Clamped Edges Research Articles

Continuum model for converse flexoelectricity in a thin plate

Flexoelectricity – the effect where inhomogeneous polarization causes deformation – substitutes symmetry-forbidden piezoelectricity in nano-scale centrosymmetric materials. Its mathematical treatment in terms of continuum mechanics represents a challenge: correct models are subjects of debates during the past decades. In the present article we formulate a model with reasonable approximations for converse flexoelectric effect in a plate. Except for very specific boundary conditions the theory inevitably include nonclassical higher order terms. These are treated using variational calculus and are essentially reduced to a modification of the boundary conditions for the elastic variables. Analytical solutions are obtained for a circular plate with round electrodes, where deflections are proportional to the inverse square of plate thickness. For mechanically free boundary conditions at plate’s edges deflection grows monotonically with the electrode radius. In contrast, a plate with clamped edge does not deform at all when fully electroded, an optimal electrode size is found rendering maximal deflection.

Time Period Analysis of Orthotropic Skew Plate with 2-D Circular Thickness and 1-D Circular Density

In the present study, the time period of vibration of an orthotropic parallelogram plate with 2 − D (two-dimensional) circular thickness under the effect of 2 − D parabolic temperature is investigated for the first time. The different edge conditions are C C C C , C C C F , C F C F , C S C F , and S F S F boundary conditions, where C, S, and F stands for clamped, simply supported, and free edges of the plate, respectively. The variation in density of plate material is considered to be 1 − D (one-dimensional) circular. The Rayleigh–Ritz technique is used to solve the differential equation and evaluate the time period for the first two modes of vibration. A convergence study of an orthotropic parallelogram, rectangle plate, and square plate for modes of frequency at various edge conditions is also carried out. The authors performed a comparative analysis of the time period and modes of frequency of orthotropic parallelogram, rectangle plate, and isotropic square plate with the available published results at various edge conditions. The main conclusion which we made from this study is that by choosing the above-mentioned plate parameters, we obtained fewer modes of frequency in comparison to other variations mentioned in the literature. Also, the study suggests that the variation in modes of frequency is less in comparison to other variations.

Analytical Solution for the Static Bending Elastic Analysis of Thick Rectangular Plate Structures Using 3-D Plate Theory

In the current work, an analytical solution for static bending analysis of the thick rectangular plate structure was obtained using three-dimensional plate theory. First, the energy equation was formulated from the static elastic principles and transformed into a compatibility equation through general variation to get the slope and deflection relationship. The solution of the compatibility equation gave rise to the exact polynomial deflection function. In contrast, the coefficient of deflection and shear deformation of the plate was obtained from the governing equation through the direct variation method. These solutions were used to obtain the characteristic expression for analyzing the displacement and stresses of a rectangular plate. This formula was used for the solution of the bending problem of rectangular plate support conditions of two clamped edges, one free edge, and a simply-supported edge (CCFS). The result of the deflection and stresses decreases as the span-thickness ratio increases. More so, the aspect ratio effect of the shear stress of isotropic plates is investigated and discussed after a comparative analysis between the present work and previous studies. The result shows that the present study differs with RPT) of assumed deflection by 2.7%, whereas exact 2-D RPT by 1.9%. This shows the efficacy of the exact 3-D plate theory for rectangular plate analysis under CCFS support and loading condition.

Free vibrations of small-scale plates with complex shape based on the nonlocal elasticity theory

Free vibrations of the orthotropic micro/nanoplate with nonclassical shape are investigated. The considered model is based on the nonlocal elasticity theory. The developed method uses the Ritz method as well as R-function theory for the construction of the system of coordinate functions. The linear frequencies are obtained for a rectangular plate with two cutouts on opposite sides, while the boundary conditions are considered of several types, including simply supported and clamped edges. The small-scale effects for various sizes of cutouts are discussed.

Roof Fractures of Near-Vertical and Extremely Thick Coal Seams in Horizontally Grouped Top-Coal Drawing Method Based on the Theory of a Thin Plate

During the mining process of the near-vertical seam, there will be movement and collapse of the “roof side” rock layer and the “overlying coal seam,” as well as the emergence of the “floor side” rock layer roof which is more complicated than the inclined and gently inclined coal seams, which causes problems with slippage or overturning damage. With the increase of the inclination of the coal seam, the impact of the destruction of the immediate roof on the stope and roadway gradually becomes prominent, while the impact of the destruction of the basic roof on the stope and the roadway gradually weakens. The destruction of the immediate roof of the near-vertical coal seam will cause a large area of coal and rock mass to suddenly rush to the working face and the two lanes, resulting in rapid deformation of the roadway, overturning of equipment, overturning of personnel, and even severe rock pressure disaster accidents, all of which pose a serious threat to coal mine safety and production. It is necessary to carry out research on the mechanical response mechanism of the immediate roof of near-upright coal seams, to analyse the weighting process of steeply inclined thick coal seam sub-level mining. A four fixed support plate model and top three clamped edges simply supported plate model for roof stress distribution are established before the first weighting of the roof during the upper and lower level mining process. The bottom three clamped edges simply supported plate model and two adjacent edges clamped on the edge of a simply supported plate model are established for roof stress distribution before periodic weighting of the roof during the upper and lower level mining process. The Galerkin method is used to make an approximate solution of deflection equation under the effect of sheet normal stress, and then roof failure criterion is established based on the maximum tensile stress strength criterion and generalized Hooke law. This paper utilizes FLAC3D finite element numerical simulation software, considering the characteristics of steeply inclined thick coal seam sub-level mining. It undertakes orthogonal numerical simulation experiment in three levels with different depths, coal seam angles, lateral pressure coefficient, and orientation of maximum horizontal principal stress, and translates roof stress of corresponding 9 simulation experiment into steeply inclined roof normal stress. We conclude that the distribution law of normal stress along dip and dip direction of a roof under the circumstance of different advancing distances and different sub-levels. The caving pace of first weight and periodical weight were counted under the effect of the roof uniform normal stress. It can better predict the weighting situation of the working face and ensure the safe, efficient, and sustainable mining of coal mines.

A separation-of-variable method for the wrinkling problems of orthotropic rectangular stretched sheets

Film structures have been extensively employed in space and soft materials. The flexural rigidity of a sheet is so small that it can hardly withstand pressure, causing instability and forming wrinkles. This work develops a separation-of-variable (SOV) method to the stretched-induced wrinkling analysis of orthotropic rectangular sheets. In this SOV method, wrinkling mode functions are in a separation-of-variable form, and two ordinary characteristic differential equations of wrinkling mode functions are derived through the minimum principle of the potential energy, then the mode functions are expressed in terms of the eigenvalues of the characteristic differential equations. Finally, closed-form mode functions and explicit equations for critical tensile strains are achieved with the boundary conditions of rectangular sheets. In this work, both stretched edges are clamped or simply supported, and another pair of edges are free, and the closed form solutions for the rectangular sheets with clamped stretched edges are achieved for the first time. The present results agree well with the analytical and numerical results in literature. In addition, the stretching process is described and several observations are found through theoretical and numerical analyses.

Solutions and applications of 3D elastic–plastic constraint parameters for clamped single edge notched tension (SENT) specimens

Single edge notched tension (SENT) specimen is a common fracture toughness test specimen now widely used in the pipeline and pressure vessels industries. In this paper, an extensive study has been conducted by three-dimensional (3D) finite element analyses (FEAs) to investigate the in-plane constraint parameters (namely A and QSSY) and out-of-plane constraint parameter, that is, Eε33 for clamped SENT specimens with side-grooves. A wide range of crack depth to specimen width ratios a/W = 0.2, 0.3, 0.4, 0.5, 0.6 and 0.7, specimen thickness to specimen width ratios B/W = 0.5, 1.0, 1.5, 2.0, 3.0 and 4.0 were included. The various strain hardening exponent (n = 4, 7, 10) and applied loading levels (from small-scale yielding to large-scale yielding) were also investigated. The results show that the effects of crack depth, specimen thickness, loading level and strain hardening index on the 3D constraint parameters show a significant coupling effect. With the increase of load level, A value increases monotonically, while QSSY and the normalized Eε33 (called V33) decrease, and the influence of specimen geometry on the constraint levels will become greater. The values of A and V33 obviously decrease with the increase of hardening exponent, but the change of QSSY is relatively small. Moreover, it can be observed that both A and QSSY can well describe the in-plane constraint effect, and V33 can better quantify the out-of-plane constraint effect under varying loading levels and n. In addition, elastic–plastic constraint parameters were also calculated for various experimental test specimens made of 16MND5 steel, with n = 9.2. It was demonstrated these constraint parameters can be applied to quantify and predict the J-R curves of 16MND5 steel with good agreements for specimens with different in-plane and out-of-plane constraint conditions.

A comparative study on constraint parameters for characterizing fracture toughness of X60 and X100 pipe steels

The fracture toughness resistance curve (e.g., J-R curve and crack tip opening displacement (CTOD) or δ-R curve) is important in facilitating strain-based design and integrity assessment of oil and gas pipelines, where influence of ground movement and presence of planar weld flaws must be accounted for. In the present study, duplicate single edge notched bend (SE(B)) and clamped single edge notched tension (SE(T)) specimens with two different initial crack lengths were tested at room temperature for longitudinally-oriented specimens machined from X60 and X100 pipe steels where the notch is along the through-thickness direction. The results showed the expected trends of higher fracture toughness resistance curves for shallow-cracked over deep-cracked specimen types and tension over bending loading modes. Eleven constraint parameters (i.e., QHRR, QSSY, QLM, QBM, A2, A2BM, h, Tz, Cp, Ap, Vp) were calculated based on 3-D finite element analysis for all tested SE(B) and SE(T) specimens. The results indicated that the shallow-cracked SE(B) specimens exhibit a constraint condition similar to the intrinsically low-constraint deep-cracked SE(T) specimens made of the X60 pipe steel. Among all of the constraint parameters, the Ap parameter based on the equivalent plastic strain isoline was found to correlate the best to the material fracture toughness for currently tested specimens. Furthermore, a linear master curve between various critical fracture toughness values and Ap parameter was obtained for all the tested specimens with different crack lengths, loading modes and pipe steels.

On the flexoelectric effect on size-dependent static and free vibration responses of functionally graded piezo-flexoelectric cylindrical shells

The flexoelectric effect depends on the strain gradient, which naturally appears in intelligent structures at micro and nano scales. Therefore, the flexoelectric phenomenon significantly affects electro-mechanical behavior of structures in these scales. In this study, the first-order shear deformation theory and the reformulated flexoelectric theory are applied to derive coupling equations of motion for a size-dependent static and dynamic responses of multilayered composite shell with simply supported and clamped edges. Constitutive equations are defined based on assumption that the internal energy function depends not only on strains and polarization tensors but also on strains and polarization gradient tensors. The piezo-flexoelectric structure is made of an isotropic functionally graded core and two orthotropic layers. Reynolds transport theorem and variational Hamilton principle are used to derive equations of motion expressed by forces and displacements, and corresponding boundary conditions. In order to check the correctness of the formulation, and solution of the problem obtained by Galerkin method, static and free vibrations analyses of macro-/micro-/nanoshell are investigated. Diagrams of deflection and electrical potential variations of the structure under the load are obtained for the direct flexoelectric effect. Additionally, the influence of various parameters, e.g., geometric properties, size effect, power-law index, and flexoelectric layer thickness, on the deflection, potential function, and natural frequency of the microshell have been investigated. The numerical results and discussion present that above parameters significantly affect a mechanical behavior of the microshell under consideration.

High-precision computation in mechanics of composite structures by a strong SaS formulation: Implementation for laminated cylindrical shells and panels with clamped and free edges

The paper focuses on the three-dimensional (3D) stress analysis of laminated composite cylindrical shells and panels with general boundary and loading conditions using the strong sampling surfaces (SaS) formulation and the extended differential quadrature (EDQ) method, recently developed by the first author. In the strong SaS formulation, the SaS parallel to the middle surface and located at Chebyshev polynomial nodes in layers are utilized to introduce displacements of these surfaces as shell unknowns. This choice of unknowns with the use of Lagrange polynomials in the approximation of displacements, strains and stresses through the thickness leads to an efficient shell formulation. The outer surfaces are not included into a set of SaS that makes it possible to minimize uniformly the error due to Lagrange interpolation. Therefore, the strong SaS formulation based on direct integration of the equilibrium equations of elasticity in the thickness direction in conjunction with the EDQ method can be effectively applied to high-precision calculations for laminated composite cylindrical shells and panels with arbitrary boundary conditions. This is due to the fact that in the SaS/EDQ formulation the displacements, strains and stresses of SaS are interpolated in a rectangular domain, which is mapped into the middle surface by using the Chebyshev-Gauss-Lobatto grid and the Lagrange polynomials are also utilized as basis functions.

Free vibration analysis for V-notched Mindlin plates with free or clamped radial edges

Free vibration analysis for V-notched Mindlin plates with free or clamped radial edges

Determination of ductile fracture properties of 16MND5 steels under varying constraint levels using machine learning methods

The current paper presents a machine learning method based on artificial neural network (ANN) model for the determination of ductile fracture properties of 16MND5 bainitic forging steel with various three-dimensional (3D) constraint conditions. A series of fracture test data with clamped single edge notched tension (SENT) specimens were used for model training and test. With the comprehensive analysis of prediction accuracy and extrapolation ability, a training strategy for ANN model was proposed including an artificially divided training set and the introduction of dropout layer. The artificial division makes the experimental samples in training set reduced by 40.7%, while the dropout layer prevents ANN model from overfitting caused by reduction of training data. Moreover, the deep nonlinear relationship between geometric dimensions (H/W, B/W, a/W) and ductile fracture properties was well learned by the ANN model. The average error of prediction is less than 11%. Finally, the proposed training strategy was extended to solve the fracture behaviors under varying thermal aging duration with saving training experimental samples by 53.8%. The results showed that the comprehensive interaction of in-plane constraint, out-of-plane constraint and thermal aging on the ductile fracture behaviors are well reproduced. Due to the good prediction performance, generalization and low training cost, the proposed training strategy can make the ANN model much more helpful for the solution of ductile fracture properties of different geometric dimensions in harsh environments.

Prediction of influences of size and locations of delamination on dynamic characteristics of laminated composite plate using particle swarm optimization and artificial neural network

AbstractIn laminated composite structures, delamination is one of the most common defects. The delamination affects the vibration characteristics of laminates, and thus these indicators can be used to detect the potentially catastrophic failures and measures the health characteristics of laminates. In this study, particle swarm optimization (PSO) and artificial neural network (ANN) are used to optimize and predict the influences of location and size of delamination on the dynamic behavior of composite plates. The classical laminated plate theory adopted principle equation based on the dynamic characteristics of composite laminate has expressed through the finite element method. The delamination behavior of laminate composite plate is modeled by considering delamination at several interfaces with variable sizes and locations. PSO methodology is recommended to optimize the decrease in natural frequency due to the delamination with varying weight fractions under completely clamped edge boundary conditions. Further, an ANN algorithm is proposed for predicting the dynamic responses of the composite plate by considering unknown ranges of parameters such as weight fraction, ply configuration, and different delamination locations.

Flow control of two tandem cylinders by a highly flexible filament: Lattice spring IB-LBM

In the current study, the Lattice Spring Method (LSM) is combined with the Immersed Boundary-Lattice Boltzmann Method (IB-LBM) to investigate fluid-elastic body interaction problems in unsteady conditions. The combined method benefits from worthy features of all three methods. As compared to the conventional fluid-solid interaction models, this model needs no arbitrary spring parameters to determine the deformation of the flexible body. The method is first verified against the previous studies in which a thin, long filament flaps in the wake of a cylinder that attached to the filament. In the numerical results section, the problem of a highly deformable filament in a tandem configuration, two series of rigid cylinders, is studied in detail. The effects of several variables such as Capillary number, the filament's length, and location of the trailing edge on the lift and drag coefficients of two tandem cylinders are investigated. Based on the numerical results, the fluctuations of the lift coefficient on both cylinders are damped when filament capillary number gets its maximum value and the filament clamped edge is located in the middle of the cylinders. This is shown by a frequency analysis of the wake fluctuations downstream of the filament and cylinders Also, it is found that increasing the filament's length can halve the drag coefficient on the downstream cylinder due to the low-pressure region created at the filament's tip.

Dynamic response of an annular circular plate made of shape memory alloy using differential quadrature method

This study investigates the dynamic response of an annular circular plate with clamped edges under harmonic load. The circular plate is analyzed based on the first-order shear deformation theory (FSDT). The Boyd-Lagoudas constitutive model is used to simulate the pseudoelastic behavior, and Hamilton’s principle is used to obtain the equations of motion. Also, differential quadrature and Newmark methods are utilized to obtain the time and frequency responses of the plate. The phase transformation effects are studied on the time and frequency responses of the plate. Then, the accuracy of these results is checked with the available literature and ABAQUS software. Results indicate that the alloy phase transformation leads to reduced material strength and the nonlinear behavior of the alloy.

Natural Frequencies of Isotropic Rectangular Plates in Improved Accuracy

The objective of this paper is to present comprehensive lists of accurate natural frequencies of isotropic thin rectangular plates. For this purpose, a simple yet very accurate analytical approach is described to study the free vibration of the plates. In numerical computations, convergence and comparison studies are conducted to check accuracy of the present solution and to demonstrate improvement of numerical solutions in the results. Twenty-one tables are then provided to give the lowest six frequency parameters for all possible combinations of three typical edge conditions (clamped, simply supported and free edges). Each frequency value in the tables is given in five significant figures for non-Levy type problem (i.e., two opposite edges not simply supported) and six significant figures for Levy type problem. These results are presented for five different aspect ratios in the same format of the past representative reference.

Limit Carrying Capacity Calculation of Two-Way Slabs with Three Simply Supported Edges and One Clamped Edge under Fire

Fire-resistance experiments were conducted on two full-scale two-way reinforced concrete slabs with three simply supported edges and one clamped edge. This paper presents the design of the furnace, specimens and clamped-end device, the test plan, and the measuring contents and methods. The cracking and failure characteristics of the tested two-way reinforced concrete slabs were introduced. The temperature field distribution of the concrete and steel in the direction of the section thickness, out-of-plane deflections, in-plane deflections, and angle of the slab edges of the two-way concrete slab under fire conditions were analyzed. Cracks form on the slab surface, which is shaped similar to the bottom of a shallow bowl, with two transverse cracks in the mid-area of the medially clamped side and intensively annular diagonal cracks on the corner side. There were several main transverse cracks in the central area of the slab surface. The results indicate that there was a significant decrease in frequency under fire conditions. Relations between the frequency and the central vertical deflection of the two slabs were analyzed by the regression method. This research generates valuable test data that can be used to validate the numerical models developed by fellow researchers in the field of structural fire engineering. Based on the plate balance method and energy method, calculation formula of the limit carrying capacity calculation formula of reinforced concrete two-way slabs with four different boundaries under fire are given, which are simply supported on three sides and clamped on one side. The influence of membrane effect under large deflection is considered in the formula, and the calculated results are in good agreement with the experimental results.

Free vibration analysis of a laminated honeycomb sandwich panel: a suggested analytical solution and a numerical validation

PurposeThe purpose of this paper is to investigate the free vibration response of a laminated honeycomb sandwich panels (LHSP) for aerospace applications. Higher order shear deformation theory (HSDT) was simplified for the dynamic analysis of LHSP. Furthermore, the effects of honeycomb parameters on the value of natural frequency (NF) of vibration were explored.Design/methodology/approachThis paper applies HSDT to the analysis of composite LHSP to derive four vibration differential equations of motion and solve it to find the NF of vibration. Two analytical models (Nayak and Meunier models) were selected from literature for comparison of the NF of vibration. In addition, a numerical model was built by using ABAQUS and the results were compared. Furthermore, parametric studies were conducted to explore the effect of honeycomb parameters on the value of the NF of vibration.FindingsThe present model is successful in simplifying HSDT for the analysis of LHSP. The first five natural frequencies of vibration were calculated analytically and numerically. In the parametric study, increasing core height or young’s modulus or changing laminate layup will increase the value of NF of vibration. Furthermore, increasing plate constraint (using clamped edge boundary condition) will increase the value of NF of vibrations.Research limitations/implicationsThe current analysis is suitable for all-composite symmetric LHSP. However, for isotropic or non-symmetric materials, minor modifications might be adopted.Originality/valueThe application of simplified HSDT to the analysis of LHSP is one of the important values of this research. The other is the successful and complete dynamic analysis of all-composite LHSP.

Strength analysis of a cutting kinked curve plate with clamped free edges of a chisel type deep tiller

Currently existing chisel type deep tillers have a number of disadvantages. The authors propose a research of the strength of a deformer system element of an experimental chisel type deep tiller in the form of kinked plates with clamped-free edges. The present paper analyses the possibility of lengthening of the horizontal elements of the deep tiller shank deformer system without loss of their strength and significant change in their thickness. A mathematical model of a thin elastic homogeneous isotropic plate with clamped-free edges was used. A modified model of the kinked plate is proposed, which has not been investigated earlier. The presented problem is solved numerically. Calculation is performed in the first approximation for the stress-strain state of two isotropic fixed kinked plates with clamped medial edges, symmetrically fixed on a vertical shank, set obliquely to the vertical axis, having equal stiffness and constant cross-section along the entire length (volume objective). It is found that the reduction of the actual plate thickness by more than 1/3 of the design thickness is the most dangerous case. This result leads to failure of a separate fragment of the kinked plate. The proposed thickness of 2 mm does not immediately result in failure of a separate fragment of the part. The plate of the investigated form, made of C255 grade steel, will not provide sufficient strength of the part and its fixing on the deep tiller’s shank. The paper proves the possibility of using kinked plates of the given shape having a variable thickness profile.

Constraint effect on fracture toughness resistance curves of an X60 pipe steel

The fracture toughness resistance curve (e.g. J-R curve and crack tip opening displacement (CTOD) or δ-R curve) is important in facilitating strain-based design and integrity assessment of oil and gas pipelines, where influence of ground movements and presence of planar weld flaws must be accounted for. In this paper, duplicate single edge bend (SE(B)) and clamped single edge tension (SE(T)) specimens with two different initial crack sizes were prepared from an X60 pipe steel. The longitudinally-oriented specimens were notched in the through-thickness direction and experimentally tested at room temperature. In general, the results showed the expected trends of higher resistance curves for both shallow-cracked versus deeply-cracked specimen types and tension versus bending loading modes. Eleven constraint parameters (i.e. QHRR, QSSY, QLM, QBM, A2, A2BM, h, Tz, Cp, Ap, Vp) were calculated based on three-dimensional (3-D) finite element analyses (FEA) for various tested SE(B) and SE(T) specimens.The results and analysis indicated that the shallow-cracked SE(B) specimens exhibited a constraint condition similar to the intrinsically low-constraint deeply-cracked SE(T) specimens. Among all of the constraint parameters, the Ap parameter based on the normalized equivalent plastic strain contour was found to have the most promising trend in terms of correlating the material’s toughness to constraint level for currently tested specimens. Further on-going investigations are underway to extend the evaluation to resistance curves for specimens made from X100, X80 and X70 pipe steels to allow for the characterization of strain hardening effect, as well as the evaluation of their associated weld metals and heat affected zones (HAZs).