Small bending of a circular bar superposed on finite extension or compression

Summary Solutions are presented for the small bending by terminal couples or by a transverse end load of a circular cylinder of incompressible isotropic material which has been initially finitely extended or compressed by axial loading. Numerical results are given for a Mooney material. The results are used to investigate the stability of a circular bar under end thrust. An elementary solution is giver for the small bending by its own weight of a stretched horizontal cylinder. The results presented in this paper were obtained in the course of research sopnsored at Brown University by the Office of Ordnance Research, U.S.Army, under Grant DA-ARO(D)-31-124-G-89     

Laminar mixed convection on a horizontal plate of finite length in a channel of finite width

The steady two-dimensional laminar mixed-convection flow past a horizontal plate of finite length is analysed for large Péclet numbers, small Prandtl numbers and weak buoyancy effects. The plate is placed in a channel of finite width, with the plane walls of the channel being parallel to the plate. The temperature of the plate is assumed to be constant. The hydrostatic pressure difference across the wake behind the plate is compensated by a perturbation of the inviscid channel flow. This outer flow perturbation affects the temperature distribution in the thermal boundary layer at the plate and the heat transfer rate, respectively. Solutions in closed form are given. The forces acting on the plate due to the potential flow perturbation are also determined.     

Elastic–plastic near field solution of an eccentric crack under shear in a finite width plate

The near crack line analysis method is used to investigate an eccentric crack loaded by shear forces in a finite width plate, and the analytical solution is obtained in this paper. The solution includes: the unit normal vector of the elastic–plastic boundary near the crack line, the elastic–plastic stress fields near crack line, variations of the length of the plastic zone along the crack line with an external loads, and the bearing capacity of a finite plate with a centric crack loaded by shear stress in the far field. The results obtained in this paper are sufficiently precise near the crack line because the assumptions of small scale yielding theory have not been made and no other assumptions have been taken. Subsequently, the present results are compared with the traditional line elastic fracture mechanical solutions and elastoplastic near field solutions under small scale yielding condition. On the basis of the minimum strain energy density (SED) theory, the minimum values of SED in the vicinity of the crack tip are determined, the initial growth orientation of crack are determined. It is found that the normalized load under large scale yielding condition is higher than those under small scale yielding condition when the length of the plastic zone is the same.     

The cantilever strip plate under torision, bending or flexure at infinity

A homogeneous, isotropic plate occupies the region 0x ₁, |x ₂|a, |x ₃|h, where the ratio h/a is sufficiently small so that the classical theory of thin plate bending applies. The short end of the plate at x ₁=0 is clamped while the long sides are free. This cantilever plate is now loaded at x ₁=+ by an applied twisting moment, by a bending moment or by flexure. Despite the fundamental nature of these problems, and the long history of thin plate theory, no solutions are to be found in the existing literature that will determine (for instance) the important unknown resultants V ₁, M ₁₁ at the clamped end x ₁=0. The main reason for this is that this combination of boundary conditions leads to severe oscillating singularities of the field in the corners (0, ±a). The fact that such singularities must exist is widely known, but we present here for the first time a method of solution that takes these singularities fully into account.Our numerical results show that the values of M ₁₁, V ₁ on x ₁=0 bear little resemblance to those of the corresponding Saint-Venant solutions, which do not fully satisfy the boundary conditions at the clamped end. Indeed, significantly large values of these resultants were found at points far enough from the corners so as to be relevant in actual engineering applications. Also of interest are certain weighted integrals of M ₁₁, V ₁ which we calculate. These constants determine the effect of the clamping at large distances (greater than 4a, say) from the cla,ped end. At such distances, the effect of the clamping is merely to impose an additional rigid body deflection on the plate.Finally, we consider the plate of finite length. Provided that the aspect ratio is 2 or more, we give accurate approximate solutions for the torsion, bending or flexure of a finite plate clamped at both ends.     

Solution for bending of a polygonal plate with cracks by the linear conjugate method

Archive of Applied Mechanics - Stress–strain state of a polygonal multiply connected plate under bending is considered. The plate under the action of different force factors (bending moment M...     

A complex potential method for plate problems with bending extension coupling

Summary Within the framework of linear elasticity a complex plate analysis method is presented which includes bending extension coupling as it appears for example in laminates. Four complex potentials are introduced from which the resultant forces and moments and the accompanying deformations can be determined in a rather simple way. The underlying bending extension coupling is reflected by the actual numerical value of some constants. The usefulness of the introduced complex potentials is demonstrated by the example of a cracked plate under inplane and bending forces.

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Eine Methode komplexer Potentiale für Scheiben/Platten-Probleme mit Biege-Dehn-Kopplung

Übersicht Im Rahmen linearer Elastizität wird eine komplexe Scheiben/Platten-Analyse-Methode vorgestellt, die Biege-Dehn-Kopplung erfaßt, wie sie beispielsweise in Laminaten vorkommt. Es werden vier komplexe Potentiale eingeführt, aus denen die resultierenden Kräfte und Momente in einer recht einfachen Weise bestimmt werden können. Die zugrundeliegende Biege-Dehn-Kopplung spiegelt sich im tatsächlichen numerischen Wert einiger Konstanten wieder. Die Nützlichkeit der eingeführten komplexen Potentiale wird am Beispiel einer rißbehafteten Scheibe/Platte demonstriert, die in ihrer Ebene und durch Biegung belastet ist.

     

A simple and efficient triangular finite element for plate bending

A simple and efficient triangular finite element is introduced for plate bending application. The element is a three-node triangular one with three basic degrees of freedom per node and two internal rotation degrees of freedom, using selective reduced integration. Numerical examples indicate that, despite its simplicity, the element is not only competitively accurate, but also useful as a thick/thin triangular plate bending element. It is also pointed out that this element using selective reduced integration is in fact a mixed element.     

Stress-intensity distributions for corner cracks emanating from open holes in plates of finite width

A series of frozen stress photoelastic tests is carried out in the present investigation. Linear Elastic Fracture Mechanics is used to analyze the problem of a part-through corner crack at an open hole in a plate of finite width loaded in tension. The photoelastic modeling capability of three-dimensional subcritical crack growth problems is assessed. This is achieved by comparing monotonically grown flaw shapes in epoxy models with crack profiles relative to fatigue crack propagation tests in a different material. Photoelastic stress-intensity factor distributions are checked against numerical results obtained for quarter elliptical geometries. The dependence of stress intensity factors on flaw shape is also discussed.     

P-version hybrid analytical finite element method for plate bending problems

Two sets of trial functions with different variables are constructed for the admissible space of the finite element analysis. The trial functions satisfy the equilibrium differential equation inside elements, while the deflections and rotations on the edges of the elements are approximated by the Peano hierarchical interpolation functions. Then, a generalized variational principle is applied to set up the p-version hybrid analytical finite element method for plate bending problems. The accuracy of finite element computation can be improved by increasing the order of the interpolation polynomials with fixed mesh. In the finite element formulation, to obtain the stiffness matrices and the load vectors, it is only necessary to perform quadrature over the edges of the elements. These matrices and vectors possess an embedding structure. The conformability between the elements can be controlled automatically.This work is supported by the Natural Science Foundation of China and the Aeronautical Science Foundation of China.     

Propagation of surface breathing cracks in shafts under quasi-static rotary bending

Nonlinear Dynamics -     

Interaction between cracks and a circular inclusion in a finite plate with the distributed dislocation method

This paper presents a numerical solution of interaction between cracks and a circular inclusion in a finite plate. Both the boundaries and the cracks are modeled by distributed dislocations. This approach will result in a set of singular integral equations with Cauchy kernels, which can be solved by Gauss–Chebyshev quadratures. Several numerical examples are given to assess the performance of the presented method. The solutions obtained by this method have been checked and confirmed by the finite element analysis results.     

Closed-form solution for two collinear mode-III cracks in an orthotropic elastic strip of finite width

The problem of an orthotropic strip containing two collinear cracks normal to the strip boundaries is considered. The Fourier series method is used to reduce the associated boundary value problem to triple series equations, then to a singular integral equation, which can be solved analytically. Under remote uniform antiplane shear loading, the stress field and the crack sliding displacement are determined analytically and stress intensity factors are also given in a closed form.     

Elastic equilibrium of a finite anisotropic plate weakened by an elliptical hole and containing cracks

Novosibirsk. Translated from Prikladnaya Mekhanika, Vol. 25, No. 8, pp. 94–100, August, 1989.     

Two arbitrarily situated cracks in an elastic plate under flexure

Based on Reissner's theory for the bending of thin plates and by replacing the cracks by dislocation arrays, the flexure problem for an unbounded plate, containing two arbitrarily situated rectilinear cracks, is reduced to a system of singular integral equations relative to six functions which characterize the density of dislocations.The solution, in the form of the product of the series of Chebyshev polynomials of the first kind and their weight function, is obtained for the cases of plain bending and uniform twisting along an arbitrarily inclined direction to the cracks.Numerical results are shown for two fundamental cases of crack configuration, i.e. a pair of equal colinear cracks and equal parallel cracks without stagger.     

Postbuckling growth of a concentric delamination in a circulal plate under axisymmetric compression and bending

Postbuckling growth of a thin circular delamination at the center of a circular plate subjected to axisymmetric edge compression and bending is analyzed. A one-parameter family of solutions of the non-dimensionalized Von Karman's nonlinear equations is obtained by using the shooting method based on the fourth-order Runge-Kutta integration formula. The expression of energy release rate is deduced by means of the path-independent M-integral, and corresponding numerical results are given. The present analysis investigates the various possibilities of bifurcation, snap-through and crack growth.     

Generation of cracks in a perforated reinforced plate

A problem of fracture mechanics on crack nucleation in a reinforced plate attenuated by a periodic system of circular holes is considered. Crack nucleation is modeled by a pre-fracture band in the plastic flow state with a constant stress, which is considered as a region of attenuated bonds between material particles. Determining unknown parameters characterizing the emerging crack reduces to solving a singular integral equation. The condition of crack emergence is formulated with allowance for the criterion of the limiting opening of the faces of the material pre-fracture band. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 6, pp. 170–180, November–December, 2008.