Page 227 - Introduction to Continuum Mechanics

212 Problems

where TJ& is the first invariant of the stress tensor T.
(a) Show that the first invariant of the deviatoric stress vanishes.
(b) Given the stress tensor

evaluate S

(c) Show that the principal direction of the stress and the deviatoric stress coincide.
(d) Find a relation between the principal values of the stress and the deviatoric stress.
4.13. An octahedral stress plane is defined to make equal angles with each of the principal
axes of stress.
(a) How many independent octahedral planes are there at each point?

(b) Show that the normal stress on an octahedral plane is given by one-third the first stress
invariant.
(c) Show that the shearing stress on the octahedral plane is given by

where Tj, T^, T^ are the principal values of the stress tensor.
4.14. (a) Let m and n be two unit vectors that define two planes M and N that pass through
a point P. For an arbitrary state of stress defined at the point J°, show that the component of
the stress vector !„, in the n- direction is equal to the component of the stress vector t,, in the
iii-direction.
(b) If m = eiand n = 62, what does the result of part (a) reduce to?
4.15. Let m be a unit vector that defines a plane M passing through a point P. Show that the
stress vector on any plane that contains the stress traction ^ lies in the M-plane.
4.16. Let t,,, and t,, be stress vectors on planes defined by the unit vectors m and n and pass
through the point P. Show that if k is a unit vector that determines a plane that contains !„,
and tn, then t,,, is perpendicular to m and n.
4.17. True or false
(i) Symmetry of stress tensor is not valid if the body has an angular acceleration,
(ii) On the plane of maximum normal stress, the shearing stress is always zero.
4.18. True or false