Differential Cubature Research Articles

Transient dynamic, energy absorption and out-phase/in-phase vibration response of coupled annular/circular SMA nanoplates assuming surface effects on frictional substrates

Shape and size are two important factors that affect the mechanical properties of nanostructures. Therefore, various forms of nano-scaled structures such as annular and circular ones have been applied in the engineering fields. Shape Memory Alloy (SMA), which can be used as the actuator because of its ability to change shapes, has potential applications in nano-electro mechanical system (NEMS). Dynamic behavior and the ability to absorb energy are two critical characteristics of advanced structures that are under static or various forms of dynamic loadings to be specific, this research numerically examined the in-phase/out-phase vibration, transient dynamic behavior under impulse load, and energy absorption in SMA annular/circular nanoplates with ferroelastic and pseudoelastic effects coupled together via damper and spring, placed between two frictional layers. In addition, an assumption of the Kuhn–Tucker phase transformation conditions was made. Moreover, the governing equations were derived based on Hyperbolic Higher Order Shear Deformation Theory (HHSDT) and solved using Differential Cubature Method (DCM). The constitutive equation of SMA proposed by Brinson was employed to model the pseudo-elastic behavior of SMA. Besides, Gurtin-Murdoch and modified couple stress theories were employed to calculate surface and micro-scale effects. As revealed by numerical achievements, under out-phase vibration mode, by increasing the surface effect coefficient to 0.4 when the annular nanoplate inner to outer radius ratio was 0.35, the natural frequency enhanced about 174%. As the material length scale parameter to thickness ratio was increased to 0.3 and the surface effects were considered, the energy absorption capacity was enhanced by nearly 28% and 23%, respectively.

Vibration smart control analysis of sandwich porous nanoshell with GPL-reinforced smart layers using surface-visco-piezo-elasticity theory

Smart nanostructures have many practical application in nano-electro-mechanical systems (NEMS) as sensor and actuator. Present research deals with the vibration and damping of advanced sandwich smart nanoshells. This nanostructure comprised of porous nanoshell coated by nanocomposite piezoelectric layers as sensor and actuator. The nanocompisite layers are reinforced by graphene platelets (GPLs) with various distributions based on Halpin-Tsai model. In order to presume this sandwich structure much more realistic, Kelvin model will be utilized and it is surrounded by visco-Pasternak foundation. Further, non-local surface piezoelasticity theory is employed for considering the small scale and surface effects. On the basis of zigzag shear-deformation theory and Hamilton principle, the motion equations are derived. Differential cubature method (DCM) is used as numerical method for calculating natural frequency and damping ratio. In order to validate the results, this paper is compared by another simplified cases and ABAQUS commercial software. Finally, the influences of porosity and its distribution, volume percent and distribution of GPLs, surface stresses, small scale parameter, and external voltage are shown on natural frequency and damping. The results show that the damping ratio and frequency of system considering surface effect with symmetric porosity distribution decrease and increases about 17% and 25% compared to the without surface effect case and asymmetric porosity respectively. In addition, the nanostructure in critically damped state can be turned into underdamped state by applying negative voltage. Furthermore, the dispersion of GPLs near the bottom and top surface of nanocomposite layers is the best choice due to high stiffness and frequency.

Energy absorption and vibration of smart auxetic FG porous curved conical panels resting on the frictional viscoelastic torsional substrate

In recent years, the utilization of various methods to absorb vibration in widely-used structures has gained significant attention to control and reduce the destructive effect of the resonance phenomenon. The increased useful lifetime and reduced maintenance and repair costs are the advantages of vibration absorption. Besides, utilization of energy absorbers is one of the most effective strategies for reducing energy supply costs for different parts of the structure, such as sensors. In this regard, a numerical study is performed in this research to evaluate the energy absorption by vibrations' absorption in a conical three-layered panel located on a viscoelastic substrate. This three-layered panel is composed of three parts: piezoelectric upper layer reinforced with Graphene nanoplatelets (GPLs), auxetic core, and functionally graded material (FGM) porous lower layer. The substrate includes spring element, shear, torsion, and substrate-structure friction factor. Hamilton's principle and energy method have been used to extract the governing equations to model the first-order shear deformation theory (FSDT) for the presented structure. The Differential Cubature Method (DCM) and the Newmark method have been used to solve the governing equations. Results show that by applying the negative voltage to the sandwich conical panel, the in-plane compression forces and natural frequency are increased, while the maximum dynamic deflection is decreased. Furthermore, increasing the inherent damping coefficient increases the damping decrement. Meanwhile, the nonlinear scattering and increased dispersion concentration of GPLs near the upper and lower surfaces of the piezoelectric layer will increase the frequency and energy absorption. Besides, the increased friction factor of the viscoelastic substrate and sandwich cone leads to more absorption energy.

A Dynamic Procedure for Time and Space Domain Based on Differential Cubature Principle

Based on the differential cubature (DC) principle, a dynamic procedure for simultaneous discretization of time and space is developed. A spatial–temporal differential cubature analysis method for dynamic problems is established with the Timoshenko shear beam; the reliability of analysis results obtained by which is verified, and the stability of the numerical scheme is studied. This method is extended to the two-dimensional structure, and the forced vibration analysis is carried out with the thin plate as an example. The research shows that the method can acquire highly accurate numerical results, and the calculated time-history numerical solution of beam displacement is extremely consistent with the analytical solution, which can adopt to the changes in beam properties and load parameters. With fewer nodes and longer time step than the finite element method (FEM), the method in this paper can still obtain stable and accurate results when solving displacement responses of plate under forced vibration. The numerical stability of this method is closely related to the grid form and the size of time step, and the increase in the number of nodes in the time domain is conducive to increasing the stability range.

Dynamic deflection and contact force histories of graphene platelets reinforced conical shell integrated with magnetostrictive layers subjected to low-velocity impact

The present paper concerns low-velocity impact for nanocomposite sandwich truncated conical shells (NSTCS). Graphene platelets (GPLs)-reinforced as core layer is covered through magnetostrictive layers as face sheets. The supposed impacts are applied over the above face layer and further, the interaction among impactors as well as NSTCS is assumed utilizing novel equivalent three-degree-of-freedom (TDOF) with spring–mass–damper (SMD) model. For modeling the core layer and face sheets mathematically, higher-order shear deformation theory (HSDT) besides first-order shear deformation or Mindlin theory (FSDT) is utilized, respectively. To presume this sandwich structure much more realistic, the Kelvin–Voigt model will be used. According to Hamilton’s principle concerning continuity boundary conditions, the governing equations are obtained. Utilizing differential cubature (DC) as well as Bolotin procedures, the governing equations will be solved. In this work, different variables covering various boundary edges, Controller, cone’s semi vertex angle, damping,​ feedback gain, the proportion of core to face sheets thickness, dispersion kinds of GPLs and their volume percent, and their effects on low-velocity impact analysis of the sandwich structure will be investigated. To indicate the accuracy of applied theories as well as methods. the results are collated with another paper. It is found that increment of GPLs volume percent leads to rise the deflection as well as maximum contact force whereas contact duration is reduced.

Dynamic stability response of truncated nanocomposite conical shell with magnetostrictive face sheets utilizing higher order theory of sandwich panels

Present research is conducted in order to assess dynamic stability behavior of a nanocomposite sandwich truncated conical shells (NSTCS). In fact, graphene platelets (GPLs)-reinforced as core layer is encompassed through magnetostrictive layers as face sheets. For modeling the core layer and face sheets mathematically, higher order shear deformation theory (HSDT) besides first order shear deformation theory (FSDT) are utilized, respectively. To presume this sandwich structure much more realistic, Kelvin-Voigt model will be used. According to Hamilton's principle with respect to continuity boundary conditions, the governing equations are obtained. Utilizing differential cubature (DC) as well as Bolotin procedures, the governing equations will be solved and the region related to the dynamic instability is achieved. In this novel work, different variables covering various boundary edges, controller, cone's semi vertex angle, damping, feedback gain, proportion of core to face sheets thickness, dispersion kinds of GPLs and its volume percent will be studied. So as to indicate the accuracy of applied theories as well as methods, the results are collated with another paper. It is found that increment of GPLs volume percent leads to rise of excitation frequency.

Application of differential cubature method for nonlocal vibration, buckling and bending response of annular nanoplates integrated by piezoelectric layers based on surface-higher order nonlocal-piezoelasticity theory

This paper deals with the vibration, buckling and bending analyses of annular nanoplate integrated with piezoelectric layers at the top and bottom surfaces. The higher order nonlocal theory for size effect and Gurtin–Murdochtheory for surface effects are utilized. The governing equations are derived based on the layer-wise (LW) theory and Hamilton’s principle. The differential cubature method (DCM) as a new numerical procedure is utilized to solve the motion equations for obtaining the frequency, buckling load and deflection. The influences of various parameters such as external voltage, boundary condition, surface stresses, nonlocal parameter, outer to inner radius ratio and core to top layer thickness ratio were shown on the vibration, buckling and bending responses of the nanostructure. The results of vibration, buckling and bending are validated with other published works. The outcomes show that the surface stresses have a significant effect on the increases of the frequency and buckling load and decrease of the deflection.

Dynamic response of auxetic honeycomb plates integrated with agglomerated CNT-reinforced face sheets subjected to blast load based on visco-sinusoidal theory

This paper deals with the dynamic response of sandwich plates subjected to blast load. The sandwich plate is composed of auxetic honeycombs core layer with negative Poisson's ratio integrated with nanocomposite at the top and bottom surfaces. The nanocomposite layers are reinforced by carbon nanotubes (CNTs) where the effective material properties are calculated by Mori-Tanaka approach considering agglomeration effects. Due to the existence of CNTs, the structure is subjected to magnetic field. Based on the Kelvin-Voigt theory, the viscoelastic properties of the structure are considered. The viscoelastic medium is simulated by orthotropic visco-Pasternak model. Utilizing sinusoidal shear deformation theory (SSDT), the motion equations are derived based on Hamilton's principle. Differential cubature method (DCM) in conjunction with Newmark method is used for obtaining the dynamic deflection of the sandwich structure for different boundary conditions. The effects of various parameters such as structural damping, viscoelastic medium constants, geometric parameters of plates, volume fraction and agglomeration of CNTs, duration of blast pulse, geometrical parameters of auxetic honeycombs core and magnetic field on the dynamic deflection of the sandwich structure are investigated. The results reveal that by reinforcing the structure with CNTs, the dynamic deflection induced by blast load decreases about 59%. In addition, with increasing the geometrical parameters of auxetic honeycombs core, the dynamic deflection will be enhanced.

A new numerical approach and visco-refined zigzag theory for blast analysis of auxetic honeycomb plates integrated by multiphase nanocomposite facesheets in hygrothermal environment

The current work suggests a mathematical model for the dynamic response of sandwich plates subjected to a blast load using a numerical method. The sandwich structure is made from an auxetic honeycomb core layer integrated by multiphase nanocomposite facesheets. The facesheets are composed of polymer–carbon nanotube (CNT)–fiber where the equivalent material properties of the multiphase nanocomposite layers are obtained using fiber micromechanics and Halpin–Tsai equations in hierarchy. The top and bottom layers are subjected to magnetic field and the material properties of them are assumed temperature and moisture dependent. The Kelvin–Voigt model is employed to consider the viscoelastic properties of the structure. The sandwich structure is rested on a viscoelastic foundation which is modeled by orthotropic visco-Pasternak medium. Based on refined zigzag theory (RZT), energy method and Hamilton’s principle, the motion equations are derived. A new numerical method, namely differential cubature method (DCM) in conjunction with Newmark method is utilized for obtaining the dynamic deflection of the structure for different boundary conditions. The effects of various parameters such as blast load, viscoelastic foundation, structural damping, magnetic field, volume fraction of CNTs, temperature and moisture changes, geometrical parameters of honeycomb layer and sandwich plate are considered on the dynamic deflection of the structure. The results show that the magnetic field to the facesheets can be considered as effective parameters to control the dynamic deflection. In addition, hygrothermal condition leads to increase of 24% in the dynamic displacement of system.

Dynamic buckling of magnetorheological fluid integrated by visco-piezo-GPL reinforced plates

This paper deals with the dynamic buckling analysis of sandwich plates with magnetorheological (MR) fluid core and piezoelectric nanocomposite facesheets. The core is subjected to magnetic field and the facesheets are exposed to 3D electric field. Due to the piezoelectric properties of the facesheets, the top and bottom layers can be used as actuator and sensor, respectively. Hence, a proportional-derivative (PD) controller is employed to control the dynamic buckling and vibration responses of the structure. In addition, the facesheets are reinforced by with non-uniform graphene platelets (GPLs) which their equivalent material properties are obtained using Halpin-Tsai micromechanics model. The structural damping of the piezoelectric layers is considered according to Kelvin-Voigt theory. The sandwich structure is rested on orthotropic viscoelastic model with normal, shear and damping forces. Based on the sinusoidal shear deformation theory (SSDT) and piezoelasticity theory, the motion equations are derived utilizing Hamilton's principle. Applying differential cubature method (DCM), the motion equations are solved for calculating the dynamic instability region (DIR) of the structure. The influences of various parameters such as magnetic field, applied voltage, structural damping, viscoelastic medium, volume fraction and distribution of GPLs, boundary conditions and geometric parameters of the structure are shown on the dynamic buckling behavior of the system. The results are validated with other published work for indication the accuracy of the obtained results. The results reveal that by applying the magnetic field to the MR fluid core, the DIR with be happened at higher excitation frequencies.

A comparative study on the bending, vibration and buckling of viscoelastic sandwich nano-plates based on different nonlocal theories using DC, HDQ and DQ methods

In this paper, bending, buckling and buckling of embedded nano-sandwich plates are investigated based on refined zigzag theory (RZT), sinusoidal shear deformation theory (SSDT), first order shear deformation theory (FSDT) and classical plate theory (CPT). In order to present a realistic model, the material properties of system are assumed viscoelastic using Kelvin–Voigt model. The elastic medium is simulated by orthotropic visco-Pasternak medium. Based on energy method and D'Alembert's principle, the derivation of the nonlinear equations of motion is presented. A novel numerical method namely as differential cubature (DC) method is applied for obtaining the static response, the natural frequencies and the buckling loads of nano-sandwich plates. The effects of different parameters such as nonlocal parameter, structural damping, viscoelastic foundation, geometrical parameters, stiffness of core and boundary conditions are shown on the bending, buckling and vibration behaviors of nanostructure. The accuracy of the proposed method is verified by comparing its numerical predictions with other published works as well as solution of system with differential quadrature (DQ) and harmonic differential quadrature (HDQ) methods. The numerical investigation shows that RZT is highly accurate in predicting the deflection, frequency and buckling load of nano-sandwich plates without requiring any shear correction factors.

Determination of isolation layer thickness for undersea mine based on differential cubature solution to irregular Mindlin plate

The differential cubature solution to the problem of a Mindlin plate lying on the Winkler foundation with two simply supported edges and two clamped edges was derived. Discrete numerical technology and shape functions were used to ensure that the solution is suitable to irregular shaped plates. Then, the mechanical model and the solution were employed to model the protection layer that isolates the mining stopes from sea water in Sanshandao gold mine, which is the first subsea mine of China. Furthermore, thickness optimizations for the protection layers above each stope were conducted based on the maximum principle stress criterion, and the linear relations between the best protection layer thickness and the stope area under different safety factors were regressed to guide the isolation design. The method presented in this work provides a practical way to quickly design the isolation layer thickness in subsea mining.

Visco-nonlocal-refined Zigzag theories for dynamic buckling of laminated nanoplates using differential cubature-Bolotin methods

Present analysis, deals with dynamic buckling of sandwich nano plate (SNP) subjected to harmonic compressive load based on nonlocal elasticity theory. The material properties of each layer of SNP are supposed to be viscoelastic based on Kelvin-Voigt model. In order to mathematical modeling of SNP, a novel formulation, refined Zigzag theory (RZT) is developed. Furthermore, the surrounding elastic medium is simulated by visco-orthotropic Pasternak foundation model in which damping, normal and transverse shear loads are taken into account. Using energy method and D′Alembert's principle, the size dependent governing motion equations are derived. In this study, the governing motion equations are solved numerically using new procedure namely differential cubature (DC) method in conjunction with Bolotin method. The effects of some remarkable parameters such as viscoelastic foundation, damping coefficient of viscoelastic plates, aspect ratio, amount of small scale effect, various boundary conditions, different values of fiber orientation of the face sheets, number of grid points and thickness-length ratio on the dynamic instability region (DIR) are investigated. The results show that considering viscoelastic property of system is essential to obtain real mechanical behavior and instability of systems. In addition, the surrounding elastic medium is an effective parameter on the DIR of SNP.

Differential cubature method for vibration analysis of embedded FG-CNT-reinforced piezoelectric cylindrical shells subjected to uniform and non-uniform temperature distributions

Vibration analysis of embedded functionally graded (FG)-carbon nanotubes (CNT)-reinforced piezoelectric cylindrical shell subjected to uniform and non-uniform temperature distributions are presented. The structure is subjected to an applied voltage in thickness direction which operates in control of vibration behavior of system. The CNT reinforcement is either uniformly distributed or functionally graded (FG) along the thickness direction indicated with FGV, FGO and FGX. Effective properties of nano-composite structure are estimated through Mixture low. The surrounding elastic foundation is simulated with spring and shear constants. The material properties of shell and elastic medium constants are assumed temperature-dependent. The motion equations are derived using Hamilton's principle applying first order shear deformation theory (FSDT). Based on differential cubature (DC) method, the frequency of nano-composite structure is obtained for different boundary conditions. A detailed parametric study is conducted to elucidate the influences of external applied voltage, elastic medium type, temperature distribution type, boundary conditions, volume percent and distribution type of CNT are shown on the frequency of system. In addition, the mode shapes of shell for the first and second modes are presented for different boundary conditions. Numerical results indicate that applying negative voltage yields to higher frequency. In addition, FGX distribution of CNT is better than other considered cases.

Differential cubature and quadrature-Bolotin methods for dynamic stability of embedded piezoelectric nanoplates based on visco-nonlocal-piezoelasticity theories

This paper is concerned with the dynamic stability response of an embedded piezoelectric nanoplate made of polyvinylidene fluoride (PVDF). In order to present a realistic model, the material properties of nanoplate are assumed viscoelastic using Kelvin–Voigt model. The visco-nanoplate is surrounded by viscoelastic medium which is simulated by orthotropic visco-Pasternak foundation. The PVDF visco-nanoplate is subjected to an applied voltage in the thickness direction. In order to satisfy the Maxwell equation, electric potential distribution is assumed as a combination of a half-cosine and linear variation. Adopting the nonlocal Mindlin plate theory, the governing equations are derived based on the energy method and Hamilton’s principle. A novel numerical method namely as differential cubature method (DCM) in conjunction with the Bolotin’s method is applied to obtain the dynamic instability region (DIR) and parametric resonance responses of the visco-nanoplate. The effects of different parameters such as nonlocal parameter, external electric voltage, structural damping, boundary condition, dimension of nanoplate and viscoelastic medium are shown on DIR and parametric resonance frequency of structure. The accuracy of the proposed method is verified by comparing its numerical prediction with other theoretical and experimental published works as well as solution of system with differential quadrature method (DQM). Results indicate that the external electric voltage increases the frequency of the system especially in thick nanoplates.

Visco-surface-nonlocal piezoelasticity effects on nonlinear dynamic stability of graphene sheets integrated with ZnO sensors and actuators using refined zigzag theory

Control and analyze the nonlinear dynamic stability of single layered graphene sheets (SLGSs) integrated with zinc oxide (ZnO) actuators and sensors are the main contributions of present work. In order to present a realistic model, the material properties of system are assumed viscoelastic using Kelvin–Voigt model. The surrounding elastic medium is simulated with nonlinear orthotropic visco-Pasternak foundation. Considering refined zigzag theory (RZT), a new formulation is developed through the Gurtin–Murdoch elasticity theory in which the effects of surface stress are incorporated. A novel numerical procedure namely as differential cubature (DC) method is applied for solution. A proportional-derivative (PD) controller is used for the active control of the system dynamic stability. The detailed parametric study is conducted, focusing on the combined effects of the nonlocal parameter, magnetic field, viscoelastic foundation, surface stress, applied voltage, controller and structural damping on the dynamic instability region (DIR) of system. The accuracy of the proposed method is verified by comparing its numerical prediction with other published works as well as solution of system with differential quadrature (DQ) and harmonic differential quadrature (HDQ) methods. Results depict that the magnetic field and external voltage are effective controlling parameters for DIR of system.

Non-Newtonian pulsating blood flow-induced dynamic instability of visco-carotid artery within soft surrounding visco-tissue using differential cubature method

The high blood rate that often occurs in carotid arteries may play a role in artery failure and tortuosity which leads to blackouts, transitory ischemic attacks, and other diseases. However, dynamic analysis of carotid arteries conveying blood is lacking. The objective of this study was to present a biomechanical model for dynamic instability analysis of the embedded carotid arteries conveying pulsating blood flow. In order to present a realistic model, the carotid arteries and tissues are assumed viscoelastic using Kelvin–Voigt model. Carotid arteries are modeled as elastic cylindrical vessels based on Mindlin cylindrical shell theory (MCST). One of the main advantages of this study is considering the pulsating non-Newtonian nature of the blood flow using Carreau, Casson, and power law models. Applying energy method, Hamilton’s principle and differential cubature method (DCM), the dynamic instability region (DIR) of the visco-carotid arteries is obtained. The detailed parametric study is conducted, focusing on the combined effects of the elastic medium and non-Newtonian models on the dynamic instability of the visco-carotid arteries. It can be seen that with increasing the tissue stiffness, the natural frequency of visco-carotid arteries decreases. The current model provides a powerful tool for further experimental investigation about arterial tortuosity.

Nonlinear surface and nonlocal piezoelasticity theories for vibration of embedded single-layer boron nitride sheet using harmonic differential quadrature and differential cubature methods

Surface stress and small-scale effects on nonlinear vibration analysis of a single-layer boron nitride sheet are investigated based on theories of nonlocal and surface piezoelasticity. The single-layer boron nitride sheet is embedded in an elastic medium which is simulated by Pasternak model. Considering electromechanical coupling, the discretize governing equations of motion are obtained by Hamilton’s principal. Two numerical methods, harmonic differential quadrature and differential cubature, are employed to determine the nondimensional nonlinear frequency of single-layer boron nitride sheet. The detailed study parameters are conducted to investigate the influences of the small-scale, temperature change, surface effects, elastic medium, vibrational mode, aspect ratio, and smartness of sheet on the nonlinear nonlocal vibration characteristics of single-layer boron nitride sheet. It is concluded that ignoring surface and small-scale effects lead to inaccurate results in vibrational response of the single-layer boron nitride sheet. Furthermore, differential cubature method yields accurate results with less computational effort with respect to harmonic differential quadrature method. The obtained results of this study may be useful for designing of nano-electro mechanical system or micro-electro mechanical system and other nano-/micro-smart structures.

Differential Cubature Method for Static Solution of Laminated Shells of Revolution with Mixed Boundary Conditions

The bending analysis of laminated shells of revolution, such as spherical, conical and cylindrical panels, is carried out utilizing the differential cubature method (DCM). To do so, a general software based on the DCM is developed which can tackle shells of revolution with symmetric and unsymmetric lamination sequence. Analysis of shells with general Loading and various combinations of clamped, simply supported, free and mixed boundary condition, may be carried out having acceptable accuracy. Using first order shear deformation theory, fifteen first order partial differential equations are obtained which contain fifteen unknowns in terms of displacements, rotations, moments and forces. Utilizing all of these equations results in the capability of the method to deal with any kinds of boundary conditions. Comparison of the results obtained by the DCM, shows very good agreement with the results of other numerical and analytical methods, while having less computational effort.

Static bending analysis of laminated cylindrical panels with various boundary conditions using the differential cubature method

Static bending analysis of laminated cylindrical panels with various boundary conditions using the differential cubature method