two opposite edges are simply supported and remains are arbitrary. The numerical examples show that the present results coincide well with the existing results in the references, thus validating that the present solving method is reliable. The higher-order theory of Reddy is simpler in calculation but has higher precision than the first-order shear deformation theory because the former has fewer unknows than the latter and requires no shear coefficients.

Higher-order electroelastic modelling of piezoelectric cylindrical nanoshell on elastic matrix

This paper develops electro-elastic relations of functionally graded cylindrical nanoshell integrated with intelligent layers subjected to multi-physics loads resting on elastic foundation.The piezoelectric layers are actuated with external applied voltage.The nanocore is assumed in-homogeneous in which the material properties are changed continuously and gradually along radial direction.Third-order shear deformation theory is used for the description of kinematic relations and electric potential distribution is assumed as combination of a linear function along thickness direction to show applied voltage and a longitudinal distribution.Electro-elastic size-dependent constitutive relations are developed based on nonlocal elasticity theory and generalized Hooke’s law.The principle of virtual work is used to derive governing equations in terms of four functions along the axial and the radial directions and longitudinal electric potential function.The numerical results including radial and longitudinal displacements are presented in terms of basic input parameters of the integrated cylindrical nanoshell such as initial electric potential,small scale parameter,length to radius ratio and two parameters of foundation.It is concluded that both displacements are increased with an increase in small-scale parameter and a decrease in applied electric potential.

Merger estimates for a disformal Kerr black hole in quadratic degenerate higher-order scalar-tensor theories

We investigate the main features of a disformal Kerr black hole merger in quadratic degenerate higher-order scalar-tensor theories.In the ringdown stage of the black hole merger,for the prograde orbit,the real part of the quasinormal modes decreases with an increase in the disformal parameter,and the imaginary part also decreases,except in the Kerr case for a large spin parameter.However,for the retrograde orbit,the real part increases with an increase in the disformal parameter,and the imaginary part always decreases with it.For the approximate final spin,regardless of an equal spin,unequal spin,or generic spin configuration merger,the final black hole spin always increases with an increase in the disformal parameter.Our results show that the disformal parameter in the disformal Kerr solution and the MOG parameter in the Kerr-MOG case have obviously different effects on the black hole merger,which suggests the differences between these two spacetime structures.

A two variable refined plate theory for the bending analysis of functionally graded plates

Bending analysis of functionally graded plates using the two variable refined plate theory is presented in this paper.The number of unknown functions involved is reduced to merely four,as against five in other shear deformation theories. The variationally consistent theory presented here has, in many respects,strong similarity to the classical plate theory. It does not require shear correction factors,and gives rise to such transverse shear stress variation that the transverse shear stresses vary parabolically across the thickness and satisfy shear stress free surface conditions.Material properties of the plate are assumed to be graded in the thickness direction with their distributions following a simple power-law in terms of the volume fractions of the constituents.Governing equations are derived from the principle of virtual displacements, and a closed-form solution is found for a simply supported rectangular plate subjected to sinusoidal loading by using the Navier method.Numerical results obtained by the present theory are compared with available solutions,from which it can be concluded that the proposed theory is accurate and simple in analyzing the static bending behavior of functionally graded plates.

SYMMETRY IN FIELD THEORY FOR SINGULAR LAGRANGIAN WITH DERIVATIVES OF HIGHER ORDER

We have derived the first Noether theorem and Noether identities in canonical formalism for field theory with higher-order singular Lagrangian,which is a powerful tool toanalyse Dirac constraint for such system. A gauge-variant system in canonical variables formalism must has Dirac constraint.For a system with first class constraint (FCC), we have developed an algorithm for construction of the gauge generator of such system. An application to the Podolsky generalized electromagnetic field was given.

Chebyshev polynomial-based Ritz method for thermal buckling and free vibration behaviors of metal foam beams

This study presents the Chebyshev polynomials-based Ritz method to examine the thermal buckling and free vibration characteristics of metal foam beams.The analyses include three models for porosity distribution and two scenarios for thermal distribution.The material properties are assessed under two conditions,i.e.,temperature dependence and temperature independence.The theoretical framework for the beams is based on the higher-order shear deformation theory,which incorporates shear deformations with higher-order polynomials.The governing equations are established from the Lagrange equations,and the beam displacement fields are approximated by the Chebyshev polynomials.Numerical simulations are performed to evaluate the effects of thermal load,slenderness,boundary condition(BC),and porosity distribution on the buckling and vibration behaviors of metal foam beams.The findings highlight the significant influence of temperature-dependent(TD)material properties on metal foam beams'buckling and vibration responses.

An advanced five-unknown higher-order theory for free vibration of composite and sandwich plates

Accurate prediction of dynamic characteristics is quite critical to understand the strength of layered structures.Nevertheless,the existing five-unknown higher-order theories encounter difficulties to forecast accurately the dynamic response of sandwich structures.Therefore,a new five-unknown higher-order theory is developed for free vibration analysis of composite and sandwich plates,which possesses the same degree of freedom as those of other five-unknown higherorder theories.The developed model can meet beforehand interlaminar continuity conditions and the free-surface conditions of transverse shear stresses.To assess capability of the proposed model,analytical solution for such composite structures with simply-supported conditions has been presented by employing Hamilton’s principle,which is utilized for analysis of mechanical behaviors of composite and sandwich plates.Compared with the three-dimensional(3 D)elasticity solutions,3 D finite element results and the results obtained from the chosen five-unknown higher-order models,the proposed model can yield accurately natural frequencies of composite and sandwich plates.Even for the thick plates,the higher-order frequencies calculated from the proposed model are in good agreement with the 3 D finite element results.By studying effect of the thickness/length ratios on natural frequencies,it is found that the proposed model is adaptable to predicting natural frequencies of the sandwich plates with the thickness/length ratios between 1/4 and 1/100.In addition,some factors influencing accuracy of five-unknown higher-order models have been investigated in detail.Finally,by means of numerical analysis and discussion,some conclusions have been drawn as well,which can serve as a reference for other investigators.

Exact solution for thermal vibration of multi-directional functionally graded porous plates submerged in fluid medium

An analytical method for analyzing the thermal vibration of multi-directional functionally graded porous rectangular plates in fluid media with novel porosity patterns is developed in this study.Mechanical properties of MFG porous plates change according to the length,width,and thickness directions for various materials and the porosity distribution which can be widely applied in many fields of engineering and defence technology.Especially,new porous rules that depend on spatial coordinates and grading indexes are proposed in the present work.Applying Hamilton's principle and the refined higher-order shear deformation plate theory,the governing equation of motion of an MFG porous rectangular plate in a fluid medium(the fluid-plate system)is obtained.The fluid velocity potential is derived from the boundary conditions of the fluid-plate system and is used to compute the extra mass.The GalerkinVlasov solution is used to solve and give natural frequencies of MFG porous plates with various boundary conditions in a fluid medium.The validity and reliability of the suggested method are confirmed by comparing numerical results of the present work with those from available works in the literature.The effects of different parameters on the thermal vibration response of MFG porous rectangular plates are studied in detail.These findings demonstrate that the behavior of the structure within a liquid medium differs significantly from that within a vacuum medium.Thereby,they offer appropriate operational approaches for the structure when employed in various mediums.

Electromechanical buckling behavior of smart piezoelectrically actuated higher-order size-dependent graded nanoscale beams in thermal environment

In the present work,thermo-electro-mechanical buckling behavior of functionally graded piezoelectric(FGP)nanobeams is investi-gated based on higher-order shear deformation beam theory.The FGP nanobeam is subjected to four types of thermal loading including uniform,linear,and sinusoidal temperature rise as well as heat conduction through the beam thickness.Thermo-electro-mechanical properties of FGP nanobeam are supposed to change continuously in the thickness direction based on power-law model.To consider the influences of small-scale sizes,Eringen’s nonlocal elasticity theory is adopted.Applying Hamilton’s princi-ple,the nonlocal governing equations of an FGP nanobeam in thermal environments are obtained and are solved using Navier-type analytical solution.The significance of various parameters,such as thermal loadings,external electric voltage,power-law index,nonlocal parameter,and slenderness ratio on thermal buck-ling response of size-dependent FGP nanobeams is investigated.

Bending and free vibrational analysis of bi-directional functionally graded beams with circular cross-section

The bending and free vibrational behaviors of functionally graded(FG)cylindrical beams with radially and axially varying material inhomogeneities are investigated.Based on a high-order cylindrical beam model,where the shear deformation and rotary inertia are both considered,the two coupled governing differential motion equations for the deflection and rotation are established.The analytical bending solutions for various boundary conditions are derived.In the vibrational analysis of FG cylindrical beams,the two governing equations are firstly changed to a single equation by means of an auxiliary function,and then the vibration mode is expanded into shifted Chebyshev polynomials.Numerical examples are given to investigate the effects of the material gradient indices on the deflections,the stress distributions,and the eigenfrequencies of the cylindrical beams,respectively.By comparing the obtained numerical results with those obtained by the three-dimensional(3D)elasticity theory and the Timoshenko beam theory,the effectiveness of the present approach is verified.

Wave Slamming on An OWSC Wave Energy Converter in Coupled Wave-Current Conditions with Variable-Depth Seabed

Coastal wave energy resources have enormous exploitation potential due to shorter weather window,closer installation distance and lower maintenance cost.However,impact loads generated by depth variation from offshore to nearshore and wave-current interaction,may lead to a catastrophic damage or complete destruction to wave energy converters(WECs).This objective of this paper is to investigate slamming response of a coastal oscillating wave surge converter(OWSC)entering or leaving water freely.Based on fully nonlinear potential flow theory,a time-domain wave-current-structure interaction model combined with higher-order boundary element method(HOBEM),is developed to analyze the coupled hydrodynamic problem.The variable-depth seabed is considered in the model to illustrate the shallow water effect on impact loads and free surface profiles in coastal zone.A domain decomposition approach is utilized to simulate the overlapping phenomenon generated by a jet falling into water under gravity effect.Through a series of Lagrangian interpolation methods,the meshes on boundaries are rearranged to avoid the mismatch between element size on free surface and body surface.The present model is validated against the existing experimental and numerical results.Simulations are also provided for the effects of wave-current interaction and uneven local seabed on the slamming responses.It is found that the length of the splash jet increases for a following current and decreases for an opposing current,and that the slamming response of the OWSC device is sensitive to the geometric features of the uneven seabed.

Higher-order vibration of thick composite and sandwich plates based on an alternative higher-order model

The transverse stretching vibration of thick sandwich plates,which is attributed to largely different stiffness at the adjacent layers,is a challenging issue,and efficient approach for such issue is less reported in the published literature.Thus,natural frequencies corresponding to stretching vibration modes are generally neglected in engineering design,which might impact structural safety as frequencies of the exciting force are close to transverse stretching vibration frequencies.This paper proposes an alternative higher-order model for dynamic analysis corresponding to the higher-order vibration modes.The proposed model is classified in the displacement-based equivalent single-layer theory,as the number of displacement parameters in the proposed model is independent of the layer number.The continuity of displacements and transverse shear stresses can be fulfilled at the interfaces between the adjacent layers of structures.To demonstrate the capability of the proposed model,typical examples are analyzed by utilizing the proposed model,the threedimensional finite element method and the chosen higher-order models.By comparing with the exact three-dimensional elasticity solutions,it is found that the proposed model can yield more accurate natural frequencies corresponding to the higher-order displacement modes than the selected models.Moreover,the factors influencing reasonable prediction of the higher-order frequencies are investigated in detail,which can provide a reference for the accurate prediction of the higher-order frequencies.

A Quadrilateral Element-based Method for Calculation of Multi-scale Temperature Field

In the analysis of functionally graded materials （FGMs）, the uncoupled approach is used broadly, which is based on homogenized material property and ignores the effect Of local micro-structural interaction. The higher-order theory for FGMs （HOTFGM） is a coupled approach that explicitly takes the effect of micro-structural gradation and the local interaction of the spatially variable inclusion phase into account. Based on the HOTFGM, this article presents a quadrilateral element-based method for the calculation of multi-scale temperature field （QTF）. In this method, the discrete cells are quadrilateral including rectangular while the surface-averaged quantities are the primary variables which replace the coefficients employed in the temperature function. In contrast with the HOTFGM, this method improves the efficiency, eliminates the restriction of being rectangular cells and expands the solution scale. The presented results illustrate the efficiency of the QTF and its advantages in analyzing FGMs.

Formal Reasoning About Finite-State Discrete-Time Markov Chains in HOL

Markov chains are extensively used in modeling different aspects of engineering and scientific systems, such as performance of algorithms and reliability of systems. Different techniques have been developed for analyzing Markovian models, for example, Markov Chain Monte Carlo based simulation, Markov Analyzer, and more recently probabilistic model- checking. However, these techniques either do not guarantee accurate analysis or are not scalable. Higher-order-logic theorem proving is a formal method that has the ability to overcome the above mentioned limitations. However, it is not mature enough to handle all sorts of Markovian models. In this paper, we propose a formalization of Discrete-Time Markov Chain （DTMC） that facilitates formal reasoning about time-homogeneous finite-state discrete-time Markov chain. In particular, we provide a formal verification on some of its important properties, such as joint probabilities, Chapman-Kolmogorov equation, reversibility property, using higher-order logic. To demonstrate the usefulness of our work, we analyze two applications： a simplified binary communication channel and the Automatic Mail Quality Measurement protocol.

Postbuckling behavior of 3D braided rectangular plates subjected to uniaxial compression and transverse loads in thermal environments

Postbuckling behavior of the 3D braided rectangular plates subjected to uniaxial compression combined with transverse loads in thermal environments is presented.Based on a micro-macro-mechanical model,a 3D braided composite may be treated as a cell system and the geometry of each cell is deeply dependent on its position in the cross-section of the plate.The material properties of the epoxy are expressed as a linear function of temperature.Uniform,linear and nonlinear temperature distributions through the thickness are involved.The lateral pressure(three types of transverse loads,i.e.transverse uniform load;transverse patch load over a central area;and transverse sinusoidal load)is first converted into an initial deflection and the initial geometric imperfection of the plate is taken into account.The governing equations are based on Reddy’s higher-order shear deformation plate theory with a von Kármán-type of kinematic nonlinearity.Two cases of the in-plane boundary conditions are also taken into account.A perturbation technique is employed to determine buckling loads and postbuckling equilibrium paths of simply supported 3D braided rectangular plates.The results reveal that the temperature rise,geometric parameter,fiber volume fraction,braiding angle,the character of the in-plane boundary conditions and different types of initial transverse loads have a significant effect on the buckling and postbuckling behavior of the braided composite plates.

Effect of noncircularity on the dynamic behaviors of particles in a disformal rotating black-hole spacetime

A disformal rotating black-hole solution is a black-hole solution in quadratic degenerate higher-order scalar-tensor theories.It breaks the circular condition of spacetime different from the case of the usual Kerr spacetime.This study investigated the dynamic behaviors of the motion of timelike particles in such disformal black-hole spacetime with an extra deformation parameter.Results showed that the characteristics of the particle’s motion depend on the sign of the deformation parameter.For the positive deformation parameter,the motion is regular and orderly.For the negative one,as the deformation parameter changes,the motion of the particles undergoes a series of transitions between the chaotic motion and the regular motion and falls into the horizon or escapes to spatial infinity.This means that the dynamic behavior of timelike particles in the disformal Kerr black-hole spacetime with noncircularity becomes richer than that in the usual Kerr black-hole case.

Impact of aperture on dynamic response of sandwich plates with aluminium face-sheets and PMI core based on an alternative finite element formulation and experiments

Apertures generally exist in the sandwich structures attributing to mechanical connection and lightweight, which might induce failure of such structures. Thus, it is required to realize the impact of aperture on mechanical behaviors of sandwich structures. If transverse shear deformations are unable to be described accurately, the reasonable prediction of dynamic behaviors of the form-core sandwich plates with apertures will meet severe challenges due to a large difference of transverse shear modulus at the adjacent layers. Thereby, such issue is less studied by using the efficient models and experimental testing, so an alternative sinusoidal-type finite element formulation is to be proposed to precisely predict dynamic response of the form-core sandwich structures with apertures. The proposed finite element formulation can meet beforehand compatible conditions of transverse shear stresses at the interfaces of adjacent laminates. In order to appraise strictly capability of the proposed model, experimental tests on natural frequencies of three groups of specimens with different apertures have been carried out. Moreover, four specimens in each group are tested to reduce the testing errors, which is less reported in the published literature. In addition,three-dimensional Finite Element Method(3-D FEM) is also selected to account for the good performance of the present model. Finally, the impact of aperture diameter on the natural frequencies of the sandwich structures is both experimentally and numerically investigated, which can serve as a reference for other researchers.

Formally Analyzing Expected Time Complexity of Algorithms Using Theorem Proving

Probabilistic techniques are widely used in the analysis of algorithms to estimate the computational complexity of algorithms or a computational problem.Traditionally,such analyses are performed using paper-and-pencil proofs and the results are sometimes validated using simulation techniques.These techniques are informal and thus may result in an inaccurate analysis.In this paper,we propose a formal technique for analyzing the expected time complexity of algorithms using higher-order-logic theorem proving.The approach calls for mathematically modeling the algorithm along with its inputs,using indicator random variables,in higher-order logic.This model is then used to formally reason about the expected time complexity of the underlying algorithm in a theorem prover.The paper includes the higher-order-logic formalization of indicator random variables,which are fundamental to the proposed infrastructure.In order to illustrate the practical effiectiveness and utilization of the proposed infrastructure,the paper also includes the analysis of algorithms for three well-known problems,i.e.,the hat-check problem,the birthday paradox and the hiring problem.