ON MAXWELL'S STRESS FUNCTIONS FOR SOLVING THREE DIMENSIONAL ELASTICITY PROBLEMS IN THE THEORY OF ELASTICITY

author

  • Charles Ike Department of Civil Engineering, Enugu State University of Science & Technology, Enugu State, Nigeria
Abstract:

The governing equations of three dimensional elasticity problems include the six Beltrami-Michell stress compatibility equations, the three differential equations of equilibrium, and the six material constitutive relations; and these are usually solved subject to the boundary conditions. The system of fifteen differential equations is usually difficult to solve, and simplified methods are usually used to achieve a solution. Stress-based formulation and displacement-based formulation methods are two common simplified methods for solving elasticity problems.This work adopted a stress-based formulation for a three dimensional elasticity problem. In this work, the Maxwell's stress functions for solving three dimensional problems of elasticity theory were derived from fundamental principles. It was shown that the three Maxwell stress functions identically satisfy all the three differential equations of static equilibrium when body forces were ignored. It was further shown that the three Maxwell stress functions are solutions to the six Beltrami-Michell stress compatibility equations if the Maxwell stress functions are potential functions. It was also shown that the Airy's stress functions for two dimensional elasticity problems are special cases of the Maxwell stress functions.

Upgrade to premium to download articles

Sign up to access the full text

Already have an account?login

similar resources

Elzaki transform method for finding solutions to two-dimensional elasticity problems in polar coordinates formulated using Airy stress functions

In this paper, the Elzaki transform method is used for solving two-dimensional (2D) elasticity problems in plane polar coordinates. Airy stress function was used to express the stress compatibility equation as a biharmonic equation. Elzaki transform was applied with respect to the radial coordinate to a modified form of the stress compatibility equation, and the biharmonic equation simplified t...

full text

The Nonlinear Bending Analysis for Circular Nano Plates Based on Modified Coupled Stress and Three- Dimensional Elasticity Theories

In this paper, the nonlinear bending analysis for annular circular nano plates is conducted based on the modified coupled stress and three-dimensional elasticity theories. For this purpose, the equilibrium equations, considering nonlinear strain terms, are calculated using the least energy potential method and solved by the numerical semi-analytical polynomial method. According to the previous ...

full text

DAMAGE IDENTIFICATION BY USING MODAL EXPANSION AND TOPOLOGY OPTIMIZATION IN THREE DIMENSIONAL ELASTICITY PROBLEMS

In this paper, topology optimization is utilized for damage detection in three dimensional elasticity problems. In addition, two mode expansion techniques are used to derive unknown modal data from measured data identified by installed sensors. Damages in the model are assumed as reduction of mass and stiffness in the discretized finite elements. The Solid Isotropic Material with Penalization (...

full text

Stress invariance and redundant moduli in three-dimensional elasticity

A three-dimensional framework is established for generating invariant stress configurations and associated shifts in the elastic compliance. Under these shifts the stress throughout an elastic body is unaltered, while the compatibility equations for the strain are automatically satisfied. The types of invariant stress fields and translations of the compliance identified here generalize the resu...

full text

A hybrid algorithm for solving inverse problems in elasticity

The paper offers a new approach to handling difficult parametric inverse problems in elasticity and thermo-elasticity, formulated as global optimization ones. The proposed strategy is composed of two phases. In the first, global phase, the stochastic hp-HGS algorithm recognizes the basins of attraction of various objective minima. In the second phase, the local objective minimizers are closer a...

full text

On Locking-free Finite Element Schemes for Three-dimensional Elasticity

In the present paper, the authors discuss the locking phenomenon of the finite element method for three-dimensional elasticity as the Lamé constant λ → ∞. Three kinds of finite elements are proposed and analyzed to approximate the three-dimensional elasticity with pure displacement boundary condition. Optimal order error estimates which are uniform with respect to λ ∈ (0, +∞) are obtained for t...

full text

My Resources

Save resource for easier access later

Save to my library Already added to my library

{@ msg_add @}


Journal title

volume 49  issue 2

pages  342- 350

publication date 2018-12-01

By following a journal you will be notified via email when a new issue of this journal is published.

Hosted on Doprax cloud platform doprax.com

copyright © 2015-2023