A Theory in Finite Differences for Computation of Interchanging Principal Stresses and Principal Axes Rotation

Abstract

Suppose consecutive ordinates of three arbitrary and nonproportional stress-time functions of plane state of stress are entered into a computer by a little finite time step. The theory proposed solves the following problem: correct computation of the ordinates of the principal stress-time functions and the angle of principal axes rotation. This problem is not as simple as researchers approached it prior to the computer era. First of all, the correct solution for the principal stresses and the principal axes rotation require correct interchange of the principal stresses while computing them, i.e., correct interchange of the plus/minus signs in the well-known equations for them. For the interchange analysis, an ellipse of stress transformation in the three-dimensional stress-coordinate space is revealed. By changing a coordinate scale, the ellipse turns into a circumference that is an analog to, but different from, a Mohr circle. The correct solution also requires treating the principal axes rotation in little finite differences per little time differences during which little finite elements appear as building the stressing path in the three-dimensional stress-coordinate space. Based on the ellipse/circumference mentioned, three interchange conditions are revealed. The third one is the most important. And, a necessity is also revealed for dividing some stressing path's elements into two subelements. Based on all the findings, the main commands of an algorithm for computing the ordinates of the principal stress-time functions and the angle of principal axes rotation are presented. The correct solution of the problem has been achieved thanks to new notions taken from the so-called integration of damage differentials (IDD) theory. In fact, the paper presents a new contribution to the variable plane stress state analysis.