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Dynamic 3D Visualization of Stress Tensors

Abstract

Sophomore and junior engineering students in majors such as mechanical, aerospace, civil, and materials engineering learn about the concept of the “state-of-stress” at a point within an object. Many engineering students have some difficulty in thoroughly grasping this concept, especially the more mathematical and visual aspects. To date, the best method we have for visualizing the state-of-stress has been to use Mohr’s circle(s), named after the famous 19th century German civil engineer, Christian Otto Mohr. Mohr’s circle applies to the case where rotations of a differential cube about a principal direction (only) are considered. While the discovery of Mohr’s circle was a brilliant accomplishment, it is somewhat non-intuitive to many students and it can take quite a bit of practice until the student has mastered the technique. Even when the student finally does grasp the concept, they may not necessarily have a complete picture of the state-of-stress at a point because Mohr’s circle only applies to rotations of a differential cube about a principal direction. In that sense it is a 2D method. Of course, in general one would be interested in viewing the stresses associated with all possible 3D orientations of the differential cube. In addition, while in recent years several education researchers have developed custom software to permit dynamic visualization of the state-of-stress as the differential cube rotates, visualization is typically static. What is needed is a true 3D dynamic visualization tool that permits one to visualize an arbitrary state-of-stress from the perspective of continuously varying and arbitrary 3D differential cube orientations, parameterized by a time varying rotation matrix, such as that driven by an Euler matrix with 3 time varying angles.

The objective of this educational research project is to: (1) develop the mathematics that permit one to arbitrarily change the orientation of a differential cube and determine the stresses in the new coordinate system (i.e. 3D tensor change of bases), (2) create a corresponding computer- aided-engineering (CAE) software tool using primarily MATLAB® and SolidWorks®, (3) generate useful simulations using MATLAB® and corresponding animations using SolidWorks®, and (4) attempt to determine their educational value with “mechanics” students. The animations in particular can be used within the engineering curriculum, specifically within the Mechanics of Materials and Machine Design & Synthesis courses where the 3D state-of-stress at a point is very important for understanding advanced mechanics concepts and failure theories which are inherently 3D in nature. In summary, this paper presents intriguing and very useful results that others, such as mechanics engineering faculty and students should find useful in enhancing their understanding of stress tensors. This has certainly been our classroom experience.

1. Introduction

Christian Otto Mohr (1835-1918), born in Germany in the coastal area by the North Sea, began his career as a civil engineer employed by the German railroad industry. During these years, Mohr began developing his theories of stress and strength of materials. At the age of 32, Mohr left the railroad industry and became a full-time theoretical engineer and a professor of mechanics. Eventually, after much investigation, Dr. Mohr developed a method for describing the state-of-stress at a point, his “Circles of Stress,” which now bear his name1-3. “Mohr’s circles” have been used extensively in modern engineering, playing significant roles in

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Hennessey, M., & Hacker, L. (2006, June), Dynamic 3 D Visualization Of Stress Tensors Paper presented at 2006 Annual Conference & Exposition, Chicago, Illinois. 10.18260/1-2--708