Vibrations of nonhomogeneous orthotropic rectangular plates with bilinear thickness variation resting on Winkler foundation

Free transverse vibrations of nonhomogeneous orthotropic rectangular plates with bilinear thickness variation resting on Winkler foundation are presented here using two dimensional boundary characteristic orthogonal polynomials in the Rayleigh-Ritz method on the basis of classical plate theory. Gram-Schmidt process has been used to generate orthogonal polynomials. The nonhomogeneity of the plate is assumed to arise due to linear variations in elastic properties and density of the plate material with the in-plane coordinates. The two dimensional thickness variation is taken as the Cartesian product of linear variations along the two concurrent edges of the plate. Effect of nonhomogeneity parameters, aspect ratio and thickness variation together with foundation parameter on the natural frequencies has been illustrated for the first three modes of vibration for four different combinations of clamped, simply supported and free edges correct to four decimal places. Three dimensional mode shapes for specified plate for all the four boundary conditions have been plotted. A comparison of results in special cases with published one has been presented.     

AN EXACT ELEMENT METHOD FOR THE BENDING OF NONHOMOGENEOUS REISSNER’S PLATE

In this paper,based on the step reduction method and exact analytic method,a new method,the exact element method for constructing finite element,is presented.Since the new method doesn’t need variational principle,it can be applied to solve non-positive and positive definite partial differential equations with arbitrary variable coefficients.By this method,a triangle noncompatible element with15 degrees of freedom is derived to solve the bending of nonhomogenous Reissner’s plate.Because the displacement parameters at the nodal point only contain deflection and rotation angle.it is convenient to deal with arbitrary boundary conditions.In this paper,the convergence of displacement and stress resultants is proved.The element obtained by the present method can be used for thin and thick plates as well,Four numerical examples are given at the end of this paper,which indicates that we can obtain satisfactory results and have higher numerical precision.     

An interaction energy integral method for nonhomogeneous materials with interfaces under thermal loading

A plane crack problem of nonhomogeneous materials with interfaces subjected to static thermal loading is investigated. A modified interaction energy integral method (IEIM) is developed to obtain the mixed-mode thermal stress intensity factors (TSIFs). Compared with the previous IEIM, the original point of this paper is: the domain-independence of the modified IEIM still stands in nonhomogeneous materials with interfaces under thermal loading. Therefore, the modified IEIM can still be applied to obtain the TSIFs of nonhomogeneous material even if the integral domain includes interfaces. The modified IEIM is combined with the extended finite element method (XFEM) to solve several thermal fracture problems of nonhomogeneous materials. Good agreement can be obtained compared with the analytic solutions and the domain-independence of the IEIM is verified. Therefore, the present method is effective to study the TSIFs of nonhomogeneous materials even when the materials contain interfaces. The influence of the discontinuity of the material properties (thermal expansion coefficient, thermal conductivity and Young’s modulus) on the TSIFs is investigated. The results show that the discontinuity of both thermal expansion coefficient and Young’s modulus affects the TSIFs greatly, while the discontinuity of thermal conductivity does not arouse obvious change of the TSIFs.     

Stress-strain state of a plate unevenly heated through its thickness

Conclusion Using the method of differential trigonometric series, distinguished by its correctness and generality and the fact that it gives exact solutions for boundary-value problems, we obtained a new solution to the problem of thermal sresses in an unevenly heated plate, as well the problem of the deformation of a freely supported plate under the influence of moments uniformly distributed over its edges. The series entering into the deflections and bending and turning moments converge, and the solution coincides with well-known solutions.Nikolaev Institute of Shipbuiding. Translated from Prikladnaya Mekhanika, Vol. 17, No. 9, pp. 69–75, September, 1981.     

可计算厚度方向应力的壳单元

在板料成形数值模拟中通常使用壳单元,然而,传统的壳单元在计算“回弹”(弹性恢复)变形时精度很差,其原因之一是由于这些壳单元忽略了厚度方向的应力而代之为平面应力条件。为了改善这个缺陷,本文提出的壳单元,通过构造厚度方向的二次形函数引入厚度方向的变形,使得三维本构关系可以不经任何改变被直接使用,并且壳上、下表面的载荷对厚度方向应力的影响也能较好地表现出来。通过几个数值算例证明该壳单元对弹塑性大变形问题分析是有效的。     

On the simple tension of a nonhomogeneous anisotropic elastic plate with cracks

Summary A two-dimensional solution expressed in finite terms is given to the problem of the extension of two anisotropic semi-infinite plates, which have different elastic properties and are bonded to each other along a finite number of straight-line segments on their boundaries. The method of solution is based on the reduction to a type of dual nonhomogeneousHilbert problems for two functions.To explain the effect of anisotropy on the stresses near the common edge, numerical computations are carried out for the case where two semi-infinite plates bonded along a single segment are subjected to some special loading conditions. Stress distributions on this common edge and isochromatic lines in its neighborhood are shown in several figures.

---

Übersicht Es wird eine zweidimensionale geschlossene Lösung für die Zugbeanspruchung zweier anisotroper Halbscheiben gegeben, die längs eines Stücks ihrer geraden Berandung miteinander verbunden sind. Die Lösung wird auf ein duales inhomogenesHilbertsches Problem für zwei unbekannte Funktionen zurückgeführt. Um den Einfluß der Anisotropie auf den Verlauf der Spannungen in der Nähe der miteinander verbundenen Randteile zu erkennen, werden numerische Berechnungen für verschiedene Belastungsfälle durchgeführt. Der Verlauf dieser Spannungen ist in einigen Diagrammen dargestellt.

     

Analysis of transient thermal stress in cylindrically orthotropic tubes

IntroductionCylindricallyorthotropictubesarenowwidelyappliedinengineering ,theresearchofthestressesfieldhasattractedalotofattentionfromscientistsandengineers .ConsiderableworksonstressfieldduetomechanicalloadinghavebeendonebyLehknitskii[1],Sherrer[2 ]andPagano[3].KalamandTauchert[4 ],HyerandCooper[5 ]studiedthethermalstressesinorthotropiccylindersduetosteaky_statetemperaturegradient.TheproblemoftransientthermalstressesinaholloworthotropiccircularcylinderwasfirstlyproposedandtreatedbyKardoma…     

Integral representations for a nonhomogeneous region in couple stress theory of elasticity

In an infinite space a Cosserat medium (with forced rotations) is subjected to a singular concentrated force. The medium is nonhomogeneous and consist of two regular regions where the elastic constants differ. Formulas are derived for the displacements in both the internal and the external region. With the help of this Green's function matrix integral representations are derived for the displacements.     

The concentration of stress and strain in finite thickness elastic plate containing a circular hole

The elastic stress and strain fields of finite thickness large plate containing a hole are systematically investigated using 3D finite element method. It is found that the stress and strain concentration factors of the finite thickness plate are different even if the plate is in elasticity state except at notch root of plate surface. The maximum stress and strain do not always occur on the mid plane of plate. They occur on the mid plane only in thin plate. The maximum stress and strain concentration factors are not on mid plane and the locations of maximum stress and strain concentration factors are different in thick plate. The maximum stress and strain concentration factors of notch root increase from their plane stress value to their peak values, then decrease gradually with increasing thickness and tend to each constant related to Poisson’s ratio of plate, respectively. The stress and strain concentration factors at notch root of plate surface are the same and are the monotonic descent functions of thickness. Their values decrease rapidly and tend to each constant related to Poisson’s ratio with plate thickness increasing. The difference between maximum and surface value of stress concentration factor is a monotonic ascent function of thickness. The thicker the plate is or the larger the Poisson’s ratio is, the larger the difference is. The corresponding difference of strain concentration factor is similar to the one of stress concentration factor. But the difference magnitude of stress concentration factor is larger than that of strain concentration factor in same plate.     

An unsteady-state thermal boundary layer on a flat isothermal plate

The Kármán-Polhausen integral method is used to investigate the problem of an unsteady-state thermal boundary layer on an isothermal plate with a stepwise change in the conditions of flow around the plate; analytical expressions are obtained for the thickness of the thermal boundary layer. A dependence is found for the rate of movement of the boundary between the steady-state and unsteady-state regions of the solution on the Prandtl number. A similar problem was solved in [1, 2] for a dynamic layer, Goodman [3] discusses the more partial problem of an unsteady-state thermal boundary layer under steady-state flow conditions. Rozenshtok [4] considers the problem in an adequate statement but, unfortunately, he permitted errors of principle to enter into the writing of the system of characteristic equations; this led to absolutely invalid results. In an evaluation of the advantages and shortcomings of the integral method under consideration, given in [4], it must only be added that the method is applicable to problems in which the initial conditions differ from zero since, in this case, approximation of the velocity and temperature profiles by polynomials is not admissible.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 64–69, July–August, 1970.     

A Green's function approach to the deflection of a FGM plate under transient thermal loading

Summary  Green's function approach is adopted for analyzing the deflection and the transient temperature distribution of a plate made of functionally graded materials (FGMs). The governing equations for the deflection and the transient temperature are formulated into eigenvalue problems by using the eigenfunction expansion theory. Green's functions for solving the deflection and the transient temperature are obtained by using the Galerkin method and the laminate theory, respectively. The eigenfunctions of Green's function for the deflection are approximated in terms of a series of admissible functions that satisfy the homogeneous boundary conditions of the plate. The eigenfunctions of Green's function for the temperature are determined from the continuity conditions of the temperature and the heat flux at interfaces. Received 9 October 2000; accepted for publication 3 April 2001     

Weight functions for a surface semi-elliptical crack in a finite thickness plate

Weight functions for the surface and the deepest point of a semi-elliptical crack in a finite thickness plate were derived from a general weight function and two references stress intensity factors. The weight functions were validated against finite element data by comparison of stress intensity factors calculated for several linear and non-linear stress fields. Differences were less than 2% for the surface point and 5% for the deepest point. The final weight functions are given in closed forms suitable for computer numerical integration. The weight functions appear to be particularly suitable for fatigue crack growth prediction of semi-elliptical cracks and fracture analysis of such cracks in complex stress fields.     

Nonlinear transient thermal stress analysis of thick-walled FGM cylinder with temperature-dependent material properties

In this paper, the problem of generalized thermoelasticity in a thick-walled FGM cylinder with one relaxation time is presented. The material properties are taking as function of temperature and graded in the radial. Due to the nonlinearity of the governing equations, finite element method is adopted to solve such problem. Both the inner and outer curved surfaces of the cylinder are stress free while the inner surface is subjected to thermal shock, while the outer surface is thermally isolated. The effects of temperature-dependent properties, volume fraction parameter and the thermal relaxation time on the physical quantities behavior are evaluated. Results confirm the efficiency of the present algorithm and reveal the significant effects of the temperature-dependent of the material properties.     

An extension of Alfrey's analogy to thermal stress problems in temperature dependent linear viscoelastic media

The elastic-viscoelastic analogy due to Alfrey (1944, 1948) is extended to incorporate thermal stress problems in nonhomogeneous linear viscoelastic media. It is shown that the thermal stresses in a body possessing temperature-dependent linear viscoelastic properties can be deduced from an equivalent elastic body.     

Dynamics stress concentrations in Ambartsumian's plate with a cutout

Based on the motion equations of flexural wave in Ambartsumian's plates including the effects of transverse shear deformations, by using perturbation method of small parameter, the scattering of flexural waves and dynamic stress concentrations in the plate with a cutout have been studied. The asymptotic solution of the dynamic stress problem is obtained. Numerical results for the dynamic stress concentration factor in Ambartsumian's plates with a circular cutour are graphically presented and discussed. The project supported by the National Natural Science Foundation of China.     

AN ANALYTICAL SOLUTION OF HEAT CONDUCTION ON A LOCALLY NONUNIFORMLY HEATED SURFACE OF A PLATE WITH FINITE THICKNESS

ＡＮＡＮＡＬＹＴＩＣＡＬＳＯＬＵＴＩＯＮＯＦＨＥＡＴＣＯＮＤＵＣＴＩＯＮＯＮＡＬＯＣＡＬＬＹＮＯＮＵＮＩＦＯＲＭＬＹＨＥＡＴＥＤＳＵＲＦＡＣＥＯＦＡＰＬＡＴＥＷＩＴＨＦＩＮＩＴＥＴＨＩＣＫＮＥＳＳ（王明锐）（靳辉）ＡＮＡＮＡＬＹＴＩＣＡＬＳＯＬＵＴＩ...