In-plane Boundary Conditions Research Articles

Vibration and thermomechanical transient response of doubly curved FGM panels with porosities and elastically restrained edges

The present paper aims to analyze the influences of porosities, elevated temperature and elasticity of in-plane boundary conditions on the free vibration and nonlinear transient response of functionally graded material (FGM) doubly curved panels subjected to uniform lateral pressure. Porosities are assumed to exist in the FGM according to even and uneven distributions. Effective properties of porous FGM are determined using a modified rule of mixture. Basic equations in terms of deflection and stress function are derived basing on the classical shell theory taking into account von Kármán–Donnell nonlinearity and initial geometrical imperfection. Analytical solutions for simply supported panels are assumed and Galerkin method along with Runge–Kutta scheme are applied to obtain the deflection–time response for nonlinear dynamic analysis. Numerical results demonstrate that tangential constraints of edges have extremely significant influences on the vibration and transient response of porous FGM panels exposed to an elevated temperature. The analysis also reveals that porosities have negative and positive effects on the dynamic response of FGM panels at reference and elevated temperatures, respectively. Furthermore, when the edges are restrained, initial imperfection importantly influence the behavior tendency of porous FGM panels exposed to preexist thermal environment.

Diatom-inspired stiffness optimization for plates and cellular solids

Diatoms, a class of aquatic autotrophic microorganisms, are characterized by silicified exoskeletons with highly complex architectures. These morphologies have been shaped by the selection pressure that the organisms have been subjected to during their evolutionary history. Two properties which are highly likely to have contributed to the evolutionary success of current diatom species are lightweightness and structural strength. Thousands of diatom species are present in water bodies today, and although each has its unique shell architecture, a strategy that is common across species is the uneven and gradient solid material distribution across their shells. The aim of this study is to present and evaluate two novel structural optimization workflows inspired by material grading strategies in diatoms. The first workflow mimics the Auliscus intermidus diatoms’ surface thickening strategy and generates continuous sheet structures with optimal boundaries and local sheet thickness distributions when applied to plate models subjected to in-plane boundary conditions. The second workflow mimics the Triceratium sp. diatoms’ cellular solid grading strategy, and produces 3D cellular solids with optimal boundaries and local parameter distributions. Both methods are evaluated through sample load cases, and prove to be highly efficient in transforming optimization solutions with non-binary relative density distributions into highly performing 3D models.

Frequency of the composite shell rested on the complex non-polynomial viscoelastic media involving friction force

In this study, a 3D-flexibility theory is employed in order to investigate the frequency behavior of the composite cylindrical shells on the elastic substrate. The current composite structure is reinforced by functionally graded graphene oxide powders (FG-GOPs). It is assumed that the system is resting on an elastic foundation, in which friction force in the horizontal direction, as well as viscoelastic substrate, are taken into account. The results are extracted by the differential quadrature method (DQM). Additionally, the results are validated by COMSOL Multi-physics software. Lastly, the effect of in-plane boundary conditions, Biot’s coefficient, circumferential as well as initial radial stresses, and FG-GOP’s weight fraction on the frequency responses of the FG-GOPRC cylindrical shells on the elastic substrate is studied. Also, when Biot’s coefficient intensifies, the negative impact of the geometry parameter on the frequency response is more significant than the positive influence from the geometry parameter.

Thermo-torsional postbuckling of CNT-reinforced composite toroidal shell segments with surrounding elastic media and tangentially restrained edges

An analytical investigation on the nonlinear stability of toroidal shell segment (TSS) made of carbon nanotube (CNT)-reinforced composite, surrounded by an elastic medium, exposed to elevated temperature and subjected to uniform torsion is presented in this paper. The properties of constituents are assumed to be temperature dependent and effective properties of composite are determined using an extended rule of mixture. CNTs are embedded into matrix phase according to functionally graded or uniform distributions. Basic equations in terms of deflection and stress function are established within the framework of classical shell theory including geometrical nonlinearity in von Kármán–Donnell sense and interactive pressure from surrounding elastic medium. Two boundary edges of the shell are assumed to be simply supported and tangentially restrained. Multi-term analytical solutions are assumed and Galerkin method is used to derive expressions of buckling load and nonlinear relation between torsional load and deflection. Parametric studies are carried out to analyze the effects of material and geometry properties, in-plane boundary condition, elevated temperature and surrounding elastic medium on the buckling resistance and postbuckling behavior of torsionally loaded TSS. Novel finding of this study is that tangential edge constraints have no and negative effects on critical torsional loads at room and elevated temperatures, respectively, and profoundly beneficial influences on postbuckling load capacity of TSSs.

Buckling and postbuckling of functionally graded porous material nearly cylindrical shells under external lateral pressure in thermal environments

ABSTRACT This paper investigates the buckling and postbuckling of nearly cylindrical shell (NCS) made of functionally graded porous material (FGPM) and subjected to external lateral pressure (ELP) in thermal environments. The variation of two material constituents through the shell thickness is modelled by a power-law distribution. Porosities are assumed to distribute into FGPM according to forms of cosine functions. Basic equations in terms of deflection and stress function are established within the framework of classical shell theory taking into account von Kármán – Donnell nonlinearity. Boundary edges of the NCS are assumed to be simply supported and tangentially restrained. Multi-term analytical solutions are assumed and Galerkin method is used to derive expressions of buckling loads and nonlinear load–deflection relation. Parametric studies are carried out to assess the effects of coefficient and distribution type of porosity, thermal environments, geometry properties and in-plane boundary condition on the critical loads and postbuckling strength of pressure-loaded FGPM NCSs.

Global buckling and wrinkling of variable angle tow composite sandwich plates by a modified extended high-order sandwich plate theory

The advanced Automated Fibre Placement (AFP) manufacturing technologies make the synthesis of Variable Angle Tow (VAT) composites, which enable the design of lightweight sandwich structures to possess variable stiffness facesheets. However, both global and local instability phenomena of VAT sandwich plates under in-plane compressive loads are rarely explored till now. The objective of this article is to fill this gap by developing a Rayleigh–Ritz procedure based on a modified extended high-order sandwich plate theory (EHSAPT). The original two-dimensional EHSAPT proposed by Phan et al. (2012) is extended to a three-dimensional case with a minor modification that the first-order shear deformation theory instead of the classical Kirchhoff–Love hypothesis is adopted for each facesheet, which brings several merits such as the conciseness in derivation process and the easiness in program implementation. The Rayleigh–Ritz approach combined with the principle of minimum potential energy is employed to derive the eigenvalue equation that governs the instability problem of VAT sandwich plates under in-plane compressive loads. Both global buckling and wrinkling patterns of VAT sandwich plates can be captured under this proposed analytical model framework. Before instability analysis, the nonuniform prebuckling stresses over the entire sandwich plate are determined under the assumption of membrane prebuckling state. The usage of Lagrange multiplier method in the prebuckling regime removes the restrictions inherent in conventional Rayleigh–Ritz formulation and thus provides a general way to model in-plane boundary conditions. The accuracy and effectiveness of the developed Rayleigh–Ritz procedure are validated by comparing with previously published results and FE solutions given by ABAQUS. Effects of core thickness, core orthotropy, and fibre orientation angle of the facesheets on the instability behaviour of VAT sandwich plates are investigated. Finally, the mechanism of steering the fibre trajectory over the facesheets to improve the buckling resistance for the sandwich plate is studied and discussed.

Size-Dependent Elastic Buckling of Two-Variable Refined Microplates Embedded in Elastic Medium

This study applies a two-variable refined shear deformation theory (TVRSDT) and a modified couple stress theory (MCST) to develop a size-dependent elastic buckling model for microplates under combined axial compression and in-plane shear and resting on Winkler–Pasternak foundation. Our model incorporates two transverse displacement variables and one material length scale parameter (MLSP); it satisfies the zero-traction boundary conditions on the top and bottom surfaces of microplates, thereby circumventing the use of a shear correction factor. The equations of motion are determined via the Euler–Lagrange equation. A closed-form buckling solution is presented for microplates with four edges simply supported. To handle the nonsimply supported microplates, a thirty-two-degree-of-freedom (32-DOF) four-node differential quadrature (DQ) finite element with [Formula: see text]-continuity is proposed. The corresponding element matrices are obtained using the minimum potential energy principle. Comparison studies are conducted to exhibit the validity of the derived formulations. Finally, the present buckling model is employed for predicting the elastic buckling behaviors of microplates embedded in an elastic medium. The effects of in-plane loading patterns, boundary conditions, aspect ratio, length-to-thickness ratio, MLSP and elastic medium parameters on the buckling load and buckling mode are elucidated.

On the dynamics of the axisymmetric circular system via an exact continuum elasticity theory and discrete singular convolution method

The vibration for the circular sector plates made of reinforced composite with carbon nanotubes based on a 3D-poroflexibility theory is presented in this research. A discrete singular convolution integration method (DSC-IM) is used. By using the renowned rule of mixtures, the effective mechanical properties are related to reinforced composites. The present results, formulations, and solution procedures are verified. The novelty of this work considers the effects of 3D- elasticity theory’s initial radial and environmental stresses and various boundary conditions via DSC-IM. Consequently, the outcomes demonstrate that in-plane boundary conditions, CNT’s volume fraction, Biot’s coefficient, and radial and circumferential initial stresses greatly impact the vibration frequency of the circular sector plates. Also, the effect of initial circumferential stress is more prominent than radial ones in determining the vibration frequency. Also, in higher open-angle, the impact of volume fraction related to CNTs on the vibrational frequency of the system is more apparent.

Computational homogenization of locally resonant acoustic metamaterial panels towards enriched continuum beam/shell structures

Locally resonant acoustic metamaterial (LRAM) panels are excellent candidates for the low-frequency flexural wave attenuation in thin structures. To enable the efficient analysis and design of LRAM beam/shell structural elements for practical applications, a computational homogenization method for modelling wave propagation phenomena in LRAM panels is presented in this work. The approach is based on the notion of a relaxed separation of scales, which tailors the methodology to the phenomena governed by local resonators embedded in a host medium. The macroscopic LRAM panel is modelled as a thin continuum beam/shell described by proper structural kinematics and momentum balance relations. At the microscale, a LRAM unit cell is considered with lamina-wise in-plane boundary conditions and zero out-of-plane tractions, representing the free top and bottom surfaces of the macroscopic LRAM panel. Under the assumption of linear elasticity, in the relaxed separation of scales regime, the microscale unit cell problem can be represented by the superposition of a static and a dynamic problem, hereby enabling a significant model reduction. As a result, the macroscale effective material properties can be computed once and for all off-line for a given unit cell. In addition, a new macroscale evolution equation emerges describing the effect of the microscale internal dynamics on the macroscopic fields through the introduction of new macroscale enrichment variables, which reflect the modal amplitudes of localized microscale modes. The proposed homogenization methodology reveals a high level of accuracy and numerical efficiency compared to the reference direct numerical simulation (DNS) for all relevant analyses: computation of real and complex dispersion spectra for an infinite LRAM panel, as well as steady-state frequency response and transient behaviour for a finite LRAM panel.

Thermal and thermomechanical buckling of CNT-reinforced composite sandwich cylindrical shells including elasticity of tangential edge restraint

This paper presents an analytical approach to investigate the buckling of sandwich cylindrical shells subjected to uniform temperature rise and external lateral pressure. Two sandwich models corresponding to carbon nanotube reinforced composite (CNTRC) face sheets and core layer are considered. The properties of all constitutive materials are assumed to be temperature dependent and effective properties of CNTRC are determined according to an extended rule of mixture. Governing equations are established using first order shear deformation theory and solved employing two-term form of deflection along with Galerkin method for simply supported edge shells. In order to account for practical situations of in-plane boundary condition, the elasticity of tangential constraint of boundary edges is included. Owing to temperature dependence of material properties, critical thermal loads are determined adopting an iteration process. Numerous parametric studies are carried out and interesting remarks are given. The study reveals that sandwich shell model with CNTRC core layer and homogeneous skins has considerably strong capacity of buckling resistance. Numerical results also indicate that tangential edge constraint has significant effects on critical loads, especially at elevated temperature. In addition, in the case of thermal load, an intermediate volume percentage of carbon nanotubes can confer the highest critical temperatures of sandwich shells.

Bending analysis of FG-GPLRC axisymmetric circular/annular sector plates by considering elastic foundation and horizontal friction force using 3D-poroelasticity theory

Composite structures encounter different types of imperfections and defects during the installation, working environment, and fabrication process. In the present research, a 3D-poroflexibility theory has been presented to study the bending responses of the functionally graded graphene nanoplatelets reinforced composite (FG-GPLRC) axisymmetric circular/annular sector plates on the elastic substrate. For modeling elastic substrate, horizontal friction force and the elastic foundation with five elastic parameters have been formulated. For obtaining the exact displacement and stress/strain responses of the current structure, a discrete singular convolution integration method (DSC-IM) has been utilized. Also, the outputs of the current research have been verified with the results of COMSOL Multi-physics software. Consequently, the outcomes demonstrate that GPL’s weight fraction, in-plane boundary condition, radial and circumferential initially stresses, and Biot's coefficient have a remarkable impact on the bending responses of the circular/annular sector plates. Also, when the normal compressive stress increases, the negative effect from normal tensile stress on the bending characteristics of the structures is more remarkable than the positive effect from normal compressive stress. Also, the effect of GPL’s weight fraction on the stress and displacement fields is more considerable at the higher value of the opening angle of the current composite structure.

Identifying all combinations of boundary conditions for in-plane vibration of isotropic and anisotropic rectangular plates

This work presents a combined study of an effective vibration analysis and a counting theory to identify all combinations of boundary conditions for in-plane vibration of isotropic, specially orthotropic and symmetrically laminated rectangular plates. First, a method of analysis is proposed to obtain the in-plane natural frequencies of plates under any in-plane boundary conditions of free, clamp and two types of simple supports, and this method makes it possible to calculate the frequencies of rectangular plates subject to 256(=4 powered by 4) sets of boundary conditions. Secondly, Polya counting theory is introduced to determine theoretically the total number of distinct combinations of plate boundary conditions, and to reduce the 256 sets into the essentially identical subsets of frequencies. In numerical experiments, all sets of natural frequencies are calculated for the rectangular plates with different aspect ratio, material property and lamination, and are sorted into the classes with identical sets of frequencies. The distinct combinations are thus obtained for in-plane vibration of the plates, and it is shown that the number of combinations from the experiment exactly agrees with prediction by Polya counting theory.

Computational modeling to predict the micromechanical environment in tissue engineering scaffolds

Cell fate in tissue engineering (TE) strategies is paramount to regenerate healthy, functional organs. The mechanical loads experienced by cells play an important role in cell fate. However, in TE scaffolds with a cell-laden hydrogel matrix, it is prohibitively complex to prescribe and measure this cellular micromechanical environment (CME). Accordingly, this study aimed to develop a finite element (FE) model of a TE scaffold unit cell that can be subsequently implemented to predict the CME and cell fates under prescribed loading. The compressible hyperelastic mechanics of a fibrin hydrogel were characterized by fitting unconfined compression and confined compression experimental data. This material model was implemented in a unit cell FE model of a TE scaffold. The FE mesh and boundary conditions were evaluated with respect to the mechanical response of a region of interest (ROI). A compressible second-order reduced polynomial hyperelastic model gave the best fit to the experimental data (C10 = 1.72 × 10−4, C20 = 3.83 × 10−4, D1 = 3.41, D2 = 8.06 × 10−2). A mesh with seed sizes of 40 µm and 60 µm in the ROI and non-ROI regions, respectively, yielded a converged model in 54 min. The in-plane boundary conditions demonstrated minimal influence on ROI mechanics for a 2-by-2 unit cell. However, the out-of-plane boundary conditions did exhibit an appreciable influence on ROI mechanics for a two bilayer unit cell. Overall, the developed unit cell model facilitates the modeling of the mechanical state of a cell-laden hydrogel within a TE scaffold under prescribed loading. This model will be utilized to characterize the CME in future studies, and 3D micromechanical criteria may be applied to predict cell fate in these scaffolds.

Effect of non-uniform in-plane bending and edge conditions on the stability behaviour of laminated panels with and without cutouts

The effect of uniaxial and biaxial non-uniform in-plane bending and cutout sizes on the buckling behaviour of laminated composite panels has been investigated by developing finite elements formulation. The effect of shear deformation is included in the formulation. Therefore, this formulation is applicable for both thin and moderately thick panels. The panel is modelled by using 9-noded heterosis plate element. The main advantage of this element is that, it offers high level accuracy, even for extremely thin plate configuration. The effect of various parameters such as type of in-plane bending, cutout size, boundary conditions and ply-orientations are also studied in this work. Finally, it is shown that the in-plane bending and cutout sizes have a remarkable effect on the buckling behaviour of a panel.

A study on buckling of simply supported plates with edges partially restrained for in-plane movement

This paper investigated experimentally the influence of size and location of a circular cut-out on the buckling load of simply supported plates with in-plane boundary conditions (and thus, induced stress pattern in the pre-buckling state). Experiments were conducted on solid and perforated stainless steel plates with different aspect ratios and edges partially restrained for in-plane movements. Results of these experiments demonstrated that the size and position of cut-out significantly affected the buckling load of the plate specimens. Further, to better interpret the experimentally obtained results, finite element analyses were conducted wherein the numerical model closely represented the experimental setup. The Riks method of analysis and eigenvalue elastic buckling analysis were carried out with the same set of experimental boundary conditions. From these analysis, it was found that the in-plane restraints significantly affected the buckling behavior of a plate. It was also observed that the buckling loads obtained from experiments matched reasonably well with the analytical values.

Buckling and post-buckling of FRC laminated beams in thermal environment using a generalized higher-order shear deformation zig-zag beam model

The thermal buckling and post-buckling of fibre-reinforced composite (FRC) beams are analyzed based on a generalized higher-order shear deformation zig-zag beam theory (RHDZT). The current zig-zag beam model satisfies the stress boundary conditions on the upper and lower surfaces and the reality that the shear strain is discontinuous at the layer interfaces. The material properties of FRCs are assumed to be temperature independent and the in-plane boundary conditions supposed immovable. By using the Hamilton variation principle, the governing equations are established based on the mid-plane of the beam. A two-step perturbation method is employed to obtain the approximate analytical solutions of the critical thermal buckling loads and post-buckling behaviors with clamped-clamped boundary conditions. The critical buckling results based on the zig-zag model and its degenerate equivalent single-layer models are compared and verified with those in the literatures. The post-buckling results of the FRC beams with different slenderness ratios, elastic modulus ratios and ply angles are investigated. Numerical results show the remarkable accuracy of the current model and also demonstrate the significant roles the above-mentioned effects play in the buckling and post-buckling problem of FRC laminated beams.

Predicting the overall response of an orthogonal 3D woven composite using simulated and tomography-derived geometry

Effects of two meso-scale geometry generation approaches on predicting effective elastic properties and thermal conductivity of an orthogonal 3D woven composite are studied in this paper. The two approaches used are fabric mechanics simulation (DFMA software, Kansas State University) and direct processing of X-ray microtomography (µCT) data. Two models are created in DFMA using two different sets of cross-sectional areas as input: nominal (based on the initial weave pattern) and adjusted (informed by volume fraction measurements obtained from microscopy data). In addition, one conformal mesh and three voxel mesh models are created from µCT data. Experimental measurements of transverse Young’s moduli are used to evaluate the accuracy of the predicted results.In each case, a unit cell with in-plane periodic boundary conditions is modeled, which has not been previously done in the case of µCT-based models. The effect of high frequency oscillations in local material orientations imparted by a wavy centerline (artifact of µCT image processing) on the elastic and thermal conductivity properties is studied. The differences in volume fractions and shapes of bundles of fibers (tows) between µCT-based and DFMA-based models are also investigated to determine sensitivity of effective thermo-mechanical properties to each of these factors.

Prebuckling and buckling analysis of moderately thick variable angle tow composite plates considering the extension-shear coupling

In this paper, an analytical model based on a generalized Rayleigh-Ritz method is developed to deal with both the prebuckling and buckling problems of moderately thick VAT composite plates under a more general in-plane boundary condition. The non-uniform in-plane stress fields of VAT composite plates under various in-plane boundary conditions is firstly determined based on the minimum complementary energy principle combined with the Airy’s stress function. Afterwards, the governing equation of buckling problem of VAT composite plates under non-uniform in-plane stress fields is derived based on third-order shear deformation theory of Reddy’s type. The innovation of this paper lies in that a linear fitting method combined with Chebyshev-Gauss-Labotto point distribution is introduced to retrieve the boundary stress coefficients from the stress constraint conditions generated by the Lagrangian multiplier method, and meanwhile an integral expression of work done by the unknown boundary force along the prescribed in-plane displacement is adopted to satisfy the displacement constraint condition along the boundary line. In so doing, three kinds of in-plane boundary conditions are taken into account, that is, pure stress boundary condition, pure displacement boundary condition and mixed stress and displacement boundary condition. In addition, no assumption is made on the extension-shear coupling terms and thus the proposed analytical model is suitable to the unbalanced layup. The accuracy and robustness of the proposed analytical model is validated by finite element analysis and with prior results. Effects of the in-plane boundary condition, extension-shear coupling and varying fibre orientation angles on both in-plane stress and buckling responses of VAT composite plates are studied by numerical examples. It is shown that for the balanced panels under uniform compression, the benign load redistribution mechanism offered by the VAT layups remain in effect even if there exists the non-uniform shear constraint along the loading edges. However, the buckling load of the panel can be reduced by the shear constraint along the loading edges to a certain degree. Furthermore, it is found that for some displacement-controlled shear loading cases, the extension-shear coupling plays a positive role in the load redistribution, which results in a sight improvement in the longitudinal load of the panel when critical buckling occurs.

Buckling of piezoelectric sandwich microplates with arbitrary in-plane BCs rested on foundation: effect of hygro-thermo-electro-elastic field

Buckling and post-buckling behaviors of simply supported microplates in complex environment are studied, where elastic foundation and hygro-thermal-electro-mechanical loads are considered. The first-order shear deformation theory is used to establish basic equations of the microplate considering the von Karman’s nonlinearity. The size-dependent effect is characterized by the modified couple stress theory. A unified boundary condition model is introduced to discuss various in-plane boundary conditions (BCs). Analytical solutions for critical mechanical/hygrothermal buckling loads and post-buckling paths of the microplate under different in-plane BCs are obtained by using the perturbation method and the Galerkin method, respectively. Results reveal that size-dependent effect and elastic foundation enhance the stiffness of the microplate. Transverse displacement of the microplate in the post-buckling stage increases with the external compressive load, temperature and moisture concentration, expressing a nonlinear curve. When the displacement constraint in the normal direction is applied on the microplate edge, the critical mechanical/hygrothermal buckling load decreases. These results can be utilized in the optimization design of the micro-electro-mechanical systems.

Buckling and stability analysis of sandwich beams subjected to varying axial loads

This article presented a comprehensive model to study static buckling stability and associated mode-shapes of higher shear deformation theories of sandwich laminated composite beam under the compression of varying axial load function. Four higher order shear deformation beam theories are considered in formulation and analysis. So, the model can consider the influence of both thick and thin beams without needing to shear correction factor. The compression force can be described through axial direction by uniform constant, linear and parabolic distribution functions. The Hamilton\'s principle is exploited to derive equilibrium governing equations of unified sandwich laminated beams. The governing equilibrium differential equations are transformed to algebraic system of equations by using numerical differential quadrature method (DQM). The system of equations is solved as an eigenvalue problem to get critical buckling loads and their corresponding mode-shapes. The stability of DQM in determining of buckling loads of sandwich structure is performed. The validation studies are achieved and the obtained results are matched with those. Parametric studies are presented to figure out effects of in-plane load type, sandwich thickness, fiber orientation and boundary conditions on buckling loads and mode-shapes. The present model is important in designing process of aircraft, naval structural components, and naval structural when non-uniform in-plane compressive loading is dominated.