1. Continuum Mechanics (MTH487)
Lecture 9
Instructor
Dr. Junaid Anjum

2. Recap … divergence theorem local equilibrium equation
stress tensor symmetry

3. Aims and Objectives Coordinate transformation (recap)
Transformation of Cartesian Tensor
some problems

4. Stress Transformation laws …
Transformations of Cartesian Tensors

5. Stress Transformation laws …
Transformations of Cartesian Tensors

7. Stress Transformation laws …
Transformations of Cartesian Tensors

8. Stress Transformation laws …
Problem: Let the stress components (in M Pa) at point P with respect to axes Ox1x2x3 be expressed by the matrix
and let the primed axes Ox’1x’2x’3 be obtained by a 45o counterclockwise rotation about the x3 axis. Determine the stress components .

10. Stress Transformation laws …
Problem: Assume the stress tensor T (in ksi) at P with respect to axes Ox1x2x3 is represented by the matrix
If the x’1 axis makes equal angles with the three unprimed axes, and the x’2 axis lies in the plane of x’1 x3, as shown by the sketch, determine the primed components of T assuming Ox’1x’2x’3 is a right handed system.

13. Stress Transformation laws …
Problem: At the point P, rotated axes Ox’1x’2x’3 is related to the axes Ox1x2x3 by the transformation
where a, b and c are to be determined. Determine if the stress matrix relative to the axes Ox1x2x3is given in MPa by

16. Stress Transformation laws …
Problem: The stress matrix referred to axes Ox1x2x3 is given by
Let rotated axis Ox’1x’2x’3be defined by with respect to axes Ox1x2x3by the table of base vectors
Determine the stress vectors on planes at P perpendicular to the primed axes; determine in terms of base vectors
Project each of the stress vectors obtained in (a) onto the primed axes to determine the nine components of
Verify the results in (b) by a direct application of
2/7
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6/7
-6/7
-3/7

17. 2/7
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6/7
-6/7
-3/7

19. Aims and Objectives
Transformation of Cartesian Tensor