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1.
S. Papargyri-Beskou A.E. Giannakopoulos D.E. Beskos 《International Journal of Solids and Structures》2010,47(20):2755-2766
Gradient elastic flexural Kirchhoff plates under static loading are considered. Their governing equation of equilibrium in terms of their lateral deflection is a sixth order partial differential equation instead of the fourth order one for the classical case. A variational formulation of the problem is established with the aid of the principle of virtual work and used to determine all possible boundary conditions, classical and non-classical ones. Two circular gradient elastic plates, clamped or simply supported at their boundaries, are analyzed analytically and the gradient effect on their static response is assessed in detail. A rectangular gradient elastic plate, simply supported at its boundaries, is also analyzed analytically and its rationally obtained boundary conditions are compared with the heuristically obtained ones in a previous publication of the authors. Finally, a plate with two opposite sides clamped experiencing cylindrical bending is also analyzed and its response compared against that for the cases of micropolar and couple-stress elasticity theories. 相似文献
2.
Peter C.Y.Lee 《Acta Mechanica Solida Sinica》2011,24(2):125-134
An infinite system of two-dimensional equations of motion of isotropic elastic plates with edge and corner conditions are deduced from the three-dimensional equations of elasticity by expansion of displacements in a series of trigonometrical functions and a linear function of the thickness coordinate of the plate. The linear term in the expansion is to accommodate the in-plane displacements induced by the rotation of the plate normal in low-frequency flexural motions. A system of first-order equations of flexural motions and accompanying boundary conditions are extracted from the infinite system. It is shown that the present system of equations is equivalent to the Mindlin’s first-order equations, and the dispersion relation of straight-crested waves of the present theory is identical to that of the Mindlin’s without introducing any corrections. Reduction of present equations and boundary conditions to those of classical plate theories of flexural motions is also presented. 相似文献
3.
Summary Free and forced vibrations of moderately thick, transversely isotropic plates loaded by lateral forces and hydrostatic (isotropic)
in-plane forces are analyzed in the frequency domain. Influences of shear, rotatory inertia, transverse normal stress and
of a two-parameter Pasternak foundation are taken into account. First-order shear-deformation theories of the Reissner–Mindlin
type are considered. These theories are written in a unifying manner using tracers to account for the various influencing
parameters. In the case of a general polygonal shape of the plate and hard-hinged support conditions, the Reissner-Mindlin
deflections are shown to coincide with the results of the classical Kirchhoff theory of thin plates. The background Kirchhoff
plate, which has effective (frequency-dependent) stiffness and mass, is loaded by effective lateral and in-plane forces and
by imposed fictitious “thermal” curvatures. These deflections are further split into deflections of linear elastic prestressed
membranes with effective stiffness, mass and load. This analogy for the deflections is confirmed by utilizing D'Alembert's
dynamic principle in the formulation of Lagrange, which yields an integral equation. Furthermore, the analogy is extended
in order to include shear forces and bending moments. It is shown that in the static case, with no in-plane prestress taken
into account, the stress resultants for certain groups of Reissner-type shear-deformable plates are identical with those resulting
from the Kirchhoff theory of the background. Finally, results taken from the literature for simply supported rectangular and
polygonal Mindlin plates are yielded and verified by analogy in a quick and simple manner.
Received 29 September 1998; accepted for publication 22 June 1999 相似文献
4.
The bending analysis of a thin rectangular plate is carried out in the framework of the second gradient elasticity. In contrast to the classical plate theory, the gradient elasticity can capture the size effects by introducing internal length. In second gradient elasticity model, two internal lengths are present, and the potential energy function is assumed to be quadratic function in terms of strain, first- and second-order gradient strain. Second gradient theory captures the size effects of a structure with high strain gradients more effectively rather than first strain gradient elasticity. Adopting the Kirchhoff’s theory of plate, the plane stress dimension reduction is applied to the stress field, and the governing equation and possible boundary conditions are derived in a variational approach. The governing partial differential equation can be simplified to the first gradient or classical elasticity by setting first or both internal lengths equal to zero, respectively. The clamped and simply supported boundary conditions are derived from the variational equations. As an example, static, stability and free vibration analyses of a simply supported rectangular plate are presented analytically. 相似文献
5.
《International Journal of Solids and Structures》2003,40(19):5187-5196
The stability of parametric vibrations of circular plate subjected to in-plane forces is analyzed by the Liapunov method. Assuming that the compressing forces are physically realizable ergodic processes the plate dynamics is described by stochastic classical partial differential equations. The energy-like functional is proposed; its positiveness is equivalent to the condition in which static buckling does not occur. Taking into account that a plate is compressed radially by time-dependent and uniformly distributed along its edge forces, a dynamic stability of an undeflected state of isotropic elastic circular plate is analyzed. The rate velocity feedback is applied to stabilize the plate parametric vibration. The critical damping coefficient has been expressed by the variance and the mean value of compressing force. The admissible variances of loading strongly depend on the feedback gain factor. 相似文献
6.
7.
S. Papargyri-Beskou D. Polyzos D.E. Beskos 《International Journal of Solids and Structures》2009,46(21):3751-3759
Analytical wave propagation studies in gradient elastic solids and structures are presented. These solids and structures involve an infinite space, a simple axial bar, a Bernoulli–Euler flexural beam and a Kirchhoff flexural plate. In all cases wave dispersion is observed as a result of introducing microstructural effects into the classical elastic material behavior through a simple gradient elasticity theory involving both micro-elastic and micro-inertia characteristics. It is observed that the micro-elastic characteristics are not enough for resulting in realistic dispersion curves and that the micro-inertia characteristics are needed in addition for that purpose for all the cases of solids and structures considered here. It is further observed that there exist similarities between the shear and rotary inertia corrections in the governing equations of motion for bars, beams and plates and the additions of micro-elastic (gradient elastic) and micro-inertia terms in the classical elastic material behavior in order to have wave dispersion in the above structures. 相似文献
8.
Using Hamilton’s principle the coupled nonlinear partial differential motion equations of a flying 3D Euler–Bernoulli beam
are derived. Stress is treated three dimensionally regardless of in-plane and out-of-plane warpings of cross-section. Tension,
compression, twisting, and spatial deflections are nonlinearly coupled to each other. The flying support of the beam has three
translational and three rotational degrees of freedom. The beam is made of a linearly elastic isotropic material and is dynamically
modeled much more accurately than a nonlinear 3D Euler–Bernoulli beam. The accuracy is caused by two new elastic terms that
are lost in the conventional nonlinear 3D Euler–Bernoulli beam theory by differentiation from the approximated strain field
regarding negligible elastic orientation of cross-sectional frame. In this paper, the exact strain field concerning considerable
elastic orientation of cross-sectional frame is used as a source in differentiations although the orientation of cross-section
is negligible. 相似文献
9.
M. R. Karamooz Ravari M. R. Forouzan 《Archive of Applied Mechanics (Ingenieur Archiv)》2011,81(9):1307-1322
Orthotropic circular annular plates have a lot of applications in engineering such as space structures and rotary machines.
In this paper, frequency equations for the in-plane vibration of the orthotropic circular annular plate for general boundary
conditions were derived. To obtain the frequency equation, first the equation of motion for the circular annular plate in
the cylindrical coordinate is derived by using the stress-strain- displacement expressions. Helmholtz decomposition is used
to uncouple the equations of motion. The wave equation is obtained by assumption a harmonic solution for the uncoupled equations.
Using the separation of the variables leads to the general wave equation solution and the in-plane displacements in the r
and θ directions. Finally, boundary conditions are exerted and the natural frequency is derived for general boundary conditions.
The obtained results are validated by comparing with the previously reported and those from finite element analysis. 相似文献
10.
Bending of strain gradient elastic thin plates is studied, adopting Kirchhoff’s theory of plates. Simple linear strain gradient elastic theory with surface energy is employed. The governing plate equation with its boundary conditions are derived through a variational method. It turns out that new terms are introduced, indicating the importance of the cross-section area in bending of thin plates. Those terms are missing from the existing strain gradient plate theories; however, they strongly increase the stiffness of the thin plate. 相似文献
11.
Mindlin, in his celebrated papers of Arch. Rat. Mech. Anal. 16, 51–78, 1964 and Int. J. Solids Struct. 1, 417–438, 1965, proposed two enhanced strain gradient elastic theories to describe linear elastic behavior of isotropic materials with micro-structural effects. Since then, many works dealing with strain gradient elastic theories, derived either from lattice models or homogenization approaches, have appeared in the literature. Although elegant, none of them reproduces entirely the equation of motion as well as the classical and non-classical boundary conditions appearing in Mindlin theory, in terms of the considered lattice or continuum unit cell. Furthermore, no lattice or continuum models that confirm the second gradient elastic theory of Mindlin have been reported in the literature. The present work demonstrates two simple one dimensional models that conclude to first and second strain gradient elastic theories being identical to the corresponding ones proposed by Mindlin. The first is based on the standard continualization of the equation of motion taken for a sequence of mass-spring lattices, while the second one exploits average processes valid in continuum mechanics. Furthermore, Mindlin developed his theory by adding new terms in the expressions of potential and kinetic energy and introducing intrinsic micro-structural parameter without however providing explicit expressions that correlate micro-structure with macro-structure. This is accomplished in the present work where in both models the derived internal length scale parameters are correlated to the size of the considered unit cell. 相似文献
12.
Marié Grobbelaar-Van Dalsen 《Journal of Mathematical Fluid Mechanics》2008,10(3):388-401
In this paper we consider a model for fluid-structure interaction. The hybrid system describes the interaction between an
incompressible fluid in a three-dimensional container with interior a fixed domain and a thin elastic plate, the interface,
which coincides with a flexible flat part of the surface of the vessel containing the fluid. The motion of the fluid is described
by the linearized Navier–Stokes equations and the deformation of the plate by the classical plate equations for in-plane motions,
modified to include the viscous shear stress which the fluid exerts on the plate as well as damping of Kelvin–Voigt type.
We establish the existence of a unique weak solution of the interactive system of partial differential equations by considering
an appropriate variational formulation. Uniform stability of the energy associated with the model is shown under the assumption
that the potential plate energy is dominated by the dissipation induced by the viscosity of the fluid. The retention of the
physical parameters in the problem is an a priori requirement in this physical condition.
相似文献
13.
The exact linear three-dimensional equations for a elastically monoclinic (13 constant) plate of constant thickness are reduced
without approximation to a single 4th order differential equation for a thickness-weighted normal displacement plus two auxiliary equations for
weighted thickness integrals of a stress function and the normal strain. The 4th order equation is of the same form as in
classical (Kirchhoff) theory except the unknown is not the midsurface normal displacement. Assuming a solution of these plate
equations, we construct so-called modified Saint-Venant solutions—“modified” because they involve non-zero body and surface
loads. That is, solutions of the exact three-dimensional elasticity equations that exhibit no boundary layers and that are
subject to a special set of body and surface loads that leave the analogous plate loads arbitrary. 相似文献
14.
D. N. Sheidakov 《Journal of Applied Mechanics and Technical Physics》2007,48(4):547-555
The stability problem of a rectangular plate undergoing uniform biaxial in-plane tensile strain is solved using the three-dimensional
equations of nonlinear elasticity. The surfaces of the plate are stress-free, and special boundary conditions that allow one
to separate variables in the linearized equilibrium equations are specified on the lateral surfaces. For three particular
models of incompressible materials, the critical curves are constructed and the instability region is determined in the plane
of the loading parameters (the multiplicities of elongations of the plate material in the unperturbed equilibrium state).
The numerical results show that for thin plates loaded by tensile stresses, the size and shape of the instability region depend
only slightly on the relation among the length, width, and thickness of the plate. Based on the results obtained, a simple
approximate stability criterion is proposed for an elastic plate under tensile loads.
__________
Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 4, pp. 94–103, July–August, 2007. 相似文献
15.
《International Journal of Solids and Structures》2005,42(3-4):1225-1251
This work presents the highly accurate numerical calculation of the natural frequencies and buckling loads for thick elastic rectangular plates with various combinations of boundary conditions. The Reissener–Mindlin first order shear deformation plate theory and the higher order shear deformation plate theory of Reddy have been applied to the plate’s analysis. The governing equations and the boundary conditions are derived using the dynamic version of the principle of minimum of the total energy. The solution is obtained by the extended Kantorovich method. This approach is combined with the exact element method for the vibration and stability analysis of compressed members, which provides for the derivation of the exact dynamic stiffness matrix including the effect of in-plane and inertia forces. The large number of numerical examples demonstrates the applicability and versatility of the present method. The results obtained by both shear deformation theories are compared with those obtained by the classical thin plate’s theory and with published results. Many new results are given too. 相似文献
16.
In this research work, an exact analytical solution for buckling of functionally graded rectangular plates subjected to non-uniformly
distributed in-plane loading acting on two opposite simply supported edges is developed. It is assumed that the plate rests
on two-parameter elastic foundation and its material properties vary through the thickness of the plate as a power function.
The neutral surface position for such plate is determined, and the classical plate theory based on exact neutral surface position
is employed to derive the governing stability equations. Considering Levy-type solution, the buckling equation reduces to
an ordinary differential equation with variable coefficients. An exact analytical solution is obtained for this equation in
the form of power series using the method of Frobenius. By considering sufficient terms in power series, the critical buckling
load of functionally graded plate with different boundary conditions is determined. The accuracy of presented results is verified
by appropriate convergence study, and the results are checked with those available in related literature. Furthermore, the
effects of power of functionally graded material, aspect ratio, foundation stiffness coefficients and in-plane loading configuration
together with different combinations of boundary conditions on the critical buckling load of functionally graded rectangular
thin plate are studied. 相似文献
17.
本文分析了各向同性封闭圆柱壳的非线性自由振动。文中采用经典的非线性弹性力学方法推导了圆柱壳的大振幅运动方程,这些方程的静态形式与冯·卡门的板理论方程具有同样的精度。文中讨论了四种基本振动模态,并且还以数学公式的形式给出了一般的最终结果,一些例子以曲线给出结果,并进行了比较。结果还表明线性振动可以作为非线性振动的一种特例。 相似文献
18.
The paper is devoted to mathematical modelling of static and dynamic stability of a simply supported three-layered beam with a metal foam core. Mechanical properties of the core vary along the vertical direction. The field of displacements is formulated using the classical broken line hypothesis and the proposed nonlinear hypothesis that generalizes the classical one. Using both hypotheses, the strains are determined as well as the stresses of each layer. The kinetic energy, the elastic strain energy, and the work of load are also determined. The system of equations of motion is derived using Hamilton’s principle. Finally, the system of three equations is reduced to one equation of motion, in particular, the Mathieu equation. The Bubnov-Galerkin method is used to solve the system of equations of motion, and the Runge-Kutta method is used to solve the second-order differential equation. Numerical calculations are done for the chosen family of beams. The critical loads, unstable regions, angular frequencies of the beam, and the static and dynamic equilibrium paths are calculated analytically and verified numerically. The results of this study are presented in the forms of figures and tables. 相似文献
19.
DUAL RECIPROCITY BOUNDARY ELEMENT METHOD FOR FLEXURAL WAVES IN THIN PLATE WITH CUTOUT 总被引:2,自引:0,他引:2
The theoretical analysis and numerical calculation of scattering of elastic waves and dynamic stress concentrations in the thin plate with the cutout was studied using dual reciprocity boundary element method (DRM). Based on the work equivalent law, the dual reciprocity boundary integral equations for flexural waves in the thin plate were established using static fundamental solution. As illustration, numerical results for the dynamic stress concentration factors in the thin plate with a circular hole are given. The results obtained demonstrate good agreement with other reported results and show high accuracy. 相似文献
20.
We present the theory of space–time elasticity and demonstrate that it is the extended reversible thermodynamics and gives the coupled model of thermoelasticity and heat conductivity and involves traditional thermoelasticity. We formulate the generally covariant variational model’s dynamic thermoelasticity and heat conductivity in which the basic kinematic and static variables are unified tensor objects (subject, matter). Variation statement defines the whole set of the initial-boundary problems for the 4D vector governing equation (Euler equation), the spatial projections of which define motion equations and the time projection gives the heat conductivity equation. We show that space–time elasticity directly implies the Fourier and the Maxwell–Cattaneo laws of heat conduction. However, space–time elasticity is richer than classical thermoelasticity, and it advocates its own equations of motion for coupled thermoelasticity. Moreover, we establish that the Maxwell–Cattaneo law and Fourier law can be defined for the reversible processes as compatibility equations without introducing dissipation. We argue that the present framework of space–time elasticity should prove adequate to describe the thermoelastic phenomena at low temperatures for interpreting the results of molecular simulations of heat conduction in solids and for the optimal heat and stress management in the microelectronic components and the thermoelectric devices. 相似文献