Representations of Fourth-Order Cartesian Tensors of Structural Mechanics
Author(s): |
Tibor Tarnai
András Lengyel |
---|---|
Medium: | conference paper |
Language(s): | English |
Conference: | 35th Annual Symposium of IABSE / 52nd Annual Symposium of IASS / 6th International Conference on Space Structures: Taller, Longer, Lighter - Meeting growing demand with limited resources, London, United Kingdom, September 2011 |
Published in: | IABSE-IASS 2011 London Symposium Report |
Year: | 2011 |
Abstract: |
This paper is concerned with geometrical and algebraic representation of fourth-order Cartesian tensors. As fourth-order tensors are four-dimensional objects, it is difficult to visualize them. A possible way of representation proposed here is based on an orthogonal projection of a four- dimensional cube into a planar octagon. Another way of geometrical visualization is possible by means of a quartic form in the three-dimensional space, though this mapping does not provide a one-to-one correspondence. Different kinds of symmetry and the existence of the inverse are also investigated, and it is established that the stiffness and compliance tensors of general Hooke's law are not inverses of each other. We show that a fourth-order tensor can be represented by a 3×3 matrix whose entries are 3×3 matrices, and also by a 9×9 matrix. The paper summarises some old facts and new results. |
Keywords: |
Fourth-order tensor geometric representation algebraic representation double contraction single contraction inverse symmetry Hooke tensors
|