Shear Deformation Theory Research Articles

A quasi-3D SinZZ model-driven multi-field Chebyshev FEM for nonlinear vibration control in multilayer multiferroic composite plates

This paper presents an advanced numerical method for modeling and mitigating nonlinear vibrations in thin multilayered fiber-reinforced multiferroic composite plates, addressing complex multi-physical interactions. The proposed methodology leverages a multi-physical coupling Chebyshev finite element formulation, utilizing high-order shape functions derived from Chebyshev polynomials. By integrating the strengths of spectral element methods and Legendre spectral finite element methods, this approach effectively overcomes challenges such as shear locking and spurious zero energy modes while ensuring high convergence rates in multi-physical problems. A quasi-3D refined model, incorporating the Murakami zig-zag model and sinusoidal shear deformation theory, is employed to accurately capture the nonlinear Von Kármán strain–displacement relationship and the magneto-electro-elastic coupling in multilayer structures with thickness-dependent material properties. To suppress nonlinear vibrations, the study utilizes a closed-loop multiphysical Chebyshev finite element model for time-domain analysis of viscoelastically damped systems, employing the Golla–Hughes–McTavish model. The results underscore the significant influence of multiferroic properties and the strategic distribution of ferroelectric fibers within the substrate on the dynamic behavior of the plate. The numerical validation, supported by rigorous verification, demonstrates the robustness of the proposed method in effectively simulating and controlling multilayer fiber-reinforced multiferroic composite plates. Additionally, this research highlights the potential for significant vibration reduction through semi-active damping mechanisms, offering valuable insights for practical applications in industries where precision and stability are critical.

A meshfree formulation for size-dependent thermal buckling and post-buckling behaviour of porous microplates on elastic foundation subjected to localized heating

The semi-analytical framework is developed for in-plane thermal stresses within the microplates under localized heating. Localized heating is modelled using Fourier Series expansions over the complete domain of the plate. These stresses within the plate are estimated after solving the in-plane thermoelasticity problem using Airy's stress approach. Utilizing these stresses, present work also studies the buckling and post-buckling behaviour of a porous metal foam microplate resting on Winkler and Pasternak elastic foundations. The plate is modelled using Reddy's third-order shear deformation theory (TSDT) and von Kármán geometric nonlinearity, and the size-dependent effects are included using the modified strain gradient theory (MSGT). Galerkin's weighted residual method converts the partial differential equations (PDEs) into algebraic equations. The post-buckling equilibrium path is obtained using the modified Newton-Raphson method. The mode shapes of the microplate for the various configurations of the plate with different boundary conditions due to localized thermal loading are plotted. Also, these mode shapes are compared with the mode shapes of the microplate due to in-plane mechanical loading. The parametric effect of porosity, elastic foundation parameters, thickness of plate, size of plate, boundary conditions and loading concentrations on the buckling and post-buckling behaviour of the plate is studied.

Two-dimensional analysis of electrical behavior in composite semiconductor shell structures with magnetic field tuning

To precisely investigate the electric properties and reveal the intrinsic interaction mechanisms between multiple fields in the composite magneto-electric-semiconductor shell structure, two-dimensional analyses were conducted within the framework of the third-order shear deformation theory and the coupled field theory. The governing equations for the composite piezoelectric semiconductor (PS) shell structures were derived using the principle of virtual work. Field distributions of the composite PS shell structures were obtained by expanding basic field quantities into Fourier series along the width direction and using the differential quadrature method. Prior to analysis, the convergence and correctness of the adopted method were evaluated. Several numerical examples were presented to demonstrate how magnetic manipulation can tune electrical behavior. It was found that inhomogeneous shear strain induced by applied magnetic fields is the key factor affecting potential variation in the composite PS shells. The appearance of potential barriers and wells in the composite PS shell structure with an opposite c axis is explained thoroughly. Additionally, the effect of magnetic field non-uniformity on the composite PS shell structure was investigated using an equivalent body force model, offering valuable insights for designing thin-film devices.

Bending and buckling responses of organic nanoplates considering the size effect

As humanity progresses, fossil fuel reserves are being increasingly exhausted, leading to increased focus on sustainable energy alternatives. Scientists are now researching solar panels that can efficiently convert solar energy into electricity for human use. This research employs a novel third-order shear deformation theory to investigate the static buckling and static bending behaviors of organic nanoplates, marking the first instance of such an approach being used. The formulas are computed using strain gradient theory to consider the impact of the size effect, where this size effect parameter is considered in both positive and negative cases. The plate's balancing equation is derived using the notion of virtual displacement, and the analytical solution is obtained using Navier's solution. The mathematical expression for deflection and critical buckling load in this study has been validated by comparing it with previously published analytical findings. This study also includes a set of numerical calculations to demonstrate the impact of certain geometric factors and the size effect on the static bending response and static buckling of organic plates. This study aims to assist designers in developing organic plate products that demonstrate optimal functionality in real-world applications

On the measurement of the nonlinear dynamics of sandwich sector plate surrounded by the auxetic concrete foundation: Introducing a machine learning algorithm for nonlinear problems

The measurement of the nonlinear dynamics of sandwich rotary sector plates is crucial for measurement engineers as it enables the precise analysis and understanding of the behavior of advanced composite structures under dynamic conditions. These measurements help in identifying and mitigating potential issues related to vibration, stability, and fatigue, which are critical in industries like aerospace, automotive, and civil engineering. Accurate data on nonlinear dynamics aids in the optimization of design, enhancing performance, durability, and safety. Furthermore, it supports the development of predictive maintenance strategies, reducing downtime and operational costs, and contributing to the advancement of engineering materials and techniques. So, in the current work, for the first time, nonlinear dynamics of sandwich rotary sector plate via a mathematical modeling and machine learning algorithm is presented. First, due to a lack of information on the nonlinear dynamics of sandwich rotary sector plates, a dataset for training, testing, and validating the machine learning algorithm is presented. For this purpose, using Hamilton’s principle, Von-Karman nonlinearity, and first-order shear deformation theory (FSDT), the nonlinear governing equations are obtained. Also, due to increasing stiffness and stability, an auxetic concrete foundation covers the sandwich structure. After that using the two-dimensional generalized differential quadrature method (GDQM) and Newmark’s time integration method (NTIM), the nonlinear equations are solved. This research provides valuable insights for engineers in designing advanced composite structures with improved dynamic properties. The results also contribute to the broader understanding of nonlinear dynamic interactions in complex material systems, paving the way for innovative applications in various engineering fields.

Robust active vibration control of flexible smart beam by μ-synthesis

This paper presents a comprehensive method for designing a robust active vibration control system to suppress low-frequency vibrations in smart structures. A novel finite element method based on the first-order shear deformation theory is used to calculate the dynamic response of a smart beam. Through a comprehensive system identification process, the uncertain model of the smart beam is extracted considering both the magnitude and phase. The model fits the experimental data successfully. In addition, a generalized low-frequency vibration control performance function is designed for the piezoelectric smart beam. Using a linear fractional transformation, the system is converted into a standard μ-synthesis control framework, and the controller K is synthesized using structural singular values μ. The effectiveness of the proposed method is experimentally validated using a setup with a piezoelectric smart beam. The experimental results suggest that the proposed control method exhibits robust stability and robust performance, effectively enhancing the performance of smart structure control in various scenarios. The proposed control framework utilizes structured singular value analysis to provide optimal robust stability margins and superior robust control performance, effectively addressing system uncertainties and non-linearities.

NURBS-based isogeometric formulation for linear and nonlinear buckling analysis of laminated composite plates using constrained and unconstrained TSDTs

Two isogeometric plate models employing Reddy's third-order shear deformation theory (TSDT) and unconstrained third-order shear deformation theory (UTSDT) are presented and compared for linear and nonlinear buckling analysis of laminated composite plates with and without imperfection and subjected to different inplane loads. Cubic non-uniform rational B-spline (NURBS) basis functions that easily satisfy C1 continuity of the IGA-TSDT model are employed. The total Lagrangian approach in conjunction with the principle of virtual work is used to derive the governing equations. The primary and secondary solutions are traced using a tangent based arc-length method with a simple branch switching technique. The performance of the models is evaluated by validation and comparison with solutions obtained using ANSYS, Navier method (for linear analysis only), and those in the literature. The buckling response is significantly affected by pre-buckling boundary conditions and strain-displacement relations. The nonlinear buckling approach, among other approaches, is observed to be the most accurate methodology for an arbitrarily laminated composite plate. Further, IGA-UTSDT with nine DOF gives marginal improvement over IGA-TSDT with five DOF at the cost of computation. The IGA-TSDT is observed to be superior to FEM-TSDT in terms of computation demand and performance.

An exact analytical method for free vibration analysis of FG-GPLRC sector cylindrical shells under Levy-type boundary conditions

In this article, an exact analytical method for the free vibration analysis of functionally graded (FG) graphene platelet (GPL)-reinforced composite (GPLRC) sector cylindrical shells is presented by considering Levy-type boundary conditions for the first time. The analysis relies on the use of the Halpin–Tsai micro-mechanical model for evaluating the material properties of the graded layers of the shell with three different grading patterns. Mathematical modeling of the Levy-type cylindrical shell is based on the Hamilton principle and the Sanders first-order shear deformation theory (FSDT). The governing equations of the composite shell are analytically solved using the state-space method. The validity of the proposed analytical method is demonstrated by the excellent agreement between the obtained results of the exact analytical solution and the results available in the literature. Furthermore, some parametric studies are conducted to reveal the effects of variations in boundary conditions, GPL distribution patterns, GPL weight fraction, and geometrical parameters such as shallowness angle, length-to-radius ratio, and thickness on the free vibration behavior of the shell structure. Natural frequencies and mode switching are reported for different mode numbers.

Global buckling response of sandwich panels with additively manufactured lattice cores

This paper deals with the analytical and numerical global buckling analysis of rectangular sandwich plates utilized AlSi10 Mg in both facesheets and lattice cores. In this study, six different strut-based lattice core models are designed. Additionally, the buckling resistance of lattice panels is compared with a typical honeycomb panel. Analytical studies were carried out using Kirchoff plate theory (CLPT), first-order shear deformation theory (FSDT) coded on MATLAB and finite element (FE) analyses in Abaqus. In the theoretical approaches, the Navier solution is derived for sandwich plates with simply supported boundary conditions at all edges. In the FE analyses, validated homogenized lattice structures models were used to avoid excessive numbers of elements and to save computational time. The parametric effects of side-to-thickness ratio, and different designed core cells on global buckling responses are investigated. As a result of comparing the analytical results with the FE model, a good agreement is obtained, and it is found that analytical buckling analyses FSDT can be used within certain size limits for the global buckling analysis of lattice core sandwich structures.

High-dimensional nonlinear flutter suppression of variable thickness porous sandwich conical shells based on nonlinear energy sink

Nonlinear energy sink (NES) is widely applied in engineering field due to the advantages of light weight, high robustness, unidirectional energy transfer, rapid and broadband vibration isolation. In this paper, nonlinear energy sink is utilized to suppress vibration of the variable thickness porous sandwich conical shells for the first time, and the high-dimensional nonlinear flutter suppression characteristics of the system of simply supported variable stiffness truncated porous sandwich conical shell coupled NES under aerodynamic force and thermal stress are investigated. By applying the first-order shear deformation theory (FSDT), Hamilton's principle and Galerkin technique, the high-dimensional nonlinear ordinary differential flutter suppression equations of the system appended with NES are established. The accuracy of the theoretical approach is ensured by the comparison of frequency results, while the NES dissipated kinetic energy ratio and the comparison of NES performance with other suppression systems are presented to prove the effectiveness of NES on nonlinear flutter suppression. The time history diagrams and limit cycle oscillation (LCO) amplitude curves, which reflect the high-dimensional nonlinear flutter suppression effect of NES, are obtained by employing the Runge-Kutta method. The effects of aerodynamic pressure, the parameters and positions of single NES, and the positions of parallel NES and series NES on the high-dimensional nonlinear flutter suppression characteristics of the system attached with NES are discussed in depth. Finally, the optimal high-dimensional nonlinear flutter suppression scheme is arrived.

Thermoelastic analysis of FG-CNTRC cylindrical shells with various boundary conditions and temperature-dependent characteristics using quasi-3D higher-order shear deformation theory

Abstract. This study focuses on performing static analysis of FG-CNTRC cylinder shells with various boundary restrictions, including thermomechanical responses. The governing equations are developed by taking into account the temperature-dependent material features, the quasi-3D high-order shear deformation hypothesis, and the normal transverse stress effect. The temperature gradient inside the thickness is expected to fluctuate, and the distribution pattern is derived by using the heat transfer equation and considering the temperature boundary limitations. A singular trigonometric series and the Laplace transform are used in an analytics solution to address basic equations. This study primarily examines the stress levels at the border region. The findings indicate that it is crucial to take into account the abrupt rise in stress at the boundary area, particularly when the shell’s relative length is small. The reciprocal impact of pressure and temperature load is also emphasized. Significant findings indicate that thermal load may either augment or diminish stress levels, contingent upon the orientation of the pressure and thermal load effect. The results of the study of this issue serve as the foundation for the calculation and design of relevant structures in practical applications. Furthermore, this serves as a foundation for the creation of more intricate issues in the forthcoming.

A CO-HSDT isogeometric analysis for free vibration of matrix cracked FG-GPLRC plates coupled with stationary fluid

This paper for the first time proposes an efficiently C0-higer order shear deformation theory (HSDT) and isogeometric analysis (IGA) for the free vibration analysis of matrix cracked functionally graphene nanoplatelets reinforced composite (FG-GPLRC) plate coupled with stationary fluid. This case represents essential components of sophisticated structures utilized in industries such as shipbuilding, nuclear, marine, and naval. The properties of the four GPLs distributions of FG-GPLRC are evaluated by using a combination of the modified Halpin-Tsai micromechanics model and the rule of mixtures, while the degraded stiffness of cracked layers is predicted by the self-consistent micromechanical (SCM) model. The fluid is assumed to be homogeneous, inviscid, incompressible and irrotational, so the free-surface waves and hydrostatic pressure effects on structures are neglected. The governing equation of motion is derived by the Hamilton's principle, where three different fluid-plate interaction systems are taken into consideration. After validating the proposed method against existing literature, the effect of various parameters such as crack density, interaction boundary conditions (IBC), fluid level, GPLs distribution pattern, total number of layers, and geometric parameter on the free vibration of matrix cracked FG-GPLRC plate are investigated. It is believed that the finding in this paper may be helpful for the accurate design and analysis of matrix cracked FG-GPLRC plate submerged in fluid.

Study and Review of Shear Deformation Theories for Analysis of Deep Beams

This work explores shear deformation theories related to beams in detail and provides a careful analytical evaluation. There are various theories related to the Deformation in Shear action that are painstakingly traced throughout the review in detail. Every hypothesis is systematically assessed, taking into account both its advantages and disadvantages. The study thoroughly examines the results generated by each theory and explores the general approaches to solving problems that result from their use. The evaluation also offers a thorough overview and recommendations for further lines of inquiry. The primary goals of this review are to encourage the continued application of higher-order theories in deep beam behavior prediction and to further our understanding of their development.

Multiquadric Collocations for Nonlinear Vibration of Functionally Graded Plates of Ti–6Al–4V Metal

While Finite Element Method (FEM) has proven reliable for analyzing linear or nonlinear stresses in materials, structures, and fluid flows, its reliance on mesh generation can escalate project simulation costs. In contrast, mesh- free techniques offer an alternative by directly generating system algebraic equations for the entire issue domain, bypassing the need for a preset mesh. This study focuses on investigating the nonlinear free vibration response of shear-deformable Functionally Graded Material (FGM) plates. These plates are expected to exhibit varying material properties, following a power-law distribution linked to the volume fractions of constituent materials as plate thickness increases. Utilizing shear deformation theories and von Karman nonlinear kinematics, the study seeks a mathematical solution for the real physical issues encountered in these plates. To address these challenges, a meshless method employing the Multi Quadric Radial Basis Function (MQRBF) is employed for evaluation. The computed results reveal insights into how different material properties and amplitudes influence the nonlinear free vibration response of composite plates. Comparing these outcomes to prior research demonstrates promising progress in this area.

Nonlinear effect on stable state and snap-through bistability of square composite laminate

Bistable structures with the elastic instability caused by buckling are confirmed to perform significantly in micro-electromechanical systems, metamaterials, energy harvester, vibration isolation and morphing structures. The target of this paper is to explore the dynamic responses of cross-ply bistable composite laminate, focusing on the nonlinear effect of the single- and double-well vibration. Both theoretical and finite element (FE) methods are employed to simulate the nonlinear vibration and dynamic snap-through of bistable composite laminate under the foundation excitation at the center. In the theoretical model, the governing equations are established via Lagrange's equation based on the first-order shear deformation theory, von Karman nonlinear strain-displacement relation and Rayleigh-Ritz method. The fourth-order Runge-Kutta method is adopted to solve the governing equations, and the numerical results are validated by FE model. Subsequently, the details of the dynamic responses are analyzed to identify the nonlinear effects in the form of bifurcation diagram, phase portrait, time history, Poincare maps and amplitude spectrum. The dynamic responses are examined for a series of excitation parameters in both time and frequency domain. Through fixed frequency and frequency sweep tests, the nonlinear phenomenon of the single- and double-well vibrations are analyzed including superharmonic resonance, stiffness softening, hysteresis phenomenon, various periodic and chaotic vibration. The diverse responses to external inputs contribute significantly to efficiently predicting mechanical behaviors in real-world conditions, thereby offering indispensable theoretical support for structural design applications.

Chaos and resonance of nonlinear FG shallow shells reinforced by oblique stiffeners

This paper presents an analysis of the nonlinear dynamics of functionally graded (FG) shallow shells that are simply supported, reinforced with oblique stiffeners, and subjected to external excitation. The properties of these oblique stiffened functionally graded (OSFG) shallow shells vary across their thickness, following a power law distribution based on the volume fractions. Employing the first-order shear deformation theory (FSDT) coupled with the von Kármán nonlinear strain-displacement relations, the derivation of the nonlinear equations of motion is accomplished through the application of Hamilton's principle. The model of FG shallow shells reinforced with oblique two-series stiffeners is developed, allowing for the angles of the two stiffeners to be similar or different from each other. In this context, the employment of the first-order shear deformation theory (FSDT) results in the enhancement of the complexity in modeling the oblique stiffeners. Galerkin's method is applied to discretize the given equations, converting them into a nonlinear dynamical system with two degrees of freedom. This system incorporates both quadratic and cubic nonlinearities and considers the effects of external excitation. The analysis of this research particularly focuses on the resonant scenario involving principal resonance and 1:1 internal resonance. Through the asymptotic perturbation method, a four-dimensional nonlinear averaged equation is obtained. For the first time, a numerical investigation of the research reveals critical insights into the behavior of the system, such as waveforms, phase diagrams, and Poincaré maps, highlighting the effects of different angles of the stiffeners and variations of material parameters on the nonlinear dynamics of these shells.

First Failure Load of Rectangular Laminated and Sandwich Plates Using Isogeometric Analysis

This paper investigates the first failure load of simply supported and clamped laminate and sandwich plates loaded by a distributed normal traction on the top surface. A third-order shear and normal deformable plate theory, the isogeometric basis functions, and five failure criteria, namely, the maximum stress, the Tsai–Wu, the Tsai–Hill, the Hoffman, and the Hashin for laminated plates, and only the Tsai–Hill criterion for sandwich plates are used in this study. Of these, Hashin’s criteria distinguish between the fiber and the matrix failure. The in-plane stresses are found from the plate displacements and Hooke’s law, and the transverse stresses are recovered by integrating the equilibrium equations through the thickness. Effects of the plate aspect ratio, the fiber angle, the face sheet materials, and the core materials on the first failure load are identified. The computed results are found to agree well with those reported in the literature. Whereas the five failure criteria for laminated plates predict nearly the same value of the first failure load, they do not provide the same location of the failure initiation.

Isogeometric Analysis of Laminated and Carbon Nanotube Reinforced Composite Stiffened Plate Using Higher Order Shear Deformation Theory

Isogeometric Analysis of Laminated and Carbon Nanotube Reinforced Composite Stiffened Plate Using Higher Order Shear Deformation Theory

The vibration characteristics of LFGMs plates with openings based on spectral shift Legendre method

This paper introduces a novel spectral shift Legendre method for analyzing the vibrational characteristics of laminated functionally graded materials (LFGMs) plates with openings. By employing shifted Legendre series as permissible displacement functions, the computational process is significantly simplified by transforming the integral and differential calculations in the vibration characteristic equation into matrix multiplication. The stability and accuracy of the method are ensured by expanding the displacement at shifted Gauss-Lobatto points (SGLP). The proposed method is then extended to study the dynamic performance of LFGMs plates with openings. To achieve this, a dynamic equation for LFGMs plates without openings is constructed based on the first-order shear deformation theory (FSDT) and LFGMs theory. Subsequently, by setting the weights of the SGLP corresponding to the opening locations to zero, a rapid and accurate solution to the vibration problems of LFGMs plates with openings is realized. To verify the reliability and accuracy of the method, vibrational analyses of several typical LFGMs plates with openings are conducted, and the results are compared with existing literature and results obtained using the finite element method (FEM). The comparisons indicate that the method is reliable and effective, offering a more direct and convenient modeling process. This work significantly expands the application scope of spectral methods and provides a new perspective for the vibrational analysis of LFGMs with openings.

On thermo-mechanical buckling of porous bi-directional functionally graded plates using isogeometric analysis

The combined thermal and mechanical loads usually lead to buckling failure of key structures in various engineering fields. It is found that porous functionally graded structures can be used to reduce and evenly prevent potential impact on the stability of structures from extremely high temperature loads. In this paper, we employ the isogeometric analysis (IGA) method to investigate the thermo-mechanical coupling buckling behaviors of porous bi-directional functionally graded (PB-FGM) plates with porosity. The material properties such as Young's modulus, Poisson's ratio, and thermal expansion coefficient are assumed to be linearly dependent on the x-axial and z-axial coordinates. Such a gradient property can be easily realized by changing the porosity distribution. Based on the first-order shear deformation theory (FSDT), the thermo-mechanical buckling behaviors of PB-FGM rectangular and circular plates under combined thermal and mechanical loads are numerically studied. This paper elucidates the mechanical mechanism through which gradient indexes, aspect ratios, loading types, and boundary conditions influence the critical thermo-mechanical coupling-induced buckling temperature.