Strain-displacement Equations Research Articles

Stability and dynamic analyses of hybrid laminates using refined higher-order zigzag theory

This article presents a comparative study on the stability and dynamic response of hybrid laminated composite plates over monolithically produced laminated composite plates. A Refined Higher-order Zigzag Theory (RHZT) is formulated and implemented using finite element method for the analysis of hybrid laminates. This formulation considers both the interlaminar shear stress continuity and shear-free boundary conditions at surfaces of the plates. The numerical implementation employs C 0 continuous eight-noded isoparametric plate elements considering geometric nonlinearity. The element stiffness matrix is formulated to consider both the linear and nonlinear terms of the strain-displacement equations. Derivation of the consistent mass matrix for a plate element is presented following the approximation of total kinetic energy and utilising Hamilton’s principle. Numerical examples are provided to demonstrate buckling and free vibration analyses (as eigenvalue problems). The results are numerically verified and validated using data from published literature. Finally, parametric studies are presented to demonstrate the effectiveness of fibre hybridisation approach on the buckling and free vibration behaviour of laminated composites.

Nonlinear dynamic response of a temperature-dependent FGM spherical shell under various boundary conditions and thermal shocks: Examination of dynamic snap-through

In this study, a comprehensive exploration of the nonlinear dynamic snap-through phenomenon observed in shallow spherical shells composed of functionally graded materials (FGMs) is examined. The analysis encompasses the interplay of thermomechanical characteristics within metal and ceramic phases, taking into account their temperature-dependent behaviors. To address this problem, the transient heat conduction equation in a nonlinear form, accounting for the temperature dependence factor is derived. To solve this complexity, the Generalized Differential Quadrature (GDQ) method is employed, coupled with the Crank-Nicolson time integrating scheme, leveraging an iterative approach for precision. The formulation of nonlinear dynamic motion equations is based on Hamilton's principle and incorporates the utilization of nonlinear strain-displacement equations and uncoupled thermoelasticity. In the spatial domain, GDQ proves invaluable in solving these nonlinear, coupled equations. In the temporal domain, the Newmark procedure in conjunction with the Newton-Raphson iterative method are utilized to navigate through dynamic equations, yielding insights into shell's behavior over time. In the initial stages of the investigation, we validate our spherical shell formulation and methodologies by comparing shell's response with existing, simpler works. This step ensures the reliability and accuracy of our model. The primary objective is to assess the presence and characteristics of the thermal snap-through phenomenon within the shell under these diverse conditions.

A simple recovery post-processing of stresses and displacements in Reissner–Mindlin analysis of anisotropic laminated plates

The bending problem of anisotropic laminated plates is considered, modeled with first order shear deformation (FSDT) kinematic model and approximations obtained from the Generalized Finite Element Method (GFEM/ XFEM). A procedure is developed to recover the transverse normal and shear stresses and all displacement components, with improved variations across the laminate thickness, generating a complete three-dimensional approximate solution of the problem. The procedure starts with the results issuing from direct computations of in-plane stresses and displacements obtained by the 2D kinematic and constitutive equations. The recovered fields are obtained to, approximately, enforce local equilibrium, constitutive and strain–displacement equations in their three-dimensional forms, and interlaminar continuity. The general procedure considers inertia forces and von Kármán non-linearity. Corrections are made to impose the necessary 3D boundary conditions in both faces of the laminate. The easy way the GFEM admits basis function enrichment, whether by singular, discontinuous or by higher order p-enrichment, on a fixed mesh, makes the entire recovery procedure straightforward and non-iterative. The recovered fields accuracy is demonstrated in standard problems against exact solutions from three-dimensional elasticity and FEM reference approximations. Up to the author’s knowledge, the presented strategy is novel in the published literature of non-iterative post-processing methods. It provides a simple mean to obtain all stress and displacement component approximations necessary to application in many complete 3D local failure theories.

Self-weight and thermal bifurcation analysis of functionally graded vertical annular plates

The present paper thoroughly investigates the influence of structural weight on the thermal buckling of standing circular and annular plates. The structure is composed of symmetrically functionally graded materials (FGM) in terms of its thickness. The first-order theory of shear deformation along with nonlinear strain–displacement equations has been employed to derive the governing equations of the FG plate. First, the asymmetric pre-buckling path of the structure under the influence of weight and heat is calculated by a semi-analytical method. The plate's stability relations are extracted using the adjacent equilibrium criterion and virtual work principle. Then, the stability equations are analyzed by the 2D generalized differential quadrature (GDQ) method. Parametric results are presented based on the analysis of the FGM power-law index, geometric ratios, and edge constraints on the buckling of the vertical plate. The study reveals the critical power-law index and thickness at which the structure becomes unstable, shedding light on the previously neglected calculation of weight-induced buckling in vertical structures. Also, the effect of weight on the buckling mode shapes of heavy plates and disturbing its symmetry is demonstrated. Additionally, this study presents the first investigation into the effect of weight on the thermal stability of annular FG plates.

Time-dependent creep analysis of ultra-high-temperature functionally graded rotating disks of variable thickness

Functionally graded materials (FGMs) are high temperature-resistant materials that can simultaneously maintain metallic tenacity and anti-corrosive properties. Nevertheless, using FGMs during a multi-year service life at ultrahigh temperatures is crucial. Hence, the time-dependent creep response of variable-thickness rotating disks made of FGM is investigated. Four different disk profiles of linear, concave, convex, and uniform are considered. The material's creep properties are defined by the Bailey-Norton creep law. Loading is a rotation-based mechanical body force and a uniform temperature throughout the disk. Simultaneous solution of equilibrium, stress-strain, and strain-displacement equations yields a non-homogenous differential equation containing variable and time-dependent coefficients. In an attempt to optimize the computation cost, Bat and Fish algorithms were used to optimize the initial strain presumptions. Semi-analytical solution of this differential equation gives radial and circumferential stress histories and displacement histories. To confirm the solution method, initial thermo-elastic radial stress, and the effective stress history are validated with the existing literature; there is a good agreement between the results. In addition, the finite element software ABAQUS was used to model the FGM disk thermo-elastic behavior, and the result was compared with the semi-analytical solution results. This study emphasizes the significance of accounting for creep effects in the design of FGM rotating disks, as remarkable changes in their displacements and stresses occur over time. This study emphasizes the significance of accounting for creep effects in the design of FGM rotating disks, as notable changes in their displacements and stresses occur over time.

Frequency response analysis of edge-cracked magneto-electro-elastic functionally graded plates using extended finite element method

This paper studies the frequency response of edge-cracked magneto-electro elastic functionally graded (ECMEE-FG) plates using the extended finite element method (XFEM). First-order shear deformation theory (FSDT), von Karman's nonlinear strain-displacement equations, and a modified power-laware used to develop the numerical model. The coupled equations are derived and analyzed using Hamilton's principle and extended finite element methods. The influence of B-rich bottom and F-rich bottom material gradation, crack orientation, crack length, and aspect ratio on the geometrically nonlinear frequency response was investigated after the current study was validated. Furthermore, crack propagation behavior in the ECMEE-FG plate was examined. The results could be helpful for the design of functionally graded magneto electro elastic structures and devices.

Nonlinear dynamic analysis of sandwich shell panels with auxetic honeycomb core and curvilinear fibre reinforced facesheets

The nonlinear dynamic behaviour of sandwich shell panels having auxetic honeycomb core and facesheets composed of composite material with curvilinear fibres is investigated for the first time. A finite element study is performed considering a third-order shear deformation theory incorporating Murakami zig-zag effects for the analysis of the variable stiffness composite laminate (VSCL) auxetic honeycomb sandwich panels. A nine-noded C0 isoparametric quadratic shell element with eleven degrees of freedom per node is considered and von Kármán nonlinear strain-displacement equations are used to include the geometrical nonlinearity. The linear free vibration and nonlinear transient response of the panels are obtained using the eigenvalue solution and Newmark's time integration scheme, respectively. Most of the auxetic sandwich structures considered in the literature are made up of aluminium metal and therefore, the authors explored the viability of the VSCL based auxetic sandwich panels. A thorough numerical study including the effects of the geometrical parameters of the honeycomb unit cell, curvilinear fibre path angles and boundary conditions on the nonlinear dynamic response of auxetic sandwich panel with negative Poisson's ratio is presented.

Simplified Closed-Form Method to Large Deflection Stability Analysis of Thin Rectangular Plate

At the moment, the analyses of postbuckling behavior of thin rectangular plates of constant thickness are generally based on von Karman’s large deflection equations which make use of stress functions and trigonometric variables. These equations are coupled, nonlinear partial differential equations of fourth order each. Getting a closed-form solution of these equations is near impossible and tedious. The present work presents a simplified closed-form general equation using a variational approach for large deflection analysis. The approach adopted here is devoid of Airy’s stress functions. A new strain-displacement equation is formulated, and the total potential energy equation is minimized. The resulting compatibility equation was solved to obtain the general governing stability equation under large deflection. This governing equation was applied to a plate simply supported all-around using polynomial displacement function and numerical results were obtained. To validate the numerical results obtained, they were compared with the values obtained by Levy whose results are generally acclaimed as exact and with two others. It was observed that the minimum and maximum percentage differences were 0% and 41.56% at stress parameters 3.66 and 21.45 respectively. Also, for deflection to thickness ratio (w/t), the present results showed a close agreement with those of Levy with a minimum and maximum percentage difference of 0.07% and 41.56% at w/t of 0 and 3.376 respectively. Importantly, the present result lies between two other research results. We, therefore, conclude that the present work is adequate and a new simple closed-form approach to understanding and predicting the postbuckling strength of thin rectangular plates.

Study on the similarity of elasticity and ideal plasticity response of thin plate under impact loading

For the impact similarity problem of the scaled model and the prototype usually have different materials with elastic and plastic properties, the differences of material properties and the coexistence of elastoplastic in different deformation stages will lead to the failure of the previous impact similarity theory. Based on the theory of the thin plate impact problem, the similarity law of impact response was derived by using the method of equation similarity analysis for the material with the linear elastic and ideal rigid-plastic properties. The basic equations of the thin plate structure, such as the energy conservation equation and the strain-displacement equation, were analyzed using equation similarity analysis methods, and the similarity scaling factor of the ideal elastic-plastic thin plate structure was derived. Based on the equation similarity analysis methods, a thickness compensation method that can simultaneously consider the similarity of elastic deformation and plastic deformation was proposed. For the impact similarity problem of the scaled model and the prototype using different ideal elastoplastic materials, this method can be used to calculate the geometric sizes and load conditions of the scaled model through the material properties when the response of the scaled model is similar to that of the prototype. Two finite element models of circular plate mass impact and circular plate velocity impact were established.The geometric sizes and load conditions of the scaled model were calculated through the thickness compensation method when the prototype uses aluminum alloy and the scaled model uses different materials such as steel, brass, etc. The applicability of the thickness compensation method was verified by the response of prototype and scaled model. The results show that the structural response of the scaled model obtained by the thickness compensation method can accurately predict the impact response of the prototype, even though the scale model and the scale model use different materials.

Modified G-L thermoelasticity theory for nonlinear longitudinal wave in a porous thermoelastic medium

This paper presents a nonlinear analysis of the transient response of a porous medium affected by external traction. The formulation is presented based on a new modified G-L theory of thermoelasticity while temperature and strain rate parameters are included in the governing equations. Based on the finite strain theory (FST), the Lagrangian strain–displacement equation and Second Piola-Kirchhoff stress are applied to express the generalized form of thermoelasticity equations including large deformation. Both mechanical and thermal shocks are applied and finally, the effects of loading rate on transient response are discussed. An updated finite element method and Newmark’s numerical time integration method are employed to solve the nonlinear and time-dependent equations. It is determined that the modified G-L model is more powerful in predicting the wave propagation phenomenon. Furthermore, the findings indicate that the external surface shock induces compressive stresses, temperature raise, and volume fraction field increase in a porous medium. It is found compared to the classic model, the modified G-L model is superior in capturing the perceived wave propagation phenomenon.

Nonlinear dynamic responses of electrically actuated dielectric elastomer-based microbeam resonators

This paper aims to investigate the chaotic and nonlinear resonant behaviors of a dielectric elastomer-based microbeam resonator, incorporating material and geometric nonlinearities. The von Kármán strain-displacement equation is utilized to model the geometric nonlinearity. Material nonlinearity is described via the hyperelastic Gent model and Neo-Hookean constitutive law. The applied electrical loading to the elastomer includes both static and sinusoidal voltages. The governing equations of motion are formulated based on an energy approach and generalized Hamilton’s principle. Employing a single-mode Galerkin technique, the governing equations are obtained only in terms of time derivatives. The governing ordinary differential equations are solved by means of the multiple scale method and a time-integration-based solver. The nonlinear resonance characteristics are explored through the frequency-amplitude plots. The nonlinear oscillations of the system are analyzed making use of visual techniques such as phase plane diagram, Poincaré section and time history, and fast Fourier transform. Based on the results obtained, the resonant behavior is the hardening type. The vibration of the dielectric elastomer based-microbeam is the quasiperiodic response.

LINEAR AND NONLINEAR FREE VIBRATION ANALYSIS OF RECTANGULAR PLATE

The major assumption of the analysis of plates with large deflection is that the middle surface displacements are not zeros. The determination of the middle surface displacements, u0 and v0 along x- and y- axes respectively is the major challenge encountered in large deflection analysis of plate. Getting a closed-form solution to the long standing von Karman large deflection equations derived in 1910 have proven difficult over the years. The present work is aimed at deriving a new general linear and nonlinear free vibration equation for the analysis of thin rectangular plates. An elastic analysis approach is used. The new nonlinear strain displacement equations were substituted into the total potential energy functional equation of free vibration. This equation is minimized to obtain a new general equation for analyzing linear and nonlinear resonating frequencies of rectangular plates. This approach eliminates the use of Airy’s stress functions and the difficulties of solving von Karman's large deflection equations. A case study of a plate simply supported all-round (SSSS) is used to demonstrate the applicability of this equation. Both trigonometric and polynomial displacement shape functions were used to obtained specific equations for the SSSS plate. The numerical results for the coefficient of linear and nonlinear resonating frequencies obtained for these boundary conditions were 19.739 and 19.748 for trigonometric and polynomial displacement functions respectively. These values indicated a maximum percentage difference of 0.051% with those in the literature. It is observed that the resonating frequency increases as the ratio of out–of–plane displacement to the thickness of plate (w/t) increases. The conclusion is that this new approach is simple and the derived equation is adequate for predicting the linear and nonlinear resonating frequencies of a thin rectangular plate for various boundary conditions.

Nonlinear vibration of conical shell with a piezoelectric sensor patch and a piezoelectric actuator patch

In this study, nonlinear vibration of a conical shell with a piezoelectric sensor patch and a piezoelectric actuator patch is evaluated. The strain displacement relation is considered to be nonlinear and based on that the equations of motion and piezoelectric sensor output signal are extracted. For the first time, nonlinear electromechanical equations of motion for the conical shell with a piezoelectric sensor layer and a piezoelectric actuator layer are extracted in detail and presented. The producer of extraction of conical shell electromechanical equations of motions is presented for further clarifications. The output signal of the piezoelectric sensor patch considering the nonlinear strain displacement equation of motion is derived. For controlling the vibration actively, a proportional controller is defined by which the applied actuator signal is determined. The nonlinear electromechanical equations of motion are expanded based on conical shell strains. By applying the Galerkin method, the nonlinear time-domain equations are extracted. The complicated attained equations are solved using the harmonic balance method. Conical shell nonlinear frequency response in uncontrolled condition and controlled condition is computed and compared with each other. The frequency response shows that the active vibration control of the conical shell increases the nonlinearity of the system and reduces the vibration amplitude. In addition, free nonlinear vibration of the proposed system is calculated and presented. The results show higher impacts of piezoelectric patches on the conical shell free vibration response and they can reduce the vibration amplitude dramatically.

Nonlinear vibration analysis of smart functionally graded plates

Abstract This paper presents the damping characteristics of nonlinear vibrations of functionally graded (FG) plates utilizing piezoelectric composite material. For this analysis the active fiber composite (AFC) is taken as piezoelectric composite material. Finite element (FE) models are made for the FG plates combined with active constrained layer damping (ACLD) patches made of AFC material. Von Karman strain–displacement equations considering the displacement fields and satisfactory boundary conditions are used. The working of the ACLD patches for nonlinear vibration damping of FG plates is explored where AFC is used for the constraining layer of the ACLD patches. The Golla-Hughes-McTavish method is used for modeling the constrained layer of ACLD patch which is considered to be made of a viscoelastic material. Simulation models are also developed using ANSYS software for smart FG plates for validating the FE model. The contribution of ACLD patches for the nonlinear vibration control of the FG plates is investigated.

Non-Linear Response of Torsional Buckling Piezoelectric Cylindrical Shell Reinforced with DWBNNTs Under Combination of Electro-Thermo-Mechanical Loadings in Elastic Foundation

Nanocomposites provide new properties and exploit unique synergism between materials. Polyvinylidene fluoride (PVDF) is an ideal piezoelectric matrix applicable in nanocomposites in a broad range of industries from oil and gas to electronics and automotive. And boron nitride nanotubes (BNNTs) show high mechanical, electrical and chemical properties. In this paper, the critical torsional load of a composite tube made of PVDF reinforced with double-walled BNNTs is investigated, under a combination of electro-thermo-mechanical loading. First, a nanocomposite smart tube is modeled as an isotropic cylindrical shell in an elastic foundation. Next, employing the classical shell theory, strain-displacement equations are derived so loads and moments are obtained. Then, the total energy equation is determined, consisting of strain energy of shell, energy due to external work, and energy due to elastic foundation. Additionally, equilibrium equations are derived in cylindrical coordinates as triply orthogonal, utilizing Euler equations; subsequently, stability equations are developed through the equivalent method in adjacent points. The developed equations are solved using the wave technique to achieve critical torsional torque. Results indicated that critical torsional buckling load occurred in axial half-wave number m = 24 and circumferential wave number n = 1, for the investigated cylindrical shell. The results also showed that with the increase in the length-to-radius ratio and in the radius-to-shell thickness ratio, the critical torsional buckling load increased and decreased, respectively. Lastly, results are compared in various states through a numerical method. Moreover, stability equations are validated via comparison with the shell and sheet equations in the literature.

Assumed stress quasi-conforming formulation for static and free vibration analysis of symmetric laminated plates

Purpose Numerous finite elements are proposed based on analytical solutions. However, it is difficult to find the solutions for complicated governing equations. This paper aims to present a novel formulation in the framework of assumed stress quasi-conforming method for the static and free vibration analysis of anisotropic and symmetric laminated plates. Design/methodology/approach Firstly, an initial stress approximation ruled by 17 parameters, which satisfies the equilibrium equations is derived to improve the performance of the constructed element. Then the stress matrix is treated as the weighted function to weaken the strain-displacement equations. Finally, the Timoshenko’s laminated composite beam functions are adopted as boundary string-net functions for strain integration. Findings Several numerical examples are presented to show the performance of the new element, and the results obtained are compared with other available ones. Numerical results have proved that the new element is free from shear locking and possesses high accuracy for the analysis of anisotropic and symmetric laminated plates. Originality/value This paper proposes a new QC element for the static and free vibration analysis of anisotropic and symmetric laminated plates. In contrast with the complicated analytical solutions of the equilibrium equations, an initial stress approximation ruled by 17 parameters is adopted here. The Timoshenkos laminated composite beam functions are introduced as boundary string-net functions for strain integration. Numerical results demonstrate the new element is free from shear locking and possesses high accuracy for the analysis of anisotropic and symmetric laminated plates.

Quasi-conforming analysis method for trimmed CAD surfaces

In this paper, the quasi-conforming method is introduced for the analysis of trimmed computer aided design (CAD) surfaces. The main benefit of the proposed method is that the boundary curves of elements are adopted for the numerical integration directly. In the quasi-conforming technique, the strains are approximated by using polynomials, and the weighted test functions are used to weaken the strain-displacement equations. The interpolation functions are introduced for strain integration. An appropriate choice of initial strain approximation and weighted test function ensures that inner-field functions are not required for strain integration, and this is used for the analysis of trimmed CAD surfaces. For example, the assumed stress quasi-conforming method is applied for the two-dimensional linear elastic problem. All the element edges are approximated by using quadratic Bézier curves for the conciseness, and this is easily incorporated into existing finite element codes and applied to Dirichlet boundary conditions. Numerical examples indicate the effectiveness and accuracy of the method.

An efficient finite strip procedure for initial post-buckling analysis of thin-walled members

An efficient procedure based on the semi-analytical finite strip method with invariant matrices is developed and applied to analyze the initial post-buckling of thin-walled members. Nonlinear strain–displacement equations are introduced in the manner of the von Karman assumption for the classical thin plate theory, and the formulations of the finite strip methods are deduced from the principle of the minimum potential energy. In order to improve the computational efficiency, an analytical integral of the stiffness matrix is transformed into matrix multiple calculation with introducing invariant matrices which can be integrated in advance only once. Three commonly employed benchmark problems are tested with proposed method and other state-of-the-art methods. The corresponding comparison results show that: (1) this finite strip method is proved to be a feasible and accurate tool; (2) compared with the calculation process of the conventional finite strip methods, the proposed procedure is much more efficient since it requires the integration of the stiffness matrix only once no matter how many iterations are needed; and (3) the advantage of time-saving is greatly remarkable as the number of iterations increases.

THE CONTACT MECHANIC FOR FUNCTIONALLY GRADED LAYER RESTING ON HALF PLANE

Bu çalışmada, yarım düzlem üzerine oturan ve üstten yayılı yükler ile bastırılan fonksiyonel derecelendirilmiş (FD) bir tabakanın eksenel simetrik değme mekaniği elastisite teorisine göre ele alınmıştır. Değme yüzeyleri sürtünmesiz olup, kütle kuvvetlerinin etkisi ihmal edilmiştir. Denge denklemlerine, bünye denklemlerine ve şekil değiştirme-yer değiştirme bağıntılarına sınır şartları uygulanıp problemde integral dönüşüm teknikleri kullanılarak değme gerilmelerinin bilinmeyen olduğu integral denklemine indirgenmiştir. İntegral denklemin sayısal çözümü, denge şartı da dikkate alınarak, Gauss-Jacobi integrasyon formülasyonuyla gerçekleştirilmiş, değme uzunlukları ve değme gerilmeleri bulunmuştur. Ayrıca çalışmada, yayılı yükün uygulama genişliği ve malzeme özelliklerinin fonksiyonel değişiminin tabakada oluşacak gerilme dağılışlarına etkisi incelenmiştir.

An efficient finite strip procedure for initial post-buckling analysis of composite laminated members

An efficient procedure based on the semi-analytical finite strip method with newly introduced invariant matrices is developed to analyze the initial post-buckling of composite laminated members. The nonlinear strain-displacement equations obtained from the Von-Karman assumption and three plate theories, which are classical thin plate theory, first-order shear deformation plate theory, and high-order shear deformation plate theory can be used to evaluate the initial post-buckling performance of the composite laminated members. According to the principle of the minimum potential energy, the formulations of the finite strip method can be deduced. In order to improve the computational efficiency, the pre-integrated invariant matrices are introduced, which can convert the complicated analytical integral calculation of the stiffness matrix into a relatively simple matrix multiplication calculation. Several benchmark problems are tested based on the proposed method and other conventional methods. The corresponding comparison results show that: (1) the proposed method is proved to be feasible and accurate for those three different theories; (2) compared with the other conventional finite strip methods, the proposed method is much more efficient since it requires the integration of the stiffness matrix only once no matter how many iterations are needed, (3) and the advantage of time-saving is increasingly remarkable as the number of iterations increases.