Higher-order Displacement Field Research Articles

Extended formulation of macro-element based modelling – Application to single-lap bonded joints

An extended formulation of the macro-element (ME) based models, representing for both adherends and adhesive along the entire overlap in only one four-node element, is presented. Compared to earlier modelling, continuum ME (CME) and discrete ME (DME) based models, the adherend parts are also modelled as plane continuum media, for which high order displacement fields are freely supposed. Both extended CME (ECME) and extended DME (EDME) based modelling can have their displacement field orders of each continuum layer be set individually, and can be enriched using springs for interfaces or boundary conditions modelling, allowing the stresses to vanish at the free edges. The methodology and formulation of the stiffness matrix for both the adherend without adhesive and the bonded overlap is presented. The assessment is performed for a single-lap joint geometry by comparison results from a plane strain finite element (FE) model with extended ME-based models. Good agreements are shown for both thin and thick adhesive case studies.

Framework of Physically Consistent Homogenization and Multiscale Modeling of Shell Structures**

A framework of physically consistent homogenization and multiscale modeling (PCHMM) for reduced-order analysis of plate/shell structures is developed in this paper. To address the inapplicability of conventional periodic boundary conditions and Hill’s condition involved in homogenization of shear-deformable shell structures, the paper proposes physically consistent boundary conditions and modified Hill’s condition for plate/shells. Unlike the PCHMM method for beams, considering the contradiction between high-order displacement fields induced by shear forces and low-order kinematic assumptions, additional constraints are applied to the plate/shell structure sectional strains during the solution of perturbation fields. The correctness and effectiveness of the proposed plate/shell PCHMM framework and method are verified by typical numerical examples. The proposed theory can also be conveniently embedded into commercial finite element software for homogenization and multiscale analysis of structures such as microscale metamaterials like lattice plates and large complex structures like aircraft fuselage sections.

Assessment of new four variables 2D and quasi-3D higher order shear theories for bending analysis of different double curved FG shells

ABSRACT This study introduces a novel theory employing four unknowns to examine bending of a double-curvature functionally graded shells experiencing both uniform and sinusoidal pressure. This investigation utilizes 2D and quasi-3D higher order shear deformation theories, integrating a fresh model of a high-order displacement field. It focuses on exploring how geometric ratios, volume fraction and power law index influence the bending behavior, specifically examining their effects on the normal and shear stresses variation, along with their amplification-reduction impacts. The effectiveness of the developed model is evaluated by comparing its performance in simulating bending characteristics of plates, cylinders, spheres and ellipses with those presented in literature.

Dynamic stiffness method and CUF-based component-wise theories applied to free vibration analysis of solid beams, thin-walled structures and reinforced panels

In this paper, an exact dynamic stiffness formulation using higher-order theories with displacement variables only is presented and subsequently used to investigate the free vibration characteristics of solid beams, thin-walled structures and reinforced panel structures. In essence, higher-order displacement fields are developed by using the Carrera unified formulation (CUF), and by discretizing the cross-section kinematics with bilinear, cubic and fourth-order Lagrange polynomials. In particular, the component-wise (CW) approach based on Lagrange expansion is applied in which the solid part and thin-walled part are considered as two independent components that can be assembled. The principle of virtual displacements is used to derive the governing differential equations and the associated natural boundary conditions. An exact dynamic stiffness matrix is then developed by relating the amplitudes of harmonically varying loads to those of the responses. The explicit terms of the dynamic stiffness matrices are also presented. The Wittrick–Williams algorithm of the dynamic stiffness method (DSM) is applied with the explicit expressions of the J0 count for beam elements under all above support conditions. In return, there is no need to refine the element in the DSM, and thus, it becomes immensely efficient. The accuracy and efficiency of the proposed methodology are extensively assessed for different solid beams, thin-walled structures and reinforced panels and the results are compared with those appearing in published literature and also checked by 3D finite element (FE) solutions.

A novel beam model suitable at the microscale

A novel beam model is proposed to address the limitations of existing microscale beam models by introducing a higher-order displacement field assumption for the modified couple stress theory. The governing equations and boundary conditions are derived using the variational principle. Validation through finite element analysis for a simply supported beam problem demonstrates the beam model successfully captures the width-dependent behavior of microbeams, which existing beam models cannot explain. Additionally, the beam model accurately represents the plane strain behavior as the beam width increases and the classical beam behavior at a small length scale parameter.

An enriched finite element for the simplified stress analysis of an entire bonded overlap: Continuum macro-element

A methodology for the formulation of the stiffness matrix of an enriched finite element, called continuum macro-element (CME), representing for the full length of a bonded overlap and both the adhesive and the adhesive in only one four-node elements, is presented. Compared to earlier macro-elements modelling the bonded overlap as beams on elastic foundation, the CME supposed the adherends and the adhesive as plane continuum media, for which higher-order displacement fields is supposed for the adhesive. The formulation of the stiffness matrix for the adherend outside of the overlap is presented as well to address the stress analysis of single-lap bonded joint for assessment purpose. The assessment is performed by comparisons with the results from (i) a plane strain finite element and (ii) two recent papers by Nguyen and Le Grognec (2021) and Methfessel and Becker (2022). Good agreements are shown.

Dynamic Response of Thick Laminated Shells using Higher-Order Theory

In this paper, the dynamic characteristics of thick cross-ply laminated composite cylindrical shells are studied using a higher-order displacement model. The formulation accounts for the nonlinear variation of the in-plane and transverse displacements through the thickness, and abrupt discontinuity in the slope of the in- plane displacements at any interface. The effect of inplane and rotary inertia terms is included. The analysis is carried out using finite element approach. The governing equations derived using Lagrange's equations of motion are solved employing Newmark's direct integration technique for transient response analysis. The influences of various terms in the higher-order displacement field on the free vibrations, and transient dynamic response characteristics of cylindrical composite shells subjected to thermal and mechanical Loads are analysed.

On exact wave propagation analysis in a beach volleyball ball

Wave propagation and vibration state in the sports balls have important effects on the stability in movement and accuracy of hits. In some sports, similar to football and volleyball, the balls are subjected to continuous strong hits from players. In the current study, the wave propagation in a beach volleyball ball under the effect of temperature change, different material characteristics, different thicknesses, and the total mass is investigated. In this regard, the ball is modeled using a spherical shell structure and a higher-order displacement field is considered. The classical theory of elasticity is employed along with Hamilton’s principle for the spherical shell. The outcome equations with different boundary conditions are solved by engaging the discrete singular convolution method (DSCM). In addition, the finite element method is used to obtain the modal shape and natural frequency. The formulations and solution method are utilized in different steps of convergence examination, validation, and parametric study to acquire a comprehensive insight into the behavior of beach volleyball balls under the effect of the dynamic loading conditions. In specific, we are more interested in the phase velocity of the ball. The results are presented for different temperature changes and geometrical and material properties.

On the impact process and stress field of functionally graded graphene reinforced composite pipes with a viscoelastic interlayer

In the paper, impact process of the fluid-conveying pipes composed of graphene-reinforced composite layers and a viscoelastic interlayer is studied. The bending stress field, midspan displacement and contact force are focused on. Based on the theory of high-order displacement field, the governing equations are derived through the Hamilton variational principle. To obtain the approximate solution for the impact dynamics of pipes, the Galerkin method is extended to expand the generalized displacements into the schemes of triangular series. Further, by utilizing the orthogonality of the trigonometric series, the differential equations of various orders for impact dynamics are established. The fourth-order Runge–Kutta method is introduced to the truncated solutions. Subsequently, the parameter sensitivity of the transient responses in the impact stage is emphatically discussed. It is revealed that the insensitive parameters to contact force and midspan displacement have a great influence on the stress field. Furthermore, the evolution characteristics of the bending normal stress field are highlighted. Numerical results illustrate that the attenuation characteristics of bending stress field are determined by the coupling effects of internal flow and structural features on structural stiffness and damping.

Nonlocal vibration of functionally graded nanoplates using a layerwise theory

This work studies the size-dependent free vibration response of functionally graded (FG) nanoplates using a layerwise theory. The proposed model supposes not only a higher-order displacement field for the core but also the first-order displacement field for the two face sheets, thereby maintaining an interlaminar displacement continuity among layers. Unlike the conventional layerwise models, the number of variables is kept fixed and does not increase for an increased number of layers. This is a very important feature compared to conventional layerwise models and facilitates significantly the engineering analyses. The material properties of the FG nanoplate are graded continuously through the thickness direction in accordance with a power-law function. The Eringen’s nonlocal elasticity theory is here adopted to relax the continuum axiom required in classical continuum mechanics and hence hopeful to capture the small size effects of naturally discrete nanoplates. The equations of motion of the problem are obtained via a classical Hamilton’s principle. The present layerwise model is implemented with a computationally efficient C0- continuous isoparametric serendipity elements and applied to solve a large-scale discrete numerical problem. The robustness and reliability of the developed finite element model are demonstrated by a comparative evaluation of results against predictions from literature. The comparative studies show that the proposed finite element model is: (a) free of shear locking, (b) accurate with a fast rate convergence for both thin and thick FG nanoplates, and (c) excellent in terms of numerical stability. Moreover, a detailed parametric analysis checks for the sensitivity of the vibration response of FG nanoplates to the aspect ratio, length-to-thickness ratio, nonlocal parameter, boundary conditions, power-law index, and modes shapes. Referential results are also reported, for the first time, for natural frequencies of FGM nanoplates which will serve as benchmarks for further computational investigations.

Nonlinear Resonance of Functionally Graded Porous Circular Cylindrical Shells Reinforced by Graphene Platelet with Initial Imperfections Using Higher-Order Shear Deformation Theory

This paper uses the higher-order shear deformation theory (HOSDT) to analyze the nonlinear resonance of functionally graded graphene platelet-reinforced porous (FG-GPL-RP) circular cylindrical shell with geometrical imperfections subjected to the harmonic transverse loading. The shell considered is surrounded by the elastic Winkler–Pasternak foundation. Based on Hamilton’s principle, the nonlinear governing equations of motion of the imperfect system considered are established using the first-order and various higher-order shear deformation shell theories that contain a unified higher-order displacement field. Four types of porosity and graphene nanoplatelet distribution patterns are considered. The modified Halpin-Tsai scheme and rule of mixture are utilized to calculate the effective properties of FG-GPL-RP materials. Explicit expressions of the nonlinear primary resonance of imperfect FG-GPL-RP cylindrical shells for simply-supported boundary conditions are achieved by the Galerkin technique and method of multiple scales. After verifying the accuracy of the model used and the results obtained, the influences of the geometric parameters, material properties and distribution and imperfection value on the nonlinear primary resonant response of the imperfect FG-GPL-RP circular cylindrical shells are examined in detail.

Buckling analysis of isotropic and orthotropic square/rectangular plate using CLPT and different HSDT models

Buckling analysis of simply supported isotropic and orthotropic rectangular plate has been performed using Classical plate theory (CLPT) and various higher-order shear deformation theories (HSDT) developed by Reddy, Karama, Shi, and Touratier. The considered HSDT models utilize parabolic, trigonometric, and exponential distribution functions for shear strain along the transverse direction. The total minimum potential energy approach has been applied to find the governing equations using a generalized higher-order displacement field. The non-dimensional stiffness parameters are calculated for all the considered HSDT models employing governing differential equations. Subsequently, the governing differential equations have been solved using the Navier method. The non-dimensional critical buckling load (Ncr) as per different HSDT models has been discussed for various plate thicknesses. The Ncr of the rectangular plate considering HSDT and CLPT models for various plate thicknesses has been plotted to analyze the numerical result. Finally, a detailed discussion has been done to compare the Ncr values obtained using different shear deformation theories for isotropic and orthotropic plates. From the above critical analysis, the authors wanted to investigate the best suitable plate thickness range for CLPT as well as HSDT models for better accuracy in bucking load calculation.

Thermomechanical Buckling Analysis of the E&P-FGM Beams Integrated by Nanocomposite Supports Immersed in a Hygrothermal Environment.

Due to the widespread use of sandwich structures in many industries and the importance of understanding their mechanical behavior, this paper studies the thermomechanical buckling behavior of sandwich beams with a functionally graded material (FGM) middle layer and two composite external layers. Both composite skins are made of Poly(methyl methacrylate) (PMMA) reinforced by carbon-nano-tubes (CNTs). The properties of the FGM core are predicted through an exponential-law and power-law theory (E&P), whereas an Eshelby–Mori–Tanaka (EMT) formulation is applied to capture the mechanical properties of the external layers. Moreover, different high-order displacement fields are combined with a virtual displacement approach to derive the governing equations of the problem, here solved analytically based on a Navier-type approximation. A parametric study is performed to check for the impact of different core materials and CNT concentrations inside the PMMA on the overall response of beams resting on a Pasternak substrate and subjected to a hygrothermal loading. This means that the sensitivity analysis accounts for different displacement fields, hygrothermal environments, and FGM theories, as a novel aspect of the present work. Our results could be replicated in a computational sense, and could be useful for design purposes in aerospace industries to increase the tolerance of target productions, such as aircraft bodies.

Numerical Nonlinear Static Analysis of Cutout-Borne Multilayered Structures and Experimental Validation

The influence of two different types of nonlinear strain-displacement kinematics (Green–Lagrange and von Kármán) and their importance are investigated in this analysis by computing the static deflection and stress (normal and shear) values of cutout abided composite structures. The structural deformation behavior under the external loading conditions is derived with a higher-order polynomial without hampering the strain and stress (in-plane and out-of-plane) continuity. The mathematical formulation of the curved structure is derived, and the governing equation order is reduced using nonlinear finite element steps. A generic computer code is prepared using the derived mathematical formulation in conjunction with the isoparametric finite element technique in a MATLAB environment. The final output (that is, the nonlinear deflection and stress data) is predicted using a robust nonlinear solution method (Picard’s iteration). The suitability of full-scale geometrical nonlinearity (Green–Lagrange strain) in the framework of a higher-order displacement field model for the laminated structure is established by comparing the experimental deflection data and published analytical solutions. Furthermore, a series of numerical examples is solved to show the influences of different individual and/or the combined parametric influences on the structural deflections and the normal/shear stress (in-plane and out-of-plane) values, including the variable geometrical shapes.

Thermal buckling analysis of cross-ply plates based on new displacement field

A higher-order displacement field is used for the analysis of the thermal buckling of composite plates subjected to thermal load; it is based on a constant ‘‘m’’, which is optimized to get results relatively close to those given by 3D elasticity theory. Adequate transverse shear strains distribution through the thickness and free stress surfaces of the plate is satisfied using this theory. Hamilton’s principle is used to derive equations of motion, which are solved using Navier-type series for simply supported plates. Thermal buckling of cross-ply laminates with various (α2 / α1) ratios, number of layers, aspect ratios, E1/E2 ratios, and stacking sequence for thick and thin plates is studied in detail. It is concluded that the obtained results using this displacement field are close to those calculated by 3D elasticity theory and other shear deformation plate theories when m=0.05.

Thermal shock response of porous functionally graded sandwich curved beam using a new layerwise theory

This paper, for the first time, presents the thermal shock response of porous functionally graded (FG) sandwich curved beams subjected to thermal shocks. The authors have used a higher-order layerwise beam theory developed for the analysis. This new layerwise theory has a higher-order displacement field for the core and a linear displacement field for top and bottom facesheets maintaining the continuity of displacement at the layer interface. The top and bottom layers of the functionally graded sandwich curved beam are made of pure ceramic and pure metal constituent, respectively, and the core is considered to be made of functionally graded material (FGM) having porosity. The top surface of the beam is subjected to thermal shocks the bottom surface is kept at a reference temperature or is thermally insulated. The solution for the nonlinear temperature profile is obtained using the Crank-Nicolson method. An eight-noded isoparametric element having seven degrees of freedom per node is used to develop the finite element formulation. The governing differential equation is obtained using the Hamilton’s principle, and the resulting transient problem is solved using the Newmark constant acceleration integration method. The effects of curvature, thickness, core to facesheet thickness ratio, intensity of the thermal shock, porosity coefficient, and boundary conditions are investigated on thermally induced vibration response of porous FG sandwich curved beams. The analysis reveals that the proposed finite element formulation is straight forward, accurate, and widely applicable for the analysis of porous functionally graded sandwich curved beams.

On the layerwise finite element formulation for static and free vibration analysis of functionally graded sandwich plates

This paper presents a novel C0 higher-order layerwise finite element model for static and free vibration analysis of functionally graded materials (FGM) sandwich plates. The proposed layerwise model, which is developed for multilayer composite plates, supposes higher-order displacement field for the core and first-order displacement field for the face sheets maintaining a continuity of displacement at layer. Unlike the conventional layerwise models, the present one has an important feature that the number of variables is fixed and does not increase when increasing the number of layers. Thus, based on the suggested model, a computationally efficient C0 eight-node quadrilateral element is developed. Indeed, the new element is free of shear locking phenomenon without requiring any shear correction factors. Three common types of FGM plates, namely, (i) isotropic FGM plates; (ii) sandwich plates with FGM face sheets and homogeneous core and (iii) sandwich plates with homogeneous face sheets and FGM core, are considered in the present work. Material properties are assumed graded in the thickness direction according to a simple power law distribution in terms of the volume power laws of the constituents. The equations of motion of the FGM sandwich plate are obtained via the classical Hamilton’s principle. Numerical results of present model are compared with 2D, quasi-3D, and 3D analytical solutions and other predicted by advanced finite element models reported in the literature. The results indicate that the developed finite element model is promising in terms of accuracy and fast rate of convergence for both thin and thick FGM sandwich plates. Finally, it can be concluded that the proposed model is accurate and efficient in predicting the bending and free vibration responses of FGM sandwich plates.

Aeroelastic analysis of foam-filled composite corrugated sandwich plates using a higher-order layerwise model

Due to the versatility of load bearing and heat insulation, foam-filled composite corrugated sandwich plates have a broad application in aerospace industry as thermal insulation structures for hypersonic vehicles. In this paper, a C0 higher-order layerwise finite element model is presented for aeroelastic analysis of foam-filled composite corrugated sandwich plates in supersonic flow with equivalent properties. The homogenization method for the foam-filled corrugated core is formulated by force and energy equivalence under different assumptions of force or moment distributions. In the proposed layerwise model, the composite sandwich plate is divided into three sections through thickness, while a higher-order displacement field is assumed for the single-layer orthotropic core section and a first-order displacement field for the two face-sheet sections made of multi-layer laminates. The effect of transverse normal strain is also taken into consideration. The unsteady aerodynamic pressure is evaluated by the first-order Piston theory. An eight-noded isoparametric element with 16 degrees of freedom per node is used for finite element method. The accuracy of the present model is verified with existing results in the literature. The influence of geometric parameters and material properties on critical dynamic pressure are discussed.

Thermal Buckling of Angle-Ply Laminated Plates Using New Displacement Function

ABSTRACT
 Critical buckling temperature of angle-ply laminated plate is developed using a higher-order displacement field. This displacement field used by Mantari et al based on a constant ‘‘m’’, which is determined to give results closest to the three dimensions elasticity (3-D) theory. Equations of motion based on higher-order theory angle ply plates are derived through Hamilton, s principle, and solved using Navier-type solution to obtain critical buckling temperature for simply supported laminated plates. Changing (α2/ α1) ratios, number of layers, aspect ratios, E1/E2 ratios for thick and thin plates and their effect on thermal buckling of angle-ply laminates are studied in detail. It is concluded that, this displacement field produces numerical results close to 3-D elasticity theory with maximum discrepancy (7.4 %).

Effect of Thickness Stretching on the Natural Frequencies of Laminate-Faced Sandwich Plates Using a New Layerwise Model

The current work investigates the effect of thickness stretching on the natural frequencies of laminate-faced sandwich plates using new layerwise finite element model. The proposed model assumes higher-order displacement field for the core and first-order displacement field for the face sheets. Thanks for enforcing the continuity of the interlaminar displacement, the number of variables is independent of the number of layers. The consistent mass matrix and the element stiffness matrix are derived using the Hamilton’s principle. The performance and reliability of the proposed formulation are demonstrated by comparing the author’s results with those obtained using the three-dimensional elasticity theory, analytical solutions and other advanced finite element models.