Relative Plate Thickness Research Articles

Natural frequencies of plates of medium thickness

A comparison is made of the frequencies obtained by the classical theory of thin plates and the refined theory of plates of medium thickness. Considered a rectangular plate with articulation along the contour. To determine the fundamental frequency, the Bubnov-Galerkin variational method is used. It is shown that with a relative plate thickness h/a = 0.2, the calculation refinement is approximately 15-20% for the values of the fundamental frequency of natural vibrations of a rectangular plate.

Influence of the plate thickness on the frequency of its natural vibration

A comparison is made of the frequencies obtained by the classical theory of thin plates and the refined theory of plates of medium thickness. Considered a rectangular plate with articulation along the contour. To determine the fundamental frequency, the Bubnov-Galerkin variational method is used. It is shown that with a relative plate thickness h/a = 0.2, the calculation refinement is approximately 15-20% for the values of the fundamental frequency of natural vibrations of a rectangular plate.

Нелинейное возбуждение гиперзвуковых колебаний в ферритовой пластине в условиях комбинированного воздействия на двух частотах. Часть 3. Вариация толщины пластины

The task about nonlinear excitation of hypersound vibrations in ferrite plate in conditions of combine influence in two frequencies is investigated. As a most important parameter which is varied it is proposed the relative thickness of plate which is determined as relation of real thickness to the thickness which correspond to elastic resonance on the difference of excitation frequencies. It is established the necessity of choosing of character value of constant field which is determined by enough effective excitation of elastic vibrations. The system of nonlinear equations of motion of magnetization and elastic displacement is described. For solving of this system, the numerical Rounge-Cutta method is applied. The results of this calculation are the time-evolvent of vibrations, dependencies magnetic end elastic vibrations amplitudes and the spectra of vibrations in permanent conditions after end of relaxation processes. It is found the multi-regime character of elastic vibrations which takes place by variation of plate thickness. In the character of development of elastic vibrations in time by the increasing of plate thickness it is found four regimes: regime №1 – regular beatings, regime №2 – established resonance, regime №3 – displacement of center of established vibrations, regime №4 – gigantic oscillations. The intervals of thickness values which are necessary, or realization of these regimes are determined. The properties of each regimes taken separately are investigated. It is found that the regime №1 is realized when the thickness of plate is more less then the thickness of resonance on differential frequency. In this case the elastic vibrations in generally repeats the vibrations of magnetization which are realized as beating between two frequencies of excitation. The regime №2 takes place when the plate thickness is near to resonance on differential frequency. When thickness is corresponds to resonance on differential frequency it is found large raising of resonance character. In the vibrations of elastic displacement, the constant component is discovered. The regime №3 takes place when the plate thickness is exceeded of resonance on several (from two to seven) times. The vibrations of magnetization in this regime are the same as in regimes №1 and №2. The elastic displacement has two components: oscillatory on differential frequency and constant which value by increasing of thickness smoothly is increased. The displacement of center of oscillatory component by thickness is increased has quadratic character. The regime №4 takes place by plate thickness exceeds resonance thickness on the order and more. The vibrations of magnetization maintain the character of beating which are the same as in regimes №1, №2 and №3. The vibrations of elastic displacement are characterized by extremely large amplitude which is more then the amplitude in regime №3 on order and more and has extremely large period which is more then period of differential frequency vibrations on two-three order and more. The amplitude of vibrations and its period by the thickness is increases also increase by linear meaning. The some quality opinions about the nature of observed phenomena are proposed. It is established the specific character of two-frequency excitation in comparison to single-frequency excitation. As the possible task it is proposed the plan of singing the part of solution as dependence of vibration amplitude from plate thickness has quadratic character with necessary appreciation of two-frequency excitation. The mechanical analogy for vibrations of hard rod which is compressed on both ends by approaching forces is proposed. This analogy allows to interpret the displacement of vibrations center and gigantic oscillations regime.

Effect of Relative Plate Thickness in the Heat Flow and Cooling Rate during Welding of Super Duplex Stainless Steel

Effect of Relative Plate Thickness in the Heat Flow and Cooling Rate during Welding of Super Duplex Stainless Steel

Efficient Higher-Order Shear Deformation Theories for Instability Analysis of Plates Carrying a Mass Moving on an Elliptical Path

The dynamic performance of structures under traveling loads should be exactly analyzed to have a safe and reasonable structural design. Different higher-order shear deformation theories are proposed in this paper to analyze the dynamic stability of thick elastic plates carrying a moving mass. The displacement fields of different theories are chosen based upon variations along the thickness as cubic, sinusoidal, hyperbolic and exponential. The well-known Hamilton’s principle is utilized to derive equations of motion and then they are solved using the Galerkin method. The energy-rate method is used as a numerical method to calculate the boundary curves separating the stable and unstable regions in the moving mass parameters plane. Effects of the relative plate thickness, trajectories radii and the Winkler foundation stiffness on the system stability are examined. The results obtained in this research are compared, in a special case, with those of the Kirchhoff’s plate model for the validation.

Influence of the ambient pressure on thin plate and film bending

The distributed lateral load on a plate located in a gas medium has been determined. The difference in the gas pressures of the plate surfaces results in a pressure drop and a lateral force, which depends on the middle surface curvature. It is shown that both these components of the lateral force must be taken into account in the general case. The influence of the second load component is small at a small ratio of the average pressure to the elastic modulus of the plate material and at a large relative thickness of the plate. At a small relative plate thickness and a large ratio of the average pressure of the medium to the elastic modulus of the plate material, the influence of the second component of the lateral load on the bending becomes significant. Cases of both linear and nonlinear plate bending are considered.

An elastic plate bending equation of second-order accuracy

A study is carried out of a thin plate of constant thickness made of linearly elastic material which is transversally isotropic and heterogeneous in the thickness direction. Asymptotic expansions in powers of the relative plate thickness are constructed, and the bending equation of second-order accuracy (the SA model) is delivered. The results of the SA model are compared with the Kirchhoff–Love classical model and with the Timoshenko–Reissner (TR) model, as well as with the exact solution. To this end, some problems for a functionally gradient plate bending, and for a multi-layer plate bending and free vibration are solved and analysed. The range of plate heterogeneity, for which the error of the approximate models is small, is established. The TR model and the SA model are proved to yields results close to each other and the exact results for a very broad range of heterogeneity. That is why the generalized TR model for one-layered homogeneous transversely isotropic plate is proposed. Parameters of this model are chosen so that the results are close to the exact results and the results by the SA model. For the Navier boundary conditions, the analytical solution of 3D problems for a rectangular heterogeneous plate is constructed.

Optimal Poisson’s ratios for laterally loaded rectangular plates

Although most materials used in practice possess positive Poisson’s ratio in the range 0 to ½, there has been increasing number of auxetic materials being developed and investigated. The counter-intuitive property of auxetic materials opens up the possibility of solids and structures that behave in ways that have been overlooked in the past but are nevertheless consistent with physical laws. This article investigates how the auxeticity of a plate material influences the stresses and moments arising from lateral loads. Using classical plate theories, the optimal Poisson’s ratio of simply supported rectangular plates under lateral loads are obtained on the basis of (a) stress minimization from the maximum bending and shearing stresses, and (b) moment minimization from the maximum bending and twisting moments. Results from stress minimization shows that the optimal Poisson’s ratio is strongly influenced by the relative plate thickness and plate aspect ratio. Using the example of a simply supported rectangular plate under sinusoidal load, the optimal Poisson’s ratio becomes more negative for higher aspect ratio. In addition, the optimal Poisson’s ratio becomes more negative, until a limit of −1 is reached, for decreasing relative plate thickness. Results from moment minimization reveal that the optimal Poisson’s ratio is strongly influenced by the load distribution. For a square plate, the optimal Poisson’s ratio is a negative value if the load distribution is more concentrated than a sinusoidal load or if the central square patch of uniform load takes up less than 43% of the plate area. These results suggest the usefulness of auxetic materials for lowering the stresses and moments in rectangular plates under most loading conditions.

Closed form solutions for free vibrations of rectangular Mindlin plates

A new two-eigenfunctions theory, using the amplitude deflection and the generalized curvature as two fundamental eigenfunctions, is proposed for the free vibration solutions of a rectangular Mindlin plate. The three classical eigenvalue differential equations of a Mindlin plate are reformulated to arrive at two new eigenvalue differential equations for the proposed theory. The closed form eigensolutions, which are solved from the two differential equations by means of the method of separation of variables are identical with those via Kirchhoff plate theory for thin plate, and can be employed to predict frequencies for any combinations of simply supported and clamped edge conditions. The free edges can also be dealt with if the other pair of opposite edges are simply supported. Some of the solutions were not available before. The frequency parameters agree closely with the available ones through pb-2 Rayleigh–Ritz method for different aspect ratios and relative thickness of plate.

Quasistatic transduction of the fundamental symmetric Lamb mode in longitudinal wave transducers

The Green’s function formalism has been applied to the analysis of the fundamental symmetric Lamb mode excited by means of longitudinal wave (LW) transducers. The Lamb wave Green’s function has been related to the properties of the effective permittivity defined as a function of the wave slowness and the relative plate thickness. An electrostatic analysis that neglects the piezoelectric coupling is employed to relate the charge density to the applied voltage. This analysis is subsequently used to calculate the radiation conductance of a LW transducer. The results obtained are consistent with the equivalent circuit approach and define the device coupling as a function of the plate thickness and the intrinsic Lamb wave coupling.

Cohesive Fracture Model Based on Necking

A linear hardening model together with a linear elastic background material is first used to discuss some aspects of the mathematical and physical limitations and constraints on cohesive laws. Using an integral equation approach together with the cohesive crack assumption, it is found that in order to remove the stress singularity at the tip of the cohesive zone, the cohesive law must have a nonzero traction at the initial zero opening displacement. A cohesive zone model for ductile metals is then derived based on necking in thin cracked sheets. With this model, the cohesive behavior including peak cohesive traction, cohesive energy density and shape of the cohesive traction–separation curve is discussed. The peak cohesive traction is found to vary from 1.15 times the yield stress for perfectly plastic materials to about 2.5 times the yield stress for modest hardening materials (power hardening exponent of 0.2). The cohesive energy density depends on the critical relative plate thickness reduction at the root of the neck at crack initiation, which needs to be determined by experiments. Finally, an elastic background medium with a center crack is employed to re-examine the shape effect of cohesive traction–separation curve, and the relation between the linear elastic fracture mechanics (LEFM) and cohesive zone models by considering the cohesive zone development and crack growth in the infinite elastic medium. It is shown that the shape of the cohesive curve does affect the cohesive zone size and the apparent energy release rate of LEFM for the crack growth in the elastic background material. The apparent energy release rate of LEFM approaches the cohesive energy density when the crack extends significantly longer than the characteristic length of the cohesive zone.

Analysis of Natural Vibrations of Round Bimorph Piezoelectric Ceramic Plates of Arbitrary Dimensions. I. Simply Supported Piezoelectric Plate

The problem of natural vibrations of a round bimorph rigidly fixed supported piezoelectric ceramic plate (piezoplate) of an arbitrary thickness with arbitrary axisymmetric electrodes is solved using the method of finite elements. The natural frequency spectra are analyzed in the resonance and antiresonance regimes. The displacement distributions over the piezoplate surfaces (vibration modes) are also analyzed and the dynamic electromechanical coupling coefficient (DCC) is investigated as a function of the relative plate thickness for various piezoelectric ceramic compositions. Round bimorph piezoplates with partial electrodes are considered; it is shown that the DCC can be significantly (by tens times) raised using partial electrodes. The results obtained make it possible to estimate the limits of applicability of the approximate one-dimensional theory and choose the optimum geometric dimensions of the plate and electrodes that ensure the maximum DCC.

Bending Analysis of Simply Supported Shear Deformable Skew Plates

This paper presents the bending analysis of a simply supported, thick skew plate based on the first-order shear deformation Reissner/Mindlin theory. Using the geometric transformation, the governing differential equations and boundary conditions of the plate are first transformed from the physical domain into a unit square computational domain. A set of linear algebraic equations is then derived from the transformed differential equations via a numerical technique, the differential quadrature method (DQM), and the approximate solutions of the problem are obtained by solving the set of algebraic equations. The applicability, accuracy, and convergent properties of DQM for the bending analysis of simply supported skew plates are carefully examined for various skew angles and various relative plate thickness. In this paper, the central deflections and moments of the plate are presented for different thickness-to-span ratios, and skew angles for future references.

Determination of elastic T-stress along three-dimensional crack fronts using an interaction integral

We introduce an effective computational method, based on an interaction integral, to evaluate elastic T-stress along three-dimensional crack fronts. Using this technique, T-stress distributions of various three-dimensional geometries are determined from finite element calculations. The computed results show that the through-thickness variation of T-stress along the crack front of edge-cracked plates is relatively small under both tension and bending load conditions. The deviation from the corresponding two-dimensional results increases with increasing relative crack length and with decreasing relative plate thickness. The increase in T-stress with decreasing thickness results from the inherent positive biaxiality of thin elastic plates. Furthermore, the value of three-dimensionally computed T-stress in these through-crack geometries increases with increasing Poisson's ratio. The T-stress distribution is also calculated along the curved crack front of a surface-flawed plate. For the particular geometry considered, the maximum T-stress along the crack front is located at the mid-point of the crack front, and there is a greater variation of T-stress along the crack front under bending loads than in tension.