Representations of Fourth-Order Cartesian Tensors of Structural Mechanics

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.