The Areolar Strain Concept Applied To Elasticity
Author(s)
I. D. Kotchergenko
Abstract
In contrast to the books that start with solutions of Lamé-Navier equations using
complex variables, the present article starts with a presentation of the
fundamentals of the complex theory of two-dimensional elasticity. A new strain
expression is derived and the compatibility conditions for these strains are given.
The fundamental equations for the isotropic and orthotropic plane elasticity are
also presented in somewhat new complex forms. The homogeneous equilibrium
equation is presented in a complex form that proved to be easily solvable.
Kolosov’s general solution for the isotropic case is obtained in a fairly
straightforward fashion. New equilibrium equations and boundary conditions for
finite rotation are also given.
Keywords: areolar strain, compatibility equation, finite rotation, complex
elasticity.
1 Introduction
The development of the theory of two-dimensional elasticity hereof is grounded
on the concept of areolar strain. This concept was first presented by
Kotchergenko [8], in 1983, though containing some errors. In this approach the
strain is obtained by the division of two complex-valued quantities associated
with 2D vectors and called areolar strain owing to the fact that its real part
represents a radial strain and its imaginary part represents either a
circumferential strain or a rotation. The quaternion concept can probably be used
to extend the areolar strain concept to the 3D vectors case. Improvements on the
linear theory were recently presented, Kotchergenko [9,10]. Equilibrium
equations for finite rotation and several other improvements are now included. A
detailed review of the main results obtained until now is provided in order to
make this article self contained.
Keywords
areolar strain, compatibility equation, finite rotation, complexelasticity.
Related Book
Other papers in this volume
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