Evaluation of mixed theories for laminated plates through the axiomatic/asymptotic method

This paper proposes variable kinematic, mixed theories for laminated plates built via the asymptotic/axiomatic method (AAM). This method has been recently developed and successfully applied to develop refined theories for multilayered plates and shells. The AAM evaluates the accuracy of each unknown variables of a structural model. The present paper extends the AAM to mixed theories based on the Reissner Mixed Variational Theorem (RMVT). The displacement and transverse stress fields are modeled by means of the Carrera Unified Formulation (CUF), and expansions up to the fourth-order are employed. Equivalent Single Layer (ESL) and Layer Wise (LW) schemes are adopted, and closed-form Navier-type solutions are considered.The AAM is exploited to determine the set of active terms of a refined plate model. The inactive terms are then discarded. The effectiveness of each variable is evaluated with respect to an LW, fourth-order mixed model. Reduced models are built for different thickness ratios, stacking sequences and displacement/stress variables.The results suggest that reduced models with significantly less unknown variables than full models can be built with no accuracies penalties. Such models are problem dependent, and full models should be preferred in the case of thick, asymmetric plates.     

Damage in MMCs using the GMC: theoretical formulation

In this work the incorporation of damage in the material behavior is investigated. Damage is incorporated into the generalized cells model (GMC), and applied to metal-matrix composites (MMCs). The local incremental damage model of Voyiadjis and Park is used here in order to account for damage in each subcell separately. The resulting micromechanical analysis establishes elasto-plastic constitutive equations that govern the overall behavior of the damaged composite. The elasto-plastic constitutive model is first derived in the undamaged configuration for each constituent of the metal-matrix composite. The plasticity model used here is based on the existence of a yield surface and flow rule. The relationships are then transformed for each constituent to the damaged configuration by applying the local incremental constituent damage tensors. The overall damaged quantities are then obtained by applying the local damage concentration factors obtained by employing the rate of displacement and traction continuity conditions at the interface between subcells and between neighboring repeating cells in the generalized cells model. Examples are solved numerically in order to explore the physical interpretation of the proposed theory for a unit cell composite element.     

Size-dependent free flexural vibrational behavior of functionally graded nanobeams using semi-analytical differential transform method

In this paper nonlocal Euler–Bernoulli beam theory is employed for vibration analysis of functionally graded (FG) size-dependent nanobeams by using Navier-based analytical method and a semi analytical differential transform method. Two kinds of mathematical models, namely, power law and Mori-Tanaka models are considered. The nonlocal Eringen theory takes into account the effect of small size, which enables the present model to become effective in the analysis and design of nanosensors and nanoactuators. Governing equations are derived through Hamilton's principle and they are solved applying semi analytical differential transform method (DTM). It is demonstrated that the DTM has high precision and computational efficiency in the vibration analysis of FG nanobeams. The good agreement between the results of this article and those available in literature validated the presented approach. The detailed mathematical derivations are presented and numerical investigations are performed while the emphasis is placed on investigating the effect of the several parameters such as small scale effects, different material compositions, mode number and thickness ratio on the normalized natural frequencies of the FG nanobeams in detail. It is explicitly shown that the vibration of a FG nanobeams is significantly influenced by these effects. Numerical results are presented to serve as benchmarks for future analyses of FG nanobeams.     

Thermo-mechanical vibration analysis of nonlocal temperature-dependent FG nanobeams with various boundary conditions

In this paper, the thermal effect on free vibration characteristics of functionally graded (FG) size-dependent nanobeams subjected to an in-plane thermal loading are investigated by presenting a Navier type solution and employing a semi analytical differential transform method (DTM) for the first time. Material properties of FG nanobeam are supposed to vary continuously along the thickness according to the power-law form. The small scale effect is taken into consideration based on nonlocal elasticity theory of Eringen. The nonlocal equations of motion are derived through Hamilton's principle and they are solved applying DTM. According to the numerical results, it is revealed that the proposed modeling and semi analytical approach can provide accurate frequency results of the FG nanobeams as compared to analytical results and also some cases in the literature. The detailed mathematical derivations are presented and numerical investigations are performed while the emphasis is placed on investigating the effect of the several parameters such as thermal effect, material distribution profile, small scale effects, mode number and boundary conditions on the normalized natural frequencies of the temperature-dependent FG nanobeams in detail. It is explicitly shown that the vibration behavior of an FG nanobeams is significantly influenced by these effects. Numerical results are presented to serve as benchmarks for future analyses of FG nanobeams.     

Determination of the out-of-plane tensile strength using four-point bending tests on laminated L-angle specimens with different stacking sequences and total thicknesses

Laminated composite materials are increasingly used for the design of aircraft primary structures subjected to complex 3D loadings. The delamination observed in curved parts ensuring the junction between the different perpendicular panels is one of the most critical failure mechanisms. The present article proposes a complete protocol to identify the out-of-plane tensile strength of specimens composed of unidirectional plies. Firstly, a method to design a four-point bending (4 PB) test on L-angle specimens has been proposed. Secondly, a test campaign on T700GC/M21 laminated L-angle specimens has been performed at ONERA. Thirdly, the analysis of these tests with different methods has been performed to demonstrate that such a test is relevant to determine the material out-of-plane tensile strength, which seems to be independent of the stacking sequence and of the total thickness of the specimen, thus allowing the use of this strength in a 3D failure criterion. Finally, the different advantages and drawbacks of 4 PB tests performed on curved beams are discussed.     

Dynamic analysis and damping of composite structures embedded with viscoelastic layers

In recent years, it has been found that composites co-cured with viscoelastic materials can enhance the damping capacity of a composite structural system with little reduction in stiffness and strength. Because of the anisotropy of the constraining layers, the damping mechanism of co-cured composites is quite different from that of conventional structures with metal constraining layers. This paper presents an analysis of the dynamic properties of multiple damping layer, laminated composite beams with anisotropic stiffness layers, by means of the finite element-based modal strain energy method. ANSYS 4.4A finite element software has been used for this study. The variation of resonance frequencies and modal loss factors of various beam samples with temperature is studied. Some of these results are compared with the closed-form theoretical results of an earlier published work. For obtaining optimium dynamic properties, the effects of different parameters, such as layer orientation angle and compliant layering, are studied. Also, the effect of using a combination of different damping materials in the system for obtaining stable damping properties over a wide temperature range is studied.     

Mixed projection-mesh scheme of the finite-element method for the solution of the boundary-value problems of the theory of small elastic-plastic strains

A mixed projection-mesh scheme for the solution of nonlinear boundary-value problems of the theory of small elastic-plastic strains has been formulated. Correctness and convergence of the mixed approximations for stresses, strains, and displacements have been analyzed. The properties of projection operators are studied in detail, and on the basis of the results obtained, a condition has been formulated, which ensures the existence, uniqueness, and stability of the solution to a discrete problem. Application of the numerical integration has been analyzed and the obtained results are presented. The correctness and convergence estimates are based on the theory of generalized functions and the functional analysis method.Translated from Problemy Prochnosti, No. 6, pp. 59–86, November–December, 2004     

Numerical and exact models for free vibration analysis of cylindrical and spherical shell panels

The present paper shows a comparison between classical two-dimensional (2D) and three-dimensional (3D) finite elements (FEs), classical and refined 2D generalized differential quadrature (GDQ) methods and an exact three-dimensional solution. A free vibration analysis of one-layered and multilayered isotropic, composite and sandwich cylindrical and spherical shell panels is made. Low and high order frequencies are analyzed for thick and thin simply supported structures. Vibration modes are investigated to make a comparison between results obtained via the FE and GDQ methods (numerical solutions) and those obtained by means of the exact three-dimensional solution. The 3D exact solution is based on the differential equations of equilibrium written in general orthogonal curvilinear coordinates. This exact method is based on a layer-wise approach, the continuity of displacements and transverse shear/normal stresses is imposed at the interfaces between the layers of the structure. The geometry for shells is considered without any simplifications. The 3D and 2D finite element results are obtained by means of a well-known commercial FE code. Classical and refined 2D GDQ models are based on a generalized unified approach which considers both equivalent single layer and layer-wise theories. The differences between 2D and 3D FE solutions, classical and refined 2D GDQ models and 3D exact solutions depend on several parameters. These include the considered mode, the order of frequency, the thickness ratio of the structure, the geometry, the embedded material and the lamination sequence.     

Numerical prediction of elasto-plastic behaviour of interpenetrating phase composites by EFGM

Interpenetrating Phase Composites (IPCs) can be defined as multiphase materials in which each phase is three-dimensionally interconnected throughout the structure. No phase can be distinguished from the other based on the states of isolation and continuity; however both the phases contribute to the strengthening and improvement of the composite. The tensile and compressive yield and ultimate strengths of IPCs are much higher than a similar particulate composite due to their interpenetrating structure. This behaviour has been numerically simulated using element free Galerkin method. Ramberg–Osgood material model has been used to model the elasto-plastic behaviour of the composite. A progressive damage model has been used to simulate the failure mechanism of each phase. Three types of models have been proposed based on the treatment of the interface. The ultimate strength and the yield strength of IPC are obtained. The ultimate strength and the yield strength of the IPC depend largely on the properties, volume fraction and interpenetration of the constituent phases. The results of the present simulations are found in good agreement with the experimental results.     

Stochastic free vibration analyses of composite shallow doubly curved shells – A Kriging model approach

This paper presents the Kriging model approach for stochastic free vibration analysis of composite shallow doubly curved shells. The finite element formulation is carried out considering rotary inertia and transverse shear deformation based on Mindlin’s theory. The stochastic natural frequencies are expressed in terms of Kriging surrogate models. The influence of random variation of different input parameters on the output natural frequencies is addressed. The sampling size and computational cost is reduced by employing the present method compared to direct Monte Carlo simulation. The convergence studies and error analysis are carried out to ensure the accuracy of present approach. The stochastic mode shapes and frequency response function are also depicted for a typical laminate configuration. Statistical analysis is presented to illustrate the results using Kriging model and its performance.     

Ultimate behaviour of FRP wrapped sections under axial force and bending: Influence of stress–strain confinement model

In the present work, the influence of the adopted confined concrete constitutive law, among those available in the technical literature, on the flexural strength and curvature ductility of reinforced concrete sections strengthened by FRP (fibre reinforced polymer) wrapping is investigated. An important issue to be underlined is that the stress–strain relationship of confined concrete depends not only on the number of layers and on the type of FRP used for wrapping, but also on the size and the shape of the section. By using the main constitutive laws proposed in the technical literature to model the confined concrete behaviour, the moment–curvature diagrams have been evaluated for a significant number of study cases by means of a specifically developed computer program based on a refined fibre model. The results show that even if the different constitutive laws exhibit large differences in the resulting stress–strain behaviour, they lead to negligible differences in terms of flexural resistance, but to very significant differences in terms of curvature ductility. Therefore, the accurate evaluation of the ultimate strain seems of paramount importance compared to the whole stress–strain curve. In addition, the influence of pre-existing loads acting on the structure at the time of the strengthening intervention has been investigated showing that it affects the knee region of the moment–curvature relationship, while the ultimate flexural resistance remains almost unaffected.     

Radial basis functions based on differential quadrature method for the free vibration analysis of laminated composite arbitrarily shaped plates

The conventional strong form collocation approach known as Differential Quadrature (DQ) method has been applied in the past to a vast type of engineering problems. It is well-known that its application is strictly limited to regular regions where derivatives are approximated along mesh lines. Generally, its accuracy increases when the number of collocation points is large and the method tends to be stable. However, for some numerical problems several points are needed in order to obtain an accurate solution. Changing the basis functions another numerical technique was developed called Radial Basis Functions (RBFs) method, which has the advantage of approximating derivatives using irregular point distributions and the basis functions depend on the mutual radial distance of the grid points. In order to extend the idea of DQ method to a general case a Radial Basis Function based on Differential Quadrature (RBF-DQ) method has been recently developed. This method merges the advantages of both techniques. Furthermore, this work proposes the application of RBF-DQ when a domain decomposition technique is considered. In this way it will be shown that, using some kind of basis functions the number of grid points per element can be reduced compared to other classical approaches. Furthermore, once the shape parameter is fixed for one case, it is not needed to calculate it again for other applications.     

Multi-sequence dome lay-up simulations for hydrogen hyper-bar composite pressure vessels

The aim of this study is to propose a calculation method for predicting the geometrical characteristics of the end-closures of composite pressure vessels used in gaseous hydrogen storage applications. We present here a method for predicting the characteristics of domes made by multi-sequence lay-up and subject to changes in winding angle and thickness. In the proposed model, winding angle and thickness calculations are based on two existing methods that have been investigated in a single dome lay-up context. Using them for multi-sequence dome lay-up calculation is the originality of this work. The first predictive results are compared with experimental pressure vessel data and discussed. A good correlation between the theoretical approach and the experiments was found, indicating that the predictions could be helpful in a procedure of overall optimisation.     

Mixed Projection-Mesh Scheme of the Finite-Element Method for the Solution of Problems of the Elasticity Theory

A mixed projection-mesh scheme for solving the boundary-value problems of the elasticity theory has been formulated. Correctness and convergence of the mixed approximations for strains and displacements are studied. Application of the numerical integration has been analyzed, and the obtained results are given. The theory of generalized functions and the functional analysis methods have been used for the convergence and accuracy estimates. Iteration algorithms for solving the discrete problems have been proposed.     

Simplified buckling and ultimate strength analysis of composite plates in compression

The work presented here concerns the ultimate strength predictions of simply supported, square plates of laminated composite material subjected to uniaxial in-plane compressive load. Plates having a range of thicknesses and initial geometric imperfections have been investigated. Several models are established, each based on first order shear deformation theory and assumption of small deflections. The approaches give reasonable but somewhat conservative estimates for the thicker plates considered, while for the thinner plates, neglect of the post-buckling behaviour makes the results very conservative. It will be necessary to use a large deflection plate theory for some of the models to realise their full potential.     

Analysis of a two-way tension-shear coupling in woven fabrics under combined loading tests: Global to local transformation of non-orthogonal normalized forces and displacements

Dependence of shear rigidity of woven fabrics on yarn pre-tension has been reported in the recent literature, however some conflict regarding the trend of this effect is observed. Sources of this conflict are discussed and resolved in the present article using a new characterization framework and a custom-design combined loading fixture. It is shown that in order to correctly characterize the tension-shear coupling behavior in woven fabrics, instead of using global measured data, local normalized forces and displacements should be driven via a non-orthogonal transformation procedure, while considering kinematic force coupling in the setup. In addition, the effect of fabric shear on the tensile behavior of yarns has been investigated, suggesting that the coupling under question is in fact two-way. In particular, results revealed that applying yarn pre-tension increases the shear resistance of the fabric reinforcement, while the tensile behavior of the material becomes more compliant when undergoing shear deformation.     

Local stress field approach for post cracking analysis of FRP strengthened RC elements

A computational framework previously presented for nonlinear analysis of RC elements, has been developed for FRP strengthened RC elements in this study. With the aim of the developed model nonlinear behavior of strengthened RC elements can be simulated based on local stresses state at the crack surface considering all stress transfer mechanisms. Moreover, the local response of each component and its effect on the global behavior of the element can be obtained which is useful for proposing rational design relations. The versatility of the proposed method is verified by comparing the analytical and experimental results. Based on the analytical results, a simple relation is proposed for shear design and assessment of FRP strengthened RC elements and members. The accuracy of the proposed design relation is verified against available experimental results on FRP strengthened RC beams.     

On the vibration and stability of shear deformable FGM truncated conical shells subjected to an axial load

The aim of present study is to investigate the vibration and stability of functionally graded (FG) conical shells under a compressive axial load using the shear deformation theory (SDT). The basic equations of shear deformable FG conical shells are derived using Donnell shell theory and solved using Galerkin's method. The novelty of this study is to achieve closed-form solutions for the dimensionless frequencies and critical axial loads for freely-supported FG truncated conical shells on the basis of the SDT. Parametric studies are made to investigate effects of shear stresses, compositional profiles and conical shell characteristics on the critical parameters. Some comparisons with the various studies have been performed in order to show the accuracy of the present study.     

Optimum shapes of scarf repairs

Adhesively bonded scarf repairs are the preferred method for repairing composite structures, limited mainly by the amount of material removal associated with scarfing. In addition to the high strength restoration, scarf repairs also enable recovery of the original external surface as required by aerodynamic and/or external mould line considerations. However, scarf repairs almost inevitably result in the removal of undamaged material to make way for the scarf insert. This can be a particularly significant issue for thick structures, because the scarf length can vary between 20 and 100 times the thicknesses of the parent structure. In this investigation, an optimisation method has been developed for determining the optimum repair shapes for a given biaxial loading condition. The optimum scarf shape is determined by numerically solving the resulting non-linear differential equation governing the scarf angle. The optimum and near-optimum shapes are presented and discussed with respect to computational modelling using the finite element method.     

Application of a genetic algorithm: near optimal estimation of the rate and equilibrium constants of complex reaction mechanisms

In iterative non-linear least-squares fitting, the reliable estimation of initial parameters that lead to convergence to the global optimum can be difficult. Irrespective of the algorithm used, poor parameter estimates can lead to abortive divergence if initial guesses are far from the true values or in rare cases convergence to a local optimum. For determination of the parameters of complex reaction mechanisms, where often little is known about what value these parameters should take, the task of determining good initial estimates can be time consuming and unreliable. In this contribution, the methodology of applying a genetic algorithm (GA) to the task of determining initial parameter estimates that lie near the global optimum is explained. A generalised genetic algorithm was implemented according to the methodology and the results of its application are also given. The parameter estimates obtained were then used as the starting parameters for a gradient search method, which quickly converged to the global optimum. The genetic algorithm was successfully applied to both simulated kinetic measurements where the reaction mechanism contained one equilibrium constant and two rate constants to be fitted, and to kinetic measurements of the complexation of Cu²⁺ by 1,4,8,11-tetraazacyclotetradecane where two equilibrium and two rate constants were fitted. The implementation of the algorithm is such that it can be generally applied to any reaction mechanism that can be expressed by standard chemistry notation. The control parameters of the algorithm can be varied through a simple user interface to account for parameter range and the number of parameters involved.