1. Continuum Mechanics (MTH487)
Lecture 5
Instructor
Dr. Junaid Anjum

2. Recap
Matrices
M=N, square matrix
M=1, row matrix
N=1, column matrix

3. Recap
Determinants
Inverse of a matrix

4. Aims and Objectives Transformations of Cartesian Tensors
Principal values and Principal Directions
Finding principal values/directions
Transformation to principal axes

5. Transformations of Cartesian Tensors
Consider two sets of rectangular Cartesian axes Ox1x2x3 and Ox’1x’2x’3 having a common origin.

6. Transformations of Cartesian Tensors
Consider two sets of rectangular Cartesian axes Ox1x2x3 and Ox’1x’2x’3 having a common origin.

7. Transformations of Cartesian Tensors
Example: Let the primed axes Ox’1x’2x’3 be given with respect to the unprimed axes by a 45o counterclockwise rotation about the X2 axis as shown. Determine the primed components of the vector given by

9. Principal values and Principal Directions….
The tensor-vector product given below can be thought of as a linear transformation on vector u
In particular for every symmetric tensor T having real components tij and defined at some point in physical space, there is associated with each direction at that point (identified by the unit vector ni) an image vector vi, given by
If the vector vi is a scalar multiple of ni, that is
then the direction defined by ni is called a principal direction, or eigenvector of T and the scalar is called the principal value or eigenvalue of T.

10. Principal values and Principal Directions….
This system of homogeneous equations will have non-trivial solution only if determinant of the coefficients vanishes
which upon expansion leads to the cubic in called the characteristic equation

11. Principal values and Principal Directions….
real roots for symmetric tensor (tij real)
if are distinct then principal directions are unique and mutually perpendicular.
if (for example) then only the direction associated with will be unique (take any two (perpendicular) directions orthogonal to )
if every set of orthogonal axes can be regarded as the principal axes and every direction is said to be a principal direction

12. Principal values and Principal Directions….
Transformation to principal axes Ox*1x*2x*3
where is a diagonal matrix whose elements are the principal values

13. Principal values and Principal Directions….
Transformation to principal axes Ox*1x*2x*3
where is a diagonal matrix whose elements are the principal values

14. Example: Determine the principal values and principal directions of the second order tensor T whose matrix representation is

16. Example: Show that the principal values for the tensor having the matrix
have a multiplicity of two, and determine the principal directions.

18. Aims and Objectives Transformations of Cartesian Tensors
Principal values and Principal Directions
Finding principal values/directions
Transformation to principal axes

19. Quiz…
Write down the equations required for finding principal values and principal directions.

20. Quiz…
Write down the equations required for finding principal values and principal directions.
how many unknowns we have? and how many equations ?