1. Continuum Mechanics (MTH487)
Lecture 10
Instructor
Dr. Junaid Anjum

2. Recap …
Transformation of Cartesian Tensor

3. Aims and Objectives principal values and principal directions (recap)
principal stresses
principal stress directions
coordinate transformation in principal directions
some problems
normal and shear stress
optimal stress values

4. Recap … 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.

5. Recap … 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

6. Recap … 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

7. principal values and principal directions …
For certain special directions at P, the stress vector does indeed act in the direction of and may therefore be expressed as a scalar multiple of that normal. Thus for such directions
where is a scalar constant. Directions designated by for which the above equation is valid are called principal stress directions, and the scalar is called a principal stress value of .

8. principal values and principal directions …
We determine principal stress values and principal stress directions in the same way as done for determining principal values and principal directions of any symmetric second-order tensor. Hence, we write
In the above system, the tensor components are assumed known, the unknowns are the 3 components of the principal normal and the corresponding principal stress To complete the system of equations for the 4 unknowns, we use the normalizing condition
for the non-trivial solution of the above homogeneous system of equations, the determinant of the coefficients must vanish. That is,

9. principal values and principal directions …
which upon expansion yields
whose roots and are the principal stress values of . The coefficients and are known as the first, second and third invariants, respectively, of and may be expressed in terms of its components by

10. principal values and principal directions … Orthogonality:
If the principal stress values and are distinct, the principal directions associated with these stresses are mutually orthogonal. To see why this is true, let and be the normalized principal direction vectors (eigenvectors) corresponding to and respectively.
which expresses orthogonality between and

11. Recap … Transformation to principal axes
Transformation to principal axes Ox*1x*2x*3
where is a diagonal matrix whose elements are the principal values

13. Transformation to principal axes
is a diagonal matrix whose elements are the principal values

14. principal values and principal directions …
Problem: The components of the stress tensor at point P are given in M Pa with respect to axes Ox1x2x3 by the matrix
Determine the principal stresses and the principal stress directions at P.

16. principal values and principal directions …
Problem: At point P, the stress matrix is given in M Pa with respect to axes Ox1x2x3 by
Determine the principal stress values and the principal stress directions for each case.
Case I:
Case II:

19. principal values and principal directions …
Problem: When referred to principal axes at P, the stress matrix in ksi units is
If the transformation matrix between the principal axes and axes is
where and are to be determined, calculate

22. Normal and Shear Stress
The stress vector on an arbitrary plane at P may be resolved into a component normal to the plane having a magnitude , along with a shear component which acts in the plane and has a magnitude as shown in the figure below P

23. Normal and Shear Stress
Optimal normal stress values:
Hence we conclude that the principal stresses include both the maximum and minimum normal stress values.
Optimal shear stress values:
We will work in the coordinate axes Ox*1x*2x*3 and order the principal stress in the sequence so that is expressed in vector form by

24. Normal and Shear Stress
Optimal shear stress values:

25. Normal and Shear Stress
Optimal shear stress values:

26. Normal and Shear Stress
Problem: At point P, the stress matrix is given in M Pa with respect to axes Ox1x2x3 by
Determine the normal stress component and shear stress component on the plane whose unit normal is

28. Aims and Objectives principal stresses and principal stress directions
coordinate transformation in principal directions
normal and shear stress
optimal stress values