Thin Plate Bending Theory Research Articles

Mechanical Mechanism Analysis of Roof Fracture Evolution in Stope with Variable Length Based on Elastic-Plastic Structure Theory

Stope with variable length is formed by length change of mining face due to the irregular distribution of geological rock pillars, which is a typical representative of complex coal seam mining. According to the geometric characteristics and mechanical boundaries of each mining stage of stope with variable length, the roof structure models with four boundary conditions were established and solved successively by using the small deflection thin plate bending theory. Combined with the simulated images of MATLAB and FLAC3D, the fracture laws and corresponding engineering phenomena were analyzed. According to the characteristics of roof rock pressure zoning, the overburden structure pressure model of “three stopes, three areas, and three structures” is constructed. The research shows that the traditional “O-X” fracture occurs in the roof of small mining stope. For the cracks generated by prolonged “O-X” fracture and drift “O-X” fracture in mutative mining stope are similar to the crack development characteristics of large mining stope, so they are integrated into full-scale mining stope. The full-scale mining stope roof is broken in “X-O” shape, and the crack continues to develop to produce extended fracture, forming the roof fracture theory of “two stopes and two laws.” The research conclusion strongly reveals the failure law of roof from tension instability to plastic fracture and abnormal ground pressure during mining in stope with variable length. It provides a basis for exploring the essence of overburden migration in stope with variable length and strengthening the roof prevention and control theory under the occurrence conditions of deep complex coal seams.

Experimental study on mechanical behavior of double-layer toughened sandwich glass with two-side simple support

To study the mechanical behavior of double-layer toughened sandwich glass with two-side simple support under out-of-plane bending, a total of sixty-four specimens with different test parameters were experimentally investigated in this paper. The main parameters of specimens include thickness of single layer, supporting width, and loading length. It was demonstrated that the specimens failed with all pieces of cracks combined together as an entire unit rather than to separate into a large number of glass fragments. What's more, the cracks appeared only in the lower layer of the toughened sandwich glass, extending in a way similar to the spider web. The ultimate bending strength and bending stiffness of the toughened sandwich glass are enhanced with the increase of the thickness of single layer, the supporting width, and the loading length. Among the above parameters, the thickness of single layer plays a leading role in enhancing the ultimate bending strength and bending stiffness of the toughened sandwich glass. Nevertheless, the supporting width only has a slight effect on improving the ultimate bending strength and bending stiffness. Based on the bending theory of thin plates, the simplified design formulas were presented to evaluate the ultimate bending strength of double-layer toughened sandwich glass with two-side simple support under out-of-plane bending, which was verified to be reasonable and accurate.

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.

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.

Deformation and Mechanical Behaviors of SCSF and CCCF Rectangular Thin Plates Loaded by Hydrostatic Pressure

Elastic rectangular thin plate problems are very important both in theoretical research and engineering applications. Based on this, the flexural deformation functions of the rectangular thin plates with two opposite edges simply supported, one edge clamped and one edge free (SCSF) and three edges clamped and one edge free (CCCF), loaded by hydrostatic pressure are determined by single trigonometric series. And the flexural deformation functions are solved via the principle of minimum potential energy. Next, the internal force and stress functions of rectangular thin plates with two boundary conditions are obtained based on the small deflection bending theory of thin plates. The dimensionless deflection, dimensionless internal force, and dimensionless stress functions of the rectangular thin plates are established as well. The analytic solution in this paper is validated by the finite element method. Finally, the influence of aspect ratio λ and Poisson’s ratio μ on the deformation and mechanical behaviors of the rectangle thin plates is analyzed in this paper. This research can provide references for the plane water gate problem in seaports and channels.

The Critical Axial Force Model in Drilling of Carbon Fiber Reinforced Composites under Different Stacking Modes

Delamination is a failure mode in which the initial crack propagates along the interlayer due to the axial force pushing the uncut material beyond the interlaminar shear strength during drilling. Therefore, it is necessary to study the effect of axial force on the delamination of carbon fiber reinforced composites (CFRP) during drilling. The critical axial force model of different stacking modes is established based on linear elastic fracture mechanics, thin plate bending theory, composite mechanics and superposition principle. The experimental results show that the theoretical values are in good agreement with the experimental values and can be used to predict the critical axial force to suppress the formation of delamination defects.

Gradient elasticity theory enrichment of plate bending theories

The influence of microstructure on macro-structural plate bending is incorporated into plate bending theories by replacing the classical elasticity theory by strain gradient elasticity. The microstructure is represented by gradients of macro-strains and one new material coefficient which is called the micro-scale length parameter. The strain gradients can be introduced either in the non-symmetrized or in the symmetrized formulation. Though the former is physically inconsistent, both these formulations are considered and compared in numerical simulations. The governing equations as well as boundary conditions are derived from the principle of variations applied to unified formulation of plate bending with assuming the assumptions of the classical thin plate bending theory (Kirchhoff-Love theory) and/or the shear deformation plate theory of the 1st and 3rd order (FSDPT and TSDPT). The derived formulation is applicable to FGM (Functionally Graded Material) plates, since the gradation of Young’s modulus is allowed in both the transversal and in-plane directions. Some numerical examples are considered in numerical simulations, in order to illustrate the influence of microstructure on the behavior of macro-structural plate subjected to static transversal loading.

An applied electro-magneto-elastic thin plate theory

The physical variational principle (PVP) in a static magneto-elastic field without source current is discussed first, and then, the PVP in a general electromagnetic field is derived. It is especially useful in the plate vibration problem. Using the PVP and the pseudo total stress principle, the electro-magneto-elastic thin plate bending theory in first order and a Mindlin-type plate bending theory for a moderate thickness plate are easily obtained. This method significantly simplifies the derivation of the governing equations of the thin and moderate plates with nonlinear electromagnetic behavior. The Maxwell stress is naturally included in the governing equation. Using the governing equation of a thin plate, the analyses of the stability and vibration of a plate in an external homogeneous magnetic field are discussed.

A fast multipole boundary element method for solving the thin plate bending problem

A fast multipole boundary element method (BEM) for solving large-scale thin plate bending problems is presented in this paper. The method is based on the Kirchhoff thin plate bending theory and the biharmonic equation governing the deflection of the plate. First, the direct boundary integral equations and the conventional BEM for thin plate bending problems are reviewed. Second, the complex notation of the kernel functions, expansions and translations in the fast multipole BEM are presented. Finally, a few numerical examples are presented to show the accuracy and efficiency of the fast multipole BEM in solving thin plate bending problems. The bending rigidity of a perforated plate is evaluated using the developed code. It is shown that the fast multipole BEM can be applied to solve plate bending problems with good accuracy. Possible improvements in the efficiency of the method are discussed.

Bending and vibration of an electrostatically actuated circular microplate in presence of Casimir force

The pull-in instability and the vibration for a prestressed circular electrostatically actuated microplate are investigated in consideration of the Casimir force. Based on von Kármán’s nonlinear bending theory of thin plates, the governing equations for the whole analysis are decomposed into two two-point boundary value problems. For static deformation of the plate, the geometric nonlinearity is involved and the pull-in parameters are obtained by using the shooting method through taking the applied voltage or Casimir parameter as an unknown. This algorithm is also used to study the small amplitude free vibration about the predeformed bending configuration following an assumed harmonic time mode, and the variation of the prestress and Casimir parameters dependent fundamental natural frequency with the applied voltage is presented. Several case studies are compared with available published simulations to confirm the proposed method. The influences of various parameters, such as the initial gap-thickness ratio, Casimir effect, prestress on the pull-in instability behavior and the natural frequency are examined.

A Novel Stress Characterization Technique for the Development of Low-Stress Ohmic Contacts to HgCdTe

HgCdTe material intended for long-wavelength infrared detection is particularly susceptible to damage from stress. As a result, an ideal ohmic contact needs to have good adhesion and low specific contact resistance. The contact should act as a diffusion barrier and induce the least amount of stress in the underlying material. In this paper we present a set of stress measurements from different ohmic contact materials deposited on short- and long-wavelength HgCdTe films grown by liquid-phase epitaxy (LPE). Using a new experimental technique we remove the substrate and measure the stress induced on single- and multilayered HgCdTe cantilevers. To interpret our results, we develop a theoretical model that describes the physics of elastic deformation in HgCdTe layers. Our model is based on classical thin-plate bending theory and explicitly takes into account the realistic boundary conditions that are present in the experimental setup by using a variational approach.

Multipoles in an elastic multilayer structure in bending and their applications (continual approximation)

We consider the equilibrium of a multilayer structure in bending. This structure is assumed to be formed by plane homogeneous and isotropic layers. The strain of each of the layers is described by the classical small bending theory of thin plates with regard to the fact that the layer is acted upon by the neighboring layers, which contact it either directly or through sufficiently compliant interlayers whose flexural rigidity can be neglected. The characteristic scales of variation in all the fields under study treated as functions of coordinates are assumed to exceed the layer thicknesses so much as to justify the description of the structure deformation in the continual approximation. We assume that the layer deflections are completely determined by the transverse loads and are not affected by tangential stresses. We consider multipoles in this structure. Attention is mainly paid to the multipole asymptotics and their applications to the study of solutions far from the regions where the action exerted on structure are mainly concentrated. We present the results of field computations with the use of these asymptotics and of the exact solutions of the corresponding problems for the plane and spatial cases. These results are used to discuss the possibilities of applying the multipole asymptotics for solving the problems approximately.

Analysis on the magneto-elastic-plastic buckling/snapping of cantilever rectangular ferromagnetic plates

An analysis of buckling/snapping and bending behaviors of magneto-elastic-plastic interaction and coupling for cantilever rectangular soft ferromagnetic plates is presented. Based on the expression of magnetic force from the variational principle of ferromagnetic plates, the buckling and bending theory of thin plates, the Mises yield criterion and the increment theory for plastic deformation, we establish a numerical code to quantitatively simulate the behaviors of the nonlinearly multi-fields coupling problems by the finite element method. Along with the phenomena of buckling/snapping and bending, or the characteristic curve of deflection versus magnitude of applied magnetic fields being numerically displayed, the critical loads of buckling/snapping, and the influences of plastic deformation and the width of plate on these critical loads, the plastic regions expanding with the magnitude of applied magnetic field, as well as the evolvement of deflection configuration of the plate are numerically obtained in a case study.

Impact of plates: Some questions on numerical precision and scaling

The deflection of an impacted plate may be treated as a time-dependent forced vibration at the impact point. The position of the impactor head can be determined from its initial velocity and deceleration on impact. The difference between the impactor position and the plate deflection is the relative approach of the contacting surfaces (Hertz’ law). This is a Volterra integral equation of the second kind which must be solved numerically. Two structures of different sizes but of the same shape will have proportional modal properties. For similar geometries, it can be shown that there are three basic variables; the fundamental frequency, the mass ratio of impactor and plate, and a contact parameter. It is of interest to determine the extent to which the distribution of the higher modes (or equivalently, the shape of the plate) governs the impact force and duration. For example, is it valid to approximate a circular plate by a rectangular one, or to use thin plate bending theory rather than thick? The benefit is the reduced precision required in the modal analysis: With a numerical modal analyzer such as finite-element method, reliable impact force levels may be determined with a low precision modal model.

The cantilever strip plate under torision, bending or flexure at infinity

A homogeneous, isotropic plate occupies the region 0≤x1≤∞, |x2|≤a, |x3|≤h, where the ratio h/a is sufficiently small so that the classical theory of thin plate bending applies. The short end of the plate at x1=0 is clamped while the long sides are free. This cantilever plate is now loaded at x1=+∞ by an applied twisting moment, by a bending moment or by flexure. Despite the fundamental nature of these problems, and the long history of thin plate theory, no solutions are to be found in the existing literature that will determine (for instance) the important unknown resultants V1, M11 at the clamped end x1=0. The main reason for this is that this combination of boundary conditions leads to severe oscillating singularities of the field in the corners (0, ±a). The fact that such singularities must exist is widely known, but we present here for the first time a method of solution that takes these singularities fully into account.

An analysis of thermal stress and distortion in bead-on-plate welding using laminated isotropic plate theory

In this study, thin-plate bending theory using the assumption of small deflection has been adopted for a two-dimensional analysis of the thermo-elasto-plastic behavior of plates in welding to overcome some difficulties arising from a fully three-dimensional approach. For simulating the temperature gradient through the plate thickness, the concept of a layered plate has been proposed, and the thermo-elasto-plastic behavior of the plate during GMA bead-on-plate welding analyzed by using the finite-element method. It was revealed that the suggested method can predict the trend of welding distortions quite satisfactorily.

Optimized axial support topologies for thin telescope mirrors

Thin and highly flexible telescope mirrors need to be supported carefully to avoid undesirable elastic deformations and a reduction of their optical quality. In this study, a wide variety of support topologies are examined to provide a basic set of optimized point supports for these telescope mirrors. This is carried out for mirrors without a central hole, with circular and annular entrance pupils. Efficient topologies introducing a small amount of zenith-angle-dependent defocusing are also proposed. The number of supporting points ranges between 3 and 36. Optimal forces and locations of point supports are calculated using thin-plate bending theory. Numerical methods include a linear least-squares method for determining the best forces, and a downhill simplex algorithm to optimize the support locations. The robustness of the proposed solutions is tested by simulated annealing. Scaling laws are briefly reviewed, and support efficiencies are given for each optimized topology. Results show that taking into account the central obscuration ratio (annular pupil) and tolerating a homologous (paraboloidal) deformation of the mirror allows an improvement of efficiency of up to 50% over the case of an unobscured pupil where defocus is not permitted. This work includes a study of support efficiencies versus the Poisson's ratio of the mirror material. Wavefront errors are also estimated in the case of a defective cell, to specify tolerances on forces and support locations.

Analysis of the Second Mixed Boundary Value Problem for a Thin Plate

The second mixed boundary value problem is solved by the classical theory of thin plate bending. The mixed boundary consists of a boundary (M) on which one respective component of external force and deflective angle are given, and on the remaining boundary the external forces are given. The boundary (M) is straight and the remaining boundary is arbitrary configuration. A closed solution is obtained. Complex stress functions and a rational mapping function are used. A half-plane with a crack is analyzed under a concentrated torsional moment. Stress distributions before and after the crack initiation, and stress intensity factors are obtained for from short to long cracks and for some Poisson’s ratio.

The semi-infinite plate on a Winkler base, free along the edge, and subjected to a vertical force

At first, a brief review of published analyzes for related problems is presented. Then a new solution is derived for the case when plate is subjected to a concentrated force at any point on the plate. As in the reviewed papers, the analysis is based on the linear bending theory of thin plates. The solution procedure utilizes the closed-form Green's function for the infinite plate, in conjunction with the Fourier integral method. The advantage of the present approach is that the characteristic response of the plate near the load is represented by the closed-form Green's function which contains the proper singularity, whereas the other terms represent the correction due to the free edge. To study the mechanical features of the problem under consideration, deflection profiles and bending moment distributions were calculated and are presented as graphs.

Point Impedances Of Rectangular Plates And Vibration Isolation Of Machines

The results of a theoretical investigation of point impedances at edges and in corners of thin isotropic and orthotropic plates are presented. These impedances for plates of reinforced concrete, steel and plywood were measured and compared with theoretical ones. The theoretical impedances at edges and in corners of plates are smaller than the measured ones, but they are in good agreement at the centres of the plates. This means probably that the classical theory of thin plate bending does not describe the motion at edges and in corners well. Vibration isolation was measured with the above-mentioned plates being installed on steel springs or rubber isolators. The measured vibration isolation agrees well with the predicted one if the impedances corrected according to the experiments are used.