In-plane Displacement Fields Research Articles

A model for laminated composite beams based on a novel zig-zag theory

We develop a new model called C-311 for laminated composite beams based on a new zig-zag theory that is firstly proposed in this paper. The in-plane displacement field of the model includes two parts: part one is regarded as the global displacement field which is made of third-order polynomial about z, part two is the local displacement field which is the new zig-zag theory. By satisfying inter-laminar transverse stress continuity and the free surface conditions, the local displacement unknown variables can be eliminated, and the final displacement field should be expressed by the fixed number of unknown displacement variables. For testing the ability of C-311, the classical example about a simply supported laminated beam is investigated under mechanical loading in this paper. In comparison with other models of laminated composite beams, the numerical results show that the present model can give more accurate analysis, and are closer to the exact solutions. Especially the accurate transverse shear stresses are able to be directly obtained from the constitutive equations without any processing, and that is never done for other zig-zag models existing in the literature and benefit for practical engineering applications in the finite element method.

Thermal Stress Analysis of Composite Laminates using Trigonometric Shear Deformation Theory and Finite Element Method

Laminated composite materials offer a versatile design approach for achieving the desired levels of stiffness and strength by selecting specific lamination schemes. The Trigonometric Shear Deformation Theory (TrSDT) effectively addresses the appropriate distribution of transverse shear strains throughout the plate thickness while maintaining stress-free boundary conditions on the plate's top surfaces. Consequently, there is no need for a shear correction factor. In this research paper, we use the Trigonometric Shear Deformation Theory (TrSDT) that takes into account the influence of transverse shear deformation. The in-plane displacement field incorporates a sinusoidal function with respect to the thickness coordinate to accommodate the effects of shear deformation. Theories that involve trigonometric functions based on the thickness coordinate in the displacement fields are collectively referred to as Trigonometric Shear Deformation Theories (TrSDTs). In the present study, we conduct a thermal stress analysis of Laminated Composite Plates using the TrSDT. This theory eliminates the need for shear correction factors and provides a more accurate distribution of interlaminar stresses compared to other methods like CPT and FOST. We assess deflection and stress at various locations and for different aspect ratios under thermal loads using TrSDT. Stress evaluations are carried out analytically, and the results are validated by comparing them with existing findings from the literature. To further verify our findings, we model a composite laminate under thermal loads using the commercial Finite Element Method tool ABAQUS, and our results are validated against those obtained with the TrSDT for plates with simply supported boundary conditions.

Efficient zigzag theory-based spectral element model for guided waves in composite structures containing delaminations

Lamb wave-based structural health monitoring offers excellent potential for real-time delamination detection in laminated structures. It necessitates fast and accurate numerical simulation of guided wave interaction with delaminations. Towards this objective, we present an efficient layerwise zigzag theory (ZIGT) based time-domain spectral element for wave propagation analysis of composite beam- and panel-type structures (strips) containing delamination at arbitrary locations. The ZIGT delivers accuracy like layerwise theories by allowing slope discontinuities in the in-plane displacement field at layer interfaces while maintaining computational efficiency like equivalent single-layer (ESL) theories, making it ideal for such analyses. The high-order elemental nodes at Lobatto points have only four displacement variables, irrespective of the number of layers in the laminate. Delamination is modelled using the region approach, adapting the recently proposed hybrid point-least squares continuity method to satisfy the nonlinear displacement field continuity at the delamination fronts. The model’s efficacy is examined vis-à-vis elasticity-based elements, ESL theory-based elements, and the ZIGT-based standard finite element for free vibration and guided wave propagation responses of composite strips featuring multiple delaminations. The present model shows superior overall efficiency, accuracy, and convergence. An application of the model is also explored for identifying delamination locations from Lamb wave velocity fields.

Accurate Dynamic Electromechanical Solution for Rectangular Piezoelectric Plate Based on Modified FSDT

In this study, the modified first-order shear deformable theory (FSDT) is applied to analyze the resonant characteristics of piezoelectric (PZT) moderately thick plate vibrators with different thicknesses. A simple and straightforward simplification method between mechanical and electrical functions is also proposed based on the modified FSDT. The natural frequencies and associated mode shapes of piezoelectric plates with completely free boundary are analyzed. The displacement fields of the flexural modes are presented based on the superposition method. This analytical method provides a strong-form novel solution for the flexural-dominated motion’s out-of-plane and in-plane coupling displacement fields. The solutions of displacement functions, which are obtained from the superposition method, show excellent convergence for numerical calculation. To verify the validity of the theoretical solution, the resonant frequencies and the corresponding mode shapes are compared with that obtained by the finite element method (FEM) and experimental measurements using the amplitude-fluctuation electronic speckle pattern interferometer (AF-ESPI) technique. According to the electromechanical formula and the derived theoretical solution, four types of electrode are designed to excite the moderately thick PZT specimen for examining the application feasibility of the proposed method. It is shown in this study that the theoretical analysis, numerical calculation, and experimental measurement are fully compared, and excellent agreements of the piezoelectric moderately thick plate are also obtained.

Full-Field Vibration Measurements by Using High-Speed Two-Dimensional Digital Image Correlation

This work developed a method that uses a single monochrome high-speed camera without sacrificing the spatial resolution to measure both in-plane and out-of-plane full-field vibrations. By using the high-speed camera and a two-dimensional digital image correlation (2D-DIC) algorithm, the method first extracts the out-of-plane displacement field from the measured virtual in-plane strains. Then it retrieves the in-plane displacement field after eliminating the out-of-plane motion-induced virtual component. For validation, in-plane and out-of-plane translation tests and single-frequency vibration experiments were carried out. The measurement results show good agreement with the reference values, indicating the effectiveness of the proposed high-speed 2D-DIC (HS-2D-DIC). Further, the natural frequencies and mode shapes of a rectangular cantilever panel were also measured successfully, exhibiting the method’s effectiveness in practical applications. Since the HS-2D-DIC requires only a single monochrome camera, no complex optical setup, and no complicated calibration process, the method can be developed as a competitive tool for full-field vibration characterizations.

Mathematical and Physical Analyses of Middle/Neutral Surfaces Formulations for Static Response of Bi-Directional FG Plates with Movable/Immovable Boundary Conditions

This article is prompted by the existing confusion about correctness of responses of beams and plates produced by middle surface (MS) and neutral surface (NS) formulations. This study mathematically analyzes both formulations in the context of the bending of bi-directional functionally graded (BDFG) plates and discusses where the misconceptions are. The relation between in-plane displacement field variables on NS and on MS are derived. These relations are utilized to define a modified set of boundary conditions (BCs) for immovable simply supported plates that enables either formulation to apply fixation conditions on the refence plane of the other formulation. A four-variable higher order shear deformation theory is adopted to present the displacement fields of BDFG plates. A 2D plane stress constitution is used to govern stress–strain relations. Based on MS and NS, Hamilton’s principles are exploited to derive the equilibrium equations which are described by variable coefficient partial differential equations. The governing equations in terms of stress resultants are discretized by the differential quadrature method (DQM). In addition, analytical expressions that relate rigidity terms and stress resultants associated with the two formulations are proved. Both the theoretical analysis and the numerical results demonstrate that the responses of BDFG plates based on MS and NS formulations are identical in the cases of clamped BCs and movable simply supported BCs. However, the difference in responses of immovable simply supported BCs is expected since each formulation assumes plate fixation at different planes. Further, numerical results show that the responses of immovable simply supported BDFG plates obtained using the NS formulation are identical to those obtained by the MS formulation if the transferred boundary condition (from NS- to MS-planes) are applied. Theoretical and numerical results demonstrate also that both MS and NS formulations are correct even for immovable simply supported BCs if fixation constraints at different planes are treated properly.

Application of UPV-Instrument in Health Monitoring of Indian Rail Section Using AE Technique

In present work, buckling analysis of laminated composite plates is carried out using recently proposed higher-order zigzag theory (HOZT). Third order variation of in-plane displacement field is taken across the thickness of the plate. For predicting the behavior efficiently for thick plates, quadratic variation of transverse displacement field is assumed for core layer and constant for face layers. The present theory satisfies inter-laminar transverse shear stress continuity condition at interface along with zero value at top and bottom surface of the plate. In present study, nine-noded finite element having eleven degrees of freedom per node is used. Present model is free from requirement of any kind of penalty function or C-1 condition.

Buckling Analysis of Laminated Composite Plates under Thermal Conditions

In present work, buckling analysis of laminated composite plates is carried out using recently proposed higher-order zigzag theory (HOZT). Third order variation of in-plane displacement field is taken across the thickness of the plate. For predicting the behavior efficiently for thick plates, quadratic variation of transverse displacement field is assumed for core layer and constant for face layers. The present theory satisfies inter-laminar transverse shear stress continuity condition at interface along with zero value at top and bottom surface of the plate. In present study, nine-noded finite element having eleven degrees of freedom per node is used. Present model is free from requirement of any kind of penalty function or C-1 condition.

Static Response of 2D FG Porous Plates Resting on Elastic Foundation Using Midplane and Neutral Surfaces with Movable Constraints

The current manuscript develops a novel mathematical formulation to portray the static deflection of a bi-directional functionally graded (BDFG) porous plate resting on an elastic foundation. The correctness of the static response produced by middle surface (MS) vs. neutral surface (NS) formulations, and the position of the boundary conditions, are derived in detail. The relation between in-plane displacement field variables on NS and on MS are derived. Bi-directional gradation through the thickness and axial direction are described by the power function; however, the porosity is depicted by cosine function. The displacement field of a plate is controlled by four variables higher order shear deformation theory to satisfy the zero shear at upper and lower surfaces. Elastic foundation is described by the Winkler–Pasternak model. The equilibrium equations are derived by Hamilton’s principles and then solved numerically by being discretized by the differential quadrature method (DQM). The proposed model is confirmed with former published analyses. The numerical parametric studies discuss the effects of porosity type, porosity coefficient, elastic foundations variables, axial and transverse gradation indices, formulation with respect to MS and NS, and position of boundary conditions (BCs) on the static deflection and stresses.

Mode conversion of Lamb waves in a composite phononic crystal plate: Numerical analysis and experimental validation

The mode manipulation of Lamb waves plays an important role in damage detection and identification of damage types, location, and size. In this paper, we propose a composite phononic crystal (PC) plate with antisymmetric and symmetric PCs for realizing mode conversion from A0 to S0 mode of Lamb waves. The theoretical analysis, numerical simulations, and experimental validation are introduced and the mode conversion mechanism of the composite PC plate is systematically investigated. The effect of geometrical parameters on band structures of antisymmetric and symmetric PCs is first discussed. Then multi-physics field simulation models are developed and in-plane displacement fields are obtained in numerical simulations, which shows that the mode conversion is enhanced when the period number of the antisymmetric PC decreases and that of symmetric the PC increases. The composite PC plate specimens are fabricated with precision wire cutting technology for experimental measurements and the self-designed polyvinylidene fluoride (PVDF) comb transducer is used to stimulate the Lamb waves. The experimental results are consistent with the numerical simulations, which demonstrate that the proposed composite PC plate can achieve the mode conversion from A0 to S0 mode of Lamb waves. Our proposed structures have applicable values for the mode manipulation of Lamb waves in damage detection.

A lightweight convolutional neural network as an alternative to DIC to measure in-plane displacement fields

Convolutional Neural Networks (CNNs) are now commonly used in the computer vision community, in particular for optical flow estimation. Some attempts to use such tools to measure displacement and strain fields from pairs of reference/deformed speckle images (like Digital Image Correlation) have been recently reported in the literature. The aim of this work is twofold. The first one is to customize a state-of-the-art CNN dedicated to optical flow estimation to reach better performance when processing speckle images. This is mainly obtained by removing the deepest levels. The second one is to further simplify the CNN by reducing as much as possible the number of filters in the remaining levels while keeping equivalent metrological performance to the original version, in order to accelerate image processing on a power-efficient compact Graphics Processing Unit (GPU).Synthetic images deformed through a suitable displacement field are used to assess the metrological performance of the different versions of the CNN tested in this study. We focus the sub-pixel part of the displacement is considered for this first attempt, this part being much more challenging to determine than integer displacements obtained at the pixel scale. The latter can be found by cross-correlation or with a rough version of DIC. Real images are tested with the simplest version of the CNN and obtained results are compared with those provided by classic subset-based Digital Image Correlation. The two main conclusions are i- that the customization procedure improves the metrological performance of the original version, and that ii- the metrological performance of the ultimate simplified version of the CNN is globally equivalent to the one of the initial version despite the drastic simplification obtained at the end of the procedure. This performance lies between that of DIC used with first- and second-order subset shape functions.

A novel hybrid technique to decompose in-plane thermoelastic displacement fields into thermal and structural displacement fields

A novel hybrid technique to decompose in-plane thermoelastic displacement fields into thermal and structural displacement fields

On the prediction of the stress field in adhesive joints using a combined analytical-numerical method

In this paper, the stress field in adhesively bonded joints is predicted by employing an analytical approach together with the finite element method (FEM). To accomplish this, the adhesive joint is considered as a bi-material notch. Then, a unified analytical formulation for the in-plane stress and displacement field of bi-material notches is presented in the form of asymptotic series. Subsequently, a numerical procedure based on an over-determined system of equations is used to obtain the coefficients of the series solution for various types of adhesive joints including single lap joint (SLJ), single fillet lap joint (SFLJ), butt joint (BJ), scarf joint (SJ), and single lap joint with an inside taper and adhesive fillet (SLJ-ITAF). By comparing the results obtained from the analytical stress field with the numerical results, the accuracy of the calculated coefficients is evaluated. It is shown that by using the proposed approach one can obtain the stress field within the adhesive layer by using the stress or displacement fields of the adherend. The obtained results also show that neglecting the higher-order terms can result in significant errors while considering the first three non-singular stress terms besides the singular term reveals very good results.

A sub-laminate based higher-order model for bending of laminated beams containing multiple delaminations

The paper presents a new sub-laminate based higher-order laminated model (HOLM) for modelling the bending response of laminated composite and sandwich beams with and without delaminations of any configuration. The variation of in-plane and transverse displacement fields through the thickness of a sub-laminate is assumed cubic and quadratic, respectively. Such variation involves specifying displacement parameters at the top and bottom surfaces of sub-laminates that provide significant benefit in conveniently modelling perfect and delaminated multilayered beams with any desired level of accuracy. The option exists to therefore trade-off between prediction and computational efficiency by adapting a suitable sub-laminate modelling scheme. A C0 continuous beam element with quadratic in-plane variation of displacements is developed for implementation of the proposed laminate model. The versatility and accuracy of the model is demonstrated by via a number of numerical examples of composite and sandwich beams, covering a range of variables such as boundary conditions, loading, and delamination configuration. A number of the newly developed results reported in this paper will serve as baseline data for delaminated structures.

Image Spherizing-Based Three-Dimensional Displacement Measurement Technique for Microscope with Single Camera

The detection of three-dimensional displacement, caused by micro-nano scale deformation, is of great significance for the industrial production of glass or quartz optical fiber products. Considering the detection cost and practicability, an image spherizing-based three-dimensional displacement measurement method for a monocular microscopic imaging system (a microscope with a normal industrial camera) is proposed. Differential theory tells us that an arbitrary large-scale deformation can be approximatively considered as the sum of slope-shaped deformations. In this paper, an ordinary slope model is presented by the idea of image spherizing and differential theory, which reveals the mathematical relation between in-plane displacement and out-of-plane displacement in the deformed images captured by an optical microscope. After obtaining the in-plane displacement field, the out-of-plane displacement field can be extracted from the in-plane displacement field by the ordinary slope model, which indicates that the out-of-plane displacement information carried by the in-plane displacement is fully utilized. Simulation and application experiments are performed to demonstrate the effectiveness and all the absolute errors of the measurement by optical microscope with a magnification of 50× are less than 0.2 μm. Unlike conventional methods, the new method does not need a phase unwrapping process during detection and has many attractive characteristics such as low-cost, simple arrangements and good applicability. Because the measurement precision of this technique can meet the detection requirement, it has been applied to the industrial detection of glass or quartz optical fiber products.

Long-range in-plane elastic displacement fields of double vacancies in graphene

An analytical model based on the force-dipole approximation for an elastic 2D continuum is examined for a description of long-range in-plane elastic displacement fields of point defects in graphene. Through atomistic numerical calculations with a particular interatomic potential (PPBE-G) the model is adjusted to a particular point defect (double vacancy). Both atomistic and continuum calculations are carried out with periodic boundary conditions. The displacement field of a single divacancy in an infinite graphene sheet is then obtained and is shown to be highly anisotropic. It is established that the elasticity theory approximates the atomistic simulations with the accuracy better than 0.3% at more than 10 interatomic distances from the defect. • The divacancy displacement field can be described by the continuum elasticity theory. • A simulation box of a few thousand atoms ensures a numerically stable result. • The divacancy displacement field in graphene is highly anisotropic.

An enhanced higher order zigzag theory for laminated composite plates under mechanical/thermal loading

In this paper, an enhanced higher order zigzag theory is developed for laminated composite plates. By satisfying inter-laminar stress continuity conditions and free surfaces conditions, the final displacement field of the present theory can be obtained from Cho’s the initial displacement field [27]. The displacement variables in the present theory are independent on the number of layer. In contrast to Cho’s higher order zigzag theory [27], only one transverse displacement variable is introduced into in-plane displacement field in the proposed theory. To assess the accuracy of this model, simply-supported laminated composite plates subjected to mechanical or thermal loading are analyzed respectively by using the Navier method. Numerical results show that the present model can accurately predict behaviors of laminated composite plates and are more suitable for multilayered plates in comparison with Cho’s higher order zigzag theory.

A high accuracy four variable displacement model for thick-faced sandwich beams

This paper presents a new four variable displacement model for thick-faced sandwich beams based on a layerwise formulation of transverse shear stress field. To account for structural characteristics of thick-faced sandwich beams, the transverse shear stress field is assumed to follow quadratic and uniform distributions in face layers and soft core respectively. A novel comprehensive approach is first proposed to determine the relation between stress variables of quadratic terms in shear stress expressions of the top and bottom layers, and thus the number of independent variables in shear stress expressions is reduced from three to two. According to the simplified stress expressions, the in-plane displacement field is determined, and the final displacement expression with four unknowns is further achieved by imposing interlaminar continuity conditions. The present model can accurately describe zigzag effects and provide ideal transverse shear stress results for sandwich beams. Then, finite element solutions for static and dynamic analyses of sandwich beams are presented. Moreover, complex eigenvalue problems for free vibration analysis of viscoelastic sandwich beams are studied, and an iterative procedure is presented for determining modal results of sandwich beams with frequency-dependent viscoelastic core. Lastly, the accuracy of the present model for static and dynamic analyses of sandwich beams is well validated by numerical cases.

Static Bending of Isotropic Circular Cylindrical Shells Based on the Higher Order Shear Deformation Theory of Reddy and Liu

In this paper, a displacement based shear deformation theory formulated on the cubic in-plane displacement field equation of Reddy and Liu is presented for the static bending analysis of isotropic circular cylindrical shells. The adopted displacement field accounts for a quadratic (parabolic) distribution of the transverse shear through the shell thickness as well as satisfies the need for a stress free upper and lower boundary surfaces of the shell. The equations of static equilibrium are obtained on application of the principle of virtual work. Numerical results of the bending analysis for the displacements and stresses are presented for the simply supported shell. A comparison made to those of the Kirchhoff-Love theory for varying shell length to mean – radius of curvature ratios, shows good agreement for thin shells irrespective of the shell length to radius of curvature ratio (<i>l</i> / <i>a</i>). The transverse sharing effect is found to be noticeable in the deformation of thick shells, however, this effect diminishes with a continuous increase in <i>l</i> / <i>a</i> ratios.

Measurement in-plane deformations in electronic speckle pattern interferometry using phase-shifting modulated by polarization

We propose to combine electronic speckle pattern interferometry with polarizing phase-shifting techniques to measure in-plane displacement fields. We present the technical and theoretical procedures for its implementation, added with the necessary calibration procedures employing a commercial polarimeter. Finally, we tested our proposal experimentally using a two-phase step algorithm to measure in-plane displacements on a latex sample.