Stress tensor in matrix form

The 9 components of the stress tensor may be written in matrix form as shown above. In the lower of the two matrices above, the first number in the subscripts to the sigmas gives the row of the matrix, and the second number gives the column. So, for example the subscript "11" is used to represent row 1, column 1, and so on.

The subscripts also have another meaning: the three x, y, and z axes of our Cartesian coordinate system can be labelled 1, 2, and 3 respectively. Then the subscript "12" stands for "xy," and so on, allowing us to write the stress matrix as it's shown in the upper matrix.

An important simplification to elasticity theory is made possible by the fact that the stress tensor for most materials is symmetric: That is, the "xy" and "yx" components are equal, and similarly for "xz" and "zx," and for "yz" and "zy. Therefore the stress tensor has only six distinct components, or "unknowns" to be determined. They are the three diagonal components, and the three (different) components off-diagonal.

Vectors can also be written as matrices, with the three vector components as either a column or row of three numbers. I'll give an example in the next post, by writing the Cauchy formula in matrix form, followed by the use of matrix multiplication to write an equation for each component of the stress vector t.