Thin Elastic Plate Research Articles

An antagonistic variable-stiffness pneumatic flexible joint.

This paper develops an antagonistic variable-stiffness pneumatic flexible joint in which two groups of artificial muscles are symmetrically distributed on both sides of the elastic thin plate. The elastic thin plate restricts the axial movement of the joint. Therefore, the joint can achieve single-dimensional and bidirectional bending by controlling the air pressure value of the pneumatic artificial muscle. Two variable stiffness elastic dampers are also symmetrically installed on both sides of the elastic thin plate, using a positive-pressure driving method to achieve real-time posture maintenance function of the joint based on particle-blockage, wedge structure, and antagonistic effect. The mathematical models for the bending angle and stiffness of flexible joints were established, and relevant experiments were conducted. When the air pressure of the pneumatic artificial muscle is 0.32 MPa, the joint bending angle reaches 62.7°. When the bending angle is 60° and the air pressure of the variable-stiffness elastic damper is 0.5 MPa, the stiffness in the bending direction of the flexible joint with the variable-stiffness elastic damper is 6.9 times that of the flexible joint without the variable-stiffness elastic damper, and the stiffness in the reverse bending direction is 10.3 times that of the flexible joint without the variable-stiffness elastic damper under the same conditions.

Two-Dimensional Wave Interaction with a Rigid Body Floating near the Marginal Ice Zone

The interaction problem of waves with a body floating near the marginal ice zone is studied, where the marginal ice zone is modeled as an array of multiple uniformly sized floating ice sheets. The linear velocity potential theory is applied for fluid flow, and the thin elastic plate mode is utilized to describe the ice sheet deflection. A hybrid method is used to solve the disturbed velocity potential; i.e., around the floating body, a boundary integral equation is established, while in the domain covered by ice sheets, the velocity potential is expanded into an eigenfunction series, and in the far-field with a free surface, a similar eigenfunction expansion is used to satisfy the radiation condition. The boundary integral equation and the coefficients of the eigenfunction expansions are solved together based on the continuous conditions of pressure and velocity on the interface between the sub-domains. Extensive results for the equivalent Young’s modulus of the ice sheet array and hydrodynamic force on the body are provided, and the effect of individual ice sheet length as well as wave parameters are investigated in detail.

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.

Wave motions due to a point source pulsating and advancing at forward speed parallel to a semi-infinite ice sheet

The Green function, or the wave motion due to a point source pulsating and advancing at constant forward speed along a semi-infinite ice sheet in finite water depth is investigated, based on the linear velocity potential theory for fluid flow and thin elastic plate model for the ice sheet. The result is highly relevant to the ship motions near marginal seas. The ice edge is assumed to be free, or zero bending moment and shear-force conditions are used, while other edge conditions can be similarly considered. The Green function G is derived first through the Fourier transform along the direction of forward speed and then by the Wiener-Hopf technique along the transverse direction across both the free surface and ice sheet. The result shows that in the ice-covered domain, G can be decomposed into three parts. The first one is that upper ocean surface is fully covered by an ice sheet, and the second and third ones are due to the free surface and ice edge. Similarly, in the free-surface domain, G contains the component corresponding to that the upper water surface is fully free, while the second and third ones are due to the ice sheet and ice edge. In both domains, the latter two are due to the interactions of the free-surface wave and ice sheet deflection, which leads to the major complication. In-depth investigations are made for each part of G, and aim to shed some light on the nature of the wave motions induced by a ship advancing along a semi-infinite ice sheet at constant forward speed. Published by the American Physical Society 2024

Shear Waves in an Elastic Plate with a Hole Resting on a Rough Base

The article is devoted to the analytical and numerical study of the pattern of propagation and attenuation, due to Coulomb friction, of shear waves in an infinite elastic thin plate with a circular orifice of radius r0 lying on a rough base. Considering the friction forces and their influence on the sample of wave propagation in extended rods or thin plates is important for calculating the stress–strain state in them and the size of the area of motion. An exact analytical solution of a nonlinear boundary value problem for tangential stresses and velocities is obtained in quadratures by the Laplace transform, with respect to time. It turned out that the complete exhaustion of the wave front of a strong rupture occurs at a finite distance r* from the center of the orifice, and an elementary formula is given for this distance (the case of tangential shock stresses suddenly applied to the orifice boundary is considered). For various ratios of the magnitude of the limiting friction force to the amplitude of the applied load, the stopping (trailing) wave fronts are calculated. After passing them, a state of static equilibrium between the elastic and friction forces with a nonlinear distribution of residual stresses is established in the region r0≤r≤r*. For the first time, a precise analytical solution was obtained for the boundary value problem of the propagation of elastic shear waves in an infinite isotropic space with a cylindrical cavity, when a tangential shock load is set on its surface.

Stand-Up Time Dependence on Protective Roof–Pillar Bearing Structure of Bauxite

The immediate roof of Shanxi sedimentary bauxite is hard clay rock, which maintain stable difficultly in goaf. It is necessary to ensure the stability of the goaf during the mine production period. The relevant research objects did not involve soft rock mass such as bauxite and hard clay and did not pay attention to the weakening characteristics of load-bearing structures under the action of weathering and rheology. This paper provides theoretical support for the safety production of bauxite and similar mines. In order to study the relationship between the stability of the protective roof-pillar bearing structure and time, this paper uses elastic thin plates and rheological theory to build the physical model of the bauxite protective roof-pillar bearing structure, and gives the calculation formula of the stand-up time of the bearing structure. The influence of factors such as the thickness of the protective roof, the uniform surface force coefficient of pillar, the span of the goaf and the thickness of the overlying rock layer on the stand-up time of the bearing structure is analyzed. The relationship between the ultimate bearing capacity and stand-up time of the bearing structure is quantified. The results show that the bearing capacity of the bearing structure is affected by the mechanical properties of the rock mass and the structural parameters of the goaf. Under the condition that the influencing factors of the mechanical parameters of the rock mass remain unchanged, the stand-up time T, which represents the bearing capacity of the bearing structure, is positively correlated with the thickness of the protective roof, positively correlated with the uniform surface force coefficient of the pillar, negatively correlated with the span of the goaf and negatively correlated with the thickness of the overlying rock layer. The engineering example verifies the rationality of theoretical calculation and provides a new idea for mining safety.

Free vibration of thin, creased elastic plates: Optimization and scaling laws

Creasing, crumpling, and folding help to increase the flexural rigidity of thin elastic plates. In this study, the fundamental frequency is used as a measure of improvement in flexural stiffness. Firstly, an analysis of an orderly, creased, thin elastic plate with nine separate pyramidal crumples is presented. Such a plate shows an increase of 124% in the fundamental frequency with only a 0.5% increase in the total mass/ surface area when compared to that of a flat configuration. Further, scaling laws are introduced for such ordered creased structures for three separate cases- scaling based on (i) the total surface area of the structure, (ii) the amplitude/ height of the ordered pyramidal crumples, and (iii) the number of unit cells within a given edge length of the plate. Further, an optimization problem is posed to achieve maximum flexural stiffness under given constraints for an additively manufactured thin, elastic plate supported on all four edges. The objective is posed as a constrained optimization problem maximizing the fundamental frequency of the structure compared to that of a flat sheet. The results are promising, with a 176% increase in stiffness with only a 0.84% increase in the total mass/ surface area.

Buckling analysis of thin plates with circular cut-outs for sustainable ventilated food packaging design

Thin plates with circular cut-outs are widely used in engineering structures such as bridges, construction, and ventilated corrugated paperboard boxes in food packaging. This study investigates the critical buckling loads of thin elastic plates comprised of a single circular hole at different locations and with different sizes. To determine the critical loads, experimental tests and finite element method simulations are conducted. The vertical location and the diameter of the hole impact the buckling results significantly. The results indicate a significant correlation between the experimental and finite element data. The study develops second and third order polynomial formulas that predict buckling loads. In this study, corrugated paperboard plates were used as samples to gain insights into ventilated food packaging designs, aiming to enhance their buckling strength while reducing packaging material consumption for packaging sustainability purposes. The solution proposed in this study can be applied to predict the buckling of other elastic thin plate structures with a circular hole, relevant to industries such as construction and aerospace.

Deformations and Wave Forces in the Motion of a Load on an Ice Cover in the Presence of a Current with Velocity Shear

The deformations of an ice cover on the surface of an ideal incompressible fluid of finite depth under the action of a pressure domain that moves rectilinearly at a constant velocity in the presence of a current with velocity shift, as well as the wave forces exerted on a moving body, are studied. Fluid flow is not potential. The ice cover is modeled by a thin elastic plate with account for uniform compression. The motion of the load can occur at an arbitrary angle to the direction of current. It is assumed that the ice deflection is steady in the coordinate system moving with the load. The Fourier transform method is used within the framework of the linear wave theory. The maximum deformations of ice cover and the wave forces are studied depending on the current velocity gradient, the direction of motion, and the compression ratio.

Three-Dimensional Bending-Gravitational Waves in a Floating Ice Sheet from a Moving Source of Disturbances

The ice cover is modeled by a thin elastic isotropic plate floating on the surface of a liquid of finite depth. The source of disturbances moves along the surface of the plate. The values of critical velocities at which the character of the wave disturbance changes are obtained. The angular zones in which the waves propagate are determined. The influence of the velocity of movement of the source of disturbances, the thickness of the ice plate, compression and stretching forces on the amplitudes of the waves formed is investigated.

Small-amplitude waves in a floating poroelastic plate forcing by vertical pitching plate

The linear two-dimensional problem of flexural-gravity waves generated by an oscillating rigid plate build-in a floating poroelastic plate is studied. The problem is coupled. The plate deflections and the hydrodynamics loads are determined at the same time. The liquid under the poroelastic plate is inviscid and incompressible. Dynamics of the floating plate is described by a thin elastic plate equation. Porosity of the floating plate is taken into account only through the liquid flux into the plate. The velocity of the inflow is assumed to be governed by Darcy's law being proportional to the hydrodynamic pressure at the plate/liquid interface. Two cases of the oscillating rigid plate with and without its part in the liquid are considered. The problems are solved by the Fourier transform method for non-zero porosity and by the vertical mode method for elastic plates with zero porosity. The deflection and strain distributions are analyzed depending on the excitation frequency and the porosity. Two models of floating plate porosity, where the hydrostatic pressure is included into Darcy's law (Zavyalova's model) and excluded (Meylan's model), are compared. Plate porosity induces damping to the system. It is shown that the damping rate is non-monotonic with respect to the plate porosity.

A mechanical model for thin sheet straight cutting in the presence of an elastic support

We study the mechanics of sheet straight cutting in terms of a linear elastic fracture mechanics (LEFM) problem for a infinite thin elastic Kirchhoff plate partly supported by a Winkler foundation. The plate features a semi-infinite crack that is located at the edge of the supported zone and that is subjected to shear and bending loads, representing the action of the cutting tool (e.g. scissors blades). The fact that the plate is only partly supported by the foundation significantly complicates the analysis for it creates a non-symmetric framework, both locally and globally. Yet, a semi-analytical solution is obtained through casting the matrix Wiener–Hopf problem in terms of a pair of convolution integral equations defined on a semi-infinite domain. Stress intensity factors (SIFs) are obtained which converge to the known limits for a symmetric and skew-symmetric free plate. This analysis reveals the fundamental role played by the support in affecting the SIFs in an opposing manner, by enhancing/decreasing the symmetric/skew-symmetric components. Consequently, changing the support stiffness is capable of shifting the failure mechanism, from bending to shear. This observation may be taken advantage of when cutting materials which are more sensitive to either of these failure mechanisms. Also, it proves that the role of the support cannot be neglected when developing mechanical models of any cutting process.

Force impact of a flow of an infinitely deep liquid on a source under ice cover

A characteristic natural factor of the polar regions of the World Ocean and freezing sea areas is the presence of ice cover. The floating ice cover, which determines the dynamic interaction between the ocean and the atmosphere, affects the dynamics of not only the sea surface, but also subsurface waters, while both the ice cover and the entire mass of liquid beneath it participate in the general vertical movement. It is assumed that the ice cover is continuous, that is, its horizontal scales exceed the lengths of the excited waves and, under fairly natural conditions, is modeled by a thin elastic plate, the deformations of which are small and the plate is physically linear. The problem of calculating the force impact of a flow of an infinitely deep homogeneous liquid on a localized source under the ice cover is solved. The problem is solved for the two-dimensional case. Integral representation of the solution for wave drag and lift is obtained, which arise due to the presence of an ice cover and act on the source. The results of calculations of the force action acting on a localized source, simulating a blunt semi-infinite body of finite width, and a dipole, simulating a cylinder, are presented for various values of the oncoming flow velocity and their immersion depth. Numerical calculations show that as the depth of the source immersion increases, the force effect of the fluid flow, which occurs due to the presence of an ice cover, decreases. The dependences of the wave resistance and lift force on the velocity of the incoming fluid flow demonstrate a qualitatively different behavior. The obtained results with different values of the physical parameters included in them make it possible to evaluate the characteristics of ice cover disturbances and its impact on various sources of natural and anthropogenic disturbances observed in real marine conditions.

How to Grow a Flat Leaf.

Growing a flat lamina such as a leaf is almost impossible without some feedback to stabilize long wavelength modes that are easy to trigger since they are energetically cheap. Here we combine the physics of thin elastic plates with feedback control theory to explore how a leaf can remain flat while growing. We investigate both in-plane (metric) and out-of-plane (curvature) growth variation and account for both local and nonlocal feedback laws. We show that a linearized feedback theory that accounts for both spatially nonlocal and temporally delayed effects suffices to suppress long wavelength fluctuations effectively and explains recently observed statistical features of growth in tobacco leaves. Our work provides a framework for understanding the regulation of the shape of leaves and other leaflike laminar objects.

Stochastic nonlinear eigenvibrations of thin elastic plates resting on time-fractional viscoelastic supports

– The main objective of this work is to investigate natural vibrations of the thin (Kirchhoff-Love) plate resting on time-fractional viscoelastic supports using the Stochastic Finite Element Method (SFEM). The mechanical behavior of the supports is deterministic and is described by the fractional order derivatives of the Riemann-Liouville type. A subspace iteration method in conjunction with the continuation method is the basis of the numerical solution carried out using the displacement formulation of the Finite Element Method. The probabilistic analysis includes independent Monte-Carlo simulation, the semi-analytical approach, and the iterative generalized stochastic perturbation method of the first two probabilistic moments. Moreover, a corresponding relative entropy and the entropy-based reliability index are contrasted with the First Order Reliability Method (FORM). It has been found that the relative entropy for analyzed eigen-vibrations is some interesting alternative to the FORM methodology in reliability assessment.

Flexural-gravity wave interaction with multiple vertical cylinders of arbitrary cross section frozen in an ice sheet

The interaction problem of flexural-gravity wave with multiple vertical cylinders frozen in an ice sheet on the surface of water with finite water depth is considered. The linearized velocity potential theory is adopted for fluid flow, and the thin elastic plate model is applied for ice sheet deflection. Each cylinder is bottom-mounted, and the shape of its cross section can be arbitrary while remaining constant in the vertical direction. The velocity potential is expanded into an eigenfunction series in the vertical direction, which satisfies the boundary condition on the ice sheet automatically. The horizontal modes, which satisfy the Helmholtz equations, are then transformed into a series of boundary integral equations along the ice sheet edges or the intersection of the ice sheet with the cylinders. The problem is then solved numerically by imposing the ice sheet edge condition together with the impermeable condition on the cylinders. The solution is exact in the sense that the error is only due to numerical discretization and truncation. Computations are first carried out for single and multiple vertical circular cylinders, and good agreements are obtained with the semi-analytical solution. To resolve the difficulty of excessive computation at a large number of cylinders, the effect of the evanescent wave of a cylinder on those at large distance is ignored. This allows for the case of a large number of cylinders in different arrangements to be simulated. Extensive results are provided. Their physics and practical relevance are discussed.

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.

A Thin Elastic Plate Model for Thermally Contracting Young Oceanic Lithosphere: Insights From Comparison With Modern Seafloor Observations

AbstractWe find at fast‐ and intermediate‐spreading seafloor that their ridge‐parallel bathymetric profiles between two neighboring fracture zones, excluding the part of the seafloor inward to fracture zone (FZ) valley, are predominantly upward concave. The temporal evolution of the bathymetric profiles from the lithosphere formed at the Chile Rise is characterized by (a) the rapid growth of the middle deflection to about 200 m relative to the ends for the first few millions of years and (b) a steady state afterward. We show that these characteristics and the upward‐concave sense of bending can be reasonably explained as the flexure of a thin elastic plate contracting thermally from the top while cooling. The best‐fitting model needs only about 10% of the thermal bending moment based on the half‐space cooling model and the free‐end assumption. Our model is consistent with the recent observations that oceanic lithosphere is cut open at a FZ valley, which disprove the previous assumption that ocean floor is bent down forming the valley walls.

Continental and island arc segmentation: Implications for retro‐arc and pro‐arc foreland basin and backarc basin geometries and sediment distributions

AbstractSegmented magmatic arcs create arc‐parallel variable loads and influence associated foreland basins through flexural isostasy along strike, in addition to standard subduction‐perpendicular foredeep/forebulge/backbulge models. Segmentation occurs in both continental and island arcs (Aleutians, Calabria, Japan, Kuril/Kamchatka, Lesser Antilles, Solomon Islands and Sumatra/Java). Some segments have variable gravity anomalies and elastic thicknesses (Te). In standard theory, the load created by an orogenic belt/magmatic arc depresses the lithosphere modelled as a thin elastic plate floating on a fluid mantle substrate, leading to an arc‐perpendicular foredeep, forebulge and backbulge. Arc‐parallel topographic and gravitational features should influence loading along strike in a similar way, resulting in a checkerboard pattern of foredeeps, forebulges and backbulges in both pro‐arc and retro‐arc foreland basins. This pattern exercises profound controls on sediment and facies distribution in a here‐to‐fore un‐envisaged manner, with implications for resource exploration in foreland basins.

Stokes wave traveling along a thin elastic plate floating at water surface

This paper addresses nonlinear deflection waves traveling along a thin elastic plate floating at the water surface. The investigations are based on both the analytical and experimental aspects. A finite amplitude wave theory for the waves traveling on the plate developed by the present authors, in which, the nonlinear wave is formulated by a perturbation with respect to the wave height/wavelength ratio, is used. The behavior of the nonlinear wave traveling on the plate is like what the Stokes wave theory predicts for the free surface. However, for the waves traveling on the plate, there is an instability frequency at which the second order wave component grows considerably. The sign of the second order wave component is inverted at the instability frequency when the incident wave frequency is changed from low to high regions. These predictions are verified in a tank test in the two-dimensional flume using a long flexural plate that was newly designed. The spatio-temporal distribution of the deflection and bending strain on the plate was measured directly using the stereo Digital Image Correlation (DIC) technique with two high-speed CCD cameras. The measured results are compared with the predictions by analytical theory. The instability frequency is found in the tank test and the nonlinear wave properties are discussed.