Page 229 - Introduction to Continuum Mechanics
P. 229
214 Problems
(a) Find the stress vector on the plane whose normal is in the direction e t + e 2 .
(b) Find the normal stress on the same plane.
(c) Find the magnitude of the shearing stress on the same plane.
(d) Find the maximum shearing stress and the planes on which this maximum shearing stress
acts,
4.25. The stress state in which the only non-vanishing stress components are a single pair of
= T an
shearing stresses is called simple shear. Take 7\ 2 = 721 d all other 7/j = 0.
(a) Find the principal values and principal directions of this stress state.
(b) Find the maximum shearing stress and the plane on which it acts.
4.26. The stress state in which only the three normal stress components do not vanish is called
tri-axial stress state. Take TU = a^ T 22 = a 2, T^ = a 3 with Oi>o 2>o^ and all other 7^- = 0.
Find the maximum shearing stress and the plane on which it acts.
4.27. Show that the symmetry of the stress tensor is not valid if there are body moments per
unit volume, as in the case of a polarized anisotropic dielectric solid.
4.28. Given the following stress distribution
find 7\ 2 so that the stress distribution is in equilibrium with zero body force and so that the
stress vector on x± = 1 is given by t = (1 + x^i + (5 - JC 2)e 2.
4.29. Suppose the body force vector is B = -#63, where g is a constant. Consider the following
stress tensor
and find an expression for Ty$ such that T satisfies the equations of equilibrium.
430. In the absence of body forces, the equilibrium stress distribution for a certain body is