Global-local Theory Research Articles

Double superposition hypothesis for smart composite plates-two approaches

Structural failure in laminated composites is primarily due to the interlaminar debonding that results due to the transverse shear stresses and such scenario arises due to their inaccurate prediction by various shear deformation theories. In order to predict such stresses accurately and in a simple yet effective manner, the global local theories are employed. In this study, two such theories have been used for static response of smart laminated plates under electromechanical loading. Though both theories use the same principle of double superposition, one of them has been observed to predict better accuracy for all laminated plates subjected to both mechanical loading and electric potential loading conditions. The effect of inclusion of the piezoelectric coefficient in the transverse displacement assumption has been observed to yield much improved response.

Numerical investigating of the effect material and geometrical parameters on the static behavior of sandwich plate

Studying the behavior of sandwich panels is very important due to their widespread use in different industries. Therefore, over the past decades, various theories have been proposed to study the behavior of these panels. In this paper a higher order global-local theory with 3D equilibrium-based corrections is presented to study behavior of thick and thin sandwich plate with flexible and auxetic core. In addition to correcting the results with 3D elasticity equations, another important advantage of the presented theory is the ability to consider the transverse core deformation of the sandwich panels. It should be mentioned that to study the behavior of thick sandwich panels, especially with soft core, the existence of this feature is very necessary and has a great effect on the accuracy of the obtained results. Comparison of the obtained results with those existing in valid references showed that the formulation of the provided finite element had a very good accuracy even for thick and thin sandwich plates. Finally, the effect of different material and geometrical parameters on the behavior of sandwich plates are carefully investigated using the presented theory.

Parametric study of Sandwich Plates with Viscoelastic, Auxetic Viscoelastic and Orthotropic Viscoelastic Core Using a New Higher Order Global-Local Theory

In this paper, sandwich plates with flexible core and composite surfaces as well as viscoelastic and auxetic core is investigated under dynamic loading. A new higher order global-local theory is used for simulation of the dynamic behavior of sandwich plate. Ability of simulation the thickness changes of the plate and calculation of exact transverse stresses which are so crucial for studying of thick sandwich plate especially by soft core are some of the important features of the presented theory. Furthermore, in terms of solving equations, an iterative incremental method based on the formulation of transient nonlinear finite element as well as a real time algorithm was employed to simulate viscoelastic behavior accurately. The results indicate a significant increase in the stiffness of the sandwich plate due to the auxetic properties of the core materials, leading consequently to the reduction of the vibration amplitude and stresses level. Some of the innovations belonging to this paper are: 1) presenting a global-local higher-order theory while considering the changes in the thickness of the sandwich plates; 2) calculating transverse stresses using the three-dimensional elasticity method as well as modifying the results obtained from displacement and inertial effects based on this method; 3) simulating sandwich plate with viscoelastic and auxetic cores; 4) taking orthotropic properties for the viscoelastic core into account.

Experiment and analysis on free vibration of sandwich plates based on an alternative sinusoidal global-local theory

This study presents an alternative and easy-to-use model to produce accurately higher-order frequencies of composite and sandwich plates. The proposed model only contains the independent displacement variables, and compatible conditions of displacements and stresses at interfaces of adjacent layers can be met beforehand. Exact solution is firstly utilized to appraise capability of present model. Subsequently, a triangular element based on the proposed model has been also established for dynamic analysis of sandwich plates with complex boundary conditions, where the refined element approach is utilized to conquer compatible requirement of first derivatives of deflection at shared edges of adjacent elements. To strictly evaluate the models, tests of natural frequencies for sandwich plates composed of aluminium face sheets and PMI foam core are carried out, which are less reported in literature. Compared to testing data, the good performance of the proposed model can be illustrated. This work can also demonstrate that compatible conditions of transverse shear stresses at interfaces have a significant impact on accurate prediction of higher-order frequencies of composite and sandwich plates, but model fulfilling compatible conditions of stresses might also fail to produce precisely higher-order frequencies.

Novel rule-based global-local theory and energy model for sandwich plates with compliant cores and unevenly-distributed anisotropic SMA wires under impulsive/impact loads

A rational new plate theory must meet specific guidelines and cannot be proposed by merely choosing a different function for the transverse variations of the displacement components. In the present research, a novel hyperbolic global-local plate theory is proposed based on 8 criteria, for the first time and implemented in a new energy formulation that takes into account the potential energy of the contact region, for behavior analysis of biaxially preloaded sandwich plates with unevenly-distributed SMA wires and soft cores under impulsive/impact loads. The proposed theory contains both odd and even functions and consequently, is more appropriate for asymmetric lamination schemes; in addition to satisfying the continuity condition of the interlaminar stresses. It is the first time that stress-strain asymmetry and anisotropy of the SMA is incorporated in the impact response analysis. Modified micro-macro-mechanical and bridging models are employed to relate the apparent properties to the local ones and vice versa. Effects of the uniform, double-linear, and double-sinusoidal distributions of the SMA wires are compared. The localized time variations of phases evolutions of the anisotropic SMA wires are predicted based on a particular algorithm, employing an iteration-based updating solution procedure for the resulting finite element formulation within each time step.

Higher-order global-local theory with novel 3D-equilibrium-based corrections for static, frequency, and dynamic analysis of sandwich plates with flexible auxetic cores

ABSTRACTIn the present article, a high-order global-local theory with three-dimensional elasticity corrections is employed to trace the local and instantaneous variations of lateral deflections and stress components of sandwich plates with auxetic (negative Poisson ratio) cores under static and dynamic loads. Effects of the auxetecity of the core material on the natural frequencies are evaluated as well. The governing equations are extracted based on Hamilton's principle. The main novelties of the present research in comparison to the available literature and previous researches of the second author of the present paper are: (i) Presenting a higher-order global-local plate theory with a novel equilibrium-based three-dimensional elasticity corrections, (ii) Incorporation of the transverse flexibility of the core; a fact that is crucial when studying behaviors of thick or soft core sandwich plates, (iii) Frequency and dynamic behavior analyses (in addition to the traditional static analysis) of sandwich plates with soft cores by means of the presented accurate global-local theory, and (iv) Investigation of the negative Poisson ratio (auxeticity) effects of the core material on the static (stress) and dynamic responses and natural frequencies. All these items are accomplished here, for the first time. Since the transverse shear stresses are extracted based on the three-dimensional elasticity theory, in contract the traditional constitutive-based theories, the inter-laminar continuity condition of the transverse shear stresses is met. The verification results show that the presented finite element formulation leads to highly accurate results, even for thick or soft core sandwich plates. A comprehensive parametric study is accomplished to evaluate effects of the auxeticity of the core material and transverse compliance of the core on the resulting displacement and stress distributions, natural frequencies, and dynamic responses. Results reveal that auxeticity of the core material decreases the global and relative stresses and lateral deflections of the face sheets and the core compliance may lead to asynchronized movements of the face sheets and strengthen the local bending and extensions.

An efficient materially nonlinear finite element model for reinforced concrete beams based on layered global-local kinematics

In this study, a finite element model with low degrees of freedom is presented for the materially nonlinear analysis of reinforced concrete (RC) beams. The kinematics considered for the RC beam is based on a layered global-local (LGL) theory. Unlike the classical one-dimensional (1D) beam kinematics (e.g. Euler–Bernoulli’s and Timoshenko’s beam theory), the effects of shear deformation and normal flexibility of the RC beams are considered in the formulation of the present LGL theory. No shear correction factor is needed in the present LGL theory. The LGL theory has only one extra displacement variable more than Timoshenko’s beam theory. Elastoplastic and fixed smeared crack models are considered for modeling the material nonlinearity of the reinforcing steels and cracked concrete, respectively. The results obtained from the present nonlinear finite element model are compared with experimental and numerical results. Comparisons show that the present model gives accurate results at a low computational cost.

Assessment of refined high-order global–local theory for progressive failure analysis of laminated composite beams

An efficient and computationally low-cost one-dimensional (1D) finite element model is developed for the progressive failure analysis of laminated composite beams. The employed finite element formulation is based on a refined high-order global–local beam (RHGB) theory which satisfies all the kinematic and stress continuity conditions at the layer interfaces. This RHGB theory also considers effects of the transverse normal stress and transverse flexibility. By using the well-known failure theories, an iterative method is adopted to predict the first ply failure load of the laminated composite beam. After the first ply failure, the material properties of the failed elements are modified using a stiffness degradation factor. Then, the applied load is increased step-by-step. This analysis is repeated for each load increment until the ultimate strength of the laminate is reached. Hashin, Maximum stress, Hoffman, Tsai–Hill and Tsai–Wu failure criteria are used to assess the possible damage at beam elements from the initial to the final step. The present failure formulation has been validated by comparison with experimental and theoretical results available in the open literature.

A model of composite laminated Reddy plate of the global-local theory based on new modified couple-stress theory

ABSTRACTIn this article, based on the global-local theory, a model for a composite laminated Reddy plate of new modified couple-stress theory is developed for the first time. This model satisfies free surface conditions and the geometric and stresses continuity conditions at interfaces. Differing from existing modified couple-stress theory, an anisotropic constitutive relation is suggested. There is only one micro material characteristic length constant in each layer of the composite laminated plate. Principle of virtual work is employed to derive the equilibrium equations and the corresponding boundary conditions. With the example of a cross-ply simple-supported Reddy plate subjected to the bending load q0 = sin (πx/L)sin (πy/L), the transverse shear stresses at interfaces are addressed. Additionally, the numerical results show that the present plate model can capture the scale effect, especially the scale effect of the transverse shear stresses at interfaces of the composite laminated plate.

Efficient coupled refined finite element for dynamic analysis of sandwich beams containing embedded shear-mode piezoelectric layers

ABSTRACTAn efficient and computationally low cost finite element (FE) model is developed for dynamic free and forced response of sandwich beams with embedded shear piezoelectric layers based on a coupled refined high-order global-local theory. Contrary to most of the available models, all of the kinematic and stress boundary conditions are ensured at the interfaces of the shear piezoelectric layers. Moreover, both the electrical-induced strains components and transverse flexibility are taken into account for the first time in the present theory. For validation of the proposed model, various free and forced vibration tests for thin and thick sandwich beams are carried out. For various electrical and mechanical boundary conditions, excellent agreement has been found between the results obtained from the proposed formulation with previously published and the coupled two-dimensional (2D) FE results.

Fully coupled thermo-mechanical analysis of bi-directional FGM beams using NURBS isogeometric finite element approach

The fully coupled thermo-mechanical behavior of bi-directional functionally graded material (FGM) beam structures is studied using a computationally low cost isogeometric finite element model. In the proposed analysis, the temperature is considered as a primary variable. The kinematical equations of the bi-directional FGM beam are described by a refined high order global–local theory. A combination of polynomial and exponential expressions is used to introduce the in-plane displacement field. This representation of the in-plane displacement field allows avoiding shear correction factors. Concerning the thermal part, a high-order temperature field is considered through the thickness direction of the bi-directional graded beam. The capability of this novel formulation has been assessed through various thermal and thermo-mechanical tests. To this aim, the obtained numerical results from the proposed isogeometric formulation have been compared with thermo-elasticity solutions or other available results in open literature. It is found that the results obtained from the present beam formulation agree quite well with the reference results.

Accurate eccentric impact analysis of the preloaded SMA composite plates, based on a novel mixed-order hyperbolic global–local theory

Various high-order polynomial, trigonometric, exponential, and hyperbolic plate theories have recently been proposed. Majority of these theories have described the displacement field through odd functions to satisfy the zero shear traction condition on the top and bottom surfaces of the plate and thus are mainly suitable for symmetric lamination schemes or material properties distributions. In the present paper, a higher-order global–local hyperbolic plate theory that includes both odd and even functions and consequently, is especially adequate for description of the general asymmetric displacement fields, is proposed. This theory is employed to investigate the complicated responses of a pre-stressed composite plate with SMA wires subjected to a low velocity eccentric impact. Non-uniform and time-dependent distributions of the phases of the SMA wires are considered instead of using the simple recovery stress or uniform martensite volume fraction distribution approaches. A refined contact law based on a proper homogenization technique is proposed instead of using the traditional Hertz law. The resulting highly nonlinear finite element governing equations are solved by an iterative algorithm within each time step. Different contact laws are considered for the loading and unloading phases. A comprehensive parametric study is accomplished to study effects of the pre-stresses, eccentricity, and lamination schemes.

A global–local theory with stress recovery and a new post-processing technique for stress analysis of asymmetric orthotropic sandwich plates with single/dual cores

While the 3D elasticity and the available local and zigzag theories encounter inaccuracies for very thin plates, results of the global theories become unreliable for thick plates with severe transverse variations in the material properties. In the present research, a quadratic finite element global–local sandwich plate theory with elasticity correction (stress recovery) is proposed for static stress and displacement analysis of sandwich plates with orthotropic face sheets and single or dual cores. Since the transverse shear stresses are derived based on the three-dimensional theory of elasticity, the continuity condition of the transverse shear stresses is satisfied at the interfaces between the layers, a priori. The presented theory not only leads to higher accuracies in comparison to the available high-order zigzag theories in some cases (especially, for soft or dual cores) but also it is computationally more economic. Results show that the results of the available zigzag and global–local theories may encounter inaccuracy problems not only for huge numbers of the sub-layers, but also for small numbers of the orthotropic layers.

Eccentric impact analysis of pre-stressed composite sandwich plates with viscoelastic cores: A novel global–local theory and a refined contact law

A novel double superposition power-exponential global–local theory and a refined contact law are developed to investigate eccentric low-velocity impact responses of rectangular sandwich plates with viscoelastic cores. The continuity conditions of the transverse normal and shear stresses at the interfaces between layers is satisfied a priori. Stiffness of the underneath layers is considered in the contact law as well, for the first time. The non-linear integro-differential governing equations are solved by a second-order finite element and a special numerical time integration procedure. Effects of the pre-stresses on the indentation and contact force are investigated for the first time. Moreover, effects of the eccentricity on the impact responses of the sandwich plates are discussed in detail, for the first time. Verification of the results is accomplished through comparing present results with experimental results of a known reference. Results show that in the eccentric impacts, the contact force and the absorbed energy increase. Therefore, the failure occurrence can be more likely in the eccentric impacts. Furthermore, by utilizing a viscoelastic core, the apparent stiffness of the contact region increases and consequently the impact force and the absorbed energy increase. Biaxial tension increases the impact force and consequently, leads to premature failures.

The Interlaminar Stresses Analysis of Composite Laminated Plates Based on the Generalized Higher-Order Global-Local Plate Theory

A generalized higher-order global-local theory was presented. The transverse shear stresses can be got directly through the constitutive equation without using the equilibrium equation. The second derivative of interpolation function was deduced. The hammer integration of triangular area coordinate method was applied to solve the multiple integration problem of the element stiffness matrix. The order choice of numerical integration was discussed and results obtained through two different integration orders were compared. The flow of how to compile a FORTRAN program was given. A moderately thick composite laminated plate was analyzed via finite element method (FEM) based on the theory and results were compared with that of Paganos three-dimensional elasticity. It shows that the interlaminar stresses are accurate for moderately thick laminated plates.

A finite element model based on coupled refined high-order global-local theory for static analysis of electromechanical embedded shear-mode piezoelectric sandwich composite beams with various widths

In this study, a new theory for the accurate simulation of the shear-mode behaviour of thin or thick piezoelectric sandwich composite beams is developed. The effects of transverse normal stress and transverse flexibility of layers are considered in the development of the proposed formulation. In order to increase the computational accuracy, all kinematic and stress continuity conditions are satisfied at layer interfaces. Moreover, for the first time, both the electrically induced strain components and the transverse flexibility are taken into account in the proposed formulation. Despite the fact that the number of unknown mechanical parameters in this theory is only one degree higher than the first order shear deformation theory, the accuracy is surprisingly more pronounced for the thicker beams.

A finite element based global–local theory for static analysis of rectangular sandwich and laminated composite plates

In the present paper, a global–local theory with three-dimensional elasticity correction is developed for stress and deformation analysis of the multilayered and sandwich plates. The governing equations are derived based on the principle of minimum potential energy. The coupled governing equations are solved based on a finite element procedure uses the second-order Lagrangian elements and a Galerkin-type formulation. Results of this theory are accurate for both thin and thick plates. The transverse shear stresses are extracted based on the three-dimensional theory of elasticity. Thus, the continuity of the interlaminar transverse shear stresses is satisfied a priori. The proposed finite element formulation is implemented using MATLAB software. Results revealed that, while the accuracy of the present theory is comparable to that of the three-dimensional theory of elasticity and sometimes (e.g., when a very soft core is used) is higher than the high-order zigzag theories, the presented solution approach is computationally more economic. Finally, a comprehensive parametric study including evaluating the influences of the lamination scheme, number of layers, plate aspect ratio, side-to-thickness ratio, and material properties variations on the in-plane stresses, transverse shear stresses, and displacement components is performed.

Global–local and zigzag-local theories for direct transverse shear stress computation in piezolaminated plates under thermal loading

Two dimensional laminate theories in which the number of primary displacement variables is independent of the number of layers are generally not capable of accurately predicting the transverse stresses in hybrid piezolaminated plates, directly from the constitutive equations. The prediction is particularly poor for the electric potential and thermal loadings. In a recent publication, the authors presented a coupled global–local theory (GLT) with 11 displacement unknowns and a zigzag-local theory (ZLT) with nine unknowns for hybrid plates, with a view to predict the transverse shear stresses accurately under electromechanical loading. It was found that the ZLT yields accurate and superior prediction compared to the GLT. In this study, we extend the two theories for thermal loading and assess their performance in comparison with the three dimensional piezothermoelasticity solution. It is found that, contrary to the electromechanical loading, the GLT performs better than the ZLT for thermal loading, in predicting transverse shear stresses, besides predicting displacements and inplane stresses accurately. The thermoelectric transverse normal strain is accounted for without introducing additional variables. Without this term, the GLT yields much worse results not only for deflection, but also for inplane displacements and stresses.

An analytical global–local Taylor transformation-based vibration solution for annular FGM sandwich plates supported by nonuniform elastic foundations

Free vibration of functionally graded annular sandwich plates resting on Winkler-type elastic foundations is investigated based on a zigzag global–local plate theory and a finite Taylor's transform whose center is located at the outer edge. Material properties of each layer may be graded in the transverse direction according to a power law. It is the first time that a global–local theory is combined with a layerwise analytical solution for analysis of the annular functionally graded sandwich plates. Various edge conditions are considered for the inner and outer edges. A parametric study including evaluating effects of the material properties distributions of the core and face sheets, symmetric and asymmetric layups, thickness to radius ratio of the plate, inner to outer radius ratio, coefficient of the elastic foundation, and the edge conditions on vibration behavior of the annular plate is carried out. Accuracy of the employed sandwich plate theory and the presented analytical solution are verified by comparing present results with those of the three-dimensional theory of elasticity extracted from ABAQUS software.

A model of composite laminated beam based on the global–local theory and new modified couple-stress theory

In this paper, a model of composite laminated beam based on the global–local theory for new modified couple-stress theory is developed. For the modified couple-stress theory, an anisotropic constitutive relation is suggested. There is only one microlength-scale parameter in each ply of the composite laminated beam. The Reddy beam model of global–local higher-order theory proposed by Chen and Wu (2005) [1], which satisfies free surface conditions and the geometric and stresses continuity conditions at interfaces, and is adopted to formulate this model. On the model of laminated beam of modified couple stress theory, the transverse shear stress with scale effect is presented for the first time. Numerical results show that the proposed beam model can capture the scale effect at interfaces of the microstructure of the composite laminated beam.