Bending Of Thin Plates Research Articles

МАТЕМАТИЧНЕ МОДЕЛЮВАННЯ В ЗАДАЧАХ ГЕОМЕТРИЧНО НЕЛІНІЙНОЇ ТЕОРІЇ ПРУЖНОСТІ

Розв’язки багатьох важливих для практики задач, що виникають в сучасній техніці, не завжди можуть бути отримані традиційними методами теорії аналітичних функцій або за допомогою інтегральних перетворень. Це відноситься, наприклад, до контактних задач, в яких враховуються скінченні розміри області хоча б в одному напрямку, або досліджуються середовища з криволінійною анізотропією тощо. Засоби математичної теорії пружності виявляються не надто ефективними для дослідження таких задач. У цьому випадку доцільно використовувати досягнення теорії потенціалу. Застосування ж асимптотичних методів при цьому, навіть в складних випадках, дозволяє отримувати обґрунтовані наближені рівняння, уточнювати якісні закономірності і отримувати аналітичні розв’язки задач. У даній роботі представлене узагальнення методу збурень, яке дозволяє звести дослідження складних задач геометрично нелінійної теорії пружності (в плоскій та просторовій постановці) до послідовного розв’язання більш простих крайових задач теорії потенціалу. Геометрично нелінійна теорія пружності містить в собі деякі особливості, завдяки яким вона відрізняється від класичної (лінійної) теорії. Головна відмінність полягає в урахуванні різниці між геометрією недеформованого та деформованого станів досліджуваного тіла, коли мають місце переміщення, які викликають значні зміни геометрії тіла. При цьому рівняння рівноваги необхідно складати з урахуванням зміни форми і розмірів конструкцій. Врахування кінцевих деформацій, які при створенні математичних моделей веде до значних труднощів при розв’язуванні задач, але в той же час наближає модель до реальної проблеми. Метод збурень, що використовується для розв’язання нелінійних рівнянь у частинних похідних, має теоретичне і практичне значення. Він універсальний і може використовуватися для аналізу різних завдань математичної фізики. Розроблений підхід може бути застосований для вирішення завдань, в яких істотну роль грають залишкові деформації. Наприклад, згин тонких пластин і оболонок. У розглянутій модельній задачі вдалося виділити вплив геометричної нелінійності на напружено-деформований стан досліджуваного тіла. Саме тому результати представленої роботи мають як теоретичне, так і прикладне значення, а дослідження є актуальним.

Релаксация поляризации при статическом прямом флексоэлектрическом эффекте в монокристалле SrTiO-=SUB=-3-=/SUB=-

The results of a study of the dielectric polarization induced by spherical bending of thin single-crystal SrTiO3 plates are presented, and the flexoelectric coefficients are estimated. The processes of relaxation of inhomogeneous deformation and induced polarization are investigated.

A Plate Bending Kirchhoff Element Based on Assumed Strain Functions

To investigate static and free vibration for thin plate bending structures, a four-node quadrilateral finite element is proposed in this research paper. This element has been formulated by using both the assumptions of thin plates theory (Kirchhoff plate theory) and strain approach. The suggested element which possesses only three degrees of freedom (one transverse displacement and two normal rotations) at each of four corner nodes is based on assumed higher-order functions for the various components of strain field that satisfies the compatibility equation. The displacement functions of the developed element are obtained by integrating the assumed strains functions and satisfy the exact representation of the rigid body modes. Several numerical tests in both static and free vibration analysis are presented to assess the performance of the new element. The obtained results show high solution accuracy, especially for coarse meshes, of the developed element compared with analytical and other numerical solutions available in the literature.

On the Boundary Value Problems of Bending of Thin Elastic Plates With Surface Effects

Abstract We modify classical thin plate theory by incorporating surface effects via the Gurtin–Murdoch surface model to accommodate the mechanical behavior of thin plates at the nanoscale. We formulate the corresponding Dirichlet and Neumann boundary value problems and establish uniqueness results in the appropriate function spaces. In addition, we obtain the fundamental solution of the governing system of equations, which is central to further studies concerning well-posedness analysis of the model by the boundary integral equation method. Finally, we validate our model by comparison with results in the existing literature.

Symplectic Solutions in Stress Analysis of Thin Plates

Aiming at the common elastic thin plate materials in engineering, the dual control differential equation and the complete solution space composed of zero eigensolution and non-zero eigensolution are established in the state space. On this basis, the non-homogeneous condition of the boundary and the solution method of the non-homogeneous equation are studied systematically. The stress distribution of thin plate bending problem is discussed with an example, and the stress concentration of material under special constraints is analyzed.

Is it always worthwhile to resolve the governing equations of plate theories for graded porosity along the thickness?

Functionally graded porous (FGP) plates have been introduced as modern structural members which open a new window to optimal and functional designs. Despite the need to study the effect of graded porosity on the mechanical behavior of FGP plates, it is necessary to consider the very extensive and valuable literature in plate field, presenting remarkable closed-form solutions. Hence, this paper aims to answer where is possible to implement the available exact solutions for the analysis of FGP plates. The special distinction of FGP plates, graded porosity, is reflected in their stiffnesses and moments of inertia coefficients. Here 12 different functionalities of porosity distribution along the thickness are considered and a set of explicit formulations for evaluating these coefficients are presented to be substituted in already provided analytical solutions. Many examples including bending and free vibration of thin and thick FGP plates are exhibited and the influence of the type of porosity distribution is discussed in details. This work can be considered as a guideline for designers to evaluate the effect of graded porosity based on the cornerstone of the huge number of solutions in the precious literature of plate theories.

A hierarchical wavelet method for nonlinear bending of materially and geometrically anisotropic thin plate

The paper aims at the application of a hierarchical wavelet homotopy methodology for solving nonlinear bending of anisotropic thin plates. The newly mechanical models of materially anisotropic plate and geometrically orthotropic combined plate at ship double bottom in large deflection have been formulated. The dimensionless coupled and nonlinear governing equations have been derived and decomposed into linear differential terms, while the solutions are expanded in wavelet series as a requirement for implementing the Homotopy recursive system of algebra equations by Galerkin method. Highly accurate Coiflet-type solutions have been obtained with good computational efficiency in excellent agreement with other numerical results in published literatures. Materially and geometrically orthotropic analysis have been conducted with different ratios of elasticity modulus or inertia moment in orthogonal directions, which indicate both orthotropy play an important role in largely deformed anisotropic plate under extreme loads. Convergent solutions over wider bending range of isotropic or anisotropic plates subjected to different boundary conditions are given which reveals our proposed wavelet methodology performs good superiority and versatility dealing with strongly nonlinear problems.

Glacial pumping of a magma-charged lithosphere: A model for glaciovolcanic causality in magmatic arcs

Glacial loading and unloading can influence the rates and timing of magmatism in tectonic settings where the lithosphere is relatively thin and hot. In continental arcs, where the lithosphere is substantially thicker, nominal loading by surface glaciers is unlikely to affect the production of subduction-related magmas. However, geologic evidence suggests that a causal linkage between volcanism and glaciations in arcs does exist. The last deglaciation of the Garibaldi volcanic belt in southwestern British Columbia, Canada saw a ∼5-fold volumetric increase in volcanism over the mean Pleistocene (background) rate that coincided with ∼400 m of rapid postglacial crustal uplift. Here, we use a cylindrical, thin plate bending approximation to calculate the horizontally oriented stresses developed in the crust during glacial loading/unloading (up to ±12 MPa). The distribution of compressive and extensional stresses is shown to modulate magma transport and storage by facilitating or suppressing magma ascent by dike propagation. We use Monte Carlo simulations to track the depth of upward migrating magma bodies in the crust over a single glacial cycle (i.e., during crustal depression and rebound). During loading, volcanism is suppressed as magma becomes trapped by compressive deviatoric stress in the upper part of the crust. During unloading, relaxation of this stress acts to release accumulated magma from the upper crust and causes an increase in eruption rate immediately following deglaciation. This model of “glacial pumping” of a magma-charged lithosphere suggests a causal linkage between volcanism and glaciation in volcanic arcs that explains the persistent observation that the volume and timing of volcanism is correlated with the major climate cycles.

Bending of nonconforming thin plates based on the first-order manifold method

Bending of nonconforming thin plates based on the first-order manifold method

Bending Analysis of Circular Thin Plates Resting on Elastic Foundations Using Two Modified Vlasov Models

The influence of soil heterogeneity is studied on the bending of circular thin plates using two modified Vlasov foundation models. The model parameters are determined reasonably using an iterative technique. According to the principle of minimum potential energy and considering transversely isotropic soils and Gibson soils, the governing differential equations and boundary conditions for circular thin plates on two modified Vlasov foundations are derived using a variational approach, respectively. The determination of attenuation parameters is a difficult problem, which has hindered the further application of the Vlasov foundation model. The equation that must be satisfied by the attenuation parameter is determined, and an iterative method is used to solve the problem. A comparative analysis is conducted between two modified Vlasov models and the traditional Vlasov model. The results show that the governing equations and boundary conditions for circular thin plates resting on two modified foundations are consistent with those for a circular thin plate on traditional two-parameter foundation after degradation. The accuracy and reliability of the proposed solutions are demonstrated by comparing the obtained results with those reported in the literature. The heterogeneity of soils, including the transversely isotropic soils and Gibson soils, has a certain effect on characteristic parameters of the foundation models as well as the deformations and internal forces of circular thin plates. The present study could be employed as a reference for future engineering designs.

Some notes on “Statically admissible FE solution of the thin plate bending problem with a posteriori error estimation” by Paulina Świa̧tkiewicz and Zdzisław Wiȩckowski (International Journal for Numerical Methods in Engineering, 2020)

Some notes on “Statically admissible FE solution of the thin plate bending problem with a posteriori error estimation” by Paulina Świa̧tkiewicz and Zdzisław Wiȩckowski (International Journal for Numerical Methods in Engineering, 2020)

Effect of solid waste ceramic on uniaxial tensile properties and thin plate bending properties of polyvinyl alcohol engineered cementitious composite

As a kind of non degradable waste, solid waste ceramics has caused serious negative impact on land resources and ecological environment. Therefore, the reasonable recycling of solid waste ceramics is of great significance to the sustainable development of society. In this study, solid waste ceramic powder (CP) was taken as a supplementary cementing material to replace cement (Cement) partly in polyvinyl alcohol-engineered cementitious composites (PVA-ECC) materials. Twelve groups of PVA-ECC materials were designed, with a CP substitution rate of 10%, 15%, 20% (proportion in the mass of cementing materials) respectively. The uniaxial tensile properties and thin plate bending properties of PVA-ECC with different substitution rate solid waste CP were compared and analyzed. The theoretical calculation model of uniaxial tensile stress-strain is established. The results show that recycling of solid waste CP is helpful to clean production of PVA-ECC materials. PVA-ECC with solid waste CP shows good tensile strain and bending toughness. Therefore, CP was taken as a supplementary cement material is feasible. When the content of fly ash (FA) is 70%, the content of Cement is 10%, and the content of solid waste CP is 20%, the uniaxial tensile strain of the material is the largest, which is 4.94%; at the same time, the ultimate bending toughness index of the material is the largest, which is 352.21. Therefore, the mix proportion of this group of specimen is the optimal mix proportion. Compared with the uniaxial tensile test results, the accuracy of the model established in this study is verified. It can effectively predict the uniaxial tensile constitutive model of PVA-ECC material with any substitution rate when the substitution rate of solid waste CP is 10%–20%. This study can play a certain role in promoting the resource utilization of solid waste CP.

Bending of Nonconforming Thin Plates Based on the Mixed‐Order Manifold Method with Background Cells for Integration

As it is very difficult to construct conforming plate elements and the solutions achieved with conforming elements yield inferior accuracy to those achieved with nonconforming elements on many occasions, nonconforming elements, especially Adini’s element (ACM element), are often recommended for practical usage. However, the convergence, good numerical accuracy, and high computing efficiency of ACM element with irregular physical boundaries cannot be achieved using either the finite element method (FEM) or the numerical manifold method (NMM). The mixed‐order NMM with background cells for integration was developed to analyze the bending of nonconforming thin plates with irregular physical boundaries. Regular meshes were selected to improve the convergence performance; background cells were used to improve the integration accuracy without increasing the degrees of freedom, retaining the efficiency as well; the mixed‐order local displacement function was taken to improve the interpolation accuracy. With the penalized formulation fitted to the NMM for Kirchhoff’s thin plate bending, a new scheme was proposed to deal with irregular domain boundaries. Based on the present computational framework, comparisons with other studies were performed by taking several typical examples. The results indicated that the solutions achieved with the proposed NMM rapidly converged to the analytical solutions and their accuracy was vastly superior to that achieved with the FEM and the traditional NMM.

Statically admissible finite element solution of the thin plate bending problem with a posteriori error estimation

SUMMARYTwo triangular elements of class C0 developed on the basis of the principle of complementary work are applied in the static analysis of a thin plate. Some techniques to widen the versatility of the equilibrium approach for the finite element method are presented. Plates of various shapes subjected to diverse types of loading are considered. The results are compared with outcomes obtained by use of the displacement‐based finite element method. By use of these two dual types of solutions, the error of the approximate solution is calculated. The lower and upper bounds for the strain energy are found.

Boundary value problems for systems of inhomogeneous polyharmonic equations with applications in the theory of thin shells and plates

A number of problems in the theory of elasticity, the theory of heterogeneous media, the theory of thin shells and plates is reduced to solving boundary value problems for systems of inhomogeneous polyharmonic equations. The paper proposes a numerical algorithm for solving systems of polyharmonic equations of the form in single-connected and multi-connected areas with a piecewise smooth contour with specified boundary conditions. Two cases are considered when the function is a known polyharmonic function and when the function is also the desired polyharmonic function. The boundary conditions can have the form similar to Dirichlet conditions, Neiman conditions, and can have a mixed form when on one part of the boundary conditions of the Dirichlet type are given, and on the other - hand, the conditions of the Neiman type. On the basis of multiple applications of the Laplace operator and the boundary element method, which is based on the green integral identity, the given system is reduced to a system of integral identities. After approximating the boundary by an inscribed n-gon and discretizing the system of integral identities, the latter is reduced to a system of linear algebraic equations, which is conveniently represented as a system of matrix equations. The existence and uniqueness of the solution follows from the existence of a unique solution of a system of linear algebraic equations. Special attention is paid to the application of the algorithm to the solution of problems on the bending of thin plates, and the bending load can be a known function, and can be an unknown polyharmonic function of an arbitrary order with given boundary conditions. The problem of bending a thin plate of elliptic shape with a known load on the surface is solved, as well as the problem of bending a thin square plate with an unknown load, which is the solution of a harmonic equation with given boundary conditions. The level lines are constructed and the forms of curved plates are given.

Printability conditions for an all-solid-state laser transfer

Several laser technologies exist capable of adding solid materials to a targeted area of a substrate, including photopolymerization, laser sintering, or laser-induced forward transfer. However, the added material normally undergoes a phase change, causing adverse effects such as shrinkage, stress, or degradation. As recently demonstrated, this issue can be addressed by using laser pulses to mechanically delaminate and eject a disk from a solid film. In this case, the laser plays the role of a catapult, with minimal thermal damage to the transferred disk. Despite proven success in micro-electronics and micro-optics, little is known about the mechanical properties of the film that lead to a crack-free all-solid-state transfer. Here, we present a theoretical and experimental study on the effects that film rigidity, elasticity, and plasticity play on laser catapulting. By combining the thermodynamic equations of the laser-generated propulsion force with the theory of thin plate bending, we derived an analytical model that fully describes the list of events responsible for disk ejection. The model is in good agreement with experiments using elastomers, polymers, and metals. A complete printability map based on the film mechanical parameters is reported, which can help to broaden the family of materials suitable for laser additive manufacturing.

Energy-Based Solution for Bending Analysis of Thin Plates on Nonhomogeneous Elastic Foundation

In this paper, the application of Ritz method to the bending of thin plates having mixed boundary conditions and resting on nonhomogeneous foundation with varied subgrade reaction is presented. A typical application of the considered problem is in the analysis and design of foundations resting on a varying soil bed. The proposed model can tackle different kinds of changes in the subgrade reaction including zero subgrade reaction at some locations (cavity zones) as well as rigid zones. The variation of the subgrade reaction modulus is represented by an appropriate mathematical expression that accounts for its possible discontinuous variation. A systematic approach for deriving Ritz trial functions that satisfy the geometric boundary conditions is illustrated for different plate supporting conditions. Three numerical examples are solved to verify the accuracy of the proposed method.

Bending of orthotropic rectangular thin plates with two opposite edges clamped

This work presents integral transform solutions of the bending problem of orthotropic rectangular thin plates with constant thickness, subject to five sets of boundary conditions: (a) fully clamped; (b) three edges clamped and one edge simply supported; (c) three edges clamped and one edge free; (d) two opposite edges clamped, one edge simply supported, and one edge free; and (e) two opposite edges clamped and two edges free. By adopting eigenfunctions of Euler–Bernoulli beams with corresponding boundary conditions for each direction of the plate, the governing fourth-order partial differential equation is integral transformed into a system of linear algebraic equations. Boundary conditions at the free edges are treated exactly by carrying out integral transform in the boundary formulations, which are incorporated in the transformed governing equations by integration by parts. The numerical difficulties with the high-order beam functions are overcome by using modified exponential forms, thus limiting the eigenfunctions to the range between −2 and 2. Analytical integration forms are used for the integrals of the coefficients of the transformed equations, further avoiding numerical difficulties with large high-order eigenvalues. The accuracy and convergence of the solutions are shown through numerical examples in comparison with available solutions in the literature and with finite element solutions obtained by using Abaqus program.

Simulating thin plate bending problems by a family of two-parameter homogenization functions

This study develops a simple and effective numerical technique, which aims to accurately and quickly address thin plate bending problems. Based on the given boundary conditions, the thin plate homogenization function is constructed and a family of two-parameter homogenization functions are derived. Then, the superposition of homogenization functions method for the thin plate, the clamped plate, and the simply supported plate is obtained, which is meshless without numerical integration and iteration with the merits of easy-to-program and easy-to-implement. Six numerical experiments are employed to verify the effectiveness, accuracy and convergence of the proposed novel strategy. The proposed method is evaluated by the comparisons with the analytical solutions and the referenced solutions. It can be observed that the proposed method is quite accurate for the thin plate, the clamped plate, and the simply supported plate problems.

A passive wireless surface acoustic wave sensor for pillar load measurement

This paper presents a passive wireless pillar load measurement method based on surface acoustic wave (SAW) technology. A strain transfer mechanical structure was designed, and the strain distribution of an elastic substrate was analyzed using elastic thin plate bending theory. In addition, the relationship between the strain and frequency shift of the SAW resonator (SAWR) was analyzed, and a strain sensitivity numerical analysis using different bonding directions was performed. Results corroborated that the differential structure of resonators, one of which is placed at the edge along the direction of a diameter and the other is placed at the center with an angle of 45° to the diameter, is a suitable form of the SAW load sensors. Moreover, the SAWR parameters were optimized to obtain a high-performance resonator. Furthermore, The SAWRs and strain transfer mechanical structures were manufactured, the pillar load measurement system was set, and the proposed SAW load measurement method was evaluated. Experimental results affirmed that the load sensitivity is 8.64 kHz/kN. The SAW pillar load method is feasible for practical application.