Determine the invariants, and principal values and directions of the following matrices. Use the determined principal directions to establish a principal coordinate system, and following the procedures in Example 1.3, formally transform (rotate) the given matrix into the principal system to arrive at the appropriate diagonal form.

a.

b. 

c.

d. 

Data from example 1.3

Equation 1.6.3

Equation 1.61

Fig 1.3

Equation 1.41

Equation 1.51

This result then validates the general theory given by relation (1.6.4), indicating that the tensor should take on diagonal form with the principal values as the elements.  Only simple second-order tensors lead to a characteristic equation that is factorable, thus allowing solution by hand calculation. Most other cases normally develop a general cubic equation and a more complicated system to solve for the principal directions. Again, particular routines within the MATLAB package offer convenient tools to solve these more general problems. Example C.2 in Appendix C provides a simple code to determine the principal values and directions for symmetric second-order tensors.

Equation 1.6.4

Transcribed Image Text:

1 1 -1 001 -1 0 1 0