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

Recap … divergence theorem local equilibrium equation stress tensor symmetry

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

Stress Transformation laws … Transformations of Cartesian Tensors

Stress Transformation laws … Transformations of Cartesian Tensors

Stress Transformation laws … Transformations of Cartesian Tensors

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 .

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.

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

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 3/7 6/7 -6/7 -3/7

2/7 3/7 6/7 -6/7 -3/7

Aims and Objectives Transformation of Cartesian Tensor