• Structural Engineering and Mechanics
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• 제81권3호
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• pp.335-348
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• 2022

It is well known that after finding the displacement in the structural mechanics, strain and stress can be obtained in the straight-forward process. The main purpose of this paper is to unify the displacement functions for solving the solid body. By performing mathematical operations, three sets of these key relationships are found in this paper. All of them are written in the Cartesian Coordinates and in terms of a simple function. Both analytical and numerical approaches are utilized to validate the correctness of the presented formulations. Since all required conditions for the bodies with self-equilibrated loadings are satisfied accurately, the authors' relations can solve these kinds of problems. This fact is studied in-depth by solving some numerical examples. It is found that a very simple function can be used for each formulation instead of ten different and complex displacement potentials defined by previous studies.

• Advances in Computational Design
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• 제9권1호
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• pp.39-52
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• 2024

If the governing differential equation arising from engineering problems is treated as an analytic, continuous and derivable function, it can be expanded by one point as a series of finite numbers. For the function to be zero for each value of its domain, the coefficients of each term of the same power must be zero. This results in a recursive relationship which, after applying the natural conditions or the boundary conditions, makes it possible to obtain the values of the derivatives of the function with acceptable accuracy. The elastoplastic analysis of an inhomogeneous thick sphere of metallic materials with linear variation of the modulus of elasticity, yield stress and Poisson's ratio as a function of radius subjected to internal pressure is presented. The Beltrami-Michell equation is established by combining equilibrium, compatibility and constitutive equations. Assuming axisymmetric conditions, the spherical coordinate parameters can be used as principal stress axes. Since there is no analytical solution, the natural boundary conditions are applied and the governing equations are solved using a proposed new method. The maximum effective stress of the von Mises yield criterion occurs at the inner surface; therefore, the negative sign of the linear yield stress gradation parameter should be considered to calculate the optimal yield pressure. The numerical examples are performed and the plots of the numerical results are presented. The validation of the numerical results is observed by modeling the elastoplastic heterogeneous thick sphere as a pressurized multilayer composite reservoir in Abaqus software. The subroutine USDFLD was additionally written to model the continuous gradation of the material.