Two-dimensional Theory Of Elasticity Research Articles

Free vibration and buckling of composite beams with interlayer slip by two-dimensional theory

The free vibration and buckling of partial interaction composite beams are investigated using the state-space method based on the two-dimensional theory of elasticity. The analytical solutions of a beam with two simply supported ends are obtained as well as semi-analytical solutions are obtained for other end conditions using the differential quadrature method coupled with the state-space method. The frequencies and buckling loads are tabulated and compared with those available in the literature. Because the plane section assumption of the classical beam theory is not used, the presented method is applicable not only to slender beams, but also to thick beams. Consequently, the presented method can be a benchmark for other approximate methods based on one-dimensional beam theories.

In-plane Free Vibration Analysis of Laminated Thick Circular Deep Arches

In order to accurately determine the natural frequencies of laminated thick circular deep arches, a solution procedure based on the two-dimensional theory of elasticity is presented. The layerwise finite element method is used to discretize the equations of motion and the related boundary conditions through the thickness of the arch. The formulations are general so that the effects of the variation of beam curvature across the cross-section, the transverse shear and normal stresses and rotary inertia can easily be implemented. Exact solutions are obtained for antisymmetric cross-ply laminated thick circular deep arches with simply supported boundary conditions. The convergence of the method is shown against the number of mathematical layers used for modeling the arches. The effects of different parameters such as opening angle, thickness-to-length and orthotropy ratios on the natural frequencies of the arches are outlined. Due to high accuracy of the method, the results can be used as benchmarks for future research.

Two-dimensional elasticity solutions for functionally graded beams resting on elastic foundations

Abstract Exact solutions for bending and free vibration of functionally graded beams resting on a Winkler–Pasternak elastic foundation are presented based on the two-dimensional theory of elasticity. The beam is assumed orthotropic at any point, while material properties varying exponentially along the thickness direction. The system of governing partial differential equations is reduced to an ordinary one about the thickness coordinate by expanding the state variables into an infinite trigonometric series. The problem is finally solved using the state space method, which is validated by comparing the present results to those available in the literature. Effects of several parameters, such as gradient index, aspect ratios, and foundation parameters on mechanical behavior of FGM beams are investigated. Numerical results are presented to serve as benchmarks for future analyses of such beams.

A mixed differential quadrature and finite element free vibration and buckling analysis of thick beams on two-parameter elastic foundations

As a first endeavor, a mixed differential quadrature (DQ) and finite element (FE) method for boundary value structural problems in the context of free vibration and buckling analysis of thick beams supported on two-parameter elastic foundations is presented. The formulations are based on the two-dimensional theory of elasticity. The problem domain along axial direction is discretized using finite elements. The resulting system of equations and the related boundary conditions are discretized in the thickness direction and in strong-form using DQM. The method benefits from low computational efforts of the DQ in conjunction with the effectiveness of the FE method in general geometry and systematic boundary treatment resulting in highly accurate and fast convergence behavior solution. The boundary conditions at the top and bottom surface of the beams are implemented accurately. The presented formulations provide an effective analysis tool for beams free of shear locking. Comparisons are made with results from elasticity solutions as well as higher-order beam theory.

Mechanical Properties of the Cell Nucleus and the Effect of Emerin Deficiency

Nuclear structure and mechanics are gaining recognition as important factors that affect gene expression, development, and differentiation in normal function and disease, yet the physical mechanisms that govern nuclear mechanical stability remain unclear. Here we examined the physical properties of the cell nucleus by imaging fluorescently labeled components of the inner nucleus (chromatin and nucleoli) and the nuclear envelope (lamins and membranes) in nuclei deformed by micropipette aspiration (confocal imaged microdeformation). We investigated nuclei, both isolated and in intact, living cells, and found that nuclear volume significantly decreased by 60–70% during aspiration. While nuclear membranes exhibited blebbing and fluid characteristics during aspiration, the nuclear lamina exhibited behavior of a solid-elastic shell. Under large deformations of GFP-lamin A-labeled nuclei, we observed a decay of fluorescence intensity into the tip of the deformed tongue that we interpreted in terms of nonlinear, two-dimensional elasticity theory. Here we applied this method to study nuclear envelope stability in disease and found that mouse embryo fibroblasts lacking the inner nuclear membrane protein, emerin, had a significantly decreased ratio of the area expansion to shear moduli ( K/ μ) compared to wild-type cells (2.1 ± 0.2 versus 5.1 ± 1.3). These data suggest that altered nuclear envelope elasticity caused by loss of emerin could contribute to increased nuclear fragility in Emery-Dreifuss muscular dystrophy patients with mutations in the emerin gene. Based on our experimental results and theoretical considerations, we present a model describing how the nucleus is stabilized in the pipette. Such a model is essential for interpreting the results of any micropipette study of the nucleus and porous materials in general.

Second-Order Fredholm Equations for the First Boundary-Value Problem in the Two-Dimensional Anisotropic Theory of Elasticity

A complete potential theory is constructed for the first boundary-value problem in the two-dimensional anisotropic theory of elasticity (the force vector is specified on the boundary) in a bounded domain on a plane with a Lyapunov boundary.

面外垂直応力下での多層だ円形介在物を有する等方性弾性体の解析および数値計算

In this paper, an analytical solution for infinite medium containing elliptical inclusions under uniform out-of-plane normal stress is presented. It is assumed that each layer consists of isotropic elastic material, and that all elliptic boundaries have common focal point. Introducing such assumptions, prescribed solution is obtained to apply the concept of two-dimensional elastic theory. Because only a normal stress whose direction is parallel to a generator of elliptical inclusion affects a matrix, it is noted that antiplane shear stresses do not occur on every points of analytical model. Some numerical examples are shown in graphic and tabular representations.

Scale and conformal invariance in field theory: a physical counterexample

In this Letter, we illustrate how the two-dimensional theory of elasticity provides a physical example of field theory displaying scale but not conformal invariance.

Free vibration of orthotropic functionally graded beams with various end conditions

Free vibration of orthotropic functionally graded beams, whose material properties can vary arbitrarily along the thickness direction, is investigated based on the two-dimensional theory of elasticity. A hybrid state-space/differential quadrature method is employed along with an approximate laminate model, which allows us to obtain the semi-analytical solution easily. With the introduction of continuity conditions at each fictitious interface and boundary conditions at the top and bottom surfaces, the frequency equation for an inhomogeneous beam is derived. A completely exact solution of an FGM beam with material constants varying in exponential way through the thickness is also presented, which serves a benchmark to verify the present method. Numerical results are performed and discussed.

Evaluation of the Dynamic Strength of Cylindrical and Conic Matrices of Finite Length for Stamping by High Explosives

The equations of two-dimensional dynamic theory of elasticity are used to study the domain of applicability of engineering one-dimensional analytic procedures developed earlier for the evaluation of the dynamic strength of cylindrical and conic matrices of finite length both with free and fastened ends subjected to stamping by high explosives. It is shown that, in view of the “buildup” effect leading, in some cases, to the violation of strength of the matrix, these procedures should be supplemented with correcting calculations performed according to the two-dimensional model for large periods of time. We propose new procedures of design-basis and verifying analyses for cylindrical and conic matrices of finite length subjected to stamping by high explosives.

2211 曲げモーメントを受ける突合せ接着接合の二次元応力解析と強度推定(J10-1 接着界面強度評価,J10 締結・接合部の力学とプロセス)

This paper deals with a two-dimensional stress analysis and strength estimation of adhesive butt joints under bending moments. Similar adherends and an adhesive bond are replaced with finite strips in the analyses. Stress distributions in adhesive joints are analyzed exactly by using the two-dimensional theory of elasticity. The effects of stiffness and adhesive thickness on the stress distributions at the interfaces are shown by numerical calculations. It is found that the stress singularity occurs at the ends of the interfaces. For verification, experiments to measure the strains and the strength were carried out. The analytical results are in fairly good agreements with the experimental ones. In addition, the analytical result is also compared with the result obtained by F.E.M. They are fairly consistent

VHF Single-Crystal Silicon Elliptic Bulk-Mode Capacitive Disk Resonators—Part I: Design and Modeling

This work, the first of two parts, presents the design and modeling of VHF single-crystal silicon (SCS) capacitive disk resonators operating in their elliptical bulk resonant mode. The disk resonators are modeled as circular thin-plates with free edge. A comprehensive derivation of the mode shapes and resonant frequencies of the in-plane vibrations of the disk structures is described using the two-dimensional (2-D) elastic theory. An equivalent mechanical model is extracted from the elliptic bulk-mode shape to predict the dynamic behavior of the disk resonators. Based on the mechanical model, the electromechanical coupling and equivalent electrical circuit parameters of the disk resonators are derived. Several considerations regarding the operation, performance, and temperature coefficient of frequency of these devices are further discussed. This model is verified in part II of this paper, which describes the implementation and characterization of the SCS capacitive disk resonators.

Differentiation of Energy Functionals in Two-Dimensional Elasticity Theory for Solids with Curvilinear Cracks

This paper considers the equations of two-dimensional elasticity theory in nonsmooth domains. The domains contain curvilinear cracks of variable length. On the crack faces, conditions are specified in the form of inequalities describing mutual nonpenetration of the crack faces. It is proved that the solutions of equilibrium problems with a perturbed crack converge to the solution of the equilibrium problem with an unperturbed crack in the corresponding space. The derivative of the energy functional with respect to the length of a curvilinear crack is obtained.

Elasticity solution for free vibration of laminated beams

Based on the two-dimensional theory of elasticity, a new approach combining the state space method and the differential quadrature method is presented in this paper for freely vibrating laminated beams. Applying the differential quadrature method to the state space formulations along the axial direction of the beam, new state equations about state variables at discrete points are obtained. Using matrix theory, the solution can be easily derived, which can very conveniently deal with the continuity conditions. Frequency equation governing the free vibration of laminated beams is then derived and the natural frequencies are obtained. No other assumption on deformations and stresses along the thickness direction is introduced, so that the present method is efficient for laminated beams with arbitrary thickness. It also can cope with arbitrary boundary conditions without applying Saint-Venant’s principle. Numerical examples of multi-layered beams and sandwich beams are performed. Results are verified by comparing them with the published results obtained from various finite element methods and shear beam theories.

In-plane free vibration and stability of rotating annular discs

The in-plane free vibration in an elastic, isotropic, rotating annular disc is investigated on the basis of the two-dimensional linear plane stress theory of elasticity. An analytical solution of the governing equations is developed. Accurate natural frequencies and mode shapes for several modes at different radius ratios and boundary conditions are determined. The computed results demonstrate the influence of rotational speed and radius ratio on the natural frequencies and elastic stability of the rotating discs for several modes. Comparisons of the results with previously established results indicate excellent agreement.

General expressions for energy–release rates for delamination in composite laminates

When twodimensional or threedimensional elasticity theory is applied to interfacial cracks, individual components of the energyrelease rate are not well defined in their classical crack closure int...

109 二次元弾性理論解に基づく Pasternak 弾性床定数の評価

A theoretical evaluation method for Pasternak foundation coefficient is presented using the twodimensional elasticity theory. A beam on the Pasternak elastic foundation is modeled as a contact problem of two materials with different elastic modulus. The Pasternak foundation model corresponds to perfect bounding of the interface of the two materials. Using the Airy's stress functions the stresses and displacements of the two materials are determined analytically. The Pasternak foundation coefficient is evaluated from these analytical results.

Lateral variations of effective elastic thickness of the subducting Philippine Sea plate along the Nankai trough

We investigated lateral variations of effective elastic thickness of the subducting Philippine Sea plate along the Nankai trough. For this purpose, we applied two-dimensional thin elastic plate bending theory to several profiles almost perpendicular to the trough axis, correcting the effect of thick marine sediments in the Shikoku Basin. The 15 topographic profiles were selected from 61 profiles and were fit to respective theoretical deflection curves by minimization of the L1 norm. Classifying the Shikoku Basin into three regions, namely, the western, the middle, and the eastern regions, we obtained the results that the effective elastic plate thickness tends to decrease gradually from the east of the Ryukyu-Palau ridge to the Kinan seamount chain. This is consistent with heat flow data, seafloor age, formation process of the Shikoku Basin, length of the slab subducted from the Nankai trough, and spatial distribution of subcrustal earthquakes associated with subduction of the Philippine Sea plate.

Strain gage method for determining stress intensities of sharp-notched strips

An efficient and simple strain gage method for determining the stress intensities of sharp-notched strips is proposed. The bisector of the notch angle is inclined to the edge so that the mixed-mode loading is created simultaneously at the notch tip. A theory of determining the stress intensities using strain gages is described on the basis of the two-dimensional theory of elasticity. Experiments on specimens with various notch shapes are carried out to verify the theoretical results. Experimental results are in good agreement with theoretical results.

Studies on Stress Transference Mechanism of Steel Fibre Reinforced Concrete

The stress transfer mechanism of steel fibre reinforced concrete is studied. The solutions for the stress and displacement were regarded as the superposition of “the elementary solutions” and “the improved solutions”. The elementary solutions were found by using two-dimensional elastic theory and the improved solutions were found by using the Love displacement function method. The calculated results indicate that the solutions possess good convergence. By comparing the three-dimensional solutions with the shear-lag solutions, evident difference may be found.