Nonuniform In-plane Research Articles

Nonlinear Stability of Curved Multiphase Composite Panels: Influence of Agglomeration in Randomly Distributed Carbon Nanotubes with Nonuniform In-Plane Loads

Nonlinear Stability of Curved Multiphase Composite Panels: Influence of Agglomeration in Randomly Distributed Carbon Nanotubes with Nonuniform In-Plane Loads

Influence of porosity and nonuniform in-plane edge loads on vibration and buckling response of power law and sigmoid function based FG sandwich plates with geometrical discontinuities

Porosity defects can emerge during the manufacturing process of Functionally Graded Sandwich Plates (FGSPs). In practice, FGSPs have geometric discontinuities/cutouts and are exposed to Nonuniform In-plane Edge Loads (NIELs). The presence of porosity, cutout and NIELs causes a significant reduction in the stiffness of the plate affecting its vibration and buckling behavior. Therefore, in this paper vibration and buckling results obtained using the Finite Element (FE) method hitherto not reported in the literature are presented for porous FGSPs with cutouts and subjected to NIELs. Four types of porosity distribution models are explored, and the porosity imperfections are modeled as the criteria of stiffness reduction. Porosity-dependent material properties of FGSPs are evaluated using modified power law and sigmoid function. The current study incorporates two distinct kinds of sandwich configuration in such a way that there is no material mismatch along the thickness direction. The application of different cases of NIELs on the plate with cutouts leads to the development of nonuniform stresses. Hence a novel dynamic approach has been proposed to evaluate buckling loads by implementing two sets of boundary conditions. The first set of boundary conditions calculates pre-buckling stress, while second set calculates critical buckling loads. The generated results from the current FE formulation are compared with existing results in the literature to ensure accuracy. Finally, the effect of porosity distribution, NIELs, cutout ratio and its positions, support conditions, volume fraction exponents and geometric parameters on the vibration and buckling response are studied and discussed to arrive at appropriate conclusion.

Buckling behaviour of composite laminates of trapezoidal panel with cutout subjected to non-uniform in-plane edge loads

Abstract Critical buckling loads of composite laminates are usually calculated by using analytical solutions based on the assumptions of uniform in-plane loads, despite of the fact that real structures are often subjected to various nonuniform loads. The present work is focused on the buckling behavior of composite trapezoidal laminates panel with cutouts, subjected to various nonuniform in-plane loads. The effect of variation is size of trapezoidal plate on the buckling behavior has been studied using finite element method. The governing differential eigenvalue equations have been solved using subspace algorithm. The analytical solution for such problems is very difficult. Hence in this work, a finite element software FORTRAN is used to study the buckling behavior of thin and thick laminated panels with cutouts subjected to various kinds of non-uniform edge loads. The panel is modeled by using eight node elements with five degrees of freedom at each node. The accuracy of the model is confirmed by comparing present results with available literature. The effect of various parameters such as cutout size, boundary conditions and ply-orientations of panel are included in this work. It is observed from this study that the non-uniform edge loads and cutout size has remarkable effect on the buckling behavior of said panels.

Buckling analysis of porous FGM sandwich plates under various types nonuniform edge compression based on higher order shear deformation theory

This study intends to model the effect of porosity type defect and analyze the consequences of porosity on the buckling characteristic of various types of FGM sandwich combinations. Here, we consider four types of FGM sandwich combinations considering the multiple arrangements of the constituent layers (i.e., two face sheets and a core). Further, the combinations of all kinds of sandwiches are considered in such a way that there is no material discontinuity or jump. The porosity defects are modeled in this work as the criteria of stiffness reduction and incorporated in the rule of mixture. Finally, results are calculated by considering two types of porosity distribution, evenly and unevenly, while modeling the FGM plate. The buckling response of FGM sandwich plates has been analyzed under a nonuniform in-plane edge load. The application of nonuniform in-plane loads causes in-plane stresses to be nonlinear. Hence, in-plane stresses need to be calculated apriori for the buckling study of the FGM plate under nonuniform load. Further, the effects of various parameters like porosity contents, sandwich combinations, aspect ratio, thickness ratio, boundary conditions on buckling load of FGM plate have been studied in this paper.

Supercurrent vortices and Majorana zero modes induced by an in-plane Zeeman field on the surface of a three-dimensional topological insulator

A nonuniform in-plane Zeeman field can induce spontaneous supercurrents of spin-orbit-coupled electrons in superconducting two-dimensional systems and thin films. In this work it is shown that current vortices can be created at the ends of a long homogeneously magnetized strip of a ferromagnetic insulator, which is deposited on the surface of a three-dimensional topological insulator. The $s$-wave superconductivity on its surface is assumed to have an intrinsic origin, or to be induced by the proximity effect. It is shown that vortices with the odd vorticity can localize Majorana zero modes. The latter may also be induced by sufficiently narrow domain walls inside the strip, which opens a way for manipulating these modes by moving the walls. It is shown that the vorticity can be tuned by varying the magnetization and width of the strip. A stability of the strip magnetization with respect to the Berezinsky-Kosterlitz-Thouless transition has been analyzed.

Buckling analysis of FGM plates under uniform, linear and non-linear in-plane loading

The paper investigates the buckling responses of functionally graded material (FGM) plate subjected to uniform, linear, and non-linear in-plane loads. New nonlinear in-plane load models are proposed based on trigonometric and exponential function. Non-dimensional critical buckling loads are evaluated using non-polynomial based higher order shear deformation theory. Navier’s method, which assures minimum numerical error, is employed to get an accurate explicit solution. The equilibrium conditions are determined utilizing the principle of virtual displacements and material property are graded in the thickness direction using simple Voigt model or exponential law. The present formulation is accurate and efficient in analyzing the behavior of thin, thick and moderately thick FGM plate for buckling analysis. It is found that with the help of displacement-buckling load curve, critical buckling load can be derived and maximum displacement due to the instability of inplane load can be obtained. Also, the randomness in the values of transverse displacement due to inplane load increases as the extent of uniformity of the load on the plate is disturbed. Furthermore, the parametric varying studies are performed to analyse the effect of span-to-thickness ratio, volume fraction exponent, aspect ratio, the shape parameter for non-uniform inplane load, and non-dimensional load parameter on the non-dimensional deflections, stresses, and critical buckling load for FGM plates.

Discrete Loading Ring-Core Method for Nonuniform In-Plane Residual Stress Analysis in Micro Area

The ring-core method is often used in residual stress analysis. It is applied to macro- and microscale stress analysis and has a unique advantage of releasing the residual stress across the core instead of at a single point, which makes it possible to measure an uneven residual stress field within a limited area, especially when the area is too small to be measured by other techniques. We developed a new layer-by-layer stress analysis method based on the ring-core method to retrieve the uneven in-plane stress, in which a nonuniform load that surrounds the core is approximated by discrete loading and then used to reveal the stress distribution within the core. The displacement–stress relationship is calibrated through finite element simulation. Because of the difficulty of preparing a standard specimen with an accurate high-gradient in-plane field stress in an area of several micrometers, the performance of the method was tested by a finite element simulation experiment. Good matches were achieved when comparing the calculated stress fields and the stress fields in the simulation experiments, including the biaxial, uniaxial and high-gradient cases. The method was applied to a piece of superconductor stand with a highly nonuniform stress by using a 4 μm-diameter-area ring core cut with focused ion beam.

Vibration analysis of nonlocal strain gradient embedded single-layer graphene sheets under nonuniform in-plane loads

This paper develops a nonlocal strain gradient plate model for vibration analysis of graphene sheets under nonuniform in-plane mechanical loads. For more accurate analysis of graphene sheets, the proposed theory contains two scale parameters related to the nonlocal and strain gradient effects. Graphene sheet is modeled via a two-variable shear deformation plate theory needless of shear correction factors. Governing equations of a nonlocal strain gradient graphene sheet on elastic substrate are derived via Hamilton’s principle. Galerkin’s method is implemented to solve the governing equations for different boundary conditions. Effects of different factors such as in-plane loading, load factor, nonlocal parameter, length scale parameter, elastic foundation, and boundary conditions on vibration characteristics of graphene sheets are examined.

Nonlinear Stability Characteristics of Composite Cylindrical Panel Subjected to Non-Uniform In-Plane Mechanical and Localized Thermal

The non-linear stability analysis of composite cylindrical panel subjected to concentrated in-plane mechanical and localized thermal loadings are reported here. The buckling of composite panel subjected to concentrated in-plane mechanical loading/ localized thermal loading is solved in two steps as the prebuckling stress distribution within the panel is not known a priori. In the first step, the semi-analytical expressions for the pre-buckling stresses within the composite cylindrical panel under in-plane mechanical/ thermal loadings are developed by solving in-plane elasticity/thermoelasticity problem. Subsequently, using these in-plane stresses within the cylindrical panel, the governing equations for nonlinear stability of layered composite panel are formulated using variational principle. The cylindrical panel is modeled based on Donnell’s shallow shell theory considering higher order shear deformation theory and incorporating von-Karman geometric nonlinearity. The Galerkin’s method is used to solve the non-linear governing partial differential equations. The influence of different types of mechanical and thermal loadings, initial geometric imperfections, and radius of curvature on the postbuckling equilibrium paths is investigated.

Nonlinear stability and dynamics of composite skew plates under nonuniform loadings using differential quadrature method

The buckling, postbuckling and postbuckled vibration behaviour of composite skew plates subjected to nonuniform inplane loadings are presented here. The skew plate is modelled using first order shear deformation theory accounting for von-Kármán geometric nonlinearity and initial geometric imperfections. The different types of nonuniform loads that have been considered in this study are concentrated load, partial load and parabolic load. The explicit analytical expressions for prebuckling stress distributions within composite skew plate subjected to three different types of nonuniform inplane loadings are developed by solving plane elasticity problem using Airy's stress function approach. It is observed that the inplane normal stress distributions within the skew plate due to above nonuniform loadings do not become uniform even at mid-section. The generalized differential quadrature (GDQ) method is used to solve the nonlinear governing partial differential equations. It is observed that the postbuckled load carrying capacity of skew plate under concentrated loading is the lowest compared to other nonuniform and uniform loadings.

Linear and nonlinear parametric instability behavior of cylindrical sandwich panels subjected to various mechanical edge loadings

ABSTRACTLinear and nonlinear dynamic instability behavior of cylindrical sandwich panels subjected to combined static and dynamic nonuniform in-plane loadings is studied in this article. The core compressibility effects are considered in the model by assuming fourth and fifth order expansions for the transverse and tangential displacement of the core. The exact stress distributions within the panel are determined by panel prebuckling analysis for the applied parabolic and partial edge loadings. Galerkin's method is used to reduce the governing partial differential equations of the shell panel into a set of nonlinear ordinary differential equations. Dropping the nonlinear term, dynamic instability regions are obtained by solving the Mathieu-type differential equation by the method of Fourier series. The characteristics feature of the stable and unstable regions are investigated by linear and nonlinear time history responses and phase plots of the shell panel in those regions using Newmark's time integration. Incremental harmonic balance (IHB) method is used to study the nonlinear frequency amplitude responses of the cylindrical sandwich panels.

Dynamic Stability of Axially Moving Viscoelastic Plates under Non-Uniform In-Plane Edge Excitations

Dynamic Stability of Axially Moving Viscoelastic Plates under Non-Uniform In-Plane Edge Excitations

2-tier in-plane motion correction and out-of-plane motion filtering for contrast-enhanced ultrasound.

Contrast-enhanced ultrasound (CEUS) cines of focal liver lesions (FLLs) can be quantitatively analyzed to measure tumor perfusion on a pixel-by-pixel basis for diagnostic indication. However, CEUS cines acquired freehand and during free breathing cause nonuniform in-plane and out-of-plane motion from frame to frame. These motions create fluctuations in the time-intensity curves (TICs), reducing the accuracy of quantitative measurements. Out-of-plane motion cannot be corrected by image registration in 2-dimensional CEUS and degrades the quality of in-plane motion correction (IPMC). A 2-tier IPMC strategy and adaptive out-of-plane motion filter (OPMF) are proposed to provide a stable correction of nonuniform motion to reduce the impact of motion on quantitative analyses. A total of 22 cines of FLLs were imaged with dual B-mode and contrast specific imaging to acquire a 3-minute TIC. B-mode images were analyzed for motion, and the motion correction was applied to both B-mode and contrast images. For IPMC, the main reference frame was automatically selected for each cine, and subreference frames were selected in each respiratory cycle and sequentially registered toward the main reference frame. All other frames were sequentially registered toward the local subreference frame. Four OPMFs were developed and tested: subsample normalized correlation (NC), subsample sum of absolute differences, mean frame NC, and histogram. The frames that were most dissimilar to the OPMF reference frame using 1 of the 4 above criteria in each respiratory cycle were adaptively removed by thresholding against the low-pass filter of the similarity curve. Out-of-plane motion filter was quantitatively evaluated by an out-of-plane motion metric (OPMM) that measured normalized variance in the high-pass filtered TIC within the tumor region-of-interest with low OPMM being the goal. Results for IPMC and OPMF were qualitatively evaluated by 2 blinded observers who ranked the motion in the cines before and after various combinations of motion correction steps. Quantitative measurements showed that 2-tier IPMC and OPMF improved imaging stability. With IPMC, the NC B-mode metric increased from 0.504 ± 0.149 to 0.585 ± 0.145 over all cines (P < 0.001). Two-tier IPMC also produced better fits on the contrast-specific TIC than industry standard IPMC techniques did (P < 0.02). In-plane motion correction and OPMF were shown to improve goodness of fit for pixel-by-pixel analysis (P < 0.001). Out-of-plane motion filter reduced variance in the contrast-specific signal as shown by a median decrease of 49.8% in the OPMM. Two-tier IPMC and OPMF were also shown to qualitatively reduce motion. Observers consistently ranked cines with IPMC higher than the same cine before IPMC (P < 0.001) as well as ranked cines with OPMF higher than when they were uncorrected. The 2-tier sequential IPMC and adaptive OPMF significantly reduced motion in 3-minute CEUS cines of FLLs, thereby overcoming the challenges of drift and irregular breathing motion in long cines. The 2-tier IPMC strategy provided stable motion correction tolerant of out-of-plane motion throughout the cine by sequentially registering subreference frames that bypassed the motion cycles, thereby overcoming the lack of a nearly stationary reference point in long cines. Out-of-plane motion filter reduced apparent motion by adaptively removing frames imaged off-plane from the automatically selected OPMF reference frame, thereby tolerating nonuniform breathing motion. Selection of the best OPMF by minimizing OPMM effectively reduced motion under a wide variety of motion patterns applicable to clinical CEUS. These semiautomated processes only required user input for region-of-interest selection and can improve the accuracy of quantitative perfusion measurements.

BUCKLING BEHAVIOR OF COMPOSITE LAMINATES (WITH AND WITHOUT CUTOUTS) SUBJECTED TO NONUNIFORM IN-PLANE LOADS

Critical buckling loads of composite laminates are usually calculated using analytical solutions based on the assumption of uniform in-plane loads, despite of the fact that real structures are often subjected to various nonuniform loads. The present work is focused on the buckling behavior of composite laminates, with and without cutouts, subjected to various nonuniform in-plane loads. The effect of the size of the cutouts, on the buckling behavior, has been studied using finite element method. Furthermore, parametric studies on the effects of plate aspect ratio, location of the cutout and the application of a nonuniform load combined with a shear load have also been studied. Higher buckling loads were observed in pure in-plane bending compared to uniformly/nonuniformly distributed loads. Consequently, it is important to consider nonuniformly distributed loading whenever applicable, to utilize complete strength of the composite laminate and to avoid premature failure of the composite laminate due to structural instability.

PARAMETRIC INSTABILITY OF LAMINATED COMPOSITE CYLINDRICAL PANELS SUBJECTED TO PERIODIC NON-UNIFORM IN-PLANE LOADS

In the present investigation, the dynamic instability regions of shear deformable cross-ply laminated and composite cylindrical panels subjected to periodic nonuniform in-plane loads are reported. Since the applied in-plane load is nonuniform, initially the static part of the nonuniform in-plane loads are applied and the stresses (σx, σy and τxy) within the panel are evaluated by the solution of cylindrical panel membrane problem. Subsequently, superposing the stress distribution due to static and dynamic in-plane loads, the stress distributions within the panel are obtained. Using these stress distributions the governing equations of the problem are derived through Hamilton's variational principle based on higher-order shear deformation theory of elastic shell theory incorporating von Kármán-type nonlinear strain displacement relations. The governing partial differential equations are reduced into a set of ordinary differential equations (Mathieu-type of equations) by employing Galerkin's method. The instability boundaries of Mathieu equation corresponding to periodic solutions of period T and 2T are determined using Fourier series. Effect of various parameters like static and dynamic load factors, aspect ratio, thickness-to-radius ratio, shallowness ratio, linearly varying in-plane load, parabolic in-plane load and various boundary conditions on the instability regions are investigated.

Stability analysis of graphene based laminated composite sheets under non-uniform inplane loading by nonlocal elasticity

Size dependent buckling of composite laminates made of isotropic graphene layers interlaid with bonding agents is considered. Nonlocal theory of elasticity is used in the buckling analysis to reflect the size scale effects on the critical buckling loads which is discussed in detail. The method is capable of predicting the relative buckling modes for non-uniform inplane loading applied through the thickness of the laminate. All modes of buckling in which the layers may displace together or opposite one another are investigated to study their scale dependent effects. Displacement or load controls are implemented through independent parameters as constraints to form special combination of buckling modes. Each graphene sheet is considered as a Kirchhoff plate model. The interlaid bonding agent is laterally treated as Winkler elastic foundation between graphene layers while neglecting their other load carrying capacities. Various numerical results are obtained reflecting the nonlocality effects. It is observed that in cases of higher load ratios and simpler buckling modes, the effect of nonlocality tends to drastically increase. The results of simpler examples studied are verified by another reference.

Buckling and Postbuckling Behavior of Cross-Ply Composite Plate Subjected to Nonuniform In-Plane Loads

This paper presents a study of buckling and postbuckling behaviour of simply supported composite plates subjected to nonuniform in-plane loading. The mathematical model is based on higher order shear deformation theory incorporating von Kármán nonlinear strain displacement relations. Because the applied in-plane edge load is nonuniform, in the first step the plane elasticity problem is solved to evaluate the stress distribution within the prebuckling range. Using these stress distributions, the governing equations for postbuckling analysis of composite plates are obtained through the theorem of minimum potential energy. Adopting Galerkin’s approximation, the governing nonlinear partial differential equations are reduced into a set of nonlinear algebraic equations in the case of postbuckling analysis, and homogeneous linear algebraic equations in the case of buckling analysis. The critical buckling load is obtained from the solution of associated linear eigenvalue problem. Postbuckling equilibrium paths are obtained by solving nonlinear algebraic equations employing the Newton-Raphson iterative scheme. Explicit expressions for the plate in-plane stress distributions within the prebuckling range are reported for isotropic and composite plates subjected to parabolic in-plane edge loading. Buckling loads are determined for three plate aspect ratios (a/b=0.5, 1, 1.5) and three different types of in-plane load distributions. The effect of shear deformation on the buckling loads of composite plate is reported. The present buckling results are compared with previously published results wherever possible.

Postbuckling analysis of cross-ply laminated cylindrical shell panels under parabolic mechanical edge loading

Postbuckling equilibrium paths of simply supported cross-ply laminated cylindrical shell panels subjected to non-uniform (parabolic) inplane loads are traced in this paper. Love's shell theory with higher order shear deformation theory and von Kármán nonlinear strain–displacement relations are used in the mathematical formulation of the problem. In the first step, the plate membrane problem is solved to evaluate the stress distribution within the prebuckling range as the applied inplane edge load is non-uniform. The governing shell panel postbuckling equations are derived from the principle of minimum total potential energy using the above stress distributions. Adopting multi-term Galerkin's approximation, the governing equations are reduced into a set of non-linear algebraic equations. Newton–Raphson method in conjunction with Riks approach is employed to plot the postbuckling paths through limit points. Numerical results are presented for symmetric (0/90/0) crossply laminated cylindrical shell panels under parabolic inplane load, lateral distributed load and initial imperfections. Limit loads and snap-through behavior of shell panels are studied.

Buckling of rectangular plates with various boundary conditions loaded by non-uniform inplane loads

In the present paper, buckling loads of rectangular composite plates having nine sets of different boundary conditions and subjected to non-uniform inplane loading are presented considering higher order shear deformation theory (HSDT). As the applied inplane load is non-uniform, the buckling load is evaluated in two steps. In the first step the plane elasticity problem is solved to evaluate the stress distribution within the prebuckling range. Using the above stress distribution the plate buckling equations are derived from the principle of minimum total potential energy. Adopting Galerkin's approximation, the governing partial differential equations are converted into a set of homogeneous linear algebraic equations. The critical buckling load is obtained from the solution of the associated linear eigenvalue problem. The present buckling loads are compared with the published results wherever available. The buckling loads obtained from the present method for plate with various boundary conditions and subjected to non-uniform inplane loading are found to be in excellent agreement with those obtained from commercial software ANSYS. Buckling mode shapes of plate for different boundary conditions with non-uniform inplane loadings are also presented.

Buckling of orthotropic plates under various inplane loads

When a laminated composite plate is subjected to a compressive edge force lying in the plane of the plate, failure can occur by buckling at a stress below the strength of the material. Critical buckling loads of orthotropic plates are usually calculated using the analytical solutions which are based on the assumption of uniform end loads, despite the fact that real structures are subjected to various non-uniform inplane loads. This paper examines the buckling behavior of the orthotropic plates under non-uniform inplane loads with different boundary conditions (all edges simply supported, two loaded edges clamped and others simply supported, andall edges clamped) using the finite element method (FEM). Numerical results show that the existing formulas for buckling load basedon the uniform load assumption leads to overestimation of buckling loads.