Elastic Plate Theory Research Articles

Theory of thin elastic plate: history and current state of the problem

The article is an analytical review and is devoted to the theory of thin, isotropic elastic plates. There are basic relations of the theory based on the kinematic hypothesis confirming that the tangential displacements are distributed linearly along the thickness of the plate and its deflection does not depend on the normal coordinate. As a result, a system of equations of the sixth order with respect to two potential functions – the penetrating potential, which determines the plate deflection, and the boundary potential which makes it possible to set three boundary conditions on the plate edge and eliminate the known contradiction of Kirchhoff's plate theory was obtained. Problems that have no correct solution in the framework of Kirchhoff's theory – cylindrical bending of a plate with a free edge, bending of a rectangular plate with non-classical hinge fixing, torsion of a square plate by moments distributed along the contour, and bending of a plate by a rigid die – are considered. In conclusion, a brief historical review of the papers devoted to the theory of plate bending was presented.

CALCULATION OF FLAT SHELLS BY THE BOUNDARY ELEMENT METHOD

This article describes the iterative process of solving the problem of the stress-strain state of a long flexible cylindrical panel. Stress-strain state and the study of the stability of flat shells taking into account the physical, mechanical properties of the material, the process of deformation change. It is based on sequential approximation methods and the boundary element method (MGE). The results of algorithmic weighting are presented in the form of a table showing what value each band represents. The equations are determined by solving twelve unknown values of functions at the ends of the segment. Linear problems of elasticity theory and plate theory fundamental solutions have a simple form, so the method is widely used here. For flat shells, the matrix of fundamental solutions is determined by complex volumetric expressions, and for flat shells - by special functions. Therefore, there is little research on solving problems in the theory of flat shells by the boundary element method. In this regard, an urgent topic of research is the development of methods of boundary integral equations for solving linear and nonlinear problems in the theory of flat shells based on the application of fundamental solutions defined by simple analytical expressions. The scientific novelty of the work consists in the development of a methodology for assessing the reliability of thin-walled spatial structures using the boundary element method.

Surface Subsidence Modelling Induced by Formation of Cavities in Underground Coal Gasification

Underground coal gasification (UCG) is an efficient method for the conversion of deep coal resources into energy. The scope of this work is to model the subsidence of four gasification cavities with a size of 30 m × 30 m × 15 m, separated by 15 m wide pillars. Two scenarios of gasification sequence are modelled, one with the gasification of cavities 1 and 2 followed by 3 and 4, and the other one with the sequence of cavities 1 and 3, followed by 2 and 4. The results show that the final surface subsidence after gasification of four cavities is 9.8 mm and the gasification sequence has an impact only on the subsidence at the intermediate stage but has no impact on the final subsidence after all four cavities are formed, when only the elasticity regime is considered. Additionally, the maximum surface subsidence for the studied cavities of different sizes ranges from 0.016 mm to 7.14 mm, and the relationship between the subsidence and the cavity volume is approximately linear. Finally, a prediction model of surface subsidence deformation is built up using the elastic plate theory, and the formula of surface deformation at a random point is given. The maximum difference between measured and calculated deformation is 4.6%, demonstrating that the proposed method can be used to predict the ground subsidence induced by UCG.

Study on Calculation Method of Bending Performance of Concrete Sandwich Composite Slab.

In order to explore the flexural behavior of a concrete sandwich panel under concentrated boundary conditions, based on Kirachhoff's elastic thin plate theory in this paper, the geometric deformation, physical conditions, and equilibrium relationship of a sandwich panel are deduced by constructing the layered analysis model of the sandwich panel, the basic differential equation of the flexural deformation of the concrete sandwich thin plate is obtained, and the mathematical expression of the internal force and displacement under the boundary condition of concentrated support is given. Combined with an engineering example, the proposed calculation method is verified. The results show that, in the arrangement of reliable connectors for concrete sandwich panels, the concrete wythes bear the load while the contribution of the core layer to the bending capacity of the structure can be ignored. When subjected to a laterally distributed load, the sandwich panel mainly experiences out-of-plane bending deformation, and the bending normal stress in the concrete panel layer shows a linear non-uniform distribution along the thickness direction of the panel. The bending deformation performance and bearing efficiency of a concrete sandwich slab with the change in concentrated support position have significant effects, and the load transfer efficiency can be improved by optimizing the arrangement of supports. Except for small local areas near the supports, the bending stress distribution and deformation behavior of the concrete sandwich panel can be accurately analyzed by the calculation method established in this paper.

A novel methodology for assessing the potential for formation bridging in coal seam gas fields: A case study from southern Queensland

The substantial growth in the development of coal seam gas (CSG) – also known as coal bed methane – fields in southern Queensland, Australia, over the past decade has been associated with various impacts on the environment and human activities. Of particular concern is the potential impact of CSG-induced subsidence on farming operations. Previous research in underground coal mining has shown that overlying competent formations can function as bridges over mined-out areas, effectively preventing subsurface deformation from fully or partially reaching the ground surface. Formation bridging can also occur in CSG producing areas. Modelling formation bridging is essential for understanding and predicting the historical and future impact of CSG production on the surface, respectively. This paper introduces a novel modelling methodology based on the elastic plate theory that helps in assessing the potential for formation bridging in overlying formations during CSG production. The applicability of this approach is demonstrated through a case study in the Surat Basin of southern Queensland. This study shows that excluding bridging for a particular formation rules it out for that formation when overlain by other formations or when it is thinner. Results of this study indicate that formation bridging in the study area may only occur during the very early phase of a CSG well's life cycle. Where the diffusivity of the coal seam is low, leading to a slower expansion of the influence area, there is also an increased likelihood of bridging taking place and persisting for an extended duration. Although the methodology introduced in this paper is mainly analytical, it offers an initial evaluation of the formation bridging probability across extensive basins where conducting detailed three-dimensional numerical modelling is challenging, if not impractical. This preliminary analysis may assist in deciding whether, and where, more comprehensive and detailed assessment of formation bridging is necessary.

Numerical investigation of non-local elasticity theory for free vibration of polymer embedded multi-layer graphene sheets

The present study investigates the free vibration behavior of multi-layered graphene sheets with various shapes embedded in a polymer matrix under different boundary conditions based on Eringen non local elasticity. The governing equations are obtained through the minimization of potential energy or Newton’s law which is not only equivalent to Hamilton’s principle and mutually can be derived from each other. Differential Quadrature and pb2 Rayleigh Ritz procedures are applied to Multi-layered Graphene sheets employing Eringen’s elasticity and classical plate theory. Results indicate that the resonant frequencies of the sheets are overestimated up to 60% using the classical elastic model. Depending on the shapes of Graphene sheets, small-scale parameters, boundary conditions, stiffness of polymer matrix, the number of layers and van der Waal’s interaction on the vibrational characteristics of graphene sheets are studied, and the results are tabulated. Conclusions are drawn and recommendations are provided for future work.

Roof movement and instability fracture characteristics in shallow-buried thin coal seam conventional mining faces

The variation in the width of the mining face significantly affects the stability of the face, leading to potential roof fracturing and collapse. Additionally, strong mining pressure can manifest, severely impeding the safe production of coal mines. This study uses the No. 16705 conventional working face of Jinda Coal Mine as its engineering background to investigate the characteristics of roof strata movement and instability under conditions of variable-width mining in shallow-buried thin coal seams. First, the dynamic load of the roof strata is estimated based on the key strata theory. Next, a mechanical model of the immediate roof strata movement in the working face is established based on the theory of elastic thin plates, which has been used to reveal the impact of different dimensions of the overhanging plate structure and residual overhanging structures in the corner on roof movement and its associated fracture mechanics. The findings indicated that the maximum bending deformation, deformation moment, and bending stress all have an exponential function relationship with the roof width. Similarly, these metrics have an exponential function relationship with the overhanging span of the roof. In addition, these parameters all have a linear functional relationship with the size of the residual overhanging structures in the corner. Finally, the effect of roof instability on overlying pressure is analyzed, and both the initial fracture step length and cyclic movement fracture step length of the roof are estimated. These insights offer valuable scientific guidance and a theoretical foundation for analyzing the adaptability of load-bearing pillars pressure in thin coal seam mining faces, bearing significant relevance to safety production.

Vibration and Small Scale Effects of Skew Graphene Sheets Using Nonlocal Elasticity Theory

Nonlocal elasticity theory is a popularly growing technique for the realistic analysis of nano structures. In the present work nonlocal elasticity plate theory has been employed and vibration analyses of skew graphene sheets are carried out. Relevant governing differential equations are reformulated using the nonlocal differential constitutive relations suggested by Eringen. The equations of motion including the nonlocal theory are derived. All edges of the skew graphene sheets are assumed to be simply supported. Naviers approach has been employed to solve the governing differential equations. Bauers skew plate analysis has been extended to include the nonlocal elasticity plate theory. Vibration response of the skew graphene sheets is studied. Effects of the (i) size of the graphene sheets (ii) modes of vibration (iii) nonlocal parameter and (iv) skew angle of graphene sheet on nonlocal vibration frequencies are investigated. It has been observed that the vibration response of the skew graphene sheets are influenced significantly by the nonlocal parameter.

Study on the Periodic Collapse of Suspended Sandstone Interlayer under Coupled Thermo–Hydro–Mechanical Environment

In the process of in situ heat injection mining of coal seams, a circular combustion chamber will gradually be formed. However, for the interlayer coal seam, the interlayer will gradually suspend in the combustion chamber. When the critical distance is reached, the initial collapse and periodic collapse will occur, which will affect the flow field distribution and extraction efficiency of combustion gas. In order to study the initial and periodic collapse steps of interlayer in real-time high temperature and high-pressure thermo–hydro–mechanical (THM) coupling environments, sandstone specimens were collected from the field, and the macroscopic mechanical parameters of sandstone interlayer were tested using a self-developed thermo–hydro–mechanical coupling tester. The mechanical parameters (e.g., triaxial compressive strength, elastic modulus, Poisson’s ratio) of sandstone under different stress conditions were obtained. Then, based on the elastic thin plate theory, the mechanical modeling of the annular sandstone interlayer is carried out and the preceding mechanical parameters are brought into the generated analysis equation. Thus, the periodic collapse step of the suspended circular sandstone interlayer is obtained. The study results show that: (1) the macroscopic physicomechanical parameters of the sandstone interlayer do not vary monotonically with temperature—a slight increase in macroscopic parameters occurs at 400°C; and (2) the periodic caving pace of the sandstone interlayer decreases with the increase in the number of collapses. With the increase of temperature, the periodic caving pace of the sandstone interlayer decreases first, then increases, and then decreases. It gradually decreases with the increase of surrounding pressure. The research results of this paper can provide theoretical guidance and technical support for in situ heat injection mining technology.

The exact solution of annular sandwich plate undercoated with radial-FGM subjected to radial pressure or transverse load

This study aims to derive the exact stretching and bending solutions of the radially FGM annular sandwich plate subjected to either radial pressure or transverse load. The radially FGM annular sandwich plate consists of two homogeneous rings and an FGM-undercoated layer with uniform Poisson’s ratio and radially graded Young’s modulus based on the power function. Although the radially FGM annular sandwich plate is nonhomogeneous, the stretching and bending problems can be analyzed separately based on the elastic plate theory. Then the exact solution corresponding to the power-law graded FGM can be evaluated by solving the simultaneous equations of the boundary and continuous conditions. The theoretical results are validated through the comparison with the finite element solutions. When degenerated, our results coincide with those of the homogeneous annular plate in the literature, making the obtained solutions reliable. Using the radial-FGM as the undercoated layer makes the strains and stresses smoothly continuous at the interfaces of the layers. The effects of the material index on mechanical behavior are under investigation.

Swelling under Constraints: Exploiting 3D‐Printing to Optimize the Performance of Gel‐Based Devices

AbstractStimuli‐responsive hydrogels that swell under constraints such as spatial geometric confinement are employed in many applications, including biomedical devices and actuators, to perform mechanical work. Due to the heterogeneous deformations that arise from these constraints, the computation of the swelling‐induced stress poses numerical difficulties. This work proposes a simple experimental method based on 3D‐printing technologies to characterize this stress. The swelling is investigated under two types of geometric confinements—transversely constraining and elastically constraining boxes. In the former, the box enforces uniaxial swelling of the gel. The results show that the longitudinal deformations decrease as the transverse stretches increase. The elastically constraining box comprises soft walls with various stiffnesses and sizes that deform in response to the swelling‐induced stress exerted by the gel. By employing elastic plate theory, a method to determine this stress is developed. The results reveal that: 1) the maximum volumetric deformation is achieved by free swelling and 2) the stress gels exert depends on the wall stiffness and non‐linearly decreases as the gel nears its freely swollen state. The insights from this work can be used to optimize the performance of swelling‐based systems and characterize the stress generated due to other stimuli such as pH and temperature.

The Moment-Membrane Theory of Elastic Flexible Plates as a Continual Geometrically Nonlinear Theory of a Graphene Sheet

The Moment-Membrane Theory of Elastic Flexible Plates as a Continual Geometrically Nonlinear Theory of a Graphene Sheet

Interaction of a Flexural-Gravity Wave with a Vertical Rigid Plate Built in a Floating Elastic Plate

The linear two-dimensional problem of interaction between an hydroelastic wave propagating along an elastic floating ice plate with built-in vertical rigid plate is studied. The fluid under the ice is inviscid and incompressible. The fluid depth is finite. The deflection of the ice plate is described by the linear theory of thin elastic plates. The flow under the ice is potential. The total velocity potential is decomposed into the potential of the incident wave, even potential caused by the vertical motion of the rigid plate, and an odd potential caused by the rotation of the rigid plate. The vertical mode method is used. The third potential is obtained by solving a mixed boundary-value problem numerically using Chebyshev polynomials. The solution is validated by analysis of its convergence. The first and second potentials, and the corresponding deflections and strains of the ice plate, are obtained analytically. The motions of the rigid plate, as well as deflection and strains in the floating plate, are numerically analyzed. It is shown that the rotation of the rigid plate due to the incident wave is the main factor of increasing strains in the ice plate.

Kaluza–Klein dimensional reduction from elasticity theory of crumpled paper

During the last century, two theories using the concept of dimensional reduction have been developed independently. The first, known as Föppl–von Kármán theory, uses Riemannian geometry and continuum mechanics to study the shaping of thin elastic structures which could become as complex as crumpled paper. The second one, known as Kaluza–Klein theory, uses Minkowskian geometry and general relativity to unify fundamental interactions and gravity under the same formalism. Here we draw a parallel between these two theories in an attempt to use concepts from elasticity theory of plates to recover the Einstein–Maxwell equations. We argue that Kaluza–Klein theory belongs to the same conceptual group of theories as three-dimensional elasticity, which upon dimensional reduction leads to the Föppl–von Kármán theory of two-dimensional elastic plates. We exploit this analogy to develop an alternative Kaluza–Klein formalism in the framework of elasticity theory in which the gravitational and electromagnetic fields are, respectively, associated with stretching-like and bending-like deformations. We show that our approach of dimensional reduction allows us to retrieve the Lagrangian densities of both gravitational, electromagnetic and Dirac spinors fields as well as the Lagrangian densities of mass and charge sources.

MOMENT-MEMBRANE THEORY OF ELASTIC FLEXIBLE PLATES AS A CONTINUAL GEOMETRICALLY NONLINEAR THEORY OF A GRAPHENE SHEET

In the present work, under the assumption of smallness of deformations, bending-torsional characteristics and angles of rotation (including the angles of free rotation) of the elements of the plate, based on the three-dimensional geometrically-nonlinear moment theory of elasticity, preserving only those nonlinear terms, that come from normal displacement (deflection) and its derivatives, a geometrically nonlinear moment-membrane theory of elastic plates is constructed as a continual theory of deformations of a flexible graphene. For the indicated nonlinear theory of elastic plates, by introducing stress functions, the resolving equations are presented also in a mixed form: these are the system of equilibrium equations for transverse-bending deformation, compiled in the deformed state of the plate, and deformations continuity equations, expressed in stress functions and deflection functions. For the geometrically nonlinear moment-membrane theory of elastic plates Lagrange-type variational principle is established.

Towards predicting shear-banding instabilities in lipid monolayers

Langmuir monolayers are advantageous systems used to investigate how lipid membranes get involved in the physiology of many living structures, such as collapse phenomena in alveolar structures. Much work focuses on characterizing the pressure-bearing capacity of Langmuir films, expressed in the form of isotherm curves. These show that monolayers experience different phases during compression with an according evolution of their mechanical response, incurring into instability events when a critical stress threshold is overcome. Although well-known state equations, which establish an inverse relationship between surface pressure and area change, are able to properly describe monolayer behaviour during liquid expanded phase, the modelling of their nonlinear behaviour in the subsequent condensed region is still an open issue. In this regard, most efforts are addressed to explain out-of-plane collapse by modelling buckling and wrinkling mainly resorting to linearly elastic plate theory. However, some experiments on Langmuir monolayers also show in-plane instability phenomena leading to the formation of the so-called shear bands and, to date, no theoretical description of the onset of shear banding bifurcation in monolayers has been yet provided. For this reason, by adopting a macroscopic description, we here study material stability of the lipid monolayers and exploit an incremental approach to find the conditions that kindle shear bands. In particular, by starting from the widely assumed hypothesis that monolayers behave elastically in the solid-like region, in this work a hyperfoam hyperelastic potential is introduced as a new constitutive strategy to trace back the nonlinear response of monolayer response during densification. In this way, the obtained mechanical properties together with the adopted strain energy are successfully employed to reproduce the onset of shear banding exhibited by some lipid systems under different chemical and thermal conditions.

Stress Analysis of Transversely Loaded Isotropic Three-Dimensional Plates Using a Polynomial Shear Deformation Theory

An exact solution for the bending attributes of a thick rectangular plate under transverse loading is modeled herein using three-dimensional (3-D) elasticity plate theory and fourth-order polynomial shear deformation function. Precluding coefficients of shear correction, this model captured the effect of shear deformation along with the transverse normal strain stress. The expression for total potential energy was derived from a 3-D kinematic and constitutive relation the equilibrium equation was developed and employed from the energy functional transformation to get the relationship between slope and deflection. Exact polynomial functions were obtained from the outcome of the equilibrium equation and with the aid of the direct variation approach, the coefficient of deflection of the plate was generated from the governing equation. The expression for computing the displacement, bending moments, and stress components along the three axes of the plate was established from these solutions for the assessment of the bending characteristics of a rectangular plate. The result of a simply supported at one edge, free at one edge and clamped at the two outer edges (SCFC) was evaluated using the obtained functions in this study. The report of this study confirms the exactness and consistency of the 3-D model unlike the refined plate theories applied by previous authors in the available literature. The value of 8.05% is the comprehensive average percentage variation of the values for center deflection obtained by Onyeka and Okeke (2020) and Gwarah (2019). It is established that at the 92 % confidence level, this model is worthy of adoption for safe, cost-effective and accurate bending analysis of thick plates of any support condition.

Study on Initial Fracture Characteristics of the Main Roof in Fully Mechanized Caving Mining of Inclined Coalbed

In view of the occurrence conditions of inclined coalbed, the deformation and failure characteristics of the main roof will affect the safe production of the working face. Therefore, the study of the deformation and failure characteristics of the main roof in the inclined coalbed has guiding significance for the control of surrounding rock. This paper takes the II1042 working face of Taoyuan Coal Mine as the research background, adopts the methods of theoretical analysis, numerical calculation, and field practice to analyze the evolutionary characteristics of the initial failure of the main roof of the working face under the background of the inclined coalbed, and explores the mechanical behavior characteristics of the working face roof during the mining of inclined coalbed. Based on the elastic thin plate theory, a mechanical model of the overlying rock roof of a large-angle coal seam is established, and the mechanical characteristics of the surrounding rock under the initial failure of the main roof under the unbalanced load are studied. The stress distribution characteristics of the lower surface are summarized, and the evolution law of the initial fracture of the main roof is summarized. According to the actual geological conditions of the II1042 working face of Taoyuan Coal Mine, the failure characteristics of the main roof and the initial breaking step distance are obtained by analysis, and the analysis results are verified by monitoring the mine pressure of each part of the target working face on site. The research results show that: ① Under the unbalanced load of the inclined coalbed, the deflection surface of the main roof of the coal seam is asymmetrical with respect to the arrangement direction of the working face, and the maximum deflection point is located at the upper middle position of the working face, namely (a/2, 1.836 b/π), and the main roof of the working face breaks for the first time when it advances to 35 m. ② With the advancement of the working face, the two long sides of the roof break first. With the deflection and deformation of the roof, the tensile stress in the middle of the main roof reaches the tensile strength of the rock and breaks, and then the two short sides of the roof break under the action of the breaking and turning of the rock, and the upper short side will break before the lower one. ③ According to the monitoring and analysis of the rock pressure at each part of the working face, it is judged that the initial pressure step distance is between 28.2 m and 34.6 m, which is consistent with the theoretical analysis results.

Failure mode analysis and validation of hierarchical sandwich structures under compression

Based on the elastic thin plate theory, the mechanical properties and failure pattern are studied, considering the local failure defects, and the nominal positive stress expression of the hierarchical sandwich structures with a second-order core were presented. Moreover, the dominant relationship of each failure mode was compared, and the structure failure mechanism diagram is constructed by MATLAB, then the influence law of different local failure defects on the structure-dominant failure mode was discussed. However, for the failure mode of the structure, the emergence of the dominant failure mode of the secondary transmission path component may not be used as the failure criterion of the structure. Only when the failure mode of the structure appears on the main transmission path components, the failure mode can be used as the failure criterion of the hierarchical sandwich structure with a second-order core. In the failure mechanism diagram of the hierarchical sandwich structure with a second-order core, the single dominant failure mode of the structure can be accurately predicted, and for the junction of multiple mixed dominant failure modes, the structure is more prone to structural/component stability failure. At the same time, in order to test the correctness of the theoretical prediction, the verification experiment models were made with photosensitive resin 9000 material and 3D printing technology. Through the uniaxial compression test on the experimental model and numerical simulation, the correctness of the theoretical prediction of structural failure mode and ultimate bearing capacity is verified. The results show that the failure mode of the structure can be accurately predicted by the theoretical analysis, and the relative error is about 20% when calculating the bearing capacity. The experimental and simulation results agree well with the theoretical prediction.

Intrinsic formulation of the Kirchhoff–Love theory of nonlinearly elastic plates

The classical Kirchhoff–Love theory of a nonlinearly elastic plate provides a way to compute the deformation of such a plate subjected to given applied forces and boundary conditions by solving the Euler–Lagrange equation associated with a specific minimization problem, whose unknown is the displacement field of the middle surface of the plate. We show that this Euler–Lagrange equation is equivalent to a boundary value problem whose sole unknowns are the bending moments and stress resultants inside the middle surface of the plate, without any reference to the displacement field in its formulation. As a result, computing the stresses inside deformed plate can be done without using the derivatives of the corresponding displacement field, usually a source of numerical instabilities.