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.     

Extension of a viscoplastic plate with a circular hole

International Applied Mechanics -     

Analysis of an infinite shape memory alloy plate with a circular hole subjected to biaxial tension

This paper presents an exact solution for the stresses in an infinite shape memory alloy plate with a circular hole subjected to biaxial tensile stresses applied at infinity. The solution obtained by assumption of plane stress is based on the two-dimensional version of the Tanaka constitutive law for shape memory materials. The plate is in the austenitic phase, prior to the application of external stresses. However, as a result of tensile loading, stress-induced martensite forms, beginning from the boundary of the hole and extending into the interior, as the load continues to increase. Therefore, in a general case, the plate consists of three annular regions: the inner region of pure martensite, the intermediate region where martensite and austenite coexist, and the outer region of pure austenite. The boundaries between these annular regions can be found as functions of the external stress. Two methods of solution are presented. The first is a closed-form approach based on a replacement of the actual distribution of the martensitic fraction by a piece-wise constant function of the radial coordinate. The second method results in an exact solution obtained by assuming that the ratio between the radial and circumferential stresses in the region where austenite and martensite coexist is governed by the same relationship as that in the encompassing regions of pure austenite and pure martensite.     

Reinforcement of a plate with a circular hole by an eccentric circular patch

We study the reinforcement of an infinite elastic plate with a circular hole by a larger eccentric circular patch completely covering the hole and rigidly adjusted to the plate along the entire boundary of itself. We assume that the plate and the patch are in a generalized plane stress state generated by the action of some given loads applied to the plate at infinity and on the boundary of the hole. We use the power series method combined with the conformal mapping method to find the Muskhelishvili complex potentials and study the stress state on the hole boundary and on the adhesion line. We consider several examples, study how the stresses depend on the geometric and elastic parameters, and compare the problem under study with the case of a plate with a circular hole without a patch. In scientific literature, numerous methods for reinforcing plates with holes, in particular, with circular holes, have been studied. In the monographs [1, 2], the problem of reinforcing the hole edges by stiffening ribs is solved. Methods for reinforcing a circular hole by using two-dimensional patches pasted to the entire plate surface are studied in [3, 4]. The case of a plate with a circular cut reinforced by a concentric circular patch adjusted to the plate along the boundary of itself or along some other circle was studied in [5, 6]. The reinforcement of an elliptic hole by a confocal elliptic patch was considered in [7].     

Flexural vibrations of a semiinfinite plate with a circular hole

Khar'kov Aviation Institute. Translated from Prikladnaya Mekhanika, Vol. 26, No. 3, pp. 69–73, March, 1990.     

Elastoplastic analysis of a bent circular plate with a central hole

A circular aluminum plate with a small concentric hole (1/10 the plate thickness) and supported on its outer edge by a ring was subjected to a concentrated load at its center, applied through a rigid ball of radius equal to the plate thickness. Strains were determined using grids, moiré, and electrical strain gages on the top and bottom surfaces of the plate for loads up to and including the one associated with the appearance of the first crack in the plate. The investigation is related to the development of specimens to be used to determine fracture characteristics of materials used in lightweight construction.     

Tensionless contact of a finite circular plate

A general formulation is developed for the contact behavior of a finite circular plate with a tensionless elastic foundation. The gap distance between the plate and elastic foundation is incorporated as an important parameter. Unlike the previous models with zero gap distance and large/infinite plate radius, which assumes the lift-off/separation of a flexural plate from its supporting elastic foundation, this study shows that lift-off may not occur. The results show how the contact area varies with the plate radius, boundary conditions and gap distance. When the plate radius becomes large enough and the gap distance is reduced to zero, the converged contact radius close to the previous ones is obtained.     

Stress analysis of a functional graded material plate with a circular hole

This paper is to study the two-dimensional stress distribution of a functional graded material plate (FGMP) with a circular hole under arbitrary constant loads. With using the method of piece-wise homogeneous layers, the stress distribution of the functional graded material plate having radial arbitrary elastic properties is derived based on the theory of the complex variable functions. As examples, numerical results are presented for the FGMPs having given radial Young’s modulus or Poisson’s ratio. It is shown that the stress is greatly reduced as the radial Young’s modulus increased, but it is less influenced by the variation of the Poisson’s ratio. Moreover, it is also found that the stress level varies when the radial Young’s modulus increased in different ways. Thus, it can be concluded that the stress around the circular hole in the FGMP can be effectively reduced by choosing the proper change ways of the radial elastic properties.     

Characteristic features of the dynamic deformation of a circular plate with a hole

Hydromechanics Institute, National Academy of Sciences of Ukraine, Kiev; and, Kiev Polytechnic Institute, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 31, No. 2, 39–42, February, 1995.     

Dynamic stress analysis of a functionally graded material plate with a circular hole

This paper is to study the two-dimensional dynamic stress of a functionally graded material (FGM) plate with a circular hole under plane compressional waves at infinity. With using the method of piece-wise homogeneous layers, the dynamic stress distribution of the FGM plate having radial arbitrary material parameters is derived based on the complex variable method. As examples, numerical results are presented for the FGM plate having given radial shear modulus, density and Poisson’s ratio. It is found that the dynamic stress around the circular hole in the FGM plate can be effectively reduced by choosing the proper change ways of the radial material parameters for different frequency incident wave.     

Elastoplastic state of flexible conical shells with a circular hole under axial tension

The elastoplastic state of thin conical shells with a circular hole is analyzed assuming finite deflections. The distributions of stresses, strains, and displacements along the hole boundary and in the zone of their concentration are studied. The stress–strain state of shells around the hole under axial tension is analyzed taking into account two nonlinear factors. The numerical results are presented as plots and tables