Semi-analytical Polynomial Method Research Articles

On a comprehensive analysis for mechanical problems of spherical structures

This paper applies a semi-analytical polynomial method (SAPM) to solve the mechanical governing equations of nonlinear static and dynamic deformations, free and forced vibrations, and buckling analyses of nano-sized spherical functionally graded structures. Due to the nano-sized structure, the constitutive equations of motion are obtained based on the nonlocal elasticity theory. The governing equations are solved to obtain the deformations, critical buckling loads, and natural frequencies of the analyzed structure. The SAPM is based on polynomial functions (with no boundary conditions), including unknown finite coefficients. The spherical coordinate system is assumed to obtain several geometrical structures: cylindrical, conical, circular, sectorial, and rectangular. The nonlinear strains and the first-order shear deformation theory (FSDT) are employed to model the structure. The research considers the von Kármán assumptions to formulate the nonlinear strain components in the spherical coordinate system. The obtained results are compared with those in other articles to examine the accuracy of the applied solution method. The formulation's generality and simplicity and the appropriate accuracy of the SAPM's results distinguish this solution method for analyzing all mechanical aspects.

Viscoelasticity in Large Deformation Analysis of Hyperelastic Structures.

In this paper, an annular/circular plate made of hyperelastic material and considering the viscoelastic property was investigated based on a novel nonlinear elasticity theory. A new approach for hyperelastic materials in conjunction with the Kelvin-Voigt scheme is employed to obtain the structure's large deformation under uniform transverse loading. The constitutive equations were extracted using the energy method. The derived partial differential time-dependent equations have been solved via the semi-analytical polynomial method (SAPM). The obtained results have been validated by ABAQUS software and the available paper. In consequence, a good agreement between the results was observed. Finally, several affecting parameters on the analysis have been attended to and studied, such as the nonlinear elasticity analysis, the boundary conditions, loading, and the material's viscosity. It can be possible to obtain the needed time for achieving the final deformation of the structure based on the applied analysis in this research.

A Novel Nonlinear Elasticity Approach for Analysis of Nonlinear and Hyperelastic Structures

There are many materials that the linear elasticity theory cannot predict their mechanical behavior against applied loads. This research proposes a comprehensive theoretical method to obtain the mechanical response of hyperelastic models (with nonlinear elastic deformations) such as polymers and rubbers. They are vital in the design phase of complicated engineering structures like engine mounts and structural bearings in aerospace and automotive industries. The presented theory is implemented in detail and has no limitations in analyzing geometrically and physically nonlinear materials. As a test case, the governing equations of a sheet made of nonlinear elastic material are derived within the framework of this new approach. The derived governing equations are completely nonlinear in all major directions. Consequently, results can be one of the most accurate mathematical simulation results for nonlinear elastic material structures. Then, the obtained equations are solved using a meshless solution method named the semi-analytical polynomial method. The stress and deformation results of the structure under uniform and non-uniform transverse external loadings are obtained for different types of boundary conditions, loading, and material properties. Consequently, this research can be widely used as a suitable essential reference for researchers studying the mechanical behavior of nonlinear elastic structures.

On the mechanical analysis of microcrystalline cellulose sheets

In this research, it has been tried to study the mechanical behavior of a micro annular circular sheet made of microcrystalline cellulose material. A new combination between the modified higher-order shear deformation, also, quasi-three-dimensional displacement fields and the modified couple stress theory has been presented for simulating the mentioned micro-sheet under the transverse loading. Consequently, two types of two and three-dimensional equations in micro-size have been obtained. By solving the obtained governing equations with semi-analytical polynomial method, deflection results have been depicted in form of several diagrams. Due to the use of modified couple stress theory, there is a micro-scale parameter in the analysis that requires laboratory experiments for determining its specific values. Due to this purpose, a micro-cellulose sheet has been tested to determine its deformations under the transverse loading, in consequence, the micro-length scale parameter has been calculated by comparison between the theoretical and test results. It is demonstrated that the size of the analyzed sheet (here the amount of its radius) has the most impact on the value of micro-length scale parameter. This research can be very useful for researchers who study the micro-scale parameter in couple stress theory as well as the interesting properties of structures made of microcrystalline cellulose.

On the shell model for human eye in Glaucoma disease

A new mechanical and analytical models of the human eye is presented in order to study Glaucoma disease and its complications. Mechanical modeling of the human eye is performed with a hollow sphere under internal pressure. The first-order shear deformation theory is used to obtain the governing equations of the model, which are the set of partial differential equations. The nonlinear von Kármán assumption is considered for strain field to obtain the more precise results. The viscoelastic effects on the material structure of the eye are also considered to be more consistent with the results. The obtained governing equations and boundary conditions are solved using Semi-Analytical Polynomial Method. The results are studied for Glaucoma and the effects of this disease on the patient's vision as well as the temporary and permanent deformities caused in the patient's eyeball are further investigated. The modeling and information obtained from this study can help clinicians to provide more appropriate therapeutic strategies. Stress concentration can be identified in the eyeball tissue of patients suffering from Glaucoma using the simulations presented in this study. Given the generality of the proposed model, not only Glaucoma disease but also a wide range of eye diseases can be studied.

Nonlocal elasticity analysis of moderately thick porous functionally graded plates in a hygro-thermal environment

This work proposes a novel quasi three-dimensional (3D) approach for the bending analysis of moderately thick plates with a functionally graded material (FGM), in presence of porosities due to some incorrect manufacturing processes. Such porosities can appear within the plate in two forms, namely, even and uneven distributions. The modeled system has a polymer matrix where both shear and transverse factors coexist. The bending equations of the problem are obtained from the Hamiltonian principle. In order to apply the quantum effects for the nanosystem, the well-known nonlocal theory of Eringen is here assumed, while checking for its numerical accuracy. A physically-consistent analysis of the nanostructures would investigate possible surrounding effects. Thus, the thermal and humidity influence is accounted for the 3D problem, whose governing equations are solved through a semi-analytical polynomial method (SAPM), as recently proposed in literature for different applications. The proposed method is based on a simple procedure with very accurate numerical outcomes, whose performance is checked against the available literature. After computing the deflection relations, a systematic study is performed for the bending response of nanoporous FGMs in a hygro-thermal surrounding environment, with promising results for practical applications.

Mechanical analysis of eccentric defected bilayer graphene sheets considering the van der Waals force

In this article, we have tried to simulate nonlinear bending analysis of a double-layered graphene sheet which contains a geometrical imperfection based on an eccentric hole. The first-order shear deformation theory is considered to obtain the governing equations. Also, the nonlinear von Kármán strain field has been assumed in order to obtain large deformations. Whereas the double-layered graphene sheet has been considered, the effect of van der Waals forces has been taken into account in the analysis. In order to implement the nanoscale impact, the nonlocal elasticity theory has been employed. The solution methodology, which is here based on the semi-analytical polynomial method solving technique presented previously by the authors, has been applied and again its efficiency has been demonstrated due to its highly accurate results. Due to the fact that this research has been done for the first time and there is no validation available, the results of the local single layer sheet are compared with ABAQUS software. The effects of some other parameters on the results have been studied such as the value of eccentricity, van der Waals interaction, and nonlocal parameter.

On the effect of viscoelasticity on behavior of gyroscopes

In the present study, the size- and time-dependent viscoelastic bending analysis of rotating spherical nanostructures made of functionally graded materials (FGMs) is performed. To simulate gyroscopic rotation, the structure is assumed to be rotated around two axes with constant angular accelerations. The nonhomogeneous nanostructure with functionally variable thickness (FVT) is assumed as FGM/FVT structure. Geometrically, the inner side of the structure has a spherical shape and the outer side is considered as an ellipse or any other arbitrary function. The size-dependent governing equations and related boundary conditions are obtained by implementing Hamilton's principle based on Eringen's nonlocal elasticity and first-order shear deformation theories. The resulting equations are solved using the Semi-Analytical Polynomial Method (SAPM). The time-dependent deflection is obtained by applying the boundary and initial time conditions. The effects of a hygro-thermal environment and porosity are also considered for a comprehensive analysis. Since no research has been conducted so far on the nonlocal viscoelastic dynamic analysis of FGM/FVT gyroscopes, the classical results are compared with the previously published works and ABAQUS software as well as the effect of many parameters on the results including nonlocality, FGM properties of the structure, thickness changes, angular velocities and accelerations, hygro-thermal environment, porosity, and viscoelastic properties are thoroughly conducted.

A Novel Method for Numerical Analysis of 3D Nonlinear Thermo-Mechanical Bending of Annular and Circular Plates with Asymmetric Boundary Conditions Using SAPM

This study is the first report of numerical solution of nonlinear bending analysis for annular and circular plates based on 3D elasticity theory with asymmetric boundary conditions using semi-analytical polynomial method (SAPM). Orthotropic annular and circular plates are subjected to transverse loading and 3D bending analysis in the presence of symmetric and asymmetric boundary conditions is studied. For asymmetry cases, the plate boundaries are divided to two or three parts and various boundary conditions such as clamped, simply-supported and free edges are defined for each part. The asymmetry in one and two directions is studied. The influence of elastic foundations, mechanical and thermo-mechanical loadings are examined. Regarding this fact that no study has been done in the case of asymmetric boundary conditions, the obtained results are compared with FEM results by ABAQUS. The results show good agreement with the literatures and FEM results, which it shows that the presented method can use to analyze the 3D bending of plates under asymmetric conditions. Also, it is observed that 3D elasticity estimates some higher deflections than other theories. But, the obtained results by 3D elasticity theory and those obtained by FEM analysis in the case of asymmetric conditions are so close.

A new approach for time-dependent response of viscoelastic graphene sheets embedded in visco-Pasternak foundation based on nonlocal FSDT and MHSDT theories

In this paper, the thermo-mechanical dynamic response of annular/circular viscoelastic graphene plates embedded in visco-Pasternak foundation has been investigated using nonlocal first and modified higher-order shear deformation theories. The modified higher-order shear deformation theory is assumed to obtain the results of thick (high ratio of thickness to size) plates too. Also, the sheet is considered in thermal environment in order to study the thermal effects on the viscoelastic analysis. The viscoelastic behavior of the plate is simulated based on the Kelvin–Voigt model and the nonlinear von Karman strains have been considered. The dynamic governing equations have been derived using the Hamilton stationary of minimum potential energy based on the nonlocal first and modified higher-order shear theories and have been solved applying semi-analytical polynomial method solving method. The solving methodology is unique and has not been used in any other dynamic analysis before and its ability for solving the dynamic governing equations has also been confirmed. The time-dependent deflection of the sheet under uniform and non-uniform loads has been obtained. Different effects on the problem, including boundary conditions, damping coefficient of visco-Pasternak foundation, viscoelastic coefficient of the plate, small-scale effects, loading, and nonlinear effects have been discussed more precisely.

On the statics of fullerene structures

In the present study, thorough research on bending analysis of the fullerene structures (such as C60, C720, C2160, and C4860) resting on the Winkler-Pasternak elastic foundation and under external loading is conducted. Small-scale effects on the results are taken into consideration via nonlocal elasticity theory. The geometry of fullerene structures is assumed as a sphere and doubly-curved structure. The static governing equations for fullerene structures are derived by implementing the principle of minimum potential energy on the basis of the nonlocal first-order shear deformation theory. Various types of boundary conditions including the free edges are considered. The elastic foundation is applied locally on the particular area of the structure, on the contrary of the most available researches that have considered it on the whole of problem geometry. For this aim, an innovative formulation is presented in which simulates the elastic foundation locally. The resulting equations are solved by using the semi-analytical polynomial method (SAPM). Since no similar research has been done before, the bending results of a spherical structure (whereas the fullerene structure can be approximated by a sphere) are compared and validated with the results of ABAQUS software. Also, the effects of several important parameters such as the type of fullerene structure, the nonlocal parameter, and the elastic foundation constants on the results are investigated in detail.

A novel approach for nonlinear bending response of macro- and nanoplates with irregular variable thickness under nonuniform loading in thermal environment

A new method has been presented for static analysis of macro- and nano sector plates with variable thickness. The thickness variations are assumed linearly, functionally, and also irregularly. The governing equations have been derived based on the nonlocal modified higher-order shear deformation theory and have been solved applying the semi-analytical polynomial method. The obtained results have been compared with the results of the other relevant articles. The effective parameters on the analysis have been investigated such as the thickness, the size of the plate under uniform and nonuniform loadings, and the boundary conditions including free edges.Communicated by Marco Amabili.

A new approach for bending analysis of bilayer conical graphene panels considering nonlinear van der Waals force

Abstract In the present study, the effects of nonlinear van der Waals forces on the bending analysis of orthotropic bilayer conical graphene panels are investigated. In order to model the nano-sized panels, the first-order shear deformation shell theory is used to obtain the governing equations by applying energy method in nonlocal form. Semi-analytical polynomial method (SAPM) is utilized to solve the resulting nonlinear governing equations. Due to cover a wide range of geometric shapes, the geometry is considered as a conical panel, so the results can be simulated for cylindrical panels, annular sectors, and even rectangular plates. The van der Waals force between upper and lower layers of graphene panel is simulated by an extension spring with the linear and nonlinear stiffness. It is easy to obtain the results of single-layer panels by eliminating the van der Waals force, and also for macroscopic plates, with neglecting the nonlocal effects. Finally, the affecting parameters on the results are examined in detail such as the plate size, orthotropic effects of material, nonlocal effects, van der Waals force including its nonlinear term and various types of boundary conditions, even the free edges.

Nonlocal analysis of single and double-layered graphene cylindrical panels and nano-tubes under internal and external pressures considering thermal effects

Mechanical behavior of a bilayer graphene cylindrical panel and nano-tube is studied based on nonlocal continuum mechanics with regard to this aim, von-Karman assumptions and nonlocal theory of Eringen are considered. Then, the governing equations and boundary con- ditions have been derived applying energy method. While analyzing the bilayer cylindrical panel, the van der Waals interaction between the layers is considered in calculations. The constitutive equations are developed for nano-tubes under internal and external pressures. In order to solve the governing equations, the semi-analytical polynomial method (SAPM), which was presented by the authors before, is utilized and bending behavior of bilayer cylin- drical panels and nano-tubes is investigated. Finally, the effects of temperature, boundary conditions, elastic foundation, loading, van der Waals interaction between the layers and single layer to bilayer analyses are studied for graphene cylindrical panels and nano-tubes.

Non-linear bending analysis of multi-layer orthotropic annular/circular graphene sheets embedded in elastic matrix in thermal environment based on non-local elasticity theory

In this paper, the large deflection of multi-layer orthotropic annular/circular graphene sheets is investigated based on the non-local elasticity theory. The plate is considered to be in thermal environment. The equilibrium equations are derived in terms of generalized displacements and rotations considering the FSDT non-local elasticity theory and the van der Waals interaction between the layers. In order to solve the governing equations, the differential quadrature method (DQM) which is an accurate numerical method and a new semi-analytical polynomial method (SAPM) are applied. By applying DQM or SAPM, the ODE's would be converted to non-linear algebraic equations. In continue, the Newton–Raphson iterative scheme is applied to solve the obtained non-linear algebraic equations. The results of DQM and SAPM are compared. Although, the SAPM's formulation is considerably simpler than DQM, however, the results of two methods are so close to each other. The results are validated with the other available researches. The effect of small scale parameter, temperature effects on non-local results, the value of van der Waals interaction between the layers for bi-layer and triple layers graphene sheet, different values of elastic foundation matrix and load for various small scale parameters, the comparison between local and non-local deflections and linear to non-linear analysis are investigated.

The effect of vacant defect on bending analysis of graphene sheets based on the Mindlin nonlocal elasticity theory

Abstract In the paper, effects of eccentric vacant defect on bending analysis of circular graphene sheets have been studied based on nonlocal elasticity theory considering the Mindlin theory of the plates. The governing equations have been derived for circular graphene sheet including eccentric vacant defect. Because of existence of an eccentric vacant defect on the plate, the anti-symmetric problem has been obtained. So, the constitutive equations are partial differential equations system. Considering this fact that there is not any analytical solution available for solving the nonlinear constitutive equations based on the first order shear deformation theory, a new semi-analytical polynomial method is applied which has been presented by the authors recently. By applying the mentioned method, the governing equations have been solved. Regarding this fact that no study has been done heretofore in case of the effects of eccentric vacant defect on bending analysis of graphene sheets, the obtained results of local analysis have been compared with the results of Abaqus software. At the following, the effects of nonlocal parameters, size and location of vacant defect on the results have been surveyed.

Nonlinear static analysis of single layer annular/circular graphene sheets embedded in Winkler–Pasternak elastic matrix based on non-local theory of Eringen

Abstract Nonlinear bending analysis of orthotropic annular/circular graphene sheets has been studied based on the non-local elasticity theory. The first order shear deformation theory (FSDT) is applied in combination with the nonlinear Von-Karman strain field. The obtained differential equations are solved by using two methods, first the differential quadrature method (DQM) and a new semi-analytical polynomial method (SAPM) which is innovated by the authors. Applying the DQM or SAPM, the differential equations are transformed to nonlinear algebraic equations system. Then the Newton–Raphson iterative scheme is used. First, the obtained results from DQM and SAPM are compared and it is concluded that although the SAPM’s formulation is considerably simpler than DQM, however, the SAPM’s results are so close to DQM. The results are validated with available papers. Finally, the effects of small scale parameter on the results, the comparison between local and non-local theories, and linear to nonlinear analyses are investigated.

A Non-linear Static Equivalent Model for Multi-layer Annular/Circular Graphene Sheet Based on Non-local Elasticity Theory Considering Third Order Shear Deformation Theory in Thermal Environment

In this paper, it is tried to find an approximate single layer equivalent for multi-layer graphene sheets based on third order non-local elasticity theory. The plates are embedded in two parameter Winkler-Pasternak elastic foundation, and also the thermal effects are considered. A uniform transverse load is imposed on the plates. Applying the non-local theory of Eringen based on third order shear deformation theory and considering the van der Waals interaction between the layers, the governing equations are derived for a multi-layer graphene sheet. The governing equations for single layer graphene sheet are obtained by eliminating the van der Waals interaction. In this study, two different methods are applied to solve the governing equations. First, the results are obtained applying the differential quadrature method (DQM), which is a numerical method, and then a new semi-analytical polynomial method (SAPM) is presented. The results from DQM and SAPM are compared and it is concluded that the SAPM results are satisfactorily accurate in comparison with DQM. Since analyzing a multi-layer graphene sheet needs a time-consuming computational process, it is investigated to find an appropriate thickness for a single layer sheet to equalize the maximum deflections of multi-layer and single layer sheets. It is concluded that by considering a constant value of the van der Waal interaction between the layers, the maximum deflections of multi and single layer sheets are equal in a specific thickness of the single layer sheet.

Nonlinear Bending Analysis of Sector Graphene Sheet Embedded in Elastic Matrix Based on Nonlocal Continuum Mechanics

The nonlinear bending behavior of sector graphene sheets is studied subjected to uniform transverse loads resting on a Winkler-Pasternak elastic foundation using the nonlocal elasticity theory. Considering the nonlocal differential constitutive relations of Eringen theory based on first order shear deformation theory and using the von-Karman strain field, the equilibrium partial differential equations are derived. the nonlinear partial differential equations system is solved using the differential quadrature method (DQM) and a new semi analytical polynomial method (SAPM). By using the DQM or SAPM, the partial differential equations are converted to nonlinear algebraic equations, then the Newton–Raphson iterative scheme is applied to solve the resulting nonlinear algebraic equations system. The obtained results from DQM and SAPM are compared and observed the SAPM results are so close to DQM. Whereas, the SAPM’s formulations are considerably simpler than the DQM. Different boundary conditions including clamped, simply supported and free edges are considered. The obtained results are validated with available researches, then the small scale effects is investigated on the results due to various conditions such as outer radius to thickness ratio, boundary conditions, linear to nonlinear analysis, nonlocal to local analysis ratio, angle of the sector and stiffness value of elastic foundation.