Effect of the stamp shape on areas of possible plate plastic deformation
Journal Title: Вісник Тернопільського національного технічного університету - Year 2015, Vol 77, Issue 1
Abstract
An elastic deformation of the parallelepiped shaped plate subject to the smooth stamp, which contacts with the plate on its entire upper base, has been considered. The lower base is free of load. The vertical displacements of the points of the lateral face are equal to zero. We got the analytical form for the components of the stress tensor and vector displacements of the points of the parallelepiped.The Saint-Venant semi-inverse method has been used. The missing boundary conditions have been obtained considering that the normal displacements can be represented in the form of the sine Fourier’s series. The obtained coefficients of the trigonometric series for the stresses and displacements determine the exact solution of the problem for arbitrarily form of the stamp. The problem connected with the distortion energy distribution over volume in the case of the general parabolic shaped stamp has been stated and solved. The expression of the closed form for the distortion energy which is used in the fourth (energy) failure hypothesis of Mises was obtained. The prediction of the safest, in the sense of the stability, zones of the plate with the square horizontal section has been made with the help of the numerical analysis. The location of the global maximum of the distortion energy function under given boundary conditions, constant value of the Poisson coefficient and the geometric parameters of the stamp has been determined with the help of the conjugate gradient method. Numerical research of a parallelepiped, that takes area , was done. The stamp surface is described by the equation . In the numerical solution the first nine harmonics were kept. The Poisson coefficient is considered to equal 0.25. Since the function of potential energy is proportional to the coefficient and shear modulus , we considered them to equal 1. The graphs, presented in the article, testify that the stamp shape effects significantly the distribution of potential energy in the plate. With varying parameter global maximum changes its position, in addition, the appearance of local maximum at the top of the plate is observed.
Authors and Affiliations
Tetyana Shtefan
Keywords
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