Free Edge Boundary Conditions Research Articles

On the mechanics of a double thin film on a finite thickness substrate

The single edge loading problem of a double thin film deposited on a finite thickness substrate is considered. The films and the substrate are assumed to be homogeneous and isotropic. The membrane analogy is adopted for the films and two dimensional elasticity is applied to the substrate. The compatibility of the tangential strain between the film and the substrate yields the governing singular integral equation in terms of the interfacial shear stress. The numerical solution is provided using the Gauss–Chebyshev discretization method. The sensitivity of the interfacial shear and the normal stresses to several effective parameters are examined. The results of current study are compatible with the finite element simulations. The lowest interfacial shear stress pertains to a substrate with a free edge boundary condition. The results of current study are applicable to the debonding failure assessment of the multi-layer joints and semiconductors which are susceptible to the interfacial shear stresses near the film edges.

Size-Effect Analysis on Vibrational Response of Functionally Graded Annular Nano-Plate Based on Nonlocal Stress-Driven Method

Dynamic analysis of functionally graded size-dependent annular nano-plate is the main concern in this study. To obtain the vibrational behavior of this plate, the stress-driven nonlocal integral elasticity, as well the strain gradient theory were used in conjunction with the classical plate theory. The resulting equilibrium equations were solved using the generalized differential quadrature rule (GDQR) and the influences of various parameters such as; size-effect parameter, material heterogeneity index, the aspect ratio of the inner to outer radii, and the effects of different boundary conditions were investigated on the vibrational behavior of the nano-plate, based on different types of boundary conditions. Results indicate that the natural frequencies increase with an increase in the heterogeneity index [Formula: see text] and the increase in size-effect parameter shows a similar effect in both models. Additionally, for the simply supported and free-edge boundary conditions (for both edges), as well as the free and knife-edges, and simply supported-free edges, the strain gradient theory predicts higher values of frequency ratios as [Formula: see text] was increased. Similar results were obtained for the remaining types of boundary conditions, with a higher sensitivity to [Formula: see text], provided the stress-driven model is used. This behavior can be interpreted as the sensitivity of the nano-plate to [Formula: see text] that is manifested by the use of the stress-driven model for the prediction of vibrational behavior of the nano-plate.

Series Solution-Based Approach for the Interlaminar Stress Analysis of Smart Composites under Thermo-Electro-Mechanical Loading

This paper introduces a new loading condition considering the combined thermo-electro-mechanical coupling effect in a series solution-based approach to analyze the free-edge interlaminar stresses in smart composite laminates. The governing equations are developed using the principle of complementary virtual work. The assumed stress fields satisfy the traction-free and free-edge boundary conditions. The accurate stress states of the composite structures are acquired through the procedure of generalized eigenvalue problems. The uniform temperature is employed throughout the laminate, and the electric field loading is applied to the symmetric piezo-bonded actuators to examine the combined effect of thermal and electrical stresses on the overall deformation of smart composite laminates. It was observed that the magnitude of the peeling stresses generated by mechanical loading was reduced by the combined thermal and electric excitation loading (up to 25.3%), which in turn resulted in expanding the service life of the smart composite structures. The proposed approach is implemented on three different layup configurations. The efficiency of the current methodology is confirmed by comparing the results with the 3D finite element (FEM) solution computed by ABAQUS.

Multi-directional Falling Weight Deflectometer (FWD) testing and quantification of the effective modulus of subgrade reaction for concrete roads

ABSTRACT Falling Weight Deflectometry (FWD) tests are performed around the centres of two rectangular concrete plates, with geophones measuring vertical deflections in eight directions. Experimental results allow for quantifying asymmetries regarding the structural behaviour. Significant asymmetries are found for a 22-year-old plate scheduled for replacement. A new plate, tested a few weeks after production, is found to behave in a virtually double-symmetric fashion. Structural analysis of the new plate is based on Kirchhoff–Love plate theory, using free-edge boundary conditions. The support of the plate is provided by a Winkler foundation. Performing a static analysis, the uniform modulus of subgrade reaction is optimised to reproduce the measured deflections. The result is not convincing. The model is extended towards consideration of a second optimisation variable: a uniform auxiliary surface load. This allows for reproducing the measured deflections. The auxiliary load is superimposed with the pressure resulting from the Winkler foundation. This yields a realistic distribution of subgrade pressure. Dividing it by the deflections results in the distribution of the effective modulus of subgrade reaction. Finally, the analysis is extended towards the consideration of inertia forces. They increase the effective moduli of subgrade reaction determined by means of static analysis by less than 3.5%.

Geometrical Dimensional Effect on Natural Frequency of Single Layer Graphene in Armchair Configuration

Graphene has remarkable strength, such as yield strength and elasticconstant. The dynamic behaviour of graphene sheet is affected bygeometrical variation in atomic arrangement. This paper introducedgraphene with armchair atomic structure for estimating fundamental naturalfrequencies. The presented analysis can be useful for the possible highfrequency nanomechanical resonator systems. The analytical formulation,based on classical plate theory and continuum solid modelling based finiteelement method have been performed for estimation of fundamental naturalfrequencies of single layer graphene sheet (SGLS) with different boundaryconditions. The free edge and clamped edge boundary conditions have beenconsidered. For simplifying analytical formulations, Blevins approach fordynamic solution has been adopted and for validating analytical results.The finite element analysis of SLGS has been performed using ANSYSsoftware. The effect of variation in geometrical parameters in terms ofwidth and length of SLGS has been analysed for realization of ultra-highfrequency based nanomechanical resonator systems

Size Effect on the Axisymmetric Vibrational Response of Functionally Graded Circular Nano-Plate Based on the Nonlocal Stress-Driven Method

In this work, the axisymmetric-vibrational behavior of a size-dependent circular nano-plate with functionally graded material with different types of boundary conditions was investigated. The analysis was performed based on the Stress-driven model (SDM) and Strain-gradient theory (SGT) in conjunction with classical plate theory. The governing equations of motion and their corresponding equations for boundary conditions were obtained based on Hamilton’s principle and solved using the generalized differential quadrature rule. Results show that this method is applicable to the vibrational analysis of such structures with a fast convergence rate; as N approaches 6 for the first mode, and 10 for the second as well as the third and fourth modes, regardless of the type of boundary condition. In both models, the influences of various parameters such as size-effect parameter Lc, material heterogeneity index n, and types of boundary conditions were obtained on the first four modes and compared with each other. Results indicate that the natural frequencies in these modes increase with an increase in the heterogeneity index n, and size-effect parameter Lc. Additionally, these parameters appear to have a stiffening effect on the nano-plate vibrational behavior. However, for a nano-plate resting on a knife or simply supported edge, in the first mode, the SDM shows a more stiffening effect on the plate behavior as compared with the SGT. Nonetheless, for the clamped and free edge boundary conditions, both models predicted the same behavior. The SGT showed a higher-stiffening effect only in the fourth mode, for all types of considered boundary conditions.

A non-classical model for circular cylindrical thin shells incorporating microstructure and surface energy effects

A non-classical model for circular cylindrical thin shells is developed by using a modified couple stress theory and a surface elasticity theory. The equations of motion and boundary conditions are simultaneously obtained by a variational formulation based on Hamilton’s principle, which provides a unified treatment of the microstructure and surface energy effects. The new non-classical shell model contains one material length scale parameter to capture the microstructure effect and three surface elastic constants to describe the surface energy effect. The current model includes the shell models considering the microstructure effect only or the surface energy effect alone as special cases. Also, the newly developed shell model reduces to the classical circular cylindrical Love–Kirchhoff thin shell model when both the microstructure and surface energy effects are suppressed. In addition, it recovers the non-classical model for Kirchhoff plates incorporating the microstructure and surface energy effects when the shell radius tends to infinity. To illustrate the new model, the static bending and free vibration problems of a simply supported closed circular cylindrical shell and of an open circular cylindrical shell with free edge boundary conditions are analytically solved by directly applying the model. For the static bending problem in each case, the numerical results show that the shell deflection predicted by the current model is smaller than that predicted by the classical model, and the difference is significant when the shell is very thin but diminishes as the shell thickness increases. For the free vibration problem in each case, it is found that the natural frequency predicted by the new model is higher than that predicted by its classical counterpart, and the difference is large only for very thin shells.

A New Analytical Method for Spherical Thin Shells’ Axisymmetric Vibrations

In order to improve the spherical thin shells’ vibrations analysis, we introduce a new analytical method. In this method, we take into consideration the terms of the inertial couples in the stress couples’ differential equations of motion. These inertial couples are omitted in the theories provided by Naghdi–Kalnins and Kunieda. The results show that the current method can solve the axisymmetric vibrations’ equations of elastic thin spherical shells. In this paper, we focus on verifying the current method, particularly for free vibrations with free edge and clamped edge boundary conditions. To check the validity and accuracy of the current analytical method, the natural frequencies determined by this method are compared with those available in the literature and those obtained by a finite element calculation.

Mesh-adapted stress analysis of multilayered plates using a layerwise model

This paper proposes a new finite-element modelling of a recent layerwise model for multilayered plates. This layerwise model is built from a specific 3D stress-field expansion along the thickness direction and involves, in particular, interlaminar transverse shear and out-of-plane stresses as generalized stresses. Its main feature is that 3D equilibrium equations and free-edge boundary conditions are directly taken into account into the stress-based construction of the model. A dual displacement-based finite-element discretization is implemented using the FEniCS software package and a remeshing strategy is proposed based on a novel error indicator. The error indicator is built based on the 3D stress field directly deduced from the layerwise generalized stresses and compared to a reconstructed stress field based on the model generalized displacements. The proposed error indicator is shown to identify the most critical parts of a laminate structure associated with complex 3D stress fields such as boundaries or stress concentration/singularity regions (near free-edges or delamination fronts). Through the combination of thickness discretization and in-plane mesh refinement in regions of interest, the proposed framework therefore offers an attractive alternative to 3D solid finite elements for an accurate prediction of stress states in composite laminates.

Nonlinear finite element analysis of lattice core sandwich plates

A displacement-based, geometrically nonlinear finite element model is developed for lattice core sandwich panels modeled as 2-D equivalent single-layer (ESL), first-order shear deformation theory (FSDT) micropolar plates. The nonlinearity is due to the moderate macrorotations of the plate which are modeled by including the von Kármán nonlinear strains in the micropolar strain measures. Weak-form Galerkin formulation with linear Lagrange interpolations is used to develop the displacement finite element model. Selective reduced integration is used to eliminate shear locking and membrane locking. The novel finite element model is used to study the nonlinear bending and linear free vibrations of web-core and pyramid core sandwich panels. Clamped and free edge boundary conditions are considered for the first time for the 2-D micropolar ESL-FSDT plate theory. The present 2-D finite element results are in good agreement with the corresponding detailed 3-D FE results for the lattice core sandwich panels. The 2-D element provides computationally cost-effective solutions; in a nonlinear bending example, the number of elements required for the 2-D micropolar plate is of the order 103, whereas for the corresponding 3-D model the order is 105.

Analytical, numerical and experimental study of the plastic buckling behavior of thick cylindrical tubes under axial compression

This paper deals with the elastoplastic buckling problem of thick cylindrical shells under axial compression. The main idea is to propose an analytical solution procedure which gives rise to reliable results in a straightforward and low time-consuming manner, for dimensioning purposes. Thick shells are thus considered as they are known to be less imperfection-sensitive than thin ones. First, experimental compression tests are performed thanks to a specific experimental set-up, which is designed so as to prevent again, as much as possible, from the imperfections due to boundary conditions. Then, numerical finite element computations are carried out, and it is shown that the choice of free edge boundary conditions (for which pre-critical deformations are supposed to be homogeneous) in the numerical model leads to the same critical load as obtained during the experimental tests. Next, the problem is solved analytically. The classical plastic buckling load, usually derived considering an homogeneous pre-critical state, is likely to agree with the numerical predictions in this particular case of free conditions, contrary to the cases of simply-supported or clamped edges actually displaying an heterogeneous pre-critical state. In return, the substantial thickness of the tubes considered experimentally makes it necessary to improve the well-known solution which appears to be in very bad agreement with the numerical/experimental critical values. On one hand, the issue of kinematics is discussed and, on the other hand, geometric non-linearities are taken into account in the model (pre-critical strains are no longer infinitesimal). Finally, the new analytical procedure derived in this paper shows a very good agreement with both numerical and experimental results.

Analytical Modeling of Multi-sectioned Bi-stable Composites: Stiffness Variability and Embeddability

Multi-stable composite laminates have been widely investigated for morphing applications due to distinct geometrical characteristics about each stable configuration. Recently, a mechanism for realizing on-demand stiffness adaptation in compliant structures has been proposed exploiting the stiffness variation from switching between the stable states of embedded bi-stable laminates. This allows for reducing the coupling between loading and morphing deformation modes, broadening the overall design space for morphing structures. However, the design of such embeddable laminates requires the study of multiple lamination domains to allow embedding within larger compliant structures. This process is currently done by computationally expensive simulations. We present an analytical model to predict the stable shapes and structural characteristics of multi-section, multi-stable composite laminates subject to two clamped boundary conditions. The analysis is conducted employing the Rayleigh-Ritz method using polynomial approximations of the displacement fields. The structural characteristics are predicted while constraining the laminate edges, thus extending previous models designed to account mainly for free edge boundary conditions. Our model is subsequently applied to explore the design space for characterizing the stability and variable stiffness properties of two different classes of bi-stable laminates. The presented model allows for the efficient design of multi-stable laminates embeddable within compliant structures.

Free edge stress prediction in thick laminated cylindrical shell panel subjected to bending moment

• A Galerkin based Layer-wise formulation is presented for thick composite panel subjected to bending moment. • An analytical solution is presented for governing equations of the panel considering free edge conditions. • Free edge stresses in bending of thick panel with general layer stacking is studied for the first time. • The predictions satisfy the free edge and free surface conditions. • The effect of geometrical parameters and layer stacking are investigated. A thick composite cylindrical shell panel with general layer stacking is studied to investigate the free edge and 3D stresses in the panel which is subjected to pure bending moment. To this aim, a Galerkin based layerwise formulation is presented to discretize the governing equation of the panel to ordinary differential equations. Employing a reduced displacement field for the cylindrical panel, the governing equations for thick panel are developed in terms of displacements and a set of coupled ordinary differential equations is obtained. The governing equations are solved analytically for free edge boundary conditions and applied pure bending moment. The accuracy of numerical results is examined and the distribution of interlaminar and in-plane stresses is studied. The free edge stresses are studied and the effect of radius to thickness ratio, width to thickness ratio and layer stacking on the distribution of stresses is investigated. The focus of numerical results is on the prediction of boundary layer and free edge stress distribution.

An analysis of the thickness vibration of an unelectroded doubly-rotated quartz circular plate.

The scalar differential equation in the thickness eigendisplacement for the doubly-rotated quartz plates is applied to analyze thickness vibrations of an unelectroded circular plate with free edges at the neighborhood of the pure thickness vibration mode. The scalar differential equation is transformed into an elliptical coordinate system. With the boundary conditions of free edges, the frequencies and the modes are solved in terms of the Mathieu function and the Modified Mathieu function. The results of frequencies of the fundamental harmonic and its third overtone of an AT-cut quartz circular plate by the present approach agree well with the existing theoretical results and the experiment results. The frequencies and modes of an SC-cut quartz circular plate are investigated by the present approach. The frequencies are close to each other when the order of the harmonics is the same. A rotation angle of the symmetric axes of the vibration modes are observed that is dependent on the anisotropic material constants and the order of the harmonics. This approach has potential applications in the design of the doubly-rotated quartz circular resonators.

Analytical solution for strain gradient elastic Kirchhoff rectangular plates under transverse static loading

Levy's analytical solution approach is extended for analysis of rectangular strain gradient elastic plates under static loading for the first time with different boundary conditions at the edges using the method of superposition. The governing equation of equilibrium and the corresponding classical/non-classical boundary conditions for strain gradient flexural Kirchhoff plate under static loading are considered. Numerical examples on static bending of Kirchhoff nanoplates involving five different combinations of simply supported, clamped and free edge boundary conditions are presented. The effect of negative strain gradient terms is of hardening nature thus resulting in decrease in the deflection. Plates with geometry comparable to the microstructural length scale show significant size effect and this size dependency diminishes with the increase in the plate size.

A simplified method for determining the damping loss factor of panels using a narrow band decay method

This paper presents a damping loss factor estimation method for structures made of solid elastic and viscoelastic materials. Using an impulsive excitation signal, an acceleration impulse response of the panel in free edge boundary conditions is measured and the damping loss factor is determined at each frequency of interest, by applying an overlapped narrow band frequency decay method. This means that, for the duration of the impulse, the decay of each frequency band is measured in time domain and, using a backward integration approach, structural reverberation time is determined. Successively, the damping loss factor is calculated using a well-established relationship between measured quantities and frequencies. Firstly, several investigations related to the post-processing procedures for the determination of structural reverberation time will be analyzed in detail. Results will then be presented and discussed for different structures and compared with different available measurement techniques.

A Galerkin Layerwise Formulation for three-dimensional stress analysis in long sandwich plates

A layerwise (LW) formulation based on the Galerkin method is presented to investigate the three-dimensional stress state in long sandwich plate which is subjected to tension force and pure bending moment. Based on the Galerkin method and the LW discretization approach, the equilibrium equations of elasticity for the long plate are written in the weak form and discretized through the thickness of the plate. The discretized equations are written in terms of displacement components of the numerical layers. The governing equations of the plate are solved analytically for the free edge boundary conditions. The distribution of stress state especially the 3D stress state in the vicinity of the edges of the sandwich plate which is subjected to tension and pure bending is studied. In order to increase the accuracy, the out of plane stresses are obtained by integrating the equilibrium equations of elasticity. The convergence and accuracy of the predictions are studied and various numerical results are presented for distribution of the in-plane and out of plane stresses in symmetric and un-symmetric sandwich plates.

Vibration analysis of composite airfoil blade using orthotropic thin shell bending theory

In this paper, starting with the thin shell theory, the governing partial differential equation of motion for the transverse deflection of a rotating pre-twisted airfoil is derived. Strain-displacement relationships include the effect of warping of the cross-section due to twist-bend coupling effect introduced as a result of varying stagger angle and camber radius of the blade. The equation of motion, thus derived, is used to formulate the free vibration of a typical turbo-machinery cantilevered airfoil by considering it as an anisotropic shell in full curvilinear coordinates subjected to a centrifugal force field. The analytical derivation considers both the stress-stiffening as well as stress-softening effects of the centrifugal forces on the spinning airfoil. The fourth-order partial differential equation characterizing the flexural motion of the airfoil is transformed into a matrix- eigenvalue form using a Rayleigh-Ritz technique. The blade deformations are represented by a set of “admissible” sinusoidal trial functions, which fully satisfy all the clamped-end constraints as well as the free-edge boundary conditions. The numerical results presented in a non-dimensional parametric form are directly applicable in determining the static and running frequencies of typical composite blades used in the fan module of an aeroengine.

Simulation of stress wave attenuation in plain weave fabric composites during in-plane ballistic impact

An analytical method is presented for the simulation of longitudinal stress wave attenuation in typical plain weave fabric composites made of E-glass/epoxy and T300 carbon/epoxy under in-plane ballistic impact. The stress wave propagation/attenuation simulation studies are carried out for the case of in-plane impact of projectiles on composite beam targets. Geometrical modeling of plain weave fabric composites and an algorithm for stress wave propagation/attenuation are presented. Studies are carried out for both free edge and rigid boundary conditions for different beams. Based on the analytical simulation, for typical plain weave E-glass/epoxy composite beams studied, it is observed that the stress intensity is about 5% of the incident value when the longitudinal stress wave reaches a distance of 110mm from the point of in-plane impact at the representative section.

Stress redistribution as an effect of non-uniform in-plane laminate stresses in laminate composite plates

This paper describes a phenomenon that significantly affects the stress distribution in notched laminate composite structures. Ply-to-ply interactions under non-uniform laminate stress fields induce interlaminar and in-plane stresses that are not described in classical laminate theory. The theory is consist of a power series expansion along the thickness-wise coordinate. The most prominent influence of this effect is a substantial reduction in stress concentrations around notches. This may explain various experimental facts that conflict with classical laminate theory. Although this phenomenon appears to be a derivative of the so-called free-edge effect, it is not based on free-edge boundary conditions. For verification of the theory, this paper also presents finite element analysis cases for 5mm diameter open hole [90/0]s cross-ply laminate plates. Reductions in stress concentration around hole edges were up to 10% of the far field stress. Insertion of low elasticity layers between 90° and 0° plies, the use of thick plies, or the adoption of ply materials with low transverse shear moduli were identified as effective means of reducing stress concentration around a hole. These findings may enable the application of new design principles to deal with stress concentrations for notched composite structures.